Title

Joseph-Louis
The Euler-Lagrange Approach Lagrange
(1736 – 1813)
for Steady and Unsteady Flows
Leonard Euler
(1707 – 1783)
M. Sommerfeld
Zentrum für Ingenieurwissenschaften
Martin-Luther-Universität
Halle-Wittenberg
D-06099 Halle (Saale), Germany
www-mvt.iw.uni-halle.de

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 Content of the Lecture

 Introduction discrete particle methods (DPM)

 Description of the Euler/Lagrange approach
 Particle tracking details
 Steady and unsteady coupling
 Convergence behaviour
 Examples of application
 unsteady particle-laden swirling flow
 particle dispersion in a stirred vessel

Conclusions/Outlook

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 Hybrid CFD – DPM
 The numerical calculation of dispersed two-phase flows is often based on a
coupled hybrid approach: CFD (fluid flow) and DPM (particle phase)

Fluid Flow Particle Phase

Euler approach based on continuum Lagrangian tracking of a large number
assumption (fixed numerical grid): of particles assuming point masses
DNS, LES, RANS or URANS with DP << ∆x (except for collisions)

Two-way
coupling

Advantages:
 particle size distribution
 detailed
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modelling
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 CFD-Methods for Fluid Flow
 For the numerical calculation of fluid flows the following methods
are applied:
Increased
 numerical methods for laminar flows Energy
modelling
 Direct numerical simulations (DNS) containing
eddies
requirements

LES
E
Regime of
dissipation
 Large eddy simulations (LES)

k 1 /∆x 1 /η

 Numerical methods based on the Reynolds-averaged conservation

equations (RANS) in connection with turbulence modelling.
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 DPM – Methods 1
 Discrete particle methods (point particles) for fluid-particle
systems:  Multiple particle contacts (finite sized particles)
DPM with soft-sphere collisions  Particle motion caused by fluid-dynamic, external
Discrete element methods (DEM) and contact forces (Van der Waals, electrostatic,.…)
 Contact model: spring, dashpot, friction slider
 Limited by the number of possible particles

β Vp  
( )
 
m p ,i
d v p ,i
=∑ Fn ,ij + Ft ,ij + (u g − u p )
dt j (1 − ε ) tangential
− Vp ∇p + m p g

d ωp ,i
( 
)
 
I p ,i =∑ ri × Ft ,ij − M p ,ij + Tp ,i normal
dt j

3 (1 − ε )  
β=
4 Dp
ρg C*D u g − u p ε −1.65 ε > 0.8 Wen & Yu for dilute regime

(1 − ε)
2
µg
+ 1.75 (1 − ε )
ρg  
β = 150
ε D 2p Dp
ug − up ε ≤ 0.8 Ergun for dense regime

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 DPM – Methods 2
 Discrete particle methods (point particles) for fluid-particle
systems:
 Particle motion driven by fluid-dynamic and
external forces as well as binary collisions
 Only one instantaneous collision of two
DPM with hard-sphere collisions particles at a time (time step: event driven)
DPM with all real particles  deterministic inter-particle collisions
standard Lagrangian tracking  stochastic inter-particle collision model
(representative particles, parcels)


d x p ,i 
= u p ,i
dt

β Vp  
m p ,i
d v p ,i
= (u g − u p ) − Vp ∇p + m p g
dt (1 − ε )

d ωp ,i 
I p ,i = Tp ,i
Solve impulse equations for dt
translation and rotation
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 DPM – Methods 3
 Remarks on the application of DPM:
 The particles need to be considerably smaller than the numerical grid
 The action of the particles should be distributed to the neighbouring cells
 Without inter-particle collisions there is no mechanism to avoid particle
concentrations larger than the closest packing

There should be no limitation in the application of the

Euler/Lagrange approach as long as all the required
physics is modelled properly !!!
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 CFD – DPM Calculations
 Conservation equations for the continuous phase with momentum
∂ (ε ρg )
coupling terms.
Continuity equation 
+ ∇ ⋅ ε ρg u g = 0
Momentum equation ∂t
(Navier-Stokes)
∂ (ε ρg u )

1 p  β Vp   
(u g − u p )
N
  
∂t
+ ∇ ⋅ ε ρ g u u = − ε ∇p − ∇ ⋅ ε τ g + ε ρ g g − ∑ 
Vcell i =1  (1 − ε ) 

Examples of applications for dense particle-laden flows:

CFD-DPM (laminar): Hoomans et al. (1996) two-dimensional fluidised bed
CFD-DPM (laminar): Helland et al (2002) two-dimensional fluidised bed
CFD-DEM (laminar): Feng and Yu 2007 binary mixture in fluidisation
RANS-DEM: Zhao et al. (2008) turbulent 2d spouted bed
LES-DEM: Zhou et al. (2004) 2d simulation of fluidised bed

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 Examples of CFD-DPM Calculations

RANS-DEM of spouted bed (Zhao et al. 2008) Averaged gas velocity

 width 152mm, depth 15 mm, static height 100 mm
 two-dimensional flow simulations (k-ε turbulence model)
 superficial gas velocity 1.58 m/s
 glass beads: 2 mm

Turbulent kinetic energy

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 Euler/Lagrange Approach 1
Two-way
The fluid flow is calculated by solving the Reynolds-averaged coupling
iterations
conservation equations (steady or unsteady) by accounting for
the influence of the particles (source terms).
 k-ε turbulence model
Turbulence models
with coupling:  Reynolds-stress model

The Lagrangian approach relies on tracking a large number

of representative particles (point-mass) through the flow field
accounting for rotation
 drag force
and all relevant forces like:  gravity/buoyancy
 slip/shear lift
Models elementary processes:  slip/rotation lift
 turbulent dispersion  torque on the particle
 particle-rough wall collision
 inter-particle collisions

Particle properties and source terms result from

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averaging for each control volume
 Euler/Lagrange Approach 2
 Fluid flow calculations: Reynolds averaged (conservation) Navier-
Stokes equations (RANS) combined with a suitable turbulence model.
∂ ∂ ∂  ∂ φ 
(σf U j φ) =  Γ  + Sφ + SφP
General form of the
conservation equations (σf φ) +
with source terms (Lain ∂t ∂xj ∂xj  ∂xj 
& Sommerfeld 2012): σ f = (1 − α P ) ρ
k-ε turbulence model

φ Sφ Sφ,P Γφ
constants
1 - 0 -

∂  ∂Uj  ∂p Cµ = 0.09
 ΓU i − µ + µt
Ui
∂xj  ∂  ∂ x + ρ gi S Ui ,P
 x i  i

µ C1 = 1.44
k Gk − ρε S k ,P µ+ t
σk
ε µ C2 = 1.92
ε (C 1 G k − C 2 ρ ε) Sε ,P µ+ t
σε
k
σk = 1.0
 ∂ Ui ∂ U j ∂ Ui 2
Gk = µt  +  , µ t = Cµ ρ
k
∂x ∂ ∂x ε
 j x i  j σε = 1.0
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 Euler/Lagrange Approach 3
The simulation of the dispersed phase by the Lagrangian approach implies
the tracking of a large number of particles through the calculated flow field
solving a set of ordinary differential equations (Sommerfeld 1996, 2008, 2010) .
particle location particle velocity particle angular velocity
  
d xp  d up  d ωp 
= up mp =∑ Fi Ip =T
dt dt dt
 Heat and mass transfer between phases requires the solution of two additional
partial differential equations for droplet diameter and droplet temperature
(Sommerfeld et al. 1993 a).
droplet size d Dp 2m  droplet temperature d Tp 6Q
= =
dt π ρl D 2p dt π ρl D 3p c p ,l
 The particles are treated as point-masses and their size must be smaller than
the dimensions of the numerical grid.
 Each parcel consists of a number of real particles with identical properties.
 Sequential (stationary flows) and simultaneous (unsteady flows) tracking of the
parcels.
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 Particle Tracking Issues 1
 Integrated version of the equation of motion to be solved numerically:
       ∆t L  2  ∆t L   ∆t L   
x new = x + u ∆t + (u − u ) τ P 1 − exp −  + τ P  − 1 − exp −   ∑ Fnon −drag
τ τ τ
P P L P
  P   P   P  

 ∆t L    ∆t L  
u p = u + (u p − u ) exp −
 new     ∑ Fnon −drag
 τ  p
+τ 1 − exp − 
 p    τ p 

 The Lagrangian time step is adjusted dynamically along the particle trajectory
 Criteria for the selection of Lagrangian time step (Sommerfeld 1996, Lain and
Sommerfeld 2008):
Collision time:
∆t L = 0.25 ⋅ min (Tcross , TL , τ P , τ K ) τ K = f (n P , D P , u Re l )

Particle response time:

4 =ρ4P Dρ2P B D B
2
Residence time in Time scale of τ
τP = B
the control volume turbulence D µ
3 Re3P cRe B cD µ

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 Particle Tracking Issues 2
Random particle injection at inlet boundary:
inlet plane profile Number of real
particles in a parcel:
f F g m , j A j t ref
N P ,k =
m m N Parcel

 Sampling of the particle starting position with a uniform distributed random

number:
y P ,k = y j,k + ∆y j ⋅ RY
0 1.01.0 • Select RN
QQ (x )
r (x0i ) i • Loop through
 Sampling of the initial particle size from a the cumulative
prescribed or measured size distribution distribution
(cumulative distribution function) • Q0(xi) > RN
particle size
x i x i+1 x
x
 Sampling of the particle velocity components from a normal distribution function
(mean value and standard deviation interpolated on particle position):
u 0P ,k = u P , j + σ P , j ⋅ RU
 Generation of the fluid velocities seen by the particle using local mean values and
turbulent fluctuations from a normal distribution function.
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 Particle Tracking Issues 3
Properties of the particle phase in a control volume (Decker 2005):
Particle volume fraction: 1 nS
π ∆t n
αP =
VKV
∑6 D
n =1
3
P ,n N P ,n f s,n
t ref
∆t *n
f s,n =
Particle velocity: 1 nS
∆t n
U P ,i =
Nt
∑u
n =1
P ,i N P , n f s , n ∆t n

Particle mean fluctuating velocity: 1 nS

σ P ,i =
Nt
∑u
n =1
2
P ,i N P ,n f s ,n ∆t n − U 2P ,i

Weighing factor: nS
N t = ∑ N P , n f s , n ∆t n ∆t *n
n =1

Particle mass flux: f = α ρ U

P ,i P P P ,i ∆t *n

 In the determination of the averaged particle properties it must be insured that

enough parcels have crossed the control volume.
reduced statistical uncertainty
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 Euler-Lagrange Coupling 1

Coupling approaches between Eulerian and Lagrangian calculations

Fully unsteady Steady calculation

calculation

Ensemble averaging of source terms

from instantaneous particle distribution Temporal and ensemble averaging of
Momentum sources from all fluid source terms from particle trajectories
dynamic forces Momentum sources from change of
Requires a large number of particles particle velocity along trajectories
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Requires less particle trajectories
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 Euler-Lagrange Coupling 2
 Temporal discretisation of Euler/Lagrange approach:
Stationary Flows are solved using the steady-state formulation of the
conservation equations (∆TE = ∞):
 ∆TL is determined from the characteristic time scales of particle motion
 Sequential or simultaneous tracking of the parcels
 Ensemble and temporal averaging of source terms during each coupling
iteration
 Under-relaxation is essential to improve convergence
Unsteady flows are calculated on the basis of the time-dependent
conservation equations (∆TL ≈ ∆TE):
 Eulererian und Lagrangian programme modules are solved sequentially
with identical time steps
 Simultaneous particle tracking
 Spatial and ensemble averaging of particle phase properties and source
terms (no under-relaxation) within a computational cell
 A large number of parcels are needed to obtain reliable sources
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 Euler-Lagrange Coupling 3

Unsteady flows with ∆TE > ∆TL are also calculated on the basis of the
time-dependent conservation equations (Sommerfeld et al. 1997;
Lipowsky and Sommerfeld 2005; Sommerfeld et al. 2010):
 The selection of ∆TE determines the temporal resolution of the flow
 Sequential calculation of both phases with different time steps
 The particles see a flow field frozen over ∆TE
 Simultaneous particle tracking
 Temporal and ensemble averaging of particle phase properties and
source terms (no under-relaxation) within a computational cell

Eulerian Calculation ∆tE

Flow field
Sources

∆tL
Lagrangian Calculation

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 Euler-Lagrange Coupling 4
 Particle phase source terms for k-ε turbulence model:
 Sampling of source terms along parcel trajectories calculated from
particle velocity change (ensemble and time averaging):
 Momentum equations, Crowe et al. 1977:
 n +1  ρL  
SU i ,P =−
1
Vcv ∆t E
m N  u (
∑k k k ∑n k ,i k ,i − g i
− u n
) 1 −  ∆t L 
  ρB  
 Turbulent kinetic energy:

S k , P = ∑ S k ,i
i

S k ,i = u i S U i P − u i S U i P
 Dissipation rate:
ε
Sε , P = C ε 3 Sk , P (Cε3 = 1.1 − 1.8)
k
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 Euler-Lagrange Coupling 5
Grid generation, Eulerian Part Lagrangian Part
Boundary conditions, Calculation of the fluid flow without Tracking of parcels without inter-
Inlet conditions particle phase source terms particle collisions,
Sampling of particle phase properties
and source terms

Eulerian Part
Calculation of the fluid flow with
Two-way coupling particle phase source terms:
• Converged Solution
approach for • Solution with a fixed number
stationary flows Coupling
of iterations

Iterations
Lagrangian Part
Tracking of parcels with inter-particle
collisions,
Sampling of particle phase properties
and source terms
Under-relaxation of source terms:
= (1 − γ ) S + γ S
no
i +1 i i +1 Convergence
SφP φP φP ( calculated ) two-way
coupling

yes
Under-relaxation of the source terms
improves convergence behaviour !!! Output:
Flow field,
(Kohnen et al. 1994) Particle-phase statistics

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 Euler-Lagrange Coupling 6

 Temporal coupling of Eulerian and Lagrangian calculations.

 Fluid flow calculated at fixed time step ∆tE

 Particles are tracked in a ‚frozen‘ fluid field over ∆tE
 Dynamic calculation of ∆tL for every Lagrangian time step
 Independent time stepping for each particle
 PSTs are written back to fluid solver for the next Eulerian time step
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 Horizontal Pipe Flow 1
Convergence behaviour of stationary particle-
laden flow in the pipe (steady-state) k-ε turbulence model
(Lain & Sommerfeld 2012) pipe length: 6 m
pipe diameter: 63 mm
Two-way coupling U0 = 20 m/s
130 μm particles
1 hor. velocity ρp = 2450 kg m−3
Normalised Residuals [ - ]

vert. velocity
0.1 mass loading η = 1.0
0.01 R0 = 2.2 µm, ∆γ = 1.4°
1E-3
1E-4
1E-5
1E-6
Four-way coupling
1E-7
0 5000 10000 15000 20000 25000 30000 35000
Number of Iterations [ - ]
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 Horizontal Pipe Flow 2
Convergence behaviour of stationary particle-laden flow in the pipe:
1.0
Profile of stream-wise
0.8 single-phase
gas velocity
1. iteration
3. iteration
Niter ≤ 7 Two-Way
y/D[-]

0.6 7. iteration
8. iteration
10. iteration
0.4 18. iteration
27. iteration 1.0
0.2
0.8
single-phase
0.0 1. iteration

y/D[-]
0.7 0.8 0.9 1.0 1.1 1.2 0.6 3. iteration
U / U0 [ - ] 7. iteration
8. iteration
0.4 10. iteration
18. iteration
27. iteration
0.2
Profile of turbulent
kinetic energy 0.0
0.000 0.002 0.004 0.006 0.008
2
k/U [-]
0
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 Examples of Application

Unsteady particle-laden swirling

flow
Lipowsky and Sommerfeld 2005;
Sommerfeld et al. 2010

Particle dispersion in a stirred

vessel
(Decker and Sommerfeld 2000;
Sommerfeld and Decker 2004)

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 Examples of Application

Particle separation devices

Pneumatic conveying

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Thank you very much for your attention !!!

We like to sit
on the particles

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