Turbulent Combustion Modeling

FLUID MECHANICS AND ITS APPLICATIONS

Volume 95

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Tarek Echekki r Epaminondas Mastorakos
Editors

Turbulent
Combustion
Modeling

Advances, New Trends and Perspectives

Editors
Tarek Echekki Epaminondas Mastorakos
Dept. Mechanical & Aerospace Engineering Department of Engineering
North Carolina State University University of Cambridge
911 Oval Drive CB2 1PZ Cambridge
27695 Raleigh UK
North Carolina em257@eng.cam.ac.uk
USA
techekk@ncsu.edu

ISSN 0926-5112
ISBN 978-94-007-0411-4 e-ISBN 978-94-007-0412-1
DOI 10.1007/978-94-007-0412-1
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To our families and mentors
Preface

The field of turbulent combustion has undergone significant progress since the first
original paradigms for modeling turbulent combustion flows emerged more than
60 years ago. In the seventies, the emergence of computational fluid dynamics
(CFD) and access to more advanced non-intrusive laser-based techniques for com-
bustion measurements have enabled further development in the field. More recently,
rapid progress in the modeling and simulation of turbulent flows has occurred.
This progress may be attributed to different factors. First, we have access to more
advanced computational and experimental resources. Advanced computational re-
sources enable the computations of more realistic combustion flows with better de-
scription of the flow and chemical reactions. The higher access to computational
resources also enabled the emergence of new paradigms in turbulent combustion
simulation that address direct computations of unresolved physics. Turbulent com-
bustion has long been considered a paradigm for multiscale problems and has long
been identified as one of the important problems to solve, hence the increasing in-
terest from the computational and applied mathematics communities.
Advanced experimental resources enabled measurements that serve as standards
for validation. We cite here the great synergy created by the Turbulent Nonpremixed
Flames (TNF) workshop and its counterpart, Turbulent Premixed Flames workshop.
More experimental data is being made available online for validation, where in the
past they can be gleamed only through plots from archival journals and reports.
A second impetus is the increasing requirements to design both efficient and
clean combustion technologies. These technologies no longer represent operation
at prescribed modes (e.g. premixed, non-premixed) or regimes (e.g. the flamelet
regime). Computer simulation of turbulent combustion flows can potentially reduce
the turn-around time for the expensive design cycle.
To cover all the progress in the field of turbulent combustion may be beyond
the scope of a single volume; and chances are, revisions on such a volume may
have to be started once it is published. However, some of the basic foundations
that every researcher in the field relies on remain timeless and are covered by some
recent textbooks and monographs. We cite here, for example, Turbulent Combustion
viii Preface

by Peters 1 ; Theoretical and Numerical Combustion by Poinsot and Veynante 2 ;

Computational Models for Turbulent Reacting Flows by Fox 3 ; and An Introduction
to Turbulent Reacting Flows by Cant and Mastorakos 4 . There are also books that
have reviewed on a regular basis progress in the field. They include in particular
the series Turbulent Reacting Flows edited by Libby and Williams. 5 6 Extensive
reviews in aspects of turbulent combustion models are also available in a number of
journals and proceedings, including Progress in Energy and Combustion Science,
Proceedings of the Combustion Institute, and Annual Reviews of Fluid Mechanics.
Having recognized the progress that has been achieved recently in the turbu-
lent combustion field, we have adopted a two-pronged approach in this book. First,
we have attempted to present to the reader the current state-of-the-art in advanced
models in turbulent combustion. Here, we have attempted to avoid duplication of
topics that are already covered in the aforementioned textbooks. Instead, we have
emphasized more recent progress and have identified the current needs and trends
associated with the state-of-the-art models in turbulent combustion.
Perhaps an important distinction between the present volume and the now classic
Libby and Williams series is a greater emphasis on the topic of computation of tur-
bulent combustion flows. This emphasis represents the second scope of the present
volume.
The primary audience for this book is graduate students in engineering, applied
and computational mathematics, and researchers in both academia and industry. It
is assumed that readers have a good knowledge of the state-of-the-art models in
turbulent reacting flows. Therefore, the book can serve as a graduate text or desk
reference.
The book is divided into four major parts and includes 19 chapters. The first in-
troductory part includes two chapters. The first chapter attempts to reassert the role
of combustion science in the current energy debate and identifies the current chal-
lenges and requirements to advance the field forward. The second chapter summa-
rizes the governing equations for turbulent reacting flows and motivates the various
approaches needed to address the closure problem.
In Part II, the state-of-the-art and current trends of advanced turbulent combus-
tion models are presented. The discussion addresses the flamelet approach (Chap-
ter 3), models for premixed combustion (Chapter 4), the conditional moment clo-
sure model (Chapter 5), transported probability density function function meth-

1 Peters, N.: Turbulent Combustion, Cambridge University Press, Cambridge, UK (2000)

2 Poinsot, T., Veynante, D.: Theoretical and Numerical Combustion, 2nd Ed., R.T. Edwards,
Philadelphia, USA (2005)
3 Fox, R.O.: Computational Models for Turbulent Reacting Flows, Cambridge University Press,

Cambridge, UK (2003)
4 Cant, R.S., Mastorakos, E.: An Introduction to Turbulent Reacting Flows, Imperial College Press,

London, UK (2008)
5 Libby, P.A., Williams, F.A. (Eds): Turbulent Reacting Flows, Springer-Verlag, Berlin Heidelberg,

Germany (1980)
6 Libby, P.A., Williams, F.A. (Eds): Turbulent Reacting Flows, Academic Press, London, UK

(1994)
Preface ix

ods (Chapter 6), and the multiple mapping conditioning approach (Chapter 7).
These chapters introduce aspects of the model formulations, illustrate the models
through examples and identify challenges and trends in advancing these modeling
approaches.
Part III addresses multiscale approaches in turbulent combustion. Chapter 8 mo-
tivates requirements for multiscale scale models in combustion and summarizes
current approaches. Chapter 9 addresses methods and strategies for the integration
and acceleration of chemistry in reacting flow computations. Chapters 10, 11 and
12 present the linear-eddy, the one-dimensional turbulence and the unsteady flame-
embedding approaches as multiscale strategies based on hybrid solutions combining
coarse-grained and fine-grained approaches. The following two chapters illustrate
multiscale strategies through mesh-adaptivity based on the adaptive mesh refine-
ment approach (Chapter 13) and the wavelet approach (Chapter 14).
The final part of the book, Part IV, presents what is termed cross-cutting sci-
ence. It samples important disciplines that are relevant to advancing the field of
turbulent combustion. The first discipline is associated with the topic of validation
and verification. Chapter 15 reasserts the role of experiment in advancing turbulent
combustion models. The second discipline is associated with requirements to man-
age large-scale computations associated with turbulent combustion. This discipline
is represented by two chapters. Chapter 16 reviews recent progress and trends in un-
certainty quantification. Chapter 17 discusses a computational framework, based on
the common component architecture, as a strategy to efficiently develop computa-
tional tools for advancing science. Examples are presented in this chapter for com-
bustion flows. The third discipline is multiscale science. An important role of multi-
scale mathematics is the construction of frameworks to couple models designed for
disparate scales. Chapter 18 illustrates the development and implementation of such
frameworks for combustion using the heterogeneous multiscale method. The fourth
discipline is associated with the choices of the governing equations for turbulent
reacting flows. The Navier-Stokes equations in their instantaneous or filtered forms
are the most popular forms of representing turbulent combustion flows. Chapter 19
discusses progress on an alternative representation of the governing equations based
on the lattice-Boltzmann method. The method may serve as an alternative to the
continuum-based Navier-Stokes equations where potentially closer coupling with
atomistic scales is needed.
This book is a collaborative effort that has involved researchers/experts from dif-
ferent disciplines. This is a reflection of how complex the theme of turbulent com-
bustion has become. It could not have been completed without the expert opinion,
the patience and the diligence of the 31 contributors.

Raleigh, NC, USA and Cambridge, UK, Tarek Echekki

July 2010 Epaminondas Mastorakos
Contents

Part I Introductory Concepts

1 The Role of Combustion Technology in the 21st Century . . . . . . . . . . . . 3

R.W. Bilger
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Sustainable Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Technology Forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Implications for Combustion Technology . . . . . . . . . . . . . . . . . . . . . 12
1.5 Prospects for Advanced Computer Modeling of Combustors . . . . . 14
1.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Turbulent Combustion: Concepts, Governing Equations and

Modeling Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Tarek Echekki and Epaminondas Mastorakos
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1 Conservation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.2 Constitutive Relations, State Equations and Auxiliary
Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Conventional Mathematical and Computational Frameworks for
Simulating Turbulent Combustion Flows . . . . . . . . . . . . . . . . . . . . . . 28
2.3.1 Direct Numerical Simulation (DNS) . . . . . . . . . . . . . . . . . . 28
2.3.2 Reynolds-Averaged Navier-Stokes (RANS) . . . . . . . . . . . . 30
2.3.3 Large-Eddy Simulation (LES) . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 Addressing the Closure Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5 Outline of Upcoming Chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
xii Contents

Part II Recent Advances and Trends in Turbulent Combustion Models

3 The Flamelet Model for Non-Premixed Combustion . . . . . . . . . . . . . . . . 43

Benedicte Cuenot
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.1 The Mixture Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.2 The Flamelet Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.3 The Counterflow Diffusion Flame . . . . . . . . . . . . . . . . . . . . 47
3.2.4 Validity of the Flamelet Approach . . . . . . . . . . . . . . . . . . . . 48
3.3 RANS Flamelet Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.1 Steady Flamelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3.2 Transient Flamelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.3 Representative Interactive Flamelets (RIF) Model . . . . . . 55
3.3.4 Eulerian Particle Flamelet Model (EPFM) . . . . . . . . . . . . . 56
3.3.5 FlameletProgress Variable (FPV) Models . . . . . . . . . . . . 56
3.4 LES Flamelet Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.4.1 Subgrid Scale Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4 RANS and LES Modelling of Premixed Turbulent Combustion . . . . . . 63

Stewart Cant
4.1 Introduction to Premixed Flames . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Modelling Framework for RANS and LES . . . . . . . . . . . . . . . . . . . . 64
4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2.2 Regimes of Premixed Turbulent Combustion . . . . . . . . . . . 65
4.2.3 Averaging and Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.4 Modelling Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3 Transport Modelling for Premixed Turbulent Flames . . . . . . . . . . . . 70
4.4 Reaction Rate Modelling for Premixed Turbulent Flames . . . . . . . . 71
4.4.1 Simple Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.4.2 Flame Surface Density Modelling . . . . . . . . . . . . . . . . . . . . 73
4.4.3 G-equation Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.4.4 Scalar Dissipation Rate Modelling . . . . . . . . . . . . . . . . . . . 83
4.4.5 Other Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.5 Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5 The Conditional Moment Closure Model . . . . . . . . . . . . . . . . . . . . . . . . . 91

A. Kronenburg and E. Mastorakos
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2 Methodological Developments in CMC . . . . . . . . . . . . . . . . . . . . . . . 93
5.2.1 The CMC Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2.2 Advances in Second Order Closures . . . . . . . . . . . . . . . . . . 96
Contents xiii

5.2.3 Advances in Doubly Conditioned Moment Closures . . . . 101

5.2.4 Premixed Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2.5 Liquid Fuel Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.3 Application to Flows of Engineering Interest . . . . . . . . . . . . . . . . . . 109
5.3.1 Dimensionality of the CMC Equation . . . . . . . . . . . . . . . . . 109
5.3.2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.3.3 Applications and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6 Transported Probability Density Function Methods for

Reynolds-Averaged and Large-Eddy Simulations . . . . . . . . . . . . . . . . . . 119
D.C. Haworth and S.B. Pope
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.2 A Baseline PDF Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.3 Recent Advances in PDF Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.3.1 Mixing Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.3.2 Hybrid Lagrangian Particle/Eulerian Mesh Methods . . . . 125
6.3.3 Eulerian Field Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.3.4 Multiscale, Multiphysics Modeling . . . . . . . . . . . . . . . . . . . 128
6.3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.4 PDF-Based Methods for Large-Eddy Simulation . . . . . . . . . . . . . . . 132
6.4.1 Spatial Filtering, FDFs, and FDF Transport Equations . . . 133
6.4.2 Equivalent Representations, Models, and Algorithms . . . . 134
6.4.3 An Alternative Interpretation . . . . . . . . . . . . . . . . . . . . . . . . 135
6.4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

7 Multiple Mapping Conditioning: A New Modelling Framework for

Turbulent Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
M.J. Cleary and A.Y. Klimenko
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.2 The Basic MMC Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.2.1 Context and Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.2.2 Mapping Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.2.3 The Deterministic MMC Model . . . . . . . . . . . . . . . . . . . . . 148
7.2.4 The Stochastic MMC Model . . . . . . . . . . . . . . . . . . . . . . . . 152
7.2.5 Qualitative Properties of MMC . . . . . . . . . . . . . . . . . . . . . . 154
7.2.6 Replacement of Reference Variables . . . . . . . . . . . . . . . . . . 154
7.3 Generalised MMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.3.1 Reference Variables in Generalised MMC . . . . . . . . . . . . . 156
7.3.2 Features of Generalised MMC Models . . . . . . . . . . . . . . . . 157
7.3.3 MMC with Dissipation-like Reference Variables . . . . . . . 159
7.3.4 DNS/LES Simulated Reference Variables . . . . . . . . . . . . . 160
xiv Contents

7.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

7.4.1 MMC in Homogeneous Turbulence . . . . . . . . . . . . . . . . . . 161
7.4.2 MMC with RANS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7.4.3 MMC with the Binomial Langevin Model . . . . . . . . . . . . . 165
7.4.4 MMC with LES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
7.5 Summary and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Part III Advances and Trends in Multiscale Strategies

8 The Emerging Role of Multiscale Methods in Turbulent Combustion . 177

Tarek Echekki
8.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
8.2 The Multiscale Nature of Turbulent Combustion Flows . . . . . . . . . . 178
8.3 The Case for Multiscale Strategies in Turbulent Combustion . . . . . 180
8.3.1 Emerging Combustion Technologies . . . . . . . . . . . . . . . . . . 181
8.3.2 Emerging Multiscale Science . . . . . . . . . . . . . . . . . . . . . . . . 182
8.4 Multiscale Considerations for Turbulent Combustion . . . . . . . . . . . 183
8.4.1 Basic Requirements for Multiscale Approaches in
Turbulent Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
8.4.2 General Frameworks for the Governing Equations for
Multiscale Models of Turbulent Combustion . . . . . . . . . . . 185
8.5 Multiscale Approaches in Turbulent Combustion and Preview of
Relevant Chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
8.5.1 Time-Step Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
8.5.2 Mesh Adaptive Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
8.5.3 Flame Embedding Approaches . . . . . . . . . . . . . . . . . . . . . . 187
8.5.4 Hybrid LES-Low-Dimensional Models . . . . . . . . . . . . . . . 188
8.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

9 Model Reduction for Combustion Chemistry . . . . . . . . . . . . . . . . . . . . . . 193

Dimitris A. Goussis and Ulrich Maas
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
9.2 Traditional Methodologies for Reduction: QSSA and PEA . . . . . . . 198
9.2.1 The QSSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
9.2.2 The PEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
9.2.3 Comments on the QSSA and PEA . . . . . . . . . . . . . . . . . . . 201
9.2.4 A Common Set-up for the QSSA and PEA . . . . . . . . . . . . 201
9.2.5 The Need for Algorithmic Methodologies for Reduction . 204
9.3 Reduction Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
9.4 Interaction of Chemistry with Diffusion . . . . . . . . . . . . . . . . . . . . . . . 208
9.5 Manifold Methods and Tabulation Strategies . . . . . . . . . . . . . . . . . . 209
9.5.1 Principles of Manifold Methods . . . . . . . . . . . . . . . . . . . . . 209
9.5.2 Calculation of Low-Dimensional Manifolds . . . . . . . . . . . 211
Contents xv

9.6 Tabulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

9.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

10 The Linear-Eddy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

Suresh Menon and Alan R. Kerstein
10.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
10.2 Triplet Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
10.3 Map Sizes and Frequency of Occurrence . . . . . . . . . . . . . . . . . . . . . . 223
10.4 Application to Passive Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
10.5 Application to Reacting Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
10.6 Application to Reacting Flows as a Subgrid Model . . . . . . . . . . . . . 228
10.6.1 The LEM Subgrid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
10.6.2 Large-Scale Advection of the Subgrid Field . . . . . . . . . . . 232
10.7 LEMLES Applications to Reacting Flows . . . . . . . . . . . . . . . . . . . . . 237
10.8 Summary and Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

11 The One-Dimensional-Turbulence Model . . . . . . . . . . . . . . . . . . . . . . . . . 249

Tarek Echekki, Alan R. Kerstein, and James C. Sutherland
11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
11.2 Constant-Property ODT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
11.2.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
11.2.2 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 255
11.2.3 Generalizations and Couplings . . . . . . . . . . . . . . . . . . . . . . 255
11.2.4 Features of the ODT Representation of Turbulent Flow . . 256
11.3 Applications of ODT in Combustion . . . . . . . . . . . . . . . . . . . . . . . . . 258
11.3.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
11.3.2 Stand-Alone ODT Simulations . . . . . . . . . . . . . . . . . . . . . . 261
11.3.3 Hybrid ODTLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
11.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

12 Unsteady Flame Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

Hossam A. El-Asrag and Ahmed F. Ghoniem
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
12.2 Historical Perspective on the Flame Embedding Concept . . . . . . . . 280
12.3 Elemental Flame Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 283
12.4 Numerical Solution for the Elemental Flame Model . . . . . . . . . . . . 286
12.5 UFE LES Sub-grid Combustion Model . . . . . . . . . . . . . . . . . . . . . . . 288
12.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
12.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
xvi Contents

13 Adaptive Methods for Simulation of Turbulent Combustion . . . . . . . . . 301

John Bell and Marcus Day
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
13.2 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
13.3 AMR Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
13.3.1 Creating and Managing the Grid Hierarchy . . . . . . . . . . . . 305
13.3.2 AMR Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
13.3.3 Hyperbolic Conservation Laws . . . . . . . . . . . . . . . . . . . . . . 307
13.3.4 Elliptic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
13.3.5 Parabolic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
13.4 AMR for Low Mach Number Combustion . . . . . . . . . . . . . . . . . . . . 315
13.5 Implementation Issues and Software Design . . . . . . . . . . . . . . . . . . . 319
13.5.1 Performance of Adaptive Projection . . . . . . . . . . . . . . . . . . 320
13.6 Application Lean Premixed Hydrogen Flames . . . . . . . . . . . . . . . . 321
13.6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
13.6.2 Models and Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
13.6.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
13.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

14 Wavelet Methods in Computational Combustion . . . . . . . . . . . . . . . . . . . 331

Robert Prosser and R. Stewart Cant
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
14.2 Wavelet Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
14.2.1 Orthogonal Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
14.2.2 Biorthogonal Wavelet Transforms . . . . . . . . . . . . . . . . . . . . 335
14.2.3 Second Generation Wavelets . . . . . . . . . . . . . . . . . . . . . . . . 336
14.3 Wavelets as a Method for DNS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
14.3.1 The Wavelet Representation of the Derivative . . . . . . . . . . 340
14.3.2 Higher Dimensional Discretizations . . . . . . . . . . . . . . . . . . 341
14.4 An Application of Wavelets to Reacting Flows . . . . . . . . . . . . . . . . . 343
14.4.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
14.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
14.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

Part IV Cross-Cutting Science

15 Design of Experiments for Gaining Insights and Validating

Modeling of Turbulent Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
A.R. Masri
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
15.2 The Turbulent Combustion Domain . . . . . . . . . . . . . . . . . . . . . . . . . . 358
15.3 Basic Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
15.3.1 Design Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
Contents xvii

15.3.2 Operational Envelopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

15.3.3 Experimental Considerations . . . . . . . . . . . . . . . . . . . . . . . . 364
15.3.4 Numerical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 366
15.4 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
15.4.1 The Swirl Stabilised Burner . . . . . . . . . . . . . . . . . . . . . . . . . 367
15.4.2 The Premixed Burner in Vitiated Coflows . . . . . . . . . . . . . 370
15.4.3 The Piloted Spray Burner . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
15.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

16 Uncertainty Quantification in Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . 381

Habib N. Najm
16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
16.1.1 Polynomial Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
16.1.2 Challenges in PC UQ Methods . . . . . . . . . . . . . . . . . . . . . . 389
16.2 Polynomial Chaos UQ in Fluid Flow Applications . . . . . . . . . . . . . . 392
16.2.1 Incompressible Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
16.2.2 Reacting Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
16.2.3 Compressible Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
16.2.4 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
16.3 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

17 Computational Frameworks for Advanced Combustion Simulations . 409

J. Ray, R. Armstrong, C. Safta, B. J. Debusschere, B. A. Allan and
H. N. Najm
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
17.2 Literature Review of Computational Frameworks . . . . . . . . . . . . . . . 410
17.3 The Common Component Architecture . . . . . . . . . . . . . . . . . . . . . . . 413
17.3.1 Features of the Common Component Architecture . . . . . . 414
17.4 Computational Facility for Reacting Flow Science . . . . . . . . . . . . . . 416
17.4.1 Numerical Methods and Capabilities . . . . . . . . . . . . . . . . . 416
17.4.2 The Need for Componentization . . . . . . . . . . . . . . . . . . . . . 417
17.5 Computational Investigations Using CCA . . . . . . . . . . . . . . . . . . . . . 420
17.5.1 Fourth-order Combustion Simulations with Adaptive
Mesh Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
17.5.2 Computational Singular Perturbation and Tabulation . . . . 425
17.6 Research Topics in Computational Frameworks . . . . . . . . . . . . . . . . 431
17.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433

18 The Heterogeneous Multiscale Methods with Application to

Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
Weinan E, Bjorn Engquist and Yi Sun
18.1 The Heterogeneous Multiscale Method . . . . . . . . . . . . . . . . . . . . . . . 439
xviii Contents

18.1.1 The Basic Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

18.1.2 The Seamless Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
18.1.3 Stability and Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
18.2 Capturing Macroscale Interface Dynamics . . . . . . . . . . . . . . . . . . . . 447
18.2.1 Macroscale Solver: The Interface Tracking Methods . . . . 447
18.2.2 Estimating The Macroscale Interface Velocity . . . . . . . . . . 448
18.3 HMM Interface Tracking of Combustion Fronts . . . . . . . . . . . . . . . . 451
18.3.1 Majdas Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
18.3.2 Reactive Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
18.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

19 Lattice Boltzmann Methods for Reactive and Other Flows . . . . . . . . . . 461

Christos E. Frouzakis
19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
19.2 The Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
19.2.1 Basic Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
19.2.2 Lattice Boltzmann Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
19.2.3 Variations on the LBM Theme . . . . . . . . . . . . . . . . . . . . . . . 470
19.2.4 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 472
19.2.5 Computational Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
19.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
19.3.1 Isothermal Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
19.3.2 Non-Isothermal Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
19.3.3 Multicomponent Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . 478
19.3.4 Reactive Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
19.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
List of Contributors

Benjamin A. Allan
Sandia National Laboratories, Livermore, CA, USA, e-mail: baallan@sandia.
gov
Rob Armstrong
Sandia National Laboratories, Livermore, CA, USA, e-mail: rob@sandia.gov
John B. Bell
Center for Computational Sciences and Engineering, Lawrence Berkeley National
Laboratory, Berkeley, CA, USA, e-mail: jbbell@lbl.gov
Robert W. Bilger
School of Aerospace, Mechanical and Mechatronic Engineering, The University of
Sydney, NSW 2006, Australia, e-mail: robert.bilger@sydney.edu.au
R. Stewart Cant
Department of Engineering, University of Cambridge, Cambridge, UK, e-mail:
rsc10@eng.cam.ac.uk
Matthew J. Cleary
The University of Queensland, School of Mechanical and Mining Engineering,
St Lucia, Queensland, 4072, Australia, e-mail: m.cleary@uq.edu.au
Benedicte Cuenot
Centre Europeen de Recherche et Formation Avancees en Calcul Scientifique
(CERFACS), Toulouse, France, e-mail: Benedicte.Cuenot@cerfacs.fr
Marcus S. Day
Center for Computational Sciences and Engineering, Lawrence Berkeley National
Laboratory, Berkeley, CA, USA, e-mail: msday@lbl.gov
Bert Debusschere
Sandia National Laboratories, Livermore, CA, USA, e-mail: bjdebus@sandia.
gov
xx List of Contributors

Weinan E
Department of Mathematics and Program in Applied and Computational
Mathematics, Princeton University, Princeton, NJ, USA, e-mail: weinan@math.
princeton.edu
Tarek Echekki
Department of Mechanical and Aerospace Engineering, North Carolina State
University, NC, USA, e-mail: techekk@ncsu.edu
Bjorn Engquist
Department of Mathematics, University of Texas at Austin, Autsin, TX, USA,
e-mail: engquist@math.utexas.edu
Christos E. Frouzakis
Aerothermochemistry and Combustion Systems Laboratory, Swiss Federal Institute
of Technology, Zurich, Switzerland, e-mail: frouzakis@lav.mavt.ethz.ch
Ahmed F. Ghoniem
Department of Mechanical Engineering, Massachusetts Institute of Technology,
Cambridge MA, USA, e-mail: ghoniem@mit.edu
Dimitris A. Goussis
School of Applied Mathematics & Physical Sciences, National Technical University
of Athens, Greece, e-mail: dagoussi@mail.ntua.gr
Daniel C. Haworth
Department of Mechanical and Nuclear Engineering, The Pennsylvania State
University, University Park, PA, USA, e-mail: dch12@psu.edu
Hossam A. El-Asrag
Department of Mechanical Engineering, Massachusetts Institute of Technology,
Cambridge MA 02139, e-mail: helasrag@mit.edu
Alan R. Kerstein
Sandia National Laboratories, Livermore, CA, USA, e-mail: arkerst@sandia.
gov
Alexander Klimenko
The University of Queensland, School of Mechanical and Mining Engineering,
St Lucia, Queensland, 4072, Australia, e-mail: a.klimenko@uq.edu.au
Andreas Kronenburg
Institut fur Technische Verbrennung, University of Stuttgart, 70569 Stuttgart,
Germany, e-mail: kronenburg@itv.uni-stuttgart.de
Ulrich Maas
Institut fur Technische Thermodynamik, Karlsruhe Institute of Technology,
Karlsruhe, Germany, e-mail: ulrich.maas@kit.edu
Assaad R. Masri
School of Aerospace, Mechanical and Mechatronic Engineering, The University of
Sydney, NSW 2006, Australia, e-mail: masri@aeromech.usyd.edu.au
List of Contributors xxi

Epaminondas Mastorakos
Department of Engineering, University of Cambridge, Cambridge, UK, e-mail:
em257@eng.cam.ac.uk
Suresh Menon
School of Aerospace Engineering, Georgia Institute of Technology, e-mail:
suresh.menon@aerospace.gatech.edu
Habib N. Najm
Sandia National Laboratories, Livermore, CA, USA, e-mail: hnnajm@sandia.
gov
Stephen B. Pope
Sibley School of Mechanical and Aerospace Engineering, Cornell University,
Ithaca, N.Y., USA, e-mail: s.b.pope@cornell.edu
Robert Prosser
School of Mechanical, Aerospace and Civil Engineering, University of Manchester,
Manchester, UK, e-mail: robert.prosser@manchester.ac.uk
Jaideep A. Ray
Sandia National Laboratories, Livermore, CA, USA, e-mail: jairay@sandia.
gov
Cosmin Safta
Sandia National Laboratories, Livermore, CA, USA, e-mail: csafta@sandia.
gov
Yi Sun
Statistical and Applied Mathematical Sciences Institute (SAMSI), Research
Triangle Park, NC, USA, e-mail: yisun@samsi.info
James C. Sutherland
Department of Chemical Engineering, The University of Utah, Salt Lake City, UT,
USA, e-mail: James.Sutherland@utah.edu
 Part I
Introductory Concepts
Chapter 1
The Role of Combustion Technology in the 21st
Century

R.W. Bilger

Abstract Continuing support for combustion research is threatened by those who

argue that continued use of fossil fuels is not sustainable. Reserves of fossil fuels are
in fact sufficient for several generations and their use can be continued if their green-
house gas emissions can be captured and sequestered or otherwise offset. Forecasts
for future energy technologies foresee considerable reductions in the role for com-
bustion technologies. It is important that combustion researchers become involved
in such forecasting so that viable combustion technologies such as carbon capture
and sequestration do not become sidelined by over-optimistic projections for photo-
voltaics and the hydrogen economy. It is evident that many energy technologies will
undergo at least two stages in transition to achieving the goals that are needed by
2050. Furthermore, there will be many factors such as geography and economic and
policy changes that will have drastic effects on the marketability of energy systems.
Combustion systems will need to be able to respond quickly to these rapidly chang-
ing markets. There is a vitally important role for advanced computer modelling in
meeting this challenge. Current combustor modelling capability is of only periph-
eral use in the development of new combustion systems. The main problem is in the
modelling of turbulence chemistry interactions. Considerable investment is needed
in the development of advanced modelling approaches and in high quality measure-
ments of an hierarchy of experimental data bases that will provide physical insights
and become a basis for model validation.

1.1 Introduction

For tens of thousands of years combustion has been the main means of providing
heating for human comfort and for the cooking of food. For about five thousand
years combustion has been the means for materials processing - producing pottery,

Department of Mechanical and Mechatronic Engineering, University of Sydney, NSW 2006, Aus-
tralia, e-mail: robert.bilger@sydney.edu.au

T. Echekki, E. Mastorakos (eds.), Turbulent Combustion Modeling, 3

Fluid Mechanics and Its Applications 95, DOI 10.1007/978-94-007-0412-1 1,
Springer Science+Business Media B.V. 2011
4 R.W. Bilger

bronze, cement, iron, steel, and many other materials associated with the rise of
human civilization. It is only a few hundred years since mechanical power from hu-
mans, animals, wind and water has been surpassed by the mechanical power from
combustion engines - steam, spark-ignition, diesel and gas turbine. Much of this me-
chanical power is now used for generation of electricity, transportation of materials
and for personal mobility.
Combustion is a means of transforming primary energy into useful secondary
energy such as heating, processed materials and mechanical and electrical power.
For most of history the primary energy consumed in combustion has been wood,
charcoal and other so-called bio-fuels. Only after the steam engine was stripping
bare the forests of Europe did coal become the main source of primary energy in
the industrialized world, later to be augmented by oil and natural gas. For some two
centuries now the main sources of primary energy have been these fossil fuels.
Fossil fuels are so named as the energy they embody is energy from the sun -
solar energy - that was originally captured in plants of various kinds and has been
stored for millions of years in fossil form. This source of primary energy has proved
to be a great boon to mankind. It soon became evident that fossil energy was far less
costly to obtain than wood, charcoal and other bio-fuels. The technology needed to
combust fossil fuels was quickly found - no doubt because these fuels are far more
energy intensive than wood, charcoal and other bio-fuels.
Coal and other fossil fuels contain sulphur and the fumes from their combustion
contain carcinogens and toxins not found in wood smoke. Wood smoke contains par-
ticulates and tars that seemingly should be very harmful - but perhaps over 100,000
years the genes of humans have been selected to provide most people with some
tolerance to this cocktail. Concerns about air pollution in the troposphere affecting
human health and amenity have largely been associated with fossil fuels. In the 20th
century laws based on torts between individuals became surpassed by air pollution
laws aimed at preserving human health and amenity in urban air basins.
From their very nature, it is certain that only finite amounts of fossil fuels have
been stored in the earths crust. Our consumption of fossil fuels has been rising
exponentially. An important question arises as to whether our consumption will
soon deplete these storages by significant amounts. This question is compounded
by the fact that the geographical locations of the storages as currently known are
some distance from the locations where their consumption is highest. This fact im-
mediately transforms questions of global sufficiency into ones of national energy
security.
The dominant constituent of fossil fuels is the element carbon. Combustion of
fossil fuels turns essentially all of this carbon into carbon dioxide. Carbon dioxide
mixed into the local atmosphere has essentially no direct effect on human health or
on the local amenity of humans or of their immediate environment. Carbon dioxide
levels in the global atmosphere have been rising. There is now widely-accepted evi-
dence that human caused emissions of carbon dioxide are directly linked to increases
in global average temperature levels, depletion of arctic ice floes and many other
climate and environmental changes of serious concern. This conclusion is strongly
disputed by a small group of climate change skeptics or deniers.
Introduction 5

It is seen that the use of fossil fuels is directly and intimately connected with
tropospheric air pollution, energy security and climate change. These are three of
the most important challenges for the world in the 21st century.
A common facile response to these challenges is for proposals to ban the use
of fossil fuels. The world has agreed that we must only have development that is
sustainable [24]. In this facile response it is argued that since fossil fuels are being
depleted they cannot be sustainable - only renewable sources of energy can be sus-
tainable. In reality, many renewable sources of energy are not sustainable solutions
to the worlds energy needs and fossil fuels can be the preferred sustainable option
for the way ahead. This is examined in more detail in the next section.
Our main concern here is with combustion: its technology and its underlying
science. With some 100,000 years of development, it might be expected to be a ma-
ture technology. In fact, it is the least developed technology of modern engineering
systems. Engineering systems with mature technology have computer models that
are routinely used in analysis and can be used in the optimization of the geometry
and operating conditions of new designs. Thus the wings of new models of jet air-
craft have subtle geometry contours that greatly increase their efficiency. The same
can be said for the new generation of fans in turbo-fan engines. Computer mod-
eling of combustors is still at a fledgling stage and is only used peripherally in the
development of new combustors. Combustor development is still largely by cut-and-
try testing in experimental rigs and in prototype and in-service engines and power
plants. Mongia [21] reviews important concepts and methodologies for usefully in-
tegrating CFD into the design process for gas-turbine combustors in a commercial
engineering environment.
The fledgling status of computer modelling of combustors comes not from the
want of trying. More than forty years of effort by some of the leading scientists
in the world has still not got us within close sight of achieving the goal of having
soundly based predictive models for combustors that can be used in industry with
confidence on a routine basis. This is because it is a multi-scale problem requir-
ing advanced models for highly turbulent recirculating and swirling fluid flows, for
gaseous and particulate fuel dispersion and mixing and for the extremely complex
chemical kinetics involved in the oxidation of hydrocarbon fuels and the formation
and burnout of pollutants such as soot, polycyclic hydrocarbons, carbon monoxide
and oxides of nitrogen. Furthermore, these highly non-linear processes are inher-
ently and strongly coupled together. A particularly difficult coupling is that between
turbulence and chemical reaction [7]. The main chapters in this book summarize the
variety of approaches that are being developed to overcome this particular difficulty
in the modeling of practical combustion systems.
The question arises as to whether efforts to improve the capabilities for com-
puter modeling of combustors are a worthwhile use of scientific resources. Some
authorities argue that combustion will quickly lose the dominant role that it has
historically had due to the widely accepted need for decarbonization of energy pro-
cesses to mitigate the causes of climate change - investment in improving such tools
for combustion systems is wasting money on a technology that has already passed
its potential usefulness. A main purpose of this introductory chapter is to explore
6 R.W. Bilger

the nature of such technology forecasting and whether it is prudent to make such
summary dismissals of combustion technology.
Another main purpose is to outline the prospects for advanced computational
modeling of combustors in practical systems and to identify the major thrusts needed
in research.

1.2 Sustainable Energy

All of the significant nations in the world have ratified the United Nations Frame-
work Convention on Climate Change [24] the so-called Rio Convention. They are
committed to stabilization of greenhouse gas concentrations in the atmosphere at a
level that would prevent dangerous interference with the climate system. The Rio
convention also defines sustainable development as development that meets the
needs of the present without compromising the ability of future generations to meet
their own needs.
In the 1970s we had two energy crises associated with wars in the Middle East.
Fear was engendered that we were running out of oil as proven reserves were only 30
times the then current rates of consumption. The movement for renewable sources of
energy was born, including solar thermal, solar photovoltaic, hydroelectricity, wind
turbines, and combustion of biomass. In the 1990s renewable energy was widely
touted as being the only truly sustainable source of energy and should play the major
role in the mitigation of climate change.
Today proven reserves of oil have not been exhausted as feared in the 1970s but
are now 40 times our much increased current rates of consumption [26]. Proven re-
serves for oil, natural gas, coal, oil shale, tar sands and other fossil fuels are such
that they could provide the worlds energy needs for more than a century at current
rates of consumption [26]. The cost of exploration for oil is high and it is not prof-
itable to spend money today to prove reserves that will not be producing income for
more than a few years ahead. Storages of fossil fuels as yet unproven are likely to
be many times those already proven. Even with the industrialization of China, India,
Brazil and other emerging economies it is certain that there is enough energy from
fossil fuels to meet world needs for several generations to come. It may be that oil
will be more short-lived, but liquid hydrocarbon fuels can readily be obtained from
coal, oil shale and tar sands at modest cost. It can be firmly concluded that fossil
fuels are not to be disqualified as being sustainable on the grounds that they will be
exhausted within a few generations.
It is clear, however, that continued use of fossil fuels is un-sustainable if the car-
bon dioxide produced upon combustion needs to be contained or offset. Without
such containment or offsetting, it is widely accepted that significant interference
with the climate system will occur and the ability of future generations to meet
their own needs will be greatly compromised. It is also clear that use of fossil fu-
els can qualify as being sustainable where carbon capture and storage technologies
[5, 14] are used or where their carbon dioxide emissions are offset by black carbon
Introduction 7

sequestration [11] or other such means of extracting and storing carbon from the
atmosphere. On the other hand, if the climate change skeptics are correct, the con-
tinued use of fossil fuels in this century will be sustainable but policy will be largely
focused on energy security issues.
Renewable energy sources appear to easily qualify as being sustainable on the
basis of their carbon dioxide emission contributions to climate change. Several of
them may not be sustainable on the basis of other aspects of sustainability associated
with meeting the needs of the present and compromising the ability of future
generations to meet their own needs. Significant early investment in photovoltaic
generation of electricity will require huge amounts of extra capital that may be more
sorely needed to fight disease and starvation in under-developed parts of the world.
Bio-fuel production requires land and water resources that may be better used for
food production or for conservation of ecosystems that sustain endangered plant and
animal species that would be lost to future generations. It has long been recognized
that hydroelectric dams do great damage to local ecosystems and even to estuarine
and coastal systems that may be many hundreds of kilometers downstream. It is
clear that many renewable sources of energy may be less sustainable than using
fossil fuels with carbon capture and storage or offset technologies.

1.3 Technology Forecasts

In 2006 the US Department of Energy held a workshop on the Basic Research

Needs for Clean and Efficient Combustion of 21st Century Transportation Fuels
[19]. The workshop focused on new fuels and engines needed to meet the challenge
of energy security. The new fuels considered included oil from coal, shale and tar
sands. New engines considered included the homogeneous charge compression ig-
nition (HCCI) engine. There was no direct consideration given to needs related to
severe mitigation of greenhouse gases (GHGs). With a change of administration in
the USA much more emphasis is now placed on mitigation of GHGs. A recent DOE
workshop [12] recognizes this need for GHG mitigation and places considerable
emphasis on the prospects for advanced bio-fuels.
Technology forecasts are very dependent on whether the already expressed gov-
ernment policies of containing carbon emissions will be followed through. The
UNCCC meeting in Copenhagen in December, 2009 failed to come up with a bind-
ing resolution extending the Kyoto protocol. Much of this failure of outcome can
perhaps be attributed to the campaign mounted by the climate change skeptics. It
seems that this campaign may have greatly defused international attempts to put
world-wide regulation of carbon emissions in place.
There is small list of scientists who dispute the conclusions of the IPCC reports
on climate change1 . Essentially all of these doubts have been put forward in news-
1 Wikipedia lists some 40 or more of them (http://en.wikipedia/wiki/:26/03/2010).

Their doubts are grouped there under the following headings: 1) Global warming is not occurring
(3 scientists), 2) Accuracy of IPCC climate projections is questionable (4 scientists), 3) Global
8 R.W. Bilger

papers, books and the general electronic media rather than in anonymously peer-
reviewed scientific journals.2
It is evident that much confusion about the validity of projections of climate
change science has been engendered in the USA, Australia and many other coun-
tries. It is far from certain that international agreement will be reached on carbon
emission targets and the international governance needed to implement them. Of
perhaps more importance is that emission trading schemes (ETS) or cap-and-trade
(CaT) legislation and other such measures intended to provide a level playing field
for the the development of GHG control technologies may not succeed. It seems
that, although they are the most sensible economic measures, they are too easily
branded as high tax in Western democracies. The failure to implement ETS or
CaT is likely to severly affect the prospects for carbon capture and storage (CCS)
technologies that need such legislation to underwrite the large capital investment
involved.
The International Energy Agency (IEA) of the OECD has made a comprehensive
study of the energy technologies that are likely to succeed in the transition to the low
carbon emissions regime that has been projected to be needed by the year 2050 [13].
They consider two scenarios: the ACT scenario uses currently available technologies
and brings world greenhouse gas emissions down to the 2005 level of 27 GtCO2e/yr;
and the BLUE scenario that brings GHG emissions down to half this level or 14
GtCO2e/yr but requires the development of new technologies that are little beyond
proof of feasibility at this stage. World growth in energy demand by 2050 would
result in baseline emissions of 62 GtCO2e/yr if the technologies of 2005 were kept
in place. The study was requested by the G8 members of OECD and has taken into
account the capital investment needed, the availability of the resources needed and
feasible rates of commercial deployment. Figure 1.1 shows the marginal emission
reduction cost in USD/tCO2 as a function of the emissions reduction relative to
baseline. It is seen that improvements in end use efficiency can, in fact, be profitable,
and that reducing emissions from the electrical power generation sector will cost up
to 30 USD/tCO2 which is about the current European carbon trading price.
Controlling GHG emissions from industry and the transport sector are seen to
be very expensive at 50500 USD/tCO2. It should be noted that GHG emission
reductions from the transport sector by downsizing cars and their engines are seen as
belonging to the increasing end use efficiency sector. The predictions for the power
generation sector see little contribution from carbon capture and storage and from
combustion of biomass as the sector becomes dominated by wind, photo-voltaic

warming is primarily caused by natural processes (24 scientists), 4) Cause of global warming
is unknown (7 scientists), 5) Effects of global warming will not be significantly negative (3
scientists)
2 There are many web sites that refute these claims - see for example: 1) http://www.

grist.org/article/series/skeptics, http://www.astronomynotes.com/
solarsys/s11b.htm, http://scientificamerican.com/article.cfm?id=
seven-answers-to-climate-contrarian-nonsense. It seems that such refutation
is also confined to the public media rather than to the anomynously peer-reviewed scientific
literature. Meehl et al. [20] is, however, quoted in the Wikipedia summary as the widely accepted
response on the effects of natural causes.
Introduction 9

Fig. 1.1: Marginal emission reduction costs for the global energy system (2050).
Reprinted from [13] with permission. Copyright 2010, OECD/IEA.

electricity and nuclear power. In the transport sector, it is noteworthy that bio-fuels
are seen as being reserved for large trucks, ships and aircraft. Fuel cells and battery
electric vehicles are seen as being the power systems for cars. It is evident that this
reflects the limitations on land, water and other resources in competition with needs
for food supply. Overall it is predicted that use of combustion technologies in energy
transformation will decline from present levels of about 80% on a world wide basis
to less than 30%. The predictions are thus supportive of those that question further
investment in improving computational aids for combustor design.
Such studies of the potential of various energy technologies are of great interest
and importance for government policy makers. The UK Energy Research Centres
mission is to be the UKs pre-eminent centre of research, and source of information
and leadership on sustainable energy systems. The Centre takes a whole systems ap-
proach to energy research, incorporating economics, engineering and the physical,
environmental and social sciences while developing and maintaining the means to
enable cohesive research in energy [25]. The Centre is based at the University of
Edinburgh with satellite contributors, mainly from other universities. Their major
project is the UKERC Energy 2050 project which is exploring how the UK can
move towards a resilient low-carbon energy system over the next forty years. Their
second report on this project [25] forecasts energy technologies for 2050 that are
similar in nature to those of the IEA predictions. They also predict a low contribu-
tion from combustion technologies.
It is apparent that there are mounting predictions against combustion having a
major role in energy systems in the middle and latter half of this century. It is evi-
dent that the combustion research community will, in the not too distant future, have
a challenging job to keep university and government administrators convinced that
this difficult area of engineering science merits further investment in research. It is
important that the combustion research community engage itself in this emerging
research area of energy technology prediction so that we do not get sidelined by en-
10 R.W. Bilger

thusiastic protagonists for energy systems that are inherently more costly to develop
to commercial viability and require distorted subsidies or market guarantees to help
them to get into the marketplace.
On a more detailed reading it seems that the IEA and UKERC studies may have
fallen into this trap. Figure 1.2 shows the projections assumed for the capital cost of
photo-voltaic systems in the UKERC study [25]. It is seen that an order of magnitude
decrease in capital costs is assumed, similar in magnitude for the semi-conductor in-
dustry. It seems to the author that the photo-voltaics community have been making
such projections for several decades now and have not been able to show such learn-
ing curves despite large government investment. Kazmerski [15] argues that such
large advances in the capital cost of solar photovoltaics will require a break-through
or tipping point in the fundamental science of photovoltaics.

Fig. 1.2: Capital costs of photo-voltaic systems. Adapted from [25].

In the IEA and UKERC studies, photo-voltaic generation of electricity is pro-

jected to replace carbon capture and storage (CCS) for power generation. Assump-
tions for CCS on the other hand see little reduction in costs over this period and
seem to be lumbered with a further assumption that they will be limited to only
90% capture which is a severe penalty in reaching the IEA BLUE scenario and the
UKERC 80% reduction scenario. They foresee no dramatic advances in CCS tech-
nology over the next 40 years.
Projections of this sort are fraught with much uncertainty beyond a decade or two.
Optimistic projections for photo-voltaic systems, batteries and fuel cells may not be
realized. There is always the prospect that technologies yet to be discovered or now
in early stages of development will emerge as winners and play a very important role
in two or three decades from now. Novel technologies for carbon capture in the CCS
approach have, in my view, a high potential for such success. These technologies
are likely to involve the development of entirely new combustion systems and thus
provide a challenge to combustion researchers.
Introduction 11

Oxy-fuel firing is a combustion technology that has promise [14] of providing

carbon dioxide capture at costs much lower than those for systems involving scrub-
bing of the power plant flue gases. Most proposals for oxy-fuel firing involve recycle
of CO2 to control combustion temperatures. In the Clean Energy Systems realization
of this technology [2, 14] it is H2 O that is recycled. Thermodynamically this system
has prospects for much higher cycle efficiencies as heat addition temperatures are
not limited by the need to transfer heat through a boiler tube wall and heat rejec-
tion temperatures can be low due to condensation of the steam at sub-atmospheric
pressure. As in the Rankine cycle, the costs of compression of the H2 O as water
are much reduced as compared with the costs of compression of N2 or CO2 as the
diluent in the Brayton cycle part of a Gas Turbine Combined Cycle (CTCC) system.
The author has given the name of the internal combustion Rankine cycle (ICRC)
to this cycle [6]. Even accounting for the energy needed for air separation, cycle
efficiencies can be expected to be not much less than those of current day GTCCs.
Furthermore, capital costs will be significantly lowered due to the use of a pump in-
stead of a N2 compressor and the elimination of the waste heat boiler. It seems clear
that this system should be capable of providing CCS at much less than $30/tCO2
when fully developed. Novel combustion systems are used for the primary steam
generator combustor and for the reheat combustor. Current designs of these com-
bustors appear to be quite rudimentary and can be expected to evolve considerably
with the use of advanced combustion modeling tools.
Reciprocating engine versions of the ICRC cycle have been studied [6, 8]. Such
engines have application at lower power ratings than for the turbine system and
could be suitable for stationary power plants up to 10MW, for combined heat and
power (CHP) applications in buildings and industry, for ships, railway locomotives,
trucks, buses, and even cars. Bilger & Wu [8] consider the application to a small car
with a 50 kW power plant. Oxygen is carried in a high pressure storage tank which
is also used to collect the CO2 that is captured. Conventional gasoline or diesel is
used, or liquids from coal, shale or tar sands or bio-fuels. Increases in power plant
volume weight and cost are predicted to be only moderate compared with those for
current uncontrolled cars and range and acceleration performance are little affected.
They will far out-perform fuel-cell and electric vehicles. Control of GHG emissions
will be close to 100% even using fossil fuels. Control costs are estimated to be about
$7/tCO2 which is almost two orders of magnitude less than is forecast for fuel cells
and electrics (see Fig. 1, above). So far these are only paper studies. A demonstration
of the engine is planned at Tongji University using a modified motor cycle engine.
The design of the combustion system involving ignition of a fuel oxygen mixture
and injection of a water spray is very challenging. Development of the engine will
be greatly aided by advances in combustion modeling tools.
It is apparent that there will be major changes in the energy technologies within
the next 40 years driven by the need to drastically reduce GHG emissions but also
for improving energy security and decreasing tropospheric air pollution. It is also
apparent that this will not be a single smooth transition from current technologies
to the technologies needed by 2050 to provide a long-term sustainable energy sys-
tem mix. This is because near-term reductions in emissions are needed as well as the
12 R.W. Bilger

achievement of a low long-term goal [22]. The near-term reductions can be achieved
with existing technologies and technologies that are well-advanced in their devel-
opment. Many of these technologies will not survive in the long term as they cannot
provide sufficient control, will not be cost effective compared with newly developed
technologies or there will not be sufficient availability of input resources.
An illustration of this can be seen for the transportation sector. This sector
amounts to some 40% of current total energy consumption and early curtailment
of GHG emissions from this sector is important. This early containment can most
readily be obtained by regulating the fleet average emissions from new cars. The
EU has proposed a limit of 130 gCO2/km for new vehicles sold in 2012 and 95
gCO2/km for 2020. These limits can be met by down-sizing of vehicle mass and
engine power, using advanced engines such as the HCCI engine [10], using hybrids,
and with the use of bio-fuel blends. The 2020 goal is only about a 35% reduction
below current total fleet, old and new, of European emissions. The 2050 goal is for
an 80% decrease overall on a world wide fleet double that of the current size, im-
plying a per vehicle reduction of 90% averaged over the then fleet, new and old. It is
apparent that by 2050 new cars sold must be essentially zero emission vehicles. By
2050 it is also apparent that bio-fuels will not be able to provide this level of control
for the world-wide transportation fleet. The IEA study [13] projects that bio-fuels
will only be viable for aeroplanes, ships and large trucks. There will be insufficient
availability of land and water resources to produce more. So, bio-fuels and new con-
ventional engine technologies will not have a major role in the 2050 transportation
energy mix. Hydrogen fuel cells and electric vehicles, both deriving their energy
from nuclear, wind and photo-voltaic systems are thus seen as the end-of-transition
mix. It is clear that the transition to the long-term energy mix for transport will be
at least a two stage process.
Multi-stage transition of energy technologies in other sectors such as power gen-
eration can also be expected. In the near term co-firing of biomass will increase
together with increasing contributions from wind power. In the mid term nuclear
power will play an increasing role together with conventional capture for CCS. In
the long term photo-voltaics may supplant advanced CCS systems, but this is far
from certain.

1.4 Implications for Combustion Technology

It is seen that a variety of market forces and government regulations will strongly
influence combustion system design in the near term and in the longer term. This en-
vironment can be expected to be highly volatile due to strong fluctuations in prices
of primary energy, economic and political cycles and a host of other factors. Further-
more, geographical factors are likely to be important with technologies suitable for
Europe or the USA possibly not being appropriate for China or Japan. It is apparent
that strategies for survival of commercial enterprises in the energy technology sec-
tor must include a broad scope of inherently different products that are each being
Introduction 13

Table 1.1: Status of current use of computational modelling in thermal systems in

commercial engineering enterprises. The number of () indicates the confidence in
computing certain physics. For example, 5 ()s indicate everyday use with very
high confidence on quantitative prediction; 3 ()s indicates frequent use with engi-
neering value for qualitative information on trends; and 1 () indicates occasional
use with only marginal input into the design and development process.

Status Physics
 Stress and deformation prediction.
 Heat transfer and solid temperature prediction.
 Fluid flow and convective heat transfer prediction in the
absence of combustion.
 General flow and mixing prediction of combustion systems.
 Prediction of finite-rate chemistry effects, such as ignition,
extinction, and pollutant formation of CO, NOx, soot, etc.

rapidly developed to keep pace with the competition. It will not be prudent to bet
the firm on a single solution, either in the near term or in the longer term.
For energy technologies involving combustion this will focus attention on the
ability to rapidly develop combustors. This will include evolutions of existing de-
signs to meet increasingly stringent regulations and market changes and for the de-
velopment of entirely new concepts such as for CCS using oxy-fuel firing in power
generation [2] and transportation [8].
The focus on combustor development will be intensified by the fact that most
other elements of thermal systems are much further advanced in being able to be
successfully modeled computationally. This can be seen from the assessment given
in Table 1.1.
This assessment is the authors based on numerous contacts with colleagues in in-
dustry, academe and government research laboratories. It is seen that computational
modeling of combustors is in only a fledgling state and its role is a secondary one in
combustor development in commercial enterprises. Combustor development is still
largely by cut-and-try testing in experimental rigs and in prototype and in-service
engines and power plants.
It is evident that the fledgling status of computational modeling of combustors
will be a severe handicap for the development of energy technologies involving
combustion in the 21st century. The 2006 DOE workshop [19] identified a single,
overarching grand challenge: the development of a validated, predictive, multi-scale,
combustion modeling capability to optimize the design and operation of evolving
fuels in advanced engines for transportation applications. Such modeling capability
is also sorely needed in the power generation and materials processing sectors of
energy technologies.
14 R.W. Bilger

1.5 Prospects for Advanced Computer Modeling of Combustors

As has already been stated in the Introduction, the fledgling status of computer mod-
eling of combustors comes despite much effort over many years. It is an extremely
complex multi-scale problem.
In terms of prescribed input length scales: overall dimensions of combustors are
of order of 0.1 to 10 meters; details of their geometry affecting the flow are of order
of 102 of this; and the dimensions of liquid or solid fuel particles are of order
a further factor of 102 lower, making them 104 of the overall dimension. Fluid
dynamic length scales, at the Reynolds numbers of practical systems, range from
these dimensions down to a further factor of 102 for each of these prescribed input
length scales. For example the boundary layers on injection orifices will be of order
102 of the orifice size, and the boundary layers on fuel particles will be of 102
of the particle size. Reaction zone length scales at the high Damkohler numbers of
practical combustion systems can be even thinner than these fluid dynamic length
scales, say 101 . This amounts to 107 overall, so far. Length scales associated with
formation of particulates such as soot are of this order or even smaller.
In terms of prescribed input time scales: the residence time in combustors is of
order 10 ms to 1 s; acoustic time scales can be of order 102 to 10 times this; and
particulate fuel combustion time scales are of order 102 of this. At the Reynolds
numbers of practical systems, fluid dynamic time scales range from these dimen-
sions down to a further factor of 102 for each of these prescribed input time scales.
Chemical time scales associated with ignition and extinction phenomena are of or-
der 102 of the residence time for the combustor as a whole and also 102 of particu-
late fuel combustion time scales for phenomena associated with ignition/extinction
phenomena associated with individual particles. Chemical time scales associated
with pollutant formation and burnout can be of order 102 of the combustor resi-
dence time.
Important physics, including chemical physics, can be associated with any com-
bination of these length and time scales.
Advances in computational power can be expected to continue. Even if the ad-
vances continue at present exponential rates, it is evident that full direct numerical
simulation (DNS) is not possible for practical combustion systems in the foresee-
able future; certainly not for the next two decades of the developments needed for
combustion technology responses to GHG reduction for climate change and energy
security in the near-term. It should be remembered that, to be an effective design
tool in industry, a 20-hr turnaround time, or better, is needed. Perhaps new devel-
opments in computers, such as quantum computing [1], will become available in
the longer term and be available to provide the computational tools in DNS that are
needed before 2050. Such developments cannot be counted on, however.
Advanced computer methods for combustors will thus, inevitably, involve mod-
eling of many of the important physical aspects of the flow that we cannot afford
to resolve in our computations. The relative importance of these physical aspects
to be modeled will depend on the application and on the information being sought.
Thus predicting the altitude ignition behavior of a helicopter gas turbine engine will
Introduction 15

require a different modeling focus from that needed to predict re-heater tube foul-
ing in a boiler co-firing coal and biomass. The prediction of NOx emissions from
the same helicopter engine will require a different modeling focus from that needed
for ignition prediction: as will the prediction of flame stability in the co-fired boiler
from that needed for tube fouling.
This diversity in application and purpose of combustion modeling of practical
systems, and the perceived economic importance of this research area, is reflected
in the very diverse range of research efforts directed at advancing combustion mod-
eling. Table 1.2 lists a categorization of those involved.

Table 1.2: Current provenenance of turbulent combustion codes.

Code categories Examples

Public domain codes KIVA.
Government lab codes NASA, CERFACS, IFP, CSIRO, DSTO.
Commercial codes Fluent/CFX, Adapco/STAR-CD.
Corporate codes Ferrari, GE, Rolls-Royce, AVL, Ricardo.
Academic codes Furnace, BOFFIN.

It is apparent that there are several hundred code developers around the world
involved in advancing computer predictions for combustion flows. It is also apparent
that most of these efforts are of a man and boy nature and there is no overall co-
ordination.
From the wider perspective of the challenges facing the world in combustion
technology in the 21st century, it would seem that a more concerted international
effort is needed. Computational modeling is a tool and so in the hierarchy of re-
search and development would seem to fall in the pre-competitive range of this
path to innovation in energy systems. If this is so, then it seems that the world would
be better served if there was open and co-operative interchange - world wide - on
developments in codes. And yet this is far from the case: computer codes are viewed
as intellectual property, are subject to copyright law and in most cases are jealously
protected and only communicated under stringent contractual arrangements at quite
high prices.
A closer look at the situation, however, suggests that efficient coding and fast
numerical algorithms are not the main limitations to advancement of combustion
codes. These aspects can be quite significant. Of more importance, however, is the
validity of the modeling approach and the sub-models that are used for the essen-
tial physics (including chemical physics) involved in the particular application. The
fledgling status of the use of combustion prediction in industry depicted in Table 1.1
above is widely attributed to inadequacies in capturing the right physics in the mod-
eling. This is often attributed to computational limitations that prohibit the inclu-
sion of more detailed modeling of important physical processes. More often, it is
acknowledged that the modeling approach is lacking in validity.
16 R.W. Bilger

An important problem area in such modeling is that associated with turbulence-

chemistry interactions. Bilger et al. [7] survey the modeling approaches and com-
bustion sub-models that were of major interest at that time. The emphasis was on
gaseous combustion - non-premixed, premixed and partially premixed. Here, the
main chapters in this book outline and update these approaches and others that may
have potential usefulness. It is noted that it is unlikely that a single modeling ap-
proach will be able to be adequate for all applications: it is more likely that different
approaches will be needed for different applications.
The most important area is undoubtedly that of validation. All too often com-
puter model predictions are validated against sparse experimental data for full-
scale combustors or for laboratory combustors that have some of the flow and other
complexities of practical systems. Such data can often be adequately matched by
choices made for boundary conditions that are unfettered by experimental measure-
ment. Little information is gained about the adequacy of the modeling approach or
of the combustion models used. A more scientific approach is to devise experiments
in which the flow is simple enough for the boundary conditions to be adequately
measured and for which the crucial physics that need modeling are dominant.
The Sandia flame series of experiments [4] using the Sydney [16, 23] burner
for piloted jet diffusion flames are an outstanding example of the sort of valida-
tion experiments that are needed. The burner was designed to provide clear optical
access to the regions of interest in the flame. The pilot was designed to be large
enough so that the region of strong coupling between mixing and chemical reaction
would be delayed beyond the near field where transitional turbulent flow associated
with coherent structures would be located. The upstream inflow boundary condi-
tions could be adequately measured. The data from these experiments has proved
to be a widely-recognized stringent test of the adequacy of the modeling of the
coupling between turbulence and chemistry in such flows. It has led to a series
of international workshops on turbulent non-premixed flames [3] in which there
has been a robust interaction between modelers and experimentalists that has led
to significant advances in both experimentation and modeling. Lifted flame stabi-
lization [9], auto-ignition and forced ignition [18] in such flows are further steps
in this hierarchy. These more complex problems still have relatively simple geom-
etry and flow structure. Their study builds on the gains in experimental accuracy
and improvements in modeling arising from the jet-flow experiments and provides
an important stepping stone to the modeling of practical combustors. Bluff-body
and swirling flames have been recognized as next steps in an hierarchy of such test
problems and measurements have been made and some modeling predictions are
becoming available [3]. Such flows where there is not a dominant advective time
scale giving a direct analogue for the chemical time scales provide an important
challenge to unsteady flamelet models and several of the new multi-scale modeling
approaches.
It is only with such an hierarchical approach to validation that sure progress on
advanced computational modeling for practical combustors can be made. Such an
approach for systems with premixed combustion is yet to be satisfactorily defined
[7]. The experiments at Sydney University [17] on spray jet mixing and combustion
Introduction 17

appear to be a firm basis for such an hierarchy for the combustion of particulate
fuels.

1.6 Concluding Remarks

It is apparent that the dominant role of combustion technology in conversion of

primary energy into secondary energy may be greatly reduced by the middle of
the 21st century. It is important that the combustion community become actively
engaged in the new research area of energy technology forecasting so that viable
combustion technologies such as CCS do not become sidelined by over-optimistic
projections for photo-voltaics and the hydrogen economy.
The fledgling status of computational modeling of practical combustion systems
will be a seriously limiting factor in the ability of combustion technology to meet the
21st century challenges of climate change, energy security and tropospheric pollu-
tion. The advances needed in such computational modeling for practical combustion
systems will require considerable investment in modeling research in this area, in-
cluding the new paradigms of multi-scale modeling. It will also need an increased
emphasis on research directed at improving experimental data bases for model val-
idation that are of an hierarchical nature and start with relatively simple problems
that encapsulate the most important physics.

Acknowledgements

This research is supported by the Australian Research Council.

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Chapter 2
Turbulent Combustion: Concepts, Governing
Equations and Modeling Strategies

Tarek Echekki and Epaminondas Mastorakos

Abstract The numerical modeling of turbulent combustion problems is based on

the solution of a set of conservation equations for momentum and scalars, plus ad-
ditional auxiliary equations. These equations have very well-defined foundations
in their instantaneous and spatially-resolved forms and they represent a myriad of
problems that are encountered in a very broad range of applications. However, their
practical solution poses important problems. First, models of turbulent combustion
problems form an important subset of models for turbulent flows. Second, the react-
ing nature of turbulent combustion flows imposes additional challenges of resolution
of all relevant scales that govern turbulent combustion and closure for scalars. This
chapter attempts to review the governing equations from the perspective of modern
solution techniques, which take root in some of the classical strategies adopted to
address turbulent combustion modeling. We also attempt to outline common themes
and to provide an outlook where present efforts are heading.

2.1 Introduction

The subject of turbulent combustion spans a broad range of disciplines. The com-
bination of the subject of turbulence on one hand and that of combustion already
reveals the daunting task of predicting turbulent combustion flows. At the heart of
the challenge is the presence of a broad range of length and time scales spanned
by the various processes governing combustion and the degree of coupling between
these processes across all scales.
Bilger et al. [4] have discussed the various paradigms that have evolved over the
years to address the turbulent combustion problem. A running theme among these

Tarek Echekki
North Carolina State University, Raleigh NC 27695-7910, USA, e-mail: techekk@ncsu.edu
Epaminondas Mastorakos
Cambridge University, Cambridge, CB2 1PZ, UK, e-mail: em257@eng.cam.ac.uk

T. Echekki, E. Mastorakos (eds.), Turbulent Combustion Modeling, 19

Fluid Mechanics and Its Applications 95, DOI 10.1007/978-94-007-0412-1 2,
Springer Science+Business Media B.V. 2011
20 T. Echekki and E. Mastorakos

paradigms is the separation of scales to overcome the coupled multiscale complex-

ity of turbulent combustion flows. In many respects, these strategies have been suc-
cessful for a large class of problems and enabled the use of computational fluid
dynamics (CFD) for the prediction and design of combustion in practical devices.
The review by Bilger et al. [4] also identified recent trends in turbulent combustion
modeling. These trends are motivated and enabled by the need to represent impor-
tant finite-rate chemistry effects and non-equilibrium chemistry effects in combus-
tion. Requirements for combustion technologies only 20 years ago are not the same
as the requirements we dictate now. A variety of alternative fuels are explored in
addition to high grade fossil fuels. Pollution mitigation also enforces additional re-
quirements on the choice of the fuel, its equivalence ratio and mixture control (e.g.
homogeneity of the charge).
Additional qualitative changes in the scope of turbulent combustion models can
be gaged from two seminal contributions in the field of turbulent combustion. They
correspond to two contributed volumes entitled Turbulent Reacting Flows, which
were edited by Libby and Williams in 1980 and 1994 [27, 28]. A comparison of the
topics covered in the two books and the present volume illustrates important expan-
sions in the scope of the field of turbulent combustion. The key areas of expansion
are outlined here:
The role of chemistry in turbulent combustion simulations has seen a tremen-
dous growth since the Libby and Williams [27, 28] volumes. Already in the
1980s software packages, such as Sandias Chemkin [20] chemistry and trans-
port libraries and associated zero-dimensional and one-dimensional applications,
have enabled important advances in the prediction of the role of finite-rate chem-
istry effects in combustion [9]. Because of the disparity of chemical scales, stiff-
integration software were becoming available for the integration of chemistry,
such as the DASSL [41] and VODE [5] software packages. These packages
played an essential role in the implementation of chemistry in combustion prob-
lems. Beyond the traditional strategies of quasi-steady state assumptions (QSSA)
for species and partial equilibrim (PE) for reactions and sensitivity analysis,
novel numerical tools have contributed to efficient strategies for the acceleration
of chemistry in numerical codes. Examples of such strategies include mecha-
nism automation strategies based on QSSA and PE [6], systematic eigenvalue
based approaches, including the computational singular perturbation (CSP) [23]
approach and the intrinsic low-dimensional manifold (ILDM) [47] approach, and
direct relation graph [24]. Chapter 9 provides ample discussion on chemistry re-
duction and integration.
Large-eddy simulation (LES) has emerged as as an alternative mathematical
framework for the solution of transport equations for momentum and scalars.
The traditional strategy, which is more common, is based on Reynolds-averaged
Navier-Stokes (RANS) and associated equations for scalar transport, is not al-
ways sufficient for complex flows. LES has seen tremendous growth in the 1980s
for turbulent non-reacting canonical flows; but, it is increasingly becoming a vi-
able modeling framework for practical combustion flows. LES potentially en-
ables accurate solutions of combustion flows incorporating unsteady flow effects.
Turbulent Combustion: Concepts, Governing Equations and Modeling Strategies 21

Successful LES simulations with advanced combustion models are increasingly

being used to model practical combustion devices [49].
A broader range of combustion modes (e.g. premixed, non-premixed, stratified)
and combustion regimes (e.g. thin or relatively thick reaction zones) are being
explored in current and novel combustion technologies. The strict classification
of combustion modes as either premixed or non-premixed, while powerful for
the development of physical models of turbulent combustion, may not be sepa-
rately adequate to represent partially-premixed combustion modes. Combustion
in stratified mixture plays key role in a number of practical combustion applica-
tions, including diesel, gas turbine and homogeneous charge compression igni-
tion (HCCI) combustion. In his textbook, Peters [40] dedicates an entire chapter
to partially-premixed combustion with the recognition of the role of this com-
bustion mode in a broad range of combustion problems, and novel modeling
strategies have been developed for this combustion regime.
Moreover, earlier important advances in turbulent combustion concerned primar-
ily phenomena in which the separation of scales can be justified, such as in the
cases of fast chemistry and in the flamelet regime. For example, both the eddy-
dissipation model (EDM) [31] and the flamelet model [39] demonstrated a broad
range of applicability in predicting combustion in practical combustion devices.
However, combustion in other regimes where both chemistry and mixing are
competitive during ignition (e.g. HCCI combustion) and flame-based combus-
tion (e.g. distributed reaction, corrugated flames), are more challenging.
Both books by Libby and Williams [27, 28] adopt the traditional view that turbu-
lent combustion modeling primarily is a physical modeling challenge. However,
the increasing availability of computational resources has enabled further and
accelerated development of direct numerical simulation (DNS) techniques for
combustion. In a recent paper, Valorani and Paolucci [53] make the observation
No longer than 10 years ago, a direct numerical simulation (DNS) [11] of a tur-
bulent flame with a four-step kinetics mechanism on a 10 mm box constituted
the state-of-the-art in combustion simulation. Nowadays, the targets are DNSs of
turbulent combustion of surrogate fuels, in half-a-meter domains. As stated in
Chapter 1 and elsewhere in this book, DNS may not be applicable to practical
combustion devices for some time to come. However, other DNS-like techniques
have been used to model laboratory-scale burners, such as recent simulations
based on adaptive mesh refinement (AMR) [1, 2].
Our current understanding of the fundamental laws governing reacting flows en-
ables us to formulate detailed physical models, with minimum empiricism, for a
large number of the processes underlying turbulent combustion. For example, atom-
istic simulations may be used to construct databases for rate constants and thermo-
chemical and molecular transport properties of reacting species. But, atomistic ap-
proaches alone may not extend to the scales relevant to practical combustion prob-
lems; yet, with the help of constitutive relations derived for molecular processes,
continuum-based formulations for reacting flows are a good starting point.
Even within the continuum limit, various strategies may be adopted. These strate-
gies may reflect the formulation of the mathematical models for the governing equa-
22 T. Echekki and E. Mastorakos

tions as well as their numerical solution in addition to inherent simplification of

these equations due to the flow regime (e.g. low Mach number formulations). They
also reflect the scope of the modeler whether she/he is interested in statistical re-
sults or fully-resolved (spatially and temporally) results. The latter scope belongs to
the realm of direct numerical simulations (DNS) where the governing equations are
solved without filtering or averaging of the solution vector and with a full account of
the required spatial and temporal resolution within the continuum limit. However,
recourse to unsteady information is progressively seen also as one of the reasons for
moving towards LES as in, for example, the effort to capture ignition or extinction
phenomena [52]. The governing equations for DNS will be the starting point for
discussing the different strategies adopted to address the mathematical models in
turbulent combustion and their numerical solutions. Our emphasis is on two mathe-
matical frameworks for representing the solution vector based on RANS and LES.
Following effort in the turbulence community, other mathematical frameworks may
be feasible as well, but RANS and LES are the most common approaches in modern
turbulent combustion modeling and will form the focus of this book.

2.2 Governing Equations

2.2.1 Conservation Equations

The governing equations for turbulent combustion flows may be expressed in differ-
ent forms; however, they normally are represented as transport equations for over-
all continuity, momentum and additional scalars that can be used to spatially- and
temporally- resolve the thermodynamic state of the mixture. These equations are
augmented by initial and boundary conditions, as well as constitutive relations for
atomistic processes (e.g. reaction, molecular diffusion, equations of state). There-
fore, in addition to density, transport equations for the evolving momentum and
composition (e.g. mass or mole fractions, species densities or concentrations) and a
scalar measure of energy (e.g. internal energy, temperature, or enthalpy). For illus-
tration purposes, we present the compressible form of the instantaneous governing
equations in non-conservative form for the mass density, momentum, species mass
fractions and internal energy. A more detailed discussion on the various forms and
their equivalence, especially for the energy equation can be found in the textbooks
by Williams [57] or Poinsot and Veynante [44].
Continuity

+ u = 0, (2.1)
t
Momentum
Turbulent Combustion: Concepts, Governing Equations and Modeling Strategies 23

Du u N
= + u u = p + + Yk fk , (2.2)
Dt t k=1

Species continuity (k = 1, , N)
DYk Yk
= + u Yk = ( VkYk ) + k , (2.3)
Dt t
Energy
De e N
= + u e = q p u + : u + Yk fk Vk . (2.4)
Dt t k=1

In the above equations, is the mass density; u is the velocity vector; p is the
pressure; fk is the body force associated with the kth species per unit mass; is
the viscous stress tensor; Vk is the diffusive velocity of the kth species, where the
velocity of the kth species may be expressed as the sum of the mass-weighted ve-
locity and the diffusive velocity, u + Vk ; k is the kth species production rate; e
is the mixture internal energy, which may be expressed as e = Nk=1 hkYk p/ ; q
is the heat flux, which represents heat conduction, radiation, and transport through
species gradients and the Soret effect. The solution vector, represented by the
above governing equations (2.1)(2.4) is = ( , u, Y, e) in its conservative
form or = ( , u, Y, e) in its non-conservative form. The governing equations may
be expressed in a more compact form as follows:
D
= F( ) (2.5)
Dt
where F( ) represents the right-hand side of the governing equations and features
terms with spatial derivatives (e.g. diffusive fluxes for mass and heat) and source
terms (e.g. reaction source terms). The material derivative D /Dt includes both
the unsteady term and the advective term in the Eulerian representation such that:
D /Dt / t +u . As can be seen, a number of terms in the governing equa-
tions are not explicitly expressed in terms of the solution vector and must rely on
constitutive relations, equations of state or any additional auxiliary relations. These
terms include expressions for the viscous stress, the species diffusive velocities, the
body forces, the species reaction rate and the heat flux. The bulk of these terms have
their origin in the molecular scales, and therefore, the role of constitutive relations
is to represent them in continuum models. In fact, the use of constitutive equations
is the first level of multiscale treatment for the modeling of turbulent combustion
flows. Alternative, but significantly more costly approaches, involve their determi-
nation using atomistic models coupled on the fly with continuum models. How-
ever, cases where such approaches are needed are very limited.
24 T. Echekki and E. Mastorakos

2.2.2 Constitutive Relations, State Equations and Auxiliary

Relations

2.2.2.1 Constitutive Relations, Transport Properties and State Equations

The constitutive relations for the conversation equations outlined above represent
primarily relations between transport terms for momentum, energy and species and
the solution vector as well as relations that describe the rate of chemistry source
terms in the species equations. They are designed to represent atomistic scale effects
of transport and reaction. Below we outline the principal terms that are represented
by constitutive relations.
The pressure and viscous stress tensor: In gas-phase flows applicable to com-
bustion problems, the Newtonian fluid assumption is reasonably valid, and the
viscous stress tensor may be represented through the following relation:
  2 
= (u) + (u)T + ( u) I (2.6)
3

In this expression, is the dynamic viscosity; is the bulk viscosity; and I is

the identity matrix. The principle of corresponding states provides generalized
curves for the viscosity of gases, liquids and supercritical fluids for a broad range
of temperature and pressure conditions. The principle states that a reduced vis-
cosity, based on the ratio of the dynamic viscosity to that at critical conditions,
may be uniquely defined in terms of a reduced temperature and pressure, both
reduced values result from the normalization of temperature and pressure with
their corresponding critical values.
The diffusive mass flux, Yk Vk : The diffusive mass flux represents the transport
of species in addition to their transport with the bulk flow, u. Diffusive mass
transport may be associated with gradients in mass or species concentration, the
so-called Fickian diffusion, temperature gradients, or the so-called Dufour effect,
and pressure gradient. A hierarchy of models for the diffusive mass flux may
be adopted. The first is based on adopting a Ficks law model using mixture-
averaged transport coefficients:

k Xk
Xk Vk = Dm (2.7)

where Dm k is the mixture-averaged mass diffusion coefficient for species k. The

mixture-averaged mass diffusion coefficient is derived, in general, using mixture
weighting rules and multi-component diffusion coefficients. A simple form of
the mixture-averaged diffusivity is based on the assumption of constant diffusion
coefficients ratios (e.g. fixed Lewis numbers or Schmidt numbers), such that the
mixture-averaged mass diffusion coefficient is expressed as follows [50]:

Dm
k = (2.8)
c p Lek
Turbulent Combustion: Concepts, Governing Equations and Modeling Strategies 25

where c p is the mixture specific heat; is the mixture thermal conductivity; and
Lek is the kth species Lewis number.
The second approach is based on a multi-component formulation [8]:
N
1 DT,k 1
Vk =
XkW W j Dk j d j
Yk T
T (2.9)
j=1, j=k

In this expression, Dk j and DT,k are the binary mass diffusivity between species
k and j and the thermal diffusion coefficient, respectively; d j is the concentration
and pressure gradients for species j:

p
d j = X j + (X j Y j ) (2.10)
p
Detailed formulations for Dk j and DT,k may be found in various textbooks (see
for example, Kee et al. [21]).
The heat flux vector, q: The heat flux vector q features contributions from dif-
ferent modes of heat transfer, including heat conduction, heat diffusion by mass
diffusion of the various species, thermal diffusion (Dufour effect), and radiative
heat transfer. A general form of the heat flux featuring the contribution of these
different heat transfer modes may be written as follows:
N N N  
X j DT,i
q = T + hiYi Vi + Ru T (Vi V j ) + qrad (2.11)
i=1 i=1 j=1 Wi Di j

In this equation, is the mixture thermal conductivity and qrad is the radiative
heat flux.
The chemical reaction term, k : This term is derived from the law of mass ac-
tion, which dictates that the rate of chemical reactions is proportional to the con-
centrations of the contributing species. The proportionality factor is primarily
a function of temperature and is denoted as the reaction rate constant. Contri-
butions to this term include statistical information about the rates of collisions,
and the fraction of collisions resulting in reactions as well as steric factor, which
take into consideration the shapes of molecules during collisions. The following
equation represents the rate of production of species k due to its involvement in
R reversible reactions:
 

R    N X   N  X p k,r
jp k,r
k = Wk k,r k,r k f ,r

kb,r
j
,
r=1 j=1 Ru T j=1 Ru T
(2.12)
where  
Ea,r k f ,r
k f ,r (T ) = Ar T r exp , kb,r = (2.13)
Ru T KC,r
In these expressions, Wk is the molecular weight for species k; j,r and j,r are the
rth reaction stoichiometric coefficients on the reactants and the products sides,
26 T. Echekki and E. Mastorakos

respectively; k f ,r and kb,r are the forward and backward rate constants for the re-
versible reaction, r. The backward reaction rate constant is related to the forward
rate constant through the concentration-based equilibrium constant, KC,r for re-
action r. In the Arrhenius form for the forward rate constant expression, Ar and
r are the pre-exponential coefficients, and Ea,r is the activation energy for the
forward reaction, r. An elementary reaction, r, is prescribed as follows:
N N
j,r A j j,r A j (2.14)
j=1 j=1

where A j is the jth species chemical symbol.

The integration of the chemical source term in the species equation (as well as in
the temperature or sensible enthalpy forms of the energy equation) poses impor-
tant and limiting challenges in computational combustion, as discussed below.
The determination of transport properties for momentum, mass and energy
remains an understated challenge. Various software packages for the evaluation
of transport properties are available, including MIXRUN [56], TRANLIB [19],
EGLIB [13] and DRFM [38]. A first challenge is to assemble reliable data for poten-
tial parameters that contribute to the evaluation of the collision integrals. Paul [38]
find that special attention needs to be made in determining the transport properties
for molecules with dipole moments (e.g. H atom, H2 molecule) and indeed numer-
ical simulations with different levels of modeling transport can lead to different
results.

2.2.2.2 Mixture Properties and State Equations

State equations enable to evaluate thermodynamic properties from known proper-

ties. A common relation involves the ideal gas law:
N  
Yj
p = Ru T Wj
(2.15)
j=1

The caloric equation of state may be used to relate a species enthalpy or internal
energy to temperature as follows:
 T
hk (T ) = hk,chem + c p,k dT (2.16)
T

and  T
ek (T ) = hk,chem + cv,k dT (2.17)
T
where hk and ek are the kth species total enthalpies and internal energies; T is
a reference temperature for the sensible enthalpy. Here, hk,chem corresponds to the
chemical enthalpy of the kth species, and the second terms on the right hand-sides
Turbulent Combustion: Concepts, Governing Equations and Modeling Strategies 27

of the two above equations corresponds to the sensible contributions; cv,k and c p,k
are the specific heats for species k at constant volume and pressure, respectively.

2.2.2.3 Other Transport Equations

Along with, or instead of, the scalar transport equations, transport equations for ad-
ditional scalars may be used. These include conserved scalars (e.g. mixture fraction,
total enthalpy), normalized reaction progress variables and flame surface variables
(e.g. flame surface density).
Conserved scalars may be found in different aspects of combustion analysis from
theory to experiment. They offer the convenience that their transport equations are
devoid of source terms. Therefore, their integration is not subject to the steep time
constraints of integrating chemistry. The Shvab-Zeldovich [57] formulation offers
an early example of the use of conserved scalars in the limit of fast chemistry in
terms of the so-called coupling functions. The same concept based on this for-
mulation resulted in one of the classic analytical solutions in combustion based
on the Burke-Schumann jet flame model [57]. However, the concepts of elemen-
tal mass fractions and mixture fractions have offered significantly more insight into
processes in turbulent combustion, especially in non-premixed combustion. From a
mixture composition, it is possible to construct an elemental mass fraction, Zl , for
element l, which may be prescribed as:
N
Zl = j,lY j (2.18)
j=1

where j,l is the mass fraction of element l in species j. The elemental mass fraction
is unaltered by reaction; and therefore, there is no source term associated with its
transport equation:

DZl Zl
= + u Zl = ( Vl Zl ). (2.19)
Dt t
Here, the diffusive velocity associated with the elemental mass fraction is expressed
as follows:
N
V j j,lY j = Vl Zl (2.20)
j=1

The mixture fraction represents a normalized form of the elemental mass fraction,
and it is a parameter of great value for non-premixed chemical systems. It measures
the fraction by mass in the mixture of the elements, which originates in the fuel.
When derived from elemental mass fractions, it may be expressed in normalized
form as:
Zl Zl,o
Fl = (2.21)
Zl, f Zl,o
28 T. Echekki and E. Mastorakos

where the subscripts o and f refer to the oxidizer and the fuel mixture conditions,
respectively. In a mixing system of fuel and oxidizer streams, values of the mix-
ture fractions based on different elements may be different because of differential
diffusion effects. Element-averaged mixture fractions, such as the Bilger mixture
fraction [3], may be adopted:

2 (ZC ZC,o )/WC + (ZH ZH,o )/(2WH ) (ZO ZO,o )/WO

FBilger =       (2.22)
2 ZC, f ZC,o /WC + ZH, f ZH,o /(2WH ) ZO, f ZO,o /WO

where the subscripts C, H and O correspond to the elements C, H and O, respec-

tively, and the symbol W refers to their corresponding molar masses. The coef-
ficients in front of the different elemental contributions serve the important role
of maintaining the stoichiometric Bilger mixture fraction value identical to the el-
emental mixture fractions. As stated earlier, the mixture fraction is an important
parameter for the modeling of non-premixed systems [3, 40, 57].

2.3 Conventional Mathematical and Computational Frameworks

for Simulating Turbulent Combustion Flows

Within the continuum limit, there are different mathematical and computational
frameworks to model turbulent combustion flows. These frameworks address the
way the governing equations are modified and the way the solutions are imple-
mented computationally (e.g. discretization). Strategies to overcome the limitations
of resolving all the time and length scales even within the continuum limit motivates
two principal classes of modeling frameworks associated with model-adaptivity or
mesh-adaptivity or both. Model adaptivity refers to a class of models in which the
governing equations, and accordingly the solution vector , are modified to a re-
duced order, a reduced dimension, or a statistical form, which effectively decouples
or eliminates ranges of scales from the solution vector. Mesh adaptivity refers to
a class of models in which the solution vector, , is resolved by adapting the grid
hierarchy or the resolution hierarchy where it is needed to meet prescribed error
criteria.
As indicated above, model adaptivity is concerned with modifying the governing
equation and the solution vector. For combustion flows, three principal strategies
have been implemented for model adaptivity; while, potentially other approaches
may be considered. They correspond to DNS, RANS and LES.

2.3.1 Direct Numerical Simulation (DNS)

DNS corresponds to the solution of the 3D unsteady governing equations (Eqs. 2.1
2.4) with the necessary resolution required to accurately integrate the solution in
Turbulent Combustion: Concepts, Governing Equations and Modeling Strategies 29

time and predict the details of velocity and scalar fields. Therefore, DNS offers
the best resolved framework for the study of turbulent combustion flows. A typical
3D unsteady DNS in combustion must span the ranges of time and length scales
discussed above (approximately 4-5 decades in length scales within the continuum
regime in a given direction), which entail resolution requirements of the order of
trillions of grid points or higher and tens of millions of time steps. Yet, the state-
of-the-art DNS of combustion have been limited to computational domains that
are approximately one order of magnitude smaller in linear scale (or three orders
of magnitude in volume) than laboratory flames or practical combustion devices.
However, achieving these length scales is fast approaching with petascale capabili-
ties and beyond. Nonetheless, DNS remains a powerful tool to understand important
turbulence-chemistry interaction processes and formulate closure models in turbu-
lent combustion [43, 54, 55]. Computational requirements for DNS may vary de-
pending on the level of description of the chemistry, molecular transport and radia-
tion as well as the representation of the governing equations (e.g. low-Mach number
formulation vs. compressible formulations). Examples of computational require-
ments may be found in a recent paper by Chen et al. [7].
A principal challenge for DNS remains the temporal integration of the conserva-
tion equations, especially those pertaining to the integration of the reactive scalars.
A temporal resolution from the fastest reactions (of the order of 109 s for hydro-
carbon chemistry) to integral scales of the flow results in hundreds of thousands to
million time steps with explicit integration schemes; accordingly, DNS simulations
remain largely CPU-limited. Lu and Law [29] present an analysis of the cost of
integrating chemistry within DNS. Their analysis shows that:
The size of a chemical mechanism (i.e. the number of reactions) increases ap-
proximately linearly with the number of chemical species considered; the scaling
factor is approximated as 5 between the chemical mechanism size and the num-
ber of species involved. This scaling is presented for hydrocarbon fuels. How-
ever, it is clear that as DNS applications are extended from hydrogen and simple
hydrocarbon fuels to more common fuels (e.g. gasoline, diesel, kerosene), addi-
tional cost is associated with both the transport of more scalar equations as well
as in the evaluation and the integration of non-linear reaction rate terms. The
task is daunting given that more than one order of magnitude separates the size
of simple and more complex fuel chemistries.
The computational cost of DNS at each grid, CDNS , also scales approximately
linearly with the number of species, N, involved: CDNS N. The proportionality
factor subsumes contributions associated with the spatial resolution and the cost
of advancing the scalar transport equations, including the evaluation of transport
properties and chemical reaction rates.
Therefore, aggressive strategies for chemistry reduction are warranted and may
need to go beyond the development of skeletal mechanisms. Additional strategies
for chemistry calculation acceleration are needed equally to overcome the stiff
chemistry. These strategies have been pursued and significant progress has been
achieved in recent years. An additional challenge is to account for the transport of
30 T. Echekki and E. Mastorakos

tens to hundreds of species in mechanisms that range in size from tens to thou-
sands of reactions. The subject of chemistry reduction and acceleration has received
increasing interest in recent years (see for example the recent reviews by Lu and
Law [29] and Pope and Ren [48]). Chapter 9 details further strategies for chemistry
reduction and acceleration.
Another equally important effort is to address spatial resolution requirements.
Spatial resolution must resolve the thinnest layers of reaction zone structures; these
layers represent the balance of reaction and diffusion and at their thinnest may be
of the order of 10 m or smaller. It is difficult to justify not to resolve these layers
if they serve a role in the combustion process; and often strategies to address the
resolution of these layers may be implemented through adapting the spatial resolu-
tions where these layers are present and coarsening the resolution when such fine
resolutions are not needed. Adaptive resolution strategies offer the most promising
strategies for addressing spatial resolution and often result in almost an order of
magnitude gain in the size of problems to be solved in comparison with DNS. Typ-
ical examples of mesh adaptivity include adaptive mesh refinement (AMR), which
is discussed in Chapter 13 and wavelet-based adaptive multiresolution strategies,
which are discussed in Chapter 14.

2.3.2 Reynolds-Averaged Navier-Stokes (RANS)

The Reynolds-averaged Navier-Stokes (RANS) formulation is based on time or en-

semble averaging of the instantaneous transport equations for mass, momentum and
reactive scalars. Within the context of scale separation, the RANS approach indis-
criminately impacts all scales. Consequently, all unsteady turbulent motion and its
coupling with combustion processes are unresolved over the entire range of length
and time scales of the problem, and closure models are needed to represent the unre-
solved physics. An additional complexity introduced by averaging is that non-linear
terms in the governing equations result in unclosed terms. The closure problem is
particularly critical for the reaction source terms in the species and some forms
of the energy equations. The treatment of these terms has been the scope of the
moment-based methods. We illustrate the closure problem using the transport equa-
tion for the conservation equations above (Eqs. 2.12.4). Before listing the conser-
vation equations, we briefly address the advantages of density-weighted averaging
or the so-called Favre-averaging [14, 18]. With Favre-averaging, all momentum and
scalars, at the exception of the density, the pressure and diffusive fluxes, are decom-
posed using a Favre-averaged means and fluctuations:

= +  (2.23)

The Favre average, may be expressed in terms of the non-weighted average as:

(2.24)

Turbulent Combustion: Concepts, Governing Equations and Modeling Strategies 31

The contribution  corresponds to the fluctuating components of the solution vec-

tor relative the Favre mean. The overbar denotes an unweighted ensemble average
over a statistically-meaningful set of realizations; the symbol denotes a density-
weighted ensemble average. Density-weighted averaging eliminates the need to
explicitly represent the density-momentum and density-scalar correlations, which
when kept in the governing equations generate additional closure terms. Based on
the above conservation equations, the Favre-averaged continuity, momentum and
scalar equations are expressed as follows:

Continuity

+
u = 0, (2.25)
t
Momentum

u N  
+ u = p + + Y
u 
k fk u u ,
  (2.26)
t k=1

Species continuity (k = 1, , N)
Y k 
u Y k = ( V
+  
kYk ) + k ( u Yk ), (2.27)
t
Energy
e N  
+
u q p u + : u + Y
e = f
k k Vk 
u  e . (2.28)
t k=1

In the RANS formulation, the solution vector is expressed as = ( , u e ).

, Y,
Both the source term and the advective term are non-linear contributions to the gov-
erning equations for the species, and invariably these terms will be unclosed since
there is no explicit transport equation used to solve them. Additional new terms
in the governing equation correspond to the Reynolds stresses and fluxes: u  u ,
  
u Yk and u e which are also unclosed and must be modeled. It is quite common
  

to treat this term as a turbulent diffusion term with a gradient model. The molecular
diffusion term is also unclosed; but, its description may depend largely on how it
is modeled and how its effects are prescribed with the reaction source term. It is
also common to assume that the turbulent diffusion term is much larger than the
molecular diffusion term in the governing equations and that the molecular diffu-
sion term is often dropped from the above governing equations. This is not strictly
true because these two transport terms may act on different scales. Therefore, the
effects of molecular diffusion may still have to be represented (typically through the
chemistry closure). Nonetheless, the most critical closure arises from modeling the
reaction source term, k .
To motivate the strategies adopted for the closure for k , the statistical represen-
tation of this term is expressed as follows:
32 T. Echekki and E. Mastorakos

k = k ( ) f ( ) d . (2.29)

In this expression, f ( ) is the joint scalar probability density function (PDF). The
vector represents components of the thermodynamic state vector, which may in-
clude for example, pressure, temperature and composition. Therefore, the vector,
, may be a subset of the solution vector, , which also includes the momentum
components. The joint PDF contains the complete statistical information about all
scalars. Therefore, the averaged scalar field, its moments and any related functions
of the field may be constructed using this joint scalar PDF:

= ( ) f ( ) d , (2.30)

and 
( ) = ( ) f ( ) d . (2.31)

A density-weighted PDF may be defined as well, which may be written as follows:

( )
f ( ) = f ( ) (2.32)

These expressions offer an important window into closure strategies in turbu-
lent combustion within the context of RANS. An accurate description of averaged
scalars, their moments and functions of these moments must involve an accurate ac-
count of the state-vector solution (i.e. the instantaneous correlation, ( ) inside the
integral) as well as an accurate account of the statistical distribution, f ( ). Often,
the modeling of the two contributions is coupled, and the choice of the combustion
mode or regime may enable strategies for the modeling of the state-vector solutions
as well as the joint scalar PDF.

2.3.3 Large-Eddy Simulation (LES)

The third approach is based on spatially filtering the instantaneous equations to cap-
ture the contribution of large scales, resulting in transport equations for spatially
filtered mass, momentum and scalars, while the effects of smaller scales are mod-
eled. This approach is known as large-eddy simulation [42]. LES relies on scale
separation between (kinetic) energy containing eddies and small scales responsible
for its dissipation (the so-called subgrid scale, or SGS). The approach is rooted in
the traditional view of turbulent flows where the bulk of turbulent kinetic energy
originates at the large scales; however, this choice has limited justification in com-
bustion flows: important physics may reside and originate at small scales.
From a modeling standpoint, LES provides a number of important advantages to-
wards the prediction of turbulent combustion flows over RANS. First, LES captures
large scale information in both the momentum and scalar fields. Therefore, it is able
Turbulent Combustion: Concepts, Governing Equations and Modeling Strategies 33

to capture the role of large flow structures on mixing and, therefore, on combustion.
These flow structures are inherently unsteady, and capturing their interactions with
combustion chemistry is very crucial in a broad range of practical applications. For
example, tumble and swirl in internal combustion engines serve to enhance the vol-
umetric rate of heat release and contribute to cycle-to-cycle variations. Moreover,
the lift-off and blow-out of lifted flames in practical burners is dependent on the
unsteady flow dynamics around the flame leading edge and the inherent instabili-
ties in the presence of shear. Another example is associated with the prediction of
thermo-acoustic and other flow-associated instabilities in gas turbine combustors.
Second, because LES is designed to capture the role of a band of scales, it can nat-
urally be implemented within the context of multiscale frameworks. In these frame-
works, simulations of the subgrid scale physics is implemented along with LES
to capture the contributions from all relevant scales. Hybrid approaches of LES
with low-dimensional stochastic models, such as the LEMLES and the ODTLES
frameworks discussed in Chapters 10 and 11 illustrate the implementation of LES
for combustion within the context of multiscale approaches. However, as outlined
in this book and briefly discussed above, LES is a promising strategy within the
context of moment-based approaches, such as non-premixed and premixed flames
(Chapters 3 and 4), the conditional moment closure (CMC) model (Chapter 5), the
transported PDF (Chapter 6), and the multiple mapping conditioning (MMC) ap-
proach (Chapter 7).
Third, since closure in LES targets primarily subgrid scale physics, a higher
degree of universality in statistics may be achieved when the contribution from
geometry-dependent large scales are eliminated from consideration.
We consider a filtering operation applied to the conservation equations. The fil-
tering operation corresponds to the implementation of a low-pass filter, which is
expressed as follows for a solution vector component, :
    
(x,t) = G x, x ; x ,t dx (2.33)

In this expression, G is a filtering function over 3D space with a characteristic scale,

, the filter size. Similarly to the RANS formulation for variable-density flows, the
filtered solution vector, at the exception of the filtered density, is based on density-
weighted filtering, such that:

( ; x,t) (2.34)

where       
= G x, x ; x , t ; x , t dx (2.35)

The Favre-filtered momentum and scalar equations are expressed as follows:

Continuity
34 T. Echekki and E. Mastorakos

+
u = 0, (2.36)
t
Momentum

u N N
+ u = p + + + Y
u k fk + [ (
uu u
u)], (2.37)
t k=1 k=1

Species continuity (k = 1, , N)
Y k   
u Y k = ( V
+ kYk + k +
k ,
uY k uY (2.38)
t

Energy
e N
+ u
q p u + : u + Y
e = k fk Vk + [ ( .
ue ue)]
t k=1
(2.39)

We have kept the same symbols for operations as above, although they corre-
spond to spatial filtering operations instead of ensemble or time-averaging as in
RANS. Here the overbar corresponds to a process of unweighted spatial filtering
and the corresponds to a density-weighted spatial filtering. Again, considering
the revised solution vector, = ( , u e ) , additional terms are present in the gov-
, Y,
erning equation, which correspond to subgrid scale stresses ( uu uu) and scalar
fluxes, ( 
uYk uYk ) and ( These terms also are unclosed and require
ue ue).
modeling. The molecular diffusion term, ( V kYk ), may be insignificant in the
governing equation relative to the scalar flux on the LES resolution, it acts at scales
that are fundamentally different from those of the scalar fluxes; and therefore, its
contribution may be closely tied to the reaction source term and its closure. The
process of averaging or filtering of the governing equations invariably leaves the
contributions of the unresolved physics unclosed, and similar challenges of closure
are found.
Similarly to the RANS formulation, an important closure term corresponds to the
reaction source terms, k . Here, a similar concept to the PDF may be used based on
the filtered-density function (FDF) [15]:

k = k ( ) F ( ) d (2.40)

and 
( ) = ( ( )) F ( ) d (2.41)

Here F ( ) is the filtered-density function.

At this point, it is important to contrast the concepts of PDF, which discussed
within the context of RANS, and FDF, which is discussed within the context of
LES. We introduce the concept of a fine-grained PDF [30], which represents the
Turbulent Combustion: Concepts, Governing Equations and Modeling Strategies 35

time- and spatially-resolved joint PDF. This fine-grained PDF may be expressed as
follows:
N
( , (x,t)) = ( (x,t)) = ( (x,t)) (2.42)
=1

The fine-grained PDF, ( , (x,t)) d represents probability that at x and t, that

+ d for all values of . Within the context of RANS, the joint
scalar PDF may be expressed as follows:

f ( ; x,t) = ( , (x,t)) (2.43)

The FDF is expressed in terms of the fine-grained PDF as follows:

 +  
F= ( , (x,t)) G x, x dx (2.44)

Therefore, the PDF represents a distribution built over time or ensembles of realiza-
tions of the scalar values at one single position, x. In contrast, the FDF represents
an instantaneous subfilter distribution of the same scalars over a prescribed filter
volume.
The closure of the reaction source term is a principal modeling challenge in com-
bustion LES; and often, strategies implemented for RANS have been extended to
LES as well, as discussed in various chapters in this book.

2.4 Addressing the Closure Problem

The scope of turbulent combustion modeling is related to the representation of re-

active scalar statistics. The traditional strategy is based on the RANS averaging
framework. However, LES is becoming a viable framework for turbulent combus-
tion models. The challenges are fundamentally similar. Averaging or filtering results
in the closure problem for key terms in the conservation equations, including pri-
marily the chemical source terms. The bulk of chapters in this book (Chapters 314)
attempt to address the different approaches to turbulent combustion closure.
In a recent review, Bilger et al. [4] discussed traditional paradigms that defined
turbulent combustion modeling over the last 40 or so years. Principal strategies re-
sulting from these paradigms are based on either a 1) separation of scales and/or
2) separation of model elements that address the model description of moments
of reactive scalars in terms of scalar description in state-space and model for the
distribution (PDF or FDF) function. Examples of models based on the separa-
tion of scales include the assumption of fast chemistry (e.g. the eddy dissipation
model (EDM) [31], the eddy break-up model (EBU) [51]) and the laminar flamelet
model [39] where the flames thicknesses are below the energetic turbulence scales.
We illustrate the second strategy by revisiting the Eqs. (2.29) and (2.40). The
mean or filtered reactions is constructed through a weighted average of the instan-
36 T. Echekki and E. Mastorakos

taneous reaction rate k ( ) and the distribution, f ( ) or F ( ). For the instanta-

neous reaction rate term, a reactor model is needed that is representative of the
state-space conditions encountered. For example, a flamelet library or CMC solu-
tions may represent such reactor models. For the distribution description different
strategies may be adopted depending on whether a reduced description of the state-
space variables is available. For example, in the standard laminar non-premixed
flamelet model and in CMC, models for the mean mixture fraction and its variance
may be used to construct presumed shape PDF functions for reactive scalars. In
the Bray-Moss-Libby (BML) model, a simple PDF function is adopted in terms of
the reaction progress variable for premixed flames, although knowledge of the PDF
shape is not always guaranteed. The more general form for the determination of the
joint PDF involves the solution for a transport equation for the PDF and the FDF.
However, intermediate strategies may be adopted as well. These include 1) the con-
struction of PDF/FDFs using independent stochastic simulations, or 2) optimally
build PDFs, such as the ones based on the statistically-most likely distribution [45].
The mapping closure approach (MMC; see Chapter 7) illustrates a strategy where
a PDF transport equation is adopted for the construction of a statistical distribution
and the CMC approach is used for the state-space relations.
Given the scope of the Bilger et al. [4] paper as related to paradigms in turbu-
lent combustion, other modeling approaches have not been discussed; they will be
presented here and the remaining chapters in the book will address them in more
detail.

2.5 Outline of Upcoming Chapters

In this chapter, we have attempted to provide a brief outline of the challenges as-
sociated with turbulent combustion modeling. These challenges may be addressed
by improved physical models of turbulent combustion processes; great strides
have been made in the last two decades since the later contribution of Libby and
Williams [28] and the more recent combustion literature. Moreover, rapid advances
in computational sciences (hardware and algorithms) have fueled important ad-
vances in high-fidelity simulations of turbulent combustion flows that provide direct
solutions of unresolved physics.
This book attempts to highlight recent progress in the modeling and simulation
of turbulent combustion flows. It is divided into four parts, which include 1) two
introductory chapters (Chapters 1 and 2) and 2) that motivate the growth of the
disciplines associated with turbulent combustion flows from a societal and tech-
nological perspectives, 2) progress and trends in turbulent combustion models, 3)
progress and trends in a new class of models based on multiscale simulation strate-
gies, and 4) cross-cutting science that may be needed to move the subject forward.
In Part II, emphasis is placed on recent progress in advanced combustion models,
including the flamelet approach for non-premixed systems (Chapter 3), approaches
for premixed combustion(Chapter 4), CMC (Chapter 5), MMC (Chapter 7) and the
Turbulent Combustion: Concepts, Governing Equations and Modeling Strategies 37

PDF approach (Chapter 6). In Part III, emphasis is placed on multiscale strategies
that seek to directly or indirectly compute subgrid scale physics. This part is pre-
ceeded by an introductory chapter highlighting the driving motivation behind multi-
scale strategies in turbulent combustion. Topics covered in this part include the role
of chemistry reduction and acceleration (Chapter 9), the linear-eddy model (LEM)
(Chapter 10), the one-dimensional turbulence (ODT) model (Chapter 11), flame-
embedding (Chapter 12), adaptive-mesh refinement (AMR) (Chapter 13), wavelet-
based methods (Chapter 14). Our coverage of existing models in Parts I and II is
admittedly incomplete; but, it provides a flavor of current state-of-the-art and trends
in turbulent combustion models. This state-of-the-art can be contrasted with the gen-
eral strategies adopted during the last three decades to gauge recent progress. Part
IV addresses cross-cutting science, which include the basic tools to advance the dis-
cipline of turbulent combustion modeling. Experiment has played, and will continue
to play, a central role in the development of new and the refinement of old strate-
gies. The role of experiment is discussed in Chapter 15. From the computational
side, two principal drivers for improving turbulent combustion modeling and simu-
lation are addressed. The first chapter (Chapter 16) deals with the subject of uncer-
tainty quantification as an emerging requirement to improve the ability of turbulent
combustion modeling and simulation tools to predict practical flows. The second
chapter (Chapter 17) addresses the need to develop effective strategies to build op-
timized software tools to predict turbulent combustion flows. Chapter 18 presents
the homogeneous multiscale method (HMM) as a mathematical multiscale frame-
work for turbulent combustion. Finally, Chapter 19 reviews the lattice-Boltzmann
method (LBM), which represents a viable alternative to the standard Navier-Stokes
equations for a large class of flows.

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 Part II
Recent Advances and Trends in Turbulent
Combustion Models
Chapter 3
The Flamelet Model for Non-Premixed
Combustion

Benedicte Cuenot

Abstract The flamelet approach for non-premixed combustion is based on the de-
scription of the turbulent flame as a collection of laminar flame elements embedded
in a turbulent flow and interacting with it. The local structure of the flame at each
point of the flame front is supposed to be similar to a laminar flamelet, while the
interaction with turbulence is reduced to the front evolution. This view is supported
by the introduction of the mixture fraction, which allows to decouple the turbulent
transport and the flame structure. One key parameter of the flamelet structure is the
scalar dissipation rate, which controls the reactant fluxes to the reaction zone and
is related to the flow velocity gradients. Probability density functions or flame sur-
face density are then used to describe the turbulent flame and relate the flamelet
description to the turbulent flame front. As unsteady effects may become signifi-
cant, various transient flamelet approaches also exist to take into account the flame
history. The flamelet approach may be used either in the RANS or LES context and
is still being developed to account for additional complexities such as heat losses
and sprays.

3.1 Introduction

Non-premixed combustion occurs when the fuel and the oxidizer are injected sepa-
rately into the combustor, and experience simultaneous mixing and burning. This is
the case in many industrial systems, mainly for safety reasons. Another advantage
is that there is no need for a specific premixing device. Non-premixed flames are
found for example in furnaces, diesel engines or gas turbines. They are also called
diffusion flames as diffusion is usually the controlling mechanism. As combustion
is a fast process compared to convective and diffusive transport mechanisms, a com-
mon assumption is then to consider the flame chemistry infinitely fast. This leads to

B. Cuenot
CERFACS, 31057 Toulouse, France, e-mail: Benedicte.Cuenot@cerfacs.fr

T. Echekki, E. Mastorakos (eds.), Turbulent Combustion Modeling, 43

Fluid Mechanics and Its Applications 95, DOI 10.1007/978-94-007-0412-1 3,
Springer Science+Business Media B.V. 2011
44 Benedicte Cuenot

important simplifications in the flame structure description. There are however cases
where finite-rate chemistry plays a role, for example when the flame is submitted to
a high stretch rate. Then non-equilibrium effects must be taken into account.
Diffusion flames do not propagate and cannot be characterized, as premixed
flames, with well determined time and length scales (such as laminar flame speed
and laminar flame thickness). This explains why most models for non-premixed
combustion use the flamelet approach, which describes the basic features of the
flame and introduces the required scales. Although the interaction of diffusion
flames with turbulence is similar to turbulent premixed flames in many aspects (e.g.
front wrinkling, effect of stretch), the absence of own dynamics makes them more
sensitive to turbulence. The critical extinction stretch rate is one order of magnitude
smaller than for premixed flames, and the increased probability of quenching make
the use of the flamelet description less justified. Extensions to transient flamelets
allow to partly overcome this difficulty but introduce additional complexity in the
computations and are more delicate to apply in complex flows.
The various flamelet models differ by their use of the flame structure in the mix-
ture fraction space, and therefore have some similarities with the Conditional Mo-
ment Closure methods (see Chapter 5). For premixed flames, the flamelet concept is
described in detail in Chapter 4 and will not be discussed further here. This chapter
reviews the fundamentals of the method for non-premixed combustion, applications
in the context of RANS, developments in the context of LES, and discusses some
future directions. It is largely inspired by the books of Peters [32] and Poinsot and
Veynante [38].

3.2 Fundamental Concepts

Following the earlier ideas of [27, 47], Peters [31] introduced a turbulent combustion
model based on the flamelet concept. This term now designates a variety of physical
models in which the turbulent flame is viewed as a collection of laminar flame ele-
ments embedded in a turbulent flow and interacting with it. The main advantage of
the flamelet concept is that it decouples the complex chemical structure of the flame
from the flow dynamics, which can be modeled independently. The flamelet struc-
ture is either described with a simplified analytical formulation or with a flamelet
library involving the detailed chemical structure of the flame. For non-premixed
combustion, it corresponds to the laminar counterflow flame, which determines a
steady diffusion flame structure. The flamelet is therefore submitted to strain only
and curvature is ignored, although it may have non negligible effects [13, 23].
The Flamelet Model 45

3.2.1 The Mixture Fraction

The decoupling of the flame structure from the flow dynamics is obtained via the
introduction of the mixture fraction variable Z. The temperature and species mass
fractions equations differ only by their chemical source terms, and one can combine
them so as to cancel the chemical source term. For a single-step reaction described
as:
F F + O O P P (3.1)
the combination sYF YO , where s = OWO /F WF and Wk is the molecular weight
of species k, is a passive scalar; its transport equation carries no source terms. This
passive scalar can then be used to describe the flame dynamics, while the flame
structure is still described by the mass fractions and the temperature. If normalized
so as to vary from 1 in the fuel side to 0 in the oxidizer side, this passive scalar
corresponds to the mixture fraction, defined as:

sYF YO +YO0
Z= (3.2)
sYF0 +YO0

where YF0 and YO0 are the fuel and oxidizer mass fractions in the unburnt fuel and
oxidizer streams. The mixture fraction Z measures the degree of premixing of both
reactants. The stoichiometric value Zst is calculated by setting sYF YO = 0 in Eq.
3.2.
There is no unique definition of the passive scalar, as multiple combinations of
the temperature and species conservation equations can be done. For example the
following quantities can also be used for the flamelet analysis:

T TO0 +YF
Z1 = (3.3)
TF0 TO0 +YF0
or
s(T TO0 ) +YO YO0
Z2 = (3.4)
s(TF0 TO0 ) YO0
where T is a normalized temperature. In fact Z = Z1 = Z2 as they are all solutions
of the same equation with the same boundary conditions.
A more general way to define the mixture fraction is related to the conservation
of chemical elements, as the mass of elements is conserved throughout the reaction.
A common definition is for example the one by Bilger [3].
Note that in case of multiple fuel injections, the problem cannot be described
with one single mixture fraction anymore but requires multiple passive scalars [20].
46 Benedicte Cuenot

3.2.2 The Flamelet Solution

The basic principle of the flamelet approach relies on the variable change (x,t) Z.
If the dynamics and flame structure are decoupled, then the space and time evolu-
tions of the temperature and species profiles are fully described by the space and
time evolution of Z.
There are different ways of expressing flame variables as functions of Z. The
most simple, called the Burke-Schumann solution, makes use of the infinitely-fast
chemistry assumption. In this case, the reaction zone reduces to an infinitely thin
layer and the two reactants never coexist. Then, the flame solution is written as:

Z Zst : Z Zst :
YF = YF0 ZZ st
YF = 0
1Zst  
YO = 0 YO = YO0 1 ZZst
T = ZTF0 + (1 Z)TO0 +YF0 Zst 1Z
1Z
st
T = ZTF0 + (1 Z)TO0 +YF0 Z

Outside the reaction zone, the temperature and species profiles are linear func-
tions of Z. If the infinitely-fast chemistry involves reversible reactions, the flame is
at equilibrium at each value of the mixture fraction.
However, real flames have finite-rate chemistry and non-equilibrium effects are
present. The variable change (x,t) (Z,t) applied to the conservation equation of
any flame variable, for example the mass fraction of species k, gives:

Yk 2Yk
= D (Z)2 + k , (3.5)
t Z2
where equal diffusivities have been assumed for all species. If chemistry is suffi-
ciently fast, the unsteady term of Eq. 3.5 can be skipped as the time dependence
of the flame is mainly due to transport, and therefore already described by the time
evolution of Z. Fast chemistry also means a thin reaction zone around the value Zst ,
and the Z-gradient can be evaluated at stoichiometry. The steady flamelet equation
is then obtained:
2Yk
D (Z)2st + k = 0 (3.6)
Z2
This equation applies to all species and the temperature and allows a full description
of the flame structure as a function of Z. The scalar dissipation rate st = 2D(Z)2st
plays a key role in diffusion flame structure and is a very important parameter in non-
premixed turbulent flame modelling. Equation 3.6 may be solved analytically for
simplified expressions of the reaction rate. In the case of sufficiently fast chemistry
(high Damkohler numbers), asymptotic analysis allows to solve the complete prob-
lem for single-step chemistry [25]. To obtain detailed chemical solutions, flamelets
can be computed with dedicated software, and stored in flamelet libraries which
can then be used in complex turbulent flame calculations. One difficulty here is to
determine the passive scalar which guarantees the uniqueness of the solution.
The Flamelet Model 47

3.2.3 The Counterflow Diffusion Flame

The counterflow diffusion flame is very often used in experiments and numerical
studies as it is the simplest configuration of a steady one-dimensional diffusion
flame. It is then used as the generic flamelet in the steady flamelet approach and
flamelet libraries are built using this configuration.
In the counterflow configuration, a planar diffusion flame stabilises at the sto-
ichiometric plane between the opposed fuel and air flows. The flame is subjected
to strain, which imposes the scalar dissipation rate. The flow is described by the
potential solution:

u = ax (3.7)
v = ay (3.8)

where a is the strain rate. The mixture fraction is the solution of the one-dimensional
stationary transport equation:
Z 2Z
u =D 2 (3.9)
x x
Replacing u with ax in the above equation, and using the boundary conditions Z = 1
on the fuel side (x ) and Z = 0 on the oxidizer side (x +), one easily finds
the solution for Z:   
1 x
Z(x) = 1 erf  (3.10)
2 2D/a

Variable-density effects can be included by replacing x with = 0x ( /0 ) dx where
0 is a reference value. The scalar dissipation rate then is written as:
a  a 
st = exp xst2 (3.11)
D
where xst is the position of the stoichiometric plane, and can be found from Zst and
Eq. 3.10. The scalar dissipation is proportional to the strain rate, i.e. the velocity
gradient along the flame front, which is useful to evaluate in turbulent flows.
This counterflow flame structure is equivalent to the unsteady unstrained flamelet,
when Z is now the solution of:
Z 2Z
=D 2 (3.12)
t x

Introducing the similarity variable = x/ 2Dt (or = / 2Dt), the solution for
Z is written:  
1
Z(x,t) = 1 erf (3.13)
2 2
Here the scalar dissipation rate becomes:
48 Benedicte Cuenot

1  a 
st = exp xst2 (3.14)
2 t D
This means that a steady flamelet with a strain rate a has the same structure in the
mixture fraction space as in the unsteady flamelet at time t = 1/2a.
Both flamelet descriptions (steady or unsteady) introduce space and time scales,
which are useful to characterize turbulent diffusion
 flames. The diffusive flame
thickness can be defined as f = (1/z)st = 2D/st , while a chemical
time scale
can be evaluated from the reaction constants, for example as c = A exp Ta /T f
where A is the pre-exponential factor and Ta the activation temperature. Asymptotic
analysis [13] leads to more precise evaluations of the chemical time scale. The reac-
tion layer thickness is much smaller than the diffusive thickness, and is also related
to the chemical parameters.

3.2.4 Validity of the Flamelet Approach

It is generally believed that the flamelet concept is valid in the range of high
Damkohler number, which compares the chemical time scale c to the flow time
scale t :
t
Da = (3.15)
c
High Da corresponds to fast chemistry compared to the turbulent time scale. In this
case, the chemistry is fast enough to adapt instantaneously to the flow changes and
unsteady effects can be neglected. Another condition is that the flame thickness is
sufficiently small compared to the turbulent length scales, so that vortices cannot
disturb the inner structure of the flame. Even if chemistry is fast, i.e. the reaction
layer is thin, the diffusive flame thickness is larger and may be perturbed by the
flow. Both space and time-scale ratios are usually recast in two non-dimensional
numbers: Da and the turbulent Reynoldsnumber Ret . It can be shown that the time
ratio condition corresponds to Da=Da0 Ret where Da0 is a critical value under
which unsteady effects appear. This value can be determined either with asymptotic
analysis or Direct Numerical Simulation of turbulent diffusion flames. A typical
regime diagram is shown in Fig. 3.1, using a log-log scale for both Da and Ret .
In
this diagram, the different regimes are separated by lines corresponding to Da
Ret [38].
There are diverse approaches to turbulent flamelet models, but they all include
the same ingredients:

A laminar flamelet description, as presented above, describing the local flame

structure,
A turbulence model, describing the unclosed terms of the averaged or filtered
equations,
A coupling model to describe the interaction of the flame with the turbulent flow.
The Flamelet Model 49

Fig. 3.1: Regimes for turbulent diffusion flames. Adapted from [38].

In the following, we will discuss the last item, which is the main differencing as-
pect among the various flamelet models. These models have been all derived in the
RANS context. Their extension to LES already have yielded good results but also
have raised new questions related to the subgrid scale motion.

3.3 RANS Flamelet Modeling

In the flamelet approach, flame dynamics are described by the mixture fraction Z
and the solution of a classical Favre-averaged transport equation for Z:

Z  
+ v Z = DZ v
 Z  (3.16)
t

where the turbulent flux v

 Z  is usually modeled via the gradient transport assump-

tion:
v t 
 Z  = D Z (3.17)
The molecular diffusivity is often neglected as it is much smaller than the turbulent
diffusivity Dt . To fully describe the mixing problem, higher moments of Z are usu-
ally required, such as the variance Z 2 , for which a conservation equation is also

written:
Z
2
+ v Z
2 = ( D Z
t  2 
2 ) + 2 D (Z)
t (3.18)
t
50 Benedicte Cuenot

where the mean scalar dissipation rate is defined as:

2
 = 2D(Z) (3.19)

This scalar dissipation rate measures the decay of the mixture fraction fluctuations
and plays for Z the equivalent role of the dissipation rate of the kinetic energy for
the velocity. The associated time scale Z 2 /
 is then often set proportional to the
flow time 
k/ , leading to the following expression for :
2
 = c Z (3.20)

k
where the constant c is of order unity.
Both equations 3.16 and 3.18 can be solved in the RANS solver, and combined
to the flamelet structure to describe the turbulent flame [32]. The most common ap-
proach uses a PDF approach, either of presumed shape or computed with a PDF
transport equation. Examples of presumed PDF shapes include the Dirac, the
and the Gaussian distributions. These distributions are constructed from a minimum
set of moments (e.g. two for the or Gaussian distributions) and may be deficient
in describing important non-equilibrium related distributions, which may require
higher moments or when the parameters that characterize these distributions are not
independent. However, intermediate strategies between presumed PDFs and trans-
ported PDFs have been developed. A statistically most likely distribution (SMLD)
[22] represents a systematic strategy for the construction of statistical distributions
based on prescribed moments. The SMLD approach is based on the maximum en-
tropy principle. Another alternative approach is based on constructed PDFs from
simulations, such as the ones based on the linear-eddy model (LEM) [18].
An alternative approach is based on the flame surface density, defined as the
flame surface area per unit volume. Several approaches exist to calculate this quan-
tity, based on algebraic models [6, 29], fractal theory [19] or on an additional trans-
port equation with sink and source terms [27, 39, 43, 44].
Although the PDF and flame surface density models are conceptually different,
they are closely linked through relationships derived by Veynante and Vervisch [45].

3.3.1 Steady Flamelets

In the PDF approach, any averaged flame variable i (being either species mass
fraction, temperature or chemical source term) can be expressed as:
 1
i = (i |Z)p(Z)dZ (3.21)
0

where the quantity (i |Z) designates the conditional average of i for a given value
of the mixture fraction Z, and p(Z) is the probability density function of Z. As
The Flamelet Model 51

Eq. 3.21 can be written either for conserved or primitive variables, or for reaction
rates, there are two possible methods to use the PDF approach. In the first method,
the mean mass fractions and temperature are calculated with Eq. 3.21, using the
flamelet formulation to express ( Yk |Z) and ( T |Z) as functions of Z. In this case
the conservation equations for these variables can be skipped, and knowledge of
p(Z) is sufficient to describe the turbulent flame. In the second method, only the
mean reaction rates are calculated with Eq. 3.21, and are introduced in the averaged
conservation equations of species and energy. The flamelet is used here for reac-
tion rates only. The first method is clearly less time-consuming, as the number of
equations to be solved is considerably reduced. On the other hand, it does not allow
to include additional flow effects on the flame structure, such as heat losses, curva-
ture or reactants inhomogeneity in the fresh gases. Both approaches are not strictly
equivalent, mainly because the impact of the turbulent species and heat fluxes on
the mean reaction rate is not taken into account in the first method.
If the infinitely-fast chemistry assumption is used, the conditional average of the
flame variables is simply:

( Yk |Z) = (Z)Yk (Z), ( T |Z) = (Z) T (Z). (3.22)

This assumption is often combined with a presumed-shape PDF. Since the mixture
fraction varies between 0 and 1, the -function is often used:

(a + b) a1
p(Z) = Z (1 Z)b1 , (3.23)
(a) + (b)
where the function is defined as:
 +
(a) = ex xa1 dx. (3.24)
0

The parameters a and b are calculated from the mean and variance of Z:
 
Z 1 Z
 with =
a = Z ; b = (1 Z) 1. (3.25)
Z
2

The -pdf takes various forms depending on the parameters a and b, and can be
either (almost) bimodal or unimodal for higher values of . It is however not able to
represent all situations, and other forms have been proposed in the literature.
The simplicity of the infinitely-fast chemistry flamelet model makes it very
attractive, but it fails to predict situations with finite-rate chemistry. The steady
flamelet analysis gives:
1 2Yk
k = (3.26)
2 Z2
This reaction rate is then a function of Z and , and a joint PDF p(Z, ) must be
used to evaluate the mean value:
52 Benedicte Cuenot
 1
1 2Yk
k = p(Z, )d dZ (3.27)
2 0 0 Z2
From Eq. 3.5, the second derivative of Yk is a Dirac-function at Z = Zst , and, for
example, one finds for the fuel:

YF0
F = st p(Zst ) (3.28)
2(1 Zst )

A similar result has been derived for infinitely-fast chemistry (see the review in
Ref. [3]). Introducing finite-rate chemistry, the flamelet approach assumes that, as
long as the reaction zone is thinner than the smallest size eddies, the flame locally
keeps the structure of a laminar flame. This corresponds to the high Damkohler
number introduced in Sec. 3.2.4. Therefore, fluctuations of flame variables inside the
reaction zone thickness can be neglected, whereas fluctuations related to turbulent
mixing in the diffusive layer must be described with the mixture fraction and its
scalar dissipation rate. The mean variables are then expressed as:
  1
Yk = Yk (Z, st ) p(Z, st ) dZ d st (3.29)
0 0
  1
T = T (Z, st ) p(Z, st ) dZ d st (3.30)
0 0
where the functions Yk (Z, st ) and T (Z, st ) are taken from the steady lami-
nar flamelet library. The difficulty again is in the determination of the joint PDF
p(Z, st ), and to simplify the problem statistical independence is often assumed:

p(Z, st ) = p(Z) p(st ) (3.31)

Assumed shapes for the PDF of st can be Dirac functions: p(st ) = (st  st ),
corresponding to a constant scalar dissipation at the mean value 
st . However, it has
been shown that this PDF is more like a log-normal distribution [15]:

1 (ln( st ) )2
p(st ) = exp (3.32)
st 2 2 2


where = ln(st ) 2 /2 and is the variance of ln(st ):
st =
2 st2 (exp 2 1),
often taken constant of the order unity. In the above expressions, the mean stoichio-
metric scalar dissipation st is still unknown. However, the mean scalar dissipation
 is known from Eq. 3.20, and may be used to evaluate st through their analytical
relation given by flamelet analysis: /F(Z) = st /F(Zst) where the function F(Z)
is: 
2 
F(Z) = exp 2 erf1 (2Z 1) (3.33)

The mean reaction rate can be calculated with the same methodology, taking
k (Z, st ) from the flamelet library:
The Flamelet Model 53
  1
k = k (Z, st )p(Z, st )dZ d st (3.34)
0 0

Alternatively, the flame surface density concept may be used [27, 39, 4144]. In this
approach, the flamelet library contains only the total (i.e. integrated across the flame
front) reaction rate, and the mean reaction rate is written:

k = k (st ) (3.35)

where k (st ) is the integrated reaction rate averaged along the flame front, and
depends only on the scalar dissipation and is the mean flame surface density
defined as:
= |Z|Z=Zst p(Zst ). (3.36)
In the case of fast chemistry (thin flame front), the following relation can be derived
[45]:
  1 
k = k (Z, st )p(Z, st )dZd st = k (st )p(st )d st (3.37)
0 0 0

showing the link between PDF and flame surface density approaches. Assuming
p(st ) as a Dirac function centered on st , one finds that:

k (st ) = k ( st ). (3.38)

The quantity k (st ) is then taken from the flamelet library. The main issue in this
approach is the determination of . Marble and Broadwell [27] were the first to
propose a conservation equation, including convective and turbulent transport as
well as sink and source terms:

v ) + (v ) = P S
+ ( (3.39)
t
where P is a production term, proportional to , where is a mean strain rate and
S is a destruction term corresponding to mutual annihilation of flame surface.
In the above formulations, one single parameter (st ) is used to describe the
flamelet library. It allows to correctly describe flames in the high Damkohler num-
ber limit, but is not sufficient to capture additional effects such as curvature, ignition
or extinction, flame history and transient effects.

3.3.2 Transient Flamelets

If the scalar dissipation rate changes rapidly compared to the flame chemical time
scale, the unsteady term in the flamelet equations must be retained as in Eq. 3.5,
where the scalar dissipation is now a function of time. Haworth et al. [21] made
54 Benedicte Cuenot

an ad hoc modification of the flamelet model, introducing an equivalent strain rate

model, to limit the effect of rapidly changing . This idea is consistent with other
studies of flames subjected to periodic oscillations of the strain rate, where it was
found that, depending on the flame response time compared to the frequency, un-
steady effects on the local flame structure may stay moderate. However, it was also
shown that, at high frequencies, the strain rate may exceed the laminar quenching
limit without flame extinction [14]. This is due to the delayed flame response, which
implies a certain time necessary to fully quench the flame. Such unsteady effects
were analysed in detail by Cuenot et al. [12].
In turbulent jet diffusion flames, unsteady effects are described by a Lagrangian
residence time of the flamelet, which characterizes the velocity at which the flamelet
is transported along the flame front, and is based on the flow velocity:
 x
1
t= dx (3.40)
0 u(x)|Z = Zst

where x is the distance from the nozzle and u is evaluated along the stoichiometric
line. Alternatively an averaged velocity may be used at each flame height, defined
along the radial direction in the zone where the flame exists (determined from a
given range of the mixture fraction):

1
< u(x) >= udAZ (3.41)
AZ AZ

The importance of transient effects in turbulent diffusion flames was also dis-
cussed by Pitsch et al. [36]. They introduced a diffusion time, that is the time needed
to transport mass and energy over a distance ( Z) in the mixture fraction space:

( Z)2
t = (3.42)
st

where ( Z) is typically the flame thickness. If this time is short compared to the
flamelet residence time, the flame is able to adapt fast enough to the flow variations,
and unsteady effects can be neglected. Generally, this is true in the first part of the
flame jet, up to 30 diameters [36], where turbulence is strong enough to maintain
high values of st . Further downstream the scalar dissipation decreases, the flame
thickens and unsteady effects appear.
Mauss et al. [28] introduced unsteady flamelets to simulate flamelet extinction
and re-ignition in a statistically-steady turbulent jet diffusion flame. They used the
residence time t as an additional parameter in the flamelet library, also viewed as a
stochastic variable with a probability density function having a mean and a variance.
Assuming a Gaussian distribution, they found the probability of the residence time
as:  
1 t t
p(t) = 1 + erf (3.43)
2 2
The Flamelet Model 55

where the variance is unity. A stochastic approach was also used by Blanquart and
Pitsch [4] to introduce a random scalar dissipation in the transient flamelet equa-
tions.
Other and more complex approaches exist to account for unsteady effects and are
described below.

3.3.3 Representative Interactive Flamelets (RIF) Model

The RIF model was proposed by Pitsch et al. [35] and is based on the interactive cou-
pling of two different solvers: a CFD code computing all necessary flow and flame-
averaged variables, as well as the averaged mixture fraction, its variance and scalar
dissipation rate, and an unsteady flamelet code calculating the unsteady flamelet in
the mixture fraction space. The averaged species mass fractions are calculated as:
 1
Yk (x) = Yk (Z,t, st )p(Z)dZ (3.44)
0

where p(Z) is a presumed-shape PDF of Z, based on the mean Z(x)  and variance
2
Z (x) calculated by the CFD code. In the above expression, a constant scalar dis-
sipation is assumed, but the model can be easily extended to account for a scalar
dissipation distribution. The flame residence time t is calculated with Eq. 3.40. The
model is illustrated in Fig. 3.2. Calculations are performed iteratively, starting with
a steady flamelet solution. Convergence is reached when the scalar dissipation con-
verges within a prescribed tolerance.

Fig. 3.2: Iterative loop in the RIF model. Adapted from [32].
56 Benedicte Cuenot

3.3.4 Eulerian Particle Flamelet Model (EPFM)

The use of a single unsteady flamelet for the whole turbulent flame limits the va-
lidity of the approach and may lead to unphysical behaviors, in particular in the
flame stabilisation zone. Multiple unsteady flamelets may be used, corresponding
to different values of the scalar dissipation describing its evolution along the flame
front [46]. The Eulerian Particle Flamelet Model proposed by Barths et al. [1, 2] in-
troduces fictitious marker particles to follow the unsteady flamelets evolution. The
mass-weighted fraction of particles corresponding to the flamelet l follows the equa-
tion: 
Il  t 
v Il )
+ ( Il = 0 (3.45)
t Scl
where Scl is the Schmidt number characterizing turbulent diffusion. The above
equation allows the determination of the probability of finding the flamelet l at each
location. The flamelet time is started from the time at which the particles are re-
leased. A time-dependent scalar dissipation is obtained from a weighted average
over the region where the particles are dispersed. Integrating the unsteady flamelet
equations over the residence time needed to reach a particular location along the
flame front, and using the presumed-shape pdf at this location, all mean flame vari-
ables can be calculated.

3.3.5 FlameletProgress Variable (FPV) Models

To fully describe the flamelet structure, and in particular the reaction zone, the
flamelet-progress variable approach combines both the mixture fraction Z and
progress variable C [24, 26]. The progress variable describes the reaction progress
and evolves through the flame from 0 in the fresh gases to 1 in the burnt gases. It
is mainly used to describe premixed flames, where mixing is already achieved be-
fore burning. In diffusion flames, the progress variable may be useful to describe
the reaction zone as the mixture fraction only describes mixing and ignores the
flame. Using the assumption that the reaction rate is the same for diffusion flames
and premixed flames at the same mixture fraction, Bradley et al. [5] built premixed
flamelet libraries to calculate the reaction rate k (Z,C), where Z is interpreted as
the premixed flame equivalence ratio, and used these libraries to compute turbulent
diffusion flames at the same value of Z.
The progress variable is constructed from the flame variables, with the constraint
of being monotonic from fresh to burnt gases, and non-dimensionalized to vary
in the range [0;1]. In simple (or reduced) chemistry, the temperature, reactants or
products may be used but not the intermediate species:
T Tu Yk Yk,u
C= or C = (3.46)
Tb Tu Yk,b Yk,u
The Flamelet Model 57

where the subscripts b and u stand for burnt and unburnt. In complex chem-
istry the definition of C is more difficult. First the temperature does not always stay
monotonic. Second the reaction zone is a superposition of multiple reactions, most
of them between intermediate species, and reactants or products do not fully repre-
sent the reacting zone. In this case combinations of species, involving for example
CO and CO2 , are built to verify monotonicity.
Introduced in LES by Pierce and Moin [33], the FPV approach may also be
viewed as an unsteady flamelet model. The unsteady evolution of the flamelet equa-
tion is then embedded in the progress variable transport equation, where the reaction
rate is kept:
C  
+ (  = Dt C +
v C) 
C (3.47)
t
The mean flame variables are then reconstructed from the expression:
 1 1

i = i (Z,C)p(Z,C)dZdC (3.48)
0 0

for a constant scalar dissipation. The above expression introduces a joint PDF, mod-
elled by first writing:
p(Z,C) = p(C|Z)p(Z) (3.49)
where p(Z) is classically assumed to be a -pdf, and the conditional p(C|Z) is taken
from a single flamelet relation:

p(C|Z) = (C C(Z, st )) (3.50)

where C(Z, st ) is the progress variable as a function of Z for a flamelet at scalar

dissipation st . The scalar dissipation st is chosen in order to satisfy:
 1
C = C(Z, st )p(Z)dZ (3.51)
0

The implementation of the model is similar to that of the SLFM, where the parame-
 This is an important improvement, as
ter st is replaced by the progress variable C.
the modeling of the scalar dissipation (Eq. 3.20) is the weak point of the flamelet ap-
proach. One issue is the definition of the progress variable. In single-step chemistry,
any species, or temperature, may be used as the system has only one degree of free-
dom in addition to the mixture fraction. For large complex chemistry descriptions,
the definition of C is not so clear and bad choices may lead to important errors. In
particular, the progress variable must guarantee a unique mapping of the flame state,
i.e. have a monotonic behavior across the flame.
The FPV model was further extended by Pitsch and Ihme [37] to account for
unsteady effects.
58 Benedicte Cuenot

3.4 LES Flamelet Modeling

Although all modelling issues have not been yet addressed, LES is now currently
applied to combustion applications. Most of the flamelet models developed in the
RANS context have been adapted to LES [34]. Filtered conservation equations re-
place averaged equations, and mixing subgrid-scale models use the turbulent vis-
cosity concept. PDF approaches can be directly applied to LES, where the PDFs
represent only the subgrid fluctuations of the variables and may also be described
with -functions, using the filtered value and subgrid fluctuations. The filtered flame
surface density may also be calculated via a filtered transport equation and adapted
source terms.
In the case of infinitely-fast chemistry, flame variables depend only on the mix-
ture fraction. Introducing the LES filter F(x), the filtered flame variables can be
written as [38]:
  1
i = i (Z ) (Z(x) Z ) F(x x ) dZ dx
V 0
 1 
= i (Z ) (Z(x) Z ) F(x x ) dx dZ (3.52)
0 V

where the quantity V (Z(x) Z )F(x x )dx represents the filtered PDF of Z.

3.4.1 Subgrid Scale Modelling

In addition to the filtered mixture fraction, and possibly filtered progress variable
or filtered flame surface density, flamelet models require the knowledge of the sub-
grid scale variance Z 2 and, in the case of finite-rate chemistry, the filtered scalar

dissipation .
The mixture fraction subgrid scale variance may be obtained with a transport
equation, which in turn requires modeling of unclosed subgrid terms. In LES, the
scale similarity assumption is often used, leading to:

 


2
Z Z Z
2 2 (3.53)

where Z denotes a test filter of larger size than the LES filter.
Using assumptions of local homogeneity and local equilibrium for the subgrid
scales, Pierce and Moin [33] derived algebraic models for the subgrid variance and
dissipation rate of Z:
Z 2
2 = C 2 |Z|
Z (3.54)
2
 = (D + Dt )|Z| (3.55)
The Flamelet Model 59

where Z is now the filtered mixture fraction, is the cutoff length of the LES fil-
ter and Dt is the subgrid diffusivity. The constant CZ may be evaluated through a
dynamic procedure.

3.5 Conclusion

A large variety of flamelet approaches exist and have been successfully applied to
various configurations. Extensions of the flamelet equation have found use even for
the study autoignition of completely premixed problems [10]. Flamelet models are
particularly adapted to LES, which provides good prediction of the filtered and sub-
grid variables, as well as their time evolution, to be used in the flamelet library.
One important feature of flamelet methods is that they give access to the fully de-
tailed chemical structure of the flame, including all minor species. Optimized and
efficient tabulation techniques (e.g. ILDM, ISAT, FPI) [11, 16, 40, 48] enable the
use of flamelet libraries in complex flow simulations. This is of major importance to
predict pollutant emissions and soot. Flamelet models are still being developed, in
particular to account for heat losses or liquid fuel sprays [17, 26] or to predict diesel
engine auto-ignition for which early examples may be found in [7, 8, 49].
Although a large majority of industrial systems are fed with separate fuel and
oxidizer flows, the flame in the chamber is usually partially-premixed, with pre-
mixed (with variable equivalence ratio) and non-premixed flame elements. In this
case unified models adapted to both premixed and diffusion flames must be devel-
oped. Flamelet methods are good candidates, as they are able to involve both the
mixture fraction and the progress variable. For example, a turbulent flame speed
method that gives ST (Z) from premixed laminar flamelets at various Z (i.e. various
equivalence ratios) and a PDF for Z was applied to turbulent jet diffusion flame
lift-off [9, 30].

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Chapter 4
RANS and LES Modelling of Premixed
Turbulent Combustion

Stewart Cant

Abstract Premixed combustion is becoming more common in practical combustion

systems in response to increasing regulatory pressure to reduce unwanted emissions.
The inherent ability of premixed flames to propagate into the unburned mixture
leads to a more active response to turbulence by comparison with non-premixed
flames. It is clear that the sheet-like reaction zone within the premixed flame is
highly resistant to disruption by turbulent eddies. Hence flamelet structure is preva-
lent in premixed combustion over a broad range of turbulent velocity fluctuation
magnitudes and turbulent eddy length scales. Modelling of turbulent transport in
premixed flames is rendered more difficult by the occurence of countergradient
transport in the presence of strong heat release. Modelling of the mean turbulent
reaction rate has involved a variety of approaches involving either algebraic expres-
sions or additional transport equations. A brief review of current modelling practice
is presented, covering some simple models together with the flame surface density
approach, the G equation and the more recent scalar dissipation rate model. The
emphasis is on models that are applicable in the context of both Reynolds-averaged
Navier Stokes (RANS) and Large Eddy Simulation (LES).

4.1 Introduction to Premixed Flames

Mixing of the fuel and air streams prior to combustion is becoming more com-
mon in practical energy-conversion devices due to greater legislative emphasis on
the reduction of unwanted emissions. A high degree of premixing provides for ef-
fective control of the stoichiometry of the flame. This brings significant benefits
by ensuring that the combustion reaction takes place under benign thermochemi-
cal conditions. As an example, it becomes possible using premixing to specify lean

Stewart Cant
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ,
United Kingdom, e-mail: rsc10@cam.ac.uk

T. Echekki, E. Mastorakos (eds.), Turbulent Combustion Modeling, 63

Fluid Mechanics and Its Applications 95, DOI 10.1007/978-94-007-0412-1 4,
Springer Science+Business Media B.V. 2011
64 Stewart Cant

fuel-air mixtures which avoid the emission of significant amounts of unburned fuel,
CO or particulates by virtue of the chemistry, and which minimise the formation
of thermal NOx by limiting the post-flame temperature. The use of well-designed
premixed burners can also help to reduce emissions of CO2 by contributing towards
greater energy efficiency of the plant.
Premixed combustion has been used in spark-ignition automotive engines for
many years. Often the fuel-air mixture strength is controlled by the engine man-
agement system to ensure that combustion takes place under stoichiometric condi-
tions for compatibility with exhaust clean-up using a catalytic converter. In recent
years, premixed combustion under very lean conditions has been widely adopted
for industrial gas turbines mainly in order to minimise emissions of NOx . Premixed
burners are also used for industrial furnaces, for example in the glass-making in-
dustry, where highly-efficient low-emission combustion is a necessity. In the field
of aerospace, premixed combustion is used in reheat systems for supersonic aircraft
but has yet to be adopted in the main combustors of gas turbine aero-engines.
The outstanding technological issues in premixed combustion centre mainly on
flame stability. Premixed burners must be carefully designed to avoid unwanted
phenomena such as blow-off and flashback, while premixed flames are notoriously
prone to convectively- and acoustically-coupled instabilities. Relatively small vari-
ations in the fuel-air mixture strength can have a major effect on the dynamics of
the flame, and the distribution of heat release must be designed wherever possible
to avoid acoustic coupling with the natural resonant modes of the combustion sys-
tem. It is rarely possible in practice to ensure that the mixing between fuel and air
is entirely complete before combustion takes place, resulting in partially-premixed
flames. Under some circumstances these are found to be more stable than a fully-
premixed flame under the same global stoichiometry, and the consequent slight loss
of emissions performance is accepted as an engineering compromise. Greater un-
derstanding of premixed flames would help to avoid this issue.

4.2 Modelling Framework for RANS and LES

4.2.1 Introduction

Premixed turbulent flames are often found to contain thin, highly-wrinkled reac-
tion surfaces in which most of the combustion chemistry takes place. Experimental
observations of premixed flames under a broad range of conditions of temperature,
pressure, mixture strength and turbulence intensity, and using a wide variety of mea-
surement techniques, have shown that such structures are remarkably robust. This
is the basis of flamelet modelling which has become well accepted for premixed
flames and which has proved successful in many different applications. This is in
contrast to the modelling of non-premixed combustion, where the flamelet concept
has proved to be less universally applicable.
RANS and LES Modelling of Premixed Turbulent Combustion 65

4.2.2 Regimes of Premixed Turbulent Combustion

Two measurable properties of laminar premixed flames, namely the laminar burning
velocity and the laminar flame thickness, have proved to be invaluable in parame-
terising regimes of behaviour for the turbulent case. The laminar burning velocity
SL is the speed at which a laminar flame will propagate normal to itself into the
reactant mixture. It is a well-defined but non-simple function of the pressure, the
reactant temperature and reactant composition. It emerges from mathematical anal-
ysis as an eigenvalue of the two-point boundary value problem that yields the flame
structure as a solution, and it can be measured accurately using modern diagnos-
tic techniques and also computed reliably given a suitably detailed treatment of the
chemical reaction mechanism and the molecular transport properties. It provides a
natural velocity scale for premixed flames and has no equivalent in non-premixed
flames. Typical values for stoichiometric hydrocarbon-air flames at ambient condi-
tions are in the region of 40-50 cm/s. The laminar flame thickness L is harder to
define, given that the flame structure merges smoothly with the surrounding vol-
umes of reactants and products. Standard measures of L include the normalized
inverse of the maximum temperature gradient or the distance between 5% and 95%
of the temperature rise through the flame. Dimensional reasoning also provides an
estimate of L as T /SL where T is the thermal diffusivity. The reaction zone thick-
ness R is known to be somewhat thinner than the overall flame structure which also
includes the preheat and equilibration zones found respectively ahead of and behind
the flame. For hydrocarbon-air flames, most estimates of R are on the order of 1
mm.
The regimes of premixed turbulent combustion can be plotted on the well-known
Borghi diagram [7] as modified by Peters [54] and shown in Fig. 15.1. The tur-
bulence is represented by the velocity fluctuation magnitude u together with the
integral length scale lT , while the thermochemistry is represented by SL and L .
This allows any given premixed turbulent flame to be plotted on the diagram, and its
location to be characterised by a set of dimensionless numbers such as the Reynolds
number Re, the Damkohler number Da and the Karlovitz number Ka, respectively
defined for the flame as
u  lT SL lT L u
Re = ; Da = ; Ka = (4.1)
SL L u L SL
where is the Taylor microscale of the turbulence.
Flamelet behaviour is seen to be prevalent except for high values of the Karlovitz
number Ka where distributed combustion occurs. The major subdivisions into the
corrugated flamelets regime and the thin reaction zones regime reflect the obser-
vation that the reaction zone can survive even in circumstances where the preheat
and/or equilibration zones have been disrupted by small-scale turbulence. An addi-
tional Karlovitz number Ka = R u /SL is required [54] to account for this.
Clearly the Borghi diagram is an oversimplification, since the turbulence in prac-
tical cases is rarely well-represented by single values of u and of lT . Moreover, the
66 Stewart Cant

Fig. 4.1: Regime diagram for premixed turbulent combustion. Adapted from Peters
[54].

transition between regimes is not sharply defined. Nevertheless the concept is useful
and can be used to guide the most appropriate modelling approach. There are exten-
sions of the diagram to account for the effects of filtering in LES [58]. Extension to
partially-premixed turbulent flames may be as straightforward as drawing an ellipse
around each point in order to represent the ranges of SL and L encompassed by the
variations in the fuel-air mixture composition, provided that the mixture remains
within the flammability limits.

4.2.3 Averaging and Filtering

The large range of length scales present in practical turbulent combustion problems
means that full resolution is unattainable at reasonable computational cost. Hence
it is necessary to treat at least the smallest scales on a statistical basis. Inevitably
this involves some form of averaging in which information is lost. The purpose of
modelling is to restore the averaged effect of the missing scales.
In the classical Reynolds Averaged Navier Stokes (RANS) approach, standard
Reynolds time averaging of any quantity Q results in the mean Q according to
 T
1
Q(x) = Q(x,t)dt (4.2)
T 0
RANS and LES Modelling of Premixed Turbulent Combustion 67

where T is the timescale for averaging, assumed to be long relative to any flow time
of interest. This approach assumes that the problem is statistically stationary, and is
often too restrictive for many combustion applications. Instead, Reynolds ensemble
averaging may be employed, using the definition

1 N

Q(x,t) = Q(x,t; n) (4.3)
N n=1

where the average is taken over a notional (and large) number of realisations N. This
approach allows for some level of unsteadiness, at least on a time scale that is slow
relative to the turbulent fluctuations. In either case the instantaneous fluctuation from
the mean is denoted by Q . Note that the averaging operation for RANS is applied
equally to all length scales. In principle no length scale is favoured over any other.
In practice, the nature of turbulence ensures that flow features that are uncorrelated
with each other are averaged out, and only features that are well-correlated over the
averaging timescale are retained. Hence, turbulent eddies of all scales are removed
leaving only the mean flow field.
In Large Eddy Simulation (LES) a spatial filtering operator is employed in order
to remove flow features that are smaller than a specified filter size . The filtering
operation involves the convolution integral


Q(x,t) = Q(x ,t)F(x x ; )dx (4.4)

where the integral is taken over the entire flow domain and F is a filter function
which must satisfy the normalisation condition

F(x ; )dx = 1 (4.5)

The function F is often chosen to have a simple top-hat form in either physical space
or wavenumber space, although there are advantages in choosing a Gaussian form
instead [61]. Note that the filtering operation is applied in space only. In contradis-
tinction to RANS averaging, the nature of turbulence then ensures that there is an
equivalent filtering operation in time, with an implied temporal filter width that is
proportional to the spatial filter width divided by the velocity scale of the turbu-
lent eddy at that length scale. This filtering timescale can be interpreted as the eddy
turnover time of the filter-scale eddy.
In turbulent combustion, density fluctuations arise due to the combined effects of
the turbulence and the thermal expansion due to heat release. It has become standard
to use Favre density-weighted averaging or filtering in order to avoid the occurrence
of density correlation terms in the governing equations. The Favre decomposition
for a quantity Q is written as Q = Q + Q where the Favre mean is given by

Q
Q = (4.6)

68 Stewart Cant

and the Favre fluctuation Q has the properties Q = 0 but Q = 0. The oper-
ation denoted by the overbar may correspond to any of the averaging or filtering
techniques outlined above.
Given that RANS and LES both have implied filtering length and time scales
that are related through the nature of turbulence, it should be possible to use similar
modelling in both approaches in order to restore the lost information at small length
and time scales. Indeed this turns out to be the case, although care must be taken
in order to interpret the different meaning of apparently similar quantities in each
approach.

4.2.4 Modelling Principles

In both RANS and LES the requirement for modelling arises from the need to pro-
vide closure of the averaged or filtered governing equations. Assuming that a suit-
able closure model already exists for the momentum equations, the remaining task
is to provide a consistent closure model for the transport equations describing the
thermochemistry. In premixed turbulent combustion, it is common practice to define
a reaction progress variable c which rises monotonically from zero in the fresh reac-
tants to unity in the fully-burned products. If the Lewis number is close to unity, the
Mach number is low and the problem is adiabatic, then the reaction progress variable
alone is sufficient to describe the complete thermochemical state of the system. The
various quantities of interest, such as the species mass fractions and temperature,
can be obtained by choosing a suitable definition for the reaction progress variable.
In the simplest case, c is treated as a scaled mass fraction of one or more product
species YP :
YP
c= (4.7)
YP
where YP is the value of YP in the fully-burned or equilibrium products. Alterna-
tively, c can be defined in terms of the fuel mass fraction YF according to:
YF YF0
c= (4.8)
YF YF0
where YF0 is the fuel mass fraction in the fresh reactants and YF is the (notional)
value of YF in the products. Clearly for lean combustion YF = 0. More complicated
definitions for c in terms of combinations of species mass fractions can be devised
to suit particular cases. The principal criteria are that c must be monotonic, bounded
between zero and unity, and provide a reasonable representation of the flame struc-
ture in terms of its thickness and location. In the rather specific case of low Mach
number adiabatic flames with unity Lewis number, it is possible to define a reaction
progress variable based on temperature according to
T T0
c= (4.9)
T T0
RANS and LES Modelling of Premixed Turbulent Combustion 69

where T0 and T are the limiting temperatures in the reactants and products respec-
tively. This definition becomes invalid whenever the temperature is affected by any
factor external to the flame, including acoustic activity, heat losses or differential
transport of heat and mass. Thus, a definition of c in terms of temperature lacks
flexibility and is not recommended for more general use.
With a suitable definition of c, a transport equation for reaction progress variable
can be derived from the relevant transport equations for species mass fraction. The
exact transport equation for c may be stated as
 
c
c + uk c = w + Dc (4.10)
t xk xk xk

in which w is the chemical production rate of c and Dc is the molecular diffusivity

of c.
For RANS, the Favre-averaged reaction progress variable transport equation is
 
c
c + uk c = w + Dc u c (4.11)
t xk xk xk xk k

The terms on the right-hand side of this equation represent the mean chemical reac-
tion rate, the mean rate of molecular transport and the Reynolds flux or mean rate of
turbulent transport. All three terms are unclosed and require modelling, although for
high Reynolds number flows the molecular transport term is often simply neglected.
For LES, the Favre-filtered transport equation for c has the form
 
c
c + uk c = w + Dc ( uk c uk c)
(4.12)
t xk xk xk xk

where now the terms on the right-hand side represent the filtered chemical reaction
rate, the filtered rate of molecular transport and the sub-grid scale contribution to
the rate of turbulent transport. Again, all of these terms are unclosed and require
modelling. Note that the form of the transport equation for c is similar for both
RANS and LES. This reflects the fact that in both approaches the combustion is
taking place at unresolved scales. This gives some hope that modelling devised for
RANS may well remain applicable in LES, and vice versa.
The molecular transport term is usually retained in LES, since the grid-scale
Reynolds number is not sufficiently large to justify its neglect relative to the turbu-
lent transport term. The simplest closure model for this term is based on the assump-
tion that the mass diffusivity Dc is uncorrelated with reaction progress variable:
 
c c
Dc = Dc (4.13)
xk xk

in which an additional assumption has been made to allow for the replacement of the
filtered value c with the Favre-filtered value c inside the spatial derivative. In reality,
70 Stewart Cant

the mass diffusivity Dc is often a function of c, but more complicated modelling

is rarely justified in view of the more pressing modelling issues associated with the
other terms in the transport equation.

4.3 Transport Modelling for Premixed Turbulent Flames

In a turbulent premixed flame the mean reaction progress variable rises from zero
to unity across the flame brush. In passing from reactants to products through a sta-
tistically one-dimensional flame, the mean density falls due to heat release and the
mean velocity normal to the flame rises due to mass continuity. Hence the condi-
tional mean velocity in the products is greater than that in the reactants, i.e uP > uR .
Fluctuations in the instantaneous velocity and reaction progress variable will occur
due to the turbulence, and heat release ensures that an upwards (downwards) fluc-
tuation in c will be associated with an upwards (downwards) fluctuation in u. Thus
the physics ensures that u and c are statistically correlated, and moreover the corre-
lation expressed by the Reynolds flux u c is positive along the direction through
the flame in which the gradient of mean reaction progress variable is positive also.
For this situation the standard gradient model for turbulent scalar transport may
be written as:
c
uk c = DT (4.14)
xk
where DT is a turbulent diffusivity. A difficulty arises at once, since the gradient
model predicts a negative value of the Reynolds flux when the mean progress vari-
able gradient is positive and vice versa. Thus the standard gradient transport model
is not valid in general for turbulent premixed flames, in which the phenomenon of
countergradient transport may well occur instead.
In physical terms, countergradient transport is a pressure-driven effect, and is
a consequence of thermal expansion due to heat release within the flame. The re-
sulting increase in the velocity through the flame brush is associated with a self-
induced favourable pressure gradient, which acts preferentially to accelerate the
lower-density product gases in preference to the higher-density reactants. This tends
to transport these gases further up the gradient of reaction progress variable, in the
opposite sense to that suggested by the gradient hypothesis [46].
In reality, turbulent transport in premixed flames involves a balance between mix-
ing due to the turbulence and acceleration due to heat release. Turbulent mixing acts
to promote gradient transport while heat release effects act to promote countergra-
dient transport. Even in turbulent premixed flames where heat release is strong and
countergradient transport prevails, there is a small region close to the leading edge
of the flame brush in which heat release effects are weak and turbulence ensures
that gradient transport still occurs [68]. From a strict interpretation of the flamelet
theory the Reynolds flux in the direction normal to the turbulent flame brush may
be expressed as [75]
RANS and LES Modelling of Premixed Turbulent Combustion 71
 
SL 2 u
u c = c (1 c) (4.15)

where is an efficiency function [52] accounting for the effects of different turbu-
lent eddy sizes. Here, countergradient transport corresponds to positive values of the
Reynolds flux. Hence this expression gives rise to a dimensionless ratio NB which
has become known as the Bray number:
SL
NB = (4.16)
2 u
A Bray number greater (less) than unity indicates that countergradient (gradient)
transport is likely to be prevalent.
Modelling of turbulent transport in premixed flames is made significantly harder
by the occurrence of countergradient transport. In RANS, several ad-hoc models
have been suggested, but the difficulty can be overcome only by resorting to a full
second-moment closure approach with separate balance equations for each compo-
nent of Reynolds stress ui uj and Reynolds flux ui c . This is computationally
expensive, and requires additional modelling effort in order to close terms in the
second-moment balance equations. The Bray-Moss-Libby (BML) model [12] re-
mains one of the most successful closure approaches in this regard.
In LES, studies of countergradient transport are rare as yet. Countergradient ef-
fects in sub-grid scale turbulent scalar transport were explored by Tullis [72] and
found to be non-negligible. Some simple models were suggested based on existing
formulations including BML.
Recent work has focussed on Lewis number effects on scalar transport in tur-
bulent premixed flames. Contrary to early expectations, the effects of differential
molecular transport of heat and mass are not masked by turbulent transport [24].
Instead, the destabilising effects on laminar flame propagation experienced at low
Lewis numbers give rise to greatly enhanced countergradient transport in the tur-
bulent flame. Locally-unstable flames become more wrinkled, leading to enhanced
production of flame surface area and higher heat release per unit volume. This leads
to greater thermal expansion and hence to a stronger self-induced pressure gradient
through the flame. A revised Bray number has been obtained taking Lewis number
effects into account [24], and new modelling has been developed for terms in the
scalar flux transport equation [25].

4.4 Reaction Rate Modelling for Premixed Turbulent Flames

4.4.1 Simple Models

Simple algebraic closure modelling for w is well worth pursuing due to the low
computational cost. An important class of models is based on the Eddy Break Up
(EBU) approach [69]. The underlying concept is that the chemistry is fast and hence
72 Stewart Cant

the mean reaction rate is governed by the rate at which turbulent mixing processes
can bring reactants and products together. In the RANS context, the rate of mixing
, leading to a
is taken to be inversely proportional to the turbulence timescale k/
reaction rate model of the form

w = CEBU c c (4.17)
k
in which the progress variable variance is used as a measure of the scalar fluctuation
magnitude. The EBU model is very simple, has very low computational cost and is
often used to obtain a first estimate of a solution as an initial guess for something
more sophisticated. The disadvantages are lack of sensitivity to chemistry, a ten-
dency to produce unphysical solutions close to walls, and the need for the model
constant CEBU to be retuned for each individual case. Straightforward extensions of
the EBU model include the Magnussen model [49] which brings in some level of
sensitivity to mixture composition. A closely-related class of models is known as
the Eddy Dissipation Concept (EDC) which has enjoyed a recent resurgence due to
its simplicity and flexibility [41].
It is entirely feasible to adapt EBU-type models for LES. The timescale is the
turn-over time for a filter-scale eddy, given by /u , and the model may be para-
phrased as [6]:
c (1 c)

w = C (4.18)

where C is a model constant and the local sub-grid variance is now evaluated in
the strict flamelet limit as with the BML approach. This form of the EBU model
still lacks sensitivity to chemistry, but it is better behaved close to walls and requires
rather less ad-hoc tuning.
Incorporation of the effects of chemistry within a simple algebraic model is dif-
ficult. One interesting approach is called the beta transformation [16]. The concept
is based on the observation that the laminar flame speed is determined
by the geo-
metric mean of the diffusivity and the reaction rate, i.e. SL Dc w where Dc is a
representative mass diffusion coefficient (or thermal diffusivity) and w is the corre-
sponding species reaction rate (or heat release rate). If the diffusivity is multiplied
by a constant denoted by , and the reaction rate is divided by the same constant,
then the flame will be thickened due to the increased rate of diffusion but the flame
speed will remain unchanged. Hence it becomes possible to resolve the flame struc-
ture on a coarse mesh without affecting the propagation speed. Furthermore, it is
possible in principle to use a fully-detailed chemical reaction mechanism to obtain
w. The approach was used for a time in RANS but was superseded by EBU and
other models. More recently the concept has been revived for LES as the thickened
flame model [30]. The thickened flame structure can be resolved over an affordable
number of LES cells, and complex chemistry can be incorporated. A disadvantage
of the approach is that it is no longer possible to represent flame-turbulence interac-
tions accurately with eddies close to the filter scale. Current thickened-flame models
make use of an efficiency function [2] in an attempt to overcome this difficulty. The
RANS and LES Modelling of Premixed Turbulent Combustion 73

thickened-flame model has been used to good effect in LES of highly-complex in-
dustrial combustion systems [14, 67].

4.4.2 Flame Surface Density Modelling

4.4.2.1 Algebraic Flame Surface Density Modelling

In its simplest form, the flame surface density (FSD) model provides an algebraic
expression for the amount of flame surface area per unit volume at each point
within the premixed turbulent flame brush. If the flamelet assumption holds then the
local flame structure remains quasi-laminar and the local propagation speed remains
close to the unstrained planar laminar burning velocity SL0 . Hence

w = R SL0 I0 (4.19)

where R is the mean density in the reactants and I0 is a factor accounting for the
change in the local burning velocity due to straining and curvature effects [9]. Since
SL0 and I0 are quantities that depend essentially on the thermochemistry, the principal
modelling problem is how to determine . Following on from earlier work on time-
series analysis [10, 11], a model was constructed [18] using spatial information as
available from flame imaging techniques. Given an instantaneous two-dimensional
image of a flame superimposed on a contour plot of the averaged flame brush, it is
possible to count the number of points at which the instantaneous flame crosses the
averaged contour, and this can be repeated over a large number of images. Using
telegraph-signal analysis, an expression for the mean number of crossings per unit
length y is obtained as
c)
gc(1
y = (4.20)
Ly

where g is a constant and L y is the integral length scale of the flame crossing process.
The mean reaction rate per crossing is found from the mass flux through the flame
at the crossing point corrected geometrically for the crossing angle :

R SL0 I0
w y = (4.21)
| cos |

Thus, the mean turbulent reaction rate is given by the product of the mean reaction
per crossing and the mean number of crossings per unit length. Assuming statistical
isotropy in the plane of the flame, the mean reaction rate per unit volume becomes

c)
gc(1
w = R SL0 I0 (4.22)
Ly | cos |

Comparing (4.19) with (4.22) indicates that the flame surface density is given by
74 Stewart Cant

c)
gc(1
= (4.23)
Ly | cos |

The quantities g and | cos | have been determined by experiment [29], while L y has
been modelled with reference to the turbulent integral length scale [18]. It is inter-
esting to note that the model predicts a symmetric distribution of FSD through the
flame brush conforming to the parabolic form c(1 c).
Observations from experi-
ment and from DNS suggest that the distribution of FSD tends to be biased towards
either side of the flame brush due to Huygens propagation effects. This can be ac-
counted for in the model by (for example) changing the exponents of c and (1 c)
in (4.23) to be different from unity. Further modelling is required in order to pa-
rameterise the factor I0 based on strain rate and curvature statistics obtained using
information available from standard turbulence models [63, 78]. Models based on
expressions similar to (4.23) have been used extensively in RANS and have proved
successful in different application areas [15, 44].

4.4.2.2 FSD transport equation

A transport equation for FSD has been derived using a kinematic approach [60]
and also using a more geometrical approach [17]. The underlying concept is based
on the earlier Coherent Flame Model developed principally for application to non-
premixed flames [51]. In common with the algebraic FSD models, the surface area in
question is that of a single isosurface of reaction progress variable at c = c where
c is a chosen constant. The reaction progress variable is a field quantity defined
everywhere in three-dimensional space, but the fine-grained flame surface density
 is non-zero only on the infinitesimally-thin two-dimensional isosurface at the
spatial locus of points corresponding to c(x,t) = c . In order to allow the FSD to be
treated also as a field variable it is necessary to define an expectation value = 
using a suitable local averaging technique. A unit vector based on progress variable
gradient may be defined as
c
N= (4.24)
|c|
which is valid at all points where the progress variable gradient magnitude is non-
zero. At c = c the vector N becomes a normal vector to the chosen isosurface, and
the velocity of the isosurface in laboratory coordinates can be expressed as

X = u + Sd N (4.25)

where u is the velocity vector of the gas mixture at the isosurface location. The
quantity Sd is the displacement speed defined as the speed of the isosurface normal
to itself relative to the local gas mixture. With these definitions the exact transport
equation for FSD may be stated as
RANS and LES Modelling of Premixed Turbulent Combustion 75

+ Xk s = S
s (4.26)
t xk

Here, Qs = Q / denotes a surface average for any quantity Q, and S is the stretch
rate defined as
1 dA Xi
S = = (i j Ni N j ) (4.27)
A dt xj
Using the definition (4.25) the stretch rate may be decomposed according to

ui Nk
S = (i j Ni N j ) + Sd = aT + 2Sd hm (4.28)
xj xk

where aT is the hydrodynamic strain rate in the plane locally tangential to the isosur-
face and hm is the geometrical mean curvature of the isosurface. With the definition
(4.25) the FSD transport may be written as

Nk
+ uk s = aT s Sd Nk s + Sd s (4.29)
t xk xk xk
For RANS applications, it is standard practice to use a Favre decomposition of the
velocity for compatibility with the usual form of the reaction progress variable equa-
tion. Hence the FSD transport equation becomes

+ uk =
t xk
Nk
u s + aT s + aT s Sd Nk s + Sd s (4.30)
xk k xk xk
Note that for further compatibility of numerical implementation it is possible to
replace with a flame surface area per unit mass = / [17]. The five terms on
the right-hand side of (4.30) are identified as (i) the turbulent transport term; (ii) the
mean strain rate term; (iii) the turbulent strain rate term; (iv) the propagation term
and (v) the curvature term. All of these terms are unclosed and require modelling.
A standard gradient model for the turbulent transport term may be written as

T
uk s = (4.31)
xk
where T is the turbulent kinematic viscosity and is the turbulent Schmidt num-
ber for FSD. Realisability arguments, valid in both RANS and LES [38], indicate
that should be taken equal to the turbulent Schmidt number for reaction progress
variable.
The strain rate terms are usually considered as production terms since flame sur-
face area is generated on average by fluid-mechanical straining. The mean strain
rate term may be written as
76 Stewart Cant

ui
aT s = (i j Ni N j s ) (4.32)
xj

where there is assumed to be no correlation between the flame normal vector and the
Favre-mean velocity. Here, a model is required for the correlation Ni N j s between
the components of the flame normal vector. There has been some DNS [21] and
experimental [74] investigation of this quantity. A suitable model may be expressed
as [19]
1
Ni N j s = ni j = Mi M j + i j (1 Mk Mk ) (4.33)
3
where M is the surface-averaged normal vector given by the exact result [19]

c
M = Ns = . (4.34)

The fluctuating strain rate term has been modelled [19] as
 
 1 Mk Mk
aT s = CA (4.35)

where is the Kolmogorov time scale. The quantity 1 Mk Mk is called the orien-
tation factor and measures the tendency of the local flame surface to align with
the direction of mean propagation. An alternative model for the strain rate term is
based on integral scaling as [31]

aT s = CA (4.36)
k
In both cases CA can be treated as a model constant, or it can be made to depend on
the local turbulent velocity scales [78]. Further investigation of the strain rate terms
is ongoing [23, 40].
The propagation term is either neglected or it is taken in combination with the
curvature term. The combined effect is usually treated as a net sink for FSD due to
the role of propagation and curvature in the destruction of surface area by Huygens
propagation effects at the trailing edge of the flame brush. A typical model is

Nk
Sd s = CH Sd s (4.37)
xk 1 c
where CH is a model constant. Again, if required, CH can be made to depend on the
local turbulence timescales as well as on the orientation factor [19].
It is worth noting that there are several outstanding issues concerning the mod-
elling of the FSD transport equation for RANS. As indicated above, the surface
average is not equivalent to a standard Reynolds average and some careful interpre-
tation is needed. Moreover, the displacement speed is not equivalent to the laminar
burning velocity and further modelling is required.
RANS and LES Modelling of Premixed Turbulent Combustion 77

In LES, it has become standard to use the generalised FSD defined as gen = |c|
[6]. In principle, using this quantity removes the need to choose a single isosurface
of reaction progress variable, although extra care is required in the modelling. With
a Favre decomposition of the filtered velocity, the filtered transport equation for the
generalised FSD may be stated as

gen
+ uk gen =
t xk
Nk
(uk s uk ) gen + aT s gen Sd Nk s gen + Sd s gen (4.38)
xk xk xk
This leaves the four terms on the right-hand side to be modelled. In order, they are
(i) the sub-grid transport term; (ii) the strain rate term; (iii) the propagation term and
(iv) the curvature term. In the turbulent transport term, the velocity difference ui s
ui gives rise to a sub-grid flux of FSD which is closely related to the turbulent scalar
flux of reaction progress variable [19]. In particular, it is clear that countergradient
transport of FSD is to be expected under similar conditions to those which promote
countergradient transport of reaction progress variable. Hence, models for turbulent
transport of FSD are of the form [26, 37]

T gen
(ui s ui ) gen = Sd Ni s gen
+ (K c) (4.39)
xi
where T is a sub-grid scale turbulent kinematic viscosity which may be evaluated
using a standard Smagorinsky or similar model, is a turbulent Schmidt number,
is the heat release parameter, K is a model constant and Sd is a modified displace-
ment speed given by Sd =  Sd s /R . The first term on the right-hand side is a clas-
sical gradient transport model which caters for the occurrence of gradient transport
of FSD. The other term embodies the effects of acceleration across the flame due
to heat release which tends to promote countergradient transport of FSD [37, 71].
It is important to note that in LES, using the generalised FSD, the surface-averaged
normal vector Ns is no longer equal to the resolved normal vector M:

c c
Ns = ; M= . (4.40)
gen |c|

These quantities each depend on the filtered reaction progress variable c and it is
necessary to relate this quantity to its Favre-filtered value c.
This can be done using
an expression derived from the Bray-Moss Libby formulation in the limit of thin
flamelets:
(1 + )c
c = (4.41)
1 + c
In LES there may well be partial resolution of the flame and the sub-grid condi-
tions may not allow the use of the simple BML approach. Instead a slightly more
complicated expression is required [26, 37]:
78 Stewart Cant

(1 + )c
c = [1 exp ( /L )] + c exp ( /L ) (4.42)
1 + c
where  0.2 is a model constant.
The strain rate term may be decomposed according to

ui
aT s = (i j Ni N j s ) + Shr + Ssg (4.43)
xj

in which the three terms on the right-hand side represent the resolved contribution,
a contribution due to heat release, and a sub-grid contribution. The correlation be-
tween the normal vector components can be treated in a similar manner to that used
in RANS:
1
Ni N j s = ni j = Ni s N j s + i j (1 Nk s Nk s ) (4.44)
3
The heat release contribution may be expressed in terms of the flame normal accel-
eration in a manner very similar to that used to derive the countergradient transport
model in (4.39) above. The resulting model is

Nk s
Sd
Shr = (K c) gen (4.45)
xk
The sub-grid contribution is commonly modelled using a scaling based on an esti-
mate of the strain rate at the LES filter scale, resulting in the form
  
u u
Ssg = K , (4.46)
SL L

where is a model constant and K is an efficiency function [2, 52] that accounts
for the variation in the effectiveness of different eddy sizes in straining the flame.
Several different forms have been proposed for the efficiency function [2, 28] and
assessment of these is ongoing [26].
For LES modelling it has been found convenient [37, 39] to combine the prop-
agation and curvature terms and to consider their decomposition into resolved and
sub-grid scale contributions as

Nk
Sd s gen Sd Nk s gen = Pmean +Cmean +Csg (4.47)
xk xk
The mean propagation term may be stated as

Pmean = Sd s Nk s gen (4.48)
xk
where the correlation between the displacement speed and the surface-averaged nor-
mal vector has been assumed to be negligible [37]. Note that further modelling is
required for the surface-averaged displacement speed. Similarly, the mean curvature
term may be stated as
RANS and LES Modelling of Premixed Turbulent Combustion 79

Nk s
Cmean = Sd s gen (4.49)
xk
This leaves only the sub-grid curvature term which is modelled in the same manner
as in RANS, resulting in the expression

gen
2
Csg = N 1 SL (4.50)
(1 c)

in which 1 is a model constant and N = 1 Nk s Nk s is an orientation factor,

also known in LES as the resolution factor [37]. It serves to quantify the extent to
which the unresolved flame surface is aligned with the mean direction of propa-
gation, and vanishes either when the surface-averaged normal vector magnitude is
unity, or when the flame is fully resolved, i.e. when the filter size tends towards zero.
A complete modelled transport equation for FSD has been suggested recently [26]
and similar approaches are proving successful in applications to real devices [65].

4.4.2.3 Displacement speed modelling

In LES modelling for the FSD transport equation it is necessary to account for the
response of the displacement speed Sd to the effects of straining and curvature. Care-
ful treatment of the displacement speed enables the extension of FSD modelling
from the corrugated flamelets regime to the thin reaction zones regime [22]. Unfor-
tunately, there is no simple relationship between the displacement speed Sd and the
laminar burning velocity SL , also known as the consumption speed.
For a one-dimensional turbulent flame brush, mass conservation indicates that
the surface-averaged displacement speed must vary through the flame according to

 Sd s
Sd s = (4.51)
 s

while the density-weighted displacement speed remains essentially constant through

the flame. This observation has been confirmed for using DNS data [20] and exper-
iment [36], and has been found to break down only when the local flame curvature
is large.
An expression for the density-weighted displacement speed can be obtained us-
ing terms from the right-hand side of the filtered reaction progress variable equation
(4.12) in the form
w + .( Dc c) = Sd |c| (4.52)
Decomposing the filtered molecular diffusion term into normal and tangential com-
ponents allows for the decomposition of the density-weighted displacement speed
into contributions arising from reaction, normal diffusion and tangential diffusion
[55]:
Sd = Sr + Sn + St (4.53)
80 Stewart Cant

where
w
Sr = (4.54)
|c|
N.( Dc N.c)
Sn = (4.55)
|c|
St = Dc .N = 2 Dc h m (4.56)

Data from DNS [20, 54] has indicated that the contributions from reaction and nor-
mal diffusion may be combined as SLS = ( Sr + Sn )s and that this combination
is insensitive to local curvature. Hence SLS can be modelled using planar laminar
flame data. The tangential contribution St is deterministically dependent on curva-
ture, and a model is required for the sub-grid mean curvature hm . A scale-similarity
approach has been suggested [22] and has been tested using DNS data.

4.4.3 G-equation Modelling

The G-equation approach is based on the use of a scalar field variable G whose
definition in principle is completely arbitrary. An isosurface of G is fixed at some
chosen value G0 , and the level set of the scalar at this value is taken to represent the
spatial location of the flame surface [76]. The location of the level set xF at any time
t is the solution of the equation G(xF ,t) = G0 . Differentiating this equation with
respect to time yields
G x
+ G. F = 0 (4.57)
t t
Noting that
xF
= u + Sd N (4.58)
t
where N is the normal vector in the direction of propagation, given by N =
G/|G|, the exact G equation can be written as

G
+ u.G = Sd |G| (4.59)
t
This is an eikonal equation which describes the advection and propagation of the
level set. It contains no explicit reaction or diffusion terms, but it can be used to
provide a marker for the location of an isosurface within a flame. In this sense the
G-equation approach is not a model per se, but instead provides a mathematical
framework for modelling.
Since the scalar variable G is arbitrary, it can be interpreted as a signed distance
from the level set at G = G0 . The incremental distance from the level set is denoted
by
RANS and LES Modelling of Premixed Turbulent Combustion 81

G
dxn = N.dx = .dx (4.60)
|G|
If the G field is assumed to be frozen for the purposes of evaluating the distance,
the increment of the scalar G is given by dG = G.dx and the incremental distance
becomes
dG
dxn = (4.61)
|G|
In practice, identity between the distance field and the G field is ensured by en-
forcing the reinitialisation condition G = 1 together with the condition xn = 0
at G = G0 . The distance field interpretation allows for the development of models
based on flame structures that are located around the level set.
An important feature of the G-equation approach is to incorporate a decomposi-
tion of the displacement speed as

Sd = SL0 La aT SL0 Lh (4.62)

where = 2hm = .N is used to represent the curvature of the level set. The three
terms of the right-hand side of (4.62) represent the unstrained planar laminar burning
velocity, the effects of straining and the effects of curvature. Linear models are used
for both straining and curvature and are parameterised by the respective Markstein
lengths La and Lh . Since the curvature can be expressed as

G
= .N = . (4.63)
|G|

the resulting G-equation becomes

G  
+ u.G = SL0 La aT |G| Dh N. (N.G) + Dh 2 G (4.64)
t
where the Markstein diffusivity Dh = SL0 Lh . The three terms on the right-hand side
represent the effects of (i) laminar flame propagation including a dependence on
strain rate, (ii) normal diffusion and (iii) curvature. It may be observed that the
curvature term involves a second derivative of G. This has an important practical
effect in providing numerical stability for the G equation and in helping to prevent
the occurence of self-intersections of the surface at G = G0 .
Applications of the G equation in RANS of practical combustion systems re-
quires the use of a Favre-averaged form. Starting with the standard G equation (4.59)
and using the Favre decomposition G = G + G yields a transport equation for the
Favre average G:

G  
+ u .G = ( Sd )|G| . u G (4.65)
t
This equation is in a more conventional form and the right-hand side consists of
somewhat familiar unclosed terms, the first representing the mean turbulent reaction
82 Stewart Cant

rate and the second representing the turbulent flux of G. These can be modelled
using a slight variation on the usual arguments. For a statistically stationary planar
turbulent flame, the averaged G equation reduces to

G
( ST ) = ( Sd ) |G| (4.66)
x
where, by conservation of mass, ST is the constant mass flow rate through the flame
and ST is the turbulent burning velocity. In essence, this relation can be viewed as
a statement of Damkohlers hypothesis, since |G| is a generalised surface density
for the level set. The increment of distance from the turbulent flame brush may be
expressed in the same manner as (4.61) as

G
dxT = (4.67)
|G|

and this expression can be used to generalise back to the three-dimensional case
yielding a model for the mean turbulent reaction rate in (4.65) as

( Sd ) |G| = ( ST ) |G|
(4.68)

It is possible to treat the turbulent flux term in (4.65) using a classical gradient
transport model in the form
   
. u G = . DT G (4.69)

where DT is the turbulent diffusivity for G. This is deemed to be undesirable in the

context of the G-equation approach [55] since it is not consistent with the math-
ematical character of the original eikonal equation. Instead, the gradient transport
model is split into its normal and tangential components according to
   
. DT G = N.
DT N.
G DT |G|
(4.70)

where N is the Favre-averaged normal vector and is the Favre-averaged curvature,

both valid on the surface at G(x,t) = G0 . Since normal diffusion has been accounted
for already by the inclusion of a turbulent burning velocity in the reaction rate term,
only the tangential diffusion component needs to be retained explicitly in the model
[55]. This results in the final modelled equation for the Favre average G:

G
+ u .G = ( ST ) |G|
DT |G|
(4.71)
t
A full closure of this equation requires additional information about the turbulent
burning velocity. Typically this is provided using an empirical correlation of the
form   n
ST u
= 1 +C (4.72)
SL SL
RANS and LES Modelling of Premixed Turbulent Combustion 83

where C and n are constants. This marks a key difference between the G-equation
approach and the FSD approach, in that the turbulent burning velocity must be spec-
ified as an input to the former, but emerges as a result from the latter. A balance
equation for the variance of G may also be derived and modelled [54].
The G-equation approach has proved popular in RANS applications to various
problems of practical interest [55]. The extension to LES appears to be straight-
forward and several formulations have been proposed [43, 77] based on essentially
the same modelling concepts as described above. Further developments have made
greater use of the special mathematical symmetries of the G-equation [53]. A recent
formulation of the G-equation for LES [56, 57] makes use of the notion that the
level set G = G0 already represents the filtered flame position, and hence G does not
have to be filtered. This results in the G equation

G
+ u.G
= (SL + S ) N.G (4.73)
t
in which N is the normal vector to the unfiltered flame surface, u is the filtered
velocity conditioned on G = G0 , SL is the laminar flame propagation speed and
S is the curvature contribution to the propagation speed. Closure modelling for
the conditionally-filtered velocity is provided in terms of the familiar Favre-filtered
velocity u [56]. Modelling for the propagation term serves to illustrate the need to
consider partial resolution of the flame. The model may be stated in the form [56]
 
(SL + S ) N = ST D + SL D N (4.74)

where N is the normal vector to the surface at G = G0 and is the curvature of

that surface. The contributions from normal propagation and curvature effects at
the unresolved length scales of the flame surface are denoted by ST and D . In
addition, there are contributions from the resolved scales as denoted by SL and D .
The sub-grid scale turbulent burning velocity ST is modelled based on previous
arguments, while D is related to the sub-grid scale turbulent diffusivity. Both of
these quantities must vanish as the filter width is reduced, leaving only the resolved
contributions which must be formulated to represent the true behaviour of the flame.

4.4.4 Scalar Dissipation Rate Modelling

The scalar dissipation rate has proved to be a very important quantity for the mod-
elling of non-premixed flames [4]. In that context the scalar dissipation rate appears
as a sink term in the balance equation for the variance of the mixture fraction. In
premixed flames the Favre-averaged scalar dissipation rate is defined in terms of the
reaction progress variable as

c c
c = Dc (4.75)
xk xk
84 Stewart Cant

It is interesting to note that the scalar dissipation rate is defined in terms of molecular
diffusion only, i.e. its definition does not include any reaction rate information. Nev-
ertheless, the strong coupling between reaction and diffusion in both non-premixed
and premixed flames ensures that it is possible to use the scalar dissipation rate
as a measure of reaction rate. The scalar dissipation rate may be interpreted as a
reciprocal time scale for molecular mixing. If the chemistry is fast and hence the
combustion is mixing-controlled, then the scalar dissipation rate may be interpreted
also as a reciprocal time scale for reaction. In the fast-chemistry limit, a relation be-
tween the scalar dissipation rate and the mean turbulent reaction rate may be stated
as [8]
2 c
w = (4.76)
2Cm 1
where Cm = cw/w and may be treated as a model constant. The scalar dissipation
rate may be modelled by analogy with the dissipation rate of turbulent kinetic energy
, using an expression of the form

c = CD c c (4.77)
k
where CD is a model constant which is normally assigned a value close to unity.
This approach leads to an algebraic closure model for the mean turbulent reaction
rate that is closely related to the EBU model [8, 45].
An alternative approach is to derive a balance equation for the scalar disspa-
tion rate [50]. This provides greater insight into the physics and allows for a proper
order-of-magnitude analysis of contributing terms [70], and for modelling of indi-
vidual terms using information from experiment [36] and from DNS [27]. A balance
between dominant terms yields the algebraic expression
  
2 S0 S0
c = 1 + Cc L1 CDc L0 +CD c c (4.78)
3 k 2 L k

in which Cc , CDc and CD are model constants. The first set of parentheses includes
a timescale associated with flame stretch while the two terms in the second set of
parentheses represent a chemical timescale and molecular dissipation timescale re-
spectively. Hence the model offers a significant advance over the classical EBU
approach, in that additional physics is captured and can be justified on a theoretical
basis. Note that a classical EBU model is recovered in the limit of intense turbulence
1
(k 2 SL0 ) and fast chemistry (L /SL0 ). The scalar dissipation rate approach can
be shown to have a direct equivalence to the FSD approach [13] and the two quan-
tities can be related according to the expression

c
= (4.79)
K SL
RANS and LES Modelling of Premixed Turbulent Combustion 85

where K = R (2Cm 1)/2. This relation allows for the derivation of improved
algebraic models for with guidance from the transport equation for the scalar
dissipation rate.

4.4.5 Other Approaches

Modelling for turbulent premixed flames has been dominated by approaches based
to a large extent on the phenomenology of flamelets. There is abundant evidence
to support their effectiveness, and the Borghi diagram supports their applicability.
Nevertheless, it is clear that flamelet structure cannot prevail everywhere. In flames
close to extinction due to turbulent straining, or close to the flammability limits,
the balance between reaction and diffusion is weakened and the identification of a
representative flame structure becomes difficult.
In such situations a more general modelling approach is desirable. The trans-
ported pdf approach [42, 59] makes no assumptions about the structure of the flame
and has proved successful in its application to turbulent non-premixed flames (see
Chapter 6). There have been applications of pdf transport modelling to premixed
flames [47, 62] in which convincing results have been obtained. The advantages
of the approach lie in its potential for generality especially in cases where direct
chemical effects may be important, but the computational cost is high and there are
technical issues concerned with the modelling of mixing processes in the presence
of the high scalar gradients typical of premixed flames.
Another approach that has proved successful in non-premixed combustion is the
Conditional Moment Closure (CMC) approach, in which the fluctuations of all vari-
ables about the conditional mean mixture fraction are assumed to be small (for more
details, see Chapter 5). In principle the CMC approach could be extended to treat
premixed flames, possibly by using the reaction progress variable as a conditioning
variable [34]. The use of a marker field variable has been suggested [5] but it is
questionable whether this would bring any advantage over the existing G-equation
approach. One promising avenue is the Conditional Source Term Estimation (CSTE)
approach [35] which is closely related to CMC and which may provide a more real-
istic way to model chemical effects in premixed flames.
In general, turbulent premixed flames well away from extinction can be modelled
without explicit consideration of the chemical reaction rate. Chemical information
is encapsulated in quantities such as the laminar burning velocity and its response to
straining. The computational costs of detailed chemistry are high, and it is preferable
to treat the chemistry separately from any turbulent flame calculation. This is true
especially in LES where computational costs are already high due to the demand
for fine spatial resolution and the need for unsteady solutions. Chemistry can be
precomputed and stored in tabulated form using a number of different approaches.
Perhaps the simplest is a premixed flamelet library, in which the laminar burning
velocity can be stored as a function of mixture strength, pressure and reactant tem-
perature. Other independent variables can be included, such as strain rate and mean
86 Stewart Cant

curvature. In order to avoid the computational costs of searching in tables, it is is

possible to make use of correlations for laminar burning velocity which have been
derived from either experimental or one-dimensional computations [1]. A more ad-
vanced approach is to use the concepts behind Intrinsic Low Dimensional Manifolds
(ILDM) [48], suitably extended to include molecular transport effects in a manner
representative of flamelet structure. An approach such as Flamelet Generated Mani-
folds (FGM) [73] or Flamelet Prolongation of ILDM (FPI) [32] allows for a reason-
able level of chemical detail that is computationally inexpensive - at least within the
turbulent flame calculation - and which is largely free from the restrictions of the
original flamelet concept.

4.5 Future

There is an increasing technological requirement to operate combustion systems in

a manner that does not correspond to either premixed or non-premixed burning. For
example, many lean premixed gas turbine combustors involve the use of a much
richer or even non-premixed pilot flame which acts to stabilise the main premixed
flame closer to the lean limit than would be possible otherwise. Modern gas tur-
bine aero-engines are being designed to pass a much greater flow of air through the
fuel injector in order to minimise rich burning and hence reduce unwanted emis-
sions. This results in flames which burn in non-uniform fuel-air mixtures and which
exhibit features such as edge flames or triple flames. Modelling efforts for such
partially-premixed flames are in their infancy and generally involve straightforward
extensions of existing models. A common assumption is that the variables used to
characterise the strength of the mixture (e.g, mixture fraction) and the progress of
the reaction (e.g, reaction progress variable) are uncorrelated, and this is unlikely
to be true in general. Improved modelling is under development using fundamental
data from DNS [33] and from experiment [3]. A more sophisticated approach is the
Libby-Williams-Poiters (LW-P) model [64] in which the conceptual basis behind
the Bray-Moss-Libby formulation is extended to capture the probabilistic behaviour
of the mixture fraction. This approach has been validated against experimental data
[66] and has been used also to derive improved models for the turbulent stresses and
scalar fluxes in RANS.
Many aspects of turbulent premixed flames have been modelled more-or-less suc-
cessfully in both RANS and LES using the flamelet concept in its different guises.
It is now clear that the FSD, G-equation and scalar dissipation rate approaches are
all essentially similar, and a great deal can be learned by comparing and contrasting
these approaches in their various forms in order to explore the various assumptions
and to elucidate additional physics. There is less certainty around the modelling of
turbulent premixed flames under intense turbulence, or close to extinction. More
work is necessary also in the immediate aftermath of ignition, where the mixture
is within the flammability limits but a flame structure is not established. Here, the
flame may become premixed or non-premixed depending on the circumstances, or
RANS and LES Modelling of Premixed Turbulent Combustion 87

it may simply fail to burn. More general models are required in order to handle such
situations, and also fully to bridge the gap between premixed and non-premixed
combustion. The inherently low computational cost of flamelet-type models may
prove to be a thing of the past as more and more chemistry is required for an ac-
curate representation. At the same time, greater computer power, for example in
the form of the much greater parallelism now becoming available from multi-core
processors, may make this irrelevant as the affordable resolution for LES is steadily
increased.

Acknowledgements Thanks are due to many people including Dr Evatt Hawkes and Dr Nilanjan
Chakraborty, and to Professor Ken Bray for all his help and for many inspirational discussions.
This chapter is dedicated to the memory of Dr Arthur McNaughtan 1956-2009.

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Chapter 5
The Conditional Moment Closure Model

A. Kronenburg and E. Mastorakos

Abstract The relatively recent Conditional Moment Closure methods for turbulent
reacting flows have advanced from application to relatively well behaved, simple
laboratory flames to complex flow geometries and flame conditions with intense
turbulence-chemistry interactions. The progress on second order closures, double
conditioning approaches, two-phase and premixed CMC is reviewed in the first part
of this chapter, while the second part is largely dedicated to numerical methods
to solve the CMC equations and to the models capability to address questions of
direct engineering interest such as the modelling of diesel engine combustion and
the analysis of flame stabilization mechanisms.

5.1 Introduction

More than ten years ago, the first and - to date - last comprehensive review of the
Conditional Moment Closure (CMC) method was published by Klimenko and Bil-
ger [35]. Yet, great strides have been made in advancing the method from its rather
typical application to relatively simple and well behaved diffusion flames to more
complex flow geometries and flame conditions.
The development of the Conditional Moment Closure (CMC) method was moti-
vated by the need to provide accurate closures for the average of the non-linear tur-
bulent reactive source term. It was conceptually derived as a mixture fraction based
approach for non-premixed turbulent combustion and has as such some similarities
with laminar flamelet methods. The basic idea is to exploit a strong correlation be-

A. Kronenburg
Institut fur Technische Verbrennung, University of Stuttgart, 70569 Stuttgart, Germany, e-mail:
kronenburg@itv.uni-stuttgart.de
E. Mastorakos
Department of Engineering, University of Cambridge, Cambridge, CB2 1PZ, UK, e-mail: em257@
eng.cam.ac.uk

T. Echekki, E. Mastorakos (eds.), Turbulent Combustion Modeling, 91

Fluid Mechanics and Its Applications 95, DOI 10.1007/978-94-007-0412-1 5,
Springer Science+Business Media B.V. 2011
92 A. Kronenburg and E. Mastorakos

tween reactive scalar species and the mixture fraction and hence fluctuations in reac-
tive scalar space can be associated with fluctuations in mixture fraction space. Con-
ditioning of the reactive species on mixture fraction then leads to relatively small
fluctuations around the conditional mean, and a simple first order closure for the
chemical source can be found. Hence, transport equations for the reactive species
mass fractions conditioned on mixture fraction have been derived, and some un-
closed terms such as the conditional velocity and the conditionally averaged scalar
dissipation need to be modelled. Relatively simple models for these terms suffice
and the solution of the temporal and spatial evolution of the conditional moments
gives relatively good predictions of finite-rate chemistry effects for a wide range of
turbulent diffusion flames [25, 31, 66, 67, 69].
During the last decade, the majority of CMC-related theoretical development has
focused on the modelling of flame regimes where local correlations between reac-
tive scalars and mixture fraction are weakened. This decorrelation will for example
occur in flames with local extinction, in lifted flames where fuel and oxidizer mix
without combustion before the stabilization point and more generally, in any flow
with regions of partial or complete premixing prior to ignition. Here, fluctuations
around the conditional mean will start to become significant, but it is very important
to realize that CMC is not rendered invalid if conditioned and conditioning quan-
tities are not well correlated. The CMC transport equations can be derived without
making assumptions on the degree of correlation. However, closures of unclosed
terms, and in particular of the conditionally averaged chemical source term, need
improvements to account for the effects of fluctuations on the evolution of the con-
ditioned moments.
The second major focus of CMC research has been on the application of the
method to more complex flow geometries and flame conditions with technical rele-
vance such as engine and gas turbine related environments. The two foci of work de-
termine the structure of the remainder of this chapter which is largely split into two
major sections. The next section is called methodological developments where
we will briefly review the standard formulation of the conditional moment closure
method and then elaborate on techniques that account for flame conditions where
reactive scalars do not correlate well with mixture fraction, namely the second order
closures and double conditioning. Further, specific subsections are dedicated to the
discussion of the first steps in CMC related premixed flame modelling and advances
in two-phase flow CMC. The second major section (Section 5.3) is dedicated to
applications of CMC to problems of engineering interest. Several research groups
have worked on CMC for diesel engine combustion, auto-ignition studies, flame
stabilization, CMC as combustion sub-model for Large-Eddy Simulations (LES) for
improved flow field modelling in complex geometries, and more generally, on pollu-
tant predictions where finite rate effects dominate. While the theoretical framework
for the application of CMC is well established, some implementation and modelling
issues arise. Section 5.3 will provide an up-to-date assessment of CMC and its capa-
bilities, but improvements to the technique are certainly needed to allow application
to all flow and flame regimes. We will attempt to outline future steps in CMC de-
The Conditional Moment Closure Model 93

velopment in the final part of Section 5.3 that should serve as encouragement and
guideline to new researchers in the field of CMC modelling.

5.2 Methodological Developments in CMC

The CMC transport equations for turbulent reacting flows were derived by Kli-
menko [33] and Bilger [4] using somewhat different methodologies and primary clo-
sure assumptions. Details on the differences are discussed in [35], and we limit our-
selves here to present one approach only that is called the decomposition method.
Its derivation requires little knowledge of combustion, it is based on the well known
instantaneous transport equations for reactive and passive scalars and only standard
mathematical techniques need to be used. The derivation in the next subsection will
be quite general and standard closures will be introduced. The further subsections
discuss more advanced closures such as second order and double conditioning ap-
proaches, but also outline CMCs potential for the modelling of premixed flames.
Modifications that need to be made for multiphase combustion are described in Sec-
tion 5.2.5.

5.2.1 The CMC Equations

We will first define the conditional mean, Qk (Zc , x,t), of a scalar Yk (x,t) as

Qk (Z, x,t) = Yk (x,t) | Yc = Zc . (5.1)

The angular brackets denote ensemble averages of Yk , conditioned on Yc = Zc ,

where Y is a multidimensional scalar space, and Z is its sample space. The choice
whether a specific scalar is a conditioned or a conditioning quantity is problem
dependent, however, the sets of conditioned and conditioning quantities are mu-
tually exclusive and no scalar should be part of both sets. The arrays to the right
of the vertical bar in Eq. 5.1 should be understood to represent a (usually small)
subset of the entire scalar space, Yc = (Y1 ,Y2 , . . . ,Ync ) and Zc = (Z1 , Z2 , . . . , Znc ),
where the number of conditioning scalars, nc , should be much smaller than the num-
ber of scalars, nsc . The scalar space of the conditioned quantities is then given by
Yq = (Ync +1 , . . . ,Yk , . . . ,Ynsc ), and the superscripts c and q are used here to indi-
cate scalar arrays with conditioning and conditioned quantities, respectively. Note
that the scalar array may include mixture fraction, scalar dissipation and an energy
related scalar such as enthalpy or temperature. The conditional mean - or conditional
expectation - is related to the joint probability density function through
 +  +
Zk P(Zk , Zc )dZk
Yk | Y = Z  =
c c
Zk P(Zk | Y = Z )dZk =
c c
, (5.2)
P(Zc )
94 A. Kronenburg and E. Mastorakos

where spatial and temporal dependencies have been omitted for clarity of presen-
tation, and the unconditional mean can be obtained from the conditional mean by
integration across the entire sample space of the conditioning variables,
 +
Yk  = Yk | Yc = Zc P(Zc )dZc . (5.3)

The starting point for the derivation of the CMC equations is the well known
transport equation for chemically reactive species, discussed in Chapter 2, which is
repeated here for the readers convenience,
 
Yk Yk Yk
+ ui Dk = wk . (5.4)
t xi xi xi

In Eq. 5.4 we have used Ficks law for the diffusive flux. The instantaneous mass
fraction Yk can now be decomposed into the conditional mean and the fluctuation
around the conditional mean,

Yk (x,t) = Qk (Z, x,t) +Yk (x,t) (5.5)

and be inserted into Eq. 5.4. The chain rule of differentiation is applied to all tem-
poral and spatial derivatives of Qk . Note, for example, the application of the chain
rule to the time derivative,

Yk Qk Qk Z j Yk
= + + , j = 1, . . . , nc (5.6)
t t Zj t t

and its application to the spatial derivatives - or repeated applications for the diffu-
sion term - obeys identical rules. The entire resulting equation needs to be condi-
tionally averaged again, and the final CMC equation results in

Qk Qk Qk
 | Zc  =  ui | Zc  + wk | Zc  w j | Zc  (5.7)
t xi Zj
Yj Yj 2 Qk Y j Yl 2 Qk
+  Dk | Zc  +  Dk | Zc 
xi xi Z j Z j xi xi Z j Zl
+ eq + ey

with
   
Qk Y j 2 Qk Y j Qk
eq  Dk + Dk + (Dk D j ) | Zc 
xi xi xi xi Z j xi xi Z j
 
Y  Y  Y 
ey  k + ui k Dk k | Zc .
t xi xi xi

All terms on the RHS need closure. The eq -term can usually be neglected under
high Reynolds number assumptions [35], but needs to be modeled if differential dif-
fusion effects are to be included [37, 38]. Similarly, the ey -term that involves all the
The Conditional Moment Closure Model 95

fluctuations around the conditional mean, must not be omitted in the presence of
significant differential diffusion effects, but can be set to zero in the absence of dif-
ferential diffusion and if a good set of conditioning variables is selected. The term
good refers to the correlation of the (unconditional) fluctuations of the conditioned
scalars with the (unconditioned) fluctuations of the conditioning scalars. A perfect
correlation would lead to Yk 0, however, correlations are hardly ever perfect, and
based on the primary closure hypothesis employed in the decomposition method
[35], ey P(Zc ) =  | Zc uYk | Zc P(Zc ) , we may propose a gradient dif-
fusion approximation for ey . The validity of the gradient diffusion model has been
corroborated by Richardson et al. [63] for relatively high turbulence levels. Devia-
tions between the model and DNS data at low turbulence levels are due to counter-
gradient transport present in the studied expanding flames that involved propagating
fronts; such phenomena may be absent in attached non-premixed flames. The im-
portance of ey , and therefore the importance of the accuracy of its closure, strongly
depends on flow and flame conditions. It is of paramount importance for flame sta-
bilization in lifted flames where flame propagation is the dominating stabilization
mechanism [15, 24].
The best modeling approaches for the conditional velocity,  ui | Zc , dissipa-
tion, N j j DY j Y j | Zc  and cross-dissipation, N jl DY j Yl | Zc , have not
yet been established for multidimensional conditioning spaces. Ample experience
exists, however, for simple, single conditioning (nc = 1), where Y1 = and Z1 =
with being mixture fraction and its sample space. Single conditioning on mix-
ture fraction leads to the standard CMC equations,

Qk Qk 2 Qk 1 
+  u i |  = N + wk |  ui Yk | P (5.8)
t xi 2 P xi

with scalar dissipation N D and subscript indicating dependence on mix-

ture fraction.
Sreedhara et al. [71] assessed different standard closures for the conditional ve-
locity, u |  and the conditional scalar dissipation, N | . The different models
for u |  were termed (1) conditional independence, (2) linear in terms of un-
conditional flux and (3) gradient diffusion in terms of local PDF. Only minor
differences between the models could be observed for the jet flames studied; such
comparison is not available for more complicated flows.
Standard models for conditionally averaged scalar dissipation have been termed
(1) AMC-, (2) Girimajis- and (3) double integration of the PDF transport
equation-model [71]. Theoretically, the AMC-model [57] requires the presence of
unmixed fluid, a condition that will not be satisfied in regions downstream of the
potential core of a jet. Girimajis model [20] has been derived for homogeneous
flows and - strictly speaking - may not hold in regions with strong mean gradients.
In contrast, the integration of the PDF transport equation ensures consistency be-
tween conditionally averaged dissipation and the scalars probability distribution in
all regions of the flow [40] and as such, should be the modellers choice. However,
the PDF transport equation requires a good model for the conditionally averaged
96 A. Kronenburg and E. Mastorakos

velocity, and accurate numerical integration of the equation is difficult in regions

of low probability. Practically, all three models provide qualitatively similar results
for conditionally averaged scalar dissipation in Sandia Flame D [2], with the AMC-
model being symmetric in -space and the PDF-model deviating most from symme-
try. 3-D measurements [2] show some asymmetry, but it would be difficult to decide
on the superiority of one specific model. Notable differences exist in a bluff body
flame, but lack of experimental data makes judgment on the quality of the different
models impossible.
Further models with great promise have been developed by Devaud et al. [14],
Cha et al. [8] and Mortensen and de Bruyn Kops [51], however, none of the models
has been validated by comparison with experimental data, and they are therefore not
discussed further in the present review.
The key to successful CMC modelling is the accuracy of the closure of the condi-
tionally averaged chemical source term, wk | . The assumption of relatively small
fluctuations around the conditional mean allows for simple first order closure of the
reaction rate term, i.e.
wk |  = wk ( Q, ). (5.9)
This closure has proven adequate for many applications, but more complex closures
may become necessary if fluctuations become significant. It is repeated here that the
CMC equation (Eq. 5.8) remains valid even if fluctuations are large, however, the
accuracy of above closures will suffer and different modelling strategies will need
to be employed. Rather unexpectedly, improved closures of the ey -term may not be
necessary, however, accurate modelling of the chemical source term is crucial [36,
43]. The latter can be achieved by multiple conditioning (see Section 5.2.3) or by
second order closures that are discussed next.

5.2.2 Advances in Second Order Closures

The reasons for fluctuations around the conditional mean are diverse. Mixture frac-
tion is a good conditioning scalar for all non-premixed flames, however, relatively
strong fluctuations in scalar dissipation due to turbulence directly affect the flame
structure, this leads locally to deviations from the conditional mean and induces
fluctuations. If scalar dissipation fluctuations are large, local extinction can occur
and conditional fluctuations are significant as shown in Fig. 5.1. Similar conditional
fluctuations can be observed in turbulent flows where fuel and oxidizer partially
premix before ignition occurs. In addition, relatively large variations of ignition lo-
cation can exist even in statistically stationary cases, e.g. Gordon et al. [21] reported
fluctuations of lift-off height of around 10 jet diameters in experiments with jets
issued into hot vitiated air. Mastorakos [47] reviewed the spatial fluctuations of au-
toignition spots in detail and found that the fluctuations of the scalar dissipation rate
are crucial for generating conditional reaction rate fluctuations.
The Conditional Moment Closure Model 97

It seems obvious that the influence of these fluctuations cannot be ignored. How-
ever, advanced closures do not necessarily aim at improved modelling of the term
that involves the fluctuations, but seek an improvement to the existing closures for
the chemical source term. The second order closures presented in this Subsection
therefore focus on a second order correction of the conditionally averaged source
term only.
The forward reaction rate of any bi-molecular elementary reaction, m, [Ck ] +
[Cl ]  [Cr ] + [Cs ] can be expressed in Arrhenius form,

YkYl bm
wm ( , Y, T ) = Am 2 T exp(Ta,m /T ) (5.10)
Mk Ml
and a Taylor expansion around the conditionally averaged first order approximation
yields
 
YkYl | 
w |  = w ( , Q, ) 1 +
m m
+ T1 + T2 (5.11)
Qk Ql
     
Ta,m Yk T |  Yl T  | 
T1 bm + +
QT Qk QT Ql QT


1 2(bm 1)Ta,m Ta,m 2
T 2 | 
T2 bm (bm 1) + + 2 .
2 QT QT Q2T

The second term in the brackets in Eq. 5.11 accounts for the correlation between
species k and l, T1 approximates the correlations between species and temperature
and T2 results from higher moments of temperature. These higher order corrections
can be significant and easily exceed the first order approximation, in particular for
reactions with high activation energies (due to the dependence of T2 on squared
Ta,m ) as can be found in the first Zeldovich step of NO formation. Closure of
Eq. 5.11 requires knowledge of the conditionally averaged correlations between all
chemically reactive species, Gkl = YkYl |  and between species and temperature

Fig. 5.1: Measurements of

temperature as function of
mixture fraction in a piloted
methane-air jet diffusion
flame with significant local
extinction and re-ignition
(Sandia Flame F). The filled
symbols indicate the condi-
tional mean of temperature,
QT = T | . Reprinted
from [19] with permission
from the Combustion Insti-
tute.
98 A. Kronenburg and E. Mastorakos

GkT = Yk T  | . A major issue is apparent: second order closure requires the so-
lution of nsc (nsc + 1)/2 additional transport equations for the conditional variance
and co-variance, a task hardly feasible when using detailed chemical kinetics for
hydrocarbon combustion. Early attempts by Mastorakos and Bilger [48] and Kro-
nenburg et al. [39] used global mechanisms that could be parameterized by two
variables, however, extension was not straightforward, and these methods therefore
lacked generality. Kim and Huh [29] introduced the idea that second order correc-
tions need to be applied to rate limiting steps only. They identified four rate limit-
ing elementary reactions in a detailed methane-air mechanism (GRI 2.11 with 49
species and 277 reactions) that control the radical pool, the CO-CO2 conversion,
initialization and chain termination. These reactions involve 9 species (and temper-
ature), and the resultant reduction from 1250 to 45 additional equations renders the
problem trackable. The conditionally averaged variance and co-variance equations
as such can easily be derived using the PDF or decomposition method [26],

Gkl Gkl 2 Gkl 1 

+ ui |  N + ui YkYl | P
t xi 2 P xi
Qk Ql
+ Yl ui |  Yk ui | 
xi xi
  
Yk Yl
= wk Yl + wl Yk |  2D | 
xi xi
2 Ql 2 Qk 1 Jgkl
+N Yk |  + N  
Y |  + (5.12)
2 l
2 P

with the transport equation for GkT being identical in form. Primary closure as-
sumptions have been invoked (i.e. eq -terms and differential diffusion effects are
neglected) and all terms on the LHS are closed or can be closed with standard clo-
sures. However, all terms on the RHS require additional models. Closures have been
mainly suggested by the Sydney [45, 72] and Pohang [26, 27, 29] research groups.
A set of relatively simple closures is given by

wk
wk Yl |  = |Y =Q Gl (5.13)
Y
  
Yk Yl 
2D |  = 2 CkCl Gkl (5.14)
xi xi k
N Yk |  = Rk N Gkk | 1/2 (5.15)
Gkl
Jgkl = Cg N P (5.16)

Equation 5.13 is a first order approximation of the correlation between the chemical
source term and the species and can be easily obtained from a Taylor expansion of
wk around wk (Q), multiplication with Yl and subsequent conditional averaging [17].
The dissipation terms (Eq. 5.14) are modelled by an equivalence of the scalar fluc-
tuation and turbulence time scales, Ck and Cl are constants to be discussed below,
The Conditional Moment Closure Model 99

and successful modelling of the dissipation-scalar correlations (Eq. 5.15) relies on

a good choice of Rk = RN Y  Nrms /N with RN Y  being the correlation coeffi-
k k
cient between N  and Yk . The turbulent transport in conserved scalar space, Jgkl , is
modelled assuming gradient transport [26].
A second set of alternative closures can be based on laminar flamelet assump-
tions [29, 72] where scalar fluctuations around their conditional means can be as-
sociated with fluctuations in scalar dissipation. The laminar flamelet solution can
be parameterized by the value of scalar dissipation at stoichiometric and further as-
sumptions of independence of P( ) and P(Nst ) and log-normality of P(Nst ) lead to
a closed form of the equations
1/2
G
wk Yl |  = ll A
l
   
A= wk Yl PN (Nst )dNst wk PN (Nst )dNst Y j PN (Nst )dNst
S S S S
(5.17)
  
Yk Yl lnYkS lnYlS
2D |  = 2D PN (Nst )dNst (5.18)
xi xi
1/2
N Gkk
N Yk |  = B
N ,st i
   
B= Nst Yk PN (Nst )dNst Nst PN (Nst )dNst Yk PN (Nst )dNst (5.19)
S S

where superscript S indicates laminar flamelet solutions and k is the RMS of YkS .
No alternative closure is suggested for Jgkl .
Data for a priori closure validation is scarce, the accuracy of some approxima-
tions has not yet been established (e.g. Eq. 5.13), and all existing studies rely on
DNS of reactive mixtures in homogeneous, isotropic turbulence for the assessment
of Eqs. 5.13-5.19. Sreedhara et al. [71] reported good qualitative and quantitative
agreement between DNS and models based on the flamelet assumption with one
notable exception: the flamelet model performed poorly for the closure of the dissi-
pation term (Eq. 5.18). This is in contrast to the findings of Swaminathan and Bilger
[72], but -in general- good performance of Eq. 5.18 cannot be expected in flames
with local extinction since the flamelet approximation imposes certain gradients in
mixture fraction space. This can best be illustrated by analysing conditionally av-
eraged temperature or mass fractions of product species such as CO2 and H2 O:
their maxima around stoichiometric impose zero gradients while extinction and re-
ignition events lead to a decorrelation of these scalars and mixture fraction resulting
in the disappearance of the local minimum of dissipation around stoichiometry. It
further needs to be emphasized that good quantitative agreement could only be es-
tablished with additional scaling factors in Eqs. 5.17 and 5.19 that account for the
mismatch between mean scalar dissipation and mean reaction rate [71]. Dissipa-
tion and temperature (and therefore reaction rate) are well correlated during extinc-
tion, however, they are rather uncorrelated during re-ignition leading to the need of
100 A. Kronenburg and E. Mastorakos

the additional parameter. These scaling factors need to be taken from DNS. Omit-
ting these additional factors, Eqs. 5.17 and 5.19 introduce modelling errors of up to
100% and 30%, respectively. Furthermore, it should be noted that flamelet assump-
tions for the closure of the conditional variance equation lead to inconsistencies with
the transport equation for the conditional mean: they could also be used to model
wk | .
Equations 5.15-5.16 provide decent agreement between models and DNS, but
errors in peak values of up to 50% may occur. Sreedhara et al. [71] assessed dif-
ferent constants for Ck in Eq. 5.14. Standard values of Ck = 1 for major species
and temperature and Ck = 2 for radicals tend to underpredict dissipation rates while
order of magnitude estimates, exploiting proportionality
 of the gradient of radicals
with Ck = 1/st and of major species with Ck = 1/ st (1 st ), lead to overpredic-
tions. In terms of qualitative behaviour, this model represents the counterpart to the
flamelet based model: it always predicts local maxima around stoichiometric and
fails to predict local minima that occur while scalar and mixture fraction are well
correlated. Last, Eq. 5.15 allows for similar freedom in choosing modelling con-
stants. The correlation coefficient is unknown, and DNS data show large variations
between -1 and +1 [48, 72]. Fortunately, RN Y  is approximately constant in mixture
k
fraction regions where the second derivative of Ql is large, and the modelling of the
third and fourth RHS terms in Eq. 5.12 with constant RN Y  = O(1) is adequate.
k
Even though a priori analysis of all suggested closures is not quite satisfactory,
predictions of the conditional variance and co-variances are promising. Good
agreement has been achieved for conditionally averaged temperature and species
RMS in several jet diffusion flames [29, 39]. However, second order closures have
had limited impact on predictions of major scalars and temperature. Peak tempera-
tures and O2 mass fractions could markedly be improved in flames with local extinc-
tion and re-ignition (Sandia Flames D, E and F) [17, 29], however, leaving room for
improvement. Temperature predictions in bluff body flames (flames HM1 and HM3)
were hardly affected by changes to the modelling of the chemical source term [70],
but effects on minor species are significant. All studies show clear improvements of
predicted OH, NO and even CO levels and examples are presented in Fig. 5.2.
In contrast to the above where the second-order correction has been applied to
only one or just a few reacting scalars, De Paola et al. [58] used second-order closure
without any reduction in the dimensionality of the second-order correlation matrix,
i.e. they solved transport equations for all YkYl |  and Yk T  | . The applica-
tion to autoignition problems with a 32-species chemical mechanism that included
low temperature autoignition was successful. The validity of some closures in the
second-order CMC equation has also been explored by comparison with DNS of
autoigniting jets [64].
It seems that some effects cannot be captured by second order closures, it is un-
clear why temperature predictions in bluff body flames are rather unimpressed by
improved source term modelling and corrections to the chemical source term may
not suffice here. It is unlikely that any - yet to be demonstrated - inaccuracy of
closures for the variance equations can be blamed; variance and co-variances are
reasonably well predicted but no sign of better temperature predictions in HM1 and
The Conditional Moment Closure Model 101

Fig. 5.2: CMC computations of temperature in Sandia Flame E (left), of temperature

in the Sydney bluff body flame HM3 (centre) and of NO in HM3 (right). Computa-
tions (solid lines denote second order closure and dashed lines are results from first
order closure) are from [29, 70] and measurements (symbols) are from [13, 19]. The
numbers 2.11 and 3.0 in the right figure refer to GRI mechanisms used for the com-
putations. Reprinted from [29, 70] with permission from the Combustion Institute.

HM3 can be detected. It is possible that the fluctuation terms themselves should not
be neglected - after all Kronenburg and co-workers [42, 43] based their statement
on the importance of ey on DNS of reacting flows in homogeneous, isotropic turbu-
lence. The strategy that is pursued in the next Section therefore aims at a reduction
of the fluctuations themselves and problems may be avoided.

5.2.3 Advances in Doubly Conditioned Moment Closures

Large fluctuations around the conditional mean indicate that the conditioning vari-
able does not sufficiently parameterize the combustion process. The introduction of
further conditioning variables should reduce the fluctuations if a strong dependence
of the reactive species on the new conditioning variables can be established, and
suitable choices for these variables, their implementation and modelling issues are
discussed next.

5.2.3.1 Basics of Double Conditioning

Traditionally, the flamelet equations suggest two quantities that parameterize the
composition field: mixture fraction and scalar dissipation. We have seen above in
Eqs. 5.17-5.19 that laminar flamelet assumptions can be invoked and instantaneous
composition fields can be parameterized by mixture fraction and scalar dissipation
if mixture fraction alone ceases to suffice as conditioning scalar. The introduction
of scalar dissipation as second conditioning variable seems a particularly adequate
choice in flames with flame extinction since large values of scalar dissipation are
the cause of extinction and lead to flame quenching. Cha et al. [9] therefore intro-
102 A. Kronenburg and E. Mastorakos

duced scalar dissipation as second conditioning variable and formulated a closed

set of equations for the computation of the evolution of the reactive scalars dou-
bly conditioned on mixture fraction and scalar dissipation at = 0.5. Their DNS
of reactive scalars in homogeneous turbulence used 1-step chemistry and showed
varying degrees of extinction and re-ignition. The results show that doubly condi-
tioned CMC predicts extinction well, but unfortunately, the onset of re-ignition is
predicted much too early. Scalar dissipation may be a good indicator of extinction
events, however, scalar dissipation does not correlate well with species composition
after flame extinction occurred. In addition, conditioning on instantaneous values
of scalar dissipation neglect the importance of relevant chemical time scales on the
flame. Very short events of very large dissipation values will not necessarily lead
to extinction due to a finite response time needed by the flame. Similarly, events
of low dissipation values (let them be long or short) will not automatically lead to
re-ignition since re-ignition requires heat flux to the mixture.

Fig. 5.3: Measurements of O2 and OH mass fractions in a piloted jet diffusion flame
(Sandia Flame F) at X/D=15. (a) O2 and OH as function of mixture fraction, where
filled symbols indicate the conditional means, and (b) O2 and OH as function of
mixture fraction ( st ) and temperature. Measurements are from [1]. Reprinted
from [42] with permission from the Combustion Institute.

The key issue here is the lack of correlation of the chemical source term with
scalar dissipation, and this results in the flamelet assumptions losing their validity.
Now, we may want to remember our original goal of finding an accurate closure
for the chemical source term. The chemical source term is largely dependent on the
provision of fuel and oxidizer and on temperature. Therefore, Bilger [3] proposed
conditioning on mixture fraction and sensible enthalpy and experimental data from
a piloted jet diffusion flame with significant extinction provide support. Figure 5.3
(left) shows instantaneous mass fractions of O2 and OH as function of mixture frac-
tion at a downstream distance of X/D=15. Apparently, mixture fraction alone does
not parameterize the flame well, and large scatter around the conditional mean can
be observed. Figure 5.3 (right) shows the same data with mixture fraction values
around = 0.351 (the stoichiometric mixture fraction) as function of temperature.
The Conditional Moment Closure Model 103

Now the fluctuations have been greatly reduced, and we may be able to find fits
through the data that correspond to the doubly conditioned means.
The governing equations for the doubly conditioned scalars result directly from
Eq. 5.7 with Zc = ( , )T , where represents the sample space of normalized sen-
 T
sible enthalpy, h s = TT0 c p dT / T0ad c p dT , and can then be written as

Qk Qk Qk 2 Qk 2 Qk
+ ui | ,  = wk | ,  whs | ,  + N11 + N22
t xi 2 2
  

Qk
2 1 , ui Yk | , P( , )
+ 2N12 (5.20)
, P( , ) xi

As above, high Reynolds numbers have been assumed, differential diffusion ef-
fects are neglected and the doubly conditioned dissipation terms are defined as
N11 D | , , N22 Dh s h s | ,  and N12 D h s | , . The
last RHS term may approximate zero due to the expected reduction in fluctuations
around the mean. In principle, the number of conditioning scalars can be increased
arbitrarily up to nc = nsc , but every addition of a conditioning variable increases the
dimension of Q, and more than two conditioning variables may not be feasible for
the computation of combustion in 2- or 3-dimensional geometries. Two condition-
ing variables should suffice to characterize a wide range of flame regimes: mixture
fraction is the key quantity for non-premixed combustion while sensible enthalpy
constitutes a kind of a progress variable that characterizes species compositions in
premixed flames. Realizations of species compositions should therefore be close to
a two-dimensional space parameterized by mixture fraction and sensible enthalpy.

5.2.3.2 Modelling Partially Premixed Flames

An approach that employs conditioning on mixture fraction and sensible enthalpy

should be ideally suited for the modelling of partially premixed flames where both
the degree of mixing and the reaction progress determine fuel conversion and flame
structure. A number of studies have dealt with reactive flows where significant lo-
cal extinction leads to unburnt, partially premixed regions that later re-ignite due
to turbulent and/or molecular diffusion of heat towards the premixed mixture. As
indicated above Cha et al. [9] used DNS of flames with extinction and reignition in
homogeneous turbulence to establish the lack of correlation between scalar dissi-
pation and a reactive mixture during re-ignition, and in similar studies, Kronenburg
[36, 43] assessed the suitability of mixture fraction and sensible enthalpy as condi-
tioning variables. The latter studies demonstrate the potential of the doubly condi-
tioned moment closure approach: the reactive species concentrations correlate well
with and h s at all times, fluctuations around the doubly conditioned mean are very
small, and CMC predictions agree very well with DNS data. The timing of extinc-
tion and the onset of re-ignition are captured accurately and the closure of chemical
conversion rates can be based on doubly conditioned values. However, it shall not be
104 A. Kronenburg and E. Mastorakos

Fig. 5.4: DNS data of doubly conditioned dissipation of normalized sensible en-
thalpy at two different times: t = 0.8 (left) and t = 2.5 (right) where t denotes
time normalized by the initial eddy turnover time. Reprinted from [36] with permis-
sion. Copyright 2004, American Institute of Physics.

forgotten that double conditioning brings a whole host of new issues that need to be
addressed. The doubly conditioned dissipation terms, N11 and N22 , and their cross-
correlation, N12 , are not closed and need modelling. Hasse and Peters [22] suggest
independence of the scalar dissipation of the respective second conditioning scalar.
This may hold if - as in their case - both conditioning scalars are passive, mixture
fraction-like. In a similar context, Nguyen et al. [56] parameterize the composition
space by mixture fraction and a progress variable. They suggest independence of
N11 of the progress variable and impose a functional dependence of N22 on mix-
ture fraction such that local maxima occur at stoichiometric. This may hold when
progress variable and mixture fraction are not well correlated, but should lead to
gross overpredictions of the dissipation rate in regions where flamelet solutions ex-
ist. This is demonstrated in Fig. 5.4 for N22 with the help of DNS data [36]. Initially,
temperature - and therefore sensible enthalpy - is a strong function of mixture frac-
tion, temperature maxima exist where = st , and N22 therefore tends to zero at
locations with stoichiometric mixture. Local extinction destroys this correlation and
the minimum of N22 at st disappears.
Accurate closures of the dissipation terms thus seem difficult and so far only been
attempted as part of DNS related studies where the mean dissipation rates of mix-
ture fraction and sensible enthalpy could be extracted from the DNS. In addition,
the joint PDF of mixture fraction and sensible enthalpy and the conditionally aver-
aged velocity do not result from the CMC equations and must be separately mod-
elled. The modelling of the doubly conditioned dissipation rates can be avoided, if
we solely focus on the modelling of the chemical source term. Bradley et al. [6]
based their closure of the chemical source term on tabulated scalar fields that were
obtained from flamelet solutions and parameterized by mixture fraction and temper-
ature. Kronenburg and Kostka [42] improved the method slightly, and the assump-
tion of a -distribution of the conditional probability density function P ,h s ( | )
The Conditional Moment Closure Model 105

Fig. 5.5: Conditionally averaged temperature (left) and CO and O2 mass frac-
tions (right) at X/D=7.5 in Sandia Flame E. Symbols represent experimental data
from [19], solid lines are CMC predictions with source term closure based on dou-
bly conditioned moments and dashed lines are CMC predictions using a standard
first order closure for wk | . Reprinted from [42] with permission from the Com-
bustion Institute.

proved accurate enough to achieve very good agreement of predicted temperature,

major species and even CO and NO with measurements of the Sandia Flame series
with progressive levels of extinction (Sandia Flames D, E and F). Examples of the
results are shown in Fig. 5.5.
Here, we would like to remind the reader that source term modelling by second
order closure has not quite led to the expected success with respect to temperature
predictions in particular in bluff body flames, but also in flames with significant ex-
tinction. This may indicate the importance of the fluctuation terms and/or diffusion
in progress variable space that is omitted in singly conditioned approaches. The very
good results achieved in [42] may therefore be rather fortunate and accurate mod-
elling of flame regimes with partial premixing may require the solution of the full
doubly conditioned moment equations.
We should also mention that such approaches may be used for problems involv-
ing autoignition, where locally an autoignition spot is developed and flames propa-
gate around it. The speed of such flames is larger than the conventional flame propa-
gation in unburnt reactants due to the fact that the region away from the autoignition
has also been reacting, albeit slowly, and is hence easier to jump to fully-fledged
combustion through the action of diffusion of species and heat from the burning
region; see Mastorakos [47] for a discussion and Wright et al. [80] for a demonstra-
tion of the importance of this quick propagation phase in diesel engines. Double-
conditioning may also be used in flame expansion problems in multiple-injection
diesel engines or following spark ignition in non-premixed systems; at present such
phenomena are captured in single-conditioned CMC (often with acceptable accu-
racy) through the ey -term [63, 76].
106 A. Kronenburg and E. Mastorakos

Fig. 5.6: Dissipation of sensible enthalpy predicted by MMC at times t = 1.0 (left)
and t = 2.0 (right). Reprinted from [41] with permission from the Combustion
Institute.

5.2.3.3 Possible Closures Using Multiple Mapping Conditioning

At this point we may want to anticipate Chapter 7 on Multiple Mapping Condition-

ing (MMC) without giving much detail on the method. We limit ourselves here to
state that MMC is PDF- and CMC-consistent, i.e. the solution of the MMC equa-
tion for the conditioning scalars provides their joint probability distribution, and the
MMC equations for the conditioned scalars are simple transformations of the CMC
equations and therefore provide identical solutions. In this subsection it shall suffice
to say that MMC implicitly provides a mapping closure for the conditionally aver-
aged dissipation terms, it succeeds in capturing the evolution of the local minimum
of N22 as shown in Fig. 5.6. In addition, the evolution of the joint probability of mix-
ture fraction and sensible enthalpy is quite well approximated. MMC correctly pre-
dicts a conditional PDF of sensible enthalpy conditioned on mixture fraction with
one peak at fully burning conditions and a second peak around h s = 0.2. A con-
ventional (conditional) -PDF would never predict the location of the second peak
away from the bounds, and MMC is clearly superior to approaches using presumed
PDFs, especially around stoichiometric.
MMC may appear as the solution to the closure issues addressed in the previous
sections, but it shall not be forgotten that all the MMC studies presented in this sub-
section refer to a comparison of MMC with DNS data of reacting flows with reduced
chemistry in homogeneous, isotropic turbulence. The feasibility of the method for
laboratory flames needs to be assessed, and we may anticipate that extension to com-
plex flow geometries may not be as simple as it first seems. Alternative stochastic
implementations of MMC may be more favourable, and this is discussed in much
more detail in Chapter 7.
The Conditional Moment Closure Model 107

5.2.4 Premixed Combustion

The conditional moment closure method has been derived for non-premixed com-
bustion where mixture fraction makes for an ideal quantity that describes flame
structure. We have seen above that the CMC method can be extended to partially
premixed flames and further application to premixed flames is straightforward.
Chemistry will be strongly linked to a progress variable, and simple first order
closure of the chemical source term conditioned on this progress variable can be
expected to yield accurate source term closures and pollutant predictions. Martin et
al. [46] applied premixed CMC to a flame stabilized on a backward-facing step. This
configuration shall mimic the flame stabilization mechanisms in a typical lean pre-
mixed gas turbine combustor where high mixing intensities justify the assumption
of distributed reaction regimes and therefore spatial homogeneity of the conditional
moments. The qualitative agreement between CMC predictions and experiments is
good, and CMC with a detailed methane-air mechanism gives a much better match
with the measurements than the reduced global mechanisms that are conventionally
used in industrial CFD work.
Martin et al. [46] have demonstrated that CMC provides a suitable source term
closure, however, CMC itself does not quite address some of the major issues asso-
ciated with modelling premixed turbulent combustion: the evolution of the progress
variable, in particular the temporal and spatial evolution of its probability distribu-
tion is one of the major research areas in turbulent combustion modelling (Chap-
ters 4 and 6). Presumed probability distributions may suffice, but the solution of the
transport equations for mean and variance require modelling of the mean scalar dis-
sipation and the turbulent scalar flux, two quantities that involve some uncertainty in
their modelling. Since the probability distribution P( ) is needed for full closure and
the conditionally averaged scalar dissipation of the progress variable and the con-
ditionally averaged turbulent scalar flux appear explicitly in the CMC equations, a
series of CMC related studies were dedicated to the analysis of these terms [73, 74].
One of the key issues is the effect of reaction on the scalar dissipation of a reactive
scalar. Often, these effects are neglected and dissipations of reactive and passive
scalars are modelled equally. Swaminathan and Bray [74] however showed, that the
dilatation effects need to be included in the modelling of N , and they suggested a
simple algebraic closure for this quantity. The derivation of conditionally averaged
dissipation rates is currently under way, but accurate modelling of the conditionally
averaged turbulent flux seems a bit further off.
The two extremes of very fast and very slow chemistry shall be mentioned here,
but will not be discussed further. Lee and Huh [44] proposed a zone conditional
modelling of premixed flames which corresponds to a CMC approach where the
PDF is approximated by two -functions at the extremes, and Bilger [5] introduced
markers fields for the mapping of the progress variable that may be limited to low
Damkohler numbers only. The reader is referred to the above references for more
detail on these methods.
108 A. Kronenburg and E. Mastorakos

5.2.5 Liquid Fuel Combustion

A further area of CMC research that has been enjoying increasing attention recently
is the modelling of reacting two-phase flows. It is postulated that liquid fuel com-
bustion is largely determined by the evaporation rate and the mixing between fuel
and oxidizer. Conditioning on mixture fraction appears as a promising concept for
the accurate closure of the chemical source term. It is well established that in the
presence of fuel evaporation, mixture fraction is no longer conserved and additional
source terms appear in its transport equations of mean and variance [65]. Mortensen
and Bilger [50] derived the fully consistent conditional moment closure equations
for spray combustion. The derivation is based on the instantaneous single phase
transport equations and a level set/indicator function technique is used to account
for the interfaces. The final form of the singly conditioned moment closure equation
for two phase flow can be written as

Qk Qk 2 Qk 1 
+ ui |  = N + wk |    ui Yk | P
t xi 2   P xi
 
Qk  | 
+ Q1,k Qk (1 )
 
1 
(1 )  Yk | P (5.21)
  P

where Q1,k denotes the conditionally averaged mass fraction of species k in the
liquid droplet. Several new terms that involve the evaporation rate, , and its fluc-
tuation appear in the final CMC formulation, and they require closure. The potential
of these new equations has not yet been evaluated and closures have not yet been
developed.
A simplified form of the CMC equations that neglected spray interactions was
applied to studies of spray autoignition under engine-like conditions with some suc-
cess. Kim and Huh [32] reported no influence of different models for the condi-
tionally averaged evaporation rate on ignition delay times, and even the effects of
different models for the closure of the mixture fraction variance equation are within
5%. The latter finding must be contrasted with the DNS studies by Reveillon and
Vervisch [62] who analysed the mixing field and showed that droplet evaporation
needs to be considered for the computation of the mixture fraction and its vari-
ance; it seems that the extra terms in the mixture fraction variance equation made
a small contribution in the flow studied by Kim and Huh [32]. Similarly, Wright et
al. [80] confirmed good agreement with measurements of ignition delay times and
spray penetration lengths in high pressure chambers using CMC and neglecting the
evaporation terms in the CMC and mixture fraction variance transport equations.
However, it seems premature to arrive at any conclusions from the above engine-
related studies on the importance of the modelling of the evaporation terms. The
modelling by Kim and Huh [32] was based on a simplistic model for the condition-
ally averaged evaporation and compared with no closure. Equally, Wright et al. [80]
The Conditional Moment Closure Model 109

used zero closures for the variance and CMC equations. These studies simply show
that auto-ignition delay times are not extremely sensitive to the correlations between
evaporation and mixture fraction, but sufficient experimental data simply does not
exist to allow any assessment of their importance on species predictions and flame
structure.
Schroll et al. [77] have corroborated the adequacy of CMC for the modelling
of droplet combustion. DNS of droplet evaporation and ignition with three differ-
ent initial droplet diameters shows the lowest conditionally averaged dissipation at
the most reactive mixture composition for the smallest droplets and CMC would
therefore correctly predict the shortest ignition times for these smallest droplets.
However, multiplication with the mixture fraction PDF and integration across mix-
ture fraction space yields the highest unconditionally averaged dissipation for the
smallest droplets that would lead - incorrectly - to the longest predicted ignition
delay times for these droplets if estimates were based on unconditional values. The
CMC concept is therefore ideally suited for the prediction of droplet combustion
but some key issues, such as the modelling of the conditionally averaged evapora-
tion rate and its correlation with species mass fractions and mixture fraction need to
be addressed. The mixture fraction PDF and the modelling of the scalar dissipation
in the presence of spray evaporation are also important to consider. It is fair to say
that CMC of two phase flows is in its infancy, and thorough investigations will be
needed for a better assessment of the viability of the method in real applications.

5.3 Application to Flows of Engineering Interest

As time progressed from the mid-90s when the first papers on CMC begun to ap-
pear, CMC simulations have developed from spatially-integrated formulations to
fully three-dimensional applications with strong temporal and spatial variations of
the conditional averages, thereby revealing the true strengths of the method. In addi-
tion, a shift from RANS to LES has begun to materialise, as well as the application
to large furnaces with important radiation effects and to new fuels such as syngas.
Some aspects of these simulations and numerical developments are discussed in this
Section.

5.3.1 Dimensionality of the CMC Equation

When using the CMC method for realistic problems, first-order CMC, with single
conditioning, is the virtually ubiquitous choice today. The key decision to be taken
by the user is the spatial dimensionality of the CMC equation, which should reflect
the physical nature of the problem, but also may be influenced by the available
computational resources. The spatial resolution needed in the CMC equation should
not be confused with the resolution needed in the calculation for the velocity or
110 A. Kronenburg and E. Mastorakos

mixture fraction fields. For statistically-steady attached flames in jets, for example,
experiment has suggested that the conditional averages are weak functions of the
radial coordinate. Therefore, ignoring streamwise turbulent diffusion and integrating
the full CMC equation (Eq. 5.8) across the jet results in:

Qk 2 Qk
 ui |  = N + wk |  (5.22)
xi 2
with the starred quantity being a PDF-weighted integral:


V | P dV
 |  = (5.23)
V P dV
with V being the region of integration (line, plane or even volume). For transient
problems, such as in diesel engines, volume integration may be performed to result
in the time derivative Qk / t replacing the convective term on the LHS of Eq. 5.22.
The computational requirements for the numerical solution of the resulting
parabolic equations is modest, even with quite detailed chemical mechanisms. This
approach has provided results of very good accuracy for many problems, especially
for jets. The similarity with transient flamelet modelling is evident, although the
modelling of N|  and the exact way the volume integration is done tends to dif-
fer. Let us denote this simplification to the full multi-dimensional CMC equation as
0DCMC, implying the lack of spatial diffusion.
For some problems, there is little or no direct experimental evidence that the con-
ditional averages are weak functions of space. In lifted jet flames, significant varia-
tions of Qk have been measured [10]. In compression-ignition engines, experimental
evidence points to a quite localised first emergence of ignition. Heat losses to engine
walls and multiple injections also lead to variations in Qk [59]. In spark ignition of
non-premixed combustors [47], the conditional distributions switch from unburnt to
burnt in different regions in space at different times. Such problems necessitate a
multi-dimensional CMC formulation, which we will denote as 3DCMC. Numer-
ical solution of the 3DCMC model has been attempted by various research groups
and some details are given next.

5.3.2 Numerical Methods

The three-dimensional CMC equation (Eq. 5.8), after modelling the spatial diffusion
term, can be written in more generic form as

Qk 2 Qk
+C(x,t, )Qk = N(x,t, ) + (D(x,t, )Qk ) + wk ( , Q2 , . . . , Qnsc )
t 2
(5.24)
where the coefficients C, N, D are, in general, functions of time, space x, and mix-
ture fraction and contain information from the fluid mechanical field. Equation
5.24 is a 5-dimensional partial differential equation with a stiff chemical source
The Conditional Moment Closure Model 111

term, making its solution at least as challenging as the numerical simulation of a

multi-dimensional laminar flame. The user of the elliptic CMC method must de-
cide: (i) whether to solve this equation in its full 3D form or whether to perform
some averaging across one of the spatial directions; (ii) the resolution to be used
(i.e. the number of grid nodes in each spatial direction); (iii) the size of the chemical
mechanism; (iv) the numerical method. Some compromises between the conflict-
ing requirements of having very fine resolution in physical space with the wish to
employ a very detailed chemical scheme may have to be made. Typically, central
differences are used for discretizing the diffusion in mixture fraction space and for
the second-order derivative in physical space, while upwind schemes have been used
for the convective term. An alternative strategy to solve the CMC equation before
dividing by the mixture fraction PDF, i.e. to solve for P Qk , has also been proposed
[11], but the comments below and the discretization issues apply to that formulation
as well.
Assuming a detailed chemical mechanism with, say, Ns = 50 species, a dis-
cretization in mixture fraction space with N = 100 points, and a physical-space
grid of, say, Nx = Ny = Nz = 25, and assuming that such a system were to be solved
by the Method of Lines, which transforms the partial differential equation into a sys-
tem of Ns N Nx Ny Nz ordinary differential equations, we would arrive at
a system of 75 106 o.d.es. Considering the stiffness involved, which necessitates
implicit schemes, the size of this system is just too large for present day solvers.
Despite this, in two physical-space coordinates (Nz = 0, Nx = Ny = 20), this solution
method has been used successfully with the GRI3.0 chemical mechanism and the
VODPK solver [24, 25].
An alternative is to solve the CMC p.d.e. by a fractional step (operator splitting)
method [16, 18, 23, 31, 59, 60, 80]. In such a scheme, one would solve the sequence

Qk
+C(x,t, )Qk = (D(x,t, )Qk ) (5.25)
t
Qk 2 Qk
= N(x,t, ) (5.26)
t 2
Qk
= wk ( , Q2 , . . . , Qnsc ) (5.27)
t
with each fractional step picking-up the solution from where the previous step left
it and advancing it for a given timestep. The chemistry fractional step is the one
that takes the most computational time; it will require separate solution of N
Nx Ny Nz stiff ordinary differential equations of the type dQk /dt = wk . This is
now more affordable, since the stiff solver will have to deal with a smaller system,
but the solver used will need to be able to deal effectively with the restart costs
when solving the chemical step while sweeping across the physical and mixture
fraction grid nodes. Splitting the various phenomena incurs an error; this has been
assessed [59, 80] and it is not negligible, but it can be controlled by a small enough
timestep. The Method of Lines is of course more accurate, but as the size of the
112 A. Kronenburg and E. Mastorakos

chemical mechanism increases, the operator splitting approach is the only one that
is practical.
It is evident that the numerical cost of solving the three-dimensional CMC equa-
tion is formidable, and experience shows that it is typically 50-90% of the total
computational cost of the whole simulation. The computational burden of a well-
resolved CMC simulation is not far from that of a low Reynolds number DNS.
For established flames, the discretization in mixture fraction space must follow
the usual practice of clustering the grid nodes around stoichiometry; 50-100 grid
nodes are the usual practice. For autoignition problems, the grid must be fine enough
across the whole mixture fraction range to resolve the reaction fronts that will prop-
agate across mixture fraction space following ignition [80]. For attached flames, the
physical-space gradients are small and hence relatively coarse physical-space grids
may be sufficient. Typical values are 10-20 grid nodes. For lifted jet flames, espe-
cially in the axial direction, a higher resolution is necessary. Note also the dangers
of numerical diffusion associated with upwind differencing and coarse grids when
having to resolve sharp transitions from Qk corresponding to unburnt fluid to Qk
corresponding to burnt fluid.
Recently, Large-Eddy Simulations have begun to appear with CMC as the com-
bustion sub-model. The CMC equation in LES is virtually identical to the RANS
formulation discussed above [55]. Various specific choices concerning the sub-
models for the conditional velocity and scalar dissipation must be made, especially
since it must be recognized from the outset that the physical-space CMC grid will
be coarser than the LES grid; this necessitates to develop a strategy concerning how
can the fine-grid fluid mechanical quantities be provided to the coarse grid used
to solve the CMC equation. Various options exist and are discussed in detail in
[75]. The LES/CMC model has been applied to attached jet flames [55], bluff-body
flames [52], autoigniting jets [53] and spark ignition problems [76]. The numeri-
cal cost here is very high. From a model development point of view, many of the
sub-models that are being used have not been properly validated yet for LES.

5.3.3 Applications and Outlook

The CMC model has been used for gaseous statistically-steady flames. Some ex-
amples include non-premixed attached jet flames [66, 67, 69], attached bluff-body
flames [28, 52, 70, 76], lifted jet flames [15, 24], autoigniting jets [53, 60, 61], and
soot formation [78, 79]. It has also been used for transient methane jets [30, 68], hep-
tane sprays [32, 80], and spark-ignition problems [76]. All these simulations have
focused on simple geometries and have, in general, produced good results com-
pared to experimental data. In addition to the DNS-based validation, these simula-
tions have also provided more detailed assessment of specific sub-models. The CMC
model has also been examined for reacting flows in porous media [34], chemical en-
gineering [49] and atmospheric flows [7], demonstrating thus the wide applicability
and usefulness of the method.
The Conditional Moment Closure Model 113

CMC is an attractive tool to model complicated geometries and problems. Multi-

dimensional CMC has been used for diesel engines [59] and compartment fires
[11, 12] and furnaces [23, 65] that have important radiation effects or low-oxygen
combustion. The analysis of the results provides very useful insights on the flame
structure and brings CMC to the point of practical calculations for design. First-
order CMC with single conditioning and proper chemistry seems sufficient for
many engineering problems (except for pollutant production, which may necessi-
tate second-order corrections). For problems with significant variations of the con-
ditional averages, such as extinctions, ignitions or quenching due to radiation or
convective heat transfer to walls, the current practice is to use refined elliptic CMC.
However, future applications should not be limited to first-order CMC. Second
order CMC provides a useful extension to first-order closures. Second-order CMC
has been applied to various laboratory flames with decent results; the theoretical
framework is well known, but a wider range of applicability would need to be es-
tablished, and future efforts should be directed towards improved modelling of the
conditional variance and co-variance equations, Eq. 5.12. Future progress in the
application of double-conditioning approaches may be more challenging. Double
conditioning is certainly attractive for (1) flames with partial premixing, for (2)
flames with local extinction in the vicinity of the walls where the second condition-
ing variable could be a wall distance parameter or for (3) engine calculations with
multiple injections. The latter has been attempted in a somewhat different frame-
work by Hasse and Peters [22] with multiple conditioning on two different mixture
fractions, however, all applications would require the closure of the doubly (or mul-
tiply) conditioned dissipation terms which is far from being trivial. Simplification
as suggested by Hasse and Peters [22] are certainly not applicable to configurations
with a reactive species as second conditioning scalar, and other alternative methods
for closure would need to be applied. MMC might offer one possible solution, but
modelling suggestions as brought forward by Nguyen et al. [56] should equally be
pursued.
Double conditioning could also be applied to liquid fuel combustion where the
second conditioning scalar describes the inter-droplet space. However, some more
fundamental issues should be assessed in CMC of multiphase flows first, such as
the effects of droplet evaporation and of the correlations between evaporation rate
and reactive scalar field on conditional moments and on mixture fraction variance.
Existing DNS studies and RANS calculations do not give a clear picture on the
importance of these terms, and we would need to know under which conditions they
become important and how they could be modelled.
But all these exciting new developments with respect to improvements of the
chemical source term closure and two-phase flow modelling should not distract from
implementation issues as addressed in Sections 5.3.1 and 5.3.2. In particular the di-
mensionality and also the CMC cell size need to be chosen carefully. Small CMC
cells lead to cells with zero probability for some of the mixture fraction bins. In case
of non-zero probability of these CMC bins in the neighbouring cells, it is not clear
yet how to model convective and diffusive flux into a cell where a certain mixture
fraction bin has zero probability. Standard practice is to neglect fluxes below a cer-
114 A. Kronenburg and E. Mastorakos

tain threshold value of the PDF, however, this may be inaccurate as new simulations
for the determination of blow-out limits of lifted flames show [54]. This example
simply illustrates that implementation can strongly affect the quality of the predic-
tions, and much more work is needed to understand the correct treatment of zero-
(or low) probability moments, in particular in LES-CMC.

5.4 Conclusion

It is of course not possible to give a complete and detailed review on all research
activities of the last ten years related to the development and application of novel
CMC techniques. The majority of research efforts had been directed towards im-
provements to the modelling of the conditioned chemical source term and towards
CMC applications to flows of practical interest. We have therefore focused in this
chapter on rather detailed summaries of second order closures, double conditioning
and issues related to CMC in more complex flow geometries. The CMC method-
ology has advanced within the last ten years, and this chapter should be therefore
viewed as complementary to Klimenko and Bilgers review [35]. We hope that it
offers some stimuli to new researchers in the field for continued work on the condi-
tional moment closure method.

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Chapter 6
Transported Probability Density Function
Methods for Reynolds-Averaged and
Large-Eddy Simulations

D.C. Haworth and S.B. Pope

Abstract Probability density function (PDF) methods provide an elegant and effec-
tive resolution to the closure problems that arise from averaging or filtering chemi-
cal source terms and other nonlinear terms in the equations that govern chemically
reacting turbulent flows. PDF methods traditionally have been associated with stud-
ies of turbulence-chemistry interactions in laboratory-scale, atmospheric pressure,
nonluminous, statistically-stationary nonpremixed turbulent flames; and Lagrangian
particle-based Monte Carlo numerical algorithms have become the predominant
method for solving modeled PDF transport equations. The emphasis in this chap-
ter is on recent advances, new trends and perspectives in PDF methods. These
include advances in particle-based algorithms, alternatives to particle-based algo-
rithms (e.g., Eulerian field methods), treatment of combustion regimes beyond low-
to-moderate-Damkohler-number nonpremixed systems (e.g., premixed flamelets),
extensions to include radiation heat transfer and multiphase systems (e.g., soot and
fuel sprays), and the use of PDF-based methods as the basis for modeling in large-
eddy simulation.

6.1 Introduction

Probability density function (PDF) methods have emerged as one of the most
promising and powerful approaches for accommodating the effects of turbulent fluc-
tuations in velocity and/or chemical composition in computational fluid dynamics
(CFD)-based modeling of turbulent reacting flows. Here the term PDF method
refers to an approach based on solving a modeled transport equation for the one-
point, one-time Eulerian joint PDF of a set of variables that describe the hydro-

D.C. Haworth
The Pennsylvania State University, University Park, PA, USA, e-mail: dch12@psu.edu
S.B. Pope
Cornell University, Ithaca, NY, USA, e-mail: s.b.pope@cornell.edu

T. Echekki, E. Mastorakos (eds.), Turbulent Combustion Modeling, 119

Fluid Mechanics and Its Applications 95, DOI 10.1007/978-94-007-0412-1 6,
Springer Science+Business Media B.V. 2011
120 Haworth & Pope

dynamic and/or thermochemical state of a reacting medium: that is, a transported

(versus presumed) PDF method.
Dopazo and OBrien [14, 15] were the first to consider a modeled equation for
the PDF of a set of scalars that describes the thermochemical state of a reacting
medium (a composition joint PDF) to model mixing and chemical reaction in turbu-
lent reacting flows. The relationship between particle models and PDF methods was
established by Pope [53], and particle methods have become the dominant approach
for modeling and solving PDF transport equations. The point of departure for PDF
methods in their modern form, including the associated Lagrangian-particle-based
solution algorithms, often is taken to be Popes 1985 paper [55]. Detailed treatments
of PDF methods are provided in the books by Pope [60] and Fox [22]. A compre-
hensive, up-to-date (2010) review can be found in [25].
In keeping with the theme of the book, this chapter emphasizes recent advances
in PDF methods. A baseline approach that is representative of the current state-of-
the-art is established in Sec. 6.2. Several recent developments with respect to this
baseline are highlighted in Sec. 6.3. In Sec. 6.4, one particularly important area of
development is discussed: the extension of PDF-based methods to model the effects
of unresolved fluctuations in large-eddy simulation. The chapter concludes with a
summary of key recent advances, outstanding issues, and promising directions for
future development. The reader who seeks more information is referred to [25].

6.2 A Baseline PDF Formulation

The velocity-composition-turbulent frequency PDF (VCF-PDF) method introduced

by Pope and coworkers represents the current state-of-the-art [7, 51, 74]. In that
case, the PDF considered is the one-point, one-time joint PDF of the three com-
ponents of velocity u, N composition variables , and a turbulent frequency .
The N composition variables are a set of scalars from which one can determine
the local thermochemical properties (mixture mass density, molecular weight, and
specific heats), species chemical production rates, and where required, the molecu-
lar transport properties and radiation properties. For a low-Mach-number, ideal-gas,
single-phase, multicomponent reacting mixture, the species mass fractions Y and
mixture specific enthalpy h usually are an appropriate set of composition variables.
The turbulent frequency is the inverse of a local turbulence time scale.
Standard notation is used to distinguish conventionally-averaged mean quanti-
ties ( ) from Favre-averaged or density-weighted mean quantities (), and to dis-
tinguish a fluctuation about a conventionally-averaged mean value ( ) from a fluc-
tuation about a Favre-averaged mean value ( ). The density-weighted VCF-PDF
is denoted as fu (V, , ; x,t). Here V, , and are the sample-space variables
corresponding to u, , and , respectively. This PDF contains complete one-point
statistical information about the turbulent velocity, composition, and frequency at
all locations x and times t. An important property of the PDF is that one-point joint
statistics can be expressed as integrals of the PDF over its sample space. For exam-
PDF Methods 121

ple, the turbulent scalar fluxes u

  are given by:
i
  
u
  =
i (Vi ui )(  ) fu dVd d , (6.1)

and integration is over the entire sample space. Analogous expressions can be writ-
ten for the Reynolds stresses u i j

 u , the scalar covariances  
, and any other one-
point statistics of interest. The velocity-composition PDF (VC-PDF), fu (V, ; x,t),
and the composition PDF (C-PDF), f ( ; x,t), also are widely used to model tur-
bulent reacting flows. These can be obtained from the VCF-PDF by integrating over
frequency space or velocity-frequency space, respectively.
Equations for fu , fu , and f can be derived from the equations expressing
conservation of mass, momentum, species composition, and energy for a reacting
system [25, 55]. The resulting PDF equations contain terms that must be modeled.
The modeling proceeds by considering a system of notional particles that are, in
some sense, analogous to fluid particles that obey the underlying governing conser-
vation equations. The idea is to devise a system of notional particles whose one-
point, one-time joint PDF evolves in the same manner as the one-point, one-time
joint PDF for the real reacting fluid system, while circumventing a key difficulty
that arises in the real system: the wide dynamic range of scales that characterizes
turbulent reacting flow at high Reynolds and Damkohler number. Here a superscript
asterisk denotes a notional particle value. For a high-Reynolds-number, low-Mach-
number, open system, neglecting body forces and radiation heat transfer, and adopt-
ing standard physical models, the equations governing notional particle properties
(position x (t), velocity u (t), composition (t), and turbulent frequency (t))
can be written as:

dxi = ui dt , (6.2)
1 p  1 3    1/2
dui = dt + C0 ui ui dt + C0 k dWi , (6.3)
( ) xi 2 4
1
d = S ( )dt C (  ) dt , (6.4)
2
 1/2
d = C3 (  ) dt S dt + 2C3C4
 dW  . (6.5)

Here all mean quantities are evaluated at the particle location: S = C 2 C 1 P/(k )
is the sink of frequency, P = u  u u
i j i / x j is the turbulence production, k =
u u /2 is the turbulence kinetic energy, and is a conditionally-averaged turbu-
i i
lent frequency given by = C  |  / .
Particle advection in physical space (Eq. 6.2) is treated exactly; this includes
transport by the mean velocity and by turbulent velocity fluctuations. Equation (6.3)
is the simplified Langevin model for notional particle velocities [28]. There p is
pressure, ( ) is the density, and W(t) is a vector of three independent stochastic
Wiener processes. The mean pressure gradient is treated exactly; that is, the mean
pressure gradient term in the notional particle equations is the same as the corre-
122 Haworth & Pope

sponding term in the fluid particle equations. The modeled terms (terms involving
) represent the effects of the fluctuating pressure gradient and molecular (viscous)
transport. Equation (6.4) governs the evolution of notional particle compositions.
There S is the chemical source term in the species mass fraction equations, which
is treated exactly. The modeled term (term involving ) represents the effects of
molecular transport (scalar mixing). Here the interaction by exchange with the mean
(IEM) mixing model [75] has been used, for simplicity. Finally, Eq. (6.5) is the
stochastic frequency model for particle turbulent frequencies [74]. There W  (t) is a
stochastic Wiener process that is independent of the Wiener process in the particle
velocity equation. There are seven model constants: C0 , C , C 1 , C 2 , C3 , C4 , and
C . Standard values are given in [51]. The VCF-PDF equation corresponding to
Eqs. (6.2-6.5) is:

  fu Vi   fu 1 p   fu S   fu

+ +
t xi xi Vi
1 3  1 2   fu
= + C0 [(Vi ui )  fu ] + C0 k
2 4 Vi 2 Vi Vi
1 S   fu
+ [C (  )  fu ] +
2
(  )  fu   fu
2
+ C3 +C3C4  . (6.6)
2
Here terms that are treated exactly are on the left-hand side, and modeled terms are
on the right-hand side. The equation is closed; all quantities that appear in Eq. (6.6)
can be deduced from fu (V, , ; x,t). (The mean pressure field is determined by
a Poisson equation, whose source can be expressed in terms of fu [25, 55].)
In a VC-PDF method, turbulent frequency is not carried as a particle property.
Then turbulence time-scale information must be specified externally. For example,
in Eqs. (6.3) and (6.4) can be replaced by /k, where is the viscous dissipation
rate of turbulence kinetic energy, and a separate modeled equation can be solved for
(e.g., the equation from a standard two-equation turbulence model). A separate
equation for k is not required, since k can be determined from fu (V, ; x,t).
In a C-PDF method, velocity and turbulent frequency are not carried as parti-
cle properties. As for a VC-PDF method, turbulence time-scale information must
be specified externally, and this can be done by writing in Eq. (6.4) as /k, and
solving separate modeled equations for k and . In this case, k cannot be determined
from f ( ; x,t). In addition, the mean velocity and mean pressure fields must be
specified externally (e.g., from a conventional CFD solver), and transport by tur-
bulent velocity fluctuations must be modeled. This usually is done by invoking a
gradient transport approximation, which is equivalent to replacing Eq. (6.2) by a
stochastic diffusion equation that corresponds to a random walk in physical space
[25, 55].
It is important to recognize that in all three formulations (VCF-PDF, VC-
PDF, and C-PDF), the principal modeling issue that needs to be addressed in
PDF Methods 123

most alternative approaches is resolved: namely, the closure problem that arises
from averaging the highly nonlinear source terms ( S( ) = S( )). For an arbitrar-


ily complex chemical mechanism, S = S ( ) f d =  S ( ) fu dVd =

S ( ) fu dVd d .
The single most important, and also probably the weakest, aspect of the physical
modeling is the scalar mixing model. Mixing models in PDF methods are closely
related to the scalar dissipation rate, a central quantity in all turbulent combustion
modeling [22, 52]. Currently the dominant scalar mixing models are IEM, variants
of Curls model (or coalescence-dispersion - CD - model) [11, 68], and the Eu-
clidean minimimum spanning tree (EMST) model [69]. Recent trends and develop-
ments in mixing models are aimed at extending PDF methods to higher-Damkohler-
number systems and flamelet regimes of combustion, at accounting explicitly for a
spectrum of turbulence scales, and at capturing differential diffusion, among other
things. Mixing models are discussed further in Sec. 6.3.1.
Most modern numerical solution algorithms for modeled PDF transport equa-
tions employ a Lagrangian particle Monte Carlo algorithm that is coupled with a
grid-based Eulerian CFD code: a hybrid Lagrangian particle/Eulerian mesh (LPEM)
algorithm. While such algorithms have been available for over 20 years [2, 27], it
has been only within the last 10 years that algorithms have matured to the point
where PDF methods can be applied to general statistically nonstationary, three-
dimensional turbulent reacting flows. Early algorithms failed to consider important
mathematical and physical consistency requirements that are implicit in the particle
representation, and that were not well understood at the time. Modern consistent
hybrid LPEM methods are robust, efficient, and accurate. This important area of
development is discussed further in Sec. 6.3.2.
The concept of statistically equivalent representations is implicit in the notional
particle representation that has been introduced above. In principle, many different
systems can give rise to the same one-point, one-time PDF equation. The trick is
to devise a system whose one-point, one-time PDF equation is as close as possible
to the PDF equation that corresponds to the real turbulent reacting flow, while reduc-
ing the dynamic range of scales that must be resolved numerically. The stochastic
particle representation and physical models of Eqs. (6.2-6.5) have been designed for
this purpose, and equivalent particle systems are the current mainstream approach
for modeling and solving PDF transport equations. Recently alternative approaches
based on equivalent stochastic or deterministic Eulerian fields have been explored,
and this line of development is discussed in Sec. 6.3.3.
PDF methods traditionally have been associated with studies of turbulence-
chemistry interactions in laboratory-scale, atmospheric, statistically-stationary, non-
luminous nonpremixed turbulent flames burning simple gaseous fuels at low-to-
moderate Damkohler numbers. These are not inherent limitations. PDF methods can
be, and have been, applied to high-Damkohler-number systems, to premixed sys-
tems, to multiphase systems (e.g., liquid fuels and sooting luminous flames), and to
practical combustion devices. The stochastic modeling and solution strategies pro-
vide a natural and powerful framework for accommodating the important nonlinear
couplings across widely disparate scales that characterize turbulent combustion and
124 Haworth & Pope

other multiscale, multiphysics systems. For example, the VCF-PDF method is in-
herently a multiscale approach that accounts for a distribution of turbulence scales at
each spatial location and time. Developments in multiscale and multiphysics mod-
eling are discussed in Sec. 6.3.4.

6.3 Recent Advances in PDF Methods

Trends in mixing models, advances in particle-based methods, alternatives to particle-

based methods, and extensions to multiscale, multiphysics systems are discussed in
the following four subsections; examples are provided in Sec. 6.3.5. A fifth impor-
tant area of development, the use of PDF-based methods to model the effects of
unresolved fluctuations in large-eddy simulation, is discussed in Sec. 6.4.

6.3.1 Mixing Models

The most essential characteristics of scalar mixing models for PDF methods are
[22, 60, 69]: 1) mean scalar quantities should not change as a result of mixing; 2)
scalar variances should decay at the correct rate; and 3) scalar quantities should
remain bounded (e.g., mass fractions should remain between zero and unity, and
should sum to unity). Additional criteria have been proposed for mixing models
used in LES [43]. In LES, it is important to treat the limit where the flow is locally
fully resolved (the DNS limit) properly. This is discussed further in Sec. 6.4.
The earliest mixing models (IEM and Curl) possess the three most essential
characteristics, but they do not perform well at high Damkohler numbers and/or
in flamelet regimes, for example. Models that enforce locality in composition
space (e.g., EMST [69]) generally perform better. A different approach for high
Damkohler numbers was proposed in [40] for premixed systems. There a modified
Curls model was used, and the mixing time scale for reactive scalars was modified
to include an explicit dependence on the ratio of the laminar burning velocity to the
Kolmogorov velocity scale. The parameterized scalar profile (PSP) model pro-
posed by Meyer and Jenny [47, 48] considers statistics of scalar profiles. Flamelet
regimes can be accommodated, and the model provides joint statistics of scalar and
scalar dissipation rate.
Mapping closures [57] provide another route to modeling scalar mixing in PDF
methods. In a mapping closure, the stochastic physical field of interest is mapped to
a Gaussian reference field. The advantage is that for a Gaussian field, all multipoint
statistics are known in terms of the mean, the variance, and the two-point correlation
function. A particular strength of mapping closures is that they cause the PDF of
conserved scalars to relax to a Gaussian distribution in statistically-homogeneous
systems. Klimenko and Pope [36] proposed a multiple mapping conditioning
(MMC) approach for mixing and reaction. MMC is based on a generalization of
PDF Methods 125

mapping closures, and combines elements of conditional moment closure (CMC)

and PDF methods. MMC has been developed subsequently by Klimenko and others
[34, 76]. In some of the most recent work [8, 9], a novel sparse-Lagrangian numer-
ical algorithm has been developed for solving PDF transport equations that have
been closed using the MMC approach.
Other approaches that have been proposed include models based on Langevin
equations [73] or the closely related Fokker-Planck equation [19, 20], and mod-
els that explicitly incorporate scale (spectral) information [21, 71]. Other mixing
models are discussed in Chapter 6 of [22]. Some models have been designed to
accommodate differential molecular diffusion [43].
It has been argued that relatively simple mixing models should suffice for LES,
because the models only need to deal with the unresolved fluctuations about the
local resolved-scale composition. This has been borne out in numerical studies, and
is consistent with the arguments presented in [35]; there it has been shown that
many mixing models are essentially the same when the physical distance between
particles becomes infinitesimally small (a DNS limit). For these reasons, IEM and
variants have been used in most LES modeling studies reported to date.

6.3.2 Hybrid Lagrangian Particle/Eulerian Mesh Methods

Three areas where important advances are being made in methods to solve the PDF
equations are: mathematical/physical consistency; numerical accuracy; and compu-
tational efficiency. Each of these is discussed in turn in the following paragraphs.
An intrinsic requirement for the notional particle representation to be consis-
tent with the PDF equation is that the spatial distribution of particle mass must
remain consistent with the spatial distribution of fluid mass in the system. The ex-
tent to which this constraint is satisfied depends on the particle-tracking algorithm,
among other things. For example, the interpolation scheme that is used to determine
the value of the local mean velocity at particle locations should guarantee a dis-
crete velocity-divergence field that is consistent with the mean continuity equation
[44, 50, 51, 79]. Even with such an interpolation scheme, a correction algorithm
may be required to redistribute particles in physical space [51, 79] to prevent the
accumulation of numerical errors. Energy consistency is another important consid-
eration. An approach that simultaneously addresses consistency and robustness of
the particle/mesh coupling was introduced in [50, 51]. There different forms of the
energy equations are solved on the mesh side and on the particle side. The mean
energy equation on the mesh side contains a chemical source term that is estimated
using particle values. This smooths the noisy particle data through an elliptic PDE,
thus enhancing robustness while still capturing the important influence of turbulent
fluctuations.
Characterization and quantification of numerical errors for LPEM methods is an-
other area where advances have been made. Errors can be categorized as statistical
error, bias error, spatial truncation error, and temporal truncation error [50]. The
126 Haworth & Pope

first two of these are peculiar to Monte Carlo methods. Statistical error refers to the
1/2
random error that results from using a finite number of particles; it scales as Npc ,
where N pc is the number of particles per cell. Bias refers to a deterministic error
that results from using a finite number of particles, and is not reduced by averaging;
1
it scales as N pc [58]. Systematic studies of numerical errors have yielded recom-
mendations for optimum values of key numerical parameters [50, 58]. Stochastic
terms in the notional particle equations (Eqs. 6.2-6.5) complicate the construction
of time-accurate methods. Higher-order implementations require extreme care in the
sequencing of operations and in the updating of coefficients; formally second-order
accurate schemes have been published [6].
Efforts to reduce computational cost are a third area where advancements are be-
ing made. Chemical kinetics usually dominate the computational effort when finite-
rate chemical mechanisms involving more than a small number (10, say) of chemical
species are used. In situ adaptive tabulation (ISAT) [41, 59] is a storage-retrieval al-
gorithm that has proved effective in improving computational efficiency, especially
for statistically stationary systems. ISAT tabulates the composition at the end of a
reaction fractional step (obtained by integrating the ODEs d /dt = S ( )). In-
creasingly effective parallelization strategies are being developed for LPEM meth-
ods, both with and without ISAT. The number of particles per cell remains approx-
imately constant in most modern algorithms. In that case, the same spatial domain
decomposition that is used to parallelize the underlying Eulerian CFD code is effec-
tive for most aspects of the particle calculations. An exception is chemical reaction;
the computational time required to integrate the stiff chemical source terms for a
particle varies widely, depending on its location in composition space. Achieving
uniform load balance across processors requires a composition-space (versus phys-
ical space) distribution strategy, and approaches are being developed to accomplish
this.

6.3.3 Eulerian Field Methods

Eulerian field representations are alternatives to the notional particle representation

that is discussed in Sec. 6.2. Both stochastic and deterministic Eulerian field meth-
ods have been proposed.
Two developments of stochastic field methods have appeared in the literature.
One originated with Valino [72] and another was introduced by Sabelnikov and
Soulard [65]. Different physical and mathematical arguments are invoked in the two
developments. For example, in the Valino formulation the stochastic fields must
remain spatially smooth, while the Sabelnikov formulation allows for discontinu-
ities. The statistical equivalence of the two approaches has been established in Ap-
pendix B of [65]. An overview of the Valino formulation for a composition PDF
follows. The turbulent reacting flow is represented by a number NF of notional Eu-
lerian fields. With superscript # referring to any one of the fields, the stochastic
Eulerian field equation is:
PDF Methods 127

# 1
d # = 
ui dt + S ( # )dt C (#  ) dt
xi 2
1
#  #
+ T dt + (2 1T )1/2 dWi# . (6.7)
  xi xi xi

Here W# denotes a vector-valued Wiener process that varies in time but is inde-
pendent of spatial location. The first three terms on the right-hand side of Eq. (6.7)
correspond, respectively, to advection by the mean flow, chemical reaction, and IEM
mixing. The terms involving the apparent turbulence diffusivity T correspond to a
gradient transport model for turbulent velocity fluctuations. Estimates of density-
weighted mean quantities are obtained by ensemble averaging over the fields. Be-
tween 10 and 30 fields have been used in modeling studies reported to date.
A deterministic approach is the multi-environment PDF (MEPDF) method pro-
posed by Fox [22, 70]. In a composition MEPDF method, a system of equations is
solved for the weights (or probabilities) and the scalar values of each of NE en-
vironments. Mean quantities are estimated as probability-weighted sums over the
environments. The system is closed by adding source terms that are prescribed to
force a subset of the moment equations from the MEPDF to match the correspond-
ing moment equations from a modeled PDF transport equation: a direct-quadrature-
method-of-moments (DQMOM) approach. The equations corresponding to gradient
transport and IEM mixing for NE = 2 and NE = 3 are given in [70]. In contrast to
stochastic Eulerian field methods, the MEPDF equations are deterministic. How-
ever, the source terms become poorly conditioned with increasing NE , and most
results reported to date have been for NE = 2. Such low values of NE correspond
to radical approximations of the PDF equations. Moreover, the approach does not
guarantee realizable compositions with multispecies chemistry.
Eulerian-field methods (stochastic or deterministic) are more straightforward to
implement into mesh-based Eulerian CFD codes than are particle methods. Deter-
ministic Eulerian-field methods are free of statistical error, and the Valino stochastic
Eulerian field formulation is free of spatially varying stochastic errors; the special
techniques that have been developed for particle/mesh coupling through the mean
density and/or energy in hybrid LPEM methods may not be necessary. On the other
hand, simple mixing models (IEM) have been used in Eulerian-field PDF modeling
studies reported to date. Mixing models that enforce locality in composition space
(e.g., EMST [69]) probably cannot be implemented effectively in the Eulerian-field
frameworks. In cases where chemical mechanisms containing more than 10 species
(say) are used, the computational effort is dominated by the chemical source terms.
In that case, the computational effort per field in an Eulerian method for a mesh
containing NC cells (the solution of a PDE) is comparable to the computational ef-
fort per NC particles in a Lagrangian particle method (the solution of NC stochastic
ODEs). Unless the number of fields that is required for an Eulerian field method
is smaller than the number of computational particles per cell that is required for
a Lagrangian particle method, little advantage can be expected in computational
time.
128 Haworth & Pope

6.3.4 Multiscale, Multiphysics Modeling

Extensions of PDF methods to deal with flamelet combustion, walls, high-speed

flows, soot, sprays, and radiation are discussed briefly in the following paragraphs.
Flamelets have been accommodated in various ways. One approach is to solve
a modeled PDF equation for a reduced set of composition variables (e.g., a mix-
ture fraction and/or a small number of progress variables) and to relate the local
thermochemical state to this reduced set using precomputed flamelet libraries [26].
Another approach is to build the flamelet structure explicitly into the formulation
and modeling. Early in the development of PDF methods, it was recognized that in
laminar premixed flames the molecular transport term in the PDF equation can be
closed by introducing a reaction progress variable [55, 63]; this was done in several
modeling studies of premixed turbulent flames [1, 63]. A third approach is to mod-
ify the scalar mixing model to include an explicit dependence on laminar burning
velocity and/or Kolmogorov scales [40] (Sec. 6.3.1).
Modifications to the PDF or particle equations have been introduced to include
wall effects, following one of two approaches. Models based on wall functions are
designed to reproduce a logarithmic mean velocity profile and other statistical fea-
tures of equilibrium turbulent boundary layers [17, 49]. In contrast, models based
on elliptical relaxation incorporate wall effects without explicitly invoking wall or
damping functions [16, 77]. Wall-function approaches are computationally more
efficient, while elliptical relaxation is more general.
In high-speed flows, the effects of pressure fluctuations on density generally can-
not be neglected. It then is necessary to include pressure and/or other thermody-
namic quantities in the composition variables. A joint PDF of species mass fractions
and enthalpy was extended to high-speed flows, including flows with shocks, in [29].
A formulation based on the joint PDF of velocity, pressure, an energy variable, and
a turbulent frequency was proposed and tested in [12, 13].
PDF-based modeling studies of sooting turbulent flames have been reported in
[39, 45, 46]. To capture the effects of turbulent fluctuations on soot processes, ad-
ditional quantities are transported on each notional particle. For example, in the
method-of-moments approaches used in [39, 45, 46], the lower-order concentration
moments of the soot particle size distribution function are transported, and a clo-
sure model is invoked to truncate the infinite series of moments (e.g., interpolative
closure). Source terms in the moment equations are designed to capture the key
physical and chemical processes involved in soot formation and oxidation.
A dual-Lagrangian formulation has been adopted in most PDF-based studies of
turbulent spray flames. There one particle ensemble represents the dispersed liq-
uid phase (e.g., the fuel spray) and a separate particle ensemble represents the car-
rier gas phase (the notional PDF particles). The principal liquid-to-gas coupling
is through the vaporization model. This is appropriate for external group combus-
tion regimes, where combustion occurs around groups of droplets. Composition-
PDF modeling studies of turbulent spray flames include [24, 31, 37, 80]. A dual-
Lagrangian velocity-composition PDF method also has been developed [4].
PDF Methods 129

PDF methods are particularly well suited for dealing with radiation heat transfer.
Radiative emission is determined by local properties, and therefore appears in closed
form in PDF equations. This includes the so-called turbulence-radiation interactions
(TRI) associated with emission: the mean radiative emission from a turbulent flame
is not the same as the radiative emission that would be determined based on the
local mean temperature and composition [38, 42, 46]. Radiative absorption requires
further modeling, and approaches that are compatible with the notional particle rep-
resentation have been developed for that purpose [45, 46, 78].

6.3.5 Examples

Three recent examples are discussed to illustrate the range of physical phenomena
that can be captured, and the diverse physical models and numerical algorithms that
have been developed.

6.3.5.1 A Statistically Stationary 1D System with Imperfect Mixing

A plug-flow-reactor often is used to study chemical kinetics, and/or as an idealiza-

tion of the combustion process in some devices. An imperfect-mixing variant was
introduced in [54] as an early test case for PDF methods, and (as now defined) was
used in [30] to study the influence of different numerical implementations of PDF
methods. A single reaction progress variable, c, characterizes this statistically sta-
tionary, one-dimensional system. Unburned reactants (c = 0) enter at x = 0, and c
approaches unity (fully burned products) at x = L. Finite-rate chemistry proceeds at
rate that is proportional to 1 c. For IEM mixing and constant values for all model
parameters, analytic solutions can be obtained for the steady-state mean and rms
progress-variable profiles (Fig. 6.1). Results from a stochastic Lagrangian particle
method, a stochastic Eulerian field method, and a deterministic Eulerian field (two-
environment MEPDF with DQMOM closure) are compared in Fig. 6.1 [30]. For the
two stochastic methods, time-accurate algorithms are used, the mean and rms values
at each x location are obtained by averaging over the local particle or field values,
and these are further averaged in time to reduce statistical error. Each individual
field in the stochastic Eulerian field method exhibits random fluctuations in time,
but remains spatially smooth. For the MEPDF method, starting from (almost) arbi-
trary initial conditions, each environment (c1 (x), c2 (x)) converges to a steady-state
profile, and each weight or probability (w1 (x), w2 (x)) converges to a uniform value
that is equal to the value that is specified at x = 0. No simple physical interpretation
can be assigned to the environments here, and there is no unique specification for
the environment weights at x = 0; these can be set to any values between zero and
unity that satisfy w1 + w2 = 1 (with c1 = c2 = 0). Here w1 = 0.1, w2 = 0.9 have
been specified. The steady-state environment profiles c1 (x) and c2 (x) would be dif-
ferent for different choices of the inlet weights, but the mean and rms profiles would
130 Haworth & Pope

Fig. 6.1: Mean and rms reaction progress variable profiles for a statistically station-
ary 1D system with imperfect mixing. Analytic mean (bold solid lines) and rms
(bold dashed lines) profiles and computed mean (filled diamonds) and rms (filled
triangles) profiles for each method are shown [30]. a) 1,000 instantaneous particle
values (open symbols) from a Lagrangian particle method. b) Five instantaneous 1D
fields (thin solid lines with open symbols) from a stochastic Eulerian field method.
c) Steady-state environment profiles (thin solid lines with open symbols) from a
two-environment MEPDF method.

remain the same. For all three methods, the mean and rms profiles agree well with
the analytic solutions. This configuration is too simple (and nonphysical) to draw
conclusions regarding the relative merits of the three methods, but it serves well to
illustrate the very different discrete representations of the PDF that are used in each
method.
PDF Methods 131

6.3.5.2 Lifted Jet flames in Vitiated Coflow

Lifted flames are useful to study autoignition and stabilization mechanisms. Labora-
tory-scale, atmospheric-pressure axisymmetric jets of H2 /N2 mixtures issuing into a
hot coflow of lean H2 -air combustion products have been studied experimentally [5].
A VCF-PDF method and a consistent hybrid LPEM solution algorithm were ap-
plied to this configuration in [7]. The study featured the simplified Langevin model
and a stochastic frequency model for particle velocities and turbulent frequencies,
respectively. The effects of coflow temperature, inlet velocity profiles, inlet turbu-
lence intensity, mixing models, and chemical mechanisms were explored. Results
are very sensitive to small variations in the temperature that is prescribed for the
coflow; a 10 K change in the temperature can change the computed liftoff height by
as much as a factor of two, and the experimental uncertainty in the coflow temper-
ature is 25 K. Computed and measured liftoff heights versus coflow temperature
are shown in Fig. 6.2. There two sets of experimental measurements are shown
(Wu and Gordon), and computed results are shown for two chemical mech-
anisms (Mueller and Li) and for three mixing models (MC - modified Curl,
IEM, and EMST). The models capture the experimental trends in all cases. Differ-
ences between model results obtained using the two chemical mechanisms are large,
while differences among the mixing models are relatively small; this suggests that
these flames are primarily controlled by chemical kinetics.

Fig. 6.2: Computed and

measured lift-off heights
H (normalized by fuel-nozzle
diameter D) versus coflow
temperature for jets of H2 /N2
issuing into a coflow of H2 -air
combustion products. Results
for two sets of experimental
measurements, two chemi-
cal mechanisms, and three
mixing models are shown.
Reprinted from [7] with per-
mission from the Combustion
Institute.

6.3.5.3 Luminous Flames

The third example illustrates the ability of PDF methods to capture complex inter-
actions of turbulence, gas-phase chemistry, soot and radiation. Here detailed soot
and radiation models have been implemented, and the models have been applied
to luminous, atmospheric-pressure, nonpremixed, ethylene- and methane/ethylene-
fueled turbulent jet flames [45, 46]. A composition PDF method and consistent
132 Haworth & Pope

hybrid LPEM algorithm are used with skeletal gas-phase chemistry (a 33-species
mechanism), a detailed soot model (method of moments with interpolative clo-
sure), and a spectral photon Monte Carlo radiation solver. Six different turbulent
flames were simulated with no changes in the model parameters, and uniformly
good agreement was observed between computed and measured mean temperature,
mean soot volume fraction, and radiative intensity (where available) profiles. One
example is shown in Fig. 6.3, for a 90% methane/10% ethylene-fueled flame with
30% oxygen in the oxidizer stream. There the equivalent soot volume fraction
was measured using a line-of-sight extinction technique. Results from two sets of
simulations are shown: in one case, turbulence-radiation interactions have been ig-
nored by computing radiative emission and absorption using cell-mean values; in the
other, turbulence-radiation interactions have been included by computing radiative
emission and absorption using particle values. In both cases, turbulence-chemistry
interactions are retained by computed chemistry at the particle level. Radiant heat
fluxes are up to 50% higher with consideration of TRI (not shown), and soot lev-
els decrease by up to a factor of three with consideration of TRI. Both heat fluxes
and soot levels are in better agreement with experimental measurements when TRI
are included. These results clearly demonstrate that TRI often ignored in model
calculations can have a major effect.

Fig. 6.3: Computed (lines)

and measured (symbols) soot
volume fraction profiles for a
90% CH4 /10% C2 H4 -fueled
flame with 30% O2 in the
oxidizer [46]. Two sets of
calculations are shown: one
where turbulence-radiation
interactions have been ne-
glected (dashed line), and one
where turbulence-radiation in-
teractions have been included
(solid line).

6.4 PDF-Based Methods for Large-Eddy Simulation

The predominant approach to extending PDF-based modeling to LES is the filtered

density function (FDF) method. The filtered density function was introduced by
Pope [56], and a transport equation for a composition FDF first was derived and
modeled by Gao and OBrien [23]. FDF methods currently are being developed
by several groups [10, 18, 32, 64, 66, 67]. The LES-FDF method is reviewed in
Secs. 6.4.1 and 6.4.2. Issues have been raised concerning the appropriateness of
PDF Methods 133

the FDF as a basis for modeling in LES. An alternative LES-PDF formulation that
addresses these issues has been proposed recently by Pope [62], and this is discussed
in Sec. 6.4.3. Examples of LES-FDF methods are provided in Sec. 6.4.4.

6.4.1 Spatial Filtering, FDFs, and FDF Transport Equations

The local spatially-filtered value of a physical quantity Q, denoted Q , is,


Q = Q(x,t) Q(y,t)G(|x y|)dy , (6.8)

where integration is over the entire flow domain. Here the low-pass spatial filter
function, G(|x y|), satisfies G(x)dx = 1, is non-negative (for FDF methods), and
has a characteristic filter width . The instantaneous value of any physical quantity
Q = Q(x,t) can be decomposed into a filtered component (the resolved field) and a
fluctuation about the filtered component (the residual field): Q(x,t) = Q(x,t) +
Q (x,t). Density weighting is useful in variable-density flows; the corresponding
fluctuation about a density-weighted spatially-filtered value (denoted Q  ) is denoted

using a double prime: Q = Q + Q . 

The FDF is the G-weighted spatial average of the fine-grained PDF in a neigh-
borhood of x [56]. For a composition FDF, for example, f , ( ; x,t)d is the G-
weighted fraction of the fluid in a neighborhood of x whose composition is in the
range < + d . Important properties of the FDF are that it is non-negative,
it integrates to unity over -space, and the local spatially-filtered value of any func-
tion of can be expressed as an integral over the FDF. These are analogous to the
corresponding properties of a PDF. However, there are important differences be-
tween FDFs and PDFs. The FDF varies on length scales down to , versus the
turbulence integral scale lT in the case of the PDF; normally is smaller than lT .
The FDF varies in time even for statistically-stationary flows, and in all three spatial
coordinates even for flows having one or more directions of statistical homogeneity.
The FDF is a random quantity. And the PDF is the expected value of the FDF in the
limit as the filter width shrinks to zero.
FDF equations can be derived following procedures that are similar to those used
to derive PDF equations, and the resulting equations have essentially the same struc-
ture as PDF equations [25]. There are two principal differences: mean quantities in
the PDF equations are replaced by spatially-filtered quantities in the FDF equa-
tions; and the mean-quantity-based turbulence scales in modeled PDF equations are
replaced by sub-filter-scale turbulence scales in modeled FDF equations. As is the
case for PDF methods, the single most compelling reason for pursuing FDF methods
is that the chemical source terms (and other important one-point processes, includ-
ing radiative emission - Sec. 6.3.4) remain in closed form even as the dynamics of
the small scales are removed by spatial filtering. For example, the filtered chemical
source term  S is closed in terms of the VCF-FDF, the VC-FDF, or the C-FDF.
134 Haworth & Pope

6.4.2 Equivalent Representations, Models, and Algorithms

The Lagrangian notional particle and Eulerian field representations for FDF meth-
ods are essentially the same as the those introduced earlier for PDF methods. Local
mean values that appear in PDF formulations are replaced by local spatially-filtered
values in FDF formulations. And in constructing models, the specification of tur-
bulence scales needs to be modified to account for the fact that, in FDF methods,
the models are responsible only for the sub-filter-scale fluctuations about the local
spatially-filtered values. The appropriate choice for a turbulence length scale nor-
mally is one that is proportional to the local filter width . With a suitable estimate
for the sub-filter-scale turbulence kinetic energy k , or for the apparent sub-filter-
scale turbulence viscosity , the local sub-filter-scale turbulent frequency is
1/2
taken to be proportional to k / or to / 2 . In most LES-FDF modeling stud-
ies to date, a composition FDF method has been used with a constant-coefficient or
dynamic Smagorinsky sub-filter-scale turbulence model. Gradient transport models
have been invoked for sub-filter-scale turbulent transport. And simple mixing mod-
els (e.g., IEM) have been used, since the models only need to deal with the local
sub-filter-scale fluctuations; the latter are expected to small compared to the fluctu-
ations about the local mean. Exceptions are the works of Givi and coworkers, who
have been developing VC-FDF [66] and VCF-FDF [67] methods.
In LES, there are likely to be regions of the computational domain where the flow
is locally fully resolved. Therefore, it is important to formulate the sub-filter-scale
models so that they recover the DNS limit. This has implications for the modeling
of molecular transport. McDermott and Pope [43] present a set of criteria for LES-
FDF models that differ somewhat from the criteria that are applied in PDF methods.
The resulting models reduce appropriately to DNS in the limit of vanishing filter
width, and are able to accommodate differential molecular diffusion.
The fluctuations about local spatially-filtered values in LES can be expected to be
smaller than the fluctuations about local mean values in Reynolds-averaged simula-
tions. For that reason, fewer particles per cell have been used in some hybrid LPEM
FDF methods, compared to PDF methods. On the other hand, LES is inherently
three-dimensional and nonstationary; this limits the extent to which spatial and/or
time averaging can used to reduce statistical error. Higher-fidelity Eulerian CFD al-
gorithms normally are used for LES compared to Reynolds-averaged simulations,
and commensurately greater care is needed in the construction of the numerical
algorithms and coupling strategies for FDF methods compared to PDF methods.
For example, the consistency issues that were raised in Sec. 6.3.2 often have been
ignored in PDF modeling studies that have been reported to date. It is likely that
LES-FDF that does not explicitly deal with these consistency and robustness issues
will be unacceptably inefficient and inaccurate, or will fail altogether.
PDF Methods 135

6.4.3 An Alternative Interpretation

Both conceptual and pragmatic issues have been raised concerning the use of
spatially-filtered velocities and filtered density functions as the basis for LES
[22, 52, 61]. For example, the filtering approach ignores the fact that there is a
distribution of physical velocity fields that corresponds to a given filtered velocity
field. It is not clear how statistics of filtered fields are related to statistics of the
corresponding physical fields, and this gives rise to ambiguities in making quantita-
tive comparisons between LES results and experimental and/or DNS data. And the
usual filter-based approaches behave inappropriately in cases where the flow locally
is fully resolved (the DNS limit) and near walls. The latter shortcoming is illustrated
in Fig. 6.4. In Fig. 6.4a, an error function (x) of width = 0.1 is plotted; this rep-
resents a scalar profile in a laminar diffusive layer. The filtered profile  (x) (ob-
tained using a top-hat filter of width = 1) also is plotted. In Fig. 6.4b are two cor-
responding variances: the second central moment of the FDF,  2 (x)  (x)2 ,
and the filtered square of the residual, ( (x)  (x) )2  . In this laminar case,
with no physical fluctations, filtering smears the profile and generates a nonphysical
variance.

Fig. 6.4: An illustration of the effects of spatial filtering on a laminar diffusive layer.
a) (x) and  (x) . b)  2 (x)  (x)2 and ( (x)  (x) )2  .

The concept of redefining LES fields in terms of conditional means originated

with Fox [22]. This idea has been developed and expanded substantially in a re-
cent paper by Pope [62] in terms of self-conditioned fields. The essence of the
self-conditioned-fields (SCF) approach is to introduce a mean velocity field that is
conditioned on quantities that can be deduced from the conditioned mean velocity
field itself; discrete values of the usual filtered velocity on a grid are appropriate
conditioning variables, for example. This bootstrapping approach provides an al-
ternative foundation for LES that resolves many of the key issues that arise with
filtering, and is readily extended to a self-conditioned composition PDF method for
turbulent reacting flows [62]. Compared to filtering, SCF explicitly account for the
136 Haworth & Pope

distribution of turbulent velocity fields, and behave appropriately in the DNS limit
and near walls. Moreover, they resolve the commutation issues that arise with fil-
tering, and the complications that arise from repeated application of spatial filters
(e.g., filtering a filtered field does not recover the original filtered field). The SCF
corresponding to (x) in Fig. 6.4 is identical to (x), and the SCF variance is zero.
In contrast to LES-FDF methods, self-conditioned fields provide a rigorous PDF-
based approach for LES of turbulent reacting flows. The approach is distinct from
LES-FDF, and will be denoted as an LES-PDF method. Equations for the self-
conditioned mean velocity and for the self-conditioned composition PDF are derived
in [62], and some initial proposals for modeling are provided there: the models are
essentially conventional sub-filter-scale models that have been modified to recover
correct limiting behaviors (e.g., the DNS limit). At the time of this writing, the prac-
tical implications for modeling and numerical implementation have not been fully
explored. Nevertheless, it is anticipated that this more precise treatment may lead to
new insights and to advances in physical modeling for LES of both nonreacting and
reacting turbulent flows.

6.4.4 Examples

Two recent examples of FDF modeling studies are discussed to illustrate the im-
portant physical phenomena that can be captured, and the different numerical ap-
proaches that are being developed.

Fig. 6.5: Computed (lines) and measured (symbols) conditional mean CO and H2
mass fractions at two different axial locations for flame E. This is Fig. 7 of [64].
Reprinted with permission from the Combustion Institute.

A series of piloted methane-air nonpremixed jet flames has been targeted for
systematic investigation of turbulence-chemistry interactions by the International
Workshop on Measurement and Computation of Turbulent Nonpremixed Flames
[3]. Sandia flames D, E, and F exhibit increasing levels of local extinction as the
PDF Methods 137

fuel-jet and pilot velocities are increased (that is, as the Damkohler number is de-
creased); flame D exhibits little local extinction, while flame F is on the verge of
global extinction. In [64], a hybrid finite-volume/particle FDF method was applied
to flames D and E. The model featured a 16-species chemical mechanism, a dynamic
Smagorinsky model for sub-filter stresses, gradient transport for sub-filter-scale tur-
bulent scalar fluxes, and a dynamic formulation to obtain the scalar time scale that
is needed in the IEM mixing model. Computational meshes of approximately one
million cells were used with 40 particles per cell. Computed radial profiles of mean
temperature and mean major species mass fractions were generally in good agree-
ment with experimental measurements (not shown). Computed conditional mean
CO and H2 mass fractions for flame E show some deviations from the measure-
ments for stoichiometric-to-rich mixtures close to the fuel nozzle (Fig. 6.5), but the
overall level of agreement is quite encouraging for this flame with strong local ex-
tinction and reignition.
In the second example, a C-FDF method is applied to the same vitiated coflow
configuration that was considered in Sec. 6.3.5. Here a stochastic Eulerian field
method is used with a detailed chemical mechanism, a Smagorinsky sub-filter-scale
turbulence model, gradient transport for scalars, and IEM mixing [33]. Sensitivities
to the number of stochastic fields and to the coflow temperature were studied, and
strong sensitivity of computed lift-off height to coflow temperature was noted, as
before. Sixteen stochastic fields and a coflow temperature of 1035 K were used
in most of the simulations. Computed and measured scatter plots of temperature
versus mixture fraction at an axial location near the flame base are shown in Fig. 6.6.
A wide range of states between pure mixing and complete combustion is evident,
including locally fuel-rich burning; these features are captured well by the model.
When sub-filter-scale fluctuations are neglected, this structure is not captured (not
shown).

Fig. 6.6: Computed (left) and measured (right) scatter plots of temperature versus
mixture fraction near the flame base. This is Fig. 23 of [33]. Reprinted with permis-
sion from the Combustion Institute.
138 Haworth & Pope

6.5 Summary and Conclusions

Significant progress has been made in PDF methods since 1985 [55]. At that time,
PDF methods were an academic research tool. Today they remain a powerful tool
for science discovery, but they also have become a mainstream modeling approach
for laboratory-scale flames, they have been implemented in multiple research and
commercial CFD codes, and they are being applied with increasing frequency to
practical combustion devices. PDF methods have been applied primarily to reacting
ideal-gas mixtures using single-turbulence-scale models. However, multiphysics,
multiscale information is readily incorporated. And while most applications to date
have been to atmospheric-pressure, laboratory-scale, statistically-stationary, non-
luminous, nonpremixed flames, PDF methods can be applied to high-Damkohler-
number systems as well as to low-to-moderate-Damkohler-number systems, to pre-
mixed systems as well as to nonpremixed and partially premixed systems, and to
practical combustion devices as well as to laboratory-scale flames. An up-to-date
(2010) review can be found in [25].
PDF methods offer compelling advantages for modeling chemically reacting tur-
bulent flows. They resolve the most important closure problems that arise from aver-
aging or filtering the highly nonlinear chemical source terms, and terms that corre-
spond to other one-point physical processes (e.g., radiative emission). PDF methods
provide a rational framework for turbulent combustion modeling, where the best
available gas-phase chemical mechanisms, soot models, and radiation models (as
established in canonical nonturbulent configurations) can be carried directly into
turbulent flames with minimal further approximations or simplifications. Complex
interactions among hydrodynamic turbulence, gas-phase chemistry, soot, and ther-
mal radiation then can be captured in a natural and direct manner. For these reasons,
PDF methods have been successful where other approaches have not. Examples in-
clude their ability to capture strong turbulence-chemistry interactions even in flames
with strong local extinction/reignition, and their ability to capture strong turbulence-
radiation interactions in luminous flames.
The somewhat unconventional stochastic methods that have been developed to
solve modeled PDF transport equations have slowed their widespread adoption.
However, stochastic methods continue to gain traction in computational physics and
engineering as their benefits are recognized: they are a powerful approach for ac-
commodating the important nonlinear couplings across widely disparate scales that
characterize turbulent combustion and other multiscale, multiphysics systems. Sig-
nificant advances in numerical algorithms have been made over the past decade, and
this is an area where substantial further progress is anticipated. This includes hybrid
Lagrangian particle/Eulerian mesh methods, and alternatives such as Eulerian field
methods.
The trend from Reynolds-averaged turbulent combustion modeling toward LES
is expected to continue. PDF-based methods have been extended to LES via the
FDF, and alternative LES-PDF formulations are being developed. Essentially sim-
ilar physical models and numerical algorithms are used in LES as in Reynolds-
averaged simulations. It has been argued that relatively simple physical models (e.g.,
PDF Methods 139

mixing models) may suffice for LES, because the models only need to represent the
unresolved (residual) component of the turbulent fluctuations.
Likely directions for next-generation clean and efficient combustion systems in-
clude higher pressures, lower temperatures, extremely lean and/or dilute mixtures,
and different fuels including reactant mixtures with high levels of H2 , O2 , syngas
(CO and H2 ), and/or exhaust-gas recirculation. The importance of finite-rate chem-
istry and turbulence-chemistry interactions and of participating-medium radiation
and turbulence-radiation interactions will increase in such systems, compared to
current lean-to-stoichiometric conventional hydrocarbon/air systems. PDF methods
are uniquely suited to meet these challenges, and it is anticipated that they will be
adopted more broadly through the 21st century.

Acknowledgements The authors are grateful for support from DOE, NASA, NSF, CD-adapco,
and GM. The contribution of SBP was supported by the Air Force Office of Scientific Research
under Grant No. FA-9550-09-1-0047. We thank Prof. W.P. Jones (Imperial College, London, UK)
for providing original files for Fig. 6.6.

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Chapter 7
Multiple Mapping Conditioning: A New
Modelling Framework for Turbulent
Combustion

M.J. Cleary and A.Y. Klimenko

Abstract Multiple mapping conditioning (MMC) is a relatively new addition to

the list of models for turbulent combustion that unifies the features of the proba-
bility density function, conditional moment closure and mapping closure models.
This chapter presents the major concepts and theory of MMC without the detailed
derivations which can be found in the cited literature. While the fundamental basis
remains the same, MMC ideas have undergone considerable evolution since they
were first proposed and the result is a generalised combustion modelling framework
which can more transparently and simply incorporate the major turbulence models
which have been developed over the past decades including LES. A significant part
of this chapter is devoted to a review of the published MMC applications comparing
model predictions with DNS and experimental flame databases. Finally, the chapter
concludes with a list of some of the advances in MMC methodology that we can
expect to see in the coming years.

7.1 Introduction

Multiple Mapping Conditioning (MMC), introduced by Klimenko and Pope in

2003 [21], is a theoretically rigorous combination of the Probability Density Func-
tion (PDF) [17, 40] and Conditional Moment Closure (CMC) [20] models incor-
porating a generalisation of mapping closure [7, 41]. The mapping closure is gen-
eralised in the sense that assumptions are not made about the type of flow being
modelled, whereas conventional mapping closures for combustion (e.g. amplitude
mapping closure [7]) are formally valid in homogeneous turbulence only. Rather
than being a specific turbulent combustion model, MMC can be viewed more as
a framework for turbulent combustion modelling. This framework contains a gen-

The University of Queensland, School of Mechanical and Mining Engineering, St Lucia, Queens-
land, 4072, Australia, e-mail: m.cleary@uq.edu.au; a.klimenko@uq.edu.au

T. Echekki, E. Mastorakos (eds.), Turbulent Combustion Modeling, 143

Fluid Mechanics and Its Applications 95, DOI 10.1007/978-94-007-0412-1 7,
Springer Science+Business Media B.V. 2011
144 M.J. Cleary and A.Y. Klimenko

eral set of principles and equations from which specific MMC based models can be
formulated to suit a particular turbulent combustion problem.
Of all the models available, the PDF models (reviewed in Chapter 6) provide
the most detailed information about the stochastic characteristics of all species in-
volved in a combustion process, and, most importantly, they permit an exact evalu-
ation of the reaction rates. However, realistic chemical processes involve hundreds
of species, ns , and the differential equations which describe those realistic kinetics
are always stiff. Therefore a direct evaluation of the joint composition PDF is ex-
pensive as it requires the solution of equations in that ns -dimensional composition
space. While the complete composition space in a turbulent flow is indeed highly
dimensional, it is not necessary in a practical model to allow all species to fluctuate
in all ways. There are constraints due to conservation of elements and other conser-
vation principles, there are fast reactions of some species forcing them to be close
to their partial equilibrium states, and (simply stated) fluctuations of some species
are unimportant to the major combustion processes [21]. This concept has lead to
the notion of an nm -dimensional reduced manifold (nm < ns ) onto which the full ns -
dimensional composition space is projected. The dimension of the manifold should
be commensurate with the effective dimension of the accessed region in composition
space for the flow under consideration and this can be expected to increase with flow
complexity [45]. From this manifold notion alternative modelling approaches have
evolved. One approach, of which the intrinsic low-dimensional manifold (ILDM)
method [32] is the prime example, involves dimension reduction by systematically
reducing the number of species in the chemical kinetics scheme. This reduced num-
ber of species then defines the low-dimensional manifold to which the eliminated
species have a functional relationship. In a numerical implementation the source
terms for the manifold species are determined from the reduced kinetics and the
eliminated species may be tabulated. While a deficiency of ILDM is that it neglects
turbulent mixing effects in obtaining the low-dimensional manifold, related kinet-
ics reduction methods such as the reaction-diffusion manifold method (REDIM) [5]
explicitly address chemistry-transport coupling. Ren and Pope [46] review ILDM,
REDIM and related methods.
A second modelling approach to evolve from the notion of reduced manifolds is
to retain arbitrarily detailed chemical kinetics schemes (i.e. with ns species) but de-
rive transport equations which effectively restrict the compositions to a certain man-
ifold. The primary example of such models in the recent literature is CMC [20] (re-
viewed in Chapter 5) which is founded on the hypothesis that, in non-premixed
combustion, there is a strong correlation between turbulent fluctuations of reactive
scalars and the fluctuations of the mixture fraction. Flamelet models [39] (reviewed
in Chapter 3) exploit this correlation also and often include parameterisation by
the scalar dissipation effectively creating a two-dimensional manifold. In CMC,
the mass fractions of the reactive scalars are conditionally averaged on the mix-
ture fraction leaving an equation which has only a single composition dimension
(i.e. mixture fraction space) in addition to the dimensions of time and space. Simple
first-order closures can be found for the conditional chemical reaction rates by mak-
ing the assumption that the conditional fluctuations (i.e. the fluctuations of reactive
Multiple Mapping Conditioning 145

species concentrations relative to mean concentrations conditioned on the mixture

fraction) are negligibly small. As a result of its low dimensionality and the simplicity
of the source term closure, the cost of first-order CMC computations is much lower
than the cost of PDF computations. In partially- or fully-premixed combustion, or in
non-premixed combustion with strong local extinction, the first-order CMC closure
of chemical source terms is less accurate and CMC models with higher order reac-
tion rate closures [26, 33] and CMC models with an additional temperature related
conditioning variable [27] have proven successful in some of these cases but they
also introduce additional terms which are difficult to model.
MMC effectively unifies the PDF and CMC approaches and allows for all of
the generality of PDF methods while also exploiting some of the advantages of
CMC. Both deterministic and stochastic MMC formulations exist. Deterministic
MMC is the natural extension of CMC and the stochastic MMC is in fact a com-
plete joint PDF method with MMC playing the role of a mixing model which en-
forces localness within a defined manifold. Since the basic MMC framework was
first proposed [21] a number of specific MMC models have been developed and
tested [8, 10, 11, 29, 5457]. The key feature common to all is the use of ref-
erence variables which are related to the physical quantities in turbulent combus-
tion (e.g. mixture fraction, sensible enthalpy, velocity and scalar dissipation). In the
original form of MMC [21] the reference variables are used as conditioning vari-
ables which form a manifold constraining the computed compositions. Fluctuations
around quantities conditionally averaged on that reference space are considered to
be small and in the basic form of MMC they are neglected for the purposes of
calculating reaction rates. These fluctuations are specific to MMC and are called
minor fluctuations. Later a more generalised interpretation of reference variables
in stochastic MMC emerged [23], whereby reference variables can take other roles
in addition to conditioning such as emulating scalar dissipation fluctuations. This
generalised interpretation allows fluctuations relative to the reference manifold to
be exploited (rather than minimised) so as to better model the physical conditional
fluctuations while keeping computational cost small.
The remainder of this review chapter is organised as follows. Section 7.2 presents
the basic MMC as it was first proposed [21]. It includes a description of the context
and concepts of MMC, a brief explanation of mapping closures, a presentation of the
deterministic and stochastic model equations, a discussion of the qualitative prop-
erties of the model and a brief discussion on the replacement of reference variables
resulting in equivalent MMC models with alternatively distributed and physically
meaningful reference spaces. Section 7.3 deals with the generalised interpretations
of MMC. Here we discuss MMC with conditioning and non-conditioning reference
variables and in the context of large eddy simulations. Section 7.4 reviews the pub-
lished MMC applications for a range of homogeneous and inhomogeneous reacting
flows. Finally in Section 7.5 we summarise the major features of MMC and the
different forms the model can take and suggest areas of research which we believe
could dominate the coming decade of research in the field.
146 M.J. Cleary and A.Y. Klimenko

7.2 The Basic MMC Framework

7.2.1 Context and Concepts

The multidimensional reacting scalar space Y (xx,t) = (Y1 , . . . ,YI , . . . ,Yns ) is governed
by the well known transport equation

YI
YI ) = wI .
+ ( vYI ) ( D (7.1)
t
Here v = v(xx,t) is the fluid velocity, D is the diffusivity which is assumed to
be the same for all species, is the density, and wI is the rate of creation of
species I due to chemical reactions. In turbulent flows the stochastic distribution
of compositions can be represented by the Favre one-time, one-point joint PDF,
PY (yy; x ,t) = nI=1
s
(yy) (yI YI )/ where the lower case y denotes the sample
space for Y , and the tilde and overline denote Favre and conventional averages,
respectively. In high Reynolds number flows the transport equation for PY derived
from (7.1) is given by [40]

PY WI PY 2 NIJ PY

+ ( uPY ) + + =0 (7.2)
t yI yI yJ
where

u(yy; x ,t)  v|Y

Y = y /Y (7.3)

WI (yy; x ,t)  wI |Y
Y = y /Y . (7.4)

YI YJ
NIJ (yy; x ,t)  D Y = y /Y
|Y (7.5)
xk xl

Y (yy; x ,t)  |Y
Y = y . (7.6)

The fundamental assumption of MMC is that the compositions which occur in

the different realisations of the flow are confined to an nm -dimensional manifold
within the ns -dimensional composition space where nm < ns . The nm species in
this manifold are called major species and their turbulent fluctuations are called
major fluctuations while the term minor species refers to the remaining set of
n = ns nm species. The word species is interpreted to include chemical species
and also other quantities related to the composition such as mixture fraction and
enthalpy. Furthermore, our terminology of major and minor species does not imply
that those species are present in large and small concentrations. A major species is
denoted by Yi (lower case Roman subscript) and Y (lower case Greek subscript) is
used to denote a minor species. The set of all major species is denoted as Y m and
Multiple Mapping Conditioning 147

the set of minor species is Y . As will be seen when the transport equations are
presented, MMC does not discriminate in its treatment of major and minor species
and this demarcation has been introduced as a conceptual aid only. Major species
are those which are permitted in the model to fluctuate in any physically realisable
way whereas minor species are restricted to fluctuate only jointly with the major
species and can be fully characterised by mean values conditionally averaged on
the major species. Provided that the major species are properly selected then the
joint PDF of all species can be effectively replaced by the marginal PDF of major
species, PY m (yym ; x,t), supplemented by the conditional means of the minor species
Q (yym ; x,t) = Y |Y Y m = ym  such that

Q y ).
PY = PY m (Q (7.7)

The reduced PDF of major species and conditional expectations of minor species
are governed by

PY m Wi PY m 2 Ni j PY m

+ ( uPY m ) + + =0 (7.8)
t yi yi y j

and
Q Q 2 Q
Q +Wi
+ u Ni j = W . (7.9)
t yi yi y j
From the assumption that the nm -dimensional major species manifold effectively
describes the accessed region in the ns -dimensional composition space, terms of the
form of PY1 Y m ), which would normally appear in CMC, have
  Y m = y m P
m (u Y |Y
been omitted from (7.9). The double prime denotes conditional fluctuations.
Like any model for the joint scalar PDF, (7.8) and (7.9) contain two unclosed
terms the conditional velocity, u, and the conditional scalar dissipation, Ni j . The
development of good closure models, particularly models for Ni j , is an ongoing area
of research in both the joint PDF and CMC communities [17, 50]. As described in
the next section MMC employs an indirect approach based on a generalisation of
mapping closure to solve the above equations consistently and in a numerically
stable manner.

7.2.2 Mapping Functions

The mapping closure concept was first introduced by Chen et al. [7] and a detailed
description of mapping closures for turbulent combustion is provided by Pope [41].
Only the main concepts are repeated here. We introduce the nr -dimensional set of
random variables = (1 , . . . , i , . . . , nr ) called the reference space whose distribu-
tion is prescribed and represented by the joint reference PDF, P ( ; x ,t). The aim
is to find a set of mapping functions X ( ; x ,t) = (X1 , . . . , XI , . . . , Xns ) such that X is
statistically equivalent to Y . In basic MMC a reference variable is assigned to em-
148 M.J. Cleary and A.Y. Klimenko

ulate the turbulence of each of the major species and therefore nr = nm . The word
emulate implies that reference variables do not model the physical scalars directly,
but that there is statistical equivalence between the mapping functions in that ref-
erence space and the physical scalar field. Statistical information about Y is then
obtained simply from the modelled mapping functions and the prescribed reference
space joint PDF:

YI (xx,t) = XI P d


Y x ,t) =
I 2(x (XI YI )2 P d (7.10)

..
.

PY (yy; x ,t) = X y )P d .
(X

The concept of mapping functions may be more readily understood by examining

the case of a single inert major scalar YZ (t) (the mixture fraction) whose mapping
XZ ( ,t) is the function of a single standard Gaussian distributed reference variable

with mean   = 0 and variance  2  = 1. Figure 7.1 shows the time evolution of
the mixture fraction PDF, PZ ( ;t) (where is the sample space variable for YZ )
and the corresponding mapping function XZ in a homogeneous field with decaying
turbulence [8]. The field is initialised at t = 0 (t is a normalised time) so that
fuel (YZ = 1) and air (YZ = 0) are mostly segregated with only a small amount of
smoothing between the two components. This corresponds to PZ being close to two
delta functions and XZ is close to a Heaviside function in reference space with XZ =
0 for < 0 and XZ = 1 for = > 0. As time evolves PZ approaches a Gaussian
distribution with a mean of YZ = 0.5 and with decaying variance. As there is always a
linear mapping between any two Gaussian distributed fields XZ approaches a straight
line with a decreasing gradient.

7.2.3 The Deterministic MMC Model

The MMC model equation governing the evolution of mapping functions in time,
and physical and reference spaces is [21]

XI XI 2 XI
+ U XI + Ak Bkl = WI . (7.11)
t k k l
This equation is valid for general inhomogeneous flows. From the assumption that
minor fluctuations are negligibly small a first-order closure of the conditional re-
X ). Equation (7.11) introduces the condi-
action rate is applied such that WI = WI (X
Multiple Mapping Conditioning 149

Fig. 7.1: Mixture fraction PDF, PZ ( ) (left) and mixture fraction mapping function,
XZ ( ) (right) in homogeneous decaying turbulence at various normalised times.
Symbols are DNS data and lines are MMC model results. Figures adapted from [8].

tional velocity, drift and diffusion coefficients U( ; x ,t), Ak ( ; x ,t), and Bkl ( ; x ,t),
respectively, whose closures are discussed below. We remind readers that the upper
case Roman subscript I represents all scalars (both major and minor) while the lower
case Roman subscripts k and l are for the major scalars only. An elegant aspect of
MMC is that, despite the conceptual division into major and minor species, a sin-
gle equation governs the transport of all species without discrimination. Mapping
functions which satisfy the deterministic model equation (7.11) are themselves de-
terministic functions and the stochasticity of the modelled scalar field results from
the stochasticity of the reference field, , whose one-point, one-time joint PDF must
satisfy the equation [21]

P Ak P 2 Bkl P
+ ( UP ) + + = 0. (7.12)
t k k l
There is not space in this review to demonstrate the compliance of the MMC
model equations (7.11) and (7.12) with equations (7.8) and (7.9) for PY m and Q .
However it is explicitly demonstrated by Klimenko and Pope [21]. We note that
this compliance is not restrictive on the dimensions involved except the trivial re-
quirement that nr ns . If nr = ns is selected then MMC is a full joint PDF model
with a generalised mapping closure for the conditional scalar dissipation. Note that
solution of (7.11) for nr >> 1 via finite difference methods is likely to be com-
putationally intractable and the efficient stochastic form of the equations presented
in Section 7.2.4 is recommended. MMC reference variables can be selected to em-
ulate fluctuations induced by variations of the mixture fraction and other scalars,
dissipation, velocity and, in principle, any other useful quantity [21]. Although a
poor selection of reference variables does not render MMC invalid it brings addi-
150 M.J. Cleary and A.Y. Klimenko

tional complexity without improving the quality of the modelling. From its roots
in CMC, practical MMC tends to focus on mixture fraction conditioning and then
on conditioning by other quantities. The CMC limit of MMC is achieved when a
single reference variable (nr = 1) is chosen to emulate the mixture fraction. Under
these conditions MMC is effectively first-order CMC plus a consistent closure of
the mixture fraction PDF and conditional mean scalar dissipation.
The MMC velocity, drift and diffusion coefficients are selected so that (7.12)
is satisfied for a prescribed (or independently determined) distribution of the joint
reference PDF, P . It is possible to determine the coefficients for any reasonable
distribution, however, following the mapping closure convention the reference vari-
ables are given standard Gaussian distributions (i.e. zero mean and unit variance):

P ( ; x ,t) = P ( ) = G(1 )G(2 ) . . . G(nr )

(7.13)
 
1 k2
G(k ) = exp .
2 2

For this reference PDF distribution, it can be shown [21] that (7.12) is satisfied when
the velocity and drift coefficients are selected to take the following forms:
(1)
U( ; x ,t) = U(0) + Uk k (7.14)

Bkl 1 (1)
Ak ( ; x ,t) = + Bkl l + ( U k ). (7.15)
l
A linear conditional velocity model is commonly used in CMC models [20] and is
also suggested for MMC. There is further discussion on this matter below. The last
term in (7.15) is particular to MMC and does not appear in conventional mapping
closures [41]. Because of that term, MMC as a PDF model is a generalised mapping
closure method that makes no assumption about the homogeneity of the flow. In
addition, of course, MMC is a CMC model for the minor species.
(1)
The velocity terms U(0) and Uk depend on the model employed for the drift
coefficient, Bkl . Selecting Bkl to be independent of gives

U(0) (xx,t) = 
v (7.16)

Uk (xx,t)k Xi  = v
(1)  
Yi (7.17)


Xi X j i j .
Bkl (xx,t) =N (7.18)
k l

In the above the averages in angular brackets are determined by integration weighted
by the reference PDF and N i j is the unconditional Favre-averaged scalar dissipa-
Multiple Mapping Conditioning 151

tion. Note that these relations are the interactions between the turbulence and re-
acting scalar fields. The great advantage of employing mapping closure is that the
turbulence-chemistry interactions are closed using the unconditional Favre-averaged
velocity, turbulent scalar flux and scalar dissipation rather than the more difficult to
model conditional averages required to close joint PDF and CMC equations (7.8)
and (7.9).
Compliance with the reference PDF does not imply that the MMC coefficients are
unique and multiple different forms are possible. The velocity U , expressed in (7.14)
in terms of , x and t, represents a model for conditional velocity v|Y
Y , which can
conventionally be approximated by v|YZ  where YZ is the mixture fraction. The
simplest, linear approximation of the conditional velocity v|YZ  =  v + YZ , where
the coefficient is linked to the turbulent scalar flux as in (7.17), is commonly
used in CMC [20]. Kuznetsov and Sabelnikov [30] introduced this approximation
and found that the joint Gaussian distribution of the velocity-scalar fluctuations im-
plied by the linear model may be too strong an assumption at the tails of the mix-
ture fraction PDF and may cause convergence difficulties. A clipped version whose
shape is an erf-like function is more stable and agrees with experiments since
larger scalar and velocity fluctuations tend to be less correlated than smaller fluc-
tuations. The same authors [30] note that the linear approximation does not yield
the same turbulent diffusivity for the first and the second moment of the mixture
fraction (Dt2 = Dt1 ). While Kuznetsov and Sabelnikov [30] believed that the dif-
ferent moments should have different turbulent diffusivities, the most common ap-
proach in RANS involves assuming the same diffusivities for both moments (i.e.
Dt2 = Dt1 ). Mortensen [37] correctly pointed out that using the linear approxima-
tion of the conditional velocity in CMC simulations is inconsistent with using the
assumption Dt2 = Dt1 in the scalar variance equation. This, of course, should not
be interpreted as a general inconsistency of the PDF and second moment equations
since a consistent equation for any scalar moment is a consequence of, and can be
derived from, the PDF equation. The problem of consistency in the common as-
sumption Dt2 = Dt1 can be resolved by using Popes gradient approximation [40]
which, as was repeatedly noted [30, 37], yields the same turbulent diffusivities for
all moments. This approximation, however, tends to have a shape which is a tan-
like function and may overestimate velocity-scalar correlations at the tails. On one
hand, linear dependence between velocity, whose distribution is close to Gaussian,
and the reference variables, which is also Gaussian, can be expected. On the other
hand, Gaussian distributions of each of the stochastic variables does not guarantee
a joint distribution, which is needed for linear dependence of the conditional ex-
pectations. Due to boundedness of the mixture fraction and unboundedness of the
Gaussian reference variable, the dependence of v|YZ  on YZ determined by (7.14)
tends to be tan-like. This overestimates dependence at the tails and may cause
difficulties in MMC simulations. Recently, Vaishnavi and Kronenburg [53] have
suggested a method that can make MMC consistent with any adopted approxima-
tion for v|YZ . It seems that this method may become important for inhomogeneous
MMC simulations.
152 M.J. Cleary and A.Y. Klimenko

7.2.4 The Stochastic MMC Model

Following the methods described by Pope [40] a stochastic form of MMC can be
derived which is equivalent to the deterministic model given by Eqs. 7.11 and 7.12.
The stochastic formulation is based on the use of Lagrangian particles. In addition
to conventional scalar properties, stochastic MMC assigns reference values to those
stochastic particles. The equivalent stochastic formulation of MMC is given by [21]
(p) (p)
dxx(p) = U( ; x ,t)dt (7.19)

(p) (p) (p) (p)

d k = A0k ( ; x (p) ,t)dt + bkl ( ; x (p) ,t)d l (7.20)

(p) (p) (p)

dXI = (WI + SI )dt (7.21)

SI | = , x = x  = 0 (7.22)

where

2 Bkl P
A0k = Ak + (7.23)
P l

bki bli = 2Bkl . (7.24)

In the above asterisks indicate stochastic quantities, the bracket index (p) indicates
a value associated with an individual particle and I are Wiener processes. As for
the deterministic model, the reference PDF, P , is prescribed and (7.20) is solved to
model the turbulent diffusion of scalars in the reference space.
Equation (7.21) governs transport in scalar space due to chemical reaction, WI ,
and a mixing operation, SI . The latter is not specified beyond the requirement in
(7.22) that it not alter the conditional expectations. The treatment of WI and SI de-
pends on ones interpretation of stochastic MMC. The first interpretation is that
equations (7.19) through (7.22) are an efficient stochastic numerical scheme for
solving the deterministic mapping equation (7.11). For nr
1 the stochastic form
will be cheaper to compute than a finite difference method applied to the determin-

istic equations. The goal is to find X I = XI | = , x = x  which can be shown to
satisfy (7.11) [21] and we therefore refer to this approach as the conditional inter-
(p)
pretation of MMC. We set WI X ) and the job of the mixing operator is to
= WI (X
(p) (p)
keep the minor fluctuations XI = XI X I small. Inevitably there will be some

scattering around X I and this is treated as stochastic error to be minimised by using
a large number of particles.
Multiple Mapping Conditioning 153

The alternative, probabilistic interpretation of MMC, is to consider the stochastic

(p)
values XI as models for the turbulent realisations of the composition; that is the
PDF PX = P(X X |xx = x) is the model for PY . Practically, stochastic MMC is almost
always used as a probabilistic (PDF) model and the recent trend is to imply proba-
bilistic when the term stochastic MMC is used. In general, the minor fluctuations
are still expected to be small when MMC conditioning is effective but deviations
(p)
from the reference space manifold are now permitted. We set WI X (p) ) and
= WI (X
SI is used to dissipate the usually small but not-negligible minor fluctuations. Note
that in conventional joint PDF methods the surrogate mixing models account for
the dissipation of all fluctuations, whereas in MMC the mixing operator dissipates
only the minor fluctuations and the dissipation of major fluctuations is modelled by
diffusion in reference space (see Eq. 7.20). Therefore MMC results are expected to
have a lower sensitivity than conventional PDF models to the form of the surrogate
mixing model.
In practice, the dissipation of minor fluctuations can occur only if mixing is be-
tween particles which are close in -space as demanded by (7.22) and this gives
MMC its localness. This localness effectively enforces a CMC-type closure on the
mixing model. Here the term CMC is quite general and refers to any method for ob-
taining conditional means according to (7.9). The probabilistic MMC is a full joint
PDF method which, through the mixing model, incorporates the ideas of CMC. Spe-
cific surrogate mixing models to dissipate the minor fluctuations can be formulated
in a variety of ways but the simplest models are those based on the conventional PDF
mixing models such as IEM (interaction by exchange with the mean) [15], IECM
(interaction by exchange with the conditional mean) [42] and Curls model [13] and
its modifications [19]. The IECM model can be seen as a special version of MMC-
IEM that involves conditioning only on velocity. Traditional MMC pays more at-
tention to conditioning on scalars than to conditioning on velocity and it is not clear
whether the true MMC regime can be achieved by IECM [23].
Here we present two alternatives, MMC-IEM and MMC-Curls. MMC-IEM is
represented by the mixing operator
(p)
(p) X I XI
SI = . (7.25)
s
Minor fluctuations are dissipated through adjustments of the relaxation timescale,

s , and X I is calculated within narrow (i.e. local) bins. This model requires that

the number of particles is large so that X I can be calculated accurately. In MMC-
Curls model particles p and q are paired on the basis that they are close to each other
in -space. During each interaction they have their values reset to the two-particle
(p)new (q)new (p) (q)
average XI = XI = (XI + XI )/2. As particles move randomly in -
space new particle pairs are formed as needed to maintain localness in that space.
Other two-particle interaction schemes, more sophisticated than the scheme shown
above, are possible and practical implementations (e.g. Refs [11, 56] tend to use
modified versions of Curls model [19].
154 M.J. Cleary and A.Y. Klimenko

7.2.5 Qualitative Properties of MMC

Being both joint PDF and CMC compliant, MMC inherits the qualitative proper-
ties of both. The reaction rates can be modelled by conditional means or by in-
stantaneous stochastic quantities. In the PDF limit convective transport is treated
exactly [40] while the convective transport of conditional quantities is modelled by
the local properties of the flow (see the dependence of the velocity in (7.14)). An
important outcome of MMC is that the PDF of the major scalars and the conditional
scalar dissipation are modelled consistently. Lists of desirable properties of condi-
tional scalar dissipation and surrogate mixing models have been suggested and ex-
panded by various authors [16, 44, 51]. MMC adheres to the most essential of these
properties [21, 22]: conservation of means, boundedness of scalars and their linear
combinations, linearity and independence, localness, equal treatment of all scalars,
decay of variances and relaxation to a Gaussian PDF distribution in homogeneous
turbulence. Additionally, as MMC in its stochastic form does not specify the form of
the surrogate mixing model, the option remains to develop mixing schemes which
also include, among other phenomena, the effects of Reynolds number, turbulence
length scales and reactions.
A key reason for the observed quality of MMC is the independence of the ref-
erence variables and the composition variables ensuring linearity of MMC mixing.
This independence does not, of course, imply that and Y are uncorrelated. In
fact, correlation is necessary for localisation in reference variables to be a useful
constraint. The independence does, however, imply that should be able to fluctu-
ate without taking the local and instantaneous value of Y into account. Practically,
a reasonable degree of independence of reference and composition scalar fields is
achieved when those fields are modelled by different processes or equations (e.g.
can be modelled by the Markov process (7.20) which is independent of the transport
of Y ). Note that this interpretation of independence allows for some quantities, such
as density, to be common to both equations.

7.2.6 Replacement of Reference Variables

The velocity, drift and diffusion coefficient closures in Sections 7.2.3 and 7.2.4 are
consistent with the reference PDF transport equation when that PDF is a joint stan-
dard Gaussian. This is convenient from a mathematical perspective but a better
physical understanding of MMC can be gained by replacement of these standard
Gaussian reference variables with random variables which more closely resemble
the physical major scalars that they emulate.
A reference space transformation from to = ( ; x ,t) is achieved by replac-
ing the velocity, drift and diffusion coefficients by [22]
Multiple Mapping Conditioning 155

i 2 i
A i = U i + Ak i Bkl
+U (7.26)
t k k l

i j
B i j = Bkl (7.27)
k l

U = U . (7.28)

Although the transformed velocity coefficient in (7.28) is unchanged this does not
imply the linear functional form in -space corresponds to a linear form in -space.
The new reference space PDF is given by

1
i
P = P det . (7.29)
k

Note that the replacement of variables is simply a mathematically equivalent trans-

formation that does not alter the physical nature of the MMC closures.
An obvious case is the replacement of a single standard Gaussian variable that
emulates mixture fraction, , with a new random variable = that has the same
distribution as the actual mixture fraction (i.e. P = PZ ). It is important to remem-
ber that the mixture fraction reference variable is not the actual mixture fraction,
YZ , which is modelled by the mapping XZ . To preserve the independence of the
reference variables, is a mixture-fraction-like variable with equivalent (or topo-
logically similar) statistics to YZ . For this special case the transformed coefficients
A i and B i j are

A = 0 (7.30)

 
XZ 2
B=B . (7.31)

It can be readily seen that replacement of by and substitution of the new co-
efficients into the mapping equation (7.11) yields the conditional moment equation
(7.9) with conditioning on the mixture fraction. The coefficient B appears in the
place of, and is therefore a model for, the conditional scalar dissipation
 2
XZ
N|  = B = B . (7.32)

Generally speaking, a higher quality emulation of the mixture fraction by the refer-
ence variable makes modelling of the mixture fraction YZ easier.
A detailed application of the replacement of reference variables for a multidimen-
sional reference space emulating mixture fraction and sensible enthalpy is contained
156 M.J. Cleary and A.Y. Klimenko

in Refs [9, 29]. With such transformations the similarities and differences between
the MMC and EMST [51] models are quite obvious. Both models treat conditional
scalar dissipation locally in composition space and use mapping closures to achieve
this. Where they differ is that EMST uses the stochastic compositions to determine
localness, but in so doing violates principles of independence and linearity. MMC,
on the other hand, uses reference variables which are formally independent of the
stochastic compositions to determine localness and thus it adheres to those prin-
ciples. However, MMC requires a model for the reference variables and finding a
suitable model may not be trivial, especially for reacting quantities.

7.3 Generalised MMC

The basic MMC framework presented in the preceding section is a rather formal
model. It assumes that the major species manifold is known and that minor fluctua-
tions are negligibly small. (The probabilistic MMC allows minor fluctuations but un-
til now they have been assumed to be small). The use of standard Gaussian reference
variables is conventional and mathematically convenient, but it also removes some
physical transparency from the model equations. In this section we present a gen-
eralised MMC which, as the name suggests, generalises the concepts of MMC and
makes them more amenable to practical implementation. Generalised MMC con-
cepts were first proposed in Ref. [23] to expand the purpose of reference variables
beyond conditioning or localisation. A series of subsequent papers [10, 11, 24, 25]
developed generalised MMC for the DNS/LES regime and replaced Markov refer-
ence variables with Lagrangian variables traced within an Eulerian field. Although
the main generalised MMC concepts are presented below, readers interested in a
detailed analysis should consult the published articles cited above.

7.3.1 Reference Variables in Generalised MMC

The basic MMC model uses nr = nm independent reference variables to emulate

each of the major species. In the stochastic form of the model mixing is localised in
the reference space effectively linking the modelled composition with the species
concentrations conditionally averaged on that space. In the conditional interpre-
tation of MMC minor fluctuations are neglected and therefore the composition is
modelled as the conditional mean. In probabilistic MMC, in which minor fluctua-
tions are permitted, the fluctuations are dissipated towards the conditional means by
the minor dissipation operator, SI . The probabilistic MMC interpretation is assumed
for the remainder of this section. Reference variables which perform a localisation
role (this is the only role we have considered until now) are now called conditioning
reference variables to distinguish them from other sorts of reference variables to be
discussed below. As before the total number of reference variables is labelled as nr
Multiple Mapping Conditioning 157

and the number of conditioning reference variables is nc with nc nr . The set of

conditioning variables = (1 , 2 , . . . , nc ) forms a subset of .
From a practical perspective it may not always be possible or desirable to have
a conditioning reference variable to emulate each of the major species. Limiting
computational cost is the major reason for this a greater number of conditioning
reference variables requires a larger number of particles to ensure adequate locali-
sation in the space of each of those reference variables. If nc < nm then, in general,
minor fluctuations are not negligibly small and their variances should be controlled
to so that the model predicts the physical conditional variances accurately.
By accepting minor fluctuations in the model we also create the possibility of
including reference variables which assist the modelling but which are not used
for conditioning purposes. Practically this means that mixing is localised only in
the space of the nc conditioning reference variables, while the non-conditioning
reference variables complement the conditioning reference variables and improve
the emulation of the physical quantities.
Until now we have only considered reference variables modelled by Markov pro-
cesses as in (7.20). However, once we allow for the possibility that nc < nr or even
for nc  nr , any physically relevant process can be used. For example, reference
variables can be Lagrangian quantities obtained with the use of DNS or LES. In-
deed a non-Markov process can be approximated well by a Markov process of much
higher dimension. Motions of Brownian or fluid particles in a turbulent flow are
deemed to be non-Markovian while DNS simulations tracing these particles repre-
sent a Markov process of a large dimension.

7.3.2 Features of Generalised MMC Models

The generalised MMC model equations were initially proposed in Ref. [23] in the
same form as the basic stochastic MMC equations (7.19) through (7.21) and with
(7.22) replaced by

SI | = , x = x  = 0. (7.33)

while SI | = , x = x may be non-zero. It is possible to demonstrate that a
failure to satisfy condition (7.22) under the conditional interpretation of MMC will
generate a spurious term in the modelled PDF transport equation (the effect of mix-
ing in the direction of non-conditioning reference variables can be interpreted as a
mixing-generated diffusion). Hence, using a conditional interpretation of general-
ized MMC is not recommended. Note that generalized MMC is a stochastic model
and it does not generally have a deterministic version. However, with a probabilistic
interpretation, generalised MMC remains compliant with the PDF transport equa-
tion. Indeed, complying with the PDF equation requires that SI |xx = x  = 0 to allow
for representation of the mixing operator in terms of the divergence of dissipation,
:
NIJ
158 M.J. Cleary and A.Y. Klimenko

P
NIJ X
PX SI |X
X = X , x = x = . (7.34)
XJ
In Ref. [21], this was shown using with the use of (7.22), which represents a suf-
ficient but not necessary condition. A weaker condition, SI |xx = x  = 0, which
can be obtained from (7.33), is sufficient for compliance with the PDF equa-
tion. Due condition (7.33) generalised MMC mixing does not alter the values of
X I = XI | = , x = x  and hence, in the absence of non-linear reacting terms,

X I is determined by the properties of the stochastic trajectories of and not by
the form or quality of the surrogate mixing operator. In other words, MMC enforces
the desired conditional properties, through the stochastic properties of the reference
variables, on any reasonable surrogate mixing operator. This, however, does not ap-
ply to higher conditional moments which are determined by the form and quality of
that mixing.
Since stochastic models aim to produce statistically equivalent fields their model
equations are not unique and alternative forms can be derived. MMC with Gaussian
reference variables is mathematically convenient but some physical transparency
of the model is lost. It is, of course, possible to transform the equations for al-
ternatively distributed reference variables according the methods described in Sec-
tion 7.2.6 and although this improves the physical transparency of the model the
transformed drift and diffusion coefficients are complex. An alternative option is
to apply the generalised MMC principles within other existing models for turbu-
lent combustion (e.g. any of the various formulations and closures of the joint PDF
models). In the broadest sense, then, generalised MMC can be interpreted as the
application of the conditioning/localisation condition (7.33) within an existing (or
maybe yet to be developed) stochastic combustion model. This interpretation has
been taken in practical hybrid binomial Langevin-MMC [57] and MMC-LES appli-
cations [10, 11].
The following three points summarise the essential features of a good generalised
MMC model [23]:
The conditioning reference variables should emulate as closely as possible the
Lagrangian properties of the key major species to ensure accurate evaluation of
conditional species expectations without compromising the independence of the
reference space. This can be done with the assistance of non-conditioning refer-
ence variables.
The surrogate mixing operator, SI , should set the dissipation of minor fluctuations
to correspond to the dissipation of physical conditional fluctuations. (Due to the
independence of reference and composition scalar fields, minor fluctuations and
conditional fluctuations are not the same thing but they are linked).
The conditioning reference variables should be selected so that minor fluctu-
ations are not too large. This ensures that scalar dissipation is predominantly
modelled by diffusion in reference space (e.g. Eq. 7.20) rather than by the surro-
gate mixing operator, SI .
Multiple Mapping Conditioning 159

7.3.3 MMC with Dissipation-like Reference Variables

We consider an MMC model governed by (7.19) through (7.21) and (7.33) with
a single conditioning reference variable emulating the physical mixture frac-
tion, YZ , via the mapping function XZ . If does not have a standard Gaussian
PDF then the coefficients are modified as described in Section 7.2.6. Conditional

fluctuations YI = YI YI |YZ  are modelled indirectly via the minor fluctuations
(p) (p)
XI = XI X I . As it is, the model does not explicitly generate minor fluc-
tuations and they are present only if they appear in the boundary conditions or if
generated by the surrogate mixing model [23]. Although generation of minor fluc-
tuations by the mixing model can in principle be used to model the conditional
fluctuations [10] it may be difficult to control. An alternative model is to intro-
duce additional non-conditioning reference variables to emulate the scalar dissipa-
tion fluctuations which are physically responsible for the appearance of conditional
fluctuations. The reference space is defined as = ( , d1 , d2 , . . .) where di are
called dissipation-like reference variables. The MMC model with dissipation-like
variables has the modified diffusion coefficients [23]

di d j
B = B , B di = Bdi = 0, Bdi d j = (7.35)
di

cdi cdi
(di ; x ,t) = exp(cdi di ) (7.36)
2

 
2 |Y 
N Z
cdi cdi = ln +1 (7.37)
N |YZ 

In the above, B is the value of the diffusion coefficient without inclusion of the
dissipation fluctuations. Each dissipation-like reference variable emulates scalar dis-
sipation fluctuations of a certain frequency, 1/di , where di spans between the Kol-
mogorov and macro time scales of the flow. Giving each di a standard Gaussian
distribution ensures that the conditional scalar dissipation N |  has a lognormal
distribution. For modelling where the ratio di1 /di is selected to be the same for all
dissipation-like variables it can be shown that the constants cdi are also equal [23].
In basic MMC, all reference variables must be used for conditioning while gen-
eralised MMC may involve conditioning only on , or and , which seems to be
more practical.
160 M.J. Cleary and A.Y. Klimenko

7.3.4 DNS/LES Simulated Reference Variables

The computational tractability of Markov processes such as the random walk given
by (7.20) have led to them being widely used in stochastic turbulence models [43].
However, the great advances in computing power mean that LES and maybe even
DNS are becoming more viable means of modelling non-reacting stochastic dif-
fusion processes and velocity fields. The cost of performing reacting DNS is still
prohibitive while LES does not resolve the thin reaction zones. Therefore hybrid
methods such as the LES/joint scalar FDF (filtered density function) model [12, 18]
have been developed whereby velocity and passive scalar fields are simulated by
conventional Eulerian LES and the reactive scalar field is simulated by a stochastic
particle scheme.
As noted in Section 7.3.1, the Markov reference variables can be replaced in gen-
eralised MMC by traced Lagrangian values within an Eulerian DNS or LES simu-
lated field (i.e. particle reference variable values are the Eulerian values observed at
the particle locations). One can note that the highest quality reference variable is the
actual physical variable simulated by a fully resolved DNS. The Eulerian reference
field is simulated according to

i
i ) = wi .
+ ( vi ) ( D (7.38)
t
For passive scalars the source term wi = 0. The mixture fraction is the most ob-
vious reference variable to be modelled in this way for non-premixed combustion
but, in principle, other passive or reactive reference variables could also be selected.
In the LES version, the filtered form of (7.38) is solved and a closure is required
for the source term of any reactive reference variables. If LES subgrid fluctuations
are filtered out a Markov process similar to (7.20) can be used to emulate the sub-

grid distribution such that i = iLES + i RW ; (RW = random walk) [23]. Practical
applications [10, 11] of MMC in LES tend to have far fewer Lagrangian particles
for the stochastic reacting species field than there are Eulerian LES grid cells (see
Section 7.4.4). Therefore the explicit inclusion of subgrid fluctuations in the formu-
lation of the reference variables is unlikely to have a significant effect on the de-
termination of localness in reference space. Of course the subgrid component of i
may have a significant effect on conditional velocity (according to the linear closure
given by (7.14)) or if some of the reference variables represent velocity components.
If the random walk component of i is neglected (as has been done in practical ap-
plications) the subgrid conditional velocity can instead be closed by the alternative
gradient model [12, 40] which manifests in the stochastic equation for the spatial
transport:  
1 
dxx(p) = U(0) + ( Deff ) dt + 2Deff d (p) (7.39)

where Deff is the sum of subgrid and (if needed) molecular diffusivities.
Multiple Mapping Conditioning 161

7.4 Examples

In the past five years an impressive list of publications have proposed and tested
specific deterministic and stochastic MMC (including generalised MMC) mod-
els in a range of idealised, homogeneous combustion conditions [8, 9, 29, 56]
and inhomogeneous, laboratory non-premixed and partially premixed flame con-
ditions [10, 11, 54, 55, 57]. Each of these specific MMC models has a refer-
ence variable to emulate mixture fraction while a few of the deterministic models
have additional reference variables to emulate scalar dissipation and/or sensible en-
thalpy [8, 9, 29]. In the inhomogeneous cases MMC has been coupled with RANS
based turbulence models [54, 55], with the binomial Langevin model to model the
joint velocity-scalar PDF [57] and LES to simulate the joint scalar FDF [10, 11].
The key features of these applications are summarised below.

7.4.1 MMC in Homogeneous Turbulence

7.4.1.1 Stochastic MMC

The first application of MMC to reacting flow conditions was the stochastic MMC
reported by Wandel and Klimenko [56]. Results are compared against DNS data [35]
in homogeneous turbulence with finite-rate, one-step chemistry and significant lo-
cal extinction. A single reference variable emulates the mixture fraction and, as
the probabilistic MMC interpretation is used, minor fluctuations of the single reac-
tive scalar, normalised temperature, are present. An MMC-Curls surrogate mixing
model dissipates the minor fluctuations and the mixing timescale, min , is set pro-
portional to the macro-mixing timescale, denoted by maj . Despite the simplicity of
the flow and chemistry this modelling demonstrates the ability of stochastic MMC
to capture heavy local extinction and subsequent reignition events more accurately
than other commonly used models such as CMC, fast chemistry, Curls, IEM and
EMST. The performance of IEM, Curls and EMST models is investigated in detail
in Mitarai et al. [36] for the same test conditions. It is instructive to compare the
scatter plots of temperature versus mixture fraction for those models and DNS in
Ref. [36] with the MMC scatter plots in Ref. [56]. IEM fails to produce the correct
physical behaviour as it cannot change the shape of the joint PDF from its initial
conditions and nor can it generate conditional fluctuations of temperature with re-
spect to mixture fraction. Curls model produces physically plausible compositions
for this case but as mixing is not local in composition space it significantly over-
predicts conditional fluctuations and the reignition is very slow compared to the
DNS. While EMST is local in composition space it violates the principles of inde-
pendence and linearity and in its basic form EMST can lead to stranding [51] or
mixing along certain preferential lines leading to non-physical behaviour. For the
test case described MMC produces physically realistic and accurate results.
162 M.J. Cleary and A.Y. Klimenko

Fig. 7.2: Mean (left) and conditional mean at stoichiometry (right) of temperature.
MMC (lowest to highest: min /maj = 1/1.04, 1/8 and 1/100), ; DNS data, ;
EMST, ; CMC, ; Curls model, ; IEM ; fast chemistry, . Figures adapted
from [56].

Figure 7.2 below shows the mean temperature (left) and the conditional mean
temperature at stoichiometry (right) as a functions of time for the DNS and the vari-
ous models. It can be seen that both EMST and MMC with min /maj = 1/8 predict
mean temperature very well while the other models either significantly overpredict
the rate of temperature rise (CMC and fast-chemistry) or significantly under predict
it (IEM and Curls). For the conditional temperature MMC with min /maj = 1/8 is
the most accurate model. The MMC results are, however, qualitatively and quantita-
tively sensitive to the parameter min /maj . By setting min /maj = 1/100 the model
rapidly dissipates any minor fluctuations, and hence conditional fluctuations, of tem-
perature which may be generated. Thus this MMC result closely resembles those
for first-order CMC. Alternatively by setting min /maj = 1/1.05 the model does not
dissipate minor fluctuations, and hence conditional fluctuations, fast enough and
produces results similar to Curls model. Although more research is required to de-
termine the best values min /maj for a range of practical combustion conditions,
the timescale ratio parameter appears to provide a useful mechanism for controlling
the level of conditional fluctuations which is not available in many other mixing
models. The authors [56] caution that the correct value of min /maj is unlikely to
be universal or constant with time. The variability with time is illustrated in the
results for min /maj = 1/8 which slightly underpredicts the rate of reignition (see
conditional temperature rise in Fig. 7.2). Attempts to reduce that timescale ratio
(for all time steps) in order to more rapidly dissipate conditional fluctuations in the
later stages of the evolution inadvertently leads to inaccuracy during the initial ex-
tinction phase. The creation of a model for min /maj would be advantageous but is
not trivial. Any such model would need to account for the rate of dissipation and
generation of conditional variance by the surrogate mixing model as analysed in
Ref. [23].
Multiple Mapping Conditioning 163

7.4.1.2 Deterministic MMC

The first application of deterministic MMC is, in fact, contained alongside the orig-
inal MMC derivation [21]. A three-stream, non-reactive, homogeneous mixing field
is modelled with the use of two reference variables emulating two independent mix-
ture fractions. It is shown that the joint PDF of the two mixture fractions is modelled
in a very realistic manner and results are in excellent agreement with the analytical
solution.
The first applications of deterministic MMC for reactive fields are found in a
series of three papers by Cleary and Kronenburg [8, 9, 29]. They propose and test
various deterministic MMC models against DNS [28] of homogeneous, decaying
turbulence with varying levels of local extinction (up to global extinction). It had
previously been established [28] that CMC with conditioning on the mixture frac-
tion alone was inappropriate for these flame conditions due to the importance of
conditional fluctuations which are neglected in first-order CMC. A number of pre-
vious CMC studies [6, 27, 28] identified scalar dissipation and normalised sensible
enthalpy as possible choices for a second conditioning variable for flames with sig-
nificant local extinction. In fact both quantities have an important role in the physics
of local extinction and subsequent reignition. While fluctuating scalar dissipation is
the primary instability which causes conditional fluctuations, those conditional fluc-
tuations tend to correlate better with temperature related quantities such as sensible
enthalpy, than with scalar dissipation.
The three MMC papers progress incrementally. The first [8] has reference vari-
ables emulating mixture fraction and multiple scalar dissipation-like quantities each
of which is associated with a certain dissipation timescale (see (7.35) through
(7.37)). As expected from the earlier CMC results [6], while conditioning on mixture
fraction and a single scalar dissipation variable is able to model the extinction phase
well, it cannot accurately predict the subsequent reignition phase which occurs after
the turbulence has sufficiently decayed. Any deterministic MMC inevitably forces
reference variables to be conditioning variables but at low temperatures there is
decorrelation of reactive species and scalar dissipation fluctuations [27] and hence
the assumption of negligible conditional/minor fluctuations and first-order reaction
rate closures are inappropriate. Although the MMC results improve modestly with
additional dissipation-like reference variables the model is illustrated to be unsuit-
able for a deterministic formulation. Note that dissipation-like reference variables
were initially suggested as non-conditioning reference variables in the stochastic
formulation of MMC (see Section 7.3).
The second paper [9] proposes an MMC model with reference variables emu-
lating mixture fraction and normalised sensible enthalpy. Like previous CMC cal-
culations with the same conditioning variables [28] results for reactive species are
impressive. This is because the manifold comprising of only mixture fraction and
sensible enthalpy adequately describes the accessed region in composition space.
However, the model does not have a mechanism for introducing the physical insta-
bilities (i.e. fluctuations in scalar dissipation) that cause extinctions to occur in the
first place. To overcome this deficiency the fluctuations are imposed on the model
164 M.J. Cleary and A.Y. Klimenko

using DNS data for the conditional PDF of normalised sensible enthalpy. The result
is a hybrid MMC / presumed PDF model which accurately describes the evolution
of minor scalars but which does not predict the joint PDF of the major scalars.
The third paper [29] describes an MMC model which is a novel combination
of the previous two. There is a reference variable emulating mixture fraction and a
second reference variable which emulates normalised sensible enthalpy but that is
also a dissipation-like variable which can generate the fluctuations leading to local
extinctions. It is explained that any reference variable may adopt the character of
a dissipation-like variable and that the physical quantity it emulates is irrelevant.
Through the dual-nature of the second reference variable the model exploits the
strong negative correlation between sensible enthalpy fluctuations and fluctuations
in scalar dissipation during the extinction process. Specifically (7.37) is replaced by

 1/2
2 | 
N
cd1 = fcorr ln +1 (7.40)
N | 

where the correlation function is simply the conditional normalised sensible en-
thalpy at stoichiometry fcorr = h s | = YZst . While this third model is complete
and follows the physics of the problem better than the previous two models on which
it is based, the quality of the results is mixed. For the flame case with heavy local ex-
tinction followed by reignition and another case with global extinction, predictions
of major and minor species are in very good agreement with DNS data. A particu-
larly impressive outcome is the models ability to accurately predict the bimodal dis-
tribution of sensible enthalpy in near stoichiometric mixtures as shown in Fig. 7.3.
This is in contrast to an assumed -PDF which cannot give a bimodal distribution
between arbitrary minimum and maximum sample space limits. Interestingly, for a
third test case exhibiting only moderate local extinction the model performs poorly
and noticeably underpredicts the extent of that mild extinction. This is blamed on a
realizability constraint which artificially restricts some of the diffusion coefficients
to positive values to ensure numerical stability.

7.4.2 MMC with RANS

Two papers by Vogiatzaki et al. document the implementation of deterministic

MMC into a RANS computer code and report on model performance for two lab-
oratory jet diffusion flames with complex hydrocarbon chemistry. The first pa-
per [54] reports on modelling of the DLR A and B CH4 /H2 /N2 flames [2, 34]
and the second paper [55] presents results and an expanded analysis for the San-
dia CH4 /O2 /N2 Flame D [1, 49]. In each case a single Gaussian reference vari-
able emulates the mixture fraction. As the flame cases exhibit low levels of lo-
cal extinction, conditioning on mixture fraction alone is appropriate as is estab-
lished by many past accurate CMC and flamelet computations. Figure 7.4 (from
Multiple Mapping Conditioning 165

Fig. 7.3: Conditional PDF of normalised sensible enthalpy at stoichiometry at two

different times. Figures adapted from [29].

Ref. [55]) shows that MMC predictions of the mixture fraction PDF for San-
dia Flame D closely resemble a -PDF. Close to the nozzle agreement with ex-
perimental data is excellent while the downstream discrepancies are linked to
the commonly observed underprediction of the mixture fraction variance by the
k turbulence closure. Conditional scalar dissipation does not appear explic-
itly in MMC but can be determined by (7.32) following a replacement of refer-
ence variables from to . Figure 7.5 shows conditional scalar dissipation pro-
files for Sandia Flame D conditions by MMC and two alternative closures based
on integration of the mixture fraction PDF transport equation [14] and amplitude
mapping closure (AMC) [7]. MMC reproduces the profile shapes and the loca-
tion of the peak value better than the integrated PDF method and is an improve-
ment over AMC which always gives the peak conditional scalar dissipation at
mixture fraction equal to 0.5. MMC compares quite well to the 1D experimen-
tal data but is unable to capture the slightly bimodal shape which is even more
apparent in the more accurate 3D experimental data. Although the obvious qual-
ities of MMC do not make a significant difference to reactive scalar predictions
in these simple flames (results are of similar good accuracy to those for CMC
with standard PDF and conditional scalar dissipation closures) the MMC com-
putations represent an important first step prior to application to more difficult
flame cases which require additional conditioning variables and for which simple
PDF shape presumptions and conditional scalar dissipation closures are not avail-
able.

7.4.3 MMC with the Binomial Langevin Model

A novel, hybrid model combining the binomial Langevin model [52] and stochastic
MMC was introduced by Wandel and Lindstedt [57] to model the joint velocity-
166 M.J. Cleary and A.Y. Klimenko

Fig. 7.4: Mixture fraction PDF at various locations for Sandia Flame D. Squares are
experimental data, solid lines are MMC predictions, and dashed lines are -PDFs.
Figure adapted from [55].

scalar statistics in an inhomogeneous, reacting scalar mixing layer that was investi-
gated experimentally by Saetran et al. [48] and Bilger and co-workers [3, 31]. The
hybrid approach overcomes implementation difficulties associated with producing
a bounded scalar field in the binomial Langevin context, while providing a simple
and accurate means for obtaining the MMC coefficients (calculation of terms in-
volving the gradient X/ can be difficult in stochastic MMC when there is a lot
of scatter). The hybrid model employs the principles of a generalised MMC closure.
In the MMC part of the hybrid model, rather than solving (7.20), the single condi-
tioning reference variable is instead modelled by inverting (7.14) and some other
manipulation to give
(p)
u u2
(p) = 2 . (7.41)
u
2
2
Here u2 is the dominant velocity component (in this case the transverse component)
and it is modelled by the binomial Langevin model. Note that (7.41) has the advan-
tage that it does not contain the diffusion coefficient B and thus it is not necessary
to calculate the gradients X/ . The model also has an additional pseudo mix-
ture fraction that is solved according to the binomial Langevin model. The pseudo
mixture fraction is a non-conditioning reference variable used only to calculate the
extent of mixing between particle pairs while mixing localisation is in -space
Multiple Mapping Conditioning 167

Fig. 7.5: Profiles of local conditional scalar dissipation in mixture fraction space for
Sandia Flame D. Squares are 1D experimental data, solid lines are the MMC model,
dotted lines are the integrated PDF model, and dashed lines are the AMC model.
Figure adapted from [55].

only. (We caution that the notation used in this chapter for conditioning and non-
conditioning reference variables is different to that used by Wandel and Lindstedt.)
The paper contains a detailed analysis of the model and makes extensive compar-
isons with experimental data. These indicate that the model is robust and provides a
similar level of accuracy to the binomial Langevin model by itself. While results for
mean quantities are in very good agreement with experimental data, the second mo-
ments are generally underpredicted signifying a need for future improvements such
as better control of the dissipation of minor fluctuations by the surrogate mixing
model. Due to the aforementioned elimination of implementation difficulties asso-
ciated with the binomial Langevin and stochastic MMC models the hybrid model is
reported to have a relatively modest computational cost. Application of this model
to inhomogeneous flows is underway and preliminary results are encouraging.

7.4.4 MMC with LES

Two recent papers have documented the application of MMC with LES [10, 11] for
the Sandia CH4 /O2 /N2 Flame D [1, 49]. The model is a generalised MMC with the
reference variable given by the LES filtered mixture fraction. The most compelling
168 M.J. Cleary and A.Y. Klimenko

aspect is the demonstration of a new very low-cost, sparse-Lagrangian scheme for

simulation of the joint scalar FDF, made possible due to the high quality of the MMC
mixing closure. Conventional FDF simulations employ an intensive-Lagrangian
particle scheme with many particles per Eulerian LES grid cell, and the terminol-
ogy of sparse-Lagrangian is introduced to refer to simulations with significantly
fewer Lagrangian particles for the joint scalar FDF than there are Eulerian grid cells.
The simulations of Flame D are performed for (on average) one particle for every
10 to 12 LES cells culminating in as few as 35,000 particles over the 70 jet-nozzle
diameter flow domain. This represents a two or three order of magnitude reduc-
tion in particle numbers and computational cost relative to conventional intensive-
Lagrangian FDF simulations of the same or similar flame conditions. As a result of
the very low cost the sparse-Lagrangian simulations are able to use detailed 219-step
chemistry, whereas previous FDF simulations of hydrocarbon flames have required
reduced or tabulated chemistry. (A list of recent FDF computations is compiled by
Haworth [17].)
The theoretical basis for sparse-Lagrangian simulations is established in two pa-
pers [24, 25] while Ref. [10] elaborates on the physical reasoning in support of
sparse methods and the reasons for the success of generalised MMC under such
conditions. Modelling aside, if it is assumed that a particle within the ensemble
representing the one-point, one-time FDF is statistically independent of all other
particles, then all those other particles can be removed while the one remaining par-
ticle continues to represent that FDF. That probability distribution exists whether
we have sufficient numbers of particles to determine it locally or not. From a mod-
elling perspective the problem of using very few particles is that numerical diffu-
sion (i.e. numerical bias in the Lagrangian fields) can become larger than is justi-
fied or needed if that diffusion performs a modelling role. Therefore modelling the
FDF with very few particles requires a high quality mixing model which specifi-
cally minimises numerical diffusion. MMC does this by enforcing localness in the
reference space. Whereas non-local mixing models such as IEM are successful in
intensive-Lagrangian FDF simulations by virtue of the high spatial resolution of
LES, it seems that only models (such as MMC), which enforce localness and ad-
here to the other desirable qualities of mixing models, are capable when particle
numbers are significantly reduced. It should be stressed that the finest details of the
fields are not available with so few particles but the published works [10, 11] demon-
strate that the major stationary statistics of the reacting scalar fields (i.e. conditional
and unconditional means and variances) are in good agreement with experimen-
tal data. Figure 7.6 (taken from Ref. [11]) shows radial profiles of unconditional
means and variances of temperature and mass fraction of carbon monoxide and the
hydroxyl radical. Results are in good agreement with the experimental data and
furthermore have a low sensitivity to a five-fold increase in the number of parti-
cles.
The model contains two tuneable parameters: a mixing timescale constant and a
localness parameter which determines the relative degree of localisation in reference
(filtered mixture fraction) and physical spaces. These parameters are adjusted to
control the small but not insignificant conditional fluctuations. More work is neces-
Multiple Mapping Conditioning 169

Fig. 7.6: Unconditional mean and rms for temperature, CO and OH at two down-
stream locations. Open symbols - experimental data, solid lines - MMC with nomi-
nally 35,000 particles, broken lines - MMC with nominally 175,000 particles. Figure
adapted from [11].

sary to determine the selection of such parameters for a wide range of conditions in-
cluding those with significant local extinction phenomena. For some complex flame
regimes, where conditional fluctuations relative to mean values conditioned on the
mixture fraction are very large, additional reference variables (e.g. to emulate sensi-
ble enthalpy or other related quantities) may be necessary. However, the need for a
filtered source term closure for reacting reference variables could pose a significant
challenge if this is attempted.
170 M.J. Cleary and A.Y. Klimenko

A principal problem of sparse-Lagrangian methods is related to the evaluation

of density which must be obtained from a small number of particles and coupled
to the more highly resolved LES flow field. The existing MMC publications do
not address this but rather have a tabulated density. Thus there is no coupling or
consistency between the Eulerian LES and Lagrangian FDF fields. Ongoing but as
yet unpublished research at The University of Queensland has successfully treated
the density coupling issue through a conditionally averaged form of the equivalent
enthalpy method of Muradoglu et al. [38] which ensures numerical stability and
consistency.

7.5 Summary and Future Directions

This chapter has reviewed the MMC concepts and theory, and their evolution, along
with the key specific MMC model applications since it was first proposed in 2003.
In its most basic deterministic form MMC represents a closed and consistent com-
bination of joint PDF modelling for a set of major species which describe the ac-
cessed region in composition space and conditional moment closure for the set of
minor species which fluctuate jointly with the major scalars. The closure and con-
sistency is facilitated by mapping closure which is generalised for inhomogeneous
flows. MMC has evolved from a deterministic to a stochastic method. Although an
equivalent stochastic formulation was introduced initially as a computationally ef-
ficient form of the deterministic model it also allows a generalised interpretation
where fluctuations of minor species relative to the major species manifold may be
exploited and where the reference variables are used to enforce desired properties by
conditioning/localisation in mixture fraction space (i.e. a CMC-type mixing model
closure) and emulation of scalar dissipation fluctuations. Further evolution of gen-
eralised MMC has occurred with the replacement of Markov reference variables by
traced Lagrangian quantities in Eulerian DNS or LES. The possibility also exists to
obtain reference variables by other simulation or modelling methods.
From the perspective of MMC the use of LES to provide reference variables is
expensive but this is more than compensated for through the demonstrated possibil-
ity of using a sparse distribution of particles to model the Lagrangian FDF. While
a PDF must describe the distributions of all turbulent scales an FDF need only de-
scribe the subgrid distributions while the large scale turbulence is resolved by the
LES. Therefore from the perspective of LES simulations, generalised MMC allows
high-quality, efficient simulations that are dramatically less expensive than conven-
tional intensive-Lagrangian FDF simulations or even LES with chemical source
terms modelled using the resolved quantities at the Eulerian grid centres. The con-
cept of sparse-Lagrangian simulations is associated with the FDF method and is
not coincident with the concepts of MMC, per se. Sparse simulations with closures
other than generalised MMC are certainly possible, however only generalised MMC
closures are currently known to work for sparse simulations.
Multiple Mapping Conditioning 171

The advances and challenges of MMC research in the coming years will include
(but are certainly not restricted to):
the application of various specific MMC models to a wide range of laboratory
combustion conditions and flame regimes to establish the best choice of model
parameters;
the application in inhomogeneous flows of MMC with multiple reference vari-
ables emulating more complex accessed composition spaces;
the expanded testing and development of hybrid MMC methods which borrow
ideas from other established models;
the establishment of criteria for assessing the optimal compromise between qual-
ity and economy in current and new sparse-Lagrangian MMC closures;
the establishment of consistent and stable density coupling methods in sparse-
Lagrangian simulations; and
the application of MMC to conditions of greater engineering and practical inter-
est.

Acknowledgements

The authors would like to thank Dr Andrew Wandel of the University of Southern
Queensland and Dr Konstantina Vogiatzaki of Imperial College for their thoughtful,
probing and critical comments on the original manuscript. This work was supported
by funding from the Australian Research Council.

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 Part III
Advances and Trends in Multiscale
Strategies
Chapter 8
The Emerging Role of Multiscale Methods in
Turbulent Combustion

Tarek Echekki

Abstract Turbulent combustion flows are governed by processes that span the range
from atomistic scales to device (e.g. engine) scales and beyond (e.g. oil pool fires,
thermo-nuclear flames in type Ia supernovae). The multiscale nature of turbulent
combustion flows poses both challenges and opportunities. The challenges arise
from the need to predict combustion phenomena that are governed by a broad range
of scales. The opportunities arise because of the emergence of the multiscale science
that permeates many fields, and which for turbulent combustion, has been motivated
by the need to predict phenomena in new and evolving combustion technologies, ad-
vances in computational and applied mathematics, and the increasing availibility of
computational resources. In this chapter, strategies and requirements for the mul-
tiscale modeling and simulation of turbulent combustion flows are discussed. The
chapter serves as an introductory chapter to Part III of this book.

8.1 Motivation

A principal challenge in predicting turbulent combustion flows arises from their

multiscale nature. In a recent review, Peters [36] highlighted some complexities that
arise in the prediction of multiscale phenemena associated with chemistry, flame
structure, and flame interactions with turbulence. While Peters review, a more re-
cent review by the author [12] and the present chapter are timely, concrete ideas
about addressing the multisale nature of turbulent combustion flows have been im-
plemented for many decades. In physical sciences, constitutive relations (e.g. vis-
cous stresses for Newtonian fluids) are used to represent processes on the molecular
scale in a form applicable to macro-scale formulations (e.g. continuum conserva-
tion equations for fluid flow). These relations effectively represent multiscale treat-
ments that eliminate the need to resolve atomistic effects. More specifically in com-

Tarek Echekki
North Carolina State University, Raleigh NC 27695-7910, USA, e-mail: techekk@ncsu.edu

T. Echekki, E. Mastorakos (eds.), Turbulent Combustion Modeling, 177

Fluid Mechanics and Its Applications 95, DOI 10.1007/978-94-007-0412-1 8,
Springer Science+Business Media B.V. 2011
178 Tarek Echekki

bustion, turbulent combustion paradigms that rest on the principle of separation of

scales have been the cornerstone of turbulent combustion models in computational
fluid dynamics (CFD) for more than three decades. The eddy-dissipation model
(EDM) [27], the eddy break-up model (EBU) [42] and the flamelet model [34] are
based on separating the chemical time scales from turbulent time scales or flame
length and time scales from the corresponding turbulence scales, respectively.
However, important advances in applied and computational mathematics and re-
sources combined with evolving requirements in novel combustion technologies are
enabling the rapid development of various multiscale strategies for the prediction
of turbulent combustion flows. These strategies represent important additions, and
quite often extensions, to traditional paradigms in turbulent combustion [3].
The present chapter serves three primary objectives: 1) motivate the need for mul-
tiscale strategies for the modeling and simulation of turbulent combustion flows, 2)
introduce some basic strategies adopted for these flows, and 3) preview the topics
covered in Part III of this book, which deal with multiscale approaches for turbu-
lent combustion. The remaining sections of the chapter are organized as follows.
In Sec. 8.2, the multiscale nature of turbulent combustion flows is illustrated. In
Sec. 8.3, the case is made for the development of multiscale approaches in turbulent
combustion based on progress in multiscale science and the need to model evolving
combustion technologies. Fundamental considerations for multiscale approaches in
turbulent combustion are discussed in Sec. 8.4. A brief survey of multiscale ap-
proaches and a preview of related chapters are presented in Sec. 8.5.

8.2 The Multiscale Nature of Turbulent Combustion Flows

Turbulent combustion flows exhibit a broad range of length and time scales, which
encompass phenomena from atomistic to device scales. The most elementary pro-
cesses at the molecular level, which are associated with the electronic structure of
molecules and the breaking of chemical bonds, may occur at time scales as short as
1015 s. However, simpler representations of chemical kinetics, such as through the
Arrhenius law for the rate constant and the law of mass action, generally reduce the
time-scale requirements to a range from approximately 1010 s for the fastest reac-
tions to a few seconds for the slower reactions. Nonetheless, the presence of a wide
range of time scales represents an important challenge to the integration of chem-
istry and the coupling of this chemistry with transport for combustion problems.
Similar disparities are present for length scales. Molecular scales start with
ranges of approximately 1010 m. Beyond this, a broad range of scales repre-
sents the mechanisms of soot particle formation at molecular scales to hundreds of
nanometers for soot particle aggregates [21]. A large gap separates these processes
from continuum scales, which start around the scale of 1 m. Beyond this range, and
within the continuum regime, structures within the reaction zones of flames range
in thickness from tens of m (e.g. within the fuel consumption layer of premixed
hydrocarbon flames) to a fraction of a millimeter corresponding to a laminar pre-
Multiscale Methods 179

mixed flame thickness (including the entire reaction zone and a good portion of the
preheat zone). Around this upper range of premixed flame thickness, there may be
overlap with turbulent scales, starting from the Kolmogorov scale in practical com-
bustion systems. The upper range of the turbulent scales may be comparable to the
device scales, which may be of the order of tens of centimeters to meters. Therefore,
a broad range of length scales represent the bulk of combustion processes from the
atomistic to the device scale.
While the above so-called chemical flames appear untameable, thermonuclear
flames, such as the ones encountered in type Ia supernovae exhibit even broader
ranges of length and time scales [41, 49]. These flames exhibit much stronger non-
linearity in their nuclear source terms (carbon burning rate during ignition is propor-
tional to the temperature to the 23rd power [49]), turbulence length scales spanning
over eleven decades, and flame thicknesses, which are comparable to those of chem-
ical flames [49].
Although, the basic processes of combustion may be described using established
physical laws, their direct simulation is beyond todays and near-future compu-
tational capabilities. Current estimates of computing resources needed to resolve
length scales place an upper limit of three decades of length scales for tera-scale
computing and four decades of length scales for peta-scale [29]. Even if computer
memory can accommodate large problems, time integration remains an important
challenge, and many computations of turbulent reacting flows remain CPU-time-
limited. Therefore, multiscale strategies are needed to capture the contribution of
processes ranging from atomistic to device scales.
It is possible to carry out computations that are designed for a particular range of
scales. For example, quantum dynamics calculations may be used to resolve the
necessary electronic structure of molecules to determine kinetic rate parameters
and heats of formations of molecules. Molecular dynamic simulations and kinetic
Monte-Carlo simulations may be used to model soot processes [21] or to deter-
mine molecular transport properties. Fine- and coarse-grained continuum simula-
tions, respectively, may be used to capture particular flame-turbulence interations or
to model an actual device. Bootstrapping results from the different computations,
which are specialized for different ranges of scales can be done through different
strategies depending on the degree of coupling between scales. Here, we must distin-
guish between two types of coupling [12]: 1) physical coupling, which is associated
with the presence of cause-and-effect between two processes occurring at different
ranges of scales, and 2) scale coupling, which is associated with the presence of
overlap between the length or time scales or both between two processes.
Many important applications in combustion involve a one-way coupling between
processes operating at disparate scales. For these applications, various analytical
and computational methods are available. Examples of such methods include:
Constitutive relations: In reacting flows, the most common constitutive laws of-
ten represent models for atomistic processes in continuum governing equations.
An obvious example is the relation between the rate-of-strain and viscous stresses
in flows for Newtonian fluids. Another example of relevance to combustion in-
cludes the formulation of expressions for species reaction rates based on the Ar-
180 Tarek Echekki

rhenius law and the law of mass action. Constitutive relations may be based on
phenomenological laws (e.g. proportionality between two quantities and scaling
relations) or may be derived from more rigorous analysis (e.g. the kinetic theory
of gases).
Time-scale reduction: In addition to constitutive relations, various methods of
time-scale reduction may be adopted to represent fast processes, which are not
coupled with transport, under the assumption of separation of scales and one-way
coupling between processes. For example, chemistry reduction using dynamical
systems approaches such as computational singular perturbation (CSP) [25] or
the intrinsic low-dimensional manifold (ILDM) approach [38] can be used to
tabulate the contribution of the fast time scales and integrate primarily the re-
maining range of slow scales. However, in contrast with traditional constitutive
relations for chemistry and molecular transport, these latter approaches (e.g. CSP
and ILDM) may depend on the mixture characteristics, the burning modes (e.g.
premixed or diffusion flames) or the combustion regime (e.g. flamelet, distributed
reaction).
Examples of problems that involve two-way physical coupling and scale sepa-
ration includes the formation of soot in fires. In this problem, soot is a major con-
tributor to radiative heat loss from the flame zone, and plays a critical role in de-
termining the flame temperature. In turn, the temperature profiles within the flame
also contribute to the formation of soot and its transport. Despite the presence of
soot-flame coupling, multiscale strategies are enabled by scale separation. Instead
of constitutive relations, which address single terms in the governing equations, mo-
ment transport equations may be adopted, which features models for transport (e.g.
thermophoeresis) and sink and source terms (e.g. condensation, surface growth).
Other examples, which involve separation of scales and strong coupling between
the scales, include flame-turbulence interactions in the flamelet regime and thermo-
acoustic coupling in a combustion chamber. When an account of the physical cou-
pling is needed to represent physics at all scales, more robust formulations are
needed that identify the dynamics of contributions across the scales.
Despite the prevalence of problems in combustion that exhibit scale separation,
there is a growing need to address problems with strong physical and scale coupling.
This need is addressed in the next section.

8.3 The Case for Multiscale Strategies in Turbulent Combustion

In recent years, important paradigm shifts have occurred in two relevant fronts asso-
ciated with turbulent combustion flows. The first is associated with the emergence
of novel combustion technologies that have pushed the envelope on the types of
combustion regimes and modes encountered in practical devices. This shift also en-
tails new requirements for the combustion performance and its prediction associated
with combustion efficiency, flexibility of the combustion fuels and operating condi-
tions, the mitigation of pollutants emissions and safety [29]. The second shift is
Multiscale Methods 181

associated with the emergence of the field of multiscale science as an important

discipline in applied and computational mathematics and its increasing implemen-
tation in physical sciences. This emergence is fueled by increasing computational
resources and technological innovations in various physical sciences (e.g. nanotech-
nologies, material science).

8.3.1 Emerging Combustion Technologies

New combustion technologies have been proposed in recent decades to address the
clean and efficient burning of a growing range of combustion fuels [29]. These new
technologies, invariably, push the envelope on the operational regimes of practi-
cal combustion devices. Two examples of combustion technologies illustrate these
trends.
In reciprocating internal combustion engines, there is a classical distinction be-
tween two competing technologies: diesel and spark-ignited engines, with the
first more efficient and the second associated with cleaner burning. Alternative
technologies have been proposed in recent years to overcome the limitations of
each engine; these technologies include gasoline direct injection (GDI) engines,
diesel low-temperature combustion (LTC) and homogeneous charge compression
ignition (HCCI) engines [7, 29]. In HCCI engines, combustion is initiated by the
compression ignition of nearly homogeneous mixtures of fuel and air, resulting
in diesel-like efficiencies and lower emissions of NOx particulate matter through
better control of the charge and mixture conditions at combustion. Various con-
trol strategies may be adopted, which impact the competition between the rates
of mixing and chemistry [7, 45]. When these rates are competitive, a separa-
tion of the chemical and transport scales is not feasible, and strategies to couple
both effects are needed. Moreover, since the mixture homogeneity may evolve
as chemistry evolves, different modes of combustion (with varying degrees of
partial premixing) may be encountered during the HCCI combustion process [7].
In gas turbine engines, strategies to lower emissions and increasing efficiency
have been developed as well [5]. One strategy includes burning at lean or ultra-
lean fuel-air mixture conditions. However, these conditions also may contribute
to the onset of different, and often coupled, combustion instabilities [19]. Impor-
tant manifestations of these instabilities may include the non-equilibrium effects
of extinction and re-ignition events. Capturing non-equilibrium effects provide
important challenges to the state-of-the-art turbulent combustion models.
Common among the above-stated problems are requirements of fuel flexibility as a
broader range of combustion fuels are considered, including in addition to the stan-
dard fuels, heavy hydrocarbons (e.g. oil sands, oil shale) and renewable fuels (e.g.
ethanol, biodiesel) [29]. With the above considerations, the traditional paradigms
of turbulent combustion based on the assumption of separation of scales may not
be applicable when mixing time scales are competitive with chemical time scales
182 Tarek Echekki

or where well-established and stable combustion fronts (i.e. flames) determine the
combustion process. More importantly, within the context of moment-based meth-
ods in turbulent combustion, three additional challenges emerge:
1. The choice and the number of appropriate transported moments is very critical to
capture the desired physics. Generally, to capture more physics, such as extinc-
tion and re-ignition, more moments are needed.
2. Closure for non-linear terms in the moment transport equations is very critical
as well, especially when different outcomes of the problem may be encountered
(e.g. burning and non-burning solutions).
3. The prediction or the assumption of particular statistical distribution (e.g. joint
scalars probability density functions) becomes an additional challenge as well.
Moment-based models have been successful in predicting problems in combus-
tion applications; however, they remain limited by their underlying assumptions,
which may be related to the combustion mode (e.g. premixed vs. non-premixed),
the combustion regime (e.g. flamelet vs. distributed reaction) or dominant chem-
istry. These assumptions determine the choice of transported moments, the imple-
mentation of closure in the moment equation and the reconstruction of the spatial
statistical distributions for these moments. Significant progress has been achieved
in all fronts as illustrated by results discussed in the various chapters of this book.
Multiscale strategies within the continuum regime may be used to replace po-
tential empiricism in the closure for unclosed terms in the moment governing equa-
tions, construct more robust statistical distributions for the moments, or replace all
moments altogether.

8.3.2 Emerging Multiscale Science

Multiscale science permeates many disciplines in the physical (e.g. solid and fluid
mechanics, biological systems, chemistry), mathematical and computational sci-
ences. It is concerned with problems and phenomena where an account for a broad
range of scales is needed. Renewed interest in multiscale science is motivated in
part by growth in computational resources, which in turn fuel interest in computing
larger problems in the physical sciences. This growth has enabled scientists and en-
gineers to go beyond scale-specific applications and tools (e.g. quantum dynamics
vs. continuum-based calculations) to address the solution of larger problems and
couple different scales within the same problem. Invariably, the attempts within the
various disciplines are different because of modeling requirements and the extent to
which scales are coupled.
Important developments in multiscale science reflect growth in a number of im-
portant areas:
The development of robust mathematical frameworks and scale coupling strate-
gies designed to accurately represent the contribution of all relevant scales
and their coupling. A mathematical framework is distinct from the individual
Multiscale Methods 183

model(s) used for the different scales and in different applications. Therefore,
they may be applicable to a broad range of problems. The choice of a partic-
ular framework depends on a number of factors, which include the extent of
coupling (physical and scale) between the different phenomena and the type of
inter-dependence between the scales. Therefore, a principal goal of the mathe-
matical frameworks is to reduce the mathematical complexity of multiscale prob-
lems associated with their high-dimensionality, high degrees of freedom and the
complex inter-dependence between phenomena at different ranges of scales. Two
mathematical frameworks appear to be very promising for the study of turbulent
combustion flows: the heterogeneous multiscale method (HMM) [9], and varia-
tional multiscale method (VMS) [20]. Chapter 18 illustrates the implementation
of the HMM for the coupling of multiple scale solutions associated with turbulent
flame propagation in turbulence. Although the implementation of VMS for non-
reacting turbulent flows has been extensively studied, only recently an extension
of VMS for variable-density flows has been formulated [18].
The development of efficient numerical tools and computational frameworks:
Computational tools may include parallel solvers for partial differential equa-
tions, linear-algebra suites, multiresolution utilities for the decomposition of the
solution vector or data analysis, adaptive mesh refinement tools and other rele-
vant applications. The various computational tools may be common among dif-
ferent disciplines. Therefore, potentially a large pool of users may contribute to
their developments. However, certain rules must be put in place to ensure the
tools portability across computational platforms and applications. Chapter 17
illustrates a computational framework that enables a collaborative environment
for various tools development based on the common-component architecture
paradigm. The chapter also illustrates the implementation of such a framework
for combustion problems.

8.4 Multiscale Considerations for Turbulent Combustion

As stated earlier, mathematical and computational frameworks for the develop-

ment of multiscale approaches may be shared among different disciplines. However,
the choice of which frameworks to adopt and the individual models that represent
physics at the different scales is different for each discipline. What drives the choice
are the characteristics of the problem, which include: 1) the extent of coupling phys-
ical and scale coupling between the different phenomena at different scales, and
2) the choice of the models that are approriate to capture physics at the different
scales. This latter consideration is very unique, as it dictates strategies for reduc-
tion of order or dimensionality of the problem and the type of closure needed to
capture unresolved physics. In the following sections, basic requirements for multi-
scale approaches and choices of the physical models are discussed. Some of these
requirements and associated multiscale approaches may be extended to a broader
range of problems beyong turbulent combustion flows [26].
184 Tarek Echekki

8.4.1 Basic Requirements for Multiscale Approaches in Turbulent

Combustion

Important aspects of the multiscale nature of turbulent combustion flows have al-
ready dictated the choice of traditional paradigms in turbulent combustion. The first
aspect is the prevalence of flames which can propagate and be wrinkled by turbu-
lent flows. These flames represent interfaces between reactants and products, fuels
and oxidizers, as well as zones of localized reaction and heat release. In numerous
applications, these flames can be thin; and therefore, strategies to couple detailed
reactions zones at the interface with fluid dynamics form one potential multiscale
approach in turbulent combustion. Indeed, flame-embedding (see Chapter 12) repre-
sents one such strategy. Mathematical multiscale frameworks for flame embedding
are concerned with the implementation of coupling strategies between the flame so-
lutions and the flow field. Illustrations of such coupling using the heterogeneous
multiscale method (HMM) are given in Chapter 18. When flames are present, po-
tential strategies to reduce the physical dimension of the flame solutions are enabled
due to the strong gradients along the flame normal.
A second aspect of the multiscale nature of turbulent combustion flows is the
clear separation between atomistic scales and continuum scales. Therefore, robust
constitutive equations for reaction and molecular transport in the continuum gov-
erning equations are a viable substitute to hybrid atomistic-continuum simulations.
Nonetheless, there is great effort to be expanded in both scale regimes to develop
chemical rate and transport data from atomistic simulations and device-scale simu-
lations from the continuum equations.
Finally, given the non-linear nature of the governing equations, the presence of
many coupled variables representing both flow and scalar quantities, and the cou-
pling between fine and coarse scales in combustion, analytical methods (e.g. ho-
mogenization, asymptotic methods, renormalization group methods) by themselves
are not sufficient to solve or separate scales in turbulent combustion flows. Instead,
these tools can be coupled with numerical techniques to develop robust multiscale
frameworks for these flows.
At this stage, it is useful to re-iterate the basic desirable attributes for multiscale
approaches in turbulent combustion. These approaches are subject to the following
constraints:
All time and length scales are resolved at least within the context of continuum-
based transport equations and including a closed-form treatment of chemistry
and molecular transport processes using constitutive relations.
The models must preserve structure (e.g. flames) involving the coupling of reac-
tion and diffusion transport.
No statistical distribution shapes may be assumed for the reactive scalars; and this
distribution must be reconstructed, if needed, as part of the multiscale solution.
Multiscale Methods 185

8.4.2 General Frameworks for the Governing Equations for

Multiscale Models of Turbulent Combustion

The description of the turbulent flow of a multicomponent reacting mixture is pre-

scribed through a set of conservation laws for mass, momentum and composition
and auxiliary equations. The most common representation of these laws in the con-
tinuum regime are extensions of the Navier-Stokes equations (Eq. 2.2).
In Chapter 2, different mathematical forms of these governing equations in the
continuum regime are presented. They include the instantaneous forms used in di-
rect numerical simulations (DNS) and forms involving ensemble/time averaging
(i.e. RANS formulations) and spatial averaging (or filtering) (i.e. LES). Averag-
ing, invariably, results in unclosed terms in the governing equations, which must be
modeled.
Within the context of multiscale frameworks, only the instantaneous and the LES
governing equations enable scale separation. In LES, coarse scales are resolved;
while, the smaller scales have to be modeled. In contrast, RANS averaging im-
pacts all scales, and closure compensates for information that is lost across all the
scales. Therefore, based on the instantaneous and the LES formulations, two prin-
cipal classes of multiscale strategies emerge. The first is based on the concept of
mesh adaptivity and involve the use of the same set of governing equations and
the adoption of a hierarchical grid structure. The level of resolution within this grid
structure depends on the spatial and temporal resolution requirements within dif-
ferent zones of the computational domain. The second strategy is based on hybrid
combinations of a fine- and a coarse-grained solutions and may adopt a LES im-
plementation for the coarse-grained solution. For the fine-grained solution, different
strategies for reducing the model complexity at the small scales may be adopted.
It is important to note that within the context of LES, various extensions of the
traditional LES approach may be adopted for turbulent combustion flows. Multi-
level strategies may be derived within various contexts, such as homogenization
approaches [15] and the variational multiscale method [20].
An alternative strategy to the extended Navier-Stokes formulations is the
Boltzmann-Maxwell equation, which models the temporal and spatial evoution of
one-particle distribution function. The lattice-Boltzmann method (LBM) is a com-
putationally efficient method to solve the Boltzmann-Maxwell equations. In contrast
to the well-established conservations laws based on the Navier-Stokes equations, the
LBM approach is more recent. Yet, it is becoming an increasingly popular alterna-
tive to the Navier-Stokes approach for the solution of complex fluids. LBM can be
used to effectively straddle sub-continuum and continuum phenomena; therefore,
it may be viewed primarily as a mesoscale approach. Moreover, LBM can natu-
rally accomodate a variety of boundary conditions, including prescribed conditions
across fluid phases. The LBM is discussed in detail in Chapter 19.
186 Tarek Echekki

8.5 Multiscale Approaches in Turbulent Combustion and

Preview of Relevant Chapters

The sections below summarizes different strategies to model turbulent combustion

flows. Although restricted to only one component of turbulent combustion models,
chemistry acceleration is included and an entire chapter (Chapter 9) is dedicated to
this topic. The present multiscale strategies in turbulent combustion may be associ-
ated closely with either established multiscale numerical/mathematical frameworks
or traditional paradigms in turbulent combustion as discussed below. The various
approaches can be classified as follows:
1. Time-step acceleration and mesh adaptive methods
2. Flame embedding methods
3. Hybrid LES and low-dimensional models
A brief discussion of the various approaches is provided below:

8.5.1 Time-Step Acceleration

Chemistry integration represents a critical bottle-neck in the time-integration of re-

active scalar equations. The disparity of time scales between the fastest and the slow-
est reactions or time-steps required to resolve transport is at the heart of the stiffness
problems in chemical mechanisms. Strategies to accelerate the time-integration of
chemistry are based on a combination of different approaches. They include:
Chemical mechanism reduction This step addresses the reduction of the size
of the representative species and reactions. Chemistry reduction may involve,
as a first step, the development of a skeletal mechanism that contains the most
relevant species and elementary reactions. In a second step, a more aggressive
reduction strategy may be adopted to develop a global mechanism with even
fewer species and global steps. This second step may render the reduction pro-
cess applicable to a more limited range of problems. More importantly, it does not
guarantee that the resulting global mechanism removes the original mechanism
stiffness. Important progress has been achieved to address mechanism reduc-
tion from the classical strategies of the quasi-steady state (QSS) and the partial
equilibrium (PE) approximations, to more systematic model reduction strategies,
such as the ones based on the ILDM and the CSP approaches. Chapter 9 provides
an extensive review of chemistry reduction strategies.
Numerical stiff and mildly-stiff integrators In the past, stiff chemistry inte-
gration have been addressed through implicit integrators, such as VODE [4] and
DASSL [37]. However, more recently, explicit as well as implicit-explicit inte-
grators have been developed that are used to efficiently and accurately integrate
stiff and mildly stiff ordinary and partial differential equations. There is inher-
ently a trade-off between the computational cost, numerical accuracy and the
Multiscale Methods 187

coupling requirements with transport operators that must be taken into consider-
ation.
Chemistry tabulation, storage and retrieval approaches Another important
strategy for chemistry acceleration is the development of storage and retrival
strategies for tabulation results of chemistry integration. Efficient strategies for
tabulation, storage and retrieval are available, including the in situ adaptive tabu-
lation (ISAT) algorithm [39], the piece-wise reusable implementation of solution
mapping (PRISM) [43] and artificial neural networks (ANN) [6].

8.5.2 Mesh Adaptive Methods

As outlined earlier, one principal consideration for multiscale approaches in turbu-

lent combustion is that combustion and important scalar and velocity gradients may
be associated with well-defined regions in the flow (e.g. flames). Mesh adaptivity is
based on adapting the spatial resolution (and accordingly, the temporal resolution)
to capture fine details of the flow and the scalar field when needed and coarsen when
not needed. Mesh adaptive methods can be implemented within the context of both
instantaneous and statistical models (e.g. LES).
Chapters 13 and 14 illustrate two adaptive mesh strategies. The first is based on
the adaptive mesh refinement (AMR) approach [2] (Chapter 13). In this approach,
a recursive refinement of the mesh is implemented until resolution requirements are
met within a region of the flow. The second approach is based on the multiresolution
wavelet approach [40] (Chapter 14). This approach is based on a decomposition
of the solution into spatially- and temporally-varying multiresolution modes (e.g.
wavelet modes) to capture the solution at different multiresolution levels.

8.5.3 Flame Embedding Approaches

Flame embedding strategies may represent an extension of the flamelet paradigm

[34]; although, some of the ideas of flame-embedding have been proposed ear-
lier [17]. A principal outcome of the flamelet paradigm is the separation of the flame
scales from the energetic flow scales, enabling a degree of decoupling of the flamelet
and the flow solutions. Therefore, three important implementation elements associ-
ated with flame embedding are needed: 1) a solution of the flowfield, 2) a flame
solution, and 3) flame tracking.
Through the decoupling of the flow and flamelet solutions, different strategies
may be adopted to resolve the flow (e.g. vortex methods) and the reactive scalar
within the flame. For the flame, two principal approaches may be identified, which
correspond to 1) flame-normal embedding, and 2) flamelet tabulation. Chapter 12
discusses primarily flame-normal strategies where 1D flames are solved along the
flame surface normal with prescribed input parameters from the flowfield (e.g. strain
188 Tarek Echekki

rate). Another strategy is based on the transport of a representative scalar for flame
tracking and the tabulation of the remaining reactive scalars, such as approaches
based on the flamelet-generated manifold (FGM) [44] and the flame-prologation of
ILDM (FPI) schemes [13].
Flame embedding strategies can be implemented within the context of the in-
stantaneous equations or statistical models. In these latter models, flamelet solu-
tions with sufficient variability in time and space are used to build statistical mod-
els of the flame response in turbulence. Examples of such strategies include the
use of the LEMLES approach based on the linear-eddy model (LEM) (see Chap-
ter 10); although, the same strategies may be adopted using the ODTLES ap-
proach based on the one-dimensional turbulence (ODT) model (see Chapter 11).
More recently, Fureby [15] proposed a homogenization-based LES (hLES) ap-
proach, which converts the governing equations into a cascade of equations at dif-
ferent scales. Fureby [15] implemented a flame-embedding strategies using one-
dimensional forms of the equations at different scales.
Tabulation strategies represent alternative approaches to in situ flame calcula-
tions. These approaches represent an extension of the strategies adopted with FGM
and FPI. Here, we cite the recent work of Vreman et al. [47] and Fiorina et al. [14].
In both studies, it is proposed that filtered reaction rate solutions of reactive scalars
are constructed by filtering of one-dimensional flame solutions.

8.5.4 Hybrid LES-Low-Dimensional Models

Potentially more versatile methods may involve a hybrid formulation involving LES
combined with a low-dimensional model for subgrid scale (SGS) physics. Statistical
flame-embedding approaches represent an important sub-class of hybrid methods;
however, the more general hybrid strategies do not require fine-grained flame solu-
tions to be attached to flame normals or tabulated via transported reactive scalars.
Hybrid methods are primarily designed to construct statistical information, such as
subgrid scale source terms or filtered scalar quantities. Because a hybrid scheme that
spans the range of scales of interest is ultimately costly, strategies for dimensional
reduction are needed. The different hybrid strategies adopted for such hybrid LES
schemes generally involve either a stochastic or a dynamical-systems formulation
incorporated with a reduced physical dimension or a lower order of the solution
vector. Among the frameworks adopted for turbulent combustion, we cite:
Low-dimensional stochastic models, including the linear-eddy model (LEM) [23]
(Chapter 10) and the one-dimensional turbulence (ODT) model [24] (Chap-
ter 11).
Transported probability and filtered density functions with structure-based mix-
ing models (e.g. the PSP approach by Meyer and Jenny [30]).
The homogenization-based LES (hLES) approach [15].
The chaotic map approach by McDonough and co-workers [28, 31].
Multiscale Methods 189

Low-dimensional stochastic models, including the LEM and ODT models, involve
the embedding of one-dimensional domains within the 3D computational domain.
The 1D transport equations for each LEM or ODT domain feature the usual terms
in the governing equations for momentum (in the case of ODT) and scalars, albeit
in 1D. The resolution requirements are DNS-like and the computational saving is
derived principally from the reduced dimension.
The chaotic map model [28, 31] is based on a fundamentally different strategy
from that adopted using 1D stochastic models. The model is based on an estima-
tion procedure for the reactive scalars based on a definition of a chaotic dynamical
system. In the chaotic map approach, residual terms of reactive scalars (i.e. instan-
taneous subtracted filtered values) are represented at each discrete grid point as the
product of an amplitude factor, which is derived from classical theories of isotropic
turbulence, an anisotropy correction factor, and components of a discrete dynamical
system (DDS), whose governing equations are derived from the original reactive
scalars transport equations [28, 31]. An additional component of the derivation is
the discretization of the solution vector for the reactive scalars into a Fourier series
in which only leading coefficients are retained and ordinary differential equations
for the Fourier coefficients are solved.

8.6 Concluding Remarks

The direct resolution of all scales in turbulent combustion flows is practically impos-
sible given their multiscale nature. In this chapter, some of the basic challenges and
opportunities towards the development of multiscale strategies for turbulent com-
bustion flows are discussed. A principal goal of these strategies is to provide vi-
able alternatives to the traditional closure problem in turbulent combustion, when
these latter models fail or are limited in their predictions. The emergence of such
strategies is dictated by technological requirements for combustion, advances in
computational sciences and the increasing availability of powerful computational
resources. A principal challenge to tackle is associated with the representation of
chemisty as a critical bottleneck in the temporal integration of solutions of reacting
flows. Multiscale approaches for turbulent combustions are based on two general
strategies based on either mesh-adaptivity or model-adaptivity. Adaptive mesh re-
finement (Chapter 13) and wavelet-based methods (Chapter 14) represent promis-
ing examples of multiscale approaches based on mesh-adaptivity. Model-adaptive
approaches primarily adopt the coupling of fine-scale reduced dimension or order
solution and coarse-scale solutions, such as LES. Chapters 10, 11 and 12 illustrate
different applications of such model-adaptive methods.
190 Tarek Echekki

Acknowledgement

Dr. T. Echekki acknowledges support from the Air Force Office of Scientific Re-
search through grants F49620-03-1-0023 monitored by Dr. Julian Tishkoff and
FA9550-09-1-0492 monitored by Dr. Fariba Fahroo and the National Science Foun-
dation through grant DMS-0915150 monitored by Dr. Junping Wang.

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Chapter 9
Model Reduction for Combustion Chemistry

Dimitris A. Goussis and Ulrich Maas

Abstract The description of chemically reacting systems leads very often to reac-
tion mechanisms with far above hundred chemical species (and, therefore, to more
than a hundred partial differential equations), which possibly react within more than
a thousand of elementary reactions. These kinetic processes cover time scales from
nanoseconds to seconds. Due to these scaling problems the detailed simulation of
three-dimensional turbulent flows in practical systems using detailed kinetics is be-
yond the capacity of even todays super-computers. Using simplified sub-models for
the chemical kinetics is a way out of this problem. The question arising in mathe-
matical modeling of reactive flows is then: how detailed, or down to which scale has
the chemical reaction to be resolved in order to allow a reliable description of the
entire process? The aim is the development of models, which should be as simple as
possible in the sense of an efficient description, and also as detailed as necessary in
the sense of reliability. In particular, an oversimplification of the coupling processes
between chemical reaction and turbulent flow should be avoided by all means to
allow a predictive character. This chapter describes different methodologies for the
reduction of chemical mechanisms for subsequent use in reacting flow calculations.

9.1 Introduction

The interest in the numerical simulation of combustion processes has grown con-
siderably [45, 66, 69, 88, 89] during the last years, and computational fluid dy-
namics (CFD) has become an important design tool in various industrial applica-
tions such as internal combustion engines, gas turbines, power plants, and small

Dimitris A. Goussis
National Technical University of Athens, GR-15780 Athens, e-mail: dagoussi@mail.ntua.
gr
Ulrich Maas
Karlsruhe Institute of Technology, D-76131 Karlsruhe, e-mail: Ulrich.Maas@kit.edu

T. Echekki, E. Mastorakos (eds.), Turbulent Combustion Modeling, 193

Fluid Mechanics and Its Applications 95, DOI 10.1007/978-94-007-0412-1 9,
Springer Science+Business Media B.V. 2011
194 Dimitris A. Goussis and Ulrich Maas

Fig. 9.1: Schematic illustration of the time-scales governing a chemically reacting

flow.

scale burners. One problem, however, is that typical applications involve complex
three-dimensional geometries, complicated fuel mixtures, and (in most cases) tur-
bulent flow characteristics. At least for practical applications a complete solution of
the governing equations (conservation equations for mass, momentum, energy, and
species mass fractions) including detailed chemical kinetics is not possible because
the chemistry introduces a large number of chemical species and reactions (typically
up to 1000) resulting in a large dimension of the underlying system of conservation
equations. In order to simplify the numerical treatment of the chemical kinetics,
many methods have been devised, such as the assumption of an infinite reaction
rate (mixed equals burnt) or one-step chemistry. On the other hand pollutant for-
mation (NO, soot, unburned hydrocarbons), as well as ignition and extinction are
kinetically controlled processes [88], and oversimplified chemical schemes are not
able to capture these processes. Thus, there is a strong need for methods which sim-
plify the chemical kinetics while still retaining the essential dynamic features of the
reaction system.
As early as one hundred years ago Bodenstein [4, 5] observed that some chemical
reactions are so fast that some chemical species in the reaction system are in a quasi-
steady state. Comparing the time-scales of the chemical reactions with the typical
physical time scales (mixing, molecular transport, residence times) one can observe
that the chemical time scales span typically a much larger range (see Fig. 9.1), and
because only the time scales of similar order couple, one can decouple the fast ones
by assuming them to be in steady state and the slow ones by assuming that they are
frozen.
Based on the ideas of Bodenstein, many attempts have been made to develop sim-
plified descriptions of chemical reaction systems, e.g., for the simulation of com-
plex combustion processes, and a variety of different approaches can be found in
the literature (see, e.g., [32, 50, 67, 76, 77] for references considering combustion
Chemistry reduction 195

processes). In principle two ways of simplifying the chemical kinetics can be distin-
guished. One is to use the knowledge about the reaction system, i.e. the information
on which species are in quasi-steady state or which reactions are in partial equi-
librium. The other is to extract exactly this information from the detailed reaction
mechanism based on mathematical methods.
The traditional methodologies for simplifying a chemical kinetics mechanism
involve the use of the Quasi Steady-State approximation (QSSA) and/or the Partial
Equilibrium approximation (PEA) [44, 88, 90]. The QSSA assumes that the rate of
production and the rate of destruction of a number of species are both much larger
than their net rate of formation. Setting equal the rates of production and destruction
of these species yields a system of algebraic relations among the elementary rates.
On the other hand, the PEA assumes that the forward and/or backward rates of some
reactions are so large that they form a number of equilibria, which are expressed by
an equal number of algebraic relations among the elementary reaction rates. The
algebraic system that results from the QSSA or PEA is used:
in order to compute some of the concentration of species (those in steady state in
the case of QSSA or those characterized as being involved in a reaction in partial
equilibrium in the case of PEA), and
to simplify the differential equations of the remaining species in the mechanism
(usually by eliminating reactions deemed fast).
It is by now well accepted that the success of the QSSA and PEA methodologies
relies on the existence of a number of dissipative time scales, which are much faster
than the rest that characterize the long-term evolution of the system [6, 75]. This
feature, which is known as stiffness, is essential for the development in phase space
of low-dimensional surfaces, known as slow manifolds, to which all neighboring tra-
jectories are attracted under the influence of the fast time scales [15, 20]. With each
additional fast time scale becoming exhausted, the dimensions of the manifolds re-
duce by one. Therefore, the more fast time scales becoming exhausted, the lower
the dimensions of the manifold. Once on the manifold, the fast time scales become
exhausted and the solution is characterized by the slow time scales. The mathemat-
ical model that describes the evolution on the manifold is usually referred to as the
reduced model. Clearly, the dimension of the manifold and the size of the reduced
model decreases as the number of fast time scales considered exhausted increases.
Starting from the 80s, the QSSA and PEA were extensively employed for the
generation of reduced chemical kinetics mechanisms, with great success [64, 65,
67, 68, 76]. When successful, the algebraic system of equations among the ele-
mentary rates provides an accurate approximation of the manifold and the reduced
mechanism provides an accurate contribution of the kinetics in advancing the solu-
tion on the manifold. However, the use of these approximations is hindered by the
difficulties in predicting correctly:
the number of QSSAs and PEAs that are applicable,
the species to be considered in steady state and, therefore, to be excluded from
the stoichiometry of the reduced mechanism,
the elementary reactions that participate in equilibria, and
196 Dimitris A. Goussis and Ulrich Maas

the elementary reactions that are fast and, therefore, must be excluded from the
rates of the reduced mechanism.
Several algorithms were developed for the successful application of the QSSA and
PEA [13, 24, 56, 79, 81]. However, problems in their use still remain, especially
when considering the reduction of large and complex mechanisms of heavy hydro-
carbons [61]. There are basically two reasons for these problems. First, a manifold is
not always approximated by the algebraic relations that develop when applying the
QSSA and/or PEA. The appropriate algebraic relations are usually more complex,
involving (i) more reactions than those participating in the assumed QSSAs and/or
the PEAs and (ii) the coefficients that multiply the elementary rates are not constant
integers, but complex functions of the species concentrations, the temperature, the
pressure, etc [28, 29, 85]. The second reason for the limits in the successful appli-
cation of the QSSAs and PEAs is that the nature and the dimension of the manifold
are not fixed, but are both functions of the local chemical and thermophysical prop-
erties of the system [29, 58]. The latter issue is demonstrated by the results shown
in Fig. 9.2, relating to the analysis of the structure of a stoichiometric, steady, 2D,
methane-air edge flame stabilized against an incoming mixing layer [58].

Fig. 9.2: Analysis of methane-air edge flame structure. The temperature field [K]
shown with superposed heat release rate (left) and the number of fast dissipative
time scales than can be neglected with superposed mass fraction of CH4 (right). The
fuel stream is on the right and the air stream is on the left. Reprinted from [58] with
permission from Taylor and Francis.

The chemical kinetics is modeled with the GRImech3.0 mechanism, which in-
volves 53 species. It is shown that the number of fast dissipative time scales that
become exhausted, and therefore can be neglected, varies from about 5 to 30, de-
pending on the position in the flame. In particular, the primary reaction zone exhibits
the lowest number of fast time scales that can be neglected, while the central diffu-
Chemistry reduction 197

sion flame region exhibits the highest number. Therefore, optimal reduction within
a pre-specified accuracy requires the use of a number of reduced models of varying
size.
Such problems in the construction of reduced models lead to the development
of algorithmic tools able to provide the appropriate simplified chemistry, without
any involvement from the investigator. Such tools can be devised efficiently if one
makes use of the facts that:
not the whole composition space (spanned by the large number of species con-
centrations) is actually accessed, and
the dynamics is confined to low-dimensional manifolds in composition space.

Fig. 9.3: Results from DNS of a hydrogen-oxygen-nitrogen non-premixed flame.

Scatter plots in the composition space.

This behavior is typically observed in combustion processes and can be seen

nicely from results of direct numerical simulations (DNS). Such DNS are based
on a direct solution of the conservation equations without applying any modeling
assumption and allow detailed studies of turbulent combustion, even taking into
account detailed chemical kinetics. Figure 9.3 shows results of a DNS for a non-
premixed hydrogen-air flame (see [55] for details). The results of the DNS have been
represented as scatter plots in the state space (here we show only a 3-dimensional
plot of the results with the mass fractions of N2 , H2 O, and H as coordinates, and in
addition the temperature is coded by the color, where blue is cold and red is hot).
The figure shows that not the whole composition space is accessed and furthermore
that the accessed states are in the neighborhood of low-dimensional manifolds.
198 Dimitris A. Goussis and Ulrich Maas

 wi 
Fig. 9.4: Dynamical behavior of the chemical kinetics in composition space Mi =
mol
kg .

This is further illustrated in Fig. 9.4, where trajectories of chemical reactions in

composition space have been plotted for different chemical systems (from syngas to
n-dodecane) and different initial conditions with the only requirement of having the
same enthalpy, pressure and element composition [3]. It can be seen that the systems
relax to lower-dimensional manifolds and finally approach equilibrium.
In the following sections of this chapter, the QSSA and PEA will be presented
and discussed. A unified view of these two approximations will lead to a reduc-
tion strategy that can represent many of the available reduction algorithms currently
available. These algorithms will then be displayed. Finally, the strategies followed
for the implementation of these algorithms will be presented.

9.2 Traditional Methodologies for Reduction: QSSA and PEA

Relatively large chemical kinetics mechanisms were systematically reduced first

with the QSSA and PEA methodologies. These methodologies will be applied here
on the Thermal NO mechanism for the formation of nitric oxide via the reactions
[88]:
k
O + N2
1
NO + N R1 = k1 [O][N2 ] (9.1)
k2
N + O2 NO + O R2 = k2 [N][O2 ] (9.2)
k3
N + OH NO + H R3 = k3 [N][OH] (9.3)
Chemistry reduction 199

where Ri and ki denote the rate and rate constant of the i-th reaction and [X] denotes
the concentration of species X. Considering a homogeneous mixture, when these
reactions are part of a larger detailed kinetics mechanism, the rate of formation of
NO is given by:
d[NO]
= R1 + R2 + R3 + rNO (9.4)
dt
while that of the remaining species by:

d[N]
= R1 R2 R3 + rN (9.5)
dt
d[O]
= R1 + R2 + rO (9.6)
dt
d[OH]
= R3 + rOH (9.7)
dt
d[O2 ]
= R2 + rO2 (9.8)
dt
d[N2 ]
= R1 + rN2 (9.9)
dt
d[H]
= R3 + rH (9.10)
dt
where the functions rX represent the possible contribution from additional reactions
in the detailed mechanism involving the species X.
Let us consider possible ways to reduce the three reactions in the Thermal NO
mechanism by considering the reaction (9.3) as fast and the species N as the one
affected the most by the related fast time scale. In particular, let us examine two
alternatives for such a reduction; either to apply the QSSA for [N] or the PEA among
the N-producing/consuming reactions (9.1), (9.2) and (9.3). These two alternatives
will be examined separately, first for the case of the evolution equation of [NO], and
then for the evolution of the concentration of all species in the mechanism.

9.2.1 The QSSA

Assuming the net rate of change of [N] is much smaller than both its rate of produc-
tion and its rate of destruction, Eq. (9.5) yields:

0 R1 R2 R3 + rN (9.11)

which, when substituted in Eq. (9.4) in order to eliminate the rate of the fast reaction
R3 , yields:
d[NO]
2 R1 + (rNO + rN ) (9.12)
dt
200 Dimitris A. Goussis and Ulrich Maas

Instead of the evolution Eq. (9.5), [N] can now be computed from the algebraic
expression:
R1 + r N
[N] (9.13)
k2 [O2 ] + k3 [OH]

9.2.2 The PEA

In its simplest version the PEA assumes that the forward and backward rates of cer-
tain reversible reactions are almost equal such that the net rate is approximately zero.
Then the law of mass action of these reactions can be used as algebraic relations
among the species. The concept of fast reactions can, however, also be applied in a
more general context, which allows to analyze the reaction mechanism (9.1)-(9.3),
which does not (in this special case) contain reversible reactions. Assuming the fast
reaction (9.3) participates in an equilibrium with the reactions (9.1) and (9.2):

R3 R1 R2 (9.14)

the PEA is applied by taking the time derivative of Eq. (9.14), substituting from
Eqs. (9.5)-(9.10), solving for the rate of the fast reaction R3 :

R3 a1 R1 a2 R2 + a3 (9.15)

and then substituting in Eq. (9.4) in order to eliminate R3 :

d[NO]
(1 + a1 ) R1 + (1 a2 ) R2 + (a3 + rNO ) (9.16)
dt
where:
k1 ([N2 ] + [O]) k3 [N]
a1 = 1 +
k3 ([N] + [OH]) + k2 [O2 ]
(k3 k2 )[N] k1 [N2 ]
a2 = 1
k3 ([N] + [OH]) + k2 [O2 ]
[N](k2 rO2 + k3 rOH ) + (k2 [O2 ] + k3 [OH])rN k1 ([N2 ]rO + [O]rN2 )
a3 =
k3 ([N] + [OH]) + k2 [O2 ]

In this case, [N] can be computed either by the algebraic expression (9.14):

R1
[N] (9.17)
k2 [O2 ] + k3 [OH]

by the algebraic expression (9.15):

a1 R1 + a3
[N] (9.18)
a2 k2 [O2 ] + k3 [OH]
Chemistry reduction 201

or by an evolution equation that can be constructed when substituting the rate of the
fast reaction R3 from Eq. (9.15) to Eq. (9.5):

d[N]
(1 a1 ) R1 (1 a2 ) R2 (a3 rN ) (9.19)
dt

9.2.3 Comments on the QSSA and PEA

The two simplified evolution equations for [NO] constructed with the QSSA and the
PEA, Eqs. (9.12) and (9.16), are related by the expression:

d[NO] d[NO]
= (1 a1 ) R1 + (1 a2 ) R2 + (a3 rN ) (9.20)
dt PEA dt QSSA
Given the relations among the rates, Eqs. (9.11) or (9.14), the QSSA and the PEA
provide the same simplified model for [NO] only when the last three terms in the
RHS of Eq. (9.20) can be neglected. This is possible only when the quantities:

k1 ([N2 ] + [O]) k3 [N]

1 a1 =
k3 ([N] + [OH]) + k2 [O2 ]
(k3 k2 )[N] k1 [N2 ]
1 a2 =
k3 ([N] + [OH]) + k2 [O2 ]
[N](k2 rO2 + k3 rOH k3 rN ) k1 ([N2 ]rO + [O]rN2 )
a 3 rN =
k3 ([N] + [OH]) + k2 [O2 ]

are sufficiently small. Otherwise, the two approximations provide different simpli-
fied equations of [NO], the accuracy of which cannot be evaluated a priori.

9.2.4 A Common Set-up for the QSSA and PEA

The QSSA and PEA can be viewed from a unified point of view as follows. For
simplicity, let us concentrate on the three reactions in the Thermal NO mechanism
and neglect the possible contribution from other reactions, which are represented by
the terms rX . In that case, Eqs. (9.4)-(9.10) can be cast in vector form as:

d
= S1 R1 + S2 R2 + S3 R3 = g( ) (9.21)
dt
where is the 7-dim. vector of concentrations, Si is the 7-dim. stoichiometric vector
of the i-th reaction and Ri is the corresponding reaction rate:
202 Dimitris A. Goussis and Ulrich Maas

[N] +1 1 1
[NO] +1 +1 +1

[O] 1 +1 0
 
=
[OH]
S = S1 S2 S3 =
0 0 1
(9.22)
[O2 ] 0 1 0

[N2 ] 1 0 0
[H] 0 0 +1

Both the QSSA and PEA amount to the resolution of the 7-dim. phase space
according to a specific set of 7-dim. basis vectors ai (i = 1, 7):

d
= a 1 f 1 + a2 f 2 + . . . + a 6 f 6 + a7 f 7 (9.23)
dt
where f i are the corresponding amplitudes, defined as f i = bi g, and the 7-dim.
vectors bi compose the dual basis (bi a j = ji ). Let us denote the set of basis vectors
(column) by a and the set of dual basis vectors (row) by b:
   T
a = a1 a2 . . . a6 a7 b = b1 b2 . . . b6 b7 (9.24)

For the QSSA, appropriate sets of basis and dual vectors are:

1 0 0 0 0 0 0
+1 +2 0 +1 0 0 2

0 1 +1 1 +1 1 +1

a = 1 1 +1 0 0 +1 1 (9.25)
0 0 1 0 0 0 +1

0 1 0 0 0 0 +1
+1 +1 1 0 0 0 +1

1 0 0 0 0 0 0
+1 0 0 0 +1 0 +1

+1 0 0 0 2 +1 +1

b =
+1 +1 0 0 0 +2 0
(9.26)
0 +1 +1 +1 +2 0 0

0 0 0 +1 0 0 +1
1 0 0 0 +1 1 1
On the basis of these vectors, Eq. (9.23) yields:

f 1 = R1 + R2 + R3 f 2 = R1 f 3 = R2 f 4 = = f7 0 (9.27)
d
= a1 f 1 + a2 f 2 + a3 f 3 (9.28)
dt
where the vanishing of the four amplitudes f 4 to f 7 is due to conservation laws;
the first three due to the conservation of the N, O and H atoms in the stoichiometric
vectors S1 , S2 and S3 .
Chemistry reduction 203

Setting f 1 0 is equivalent to invoking the QSSA for [N], Eq. (9.11), so that the
resulting simplified system:

d
a 2 f 2 + a3 f 3 (9.29)
dt
can be interpreted as the appropriate one for a detailed mechanism consisting of the
following reactions and reaction rates:

O + OH + N2 2NO + H f 2 = R1 (9.30)
O2 + H OH + O f3 = R2
(9.31)

as can be verified by inspecting the elements of the a2 and a3 vectors. Of course,

these are not physical reactions; their stoichiometry and rates being a linear combi-
nation of the stoichiometry and the rates of the elementary reactions in the original
chemical kinetics mechanism (9.1)-(9.3). Note that both the concentration of the
fast species [N] and the rate of the fast reaction R3 are missing in Eqs. (9.30)-(9.31).
[N] can be computed from Eq. (9.13), which involves R3 .
For the PEA, appropriate sets of basis and dual vectors are:

A B C D E F F
A 2 B C D E F F

0 1 +1 0 0 0 0

a =
A 1 + B 1 +C 1 + D 1 + E 2 + F 2 + F (9.32)
0 0 1 0.5 +1 1 1

0 1 0 +0.5 0 0 0
A 1 B 1 C 1 D 1 E 1 F 2 F

+1 0 k1 [N2 ]G k3 [N]G k2 [N]G k1 [O]G 0
+0.5 +0.5 0 0 0 0 0

+0.5 +0.5 +1 0 0 0 0

b = +1 +1 0 0 0 +2 0 (9.33)
0 +1 +1 +1 +2 0 0

0 0 0 +1 0 0 +1
1 0 0 0 +1 1 1
where:
k2 [O2 ] + k3 [OH] k3 [N] k1 ([N2 ] + [O])
A = B =
k2 [O2 ] + k3 ([N] + [OH]) k2 [O2 ] + k3 ([N] + [OH])
k1 [N2 ] + (k2 k3 )[N] 1 k1 [O] + (k2 2k3 )[N]
C = D =
k2 [O2 ] + k3 ([N] + [OH]) 2 k2 [O2 ] + k3 ([N] + [OH])
(k2 + k3 )[N] (k2 2k3 )[N]
E = F =
k2 [O2 ] + k3 ([N] + [OH]) k2 [O2 ] + k3 ([N] + [OH])
G = (k2 [O2 ] + k3 [OH])1
204 Dimitris A. Goussis and Ulrich Maas

Setting f 1 0 is equivalent to invoking the PEA among reactions (9.1)-(9.3),

Eq. (9.15), so that the resulting simplified system:

d
a 2 f 2 + a3 f 3 (9.34)
dt
can be interpreted as the appropriate one for a detailed mechanism consisting of the
following reactions and reaction rates:

O + (1 B)OH + N2 (2 B)NO + (1 B)H + (B)N f 2 = R1 (9.35)

O2 + (1 +C)H + (C)NO (1 +C)OH + O + (C)N f 3 = R2 (9.36)

as can be verified by inspecting the elements of the a2 and a3 vectors. Neither of

these reactions are physical ones; their stoichiometry and rates being a linear com-
bination of the stoichiometry and the rates of the elementary reactions in the original
chemical kinetics mechanism (9.1)-(9.3). Now, the concentration of the fast species
[N] is present in Eqs. (9.35)-(9.36), while the rate of the fast reaction R3 is missing.
[N] can be computed either from the algebraic Eqs. (9.17) or (9.18) or from the first
component of Eq. (9.34).
The reactions and reaction rates in Eqs. (9.30)-(9.31) and (9.35)-(9.36) represent
2-step reduced models constructed with the QSSA and PEA, respectively. A signif-
icant difference between the two reduced models is the fact that the stoichiometry
of the QSSA model involves constant coefficients, while that of the PEA model in-
volves variable coefficients. However, the stoichiometry of the two reactions in both
reduced models conserves the elements, as in the case of the elementary reactions.
Note that the reduced model constructed with the PEA can be reduced to that de-
veloped by the QSSA by setting B = C = 0. At this point it must be emphasized
that the form of the 2-step reduced mechanisms produced by both QSSA and PEA,
as stated in Eqs. (9.30)-(9.31) and (9.35)-(9.36), is not unique. This can easily be
easily demonstrated by casting Eq. (9.29) or (9.34) as:

d
(a2 + q( )a3 ) f 2 + a3 ( f 3 q( ) f 2 ) (9.37)
dt
where q( ) is an arbitrary function of .

9.2.5 The Need for Algorithmic Methodologies for Reduction

In order to validate the reduced models constructed with the QSSA and PEA in the
previous section for the case of the three reactions in the Thermal NO mechanism,
let us consider a case where the mixture is at constant temperature T = 1956 K and
initial composition:
Chemistry reduction 205

[NO] = 0.9524 1010 [O] = 0.1984 107 [OH] = 0.5028 107

[O2 ] = 0.1165 106 [N2 ] = 0.4273 105 [H] = 0.4217 107

where the units of concentration are in mol/cm3 and [N] is computed from the re-
lations R1 + R2 = R3 , so that Eqs. (9.11) and (9.14) (which are identical in the case
considered here), pertinent to the QSSA and PEA, are initially fully satisfied. It can
be shown that for this case, 1 + a1 0.5 and 1 a2 2.5. Since these quantities
are not negligible, according to the discussion in Section 9.2.3 the QSSA and PEA
are expected to provide different results.
The evolution in time of [NO] is displayed in Fig. 9.5, as computed from:
the detailed model; i.e., Eqs. (9.4)-(9.10),
the QSSA reduced model; i.e., Eq. (9.29) for all concentrations except [N] and
Eq. (9.11) for [N],
a first PEA reduced model; i.e., Eq. (9.34) for all concentrations (including [N]),
denoted as PEA (1),
a second PEA reduced model; i.e., Eq. (9.34) for all concentrations except [N]
and Eq. (9.14) for [N], denoted as PEA (2).

Fig. 9.5: Left: The evolution of [NO] (moles/cm3 ) with time (s), determined from
the full model, the QSSA model and two PEA models. Right: the evolution of the
residuals h1 and h2 , Eq. (9.38).

It is shown that both PEA models provide excellent accuracy, while the QSSA
model is shown to provide poor accuracy. Noting that the construction of both PEA
reduced models are based on Eq. (9.15), while the QSSA and second PEA mod-
els invoke the same algebraic relation among the elementary rates, Eqs. (9.11) and
(9.14), for the computation of [NO], the magnitude of the following ratios is then
examined:
R1 R2 R3 a1 R1 a2 R2 R3
h1 = h2 = (9.38)
R 1 + R2 + R 3 |a1 R1 | + |a2 R2 | + |R3 |
206 Dimitris A. Goussis and Ulrich Maas

where h1 measures the cancelations occurring in Eqs. (9.11) or (9.14) and h2 mea-
sures the cancelations occurring in Eq. (9.15). Clearly, the larger the cancelations
the greater the validity of the corresponding algebraic relation. Figure 9.5 clearly
demonstrates that the reduced model, Eq (9.34), constructed on the basis of the
refined Eq. (9.15) is much more accurate than the reduced model, Eq. (9.29), con-
structed on the basis of Eqs. (9.11) or (9.14), providing thus an explanation for the
greater success of the first PEA model.
The treatment of this simple example clearly demonstrated the limitations in the
use of the QSSA and PEA approximations. It is not known a priori if either of
the two approximations are valid, nor the extent of the accuracy they provide. In
addition, for the case of the PEA, it was shown how difficult its use is, even for this
very simple case. The algorithms to reported next, resolve all these problems.

9.3 Reduction Algorithms

Consider a detailed mechanism, consisting of K reactions, N species and E ele-

ments. Considering a homogenous and isothermal mixture, the governing equations
for the concentration of the species can be cast as:

d
= S R( ) = S1 R1 + S2 R2 + . . . + SK RK = g( ) (9.39)
dt
where is the N-dim. vector of concentrations, S is the (N K)-dim. matrix of the
N-dim. stoichiometric vectors Si and R is the K-dim. vector of the reaction rates Ri :
1
R
 
S = S1 S2 . . . SK R = ... (9.40)
RK

Let us assume that for a certain period of time there exist M dissipative fast time
scales, which are much faster than the remaining time scales that locally characterize
the evolution of the system. Let us measure this time scale gap with the ratio of the
slowest of the fast time scales to the fastest of the slow ones: = M /M+1 . Let us
further introduce the N-dim. basis (column) vectors ai :
     
aM = a1 . . . aM aS = aM+1 . . . aNME aC = aNME+1 . . . aN
(9.41)
on the basis of which Eq. (9.40) is cast as:

d
= aM fM + aS fS + aC fC (9.42)
dt
where bi a j = ji are the dual basis (row) vectors and:
Chemistry reduction 207

fM = bM g( ) fS = bS g( ) fC = bC g( ) (9.43)
1 M+1 NME+1
b b b
.. .. ..
bM = . S
b = .
C
b = . (9.44)
bM bNME bN
The elements of the vectors fM , fS and fC are the amplitudes of the projection of the
vector field g( ) along the directions of the vectors in aM , aS and aC , respectively.
Let us assume that the vectors in aM and their dual vectors bi (i = 1, M) define the
subspace in which the M fast time scales act. Therefore, when these time scales
become exhausted:
f M = bM S R 0 (9.45)
where the symbol denotes the possibility that the basis vectors in aM and their
corresponding dual vectors provide an approximation only of the fast subspace. Let
us further assume that the basis vectors in aC define the subspace in which conserva-
tion laws act (i.e., the conservation of elements in the elementary reaction), so that
fC 0. Therefore, when the fast time scales are exhausted the evolution Eq. (9.42)
reduces to:
d  
aS fS = I aM bM g( ) (9.46)
dt
where I is the unit matrix. This reduced system is free of the fast time scales and
is valid within the constraints imposed by the Eq. (9.45), which defines the slow
manifold.
Most of the existing algorithms for the generation of the manifold and the re-
duced system produce results in the form of Eqs. (9.45) and (9.46); i.e., directly or
indirectly they generate a coordinate system for the vector field g( ), on the basis
of which the system of differential equations reduces to normal form.
In particular, the iterative procedure for defining the slow manifold of Fraser and
Roussel through the invariance equation [74], can be cast in the form of Eq. (9.43)
[30]. This iterative procedure can be extended to produce the reduced model as well
[30]. Both the manifold and the reduced model can provide any order of accuracy,
since the iterative procedure generates, term by term, the asymptotic expansion of
the asymptotic expansion of the slow manifold given by geometric singular pertur-
bation theory [30, 37].
The Computational Singular Perturbation (CSP) algorithm of Lam and Goussis
provides two refinement procedures for the computation of the basis vectors ai and
their dual bi [41, 42]. The so called br -refinement generates, order by order, the
asymptotic expansion of a slow manifold, while the ar -refinement approximates the
tangent spaces to the fast fibers along which solutions relax to the slow manifold
[92, 93]. CSP detects the dimensions of the manifold, subject to a pre-specified
accuracy, and can also identify the fast variables and fast processes in the detailed
model [39, 43].
The Intrinsic Low-Dimensional Manifolds (ILDM) algorithm of Maas and Pope
uses the right and left eigenvectors of the Jacobian of the vector field g( ) in or-
208 Dimitris A. Goussis and Ulrich Maas

der to approximate the ai and bi vectors [52, 53]. With this approach, the manifold
and the reduced model provide leading order accuracy, when compared with that
provided by geometric singular perturbation theory [37]. ILDM incorporates a tab-
ulation strategy that makes the algorithm very efficient. A variant of the method
is the GQL (global quasi-linearization) method, which uses the eigenspaces of a
global quasi-linearization matrix instead of those of the local Jacobian [9, 11] or
the TILDM method which uses the eigenspaces of JJ T , where J denotes the local
Jacobian ([8].
The Zero Derivative Principle (ZDP) algorithm of Gear and Kevrekidis yields
successive approximations to the slow manifold by setting derivatives of succes-
sively higher order of the subset of fast variables of the state variables equal to
zero [21]. This algorithm was later extended to provide approximations of the basis
vectors ai and their dual bi , so that it can now provide a reduced model as well [94].
As with the CSP method, the ZDP algorithm in its extended form provides, term
by term, the results of the geometric singular perturbation theory. A variant of this
algorithm is the one developed by Contou-Carrere and Daoutidis [16].
The Method of Invariant Manifolds (MIM) and the Method of Invariant Grids
(MIG) of Gorban and Karlin [14, 26, 27], are based on the introduction of a projec-
tor P = (I aM bM ), which is defined on the basis of thermodynamics: the entropy
should grow in the fast motion. With this approach, both the manifold and the re-
duced system can be constructed, in the form of Eqs. (9.45) and (9.46). The slow
manifold is considered as the stable fixed point of a relaxation equation or of an
algebraic equation solved iteratively with the Newton method with incomplete lin-
earization.
The Rate-Controlled Constrained Equilibrium (RCCE) methodology, originally
proposed by Keck [38] and developed further by Jones and Rigopoulos [35, 36], is
also used for the construction of reduced models. The basis of RCCE is that, in a
system that exhibits substantial time-scale separation, the species governed by the
fast time scales will be in a constrained equilibrium state, led by the concentrations
of the leading species. Chemical kinetics can then be employed for computing the
evolution of the leading species only, and the fast species can be calculated via a
constrained minimisation of the Gibbs free energy, such that the leading species re-
tain their values computed by kinetics. A system of differential-algebraic equations
(DAEs) results, which is readily parametrised by the selection of leading species.

9.4 Interaction of Chemistry with Diffusion

In the presence of diffusion, the governing Eq. (9.39) is modified as:

d
= L( ) + S R( ) (9.47)
dt
where the N-dim. operator L accounts for the diffusion terms. In this case, the man-
ifold and the reduced model are stated as:
Chemistry reduction 209

fM = bM (L + S R) 0 (9.48)

d  
aS fS = I aM bM (L + S R) (9.49)
dt
where fS = bS (L + S R) [33, 39, 53, 73]. In this case, Eqs. (9.48)-(9.49) can
provide leading order accuracy only, provided the characteristic time scale due to
transport is of the order of the fastest of the slow chemical time scales [33, 40].
Under certain
 conditions
 Eq. (9.48) can be simplified to Eq. (9.45), however the
projector I aM bM must always operate on both the chemistry and the diffusion
terms [33, 40, 52, 73].
In general, there are two alternatives to proceed in the case where transport is
present:
obtain the reduced model, as in Eqs. (9.48)-(9.49) directly for the PDE;
discretize in space the original PDE, Eq. (9.47), cast the system in the form of
Eq. (9.39), and construct the reduced system in the form of Eqs. (9.45)-(9.46) for
the resulting ordinary differential equation system.
With the first approach, the number M of fast dissipative time scales that can be
considered exhausted might be a function of space, making the time integration
not as optimal as the largest simplification possible. The first approach allows for a
simpler construction of the simplified system, while the second allows for a simpler
time marching scheme [31].
The coupling of the chemical kinetics with diffusion can be handled very ef-
ficiently in the context of manifold methods by taking into account the coupling
directly in the construction of the manifolds (see next section).

9.5 Manifold Methods and Tabulation Strategies

9.5.1 Principles of Manifold Methods

Manifold methods use consequently the fact that the actual thermokinetic states are
confined to or at least in the vicinity of low-dimensional attractors in composition
space (cf. Section 9.3). Therefore the actual n = ns + 2- dimensional state space
(which is, e.g. given as = (h, p, w1 /M1 , . . . , wns /Mns )T , where ns is the number
of species, h is the specific enthalpy, p the pressure, wi and Mi the mass fraction
and the molar mass of species i, respectively) can be described by a much smaller
number m of so-called reduced coordinates or progress variables :

= ( ) (9.50)

This function can be obtained e.g. using steady-state assumptions, ILDM equations,
GQL equations, etc., which has been described above. For an implementation of the
developed method of system reduction into a reactive flow calculation it is necessary
210 Dimitris A. Goussis and Ulrich Maas

to derive reduced set of conservation equations. Let us start from a detailed equation
system of a typical reacting flow process in symbolic vector form:

1
= F ( ) v grad ( ) div (D grad ( )) (9.51)
t
where v represents the velocity field, the density and D is the (n by n)-dimensional
matrix of the transport coefficients [19].
According to the basic assumption of the manifold methods, the state is, at any
point of the flow and at any time, close to the manifold (i.e. the system dynamics
in the state space are completely described as a movement within the manifold).
If all thermochemical states everywhere in chemical reacting system are elements
of the manifold, neither the convective term causes movements in the state space
perpendicular to the manifold [52], nor the chemical source term causes movements
off the manifold for invariant manifolds and marginal movements off the manifold
for manifolds that are not invariant. However, a projection of the transport term is
required and can be done by a transformation of the transport term into the local
coordinates of the invariant slow subspace [52]:

1
= F ( ) v grad ( ) P div (D grad ( )) , (9.52)
t
where P denotes the (n by n)-dimensional projection operator for the diffusion term.
It should be noted that this projection can be omitted as an approximation if the re-
duced coordinates are chosen such that the fast relaxing chemical processes cause
only a small marginal change of these progress variables. Accordingly, the system
(9.52) calculates the full n-dimensional state , but it restricts the evolution to a
movement tangent to the manifold. This equation system contains now redundant
information, because is a function of only m variables and the evolution of
is confined to the low-dimensional manifold. This can be attributed by changing the
equation system into an equation system for . In principle there are two possible
choices. The first one is based on a representation of the reduced coordinate in terms
of original species in the mechanism, i.e.:

= C , (9.53)

where C is an (m by n)-dimensional parameterization matrix. If e.g. the species in

the detailed mechanism are CH4 , O2 , CO2 , H2 O, OH, O, . . ., and the progress vari-
ables are chosen to be CO2 and H2 O, then the matrix is simply given by:


0 0 1 0 0 0
C= (9.54)
0 0 0 1 0 0

Pre-multiplying the equation system with C then yields:

1
= CF ( ) v grad ( ) CP div (D grad ( ( ))) , (9.55)
t
Chemistry reduction 211

Details can be found e.g. in [54]. A more elegant way is to allow an arbitrary pa-
rameterization. This can avoid problems with uniqueness or condition of the pa-
rameterization. In this case the n-dimensional governing equation system (9.51) is
transformed by premultiplying with the Moore-Penrose-inverse (see e.g [25]). It is
given for a regular matrix T by
 T 1
+
= T , (9.56)

yielding

1 +
t F ( ) v grad ( ) P div (D grad ( )) .
= + (9.57)

If we denote + F ( ) by S ( ), P by ( ), and D by D ( ), we can

+ 

see that now the governing equation system for the scalar field of the reacting flow
is a partial differential equation system for the m-dimensional reduced cordinates
vector :
1
= S ( ) v grad ( ) ( ) div (D ( ) grad ( )) . (9.58)
t
This equation system is suitable for a coupled solution together with the equations
for the flow field.

9.5.2 Calculation of Low-Dimensional Manifolds

Various methods exist to calculate low-dimensional manifolds. In principle all the

methods described above can be used for a generation of low-dimensional mani-
folds, and various methods have been devised for an efficient calculation and stor-
age. These shall be described in the following.

9.5.2.1 Slow manifolds

Slow manifolds [74] have the advantage that they represent invariant manifolds and
represent the overall slow dynamics. Nevertheless their calculation is based on a
functional iteration which may have convergence problems. Therefore, Davis and
Skodje [17] had the idea to start the iteration from ILDMs as starting guess, which
improved the convergence considerably. Furthermore, a solution process in terms
of an evolution equation for the manifolds is possible [57]. The problem with this
technique is that the governing equation for the slow manifold is a partial differential
equation system which is difficult to solve. Lately, the convergence problems were
eliminated with the formulation of the method along the CSP ideas [30]. Similar to
these approaches is the method of invariant grids [26]. Another method of approxi-
212 Dimitris A. Goussis and Ulrich Maas

mating slow manifolds is the calculation of minimal entropy production trajectories

[46].

9.5.2.2 Computational Singular Perturbation Manifolds

CSP generated manifolds can account for the variation on their dimensionality and
a pre-specified accuracy [41, 42]. These manifolds are generated using an iterative
algorithm, each iteration providing a higher order correction term [83, 92]. These
terms have a magnitude of increasing powers of the fast/slow time scale ratio. There-
fore, CSP is most successful when this time scale ratio is small [39, 43]. The dis-
advantage on the implementation of CSP is the computational cost of computing
on-the-fly the appropriate basis vectors. Lately, a new computational technique for
the computation of the basis vectors was developed, which is very efficient when
large reductions are possible [30]. In addition, a new methodology was developed
which employs efficient tabulation in order to avoid the basis vectors calculations
[47].

9.5.2.3 Intrinsic Low-Dimensional Manifolds

Low-dimensional manifolds can also be constructed using the ILDM [51, 52]
TILDM [8] or GQL [9, 11] technique. The major problem here is that ILDMs of
very low dimension ( 2) exist only in the domain of fast chemistry close to equi-
librium, making a tabulation difficult (see below). A splitting technique, however,
allows to extend the ILDM into the domain of slow chemistry [10], thus keeping the
overall dimension small. Furthermore, a flamelet prolongation [23] can be used too
(see below).

9.5.2.4 Repro-modeling

The idea of repro-modeling is to calculate a large number of combustion scenarios

(homogeneous reactor calculations, flame calculations, etc.) [80] to yield a large set
of accessed compositions . Because of the existence of low-dimensional attractors
in composition space the results will lie in the vicinity of these attractors. Therefore
the accessed states are represented by a low-dimensional manifold, e.g. by approx-
imating them by a multi-variate least squares spline. The advantage of this method
is that the calculation of the accessed composition is quite straightforward. The dis-
advantage is that the choice of the controlling variables and the construction of the
approximating function is not trivial. This method can also be combined with a re-
laxation process according to the ILDM-equations, yielding a simple to implement
variant of the ILDM method [7].
Chemistry reduction 213

9.5.2.5 Trajectory Generated Manifolds

Another possibility is to construct the manifolds based on homogeneous reactor cal-

culations. An m-dimensional manifold in composition space is constructed by solv-
ing the chemical rate equations for initial conditions lying on an m 1 dimensional
manifold of initial conditions. If is the set of initial conditions, then if (t; 0 ) is
the solution of d /dt = F ( )

M = { (t; 0 ) |0 t { 0 } = } (9.59)

There are several variants to choose the manifold generator . One of them is to
choose initial conditions at the boundary of the domain [71], newer variants such as
the ICE-PIC method use constrained equilibrium manifolds or minimum-curvature
pre-image curves (see [73]). The advantage of these methods is that there are well
established methods for the calculation of the trajectories and the pre-image curves.
A disadvantage is the the subsequent parameterization of the manifold is not trivial.

9.5.2.6 Flamelet Generated Manifolds, Flamelet Prolongation of ILDM

If reduced schemes for subsequent flame calculation are devised, trajectory gener-
ated manifolds have the disadvantage that for initial conditions in the domain of
slow chemistry (e.g. the composition of the unburned mixtures) the rate of reac-
tion tends to zero. In addition the coupling with molecular transport processes has
to be included by a suitable projection, and in the domain of slow chemistry the
diffusion processes govern the low-dimensional attractors. Therefore, following the
idea of the flamelet concept, manifolds can be generated similar to trajectory gener-
ated manifolds by solving flamelet equations (here for varying boundary conditions
instead of initial conditions) [87] or to extend ILDM into the domain of slow chem-
istry by calculating flamelets between a composition on the ILDM and the unburned
mixtures [23]. Several variants exist, which are based on premixed or non-premixed
flamelets. The advantage of this method is that it is quite robust to implement and
accounts for the coupling of chemical kinetics with molecular transport already in
the construction of the manifold. One disadvantage is the fact that an extension to
higher dimensions is not straightforward, although recently such an extension has
been reported [59].

9.5.2.7 Reaction-Diffusion Manifolds

We suppose that the system solution in the state space is close or belongs to an
ms -dimensional manifold defined by an explicit function ( )

M = { : = ( ) , : Rms Rn } , (9.60)
214 Dimitris A. Goussis and Ulrich Maas

here is the m-dimensional vector of reduced coordinates. M is an invariant m-

dimensional system manifold if we solve the partial differential equation system

 
( )  1
= I F ( ) v grad ( ) div (D grad ( )) = 0
+
t
(9.61)
until a stationary solution is obtained. In this case the normal component of the vec-
tor field on the invariant manifold vanishes. Here + is the Moore-Penrose pseudo-
inverse of (see above).
Rewriting the gradients in terms of the gradients of the reduced coordinates one
obtains (see [73] for a detailed discussion of these diffusion terms)
 
( )  1
= I + F ( ( )) div (D grad ( )) = 0 (9.62)
t

This equation system can be solved if the spatial gradients are known as functions
of the reduced coordinates. There are three limiting cases:
The gradients are zero, which means that a homogeneous system is considered
only with no molecular transport. In this case the equation simplifies to the PDE
for the slow manifolds [57].
The gradients are taken from a laminar flamelet. In this case the equation reduces
for a one- dimensional manifold to the flamelet equation.
There is no chemical reaction. In this case the equation represents an evolution
equation which yields the minimal surface in composition space governed by
diffusion (a linear surface in the case of equal diffusivities.
One drawback of this method is that an estimate grad( ) = f ( ) for the gradient
in terms of the reduced coordinates is needed. But there a several ways out of the
problem:
It has been shown that at least for manifolds of higher dimensions the manifold
does not depend much on the gradient estimate, and rough estimates can be used.
Known estimates for diffusion flames or premixed flames from the literature can
be used (see e.g. [59]).
Gradient estimates can be refined during the solution of the reduced set of equa-
tions [12].

9.6 Tabulation

In principle there are two different strategies for implementing reduced chemistry
in reacting flow simulations. One is e.g. to use the derived quasi-steady state as-
sumptions and partial equilibrium assumptions directly during the integration of the
reacting flow equations. This can, however, be computationally demanding, because
Chemistry reduction 215

in this case partial differential equations are replaced by non-linear algebraic equa-
tions, which are in many cases not easier to solve than the differential equations.
Thus, despite of a considerable reduction of the chemical kinetics, very often the
computational gain is much smaller than expected. The other strategy follows in a
straightforward way from the concept of manifold methods. These methods make
use of the fact that the complete thermokinetic state is a known function of the re-
duced coordinates (see above). Therefore it is natural to store this information for
subsequent use in reacting flow calculations. The necessary steps are:
Calculate the low-dimensional manifolds using the various techniques described
in Section 9.6.
Store the results (thermokinetic state, reaction rates, information on the coupling
with molecular transport) using a sophisticated tabulation scheme.
Use an efficient table-lookup to retrieve the information during the reacting flow
calculation
Although this tabulation strategy is very simple, the actual implementation is
very challenging due to several reasons:
The storage effort of most straightforward tabulation schemes increases exponen-
tially with the dimension of the reduced coordinates (curse of dimensionality
[2]).
As can be seen in Fig. 9.3 only a small domain of the whole reduced coordi-
nate space is actually accessed, and storage capacity should not be wasted by
tabulating domains which are never accessed in practical applications.
There is a need to minimize the error associated with the tabulation table-lookup
The table-lookup has to be extremely fast and efficient.

In order to cope with these problems several strategies have been developed. In
principle they can be classified in four groups.
Tabulation in terms of a multi-dimensional grid: In this case a multi-dimensional
grid is constructed and the information is stored at the different nodes. Several
variants exist from regular tensor product grids over local mesh refinement to the
use of so-called sparse grids (see e.g. [18, 23, 52, 87]).
Piecewise storage of the results. These methods separate the composition space
into hypercubes or polyhedrons and approximation of the composition space in
the polyhedrons by a polynomial representation. One of these approaches is the
PRISM (piecewise reusable implementation of solution mapping) method [78],
while another is a piecewise tabulation strategy based on orthogonal polynomials,
which has been used to store ILDMs [60].
In situ adaptive tabulation (ISAT) [70]: this method stores the thermokinetic state
in terms of a binary search tree and is very efficient, because it tabulates only the
actual accessed composition space.
Generalized coordinates: This method uses a regular rectangular mesh con-
structed for generalized coordinates which is then mapped to the actual com-
position space. The advantage is that the mesh is adapted in a natural way to
216 Dimitris A. Goussis and Ulrich Maas

the problem (typically the mesh is constructed during the identification of the
low-dimensional manifolds), the disadvantage is that the equation system for the
reacting flow has to be projected on to the generalized coordinates [1].
Adaptive chemistry tabulation: A methodology based on Common Component
Architecture is employed in order to store all CSP quantities needed for the con-
struction of the appropriate reduced model, that delivers a pre-specified accuracy
and can be incorporated into an explicit solver [47, 62]. The composition space is
separated into hypercubes and a different reduced model is stored for each one.
According to this method, the reduced model of two adjacent hypercubes might
be of different dimensions and form.
Although much progress has been made during the last years, there is still a need
to improve the existing methods.

9.7 Concluding Remarks

In this chapter we have made the attempt to outline the existing strategies for mech-
anism reduction, making no claim to be complete. Several topics such as the produc-
tion of skeletal mechanisms (see e.g. [49, 63, 82]), the analysis of detailed reaction
mechanisms using sensitivity analysis or reaction flow analysis [77, 88], or lumping
techniques [34, 48, 72] have not been addressed. The numerical aspects of reduced
chemistry modeling, like their use with operating splitting techniques, the explicit
solvers, the numerical solution of steady state relations [22, 50, 84, 86, 91] were
also not addressed.
Although much progress has been made in the development of strategies and
tools for mechanism reduction, there are still many open questions (in particular
with respect to accuracy and efficiency of the reduced mechanisms). However, as it
was shown above, all strategies for mechanism reduction rely on common ideas, and
in particular their combination will improve the quality and efficiency considerably
in the future.

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Chapter 10
The Linear-Eddy Model

Suresh Menon and Alan R. Kerstein

Abstract Regime-independent modeling is important for accurate simulation of

the complex combustor designs needed to meet increasingly stringent performance
requirements. One strategy for achieving robust yet affordable predictive capability
is to resolve, in space and time, the relevant advection-diffusion-reaction couplings
using a low-dimensional representation of turbulent advection. In the linear-eddy
model (LEM), this is accomplished in one spatial dimension by introducing an in-
stantaneous map, the triplet map, that emulates the effect of an eddy turnover on
property profiles along a notional line of sight. The map preserves the continuity
of these profiles and obeys applicable conservation laws. Details and representative
applications of the model are presented for passive and reactive scalar mixing, with
emphasis on its use as a mixing-reaction closure for large-eddy simulation (LES)
based on the embedding of an LEM domain in each LES control volume.

10.1 Motivation

Regime-independent modeling is a widely recognized goal of turbulent combus-

tion modeling. This goal is driven by the need to model configurations involving
various combinations of regimes such as partial premixing, extinction, re-ignition,
recirculation, stratified premixed combustion, compression ignition, multi-stage ig-
nition, and transition to detonation. Techniques involving coarse-graining, ensem-
ble averaging, or state-space modeling face difficulties in this regard due to the lack
of detailed representation of regime-specific advective-diffusive-reactive couplings.

S. Menon
School of Aerospace Engineering, Georgia Institute of Technology, USA, e-mail: suresh.
menon@aerospace.gatech.edu
A. R. Kerstein
Combustion Research Facility, Sandia National Laboratories, Livermore, CA, USA, e-mail:
arkerst@sandia.gov

T. Echekki, E. Mastorakos (eds.), Turbulent Combustion Modeling, 221

Fluid Mechanics and Its Applications 95, DOI 10.1007/978-94-007-0412-1 10,
Springer Science+Business Media B.V. 2011
222 Suresh Menon and Alan R. Kerstein

Although progress in addressing these difficulties can be anticipated, the challenge

of regime-independent modeling suggests the concurrent pursuit of an alternative
approach, namely, the development of a conceptually and computationally minimal
model that resolves, in space and time, some plausible representation of the coupled
advancement of advection, diffusion, and reaction in turbulent reacting flow. The
potential advantages of this strategy motivated the formulation, two decades ago, of
the linear-eddy model (LEM) [28].
A useful starting point for introducing LEM is the numerical advancement of
equations governing a 1D unsteady opposed-flow flame. This formulation can cap-
ture some but not all of the salient features of turbulent combustion. One important
feature that it does not capture is the effect of rotational folding of the flame by
turbulent eddy motions. Successive folding and compressive motions can cause an
initially monotonic profile of, say, mixture fraction to develop multiple extrema with
stoichiometric points in between, each corresponding to a flame location. At high
turbulence intensity, the time scale of folding, compression, and diffusive mixing
might become short enough relative to chemical time scales so that broad reaction
zones, stirred internally by small eddies, are formed.
The impossibility of representing these effects in 1D by continuous-in-time mo-
tions while obeying the applicable conservation laws motivated the introduction, in
LEM, of instantaneous maps. Each map can be conceptualized as representing the
outcome of an individual eddy motion, although such a literal connection between
maps and eddies is not required.
An LEM simulation time advances the 1D unsteady diffusion-reaction equations,
including associated dilatations along the 1D domain. This advancement is punctu-
ated by instantaneous rearrangements of property profiles by mapping operations of
a specified form (see Sec. 10.2). In effect, the outcome of each map constitutes a
new initial condition for further time advancement.
A numerical consequence of this procedure is that a time of map occurrence
cannot be contained within a time step for advancement of the governing equations.
This limits the potential advantage of implicit diffusion, so in LEM, diffusion is
typically implemented explicitly, possibly with implicit chemistry depending on the
stiffness of the kinetics.
Implementation of LEM as a subgrid model for large-eddy simulation (LES) has
also been accomplished with the goal of applying it to more complex high Reynolds
number flows. In this approach the LEM is coupled with a large-scale scalar trans-
port method to capture both large-scale flame structures and subgrid wrinkling ef-
fects.

10.2 Triplet Map

It is convenient to define a map symbolically using the notation

s(y) s(M(y)), (10.1)

Linear-eddy model 223

where s(y) is any property profile, y is the 1D spatial coordinate, and M is the inverse
of the map, i.e., the property value at M(y) is mapped to location y. The map used
in LEM, termed the triplet map, has the form


3(y y0 ) if y0 y y0 + 13 l,

2l 3(y y0 ) if y0 + 13 l y y0 + 23 l,
M(y) = y0 + (10.2)

3(y y0 ) 2l if y0 + 23 l y y0 + l,

y y0 otherwise.

This map shrinks property profiles within an interval [y0 , y0 + l] to a third of their
original extent, and then fills the interval with three compressed copies of the pro-
files. The middle copy is reversed, which maintains the continuity of mapped prop-
erties and introduces the rotational folding effect of turbulent eddy motion (see
Sec. 10.1). Property fields outside the size-l interval are unaffected.
On an unstructured adaptive 1D mesh, this spatial continuum definition of the
triplet map can be implemented numerically as stated, as in an ODT adaptive-mesh
implementation described in Chapter 11. Published LEM studies to date use a fixed
uniform mesh, on which the triplet map is approximated as a permutation of mesh
cells. It is convenient to focus on this implementation because its consistency with
property conservation laws is self-evident.
Spatially discrete implementation of the triplet map is illustrated in Fig. 10.1,
involving a permutation of cell indices j through j + l 1. Taking the map range l
to be a multiple of 3 cells, the triplet map permutes the cell indices into the new order
j, j + 3, j + 6, . . . , j + l 3, j + l 2, j + l 5, j + l 8, . . . , j + 4, j + 1, j +
2, j + 5, j + 8, . . . , j + l 4, j + l 1. This operation reduces the separation of any
pair of cells by no more than a factor of three, consistent with the scale locality of
length-scale reduction by eddy motions in the turbulent cascade (see Chapter 11).
It is interesting to note that the triplet map and its uses for turbulence simulation
were discussed in a recent paper by authors who developed this concept without
knowledge of prior work on LEM [27]. This is perhaps indicative of the degree
to which map-based advection using the triplet map is a uniquely advantageous
approach for economical turbulent-mixing simulation.

10.3 Map Sizes and Frequency of Occurrence

As explained in Sec. 10.1, LEM is intended to be an alternative to models involving

coarse-grained constructs such as eddy diffusivity DT . Although eddy diffusivity
is not the modality of turbulent transport implementation in LEM, it is central to
LEM formulation. Eddy diffusivity parameterizes the aggregate turbulent transport
induced by an ensemble of turbulent eddies. Accordingly, the ensemble of triplet
maps during an LEM simulation induce aggregate transport that is quantifiable as an
eddy diffusivity, here denoted DT . Based on the triplet-map property that it induces
4 2
a mean-square displacement 27 l within the mapped interval, random-walk theory
224 Suresh Menon and Alan R. Kerstein

Fig. 10.1: Application of a triplet map with l = 9, formulated as a permutation matrix

multiplying a vector representing a flow state. Here, the map is applied to a column
vector with vertically increasing cell indices. For clarity, unity matrix elements are
boldface and cells are shifted horizontally in proportion to their index values. The
shifts are intended to suggest the 1D profile of a mapped variable.

Fig. 10.2: Scalar field evolution with LEM. (a) initial field, (b) after triplet map and
(c) after diffusion-reaction equation.

allows DT to be expressed as

2
DT = l 3 f (l) dl, (10.3)
27 0

where f (l) is the probability density function (PDF) of map sizes (a model input)
and is the frequency of map occurrences per unit domain length, so that times
the domain size is the rate of map occurrence on a notional spatially homogeneous
domain [32]. (In previous model descriptions, the notation rather that was used
in Eq. 10.3. Here, the notation is changed for consistency with Chapter 11.) Equa-
Linear-eddy model 225

tion 10.3 is used to solve for given a DT value and a size distribution f (l) that
correspond to the turbulent flow that is being simulated.
The functional form of f (l) is based on the following familiar consequence of
internal-range turbulence scaling. The eddy diffusivity DT (l) associated with turbu-
lent motions of size l or less scales as lv(l), where the eddy velocity v(l) scales as
l 1/3 , so DT (l) l 4/3 . The LEM analog of DT (l) is obtained by taking l to be the up-
per bound of the integral in Eq. 10.3 (to be distinguished from the dummy variable
l in the integrand). The map-size PDF that yields this scaling is f (l) l 8/3 .
The inertial range extends from the Kolmogorov microscale to the inte-
gral scale L, which obey the scaling L/ Re3/4 , where Re is the integral-scale
Reynolds number, Re = u L/ , and u is the turbulence intensity. Therefore, the
stated scaling of f extends from to L, and f = 0 outside this range. This deter-
mines the prefactor of f by the requirement that its integral over l is unity.
Here, and L do not have precisely the same meanings in LEM as in turbulent
flow, so strictly speaking these LEM quantities are related to their physical counter-
parts by empirical coefficients. Likewise, turbulent diffusivity values inferred from
flow data typically do not precisely conform to the mathematical definition of a dif-
fusivity, so an empirical coefficient may be needed to relate these values to the LEM
parameter DT , which has a precise definition within the model.

10.4 Application to Passive Mixing

For passive mixing involving a single scalar property s with constant molecular
diffusivity , time advancement between map occurrences is governed by the heat
equation,
s 2s
= 2. (10.4)
t y
The equation set for combustion is shown in Sec. 10.5.
For passive mixing, the family of scalar power spectra parameterized by the
Schmidt number Sc is a useful comparison data set for LEM because the spectra
exhibit universal behaviors with a sufficient number of features to overdetermine
the model and test its performance. Spectrum comparisons have been used to set the
free parameters in LEM and to demonstrate that LEM captures the salient features
of turbulent mixing [5, 32]. Figure 10.2 shows a typical implementation and com-
petition between molecular diffusion and turbulent stirring by the triplet mapping.
Turbulent stirring on an initial scalar gradient mimics the folding effect of an eddy
of size l with the subsequent molecular diffusion smoothing the gradient. These
processes interact over a range of eddy sizes in a high Re flow resulting in statisti-
cal features of scalar mixing in excellent agreement with classical scaling laws and
experimental observations.
For example, LEM as described above has been applied to mixing in grid tur-
bulence (including three-stream mixing) [28, 33], in planar mixing layers (focusing
on Sc dependence) [29], and in multi-scalar jet and homogeneous flows that exhibit
226 Suresh Menon and Alan R. Kerstein

differential molecular diffusion effects [5, 30, 35]. An application to mixing in ho-
mogeneous turbulence [37] revealed unexpected large-scale mixing behaviors that
were subsequently verified by a pipe-flow experiment motivated by the LEM results
[2123]. Other homogeneous-turbulence studies compared scalar fluctuation decay
in LEM and direct numerical simulations (DNS) [6, 43] and used LEM to study
fractal scaling properties [31] and other intermittency properties [26, 32] of scalar
fields mixed by turbulence.
The mixing of dry and moist air in clouds has been simulated using LEM in
order to study temperature changes induced by cloud droplet evaporation and con-
densation and associated dynamical consequences [41, 42]. These studies, which
were extended to simulate the evolution of droplet-size distributions [75], did not
incorporate buoyancy effects but provided indications of larger-scale consequences
of buoyancy changes during the simulations. It has been shown mathematically that
triplet maps, in conjunction with a simple represention of eddy-induced motion of
inertial particles, capture clustering effects thought to be crucial to the process of
cloud-droplet coagulation to form raindrops [36].

10.5 Application to Reacting Flows

For reacting flows, the LEM employs more general equations for the species and
temperature within the 1D context. An extension of the LEM to multi-component
reacting flows can be written as [52, 53, 71, 74, 78]:

Yk
= ( YkVk ) + k
t y
 

T T Ns T
Cp mix
t
= Cp,kYkVk + y y + T .
y k=1
(10.5)

Here, Yk is the kth species mass fraction, k is the reaction rate, and Vk is the kth
species diffusion velocity.
Any type of kinetics (global or multi-step) can be included within this approach.
Heat release related effects such as thermal expansion and mass transport to main-
tain global conservation have to be included but their specific implementation issues
depend on the problem of interest [1, 47, 71, 78]. An example of the advantages of
LEM reactive-flow simulation relative to other approaches is an application [13]
to two simple reaction mechanisms that are widely used as test cases for closure
models. LEM was shown to be in agreement with DNS of these reactions in a non-
premixed configuration. LEM was then used to investigate a much wider parameter
range than DNS can access affordability, yielding a data set suitable for stringent
tests of proposed closures. None of the tested closures reproduced LEM product-
selectivity results.
Linear-eddy model 227

LEM has likewise been used to evaluate proposed nonpremixed-combustion clo-

sures [7, 8]. A priori analysis of the LEM-predicted PDF of the scalar properties has
been shown to agree very well with experimental PDFs for jet diffusion flames [19].
It has also been used to explore regimes that closure models cannot yet address ade-
quately, such as soot-radiation-turbulence coupling [79], and in applied studies such
as NOx prediction in dual-stage combustion [51], jet flames [1, 47], aerosol mixing
in engine exhaust [52, 78], and transient behaviors during incineration [65].
More recent studies include the study of extinction/re-ignition for a non-premixed
syngas flame [6770] for a test case simulated earlier by DNS [24, 25] using the
same 21-step, 11-species skeletal syngas mechanism. Results showed the LEM can
capture both the extinction and re-ignition process at approximately the same turbu-
lent Reynolds number with the scalar dissipation rate at the stoichiometric surface
predicted to be 2394 1/s, which is very close to the DNS value of 2100 1/s.
Many canonical premixed flames have also been investigated using the LEM.
Freely propagating turbulent premixed flames in the flamelet regime [48, 71, 72]
and in the thin-reaction-zone regime [60] have demonstrated that the changes in the
flame structure can be captured without any change to the LEM. Recent stand-alone
premixed LEM studies have focused on premixed combustion far from the flamelet
regime. LEM has been applied to regimes of hydrogen [55] and hydrocarbon com-
bustion relevant to homogeneous-charge compression-ignition (HCCI) combustion.
An application to astrophysical flames [77] validated LEM using DNS of the well-
stirred-reactor regime, then explored the more challenging stirred-flame regime
[34] to identify possible mechanisms for transition to detonation resulting in super-
nova explosions. Additionally, this study showed that the LEM turbulent burning
velocity in the flamelet regime is 18DT /L for high Re, a result that is useful for
detecting departures from flamelet or high-Re scaling.
Recent studies employed the LEM to investigate flame-turbulence interactions
for a range of conditions and fuel mixture with relatively detailed reduced kinet-
ics [69, 70]. We use some results from this study to highlight key features of the
LEM when implemented for reacting flows. For this study, the LEM scalar fields
are initialized by the steady, laminar, 1D flame solution obtained for a CH4 /Air
mixture at an equivalence ratio of 0.6, and a 12-step, 16-species reduced methane/air
mechanism is used for the calculations. Thermally perfect gas with detailed multi-
component transport properties is employed for these simulations.
Figure 10.3 compares a typical H2 mass fraction profile that has evolved from the
initial laminar profile with stirring alone (Fig. 10.3, top left), with profiles evolved
by stirring and diffusion (Fig. 10.3, top right) together, and with stirring, diffusion
and reactions interacting (Fig. 10.3, bottom), which is the full LEM. It can be seen
that the profile is wrinkled and exhibits high level of scalar gradients as a result of the
turbulent stirring (Fig. 10.3, top left) but once the diffusion is included the gradients
created by the triplet-maps are smoothed out (Fig. 10.3, top right). The peak value
for the H2 intermediate species decreases as well, which indicates that the flame is
extinguishing, since reaction rates are not computed and the combined action of the
turbulent stirring and diffusion pushes the flame towards extinction. However, when
kinetics is also included (Fig. 10.3, bottom) the flame maintains its peak value while
228 Suresh Menon and Alan R. Kerstein

Fig. 10.3: Evolution of a scalar field (H2 ) by simulating the effect of stirring (top
left), stirring and diffusion (top right), and stirring, diffusion and reaction (bottom).

also showing the effect of stirring and diffusion. Thus, turbulent stirring, molecular
diffusion and reaction kinetics can interact at their respective time scales in the LEM
in a consistent manner. This is the unique nature of the LEM.

10.6 Application to Reacting Flows as a Subgrid Model

Extension of the LEM as a subgrid model requires some rethinking of the turbulent
stirring approach described above. In the classical LES methodology large scale
structures are fully resolved whereas small scale structures (subgrid-scale) are mod-
eled by using appropriate subgrid momentum and combustion models. Thus, clo-
sures for momentum, energy and scalar transport at the resolved scales are required.
This requirement is no different from the closure requirement for RANS modeling,
and an earlier effort [17, 18] demonstrated a LEM based approach for RANS ap-
plication. In the following, we focus primarily on the LES implementation of LEM
(called LEMLES hereafter).
Linear-eddy model 229

For LES of reacting flows, both low-Mach number [2, 3] and fully compress-
ible approaches have been used in the past. Closure for the momentum and en-
ergy equations is achieved using a localized dynamic subgrid kinetic energy ksgs
model (LDKM) [16, 3840, 49, 53, 54] in most of the studies discussed here. In
addition to allowing for non-equilibrium between production and dissipation at the
subgrid scales, the LDKM offers two unique advantages for LEMLES that is ab-
sent in the classical Smagorinsky type algebraic eddy viscosity mode: (a) the ksgs
distribution
 can be used to provide an estimate of the subgrid turbulence intensity
u 2ksgs /3, which can then be used for estimating turbulent stirring time scale
and frequency in the subgrid LEM, and (b) for two-phase modeling of spray com-
bustion, a stochastic component to the particle motion can be included to account
for the effect of subgrid turbulence on particle transport [45, 53, 57]. The govern-
ing LES equations for mass, momentum and energy are obtained by using density-
weighted Favre-filtering and the following discussion, although restricted to com-
pressible flows is equally applicable in low-Mach number flows [3, 4].
The LEM-based subgrid closure for mixing and chemistry was developed [44,
50] for both premixed and non-premixed applications, and has been used for LES
closure in many subsequent studies, e.g., scalar mixing [5, 46], non-premixed com-
bustion [10, 46, 67, 69], premixed combustion [3, 4, 60, 62, 73, 76], two-phase com-
bustion [53, 56, 57], soot formation [11, 12] and supersonic mixing and combus-
tion [61]. This closure introduces an LEM domain in each control volume of the
3D mesh. The LEM domain size is of the order of the resolution scale of the 3D
mesh, and evolves a 1D profile of the thermochemical state that serves as a repre-
sentative sample of evolution within the control volume that contains it. Special-
izing to LES closure, the LES time-advancement cycle begins with an update of
the coarse-grained LES flow state, consisting of velocity components and density.
(All thermochemical information resides solely within the LEM domains.) Next,
diffusive transport, chemistry, and triplet maps representing subgrid advection are
implemented within each subgrid LEM domain for a time interval equal to the LES
time step. Then the LEM domains communicate with each other by means of a
splicing procedure (see Fig. 10.6). The mass transfer across each LES mesh face
during the LES time step is computed based on the LES-resolved velocity and den-
sity. The prescribed transfer is applied to the affected LEM domains by excising a
piece of the domain that is upwind of the LES face and inserting it into the LEM do-
main that is downwind of the LES face. These processes are schematically shown in
Fig. 10.4 and indicate that the LEM subgrid model can be implemented within any
flow solver without any major revision to the original structure of the fluid dynamics
solver.
To describe this model mathematically, we split the velocity field as: ui = ui +
(ui )R + (ui )S . Here, ui is the LES-resolved velocity field, (ui )R is the LES-resolved
subgrid fluctuation (obtained from ksgs ) and (ui )S is the unresolved subgrid fluc-
tuation. Then, consider the exact species equation (i.e., without any explicit LES
filtering) for the kth scalar Yk written in a slightly different form as:
230 Suresh Menon and Alan R. Kerstein

Fig. 10.4: Schematic of the LEMLES approach.

Yk Yk
= [ui + (ui )R + (ui )S ] ( YkVi,k ) + w k + Sk . (10.6)
t xi xi

Here, Vi,k are the kth species diffusion velocity and Sk is a source term if there
is phase change as in vaporization of liquid fuel. In LEMLES, the above equation is
rewritten in a two-step process as:
 t+ tLES
1 Yk n

Yk Yk = n
[ (ui )S + ( YkVi,k )n w nk Skn ]dt  (10.7)
t xi xi

Yk n+1 Yk Yk n
= [ui + (ui )R ] . (10.8)
tLES xi
Here, tLES is the LES time-step. Equation 10.7 describes the subgrid LEM model,
as viewed at the LES space and time scales. The integrand includes four processes
that occur within each LES grid cell, and represent, respectively, (i) subgrid molec-
ular diffusion, (ii) reaction kinetics, (iii) subgrid stirring, and (iv) phase change of
the liquid fuel. These processes are modeled on a 1D domain embedded inside each
LES grid where the integrand is rewritten in terms of the subgrid time and space
scales. Equation 10.8 describes the large-scale 3D LES-resolved convection of the
scalar field, and is implemented by a Lagrangian transfer of mass across the finite-
volume cell surfaces [3, 46]. Descriptions for the subgrid processes (in Eq. 10.7) and
the 3D advection process (in Eq. 10.8) are presented elsewhere [53] but are repeated
here for completeness.
As shown in Eqs. 10.7 and 10.8, and in Fig. 10.4, there are two different elements
to the LEMLES. We will consider the subgrid LEM (the integrand in Eq. 10.7) and
the resolved-scale transport (Eq. 10.8) separately.
Linear-eddy model 231

Fig. 10.5: Schematic representation of the LEM domain and the flame for (a) stand-
alone LEM, (b) LEMLES.

10.6.1 The LEM Subgrid Model

The LEM is implemented within the LES cells as a subgrid model and Eqs. 10.5 are
solved within each LES cell. In stand-alone LEM simulations the one-dimensional
domain extends across the flame, whereas in the LEMLES approach they are embed-
ded within each LES cell. This is schematically shown in Figs. 10.5a-b. The LEM
domains are independent from each other and the 1D line is notionally aligned in the
flame normal or the maximum scalar gradient direction and thus, does not represent
any physical Cartesian direction.
Since all the turbulent scales below the grid are resolved in this approach, both
molecular diffusion and chemical kinetics are closed in an exact manner. As a result,
scalar subgrid terms do not have to be closed or modeled. The subgrid pressure,
pLEM is assumed constant over the LEM domain, and equal to the LES value, p,
which is a valid assumption in the absence of strong pressure gradients. Hence, the
subgrid density is computed from the equation of state at the subgrid level:
Nspecies
Ru
pLEM
= LEM
T LEM
YkLEM
Wk
. (10.9)
k=1

Here, Wk is the kth species molecular weight. Conservation of mass, momen-

tum and energy (at the LES level) and conservation of mass, energy and species (at
the LEM level) are fully coupled. Chemical reaction at the LEM level determines
heat release and thermal expansion at the LEM level, which at the LES level gen-
erates flow motion that, in turn, transports the species field at the LEM level. Full
coupling is maintained in the LEMLES to ensure local mass conservation.
The reaction-diffusion equation on the LEM domain is solved within each LES
cell with an explicit scheme. The time integration is achieved by using an operator
splitting technique to account for four distinct physical processes and time scales in
the LEM (molecular diffusion, chemistry, turbulent stirring and thermal expansion).
232 Suresh Menon and Alan R. Kerstein
2s
The diffusion time step is calculated as: tdi f f = Cdi f f max(D . The maximum of Dk
k)
is used for tdi f f in order to maintain the stability of the diffusion of the lightest
species. The Cdi f f in the equation is a model constant, set here to 0.25 for numer-
ical stability [63]. The chemical time step size is determined by the stiffness of the
reaction mechanism. The stiffness increases as the number of radical species in a
chemical kinetics mechanism increases. The chemistry is integrated for the given
diffusion time step size ( tdi f f ). A stiff ODE solver which uses adaptive time step
size is employed for the integration process so that the chemical processes are re-
solved in their respective time scales. This approach, nevertheless, is very time con-
suming, and therefore, novel methods such as in-situ adaptive tabulation [11, 12]
and artificial neural network [67, 69, 70] have been implemented within LEM to
achieve speed-up.
The thermal expansion time scale is associated with the volumetric expansion
induced by the increase in temperature through chemical heat release. In the current
implementation texpansion tdi f f , and turbulent stirring is implemented as dis-
crete time events during the reaction-diffusion time integration. The time interval
between each stirring event is: stir (x) = 1/ , where is the stirring frequency
described earlier, and is the LES grid filter size. Overall, for the given LES time
step, the number of stirring events is equal to Nstir tLES / tstir .
In different implementations of this closure strategy, LEM domains can have
either periodic [50] or Neumann boundary conditions [53]. In the former method
the spliced pieces can be excised and inserted at arbitrary locations or a type of first-
in-first-out criterion can be applied. In the latter approach the LEM domain has an
input side and an output side. A common characteristic of these implementations is
that the LEM domains are Lagrangian objects that have no unique spatial location
or orientation within the LES control volume that contains them.

10.6.2 Large-Scale Advection of the Subgrid Field

The large scale advection is implemented in LEMLES to account for the advec-
tion of the scalar field on the resolved level between the LES cells according to Eq.
10.8. This process accounts for species transport in the 3D domain due to both the
resolved LES velocity field ui and the resolved subgrid kinetic energy ksgs . Since
the scalar structure is inherent in the subgrid LEM cells, they are transported across
the LES fields by a Lagrangian transport process. This is in contrast to a conven-
tional finite-volume or finite-difference approximation of the filtered scalar gradi-
ents across LES cells. Thus, the right hand side of Eq. 10.8 is not discretized using
a conventional difference operator but rather, the convection of the subgrid scalar
field (and hence the subgrid scalar gradient, Yk / s) is explicitly carried out by the
Lagrangian splicing approach.
The advection process is implemented once the subgrid evolution of the scalar
field as a result of the turbulent stirring, diffusion, reaction and thermal expansion
are completed at their respective time steps. The large scale advection is a result of
Linear-eddy model 233

Fig. 10.6: Illustration of the splicing strategy, in which thermochemical information

resides solely within LEM domains (tilted line segments) contained with control
volumes of the 3D solver, but flow evolves on the coarse 3D mesh. Solid lines with
arrows are flow velocities, evolved on the coarse mesh, that determine volume trans-
fers between 1D subgrid domains. The splicing mechanism that implements these
transfers is illustrated. Each LEM domain has an input end (circle) and an output
end (square). Open and filled symbols demarcate the LEM domains before and after
splicing. Portions transferred during splicing are separated by tick marks. Dashed
curves with arrows indicate transfers between 1D domains in different control vol-
umes. An alternative to specified input and output locations is to use periodic LEM
domains. Then there are no preferred locations at which to remove and insert do-
main segments unless a first-in-first-out type of criterion is introduced.

both the resolved large scale (ui ) and the modeled subgrid scale velocities ((u i )R ).
In the current implementation, based on the assumption that the velocity field is
isotropic on the small scales, (u i )R is estimated: (u i )R = 23 ksgs . However, if ksgs
is not available this additional flux cannot be included. Regardless, this contribution
is very small and in most cases can be neglected.
Equation 10.8 can be written in a finite volume discrete form as follows,
N
( Yk )n+1 V ( Yk ) V f
 
= j ui + (ui )R j Yk A j (10.10)
tLES j=1

where V is the volume of a finite volume cell, N f is the number of cell faces, and
A j is the cell face area. The large-scale advection of the subgrid scalar structure, Yk ,
is based on the mass flux across each cell face. Therefore, Eq. 10.10 can be rewritten
by defining m = V and rearranging,
Nf
(mYk )n+1 = (mYk ) tLES ( mYk ) j (10.11)
j=1
234 Suresh Menon and Alan R. Kerstein
 
where m = ui + (ui )R A j , which is the mass flux that crosses a cell face, A j .
The last term in (10.11) is further decomposed into influx and outflux components.
Therefore, Eq. 10.11 becomes,
 
(mYk )n+1 = (mYk ) tLES ( mYk )in + ( mYk )out (10.12)
Nin Nout

where Nin and Nout are the number of influx and outflux faces surrounding a finite
volume cell. Here, ( mYk )in is computed by taking portions of the mass contained
in LEM cells in the neighboring finite volume cells. The mass in Nsplice LEM cells
is collected based on the net in-flux and added to the LEM cells in the current finite
volume cell. At the same time, ( mYk )out is computed from the mass contained in
LEM cells in the current finite volume cell and distributed to the neighboring LES
cells based on the out-flux. With proper care to advect the total mass based on con-
vective flux, proper mass conservation can be strictly enforced during this process.
Since this process is also in full 3D, the advection of the subgrid scalar gradients by
this Lagrangian process allows transport of both co-gradient and counter-gradient
subgrid structure across LES cell faces. This is a unique strength of this advection
process when compared to conventional gradient diffusion modeled at the LES level
by a standard finite-difference or finite-volume method.
Due to compressibility, volumetric expansion and grid-stretching, Nsplice may be
different than NLEM (the number of LEM cells in each LES grid volume) and this is
included in the formulation. Furthermore, Nsplice need not be an integer and the cur-
rent algorithm allows for fractions of individual subgrid volumes to be transported,
whereas the predetermined NLEM is an integer multiple of 3, and greater than 9 to
accommodate the triplet mapping procedure.
Once all the mass is transferred between the LES cells, the new LEM field can
have more or less cells than that it had before the splicing started, as shown in Fig.
10.7. Also, the volume of the cells can be different from each other based on the
transferred mass. Re-gridding is applied to uniformly divide all the mass between
LEM cells. This process is schematically shown in Fig. 10.8. Re-gridding is strictly
not needed if a variable subgrid domain is employed and is done primarily for nu-
merical expediency. As in any simplification resulting errors have to be considered.
For example, if re-gridding changes the composition in the LEM cells then it is con-
sidered artificial diffusion. This artificial diffusion can be minimized by increasing
the number of LEM cells or by splicing smaller amounts of mass. While the former
is essentially a grid refinement strategy, the latter is constrained by the time-step
of the flow solver and cannot be arbitrarily changed. In compressible explicit flow
solvers, due to the small LES time-step, the numerical diffusion effect is indeed very
small. Nevertheless, this artifact of the LEM implementation should be eliminated
for overall accuracy, and therefore, a more general implementation of the LEM (with
variable subgrid resolution) is being pursued.
The Lagrangian advection process can track interfaces accurately. For example,
Fig. 10.9 shows advection of an annular box without loss of its integrity in a uniform
flow oriented along the diagonal of the simulation domain. The mass-flux based
Linear-eddy model 235

Fig. 10.7: Species field before and after the splicing of the cell (i, j).

transport process allows counter-gradient transport of scalars (since subgrid scalar

structures are transported by splicing), enabling the model to avoid the pit-falls of
those based on the gradient-diffusion hypothesis. Figure 10.10 shows the propaga-
tion of a circular burning flame, which includes both subgrid burning and thermal
expansion effects. The circular flame is resolved on a Cartesian grid with reasonable
accuracy [3].
Implementation of splicing is relatively straightforward and visually summarized
in Fig. 10.7. Splicing is composed of the following steps: (1) calculate the LES flux,
(ui +(u i )R ) on each of the six faces of the volume, (2) determine the amount of
mass to send as well as the amount of mass to receive from the neighboring LES
cells, (3) arrange the LES fluxes with the largest out-flux carried out first in accor-
dance with the premise that the 1D scalar fields are always aligned in the direction
of the maximum scalar gradient, and (4) after the receipt of mass from each face
commensurate with step 2, rearrange the scalar field in each cell accordingly. Re-
gridding then follows, if there are heat release effects. As noted earlier, re-gridding
is a numerical artifact intended for simplicity and can be eliminated with a more
general formulation.
The ability of the Lagrangian advection to capture the flame structure is summa-
rized in Fig. 10.11, which compares a 3D simulation of a turbulent premixed flame
conducted using DNS and LEMDNS [62]. The latter approach implies that the LES
grid is as fine as the DNS grid but the LEM was included within the subgrid. Thus, in
the DNS limit the splicing process occurs only due to the resolved velocity field and
the LEM subgrid processes are reduced to only diffusion and kinetics (i.e., stirring
is turned off). The excellent agreement between a DNS and the LEMDNS shows
the physically consistent and correct implementation of the scalar evolution by the
two-step procedure (Eqs. 10.7 and 10.8).
A recently developed alternative LEM-based subgrid closure called LEM3D [64]
has a well-defined spatial structure. In fact, its structure is the same as in ODTLES
[66], a method for 3D flow advancement involving an array of coupled 1D domains,
using the one-dimensional-turbulence (ODT) model (see Chapter 11).
236 Suresh Menon and Alan R. Kerstein

Fig. 10.8: Schematic representation of the subgrid scalar field after thermal expan-
sion and re-gridding. Here u and b indicate an unburned and a burned LEM cell,
respectively.

Fig. 10.9: Propagation of an annular box in a velocity field aligned along the diago-
nal.

An advantage of LEM3D relative to the splicing strategy is that it avoids the im-
position of Neumann boundary conditions on LEM within each 3D control volume,
which is the currently preferred splicing formulation. A related advantage is that
a triplet map need not be contained within one 3D control volume, nor need it be
limited in size relative to the control volume. An advantage of the splicing strategy
is that it is readily implemented within an arbitrary structured or unstructured mesh.
LEM3D is most easily implemented on a Cartesian mesh, with some possibility of
generalization to generalized curvilinear coordinates. Owing to the novelty and lim-
ited evaluation of LEM3D to date, further discussion of LEM-based subgrid closure
focuses on the splicing strategy.
Linear-eddy model 237

Fig. 10.10: Propagation of a circular flame front. Reprinted from [3] with permission
from Taylor and Francis.

Fig. 10.11: DNS and LEMDNS of premixed flame-turbulence interaction. Reprinted

from [62] with permission from the Combustion Institute.

10.7 LEMLES Applications to Reacting Flows

As cited above, the LEMLES has been used quite extensively for a range of prob-
lems from simple canonical flame-turbulence interactions to complex flows in gas
turbine combustors. In the following we touch upon some key results primarily to
highlight the predictive ability of the LEMLES approach. An underlying theme in
all these comparisons is that the basic LEMLES approach is identical for all cases.
The only changes occurring are the changes in the LES geometry, test conditions
and appropriate boundary conditions. In the following, we summarize some key re-
sults primarily to highlight the capability of the approach. Cited references have
more detailed analysis and interpretations.
We begin with application to premixed combustion behind a triangular bluff body
[59] that has been extensively studied in the past by various LES methods and clo-
sures. Here, we compare a simple subgrid EBU closure with the LEMLES to high-
238 Suresh Menon and Alan R. Kerstein

Fig. 10.12: Flame structure predicted behind a triangular bluff body. (a) LEMLES
and (b) EBULES. Reprinted from [59].

light some features. EBULES is a very simple closure for the reaction kinetics and
therefore is used extensively in the literature. It strengths and limitations are well
known and it is not the intent here to focus on these issues. Furthermore, other
flame closure methods [14, 15] have also proven quite accurate for this test case
and therefore, the current comparison is primarily to provide a reference. As shown
in Fig. 10.12 and Fig. 10.13 show some key differences between the two predic-
tive methods. The LEMLES flame structure is more wrinkled and its spreading is
increased due to its increase burning rate. This is reflected in the mean tempera-
ture radial profile and the better agreement with the Reynolds stress measurements.
Other premixed studies [3, 4, 9, 60, 62] have demonstrated that LEMLES has the
ability to not only capture the flame propagation speed and structure, it also enables
better prediction of the flame-turbulence interactions.
A more recent study focused on the ability of the LEMLES approach to cap-
ture complex flame structures in both premixed and non-premixed combustion in a
special combustor. The Stagnation Point Reverse Flow (SPRF) is a low NOx com-
bustor [76] that employs exhaust gas recirculation to achieve stable combustion in
lean conditions while minimizing NOx and CO emissions. In this combustor, flame
stabilization is achieved via a high-temperature downstream stagnation region. The
schematic of the device shown in Fig. 10.14 consists of a generic can combustor
with the inlet and the exhaust on the same end of the combustor. This design en-
ables the EGR-preheated air to provide a high temperature and stable combustion
environment. Modeling the flow and flame structure in this device is complicated
by the interaction between various features of flow confinement, stagnation, jet en-
trainment and product preheating and dilution through interaction of the incoming
mixture with reverse co-flow. Experimentally, this combustor was shown to oper-
ate in both premixed and non-premixed combustion mode with low emissions but
Linear-eddy model 239

Fig. 10.13: The time-averaged radial profiles behind the bluff body. (a) Reynolds
stress, and (b) mean temperature at three axial locations behind the bluff body.
Reprinted from [59].

with very different flame structures, and therefore, it is a challenge for numerical
prediction.
The recent study [76] simulated both premixed and non-prmeixed combustion
using LEMLES and compared with other subgrid models such as the subgrid eddy
break-up, artificially thickened flame and steady flamelets. Here, only representative
results of the EBULES and LEMLES are discussed, although comparison with other
models were reported earlier. Figure 10.15 compares the predictions with the aver-
aged chemiluminescence field (note that although not clear here, the data does show
that this flame is attached [20]). The EBULES flame structure is not clearly defined
whereas the LEMLES predicts roughly the same flame length and reveals a flame
240 Suresh Menon and Alan R. Kerstein

Fig. 10.14: Schematic and typical dimensions of the SPRF combustor.

anchored to the injector with heat release along the shear layer and in the down-
stream region of the combustor in agreement with the data. Comparison with data
is reported elsewhere [76] and shows excellent agreement at nearly all locations.
When the combustor is operated in the non-premixed mode (with the fuel injected
through the inner tube) the flame is lifted (Fig. 10.16). The LEMLES prediction is
also a lifted flame whereas the EBULES did not capture this effect. The LEMLES
flame structure and shape is somewhat different from the measurements but this
is attributed to the reduced 1-step kinetics employed [76]. Regardless, the overall
agreement with data is encouraging considering that the same model is employed
for both premixed and non-premixed combustion without any ad hoc adjustments.

Fig. 10.15: Heat release in the premixed SPRF configuration. (a) Experiments, (b)
LEMLES and (c) EBULES. Reprinted from [76] with permission from the Com-
bustion Institute.
Linear-eddy model 241

Fig. 10.16: Heat release in the non-premixed SPRF configuration. (a) Experiments,
(b) LEMLES and (c) EBULES. Reprinted from [76] with permission from the Com-
bustion Institute.

Finally, an application in spray combustion is discussed briefly although more de-

tails are given elsewhere [53, 56, 57]. The LDI combustor consists of a 60-degree,
six helical swirl-vaned inlet that leads to a venturi, followed by a short divergent
diffuser section that ends at the dump plane of a square combustion chamber. The
swirler has outer diameter of 22.5 mm with the inner diameter of 8.8 mm. The
calculated swirl number is 1.0 and Fig. 10.17(a) shows the schematic of the swirl
generating blades in this combustor. The entire combustor (including the swirler as-
sembly) is simulated with and without break-up modeling of the kerosene-air com-
bustion system [56, 57]. Figure 10.17(b) shows the spray in the vicinity of the injec-
tor showing the breakup process. Details are in these cited references but here we
focus on the nature of flame holding and the typical characteristics of the flame to
highlight the ability of the LEMLES approach.
Centerline mean streamwise velocity is shown in Fig. 10.18(a). A prominent cen-
tral re-circulation zone (CRZ) along the axis is observed for the non-reacting [56]
(not shown) and reacting cases. This CRZ is created by the swirling inflow due to
a radial pressure gradient caused by the centrifugal effect, which in turn gives rise
to axial (and adverse) pressure gradient. For high swirl numbers, a strong coupling
between axial and tangential velocity occurs and the adverse pressure gradient is
strong enough to overcome the axial motion of the fluid. This establishes a recircu-
lation zone, a form of vortex breakdown in the central region. This is the primary
aerodynamic flame holding and stabilizing mechanism in gas turbine combustors.
242 Suresh Menon and Alan R. Kerstein

Fig. 10.17: The LDI swirler assembly and an instantaneous view of the droplets in
the near -vicinity of the injector showing breakup. Reprinted from [57] with permis-
sion from the Combustion Institute.

Fig. 10.18: Centerline velocity decay and flame structure in the LDI combustion. (a)
Centerline axial velocity, (b) Flame index contours, with the black lines indicating
the stoichiometric mixture fraction contour. Reprinted from [56] with permission
from the Combustion Institute.

Instantaneous flame structure is analyzed using the flame index, FI=YF .YO2 .
To determine the flame regime, an indexed reaction rate is defined based on the
flame index as: F = | F | |FI|
FI
, and is shown in Fig. 10.18(b). Here, F is the fuel
destruction rate. The stoichiometric equivalence ratio is shown as a thin line in the
same figure. The flame is premixed when the FI (and consequently F ) is positive
and diffusion when the FI is negative. In the central region, presence of fuel vapor in
Linear-eddy model 243

proximity of recirculating hot gases devoid of oxidizer generates a diffusion flame,

as seen by light colored V shaped flame surface. This is confirmed by the coinci-
dence between the flame and the stoichiometric line. Further outwards in the radial
direction, significant dark colored contours are seen, indicating a premixed flame.
Along the outer edges there is sufficient time for fuel-air mixing to complete before
ignition, and thus, both non-premixed and premixed flames occur adjacently near
the top half of the combustor with diffusion burning of fuel vapor evaporated from
the particles that have gone through the primary flame without completely losing
their identity. Since the LEMLES approach does not make any a priori assump-
tions regarding the nature of the flame it is able to capture the multi-faceted flame
structure in these combustors.

10.8 Summary and Future Prospects

As summarized here the LEM has been used both as a stand-alone model and as
a subgrid model for LES. Application of LEM as a RANS subgrid model has also
been reported in some earlier and recent studies. The ability of this modeling strat-
egy to study a wide range of reacting flows without any model adjustments (when
combined with the LDKM closure for momentum there are no ad hoc adjustable pa-
rameters in the method) is one of the key strengths of this modeling approach. The
robustness for a wide range of applications comes with an increase in cost. Efficient
parallel implementation (the LEM subgrid model is highly parallel in its nature)
can reduce the cost significantly but it is still expensive when compared to a steady
flamelet or EBULES approach. On the other hand, the LEM-based closure can be
used to go from one simulation regime to another without changing the model and
this is a critical strength essential to study complex flame dynamics as in combus-
tion instability, LBO and extinction-re-ignition processes. More recent successes in
parametrizing the LEM in a neural network [67, 68, 70] offers a new potential for
incorporating this models ability in a cost effective LES strategy.

Acknowledgement

The research at Georgia Tech is supported in part by Office of Naval Research,

NASA Glenn Research Center, Air Force Office of Scientific Research, General
Electric Aircraft Engine Company and Pratt & Whitney. This work in SNL is par-
tially supported by the U.S. Department of Energy, Office of Basic Energy Sciences,
Division of Chemical Sciences, Geosciences, and Energy Biosciences. Sandia Na-
tional Laboratories is a multi-program laboratory operated by Sandia Corporation,
a Lockheed Martin Company, for the United States Department of Energy under
contract DE-AC04-94-AL85000.
244 Suresh Menon and Alan R. Kerstein

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42. Krueger, S.K., Su, C.W., McMurtry, P.A.: Modeling entrainment and finescale mixing in cu-
mulus clouds. J. Atmos. Sci. 54, 26972712 (1997)
43. McMurtry, P.A., Gansauge, T.C., Kerstein, A.R., Krueger, S.K.: Linear eddy simulations of
mixing in a homogeneous turbulent-flow. Phys. Fluids A 5, 10231034 (1993)
44. McMurtry, P.A., Menon, S., Kerstein, A.R.: A new subgrid model for turbulent combustion:
Application to hydrogen-air combustion. Proc. Combust. Inst. 24, 271278 (1993)
45. Menon, S.: Computational and modeling constraints for large-eddy simulations of turbulent
combustion. Int. J. Engine Res. 1, 209227 (2000)
46. Menon, S., Calhoon, W.H.: Subgrid mixing and molecular transport modeling for large-eddy
simulations of turbulent reacting flows. Proc. Combust. Inst. 26, 5966 (1996)
47. Menon, S., Calhoon, W.H., Goldin, J.R., Kerstein, A.R.: Effects of molecular transport on
turbulent-chemistry interactions in hydrogen-argon-air jet diffusion flame. Proc. Combust.
Inst. 25, 11251131 (1994)
48. Menon, S., Kerstein, A.R.: Stochastic simulation of the structure and propagation rate of tur-
bulent premixed flames. Proc. Combust. Inst. 24, 443450 (1992)
49. Menon, S., Kim, W.W.: High Reynolds number flow simulations using the localized dynamic
subgrid-scale model. AIAA-96-0425 (1996)
246 Suresh Menon and Alan R. Kerstein

50. Menon, S., McMurtry, P., Kerstein, A.R.: A linear eddy mixing model for large eddy simula-
tion of turbulent combustion. In: B. Galperin, S. Orszag (eds.) LES of Complex Engineering
and Geophysical Flows, pp. 287314. Cambridge University Press, Cambridge, UK (1993)
51. Menon, S., McMurtry, P.A., Kerstein, A.R., Chen, J.Y.: A new mixing to predict NOx produc-
tion in turbulent hydrogen-air jet flame. J. Prop. Power 10, 161168 (1994)
52. Menon, S., Pannala, S.: Subgrid combustion simulations of reacting two-phase shear layers.
AIAA-98-3318 (1998)
53. Menon, S., Patel, N.: Subgrid modeling for LES of spray combustion in large-scale combus-
tors. AIAA J. 44, 709723 (2006)
54. Menon, S., Yeung, P.K., Kim, W.W.: Effect of subgrid models on the computed interscale
energy transfer in isotropic turbulence. Computers Fluids 25, 165180 (1996)
55. Oevermann, M., Schmidt, H., Kerstein, A.R.: Investigation of autoignition under thermal strat-
ification using linear eddy modeling. Combust. Flame 155, 370379 (2008)
56. Patel, N., Kirtas, M., Sankaran, V., Menon, S.: Simulation of spray combustion in a lean direct
injection combustor. Proc. Combust. Inst. 31, 23272334 (2007)
57. Patel, N., Menon, S.: Simulation of spray-turbulence-flame interactions in a lean direct injec-
tion combustor. Combust. Flame 153, 228257 (2008)
58. Pope, S.B.: Pdf methods for turbulent reactive flows. Prog. Energy Combust. Sci. 11, 119192
(1985)
59. Porumbel, I., Menon, S.: Large-eddy simulation of bluff body stabilized premixed flames.
AIAA-2006-0152 (2006)
60. Sankaran, V., Menon, S.: Structure of premixed flame in the thin-reaction-zones regime. Proc.
Combust. Inst. 28, 203210 (2000)
61. Sankaran, V., Menon, S.: LES of scalar mixing in supersonic shear layers. Proc. Combust.
Inst. 30, 28352842 (2005)
62. Sankaran, V., Menon, S.: Subgrid combustion modeling of 3-d premixed flames in the thin-
reaction-zone regime. Proc. Combust. Inst. 30, 575582 (2005)
63. Sankaran, V., Porumbel, I., Menon, S.: Large-eddy simulation of a single-cup gas turbine
combustor. AIAA-2003-5083 (2003)
64. Weydahl, T.: A framework for mixing-reaction closure with the linear-eddy model. Ph.D. The-
sis, Norwegian University of Science and Technology, Trodheim, Norway (2010)
65. Schenck, H.W., Wendt, J.O.L., Kerstein, A.R.: Mixing characterization of transient puffs in a
rotary kiln incinerator. Combust. Sci. Technol. 116, 427453 (1996)
66. Schmidt, R.C., Kerstein, A.R., McDermott, R.: ODTLES: A multi-scale model for 3D turbu-
lent flow based on one-dimensional turbulence modeling. Comput. Meth. Appl. Mech. Engg.
199, 865880 (2009)
67. Sen, B.A., Hawkes, E., Menon, S.: Large eddy simulation of extinction and reignition with
artificial neural networks based chemical kinetics. Combust. Flame 157, 566578 (2010)
68. Sen, B.A., Menon, S.: Artificial neural networks based chemistry-mixing subgrid model for
LES. AIAA-2009-0241 (2009)
69. Sen, B.A., Menon, S.: Turbulent premixed flame modeling using artificial neural network
based chemical kinetics. Proc. Combust. Inst. 32, 16051611 (2009)
70. Sen, B.A., Menon, S.: Linear eddy mixing and artificial neural networks for LES subgrid
chemistry closure. Combust. Flame 157, 6274 (2010)
71. Smith, T., Menon, S.: One-dimensional simulations of freely propagating turbulent premixed
flames. Combust. Sci. Technol. 128, 99130 (1996)
72. Smith, T.M., Menon, S.: Model simulations of freely propagating turbulent premixed flames.
Proc. Combust. Inst. 26, 299306 (1996)
73. Smith, T.M., Menon, S.: Large-eddy simulations of turbulent reacting stagnation point flows.
AIAA-97-0372 (1997)
74. Smith, T.M., Menon, S.: Subgrid combustion modeling for premixed turbulent reacting flows.
AIAA-98-0242 (1998)
Linear-eddy model 247

75. Su, C.W., Krueger, S.K., McMurtry, P.A., Austin, P.H.: Linear eddy modeling of droplet
spectral evolution during entrainment and mixing in cumulus clouds. Atmos. Res. 47, 4158
(1998)
76. Undapalli, S., Menon, S.: LES of premixed and non-premixed combustion in a stagnation
point reverse flow combustor. Proc. Combust. Inst. 32, 15371544 (2009)
77. Woosley, S.E., Kerstein, A.R., Sankaran, V., Aspden, A.J., Roepke, F.K.: Type Ia supernovae:
Calculations of turbulent flames using the linear eddy model. Astrophys. J. 704, 255273
(2009)
78. Wu, J., Menon, S.: Aerosol dynamics in the near-field engine exhaust plumes. J. Appl. Meteo.
40, 795809 (2001)
79. Zimberg, M.J., Frankel, S.H., Gore, J.P., Sivathanu, Y.R.: A study of coupled turbulent mixing,
soot chemistry, and radiation effects using the linear eddy model. Combust. Flame 113, 454
469 (1998)
Chapter 11
The One-Dimensional-Turbulence Model

Tarek Echekki, Alan R. Kerstein, and James C. Sutherland

Abstract The one-dimensional turbulence (ODT) model represents an efficient and

novel multiscale approach to couple the processes of reaction, diffusion and turbu-
lent transport. The principal ingredients of the model include a coupled determin-
istic solution for reaction and molecular transport and a stochastic prescription for
turbulent transport. The model may be implemented as stand-alone for simple tur-
bulent flows and admits various forms for the description of spatially developing
and temporally developing flows. It also may be implemented within the context of
a coupled multiscale solution using the ODTLES approach. This chapter outlines
the model formulation, and applications of ODT using stand-alone solutions and
ODTLES.

11.1 Motivation

In his survey of turbulent combustion modeling, Norbert Peters [25] emphasizes the
difficulties that can arise due to the interactions between turbulence and chemistry
over a wide range of length and time scales. Assumptions about inertial-range scal-
ing of the turbulent cascade are not necessarily applicable, and there are few if any
formal or conceptual constructs to which the modeler can turn when these scalings
do not apply. He notes important empirical evidence that gross features of turbu-
lent combustion often conform to inertial-range phenomenology, particularly with
regard to its most important consequence for combustion: the length and time scale

Tarek Echekki
North Carolina State University, Raleigh, NC, USA, e-mail: techekk@ncsu.edu
Alan R. Kerstein
Sandia National Laboratories, Livermore, CA, USA, e-mail: arkerst@sandia.gov
James C. Sutherland
The University of Utah, Salt Lake City, UT, USA, e-mail: James.Sutherland@utah.edu

T. Echekki, E. Mastorakos (eds.), Turbulent Combustion Modeling, 249

Fluid Mechanics and Its Applications 95, DOI 10.1007/978-94-007-0412-1 11,
Springer Science+Business Media B.V. 2011
250 T. Echekki, A.R. Kerstein and J.C. Sutherland

separation between the dominant large-scale turbulent motions and the small scales
of molecular transport and chemical kinetics in flames. Many existing models rely
on this scale separation, while the linear-eddy model (LEM: see Chapter 10) and
one-dimensional turbulence (ODT) do not. To the extent that scale separation is not
obeyed in turbulent combustion processes, LEM and ODT can be especially suitable
for modeling these processes.
Peters discusses the role of scale invariance as well as scale separation in the
inertial range. The latter is more general: scale separation is possible in the absence
of scale invariance, but there is no scale invariance without scale separation (see
below). Scale invariance of spectral energy transfer in turbulence, in conjunction
with (and in fact, relying on) the negligible influence of viscous dissipation in the
inertial range (an idealization, but valid in this context), imply that the characteristic
eddy time scale (l) at length scale l obeys the dependence (l) l 2/3 . This implies
power-law dependences of eddy velocity, energy, and diffusivity, and ultimately,
scale separation between inertial-range and dissipation-range processes.
LEM resolves all advective, diffusive, and chemical length and time scales of tur-
bulent combustion, and hence does not rely on scale separation. It represents thermal
expansion by means of dilatation of the 1D domain, but it does not model other as-
pects of feedback from combustion to turbulent motions, such as viscosity variations
and turbulence generation by expansion. In particular, the probability distribution
function (pdf) from which the maps representing turbulent eddies are sampled has a
fixed functional form whose construction is guided by the inertial-range diffusivity
scaling (Chapter 10). To the extent that turbulence-chemistry interactions result in
deviations from this scaling, the fidelity of LEM can be impaired. Other influences,
such as rapid transients caused by initial conditions or time-varying boundary con-
ditions, can also cause significant deviations from inertial-range scalings.
As noted, there is little theoretical guidance on how to model the deviations from
inertial-range scalings that might be caused by these influences or the effects of
these deviations on combustion dynamics. Within the 1D stochastic approach based
on the triplet-map representation of turbulent eddies (see Chapter 10), incorporation
of these influences is therefore approached from an entirely different perspective.
This perspective is introduced by first considering direct numerical simulation
(DNS) of turbulent combustion. In DNS, no theory or guiding principle is needed
to capture combustion-induced deviations from inertial-range scaling because the
underlying equations of motion are solved. Thus, the consequences of turbulence-
chemistry interactions are outcomes of the simulated flow evolution that do not re-
quire prior analysis or modeling. To the extent that the local, time-resolved interac-
tions between turbulent eddy motions and combustion processes can be represented
robustly with the 1D stochastic framework, that framework can likewise capture
their consequences as outcomes of simulated evolution.
A formulation with this capability becomes, in effect, a broadly predictive model
of turbulent flow evolution rather than a model focused, as LEM is focused, on sim-
ulation of mixing and reaction in a parametrically specified turbulent environment.
Thus, the quest for a robust turbulent combustion model leads ultimately to recon-
sideration of turbulence modeling more generally. This is the context in which ODT
One-Dimensional Turbulence 251

was originally formulated. The stages of its development for application to free and
confined shear flows, variable-density flows, buoyant stratified flows, and multi-
phase flows, as well as combustion and other reacting flows, have been documented
in references cited below.
To introduce ODT, a formulation applicable to constant-property flow is outlined.
Algorithmic as well as physical modeling considerations are discussed. Extensions
needed for combustion applications and representative examples are described. Use
of ODT for subgrid modeling in 3D simulations of constant-property flow and of
combustion is discussed, and future prospects are assessed.

11.2 Constant-Property ODT

11.2.1 Model Formulation

ODT is introduced with reference to LEM by formulating LEM in a notation that

carries over directly to ODT. An eddy rate distribution (y0 , l;t) is defined by setting
(y0 , l;t) dy dl equal to the expected number, per unit time, of eddies for which the
lower boundary of the eddy is in [y0 , y0 + dy] and the eddy size is in the range
[l, l + dl]. In terms of and f (l) defined in Chapter 10, (y0 , l;t) = f (l). Thus,
the units of are 1/(length2 time) and its integral over eddy sizes l is .
In LEM, has no dependence on y0 or t unless one chooses to hard-wire such
dependence into or f (l), which has been done in some instances, e.g., [13]. ODT
is formulated to incorporate such dependence in a way that reflects the relationship
between the likelihood of an eddy, quantified by the eddy time scale , and the local
flow state. This requires the introduction of a local, instantaneous representation of
the flow state in ODT, causing ODT to be a fundamentally different type of model
than LEM, whose turbulent state is characterized parametrically.
The flow representation in ODT consists of 1D profiles of one or more veloc-
ity components whose evolution, in the simplest constant-property formulation, is
qualitatively the same as in LEM applied to a constant-density passive scalar. For
example, with one velocity component u(y,t), time advancement is governed by

u 2u
= 2, (11.1)
t y
where is the kinematic viscosity. This advancement is punctuated by eddy
events, each of which consists of a triplet map (defined and explained in Chap-
ter 10) applied to the u profile, possibly (depending on the formulation) followed
by another operation that is described shortly. Absent the latter, the evolution of u is
equivalent to the evolution of a constant-property passive scalar in LEM, except for
the crucial distinction that (y0 , l;t) is now a specified function of the current flow
state u(y,t). Several formulations of this dependence have been introduced during
the course of ODT development. The formulation discussed here,
252 T. Echekki, A.R. Kerstein and J.C. Sutherland

 2
C uK l
= Z, (11.2)
l4

is a specialization of the formulation described in [17], which has a more detailed

explanation of the motivation and features of ODT than can be provided here.
In Eq. 11.2, C and Z are free parameters whose roles are explained shortly, and
for any property profile s(y,t),

1
sK s(M(y))K(y) dy (11.3)
l2
defines sK . Here, M(y) is defined by the formal mathematical representation of the
triplet map,
s(y) s(M(y)), (11.4)
which indicates that the value of property s at M(y) is mapped to location y. Thus,
M is the operational inverse of the triplet map, which is formally convenient because
M is a single-valued map but the triplet map is triple-valued. The kernel K(y)
y M(y) is the map-induced displacement of the point that is mapped to y by the
triplet map. K(y) thus depends on map parameters y0 and l, but this dependence is
suppressed in the condensed notation used here.
To explain the role of the kernel, the more general form of the eddy event in ODT
is introduced. Namely, Eq. 11.4 is generalized to

s(y) s(M(y)) + cs K(y), (11.5)

which indicates that the eddy event applies the triplet map to property s and then
adds the kernel times a coefficient cs to s(y). The kernel addition is applied only
to velocity components and is intended to add or substract kinetic energy with-
out changing the total momentum, which is assured for constant-density flow be-
cause K(y) integrates to zero. This provides a mechanism for energy redistribution
among velocity components when the formulation contains more than one compo-
nent, enabling the model to simulate the tendency of turbulent eddies to drive the
flow toward isotropy. Total energy must be conserved, imposing a constraint on the
values of the coefficients cs . The additional constraints needed to uniquely deter-
mine all the coefficients are obtained by requiring the kernel addition to produce an
energy distribution within the eddy interval [y0 , y0 + l] that is as close to isotropic
as possible. There are other reasonable criteria for determining the coefficients that
might be advantageous in some cases [19]. In applications such as buoyant strati-
fied flows or turbulent advection of immiscible liquids, eddy events might induce
changes of the gravitational or surface-tension potential energy, requiring equal-
and-opposite changes of kinetic energy. Through the kernel operation, conservation
of total energy in ODT couples flow evolution to dynamically active scalars such
as density in buoyant flows, as demonstrated in ODT studies of these flows [5, 15
17, 20, 35, 36, 3840].
One-Dimensional Turbulence 253

By construction, Eq. 11.5 conserves momentum for constant-density flow. For

variable-density flow, a more general treatment is needed. Momentum conservation
is no longer automatic, so extra degrees of freedom are introduced to enforce mo-
mentum conservation, further generalizing the eddy event as follows:

s(y) s(M(y)) + bs J(y) + cs K(y), (11.6)

where J(y) |K(y)| and the additional coefficients bs are determined by requiring
the eddy event to conserve the y-integrated momentum of all velocity components s.
For details, see [1].
Most combustion applications of ODT pre-date this variable-density formula-
tion, and several pre-date the introduction of the kernel formalism in Eqs. 11.2 and
11.5, so they do not include the kinetic-energy and variable-density phenomenology
that can now be incorporated into ODT. The less complete treatment is adequate
for many combustion applications, much as LEM, which is considerably simpler,
is broadly useful for turbulent combustion simulation. The variable-density formu-
lation is not discussed further here, but its future use, where appropriate, in ODT
simulations of turbulent combustion is encouraged.
Before proceeding further, the guiding principle that motivates the model con-
structs introduced thus far is explained. Empirical evidence and formal analysis
support the viewpoint that the turbulent cascade tends to be local in scale space,
meaning that individual eddy motions such as vortex stretching typically shrink flow
features in turbulence by order-one geometrical increments, such that the wide range
of flow scales in turbulence is the cumulative outcome of many incremental scale
reductions rather than a smaller number more drastic reductions.
Enforcement of this scale-locality principle in ODT is the basis of much of the
model formalism. The triplet map decreases flow scales by no more than a factor
of three. No other measure-preserving map induces less scale reduction. (Measure
preservation assures that applicable conservation laws are obeyed.)
Energy changes during the eddy event are likewise consistent with scale locality.
The kernel used for this purpose must be zero at the eddy endpoints (to prevent
discontinuities) and must integrate to zero, so it must have at least two extrema.
The function K(y) consists of three linear segments over size-l/3 spatial intervals.
Thus it introduces structure consistent with the scale reduction by the triplet map.
The map is applied before the kernel because the kernel followed by the map would
introduce structure at scale l/9.
Eddy selection as well as eddy implementation is guided by scale locality, in
this case meaning that size-l motions are driven by size-l influences. Eddy phe-
nomenology (i.e., mixing-length concepts applied to an individual eddy) suggests
that should be of order 1/(l 2 ), where is the eddy turnover time, or equiva-
lently, V (l)/l 3 , where V (l) is the velocity difference between the eddy endpoints.
The latter estimate, with numerical coefficients absorbed in the free parameter C
of Eq. 11.2, was used in the original ODT formulation [15] and many subsequent
applications. When kernels were introduced, the estimate V (l) uK was adopted,
which connects eddy selection to energy-based eddy implementation using an ex-
254 T. Echekki, A.R. Kerstein and J.C. Sutherland

pression that measures velocity variations over an order-l distance, and hence is
consistent with scale locality. In Eq. 11.2, the expression for involves a square
root that contains a term proportional to u2K , hence a kinetic-energy term. In formu-
lations involving multiple velocity components or other energy contributions such
as gravitational potential energy, these contributions are additive under the square
root.
A specific connection between uK and eddy energetics is identified through con-
sideration of the possible range of kernel-induced energy changes. Kernel addition
can reduce the u kinetic energy within the eddy to zero only if the spatial profile of
u within the eddy is proportional to the kernel function, so that kernel addition can
make u identically zero within the eddy. Otherwise, there is a maximum amount of
energy that can be extracted from the u profile by kernel addition that is less than the
total u kinetic energy within the eddy. This maximum, termed the available energy
of the u component, is (27/8) lu2K , where is the density [19]. This connects uK to
flow energetics in various ways. For example, implementation of the isotropy crite-
rion involves assignment of the coefficients cs so as to equalize component available
energies. In buoyant stratified flows, the available energy is the maximum kinetic en-
ergy that can be extracted in order to compensate for an equal-and-opposite change
of gravitational potential energy. If there is less than the needed amount of available
energy, then the eddy is energetically prohibited, so its value is set equal to zero.
The indication of a prohibited eddy is that the quantity in the square root in
Eq. 11.2 is negative. The parameter Z is introduced so that an eddy can be prohibited
even if there is net available energy. As indicated by the normalization of uK in that
equation, Z in effect sets a threshold Reynolds number for eddy turnover. Nonzero
Z is not always required for good model performance, but in some instances it im-
proves the results sufficiently to justify the introduction of an additional adjustable
parameter. In some instances, Z is assigned a small positive value solely for compu-
tational efficiency. It prevents the implementation of unphysically small eddies that,
if implemented, would have no noticeable effect on results of interest.
C is the main adjustable parameter of ODT. It scales the eddy event rate, and
hence the simulated turbulence intensity, for a given flow configuration. In transient
flows, it controls overall time development, e.g., the spreading rate of free shear
flows.
Just as there can be a need to assign a positive Z value to suppress small ed-
dies, there can be a need to suppress unphysically large eddies that would otherwise
occur. This need arises primarily in simulations of free shear flows with laminar
co-flows or free streams. Eddies much larger than the width of the turbulent region
can have enough available energy to enable their occurrence. (For a planar mixing
layer, the difference between the free-stream velocities can provide enough avail-
able energy irrespective of any turbulence.) Such eddies violate the scale-locality
requirement that the scale of the flow features that provide the available energy for
eddy occurrence should be of the order of the eddy size.
Several large-eddy-suppression procedures have proven useful. One that is found
to work particularly well for jets and jet diffusion flames [7, 27] requires that
(which is l/uK in the formulation described here) must be less than the elapsed flow-
One-Dimensional Turbulence 255

advancement time times an adjustable coefficient, otherwise the eddy is prohibited.

The introduction of an additional free parameter is found to be well justified by
the resulting model performance. An alternative that was used to simulate mixing
layers [1] and buoyant fire plumes [31] is the scale-reduction method, in which
the eddy is divided into three equal parts and each of these must be found to have
enough available energy for eddy turnover (based on a small Z value that negates any
numerical noise contribution; results are insensitive to the chosen value), otherwise
the eddy is prohibited. If not prohibited based on this test, the eddy is processed in
the usual manner.

11.2.2 Numerical Implementation

Based on Eq. 11.3, uK and thus depend on y0 and l, and this dependence is time
varying due to the time advancement of u(y,t). Therefore at each instant there is
a new eddy rate distribution from which individual eddy events are to be sampled.
Computation of, and sampling from, this two-parameter distribution on an ongoing
basis is computationally unaffordable. Therefore, the thinning algorithm [22] for
efficient sampling from nonstationary Poisson processes (which is a generalization
of the von Neumann rejection method) is employed. A fixed eddy rate distribution
is constructed so as to oversample all eddies, i.e., it exceeds the true value for all
y0 , l, and t. When an eddy is sampled from the fixed distribution, the true value for
that eddy based on the flow state at that instant is computed and the eddy is accepted
with probability / , otherwise rejected. This approach strongly influences many
aspects of algorithm formulation and coding.
ODT has been implemented numerically using both uniform and adaptive meshes.
On a uniform mesh, the triplet map is implemented as a permutation of mesh cells.
On an adaptive mesh, the mathematical definition of the triplet map on the spa-
tial continuum is applied. Properties are assumed constant within each cell, so the
continuum triplet map is applied to piecewise constant continuum property profiles.
This involves mapping cell faces, which creates new faces because the map is multi-
valued, and assigning cell property values accordingly.
The uniform-mesh implementation is described in many of the publications that
have been cited, and a uniform-mesh code and documentation are available for
download [9]. The adaptive-mesh implementation is explained and applied in [31].

11.2.3 Generalizations and Couplings

The adaptive mesh facilitates several generalizations of ODT that are difficult to
implement on a uniform mesh. It allows Lagrangian rather than Eulerian implemen-
tation of advection (in the conventional sense), which is useful for incorporating
thermal expansion and for implementing spatial (streamwise) rather than temporal
256 T. Echekki, A.R. Kerstein and J.C. Sutherland

advancement of ODT. Spatial advancement conserves property fluxes rather than

properties. Flux conservation requires an implementation of continuity that involves
dilatations along the 1D domain, which are convenient to implement in a Lagrangian
manner. The numerics of Eulerian spatial advancement are especially challenging
for variable-property flows. This motivated the introduction of an adaptive mesh
to simulate the vertical spatial development of a fire plume [31]. Variable-property
spatial advancement has been implemented on a uniform mesh by performing a
Lagrangian sub-step and then interpolating the displaced mesh back to the fixed
uniform mesh [1].
The adaptive mesh also facilitates ODT implementation in cylindrical geometry,
in which triplet maps must conserve r dr rather than dy. In this case, the triplet map
is not readily approximated by permuting the cells of a uniform mesh. An adaptive-
mesh implementation of a cylindrical spatial formulation has been used to simulate
round jet diffusion flames [21]. An earlier cylindrical LEM formulation on a uniform
mesh [14] conserves ensemble averages but is not locally conservative, which is less
desirable but adequate for some purposes.
A spatially advancing ODT realization can be interpreted as a 2D flow snap-
shot. On this basis, a spatially advancing fire-plume simulation [31] has been used
to compute 2D radiation fields, which are then used to specify the background ra-
diation field for the next simulated realization. This alternation between ODT and
the radiation computation was iterated to statistical convergence to obtain a coupled
flow-radiation solution.
Another physical process that has been coupled to the ODT flow simulation is
inertial-particle response to turbulent motions (one-way coupling). This formulation
has been used to simulate wall deposition in channel flow [34].
For some applications, full spatial resolution is unaffordable even in 1D. There-
fore various approaches to subgrid closure in ODT have been developed and applied
[20, 24].
Chapter 10 describes ways in which LEM domains have been coupled to under-
resolved 3D flow simulations to provide mixing and chemistry closure. ODT has
the capability to provide subgrid momentum closure as well, as demonstrated in ap-
plications to channel flow [33] and homogeneous decaying turbulence [32]. Various
formulations of ODT-based 3D flow simulation have been proposed [17, 18, 23].
Formulations that have been used for combustion simulation are described in
Sect. 11.3.3.

11.2.4 Features of the ODT Representation of Turbulent Flow

The ODT representation of a time-developing KelvinHelmholtz instability, illus-

trated in Fig. 11.1, indicates some of the flow features captured by the model. This
illustration is based on the ODT formulation of [19].
The rendering shows that the width of the active mixing zone grows primarily by
the relatively infrequent occurrence of a large event extending beyond the current
One-Dimensional Turbulence 257

range of the mixing zone, with some additional contribution by the more numerous
small events. This process is consistent with the dominant role of large engulfing
motions and the secondary role of small-scale nibbling in turbulent entraining flows
under neutral-buoyancy conditions. (The effect of density stratification on the ODT
representation of turbulent entrainment has been investigated [1, 15].)

Fig. 11.1: Graphical representation of the sequence of eddy events during a sim-
ulated ODT realization of a time-developing KelvinHelmholtz instability (left
panel) and a time-developing planar wake (right panel) [19]. The KelvinHelmholtz
and wake simulations are initialized using step-function and top-hat initial velocity
profiles, respectively. The space and time units in this illustration are arbitrary. In
the plots, each eddy is represented by an error bar whose vertical span corresponds
to the eddy range [y0 , y0 + l], and whose horizontal location corresponds to the time
of eddy occurrence. Reprinted from [19] with permission. Copyright 2001, Cam-
bridge University Press.

Bunching of events, especially after the occurrence of a large event, reflects the
interactions between the eddy events and the evolving velocity profile that induce
the model analog of the turbulent cascade. Each eddy event compresses and folds
the velocity profile within the range of the eddy. This increases the local shear and
thus the available energy that determines the frequency of subsequent eddies within
that range. A feedback process is thus induced that promotes the occurrence of suc-
cessively smaller eddies. Eventually, velocity fluctuation length scales are reduced
sufficiently so that damping of the fluctuations by concurrent viscous transport dom-
inates the production of fluctuations by eddies. Viscous damping thus terminates the
local burst of eddy activity.
A planar-wake simulation is also shown in Fig. 11.1. In the KelvinHelmholtz
simulation, vigorous turbulence, indicated by the number and size range of eddies
as the flow evolves, is sustained by the shear imposed on the flow by the free-stream
conditions (far-field velocity difference). The wake, however, evolves in a uniform
background. As the initial velocity perturbation is dispersed by eddies and dissi-
258 T. Echekki, A.R. Kerstein and J.C. Sutherland

pated by concurrent viscous evolution, the turbulence intensity decreases, affecting

the eddy frequency and size range and slowing the growth of the turbulent zone.
These qualitative impressions are supported by the quantitative consistency of ODT
simulation statistics with the known similarity scalings for these flows [19].

11.3 Applications of ODT in Combustion

Like its predecessor, LEM (see Chapter 10), the ODT model may be implemented
either as a stand-alone model or within the context of a 3D solution, such as LES.
The stand-alone ODT model may also serve a similar role to direct numerical sim-
ulations (DNS) for the construction of libraries for turbulence-chemistry interac-
tions [2830]. ODT stand-alone models are limited in scope to flows with one dom-
inant flow direction, where a boundary-layer like solution may be adopted. Similar
to LEM, the implementation of ODT within the context of more complex flows
may be achieved through the coupling of ODT with a coarse-grained simulation
approach, such as LES [3].
There are many variants of the ODT model in the literature. The modeling ap-
proach is typically explained by combining the discretization, solution algorithm,
and governing equations. In Sect. 11.3.1, we present a unified method by which all
of the approaches in the literature may be derived, along with a brief discussion of
the equations ultimately used by in the various approaches.
Section 11.3.2 then presents a sampling of results from stand-alone ODT simu-
lations of turbulent combustion.

11.3.1 Governing Equations

This section presents a brief discussion of the various forms of the governing equa-
tions presently solved in ODT. A more detailed exposition can be found in [37].
A generic balance equation for an intensive property in a control volume (CV)
V enclosed by surface S can be written as
   

dV + (vr + vs ) adS = adS + dV, (11.7)
V(t) t S(t) S(t) V(t)

where vs is the velocity of the surface S, v is the mass-averaged velocity, vr = v vs

is the velocity of the fluid relative to the surface, is the density, is the mass
diffusive flux of , and is the volumetric rate of production of . Table 11.1
defines the terms , , and for various quantities. These equations are closed
with an appropriate equation of state relating the local pressure to the composition,
density and temperature.
One-Dimensional Turbulence 259

Table 11.1: Definition of terms in (11.7) for some common governing equations.
Here is the stress tensor, g is the gravitational acceleration vector, Yk is the mass
fraction of species k, jk is the species diffusive flux vector, and q is the heat flux vec-
tor. Other equations may be added as necessary. This is just a partial representation
of commonly solved equations.

Equation Non-convective Flux, Source Term,

Continuity 1 0 0
Momentum v pI g
Species Yk jk k
Total Internal
e0 pv + v T + nk=1 hk jk g v
Energy
Internal Energy e q : u p u
p
Enthalpy h q t + u p + : u

In the following, we present the various forms of the governing equations in use
for ODT. Much of the treatment of the governing equations for ODT in the literature
combine the governing equations with the numerical algorithm. The following does
not address numerical solution techniques for the equations; rather, we focus on a
unified approach for arriving at the various forms of the governing equations implied
by present ODT approaches in the literature.

11.3.1.1 Temporally Evolving Lagrangian Formulation

The first ODT formulations employed a temporally evolving formulation in a La-

grangian frame of reference. In this case, we have v = vs so that vr = 0. Writing
(11.7) in one dimension and using the continuity equation ( = 1) to convert it to
the weak form yields  
d 1 ,y
= + . (11.8)
dt y
d ,y
In (11.8), dt represents the local change of as it moves at velocity vs , 1 y

is the change in due to diffusion, and is the change in due to consump-
tion/production.
Implementations of this approach use moving meshes and finite-volume schemes.
The CV surface positions can be determined by solving (11.8) for = v (the lateral
fluid velocity) and an ODE for position,

dy
= v = vs . (11.9)
dt
Rather than solving (11.8) for = v, however, most ODT formulations employing
the temporally evolving Lagrangian formulation instead use a discrete form of the
260 T. Echekki, A.R. Kerstein and J.C. Sutherland

continuity equation written in Eulerian coordinates, t = vys together with the

assumption of constant pressure and an equation of state to solve for vs for use in
(11.9).

11.3.1.2 Temporally Evolving Eulerian Formulation

Recently, an Eulerian temporally evolving formulation for the ODT equations was
proposed [26, 27]. In the Eulerian frame of reference, we fix the CV surface posi-
tions so that vs = 0 and vr = v so that (11.7) becomes, in differential form,

= v + . (11.10)
t
The one-dimensional differential form of (11.10) is

v ,y
= + . (11.11)
t y y
The velocity v in (11.11) represents the local fluid velocity in the y-direction, and

t represents the local change in at a given point in space and time. Current
approaches using the Eulerian form have solved the compressible form of these
equations [26, 27]. The equations solved are given by (11.10) and Table 11.1, where
= 1 is solved for , = v and = u are solved for the lateral and streamwise
momentum components, = e0 is solved for the total internal energy, = Yk is
solved for the species mass fractions, and an equation of state is used to relate T , p,
, and Yk .

11.3.1.3 Space-Time Mapping

The equations discussed above (both the Lagrangian and Eulerian forms) provide
solutions with (t, y) as independent variables. Frequently, however, we require solu-
tions that evolve spatially (e.g., when comparing with data from a spatially evolving
jet). This requires a space-time mapping, achieved by solving an ODE for stream-
wise position,
dx
= u,
(11.12)
dt
where u is a suitably chosen average velocity for advection of the ODT domain
in the streamwise direction. This creates an approximate average location of the
line in space. Of course, the line would actually tend to bend due to variations in u.
This is explicitly ignored by adopting an average velocity, u, that advects the line
downstream. One possible choice for u is

(u u )2 dy
u (t) = u + 
. (11.13)
(u u ) dy
One-Dimensional Turbulence 261

An alternative approach to using the approximate space-time mapping (11.12) is

to reformulate the governing equations with (x, y) as independent variables rather
than (t, y). This is considered in the following sections for both the Lagrangian and
Eulerian frames of reference.

11.3.1.4 Spatially Evolving Lagrangian Formulation

The spatially evolving Lagrangian formulation is obtained from (11.7) by choosing

vs x = 0 and vs y = v. Then assuming steady state, 2D, and writing the differential
equation in weak form (using the continuity equation), (11.7) becomes
 
d 1 ,y
= . (11.14)
dx u y

In deriving (11.14), we have neglected the streamwise diffusive term, x ,x . This
term is neglected primarily for practical algorithmic reasons. However, in applica-
tion to spatially evolving jets, diffusion in the lateral direction will likely dominate
any diffusion in the downstream direction. Nevertheless, this is an assumption in the
spatial ODT formulations.
This approach has been adopted by Ricks et al. [31] to perform spatially evolv-
ing simulations of buoyant pool fires including soot transport and radiation (see
Sec. 11.3.2).

11.3.1.5 Spatially-Evolving Eulerian Formulation

The spatially evolving form of the equations for ODT in Eulerian form is obtained
from (11.10) by assuming steady state and variation only in x and y. Then using

the continuity equation to write it in weak form and neglecting x ,x as we did in
(11.14), we find  
1 ,y
= v + . (11.15)
x u y y

11.3.2 Stand-Alone ODT Simulations

As a stand-alone model, ODT has been implemented for the study of jet diffusion
flames [7, 11, 12, 21, 2730], buoyant fire plumes [31], flame spread [35], and au-
toignition in jet flows [6]. In these studies different model formulations have been
implemented, which illustrate the versatility of the ODT modeling framework. We
start with the temporally-evolving Lagrangian formulation, which has been adopted
for high-Reynolds number jets, but may be implemented as well for compressible
262 T. Echekki, A.R. Kerstein and J.C. Sutherland

turbulent shear layers and wall-bounded flows. We then illustrate the temporally
evolving Eulerian formulation for temporally evolving shear layers [27] and the
temporally evolving Lagrangian formulation for buoyant fire plumes [31]. In all
formulations, a deterministic solution, involving diffusion, reaction and advective
transport operators in the Eulerian formulation, in conjunction with a stochastic im-
plementation of turbulent advection, is implemented.
The temporally evolving Lagrangian formulation is based on the solution of
equation (11.8) for the streamwise (x) component of momentum along with equa-
tions for energy and species. Figure 11.2 shows temperature contours for a wall fire
from [35] from a single (left) and 300 (right) realizations. This was solved using the
temporal formulation with (11.12) to provide an approximate space-time mapping.
Close examination reveals that small scale triplet mapping events are first observed
at approximate heights of 25 cm and at a distance of approximately 2 cm away from
the wall corresponding to the high values of temperature (and velocity, not shown).
As the flow further accelerates, progressively larger eddy stirring events are shown
to occur causing larger scale macro-mixing and the engulfment of the surrounding
air. This transition of energy from small to large scales of motion is also consis-
tent with recent observations from experiments and LES predictions of large scale
plumes [4].

Fig. 11.2: 2D renderings of temperature corresponding to a single realization (left)

and averaged (right) over 300 realizations of a wall fire. From [35].

Results from a piloted jet flame simulation with extinction and reignition are
shown in Figure 11.3 for the same formulation. The results illustrate how the ODT
model is able to predict extinction and reignition in piloted turbulent non-premixed
flames. Two zones may be identified in the jet flame. The first corresponds to a
region extending approximately fifteen diameters downstream from the inlet that il-
lustrates a transition from piloted burning to extinction. This transition is followed
by a gradual reignition as shown by the increased OH mass fraction. The 2D ren-
dering of stirring events also shows that stirring events are initiated at the interfaces
One-Dimensional Turbulence 263

between the fuel jet and the pilot flow and the interfaces between the pilot flow
and the co-flow air, and that the size of eddies progressively increases as a function
of downstream distance, emulating the progressive growth of the shear layers. Evi-
dence of the existence of an energy cascade in the ODT solutions is demonstrated
by the presence of smaller eddies that trail larger eddies, with regions of spatial
intermittency, as these smaller eddies dissipate (see also Fig. 11.1).

Fig. 11.3: 2D renderings of OH mass fraction corresponding to a single realization

(left) and averaged (right) over 100 realizations of Sandia piloted methane-air flame
F. The 1D domain corresponds to the horizontal axis. Its temporal evolution is con-
verted to a downstream distance based on (11.12) and (11.13). Reprinted from [30]
with permission from Taylor and Francis.

The formulation proposed by Punati et al. [26, 27] solves the Eulerian form of
the governing
equations
described in Sec. 11.3.1. Specifically, (11.10) is solved with
= v u e0 Yi , where u is the streamwise velocity and v is the velocity
component in the direction of the the ODT line orientation. See Table 11.1 for defi-
nitions of the diffusive fluxes in these equations. These equations are solved together
with the ideal gas equation of state, detailed CO/H2 oxidation kinetics, and mixture-
averaged transport to make direct comparison with DNS data of a planar, temporally
evolving CO/H2 -air nonpremixed jet [10]. The DNS dataset includes extinction and
reignition, with the onset of extinction at a characteristic jet time of 20 and
reignition occurring at around 30. This calculation allowed direct comparison
between the ODT and DNS data. Initial conditions were extracted directly from the
DNS data, and all treatment of diffusion, thermodynamics, and chemical kinetics
was equivalent.
264 T. Echekki, A.R. Kerstein and J.C. Sutherland

Figure 11.4 shows the average and RMS velocity and mixture fraction profiles
for the ODT and DNS simulations. The spreading rate is well captured by ODT.
The RMS profiles are captured reasonably, but the ODT under-represents the mag-
nitude of the RMS fluctuations. Similar trends hold for all species (including minor
species), with the exception that extinction is over-predicted by the ODT simula-
tions.
Figure 11.5 shows the evolution of the probability density functions conditioned
on lean and rich mixture fractions for the temperature and scalar dissipation rate.
The temperature PDF illustrates that the ODT predicts an earlier onset of extinction
than the DNS. Specifically, at = 6 there is already evidence of extinction in the
ODT data. However, the extinction-reignition process is captured relatively well
despite these differences.

Fig. 11.4: Evolution of the streamwise velocity (top) and mixture fraction (bottom)
showing the mean (left) and RMS (right) for the ODT (lines) and DNS (circles)
data. Results are shown for characteristic jet times from = 6 to = 40. From [27].

The data shown in Figs. 11.4 and 11.5 were obtained from 400 ODT realizations,
and each realization required approximately two CPU hours. In contrast, the DNS
calculations (which were three-dimensional) required several million CPU hours.
Although ODT cannot capture uniquely multidimensional effects that DNS can, it
does represent many of the physical processes present in true three-dimensional
turbulent flow at a fraction of the cost of DNS, and thus serves as a very useful tool
in combustion modeling.
Ricks et al. [31] simulated a buoyant fire plume using ODT by solving (11.14)
(the spatially evolving form of the governing equations in Lagrangian form) as out-
lined in Sect. 11.3.1, including transport equations for solid-phase soot particles as
One-Dimensional Turbulence 265

Fig. 11.5: Conditional pdf of temperature (left) and scalar dissipation rate (right).
Results are shown for lean and rich mixture fractions and three different times dur-
ing the jet evolution. DNS data is shown dot-dash lines and ODT data is shown with
solid lines. From [27].

well as gas-phase transport equations. However, Ricks et al. [31] adopt a simplified
approach for the representation of the gas phase species using the flamelet assump-
tion and transporting the gas phase mixture fraction. In this formulation, the two
independent variables correspond to the lateral (along the ODT domain) coordinate,
y, and the streamwise spatial coordinate, x. The 1D nature of the solution enables the
implementation of a host of models for soot evolution (including soot oxidation by
OH), and transport (including thermal diffusion), radiation and gas phase chemistry
on large-scale computational domains.
Figure 11.6 shows 2D renderings of the temperature corresponding to two sep-
arate realizations of the ODT simulation of a fire plume by Ricks et al. [31]. The
ODT domain is aligned with the horizontal direction (x); a marching algorithm is
implemented to evolve the ODT solution in the vertical (y) direction on a computa-
tional domain of 2 m 3 m. The necking just above the base of the flame is due to
the spatial form of the continuity enforcement, which induces inward lateral flow in
order to compensate for the buoyancy-induced increase in the streamwise mass flux.
Additional statistics on soot evolution and radiation effects may be found in [31].

11.3.3 Hybrid ODTLES

Similarly to LEM, ODT may be coupled to a 3D coarse-grained simulation ap-

proach, such as LES, for chemistry and mixing closure. Moreover, there are differ-
ent strategies for LES and ODT coupling based on Eulerian and Lagrangian for-
266 T. Echekki, A.R. Kerstein and J.C. Sutherland

Fig. 11.6: 2D renderings of temperature (in K) corresponding to two separate real-

izations of a buoyant fire plume simulation. Reprinted from [31] with permission
from Taylor and Francis.

mulations. In both formulations, ODT domains or elements are embedded in 3D

solutions to resolve subfilter-scale momentum and scalar statistics. In the Eulerian
formulation, ODT elements are fixed in space. Advective transport contributions
in this formulation are represented by both large-scale transport resolved by LES
and subfilter-scale transport represented by ODT stirring events. The simplest La-
grangian formulation may be implemented by attaching ODT solutions along the
normal to the flame brush. In this formulation, the ODT elements are advected along
with this brush. Similar strategies have been successfully adopted with LEM as dis-
cussed in Chapter 10.
In contrast to LEM, the coupling of LES with ODT may present a number of
additional advantages:
ODT has the capability to provide closure for momentum. However, one may
choose to provide closure for scalars only and allow for a standard model for
momentum closure (as discussed below).
In ODT, the coupling of momentum and scalars is implemented on the fine time
and length scales of ODT solutions; this coupling is very crucial near physical
boundaries (e.g. walls) where both scalar and momentum boundary conditions
may be implemented.
Historically, large-scale transport with LES-LEM has been implemented using
splicing events, which extract segments from a LEM solution in one LES grid
and transfers them to another LEM solution in a neighboring LES cell. The LES-
ODT formulation of Cao and Echekki [3] proposes an alternative representation
for large-scale transport based on ODT domains extending beyond a single LES
cell.
One-Dimensional Turbulence 267

The ODTLES model formulation is illustrated using the Eulerian formulation by

Cao and Echekki [3]. The ODTLES formulation is based on two simulations that
are implemented in the same computational domain. The first is a 3D LES for mass
and momentum transport. The second is based on fine-grained simulations imple-
mented on an ensemble of 1D ODT elements, which are embedded in the LES
domain. Here, we describe a formulation in which ODT elements are distributed in
a 3D Cartesian lattice as shown in Fig. 11.7 where the LES is solved on a structured
Cartesian grid as well. However, a more complex layout may be adopted. More-
over, the formulation is used primarily for reactive scalars closure. For momentum
closure, a standard LES closure model for subgrid stresses may be adopted.

Fig. 11.7: Layout of ODT elements on a Cartesian grid in LES. Adapted from [3].

The ODT governing equations are solved on each individual ODT element. The
temporal and spatial resolution requirements in ODT are similar to those needed for
direct numerical simulations. The coordinate system on which the governing equa-
tions are based is a Cartesian coordinate system with one component along the ODT
domain, x1 , and two additional orthogonal components, x2 and x3 . The spatial coor-
dinate, x1 , replaces the ODT domain coordinate, y, in previous discussions. When
laid out on a Cartesian lattice, the direction x1 represents the axis that is aligned
with the ODT element; while the other coordinates x2 and x3 represent the remain-
ing axes. The velocity field is split into a filtered (resolved in LES) component and
a residual component:
ui = ui + ui (11.16)
where ui is the filtered velocity in the ith direction. The contribution of transport due
to this velocity component is denoted as large-scale transport. The second term on
the right-hand side, ui , is the residual term of the velocity field in the ith direction.
This latter term is modeled using the stochastic turbulent stirring events in ODT.
The contribution of transport due to this velocity component is denoted as subfilter-
scale transport. The variable-density governing equations on each ODT element of
momentum, temperature, and species mass fractions are:
268 T. Echekki, A.R. Kerstein and J.C. Sutherland

Momentum
    
ui 1 i1 1 p ui 1 i2 i3
=+ + u j + + (11.17)
t x1 xi x j x2 x3

Temperature
    
T 1 q1 N
T 1 q2 q3
= + hk k u j + +
t c p x1 k=1 x j c p x2 x3
(11.18)
Species (k = 1, . . . , N)
     
Yk 1 jk1 Yk 1 jk2 jk3
= + k u j + + (11.19)
t x1 x j x2 x3

In equations (11.17)-(11.19), the index j represents the sum over all three directions
of the advective terms. The diffusive fluxes, jk1 , jk2 and jk3 correspond to mass
diffusion fluxes of the k species in the x1 , x2 and x3 directions, respectively. They
may be expressed as Vk1 , Vk2 and Vk3 , where Vki is the mass diffusion velocity of
species k in the ith direction. q1 , q2 , and q3 correspond to the components of the heat
flux vector in the x1 , x2 and x3 directions, respectively. These components represent
the contributions of heat conduction, heat transport by mass diffusion, the Dufour
effect and radiative heat transport. The ODT governing equations feature contribu-
tions which are resolved on the ODT domain (terms inside brackets [ ]). These are
the same source and transport terms present in the stand-alone ODT equations. The
resolved contributions include (1) molecular transport with gradients along the 1D
elements, (2) chemical and heat source terms, and (3) the subfilter-scale momentum
and scalar transport; this latter term is represented by the ODT stochastic stirring
events discussed in Sect. 11.2.1 and implemented on a range of length scales. Other
contributions require gradients along the normal components to the ODT domain
(terms inside brackets {}). The unresolved contributions include: (1) large-scale
transport (advective transport based on the filtered velocity components), (2) molec-
ular diffusion with gradients along x2 and x3 , and (3) the pressure gradient terms in
the momentum equations.
The coupling of LES and ODT solutions is implemented both temporally and
spatially. The ODT integration treats reaction-diffusion, subfilter-scale transport
(stirring events) and filtered-advection as parallel events that are integrated with
their own time steps, and which are fractions of the LES time step. During the tem-
poral integration of the two solutions, statistics are transmitted from one solution
scheme to another. For ODT, the LES velocity field u is evaluated from the LES
solution of the momentum equations and interpolated onto the ODT elements finer
grids. For LES, a number of variables may be filtered from ODT solutions, including
closure for the mass density, . In what follows strategies adopted for the integration
of the various terms in the ODT equations are briefly discussed:
One-Dimensional Turbulence 269

11.3.3.1 Molecular Processes

Molecular processes in the ODT governing equations include (1) reaction, (2) dif-
fusion along the ODT elements directions, and (3) diffusion along the directions
normal to the ODT elements. The integration of the first two contributions is similar
to their implementation in the stand-alone ODT formulation. The third contribution
must be modeled. The treatment of the unresolved terms may be implemented either
deterministically or stochastically. In the Cao and Echekki [3] work, the representa-
tion of non-resolved diffusive contributions is achieved deterministically by scaling
the resolved diffusive transport terms by a factor to represent the filtered contribu-
tion of mass transport from the unresolved transport. For example, if there is no
preferred gradient, such as in the presence of a flame brush, a factor of 3 is adopted.

11.3.3.2 Representation of Subfilter-Scale Stresses and Scalar Fluxes

The stochastic contributions represent 3D subfilter-scale advective transport of mo-

mentum and scalars (i.e. subfilter-scale stresses and scalar fluxes) resulting from the
residual velocity components. For momentum additional contributions may be at-
tributed to pressure scrambling [19]. The stochastic terms are implemented through
discrete triplet map events, which are implemented concurrently with other pro-
cesses within ODT. The rules for stirring events are identical to those applied in
stand-alone ODT. The range of length scales for the selected eddies is prescribed
prior to the simulation based on a choice of Lmin and Lmax , which represent the
smallest and largest eddies allowed. The value of Lmin plays a similar role to the
Kolmogorov length scale, and corresponds to length scales where viscous dissipa-
tion is predominant and stirring events are less likely to occur. The value of Lmax de-
termines the cut-off length scale beyond which turbulent advective transport is rep-
resented using the filtered advective terms. These parameters are additional model
parameters to C and Z prescribed earlier for the ODT-implementation. The cumula-
tive contribution from stirring events over time represents the subfilter-scale stresses
and fluxes.

11.3.3.3 Large-Scale Transport

The large-scale transport of momentum and scalars in ODT is represented by the

operators u j xuij , u j xTj and u j Yxkj in the ODT governing equations. The implemen-
tation of large-scale transport represents a fundamental challenge for the following
reasons: 1) Advective transport is a 3D process; thus, at least two directions are not
resolved on the ODT time scale or on the ODT 1D elements, 2) Non-linear contribu-
tions from advection processes pose important constraints on scalar boundedness. In
the Cao and Echekki [3] formulation, advective fluxes are constructed at nodes that
represent the intersection in space of three or more ODT elements. At these nodes,
the solutions of the velocity and scalar equations are updated; then, ODT solutions
270 T. Echekki, A.R. Kerstein and J.C. Sutherland

between these nodes are updated through single-component advection along the cor-
responding ODT element. In a Cartesian lattice of ODT elements, these nodes rep-
resent the intersection of three orthogonal 1D elements. Below, we describe the im-
plementation based on this Cartesian lattice configuration. The large-scale transport
process is implemented as a separate process concurrently with reaction-diffusion
and stirring in two steps: 1) node advection, and 2) inter-node relaxation.
Node Advection is implemented as follows. At the prescribed time step for large-
scale transport, the solution at each node is evaluated based on gradients represented
by the 3 ODT elements intersecting at the node. This process is implemented in two
steps. First, the solution is updated at each node for the momentum and scalars using
the following governing equations:

l 1 2 3
= u11 u22 u33 (11.20)
t x1 x2 x3
In this expression, the dependent variable corresponds to any one of the variables
in the solution vector. The subscripts, 1, 2 or 3, correspond to the components of
the velocity vector; while, the superscripts, 1, 2 or 3, correspond to the direction of
the ODT domain. Although each node is updated with the same right-hand side that
represents contributions from the three directions of ODT elements, its value at the
end of the update is different because of the values of l are different at the start
of the update from the 3 contributing ODT elements. A second step involves the
averaging of these 3 solutions at the nodes as follows:

1 +2 +3
l = (11.21)
3
Inter-Node Relaxation involves a relaxation of the solution between the nodes
based on the updated internal boundary conditions at the nodes. This relaxation is
accomplished through an integration of the solution at grid cells between the nodes
using a single-component advective flux according to the following relation:

l l
= ull (11.22)
t xl
In this expression, represents a relaxation coefficient, which governs the rate
at which the inter-node solution is updated to reflect changes at the nodes. Because
ODTLES is a statistical approach, a range of values for may be adopted to yield
reasonable statistics for the scalars and momentum solutions.
Similarly to the approach by Schmidt et al. [32], a correction to the ODT veloc-
ity solution is implemented such that the filtered ODT velocity field matches the
solution from LES.
The above Eulerian formulation was implemented by Cao and Echekki [3] for
the modeling of non-homogeneous ignition in a random mixture-fraction field with
preheated oxidizer and of the same configurations show that the model represents
adequately turbulent transport through the contributions of subfilter-scale transport
One-Dimensional Turbulence 271

and large-scale transport. Important effects of preferential diffusion as well as tur-

bulence intensity are identified in reactive-scalar conditional statistics.
Figure 11.8 shows isocontours of filtered reaction progress variable (normalized
temperature) at a prescribed value of 0.5 (for flame tracking) at different times based
on the ODTLES simulations. The figure shows the formation of discrete ignition
kernels at favorable mixture conditions, their growth and their merger at later times.
The reaction progress variable is obtained by filtering the ODT solutions; the size
of the initial nascent kernels is smaller than the LES grid.

Fig. 11.8: Evolution of flame kernel based on filtered reaction progress variables
during non-homogeneous mixture ignition in isotropic turbulence. The figure shows
the formation of ignition kernels at conditions favorable to the onset of ignition. Ad-
ditional kernels are formed at less favorable conditions for autoignition after more
delay. The kernels eventually grow to interact at later stages and merge to form larger
kernels, until the entire mixture is burned. Reprinted from [3] with permission from
Taylor and Francis.

Figure 11.9 shows the evolution of the heat-release rate conditional statistics at
two Lewis numbers, 0.5 and 2, representing the ratio of the thermal diffusivity to
the species mass diffusivities. The lower Lewis number heat-release rate profiles
exhibit higher peaks initially and then lower peaks eventually as the combustion
progresses from fuel-lean conditions to richer conditions. The difference between
the two cases reflects the strong dependence of the heat release rate on temperature,
which is affected by the Lewis number. Lower Lewis numbers indicate slower diffu-
sion of heat relative to species. Therefore, the initial formation of the corresponding
kernels favors kernels that shielded from heat loss. However, the same mechanism
272 T. Echekki, A.R. Kerstein and J.C. Sutherland

may prevent the ignition of the unburned layers next to the ignition kernels and their
propagation. Cases a and b shown in comparison with DNS statistics correspond
to two different and coarse LES resolutions (case a is twice as resolved as case b).
The two cases are in very good agreement with the DNS statistics and show that the
ODTLES formulation predicts reasonably well the contributions from large-scale
and subfilter-scale transport.
The ODTLES formulation has been extended recently to the study of turbulent
premixed flames by Echekki and Park [8]. A Lagrangian formulation has been im-
plemented more recently by Balasubramanian [2] for the study of a buoyant fire
plume. In this formulation, the ODT elements are attached to a filtered mixture
fraction surface with a fixed value corresponding to the stoichiometric value.

11.4 Concluding Remarks

Here and in Chapter 10, a strategy for turbulent combustion modeling has been out-
lined that involves a conceptually and computationally minimal representation of
the local unsteady evolution of the coupled processes of advection, diffusion, and
reaction. ODT, described in this chapter, incorporates a representation of the de-
pendences of the occurrence of eddy motions on the mechanisms that drive these
motions. In addition to capturing important effects of the unsteady couplings, this
feature results in a formulation that is, in many respects, a self-contained predic-
tive model of turbulent flow. This is perhaps a natural consequence of the effort to
capture the couplings relevant to combustion; for a model to do this well, it must
capture much of the phenomenology of turbulence.
The main limitation of ODT in this regard is its restriction to one spatial dimen-
sion. It is thus complementary to LES, which captures large-scale 3D motions but
does not resolve flame structure and evolution. Coupling of ODT to LES has been
described. The successes of the LEMLES formulations for the modeling of practical
combustion flows (see Chapter 10) also support the potential of ODTLES as a vi-
able modeling approach for similar problems. More importantly, both LEMLES and
ODTLES may be viewed as frameworks with which multiphysics and multiphase
problems may be addressed. In addition to the momentum and standard scalar equa-
tions for combustion problems, additional transport equations may be implemented
within these ODTLES frameworks, including particle transport and multiscale de-
scriptions of radiative transport in participating media.
One focus of current efforts is the coupling of arrays of ODT domains so as to
obtain a self-contained 3D flow simulation (with the smallest scales resolved only in
1D), thus eliminating the need for a distinct coarse-grained 3D flow solver [17]. This
modeling strategy is termed autonomous microstructure evolution (AME). Another
focus involves Lagrangian implementation of the ODTLES framework based on
ODT elements attached to the flame brush.
One-Dimensional Turbulence 273

Fig. 11.9: Evolution of the conditional means of the heat release rate conditions
during the ignition of a non-homogeneous mixture of fuel and preheat oxidizer at
Le = 0.5 and 2.0. Reprinted from [3] with permission from Taylor and Francis.
274 T. Echekki, A.R. Kerstein and J.C. Sutherland

Acknowledgement

Dr. T. Echekki acknowledges support from the Air Force Office of Scientific Re-
search through grants F49620-03-1-0023 monitored by Dr. Julian Tishkoff and
FA9550-09-1-0492 monitored by Dr. Fariba Fahroo and the National Science Foun-
dation through grant DMS-0915150 monitored by Dr. Junping Wang. Dr. A. Ker-
steins research was partially supported by the U.S. Department of Energy, Office
of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and En-
ergy Biosciences. Sandia National Laboratories is a multi-program laboratory op-
erated by Sandia Corporation, a Lockheed Martin Company, for the United States
Department of Energy under contract DE-AC04-94-AL85000. Dr. J. Sutherland ac-
knowledges support from the Department of Energy under award number FC26-
08NT0005015.

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Chapter 12
Unsteady Flame Embedding

Hossam A. El-Asrag and Ahmed F. Ghoniem

Abstract Direct simulation of all the length and time scales relevant to practical
combustion processes is computationally prohibitive. When combustion processes
are driven by reaction and transport phenomena occurring at the unresolved scales of
a numerical simulation, one must introduce a dynamic subgrid model that accounts
for the multiscale nature of the problem using information available on a resolvable
grid. Here, we discuss a model that captures unsteady flow-flame interactions
including extinction, re-ignition, and history effectsvia embedded simulations at
the subgrid level. The model efficiently accounts for subgrid flame structure and
incorporates detailed chemistry and transport, allowing more accurate prediction of
the stretch effect and the heat release. In this chapter we first review the work done
in the past thirty years to develop the flame embedding concept. Next we present
a formulation for the same concept that is compatible with Large Eddy Simulation
in the flamelet regimes. The unsteady flame embedding approach (UFE) treats the
flame as an ensemble of locally one-dimensional flames, similar to the flamelet ap-
proach. However, a set of elemental one-dimensional flames is used to describe the
turbulent flame structure directly at the subgrid level. The calculations employ a
one-dimensional unsteady flame model that incorporates unsteady strain rate, cur-
vature, and mixture boundary conditions imposed by the resolved scales. The model
is used for closure of the subgrid terms in the context of large eddy simulation. Di-
rect numerical simulation (DNS) data from a flame-vortex interaction problem is
used for comparison.

Hossam A. El-Asrag
Massachusetts Institute of Technology, Cambridge MA 02139, e-mail: helasrag@mit.edu
Ahmed F. Ghoniem
Massachusetts Institute of Technology, Cambridge MA 02139, e-mail: ghoniem@mit.edu

T. Echekki, E. Mastorakos (eds.), Turbulent Combustion Modeling, 277

Fluid Mechanics and Its Applications 95, DOI 10.1007/978-94-007-0412-1 12,
Springer Science+Business Media B.V. 2011
278 El-Asrag and Ghoniem

12.1 Introduction

Even with the current trend of exponential growth in computing capabilities [64],
numerical simulation of multi-dimensional turbulent reacting flows is computation-
ally prohibitive making it necessary to rely on under-resolved simulations and the
implementations of local models that capture the unresolved processes as well as
their interactions with the resolved ones. The increase in the hardware computa-
tional power allows us to improve our models by including more physics and more
detailed kinetics. Detailed numerical simulation of turbulent reacting flows is chal-
lenging because of the unsteadiness of the flow and the multiscale nature of the dif-
ferent interacting physical processes. The wide spectrum of length and time scales
in turbulent reactive flows limits the utility of direct numerical simulations (DNS)
to being a research tool. The practical alternative is under-resolved calculations in
which the large scales are resolved on a coarse grid and the impact of the small
scales is modeled.
Large eddy simulation (LES) [51] is one such approach. The extension of these
approaches to combustion poses a tremendous modeling challenge, since the scales
where the reactants mix and burn are often below those resolved directly. This is
especially true for premixed flames, where the flame thickness is often of the order
of the smallest turbulent length scales [43]. In particular, it is often important to
account for the interactions between unsteady processes and the premixed flame,
e.g., in modeling the acoustic instabilities [1].
Furthermore, unsteady flow features in the vicinity of the flame surface affect
its propagation speed significantly. For example, the flame surface wrinkling is
found to be a strong function of the hysteresis and memory effect generated by the
boundary conditions and the flame/turblence interactions [8]. Moreover, unsteady
effects can introduce scalars counter-gradient or gradient transport, depending on
the flame structure and its interaction with the vortices. For example, low Lewis
number flames are found to exhibit counter-gradient transport [3]. The Lewis num-
ber represents the relative contribution of thermal and molecular diffusivity. The
interaction between these two transport phenomena is more significant under time-
dependent conditions and controls the flame local temperature in the presence of
the curvature and strain rate. The local curvature and strain rate are functions of
turbulence and hence are unsteady. In flames with Lewis number larger than unity
and local positive curvature, reactants diffuse towards the reaction zone at slower
rate than heat is conducted out, which decreases the burning rate and might lead to
non-equilibrium phenomenon as flame extinction. On the other hand, flames with
Lewis number smaller than unity are subjected to higher thermo-diffusive instabili-
ties [3, 32].
Detailed kinetics simulations [9] confirm that species with low diffusion coeffi-
cient are more susceptible to unsteady effects. Moreover, experimental [58] as well
as numerical [27] data confirm that as turbulent intensity increases the strain rate
effect is more predominant and the curvature distribution has a broader standard
deviation than for lower strained flames. These flame/turbulence interactions occur
mainly on the small scales which are unsteady.
Unsteady Flame Embedding 279

Multi-scale combustion modeling has been applied successfully in the context of

LES. The one-dimensional turbulence (ODT) model [10, 29] and linear eddy model
(LEM) [11, 15, 16, 28, 38, 57] have been employed in premixed and non-premixed
configurations (see Chapters 10 and 11 for more details). Other combustion clo-
sure models include the transported filtered density function (FDF) methods [54]
for modeling turbulent non-premixed flames, the artificially thickened flame (ATF)
model [62] for premixed flames, the flame-surface density models (FSD) [2, 24],
and the presumed PDF flamelet approaches [42, 51].
In the flamelet approach, and because of the fast chemistry assumption, scalar
transport in the tangential direction is neglected with respect to the normal direc-
tion. Using this assumption, the scalar transport equations are transformed in the
mixture fraction space and the flamelet equations are derived by asymptotic analy-
sis [51]. A set of 1-D solutions, based on solving the 1-D steady flamelet equations,
are tabulated then mapped into the fully 3-D turbulent environment by convolut-
ing the solutions with a PDF distribution. The flamelet approach is widely used for
premixed flames combined with the level set approach [31, 50, 52]. Other flamelet
type combustion subgrid models are the Flamelet Prolongation of ILDM (FPI) [19],
the Flamelet Generated Manifolds (FGM) [63], and the Flamelet Progress Variable
(FPV) [26, 47].
Few attempts, however, have been made to extend the flamelet approach to in-
clude unsteady effects. The steady state assumption usually employed in conven-
tional flamelet type approaches is not suitable if slow physical and chemical pro-
cesses are involved such as NOx formation or radiation, or if unsteady processes are
encountered as flame extinction and re-ignition. The two known models that account
for the flamelets unsteady effects are the Lagrangian Flamelet Model (LFM) [48]
and the Eulerian Flamelet Model (EFM) [49]. In LFM a single flamelet is intro-
duced at the nozzle exit and is allowed to transport downstream with the flow. The
flamelet in the mixture fraction space is solved simultaneously with the flow field
calculations. And is integrated using a Lagrangian time scale that is computed from
the filtered flow velocity conditioned on the stoichiometric mixture fraction. This
approach was further modified as the Eulerian Flamelet Model (EFM) [49], where
the flamelet equations are solved in the Eulerian form rather than in the mixture
fraction space.
The unsteady flame embedding approach (UFE), described in this work models
the unsteady interaction between the resolved and the unresolved scales using un-
steady strained flames computed along a set of 1-D lines within the coarse grids
of the LES. The unsteady strain rate and the boundary conditions acting on the lo-
cal flame are updated temporally with the solution advancement. This allows for
consideration of history effects, as well as accurate modeling of the unsteady flame
stretch at the small scales. The model utilizes the flame embedding concept intro-
duced earlier [20, 23].
In Sec. 12.2 we review the work done in the past thirty years to develop the
flame embedding concept. In Sec. 12.3 the elemental flame model formulation is
described. Following that the numerical approach used to solve the flow along the
one-dimensional flame is shown in Sec. 12.4. In Sec.12.5 we introduce a new UFE
280 El-Asrag and Ghoniem

subgrid model components for LES applications. Validation of the subgrid model is
then described in Sec. 12.6. The chapter ends with the conclusions and future work
in Sec. 12.7.

12.2 Historical Perspective on the Flame Embedding Concept

According to the laminar flamelet concept [65], if the smallest turbulent length scale
is larger than the flame thickness, the flame can be regarded as a thin sheet that is
convoluted by turbulent eddies. Early work by Chorin [6] and Ghoniem et al. [20]
utilized this concept and implemented a flame interface tracking scheme to model
premixed combustion. In these papers the elementary turbulent eddies and their cu-
mulative effect was modeled by Chorins random vortex method [5] and the position
of the interface between the burned and unburned gases was tracked by a flame prop-
agation and advection algorithm. A volume-of-fluid approach was used to define the
volume ratio of the burned and unburned gases ( f ) inside each cell, and the flame in-
terface was constructed based on these ratios. The two fluids were advected accord-
ing to the velocity field at the cell boundaries. The impact of the self-propagation
of a laminar flame normal to itself was approximated using Huygens principle to
compute the normal propagation speed due to consumption of the unburned gases.
This approximation was extended [21] to include the effect of exothermicity and
volumetric expansion. The model was applied to a backward facing step stabilized
flame in a combustion tunnel and compared well with the experimental data.
The early work used a constant laminar flame speed. However, in highly turbulent
flows the effect of curvature induces different hydrodynamic stability effects based
on the flame topology and its interaction with the surrounding vortices. The above
model was then extended [22] by computing the burning velocity as function of the
consumption speed and the flame front curvature. The numerical results are com-
pared with the experimental data on the flame propagation in a quiescent mixture
inside a circular chamber, initiated by a central ignition source. The flow was as-
sumed inviscid, two-dimensional with adiabatic walls. The results show that the heat
conduction, coupled with front curvature, leads to the stabilization of short wave-
length perturbations. However, long wavelength perturbations cannot be damped by
this mechanism, and the coherence of the flame front must be attributed to other
mechanisms. Two mechanisms were identified: the stability of the front due to the
interaction between the distortion caused by the perturbations growth with the con-
vexity of the flame front and the expansion of the wavelength as the flame expands,
and the consumption of the concave parts of the flame.
Up to this point, the above modeling approach had not considered the unsteady
effect the turbulence exerts on the reaction zone. Under realistic conditions, the
amplitude and frequency of an oscillating strain can impact unsteady chemical pro-
cesses, such as ignition and quenching. To study these unsteady effects, Ghoniem
et al. [23] used a one-dimensional flame model and one-step Arrhenius-kinetics to
quantify the effects of steady and periodic strains on a diffusion flame, where the
Unsteady Flame Embedding 281

one-dimensional unsteady flame equations [41] were solved. A boundary-layer anal-

ysis for the flame structure is performed, i.e, due to the small flame thickness, normal
gradients are much stronger than tangential gradients, so that the latter are ignored
in the diffusion terms. The resulting set of equations represent the one-dimensional
flow for an axisymmetric stagnation point flame configuration. Applied to a flat
one-dimensional non-premixed flame configuration, the results show that the igni-
tion delay time required to reach steady burning increases exponentially with strain.
With a more sustained ignition source, the delay time is reduced, and the quench-
ing strain increases. The burning rate was also found to depend on the strain rate
fluctuations amplitude and frequency. These fluctuations were found to cause local
or complete quenching based on the amplitude and frequency. Another interesting
observation is that compressive strain inhibits reactants diffusion into the reaction
zone, and leads to partial extinction of the flame, which soon reignites as positive
strain resumes. The study shows that modeling the unsteady processes is impor-
tant especially in turbulent flows, in which rapidly varying modes of instability may
develop simultaneously.
The combustion model was generalized later by Kino et al. [30] to simulate a
two-dimensional reacting shear layer at the conditions of fast chemistry. The com-
bustion model consolidated the two-dimensional unsteady strained flame equations
described earlier, with the vorticity equation and acted as one-step forward towards
generalizing the flame model to multi-dimensional flow conditions. The resulting
model is incorporated into adaptive, Lagrangian vortex element techniques. The
evolution of external flow field is computed by tracking the motion of the vortex
elements, while flame front topology is described by a collection of thin flamelets,
which are used to compute the unsteady response of the local flame structure to the
imposed strain. This elemental flame model embedded in a Lagrangian vortex ele-
ment technique was further extended to account for detailed chemical kinetics and
multi-species transport with different Lewis numbers by Petrov and Ghoniem [46].
A reduced four-step mechanism for methane was employed to study the evolution
of a reacting shear layer. An example of these calculations is shown here. The com-
bustion was initiated at time zero by embedding elemental flames whose structures
correspond to a steady strain of 100 1/s. The results at t = 2.25 and 3 ms, i.e., after
two and three eddy turnover times, are shown for the temperature, heat release rate
in Fig. 12.1, and the production rates for CO and CO2 in Fig. 12.2.
The unsteady strained flame model was used to examine the effect of temporal
gradients on combustion in a premixed methane/air mixture [36]. The efficiency of
the numerical solution was improved by using an inexact Newton backtracking al-
gorithm to solve the nonlinear algebraic equations. This algorithm will be discussed
later in details. The results show that equivalence ratio variations with timescales
lower than 10 ms have significant effects on the burning process, including reaction
zone broadening, burning rate enhancement, and extension of the flammability limit
toward leaner mixture. The study stresses the importance of modeling the unsteady
effects to simulate turbulent flames.
The flame embedding approach was validated by comparing its results with data
extracted from a set of two-dimensional simulations of premixed flame-vortex inter-
282 El-Asrag and Ghoniem

Fig. 12.1: The evolution of the temperature and heat release rate inside a large vortex
in the spatially developing reacting mixing layer: top stream: air moving at 100
m/sec, bottom stream methane moving at 50 m/sec. From [46].

Fig. 12.2: The evolution of the CO and CO2 mass fractions inside a large vortex in
the spatially developing reacting mixing layer: top stream: air moving at 100 m/sec,
bottom stream methane moving at 50 m/sec. From [46].

actions for a matrix of vortex strengths and length scales [37]. Elements along the
flame front, or elemental flames from the direct solution were selected and compared
with the solution obtained from the flame embedding results. The results show good
agreement when the actual strain rate at the reaction zone of the one-dimensional
flame was made to match that of the two dimensional flame.
Figure 12.3 (from Marzouk et al. [37]) compares the instantaneous structure of
the elemental flame with that of the two-dimensional flame vortex interaction solu-
tion along the centerline at t = 2.0 ms. In the elemental flame calculation, the time-
dependent strain rate a(t) was chosen to ensure matching of mean reacting strain
Unsteady Flame Embedding 283

rates. This comparison highlights the challenge inherent in using the elemental flame
approach to predict the two-dimensional flame-vortex interaction. Though the mean
strain rates in the two flames are equal, the strain rate distributions are qualitatively
different, with the strain rate ( ) across the flame falling through a plateau in the two-
dimensional flame but rising monotonically in the one-dimensional flame. While
this disagreement may cause the remaining profiles to differ slightly, the elemen-
tal flame model is able to characterize the heat release rate in the two-dimensional
reacting flow with good accuracy. A further comparison between the results of the
elemental flame structure and the two-dimensional solution along the centerline is
shown in Figure 12.4; here, selected major species and radicals are shown at t = 1.0
ms (Case 4 in [37]), while matching the mean reacting strain rates in both solutions.
The strain rate, temperature, and heat release profiles are qualitatively similar to
those in Figure 12.3. The species profiles match well, with slight differences visible
mostly in the preheat zone of the flame.

Fig. 12.3: Structure comparison between the one- and two-dimensional flame el-
ements, t = 2.0 ms, showing the temperature T , the heat release rate w T , and the
strain rate. Both elements are at the same mean reacting strain. From [37].

12.3 Elemental Flame Model Formulation

The flame strained in a stagnation point flow, shown in Fig. 12.5, has long been
proposed as a reference flame model for turbulent combustion in the flamelet regime
[7, 12, 13, 23, 45, 56, 59]. We use this model here but allow the strain rate to vary
as a function of time, to capture the physics of unsteady flame response. This model
284 El-Asrag and Ghoniem

Fig. 12.4: Structure comparison between the one- and two-dimensional flame ele-
ments, t = 1.0 ms, showing various major and minor species. Both elements are at
the same mean reacting strain. From [37].

is labeled the elemental flame [44, 45] to distinguish it from the conventional quasi-
steady flamelet approach used in closure modeling.
One-dimensional governing equations for the elemental flame are obtained as
follows. A boundary layer approximation is applied across the flame, and a solution
is considered along the stagnation streamline, x = 0; y is the coordinate normal to
the flame surface. The outer flowa stagnation point potential flow with velocity
field u = a(t) x, v = ( j + 1)a(t) yyields the pressure gradient as a function
of the imposed strain rate:

p u da
= u u a u (12.1)
x a dt
Note that j = 0 for a planar flow and j = 1 for an axisymmetric flow, in which case r
may be substituted for x in the expressions above. The notation u in (12.1) empha-
sizes that the density of the unburned mixture is used to define the pressure gradient.
If the densities of both incoming streams are equal, u = = , this distinc-
tion is moot. In the premixed flame, however, expansion resulting from heat release
within the flame requires a products-side stream entering with lower densityand
to maintain a constant pressure gradient, a higher effective strain rate. Defining the
strain rate parameter a(t) on the unburned side, in accordance with the expression
for pressure gradient above, thus insures consistency across all flame configurations.
Introducing the similarity variable U u/u , the notation V v, and substi-
tuting the pressure gradient expression into the equation for momentum conserva-
tion inside the boundary layer, we obtain the following equations for species, energy,
momentum, and mass conservation, respectively:
Unsteady Flame Embedding 285

Yk Yk
+V + ( Yk Vk ) w kWk = 0 (12.2)
t y y
   
T T 1 T 1 K T 1 K

t
+V
y cp y

y
+
cp k c p,kYk Vk + w k Hk = 0
y cp k
(12.3)
   
U 1 da U U 1 da
+ U + U 2 a +V u + a = 0 (12.4)
t a dt y y y a dt
V
+ + ( j+1) Ua = 0 (12.5)
t y
where the diffusion velocity is:

1 Xk
Vk = Dkm (12.6)
Xk y
and the mixture-averaged diffusion coefficient Dkm is defined in terms of binary
diffusion coefficients D jk as [65]:

1 Yk
Dkm = (12.7)
j=k X j /D jk
K

Here Yk is the mass fraction of species k, while Wk and w k are the molar weight
and molar production rate, respectively. In the remaining equations, c p is the spe-
cific heat of the mixture, is the thermal conductivity, Hk is the molar enthalpy of
the k-th species, u is the density of the reactants mixture, and is the dynamic
viscosity of the mixture. Note that thermal diffusion velocities are neglected. The
low Mach number assumption has been employed, and hence density is calculated
as a function of the temperature, species mass fractions, and the spatially uniform
thermodynamic pressure via the ideal gas equation of state.
While the velocity u is zero along the stagnation streamline, the momentum equa-
tion (12.4) in U is retained to govern variation of the strain rate through the flame,
where, by definition, 
u 
aU = (12.8)
x x=0
yields the strain rate profile.
Boundary conditions for the species and energy equations consist of defining the
composition and temperature of the two incoming streams of the stagnation point
flow:
y = : Yk = Yk, (t), T = T (t). (12.9)
The continuity equation requires only one boundary condition; typically, this bound-
ary condition would specify zero velocity at the stagnation point, V (y = 0) = 0.
Numerical considerations discussed in the next section, however, suggest that we
286 El-Asrag and Ghoniem

impose a boundary condition at y = , and leave the stagnation point definition to

fix the origin of the y-axis.
The momentum conservation equation requires two boundary conditions. At an
unburned stream, u = u , so the boundary condition is by definition U = 1. Setting
the spatial gradients in (12.4) to zero yields an ODE for U at the burned-stream
boundary, denoted by Ub :
   
dUb 1 da u 1 da
= Ub a Ub
2
+ +a (12.10)
dt a dt b a dt

Here b is the density of the burned mixture. For a premixed flame, the strain rate
in the incoming products-side mixture thus responds dynamically to the imposed
strain rate. This far-field boundary condition also places an important requirement
on the size of the computational domain; the flame must be far enough from the +
and boundaries for spatial gradients in U to vanish. In the case of steady strain
rate, the burned-stream boundary condition on U reduces to

u
Ub = . (12.11)
b

Fig. 12.5: Premixed flame in a stagnation point flow; the elemental flame model.

12.4 Numerical Solution for the Elemental Flame Model

Numerical solutions of the governing equations are obtained via a fully-implicit

finite difference method, as necessitated by the stiffness of detailed kinetics. A first-
order backward Euler formulation is used. All the governing equations are solved
Unsteady Flame Embedding 287

simultaneously, and thus the continuity equation (12.5) is in a sense an algebraic

constraint on the implicit system.
A first-order upwind discretization is applied to all convective terms, while dif-
fusion terms are discretized to second-order accuracy using centered differences.
An upwind discretization is used in the continuity equation as well, although it
is not properly a transport equation and V / y is not a convective term per se.
Nonetheless, taking the positive sign outside V / y to suggest a positive upwind
velocity, the following discretization of the continuity equation adds dissipation of
the appropriate sign:

V n+1
j nj V jn+1 V j1n+1
+ + Ua = 0  + + n+1
j Uj
n+1 n+1
a = 0 (12.12)
t y t y j y j1

A boundary value on the mass flux V must be chosen at y = . The boundary value
can be set to any reasonable number based on the size of the computational domain
and the strain rate. The solution to the problem matches the mass flux profile to the
flame location, as reflected in the profiles of T , Yk , , and U.
In computations with unsteady strain rate, this boundary condition on V must be
updated intermittently. The strain rate parameter a can easily vary one or two orders
of magnitude in a given computation. The following method is applied to update the
mass flux profile in this case. At the start of time step n + 1, an initial guess for V n+1
is obtained by integrating the continuity equation with an+1 , U n , and n :

nj n1
j V jn+1, guess V j1
n+1, guess
+ + njU jn an+1 = 0 (12.13)
t y j y j1

In one step, this expression updates the boundary value on V at y = and gener-
ates a new guess for V n+1 . All the spatial discretizations are performed on a non-
uniform adaptive grid, permitting a dynamic clustering of grid points in regions
where spatial gradients are strong, and thus ensuring adequate resolution through
the reaction-diffusion zone over all the integration time.
The time step for integration is constant, typically chosen on the order of 1 s.
At each time step, discretization reduces the governing PDEs to a set of nonlinear
algebraic equations. The nonlinear system can be written as:

F(x) = 0, F : Rn Rn (12.14)

The output of the function F is a vector containing residuals of the discretized gov-
erning equations, while x is the solution vector, containing profiles of each funda-
mental variable: Yk , T , U, and V .
A set of numerical methods to solve (12.14) efficiently and robustly [35] using an
inexact Newton iteration [40] is used to accelerate convergence. The inexact Newton
condition essentially restates the exact Newton condition from the perspective of an
iterative linear solver. An iterative method is used to find an approximate solution
288 El-Asrag and Ghoniem

to the exact Newton condition, F  (xi )si = F(xi ), and i specifies the tolerance to
which this solution (si ) is found.
In our implementation, the inexact Newton method is coupled with a safeguarded
backtracking globalization to improve its domain of convergence [14]. If the step
si of the inexact Newton condition does not sufficiently reduce F, the step is
reduced by a scalar factor , essentially backtracking along the search direction.
Backtracking continues until the condition on F is met, for in a sufficiently small
neighborhood of the trial solution xi , the linear model must indicate the correct
downward path; the Newton equation is consistent.

12.5 UFE LES Sub-grid Combustion Model

In this section, the elements of the UFE sub-grid combustion model along with the
closure for the filtered LES equations are introduced.
The flame embedding approach is broadly compatible with the flamelet assump-
tion [42]. If chemistry is fast relative to mixing, reactions occur within a thin zone
and the flame acquires a well defined burning velocity. In this limit, turbulent eddies
do not penetrate the thin reaction zone, and a laminar flame structure can be as-
sumed. Thus, a turbulent flame can be regarded as a stretched laminar flame that is
embedded in an ensemble of vortices of different size and intensity. The net impact
of these vortices is to convolute the flame surface on multiple scales while exerting
a time-dependent strain on it. The components of the UFE model are shown in the
schematic diagram in Fig. 12.6. A high order accurate level set algorithm is used to
track the motion of the flame front. The solution of the level set equation provides
information regarding the flame surface location and orientation inside each cell.
The level set that defines the flame surface is specified using a surrogate progress
variable C = C . For this purpose, a progress variable equation is filtered and solved
along with the filtered Navier Stokes equations. The closure for the source terms
in the progress variable equation, the energy equation, and the level set equation
is implemented by solving directly on the subgrid level a set of unsteady strained
stagnation point flames. The level set subroutine and the 1-D strained flame code
provide the information needed for closure of the source terms for the LES solver.
Consequently, in our approach the level set is used in two distinct ways; numerically
to track the flame surface and as a tool to construct a subgrid model for combustion
closure. The coupling is done by integrating within each cell the solution of the
strained flame along the flame surface. A systematic diagram explaining the above
UFE closure algorithm is shown in Fig. 12.7. At each time step the above procedure
is applied for each cell in the LES mesh. More details on the LES approach and the
subgrid model can be found in El-Asrag et al. [18].
A direct subgrid closure algorithm is proposed for the turbulent flame speed,
the filtered energy, and progress variable source terms. At each time step, an av-
erage sub-filter flame front is constructed. First, we introduce an inner grid inside
each cell. Then, by using the filtered level set-field values G at the cell corners,
Unsteady Flame Embedding 289

Fig. 12.6: The components of

the unsteady flame embedding
algorithm.

Fig. 12.7: A systematic diagram explaining the UFE closure algorithm. For each
cell in the LES mesh in the upper left corner, the level set values on the cell corners
are used to provide information about the flame structure as well as the normal
directions. One-dimensional lines are constructed at the location C = C and a 1-D
strained flame is solved along these lines. The information along this 1-D line is
then used for closure of the LES equations.

we undergo bilinear interpolation on the inner mesh using a second order formula.
The flame sub-filter structure is constructed by joining the values corresponding to
G = Go . The next step is to construct a set of 1-D lines at the intersections of the
flame front within the inner mesh points in the direction normal to the flame front
defined as n = G/| Along each line a stagnation point flow problem is
G|.
solved for the elemental flame model described in Sec. 12.3.
sgs
For a general scalar , the sub-grid contribution to the filtered value
over the flame surface will be computed in two steps. In each cell, after constructing
the 1-D lines normal to the flame 2-D surface, the stagnation point flow problem
290 El-Asrag and Ghoniem

is solved using the local filtered strain rate and boundary conditions. The scalars
profiles are then integrated numerically over each line:
n1
1
i = 2 [ ( j + 1) + ( j)] [x( j + 1) + x( j)] , (12.15)
j=1

where n is the number of grid points along the 1-D line, x is the local coordinate,
and i is an index that refers to the line number inside the cell volume. The num-
ber of 1-D lines inside each cell can be considered as equivalent to the number of
flamelets generated a priori for tabulation in the laminar flamelet approach. The
second step is to integrate over the flame surface to get the flame surface integrated
contribution to the filtered field for each cell volume conditioned on C = C . The
conditioned values are computed using line integral in 2-D and surface integral in
3-D space. Consequently, for m 1-D lines along the flame surface, the line integral
can be evaluated as follows using Riemann sum directly over the flame surface.
m
|C = (i Si ) (12.16)
i=1

where Si is the distance between two points on the flame surface at the sub-grid
level and i is computed from Eq.12.15. To account for the unresolved subgrid
fluctuations, the conditioned integral solution over the flame surface |C is con-
voluted with a PDF that describes the subgrid fluctuations for the filtered progress
 A Beta distribution is adapted here following the work in the litera-
variable C.
ture [39, 47, 51, 60]. Finally the subgrid contribution to the filtered flow field is
computed as:
1
|C P (C ; x, t) dC ,
sgs
= (12.17)
0
 (C ; x, t) is the mass weighted filtered probability density function of the
where P
progress variable C. The mean of the Beta distribution is computed from filtered
progress variable c equation:

c uc c
+ + (uc
 uc) = + DC , (12.18)
t xj xj xj xj

where DC is the progress variable diffusivity and the PDF variance is computed
algebraically [47, 51]:
C
C
2 = C 2
V , (12.19)
xi
where the coefficient CV is computed dynamically [47].
To this end, the only remaining task is to compute for the closure of filtered
progress variable, energy, and level set equations. Along each 1-D embedded line
the flame location that corresponds to C = C is chosen as the location of maximum
heat release. For closure of the level set equation source term, is the filtered flame
speed S conditioned at C = C . The filtered flame speed S is expressed as [4, 9]:
Unsteady Flame Embedding 291

S = Srsgs + Snsgs + Stsgs , (12.20)

where the subgrid reaction rate, normal diffusion, and curvature contributions to the
displacement speed are Sr , Sn , and St , respectively. These values are defined on
sgs sgs sgs

the flame surface conditioned on C = C as follows:


 

Sr =
sgs

 
|C|
 C=C 
n Dn C 
Snsgs = 
 
|C| 
C=C

Stsgs = 2DC=C
 , (12.21)

where the filtered curvature is defined as  = 12 n. In Eq. 12.21  and  are

the progress variable production rate and density along the 1-D line, and D is the
mean diffusivity. The conditioned flame speed is then integrated using a similar
expression to Eq.12.16. For closure of the energy and the reaction progress variable
source terms, i will be equal to the production rate equivalent to the set of species
that represent the reactive progress variable. Finally the heat release is computed as
h = Nk=1
S
k hk , where hk is the species enthalpy.

12.6 Numerical Results

In this section the model validation is introduced. Flame/vortex interaction is a well

known canonical problem frequently used for the validation of turbulent premixed
flame models. The flame stretch is essentially governed by the flow characteristics
in the normal direction to the flame front. Therefore this setup, even in 2-D, is an
efficient model to describe flame stretch and curvature [53]. Here, we first describe
briefly the numerical approach employed for solving the Navier Stokes equations
and the problem setup. Then finally the comparisons between the UFE model and
the DNS data.
For accurate prediction of the sub-grid scalars, high-fidelity codes based on
higher order stable numerical schemes are needed to avoid contamination of the
resolved scales by the numerical error. An implicit fully compressible fifth order
accurate in space and time code is developed [17, 18] for this purpose. A hybrid ex-
plicit Runge-Kutta/Implicit algorithm is employed [34, 55]. This scheme has been
shown to reduce the stiffness produced by highly stretched grids [61], an important
property for wall-bounded flows where grid stretching is vital to capture the viscous
and heat transfer effects near the wall. Consequently, this allows for higher stability
and CFL condition. In our validation studies, up to 100 times the explicit CFL is
achieved for simple test cases. The algorithm employs collocated finite-difference
discretization on a structured grid. Time integration is performed by combining an
292 El-Asrag and Ghoniem

Fig. 12.8: The flame vortex interaction initial setup. From [18].

explicit Runge-Kutta scheme with the solution of an implicit scheme for the govern-
ing equations. The convective nonlinear Euler terms are discretized by a fifth order
WENO algorithm [25] that relaxes to third order upwind on the boundaries. The
viscous terms are discretized using central second/fourth order accurate finite differ-
ence formulation. The Navier-Stokes Characteristic Boundary Conditions (NSCBC)
approach [53] is employed for the inflow/outflow boundary conditions.
The problem setup is shown in Fig. 12.8. An inflow/outflow configuration is used
in the axial direction and symmetric boundary conditions for the upper and lower
boundaries. The inflow speed matches the flame laminar burning speed SL . The
vortex strength Umax and size RC are adjusted to operate the flame in the thin reaction
zone regime. The vortex is initialized using the following stream function [53]:
 2 
x + y2
= Umax exp . (12.22)
2RC2

The flame is initialized using a planar premixed flame solution for a stoichiomet-
ric equivalence ratio. A single step chemistry is used R P, where the production
rate is computed from the following expression:
 
(1 )
= B YR exp , (12.23)
1 (1 )

here B is the pre-exponential factor, is the temperature factor, and is the re-
duced activation energy. The pre-exponential factor is adjusted to achieve the re-
quired flame speed. The set of non-dimensional parameters that characterize the
problem is shown in Table 12.1.
Unsteady Flame Embedding 293

Table 12.1: Flame/vortex interaction operating conditions and parameters a .

Umax Rc
Le Sc Pr ReSL Rev SL Ka Da SL F

1.0 0.73 0.73 48 259 38 74 0.263 0.48 10

a Where Le is the Lewis number, Sc is the Schmidt number, Pr = Sc/Le is the
 0.5
Prandtl number, Ka = (Umax /SL )3 /(RC /F ) is the Karlovitz number, and
Da = (SL RC ) / (Umax F ) is the Damkohler number. Here F is the laminar flame
thickness computed based on the adiabatic flame temperature Tad , the unburned
Tad To
gas temperature To and the maximum temperature gradient | dTdx |max as F = dT .
| dx |max
ReSL is the Reynolds number based on the inflow or the laminar flame speed,
Rev = (Umax RC )/(2 ) is the turbulent Reynolds number based on the vortex size
RC and core velocity Umax .

Two test cases are compared. Test case(1) is fully resolved DNS, where the grid
spacing x is adjusted to resolve the flame thickness by 8 grid points and the ratio of
x/ = 1.2. These ratios are recommended for DNS simulations in the thin reaction
zone (TRZ) regime [33]. The second test case uses UFE as a sub-grid closure for an
under-resolved mesh, where the grid spacing to the Kolmogorov length scale ratio
is around 10.
Some instantaneous results are shown next. Figure 12.9 shows the reaction
rate contours combined with the vorticity spectrum at two time instants for DNS
and UFE. The general turbulence/flame interaction is well captured. The level-set
(G = Go ) is shown by the middle thick line. This line represents the flame-surface
at C = C . As the figure shows the flame reaction zone is well captured and the
flame displacement speed is well predicted by the UFE model. This is attributed to
the model ability to account for the history effects by computing the displacement
speed dynamically on the flame surface.
Figure 12.10 shows four consecutive time instants for the flame/vortex interac-
tion and the heat release contours. Vorticity is generated by the dilatation effect as
the flame propagates normal to itself and the misalignment of the density and the
pressure contours as the flame propagates towards the fresh gases (i.e. baroclinic
vorticity). As the vortex hits the flame surface, vorticity is generated with an oppo-
site direction downstream the flame surface.
Figure 12.11 shows the velocity vector plot combined with the progress variable
iso-contours. As shown, the turbulent vortices of length scales smaller than the flame
thickness penetrate in the preheat zone and cause distortion of the progress variables
iso-contours. The contours are not parallel since the displacement speed is different
for different iso-levels. This observation confirms that the flame operates in the thin
reaction zone regime.
The averaged volumetric consumption speed of the flame at each  time
  instant is
plotted in Fig. 12.12. The burning speed is computed from ST = u1A p V dV [3],
294 El-Asrag and Ghoniem

Fig. 12.9: Comparison of the reaction rate contours combined with the vorticity
spectrum (background) at two time instants between DNS and UFE. The contour of
G = Go line is shown in the UFE case. From [18].

where A p is the projected area/length of the flame surface in the mean flow direction.
This quantity is averaged over space and plotted at each time step. The figure shows
good comparison with the DNS data. The flame speed increases monotonically with
time due to the stretch effect and the increase in the flame surface area. However, this
rate declines with time due to the viscous dissipation effects as no flame quenching
is observed here for a stoichiometric unity Lewis number flame.
The PDF of the progress variable shows the well-known bi-modal Brass-Moss-
Libby (BML) shape in Fig. 12.13. The data compares well with the DNS results.
For premixed flames the most two common states are the fully burned and the un-
burned [53]. However, only finite-rate models can capture the intermediate burned
states. The PDF of curvature is also shown in Fig.12.14. In agreement with the lit-
erature [4] the curvature has a global zero mean value, with the burned side more
skewed towards negative curvature and the unburned side towards positive curva-
ture.
The above curvature behavior shown in Fig. 12.14 is expected to affect the sign
of the displacement speed, where more probability of a negative displacement speed
is observed towards the unburned side since the upper flame part is propagating to-
Unsteady Flame Embedding 295

Fig. 12.10: Vorticity spectrum (background) and heat released contours at four con-
secutive time scales. From [18].

wards the inflow. This is shown in Fig. 12.15(b). In agreement with the curvature
behavior, the displacement speed also shows zero mean value, and as the progress
variable increases (moving towards the unburned gases), the displacement speed
has more negative values. Figure 12.15(a) shows the ensemble average displace-
ment speed normalized by the laminar flame speed in the mean progress variable
space. The figure shows that the displacement speed decreases as we go towards the
unburned side due to the curvature effect as indicated earlier. When the displace-
ment speed is density weighted, however, it shows nearly constant behavior, as the
effect of expansion due to heat release compensates the increase/decrease of the
displacement speed by the curvature effect [4].
296 El-Asrag and Ghoniem

Fig. 12.11: The velocity vector plot combined with progress variable contours show-
ing the penetration of vortices in the preheat zone. From [18].

Fig. 12.12: Comparison of the volumetric mean flame speed (m/s) between
DNS () and UFE (). From [18].

12.7 Conclusions

An unsteady sub-grid combustion closure model is developed based on the flame

embedding concept. The UFE model solves directly on a set of 1-D problems em-
bedded in the normal direction to the flame surface using the resolved information
on the LES mesh. The model problem for the 1-D line is the stagnation point flow
configuration, where a formulation of variable strain rate is solved along the stagna-
tion line. The model is validated with a vortex/flame interaction problem. Promis-
ing results were shown by comparing with the DNS data. The results show that the
Unsteady Flame Embedding 297

Fig. 12.13: Comparison of the progress variable PDF(C) between DNS () and
UFE (). From [18].

Fig. 12.14: The curvature PDF( ) for different progress variable iso-contours. From
[18].

unsteady turbulent flame propagation and the flame turbulence interaction is well
captured. Future work will investigate the role of the scalars subgrid fluctuations
and the model extension to simulate problems that exhibit extinction and reignition
events.

Acknowledgment

This work is supported by the King Abdullah University of Science and Technology
(KAUST), Award No. KUS-11-010-01. The authors also gratefully acknowledge
the inspiring discussions and valuable comments from Suresh Menon, Heinz Pitsch,
Tarek Echekki, Evatt R. Hawkes, R. Stewart Cant, Christopher Rutland, Youssef
Marzouk, and Jean-Christophe Nave.
298 El-Asrag and Ghoniem

Fig. 12.15: The ensemble average displacement speed in the progress variable space
(a) and the PDF of the displacement speed (b). From [18].

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Chapter 13
Adaptive Methods for Simulation of Turbulent
Combustion

John Bell and Marcus Day

Abstract Adaptive mesh refinement (AMR) is an effective approach for simulating

fluid flow systems that exhibit a large range of numerical resolution requirements.
For example, an AMR simulation could dynamically focus maximum numerical
resolution near a propagating flame structure, while simultaneously placing coarser
computational zones near relatively large flow structures in the exhaust region down-
stream of the flame. However, since turbulent reacting flow applications already
tend to be significantly complex, an AMR implementation might quickly become
prohibitively intricate. In this chapter, we discuss basic AMR algorithm design prin-
ciples that can be applied in a straightforward way to build up extremely efficient
multi-stage solution strategies. As an example, we discuss an adaptive projection
scheme for low Mach number flows, which was used to analyze flame-turbulence
interactions in a full-scale simulation of a turbulent premixed burner experiment
using detailed chemistry and transport models.

13.1 Introduction

We have entered a new era in turbulent combustion calculations; we can now sim-
ulate laboratory-scale turbulent reacting flows with sufficient fidelity that the com-
puted time-dependent multi-dimensional solution should be expected to agree with
experimental measurements. Such calculations are possible only through the effec-
tive use of modern high-performance computing architectures, and even then only
by exploiting many of the multi-scale aspects of the reacting flow system.

John B. Bell
Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory,
Berkeley, CA, USA, e-mail: jbbell@lbl.gov
Marcus S. Day
Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory,
Berkeley, CA, USA, e-mail: msday@lbl.gov

T. Echekki, E. Mastorakos (eds.), Turbulent Combustion Modeling, 301

Fluid Mechanics and Its Applications 95, DOI 10.1007/978-94-007-0412-1 13,
Springer Science+Business Media B.V. 2011
302 John Bell and Marcus Day

Adaptive mesh refinement provides a mechanism for exploiting spatial and tem-
poral variability in resolution requirements for given problem. With an adaptive
algorithm, resolution can be changed dynamically to match local features of the
flow. For example, finer resolution can be employed in a neighborhood of a flame
to resolve the details of the flame structure. Fine resolution can also be used around
complex fluid dynamical features such as shear layers or regions of intense turbu-
lence while using coarse resolution where the flow has little structure such as in the
products region of a premixed flame.
This chapter discusses the construction of structured, hierarchical adaptive mesh
refinement schemes for modeling reacting flows. We will consider both a fully com-
pressible formulation that is applicable to generic reacting flow simulations and a
low Mach number formulation that exploits the temporal separation of scales be-
tween acoustic wave propagation and fluid motion typical of a broad range of tur-
bulent combustion scenarios. The discussion will focus on issues related to AMR
for the low Mach number model; developing an adaptive compressible flow solver
requires only a subset of the ideas. We will restrict our consideration to single-phase
gaseous combustion and, for the sake of exposition, assume an ideal gas equation of
state. The generalization to more complex equations of state is discussed in Bell et
al. [9].
The objective here is to provide a pedagogical overview of the basic design con-
cepts used to develop block-structured AMR algorithms. The mathematical formu-
lation is presented first, followed by key AMR concepts. Low Mach number AMR
is then presented, followed by a discussion of implementation issues and software
design. Finally, applications are discussed.

13.2 Mathematical Formulation

Gas phase combustion problems can be modeled by the multicomponent reacting

compressible Navier-Stokes equations. Ignoring external body forces and assuming
an ideal gas law, the governing equations expressing conservation of species mass,
momentum and total energy are:

Ym
+ YmU = Fm + m ,
t
U
+ UU + p = ,
t
E
+ ( E + p)U = (Q + U)
t
where is the density, U is the velocity, E = em (T )Ym + 1/2U U is the total en-
ergy, Ym is the mass fraction of species m, T is the temperature, and m is the net
mass production rate for species m due to chemical reactions. Also, Q is the heat
flux, is the stress tensor, and Fm is the diffusion flux of the mth species. Note
Adaptive Mesh Refinement 303

that the internal energy, em , incorporates the potential chemical energy that is re-
leased by exothermic reactions in the flame. For these equations, we have Ym = 1,
Fm = 0 and m = 0, so that the sum of the species transport equations gives the
conservation of total mass

+ U = 0
t
These evolution equations are supplemented by an equation of state, p = p( ,Ym , T ).
For the mixture of ideal gases considered here the equation of state is given by
Ym
p = RT = RT
m Wm

where Wm is the molecular weight of species m. We note that the numerical solution
approaches discussed here have been applied to a much more general equation of
state (e.g., see [9]).
The compressible flow equations admit two types of waves, material waves that
propagate at the fluid velocity U, and acoustic waves that propagate relative to the
moving fluid at the speed of sound, c. One possible approach to simulating reacting
flow is to discretize the compressible system directly and numerically resolve all of
the time scales. However, for a wide range of reacting flow phenomena, including
most practical combustion systems, the fluid velocity is considerably smaller than
the sound speed, and this disparity in scales can be exploited to compute much
more efficiently. For typical laboratory flame experiments U 3 30 m/s, while
the sound speed in the hot product gases is about 1000 m/s. Flows in this regime are
referred to as low Mach number since the Mach number, M = |U|/c  1.
The low Mach number combustion formulation was first introduced by Rehm
and Baum [34] and was later derived rigorously from low Mach number asymptotic
analysis by Majda and Sethian [24]. The basic steps of the analysis are to first nondi-
mensionalize the system with respect to the time and length scales of the flow and
then to expand the resulting terms in powers of the Mach number M. Examining the
behavior as M 0, one can show that in an unconfined domain, the pressure can be
decomposed as
p(x,t) = p0 + (x,t)
where p0 is the ambient thermodynamic
  pressure and is a perturbational pressure
field that satisfies /p0 O M 2 . (In a more general setting, p0 may be a function
of time.) With this decomposition, p0 defines the thermodynamic state; thermody-
namic quantities are independent of . The flow model in this regime becomes

Ym
+ UYm = Fm + m ,
t
U
+ UU + = , (13.1)
t
h
+ Uh = Q,
t
304 John Bell and Marcus Day

where the equation of state constrains the evolution. Note that the energy vari-
able used in this formulation is the mass-averaged enthalpy, defined as h(T,Ym ) =
m Ym hm (T )Ym , where
 T
hm (T ) = c p,m (T )dT + hm (298K). (13.2)
298K

Here, hm (298K) is the standard heat of formation of species m at 298 K, and c p,m is
its heat capacity at constant pressure. Empirical fits for both hm (T ) and c p,m (T ) are
readily available. By combining the enthalpy equation and the species conservation
equation, we can derive an alternative form of the energy equation in terms of T
given by
T   
cp +U T = Q hm Fm + m
t m

where c p = m Ym c pm is the specific heat of the mixture at constant pressure. Al-

though we advance the energy equation in terms of enthalpy, the temperature equa-
tion is useful in defining the constraint.
This low Mach number model retains compressibility effects due to chemical
heat release and other thermal processes, but eliminates acoustic wave propagation
entirely. The perturbation pressure, , plays the role of a Lagrange multiplier to con-
strain the evolution so that the thermodynamic pressure is equilibrated everywhere
instantaneously. Note that the form of these equations is no longer an initial value
problem; the constrained system forms a differential algebraic equation (DAE) sys-
tem that is considerably more difficult to evolve numerically.
For the low Mach number combustion model, we do not work with the constraint
given by the equation of state directly. Instead, we differentiate the equation of state
in the Lagrangian frame and use the evolution equations for , Ym and the auxiliary
equation for T to define a constraint on the velocity:

1 D 1 DT R 1 DYm
U = = +
Dt T Dt R m Wm Dt
 
1 1 W
= Q hm Ym hm + ( Dm Ym )
c pT m m Wm
 
1 W hm (T )
+ m
m Wm c pT
S

In section 13.4 we will outline an approach to solving the constrained low Mach
number system based on a projection formulation. However, we first present a num-
ber of basic concepts inherent to our approach to adaptive-grid discretizations.
Adaptive Mesh Refinement 305

13.3 AMR Basic Concepts

There are several distinct approaches to developing adaptive mesh refinement al-
gorithms. The approach taken here uses a block-structured hierarchical form of re-
finement, hereafter referred to simply as AMR. AMR was first developed by Berger
and Oliger [12] for hyperbolic partial differential equations. A conservative version
of this methodology for gas dynamics was developed by Berger and Colella [11]
and extended to three dimensions by Bell et al. [5]. This approach was extended to
variable-density incompressible flow by Almgren et al. [1]. Pember et al. [31] gen-
eralized the approach to low Mach number combustion with simplified chemistry
and transport. Day and Bell [19] extended the method to treat detailed chemistry
and transport.

13.3.1 Creating and Managing the Grid Hierarchy

In AMR, the local mesh refinement strategy is designed to exploit the advantages of
uniform-grid PDE discretizations (uniform well-characterized errors, high accuracy
per floating-point operation, memory-efficient stencil evaluations, etc.). The funda-
mental data object is a logically rectangular subdomain of uniformly spaced grid
cells. A given level of refinement, , in the AMR hierarchy is represented as a union
of such subdomains with a common x . The refinement ratio, r = x / x+1
(typically 2 or 4) defines the reduction in grid spacing with increasing levels in the
AMR hierarchy. The computational domain for an AMR calculation is rectangular

Fig. 13.1: Typical AMR grid structure in 2D surrounding a representative feature.

The computational domain (gray box) is tiled by the union of the coarsest (black)
grids, 1 = 1,i . The finest level, 3 = 3,i encloses the feature of interest, and
is properly nested within 2 .
306 John Bell and Marcus Day

and is completely tiled by the coarsest AMR level, 1 . Finer levels tile successively
smaller regions of the domain and satisfy the following proper nesting requirements
(see Figure 13.1):
Each subdomain at level  j , j,i , starts and ends at the corners of a cell in the
next coarser level,  j1
There must be at least nbu f > 0 level  j cells between the bounding edge of level
 j+1 and any uncovered  j1 cell.
where a cell in level  j is said to be uncovered if there are no cells in level  j+1
at that location. Note that fine levels may extend to the domain boundary and that
there is no requirement that level  j+1 grids be fully contained within a single level
 j grid (e.g., see 3,2 in Figure 13.1).
A key feature of the AMR approach is that interfaces between adjacent levels
in the hierarchy are aligned with the coordinate directions and coincide with cell
boundaries on both levels. As we will see, this greatly simplifies the construction
of composite (multi-level) solution methods. Also, the subdomains, , j at each
refinement level typically contain a large number of cells, typically 16-64 cells in
each direction. Compared with cell-by-cell local refinement strategies, AMR af-
fords significant advantages in terms of access to a wide range of well-characterized
discretizations for uniform grids over most of the computational domain. Nonuni-
form regions are limited to the substantially smaller codimension-one regions at the
interfaces between levels. Finally, the AMR grid structure presents a natural paral-
lelization strategy for distributed-data computing systems, which we discuss briefly
later in this chapter.

Fig. 13.2: Grid generation to refine a structure (blue line). Cells tagged for refine-
ment include those containing the structure (blue cells) and those in a buffer zone
surrounding them (pink cells). New rectangular grid boxes are constructed to en-
close the tagged cells with a gridding efficiency, =(# cells tagged)/ (# cells newly
boxed).

Although the AMR grid hierarchy does not move with respect to the Eulerian
frame, the structure is dynamic in time. At discrete intervals between time steps,
subsets of the hierarchy are destroyed and regenerated in response to the evolving
solution (see Figure 13.2). During such a re-grid operation, an error estimation pro-
cedure identifies individual cells at level  j requiring additional refinement (blue
Adaptive Mesh Refinement 307

cells in Figure 13.2, for example). These cells are clustered together and surrounded
with a buffer layer (pink cells in Figure 13.2), and then grouped into rectangular
boxes and refined to generate new grids at level  j+1 . Data on the new patches is
filled by a copy-on-intersect operation from data on the previous level  j+1 grids. In
regions where the procedure generates entirely new cells at that level, data is filled
by interpolation from next coarser level. This re-grid procedure must simultaneously
recompute the grid structure at all finer levels,  j+2 . . . max in order to guarantee that
the new hierarchy does not violate proper nesting requirements. In a typical appli-
cation regeneration of the AMR grid hierarchy accounts for less than 1% of the total
run time. The initial grid hierarchy is created using a similar strategy, except that
the state data is filled by user-supplied functions for initial data at each level. The
re-grid frequency required for a specific application is determined by the dynam-
ics of refined features on the Eulerian grid, and is controlled by the error tagging
procedure and through a number of adjustable parameters, including buffer zone
thickness, nesting buffer width, nbu f , and grid efficiencies ( in Figure 13.2).

13.3.2 AMR Discretization

There are several approaches to solving partial differential equations on a hierar-

chical AMR grid structure. One approach would be to simply write discrete ap-
proximations that operate directly on the irregular grids. Here, we consider a differ-
ent paradigm in which the levels are advanced independently with a uniform-grid
scheme, and then synchronized to account for the irregular interface between levels.
In the remainder of this section, we will discuss how to develop finite-volume
discretization schemes on AMR grids. The key ideas will be discussed for two-
level systems in one spatial dimension with and without subcycling in time. We will
briefly discuss the issues related to generalizing to multiple space dimensions and
extensions to more than two levels.

13.3.3 Hyperbolic Conservation Laws

We first consider a system of hyperbolic conservation laws

Ut + Fx = 0

discretized with a time-explicit finite volume scheme:

n+1/ n+1/
Uin+1 Uin Fi1/ 2 Fi+1/ 2
2 2
= (13.3)
t x
308 John Bell and Marcus Day

where the numerical fluxes are explicitly computed from the solution at time t n , i.e.,
F n+/2 = F(U n , t). The explicit treatment of fluxes implies that t is restricted by
1

the CFL limit. The ideas presented here can easily be generalized to any other dis-
cretized PDE systems that are based on flux differencing. Conservative flux-based
methods for the compressible Navier-Stokes equations, for example, fit within this
framework.
We want to consider how to modify the basic uniform-grid scheme to update
U n on a locally refined grid, as depicted in Figure 13.3. The goal is to define a
composite (multi-level) solution using a numerical flux algorithm that is unaware
of the multi-level nature of the data. Initially, we will assume that both the coarse and
fine grid use the same time step. In this case, we need to specify how to compute
the numerical flux at each interface in Figure 13.3. We define an averaging (or

Fig. 13.3: A two-level 1D system with a coarse/fine interface, c f , between fine

cell j, and coarse cell J.

restriction) operator A that maps data at the fine grid resolution to the coarse grid
resolution. We also define an interpolation operator I that interpolates data from
the coarse resolution to fine resolution. With these operators, we can define the flux
on the coarse grids away from the coarse / fine grid interface by using A (U f ,n ) to
define data needed to compute the coarse fluxes in regions covered by fine grid. For
example, if the numerical flux uses two values on each side of the interface, then
to compute the flux at edge J + 1/2 we define an effective coarse value at a fictitious
coarse cell J 1 and use that value to compute the numerical flux. Similarly, we
can use I (U c,n ) to construct data at the fine resolution from the coarse data near
the boundary of the fine grid as needed to compute fluxes at the fine resolution. To
complete the specification of a composite grid solution, we only need to specifiy the
flux at the coarse/fine boundary. We define that flux to be the flux computed using
the fine grid data and interpolated coarse grid data. This completes the specification
of a well-defined flux at each interface that enable us to advance the composite
solution.
We can reinterpret this algorithm in such a way that the coarse and fine grids
can be advanced independently and subsequently synchronized. This interpretation
provides an algorithm to advance the composite solution in terms of an AMR grid
hierarchy as defined earlier. Recall that the coarsest AMR level tiles the entire com-
putational domain, including region that is tiled by the finer grids. Although this
region is covered and thus not formally part of the composite solution, it provides
a convenient location to store A (U f ,n ). Operationally, we store A (U f ,n ) over the
entire covered portion of the coarse grid so that the coarse grid integration need
not be aware of where the fine grid is located. We can then apply the uniform grid
algorithm to advance the solution on the entire coarse grid in Figure 13.4.
Adaptive Mesh Refinement 309

Fig. 13.4: The coarse integration. Covered cells (indexed in parentheses) are first
filled using A (U f ,n ), and then become part of the uniform-grid data at the coarse
level.

In order to advance the fine level, we use I to generate ficticious fine data in a
buffer zone along the boundary of the fine grids. These cells are often referred to as
ghost cells (see Figure 13.5). The ghost cells provide sufficient data to compute
fluxes needed advance the solution on the fine grid, including fluxes at the coarse /
fine boundary needed to advance fine cells adjacent to the boundary.

Fig. 13.5: The fine grid integration. Ghost cells (indexed in parentheses) are filled
using I (U c,n ).

At this point we have values for U n+1 that are defined on both the coarse and
fine levels. The only difference between this provisional solution and our desired
composite solution is that the two levels have not been computed with the same flux
at the c f interface. In particular, the coarse grid cell UJn was advanced with the
n+1/ f
coarse flux FJ1/ 2 instead of the fine grid flux Fj+1/ as specified in the definition of
2 2
the composite solution given above. We can correct this discrepancy by modifying
UJn+1 by
xcUJn+1 := xcUJn+1 t f FJ
c
1/ + t Fj+1/ .
2
f f
2

This inter-level synchronization step is referred to as refluxing.

13.3.3.1 Sub-cycling

We can extend the notion of refluxing to enable us to subcycle in time, provided that
t c is an integer multiple of t f . First we advance the coarse grid with t c using
the uniform grid algorithm as shown in Figure 13.6. Next, we advance the fine grid
r steps with t f = t c /r. Ghost data (in parentheses) is required on the boundary
of the fine grid for each subcycled step. Here the data must be interpolated in time
and space in order to provide consistent boundary conditions for the fine levels at
intermediate times, as shown in Figure 13.7. The only modification to the algorithm
that is necessary for sub-cyling is that the refluxing correction to the coarse cell at
the boundary of the fine grid is now summed over the r fine grid time steps
310 John Bell and Marcus Day

Fig. 13.6: Time advance of the coarse level near a coarse/fine interface in a subcy-
cled integration. Ghost data (in parentheses) is obtained by the restriction operator,
A (U f ,n ).

Fig. 13.7: Time advance of the fine level near a coarse/fine interface in a subcycled
integration. Ghost data (in parentheses) is obtained by the interpolation operator,
I (U c,n ,U c,n+1 ), and interpolation in space and time.

xcUJn+1 := xcUJn+1 t c FJ 1/ + t Fj+1/

c f f
2
(13.4)
2

From this discussion, we can abstract out a process for developing PDE dis-
cretizations on a locally refined grid using a uniform-grid flux-based discretization
method, and arrive at a set of discretization design principles:
1. Define what is meant by the solution on the composite grid hierarchy.
2. Identify the errors that result from solving the equations on each level of the
hierarchy independently.
3. Solve correction equation(s) to fix the solution.
4. For subcycling, average the correction in time.
Within this setting, the coarse grid supplies Dirichlet data as boundary conditions
for the fine grids. The errors then take the form of a flux mismatch at the coarse/fine
interface. These ideas can be extended to multiple dimensions simply by defining
suitable operators for A and I . Because of the locality property of the refluxing
operation, the algorithms discussed above can be applied recursively when there are
more than two levels.
Adaptive Mesh Refinement 311

13.3.4 Elliptic

Next we illustrate how to apply the design principles described above to construct
an algorithm to solve an elliptic equation on an AMR grid. We first consider how to
define an appropriate composite solution to

xx =

on the AMR grid shown in Figure 13.3. A second-order discretization at all fine grid
points i = j can be written based on centered difference approximations as
 
1 (i+1 i ) (i i1 )
= i ,
xf xf xf

and at all coarse grid points I = J as

 
1 (I+1 I ) (I I1 )
= I .
xc xc xc

These discretizations are analogous to the flux form used in Equation 13.3 with x
playing the role of the numerical flux. At the c f boundary, the computation x is
cf

slightly more complicated. The strategy is to build a polynomial interpolant, I e ( )

using both coarse and fine data neighboring the c f interface to define an effective
value for I e at j + 1. Then approximate
I e j
xcf = (13.5)
xf

We can then discretize the equation using

 
1 ( j j1 )
x
cf
= j
xf xf

at i = j and  
1 (J+1 J )
xcf = J
xc xc
at I = J. Provided the interpolant I e ( ) is sufficiently accurate, this set of expres-
sions over the composite grid defines a suitable discrete approximation to the elliptic
equation. In particular, in the case of the second-order stencil used away from the
c f boundary, the interpolant I e needs to be sufficiently accurate that the local trun-
cation error at the coarse fine boundary is first-order accurate to ensure second-order
accuracy of the composite solution.
As before, we now consider what happens when we discretize the system on the
coarse and fine grids in two separate steps, and we use our composite discretization
to construct the appropriate synchronization. In particular, we first solve the coarse-
grid problem
312 John Bell and Marcus Day

1 ( I+1 I ) ( I I1 )
= I
xc xc xc
at all coarse grid points I, including those covered by the fine cells. We then solve

1 ( i+1 i ) ( i i1 )
= i
xf xf xf

at all fine grid points, i = j. At i = j, we use the correct stencil defined using the
Eq. 13.5.
c f
The composite solution, and , obtained from solving the levels separately
represents the provisional composite solution, which satisfies the composite equa-
tions everywhere except at J. As before, the error is manifest in the difference be-
( J J1 )
tween xcf and xc . If we let e = , then

cf e = 0

except at I = J, where

1 ( J1 )
e = c
xcf J
xc xc

where cf is the composite discrete Laplacian defined above.

We solve this linear system for the error, e, and correct the provisional solution
by setting
c
c = + ec
f
f = + ef
Note that while the flux mismatch is localized to the c f interface, the correction,
e is not. However, because our model problem is linear, this multi-step approach
exactly recovers the solution to the original composite system. Also note that the
correction equation is an elliptic system on the composite grid, and is in fact no
simpler to solve than the original composite problem. The benefit to this formulation
however, is that it allows us to use subcycling in time for algorithms that solve an
elliptic equation as part of the more complex discretization. An example of this will
be discussed in the next section when we construct a semi-implicit discretization
of a parabolic PDE. Also, in many circumstances we appeal to the properties of
elliptic regularity to reduce the complexity/cost of the composite approximation. In
particular, the right hand side of the synchronization equation vanishes on the region
covered by the fine grid so that on the fine grid e is a discrete harmonic. Thus, e is
very smooth in the fine grid region and often will be well-represented on the coarse
grid. In practice, this means that the synchronization need only be computed at the
coarse level and interpolated to the fine resolution.
Adaptive Mesh Refinement 313

Fig. 13.8: The left image illustrates the interpolation stencils used to interpolate
data to the location needed to evaluate x on the fine grid, denoted by the green dot.
The image on the right shows how a composite elliptic operator is defined in terms
of a flux integral at the coarse / fine boundary.

These options also extend to more than two levels of refinement. As above one
can form and solve a composite correction equation over the entire hierarchy or one
can solve only on the coarse grid and interpolate to the finer grids. An additional
option is to solve the composite operator on levels  and  + 1 and interpolate to
finer levels.
In defining the composite solution for the elliptic case, we have used a slightly
different paradigm than in the explicit hyperbolic case. In particular, the composite
solution does not have any dependence on the coarse data covered by the fine grid.
Although developing a discretization in this form in possible, the implicit coupling
would require that the dependencies associated with averaging operator A be re-
flected in the overall discretization, making for extremely large stencils, particularly
in multiple dimensions. In multiple dimensions, we use the interplation scheme de-
picted in Figure 13.8. We first interpolate values from coarse grid cells indicated in
red to define values at the blue points using quadratic interpolation. We then inter-
polate from the blue points and the fine cells marked to define values at the green
points needed to evaluate the stencil, again quadratically. We note that in this form,
the boundary flux x is a function of coarse value at the red points and the fine val-
cf

ues at the points on the fine grid. The multidimensional composite discretization
at the coarse cell adjacent to the fine grid is then defined by the divergence of the
composite flux at the boundary and standard fluxes at the other interfaces as shown
in Figure 13.8. With these definitions, the local truncation error is uniformly first
order accurate and the solution is uniformly second order.
314 John Bell and Marcus Day

13.3.5 Parabolic Systems

In this section we discuss how to combine AMR discretizations for hyperbolic and
elliptic systems in order to develop an AMR discretization of the parabolic model
equation
ut + fx = uxx
on the AMR grid depicted in Figure 13.3. We consider a semi-implicit algorithm
in which we combine a time-explicit treatment advective terms and Crank-Nicolson
treatment of diffusion. In particular,
n+1/ n+1/
un+1 uni fi+1/ 2 fi1/ 2  cf n+1 
i
+ 2 2
= ( u )i + ( cf un )i
t xloc 2

where xloc is the local value of x. Here the hyperbolic flux f n+/2 = f n+/2 (un )
1 1

is computed using the composite hyperbolic flux defined above and cf denotes
the composite discrete Laplacian. This defines a suitable definition of a composite
solution.
As before, we consider what happens if we advance the coarse and fine grids
separately. We first advance the coarse solution, and then use the result to generate
Dirichlet conditions for the fine grid advance. Let un+1 denote the provisional com-
posite solution formed in this way. Let en+1 = un+1 un+1 represent the difference
between the provisional and exact composite solutions. Substituting un+1 into the
composite discretization, we see that en+1 satisfies
 
t cf t
I en+1 = ( f + D)
2 xc

where
 
t f = t f J1/2 + f j+1/2
t  c,n  
t D = ux,J1/ + uc,n+1
x,J1/2
(u cf ,n
x + ucf ,n+1
x )
2 2

As before, the source term is localized to the coarse cell at c f boundary and takes
the form of a mismatch in coarse and fine fluxes. Updating un+1 = un+1 + en+1
recovers the exact composite solution. As in the elliptic case, the implicit coupling in
the parabolic equation leads to an implicit coupling in the synchronization equation.
The parabolic algorithm can also be extended to incorporate subcycling in time.
To define the provisonal solution we first advance coarse grid over the time interval
t c . We then advance fine grid over the interval t f a total of r times, using Dirichlet
boundary data interpolated from the coarse grid. As in the explicit hyperbolic case
discussed above, this interpolation must be in space and time in order to provide
consistent boundary data for the fine grid at intermediate time levels during the sub-
cycling.
Adaptive Mesh Refinement 315

The synchronization equation for the subcycled case now corrects the solution
over the entire coarse interval; the source terms for the refluxing are summed over
the r fine grid time steps. The parabolic synchronization equation is given by
 
t c c tc
I en+1 = ( f + D)
2 xc

where c is the coarse grid Laplacian.

t c f = t c f J1/2 + t f f j+1/2
t c  c,n 
tc D = ux,J1/ + uc,n+1
x,J1/
2 2 2

t f  cf ,n 
ux + uxcf ,n+1
2
In this case, for simplicity, we are only solving the correction equation on the coarse
grid; corrections to the fine grid are interpolated from the coarse data. This reflects
the notion that how we define the composite solution is not so straightforward and
the refluxing equation does not exactly recover an exact composite solution. How-
ever, it does provide a second-order accurate, conservative and stable solution when
the underlying basic operators are themselves second-order accurate and stable. Ex-
tension of these ideas to multiple dimensions and more than two levels of refinement
are a straightforward extension of idea discussed earlier.
Summarizing the construction of the AMR discretization, there are a couple
of key design goals. For the first-order hyperbolic and second-order elliptic and
parabolic equations, the coarse grids provide Dirichlet boundary conditions to the
fine grids. The synchronization steps ensure proper mathcing of the fluxes, and take
the mathematical form of the original equation; e.g., a simple hyperbolic scheme
leads to a simple local reflux corrections; an implicit parabolic equation leads to an
implicit parabolic correction.

13.4 AMR for Low Mach Number Combustion

An AMR implementation for an explicit compressible Navier-Stokes solver is a

straightforward application of some of the ideas discussed above. Developing an
AMR algorithm for the low Mach number equations is considerably more complex.
Here we will only sketch the basic ideas and refer the reader to Almgren et al. [1] and
Day and Bell [19] for the details. We first discuss a single grid integration algorithm
and then discuss some of the key issues associated with an AMR implementation.
Our basic discretization strategy is based on a projection formulation in which
we evolve the system without strictly enforcing the constraint and then project the
resulting solution back onto the constraint. Structurally, the low Mach number equa-
tions evolve momentum, density, species and enthalpy subject to a divergence con-
316 John Bell and Marcus Day

straint on the velocity field, which constrains the evolution of the thermodynamics
variables so that the equation of state is satisfied. Thus, the constrained evolution
in the low Mach number model is similar in structure to the incompressible Navier
Stokes equations. For incompressible flows, projection-based fractional step meth-
ods, which parallel standard DAE methodologies [4, 13], have proven to provide an
efficient discretization strategy [3, 8, 17]. Our goal then is to define a generalized
projection methodology for low Mach number reacting flows. This generalization
requires that we address two key differences between the incompressible flow equa-
tions and the low Mach number system. First, the low Mach number system includes
finite amplitude density variations; and, second, the constraint on the velocity field
is inhomogeneous.
Two different projection-based sequential algorithms have been proposed. One
of these approaches, developed by McMurtry et al. [26] and Rutland and Ferziger
[36], advances the thermodynamic variables and then uses the conservation of mass
equation to constrain the evolution. Imposing the constraint in this form requires
the solution of a Poisson equation. Although this approach does not fit within a
mathematical projection framework, is has been successfully used by a number of
authors to model reacting flows. See, for example, [2729, 33, 37].
We use a different approach based on a generalized projection framework first
introduced in Bell and Marcus [10]. The basic idea is that, subject to boundary
conditions, any vector field, V can be decomposed as
1
V = Ud +

where Ud is divergence free and Ud and 1 are orthogonal with respect to a suit-
able inner product. This decomposition is an exact analog to the standard Hodge
decomposition in a -weighted inner product; e.g.,

1
(Ud ) dm = 0

Using this inner product, we can define a -based projection, P such that P V = Ud
with ||P || = 1 and P2 = P .
This projection operator allows us to address the finite amplitude density varia-
tions in the low Mach number system. The Majda and Sethian analysis [24] shows
the flow compressibility, S, can be represented in terms of the gradient of a potential,

= S

Using this form, we can generalize the -weighted vector field decomposition to
write any velocity field as
1
V = Ud + +

We can then define
U = P (V ) +
Adaptive Mesh Refinement 317

so that U = S and P ( 1 ) = 0. This construction, which specifically addresses

finite amplitude density variations, provides the basis for a robust projection algo-
rithm to evolve the low Mach number equations.
The basic idea of the variable- projection algorithm is to advance the thermody-
namic variables using discretized conservation equations, and generate a provisional
velocity field with lagged approximation to the constraint. We then use the vector
field decomposition to extract the component of the velocity update that satisfies
the modified constraint on the velocity divergence based on the time-advanced ther-
modynamic variables. More specifically, we advance the species mass densities and
enthalpy equations using a flux-based conservative discretization,

n+1 n+1 n n 1
+ ( U ADV )n+/2 = D + R for = h,Ym .
t
Here we use a Crank-Nicolson discretization of the diffusion terms, and a special-
ized second-order Godunov algorithm to compute the advective derivatives. As part
of the Godunov algorithm we compute an advective velocity field U ADV on cell in-
terfaces that has been projected so that it satisfies the constraint as well. The chem-
ical rate equations are decoupled in an operator split form. In particular, we first
advance the chemistry over the interval, t/2. We then advance the advection and
diffusion components over t, and then finish with a second advance of the chem-
istry by t/2. This operator-split strategy allows us to decouple the pointwise chem-
ical kinetics so that we can use stiff ODE integration methodologies to advance the
kinetics equations, while preserving the second-order accuracy of the integration.
In a similar way, we compute a provisional velocity based on a lagged approxi-
mation to the dynamic pressure,

U U n 1 1 1 1 n +
= [U ADV U]n+/2 n/2 + .
t n+/2
1
n+/2
1
2

The updated thermodynamic variables are then used to compute the constraint
at the new time level Sn+1 . To extract the component satisfying the divergence con-
straint we solve the variable-coefficient linear system
 
1
= V Sn+1

for , where V = U + ( t/ n+/2 ) n/2 , and set

1 1

t
Un+1 = V
1
n+ /2 = and 1 (13.6)
n+/2
This procedure implements the generalized vector field decomposition discussed
above but exploits linearity to perform only a single elliptic solve to enforce the
constraint. This final projection is based on node-centered data and a conformal
bilinear finite element construction on a locally refined grid. As with the finite-
318 John Bell and Marcus Day

volume elliptic solvers discussed above, the nodal solver can be decomposed into
a solution operator on each of the refinement levels independently, followed by a
correction procedure. See Reference [1] for details. The projection operator defined
in this way is not a discrete projection; such approaches are referred to as approx-
imate projection algorithms. For approximate projections, the projection step can
be constructed using a number of analytically equivalent forms for V . The choice
identified in Equation 13.6 was demonstrated to be best from a perspective of ro-
bustness and accuracy (see Reference [2] for details).
The algorithm presented above discretely conserves species mass (up to reac-
tions) and enthalpy. However, because we use a linearized form of the constraint,
the solution can drift off the constraint surface, p0 =constant. To correct for this
drift, we include a relaxation term to the compressibility expression used to com-
pute U ADV . This relaxation term is constructed to force the solution back toward the
constraint without violating discrete conservation.
The adaptive time-step algorithm advances grids at different levels using time
steps appropriate to that level based on CFL considerations. The procedure can most
easily be thought of as a recursive algorithm, in which to advance level , 1 
max the following steps are taken:
Advance level  in time as if it is the only level. Supply boundary conditions for
U, ,Ym , h and from level  1 if level  > 1, and from the physical domain
boundaries.
Compute U ADV including projection to enforce the constraint
Advance ,Ym , h
Compute provisional U
Evaluate constraint, Sn+1
Apply generalized projection to compute U n+1 and n+/2
1

If  < max
Advance level ( + 1) r times with time step t +1 = 1r t  as indicated above.
Synchronize the data between levels  and  + 1, and interpolate corrections
to higher levels if  + 1 < max .
Solve elliptic synchronization equation for U ADV
Compute changes for advection terms from explicit reflux and changes to
U ADV
Solve parabolic synchronization with right hand side representing reflux
corrections from advection and diffusion
Apply synchronization projection to correct final velocity field. The right
hand side includes terms representing the discrepancy from projecting lev-
els independently and corrections to the velocity from the advection and
diffusion synchronization.
Adaptive Mesh Refinement 319

13.5 Implementation Issues and Software Design

The combination of adaptive mesh refinement and the projection-based low Mach
number formulation can considerably reduce the computational cost of reacting flow
simulations; however, the computational demand can still be significant, particularly
when modeling turbulent flame phenomena with detailed chemistry and transport.
Consequently, we must be able to implement the adaptive low Mach number algo-
rithm described above so that we can effectively utilize high-performance parallel
computers. Before discussing the implementation in detail, we first comment on the
impact of some of the choices we made in developing the basic algorithm on the
design of the software. Our basic discretization strategy decomposes the problem
into different mathematical components to treat advection, diffusion, chemical reac-
tions, and projections. We use an explicit treatment of advection so that the implicit
solves needed for diffusion and the projection represent discrete approximations to
self-adjoint elliptic partial differential equations. Consequently, we can solve the
requisite linear systems using geometric multigrid. Also, we have decomposed the
dynamics so that the chemistry is advanced independent of the other processes. As
a result the chemistry can be treated locally on a point-by-point basis.
Our choice of AMR strategy has a significant impact on software design. By
adopting a block-structured form of AMR, the solution at each level in the hier-
archy is naturally represented in terms of data defined on a collection of logically
rectangular grid patches each containing a large number of points. Thus, the data is
represented by a modest collection of relatively large regular data objects as com-
pared to a point-by-point refinement strategy. This type of approach allows us to
amortize the irregular aspects of an adaptive algorithm over large regular operations
on the grid patches. This organization of data into large aggregate grid patches also
provides a model for parallelization of the AMR methodology.
Our adaptive methodology is embodied in a hybrid FORTRAN software system.
In this framework, memory management and flow control are expressed in the C++
portions of the program and the numerically intensive portions of the computation
are handled in FORTRAN. The software is written using a layered approach, with
a foundation library, BoxLib, that is responsible for the basic data container ab-
stractions at the lowest levels, and a framework library, AMRLib, that marshals the
components of the AMR discretization. Support libraries built on BoxLib are used
as necessary to implement utility components, such as interpolation of cell and in-
terface data between levels, and linear solvers used in the projections and diffusion
solves.
In BoxLib, the fundamental parallel data abstraction is a MultiFab. A MultiFab is
the union of a set of distributed blocks of FORTRAN-compatible data arrays. Each
block is defined on a Box, which is defined by a pair of integer coordinate tuples
that identify the lower and upper bounds of the region in a global index space.
MultiFabs at each level of refinement are distributed independently. The software
supports two data distribution schemes, as well as a dynamic switching scheme
that decides which approach to use based on the number of grids at a level and the
number of processors. The first scheme is based on a heuristic knapsack algorithm
320 John Bell and Marcus Day

as described in Crutchfield [18] and in Rendleman [35]. The second is based on the
use of a Morton-ordering space-filling curve. MultiFab operations are performed
with an owner computes rule with each processor operating independently on its
local data. For operations that require data owned by other processors, the MultiFab
operations are preceded by a data exchange between processors.
Each processor contains meta-data that is needed to fully specify the geometry
and processor assignments of the MultiFabs. At a minimum, this requires the storage
of an array of boxes for each AMR level, the refinement ratio between levels, and
the physical location of the origin of the base grid coordinate indices. In the parallel
implementation, meta-data also includes the processor distribution of data, which
is used to dynamically evaluate the necessary communication patterns to share data
amongst the processors.

13.5.1 Performance of Adaptive Projection

Compared to a time-explicit numerical integration of the compressible reacting flow

equations, the adaptive projection methodology for low Mach number reacting flows
is enormously complex. The integration algorithm is a multi-stage procedure in-
volving a number of elliptic/parabolic solves each time step. Dynamic regridding
operations track developing features in the flow, and require frequent redistribution
of the data and workload, and the workload itself is non-uniform due to the inho-
mogeneous nature of the reaction chemistry. Not only is it interesting to assess the
parallel scalability of this algorithm to thousands of processors, it is also important
to gauge the effectiveness of the low Mach number algorithm with respect to time-
explicit simulation approaches directly, which historically have shown nearly ideal
scaling behavior.

Fig. 13.9: Parallel performance of the low Mach number adaptive algorithm. (a)
Weak scaling behavior of the adaptive low Mach number algorithm. (b) Scaling of
the time-to-solution, relative to a compressible solver.
Adaptive Mesh Refinement 321

Here we consider a weak scaling study in which the relative amount of work per
processor remains constant as we increase the number of processors. The study here
is based on the propagation of a doubly-periodic wrinkled premixed methane-air
flame in three dimensions using the GRI-Mech 3.0 [23] chemical mechanism for
methane combustion. By replicating the problem in the periodic directions, we are
able to scale the problem size without modifying the problem characteristics so that
we can reliably test the behavior of the full AMR algorithm. Normalized computa-
tional times versus number of processors is presented in Figure 13.9(a). This figure
shows a modest increase in execution time of approximately 20% as we increase
the number of processors from 64 to 4096 on the XT4-type architecture. A more in-
teresting metric than simple scalability is the relative performance of the low Mach
number methodology to that of a time-explicit scheme. Figure 13.9(b) shows the
time-to-solution for the same range of processor counts using our low Mach number
algorithm compare to the idealized performance of a non-adaptive compressible re-
acting flow solver [6]. Note that for this study, we did not run the compressible code
at all of the resolutions; we simply extrapolated the performance from 64 processors
of assuming ideal scaling. The data shows that the low Mach number methodology
is more than a factor of 200 faster than the compressible method. It is important
to point out additionally that this estimate is quite conservative, since for many in-
teresting 3D problems the finest level typically covers a far smaller fraction of the
domain (the example presented below, for example, required the highest resolution
on less than 4% of the domain).

13.6 Application Lean Premixed Hydrogen Flames

The AMR algorithm design principles and BoxLib software library have been used
to build a wide variety of PDE integration schemes, with applications in compress-
ible and incompressible turbulence (jets, shear layers), astrophysics, porous media,
fluctuating hydrodynamics, and reacting flows. The adaptive projection scheme has
been particularly successful in studies of low Mach number reacting flows, such as
astrophysical explosions and terrestrial combustion. In this section, we summarize
a recent combustion study that serves to highlight the applicability of this AMR
scheme to realistic systems with complex transport and chemistry models at the
laboratory scale.

13.6.1 Background

Within the combustion community there is considerable interest in developing fuel-

flexible burners that can be used to stabilize lean premixed flames in a stationary
turbine designed for power generation. Low-swirl burner technology, originally in-
troduced by Cheng and co-workers [14] as tool for studying the fundamental prop-
322 John Bell and Marcus Day

erties of lean, premixed turbulent flames, has the potential for meeting this need.
Burners based on modifications of the original design have been used by a number

Fig. 13.10: (a) Low-swirl nozzle, showing vanes and turbulence generation plate.
(b) typical turbulent low-swirl methane flame stabilized in divergent flow above the
low-swirl nozzle. Images courtesy of Robert Cheng, LBNL.

of research groups [15, 25, 30, 32], and have the potential for use in the design of
next-generation, lean premixed combustion systems, including those burning lean
hydrogen at both at atmospheric and elevated pressures [16].
The low-swirl burner concept is extremely simple: premixed fuel exits a pipe
after passing through a turbulence generation plate and an annular set of curved
vanes as shown in Figure 13.10. The vanes impart a swirl component to the flow
over a narrow layer near the pipe wall, and a detached premixed flame anchors
in the diverging flow above the pipe exit. Turbulence in fuel stream wrinkles the
flame, which enhances the overall rate of combustion in the device; the flames sta-
bilize where the mean burning speed matches the axial flow velocity. Application of
these types of burners, particularly for alternative fuels, depends on improving our
understanding of basic flame structure, stabilization mechanisms, emissions and re-
sponses to changes in fuel. Numerical modeling has the potential to address some
of these issues, but simulation of these types of burners has proven to be difficult
because of the large range of spatial and temporal scales in the system; the bulk of
the analysis to date has been experimental.
As noted in Bell et al. [7], the detailed structure of lean premixed flames be-
comes particularly important and difficult to simulate when burning hydrogen. Lean
hydrogen-air flames burn in cellular structureslocalized regions of intense burn-
ing, separated by regions of local extinction. In this regime, the flame surface is bro-
ken into discontinuous segments. This type of structure introduces severe difficul-
ties in applying standard turbulence/chemistry interaction models, which are based
on the presence of a highly wrinkled but continuous flame surface that propagates
locally as an idealized laminar flame structure. In the absence of a suitably gen-
eral model for the turbulent combustion of lean hydrogen-air mixtures, numerical
simulations must incorporate sufficient detail in the chemical kinetics, differential
species transport and turbulent fluid dynamics to capture all the important couplings
in these flows.
Adaptive Mesh Refinement 323

13.6.2 Models and Setup

For this problem, we treat the fluid as a mixture of perfect gases. We use a mixture-
averaged model for differential species diffusion, ignoring Soret, Dufour and radia-
tive transport processes (see [21] for a complete discussion of this approximation).
In this case, the diffusive fluxes in Equations 13.1 can be written
2
Fm = Dm Ym , = 2mix S mix U, Q = mix T hm Fm
3
where S is the symmetric part of strain tensor, Dm is the mixture-averaged mass
diffusion coefficient for species m, and mix and mix are the mixture-averaged vis-
cosity and thermal conductivity, respectively. A lean hydrogen-air inlet fuel mixture
( =0.37) was modeled with the hydrogen sub-mechanism of GRI-Mech 2.11. There
are 9 chemical speices and 27 fundamental Arrhenius chemical reactions. The trans-
port coefficients and thermodynamic relationships are obtained from EGLib [22].
An idealized flat unstretched steady 1D (the so-called laminar flame) config-
uration provides scale factors that characterize this hydrogen-air flame: the thermal
thicknesses is approximately 800 m; the full half-width maximum of the fuel con-
sumption layer is approximately 0.5 mm. The computational domain for this study
measures 25 cm3 (we aassume that this will place the computational boundaries
sufficiently far from the flame as to not significantly affect its dynamics). The base
mesh for the simulation is a uniform grid of 2563 cells and we use 3 additional levels
of factor-of-two grid refinement to track regions of high vorticity (turbulence) and
reactivity (combustion). The flame is contained entirely within the finest level with
an effective resolution of 20483 . In previous work, we demonstrated that this level
of resolution is adequate to capture the detailed structure of the flame including the
peak fuel consumption, the thermal field and major species. Note that this level oc-
cupies less than 4% of the entire computational domain. Although evolution of the
flow outside of the fine grid region certainly impacts on the flame dynamics, there
are no features in the flow that require high resolution.

Fig. 13.11: Vertical slice of solution centered on nozzle axis (Frames (b)-(e) over
2.5 cm white box in (a)). (a-b) Mole fraction of H2 , (c) H2 Consumption rate, (d)
Mole fraction of OH, (e) temperature, and (f) Composite image (width = 8cm) de-
picts OH concentration (in orange) and the vorticity magnitude (in grey).
324 John Bell and Marcus Day

Flow from the low-swirl nozzle enters our computational domain centered on
the bottom boundary. Profiles for the velocity components at the nozzle incorpo-
rate an experimental characterization provided by Petersson et al. [32]. On the inlet
face outside the nozzle, a 35 cm/s upward coflow of cold air is specified. Turbulent
fluctuations in the nozzle flow were prepared in an auxiliary simulation to have the
experimentally measured intensity and integral length scale, and were added to the
mean flows as a time-dependent boundary function. The remaining boundaries are
outflow. The domain was intitially filled with air at standard conditions, except for
a small volume of hot air above the nozzle. As the simulation progresss, the flame
ignites, and propagates downstream in the flow until reaching a quasi-stationary po-
sition in the radially divergent flow field. The additional levels of grid refinement
were added and the simulation continued until reaching a new quasi-steady config-
uration.

13.6.3 Simulation Results

In Figure 13.11, we show a cross-section through the middle of the simulation that
provides a picture of the overall structure of the flame. In the upper part of the slice
plane in Figure 13.11(a), one sees a faint cloud of unburnt fuel at low concentrations
(blue). This represents fuel that has been sufficiently diluted with air that it is below
the flammability limit. Figure 13.11(b-e) show various other fields near the flame
surface. Turbulent fluctuations from two primary sources interact with the flame (see
Figure 13.11(f)): plate turbulence from the inlet advects downstream and wrinkles
the flame in the central core; a shear layer from the swirl entrains coflowing air from
the sides and reduces fuel concentrations below flammability.
One of the principal diagnostics used in the experiments is OH-PLIF (planar
laser-induced fluorescence) based on imaging the fluorescence of OH radicals ex-
cited by a tuned laser sheet. In Figure 13.12, we show a typical vertical slice of
the OH concentration from the simulation alongside typical PLIF images from the
low-swirl burner experiment at similar conditions. Figure 13.12(a) shows the pro-
file of OH over the (25 cm)2 slice, while Figures 13.12(b) and (d) show progressive
enlargements of the data corresponding to the field of view (FOV) of the typical
OH-PLIF data shown in Figures 13.12(c) and (e). The figure shows that the simula-
tion captures with remarkable fidelity the primary features of these flames, including
their distribution of sizes, shapes and global structure. The simulation also captures
the observed variability of the OH signal (brightness on the experimental images)
along the flame surface. Previous work on turbulent hydrogen flames shows that the
high diffusivity of H2 can lead to local enrichment of the fuel mixture along the
front. This local enrichment can lead to intensification of local burning, which leads
to increased OH that can be seen experimentally as a brigher signal in the OH-PLIF
(and red regions in the simulation slice data).
From analysis of the simulation data, we can obtain a detailed characterization
of the local flame structure that is not feasible from the experimental data. We begin
Adaptive Mesh Refinement 325

Fig. 13.12: (a) Typical slice of OH concentration on vertical midplane from the sim-
ulation. Red indicates high relative values, and are correlated with the flame front.
The white box (FOV1) represents the field of view of OH-PLIF measurements
from the representative low swirl experiment focused on the large-scale flame struc-
ture. The smaller yellow box (FOV2) represents the experimental field-of-view
OH-PLIF measurements focused more tightly on the local structure of the flame
surface (width of FOV2 is approximately 26 mm). (b) Zoom of profile in (a) over
FOV1, (c) typical experimental OH-PLIF image over FOV1, (d) Zoom of profile in
(a) over FOV2, and (e) typical experimental OH-PLIF image over FOV2.

Fig. 13.13: Flame isotherm, T = 1144 K, colored by the local rate of H2 consump-
tion. (a) Low-swirl flame isotherm, conditioned on r < 25mm, z <8mm. (b) Rep-
resentative isotherm of an idealized =0.37 H2 -air flame propagating freely in uni-
form flow.

with an examination of the fuel consumption rate. As with the OH-PLIF data, the
fuel consumption shows considerable variability along the flame front. To quantify
this behavior, we extract the T =1144 K surface, which corresponds to that of the
peak fuel consumption in the flat laminar flame at =0.37. In Figure 13.13, this
isotherm is colored by local fuel consumption in the core of the burner. For com-
326 John Bell and Marcus Day

parison, we also show the analogous image for a freely propagating hydrogen flame
(taken from the study in [20]). Both images are based on the same color map, and
the length scales are as indicated in the figure. Comparing these images, one can
see that in the turbulent flame, there are finer structures than in the non-turbulent
case. The turbulent flame shows more ridge-like, cylindrical structures compared to
the freely propagating flame in which the features are more spherical. Finally, the
burning rate over most of the flame surface is considerably higher in the turbulent
configuration, particularly where the flame is tightly folded.
To quantify the relationship between flame curvature and local burning speed we
compute a local consumption-based flame speed along the isotherm. For this con-
struction, we triangulate the isotherm, trimming away sections whether the fuel con-
sumption is less than half the laminar flame value, and construct a local coordinate

Fig. 13.14: Joint probability distribution function of sc , with mean curvature.

system in a neighbor hood of the flame. We can then define a local consumption-
based flame speed on the triangulated isotherm surface by integrating fuel consump-
tion through a region normal to the flame. (See [20] for a detailed description of this
construction.) In Figure 13.14 we show a joint PDF of mean curvature and sc , nor-
malized the laminar burning speed, sL , of a flat steady hydrogen-air flame at =0.37.
This figure shows a strong correlation between local burning speed and curvature.
Regions of intense burning correspond to regions of positive curvature (positive cur-
vature corresponds to regions where the center of curvature is on the products side
of the flame interface). Also note that the dominant behavior corresponds to local
burning speeds that range from 2 to 4 times the laminar flame speed. Even at zero
curvature, the most probable local burning speed is three times the laminar flame
speed. This type of behavior illustrates the challenges in developing suitable turbu-
lence / chemistry interaction models for lean hydrogen flames and underscores the
need to perform simulations of laboratory-scale turbulent flames.
Adaptive Mesh Refinement 327

13.7 Summary

In this chapter, we have discussed the development of parallel, adaptive solvers for
reacting flows. The particular class of adaptive methods we consider are hierarchi-
cal block-structured methods that support subcycling in time. We have discussed
the key ideas needed to design adaptive methods within this context and illustrated
how those approaches are instantiated for various simple classical partial differential
equations. We then showed how those basic discretization concepts can be interwo-
ven to construct an adaptive low Mach number simulation capability. The basic al-
gorithmic constructs associated with block-structured AMR motivate the design of
a software framework to support parallel implementation of these algorithms. The
overall synthesis of these ideas leads to an enhanced simulation capability that has
made it possible to simulate realistic turbulent flames without using explicit models
for turbulence or turbulence/chemistry interaction.

Acknowledgments

This work was support by the DOE Office of Advanced Scientific Computing
Research under the U.S. Department of Energy under contract No. DE-AC02-
05CH11231. The computations presented were performed on Franklin at NERSC
as part of an INCITE award.

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Chapter 14
Wavelet Methods in Computational Combustion

Robert Prosser and R. Stewart Cant

Abstract Discretisation schemes based on the use of wavelet methods offer many
potential advantages for the numerical simulation of combustion. In many cases of
interest, flame structures are thin relative to the largest length scales of the prob-
lem and most length scales of the flow field, and so lend themselves to simulation
using adaptive-mesh methods. Wavelet methods are naturally adaptive, in that the
coefficients of the wavelet transform are non-zero only in regions where there is
significant variation present in the solution. Hence, simple thresholding can be em-
ployed to make valuable savings in storage and in execution time. In this chapter,
the basic principles of wavelet methods are established. Orthogonal and biorthogo-
nal wavelet formulations are described and their advantages and disadvantages are
discussed. An illustration of a wavelet-based discretisation scheme is provided using
the Navier-Stokes momentum equation as an example. The same wavelet approach
is applied to the simulation of a one-dimensional laminar premixed flame for which
an asymptotic solution exists. Comparisons are made between the computational
and analytical results and the accuracy of the wavelet approach is assessed. Exten-
sions to higher dimensions are discussed. Finally, the current state of development
of wavelet methods is outlined and conclusions are drawn.

14.1 Introduction

High-fidelity numerical simulation of turbulent combustion is a very demanding

task. Techniques such as Direct Numerical Simulation (DNS) and Large Eddy Sim-
ulation (LES) require the flow field to be represented in three spatial dimensions

Robert Prosser
School of MACE, University of Manchester, Manchester M60 1QD, UK, e-mail: robert.
prosser@manchester.ac.uk
R. Stewart Cant
Cambridge University, Cambridge, CB2 1PZ, UK, e-mail: rsc10@eng.cam.ac.uk

T. Echekki, E. Mastorakos (eds.), Turbulent Combustion Modeling, 331

Fluid Mechanics and Its Applications 95, DOI 10.1007/978-94-007-0412-1 14,
Springer Science+Business Media B.V. 2011
332 Robert Prosser and R. Stewart Cant

and cannot make use of statistical symmetries of the problem in the same manner as
the traditional Reynolds-Averaged Navier Stokes (RANS) approach. Moreover, the
flow field is evolving in time, and hence it is essential to use time-accurate solution
methods in order to capture the flow, the flame and the full complexity of their inter-
actions. All of this is computationally expensive, especially when the requirement
is for full spatial and temporal resolution of all relevant phenomena, as is the case
in DNS.
In turbulent flow, the computational expense follows directly from the range of
length and time scales that must be represented. In combustion problems, it is most
often the case that there are further length and time scales even shorter than those of
the turbulent flow field. The very smallest scales are associated with the diffusion-
reaction layers deep within the flame structure, and these tend to be highly localised
in space, at least on an instantaneous basis. Here the computational expense arises
mainly from the tendency of the flame structure to move around within the domain
due to the effects of advection by the flow field and propagation due to heat conduc-
tion and molecular diffusion. This means that sufficient computational mesh support
must be provided to ensure proper spatial resolution everywhere in the domain. The
time-advancement algorithm must be applied at every spatial mesh point, and hence
the computational cost is set.
There have been many attempts to exploit the localised nature of the flame struc-
ture using adaptive mesh refinement (AMR) techniques [9, 28]. Here, the compu-
tational mesh is refined locally in order to provide high resolution only in regions
of the domain (such as within the flame) where there are steep spatial gradients.
Conversely, the mesh can be made less dense in regions where there is little activ-
ity in the solution. Since the flame structure typically occupies only a small part
of the total volume of the domain, considerable computational cost savings can be
made without a net loss of resolution. Considerable progress has been made and
the utility of the approach has been demonstrated. Nevertheless it is difficult to ap-
ply such techniques while retaining high-order accuracy, and there are major issues
concerning their efficient implementation on massively-parallel computers.
Ideally, an adaptive spectral method is required which would allow for very high
spatial accuracy coupled with a capability for dynamic local mesh refinement. Meth-
ods based on Fourier transforms have been used for many years in simulations of
turbulence, offering excellent accuracy but lacking the flexibility and spatial local-
isation necessary for use in an adaptive manner. More recently, wavelet transforms
have emerged as a possible framework in which to build a class of numerical meth-
ods offering high accuracy combined with solution adaption. Wavelets, unlike the
more familiar Fourier basis functions, are localised in both the spatial and spectral
domains, and have many other interesting properties which make them an attractive
prospect for use in future combustion simulations.
This article introduces wavelet analysis in the context of combustion DNS, out-
lines the relevant mathematical background and describes the application of wavelet
techniques within a suitable numerical solution method. Results are presented which
indicate the power of the approach, and conclusions are drawn.
Wavelet Methods in Computational Combustion 333

14.2 Wavelet Transforms

14.2.1 Orthogonal Wavelets

A wavelet transform can be interpreted in much the same way as a Fourier transform.
Where the two differ is in the choice of the basis function employed. In the Fourier
setting, the basis is chosen to be the complex exponential. In the wavelet transform,
there are actually two basis functions: the scaling function and the wavelet. The
scaling function can be interpreted as a low resolution band pass filter, while the
wavelet represents the complementary high resolution band pass filter [21].
The wavelet decomposition is based on the repeated application of a two-scale
relationship. In particular, we have [8]


Wi = L2 (R)
i=
VJ = VJ1 WJ1
Vi Wi (14.1)

where represents the direct sum. Equation 14.1(a) indicates that L2 (R) can de-
composed into a family of wavelet spaces Wi . Each wavelet space forms half of
a partnership with a scaling function space Vi . i characterizes the resolution of
the spaces with i as the resolution
 is refined. The basis for the scaling func-
tion space is denoted i,k (x) 2i x k , and the basis for the wavelet space is
 
i,k 2i x k . Factors of 2 can appear in these definitions, depending on the
choice of normalization. On the real line, both the wavelet and scaling function ex-
hibit scale and translation invariance.
The projection of f (x) L2 (R) is accomplished by taking the inner product with
respect to the basis, i.e.

f (x) = Qi ( f ) (x) ,
i=

where Qi is the projector onto Wi which, in the case of orthogonal basis functions
can be written as

Qi ( f ) (x) = < f (u) , i,k (u) > i,k (x) . (14.2)

kZ

The two scale relation defined by Eq. 14.1 allows the wavelet projection to be writ-
ten in terms of scaling function projections;

Qi ( f ) (x) = (Pi+1 ( f ) Pi ( f )) (x)

where
Pi ( f ) (x) = < f (u) , i,k (u) > i,k (x) . (14.3)
kZ
334 Robert Prosser and R. Stewart Cant

Equation 14.1(b) provides the basis for the practical implementation of a wavelet
decomposition for finite domains. Assume for simplicity that we have a finite di-
mensional representation of some periodic function f (x) [0, 1), which is sampled
on 2J grid points xJ,k = k2J , 0 k < 2J . If f (x) is approximated by PJ ( f ) (x) then
an application of the two scale relation allows us to write

PJ ( f ) = (PJ1 + QJ1 ) ( f ) (x) . (14.4)

The orthogonality of Vi and Wi implies that the two projectors PJ1 and QJ1 lead
to complementary representations of PJ ( f ) on reduced dimensional spaces (in this
case, each representation is defined on grids of resolution 2J1 ). In the first pass of
the transform then, a vector of length 2J is replaced with 2 vectors of length 2J1 , but
the key observation here is that the wavelet projection QJ1 ( f ) is sparse, by which
we mean that many of the 2J1 coefficients arising from the wavelet projection are
close to zero. The defining property of the wavelet transform is that the resulting
coefficients are only non-trivial when the analyzing wavelets are close to regions of
rapid change. Such a feature makes wavelets a natural tool with which to explore
flame structures, wherein two relatively static regions (reactants and products, or
fuel and oxidiser) are separated by a region of rapid change (the flame). The sparsity
of the wavelet representation is exploited via thresholding, in which those wavelet
coefficients with a magnitude less than a user specified threshold are discarded
with a minimal loss of accuracy (in a sense that can be made precise, i.e. see [12]).
Hence, the two scale representation with thresholding
  allows us to replace a vec-
tor of dimension 2J , with one comprising O 2J1 componentsPJ1 ( f ) contains
2J1 non-zero components, and QJ1 typically contains a much reduced number of
large wavelet coefficients. The actual number of retained coefficients depends on
the smoothness of the analyzed function f (x) .
Equation 14.4 can be repeatedly applied to PJ1 ( f ) to obtain

PJ ( f ) (x) = (PJ2 + QJ2 + QJ1 ) ( f ) (x)

..
.
J1
= P0 ( f ) (x) + Qi ( f ) (x) (14.5)
i=0

Equation 14.5 is the finite dimensional equivalent of Eq. 14.1(a). In practical cal-
culations involving wavelets on the interval, the lower limit appearing in the sum
of Eq. 14.5 is larger than zero. This reflects the need to keep the support of the
wavelets smaller than the discretized interval. One final point to note is that the
mapping f (x) PJ ( f ) (x) requires an initial projection quadrature. This quadra-
ture is usually approximatemany authors choose instead to use the sample values
of the initial discretisation as the set of scaling function coefficients.
The implementation of the wavelet transform can take a number of forms. For
some basis functions, the transform can be accomplished by a modified FFT [23].
In the most common approach (which makes use of Daubechies compact wavelet
Wavelet Methods in Computational Combustion 335

[8]), the transform takes the form of a repeated finite difference-like operation (once
for each space on each resolution). The weights associated with these operations
are the quadrature mirror filter coefficients defining the wavelet. Examples of the
implementation of wavelet transforms can be found in e.g. [24].

14.2.2 Biorthogonal Wavelet Transforms

Much of the utility of the wavelet transform emerges from the unique properties of
the chosen wavelet, but there are also a number of problems with the orthogonal
representations described in the previous section. In the orthogonal setting, many
different choices of wavelet exist, some of which are more suitable for CFD appli-
cations than others. From a purely practical point of view many choices of wavelets
do not have compact support (i.e. [23])this implies that the inner products in Eqs.
14.2 and 14.3 effectively contain infinitely many quadrature filter coefficients and
become difficult to evaluate. Such wavelets can be approximated as having compact
support, but the resulting approximations lose their exact (to machine precision) or-
thogonality. Families of orthonormal wavelets with compact support do existthe
most famous examples being those of Daubechies [8]but the orthogonal restraint
leads to wavelets which are asymmetric.
For the simulation of fluid mechanics problems, asymmetric bases are undesir-
able as they introduce chirality into the numerical approximations for the govern-
ing equations [19, 20]. In addition the initial projection quadrature, which maps
f (x) PJ ( f (x)) , is non-trivial and usually irreversible (to machine precision). In
the setting of a collocation numerical scheme (or indeed any approach to the ap-
proximation of non-linear PDEs) the continual mapping to and from transformed
representations leads to strong chirality andeventuallyinstability in the numer-
ical solution [25].
One approach to bypass the problems associated with chirality is to modify
the wavelets by relaxing the constraint on orthogonality, to produce biorthogonal
wavelet systems. Most of the relations described in the previous section hold for
biorthogonal systems, but two sets of basis functions are requiredthe so-called
primal and dual bases [6]. The scaling function and wavelet projectors are then
written as

PJ ( f ) = < f (u) , J,k (u) > J,k (x)

kZ
QJ ( f ) = < f (u) , J,k (u) > J,k (x) ,
kZ

where the tildes refer to the dual quantities. Biorthogonal wavelets contain sufficient
flexibility to provide compact bases with symmetrysuch considerations are par-
ticularly important in the construction of edge wavelets for bounded intervals [7].
The problem of the initial projection quadrature remains.
336 Robert Prosser and R. Stewart Cant

14.2.3 Second Generation Wavelets

A variant of the biorthogonal wavelet can be obtained via so-called second gen-
eration wavelets, pioneered by Sweldens and Donoho [12, 14, 27, 29, 30]. These
wavelets are derived without recourse to Fourier transforms, and provide a simpler
framework in which to define wavelets for more general settings than the real line.
In addition, the classical biorthogonal wavelets of Cohen et al. [7] can be derived us-
ing the second generation approach. Finally, second generation wavelets circumvent
the problems associated with an inexact initial projection quadrature.
There are two classes of wavelets proposed in the second generation frame-
work: interpolating wavelets and average interpolating wavelets (i.e. see [27]) In
this work, we choose the family of interpolating wavelets derived from the funda-
mental solutions discussed by Deslauriers and Dubuc [11]. In the second generation
interpolating wavelet approach, the dual scaling function is defined by
 
J,k = x k2J ,

where () is the Dirac delta function. The scaling functions and wavelets have
compact support and are symmetric in the interior of a domain, in particular
   
support j,k = 2 j (m (N 1)) , 2 j (m + (N 1)) ,
 
support j,k = [2( j+1) ((2k + 1) (N 1)) , 2( j+1) ((2k + 1) + (N 1))].

In this sense, we say that the coefficients s j,k and d j,k are respectively associated with
the grid point k21 and (2k + 1) 2( j+1) . Similar arguments can be made for the near
boundary constructions, but the definitions become more algebraically complicated.
The dual scaling function (and indeed, the dual wavelet) are not members of
L2 (R) , and the resulting transforms are limited in their range of application to
smooth functions [12]. An alternative way of saying this is that the resultant wavelets
do not strictly qualify as wavelets, since they do not satisfy the admissibility condi-
tion [8]
(x) dx = 0.

The result of this inequality is that the wavelets are prone to aliasing, and do not
conserve the integral of the original function:

Pi ( f ) dx = Pm ( f ) dx i = m. (14.6)

For classes of smooth function, this problem does not appear to be especially sig-
nificant [12]; such conditions are typically encountered in the DNS of reacting flow
wherein the smoothness of the resolved profilescoupled with the limited number
of allowable subspace decompositionsappears to minimize the effect of aliasing.
The projection of f (x) onto the scaling function space leads to scaling function
coefficients that are samples of the original function;
Wavelet Methods in Computational Combustion 337
 
PJ ( f ) (x) = < f (u) , u k2J > J,k (x)
kZ
 
= f k2J J,k (x) = fJ,k J,k (x) .
kZ kZ

It follows that fJ1,k = f J,2k and hence successive scaling function approximations
are subsamples of the original discretisation. The wavelet transform is given by

Qi ( f ) (x) = di, i, (x)

kZ
= (Pi+1 ( f ) Pi ( f )) (x)
= fi+1,k i+1,k (x) f i,m i,m (x)
kZ mZ
di, = Qi ( f )|x=(2 +1)2(i+1)


1
= f i+1,2a+1 + m fi,m . (14.7)
mZ 2

Clearly, the sparsity of the representation is governed by the properties of (). Fur-
thermore, if the scaling function has compact support, it follows that ( + 1/2 m)
comprises only finitely many non-zero entries, and Eq. 14.7 can be implemented as
a finite difference-like operation.

14.3 Wavelets as a Method for DNS

We describe a collocation approach to the integration of the governing equations.

The collocation strategywhere the wavelets are used merely to calculate deriva-
tives [5]provides a natural approach, since solution algorithms couched in terms
of the transformed variables face formidable difficulties in evaluating the nonlinear
terms associated with both convection and chemical reaction. For a 1-D flow domain
, we initially discretize the governing equations on to a regular mesh G . As
the solution evolves, we want to solve the equations only on those parts of the do-
main G G where strong variations in the flow behaviour occur. Such behaviour
might be found in, say, turbulent shear layers, or rapid temperature changes associ-
ated with chemical reaction. Wavelets are very good at picking up these regions of
change, so a natural choice for the unstructured grid might be

G = {xi, : |di, | > }

where di, is the wavelet coefficient associated with the point xi, . If we try to cal-
culate the derivatives on this mesh using traditional means, then we are left with the
classical hanging node problemthose points at the outskirts of the unstructured
mesh do not have sufficient neighbours to calculate their associated derivatives ac-
curately. In the multiresolution approach however, we first calculate the wavelet ex-
338 Robert Prosser and R. Stewart Cant

pansion of the solution on G . By construction, the coefficients on G\G are small

in magnitude (defined here as |di, | < ) and may be omitted when calculating the
approximate derivative. The resulting derivative can then be inverted on to the un-
structured grid, and the solution time advanced as normal. The difficulty in this
approach lies now in calculating the wavelet transform on the unstructured mesh.
Methods for achieving this are available (i.e. [26]), but are too involved to describe
here.
For ease of discussion, we will describe a wavelet collocation discretization of the
momentum equation in what follows. The principles remain quite general however,
and can be applied to the entire coupled reacting Navier-Stokes system. In one-
dimensional problems and neglecting density variations, the momentum equation
can be written as
u u p xx
+u + 1 = 1 .
t x x x
Our goal is to express this equation in terms of projected variables PJ ( ) and PJ (u) .
Projecting the entire equation directly onto VJ leads to




PJ (u) u 1 p 1 xx
+ PJ u + PJ = PJ . (14.8)
t x x x

The unsteady term does not provide any difficulties, since the time derivative com-
mutes with the projector. Problems arise with the convective terms because: (a) we
require an approximation for the derivative itself and (b) there is a non-linear prod-
uct to incorporate.
As a model for the effect of non-linearities, consider a one dimensional velocity
field u (x) projected onto VJ :

PJ (u) = uJ,m J,m (x) .

mZ

Consider further the non-linear term PJ (u) PJ (u) , which can be obtained as part of
the convective term in Eq. 14.8. The explicit representation of this term is

PJ (u) PJ (u) = uJ,m uJ,n J,m (x) J,n (x) / VJ . (14.9)

mZ nZ

In order to be consistent with the time derivative term, we must re-project Eq. 14.9
to obtain

PJ (PJ (u) PJ (u)) = uJ,m uJ,n < J,m (x) J,n (x) , J, (x) > J, (x) ,
Z mZ nZ
= uJ, uJ, J, (x) . (14.10)
Z

More important is the effect of this non-linearity on spaces other than VJ. For ex-
ample, projecting Eq. 14.9 on to WJ , we find
Wavelet Methods in Computational Combustion 339

QJ (PJ (u) PJ (u)) = uJ,m uJ,n < J,m (x) J,n (x) , J, (x) > J, (x) .
Z mZ nZ
(14.11)
The normal consequence of the biorthogonality between VJ and WJ (i.e. the relation
QJ (PJ ()) = 0) does not hold here, since we have two scaling functions. Hence

< J,m (
x) J,n (
x) ,
J, (
x) > = 0 (14.12)

in general, even without explicitly evaluating Eq. 14.11, we see that the nonlinearity
has extended the representation of u (x) from VJ to VJ+1 by populating WJ . Similar
arguments can be made for other wavelet spaces. This behaviour is the wavelet ana-
logue of the spectral spreading caused by the convection driven convolution arising
in Fourier based pseudo-spectral methods [5], and has been studied in some detail
by Beylkin [3].
If u (x) is sufficiently smooth, then the high resolution wavelet coefficients will
be small in magnitude. Furthermore, their interactions (governed by the coupling
matrix in Eq. 14.12) will also produce only small effects. It is natural then for a DNS
exploiting wavelet thresholding to assume VJ is of a sufficiently high resolution that
the additional components arising from the non-linearity are small with reference to
the size of the non-trivial coefficients. We can then replace each of the terms in Eq.
14.8 by their projected counterparts i.e.



u u
PJ u  PJ (u) PJ
x x


1
 PJ (u) PJ P PJ (u) (14.13)
x J

where the identity operator has been decomposed as I = PJ PJ1 . As before, this
expression will contain terms in both VJ and WJ . To express the resulting term
consistently, then, we write equation 14.13(b) as



 

PJ PJ (u) PJ PJ1 PJ (u) .
x

Using the interpolating properties of the basis, this becomes

 

 
1 1
PJ PJ (u) PJ P PJ (u) = PJ (u) PJ P PJ (u) ,
x J x J

and we acknowledge that some information transmitted by this operation to WJ is

lost. Similar expressions can be derived for the simpler, linear viscous and pressure
terms.
340 Robert Prosser and R. Stewart Cant

14.3.1 The Wavelet Representation of the Derivative

The term


PJ P1 (14.14)
x J
provides
  the representation of the differential operator on VJ ; Fig. 14.1 shows the
O 2J non-zero elements for this operator.

Fig. 14.1: Structure of D approximating dx

d
on VJ . Reprinted from [26] wither per-
mission from the Institution of Mechanical Engineers.

The derivative itself can be calculated as



PJ PJ1 PJ ( f )|xJ, = < J, , J, > PJ ( f )|xJ, ,
x x
= D , PJ ( f )|xJ, .

In practice, the band diagonal structure of D , lends itself to a finite difference like
implementation.
In order to take advantage of the sparsity in the wavelet representation of the flow
field, the representations of both the operator and the flow field need to be expressed
in terms of a multi-scale decomposition. Repeatedly applying Eq. 14.1 to Eq. 14.14
leads to the so-called standard decomposition of / x [2, 4]:

 1
J
J
Dw = Pj + Qi Pj + Qi .
i= j x i= j

This operator comprises a family of band diagonal sub-matrices which are ordered
as in Fig. 14.2.
In terms of implementation, we have
Wavelet Methods in Computational Combustion 341

Fig. 14.2: Structure of Dw approximating (PJ2 + i=J2

J1 d
Qi ) dx (PJ2 +
1
i=J2 Qi ) . Reprinted from [26] with permission from the Institution of
J1

Mechanical Engineers.

   
J
f  J
Pj + Qi  D , Pj + Qi ( f )|xJ, .
w
i= j x xJ, i= j

Dw , is referred to as the standard decomposition of x [4] and typically comprises
   
O 2J log2 2J = J log 2J non-zero elements.

14.3.2 Higher Dimensional Discretizations

A natural generalisation to the one dimensional wavelets explored in this paper is

achieved by deriving multidimensional analogues from first principles (i.e. [22]).
For most practical purposes, however, multidimensional generalisations are derived
via tensor products of one dimensional transforms, i.e. for a two dimensional dis-
(2)
cretisation, we define VJ using

(2) (2) (2)

VJ = VJ1 WJ1
(2) (x) (y)
VJ = VJ VJ . (14.15)

By applying a two scale decomposition in each spatial direction to Eq. 14.15(b), we

obtain    
(2) (x) (x) (y) (y)
VJ = VJ1 WJ1 VJ1 WJ1 (14.16)

which when expanded, leads to the multidimensional wavelet space

342 Robert Prosser and R. Stewart Cant
     
(2) (x) (y) (x) (y) (x) (y)
WJ1 = WJ1 VJ1 VJ1 VJ1 WJ1 WJ1 . (14.17)

Clearly, the wavelet space actually contains three contributions; the pure wavelet
 (x) (y)   (x) (y)   (x) (y) 
space WJ1 WJ1 , and two cross spaces: WJ1 VJ1 and VJ1 WJ1 .
(x) (y)
Equation 14.16 may be further decomposed by splitting VJ1 and VJ1 into their
respective scaling function and wavelet spaces of resolution J 2. The appearance
of the two dimensional decomposition shares many similarities to the standard de-
d
composition of dx . An example is provided in Figs. 14.3 and 14.4, which show an
instantaneous realisation of the kinetic energy from a two dimensional turbulent
flow, and its associated multiresolution decomposition. In this case, J = 8, and there
are 4 subspaces in each of the x and y directions. The lower left corner of Fig.
(2)
14.4 depicts an approximation to the original signal on V4 . Each of the subsequent
(x) (y)
subblocks growing out to the top right of the figure are associated with Wi Wi .
(x) (y)
The blocks above this diagonal correspond to spaces of the form Vi Wm while
(x) (y)
those below correspond to Wi Vm . The particular point to note is the sparsity in
the wavelet representation: away from the coarsest representations, the magnitudes
of the wavelet coefficients are close to zero.

Fig. 14.3: Kinetic energy contours for 2D turbulence.

The practical application of wavelets to the Direct Numerical Simulation of re-

acting flows faces a number of difficulties. The non-linear terms associated with
the chemical reaction rate (and, to a lesser extent, the convective term) effectively
preclude the evolution of the simulation in the transform domain. Consequently, the
transport equations must be time advanced in physical space. The wavelets are then
used as part of a collocation strategy: (a) to inform the grid tracking algorithm of
where points are required, and; (b) to calculate the derivatives on this reduced grid.
The algorithmic difficulty is then one of correctly calculating the wavelet transform
on a generally sparse grid.
Wavelet Methods in Computational Combustion 343

Fig. 14.4: 2D Wavelet transform of turbulent kinetic energy.

Most wavelet transforms are top-down algorithms; the wavelet transform is

obtained by the repeated application of quadrature mirror filters in a finite differ-
ence like algorithm. Each application of the filters to a high resolution signal pro-
duce two complementary low resolution signals (the scaling function coefficients
and the wavelet coefficients). On a sparse grid, the so-called hanging node problem
appearsgrid points retained at the edge of a region of rapid change do not contain
sufficient neighbour nodes to calculate their wavelet coefficients. The hanging node
problem can be circumvented by a number of methods in 1-D (i.e. [31]), but the
extension to 2-D is difficult. In the latter case, there is no unique way in which to
order the wavelet coefficients, and multidimensional transforms lose their commu-
tivity in the presence of a hard non linear threshold (i.e. a 2-D transform comprising
x followed by y transforms will yield different coefficients to a 2-D transform
comprising y followed by x transforms). Practical experience has shown that
particularly in low Mach number flowsthe error induced by the hanging node
problem manifests itself as a small perturbation in the dependent variables. These
perturbations are picked up by the grid tracking algorithm as physical entities, and
subsequently allowed to evolve. On the new hanging nodes thus formed, new pertur-
bations are introduced and the grid grows again to accommodate these new features.
The process repeats until the full grid is retained. Circumventing this problem forms
part of ongoing research efforts.

14.4 An Application of Wavelets to Reacting Flows

14.4.1 Governing Equations

The equations governing a chemically reacting flow in one dimension are given by
344 Robert Prosser and R. Stewart Cant

+ ( u) = 0
t x
u
+ ( uu + p) = (xx )
t x x
E q
+ (u ( E + p)) = + (uxx )
t x x
x 
Yl Yl
+ ( uYl ) = l + D l = 1, 2, . . . , Ns 1. (14.18)
t x x x

where Ns > 1 is the number of species, xx is the viscous stress and the heat flux
vector q is defined as
T Ns
Yl
q = hl D . (14.19)
x l=1 x
For simplicity, the chemical reaction is assumed to comprise the single step

Reactants Products

in which case, Ns = 2 and the thermochemical state of the gas is characterized by a

progress variable. The progress variable is interpreted as a normalized product mass
fraction, and takes a value of 0 in the reactants and 1 in the products. c = Y1 here
and Y2 = 1 Y1 . The reaction rate controlling the production of c is given by [32]
 
1 T
1 = B (1 c) exp  ,
1 1 T

with 2 = 1 . B is the pre-exponential factor (taken to be 285.1 103 s1 here),

(= 6) is the Zeldovich number and T is the reduced temperature;
T T0
T = .
Tad T0
T0 and Tad are the unburned reactant and adiabatic product temperature, respectively.
is related to the heat release of the fuel, and is set here to 0.8. This corresponds
to an adiabatic flame temperature of 1500 K for an inlet temperature of 300 K.
The reaction rate has been adjusted to give a laminar flame speed of 30 cm/s, a
value typical of many hydrocarbon-air flames. The inlet velocity is set equal to the
laminar flame speed, so that a stationary
 flame profile is obtained. This specification
gives a simulation Mach number of O 103 based on the laminar flame speed. The
thermal conductivity is a modified form of the equation proposed by Echekki et al.
[13]:

T
= 0 c p ,
T0
where 0 = 2.58 105 kg/(ms). The temperature dependence of the conductivity
has been chosen such that the resulting temperature and species profiles can be de-
Wavelet Methods in Computational Combustion 345

rived analytically using asymptotic methods [32]. The viscosity and mass diffusion
are calculated via the joint assumptions of constant Prandtl number (= 0.75) and
constant unit Lewis number. The other thermodynamic quantities are assumed to be
constant and are set equal to the values given for air. The stagnation internal energy
is obtained using
Ns
u2
E = el Yl + ,
l=1 2
where el is the species internal energy, comprising the internal energy of formation
e0l , and a sensible component:
T  
el = e0l + cv T  dT  .
T0

The specific heats and the molecular weights of the components are assumed to be
constant, with c p = 1005 J/(kg K), = 1.4 and W = 28.96 kg/kmol. The pressure is
calculated from the thermal equation of state
Ns
Yl
p = R0T Wl , (14.20)
l=1

where R 0 (=8314.5 J/(kmol K)) is the universal gas constant.

The governing equations retain full compressibility, and hence acoustic waves
will need to be accurately captured by the wavelets. This has been done deliberately,
since the accurate resolution of the pressure profile provides a stringent challenge
for the discretization.

14.5 Results

Figures 14.5(a)-(d) show a benchmark solution for a 1-D laminar flame. The so-
lution has been calculated using N = 4 interpolating wavelets and initially with
wavelet thresholding switched off (i.e. using a full grid). The discretization com-
prises initially 28 + 1 (i.e. VJ = V8 ) grid points; the extra grid point comes from end
effects introduced via discretizing an interval.
Figure 14.6 shows a comparison between the benchmark solution and an analytic
profile obtained by high activation energy asymptotics [32]; the agreement between
the two is excellent, and thereby establishes the credibility of wavelets as a means
of DNS.
Figure 14.7 shows the reaction rate profile and the retained grid points, the latter
being obtained by thresholding the wavelet series based on the magnitudes of the
respective dependent (conservative) variables. The automatic clustering of the grid
points in the region of the flame are the principal driver for the development of
wavelet methods.
346 Robert Prosser and R. Stewart Cant

Fig. 14.5: Benchmark laminar flame profile.

Fig. 14.6: Progress variable profile obtained analytically and numerically with = 0.
Reprinted from [26] wither permission from the Institution of Mechanical Engi-
neers.
Wavelet Methods in Computational Combustion 347

Fig. 14.7: Reaction rate. denotes location of retained grid points.

The effects of increasing threshold on the temperature profile have been found
to be essentially invisible. Of more interest is the effect of thresholding on the reso-
lution of the pressure profile. It is well known that low Mach number systems exhibit
stiffness (i.e. see Klein [18]); this stiffness manifests itself here as a pressure profile
that is extremely sensitive to numerical noise. Figures 14.8, 14.9 and 14.10 show the
pressure profile obtained with increasing thresholds. The dynamic pressure change
is itself very small, and so consequently is quite sensitive to any perturbation.

Fig. 14.8: Dynamic pressure profile for thresholded solution, with = 106 .
Reprinted from [26] wither permission from the Institution of Mechanical Engi-
neers.

An estimate for the effect of wavelet thresholding on pressure can be established

for low Mach number systems by examining the thermodynamic relationship be-
tween the conserved variables and the pressure. Using standard thermodynamic re-
lations, the change in pressure for a mixture of Ns ideal gases can be related to
changes in the conserved variables by
348 Robert Prosser and R. Stewart Cant

Fig. 14.9: Dynamic pressure profile for thresholded solution, with = 105 .
Reprinted from [26] wither permission from the Institution of Mechanical Engi-
neers.

Fig. 14.10: Dynamic pressure profile for thresholded solution, with = 104 .
Reprinted from [26] wither permission from the Institution of Mechanical Engi-
neers.

  Ns
1
d p = ( 1) d ( E) ud ( u) + u2 d (( 1) e R T ) d ( Y ) .
2 =1
(14.21)
A better understanding of the relative effects can be obtained by non-dimensionalising
Eq. 14.21 to obtain


1 2
d p = ( 1) d ( E) ( 1) M ud ( u) u d
2
2
Ns

R T
e d ( Y ) , (14.22)
=1 ( 1)
Wavelet Methods in Computational Combustion 349

where the pressure has been non-dimensionalized with respect to 0 RT0 ; the re-
maining quantities have been non-dimensionalized with respect to u0 , (c p )0 and T0 .
Consider a discretization which decomposes VJ onto VJ1 and WJ1 for clarity. At
some point during the integration, the solution is transformed onto VJ1 and WJ1 ,
and then a threshold is applied. The resultant reduced series is then used to recon-
struct the dependent variables. This procedure is a model for an adaptive solution
algorithm where in this case, the missing coefficients are replaced with a zero in the
transform domain.
By construction in this simple example, errors of O ( ) are incurred in the energy,
the momentum, the density and the species mass fraction profiles. Using equation
14.22, we write


1
p = ( 1) ( E) ( 1) M 2 u ( u) u2
2
Ns

R T
e ( Y ) (14.23)
=1 ( 1)

where the symbol is used to denote the departure from the benchmark solution as
a result of thresholding:

p = p p
( E) = ( E) ( E)
( ) =
( Y ) = ( Y ) ( Y ) .

It
 is clear
 that both the momentum and density driven effects will be very small
O M 2 106 in comparison to those effects induced by errors in the energy
and species mass fraction, which are both of O ( ) . Consequently, in any adaptive
numerical strategy it is important to use at least E and ( Y ) as the key variables
upon which the unstructured grid is built. If the previous analysis is extended to a
multiresolution decomposition, essentially the same argument holds, but with the
magnitudes of the errors () increased by a factor of approximately O (J) where J
refers to the index of the original discretisation.
Interestingly, a similar problem is observed if asynchronous time evolution meth-
ods are employed (i.e. the schemes proposed by Bacry et al. [1]). It appears that the
coupling engendered by the pressure (through the energy equation) links phenom-
ena across many different scales, all of which interact with equal importance.

14.6 Conclusions

Wavelets have enjoyed considerable popularity in recent years. Their utility stems
from the scale and position dependent decompositions they provide, which in turn
350 Robert Prosser and R. Stewart Cant

furnishes flow descriptions with a level of fidelity beyond that offered by Fourier
transforms (which are spatially delocalized). Considerable effort has gone into the
exploitation of wavelets for reacting flow simulation. They provide a natural frame-
work to produce adaptive computations, although some technical challenges remain
regarding inertial and chemistry driven non-linearity (this problem however is preva-
lent to some extent with any numerical scheme). We have discussed in some detail
one possible line of attack for reacting flows involving interpolating wavelets. Oth-
ers are available (i.e. the vaguelettes of Schneider et al. [1517]). We have chosen to
describe the interpolating wavelet approach because of the closeness of the discreti-
sation to existing engineering calculation methods (i.e. it needs no initial projection
quadrature, nor operator modified vaguelettes, etc.)
In addition, the discussions presented here regarding non-linearities and differen-
tial operators are not restricted to the arena of DNS. We have seen how new wavelet
spaces may be populated during a nonlinear interaction. Furthermore, there is a clear
path from this emergent population to all other scales through the medium of the dif-
ferential operator. We may interpret mappings of the form Wi Wk k > i within
the operator as being akin to an inertial cascade processes, while the converse rep-
resents backscatter. Hence, wavelets provide a scale and position dependent method
of identifying and predicting the emergent phenomena in a turbulent flow and of es-
timating how that phenomena affects the resolved field. A modelling strategy may
exploit this by estimating the magnitudes of the wavelet coefficients emerging from
the non-linearity, and using the (known) values of the operator to estimate the effects
on the resolved scales. Such studies have been undertaken by a number of workers
(i.e. [10]), but further work is yet required.

References

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SIAM J. Num. Anal. 6, 17161740 (1992)
3. Beylkin, G.: On the fast algorithm for multiplication of functions in the wavelet bases. Inter-
national Conference on Wavelets and Applications (1992)
4. Beylkin, G., Coifman, R., Rokhlin, V.: Fast wavelet transforms and numerical algorithms I.
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10. de la Llave Plata, M., Cant, R.S.: On the application of wavelets to LES subgrid modelling.
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11. Deslauriers, G., Dubuc, S.: Symmetric iterative interpolation processes. Constr. Approx. 5,
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 Part IV
Cross-Cutting Science
Chapter 15
Design of Experiments for Gaining Insights and
Validating Modeling of Turbulent Combustion

A.R. Masri

Abstract This chapter addresses some key issues for consideration in the design
and development of experiments that provide a deeper understanding of combustion
physics and become benchmarks for the validation of calculations. Close interaction
between numerical and experimental approaches has proven to be a key ingredient in
advancing the predictive capabilities. A good experiment (i) addresses one or more
specific scientific issues, (ii) has well-defined boundary conditions, (iii) is amenable
to advanced diagnostics (iv) provides a range of conditions to test trends as well
as absolute quantities, (v) makes the detailed data available including information
about the accuracy of the measurements, and (vi) responds to the changing needs
of modelers as computational approaches change and evolve. A regime diagram for
turbulent combustion is first introduced followed by a section that details a series of
important considerations in the design and conduct of combustion experiments. The
last section provides details of three key burners that stabilize flames spanning most
of the turbulent combustion regime from premixed to non- premixed. Highlights and
pitfalls in the design of these burners are addressed in some detail.

15.1 Introduction

Progress in combustion science and engineering has been and continues to be evolu-
tionary rather than revolutionary with incremental and somewhat slow advancement
considering the long history of fires and flames. Developments in combustion the-
ory, experiments and more recently numerical calculations have hardly been syn-
chronized and do not necessarily occur in an ordered sequence. It is well known
however, that combustion science has historically lagged behind engineering ad-
vancements that have largely relied on experience and expensive cut-and-try meth-

A.R. Masri
School of Aerospace, Mechanical and Mechatronic Engineering, University of Sydney, NSW 2006
Australia, e-mail: a.masri@usyd.edu.au

T. Echekki, E. Mastorakos (eds.), Turbulent Combustion Modeling, 355

Fluid Mechanics and Its Applications 95, DOI 10.1007/978-94-007-0412-1 15,
Springer Science+Business Media B.V. 2011
356 A.R. Masri

ods. It is now evident that combustion science is needed to guide further develop-
ments in engineering to overcome long-standing problems of high pollution, lower
than optimal efficiency and diminishing reserves in fossil fuels. In the last century
there have been significant advances in the understanding of, and the ability to com-
pute thermo-chemistry, chemical kinetics and laminar flames; all of which are nec-
essary platforms for furthering research in turbulent combustion which is the subject
of this chapter.
The advent of computational methods [28] and advanced experimental diagnos-
tics techniques [27, 47] has initiated a healthy competition between experimental
and numerical scientists in combustion. In the second half of the twentieth century,
computational fluid dynamics (CFD) has evolved developing numerical solutions
for the Reynolds averaged Navier-Stokes (RANS) equations and hence reproducing
the basic statistical features of turbulent flames such as mean and rms fluctuations
of velocity, density, temperature and even composition fields [9]. It was not un-
til years later that experimental methods were able to generate the necessary data
for validating such calculations. For example, Raman scattering techniques capable
of measuring the concentration of some stable species became available to turbu-
lent combustion applications in the eighties [15, 16, 51]. The application of Raman
scattering to hydrocarbon flames [20] has reversed the leadership roles and led to
the discovery of local extinction and re-ignition events in non-premixed flames ap-
proaching global flame blow-off [57, 60, 61]. This feature of turbulence-chemistry
interaction was not computed until a decade later [49, 50, 79].
The series of International Workshops on Measurements and Calculations of Tur-
bulent Non- premixed Flames (TNF) which was initiated in the nineties [37] be-
came a very successful venue for bringing experimental and numerical researchers
together to advance the science of turbulent combustion. One of the basic ingre-
dients for this success is the collective focus on generic test cases or model prob-
lems that embody one or more specific issues being researched. The attention of
the first couple of workshops was directed to the study of turbulence-chemistry in-
teractions in non-premixed flames with relatively simple flows. Extensive data for
two burners with streaming flows, namely the simple jet flame and the piloted jet
burner were made freely available including sufficient details on the boundary con-
ditions [37, 55]. Calculations were then attempted by many groups using various
approaches and the results were compared, analysed and discussed at subsequent
workshops. This healthy exchange of information led to an enhancement in the ca-
pabilities of modeling approaches as well as to further sharpening in the design
of burners and the presentation of experimental data. A clear outcome here is the
breakthrough in the capabilities of the probability density function (PDF) methods
to compute local extinction in turbulent non-premixed flames [49, 50, 79].
The lead of experiments in exploring new aspects of turbulence-chemistry in-
teractions and hence further challenging modelers continued on many fronts. With
non-premixed combustion, the next generation of burner designs, namely the bluff-
body [18, 56, 59] and swirl [13, 41] burners were introduced bringing additional
complexities to the flow. Auto-ignition in turbulent flows was investigated experi-
mentally using two burners: one employs preheated air [54, 63] and the other uses
Experiments 357

hot combustion products in a vitiated co-flow [13]. A broadening in the scope of

the TNF from non-premixed toward premixed flames has led to the introduction of
the turbulent piloted premixed jet burner in vitiated co-flows [23]. Measurements
in these highly sheared flames with distributed reaction zones revealed a gradual
reduction in reactedness rather than sudden local extinction as was observed in non-
premixed flames. This mode of turbulence-chemistry interaction needs to be ac-
counted for in the modeling of premixed combustion. Partially premixed and strat-
ified flames are now starting to receive attention and a few burners to study such
flows are now being developed [4, 23, 72, 75]. It is only natural that the TNF ap-
proach of close interaction between modelers and experimentalists be extended to
advancing knowledge in other fields of turbulent combustion. As a case-in-point,
another workshop series, named the International Workshops on Turbulent Com-
bustion of Sprays (TCS), has recently been initiated to replicate the successes of the
TNF, initially in dilute, turbulent spray flames [38].
Good experiments should also benefit from the evolving requirements of ad-
vanced numerical approaches such as Large Eddy Simulations (LES) [7, 21, 44,
48, 71, 73] that is most likely to become a standard tool in the design of practical
combustors. One such requirement is that means and rms fluctuations are no longer
sufficient quantities to compare with LES which are producing time-sequences of
the entire flow, temperature, and composition fields. In response to this, high-speed
diagnostic methods have emerged to enable the imaging of velocity fields (using
high-speed particle imaging velocimetry, HS-PIV) and selected reactive scalars (us-
ing high-speed laser induced fluorescence, HS-LIF) [8, 10, 78]. Admittedly, the
imaging of true time-sequences are still limited by the repetition rates which are
gradually improving with the evolution of higher speed lasers, intensifiers and cam-
eras. The resulting time-dependent information, both numerical and experimental,
pose new challenges not only in data storage and analysis but also in the develop-
ment of smart methodologies for comparing measurements and calculations.
The improved capabilities of numerical approaches have gradually enabled the
reliance on numerical experiments to complement the limited flexibility of labora-
tory experiments. On one extreme, Direct Numerical Simulation (DNS) offers the
ultimate solution where no modeling is required but it is computationally prohibitive
and is likely to remain limited to simple flows with modest Reynolds numbers. How-
ever, DNS is extremely useful in the development, testing and validation of specific
physical processes or sub-models that are difficult to resolve with experiments. Two
examples of such uses are the development of models for the scalar dissipation rate
and its fluctuation and the DNS of auto-ignition to identify ignitable mixtures and re-
solve the composition of the radical pool prior to thermal run-away [26, 36, 64, 76].
DNS will continue to have an increasing and more effective role in the development
and validation of turbulent combustion calculations.
The use of numerical experiments is also becoming accepted from closure-
based approaches some of which are described in this book. However, care should
be exercised here to limit these applications to areas where the level of confidence in
the calculations is high. Such a healthy interaction between experiments and calcu-
lations can extend the applicability and capability to probing new areas of combus-
358 A.R. Masri

tion science. As an example, the particle-based PDF approach has been used in con-
junction with imaging experiments in auto-igniting flames of hydrocarbon fuels to
identify transport processes as well as markers for auto-ignition [31, 32]. Similarly,
LES methods are gradually advancing to enable the calculation and optimization
of practical combustors [7, 21, 48, 73]. These enhanced linkages and the growing
interdependence between experiments and calculations in turbulent combustion is a
common theme in this book.
This chapter draws on years of experience in the design of burners that have
advanced knowledge of turbulent combustion and provided deep insights into out-
standing issues such as turbulence-chemistry interactions in flames. The data result-
ing from experiments on these burners formed an essential platform for the devel-
opment and validation of calculations. Section 15.2 describes a regime diagram for
turbulent combustion ranging from premixed to non-premixed. This diagram may
also be applicable to combustion of sprays and forms the challenge space for mod-
elers and experimentalists alike. Section 15.3 addresses key issues that need to be
considered and addressed in the design of burners as well as related experiments.
Section 15.4 gives an account of selected case studies highlighting major discov-
eries and advances as well as specific problems and issues that could be useful as
potential guides for designers. The chapter closes with some concluding remarks.

15.2 The Turbulent Combustion Domain

The broad and historical classification of combustors into premixed and non-
premixed is convenient but rather not representative of practical devices consid-
ering that in most modern engines flames crossover a multitude of regimes within
the same combustion chamber. A typical example here is spray combustion where
the broad classification is for non-premixed flames while in actual fact the droplets
within the flow provide a source of fuel for partial premixing and stratification.
Nonetheless, such broad classification is very useful as it enables separate physi-
cal sub-models to be developed independently for both premixed and non-premixed
regimes before merging them together to account for the mixed modes of com-
bustion where partial premixing dominates. A classical example of such merging is
the development of the mixture fraction- reaction progress variable models which
are now applied to partially premixed flows [39, 46, 70].
The use of mixture fraction and some measure of reaction progress to describe
the regimes of turbulent combustion seems natural given the relevance of these con-
cepts in the extremes of premixed and non-premixed flames. Figure 15.1 shows a
realization of the regime diagram for turbulent combustion plotted in terms of the
flow conditions at the inlet to the combustor in a two-stream configuration. One
axis represents the inlet condition in stream 1 in terms of the overall fuel mass frac-
tion (the balance being oxidant) while the second axis represents the mass fraction
of oxidant in the second stream (the balance being fuel). Note here that the stoi-
chiometric as well as lean, and rich flammability limits marked respectively by St,
Experiments 359

Fig. 15.1: Regime diagram for turbulent combustion in a two-stream flow. Piloted
flames (L, B, M [57, 61] and D,E,F [5, 43]), bluff-body stabilised flames (HM1,
HM2, HM3) [18, 59], swirl-stabilised flames (SMH1, SMH2 and SMH3) [13, 41]
as well as piloted premixed vitiated coflow flames (PM50-PM200) [23, 25] are
shown.

L and R are only indicative and their actual location on the diagram varies with
fuel mixture. The third axis represents overall departure from fast chemistry which
may be indicated by an overall measure of burning index, B, reaction progress, C
or a Damkohler number, Da. The overall burning index is used here as a parame-
ter between zero and one where zero is for non-reacting and 1 is for fully reacting
conditions at the fully burnt limits.
Most burners and flames that have been studied so far may be positioned on this
regime diagram. Sydneys piloted flames L, B, and M have 100% fuel in stream 1,
100% air in stream 2 and range in burning index from high for flame L to almost
zero in flame M [20, 57]. Piloted Sandia flames D, E and F are similar but partially
premixed in the fuel stream [5, 43]. The complexity in the fluid mechanics is not
reflected in this diagram so Sydneys bluff-body flames HM1, HM2 and HM3 [18,
59] and swirling flames with 100% fuel in the jet stream (flames SMH1, SMH2
and SMH3) [13, 41] would be on the same traverse as flames L, B, M. Flames
positioned in a vitiated co-flow such as the Cabra and Dibble configuration [13]
have, by definition a reduced amount of air in the co-flow as shown in Fig. 15.1 by
the solid bold line with arrows that may extend further on the lean or rich sides. For
example, the auto-igniting flames of Cabra et al. [13] and Gordon et al. [30] with
partially premixed fuel would be positioned on the rich (right hand) side of the plot.
The premixed flames of Dunn et al. [23, 25] are on the lean side and are indicated by
PM-50 to PM-200 with a range of burning indices. The two-stream stratified flames
introduced recently [4, 72, 75] would fit within the dashed box depending on the
equivalence ratios used within each stream. Spray flames may also be represented
360 A.R. Masri

on this diagram although the location may vary depending on the injection condition
and the volatility of the parent liquid fuel. Direct injection of liquid into air yields
diffusion-like flames initially while air-assisted injection may result in premixed
or partially premixed depending on the volatility. Additionally, the evaporation of
droplets within the flow may lead to some stratification.
Most modeling approaches are generally adaptable to one or more modes of com-
bustion, such as premixed or non-premixed but not sufficiently general to apply to
the entire spectrum described in this regime diagram. A generalized combustion
modeling approach is one with a proven ability to compute the structure of turbu-
lent gaseous flames over a broad range of conditions: premixed to non-premixed,
low to high strain, thin to broad reaction zones, auto-igniting to flaming. This would
be compounded by additional complexities associated with strong transients and in-
stabilities such as those found in swirling flows and/or with the presence of droplets
in sprays. These are significant challenges that may well be met by one or more of
the approaches described in this book.

15.3 Basic Considerations

Good experiments are generally designed to explore specific scientific issues and
advance knowledge to the point where the resulting findings can be used to chal-
lenge, develop and validate numerical approaches. Good interaction with modelers
is essential here since an appreciation of the limitations and capabilities of numerical
methods may result in improved burner designs that are more representative while
still amenable to modeling. A few basic considerations are discussed in this section
as aids rather than prescriptions for the design and conduct of future experiments.

15.3.1 Design Issues

The distinction is made at the outset that the concern here is with generic laboratory
research burners rather than scaled-down versions of practical combustors. While
scaling issues are not considered, attention should be given to the relevant dimen-
sionless parameters to ensure that the relevant domain of the burners operation is
satisfied. Here are the three main issues that govern the design of research burners:
Research focus: The principle here is well accepted and entails identifying and
separating research issues that form the design focus of the burner. In the well-
known piloted burners, the focus was on turbulence-chemistry and hence soot, liquid
fuels and complex flows were initially eliminated. The latter condition was later
relaxed in the bluff-body and swirl stabilized burners and, later, dilute sprays were
introduced but in simple jet flows. The study of auto-ignition also uses rather simple
gaseous flows in a hot vitiated co-flow as in the burner of Cabra et al. [13]. Turbulent
stratified combustion requires a burner that allows flexibility in changing the mixture
Experiments 361

fraction and mixture fraction gradients and preferably the attainment of high levels
of u /SL . In premixed flames, most burner configurations, such as those stabilised by
low swirl and bluff bodies [6, 14, 52, 53] are generally limited to values of u /SL of
about 10. The provision of a hot vitiated co-flow enables much higher shear rates and
hence a transition of the flames to the distributed reaction regime. Representative
flames stabilized on these vitiated co-flow burners are located on the regime diagram
of Fig. 15.1.
Boundary conditions: The issue of boundary conditions is of extreme importance
and impacts directly on the final configuration of the burner. Modelers generally
require as much information as possible but the level of detail and the meaning of
boundary conditions depend on the numerical approach and location at which mod-
elers start their calculations. For RANS type calculations, profiles of the mean and
rms fluctuations would be sufficient, while time series may be required for LES. It is
not unusual for modelers to commence their calculations well upstream of the exit
plane of the burner. The case studies detailed later in this chapter will provide exam-
ples and highlight some of the associated issues. Figure 15.2 presents a sample of
some basic details required for boundary conditions with profiles of means and rms
fluctuations of velocity at the exit plane of three commonly used burners, namely
the piloted, the bluff-body and the swirl stabilized burners.
Another issue of importance is the sensitivity to boundary conditions. This is
harder to investigate experimentally and lends itself more easily to calculations once
a certain level of confidence is reached. The boundary layer associated with jet flows
may have impact on the flame. It has been shown for simple jet flames, that increas-
ing the co-flowing velocity leads to a laminar to turbulent transition [58] in the
external boundary layer hence affecting transport of the reactive scalars in the jet.
Heat transfer to the burner may also be significant as is the case in bluff-body flows.
Specifying the temperature of the bluff-body as well as the construction material
will be important in accounting for such losses.
Stabilization with a pilot is simple and effective but adds a subtle level of com-
plexity to the flow. The pilot is essentially another flow stream that needs to be
accounted for. The level of difficulty associated with modeling the pilot stream is
generally modest but depends on the approach. Accounting for the additional heat
release from the pilot is generally easy. The composition of the pilot flame gases
may be assumed to be at equilibrium or may be computed from laminar premixed
flame codes or as a plug-flow reactor. When studying pilot-stabilized turbulent pre-
mixed flames, the stoichiometry of the pilot may be different from the rest of the
flow and this tends to make the definition of the reaction progress variable no longer
uniform as is the case with the piloted premixed burner in vitiated co-flows [25].
Optical access: This is a key requirement for experimental burners given that almost
all advanced diagnostics methods will require some sort of optical access. Enclosed
burners will use windows that are generally made of quartz if ultraviolet light trans-
mission is needed. Flat windows are preferred to avoid reflection associated with
curvature. Applications that involve high pressure are much harder to handle but
these are outside the scope of this chapter.
362 A.R. Masri

Fig. 15.2: Mean and rms fluctuations of the axial velocity measured at the jet exit
plane of three burners: (Top LHS) piloted [61], (Top RHS) bluff-body [59], and
(Bottom four plots) swirl [2] which also shows the tangential component of velocity.

15.3.2 Operational Envelopes

Understanding the burners operational envelope is very important since this dictates
the limiting conditions and the combustion ranges that the burner can access. Three
considerations are noted here:
Controlling parameters: Minor and major parameters that affect the flame and
its stability need to be identified for fixed geometry parameters such as the size
of the bluff-body or the jet diameter. Initial parametric studies may be needed to
identify the relative importance of various parameters and this process is not always
straight forward. In the swirl burner for example, the swirl number is essential but
the ratio of the momentum of the co-flowing stream to the central stream is also very
important. Note also that in bluff- body stabilised burners, the co-flow velocity is a
major parameter but not so in the piloted flames.
Experiments 363

Fig. 15.3: Stability limits of the bluff-body burner [56] (top) and auto-ignition burner
in vitiated coflows [31] (bottom).

Stability limits and test cases: While relatively easy to determine once the major
controlling parameters are identified, these are important in setting the overall oper-
ational envelopes of the burner and hence selecting relevant flames for further inves-
tigation. Figure 15.3 shows typical stability limits for two burners: (i) the bluff-body
burner [56] where the two controlling parameters are the velocities of the jet fuel,
U j and the co- flowing air, Ue , and, (ii) the auto-igniting flames in hot, vitiated co-
flow [31, 32] where the lift-off height is strongly affected by the temperature of the
hot co-flowing stream. It is worth noting that in selecting the test cases, groups of
flames are chosen in particular sequences to test the importance of each of the con-
trolling parameters. For example, bluff-body flames HM1, HM2, and HM3 would
approach blow-off by increasing the jet velocity at the same co-flow velocity (move
vertically up in Fig.15.3). Similarly, piloted flame sequences L, B, M and D, E, F,
as well as swirl flames SMH1, SMH2, SMH3 of methane/hydrogen fuel mixtures.
Such sequences have proven to be extremely useful in showing relevant trends and
hence in the validation of computational models.
364 A.R. Masri

Sensitivity to boundary conditions: In some flow configurations, the flames may

be very sensitive to boundary conditions leading to broad variations when the same
experiments are repeated at different times or different laboratories. A common ex-
ample of this is the standard lifted flame where small variations in the surrounding
air velocity or turbulence level causes significant changes in the lift-off height. Other
illustrations of the sensitivity to boundary conditions are the auto-ignition burner
which is strongly affected by the temperature of the co-flowing vitiated air [13], and
the bluff-body burner which may be affected by the outer boundary layer on the air
side of the bluff-body [18, 59]. Investigating such effects is possible experimentally
but the difficulty depends on the parameters being studied. With sensitivity to the
boundary layer, it may be easier to perform such studies numerically since changing
the boundary layer experimentally is do-able but not trivial.

15.3.3 Experimental Considerations

Good planning of the experiments will improve the chances to strike a breakthrough
in the understanding of the physics or at least to just provide a useful and reliable
data base that will be used for model validation. Here is a brief description of the
issues that are worth considering:
A non-reacting test case: Along with the sequence of test cases discussed earlier,
the selection of a base case is extremely useful for model validation. This could be a
simple flame or even a non-reacting case that possesses similar characteristics to the
reacting cases. Modelers normally start with such cases to test the relatively simpler
features of their calculations such as flow and mixing fields. In the investigation of
sprays, it is extremely important to perform measurements in non-reacting case that
will yield information about the dynamics and evaporation of droplets. Selecting test
cases where the liquid fuels have different vapor pressures will also be extremely
useful in optimizing the evaporation model.
Measurements:
Flow fields: The most common techniques here are Laser Doppler Velocimetry
(LDV) and Particle Imaging Velocimetry (PIV) both of which rely on seeding
micron-sized of particles into the flow. LDV methods are well-developed but
are inherently limited to single-point measurements. PIV methods are simple
in principle and have also been well developed for low repetition rate. Recently,
there have been two major developments with PIV: (i) High-speed PIV (HS-PIV)
made possible by the development of high-speed lasers and cameras (around 20
kHz at the time of writing this chapter) and it is only a matter of time before the
laser and camera technology develops even further to enable true time evolution
measurements, (ii) Three dimensional PIV based on volume illumination and
three-aperture-three-camera detection systems (V3V) [40]. This technology is
relatively new and is currently limited to low repetition rates but is also expected
to evolve to a stage where full 3D-measurements of the time evolution of the flow
field becomes possible.
Experiments 365

Mixing fields: The use of combined Raman scattering and LIF to measure the
concentration of a sufficient number of species to obtain a conserved scalar is
limited by many factors including the complexity of the experiment and severe
interferences on the Raman signals [61]. An alternative approach is to identify in-
ert tracers that can fluoresce and withstand the flame temperatures without being
affected. Despite testing a large number of tracers [74], this approach remains in-
conclusive for high temperature applications due to flame interferences resulting
in the corruption of the original concentration of the tracer or due to significant
quenching of fluorescence from the tracer. The use of noble gases as tracers is
promising if the quenching issue is overcome since these species can easily sur-
vive the flame temperatures unscathed so they have a good chance of enabling a
simple and reliable measure of mixture fraction.
Temperature and reactive species fields: The use of Rayleigh scattering for
temperature measurements is limited to non-sooting gaseous flames but still re-
quires knowledge of the Rayleigh scattering cross-section, and hence a measure
of species concentrations which is difficult due to the complexities associated
with Raman methods. A relatively easy alternative here is to modify the fuel
mixture such that the Rayleigh scattering cross section remains as uniform as
possible across the entire mixture fraction space [19]. Combined imaging of
Rayleigh-LIF is proving to be extremely useful in flames where Raman tech-
niques are hard or inaccessible. Multi-species-LIF techniques are expensive since
they require numerous laser sources depending on the species being detected but
are extremely informative and do not suffer from the low signal levels associated
with Raman measurements. High-speed LIF (HS-LIF) methods are now becom-
ing more common albeit still limited to common species such as OH [8, 10, 78]
and speeds of about 10 kHz. Such methods are extremely informative and are
evolving quickly so it is only a matter of time before developments in the laser
and camera technology enables higher speed and a broader selection of species
that can be measured either solely or jointly.
Data presentation and availability: Two issues need urgent attention now that
high-speed imaging methods are becoming commonplace along with the widespread
use of LES for calculations: (i) presentation of huge quantities of space- and time-
resolved data represented by time-sequenced images and (ii) relevant and tractable
methods of processing and analyzing these images as well as providing meaning-
ful comparisons with LES. Figure 15.4 shows sequences of high speed images (5
kHz, hence 200 s separation) of LIF-OH collected at the base of a lifted flame
of CNG with a jet diameter of 3.85mm and a Reynolds numbers of 4800. The top
sequence illustrates the presence of what appears to be edge flames shaped in the
form of hooks while the second shows blobs that grow at the same location indi-
cating that these are coming into the page from another radial location at the base.
Both modes appear to play a significant role in stabilising the flame. Thousands of
these images are collected so that presenting the information in a sensible way is
not straightforward. The other issue is the processing of the image sequences such
that spatio-temporal information such as rate of change of spatial flow structures can
be extracted in a sensible way. Proper orthogonal decomposition, (PoD) techniques
366 A.R. Masri

Fig. 15.4: Sequences of high-speed LIF-OH images collected at the base of a lifted
methane flame. The fuel issues at 35 m/s into still air from a round nozzle with a
diameter D=3.85 mm. The measurements are centred at x/D=30. The top sequence
shows the edge flame hooks and the bottom sequence shows flame kernels likely
to be flowing in or out of the page. From [67].

[17, 22] appear to be promising in this regard although significant development is

needed before these are established as standard tools for data analysis.

15.3.4 Numerical Considerations

The focus on providing detailed and simple boundary conditions has already been
identified as one of the benefits of the strong interactions between modelers and
experimentalists. There are two others issues that are worth focusing in this section:
The starting point of the calculations: Commencing calculations at the exit plane
of the jet requires details that enable the modelers to initiate the same condition for
which data are available. Such information is generally in the form of means and rms
profiles of velocity and of scalars such as, temperature and species mass fractions.
Starting the calculation further upstream, requires details of the inner design of the
burner but then only bulk flow conditions need be provided. Measurements provided
at the exit plane can then be used to test and tune the calculations. Examples of such
scenarios are given in the next section. Another consideration here is the need to
compute the flow close to surfaces used to pipe the fluid. These are generally hard
to reproduce particularly when spray flows are used. The interaction of droplets with
solid surfaces is a difficult field that is not well modeled.
The requirements of LES: Large eddy simulations are providing an effective plat-
form for modeling the entire spectrum of turbulent reacting flows. Relevant sub-
grid-scale models for combustion are being developed but a universally accepted
model is not yet available. The use of LES imposes stringent requirement on ex-
perimental methods in two key areas: (i) more details are needed in the boundary
conditions (such as time series of say velocity fields), and (ii) the provision of time-
sequenced images of flame structure to compare with the calculations. This latter
Experiments 367

requirement may now be satisfied thanks to the advent of high-speed imaging tech-
niques [8, 10, 78] although methods to provide adequate and meaningful compar-
isons between the numerical and experimental data sets are yet to be devised.

15.4 Case Studies

This section presents three cases studies that span a broad region of the turbulent
combustion diagram shown in Fig. 15.1. These are the swirl burner, the piloted
premixed burner in vitiated co- flows and the piloted spray burner. While the first
two are established burners in the TNF, the spray burner is now established in the
new Turbulent Combustion in Sprays (TCS) [38] workshops that are attempting
to duplicate the success of the TNF in sprays, starting with dilute sprays. While
extensive details about these burners are well known from earlier publications, the
focus of this section is to summarise the scientific advances and relay highlights and
pitfalls in their design for future reference. A brief summary of the major insights
and scientific advances resulting from each burner are mentioned at the beginning
of each section.

15.4.1 The Swirl Stabilised Burner

This represents the third generation burner (after the piloted and bluff-body sta-
bilised) in the TNF series of model problems aimed at studying turbulence-chemistry
interaction in non- premixed flames. While the design maintains the features of a
laboratory research configuration with simple, well-defined boundary conditions in
an open flow, it is most representative of practical combustors with flow recircu-
lation and instabilities. A comprehensive data bank is made available for a series
of flames and three fuel mixtures as shown in Table 15.1. The results confirm ear-
lier findings about the occurrence of local extinction in non-premixed flames even
in such highly recirculating and unstable flows. There have been serious attempts
to compute some of these flames [45, 62, 66] but with mixed success due to the
complexity of the fluid dynamics. Figure 15.5 shows a detailed drawing of the full
burner assembly. The following key features have not been addressed before in the
literature and are worth discussing here for the record:
The start of the calculations at the exit plane of the burner requires details of
the boundary conditions of the velocity time series in the central fuel jet and in
the boundary layer. While the former may be assumed to conform with fully de-
veloped turbulent pipe flow, the co-flowing boundary layer is less certain. Using a
quartz shroud for the swirl annulus as shown in Fig. 15.6, extensive measurements
of the axial velocity were made just upstream of the exit plane. These have resulted
in time series for the axial velocity a sample of which is also shown in Fig. 15.6.
Also shown in Table 15.2 is a listing of the mean temperature of the ceramic on the
368 A.R. Masri

Fig. 15.5: Drawing of the swirl burner. Reprinted from [2] with permission from
Taylor and Francis.

face of the bluff-body measured using a two-color pyrometer in the various swirling
flames studied on this burner. To circumvent the problem of duplicating the mea-
sured conditions at the jet exit plane, some groups chose to start their calculations
well upstream and hence the details shown in Fig. 15.5 become extremely useful for
such calculations. The main difficulty with such approach for LES is the fact that
fine resolution is required near the inner and outer surfaces to resolve the bound-
ary layers. The benefit is that the calculations, when performed reliably, can yield
extremely useful information about the flow sensitivity to the boundary layers.
Instabilities are inherent to swirling flows and at least two modes of instabilities
have been identified here: (i) jet precession which is observed both in non-reacting
and reacting flows, and (ii) puffing which is limited to flames. Figure 15.7 shows
Experiments 369

Table 15.1: Flame conditions studied in the swirl burner.

Flame Fuel Us Ws U j Res Re jet Sg UBO L f W

Mixture (m/s) (m/s) (m/s) (-) (-) (-) (m/s) (m) (kW)
(vol.
ratio)

SM1 CNG 38.2 19.1 32.7 75900 7200 0.5 166 0.12 11.1
SM2 38.2 19.1 88.4 75900 19500 0.5 166 0.18 30

SMA1 CNG- 32.9 21.6 66.3 65400 15400 0.66 241 0.2 11.5
air (1:2)
SMA2 16.3 25.9 66.3 32400 15400 1.59 216 0.23 11.5
SMA3 16.3 25.9 132.6 32400 30800 1.59 216 0.3 23

SMH1 CNG- 42.8 13.8 140.8 85000 19300 0.32 267 0.37 104.1
H2
(1:1)
SMH2 29.7 16 140.8 59000 19300 0.54 281 0.4 104.1
SMH3 29.7 16 226 59000 31000 0.54 281 0.5 167.1

Fig. 15.6: Time series of axial velocity measured upstream of the exit plane of the
swirl burner (using the quartz shroud shown). Reprinted from [3] with permission
from the Combustion Institute.

the latter mode of puffing instability in flame SMH3. Such modes add a significant
level of complexity to the task of computing these flows and impose an additional
difficulty with measurements due to the need of phase locking. It should be noted
that the data reported for these swirling flames were not phase-locked with any
370 A.R. Masri

Table 15.2: Measured mean temperature on the face of the ceramic for the various
swirl flames investigated in Fig. 15.6 and Table 15.1.

Flame SM1 SM2 SMA1 SMA2 SMA3 SMH1 SMH2 SMH3

Ave. temp, 557 475 585 614 581 685 637 556
C
Range, 10 19 35 35 33 24 11 28
C

Fig. 15.7: Shadowgraph plots showing puffing instabilities in swirl stabilised

flames of methane/hydrogen. This sequence is for flame SMH3 [3]. Reprinted from
[3] with permission from the Combustion Institute.

frequency mode and were presented in the form of instantaneous scatter plots as
well as means and rms fluctuations.

15.4.2 The Premixed Burner in Vitiated Coflows

There are two configurations for this burner (i) the unpiloted version which is es-
sentially the basic layout as developed by Cabra and Dibble to study auto-ignition
[13] and (ii) the piloted burner which involves an additional stoichiometric stream
that maintains the flame connected to the exit plane. A layout of the piloted burner
and its stability limits are shown in Fig. 15.8. Four piloted flames were selected for
further studies and these are also shown in Fig. 15.8 along with further details in
Table 15.3. A comprehensive data base exists for these flames which are now being
computed [24].

Table 15.3: Details of premixed flames stabilized in hot coflowing pilot.

Flame U0 (m/s) Re q /SL Ret Da

PM1-50 50 12500 40 720 0.0690
PM1-100 100 25000 230 3100 0.0089
PM1-150 150 37500 300 3700 0.0063
PM1-200 200 50000 390 5200 0.0053
Experiments 371

Fig. 15.8: The premixed piloted vitiated coflow burner and its stability limits. Also
shown are the flames selected for further investigation (PM1-50, PM1-100, PM150,
PM1200). Reprinted from [23, 25] with permission from the Combustion Institute.

The flames listed in Table 15.3 are gradually approaching blow-off and fit in the
distributed reaction region of the premixed combustion regime. Note that q /SL in-
creases from 40 to 390 while the Damkohler number, defined by the ratio of chem-
ical and mixing time scales, decreases by more than an order of magnitude from
flame PM1-50 to PM1-200 which is close to blow-off. Here q is the fluctuation in
the turbulent kinetic energy. A major finding revealed by the measurements is that
the finite rate chemistry effects feature in a gradual reduction of reactedness rather
than in localized extinction as was observed non-premixed flames [57, 60, 61]. Gen-
eralised models of turbulent combustion must account for this diversified behaviour.
There are two special features in this burner design that deserve further elaboration:
The extent of the hot coflow is a key consideration because the region of interest
in the flames studied has to be fully within the hot co-flow and not affected by
the outside air. Figure 15.9 shows mean temperature profiles measured at various
axial locations in the co- flow which has a nominal temperature of 1500K. Flat-
top profiles peaking at this temperature where measured down to x/D 50 (where
D=4mm) and this corresponds to a valid cone that extends for about 200 mm
downstream of the jet exit plane. Matching the co-flowing air velocity with the
hot co-flow tends extends this region slightly.
The additional pilot is essential for flame stabilization and for enabling suffi-
ciently high rates of shear to be applied to the flame without inducing lift-off.
For this to occur, a stoichiometric pilot composition was deemed necessary. The
pilot has two main disadvantages: (i) its different stoichiometry and hence dif-
372 A.R. Masri

ferent temperature and composition fields in the pilot stream makes it difficult to
define a universal reaction progress variable in terms of temperature or species
mass fractions, (ii) the pilot adds another stream to the flow making it a three-
stream problem. While this is not a problem for most modeling approaches, it
may be so for others hence making this configuration less attractive. An alter-
native layout for this design is the use of bluff-body stabilisation which adds
complexity to the flow but reduces the number of flow streams and eliminates
the problems associated with the definition of the reaction progress variable.

15.4.3 The Piloted Spray Burner

Spray flows are now receiving significant attention and this is largely driven by the
lack of progress in this area up to date, the continued interest in liquid fuels and the
recent advances in computational and experimental capabilities. The piloted spray
burner shown in Fig. 15.10 is designed to enhance current understanding of reacting
and non-reacting dilute sprays and to provide a platform for the development and
validation of codes. In this burner, the spray is prepared upstream of the jet exit
plane and carried in a co-flowing stream of air or nitrogen to the jet exit plane.
Studies reported to date [34, 77] revealed the following interesting but challenging
features: (i) Jet spray flames using of acetone or ethanol fuel reveal different modes
of combustion largely due to the different volatility of the fuel. Acetone flames
are non-premixed near the jet exit plane due to the high vaporization rates while
ethanol flames may have dual reaction zones with a premixed flame core and an
outer diffusive layer. (ii) While single droplet combustion is occasionally observed
in these flames, the general tendency is for combustion to occur around droplet
clusters. Additional research is required to further explore these phenomena.
There are three underlying design principles in this burner: (i) simple streaming
flows, (ii) well known boundary conditions and hence the preparation of the spray
upstream of the exit plane (iii) ability to stabilize flames over a range of Damkohler
numbers. While most of these objectives have been achieved in this burner design,
there are a few issues that are worth addressing here:
Measurement of velocity and droplet fields: Phase Doppler Anemometry, PDA
coupled with LDV methods are extremely useful here but are limited to single-point
measurements and tend to fail when the concentration of droplets in the measure-
ment probe is high. This tends to limit the LDV/PDA application to cases with
relatively low droplet loadings and/or dilute regions of the flow. An alternative to
LDV/PDA is high- speed PIV-based methods which have the potential of yielding
true time evolution of the flow fields with the droplets being the scattering agent
(instead of seeded particles). The limitation of the approach lies in the sparsity of
droplets, particularly in dilute sprays. Classical PIV-processing tools expect large
number of seed particles to produce valid cross-correlations. Expanding the win-
dow of interrogation is not really a useful option and hence the PIV processing tools
Experiments 373

Fig. 15.9: Measured mean temperature profiles in the vitiated co-flow burner show-
ing the extent of the valid cone marked by a region of uniform peak temperature.
From [29].

need to be modified to handle small numbers of scattering elements in a reasonably

small interrogation window.
Measurement of mixing, temperature, and reactive scalar fields: The existence
of droplets limits the application of laser diagnostics methods such as Raman and
Rayleigh scattering due to the inability to distinguish between gaseous and liquid
374 A.R. Masri

Fig. 15.10: Piloted spray burner showing details of spray inlet condition. Reprinted
from [34] with permission from Taylor and Francis.

phases and the interferences of Mie scattering from particles. Thankfully, laser in-
duced fluorescence works well here and these methods have been employed in spray
flames to yield some very useful results for the measurements of reactive scalars
such as OH, CH2 O and others. The potential of LIF-based methods to yield a mea-
sure of mixture fraction and temperature is a challenge that is not yet fully realised.
This is either due to the inability of the tracer to survive higher temperatures with-
out flame interference or due to significant quenching on the LIF signal. The use
of NO as a tracer even in non-reacting sprays has proven to be problematic due to
LIF-quenching or to Raman interference from droplets [35]. Figure 15.11 shows the
fluorescence quenching of acetone and ethanol on NO. While n-hexane was identi-
fied as a non-quencher for NO, it had another problem in that the Raman interference
from droplets on LIF-NO was significant, rendering it unviable. Other tracers such
as dyes or even acetone may turn out to be more useful than NO. The measurement
of gaseous temperature in reacting sprays is another serious challenge that is cur-
rently receiving attention. The use of elemental tracers such as indium [42, 65] or
phosphorous [11, 12, 68, 69] appears to be promising although a lot more develop-
ment is needed before such techniques can yield reliable quantitative information.
Tube wall effects: Measurements of velocity and turbulence levels at the jet exit
plane have shown that when conditioned with the droplet size, the rms fluctuations
of large droplets are higher that those of small droplets near the inner edge of the
pipe. This is counter-intuitive and was referred to in earlier reports as the pipe-flow
anomaly. Scatter plots of velocity versus droplet size collected near the inner pipe
Experiments 375

Fig. 15.11: Reduction in the intensity of LIF-NO due to quenching by (a) acetone
and (b) ethanol vapor. Three profiles refer to different spray carries (air, nitrogen
and agron). From [35].

wall have shown a bimodal distribution as seen in Fig. 15.12. The reason for such
behaviour is now understood and explained as follows: droplets interact with the
wall of the pipe and form a boundary layer of liquid. At the exit plane, such boundary
layer sheds droplets in the form of ligaments which will quickly disintegrate into
droplets as confimred by recent high- speed imaging of Mie scattering from droplets.
Such droplets are relatively slow compared to other droplets of similar size that are
not affected by this boundary layer and hence the bimodal distribution. The impact
of such bimodality on the structure of the spray jet is not really known. Removing
such bimodality requires a reduction in the length of pipe to about 30mm which
is region where the droplets just start to impact on the wall and build the liquid
boundary layer.
Non-reacting sprays and a variety of liquid fuels: Selecting a series of non-reacting
cases is particularly important in this study given the importance of evaporation and
the need to resolve uncertainties surrounding current models of evaporation. A ref-
erence case that represents a fuel with low evaporation, namely mineral turpentine
which has a boiling point in the range of 140-190 C has proven to be useful. Measur-
ing the liquid flux at the jet exit plane of a non-reacting turpentine spray ascertains
whether the mass flux is measured correctly. Moreover, performing a series of mea-
surements in non- reacting jets of sprays of various volatilities gives the necessary
platform for validating the flow, droplet fields as well as the evaporation models.

15.5 Concluding Remarks

This chapter gives an overview of the principles associated with the design of
generic laboratory burners that form a platform for exploring aspects of combus-
tion science and gaining insights into outstanding research issues. The data resulting
376 A.R. Masri

Fig. 15.12: Upper: Scatter plot of instantaneous velocity vs. diameter at r/D = 0.057
and 0.41. Lower: Velocity pdf for droplets in the diameter range 30< d <40 m.
From [33].

from such experiments also form an international benchmark and validating and ad-
vancing modeling capabilities. A regime diagram that highlights a broad workspace
in turbulent combustion is presented. Such a parameter space is huge and only rep-
resentative sections of it can be studied experimentally. The challenge to the com-
bustion community however, is to develop generalized modeling capabilities within
this entire workspace. Examples discussed in this chapter cover scattered regions
of this diagram and address a range of considerations and issues that would hope-
fully be of benefit in the design and conduct and future experiments in combustion.
Good experiments may lead to major breakthroughs as was demonstrated in some
of the case studies presented here. It is evident that continued interaction between
modelers and experimentalists is extremely beneficial leading to a mutual under-
standing of needs and limitations, a better design of burners and experiments as
well as enhanced computational methods. Other fields of combustion science are
also benefiting from this approach.
Experiments 377

Acknowledgements

The author is indebted to Professors R.W. Bilger and E. Mastorakos for their en-
lightening feedback on earlier drafts of this manuscript. The author acknowledges
the support of Professor R.W. Dibble and Dr R.S. Barlow for enabling a long stand-
ing collaboration with Sandias Combustion Research Facility at Livermore, CA.
Thanks to Drs Sten Starner, Robert Gordon, Matthew Dunn and James Gounder as
well as to Mr Mrinal Juddoo and William OLoughlin for their help with the vari-
ous experiments. Thanks also to the Australian Research Council for the continued
support of his research.

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Chapter 16
Uncertainty Quantification in Fluid Flow

Habib N. Najm

Abstract This chapter addresses the topic of uncertainty quantification in fluid

flow computations. The relevance and utility of this pursuit are discussed, outlining
highlights of available methodologies. Particular attention is focused on spectral
polynomial chaos methods for uncertainty quantification that have seen significant
development over the past two decades. The fundamental structure of these methods
is presented, along with associated challenges. We also discuss demonstrations of
their use in a number of fluid flow applications covering a range of complexity that
is inherent in turbulent combustion.

16.1 Introduction

Uncertainty Quantification (UQ) broadly refers to the quantitative estimation of un-

certainty in computational modeling of physical processes. Recent years have seen
increasing interest in this topic, particularly in the context of computational mod-
eling of fluid systems [29, 52, 80, 125, 145]. This is due in part to the increased
complexity of computational models and our increased reliance on computational
predictions for engineering design, scientific discovery, and decision support. It is
also due to the development, over the past couple of decades, of efficient and effec-
tive UQ methods based on probability theory. In this chapter, we outline the basics
of these methods, as well as their key challenges, and discuss their utility and ap-
plication in computations of a range of fluid systems involving various facets of the
complexity inherent in turbulent reacting flow.
In general, uncertainty quantification is useful from both the engineering and
scientific points of view. In the engineering context, confidence intervals on pre-
dicted system behavior are necessary for design optimization, for reliability assess-
ment, for the determination of safety factors in engineered systems, and for decision

Habib N. Najm
Sandia National Laboratories, Livermore, CA 94550, USA, e-mail: hnnajm@sandia.gov

T. Echekki, E. Mastorakos (eds.), Turbulent Combustion Modeling, 381

Fluid Mechanics and Its Applications 95, DOI 10.1007/978-94-007-0412-1 16,
Springer Science+Business Media B.V. 2011
382 Habib N. Najm

support purposes. In the scientific context, model validation with respect to exper-
imental measurements requires quantification of uncertainty in both experimental
measurements and computational predictions. In the absence of such uncertainty
error bars, it is not possible to state whether any disagreements between model
predictions and experimental observations indicate lack of validity of a model, or
whether they are within the inherent range of uncertainty in the predictions and the
observations.
Further, from a broad perspective, UQ is an important element of the overall Ver-
ification and Validation (V&V) challenge in computational modeling [90, 92, 94].
V&V is of key importance for computational code certification [91, 121]. Starting
from a given mathematical model of a physical system, any subsequent discretiza-
tion of the governing equations for computational purposes necessarily introduces
discretization errors. It is imperative that these errors be quantified and minimized
such that their impact on computational predictions is small. Once computational
predictions are shown to be consistent with the underlying mathematical model,
then the associated code is verified [14, 29, 91, 110]. This verification challenge
is a significant undertaking, and there are many examples of substantial code-to-
code scatter in computational predictions of fluid flow problems [18, 42]. However,
verification is outside the scope of this UQ discussion. Rather, the UQ context of in-
terest here is of relevance in validation, where the focus is on the degree to which
computational predictions agree with empirical observations [90, 94].
Potential sources of uncertainty in computational modeling include any inputs,
initial/boundary conditions, and parameters that enter the governing equations, as
well as those relevant in the specification of the geometry of the computational
domain. In the present discussion, we refer to these broadly as model parameters.
Other, non-parametric sources of uncertainty, such as modeling assumptions, some
aspects of model structure, or choices of constitutive laws, can be dealt with using
information theoretic methods, e.g. Bayesian model averaging [43], but are out-
side the present scope. In the following, we focus primarily on parametric uncer-
tainty.
Moreover, we adopt the Bayesian view of probability [47], where probability is
inherently the degree of belief in a proposition, and does not necessarily derive from
sampling or observation. This viewpoint provides the means of handling both epis-
temic and alleatoric uncertainty [92] in the context of probability theory. More-
over, it provides the means of seamlessly merging subjective prior information and
empirical data in probabilistic inference, and is a sound foundation for sequential
learning. Particularly, in the absence of data, prior knowledge, e.g. based on expert
elicitation, serves to provide the requisite probabilistic representation. The interested
reader may consult numerous sources on alternative, non-probabilistic, viewpoints
on this topic [41]. These include Evidence theory [8, 40, 93, 118], Possibility the-
ory [22], Fuzzy set theory [23, 124], and Imprecise Probability theory [54]. Note
that a key advantage of probabilistic UQ methods is the utility of the concept of
measure in probability theory. In the following, we presume the existence of suffi-
cient information/data about model parameters to allow assigning them PDFs and/or
evaluating their statistics, enabling the use of probabilistic UQ methods.
Uncertainty Quantification in Fluid Flow 383

Further, the present discussion is focused on the forward propagation of uncer-

tainty in computational models, i.e. the propagation of uncertainty from parameters
to outputs/observables of interest. On the other hand, the quantification of the para-
metric uncertainties themselves is an equally important task. When experimental
data is used to estimate uncertain parameters, one is left with parametric uncertainty
that reflects a number of sources of uncertainty in experimental measurements, e.g.
limited data and instrument noise. Further, recall that experimental measurements
rarely provide direct measurement of a given parameter. Rather, an experiment mea-
sures a given observable that is the result of some instrument/experiment forward
model to which the parameter of interest is an input. Consequently, parameter es-
timation from empirical observations involves inference and the solution of an in-
verse problem. Given that inverse problems are frequently ill-conditioned and a-
causal [48], this leads to significant challenges in parameter estimation and ensuing
amplification of any noise effect, leading to potentially large parametric uncertainty.
There are numerous strategies for parameter inference, and estimation of paramet-
ric uncertainties, including e.g. least-squares methods and Bayesian methods. If
the chosen (forward) UQ strategy is a probabilistic one, requiring full specifica-
tion of the uncertain parameters as stochastic quantities, then an inference method
that provides full probabilistic characterization of the uncertain parameters, such as
Bayesian inference, is necessary. Whether the resulting specification of uncertain
parameters is in terms of confidence intervals or probability distributions, however,
it is important to note that the solution of the inverse problem generally also iden-
tifies correlations between multiple model parameters. Such correlations, if they
exist, need to be included in the forward UQ problem, for accurate evaluation of
uncertainties in model predictions [81]. In the present context, we will presume that
model parameters are specified probabilistically a priori, whether correlated or not,
and will not discuss the inference problem. Bayesian methodologies have been used
in general UQ studies [39, 49, 50, 72, 73, 88, 89], for model validation under uncer-
tainty [109], and, more specifically, in computations of fluid systems [28].
There are in general a number of UQ strategies that have been employed in the
literature. Before outlining available techniques however, it is useful to point out that
the curse of dimensionality, and the complexity of the forward model, are significant
challenges to all such strategies. Thus, a likely good first strategy before proceed-
ing to quantification of uncertainty is a screening strategy [16] for identifying those
parameters with negligible influence on the model predictions. In this approach, the
forward model is run repeatedly with significant individual changes in each of the
input parameters values, allowing a rough ranking of the parameters in terms of rel-
ative importance. This information can provide guidance as to what parameters may
be safely dropped from any more detailed UQ studies of this model, given their neg-
ligible influence on the model outputs. With the likely set of important parameters
identified, more precise UQ approaches can be subsequently pursued on a relatively
small set of parameters. Of course, care must be taken using these methods in the
context of models with significant non-linearity, and potential bifurcation behavior.
Historically, the most commonly used UQ approach is based on local sensitiv-
ity analysis and variance propagation [13, 16]. In this context, sensitivity analysis
384 Habib N. Najm

is done at the nominal value of model parameters, arriving at sensitivity coefficients

quantifying the partial derivative of each model output with respect to each pa-
rameter. Given this information, and presuming sufficiently small degrees of un-
certainty, the variances of the model output due to each parameter variation over
its respected range are added to arrive at the total variance in model output [129].
This approach provides a reasonable representation of output uncertainties as long
as the uncertainties in the input parmeters and the resulting model outputs are suf-
ficiently small. Moreover, it does not allow for examination of bifurcation of the
solution over the parameter uncertainty ranges. Further, it clearly does not make
use of specific known probability distributions of model parameters, except in using
the first and second moments; and does not provide probabilistic characterization of
model outputs, again except in terms of the first and second moments. Walters and
Huyse [125] discuss various UQ methods, including the propagation of error us-
ing sensitivity derivatives, and higher fidelity approaches such as moment methods,
similarly based on Taylor series expansions about the mean input values, as used in
CFD (see e.g. [105]).
Global sensitivity analysis techniques have also been extensively used in the UQ
literature. These methods employ sampling of the model parameters, given their
specified probability density functions (PDFs), and evaluate the output model pre-
diction for each sample. The statistics of the model outputs provide the probabilistic
characterization of their resulting uncertainty. These sampling-based methods fall
squarely within the probabilistic context. We will discuss sampling methods below
in relation to Polynomial Chaos (PC) UQ methods, and will address alternative sam-
pling strategies. Beside this, however, we will not address other statistical sampling
methods for global sensitivity analysis, and we refer the reader to the extensive lit-
erature in this area adequately reviewed by Saltelli [16]. Walters and Huyse [125]
present an exposition of this approach, using random Monte Carlo (MC) sampling,
with demonstrations in various fluid flow problems.
In the following, we begin with a discussion of the basics of the PC construc-
tion, its use in UQ, and associated challenges due to high dimensionality and non-
linearity. We then discuss the application of PC UQ methods in a range of fluid flow
applications, starting with laminar incompressible flow, and proceeding through
chemically reacting and compressible flow applications. These discussions lead up
to the issue of UQ in turbulence, thereby rounding up the full range of complexity
of the UQ problem in computations of turbulent combustion.

16.1.1 Polynomial Chaos

The key challenge with probabilistic UQ based on random sampling is that typi-
cal random sampling methods, which are largely variants of MC, have very slow
convergence with respect to the number of samples. In other words, large numbers
of samples are required to achieve adequate accuracy in computed moments, e.g.
means and standard deviations, of uncertain quantities of interest. This is an in-
Uncertainty Quantification in Fluid Flow 385

surmountable hurdle when the model is computationally expensive, requiring mas-

sively parallel large-scale computational effort for each sample. Significant progress
has been made in this regard over the past couple of decades through the develop-
ment of spectral methods for representation of random quantities, specifically PC
methods. The underlying formulation for the PC representation of random variables
is presented below.
Let ( , G, P) be a probability space, where is an event space, G is a -algebra
on , and P is a probability measure on ( , G). Any RV X : IR, that has
finite variance, i.e. is in L2 ( ), can be represented as an expansion in terms of any
complete set of functions that are orthogonal with respect to the probability measure
on L2 ( ). Given this, one introduces a set of n coordinates in IRn , being the images
of n standard RVs : IRn with known distributions, to provide a convenient
expansion in IRn . Accordingly, any L2 RV can be represented as an expansion in
terms of a complete set of functions of that are orthogonal with respect to the
density of .
Wiener [130] presented the first use of this construction, employing Hermite
polynomial functionals of standard normal RVs, for representation of Gaussian ran-
dom processes, namely the Wiener process model for Brownian motion, terming
the resulting expansion Polynomial Chaos (PC)* . Ghanem and Spanos [38] em-
ployed this Wiener-Hermite (WH) PC for representation of any L2 random process
X( , x ,t):

X( , x ,t) = Xk (xx,t)k ( ) (16.1)
k=0

where the k are multivariate Hermite polynomials, and = {1 , . . . , n } = ( ) is

a vector of n independent standard normal RVs. The PC expansion (PCE) is equally
applicable to random processes/variables. For use in a computational setting, we
truncate the PCE to finite order, giving
P
X Xkk ( ). (16.2)
k=0

By construction, orthogonality of the Hermite polynomials with respect to the den-

sity of the -basis leads to
 
Xk  1 1
Xk = = X( )k ( ( )dP( ) = X( )k ( )p ( )d
k 
2 k2  k2 
(16.3)
for k = 0, . . . , P, where

* This PC representation of nonlinear functionals of the Wiener process was shown to be con-
vergent in the limit of n [15]. Ogura [95] extended this proof for representation of nonlin-
ear functionals of a Poisson process, employing expansions in terms of Charlier polynomials and
Poisson-distributed RVs. These are the only two sets of orthogonal functions for which the con-
vergence of PC in the infinite dimensional limit has been proven. In the present context we will
handle finite n cases exclusively.
386 Habib N. Najm
n
p ( ) = pi (i ) (16.4)
i=1

and i N(0, 1), i.e. pi (x) = exp(x2 /2)/ 2 . With this expansion, one can eas-
ily construct the PDF of X by generating random samples of , and binning the
resulting X samples. Further, for any f (X), we have
 
 f (X) = f (X( ))dP( ) = f (X( ))p ( )d . (16.5)

Thus, any  f (X), including moments of interest, can be evaluated as long as X( )

is known. Given the PCE for X, the integral can be evaluated analytically for simple
moments. For example, X = X0 , and X2 = Pk=1 Xk2 k2 .
To illustrate the use of PC for UQ, let us consider a model

F(u, x,t; ) = 0 (16.6)

where x IR3 is the spatial coordinate, t is time, IR is a parameter, and u IR is

the model output. The use of a single parameter and one model output is for conve-
nience. The analysis applies equally to models with multiple outputs, and many un-
certain parameters. Consider further that is uncertain, and is modeled as a random
variable. Therefore the model output, u, is also a random variable. We are interested
in evaluation of the distribution of u, its moments, or any  f (u) of interest.
We presume that either the PDF p of is known, or it can be estimated from
available empirical observations of . In the MC context, we rely on some random
sampling strategy to generate samples i from p . For each such sample, we solve
the forward model to evaluate the corresponding ui , from which we have

1 N
 f (u) = f (ui ).
N i=1
(16.7)

Note that MC convergence is insensitive to dimensionality and does not rely on any
smoothness of the integrand. However, this robustness is achieved at the cost of
significant computational expense.
Employing PC, on the other hand, we begin by constructing the PCE for ,
P
= kk ( ) (16.8)
k=0

where is n-dimensional. This PCE can be constructed using the PDF of . Note
that, in general, the PCE constructed from the PDF is not unique, since e.g. both
expansions = 0 1 have the same Gaussian PDF N(0 , 12 ). For n = 1, the
PCE coefficients can be simply found using
 1
1
k = 1 (y) k (1 (y)) dy (16.9)
k2  0
Uncertainty Quantification in Fluid Flow 387

where () and () are the cumulative distribution functions (CDFs) of , and

, respectively [141]. Doing this for n > 1 is typically achieved employing a fit-
ting procedure, inferring the PCE coefficients that result in a PDF for that best
approximates the target PDF. Finally, in the case where a joint PDF is known for
a parameter vector , the Rosenblatt transform [111] can be used to map to a
vector of independent uniformly distributed RVs each on [0, 1], whose PCEs can be
estimated per the above procedure.
Having determined the PCE for , the key objective is then to construct the PCE
for u,
P
u= ukk ( ) (16.10)
k=0

by evaluating the mode strengths given by the Galerkin projection


uk  1
uk = = u( )k ( )p ( )d . (16.11)
k2  k2 

This can be done generally in two ways, as follows.

16.1.1.1 Intrusive PC UQ

The first approach employs Galerkin projection of the governing equations,

     
P P
F uii , x,t; jj k = 0, k = 0, . . . , P (16.12)
i=0 j=0

thereby arriving at a reformulated set of governing equations for the uk . This ap-
proach discards the original deterministic model code, requiring discretization, al-
gorithms, and associated code development for the new set of equations, and has
been consequently termed intrusive.
As a simple example, consider the ODE problem
du
= u, u(t = 0) = u0 (16.13)
dt
where = Pk=0 kk ( ) is an uncertain parameter, with a chosen PCE dimension-
ality and order. Applying the above Galerkin projection for any mode k, leads to
     
d P P P
k ull ( ) = k ii ( ) u jj ( ) (16.14)
dt l=0 i=0 j=0

and since, by construction, ij  = i j i2 , we have

duk P P
ijk 
= i u j , k = 0, . . . , P (16.15)
dt i=0 j=0 k2 
388 Habib N. Najm

where Ci jk = ijk /k2  is a tensor that can be evaluated once and stored. Thus,
we have a new, larger, ODE system to solve for PCE mode strengths {uk }Pk=0 , which
determine the PCE u = Pk=0 ukk ( ). With this in hand, one can generate samples
of u by sampling , to evaluate moments or distributions, or any other purpose, such
as evaluating any  f (u).

16.1.1.2 Non-intrusive PC UQ

Alternatively, one can evaluate the Galerkin projection integrals (Eq. 16.11) numer-
ically. This can be done in a number of ways. To begin with, of course, one can do
this using any variant of MC. There is no inherent computational advantage here,
except that the projection of the statistics of u onto its PC modes immediately pro-
vides, for no additional cost, sensitivity information regarding the dependence of the
uncertain u on any one of multiple model parameters, linked via derivatives on . Of
course, relying on the above smooth global WH PC constructions, where has in-
finite support, this approach does presume a smooth dependence u( ), although the
use of local PC methods, outlined further below, addresses this issue. Using this ap-
proach, with sampled from its PDF, employing N samples of and corresponding
computed realizations of u, the integrals in Eq. 16.11 become

1 1 N i
u k ( ),
i
uk = k = 0, . . . , P (16.16)
k2  N i=1

On the other hand, the key advantage of non-intrusive PC methods is to enable

the use of deterministic sampling of u, based on quadrature formulae, to provide
estimates of the projection integrals. The central problem is the evaluation of
 
 
n
uk  =

...

u( )k ( ) pi (i ) d1 dn (16.17)
i=1

which can be done using n-dimensional Gauss-Hermite quadrature [1, 27, 64]. With
m quadrature points in each dimension, this integration is exact if f (x) has a poly-
nomial degree smaller than 2m. For a PCE order p, the maximum degree of (uk ) in
each i is 2p, such that a quadrature discretization in each dimension with m = p + 1
is sufficient. Thus, the full tensor product Gauss-Hermite quadrature rule for the
n-dimensional integral requires m = (p + 1)n points. This exponential rise in the
number of requisite evaluations of the forward model is quite detrimental in high
dimensional problems, as is further discussed below.

16.1.1.3 Arbitrary basis

As already indicated, the above PC construction, with finite n, can be implemented

using any set of orthogonal functions and their corresponding measure on L2 ( ). In
Uncertainty Quantification in Fluid Flow 389

general, given a chosen random basis of arbitrary distribution, an associated set

of orthogonal functions can be generated, which is then readily usable to construct
an expansion to represent any RV in terms of [33, 120]. This has been demon-
strated, for example, using a range of orthogonal polynomials, members of the
Askey scheme of hypergeometric orthogonal polynomials [3, 115], and their associ-
ated random bases, and termed a generalized PC (gPC) [141, 142]. This set includes,
besides the Hermite-Gaussian pair, Legendre-Uniform (LU), Laguerre-Gamma,
Jacobi-Beta constructions for continuous RVs, and another set of polynomial-basis
pairs for discrete valued RVs. The generation of tailored orthogonal functions has
also been demonstrated in the construction of multiwavelet chaos expansions [61].
One advantage of this flexibility is the ability to choose an optimal basis under
some conditions. In particular, given a RV Z whose distribution pZ () is known, then
an optimal choice of the specific chaos expansion X = Xk k ( ) is that whose
basis has a PDF that is nearest to pZ () [12, 141, 142]. For example, the optimal
representation of a Gaussian RV is in terms of WH PC, being exact employing a 1D
first-order expansion. Similarly, a Uniform RV is best represented using LU PC. Of
course, while the parameter PDFs may be known, allowing this selection, in general
the distribution of an RV that is an output of a general model is not known a priori.

16.1.2 Challenges in PC UQ Methods

There are key challenges for the PC UQ approach associated with requirements
for (a) high dimensionality n, and (b) high order p. These arise from a number of
practical requirements in physical systems, and have motivated significant research
in recent years. We outline these briefly in the following.

16.1.2.1 High dimensionality

The necessary dimensionality of the PC system in the UQ context represents the

number of degrees of freedom that are required to represent the input space, i.e. the
uncertain parameters, initial/boundary conditions, etc. Each independent uncertain
parameter introduces at least one associated degree of freedom, while known de-
pendences among parameters can constrain this number. Input parameters that are
random processes, as opposed to random variables, can require many degrees of
freedom to represent, according on their known correlation structure [38]. At the
same time, the dimensionality is generally a modeling question. The efficient (i.e.
low order for given accuracy) PC representation of the random variables/fields that
are either inputs or components of the system state may well require introduction of
a higher number of independent degrees of freedom. For example, by definition, the
RV Z = nk=1 k2 , where the k s are independent standard normal RVs, has a chi-
square distribution with n degrees of freedom, i.e. Z k2 . Clearly, for any given
Y k2 , a second order WH PCE Y = 2k=0 Zkk (1 , . . . , n ) is exact with the proper
390 Habib N. Najm

choice of coefficients, and is a more efficient model for Y than a PCE with n < k, for
which higher order terms (> 2) would be required to approximate the exact model
with high accuracy.
In general, the resulting dimensionality n of the PC system can be quite large.
The limit of large n can lead to significant computational expense. To begin with, P,
the overall size of the PCE increases with n, following

(n + p)!
P+1 = (16.18)
n!p!

where p is the order. Thus, a 4th order PCE with n = 4 requires P = 69, while n = 8,
requires P = 494. In the intrusive context, the size of the Galerkin-projected equa-
tion system increases quickly with n. At the same time, in the non-intrusive con-
text, dense full-tensor product quadrature formulae, e.g. employing Gauss-Hermite
quadrature as outlined above, require a number of evaluations of the forward model
that goes up exponentially with n, specifially m = (p + 1)n . With p = 4, this re-
quires m = 625 for n = 4, and m = 390, 625 for n = 8. This can quickly become
prohibitively expensive, more so than MC.
Recent developments have targeted these challenges. Reducing the number of
terms in the expansion has been addressed using PCE constructions based on sparse
tensor products [11, 30, 65, 116, 117, 123], and employing arbitrary adapted prob-
ability measures [120]. At the same time, reducing the number of forward model
evaluations, in the non-intrusive/collocation context, has been addressed employ-
ing Smolyak [98100, 119] or other sparse-quadrature/cubature integration for-
mulae in general [9, 32, 34, 35, 77, 8587, 140]. There has been significant
progress in the application of these methods in UQ, such that integration of sys-
tems with substantial dimensionality is feasible, and more efficient than MC sam-
pling [83, 84].

16.1.2.2 High order

The requisite PC order for representing a given RV corresponds to the highest neces-
sary degree in the k ( ) polynomials required to capture the inherent u( ) depen-
dence, with sufficiently small truncation error in high-order terms. This requisite
order can be high, even with mild uncertainty in the input space, depending on the
non-linearity of the forward model. Two important classes of problems exhibit this
challenge, with particular relevance in fluid systems. These are systems exhibiting
(1) long time horizon dynamics with large growth rate of phase-errors, i.e. strong
rate of loss of correlation, or (2) sharp/discontinuous dependence on uncertain pa-
rameter values. These are considered in some detail below.
Uncertainty Quantification in Fluid Flow 391

Long Time Horizon

A key characteristic of non-linear dynamics is the large rate of growth of small

differences in initial conditions. This has direct consequences to the UQ problem.
Small parametric/input uncertainties can translate to large and fast-growing phase-
variances in the solution. In the PC context, representing the resulting uncertain
system state as a function of the basis used to characterize the input space, and
maintaining a specific accuracy in the representation, requires progressively higher
PC order as the time horizon of interest is increased. Pettit and Beran [101] provided
a concise illustration of this challenge employing a sinusoidal function with an un-
certain frequency, where the resulting oscillatory frequency of the system state as a
function of is observed to increase with the observation horizon.
This is a serious challenge for the application of PC methods in fluid systems
exhibiting non-linear oscillatory dynamics or convective dynamics of vortex struc-
tures. This challenge applies to both intrusive and non-intrusive implementations
as long as the system state observable of interest has a high degree of dependence
on initial conditions, therefore exhibiting a large rate of loss of correlation, and
growth of phase variances, in time. On the other hand, one easy resolution of this
challenge is available conveniently in the non-intrusive context, by focusing on ob-
servables that are well-behaved with respect to initial conditions, even though the
detailed system state is not. For example, in the context of DNS of turbulent flow,
the instantaneous detailed state of the system, comprised of the velocity vector as
a function of space and time, is highly sensitive to small changes in initial con-
ditions. Representing this uncertain field over long time horizons, accounting for
parametric/input uncertainties using PC, is a losing proposition, irrespective of the
means employed for evaluating the associated PCE. On the other hand, the depen-
dence of averaged quantities, such as mean/RMS profiles or energy spectra, is much
smoother with respect to initial conditions or model parameters. Of course, without
employing some means of turbulence modeling, there are no a priori equations for
these smooth observables, so their use in the intrusive DNS context is not possible.
However, they are usable in the non-intrusive context, where they, as opposed to the
detailed system state, are expressed as PCEs. The associated dependence on is
smooth, and the requisite order is small. This is a clear argument for non-intrusive
methods in this context. This approach has been employed for development of ef-
fective collocation-based long time-horizon PC and local Lagrange-interpolant UQ
strategies for a range of transient oscillatory systems, including structural and/or
flow-structure dynamics, employing smooth observables that parametrize the tran-
sient oscillatory system response [133139].

Discontinuities

Another common feature of non-linear systems, frequently observed in fluid sys-

tems, is the existence of high-rates of change of the solution in space/time, dis-
continuities, and bifurcations over parametric space. Examples include shocks in
392 Habib N. Najm

compressible flow, ignition/flame-fronts in reacting flow, as well as flow instability

and transitions between laminar/turbulent flow regimes. Discontinuity of the sys-
tem state as a function of the input/parametric space, and therefore as a function of
, is a serious challenge for the above introduced PC construction. Sharper depen-
dence on , in principle requires higher PC order to maintain a given accuracy in
the representation. However, as a discontinuous structure is approached, the above
global spectral PC constructions, where the -basis has infinite support, will fail
to maintain accuracy for any finite order. Obviously, a polynomial representation in
terms of a smooth basis is not useful for representing a step-function, leading to the
familiar Gibbs phenomenon. This challenge led to the development of local mul-
tiresolution analysis (MRA) PC methods employing multiwavelet bases with com-
pact support [58, 61]. In this context, the input space is subdivided into a number
of blocks/elements. Distinct PC representations are constructed over each element
using local bases with compact support, and propagated through the forward model
following conventional PC UQ procedures. Statistics and moments of the system
state can be computed by combining associated elemental statistics/moments. In
the simplest version of MRA, a PC construction comprised of Legendre polynomi-
als along with first-level 1D wavelet details, both functionals of a local uniformly-
distributed RV basis, are used on each element. The Legendre subset of the con-
struction is the Legendre-Uniform gPC with a local basis, while the directional
wavelet details provide useful information on the need for refinement of the ele-
ment mesh in each stochastic dimension. The Multi-Element gPC (ME-gPC) con-
struction employs the same above local LU PC representation, without the wavelet
details [128]. Moreover, a range of stochastic finite element constructions have been
developed employing local interpolants, as opposed to local PCEs, in numerous
applications [2, 57, 25, 32, 71, 82, 133, 137]. In general, local constructions are
useful and necessary in both the intrusive and non-intrusive contexts, where dis-
continuous/bifurcation behavior is present. Note, in particular, that even smooth ob-
servables can exhibit bifurcative behavior as a function of model parameters, e.g.
due to flow instability and change of overall flow character at a critical parameter
value.

16.2 Polynomial Chaos UQ in Fluid Flow Applications

There have been many demonstrations of the application of PC UQ methods

in a range of fluid flow applications, the earliest involving flow in porous me-
dia [36, 37]. The discussion below addresses demonstrations in general incom-
pressible, compressible, and reacting flow; with a last section devoted to turbulent
flow.
Uncertainty Quantification in Fluid Flow 393

16.2.1 Incompressible Flow

The incompressible Navier-Stokes equations have been solved in a number of flow

contexts, using a range of numerical methods, employing PC UQ. We discuss
these in the following, considering the general classes of Eulerian and Lagrangian
schemes.

16.2.1.1 Eulerian methods

The intrusive PC reformulation of the incompressible Navier-Stokes equations in an

Eulerian framework, employing Galerkin projection, leads to a set of coupled partial
differential equations for the mode-strengths of the velocity field PCE, along with
associated continuity constraints [60]. These are, for k = 0, . . . , P,

uk P P P P
+ (ui )u jCi jk =
pk + i 2 u jCi jk (16.19)
t i=0 j=0 i=0 j=0
uk = 0, (16.20)

where uk , pk , and i are the PC modes for the velocity, pressure, and viscosity, re-
spectively; and Ci jk has been already introduced. Using a projection scheme [20, 51]
to solve these equations leads to a decoupled set of elliptic solves for the pressure
field. Specifically, rewriting Eq. 16.19 as

uk
p k + Hk ,
= (16.21)
t
a projection scheme employing, e.g. a multi-step time integration strategy proceeds
with a first fractional step at time t n , giving

uk unk
= F(Hkn , Hkn1 , . . .), k = 0, . . . , P (16.22)
t
where uk are the provisional velocity modes. In the second fractional step, a pressure
correction is applied to the provisional velocity in order to satisfy the divergence
constraints, arriving at the velocity at time t n+1 ,

un+1 uk
k pk ,
= k = 0, . . . , P (16.23)
t
where the pressure fields pk are determined so that the fields un+1
k satisfy the diver-
gence constraints in (16.20), i.e.

un+1
k = 0, k = 0, . . . , P (16.24)
394 Habib N. Najm

Combining equations (16.23) and (16.24) results in the following system of decou-
pled Poisson equations:
1
2 pk = uk k = 0, . . . , P. (16.25)
t
Each of the above independent Poisson equations is solved subject to Neumann
boundary conditions, as in the conventional projection scheme. Presuming N mesh
points over the physical domain, the solution of the decoupled pressure equations,
requiring (P + 1) Poisson equation solutions, each involving an N N matrix, is
computationally efficient compared with the alternative, namely the solution of a
fully coupled (P + 1)N (P + 1)N system.
This construction, based on a global WH PC implementation and a finite differ-
ence discretization has been used to solve several variants of the laminar channel
flow, including flows with uniform and temperature-dependent viscosity, and uncer-
tain temperature boundary conditions [60]. It has also been used for computations of
stable natural convective flow in a differentially heated cavity with specified uncer-
tain wall temperature [63, 64]. When handling large temperature differences, in the
non-Boussinesq limit, particular attention is necessary regarding mass conservation
for ensuring numerical stability, requiring the use of a highly accurate stochastic
inversion algorithm.
Application of these techniques in Rayleigh-Benard flow, accounting for natural
convection instabilities, requires adequate handling of bifurcations [58]. The stabil-
ity of this flow is governed by the value of the Rayleigh number Ra, with a critical
value Ra delineating the boundary between a stable regime with conductive heat
transfer and an unstable regime with predominantly convective heat transfer. Thus,
considering natural convection in a 2D cavity, when the specified uncertain bottom
wall temperature results in an uncertain Ra spanning a range that straddles Ra , the
spectral expansion has to represent a step-function dependence on the stochastic
dimesion , clearly infeasible for global polynomial expansions in terms of smooth
functions. Studies of Rayleigh-Benard flow have shown that the bifurcation at Ra
challenges the utility of global LU PC, necessitating the use of of a local Wavelet-
Uniform PC construction instead [58]. The local construction deals with this bifur-
cation effectively, exhibiting robust physical behavior.
The PC representation fits quite naturally in the context of finite or spectral el-
ement flow solution methods, and has been applied in this context for the study
of a range of uncertain incompressible flows. Spectral element implementations
have been applied, along with a high-order projection scheme, using global Wiener-
Hermite PC, for studying flow-structure interactions in flow around a circular cylin-
der at low Reynolds number [143]. Global gPC implementations have also been used
in this context, examining incompressible 2D channel and cylinder flows [142], and
for studies of spatially developing shear layers [53]. Further, this construction has
been used for computations of flow around fixed cylinders with multiple vibrational
degrees of freedom, as well as oscillating cylinders where shedding mode switching
resulting from noisy inflow was evident [67, 69, 70].
Uncertainty Quantification in Fluid Flow 395

In the finite element context, global gPC UQ has been implemented with a
variational multiscale stabilized finite element method for solving the incompress-
ible Navier-Stokes equations in a number of flow configurations including channel,
driven cavity, and cylinder flows [82]. The convergence rate of the GMRES solver
used in this construction is reduced significantly in going from the deterministic
problem to the stochastic one, suggesting the need for improved preconditioning
strategies for the Galerkin-projected system of equations. This methodology has
also been applied in studies of natural convection in a differentially heated cav-
ity, where bifurcative behavior also necessitates use of local interpolant-polynomial
constructions [4]. Non-intrusive adaptive sparse-grid collocation UQ approaches,
employing local interpolants over stochastic space, have also been used in this natu-
ral convection context in the Boussinesq limit, considering differentially heated cav-
ities with uncertain rough wall topology [32]. Surface roughness leads to significant
enhancement of the flow and heat transfer in this system, and alteration of Rayleigh-
Benard convective dynamics. Different realizations of the random configuration of
the rough-wall topology result in switching between two stable flow-patterns, and
associated bimodal PDFs for the uncertain flow variables. Wall roughness and geo-
metric uncertainty can also be handled using domain transformations, allowing flow
solutions on a regular mesh [122, 144].
As indicated above, particular attention to accuracy needs to be considered in
the context of UQ computations of oscillatory flow dynamics and unsteady vortical
motions over long time horizons, due to the growth of phase errors in time. These
issues are relevant in shear layer and bluff-body/recirculation wake flows, the latter
including e.g. both cylinder flow and flow over an open cavity. Considerable ampli-
fication of the inflow uncertainty occurs in cylinder flows, highlighting the need for
ensuring good accuracy in the high-order PC modes [82]. Analysis of both global
and local gPC methods in spectral element solutions of incompressible flow over
an open cavity at high Reynolds number indicate better efficiency with the local
ME-gPC methods [127]. Studies of gPC in the context of a 1D oscillatory advection
model illustrate that, in order to maintain a fixed accuracy, the requisite PC order
of the global method must be increased linearly in time. On the other hand, the
time horizon corresponding to a given error threshold and PC order is extended by
a factor of N when a uniform mesh of N stochastic elements is used in the ME-gPC
context [126]. Studies of vortex shedding around a circular cylinder indicate that,
while gPC fails to converge at early integration times, ME-gPC captures the flow
accurately over a significantly longer time horizon. The use of local methods, how-
ever, does not fully resolve the challenges with representing vortex dynamics over
long time horizons, particularly for high-dimensional random inputs, as the increase
in the number of elements can be prohibitive.

16.2.1.2 Lagrangian vortex methods

Lagrangian formulations of the incompressible Navier-Stokes equations have also

been extended to account for uncertainty employing PC UQ methods. From an al-
396 Habib N. Najm

gorithmic point of view, the simplest construction in this regard models both par-
ticle locations and strengths as uncertain. Implementations along these lines have
been demonstrated in flow around airfoils with uncertainty in the inflow velocity,
where Lagrangian particles are used to represent the vorticity field. Both intrusive
and non-intrusive PC constructions have been used in this context [57, 102]. How-
ever, the approach of representing particle locations as random variables is fraught
with difficulty in general, due to the growth of phase errors and loss of correlation
for the non-linear N-body problem. Alternate developments to deal with this issue
have led to constructions employing deterministic particle locations with uncertainty
lumped in particle strengths. This construction has been developed employing PC
UQ with a stochastic hybrid particle-mesh vortex method for incompressible flow
natural convection computations in the Boussinesq limit [59]. Vorticity diffusion is
implemented using a particle strength exchange scheme with remeshing; an optimal
choice given the uncertainty formulation. A mesh is introduced for the evaluation
of the velocity field for a given vorticity distribution, as a means of accelerating the
velocity computations, and as an alternative to Biot-Savart N-body computations.
Particles are displaced with the mean velocity field, while their uncertain strengths
are computed employing PC algebra due to both diffusion and convection terms.

16.2.2 Reacting Flow

Computations of reacting flow provide a strong case for the need for uncertainty
quantification, primarily because of the proliferation of empirically-determined re-
action rate and thermodynamic parameters in chemical kinetic models. A detailed
chemical model for the oxidation of simple hydrocarbons, e.g. CH4 , involves lit-
erally hundreds of empirical parameters with significant associated uncertainties.
Models of complex fuels involve O(104 ) uncertain parameters. Moreover, in gen-
eral, there is the equally important uncertainty in the structure of the chemical reac-
tion network itself.
One significant issue that has to be stressed in this context is the need for proper
characterization of the input space. This is important in general of course, but it
gains even added significance here because of the large number of uncertain inputs
and their internal correlations. The large impact of the dependence relationship be-
tween the parameters on the resulting uncertainty in chemical model predictions has
been illustrated in a simple ignition system with two uncertain parameters [80]. The
fact that such parametric correlations, and in general the joint probabilistic distribu-
tion of uncertain chemical model parameters, is largely unavailable in the literature
is a significant challenge for non-perturbative stochastic UQ studies in reacting flow.
These distributions can be established based on experimental data, however raw data
is largely unavailable. Recent efforts are making headway in terms of warehousing
and accessibility of data on uncertainties and correlations in parameters of thermo-
chemical models [31, 112].
Uncertainty Quantification in Fluid Flow 397

Laying aside the unavailability of proper probabilistic characterization of the in-

put space of reacting flow models, two other aspects of these systems present dif-
ficulties associated with the above-discussed challenges to PC UQ. Chemical mod-
els exhibit (1) high-dimensionality associated with many uncertain parameters; and
(2) strong nonlinearity associated with the rates of chemical reactions, particularly
those with high activation energy. The dimensionality challenge highlights the need
for adaptive sparse tensor product constructions for representation of random vari-
ables, as discussed above, while also recalling the necessity of identifying the set
of important uncertain parameters by sensitivity analysis or other means, before
addressing the propagation of uncertainty. This should be done with care however,
given the second challenge of strong nonlinearity and the associated potential of
non-gaussian and multimodal distributions of uncertain solutions.
Strong nonlinearity is a hallmark of exothermic reaction systems and of para-
mount importance in models for oxidation of hydrocarbon fuels. Nonlinearity leads
to a strong dependence on, and strong sensitivity to, small changes in initial con-
ditions. It also leads to large rates of change of the solution over space/time, and
similarly to bifurcations over parametric space. Arrhenius reaction rate expressions,
and associated uncertain parameters, present a clear example of the inherent UQ
challenge in these systems. Ignition computations with a single-step irreversible
global reaction model for methane-air combustion, with small uncertainties in the
Arrhenius rate expression pre-exponential constant and activation energy, highlight
the strong dependence of ignition time on these parameters, the high amplification
of input uncertainties, and the observed bimodal structure of the probability distri-
bution of the temperature or other state variables [80]. These complexities clearly
indicate the inadequacy of perturbative linear methods for UQ in these systems, and
the need for full probabilistic accounting for uncertainty.
Probabilistic PC UQ was first implemented in chemical systems, using a col-
location approach, in a zero-dimensional isothermal context, focusing on super-
critical water oxidation (SCWO) chemistry [104]. This particular collocation ap-
proach, where (P + 1) linear algebraic equations are solved for the (P + 1) PCE
mode strengths, while ensuring accuracy of the PC representation at the colloca-
tion points, says nothing about its accuracy elsewhere in the parametric space. This
same chemical model has also been used in PC UQ studies of ignition and 1D pre-
mixed flames, employing non-intrusive spectral projection where MC Latin Hyper-
cube Sampling (LHS) was used for evaluation of the projection integrals for the
PC mode coefficients, allowing for 13 uncertain parameters [107]. The 1D flame
structure exhibits high amplification of uncertainty due to the strong non-linearity
of the model. As a result, particular attention has to be paid to convergence with PC
order, and the number of LHS samples, especially when considering higher-order
statistics [108]. By the same token, global intrusive PC UQ methods are severely
challenged with time-stability concerns in these systems, even with implicit stiff
ODE integrators [106] suggesting the development of positive eigenvalues due to
poor spectral resolution.
These challenges motivated the development of MRA constructions [58, 61]. In-
trusive MRA PC UQ with adaptive multi-block decomposition of parametric space
398 Habib N. Najm

has been demonstrated in isothermal SCWO chemical ignition [62]. More recently
exothermal ignition employing this intrusive multiwavelet construction has also
been demonstrated [80].
Isothermal reacting flow in microchannels, with electroosmotic pumping, has
also been studied using intrusive WH PC [26]. This construction couples the Navier-
Stokes equations with slip velocity due to the wall double-layer zeta potential,
species conservation equations including the electrokinetic body force, and elec-
trostatics. Both fast equilibrated electrolytic reactions and slow finite-rate reactions
are used in the chemical model.

16.2.3 Compressible Flow

Compressible flow equations, being hyperbolic, present unique challenges to un-

certainty quantification efforts. These include wave propagation phenomena, and
associated characterstics, which dictate specific numerics, both in the bulk and at
domain boundaries. They also include shocks, which, being discontinuities, also
demand specialized numerics.
Shock discontinuities lead to significant difficulties in the application of intru-
sive global PC UQ methods. Studies in model 1D nozzle flow have been successful
in the absence of strong discontinuities [76, 125]. The consequences of shocks are
significant, however, to the utility of the global PC construction. The discontinu-
ity in the primitive variables over physical space translates to a discontinuity of the
PC modes dependence over stochastic space [19]. While numerical stability of the
global construction is possible, employing suitable filters and collocation schemes,
convergence can be slow, as would be expected for the representation of discontinu-
ities using global polynomial expansions. Alternatively, local PC constructions em-
ploying non-intrusive stochastic collocation have been used for shocks in quasi-1D
nozzle flow [74, 75]. Shocks in a 1D piston problem have also been studied em-
ploying a collocation UQ approach relying on local polynomial interpolants, adap-
tive decomposition of stochastic space into simplex elements, and Newton-Cotes
quadrature [137].
In a 2D flow context, supersonic flow over a wedge, exhibiting an oblique shock,
and that over an expansion corner, exhibiting a Prandtl-Meyer expansion wave, are
canonical compressible flow problems. These problems have been studied using
intrusive PC UQ, with uncertainty in the wedge/expansion angle, employing an im-
plicit 2D finite volume solver for the Euler equations [97]. The supersonic wedge
problem has also been studied using intrusive gPC/ME-gPC constructions employ-
ing a WENO scheme in physical space [66]. In this construction, the signs of the
(uncertain) eigenvalues associated with characteristics are required to choose the
proper spatial discretization stencils. The increased size of the hyperbolic govern-
ing equation system leads to an increased computational burden associated with
the eigenproblem solution, which necessitates attention to optimal numerical strate-
gies. Improved performance with the local ME-gPC construction has been reported
Uncertainty Quantification in Fluid Flow 399

in this context, which is consistent with better resolution of the discontinuity in

stochastic space, and the improved performance of local PC methods in bifurcative
flows in general. Both the supersonic wedge and expansion corner problems have
also been studied using non-intrusive collocation PC methods [45]. Deterministic
solutions were computed using NASAs CFL3D Euler code [56], presuming uncer-
tainty in the wedge/corner angle. As might be expected, the wedge flow was found
to be more challenging than the expansion corner problem, in terms of both PDF
skewness and collocation error behavior.
Three-dimensional compressible flow around a transonic wing has been stud-
ied employing collocation PC UQ methods [44]. The compressible Euler equations
were used, coupled with aeroelastic analysis of the wing structure, allowing for
uncertainty in the free-stream Mach number and the angle of attack. Further, 2D
and 3D transonic wing computations have been demonstrated, with uncertain free
stream velocity, using adaptive collocation employing local polynomial interpolants
on simplex elements [135, 137]. Coupled fluid-structure oscillations were studied in
this context, using an Arbitrary Lagrangian-Eulerian formulation.

16.2.4 Turbulence

The application of polynomial chaos expansions in turbulent flow has a long history,
although not in the UQ context. Wiener-Hermite PC expansions have been consid-
ered as potential means for representing turbulent motions. Initially, Wiener [131]
had proposed the use of his chaos expansions for analyzing turbulence. This pro-
posal did not get much further attention however till after it was further elaborated
in [132]. The following decade saw significant work in this area. However, despite
some encouraging signs, the idea eventually proved ineffectual. The central prob-
lem is that, while the infinite expansions can indeed be used to represent turbu-
lent motions starting from some initial state, any truncated/finite expansion fails,
except in a limited time window. Orszag and Bissonnette [96] pointed out early-
on deficiencies of this approach in the context of the turbulent Burgers equation,
and using Kraichnans simple three-body problem [55]. In the latter case, it was
found that non-constant time-correlation was the Achilles heel of this approach,
even for stationary Gaussian processes, as non-linearities propagate energy into
higher-order terms. Crow and Canavan [24] further highlighted the failure of the
Wiener-Hermite expansions in representing the energy cascade of trubulent mo-
tions. Moreover, Canavan [17] showed that using the convected Gaussian basis, an
idea originally proposed by Wiener, did not resolve the problem. Chorin [21] further
analyzed the failure of these expansions, highlighting the significance of the high-
order terms given the turbulent energy cascade, and suggesting that the construction
may find some use only in the description of large-scale motions.
With the recent utilization of PC methods for UQ, this topic has regained further
relevance prompting further inquiry. These issues have been examined via analysis
of the 1D Burgers and 2D Navier-Stokes equations with a Brownian motion forc-
400 Habib N. Najm

ing term [46], employing Karhunen-Lo`eve and PC representations with adaptive

sparse tensor-product constructions [30, 65]. Despite significant demonstrations of
interesting short-term dynamics, however, long-term dynamics remain a challenge.
The necessary growth in the number of terms in the expansion, required to maintain
a given accuracy threshold, frustrates the utility of the construction for long time
horizons.
Limiting the scope to large-scale oscillatory dynamics, significant progress has
been made, even though the time horizon challenge generally requires special treat-
ment in the presence of non-linearities. Computations of vortical shedding behind
circular cylinders accounting for parametric uncertainty have met with success over
short time horizons [69, 82, 126, 143]. Further, vortex dynamics have been studied,
using non-intrusive gPC UQ employing Gauss-Legendre quadrature, in a laminar
2D shear layer with uncertain inflow forcing modes [53]. The shear layer flow struc-
ture reveals distinct regions where the influence of the fundamental or subharmonic
uncertain forcing modes dominates, largely coinciding with zones with high prob-
ability of vortex interactions. Further, localized regions of high uncertainty are evi-
dent, highlighting local flow features that are most affected by the uncertain inputs.
As is typical of shear/recriculating flows, the most probable solution is more mean-
ingful than the mean flow in this system, given that multiple dominant solutions
have significant probability. Large-scale dynamics have also been studied using PC
UQ methods in the context of airfoil limit-cycle oscillations [10, 78, 79, 101, 103]
where local PC methods are found to adequately handle the Hopf bifurcation as-
sociated with the onset of flutter. In general, however, the accurate computation of
the uncertain time evolution of oscillatory flow over arbitrarily long time horizons
remains a challenge.
One means by which this challenge has been addressed, has been to use: (1)
a non-intrusive stochastic collocation approach, and (2) a parameterization of the
oscillatory response in terms of well-behaved observables, having smooth depen-
dence on the uncertain inputs [133139]. This approach has been developed and
demonstrated on systems of increasing complexity, with recent demonstrations in-
volving transient multifrequency aeroelastic response of 2D/3D structures in fluid
flow [135]. It employs simplex elements with adaptive refinement in stochastic
space and multi-frequency parameterization of oscillatory response functions. The
resulting observables, being amplitudes, phases, and frequencies of sinusoidal func-
tions capturing the transient oscillatory response, are themselves quite smooth and
easily represented in terms of low order polynomials.
In the same spirit, focusing on smooth observables, non-intrusive gPC has been
used effectively for investigation of the uncertainty in LES computations of statis-
tical moments of decaying homogeneous isotropic turbulence, resulting from para-
metric uncertainty in the underlying Smagorinsky subgrid model constant C [68].
This uncertainty is found to have the largest impact on the smallest resolved scales
in the flow. Of course the determination of the correct range and PDF of C is a key
issue. This PDF can be inferred employing Bayesian methods based on experimental
measurements, and/or DNS computations, of target smooth observables. Ultimately
the utilization of robust models, guaranteeing a consistent level of accuracy over a
Uncertainty Quantification in Fluid Flow 401

range of conditions is of interest [114]. The validation of LES or RANS-LES com-

putations, employing a range of useful observables, is increasingly feasible given
computational resources. However, there has been little such validation taking into
account uncertainties resulting from incomplete information in complex flow sys-
tems, the typically different time spans of experimental and computational results,
and experimental measurement noise [113].

16.3 Closure

This chapter has addressed the topic of uncertainty quantification in fluid flow com-
putations, focusing on the fundamentals and application of spectral Polynomial
Chaos UQ methods. The availability of these methods, along with the continuing
increase in computational capabilities, has led to many successful demonstrations of
uncertainty quantification in a range of fluid flow computations. Ongoing algorith-
mic developments provide hope in the continued expansion of the range of difficulty
of uncertain fluid problems that can be efficiently handled. Above all, challenges
with dimensionality and nonlinearity demand ongoing attention, as do challenges
with the accurate and self-consistent description of the uncertain input space.

Acknowledgement

This work was supported by the US Department of Energy (DOE), Office of Ba-
sic Energy Sciences (BES) Division of Chemical Sciences, Geosciences, and Bio-
sciences. Sandia National Laboratories is a multiprogram laboratory operated by
Sandia Corporation, a Lockheed Martin Company, for the United States Department
of Energy under contract DE-AC04-94-AL85000.

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Chapter 17
Computational Frameworks for Advanced
Combustion Simulations

J. Ray, R. Armstrong, C. Safta, B. J. Debusschere, B. A. Allan and H. N. Najm

Abstract Computational frameworks can significantly assist in the construction, ex-

tension and maintenance of simulation codes. As the nature of problems addressed
by computational means has grown in complexity, such frameworks have evolved to
incorporate a commensurate degree of sophistication, both in terms of the numeri-
cal algorithms that they accommodate as well as the software architectural discipline
they impose on their users. In this chapter, we discuss a component framework, the
Common Component Architecture (CCA), for developing scientific software, and
describe how it has been used to develop a toolkit for simulating reacting flows. In
particular, we will discuss why a component architecture was chosen and the philos-
ophy behind the particular software design. Using statistics drawn from the toolkit,
we will analyze the code structure and investigate to what degree the aims of the
software design were actually realized. We will explore how CCA was employed to
design a high-order simulation code on block-structured adaptive meshes, as well
as a simulation capacity for adaptive stiffness reduction in detailed chemical mod-
els. We conclude the chapter with two reacting flow studies performed using the
above-mentioned computational capabilities.

17.1 Introduction

Computational science has come to be regarded as the third leg of science, after
theory and experimentation. With the advent of massively parallel computers, sim-
ulations have been used to investigate extremely challenging problems. However,
as the problems have increased in complexity, the tools used to investigate them
computationally numerical algorithms and their software implementations have
developed a commensurate sophistication and intricacy. Software complexity, with

J. Ray, R. Armstrong, C. Safta, B. J. Debusschere, B. A. Allan and H. N. Najm

Sandia National Laboratories, Livermore, CA, e-mail: [jairay,rob,csafta,bjdebus,
baallan,hnnajm]@sandia.gov

T. Echekki, E. Mastorakos (eds.), Turbulent Combustion Modeling, 409

Fluid Mechanics and Its Applications 95, DOI 10.1007/978-94-007-0412-1 17,
Springer Science+Business Media B.V. 2011
410 J. Ray et al.

its detrimental impact on software maintainability and extensibility, is regarded as

a large drain on time and effort and computational frameworks are intended to be a
solution to this problem. One means of addressing complexity is to provide shrink-
wrapped functionality, in essence transferring software and algorithmic complexity
to the implementer of the framework from its user. A second way of addressing com-
plexity is through some form of modularization. Component frameworks fall in the
second category.
The word framework is an extremely overused and consequently, confusing
term. It can refer to an architecture, a set of specifications that, when followed, im-
poses some standard on the software that adhere to it. By this token, a component
framework is a set of specifications that permit software to be constructed by mod-
ular composition. The modules are referred to as components. Confusingly, frame-
work can also refer to a software framework that is written to such a specification
and is meant to locate, instantiate and compose components. The aim of component
frameworks is to use modularity to divide and conquer complexity, by composing
programs out of software building blocks. Studies have shown that the capability
and complexity of a program is proportional to its size, which determines the num-
ber of software developers required to maintain the software [36, 63]. Component
frameworks can be used to design an organizing principle for the efficient deploy-
ment of programming resources. Individual programmers have cognizance over one
or more components of the overall program, which provides a natural division of
responsibilities. It makes obvious that the interfaces between components are where
programmers must negotiate. Interfaces exported to, or imported from components
is functionality that is needed by, or provided to a programmer as part of his/her
domain of responsibility. The structure provided by component-based software en-
gineering is one of the main reasons for using it and defines the workflow in a large
software project.
In this chapter, we will limit the discussion to the Common Component Architec-
ture (CCA), a set of specifications for developing and composing component-based
scientific simulation software. We will devote some time to the CCA frameworks
used in scientific computings and provide examples of how CCA has been used for
software for the simulation of flames.

17.2 Literature Review of Computational Frameworks

Scientific computational frameworks proliferate. Some were initially designed to as-

sist in the creation of simulation codes in a certain scientific field and consequently
have been adopted/adapted rather sporadically elsewhere. Others were intended to
assist in nothing more than the implementation of a particular software design, ir-
respective of the nature/aim of the software being developed. It is impossible to
review the myriads of scientific simulation frameworks that lie between these two
extremes and this will not be attempted here. Since this chapter describes the design
of a component-based simulation facility for reacting flows on adaptively refined
Computational Frameworks for Advanced Combustion Simulations 411

meshes, we will devote the majority of our review to frameworks that assist in com-
putations on block-structured adaptively refined meshes, and implementations of
CCA that are relevant to scientific computing. We will, however, describe a basic
categorization before proceeding with the details.
Scientific computing frameworks are application development frameworks which,
at the very least, support parallel computing and pay particular attention to high per-
formance in their design. This manifests itself in their ability to address operations
on large arrays and the elimination/reduction of overheads that scale with the size
of the data or with the number of processors on which the framework is expected to
run. Further, they rarely contain any support for languages outside FORTRAN and
C/C++, and make sparing use of remote method invocation. Within these confines,
scientific simulation frameworks can be divided into three categories.
In the first category are frameworks that were originally designed to support a
particular field of science. The Earth System Modeling Framework [30, 32] is one
such example, developed for constructing climate simulation codes; Cactus [15, 37]
is another, developed for numerical relativity studies while FLASH [31, 35], de-
veloped for astrophysical simulation, forms a third example. OpenFoam, developed
initially for finite-volume simulation of fluid flow (but now containing extensions
for solid mechanics and Direct Simulation Monte Carlo calculations as well) [80]
is in widespread use. Such application-specific frameworks often contain numerical
schemes and utility scientific models that find routine use. They are modular in
design and implement many of the ideas (e.g., involving a separation of implemen-
tation from the interface of a module) that are formalized in CCA. Note that these
frameworks do not intend to impose any design patterns on the resulting simulation
code; rather one adopts the data structures and design patterns of the framework
itself and reaps the benefits of using validated solvers and models in ones simu-
lations. These frameworks allow one to rapidly develop simulation codes as long
as the facilities provided by the framework (e.g. solvers and models) are profitably
leveraged by the simulation.
The second category of frameworks consists of those that enable the use of a
particular solution methodology e.g.. block-structured adaptive mesh refinement
(AMR) in simulation codes. These frameworks are more general than those de-
scribed in the previous paragraph they provide data-structures and solvers (or in-
terfaces to those implemented by external libraries) that are useful while developing
simulation software. Again, they make no attempt to promote any particular design
pattern. However, given that one makes heavy use of the framework-provided data
structures (which are usually implemented in the form of objects), the design pat-
terns employed for simulation codes bear a strong resemblance to those employed
for the framework itself. We will review a few such frameworks/packages that en-
able the use of AMR in simulation codes.
Simulations using block-structured adaptively refined meshes are generally con-
ducted where the spatial domain is rectangular/cuboid or can be logically described
as such. One begins with a Cartesian mesh with a resolution that is insufficient to
capture many of the features of the solution. Regions in the domain that require
refinement are identified and collated into rectangles/cuboids and a finer mesh (usu-
412 J. Ray et al.

ally, finer by a constant factor) is overlaid on the refined part of the initial mesh.
This is performed recursively leading to a mesh hierarchy of a few (usually less
than 10) levels. GrACE [5, 17, 25, 38, 67, 69, 70] and AmrLib [4] are a few frame-
works that implement such an adaptive meshing strategy. These frameworks provide
the infrastructure required to manipulate such meshes as well as data structures that
allow fields to be described on them (on distributed memory computers, this in-
volves very intricate book-keeping). The data structures also allow the use of time-
refinement [13], a time-integration technique that allows one to mitigate the effect
of having time-steps be CFL-constrained by the finest mesh in a mesh-hierarchy.
While CHOMBO stores the data on a given level of the mesh hierarchy in a sepa-
rate data-structure, both AMROC and GrACE present a data object that is described
on the entire hierarchy. The three frameworks also largely automate the work of
refining/coarsening the mesh hierarchy periodically (called regridding), based on a
user-calculated error metric and perform load-balancing of the mesh and data after
each regridding operation. While GrACE only provides a parallel block-structured
adaptive mesh and the associated data object, CHOMBO and AMROC also provide
a set of commonly used solvers.
All the three frameworks have been used for simulating reacting flows. AMROC
has its origins in the simulation of shock-laden flows [26, 66], but has been ex-
tended to shock-fluid interactions [18, 28, 29] and shock-induced combustion [27].
CHOMBO (and its predecessors) have been used for an immense variety of simula-
tions [69], including solutions of variable density formulation of the Navier-Stokes
equations [3], embedded boundary methods [21], and fourth-order-in-space dis-
cretizations [9, 20] for AMR simulations. AmrLib, which has similar foundations
as CHOMBO, has been used to develop algorithms for low Mach laminar flames
simulations [24] as well as for turbulent flames [10, 11]. GrACE was initially de-
veloped to investigate load-balancers for block-structured adaptive meshes [54, 55],
but has been used to develop efficient numerical schemes (e.g., extended stability
time integrators [52] and fourth-order spatial discretizations [72]) for the simulation
of flames [75]. These frameworks have been investigated for their scalability on par-
allel machines; see [61] for a discussion of scalability issues in GrACE and [78] for
CHOMBO.
Overture [65] had its beginnings in simulations using overset meshes i.e. a grid-
ding scheme, generally applied to complex geometries, where the domain is dis-
cretized using patches which could employ meshing schemes that were best suited
to the geometry at hand. Thus it was possible, for example, to embed a small cir-
cular patch, employing a cylindrical mesh, within a larger mesh discretized in a
Cartesian manner. It was used to develop fourth-order-in-space discretizations [39],
as well as in simulations of high-speed reactive flows [40]. It has been extended to
adaptive mesh refinement [41], multi-material flows [8] and has recently been par-
allelized and used for 3D calculations [42]. A full listing of Overture-related pub-
lications, including investigation of detonations etc. can be found on the Overture
homepage [65].
In the third category are frameworks that primarily seek to assist software de-
velopers implement a certain design pattern. CCA defines one such design pat-
Computational Frameworks for Advanced Combustion Simulations 413

tern; UINTAH [77] and CCAFFEINE [1, 14] are frameworks that implement it.
Thus components designed to operate in the UINTAH and CCAFFEINE frame-
works comply with the CCA architecture and bestow its advantages modular
design, reduction of software complexity, plug-and-play experimentation and mul-
tiple language interoperability on simulation codes that follow the architecture.
The framework is not required to provide any numerical or simulation capability
(which renders the framework usable in diverse scientific applications) and indeed
CCAFFEINE does not; however, UINTAH provides its users with a mesh and the
associated data structures for describing fields on a discretized domain. A detailed
discussion of CCA follows in Sect. 17.3.

17.3 The Common Component Architecture

The Common Component Architecture (CCA) specification defines a software stan-

dard allowing plug-and-play composition of scientific applications. Being compo-
nent-based, it is necessarily object-oriented. Its development was driven by the ben-
efits of modularization. The competitive advantages of modularization were recog-
nized by the commercial establishment in the mid-1990s, which fashioned a solu-
tion in the form of component architecture; CORBA [22], Visual Basic [85] and Java
Beans [33] are some industry-standard component architectures. These architectures
employ certain subsets of object-oriented software design principles to realize mod-
ularity and interoperability of modules under a large variety of conditions. However,
commercial component standards are ill-suited for scientific computing [1], the pri-
mary drawbacks being the lack of support for parallel computing. Starting from
the concepts common to most component standards, the CCA retains most of the
relevant characteristics while simultaneously allowing (but not stipulating) parallel
computing in an SPMD fashion. CCAFFEINE [1] and UINTAH [77] are CCA-
compliant frameworks that support parallel computing, while XCAT [87] does not.
Within CCA one considers components, modules (or objects) that implement a
particular scientific or algorithmic functionality, and a framework e.g. CCAFFEINE
or UINTAH, that wires components together into a functioning simulation code.
Components implement interfaces (Ports in CCA parlance) through which the com-
ponents provide their functionality; these ports are designed by the programmer
implementing the component. Components are peers, i.e., they do not inherit im-
plementation from other components, and are capable of being used independently.
Since components implement interfaces, they can be developed without a tight cou-
pling to a software development team. CCA stipulates that all components imple-
ment a particular interface (called the Component interface) through which indi-
vidual components can interact with the framework (specifically, using the Services
interface). Typically, this involves registering the Ports (interfaces/functionalities)
that they provide (these interfaces are called the ProvidesPorts of the component)
as well as the Ports that they use (i.e., the UsesPorts of the component). Driven by
a user script, the framework matches the ProvidesPort and UsesPort registrations.
414 J. Ray et al.

The individual components then fetch the matched-up Ports and use them by mak-
ing calls on the methods on those interfaces The Port interface is a caller-callee
boundary, not a data-flow abstraction.

17.3.1 Features of the Common Component Architecture

The initial development of the CCA specification took place using a compiler-
independent, simple subset of C++ syntax (pure virtual classes) as the interface
definition language. Subsequent CCA development (version 0.6 and beyond) use
the Scientific Interface Definition Language (SIDL) and its compiler Babel [23]
to enable language interoperability i.e. components written in the Babel-supported
languages (C, FORTRAN, Python, or Java) can call each other transparently, with-
out manually performing the data translations and language-bindings that such a
mixture of programming languages would entail. When using Babel, the interface
construct of the SIDL language replaces the pure virtual abstract class of C++ (of
the initial design). In addition to language translation, SIDL/Babel provides modern
software engineering functionality in each of the languages it supports e.g., pass-
ing and accessing multidimensional array data and C structs as references to pre-
serve performance, support for exceptions, object-oriented design, tunably enforced
programming-by-contract conditions when entering and leaving method calls etc.
The C++ specification continues to exist and provide interfaces equivalent to those
of the SIDL definition.

Fig. 17.1: Mechanics of any-to-any language calling with Babel. Client binding and
interoperability layers (solid shapes) are entirely generated code. The implementa-
tion is partially generated in the implementors choice of programming language,
X, chosen from among the supported languages. The implementor fills in method
bodies and private data structure definitions that are inaccessible from the clients.
Lines indicate function calls.
Computational Frameworks for Advanced Combustion Simulations 415

Language interoperability: SIDL/Babel enables language-interoperability by

interposing an Intermediate Object Representation (IOR) between components, with
an independent client-side binding to each language. This is illustrated in Fig. 17.1.
In order that a component, written in language X, may be called from a host
of other languages (in Fig. 17.1, these are various dialects of FORTRAN, C/C++,
Java and Python), the programmer first expresses/writes the interface i.e., func-
tion calls, and their arguments (which may use the Babel-supported data-types like
multi-dimensional arrays, exceptions etc. listed above) in a language-neutral man-
ner using SIDL. The SIDL interface is then compiled using the Babel compiler
to generate equivalent interfaces in C/C++, FORTRAN etc, collectively referred to
as client bindings in Fig. 17.1 (left). It also generates a wrapper, in language X,
(shown in dashed lines on the right side of Fig. 17.1) in which one implements
ones components. The wrapper can also include calls to an external library, and
in fact, provides a very simple route to making a library directly callable from a
multitude of programming languages. The most significant, and intricate, piece in
the cross-language orchestration is the IOR, shown in the middle of Fig. 17.1 (C
interoperability layer). This IOR, essentially a C struct containing a function ta-
ble and data pointers, maps function calls in the various client languages (left
side of Fig. 17.1) into the equivalent functions/methods on the server side (the
dashed structure on the right side of Fig. 17.1). The need for translation of data
(e.g from FORTRAN multi-dimensional arrays on the client side to perhaps C
on the server/implementation side), the throwing of exceptions and other func-
tionalities that might be needed to enable cross-language operation are detected by
Babel when compiling the SIDL interface, and is encoded into the IOR. There
are a substantial number (as seen in Fig. 17.1) of code objects which are automati-
cally derived from SIDL by Babel and then compiled. Most of the rote work needed
to build and deploy SIDL-enabled applications has been automated with the Bocca
tool [2], enabling entire component application structures to be prototyped within
minutes.
Parallel computing: Since the use of parallel computing is critical to perfor-
mance, the CCA standard is carefully crafted to avoid forcing the choice of any
particular parallel programming model on the user. Component developers may use
any communication model. A survey of the impact of using CCA on application
performance is provided in Section 6 of [14]. The typical scientific application has
a single-program-multiple-data structure (SPMD). This programming model is well
supported by the CCAFFEINE framework [1] and default driver, which supports the
component-based programming model analog, the single-component-multiple-data
(SCMD) model. In the SCMD model, all components have a representative instance
on all processes. For a single component, we call the group of parallel instances
a cohort. Message passing, by whatever means, is restricted to exchanges within a
cohort, as one component implementation cannot make any assumptions about the
communication internals of a different component implementation. CCAFFEINE
has also been used as a library to compose MCMD (multi-component-multiple-data)
applications [46].
416 J. Ray et al.

17.4 Computational Facility for Reacting Flow Science

The Computational Facility for Reacting Flow Science (CFRFS) is a toolkit for con-
ducting high-fidelity simulations of laboratory-scaled flames. The toolkit adheres
to the CCA specification (using C++, not SIDL/Babel as its interface language of
choice) and is typically run using the CCAFFEINE framework. The CFRFS toolkit
implements a set of novel numerical techniques, most of which were developed as
the toolkit was being constructed. In this section we describe the toolkit and its nu-
merical structure and discuss why a component architecture was necessary for its
implementation. We conclude with an analysis, using data drawn from the toolkit,
to determine to what degree the original aims of the software design have been re-
alized.

17.4.1 Numerical Methods and Capabilities

The CFRFS toolkit is used to perform computations of flames using detailed chemi-
cal mechanisms. It solves the low Mach number approximation of the Navier-Stokes
(NS) equations [86], augmented with evolution equations (convective-diffusive-
reactive systems) for each of the chemical species in the chemical mechanism. It
employs fourth-order finite-difference schemes for spatial discretizations within the
context of AMR meshes [13]. It adopts an operator-split construction to enable the
use of efficient integrators when time-integrating different physical processes in
the system being solved. It uses extended-stability Runge-Kutta-Chebyshev (RKC)
time-integration [76] for time-advancing the diffusive transport terms and an adap-
tive, backward difference stiff integrator for the chemical source terms. The numer-
ical details of the high-order (spatial) methods can be found in [72] and those of
RKC on block-structured adaptive meshes in [52]. The divergence constraint in the
low Mach NS equations necessitates a projection scheme, which gives rise to a non-
constant coefficient Poisson equation; this is solved using the conjugate-gradient
method with a multigrid solver (employing high-order spatial stencils) as a precon-
ditioner.
The CFRFS Toolkit, then, is an integration of many advanced numerical tech-
niques. Many of them, e.g., RKC, had been demonstrated on uniform meshes [64]
but had to be augmented to enable the solution of partial differential equations
(PDEs) on AMR meshes. For example the RKC schemes had to be modified
to preserve its order of accuracy when used with time refinement [52] on AMR
meshes; further tests were required to determine various free parameters (in the
RKC scheme) when time-advancing a convective-diffusive system [71]. The high-
order spatial discretizations required appropriate interpolation schemes, and in cer-
tain cases, needed the solution to be filtered, to remove high-wavenumber content
and prevent Gibbs phenomenon [72]; the correct pairings of discretization, inter-
Computational Frameworks for Advanced Combustion Simulations 417

polation and filter order were determined as a part of the implementation of the
toolkit. The projection scheme adapts a fourth-order finite-volume formulation [44]
for use in the context of a finite-difference approach [74]. Thus the construc-
tion of the high-order AMR simulation capability, as implemented in the CFRFS
Toolkit entailed a significant amount of development of advanced numerical meth-
ods.
The CFRFS Toolkit makes copious use of external software. The adaptive mesh-
ing and load-balancing is currently provided by the GrACE package [38]; coupling
to CHOMBO [17] is in progress. The stiff integration capability is provided by
CVODE [19], while the elliptic solvers in Hypre [34] are used for the pressure
solve. Legacy codes are used to provide implementations of various constitutive
models (transport coefficients, gas-phase reaction and thermodynamics models etc).
Being able to leverage existing, validated software (e.g., legacy codes) has saved
much implementation effort, while numerical libraries (e.g. Hypre, CHOMBO) al-
low the CFRFS toolkit to take advantage of optimized and specialized capabilities
in a facile manner.

17.4.2 The Need for Componentization

The goal of the CRFRS development effort was to develop a flexible, reusable
toolkit. At the very outset it was expected that many of its advanced features would
be contributed piecemeal by experts or incorporated using legacy software and the
necessity of an extremely modular design was recognized very early.
The componentization in the CFRFS toolkit follows strictly along functional
lines. Each component implements a physical model, a numerical scheme, or a
computational capability like a data object that stores and manages domain-
decomposed fields (e.g., a temperature field) on multiple levels on an AMR mesh.
The functionality is expressed in the Port/interface design; components implement
the functionality. As there are may different ways to provide a functionality e.g.,
one may time-integrate using many different algorithms or calculate transport prop-
erties using diverse models, a single Port may find disparate implementations. Each
component is compiled into a shared library (also known as a dynamically load-
able library); a simulation code is composed by loading a number of them into
the CCA framework and wiring them together. Figure 17.2 shows a wiring di-
agram, assembling approximately 40 components into a low-Mach number flame
simulation, whose results are discussed in Section 17.5.1. The components in the
wiring diagram implement flow models (fourth-order discretizations for convective
and diffusive fluxes, detailed chemical models etc), numerical schemes (the pressure
solution, sixth-order interpolation schemes), the AMR mesh and the associated data
object and miscellaneous components for I/O etc. The components can be approx-
imately collated into 3 sub-assemblies, responsible for scalar transport, momentum
transport (including the projection required for solving the low-Mach number ap-
418 J. Ray et al.

proximation of the Navier-Stokes equation) and for advancing reactive terms. The
components are dynamically loadable, and so, the code is composed at runtime.
The components and wiring connectivity are specified in an input file to the frame-
work; components can be exchanged simply by changing a single line in this input
file.
The aim of componentization was the taming of complexity. One measure of
complexity is the pattern in which different components might use each other. A
good design would exhibit modularity, where connectivity between components
is sparse and connections are arranged in some regular manner e.g., if compo-
nents are collected/connected into sub-assemblies, which are hierarchically com-
posed into the simulation code. Figure 17.2 shows 3 separate sub-assemblies con-
sisting of components that address the transport of species, chemical reactions and
the momentum solve (including the pressure solution). The size of each compo-
nent is a second measure of complexity; smaller components are easier to under-
stand and maintain. In Fig. 17.3 (left), we plot a histogram of the size (lines of
code) in a component; it is clear most components are small, less than 1000 lines
of code. Since components implement Ports, this strongly suggests that individ-
ual Ports do not embody much complexity either i.e., they have few methods that
need to be implemented. This is shown in Fig. 17.3 (right) where we plot a his-
togram of the number of methods in each port. Most of the ports have 10 or fewer
functions/methods. Figures 17.4 (left) plots the histogram of the number of Pro-
videsPorts i.e, the number of Ports a given component implements. It is clear that
the bulk of the components implement less than five ports each, which explains
their small size; recall that most ports have fewer than 10 methods. Another mea-
sure of a components complexity or importance is the number of UsesPorts it has.
Components that link disparate sub-assemblies together tend to have many Uses-
Ports distributed among the sub-assemblies. In Fig. 17.4 (right) we plot the number
of UsesPorts per component, which is proportional to the number of other compo-
nents a given component requires to perform its functions. We see that almost all
components have less than 10 UsesPorts. Since components provide, on an aver-
age, less than 5 ports each (see Fig. 17.4, left), a component is connected to ap-
proximately 2-3 other components, leading to sparse connectivity between compo-
nents. These statistics show that a relatively sparse specification of functionalities
and interconnections may suffice for the construction of quite complex scientific
software.
 Computational Frameworks for Advanced Combustion Simulations

Fig. 17.2: Wiring diagram of the approximately 40 components that are used for simulating reactive flows in the CFRFS toolkit. The
components are shown in white; different ports are colored. The ProvidesPorts appear on the left of a component; the UsesPorts on the
right. The components are assembled, roughly, into 3 sub-assemblies, responsible for scalar transport (blue dash-dotted box), momentum
transport (red dash-dotted box) and for modeling reactive processes (green dash-dotted box). The components can be strictly numerical
(e.g., interpolation, pressure solver), models (detailed chemistry, diffusion coefficients) or computer science e.g., the AMR Mesh and
the data object.
419
420 J. Ray et al.

Fig. 17.3: Left: Histograms of the logarithm (to base 10) of component sizes, mea-
sured as the total number of lines, including comments and blank lines. The his-
togram clearly shows that half the components are less than 1000 lines long, and
almost all are less than about 3000 lines. Components, therefore, are generally quite
small. Right: Histogram of the number of functions/methods per port. We see that
most ports have less than 10 methods each. There are a couple of ports, related to the
mesh and the data object that have approximately 40 methods each. These statistics
were extracted from a population of 100 components.

17.5 Computational Investigations Using CCA

In the previous section we described how CCA was used to architect and imple-
ment the CFRFS toolkit whose design philosophy stressed small, simple compo-
nents, sparse connectivity between components and their hierarchical composition,
via sub-assemblies, into functioning simulation code. In this section, we demon-
strate two different ways in which the components of the the CFRFS toolkit are
used.
The CFRFS toolkit consists of two sets of components. The first set, by far the
bigger one, consists of components that address the numerical issues surrounding
the use of fourth- (and higher) order spatially accurate methods on block-structured
AMR meshes. These allow efficient resolution of fine flame structures without the
necessity of overwhelming computational resources. The second set of components
addresses the identification of low-dimensional manifolds in the chemical dynam-
ics, so that chemical source terms may be tabulated and thus evaluated inexpensively
within the context of spatially resolved flame simulations. This is done using Com-
putational Singular Perturbation [51]. Many components, for example, those mod-
eling chemical reactions, thermodynamics and constitutive models find use in both
Computational Frameworks for Advanced Combustion Simulations 421

Fig. 17.4: Left: Histogram of the number of ProvidesPorts implements by the com-
ponents in the CFRFS toolkit. We see that most of the components have 4 or fewer
ProvidesPorts i.e., they implement very few ports. The distribution of the number of
methods in each port is plotted in Fig. 17.3 (right). Right: Histogram of the number
of UsesPorts a component uses. This is one measure of the complexity of the algo-
rithm/functionality a component implements. Components that link sub-assemblies
of components also tend to have many UsesPorts. We see that most components
have less than 10 UsesPorts.

the efforts. The final goal is to replace/augment the reactive subsystem, consisting
of a stiff-integrator and the chemical source terms (as described in Sec. 17.4), with
an inexpensive tabulation scheme that would allow the toolkit to be used with large
(and stiff) chemical mechanisms typically associated with higher hydrocarbons.

17.5.1 Fourth-order Combustion Simulations with Adaptive Mesh

Refinement

Chemically reacting flow systems based on hydrocarbon fuels typically exhibit a

large range of characteristic spatial and temporal scales. The complexity of kinetic
models, even for simple hydrocarbon fuels, compounds this problem, making mul-
tidimensional numerical simulations difficult. This is true even for laboratory scale
configurations.
These difficulties are commonly addressed in a variety of ways. For low speed
flows, one may adopt a low Mach number approximation [58] for the momentum
transport. This approximation assumes that acoustic waves travel at infinite speed, a
justifiable assumption in many low-speed flows. One can also exploit the structure
of the governing equations and adopt an operator-split mechanism, performing the
transport and reactive time-advancement via specialized integrators [64]. In prob-
422 J. Ray et al.

lems where fine structures exist only in a small fraction of the domain e.g., in lam-
inar jet flames, one may employ AMR [13] to concentrate resolution only where
needed [6, 12, 24, 73], while maintaining a coarse mesh resolution elsewhere.
The CFRFS toolkit implements a numerical model that can efficiently simulate
flames with detailed chemical mechanisms. The use of AMR is not without its chal-
lenges, beyond just programming complexity. In order to reduce the number of grid
points and the number of refinement levels in the computational mesh hierarchy we
employ high-order stencils to discretize the governing equations and to interpolate
between the computational blocks on adjacent mesh levels. A projection scheme is
employed for the momentum transport. Since mesh adaptivity is driven by the nar-
row flame structure rather than the velocity field, we solve the momentum transport
on the lowest level mesh in the AMR mesh hierarchy i.e., on a uniform mesh. This
further enhances the efficiency of the model since the elliptic solver required by the
pressure equation is more efficient on a uniform mesh, compared to a multilevel
one [59]. The numerical approach and results obtained for canonical configurations
are presented below.

17.5.1.1 Formulation

In the low-Mach number limit, the continuity, momentum and scalar transport equa-
tions for a chemically reacting flow system are written in compact form as
1 D
v = (17.1a)
Dt
v 1
= p +CU + DU (17.1b)
t
T
= CT + DT + ST (17.1c)
t
Yk
= CYk + DYk + SYk k = 1, 2, . . . , Ns . (17.1d)
t
Here v is the velocity vector, the density, T the temperature, Yk the mass frac-
tion of species k, p is the hydrodynamic pressure, and Ns is the number of chemical
D
species. The Dt operator in the continuity equation represents the material deriva-

tive, Dt = t + v . The system of equations is closed with the equation of state for
D

an ideal gas. The thermodynamic pressure spatially uniform and is constant in time
for an open domain in the low-Mach number limit. NASA polynomials are used
to compute thermodynamic properties [60]. The transport properties are based on a
mixture-averaged formulation and are evaluated using the DRFM package [68].
The equation of state is used to derive an expression for the right hand side of the
continuity equation (Eq. 17.1a)
Computational Frameworks for Advanced Combustion Simulations 423

Fig. 17.5: Schematic of the momentum solver components in CFRFS.

Ns  
DP0 1 D 1 W
=0 = (DT + ST ) DYk + SYk (17.2)
Dt Dt T k=1 Wk

17.5.1.2 Implementation in the CCA Framework

The numerical integration of the system of equations (17.1a-17.1d) is performed in

three stages. In the first stage, a projection scheme is used to advance the velocity
field based on Eqs. 17.1a, 17.1b. Figure 17.5 shows the main CCA components in-
volved in the momentum solver. The Momentum Driver component advances the
velocity field to an intermediate value based on convection RHSconv and diffusion
RHSdi f f contributions to the right-hand-side (RHS) term Velrhs of the momentum
equation (17.1b). This is followed by an elliptic solve in the Pressure Solver com-
ponent for the dynamic pressure p. The RHS values for the elliptic pressure equation
are computed in PressureRHS . Transport and thermodynamic properties are provided
by Transport Properties and Thermo & Chemistry components, respectively. The
gradient of the pressure field is used to correct (Velcorr ) the intermediate veloci-
ties above to obtain a field that satisfies both the continuity and momentum equa-
tions (17.1a, 17.1b). The components shown at the top of Fig. 17.5 (AMR Mesh,
Boundary Conditions, Interpolations and Derivatives) are generic components that
handle the adaptive mesh refinement library, boundary conditions, interpolations,
and derivatives. External libraries are shown in ellipses.
In the second stage, sketched in Fig. 17.6, temperature and species mass frac-
tions are advanced using an operator split approach that separates the convection,
CT ,CYk , and diffusion, DT , DYk , contributions from the ones due to the chemical
source terms, RT , RYk , in Eq. (17.1c, 17.1d). Symmetric Strang splitting is employed,
beginning with the chemical source term contribution for half the time step, fol-
424 J. Ray et al.

Fig. 17.6: Schematic of the scalar solver components in CFRFS.

lowed by the contributions from convection and diffusion terms for a full time step,
and concluded by the remaining contribution from the reaction term for half the time
step. The scalar advance due to the chemical source term is handled by Chemistry
Integrator. The convection (Scalarconv ) and diffusion (Scalarconv ) contributions are
combined by ScalarRHS component and provided to RKC2 Integrator which uses
a Runge-Kutta-Chebyshev (RKC) algorithm [84] for time advancement. A Switch-
board component is used to ensure that velocities are available at intermediate times
during the multi-stage RKC integration.
The third stage repeats the projection algorithm from the first stage using the
updated scalar fields from second stage. The overall algorithm is 4th -order accurate
in space and 2nd -order in time.
Adaptive mesh refinement We employ an AMR approach where the computational
domain is split into rectangular blocks. The advancement in time of the AMR so-
lution is based on Berger-Colella time refinement [13, 52]. Figure 17.7 shows a
schematic of this recursive time integration algorithm. Consider the solutions on
levels L and L + 1 at time tn . Level L is first advanced to tn + t, then the solution on
L + 1 is advanced in two half steps, t/2 to ensure numerical stability on the finer
grid. During time advancement on L + 1, boundary conditions are computed by in-
terpolation using the solution on L. At tn + t the solution on L + 1 is interpolated
down to the corresponding regions on level L. In order to preserve the 4th -order spa-
tial convergence of the numerical scheme, the interpolations between adjacent grid
levels use 6th -order spatial stencils [72].

17.5.1.3 Application to Flame-Vortex Interaction

A canonical vortex-flame configuration [64] was chosen to explore the performance

of the numerical construction. The computational domain is 1.5cm 0.75cm. The
Computational Frameworks for Advanced Combustion Simulations 425

Fig. 17.7: Schematic of the time refinement in the context of AMR.

velocity field corresponding to a periodic row of counter-rotating Lamb-Oseen vor-

tices is superimposed over the premixed 1D flame solution discussed above. A rel-
atively coarse mesh was used for the base mesh, with a cell size of 50 m in each
direction. Additional, finer, mesh levels were added in the flame region during the
simulation.
A one-step, irreversible Arrhenius global reaction model is used in addition to
a C1 kinetic model to study the vortex-flame interaction. Figure 17.8 shows freeze
frames of the vorticity and heat release rate fields. The vortex pair is initially located
2mm upstream of the flame and propagates with approximately 10m/s towards it.
As the vortex pair impinges into the flame, the flame intensity decreases on the
centerline for the C1 model while the one-step solution shows little change in the
interaction region. Similarly, at locations off-centerline the flame intensity for the
C1 model decreases significantly as it stretched and rolled around the vortex pair.
The last frames show a significantly contorted flame, and the relative increase in the
overall burning rate is about about 50% more for the one-step reaction simulation
compared to the simulation using the C1 model.

17.5.2 Computational Singular Perturbation and Tabulation

In the previous section, we described our experience with simulating flames using
one-step and C1 chemical mechanisms. The primary challenge in going from sim-
ple one-step chemistry to a C1 mechanism was the steep increase in the stiffness of
the dynamical system composed of the reactive processes; its main effect was to re-
duce the size of the time-step one could take without unacceptable splitting errors.
Matters are further compounded when one considers C2 (or even more detailed)
chemical mechanisms. This stiffness of detailed chemical mechanisms is due to the
wide range of time scales that they model. It leads to considerable difficulties when
time-advancing them in an efficient manner. Chemical model simplification and re-
duction strategies typically target these challenges by reducing the number of reac-
tions and/or species in the model, with associated reduction in model complexity.
When done properly, this strategy also reduces the system stiffness. Alternatively,
426 J. Ray et al.

Fig. 17.8: Vorticity (white solid contours) and normalized heat release rate (hrr,
shaded contours) for simulations using the one-step reaction model (upper row) and
a C1 kinetic model consisting of 16 species and 46 reactions (lower row).

the Computational Singular Perturbation (CSP)-based time integration construction

of [83] uses CSP analysis to project out the fast time scales from the detailed chem-
ical source term, thereby rendering the equations non-stiff. The promise of this ap-
proach is that explicit time integrators can be used for large-time step integration
of the resulting non-stiff source terms, and could potentially eliminate the need for
operator-split time integration of reaction-diffusion source terms. More details are
given in Chapter 9.
The key challenge with this time integration approach, however, is the large
computational cost of solving for the requisite CSP information and the resulting
projection matrices. An approach to mitigate this computational cost is tabulation.
By adaptively storing and reusing the CSP information, the significant CSP over-
head can be drastically reduced (by amortization), leading to an efficient overall
implementation. Such a tabulation strategy has been explored for elementary model
problems [50, 51]. This section describes the implementation of a CSP tabulation
approach, relying on kd-trees [7] to efficiently store and retrieve CSP information
along manifolds in the chemical configuration space.
In the following, the CSP time integration and tabulation approach is formulated,
followed by a discussion of its implementation in the CCA framework for react-
ing flow simulation. Next, the approach is illustrated on the simulation of H2 air
ignition.

17.5.2.1 Formulation and Implementation

Consider the chemical system described by dyy/dt = g (yy), where y IRN , and g (yy) is
the chemical source term. The CSP basis vectors {aak }Nk=1 and covectors {bbk }Nk=1 , all
Computational Frameworks for Advanced Combustion Simulations 427

in IRN , enable the decoupling of the fast and slow processes, and the identification
of low dimensional slow invariant manifolds (SIMs) [48]. Thus, we have
dyy
= g = g fast + g slow = a 1 f 1 + a 2 f 2 + + a N f N (17.3)
dt

where f i = b i g , for i = 1, 2, . . . , N. In this equation, g fast corresponds to the modes

with fast transients, which are rapidly exhausted. After relaxation of fast transients,
with M modes exhausted, g fast = M r=1 a r f 0 and g slow = s=M+1 a s f = (I
r N s
r
r=1 a r b )gg = P g . In practice, the number of exhausted modes is determined as the
M

maximum M for which M+1 M r

r=1 a r f is less than a user-specified threshold, where
M+1 is the time scale corresponding to the (M + 1)st mode.
The CSP integrator [83] proceeds in each time step by first integrating the slow
dynamics of the system, followed by a homogeneous correction (HC) to correct for
the fast time scales:
 t+ t
y (t + t) = y (t) + P g dt  (17.4)
t
M
y (t + t) = y (t + t) a m nm |t fn (17.5)
m,n=1

fn = b n g [y (t + t)] (17.6)

where nm is the inverse of nm , given by

 m 
dbb
n =
m
+ b m
J an (17.7)
dt

and J is the Jacobian of g . The matrix nm is diagonal with entries the time scales
{k }Nk=1 when the CSP basis vectors are chosen to be the eigenvectors of J and the
curvature of the SIM is neglected, i.e. dbbm /dt = 0.
The procedure outlined above separates the fast, exhausted modes from the slow
modes that drive the evolution of the system along the SIMs. As discussed in [47],
CSP also identifies the species that are associated with these fast modes as CSP
radicals. (These are the species whose concentration can be determined from the
algebraic equations resulting from setting f i = 0, i = 1, . . . , M.) Accordingly, the
species space can be separated into the CSP radicals and non-CSP radicals.
To improve the efficiency of the CSP integrator, a tabulation approach has been
developed to enable reuse of the essential CSP quantities: the M fast CSP vectors
and covectors, as well as the M + 1 fastest time scales, which are sufficient to as-
semble the slow-manifold projector P needed for the HC and CSP integration, and
to select the time step along the slow manifold. As the CSP vectors, covectors and
time scales can be modeled as functions of the non-CSP radical species only, it is
sufficient to tabulate these quantities in an N M dimensional table, rather than
having to cover the full N-dimensional state space.
428 J. Ray et al.

In the work presented here, a table with manifold conditions is constructed off-
line, by performing full CSP analysis on a number of design points in state space.
We first randomly sample a set of initial conditions over a range of initial tempera-
tures, equivalence ratios and N2 dilution factors (extra mole of N2 per mole of air)
and integrate them forward, with CVODE [19], using detailed reaction kinetics. A
set of design points is constructed from the system states encountered during those
simulations. For each of these design points, a CSP analysis is performed to iden-
tify associated SIMs. If a design point has exhausted modes, then successive HCs
are applied to project that design point onto the corresponding SIM. Each SIM is
characterized by a unique value of M and the associated CSP radicals.
For each identified SIM, the tabulation of the associated CSP information relies
on a nonparametric regression approach. For this purpose, the CSP information is
stored in kd-tree data structures over the the N M dimensional space of the non-
CSP radical species. Note that, in order to give equal weight to all dimensions of the
state vector in the computation of distance measures, all coordinates of the manifold
points are first rescaled and shifted to range between 0 and 1 before being stored
in the kd-trees. During time integration, the manifold that best corresponds to the
current condition in the chemical configuration space is determined by finding the
nearest neighbor, as measured by the Euclidean distance measure, in all of the man-
ifolds in the table. If the manifold point that is closest to the current condition over
all manifolds is within a maximum allowable distance d, then the associated man-
ifold is assumed to be the one that is currently being followed by the system. The
CSP information at the current condition is then approximated with the correspond-
ing values at the nearest neighbor point in the table, which amounts to a 0th -order
interpolation. Higher order interpolations, relying on interpolation between nearest
neighbors or on polynomial response surfaces [49, 81], are the subject of ongoing
work. In case none of the nearest neighbors in the tabulated manifolds are within the
maximum allowable distance, then a full CSP analysis is performed on the current
condition instead.
To implement the CSP integration approach, extensive use was made of existing
components in the CCA framework. For example, the evaluation of the chemical
kinetics source term and its Jacobian rely on the AMR set of components from
CFRFS toolkit discussed in Sect. 17.4. Time integration relies on a CVODE com-
ponent, part of the CFRFS toolkit. New components were developed to perform the
CSP analysis as well as the table construction and interpolation for the tabulation
approach. These components were joined together through the use of driver compo-
nents that organize the overall algorithmic sequence of operations.

17.5.2.2 Application to H2 air ignition system

The CSP integration method outlined in the previous section was applied to the sim-
ulation of ignition of a stoichiometric homogeneous H2 air mixture at a temperature
of T = 1000 K. The system is modeled using a 9 species reaction mechanism, re-
sulting in a total state space dimension of N = 10 (9 species + temperature) [88].
Computational Frameworks for Advanced Combustion Simulations 429

(a) Temperature Profile (b) Exhausted Modes

(c) Temperature Detail (d) HO2 Detail

Fig. 17.9: a) Evolution of temperature in an igniting stoichiometric H2 air system,

simulated using the detailed reaction mechanism, the CSP solver, and the CSP solver
with tabulation. b) Evolution of the number of exhausted modes, M, as obtained by
CSP analysis. All approaches are in good agreement, except for minor differences in
the ignition time delay, as shown in the close-up of the ignition zone for c) tempera-
ture and d) one of the trace species, HO2 . The initial conditions were: T = 1000 K,
YH2 = 0.0285, YO2 = 0.2264 and YN2 = 0.7451.

Figure 17.9 compares the predicted temperature evolution obtained by integrating

the detailed reaction kinetics (with the implicit solver CVODE), to the solution ob-
tained with the CSP integrator (using the explicit fourth order Runge-Kutta (RK4)
integration scheme), and with the CSP + tabulation approach.
For the tabulation approach a CSP table was constructed by sampling 100 ini-
tial conditions with Latin Hypercube Sampling over a range of equivalence ratios
between 0.9 and 1, initial temperatures between 980 and 1020 K, and dilution fac-
tors between -0.005 and 0.005. From the design points extracted from these runs,
close to 1 million states were identified on 9 different manifolds, with a number of
exhausted modes ranging from 1 to 5.
The CSP integrated solution, both with and without tabulation is in good agree-
ment with the full solution, except for a small difference in the ignition time delay,
as is shown in detail in Fig. 17.9(c) and 17.9(d).
430 J. Ray et al.

Fig. 17.10: The number of table hits, as a fraction of the total number of table
lookups, increases as the maximum allowed distance to the nearest neighbor on a
manifold is increased, resulting in more efficient usage of the tabulated data.

Note that, as the reaction progresses, the number of exhausted modes M, and the
associated CSP radicals change according to the reaction dynamics. Figure 17.9(b)
indicates that the system initially has two exhausted modes, followed by a time win-
dow during ignition where all modes are active, after which M gradually increases
up to five at late time, as more and more modes become inactive. Accordingly, as
the number of exhausted modes increases, the tabulation approach becomes more
efficient in terms of storage and lookup times as the CSP information for each mani-
fold is tabulated in N M dimensional kd-trees. For example, for the H2 air system
studied here, tabulation in a 5-dimensional table is sufficient for the section(s) of the
10-dimensional state space where 5 modes are exhausted (see Fig. 17.9(b)).
In terms of efficiency of the table usage, Fig. 17.10 shows the number of success-
ful table hits as a fraction of the total number of table lookups. As a table lookup
is performed in every time step, this number indicates how efficient the tabulation
approach is at avoiding full CSP analyzes by providing tabulated CSP information
instead. For the current table and initial condition, the table lookup success rate
increases from 25 % to about 65 % as the maximum allowed nearest-neighbor dis-
tance is increased from 0.001 to 0.03, while the accuracy of the integration does
not noticeably change (not shown here). Other numerical experiments indicate that
this table hit success rate and the accuracy of the tabulation assisted simulations
also depends on the density of the table in state space. A quantitative relationship
between the table density, the maximum allowed distance in the nearest neighbor
table lookup, and the accuracy of the CSP tabulation approach is the subject of on-
going work.
A comprehensive evaluation of the overall numerical performance of the CSP
integration scheme with tabulation, as a function of table size, system and manifold
dimensionality, degree of stiffness, and desired accuracy, is currently in progress.
However, preliminary performance measurements show the CSP tabulation scheme
to be competitive with direct CVODE integration for the cases studied in this paper.
Computational Frameworks for Advanced Combustion Simulations 431

To summarize, CCA provides a flexible framework for the implementation of

several components for the integration of stiff chemical kinetics. The modularity of
the framework allows easy reuse of components that were developed elsewhere in
the project for operations such as time integration, or the evaluation of source terms
or Jacobian. This flexibility allows the rapid development of codes to test various
integration approaches and easy switching between them for method comparison
and validation.

17.6 Research Topics in Computational Frameworks

The previous sections have described and demonstrated how modularity, obtained as
a result of adopting a component-based design, may be used to mitigate the effects
of software complexity. They have also shown the sophistication of the scientific
software that can be designed using a component-based approach. However, the
experience with CFRFS, as well as other componentization efforts [45, 46], have
revealed a number of difficulties, Some solutions have recently been crafted, which
we discuss below.
The Learning Curve: A significant challenge in adopting CCA has been the
learning curve associated with using SIDL/Babel. While this was not encountered
when developing the CFRFS toolkit (which uses the original C++-interface ap-
proach, not SIDL/Babel), it was observed in other componentization efforts [45, 46].
The process of generating client- and server-side code, as described in Sec. 17.3.1, is
prone to error if performed manually, but can be automated. An integrated develop-
ment environment, called Bocca [2], has been developed (see [16] for Bocca tutori-
als) for this purpose. Given the interfaces that a component uses and provides, Bocca
automatically invokes Babel, creates the client- and server-side auto-generated code
and constructs a build system to compile the resulting component skeleton. It min-
imizes what a CCA developer has to learn, enabling him/her to focus on more pro-
ductive tasks.
Reluctance to Abandoning Working Software: The process of componentiza-
tion could be significantly simplified if one could automatically derive components
out of a non-component codebase. A concept, called OnRamp [43], is being inves-
tigated by CCA researchers to enable such an automated derivation of components.
OnRamp is driven by annotations which are inserted into the codebase (indicat-
ing interfaces, code-blocks that will reside in components etc.), from which is it
possible, under certain restrictions, to automatically generate components and its
associated build system. This preserves the original code and most of the software
development practices that the programmer is familiar with, while bestowing the
benefits of componentization on the software in question.
Components also confer benefits beyond the fundamental requirement of con-
straining software complexity. Since components are a black box regarding imple-
mentation but adhere to a specified convention for communicating with the outside
world, they are ideal for automating computation at a high level; CCA has focused
432 J. Ray et al.

on performance improvements as the aim of automation. Below, we mention some

of the recent advances in this arena.
Automatic Proxy Generation: The collection of performance characteristics on
a method-call basis is a required, but laborious task in performance modeling. In
a component environment, this can be achieved quite easily by exploiting the fact
that components publish the interfaces that they use and provide. The collection of
performance data can be done by interposing a proxy component between an inter-
face provider component and an interface using component. The proxy component
serves to trap calls between components and switch-on/terminate the collection of
performance metrics (elapsed time, cache misses, page faults etc) by a performance
measurement tool e.g. TAU [79]. Such performance-measurement proxies can be
generated automatically [82]. They can be used to collect performance data and
identify bottlenecks; they can also be used to monitor an executing simulation and
optimize it during runtime. This is described below.
Computational Quality of Service: Since the framework has a holistic view of
the entire application, and proxy components can monitor the performance of indi-
vidual components, it becomes possible to manipulate their behavior, with a view
to ensuring robustness, celerity of computation etc. This is commonly referred to as
Computational Quality of Service [62]. The manipulation of components may be
performed by changing parameters that a component may provide or by replacing
entire components [56, 57]. See [62] for how this may be performed without modify-
ing any components; a working example, using a shock-hydrodynamics simulations
in CCAFFEINE, can be found in [56].

17.7 Conclusion

In this chapter, we have investigated how a component architecture may be used

to design and implement scientific simulation software. The Common Component
Architecture was chosen because of its ability to accommodate parallel computing.
Unlike many computational frameworks, a component framework does not require
one to marry into a prescribed set of data and code structures; in many cases, such
marriages lead to ones dependence on the framework for the integration of ex-
ternal libraries and/or legacy software. The peer nature of components (whereby all
components are independent) prevent such dependencies from arising. However, it
is to be noted that a software architecture merely lays down a few software devel-
opment principles; their judicious use is a matter of software design. The design, in
turn, is dictated by where one starts from (i.e., whether one starts with a tabula rasa,
which was our case, or whether one starts componentizing a legacy code) and what
one wishes to achieve with the particular design. Any lack of clarity regarding the
second aspect invariably leads to an unsatisfactory end.
In Sect. 17.4 we described the particular ends that wished to achieve with
our component-based design, particularly maintainabiliy and reduction in software
complexity; the statistics drawn from the 100 components in our toolkit provide
Computational Frameworks for Advanced Combustion Simulations 433

some confidence that we have largely succeeded. In Sect. 17.5 we showed how the
toolkit is used, including a few example of component reuse. Though not described
here, some of the AMR components used in Sect. 17.5.1 have also been used to
simulate problems in shock-hydrodynamics [53]. Thus there is some evidence to
indicate that the plug-and-play promise of component software, widely realized in
non-scientific software, may be replicated in our field too.
This chapter has drawn examples from the CFRFS effort, which emphasized
small, manageable components designed without any constraints imposed legacy
software. There are other efforts where legacy software has dictated both the aims
and the course of componentization (see [45] and references within), and still oth-
ers, usually involving the componentization of libraries, where users played a role
(see Sect. 11 in [14]). Component-based design, and CCA in particular, is a versa-
tile methodology for designing and developing maintainable software, and is most
profitably used when one has a clear idea of why one wishes to use it. Unlike many
frameworks developed to enable the rapid prototying of codes, its attractiveness lies
in the long term.

Acknowledgements: The work documented in this chapter was funded by the

Department of Energy, under its Scientific Discovery through Advanced Comput-
ing (SciDAC) program. Some of the computations were performed at the National
Energy Research Supercomputing Center (NERSC) in Oakland, CA. The work was
performed in Sandia National Laboratories, CA. Sandia is a multiprogram labora-
tory operated by Sandia Corporation, a Lockheed Martin company, for the United
States Department of Energys National Nuclear Security Administration under
Contract DE-AC04-94-AL85000.

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Chapter 18
The Heterogeneous Multiscale Methods with
Application to Combustion

Weinan E, Bjorn Engquist and Yi Sun

Abstract The framework of the heterogeneous multiscale methods (HMM) is

briefly reviewed. Both the original HMM and the seamless HMM are discussed.
Applications to interface capturing and flame front tracking are presented.

18.1 The Heterogeneous Multiscale Method

The heterogeneous multiscale method (HMM) is a general framework for designing

multiscale methods for a wide range of applications [8, 10]. HMM is a top-down
strategy: it starts with an incomplete macroscale model, the missing data in the
macroscale model is estimated on the fly using the microscale model. As such, it of-
fers a platform for designing numerical algorithms that makes use of the knowledge
we have at both the macro- and micro-scales. Here we will briefly review the main
ideas, its advantages and limitations. We will also discuss possible applications to
combustion.

Weinan E
Department of Mathematics and Program in Applied and Computational Mathematics, Princeton
University, e-mail: weinan@math.princeton.edu
Bjorn Engquist
Department of Mathematics and Institute for Computational Engineering and Sciences, University
of Texas, Austin, e-mail: engquist@ices.utexas.edu
Yi Sun
Statistical and Applied Mathematical Sciences Institute, e-mail: yisun@samsi.info

T. Echekki, E. Mastorakos (eds.), Turbulent Combustion Modeling, 439

Fluid Mechanics and Its Applications 95, DOI 10.1007/978-94-007-0412-1 18,
Springer Science+Business Media B.V. 2011
440 Weinan E, Bjorn Engquist and Yi Sun

18.1.1 The Basic Framework

The general setting is as follows. At the macroscopic level, we make an assumption

about the form of the macroscale model, which can be expressed abstractly as

F(U, D) = 0, (18.1)

where D denotes the data needed for the macroscale model to be complete. This
data is estimated using a microscopic model, say in the form:

t u = L (u;U), Qu(0) = U0 . (18.2)

or
f (u, d) = 0, d = d(U). (18.3)
Here the macroscale variable U may enter the system as constraint, in order to make
sure that the data does correspond to the macroscale state we are interested in, Q is
the compression (or projection) operator that maps micro-state variables to macro-
states variables [8]. Once the form of the macroscale model and the detailed mi-
croscale models are chosen, we select a macroscale solver. We also need a con-
strained microscale solver and a way of estimating the needed macroscale data from
the results of the microscale simulation. Therefore the three main ingredients of
HMM are (Figure 18.1):
1. macroscale solver
2. constrained microscale solver
3. data estimator

Fig. 18.1: Schematics of the HMM framework.

Take the example of incompressible fluids for which the macroscale model is a
continuum model for the macroscale velocity field U in the form:
The Heterogeneous Multiscale Methods with Application to Combustion 441

t U + (U )U = ,
U = 0.

These are simply statements of the conservation of momentum and mass. The un-
known data is the stress : D = .
Let us assume that the micro model is a molecular dynamics model for the parti-
cles that make up the fluid:

d 2 yi
mi = fi , i = 1, 2, , N. (18.4)
dt 2
Here mi , yi are respectively the mass and position of the i-th particle, fi is the force
acting on the i-th particle.
Given that the macroscale model is in the form of an incompressible flow equa-
tion, it is natural to select the projection method as the macro-solver [4]. In the
implementation of the projection method, we will need the values of at the appro-
priate grid points. These are the data that need to be estimated. At this point, we have
to make an assumption about what depends on, and this enters in the constraints
that we put on the microscale model. Let us assume that

= (U). (18.5)

We will constrain the molecular dynamics in such a way that the average strain rate
is given by the value of U at the relevant grid point. In general, implementing such
constraints is the most difficult step in HMM. For the present example, one possible
strategy is discussed in [27].
From the results of the microscale model, we need to extract the values of the
needed components of the stress. For this purpose, we need a formula that expresses
stress in terms of the output of the molecular dynamics. This can be obtained by
modifying the Irving-Kirkwood formula [27]. These details can be found in [27].
In this example, the three ingredients are:
1. macro-solver, the projection method;
2. micro-solver, the constrained molecular dynamics;
3. data estimator, the modified Irving-Kirkwood formula.
We can write down the HMM procedure formally as follows. At each macro time
step,
1. Given the current state of the macro variables Un , reinitialize the micro-variables:

un,0 = RUn ; (18.6)

2. Evolve the micro variables for some micro time steps:

un,m+1 = S t (un,m ;Un ), m = 0, , M 1; (18.7)

3. Estimate D:
442 Weinan E, Bjorn Engquist and Yi Sun

Dn = DM (un,0 , un,1 , , un,M ); (18.8)

4. Evolve the macro-variables for one macro time step using the macro-solver:

Un+1 = S t (Un ; Dn ). (18.9)

Here R is some reconstruction operator that reinitializes the micro model in a way
that is consistent with the current state of the macro variables, S t is the micro
solver, which also depends on Un through the constraints, as indicated. DM is some
data processing operator which in general involves spatial/temporal/ensemble aver-
aging. This is sometimes referred to as the data estimator. Finally S t is the macro
solver.
There are two important time scales that we need to consider. The first, denoted
by tM , is the time scale for the dynamics of the macro-variables. The second, denoted
by , is the relaxation time for the microscopic model. We will need to distinguish
two different cases. The first is when there is no time scale separation, i.e. tM . In
this case, from the viewpoint of numerical efficiency, there is not much room to play
with as far as time scales are concerned. We just have to evolve the microscale model
along with the macroscale model. The second case is when  tM . This is the case
we will focus on. The general guideline in this case is to choose t to accurately
resolve the tM time scale, and to choose M such that M t covers sufficiently the
time scale for equilibration to take place in the micro model.
One main difference between HMM and other multiscale methods such as sys-
tematic upscaling and the equation-free approach [2, 19] is that HMM starts
with a macroscale model. This does represent a compromise in the sense that we
need to work with a preconceived form of the macroscale model. If the correct
macroscale model is a stochastic differential equation, then clearly an HMM based
on an assumption that the macroscale model is a deterministic equation leads to
wrong results. On the other hand, there are two important reasons for starting from
a macroscale model. The first is that even for problems for which adequate closed-
form macroscale models are available, designing a stable and accurate numerical
method for that macroscale model might be a significant task. The second is that for
most problems of practical interest, we often have already accumulated some knowl-
edge about the macroscale behavior of the problem. We should make use of such
information in designing multiscale methods. In cases when one makes a wrong as-
sumption, one can still argue that HMM produces an optimal approximation for
the macroscale behavior of the solution in the class of models considered. In this
sense, HMM is a way of addressing the following question: What is the best we can
do given the knowledge we have about the problem at all scales? This was indeed
the underlying designing principle for HMM.
The Heterogeneous Multiscale Methods with Application to Combustion 443

18.1.2 The Seamless Algorithm

The version of HMM described above requires converting back and forth between
the macro- and micro-states of the system. This can become rather difficult in actual
implementations, particularly when constructing discrete micro-states (needed for
example in molecular dynamics) from continuous macro variables. The seamless
strategy proposed in [12] (see also [11, 16, 26]) is intended to bypass this difficult
step.
To motivate the seamless algorithm, let us consider the trivial example of stiff
ODEs:
dx
= f (x, y),
dt (18.10)
dy 1
= (y (x)).
dt
One simplest idea is to change the small parameter to a bigger value  , the size
of which is determined by the accuracy consideration:

dx
= f (x, y),
dt (18.11)
dy 1
=  (y (x)).
dt
This is then solved using standard methods.
We can look at this differently. Instead of changing the value of , we may change
the clock for the microscale model, i.e. if we use = t/  in the second equation
in (18.11), then (18.11) can be written as:
dx
= f (x, y)
dt (18.12)
dy 1
= (y (x)).
d
If we discretize this equation using standard ODE solvers but with different time
step sizes for the first and second equations in (18.12), we obtain the following
algorithm:

n
yn+1 = yn (y (xn )), (18.13)

Dn+1 = yn+1 , (18.14)
xn+1
= xn + t f (xn , Dn+1 ). (18.15)

Here yn y(n ) and xn x(n t). The value of is the time step size we would
use if we attempt to solve (18.10) accurately. The choice of t is more tricky. It
should not only resolve the macro time scale, but also allow the micro state to relax,
i.e. to adjust to the changing macroscale environment. For example, if t is the time
step size required for accurately resolving the macroscale dynamics and if is the
444 Weinan E, Bjorn Engquist and Yi Sun

relaxation time of the microscopic model, then we should choose t = t/M where
M > / .

Fig. 18.2: Illustration of HMM (upper panel) and the seamless algorithm (lower
panel). Middle panel: rescaling the micro time scale.

In a nutshell, we can formulate the strategy as follows:

1. Run the micro solver using its own time step .
2. Run the macro solver on a slower pace than a standard macro model: t = t/M.
3. Exchange data between the micro- and macro-solvers at every step.
The intuitive idea is illustrated in Figure 18.2. What one does is to force the mi-
croscale model to accommodate the changes (here the change in x) in the macroscale
environment at a much faster pace. For example, assume that the characteristic
macro time scale is 1 second and the micro time scale is 1 femtosecond = 1015
second. In a brute force calculation, the micro model will run 1015 steps before
the macroscale environment changes. HMM makes use of the separation of the time
scales by running the micro model only until it is sufficiently relaxed, which requires
much fewer (say M) than 1015 steps, and then extracts the data in order to evolve
the macro system over a macro time step of 1 second. That means that HMM skips
1015 M micro steps of calculation, and it exchanges data between the macro and
micro solvers after every 1015 micro time step interval. The price one has to pay is
that one has to reinitialize the microscale solver at each macro time step, due to the
gap created by skipping 1015 M micro steps. One can take a different viewpoint.
If we define
t + t n ) = yn,k , k = 1, , M,
y(k (18.16)
The Heterogeneous Multiscale Methods with Application to Combustion 445

where t = t/M and yn,k is the k-th step solution to the microscale model at the
n-th macro step, the variable y is defined everywhere in the new rescaled time axis.
By doing so, we change the clock for the micro model, but we no longer need to
reinitialize the micro model every macro time step, since the gap mentioned above
no longer exists. Because the cost of the macro solver is typically very small com-
pared with the cost of the micro solver, we may as well run the macro solver using a
smaller time step (i.e. 1/M sec) and exchange data every time step. In this way the
data exchanged tend to be more smooth.
Using the setup discussed earlier, we can write the seamless algorithm as follows:
1. Given the current state of the micro variables u( ) and the macro variables U(t),
evolve the micro variables for one time step

u( + ) = S (u( );U(t)); (18.17)

2. Estimate D:
D = D(u( + )); (18.18)
3. Evolve the macro variables

U(t + t) = S t (U(t); D). (18.19)

In this algorithm, we alternate between the macro- and the micro-solvers, each run-
ning with its own time step (therefore the micro- and macro-solvers use different
clocks). At every step, the needed macroscale data is estimated from the results of
the micro-model (at that step) and is supplied to the macro-solver. The new values
of the macro-state variables are then used to constrain the micro-solver.
Example: SDEs with multiple time scales. Consider the stochastic ODE:
dx
= f (x, y),
dt  (18.20)
dy 1 1
= (y (x)) + w.

dt
where w(t)
is the standard white noise. The averaging theorems suggest that the
effective macroscale equation should be of the form of an ODE:

dx
= F(x). (18.21)
dt
The original HMM with forward Euler as the macro-solver proceeds as follows:
1. Initialize the micro-solver, e.g. yn,0 = yn1,M ;
2. Apply the micro-solver for M micro steps:

t t n,m
y n,m+1
=y n,m
(yn,m (xn )) + , (18.22)

446 Weinan E, Bjorn Engquist and Yi Sun

for m = 0, 1, , M 1. Here { n,m } are independent normal random variables

with mean 0 and variance 1;
3. Estimate F(x):
1 M
Fn = f (xn , yn,m );
M m=1
(18.23)

4. Apply the macro-solver:

xn+1 = xn + t F n . (18.24)
In contrast, the seamless HMM with forward Euler scheme is simply:

n n
yn+1
=y
n
(y (x )) +
n
, (18.25)

xn+1 = xn + t f (xn , yn+1 ), (18.26)

where { n } are independent normal random variables with mean 0 and variance 1.
Note that for HMM, we have xn x(n t), but for the seamless algorithm, we have
xn x(n t) = x(n t/M).

18.1.3 Stability and Accuracy

One of the main advantages of HMM is that it comes with a general framework for
conducting error analysis. The basic idea, as was explained in [8], is to compare the
HMM solution with the solution of the selected macroscale solver for the effective
macroscale model. Their difference is caused by an additional error in the HMM
solution due to the error in the data estimation process. This new error term is called
the HMM error, denoted by e(HMM). Assume that both the HMM and the macro-
solver for the effective macroscale model can be expressed in the form (note that
this is not necessarily the forward Euler scheme):
n+1
UHMM n
= UHMM + tF (UHMM
n n1
,UHMM , ), (18.27)

U Hn+1 = U Hn + t F(
U Hn , U Hn1 , ). (18.28)
Note that

||Qu UHMM || ||Qu U||

+ ||U H U||
+ ||UHMM U H ||, (18.29)

where U is the solution of the macroscale model, and U H is the numerical solution
to the effective macroscale model computed using (18.28). The first term on the
right hand side of (18.29) is due to the error of the effective model; the second term
is due to the error in the macroscale solver; the third term is the HMM error, due to
the error in the estimated data. Normally we expect that estimates of the following
type hold:
The Heterogeneous Multiscale Methods with Application to Combustion 447

||Qu U|| C , (18.30)

||U U H || C( t)k , (18.31)

where k is the order of accuracy of the macro-solver. In addition, define

n ,U n1 , ) F (U n ,U n1 , ).
e(HMM) = max F(U (18.32)
U

Then under suitable stability conditions, one can show that [8]:

||UHMM U H || Ce(HMM) (18.33)

for some constant C. Therefore, we have

||Qu UHMM || C( + ( t)k + e(HMM)). (18.34)

The key in getting concrete error estimates and thereby giving guidelines to design-
ing multiscale methods lies in the estimation of e(HMM). This is specific to each
problem. Explicit examples can be found in [10].

18.2 Capturing Macroscale Interface Dynamics

In this section we discuss how the HMM philosophy can be applied to the study
of interface motion in a multi-scale setting. This discussion follows that of [3]. Our
main interest is to capture the macroscale dynamics of the interface in cases where
the velocity is not explicitly specified. Instead, it has to be extracted from some
underlying microscale model.

18.2.1 Macroscale Solver: The Interface Tracking Methods

We can select as the macroscale solver any conventional methods for interface dy-
namics such as the level set method [25], the front tracking method [17], or the
segment projection method [30]. Here we will follow [3, 29] and focus on the level
set method and the front tracking method.
In the framework of the level set method, an interface is described as the zero
level set of a globally defined function, , called a level set function. All operations,
in particular evolution, are then performed on this function in place of the interface
of interest. The level set function satisfies the PDE:

t +V = 0,

where V is a globally defined velocity field. On the interface, this is equivalent to

448 Weinan E, Bjorn Engquist and Yi Sun

Fig. 18.3: A pictorial description of the HMM setup showing the link between the
macroscopic and microscopic levels.

t Vn | | = 0, (18.35)

which involves the normal velocity only (Figure 18.3). In many problems, though,
the velocity is only naturally specified at the interface.
The front tracking method is based mainly on the point set definition, in which an
interface is simply defined by a set of discrete points, together with an interpolation
rule connecting these points (Figure 18.3). The motion of the interface is done by
moving each point individually as a Lagrangian marker. Given a velocity field v(x,t)
by which the interface should be moved, each discrete point xi is advected by v(x,t),

dxi
= v(x,t), (18.36)
dt
After the points have been moved, a new parametric connection need to be calcu-
lated. Moreover, any connectivity data structure must be updated continuously to
avoid clustering or depletion of points. New points can be added where necessary
by using the calculated parameterization, and crowded points on other parts of the
interface can be removed.

18.2.2 Estimating The Macroscale Interface Velocity

The data that need to be estimated from the microscale models are the normal ve-
locities of the macroscale interface at each macroscale grid point. The first step is to
locally reconstruct the interface. If the normal velocity of the macroscale interface
is known to only depend on the orientation of the local tangent plane of the interface
The Heterogeneous Multiscale Methods with Application to Combustion 449

then we may approximate the interface locally by a hyperplane. On the other hand,
if the normal velocity is also known to depend on the local curvature, a quadratic
approximation is needed.
As illustration, we first consider the homogenization problem. The level set rep-
resentation of the microscale model in this case takes the form
 x
t + c x, | | = 0, (18.37)

which describes motion in the normal direction at speed c. In this case, we can use
the hyperplane reconstruction. Cheng and E [3] suggests working on a transformed
coordinate through a change of variables so that this hyperplane coincides with the
{xn = 0}-plane. The original microscale model (18.37) is then solved in a domain
in the transformed space, rectangular with sides orthogonal to the coordinate axes,
that should be larger than the size of the periodic cell or the correlation length of
c. Periodic boundary conditions are imposed in the x1 , x2 , . . . , xn1 -directions and a
periodic jump condition can be imposed in the xn -direction.
To extract the quantity of interest, at each microscale time step, the microscale
Hamiltonian is averaged in the central region of the domain to reduce spurious ef-
fects that may arise from the boundary and the chosen boundary conditions. Denote
this value by h (t) 
1
h (t) = c (x)| (x,t)|dx, (18.38)
| |
the velocity of the front at the particular location is obtained from
 t t 
1
Vn = h (t)K dt. (18.39)
| | 0 t

Here K is an averaging kernel, as discussed earlier. Having obtained Vn at all the

macroscale grid points at the interface, the interface can be evolved using standard
procedures in the level set method.
Cheng and E [3] also considered the case where the microscale model is de-
scribed by a phase-field equation:
  x    x   1 V
ut = b x, u + a x, u (u), (18.40)
u
where a(x, y) > 0 and b(x, y) are smooth functions that are either periodic in y or
stationary random in y with rapidly decaying correlation at large distances and V
is a double well potential with minima at u = , . For the details concerning this
example, as well as the numerical results, we refer to [3].
In [29], the authors used the HMM framework to design a multiscale method for
solving the numerical difficulty due to rapid microscale transition at the interface,
which is modeled by a stiff reaction-diffusion-convection equation [15, 21]

u
+ f(x, y) u = (u) + u (18.41)
t
450 Weinan E, Bjorn Engquist and Yi Sun

with the source term  

1 1
(u) = u(u 1) u . (18.42)
2
When the numerical resolution is not sufficiently high, the interface or discontinuity
may propagate at a nonphysical velocity. This problem rises from the smearing of
the discontinuity caused by the transport, which introduces a nonequilibrium state
into the calculation. Then the stiff source terms turn on and immediately restore
equilibrium, thus shifting the discontinuity to a grid boundary. To overcome this
difficulty, we can evolve the model locally around the interface on a much finer grid
x with a time step t that resolves the smallest scales enforced by . As shown in
[29], since the velocity quickly relaxes to a quasi-stationary value after , which is
much shorter than the macro step tM , we may estimate the velocity after the relax-
ation time and use it to move the interface in the front tracking method on the
macroscale. In this way we can save much computational cost. For the details and
the numerical results, we refer to [29].
In summary, here are the steps of the HMM technique for interface tracking:
1. The initial macroscale interface location (t) represented by a level set function
or a set of marker points {Xi (t)} in the front tracking method is given.
2. For each of these macroscale points, choose the microscale domains Di and re-
construct the initial states for the microscale model.
3. Evolve the microscale model in the domains Di for some micro time steps. Time
or ensemble averaging may be used to estimate the velocities vi .
4. Move the interface for one macroscale time step by taking {vi } as the normal
velocity of the interface.
5. Solve the macroscale system for one macroscale time step with the level set
method or the front tracking method. Perform the reintialization of the level set
function or the redistribution of the marker points in the front tracking method if
needed.
6. Stop if the desired time is reached in the macroscale model. Otherwise, repeat
from step 2 using the newly evolved interface location.
This approach is closely related to the general philosophy of the adaptive mesh
refinement (AMR) methods [1]. As in AMR methods, the end result is to work with
a mesh which is locally refined at the interface and coarsened away from the inter-
face. However, there are some important differences on how the mesh refinement
is carried out. The first is that AMR would refine everywhere near the interface,
whereas HMM would refine only around the macroscale grid points on the inter-
face. The second is that AMR methods aim at computing accurately the local so-
lutions for all times, and for that purpose local time stepping is carried out for all
times. In contrast, HMM aims only at computing a macroscale quantity, the velocity
of the interface, and this does not require following the microscale solution for all
times.
The Heterogeneous Multiscale Methods with Application to Combustion 451

18.3 HMM Interface Tracking of Combustion Fronts

HMM has been applied to several combustion models in [29], such as 1-D Majdas
model and the reactive Euler equations in one and two dimensions. In these applica-
tions, the goal is to capture the front dynamics on the macroscale with the velocity
of the front computed in the microscale region.

18.3.1 Majdas Model

We consider a simplified combustion model derived by Majda in [23]. This model

is a 2 2 system, which couples Burgers equation to a chemical kinetics equation:
 
1 2
Ut + U q0 Z = 0, (18.43)
2 x
Zx = K(U)Z, (18.44)

where U is a variable with some features of pressure or temperature. Z is the mass

fraction of unburnt gas, where Z = 1 describes the unburnt gas and Z = 0 the com-
pletely burnt state. q0 > 0 is the heat release and K(U) is the reaction rate. If U
represents the temperature T , the function K(T ) is typically given by the Arrhenius
form
K(T ) = K0 eA/T , (18.45)
where K0 is the rate constant and A is the activation energy [24]. The reaction rate
is negligible at low temperature and grows exponentially fast if the temperature is
high enough. For computational purposes, the reaction rate (18.45) may be replaced
by a discrete ignition temperature kinetics model

K0 if T Tign ,
K(T ) = (18.46)
0 if T < Tign

where Tign is the ignition temperature. The numerical difficulty in solving this model
problem comes from the stiffness of the chemical source term when K0 = 1 is very
large.
Colella, Majda, and Roytburd [5] applied the Godunovs method and its high res-
olution extension [6] to this problem, using a fractional step method. Given (Uin , Zin ),
in the first step, we compute Zin+1 via numerical integration of the ODE (18.44) with
the formula  n )
K(Uin ) + K(Ui1
Zi1 = Zi exp x
n+1 n+1
(18.47)
2
and the initial condition Zin+1 = 1 for i large enough. In the second step, we com-
pute Uin+1 from {Uin }, {Zin+1 } by applying the EngquistOsher (EO) scheme [13]
452 Weinan E, Bjorn Engquist and Yi Sun

to (18.43). The formula for computing Uin+1 is given by

Uin+1 = Uin ( f+ (Uin ) f+ (Ui1

n n
)) + ( f (Ui+1 ) f (Uin ))
x
t
+ q0 (Zin+1 Zi1 n+1
), (18.48)
x
 
0 if U 0, U 2 /2 if U 0,
with f+ (U) = and f (U) =
U 2 /2 if U 0, 0 if U 0.

Fig. 18.4: Numerical results of Majdas model using the fractional step method
(18.47) and (18.48) with the initial data in (18.49) for different values of the re-
action rate constant K0 . We take Tign = 0, q0 = 0.8, x = 0.01, and CFL number
= 0.5. Reprinted from [29] with permission. Copyright 2006, Society for Indus-
trial and Applied Mathematics.

Figure 18.4 shows numerical results of q0 = 0.8 and different values of the reac-
tion rate constant K0 with the initial data
The Heterogeneous Multiscale Methods with Application to Combustion 453

Ul = 1.0 if x 0,
U(x, 0) = (18.49)
Ur = 0.5 if x > 0.

As shown in Figure 18.4(a), for small value of K0 t (= 0.05), the fractional step
method gives the results of the detonation profiles of U and Z that match the true
solutions well. For larger K0 t (= 0.5), the spike of the detonation profile of U
drops down due to the underresolution around it, but the location of the detonation
is still correct, as shown in Figure 18.4(b). However, for the case K0 t = 5, shown in
Figures 18.4(c) (t = 0.4) and 18.4(d) (t = 0.8), the scheme produces a nonphysical
solution, which is propagating with a velocity of one grid per time step as mentioned
before.

Fig. 18.5: Numerical results of Majdas model at time t = 0.4 using HMM with
initial data (18.49) for the reaction rate constant K0 = 1000. We take Tign = 0,
macroscale grid size x = 0.01, microscale grid size x = 0.01/50 = 2 104 ,
and a fixed CFL number 0.5. Reprinted from [29] with permission. Copyright
2006, Society for Industrial and Applied Mathematics.

Figure 18.5 shows numerical results at time t = 0.4 using the HMM interface
tracking technique introduced in previous section with the initial data in (18.49)
for the reaction rate constant K0 = 1000. The locations of detonation match the
true solution quite well. We used 200 macroscale grids on the interval [1, 1]. The
refinement ratio between the macroscale and microscale grid sizes is 50. When K0 =
104 , a higher refinement ratio of 500 can achieve the same expected result. As shown
in the right panel in Figure 18.5, the spiked strong detonation profile is resolved
on the microscale grids. The interface condition on the macroscale enters into the
flux evaluation in the EO scheme (18.48) only by inserting the new extrapolated
values.
454 Weinan E, Bjorn Engquist and Yi Sun

18.3.2 Reactive Euler Equations

In this section, we extend our HMM technique to 2-D reactive Euler equations.
The results for 1-D multispecies reactive Euler equations can be found in [29]. The
system of equations in 2-D is given by

U F(U) G(U)
+ + = S(U) (18.50)
t x y
with

u v 0
u u2 + p uv 0
2

U = v , F(U) = uv , G(U) = v + p , S(U) = 0

E u(E + p) v(E + p) 0
Z uZ vZ K(T ) Z

where u is the velocity of the gas in the x-direction and v is the y-component of the
velocity. The total energy is given by the equation of state
p 1
E= + (u2 + v2 ) + q0 Z.
1 2
where q0 is the heat release and is the ratio of specific heat for ideal gas. The
reaction rate K(T ) is described by the discrete ignition temperature kinetics model
(18.46). The temperature is given by T = p/ R, where R is the specific gas con-
stant.
The fractional step method for approximating the solutions of (18.50) is given by
t
U n+1 = LODE Ly Roe
t
Lx Roe
t
U n. (18.51)
t and L t denote Roes scheme [28] for solving the nonreactive system
where Lx Roe y Roe
t denotes the ODE-solver for the source
in the x- and y-directions, respectively. LODE
term.
Here we consider a radially symmetric detonation wave in two dimensions. The
initial values consist of totally burnt gas inside of a circle with radius 0.3 and to-
tally unburnt gas everywhere outside of this circle. The burnt and unburnt states are
chosen as follows (same as Example 3 in [18]):

b = 1.4, ub = vb = 0, pb = 1, Zb = 0 if x2 + y2 0.32 ,

uu = 0.5774 cos ,
u = 0.8876, pu = 0.1917, Zu = 1 elsewhere.
vu = 0.5774 sin ,
(18.52)
The Heterogeneous Multiscale Methods with Application to Combustion 455

where is the angle in polar coordinates. The ignition temperature is set to Tign =
0.26. The other parameters are set to = 1.4, q0 = 1, and R = 1. The computational
domain is [0, 1] [0, 1].

Fig. 18.6: Numerical solutions of 2-D reactive Euler equations at time t = 0.5 using
fractional step method (18.51) with Roes scheme and the initial data in (18.52) for
K0 = 100 and Tign = 0.26. We take x = y = 0.01 and CFL number 0.5. Reprinted
from [29] with permission. Copyright 2006, Society for Industrial and Applied
Mathematics.

Figure 18.6 shows numerical results at time t = 0.5 for the reaction rate constant
K0 = 100. The combustion front is a quarter of a circle that should expand as the
system evolves. The distance from the center to the front should be 0.8 at this time.
However, for a larger K0 = 1000, the combustion front moves too fast, and there are
nonphysical intermediate states for the density and the pressure, as evidenced by
the top panel of Figure 18.7. Moreover, the circular geometry cannot be resolved, as
shown in the bottom panel.
By using the HMM interface tracking technique introduced in previous section,
we resolve the system (18.50) with the initial data in (18.52) at time t = 0.5 for
the reaction rate constant K0 = 1000. We used 100 100 macroscale grids on the
domain [0, 1] [0, 1]. The refinement ratio between the macroscale and microscale
grid sizes is 50. We note that we evolve the 1-D system along the normal line in-
stead of 2-D system (18.50) as our microscopic solver. This idea can achieve both
accuracy and efficiency. The spiked strong detonation profiles of all components are
456 Weinan E, Bjorn Engquist and Yi Sun

Fig. 18.7: Numerical solutions of 2-D reactive Euler equations at time t = 0.5 using
fractional step method (18.51) with Roes scheme and the initial data in (18.52)
for K0 = 1000 and Tign = 0.26. We take x = y = 0.01 and CFL number 0.5.
Reprinted from [29] with permission. Copyright 2006, Society for Industrial and
Applied Mathematics.

resolved on the microscale grids that are similar to the profile shown in the right
panel of Figure 18.5. Figure 18.8 shows that the combustion front moves with the
correct velocity and the circular geometry is resolved quite well on the grid. The
small oscillation close to the front is typical of high resolution shock capturing with
steep front. The accuracy of the whole method is first order since it is based on the
time-splitting method and the upwind scheme.

18.4 Conclusions

The examples presented here are admittedly quite simple. However, they do show
the promise of HMM. In principle, the philosophy also applies to more complex
situations. In practice, many other issues have to be dealt with, such as the handling
of intermediate scales. It is likely that HMM should be combined with more tradi-
tional techniques such as AMR in order to produce numerical algorithms that are
truly effective for problems of practical interest.
The Heterogeneous Multiscale Methods with Application to Combustion 457

Fig. 18.8: Numerical solutions of 2-D reactive Euler equations at time t = 0.5 us-
ing HMM with Roes scheme and the initial data in (18.52) for K0 = 1000 and
Tign = 0.26. We take macroscale grid size x = y = 0.01, microscale grid size
x = y = 0.01/50 = 2 104 , and a fixed CFL number 0.5. In the contour plot of
mass fraction Z, we also show two sets of marker points at different times. There
are 8 segments between the points represented by at time t = 0.2 and 16 seg-
ments between the points represented by at time t = 0.5 since we perform a
redistribution of the marker points after time t = 0.2 due to the increasing distance
between the neighboring points. Reprinted from [29] with permission. Copyright
2006, Society for Industrial and Applied Mathematics.

Acknowledgements W. E is supported in part by ONR Grant N00014-01-0674 and DOE Grant

DE-FG02-03ER25587. B. Engquist is supported in part by NSF Grant DMS-0714612. Y. Sun is
supported by NSF Joint Institutes Postdoctoral Fellowship through SAMSI.

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Chapter 19
Lattice Boltzmann Methods for Reactive and
Other Flows

Christos E. Frouzakis

Abstract The lattice Boltzmann method (LBM) is receiving increasing attention in

recent years as an alternative approach for computational fluid dynamics. Through
its kinetic theory origin, the method inherits the physically appealing particle pic-
ture that can be adapted to simulate multiscale and multiphysics systems with sizes
ranging from the microscale (where the continuum hypothesis may break down) to
macroscale applications. The method is characterized by its straightforward imple-
mentation in complex geometries and the fact that it involves only nearest neighbor
interactions without global operations, making LBM algorithms ideally suited for
parallelization. However, the method in general employs a larger number of degrees
of freedom per grid point than classical CFD approaches, and parallel implemen-
tation may be essential in order to meet the higher memory requirements. In this
chapter, an overview of the method and its applications is presented focusing on re-
cent model developments for the description of the averaged macroscopic behavior
of isothermal and non-isothermal, single- and multi-component and reactive flows.

19.1 Introduction

The conventional approach for modeling most scientific and engineering flows is
based on the continuum description of macroscopic behavior formulated as partial
differential equations for a few fields (e.g. continuity and Navier-Stokes equations
for the density and momentum, respectively, in the case of isothermal flows). A
numerical technique (finite difference/volume/element, spectral or spectral element
method) is then employed to obtain the discretized set of equations on a topologi-
cally connected mesh that can be solved numerically on the computer.

Christos E. Frouzakis
Aerothermochemistry and Combustion Systems Laboratory, Swiss Federal Institute of Technology,
Zurich, Switzerland, e-mail: frouzakis@lav.mavt.ethz.ch

T. Echekki, E. Mastorakos (eds.), Turbulent Combustion Modeling, 461

Fluid Mechanics and Its Applications 95, DOI 10.1007/978-94-007-0412-1 19,
Springer Science+Business Media B.V. 2011
462 Christos E. Frouzakis

Alternatively, since on the microscopic level fluids consist of discrete particles

(atoms or molecules interacting via classical or quantum mechanics potentials and
following the Newton equations of motion), one can opt for a description based on
physical particles and obtain the evolution of macroscopic variables as collective
averages over an enormous number of individual trajectories. Introduced about fifty
years ago, molecular dynamics (MD) is probably the most widely used family of
particle methods [2]. It provides the most detailed description by solving Newtons
equations of motion to track the position and velocity of each atom or molecule in
the system. While MD yields the correct description of fluids on the microscopic
as well as the hydrodynamic scales, the typical length and time scales that can be
simulated in practice are of the order of a few tens of nanometers and a few hundred
nanoseconds, respectively.
A large variety of (physical as in MD or notional) particle methods have been
developed for the simulation of fluid flows at different scales. They include Monte
Carlo and direct simulation Monte Carlo methods (DSMC) [17] at the microscopic
level, lattice gas models [41], dissipative particle dynamics [36, 53] at the meso-
scopic, and smoothed particle hydrodynamics [67], vortex particle methods [34],
and the fluid particle model [37] at the macroscopic level.
This chapter focuses on the lattice Boltzmann method (LBM), a mesoscopic ap-
proach where notional particles exist on a set of discrete points spaced at regular
intervals to form a lattice. Time is also discrete and during a time step the particles
move between neighboring (conventionally, nearest neighbor) lattice sites along pre-
defined directions, and then scatter according to carefully chosen kinetic rules that
satisfy local conservation laws.
The lattice Boltzmann methods evolved from the FHP model [41], the first accu-
rate lattice gas cellular automaton (LGCA) to simulate fluid flow proposed in 1986
by Frisch, Hasslacher and Pomeau, who realized the importance of the lattice sym-
metry for the recovery of the Navier-Stokes equation. Unfortunately, the advantage
of the exact representation of the system state through Boolean variables marking
the presence or absence of a particle at a certain lattice location with a certain veloc-
ity were overshadowed by major drawbacks: a relatively high fixed viscosity, lack of
Galilean invariance, and the intrinsic stochastic noise which necessitates averaging
over long time and large areas. The suggestion of McNamara and Zanetti [63], and
Higuera and Jimenez [52] to replace the particles (represented by a Boolean number)
by the single-particle probability distribution function led to the lattice Boltzmann
model (LBM) which turned out to be suitable for the efficient simulation of a broad
class of flows. For a detailed description of LGCA and for the development of the
LBM from LGCA the reader is referred to [82, 90].
Due to its capability to describe different physics in an algorithmically simple
and elegant way, the lattice Boltzmann method is attracting increasing interest from
the application, the implementation and the theoretical sides alike. Additional ad-
vantages include the linear and exact description of the advection term, eliminating
some of the difficulties plaguing conventional CFD solvers, the local character of
the update rules, resulting in nearly ideal amenability to parallel implementation,
and the ease with which complex geometries can be set up.
Lattice Boltzmann methods 463

This chapter reviews some recent developments of stable and efficient models
addressing the physico-chemical processes relevant to turbulent combustion (turbu-
lent flow, mixing of multicomponent systems, strongly non-isothermal effects and
reaction). It is practically impossible to present fully the method, its foundation and
review the rapidly expanding literature on lattice Boltzmann methods within the
limits of this chapter. Additional information on the method and its applications can
be found in the review of Chen and Doolen [22], in the books of Wolf-Gladrow [90],
Succi [82], Sukop and Thorne [87], and the references therein.

19.2 The Boltzmann Equation

19.2.1 Basic Considerations

The microscopic description of flows is based on moving and colliding parti-

cles. Instead of following the enormous number of individual particles, Boltzmann
proposed to track the temporal evolution of the one-particle distribution function
f (x, c,t) prescribing the probability of finding the center of a particle with velocity
c at location x at time t. For a sufficiently dilute gas in the presence of an external
force F acting on the particles, the temporal evolution of f (x, c,t) is described by
taking into account only two processes: the free flight of the particles (second term
in the left hand side) and their collisions (right hand side)

f
+ c x f + F c f = Q( f ). (19.1)
t
Assuming only binary collisions and that the velocity of a particle is uncorrelated
with its position (the molecular chaos assumption), Boltzmann expressed the colli-
sion integral Q( f ) as a nonlinear integral.
The resulting nonlinear integro-differential Boltzmann equation is difficult to
solve and various simplifications have been proposed, resulting in kinetic models
that preserve certain features of the Boltzmann equation. The most commonly used
one is the single-relaxation time model BGK model proposed by Bhatnagar, Gross
and Krook
1
QBGK = ( f f eq ), (19.2)

describing the relaxation of f to the local Maxwellian distribution function
 
(c u)2
eq
f = exp (19.3)
(2 RT )D/2 2RT

with a characteristic time scale . Local Maxwellians are parametrized by the val-
ues of the fluid density , mean velocity u, and temperature T (D is the spatial
dimension, and R is the gas constant.)
464 Christos E. Frouzakis

H theorem. One of the most important contributions of Boltzmann is the cel-

ebrated H theorem, which roughly states that entropy does not decrease in time.
Related to entropy by S = H, the H-function

H(t) = f ln f dx dv

therefore satisfies
dH
0,
dt
with the equality holding at equilibrium. The H-theorem then dictates that the re-
laxation to the unique global Maxwellian distribution must be accompanied by a
monotonic increase of the entropy. The local Maxwellian can also be specified as
the maximizer of the Boltzmann entropy function, subject to the constraints of local
conservation laws, and is also called the local equilibrium distribution function.
Link to hydrodynamics. From the point of view of the Boltzmann equation,
the macroscopic hydrodynamic fields are the first few moments of the distribution
function
  
DRT (c u)2
= f dc, u = c f dc, = f dc. (19.4)
2 2
Methods of reduced description. The closed set of equations for the macro-
scopic hydrodynamics fields can be obtained by reducing the Boltzmann equation
using one of the classical reduction methods: the Hilbert method, the Chapman-
Enskog method, and the Grad moment method, as described briefly in e.g. [42].
The Chapman-Enskog (CE) method approximates the solution of the Boltzmann
equation as a formal series expansion in powers of the Knudsen number Kn, the ra-
tio of the mean free path to the scale of variations of the macroscopic hydrodynamic
fields. It starts from the known local equilibrium, and proceeds to derive macro-
scopic equations at different orders of truncation: the Euler hydrodynamics at Kn0 ,
the Navier-Stokes hydrodynamics at Kn1 , the Burnett hydrodynamics at Kn2 , etc.
The equations past the Navier-Stokes description become ill-conditioned making
it difficult to extend the hydrodynamic description into the highly non-equilibrium
regime. In addition to enabling the derivation of macroscopic equations without a
priori guessing, the method allowed to express transport coefficients in terms of
particle interactions (see, for example, [19, 20]).
Grads method is based on the assumption of the decomposition of the moments
into a set of slowly evolving moments M  that does not change significantly during
a time of order (slow dynamics) in comparison to the rest of the moments M  (fast
dynamics); M  includes density, momentum, energy and a subset of higher-order
moments. Towards the end of the fast evolution, the values of M  become functions
of M  , and, for t  , the dynamics are fully determined by the evolution of the
slow moments. Grad [44] approximated the solution to the Boltzmann equation as a
series expansion in terms of Hermite orthogonal polynomials in velocity space and
Lattice Boltzmann methods 465

kept thirteen of the leading terms corresponding to the five macroscopic variables
(density, momentum and energy) and their fluxes.
The Chapman-Enskog and Grads methods are also employed in the derivation
of lattice Boltzmann schemes.

19.2.2 Lattice Boltzmann Model

19.2.2.1 Original Lattice Boltzmann

The lattice Boltzmann model is based on the realization that the continuum of mi-
croscopic velocities can be reduced to a finite set without sacrificing the accurate
description of the macroscopic dynamics. Lattice Gas and Lattice Boltzmann ap-
proaches construct computationally efficient discrete kinetic models obtained by
discretizing the velocity space on a set of q discrete velocities,

C = {c1 , c2 , . . . , cq },

where q is as small as possible. The distinguishing characteristic of these methods is

that the discrete velocities define the links of a lattice on which notional particles are
allowed to propagate. The task is to find a set that recovers the target macroscopic
hydrodynamic behavior.
Denoting by fi (x,t) the populations corresponding to the discrete velocities
ci , i = 1, . . . , q, the continuous in space and time standard lattice Boltzmann model
adopts the BGK model and assumes that during collision the populations relax to
their equilibrium value with a single relaxation time

fi fi 1
+ ci = Qi ( fi ) = ( fi f ieq ) + Fi , i = 1, . . . , q, (19.5)
t x
where = 1, . . . , D is the index for the spatial dimensionality of the system (D=1,2,
or 3 for one-, two-, or three-dimensional geometries, respectively) over which sum-
mation is assumed. The source term Fi represents the force term of the Boltzmann
equation and provides an elegant way to construct models for complex problems
(fast flows, heat transfer, chemical reactions, phase transitions etc. [82]).
The easiest way to discretize the configuration (i.e. physical) space is to use a
regular lattice so that for every grid node x, x + c t is also a grid node. This way,
the information at all grid nodes is automatically known at the next time step. The
lattices used in LBM are typically characterized as DdQq, where d denotes the spa-
tial dimension and q the number of discrete velocities. Commonly used lattices for
one- (D1Q3), two- (D2Q9) and three-dimensional simulations (D3Q19) are shown
in Fig. 19.1.
In the original heuristic LB formulations, the equilibrium function was cho-
sen as low-order (typically second) polynomials of the hydrodynamic fields (e.g.
fluid density and velocity in the isothermal case), and the polynomial coefficients
466 Christos E. Frouzakis

Fig. 19.1: Commonly used lattices for (a) 1-D, (b) 2-D, and (c) 3-D simulations.

ci wi
D1Q3 {0, 1, 1} { 46 , 16 , 16 }
D2Q9 {(0, 0), { 16
36 ,
(1, 0), (0, 1), (1, 0), (0, 1), 4
36 ,
(1, 1), (1, 1), (1, 1), (1, 1)} 36 }
1

D3Q19 { (0, 0, 0), { 12

36 ,
(1, 0, 0), (0, 1, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), (0, 0, 1), 2
36 ,
(1, 1, 0), (1, 1, 0), (1, 1, 0), (1, 1, 0), 1
36 ,
(1, 0, 1), (1, 0, 1), (1, 0, 1), {1, 0, 1}, 36
1

(0, 1, 1), (0, 1, 1), (0, 1, 1), (0, 1, 1) } 36 }

1

Table 19.1: Discrete velocities grouped in energy levels that carry the same
weights for the D1Q3, D2Q9 and D3Q19 LB models.

were determined a posteriori using the Chapman-Enskog procedure so that the cor-
rect target hydrodynamics (e.g. the Navier Stokes description) could be recovered
in the large-scale long-time limit [82]. The way of constructing kinetic models is
not unique, and different equilibrium functions and/or discrete velocity sets can be
used. It was found, however, that not all models constructed this way result in stable
numerical schemes. While for isothermal hydrodynamics relatively stable schemes
emerged essentially by trial and error, the stability problem was particularly severe
when temperature was allowed to vary.
LBM was freed from its LGA origin when it was shown that it can be obtained
directly from a special finite-difference discretization of the continuous BGK form
of the Boltzmann equation [1, 48, 49]. The discrete velocities were related to the
roots of Gauss-Hermite quadratures, which ensure accurate evaluation of the low-
order moments of the distribution function defining the desired macroscopic hydro-
dynamics, while the equilibrium function was taken to be a second-order expansion
of the Maxwellian in terms of low fluid velocity,
Lattice Boltzmann methods 467
 
c u (c u)2 u u
fieq = wi 1 + 2 + ,
cs 2c4s 2c2s

with cs = RT being the speed of sound of the model, and wi related to the weights
of the quadrature [48]. The discrete velocities and the corresponding weights for
three commonly used LB lattices are given in Table 19.1. It was further shown that
the resulting fieq was the one obtained by the heuristic approach.
The second-order expansion of the equilibrium function restricts application to
hydrodynamics at the Navier-Stokes level. Shan and He [78] realized that if discrete
velocities are constructed from the zeros of the Hermite polynomials, the method
of discrete velocity is essentially equivalent to Grads moment method [44]. They
proposed expanding the equilibrium to higher orders and adopting higher degree
Gauss-Hermite quadratures (i.e. lattices with larger number of discrete velocities),
so that higher-order approximations to the Boltzmann equation important for non-
isothermal and more complex flows can be constructed. A recent presentation of this
systematic approach can be found in [79]. For orders higher than three, however,
the ratios of the roots of the Hermite polynomials defining the discrete velocities are
not in general integers and do not coincide with the lattice nodes such that the exact
space discretization of the advection step offered by the lattice has to be abandoned.
As in the continuous Boltzmann equation case, the hydrodynamic fields are the
first few moments of the populations:
q q q
= fi , ua = fi ci , DT + u2 = fi c2i . (19.6)
i=1 i=1 i=1

19.2.2.2 Entropic Lattice Boltzmann

In kinetic theory, the equilibrium distribution function is the maximum entropy state,
and Boltzmanns H-theorem, which ensures the increase of entropy, also ensures sta-
bility. Entropic Lattice Boltzmann models are constructed with the target of restor-
ing the H-theorem that is not satisfied by the standard LBM schemes discussed so
far, thereby ensuring numerical stability of the scheme [18, 54].
From the perspective of kinetic theory, local equilibria minimize the convex H-
function subject to the constraints of the locally conserved hydrodynamic fields. In
the entropic construction of LB models, evaluation of Boltzmanns H-function by
the Gauss-Hermite quadrature defines the discrete form of the H-function
q  
fi
H = fi ln , (19.7)
i=1 wi

where the weights wi associated with the discrete velocities ci must be chosen ap-
propriately. The equilibrium populations fieq are then found as the minimizers of the
H function under the constraints of the locally conserved quantities, e.g.
468 Christos E. Frouzakis
q q q
f ieq = , ci fieq = u , c2i fieq = DT + u2
i=1 i=1 i=1

Analytical solutions for f eq

have been found in some cases [5]. For example,
the isothermal 1-D Navier-Stokes description at some reference temperature T0 was
obtained for the D1Q3 model. In two and three dimensions, the discrete velocities
and the weights of the D2Q9 and D3Q27 models were constructed by taking tensor
products of the one-dimensional lattice, and the explicit solution of the constrained
minimization problem in D dimensions is the product of D one-dimensional solu-
tions [4, 5]


D    2 u + 1 + u2
ci / 3cs
3
fi = wi 2 1 + u
eq 2 , (19.8)
=1 1 + u 3

where the speed of sound is cs = T0 in lattice units and the fluid velocity is non-
dimensionalized by cs . It is worth noting that the second-order Taylor series expan-
sion of the equilibria (19.8) coincides with the polynomial equilibria used in the
standard D2Q9 LBM. This implies that this model possesses an H-function and
explains its stability in comparison to other LB models. However, the equilibrium
expressions (19.8) are preferable because they approximate more accurately higher-
order moments needed to establish isothermal hydrodynamics. A tutorial presenta-
tion of the entropic lattice Boltzmann method for 1-D hydrodynamics can be found
in [56].
More generally, the solution to the minimization problem can be expressed in
product form as
D
f ieq = wi ci , (19.9)
=1

where ( , u), ( , u) are Lagrange multipliers associated with the mass and mo-
mentum conservation constraints, respectively, and can be obtained analytically by
perturbation around the zero-velocity solution ( , 0) = , ( , 0) = 1 to any de-
sired order in velocity [28]. The product form is particularly useful in reducing the
computational complexity of LB models with large number of discrete velocities
significantly.
In addition to the constraints imposed by local conservation, f eq must also re-
spect several other higher-order non-conserved moments. Failure to do so results
in incomplete Galilean invariance, and the macroscopic hydrodynamic description
cannot be completely recovered. In the non-isothermal case, the high-order mo-
ments of interest are the stress tensor, the energy flux tensor, and the rate of energy
flux tensor
Lattice Boltzmann methods 469
q
eq
P = fieq ci ci = T0 + u u + P

i=1
q
fieq ci ci ci = T0 (u + u + u ) + u u u + Q
eq
Q =
i=1
q
eq
R = fieq ci ci ci ci = (D + 2) T02 + T0 u2 + (D + 4) T0 u u +
i=1
u2 u u + R ,

respectively, where the right-most sides are the target moment expressions obtained
from kinetic theory when the primed terms become equal to zero.
It should be stressed that it is impossible to exactly match the Maxwell-Boltzmann
expressions and achieve full Galilean invariance with LBM. What can be achieved
is to approximate these expressions to such high orders in velocity, that numeri-
cally the deviations become negligible. This can be achieved by using high-order
LB models, i.e. models with a large number of discrete velocities, but the use of
quadratures of higher order [79] faces the difficulties associated with off-lattice ve-
locities and cannot, in general, guarantee stable numerical schemes. A systematic
approach to construct stable approximate models that are on the lattice was pro-
posed recently in the framework of the entropic construction [29, 30].
Stability can be further enhanced by employing either the entropic time stepping
instead of the the time stepping of standard LB schemes [4], or the Multi Relaxation
Time (MRT) formulation described below.

19.2.2.3 Lattice Boltzmann algorithm

The LBGK scheme is derived by integrating eqn. (19.5) using a second-order trape-
zoidal scheme to approximate the time integral of the collision term and a trans-
formed set of populations, gi (x,t) = fi (x,t) 2t Qi ( f j (x,t)) satisfying gi = fi to
eq eq

obtain an explicit second-order integration scheme [47, 56]

2 t
gi (x + ci t,t) = gi (x,t) + ( ) Qi (g j (x,t)) (19.10)
2 + t
 
= gi (x,t) + gi (x,t) gi (gi (x,t)) .
eq

Here, ( , t) = 22+ t t is the discrete inverse relaxation time which, for fixed t, is
restricted in the linear stability interval [0, 2]: 0 when ; 2 when
0. The relaxation time is related to the kinematic viscosity = / of the fluid via

= .
c2s

LBM operates on a highly efficient stream-along-links-and-collide-at-nodes sched-

ule, making the method almost ideally suited for parallel implementations. The al-
470 Christos E. Frouzakis

gorithm shown in Fig. 19.2 uses intermediate populations f to store the result of
the collision step, doubling the memory requirements. By carefully combining the
collision and the streaming steps, a single set of populations can (and should) be
used.

Fig. 19.2: Stream-along-links-and-collide-at-nodes LBGK algorithm.

19.2.3 Variations on the LBM Theme

LBM can be viewed as a reduced kinetic model that can reproduce targeted macro-
scopic behaviors. In this sense, many different approaches have been (and continue
to be) developed that differ in the form of the collision term, the type of the lat-
tice (choice of discrete velocities), the equilibrium function fieq , or any combination
of the above. The choice of the discrete velocities and of fieq was discussed in the
previous section. Here, we turn to some of the options available for the collision
operator.
Matrix form of the collision operator. The most common form of the collision
term is the single relaxation time BGK form, but other forms have also been pro-
posed. The LB formulation was actually born in matrix, multi-time relaxation form
[51, 52, 82],
Lattice Boltzmann methods 471

fi fi
x
eq
+ ci = Ai j ( f j f j ), i = 1, . . . , q.
t j

The components of the scattering matrix Ai j were considered as free parameters that
could be tuned to match the macroscopic fluid viscosity [82].
Multi-relaxation time. The simplicity of the single relaxation time (SRT) em-
ployed in the popular BGK model comes at a price: bulk and shear viscosity are
equal and the Prandtl number cannot be independently controlled. Driven from a
systematic stability analysis, alternative formulations known as multi-relaxation-
time (MRT) models map at each time step the distribution function fi from the
discrete velocity space to the moment space m = M f, where the velocity-moment
transformation matrix M is a critical component of MRT. Relaxation is done on
these moments using different relaxation rates

f(x + ci t,t + t) = f(x,t) M 1 S (m meq ),

S being the q q diagonal matrix of relaxation rates si which determine the trans-
port coefficients of the system. Before performing the streaming step, the values are
transformed back to discrete velocities and the propagation and boundary condition
implementation is performed as in the SRT model. In addition to more flexibility
in describing richer physics through individual control of all moments, MRT mod-
els can achieve better numerical stability properties in comparison to the original
BGK model by appropriate tuning of the eigenvalues of the transformation matrix.
Additional details can be found in e.g. [50].
Quasi-equilibrium LBM. A number of LB models for non-isothermal and
multi-component flows have been derived based on a representation of the fast-slow
decomposition of moment relaxations near a quasi-equilibrium (QE) state. Here,
distribution functions approaches equilibrium in two steps: first from the initial state
to the QE state, f , and then from the QE to the equilibrium, f eq each with its own
relaxation time [9]:

fi fi 1 1
+ ci = ( fi fi ) ( fi fieq ), i = 1, . . . , q,
t x 1 2
Although a single distribution function is considered in the previous equation, addi-
tional such functions can be introduced to describe flows where different transport
processes take place. In addition to the set of conserved fields, the QE LBM assumes
that there is a set of quasi-conserved slow fields, and the QE state is defined as the
minimizer of the H-function under the constraints of local conservation and of fixed
values of the slow fields. The choice of the constraints depends on the particular
problem [9, 12, 69, 73].
Force terms. Due to insufficient isotropy of the lattices, additional physics can-
not be included in the equilibrium terms. By introducing force terms to the right
hand side of the LB equation, a number of new fluid phenomena, including non-
isothermal and multiphase flows, can be simulated [82].
472 Christos E. Frouzakis

Lattice Boltzmann methods without a lattice. Attempts were also made to gen-
eralize the lattice to unstructured grids and employ a finite difference of finite vol-
ume method for the spatial discretization. The experience with these approaches
were not so positive, since the methods lose their essential advantage of simple al-
gorithm and high accuracy [46]. In addition to the increased numerical dissipation
introduced by the interpolation of the populations on the lattice, stability imposes a
significantly shorter integration time step [16].

19.2.4 Initial and Boundary Conditions

In LBM, initial and boundary conditions must be specified for the whole set of dis-
tribution functions. What is available from the macroscopic description are the con-
served low-order moments of the distribution functions (density, momentum etc.),
while the non-conserved ones (stresses and other high-order fluxes) are unknown.
In many cases, the higher-order moments will relax and become slaved to the low-
order ones that determine the dynamics after a short transient. It then suffices to
initialize populations at equilibrium, which is a function only of the macroscopic
fields. For incompressible flows, Mei et al. [64] proposed an iterative procedure to
generate consistent initial conditions that takes into account the equivalent set of
moments that can be generated by the populations.
There is no generally accepted formulation for the boundary conditions and dif-
ferent approaches exist. At the inflow and outflow boundaries, the incoming and out-
going populations, respectively, are unknown and have to be approximated by other
means (see, for example, Chikatamarla et al. [29] for a method based on Grads
moment closure). On flat walls, the simplest and most commonly used boundary
condition is the bounce-back scheme that was taken from the LG method [22] and
exists in different forms, which can influence the accuracy and stability of the com-
putation.
LBM has received increasing attention recently for microfluidic simulations [3,
8, 75, 76, 81, 83]. The need for a boundary condition, that can describe properly
the flow at non-vanishing Knudsen numbers, led to the introduction of the diffusive
wall boundary condition [3]. Originating from the continuum kinetic theory, this
boundary condition appears to be more appropriate for microfluidic simulations [8].
LB is particularly well-suited for complex geometries which can be easily de-
scribed by marking the character of each lattice node as fluid or solid. For curved
boundaries this results in jagged (staircase) boundaries, which can introduce addi-
tional errors. More accurate descriptions can be obtained by grid refinement [38],
or by extrapolating the non-equilibrium part of the distribution function at the wall
nodes [45].
Lattice Boltzmann methods 473

19.2.5 Computational Cost

Most of the computational effort in LBGK simulations arises from the evaluation
of the equilibrium expressions. Other computational costs include evaluation of the
local conserved quantities, and the application of the BGK collision step. One of
the biggest advantages of LBM is the locality of the collision step, making the com-
putations local and the scheme ideally suited for parallelization. In addition, since
the method is explicit, a priori estimation of the computational costs is possible,
and domain decomposition for parallel computations can be easily and efficiently
implemented. The effort of evaluating the equilibrium, however, grows non-linearly
with the order of equilibrium (powers of velocity) and linearly with the number of
discrete velocities involved. The non-linear effort involved in evaluating the equi-
librium evaluation can be circumvented for low Mach numbers using the product
form of the equilibrium (eqn. (19.9) [28].) In this case, the Lagrange multipliers are
first evaluated to high powers of velocity and the equilibrium is then obtained as a
product of these Lagrange multipliers. Since the Lagrange multipliers are few (equal
to the number of local conservation laws) and independent of the lattice used, the
computational costs of higher-order lattices with large number of discrete velocities
can be reduced by more than an order of magnitude [28].

19.3 Applications

Lattice Boltzmann models continue to be developed not only to address new

physics, but also to overcome shortcomings (mainly numerical instability and lack
of Galilean invariance) of currently available models. At the same time, the rapidly
accelerating list of applications cover classical CFD areas (incompressible, non-
isothermal, single- and multi-phase, single- and multi-component flows etc.), as well
as nano- and micro-fluidics, biological flows, magnetohydrodynamics, in computer
graphics and visualization, etc. Model development for other applications will con-
tinue to expand the range of applicability of the method.

19.3.1 Isothermal Flows

19.3.1.1 Turbulent flows

The LBM has been shown to be an efficient and accurate alternative CFD method
for turbulent flows, and has been employed in direct and large eddy simulations as
well as for the construction of turbulence models.
Direct numerical simulations using LBGK models and comparison with results
obtained with a spectral solver were performed quite early after the introduction of
the method (e.g. [21, 62]). In a careful study of a 2-D turbulent shear layer [62], for
474 Christos E. Frouzakis

example, the time histories of global quantities and wavenumber spectra were found
to be very close to those obtained with spectral methods, although discrepancies in
the small scale features of the flow (spatial locations of vortical structures) were
observed at late stages. Similar works up to the mid nineties are listed in the review
paper of Chen and Doolen [22].
More recent applications include the DNS of grid-generated turbulence [35],
fully-developed turbulence in incompressible plane channel flow [60], and the de-
tailed comparison of the results obtained with a high-order entropic LB simulation
and a spectral element solver of a 3-D Kida flow [30, 32]. Using a second-order
D3Q19 LBGK model with second-order bounce back boundary conditions to sim-
ulate plane channel turbulence at Re = 180, Lammers et al. [60] concluded that
their parallel LB code produces single-point statistics of the same quality as pseudo-
spectral methods at comparable resolutions and with a competitive computational
cost.

Fig. 19.3: Simulation of the Kida flow for Re = 4000. Comparison of (a) the energy
spectrum, (b) two-point longitudinal and transverse correlation functions (DNS:
spectral element solution, 15v and 41v LBM solutions using the D3Q15 and D3Q41
models). From [30].

Chikatamarla et al. [32] used entropic D3Q15 and D3Q41 LB models to perform
direct numerical simulation of the Kida flow in a triply periodic box. Both models
were obtained by systematically reducing (prunning) a D3Q125 model by remov-
ing discrete velocities contained in the same energy level [31]. The D3Q41 lattice
(a high-order lattice with respect to the discretization of the velocity space) ensures
eq eq
Galilean invariance of the stress (P ) and energy flux (Q ) tensors, whereas the
lower-order LB model recovers Qeq 3
only up to O(u). The remaining O(u ) devi-
ation of the D3Q15 model results in a spurious dependence of the kinematic vis-
cosity on the fluid velocity and limits its range of applicability. It is worth noting
Lattice Boltzmann methods 475

here that the efficient evaluation of the equilibrium populations using the product
form becomes essential for efficiency as the order of the lattice is increased [28].
Comparison of enstrophy, kinetic energy, energy spectrum, skewness factor, second
and higher-order structure functions and two-point correlation functions with those
obtained with a spectral element solver, showed that DNS-quality results can be
obtained by LB models (Fig. 19.3, [32]).
Large eddy simulations and turbulence models. Smagorinsky-type large eddy
simulation LB models can be easily obtained by modifying the relaxation time using
an effective viscosity obtained from some version of Smagorinskys eddy viscosity
model instead of the fluid viscosity. Model derivations and recent applications can
be found in [6, 22, 58, 65, 74, 92] and the references therein. Similarly, standard
turbulence models like the algebraic or two-equation models can be incorporated
into LB models to compute the eddy viscosity and model high Reynolds number
turbulent flows (see, e.g. [23, 82, 88].)

19.3.1.2 Microfluidics and porous media

Typical flows in microdevices are characterized by (a) Knudsen numbers, Kn, rang-
ing from Kn  1 (continuum flows) to Kn 1 (weakly rarefied flows), and (b)
highly subsonic velocities. As Kn is increased the continuum (Navier-Stokes) de-
scription breaks down, first in the slip-flow regime (0.01  Kn  0.1), and then in
the transition regime (0.1  Kn  10). There is no well established and reliable
macroscopic equation for the description of non-equilibrium physics beyond the
Navier-Stokes regime, and LB with its kinetic origin is in principle better suited to
model flows until the early transition regime [3, 8, 75, 76, 81, 83].
For a correct microscopic description, the diffusive wall boundary condition ap-
pears to be more appropriate. In contrast to the bounce-back boundary condition,
in essence it allows the particles to forget the incoming direction after reaching
the wall by redistributing the populations in a way that is consistent with the mass-
balance, the normal-flux conditions, and the Maxwellian distribution at the wall
[3, 7]. A direct comparison with DSMC and several convergence analyses clearly
indicate that the capability of the LB models to describe microflows is not an arti-
fact of numerics or a discretization error, as some authors have claimed. Analytic
solutions obtained for flows, like the micro-Couette flow, suggest that the accuracy
of the method in the transition regime is improved when higher-order lattices are
used [8, 57].
Combining the microflow capability with the flexibility to handle very complex
geometries, the method is suitable for the study of flows through porous media
like the electrodes used in solid oxide fuel cells (SOFCs). Figure 19.4 shows the
computed velocity contours of a methane/water mixture flowing from left to right
through a 2-D slice extracted from the 3-D geometry of a real SOFC anode.
476 Christos E. Frouzakis

Fig. 19.4: Flow through a porous medium. Velocity contours in m/s are plotted for
a square domain of 100 100 m (Courtesy of N. Prasianakis and J. Mantzaras).

19.3.2 Non-Isothermal Flows

Flows with variable temperature proved much more difficult to tame with LB mod-
els, exhibiting severe numerical instabilities. In order to account for variable tem-
eq
perature, the equilibrium populations must satisfy in addition to the P and Qeq

eq
moments for any temperature, the R moments. The D2Q9 lattice, due to geo-
metric constraints, cannot fully satisfy even the Qeq
moments for a constant tem-
perature. In addition, when the single-relaxation BGK collision model is used, all
the higher-order moments relax towards equilibrium at the same rate. Since the re-
laxation rate of the higher-order moments controls the kinematic viscosity and the
thermal conductivity, the Prandtl number has a constant value Pr = 1. For all these
reasons, new degrees of freedom have been introduced leading to a multitude of ap-
proaches for heat transfer simulations. The most important ones are the following:
Extended lattices. Although it was shown theoretically that it is possible to sim-
ulate thermal flows using additional discrete velocities and including higher-order
velocity terms in the equilibrium distribution function, early implementations suf-
fered from severe numerical instabilities and the range of temperature variations
that could be tolerated was narrow [66]. Recent results towards this direction are
more encouraging (see, for example, [68, 80]). Even though the stability compared
to their predecessors is improved, the range of allowable temperature variations re-
mains quite limited. Also, the increased number of velocities and the complexity of
the equilibrium expressions inevitably lead to less efficient schemes.
Hybrid models employ an LB model to compute the flow and couple it to a finite
difference or finite element solver for the heat equation (e.g. [59]).
Passive-scalar approaches employ a separate distribution function, independent of
the distribution fi for the density and the momentum fields to simulate the tempera-
ture, which is treated as a passive scalar. The two distributions are coupled through
a buoyancy forcing term added to the equation for fi . Compressible flows where
density, momentum and energy are tightly coupled cannot be simulated with this
approach [77].
Lattice Boltzmann methods 477

Fig. 19.5: Temperature profile of a model premixed flame using the enhanced en-
tropic thermal LB model (lines: analytic solution for decreasing flame thicknesses;
symbols: LBM results). Temperature ratios of more than ten can be achieved.
Reprinted from [72] with permission. Copyright 2008, American Physical So-
ciety.

Multiple distribution functions also employ two distributions: one for the density
and momentum and another for the internal energy [47]. The standard isothermal
model is used on the first lattice and the coupling is through force terms designed to
recover the correct viscous heat dissipation and the correct compression work done
by the pressure.
Enhanced entropic LB. The entropic construction was also used to construct sta-
ble and thermodynamically consistent LB schemes which are off-lattice (e.g. the
D1Q4 model in [5]), or on standard lattices (D2Q9 in [10]). The Prandlt number
was allowed to vary by improving the collision process [9, 70]. In comparison to the
standard lattices, the method has been enhanced in two ways. First, the accuracy of
the equilibrium populations was improved by including a larger set of minimization
constraints, leading to higher accuracy in higher-order moments. Second, forcing
terms were introduced in the lattice BGK equation in order to cancel the effect of
the remaining terms in the higher-order moments of the equilibrium populations
that deviate from the corresponding Maxwell-Boltzmann expressions [69, 71]. The
resulting model offers some highly desirable properties without increasing signifi-
cantly the computational cost. The speed of sound is described correctly for a large
range of temperatures, and shock tube simulations verified the stability of the al-
gorithm. The model was used to solve a 1-D problem mimicking premixed flame
propagation [89], showing that large temperature and density variations pertinent
to combustion applications can be considered and that heat sources and sinks can
easily be included (Fig. 19.5, [72]). Finally, the model is Galilean invariant thus re-
laxing the low Mach number restriction as long as the flow remains subsonic [73].
478 Christos E. Frouzakis

19.3.3 Multicomponent Mixtures

Combustion of gas mixtures typically involves a large number of chemical species,

even for relatively simple fuels. Many attempts have been made to extend LB mod-
els to account for both the self collisions between particles of the same type and the
cross collisions involving different particles in multicomponent mixtures (see, for
example, the references in [12, 14]). Multiple fluid approaches using different BGK
collision terms and allowing the species to relax to equilibrium with different relax-
ation times are more suitable than single-fluid based models [14]. However, most of
the models fail to satisfy important physical properties such as the indifferentiabil-
ity principle (i.e. they do not reduce to the single-component BGK fluid when the
species become identical), or the H-theorem, facing numerical instabilities at large
molecular mass ratios [12].
The entropy-based quasi-equilibrium LB approach was employed to construct
isothermal models for binary [11] and multicomponent mixtures [12] with different
Schmidt numbers and arbitrary molecular weights. The kinetic equations for species
j in a mixture with M components take the form

f ji f ji 1 1
= ( f ji f ji )
eq
+ c ji ( f f ji ) + Fji , j = 1, . . . M, i = 0, . . . q,
t x 1 j 2 j ji

where the associated two relaxation times 1 j , 2 j are related to the viscosity of each
component and to the mixture-averaged diffusion coefficient, respectively. Fji is a
forcing term acting on species j in order to maintain the momentum balance when
the mixture-average approximation for the diffusion coefficients is used [12]. The
different molecular weights of the species result in different lattice speeds, and pop-
ulations corresponding to species with lowest molecular weight are streamed on the
lattice, while heavier components with lower speeds stream off-lattice. The corre-
sponding lattice values are calculated by interpolating the off-lattice neighbors, and
the interpolation scheme becomes a critical component of the model in terms of ac-
curacy as well as computational efficiency. The models recover the Navier-Stokes
and the Stefan-Maxwell diffusion equations at the hydrodynamic limit assuming
that the diffusion coefficients of the species are given by the mixture-averaged for-
mulation. The results of the simulations of a mixing layer, an opposed-jet flow, and a
micro-Couette flow were found to be in good agreement with continuum and direct
simulation Monte Carlo [12]. In the opposed-jet mixing case, viscosities and species
diffusivities computed with CHEMKIN were used to evaluate the local relaxation
times 1 j and 2 j . Figure 19.6(a) shows the hydrogen mole fraction distribution be-
tween opposed jets ejecting different mixtures of H2 , O2 , N2 , and H2 O at a fixed
temperature of 300 K; along the axis of symmetry the LB results agree very well
with the 1-D solution obtained with the OPPDIF package of CHEMKIN [61].
An MRT lattice Boltzmann model for mixtures was also recently derived from
the linearized Boltzmann equations and was used to simulate active and passive
scalar mixing in 3-D homogeneous decaying isotropic turbulence [15].
Lattice Boltzmann methods 479

Fig. 19.6: (a) Hydrogen mole fraction distribution obtained by mixing a stream
with mole fractions XH2 = 0.1, XN2 = 0.85, XH2 O = 0.05 (left boundary) and XN2 =
0.90, XO2 = 0.1 (right boundary) in an opposed-jet configuration. (b) Comparison
of the LB (symbols) with the OPPDIF solution (lines). Reprinted from [12] with
permission. Copyright 2007, American Physical Society.

19.3.4 Reactive Flows

In principle, once lattice Boltzmann models can properly account for large temper-
ature variation and mixing, extension to reactive flows essentially involves adding
appropriate source terms in the species equations to account for the reaction rate.
However, no reactive LB model has been proposed so far that capitalizes on the
latest developments.
Existing models can be classified in two groups: (a) hybrid models where the
flow is solved by an LB solver coupled to a classical CFD solver for the macroscopic
energy and species equations [40, 46], and (b) reduced model descriptions either via
a tabulated mixture fraction formulation [84, 91], or by using a reduced mechanism
obtained by the method of invariant grids [27].
Succi et al. [84] proposed the first extension of LBM for combustion applications
by considering mixture fraction and temperature as passive scalars to construct non-
premixed combustion models for infinitely fast reaction and constant density. The
constant density assumption was also employed by Yamamoto et al. [91] in D2Q9
LBGK models using separate populations for the incompressible flow field, the tem-
perature, and each of the species concentrations fields to simulate premixed flames
in an opposed-jet setup with constant transport properties and single-step kinetics.
The results agreed well with the solution of an incompressible macroscopic model,
but, as expected, significant deviations were found when compressibility was taken
into account.
480 Christos E. Frouzakis

The presence of a large number of chemical species in detailed combustion mech-

anisms results in high computational requirements for any LB model when each
species is taken into account by a different population. The requirements become
more demanding in higher dimensions and for higher-order lattices (i.e. lattices with
larger number of discrete velocities like the common D3Q27, or the more recently
proposed D3Q41 models).
The hybrid models proposed by Filippova and Hanel [39, 40] and Hanel et al.
[46] for the simulation of low Mach number reactive flows employ additional free
parameters and correction terms to couple LBGK models used to solve for the flow
field to a finite difference solver for the macroscopic energy and species conserva-
tion equations. These models are not only more memory-conscious in dealing with
large number of species, but also remove the constant density limitations of other
LB approaches. Recently, the low Mach number formulation was also used by Chen
et al. [24] to construct purely LB models for low speed combustion.

Fig. 19.7: Ignition and stabilization of a H2 /air premixed gas in an opposed-jet ge-
ometry: (a) Schematic of the computational setup, (b) temporal evolution of the
distribution of the O mass fraction profile. Courtesy of E. Chiavazzo [25].

Reduced kinetic schemes offer another way to lower the computational require-
ments. The LB model of Yamamoto et al. [91] was employed to simulate the propa-
gation of a 1-D H2 /air premixed flame propagation [27] and ignition/stabilization of
a premixed flame in a 2-D opposed-jet geometry [25] (Fig. 19.7). Detailed as well
as a reduced two-dimensional invariant manifold description [26] obtained with the
method of invariant grids [42, 43], were used to describe the kinetics. The reduced
description requires that only two additional equations for the variables parameter-
izing the manifold need to be solved by the LB model. After each LB step consisting
of collision, streaming and reaction, the species populations are projected back to
the invariant manifold along the local fast directions. In the premixed flame propa-
gation case, the laminar flame speeds obtained with the full and the reduced scheme
were found to agree well with the measured value. Incorporation of accurate re-
duced chemistry description into more elaborate multicomponent [12] and thermal
Lattice Boltzmann methods 481

LB models [71] will pave the way to the simulation of combustion applications
using the lattice Boltzmann method.
Since LB models can easily represent complex geometries, they are particularly
well suited for the simulation of reactive flows in porous media and catalyst pores.
An exploratory study of complex physics in catalytic devices was presented in [85].
More recently, the multicomponent model [12] was extended to simulate catalytic
surface reactions described via appropriate diffusive-wall boundary conditions [13].
Coupling the code with surface CHEMKIN [33] allowed for the easy implemen-
tation of realistic surface kinetics. The results of the simulation of a single-step
catalytic oxidation of methane in a planar channel at a constant temperature of 1200
K and atmospheric pressure were in very good agreement with a steady-state finite
volume macroscopic equation solver (Fig. 19.8).

Fig. 19.8: Comparison of the velocity and species profiles obtained with the LB
model (symbols) and a CFD solver (lines) in the wall-normal direction in an isother-
mal catalytic channel. Reprinted from [13] with permission. Copyright 2008,
American Physical Society.

19.4 Conclusions

The lattice Boltzmann method offers a theoretically sound and flexible basis for the
construction of multiscale and multi-physics models that are simple to implement
into numerical tools. Since the operations are local, LBM is ideally suited for paral-
lel implementation and for the large-scale computation of complex flows at different
scales.
Since its inception about 20 year ago, LBM has gone a long way to become an
accurate and competitive fluid dynamics solver for complex flows within as well
as beyond the regime of the Navier-Stokes hydrodynamics and has already proven
482 Christos E. Frouzakis

its versatility in modeling a wide range of problems. Recently, stable and accurate
lattice Boltzmann models have been developed to simulate the different processes
relevant to turbulent combustion (turbulent flows, multi-component mixing, strongly
non-isothermal and multiphase flows which were not covered here). Particularly
interesting are models that can cope with large temperature variations pertinent to
combustion without suffering from numerical instabilities, and accurate models for
multi-component mixing.
For detailed chemistry reactive flow simulations, LBM is confronted with the
high computational cost associated with the large number of chemical species
present in combustion kinetics. Already a problem for conventional CFD solvers,
this is magnified in LBM since each additional species introduces a new set of pop-
ulations. Particularly for the high-order lattices that are currently considered to ad-
dress stability and lack of Galilean invariance issues, the computational toll can then
become excessive. When passive scalar approaches are not an option, hybrid mod-
els coupling an LBM solver for the flow and a conventional solver for the energy
and species lead to significant savings, but sacrifice the algorithmic simplicity of
the method. Another option is offered by reduced kinetic descriptions which have
already been shown to work in reactive constant density LB models.
The lattice Boltzmann method is still in rapid development. Additional system-
atic validation studies, similar to the ones that other CFD approaches have under-
gone, are required before recently proposed approaches can be combined into mod-
els capable of simulating reactive flows. The impressive progress to date supports
that reaching full maturity is just a matter of time and labour, no conceptual hurdles
in sight [82] holds not only for the method in general, but also for its extension for
the simulation of turbulent reactive flows.

Acknowledgements

Many thanks to I.V. Karlin for enlightening discussions about the lattice Boltzmann
method and the opportunity to follow developments in the field during the past six
years. Special thanks to N. Prasianakis, S.S. Chikatamarla, S. Arcidiacono, and J.
Mantzaras for the input they provided for this manuscript.

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Index

adaptive mesh refinement (AMR) conditional average, 93

applications, 321 conditional flux, 95
BoxLib library, 319 conditional velocity, 95
creating grid hierarchy, 305 dimensionality of CMC equations, 109
discretization, 307 doubly-conditioned moment closure, 101
elliptic systems, 311 liquid fuel combustion, 108109
hyperbolic systems, 307 numerical methods, 110
low Mach number combustion, 315 partially-premixed combustion, 103105
parabolic systems, 314 premixed combustion, 107
performance of adaptive projection, 320 second-order closure, 96
principles, 305 closure models, 98, 99
restriction, 308 variance and co-variance equations, 98
sub-cycling, 309 consumption speed, 79, 280, 293, 326
autoignition, 96, 105, 109, 357, 360 continuity (mass) equation, 22, 285
continuity (species) equation, 23, 284
Boltzmann countergradient transport, 71
BGK model, 463
Boltzmann equation, 463 Damkohler hypothesis, 82
H theorem, 464 Damkohler number, 46, 48, 52, 53, 65, 119,
Borghi diagram, 65 121, 123, 124, 137, 138, 293, 359, 371,
Bray number, 71 372
Bray-Moss-Libby (BML) model, 36, 71, 72, direct numerical simulation (DNS), 28, 226
77, 294 displacement speed, 7679, 81, 291, 294
burning index, 359 definition, 74
modeling, 7980
Chapman-Enskog method, 464
coagulation, 226 EBULES, 238240, 243
coherent flame model (CFM), 74 eddy break-up (EBU) model, 35, 72, 178
common component architecture (CCA) eddy diffusivity, 223
features, 414 eddy dissipation concept (EDC), 72
toolkits eddy dissipation model (EDM), 178
computational facility for reacting flow energy equation, 23, 285, 304
science, 416418 experimental burners
conditional moment closure (CMC) piloted spray burner, 372
applications, 109 premixed burner in vitiated coflows, 370
CMC equation, 94, 95 swirl-stabilized burner, 367

T. Echekki, E. Mastorakos (eds.), Turbulent Combustion Modeling, 487

Fluid Mechanics and Its Applications 95, DOI 10.1007/978-94-007-0412-1,
Springer Science+Business Media B.V. 2011
488 Index

filtered density function (FDF), 34 D3Q19, 465

definition, 133 discrete velocity, 465
examples, 136137 Entropic LB, 467
models and algorithms, 134 equilibrium function, 467
flame curvature, 44, 51, 53, 7383, 86, 277, initial conditions, 472
278, 280, 291, 294, 326 LBGK model, 465
flame regimes, 358 local conservation, 468
flame strain rate, 44, 47, 53, 54 matrix form, 471
flame stretch, 44, 75, 84, 279, 291 multi-relaxation time (MRT), 471
flamelet (non-premixed) models Quasi-equilibrium, 471
counterflow diffusion flame, 47 Lattice Gas, 462
Eulerian particle flamelet model, 56 LDV, 364
flamelet-progress variable models, 56 LEMLES
LES modeling, 58 applications, 237243
RANS modeling, 49 large-scale advection, 232236
representative interactive flamelets, 55 subgrid model, 231232
steady flamelets, 50 LIF, 365
transient flamelets, 53 linear-eddy model (LEM)
validity, 48 combustion
flamelet (premixed) models large-eddy simulation, see LEMLES
flame surface density, 73 stand-alone combustion, 226
algebraic models, 73 stand-alone scalar mixing, 225226, 228
filtered transport equation, 77 stirring events, 223
instantaneous transport equation, 74 triplet map, 222223, 225
surface-averaged normal vector, 77 local extinction, 102, 104, 357, 371
turbulent strain rate model, 78 low Mach number formulation, 303
turbulent transport, 77
G-equation manifold methods for chemistry reduction
eikonal equation, 80 calculation of low-dimensional manifolds,
Favre-averaged transport equation, 81, 82 211
mean turbulent reaction rate model, 82 computational singular perturbation (CSP),
turbulent flux model, 82 207, 212
flame generated manifolds (FGM), 213
global extinction, 371 flame prolongation of ILDM (FPI), 213
Grad method, 464 intrinsic low-dimensional manifolds
(ILDM), 207, 212
HCCI combustion, 7, 21, 227 method of invariant grids (MIG), 208
method of invariant manifolds (MIM), 208
ignition, 428430 principles, 209
inertial range, 225 rate-controlled constrained equilibrium
integral scale, 225 (RCCE), 208
reaction-diffusion manifolds, 213
Karlovitz number, 65 repro-modeling, 212
kinetic models, 463 slow manifolds, 211
Kolmogorov microscale, 225 trajectory generated manifolds, 213
zero-derivative principle (ZDP), 208
large-eddy simulation (LES) mapping, 148
equations, 35 Markov process, 154, 157, 160
equations , 32 Markstein diffusivity, 81
Lattice Boltzmann Markstein length, 81
algorithm, 469 Maxwell distribution, 463
computational cost, 473 mixture fraction
D1Q3, 465 Bilger, 28
D2Q9, 465 elemental, 27
Index 489

measurement, 365 probability density function (PDF), 32

single-step reaction, 45 progress variable, 69
momentum equation, 22, 285 countergradient transport, 70
multiple mapping conditioning (MMC) definitions, 68
applications, 161170 LES transport equation, 69
basic concepts, 146 mass diffusivity in transport equation, 69
conditional fluctuation, 147 RANS transport equation, 69
deterministic MMC, 148151 reaction rate
generalized MMC, 156160 EBU-type model, 72
mapping functions, 147148 flame surface density, 73
MMC equations, 147 Reynolds flux model, 70
stochastic MMC, 152153
multiple scales of combustion quasi steady-state approximation (QSSA), 195,
length scales, 178, 194 199
modeling considerations, 183186
time scales, 178, 194
random walk, 224
multiscale approaches
Rayleigh scattering, 365
flame-embedding methods, 188
regime-independent modeling, 221
hybrid LES-low-dimensional models, 188
Reynolds number, 225
mesh adaptive methods, 187
Reynolds-averaged Navier-Stokes (RANS), 30
ODTLES, 265272
one-dimensional turbulence (ODT) model scalar dissipation rate
combustion CMC
large-eddy simulation, see ODTLES conditionally-averaged, 95, 96, 99
space-time mapping, 260261 double conditioning, 101, 102, 104
spatially-evolving Eulerian, 261 fluctuations, 96, 99
spatially-evolving Lagrangian, 261 passive scalars, 107
stand-alone simulations, 261265 reactive scalars, 107
temporally-evolving Eulerian, 260 spray evaporation, 109
temporally-evolving Lagrangian, 259260 MMC
model formulation, 251255 conditional averages, 147, 149, 155, 156,
model representation of combustion, 258 163, 165
model representation of free shear flows, Favre averages, 151
256258 fluctuations, 145, 159, 163, 164, 170
non-premixed
partial equilibrium approximation (PEA), 195, counterflow flame, 47
200 definition, 46, 47
partially-premixed combustion, 21, 59, 64, 66, LES model, 58
86, 103, 138, 145, 161, 357, 358 LES subgrid variance, 58
particle methods, 462 mean, 50
PDF methods premixed, 8385
Eulerian methods, 126 scale locality, 223
examples, 129132 slow manifold, 195
Fokker-Planck equations, 121 soot, 5, 13, 14, 128, 131132, 178180, 229,
Lagrangian methods, 125 261, 264, 265
mixing models, 123, 124 soot-radiation-turbulence coupling, 227
multiphysics modeling, 128 spray, 109, 360
PDF transport equation, 122 stability limits, 363
PIV, 364 strain rate, 75
Poisson equation, 122, 316, 394, 416 stratified mixture combustion, 357
Poisson process, 255, 385 swirl, 357, 367
power spectrum, 225
premixed flame regimes, 65 tabulation (chemistry), 214
490 Index

in situ adaptive tabulation (ISAT), 126, 187, polynomial chaos

215 applications, 401
artificial neural networks (ANN), 187 arbitrary basis, 388
piece-wise reusable implementation of challenges, 389
solution mapping (PRISM), 187 formulation, 384387
triplet map, 223, 250, 251, 253 intrusive PC, 387
turbulent burning velocity, 82
turbulent transport, 223 vitiated air, 96, 357, 360

uncertainty quantification (UQ) wavelets

applications, 392 biorthogonal wavelets, 335
compressible flow, 398 higher-dimensional discretization, 341
incompressible flow, 393 orthogonal wavelets, 333
reacting flow, 396 representation of derivatives, 340
turbulence, 399 second-generation wavelets, 336
challenges, 392 wavelet methods
polynmonial chaos direct numerical simulations, 337
non-intrusive PC, 388 Wiener process, 121, 122, 127, 152, 385