Computationally Efficient Data-Enhanced

Manifold Modeling of Multi-Modal

Turbulent Combustion

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Cristian Estremera Lacey

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A Dissertation
Presented to the Faculty
of Princeton University
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in Candidacy for the Degree

of Doctor of Philosophy

Recommended for Acceptance

by the Department of
Mechanical and Aerospace Engineering
Adviser: Professor Michael E. Mueller

May 2023
© Copyright by Cristian Estremera Lacey, 2023.
All rights reserved.

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 Abstract

The design of improved energy conversion devices may be facilitated by Large Eddy
Simulation (LES) – a computationally efficient modeling approach for simulating turbulent
flows. Brute-force combustion modeling approaches that directly transport up to hundreds
or even thousands of chemical species are generally applicable but intractable in simulations
of realistic systems. Projecting the high-dimensional thermochemical state onto a reduced-
order manifold provides an efficient alternative for modeling the unresolved combustion pro-
cesses but does not traditionally generalize to the multi-modal combustion regimes present
in practical engineering devices, introducing a fundamental modeling trade-off between com-

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putational cost and model generality. Though more general, higher-dimensional manifold
models capable of breaking this trade-off exist in theory, their implementation is impeded
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by large computational cost and memory requirements associated with pretabulating the
thermochemical state as well as unclosed terms that appear in the manifold equations.
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A novel algorithm termed In-Situ Adaptive Manifolds (ISAM) is developed to enable LES
implementations of more general, higher-dimensional manifold models by computing mani-
fold solutions ‘on-the-fly’ and reusing them with In-Situ Adaptive Tabulation (ISAT). ISAM
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is verified and evaluated via LES of two canonical turbulent nonpremixed jet flames and
extended to two higher-dimensional manifold models capable of capturing multiple and/or
inhomogeneous stream mixing and multi-modal combustion. The computational cost of
ISAM rapidly reaches parity with traditional pretabulation approaches independent of the
chemical mechanism size and model complexity while requiring up to seven orders of mag-
nitude less memory.
Then, data-based approaches are leveraged to augment physics-based manifold models –
namely, to provide closure for unclosed dissipation rates that parameterize the solutions to
the manifold equations. The instantaneous dissipation rate profiles in both premixed and
multi-modal turbulent combustion are extracted from Direct Numerical Simulation (DNS)
databases, and deep neural networks (DNNs) are trained to accurately capture the previously
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unconsidered spatiotemporal variation of the profile shapes. Quantitative predictions of
flame stabilization, ignition, and pollutant formation are shown to be particularly sensitive
to the shape of the dissipation rate profiles. In conjunction with ISAM, the hybrid physics-
and data-based models developed in this dissertation represent a critical advancement in
multi-modal turbulent combustion simulations – tools essential for developing cleaner, more
efficient power generation technology.

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 Acknowledgements

Attending graduate school has been an enormously rewarding experience – both from
a professional and personal perspective – entirely due to the people that I have met and
that have shaped my life over the past five years. Foremost among them is my adviser,
Prof. Michael E. Mueller, whose continued support and guidance have been central to
both my success as a researcher as well as my well-being as a graduate student. Michael
is without a doubt a world-class researcher and lecturer, but more importantly he is an
incredibly supportive and compassionate mentor. Despite his many other responsibilities,
Michael always makes time for his graduate students, whether it be to answer technical

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questions, carefully review written documents, or just chat about life. There are a lot of
smart people in the world, but in my experience it is exceedingly rare to simultaneously be
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such an exceptionally caring human being. I find it difficult to convey just how grateful I am
to have had such an individual as an adviser. I consider Michael not just a mentor whom
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I greatly admire but also my friend, and I think that is the greatest compliment a student
can offer.
I would like to further acknowledge the many other faculty members and external re-
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searchers, whose efforts have informed my academic growth. Through their coursework and
lectures, Prof. Chung K. Law and Prof. Howard A. Stone have provided me with a strong
theoretical foundation in the topics of combustion and fluid mechanics, while also deepen-
ing my appreciation for fundamental science. Prof. Sankaran Sundaresan set aside many
hours to discuss my research progress, always posing thoughtful questions that proved in-
valuable, for which I am very thankful. I also deeply appreciate Dr. Jacqueline H. Chen
and Dr. Martin Rieth at Sandia National Laboratories for generously donating their time
to our collaboration, consistently providing insightful feedback, and supplying a wealth of
high-fidelity simulation data – without which much of the work of this dissertation would
have been impossible. I would also like to thank my Ph.D. committee members Prof. Luc
Deike and Prof. Yiguang Ju, my dissertation readers Prof. Chung K. Law and Prof. Luc
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Deike, and my Final Public Oral examiners Prof. Sankaran Sundaresan and Prof. Marcus
Hultmark.
This section would be incomplete without recognizing all the wonderful people in the
Computational Turbulent Reacting Flow Laboratory (CTRFL) that I have had the privilege
to spend my graduate school years with. Thank you to Bruce Perry and Cody Nunno for
setting the tone in the lab and the MAE department as a whole when I was first starting
out, ensuring both were (and continue to be) warm and inviting places. Alex Novoselov,
thank you for playing such a central role in my development as a researcher, relaying all
the subtle intricacies of turbulent combustion modeling with clarity and patience. I know
I have told you this before, but it genuinely makes me so happy that you have become a

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professor. The world of academia needs more advisers like you. Kerry Klemmer, thank you
for teaching me everything I know about turbulence and crossword puzzles. I am honestly
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not sure which of the two has been more important to life in the lab. Above all, thank you
for being such a good friend and being there for me when I need it most. Jinyoung Lee, you
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carry yourself with a humility and kindness that I find truly admirable. I greatly enjoyed
being your labmate and conference roommate during my time at Princeton. I would also
like to say a special thank you to Katie VanderKam, Hannah Williams, Israel Bonilla, Agnes
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Robang, and honorary CTRFL member Megan Mazzatenta for being such tremendously
thoughtful and caring friends to me this past year. You are all so dear to me, and I will
greatly miss seeing you in lab every day. There is no group of people I would rather descend
into LEGO Town mode with more than you. Katie, thank you for single-handedly keeping
crossword time alive. Without you, the rest of us would be struggling with Thursdays as if
they were Saturdays. Hannah, I appreciate your selfless willingness to design such beautiful
lab graphics, knowing full well we will force you to make them your least favorite color
regardless. Izzy, thank you for being the new Cristian. I am happy and proud that the
role has been passed to someone as wonderful as you. Agnes, I so deeply appreciate the
unmitigated chaos you bring to lab. Truly though, it has been a pleasure to get to know

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you, and I am very impressed with all the work you have done. Megan, your callbacks
are masterful, and you will never convince me otherwise. Thank you for always making
me smile. To the newest members of the CTRFL – Sydney Rzepka, John Boerchers, and
Trevor Fush – I am glad I got the chance to get to know you all better over these past few
months. The future of the lab is certainly bright. Finally, I would like to thank current and
former postdoctoral researchers Giuseppe D’Alessio, Hernando Maldonado Colmán, Esteban
Cisneros Garibay, Pavan Prakash Duvvuri, Aditya Aiyer, and Pierre-Yves Taunay for their
many helpful discussions and suggestions over the years.
I am also very grateful to the exceptional people that make up my cohort – in particular,
thank you to Rory Conlin, Paul Kaneelil, and Amlan Sinha. Rory, thank you for being such

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a loyal friend and grumpy dungeon master. I always look forward to catching up with you
during our lunch and dinner dates, and I hope you know I consider you a very important
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friend. Paul, I do not see you as often as I would like to these days, but it is always nice to
catch up with you when I do. You are such a kind-hearted person. Amlan, I feel the same
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about you. I should also thank you for introducing me to the Sanchez and ruining an almost
twenty-three-year streak of devout pescetarianism. Also, I would be remiss not to mention
at least a few of the many wonderful humans that occasionally or consistently grace the
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atrium lunch table with their presence: Daniel Pardo, Andy Rothstein, Sasha Bodrova, Dan
Shaw, Marcel Louis, and Satya Butler. Chatting, coming up with new axes, and playing
board games with you all has been such a great time. You all make the MAE department a
better place.
I must also thank my family, whose love and support form the foundation for all that I
am. Juliet, thank you for being such a wonderful and supportive friend. You are and always
will be my best little buddy. You understand me in a way that few people do, and I feel so
fortunate to have you as a sibling. I find it deeply reassuring knowing that we will always
have each other. Mama, you are the ideal mother and caregiver. You are selfless and warm
in a way that makes it seem so easy and obvious, even when I am sure that it is not. You

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make me feel profoundly loved, and I am so very grateful to be your son. Papa, thank you
for always believing in me. From an early age, you instilled in me the confidence required to
face the many academic challenges I have encountered over the years. For that, I am very
grateful. I hope that this dissertation makes all of you proud.
I gratefully acknowledge funding provided by the Gordon Y. S. Wu Fellowship in Engi-
neering, Daniel and Florence Guggenheim Foundation Fellowship, and the Charlotte Eliza-
beth Procter Fellowship through Princeton University. I also acknowledge funding from the
U.S. Department of Energy Office of Science Graduate Student Research (SCGSR) fellow-
ship program, the Army Research Office (ARO) Young Investigator Program (YIP) under
grant W911NF-17-1-0391, the National Aeronautics and Space Administration (NASA) un-

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der grant NNX16AP90A, the University Coalition for Fossil Energy Research through the
U.S. Department of Energy’s National Energy Technology Laboratory (DE-FE0026825),
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and the Schmidt DataX Fund at Princeton University made possible through a major gift
from the Schmidt Futures Foundation. The work from Sandia National Laboratories was
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supported by the Exascale Computing Project (ECP), Project Number: 17-SC-20-SC, a
collaborative effort of two DOE organizations – the Office of Science and the National Nu-
clear Security Administration. Sandia National Laboratories is a multimission laboratory
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managed and operated by National Technology and Engineering Solutions of Sandia, LLC.,
a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of
Energy’s National Nuclear Security Administration under contract DE-NA-0003525. The
simulations presented in this dissertation were performed on computational resources sup-
ported by the Oak Ridge Leadership Computing Facility (OLCF), the Princeton Institute
for Computational Science and Engineering (PICSciE), and the Office of Information Tech-
nology’s High Performance Computing Center and Visualization Laboratory at Princeton
University.
This dissertation carries T#3442 in the records of the Department of Mechanical and
Aerospace Engineering.

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 To my family and friends

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Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

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List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

1 Introduction
1.1
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Asymptotic Combustion Modes . . . . . . . . . . . . . . . . . . . . . . . . .
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3
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1.2 Simulation of Turbulent Combustion . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Thermochemical State . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 Chemical Kinetics Modeling . . . . . . . . . . . . . . . . . . . . . . . 6
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1.2.3 Manifold Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.4 Turbulence Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.5 Filtered Chemical Source Term Modeling . . . . . . . . . . . . . . . . 12
1.3 Augmenting Physics-Based Models with Data . . . . . . . . . . . . . . . . . 16
1.4 Contributions and Organization . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Governing Equations 21
2.1 Transport Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Chemical Source Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Large Eddy Simulation (LES) . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.1 Subfilter Stress Closure . . . . . . . . . . . . . . . . . . . . . . . . . . 27

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 2.3.2 Subfilter Scalar Flux Closure . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.3 Extension of Manifold Modeling to LES . . . . . . . . . . . . . . . . 31
2.4 Deep Neural Networks (DNNs) . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.1 DNN Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.2 DNN Hyperparameter and Architecture Selection . . . . . . . . . . . 36

3 In-Situ Adaptive Manifolds (ISAM): A Novel Algorithm for Manifold

Modeling 38
3.1 Nonpremixed Manifold Model . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Manifold Modeling Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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3.2.1 Precomputation, Preconvolution, and Pretabulation (PPP) . . . . . . 42
3.2.2 Convolution-on-the-fly (COTF) . . . . . . . . . . . . . . . . . . . . . 43
3.2.3
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In-Situ Adaptive Manifolds (ISAM) . . . . . . . . . . . . . . . . . . . 43
3.3 Test Configurations: Turbulent Nonpremixed Jet Flames . . . . . . . . . . . 46
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3.3.1 Computational Infrastructure . . . . . . . . . . . . . . . . . . . . . . 46
3.3.2 Configuration I: Hydrogen Jet Flame . . . . . . . . . . . . . . . . . . 46
3.3.3 Configuration II: Sandia Flame D . . . . . . . . . . . . . . . . . . . . 47
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3.4 ISAM Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4.1 Verification: Hydrogen Jet Flame . . . . . . . . . . . . . . . . . . . . 47
3.4.2 Verification: Sandia Flame D . . . . . . . . . . . . . . . . . . . . . . 48
3.5 Performance Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.6 Numerical ISAT Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.7 Tolerance Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.8 Improving ISAM Performance . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.8.1 MPI Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.8.2 Speeding Up Manifold Calculations . . . . . . . . . . . . . . . . . . . 58
3.9 Outcomes: ISAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

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4 Extension of ISAM to Multiple and/or Inhomogeneous Inlet Streams 61
4.1 Test Configuration: Sydney Flame . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Nonpremixed Manifold Model: Modification for an Inhomogeneous Stream . 62
4.3 Sydney Flame: Constrained Dissipation Rate . . . . . . . . . . . . . . . . . . 64
4.4 Sydney Flame: Unconstrained Dissipation Rate . . . . . . . . . . . . . . . . 67
4.5 Outcomes: Sydney Flame . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5 Extension of ISAM to Multi-Modal Combustion 71

5.1 Multi-Modal Manifold Model . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2 Test Configuration: Cabra Flame . . . . . . . . . . . . . . . . . . . . . . . . 77

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5.3 Evaluation of Multi-Modal Model Performance . . . . . . . . . . . . . . . . . 78
5.3.1 Feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3.2
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Interpretability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.4 Outcomes: Cabra Flame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
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6 Data-Based Dissipation Rate Modeling for Premixed Combustion 84
6.1 Premixed Manifold Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.2 DNS Data and Profile Extraction . . . . . . . . . . . . . . . . . . . . . . . . 87
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6.2.1 DNS Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.2.2 Extraction Methodology . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2.3 Evaluating Profile ‘Universality’ . . . . . . . . . . . . . . . . . . . . . 89
6.3 Data-Based Modeling Approach . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.3.1 DNS Filtering and Training Data Generation . . . . . . . . . . . . . . 92
6.3.2 Input Feature Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.3.3 Network Hyperparameter Selection . . . . . . . . . . . . . . . . . . . 93
6.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.4.1 DNN Model Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.4.2 A Priori Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

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 6.4.3 Data-Based Variable Importance: Integrated Gradients . . . . . . . . 100
6.4.4 Physics-Based Variable Importance: Conditional Generalized Progress
Variable Dissipation Rate Transport Equation . . . . . . . . . . . . . 105
6.5 Outcomes: Premixed Dissipation Rate Profile Modeling . . . . . . . . . . . . 108

7 Data-Based Dissipation Rate Modeling for Multi-Modal Combustion 110

7.1 DNS Dataset Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.2 Multi-Modal Manifold Model: Modification for Temporally Evolving Flames 112
7.3 Nonlinear Reference Species Definition . . . . . . . . . . . . . . . . . . . . . 113
7.4 Dissipation Rate Modeling Approach . . . . . . . . . . . . . . . . . . . . . . 114

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7.5 Filtered Dissipation Rate Modeling . . . . . . . . . . . . . . . . . . . . . . . 115
7.5.1 Algebraic Models: Pierce and Linear Relaxation Models . . . . . . . 116
7.5.2
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Enforcing Physical Constraints in DNN Models . . . . . . . . . . . . 117
7.5.3 Training Data Generation . . . . . . . . . . . . . . . . . . . . . . . . 118
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7.5.4 DNN Model Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.5.5 DNN Model Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.6 Instantaneous Dissipation Rate Profile Modeling . . . . . . . . . . . . . . . . 122
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7.6.1 Algebraic Dissipation Rate Profile Models . . . . . . . . . . . . . . . 124

7.6.2 Profile Extraction Methodology . . . . . . . . . . . . . . . . . . . . . 125
7.6.3 Profile Regression Methods: PLI and GPR . . . . . . . . . . . . . . . 127
7.6.4 A Priori Evaluation of Regression Methods . . . . . . . . . . . . . . 130
7.6.5 Evaluating Profile ‘Universality’ . . . . . . . . . . . . . . . . . . . . . 132
7.6.6 Training Data Generation . . . . . . . . . . . . . . . . . . . . . . . . 135
7.6.7 DNN Model Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.6.8 DNN Model Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.6.9 A Priori Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.7 Outcomes: Multi-Modal Dissipation Rate Modeling . . . . . . . . . . . . . . 144

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8 Physics-Based Dimensionality Reduction (PBDR) 146
8.1 Input Feature Dimensionality Reduction . . . . . . . . . . . . . . . . . . . . 147
8.1.1 PCA-Based Dimensionality Reduction . . . . . . . . . . . . . . . . . 147
8.1.2 Physics-Based Dimensionality Reduction (PBDR) . . . . . . . . . . . 149
8.2 DNN Model Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
8.2.1 DNS Filtering and Training Data Generation . . . . . . . . . . . . . . 151
8.2.2 Input Feature Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 151
8.2.3 Network Hyperparameter Selection . . . . . . . . . . . . . . . . . . . 154
8.3 DNN Model Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.3.1 ODI versus Algebraic Models . . . . . . . . . . . . . . . . . . . . . . 155

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8.3.2 ODI versus PBDR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.3.3 Sensitivity to the Dimensionally-Homogeneous Set . . . . . . . . . . . 159
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8.3.4 PBDR versus PCA-FE and PCA-FS . . . . . . . . . . . . . . . . . . 161
8.4 Outcomes: Physics-Based Dimensionality Reduction . . . . . . . . . . . . . . 163
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9 Conclusions 165
9.1 Opportunities for Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 170
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A Conditional Progress Variable Dissipation Rate Transport Equation

Derivation 174

Bibliography 177

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List of Tables

6.1 DNS parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

7.1 DNS time snapshot parameters. The time snapshot selected in this work is

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highlighted in light gray. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

8.1 Summary of input features used to train each DNN model. . . . . . . . . . . 152
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8.2 Summary of training, validation, and testing losses for each DNN model (eval-
uated based on datasets corresponding to filter stencil sizes of ∆α /hα = 2, 8,
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16). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
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List of Figures

1.1 Comparison of traditional turbulent combustion modeling approaches and

their associated regimes of applicability. Models further to the left are less
computationally expensive and less general, while models further to the right

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are more computationally expensive and more general. . . . . . . . . . . . . 15

2.1
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Graphical representation of a deep neural network. Figure adapted from
Neutelings [84]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
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3.1 Time-averaged statistics at x/D = 22.5 for the hydrogen jet flame. The
vertical dashed line denotes the stoichiometric mixture fraction Zst = 0.0285.
The solid line corresponds to COTF, the dashed line to ISAM, and symbols
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to experimental measurements with estimated uncertainty [95]. . . . . . . . . 48

3.2 Time-averaged statistics at x/D = 15 for Sandia Flame D. The vertical dashed
line denotes the stoichiometric mixture fraction Zst = 0.353. The solid line
corresponds to COTF, the dashed line to ISAM, and symbols to experimental
measurements with estimated uncertainty [99]. . . . . . . . . . . . . . . . . . 49
3.3 Baseline computational cost per time step for the hydrogen jet flame and
Sandia Flame D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

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3.4 Time-averaged statistics at x/D = 15 for Sandia Flame D. The vertical dashed
line denotes the stoichiometric mixture fraction Zst = 0.353. The solid red
line ( ) corresponds to an ISAT Jacobian delta of 10−2 , the dashed blue line
( ) to 10−3 , the dotted cyan line ( ) to 10−4 , and symbols to experimental
measurements with estimated uncertainty [99]. . . . . . . . . . . . . . . . . . 52
3.5 Sensitivity of the baseline computational cost per time step with respect to
the Jacobian delta ∆ ln χZZ,ref for Sandia Flame D. . . . . . . . . . . . . . . 52
3.6 Time-averaged statistics at x/D = 15 for Sandia Flame D. The vertical dashed
line denotes the stoichiometric mixture fraction Zst = 0.353. The solid red
line ( ) corresponds to an ISAT relative error tolerance of 10−2 , the dashed

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blue line ( ) to 10−3 , the dotted cyan line ( ) to 10−4 , and symbols to
experimental measurements with estimated uncertainty [99]. . . . . . . . . .
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3.7 Time-averaged statistics at x/D = 15 for Sandia Flame D. The vertical dashed
line denotes the stoichiometric mixture fraction Zst = 0.353. The solid red
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line ( ) corresponds to a PDRs relative error tolerance of 10−6 , the dashed
blue line ( ) to 10−7 , the dotted cyan line ( ) to 10−8 , and symbols to
experimental measurements with estimated uncertainty [99]. . . . . . . . . . 54
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3.8 Sensitivity of the baseline computational cost per time step to ISAT and PDRs
relative error tolerances for Sandia Flame D. . . . . . . . . . . . . . . . . . . 55
3.9 Computational cost measured as cumulative wall time and time per time step
for LES of Sandia Flame D, comparing COTF, ISAM without MPI synchro-
nization, and ISAM with MPI synchronization after every time step. The
time per time step versus time steps plot limits the vertical range to visualize
the fluctuations in computational cost. . . . . . . . . . . . . . . . . . . . . . 57

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3.10 Computational cost measured as cumulative wall time and baseline cost for
LES of Sandia Flame D, comparing COTF, ISAM without MPI synchro-
nization, ISAM with MPI synchronization after every time step, ISAM with
constant synchronization, and ISAM with exponential synchronization. . . . 58
3.11 ISAM performance improvements for Sandia Flame D corresponding to vec-
torizing the manifold solver PDRs (a) and leveraging hybrid MPI/OpenMP
parallelization (b). Figures adapted from Bonilla et al. [103]. . . . . . . . . . 59

4.1 Time-averaged radial statistics for mixture fraction and temperature at x/D =
10 for the Sydney flame with χZZ,ref = 100 s−1 . . . . . . . . . . . . . . . . . . 65

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4.2 Nonpremixed manifold model solutions for F = 0.2, χZZ,ref = 100 s−1 ; F =
0.2, χZZ,ref = 200 s−1 ; and F = 0.3, χZZ,ref = 100 s−1 . . . . . . . . . . . . . . 66
4.3
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Computational cost measured as cumulative wall time, time per time step, and
baseline cost for LES of the Sydney flame with χZZ,ref = 100 s−1 , comparing
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COTF, ISAM with MPI synchronization after every time step, and ISAM
with two forms of exponential synchronization. . . . . . . . . . . . . . . . . . 67
4.4 Single time snapshot of Sydney flame LES data plotted versus χZZ,ref and F ,
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colored by the MPI rank in (a) and the filtered mixture fraction in (b). . . . 68

5.1 Baseline computational cost per time step for the Cabra flame. The dashed
line ( ) denotes that ‘error checking’ is disabled in ISAT after 100 time steps.
Figure taken from Novoselov et al. [70]. . . . . . . . . . . . . . . . . . . . . . 79
5.2 Time-averaged centerline profile of temperature ( ). The flame is found to
be statistically lifted at x/Djet = 37, denoted by a dashed line ( ). Figure
taken from Novoselov et al. [70]. . . . . . . . . . . . . . . . . . . . . . . . . . 80

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5.3 Instantaneous centerline snapshots of the three inverse time scales SṁΛ ( ),
SχΛΛ ( ), and SχZZ ( ). Snapshots (a)-(d) are presented in chronological
order, each separated by approximately 2 milliseconds in time. Provided insets
share the same linear colorbar and display the generalized progress variable
source term fields centered around x/Djet = 40. Figure taken from Novoselov
et al. [70]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.4 Instantaneous radial snapshot of the three inverse time scales SṁΛ ( ), SχΛΛ
( ), and SχZZ ( ) is shown on top. The instantaneous radial snapshot of the
alignment of the mixture fraction and generalized progress variable gradients
is shown on bottom. Both snapshots correspond to that of Fig. 5.3 (a). Figure

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taken from Novoselov et al. [70]. . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.1
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Instantaneous conditional dissipation rate profiles extracted from DNS data
for both Karlovitz number cases, normalized by the reference dissipation rate.
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Solid black lines denote 250 normalized dissipation rate profiles randomly
sampled from the extracted profiles for each Karlovitz number case; dotted
cyan lines denote the conditional mean of the DNS (averaged over 500,000
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samples for each Karlovitz number case); dashed red lines denote the algebraic
expression for the stretched model in Eq. 6.7; and dashed blue lines denote
the unstretched flat-flame solution in Eq. 6.7, normalized by the dissipation
rate at the reference generalized progress variable. . . . . . . . . . . . . . . . 90
6.2 Maximum value of the normalized dissipation rate profile gmax versus the
reference dissipation rate χΛΛ,ref colored by Λpeak , the generalized progress
variable at which gmax occurs. . . . . . . . . . . . . . . . . . . . . . . . . . . 91

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6.3 Conditional mean of the normalized dissipation rate profiles for datasets cor-
responding to ∆α /hα = 4. Black dotted lines denote the mean of the normal-
ized dissipation rate profiles from the DNS data; cyan dot-dot-dashed lines
denote the mean of the DNN normalized dissipation rate profile predictions;
red dashed lines denote the stretched model from Eq. 6.7; and blue dashed
lines denote the unstretched model from Eq. 6.6. Note that black dotted lines
nearly exactly coincide with cyan dot-dot-dashed lines. . . . . . . . . . . . . 95
6.4 Conditional average instantaneous model error for datasets corresponding to
∆α /hα = 4. Black dotted lines denote the average error associated with the
mean of the normalized dissipation rate profiles from the DNS data; cyan

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dot-dashed lines denote the average error in DNN instantaneous normalized
dissipation rate profile predictions; red dashed lines denote the average error
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in the stretched model from Eq. 6.7; and blue dashed lines denote the average
error in the unstretched model from Eq. 6.6. . . . . . . . . . . . . . . . . . . 96
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6.5 Manifold solutions corresponding to profiles with χΛΛ,ref = 26, 327 s−1 and
Λpeak < 0.5 in (a)-(c), χΛΛ,ref = 14, 514 s−1 and Λpeak > 0.5 in (d)-(f), and
χΛΛ,ref = 29, 197 s−1 and Λpeak ≈ 0.5 in (g)-(i). Black solid lines, black dotted
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lines, cyan dot-dashed lines, red dashed lines, and blue dashed lines denote so-
lutions corresponding to instantaneous normalized dissipation rate profiles ex-
tracted from DNS data, the mean DNS model, the DNN model, the stretched
model, and the unstretched model, respectively. Note that black solid lines
nearly exactly coincide with cyan dot-dashed lines. . . . . . . . . . . . . . . 97

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6.6 Set of integrated gradients {IGij }m
j=1 for the filtered strain rate magnitude and

generalized progress variable subfilter variance conditionally averaged with

respect to the filtered generalized progress variable, where Λj denotes the
generalized progress variable at the discrete value of the normalized dissipation
rate profile corresponding to index j and xi denotes the corresponding input
feature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.7 Set of integrated gradients {IGij }m
j=1 for the principal rates of strain condi-

tionally averaged with respect to the filtered generalized progress variable,

where Λj denotes the generalized progress variable at the discrete value of the
normalized dissipation rate profile corresponding to index j and xi denotes

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the corresponding input feature. . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.8 Probability density function of the principal rates of strain α and γ present
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in the DNS data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.9 Set of integrated gradients {IGij }m
j=1 for the filtered generalized progress vari-
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able conditionally averaged with respect to the filtered generalized progress
variable, where Λj denotes the generalized progress variable at the discrete
value of the normalized dissipation rate profile corresponding to index j and
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xi denotes the corresponding input feature, xi = Λ̃. Integrated gradients

associated with the deep neural network trained on all input features are pre-
sented in (a), and those of the reduced deep neural network trained without
the principal rates of strain or alignments are presented in (b). . . . . . . . . 104
6.10 Conditional velocity gradient conditioned on generalized progress variable,
colored by ∆Λpeak = Λpeak − Λref . The conditional velocity gradient from
the DNS (∆α /hα = 0) is shown in (a), and the resolved conditional velocity
gradient from the filtered DNS (∆α /hα = 16) is shown in (b). . . . . . . . . 107

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7.1 Unconstrained DNN predictions ( ) of the filtered mixture fraction dissi-
pation rate, filtered cross-dissipation rate, and filtered generalized progress
variable dissipation rate, each conditionally averaged on the filtered mixture
fraction in (a) or on the filtered generalized progress variable in (b). DNN
predictions are compared to conditionally averaged DNS data ( ), condition-
ally averaged Pierce model predictions ( ), and conditionally averaged LRM
predictions ( ). Conditional averages are reported for withheld testing data
with ∆α /hα = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.2 Constrained DNN predictions ( ) of the filtered mixture fraction dissipation
rate, filtered cross-dissipation rate, and filtered generalized progress variable

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dissipation rate, each conditionally averaged on the filtered mixture fraction
in (a) or on the filtered generalized progress variable in (b). DNN predictions
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are compared to conditionally averaged DNS data ( ), conditionally averaged
Pierce model predictions ( ), and conditionally averaged LRM predictions
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( ). Conditional averages are reported for withheld testing data with ∆α /hα
= 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.3 Parity plots for the filtered mixture fraction dissipation rate, filtered cross-
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dissipation rate, and filtered generalized progress variable dissipation rate,

comparing DNN predictions to corresponding filtered DNS data. (a) demon-
strates the results of the unconstrained DNN, and (b) demonstrates the results
of the constrained DNN, both as two-dimensional histograms shaded on a log-
scale according to the number of data samples N at each point. The dashed
red line ( ) denotes exact agreement between DNN predictions and the fil-
tered DNS data. Parity plots are reported for withheld testing data with
∆α /hα = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

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7.4 Mixture fraction dissipation rate, cross-dissipation rate, and generalized
progress variable dissipation rate profiles plotted as color contour plots
versus mixture fraction and generalized progress variable. (a) shows raw
unstructured data extracted from the DNS along a particular representative
manifold stream-surface computed from Eq. 7.19; (b) shows the structured
data interpolated with piecewise linear interpolation; and (c) shows the
structured data interpolated with Gaussian process regression. . . . . . . . . 128
7.5 Unfiltered thermochemical profiles computed as solutions to Eq. 5.4, given
dissipation rate profiles interpolated with either piecewise linear interpolation
(PLI) or Gaussian process regression (GPR). . . . . . . . . . . . . . . . . . . 130

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7.6 Comparison of 90 filtered thermochemical quantities ( ) computed from Eq.
5.15 (∆α /hα = 32), given dissipation rate profiles interpolated with either
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piecewise linear interpolation (PLI) or Gaussian process regression (GPR)
with outlier profiles removed. One-to-one correspondence is denoted by the
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red dashed line ( ) and ± 10% error bounds are denoted by the solid red
lines ( ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.7 Instantaneous normalized mixture fraction dissipation rate, alignment, and
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normalized generalized progress variable dissipation rate profiles plotted as

color contour plots versus mixture fraction and generalized progress variable.
(a) corresponds to instantaneous DNS profiles at a primary location on the
DNS grid; (b) corresponds to instantaneous DNS profiles at a secondary loca-
tion on the DNS grid; (c) corresponds to the ‘universal’ conditional mean of
the extracted DNS profiles; and (d) corresponds to the ‘universal’ algebraic
models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

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 7.8 Conditionally averaged instantaneous error profiles plotted as color contour
plots versus mixture fraction and generalized progress variable. (a) corre-
sponds to the DNN model; (b) corresponds to the mean DNS model; and (c)
corresponds to the algebraic models. . . . . . . . . . . . . . . . . . . . . . . 138
7.9 Instantaneous mixture fraction dissipation rate, cross-dissipation rate, and
generalized progress variable dissipation rate profiles plotted as color contour
plots versus mixture fraction and generalized progress variable. (a) corre-
sponds to instantaneous DNS profiles; (b) corresponds to instantaneous DNN
model predictions; (c) corresponds to the ‘universal’ conditional mean of the
extracted DNS profiles; and (d) corresponds to the ‘universal’ algebraic models.140

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7.10 Temperature solutions plotted as color contour plots versus mixture fraction
and generalized progress variable. Each subfigure corresponds to the multi-
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modal manifold solution for a set of dissipation rate profiles determined via a
different modeling approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
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7.11 Carbon monoxide mass fraction solutions plotted as color contour plots ver-
sus mixture fraction and generalized progress variable. Each subfigure cor-
responds to the multi-modal manifold solution for a set of dissipation rate
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profiles determined via a different modeling approach. . . . . . . . . . . . . . 142

7.12 Generalized progress variable source term solutions plotted as color contour
plots versus mixture fraction and generalized progress variable. Each subfigure
corresponds to the multi-modal manifold solution for a set of dissipation rate
profiles determined via a different modeling approach. . . . . . . . . . . . . . 143

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