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Coping with complexity: Model Reduction for
the Simulation of Turbulent Reacting flows
V. Bykov, U. Maas (Karlsruhe Institute of Technology)
V. Goldsh‘tein (Ben Gurion University)
Institut für Technische Thermodynamik
KIT – University of the State of Baden-Wuerttemberg and
National Research Center of the Helmholtz Association
www.kit.edu
Overview
Introduction
Manifold-Based Concepts for Model Reduction
Dimension reduction for reaction/diffusion systems
Implementation
Conclusions
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Institut für Technische Thermodynamik
Conservation Equations
equation for the scalar field


1
 F    v  grad  divD grad  F      , ,2
t



1
 F    v  grad  divD grad  F      , ,2 
t

chemistry convection
filtered or averaged
transport


  h,p,w1,w 2 ,K ,wns
T
Problems:
• extremely high dimension of the system!
• non-linear chemical source terms
• strong coupling of chemistry with molecular transport
• stiffness of the governing equation system
On which level of accuracy does this equation system have to be solved?
Reduce the dimension of the governing equation system!
Note: Chemistry has to be analyzed in the context of a reacting flow!
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Chemical Source Terms
describe temporal evolution of the species concentrations in chemical
reactions
needed for modeling reacting flows
species conservation equations 
w i
 v grad w i   div j i  Q  Mi  i
t
Q  M i i
averaged species conservation equations
FDF/PDF-transport equation
Q

 S  f 

source terms are functions of the thermokinetic state

 i  i T , p,c 1, c 2 ,
,c n s


i   i h, ,w 1,w 2 ,
,w ns

concept of elementary reactions
rl  Al T
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l
ns
exp E a,l / RT  c j a j ,l
j 1
nr
 i   rl a˜ i,l  ai,l 
l 1
Institut für Technische Thermodynamik
Points of View
detailed chemistry
infinitely fast chemistry
equation for the scalar field comprises
ns + 2 equations
equation for the scalar field reduces
to an equation system for h, p, ci
all species concentrations and the
temperature are known as funcions of
these variables
H2
Oxidation
CO
Oxidation
CH3OH
Oxidation
CH4/C2H6
Oxidation
C nH2n+2
Oxidation
Warnatz, Maas, Dibble: Combustion 2001
detailed and accurate, but enormous
computational effort
enormous amount of unimportant
information
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Observation:
Stiff chemical kinetics as well as molecular transport processes cause
the existence of attractors in composition space
ILDMs of higher hydrocarbons
Correlation analysis of DNS-Data (Maas
(Maas & Pope 1992, Blasenbrey & Maas 2000) & Thevenin 1998)
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Decomposition of Motions

1
2
 F   v  grad  divD grad  F      , , 
t


chemistry convection

transport
Decomposition into “very slow, intermediate and fast subspaces”
F  Zc
Zc
Zs
Nc

Zf  


Ns
 Z˜ c 
  ˜ 
 Z s 
  ˜ 

Nf  
Zf 

1
 Z%c F    Z%cv  grad  Z%c divD grad
t


1
Z%s
 Z%s F    Z%sv  grad  Z%s divD grad
t


%F    Z%v  grad  Z% 1 divD grad
Z%

Z
f
f
f
f
t

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i Nc    c
real
real
i Nf    s  i Ns 
diffusion-convection equation
for “quasi conserved” variables
evolution along the LDM
ILDM-equations
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Low-Dimensional Manifold Concepts
system equation
manifold equation

 F  
t
˜ f   F    0
Z
QSSA (Bodenstein 1913)
 0 1 0
Set right hand side for qss species to zero Z f  
 0 0 1
ILDM (Maas & Pope 1992)
Use eigenspace decomposition of Jacobian F  Z s
GQL (Bykov et al. 2007)
Use eigenspace decomposition of global
quasilinearization matrix
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N s
Z f   
 0
 |
T  F    1 L

 |
˜ s 
0  Z
  
˜ f 
Nf  Z
| 
 1

| 
1
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Reduction - decomposition of motions
the system is transformed into fast/slow subsystems
slow subsystem:
fast subsystem:
~
 dz
  Qs Qs Fz 
 dt
~
 Q
f Fz   0

~
 dz

Q
Q

f f Fz 
 dt
~
~
Q
 s z  Qs z 0
Projection of the state space of the CO-H2-O2 system
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GQL application
red mesh: ILDM, green mesh: manifold, symbols: reference points
blue curve: detailed system solution, cyan curve: fast subsystem solution
magenta curves: detailed stationary system solution of flat flames
Bykov, Goldshtein, Maas 2007
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GQL for an Ignition Problem
Red curve: detailed solution
green mesh: 2D GQL manifold
red cubes: reference set,
Spheres: reduced solution
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Temperature dependence of the ignition
delay time
Circles: reduced model (ms = 14)
red dashed curve: detailed model
(md=31)
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Reaction-Diffusion-Manifolds (REDIM)
Evolution of a manifold according to reaction and diffusion

1
 F    v  grad  div D grad
t






d
 I       F     o  o 




(Bykov & Maas 2007)
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KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH)
Principle of the Evolution equation



 I     F   

equilibrium curve
Zur Anzeige w ird der QuickTime™
Dekompressor „BMP“
benötigt.



 I     d  o  o

mixing line
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Institut für Technische Thermodynamik
Principle of the Evolution equation



 I     F   

equilibrium curve
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Dekompressor „BMP“
benötigt.



 I     d  o  o

mixing line
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Extension to detailed transport
Evolution equation for the manifold


  
1
1
 I       F    D       oD     o 






 =grad

 =div grad
Basic Procedure:
•
•
•
•
formulate initial guess
specify boundary conditions
estimate the gradient
solve the evolution equation (PDE)
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KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH)
Comparison ILDM-REDIM
Premixed syngas/air system
Left: red mesh: ILDM, green mesh: REDIM
Right: reaction rate of CO2, mesh: domain of existence of the 2D ILDM
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Estimation of the gradient
It has been shown (Bykov & Maas 2007) that a good estimate gets more
and more unimportant for increasing dimension
In this work: use gradients from typical flamelets
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KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH)
Results: Non-Premixed Syngas Flame
symbols: reduced solution; curves: detailed solution
green: Le=1, equal diffusivities
blue:
detailed transport, no thermal diffusion
red:
detailed transport
very good gradient estimates used from flamelets (cf. Bykov & Maas 2008)
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Institut für Technische Thermodynamik
KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH)
Results: Stoichiometric Premixed Syngas Flame
symbols: reduced solution; curves: detailed solution
green: Le=1, equal diffusivities
blue:
detailed transport, no thermal diffusion
red:
detailed transport
very good gradient estimates used from flamelets (cf. Bykov & Maas 2008)
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Institut für Technische Thermodynamik
KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH)
Results: Stoichiometric Premixed Syngas Flame
symbols: reduced solution; curves: detailed solution
green: Le=1, equal diffusivities
blue:
detailed transport, no thermal diffusion
red:
detailed transport
very good gradient estimates used from flamelets (cf. Bykov & Maas 2008)
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KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH)
2-D Manifold for a Non-Premixed Syngas Flame
stoichiometric syngas-air flat flame, detailed transport
curves: detailed solution, mesh: REDIM
Left: starting guess (linear interpolation between flamelets)
Right: REDIM
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KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH)
Attracting Properties of the REDIM
For simpolicity: use
visualization to monitor
the movement towards
the manifold.
Zur Anzeige w ird der QuickTime™
Dekompressor „“
benötigt.
2D REDIM (mesh) and convergence of an unsteady flame (cyan lines) towards
the REDIM
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Implementation
reduced states
ILDM
GQL
REDIM
,h,p,,h,
CFD-code
reduced variables

2
S   P   ,  ,   
t

reaction

transport


mass
t
 v

momentum
t
u

energy
t
interpolation
S  ,   ,T  ,  ,P  
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Example: LES of a premixed flame
Instantaneous contours of temperature,
red line: ZH=0.7. An event of local
extinction is seen around x/R=8, r/R=1.
Scatter plot of temperature vs. hydrogen
mass fraction.  = 0.71 at one time step,
calculated from LES resolved values.
Large eddy simulation and experimental studies of turbulent premixed combustion
near extinction
P. Wang, F. Zieker, R. Schießl, N. Platova, J. Fröhlich, U. Maas
European Combustion Meeting 2011
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Conclusions
Efficient methods for kinetic model reduction and its subsequent
implementation in reacting flow calculations have been presented.
GQL and ILDM allow an efficient decoupling of fast chemical processes
The slow chemistry domain can be treated efficiently by the REDIM
(REaction-DIffusion-Manifold, REduction of the DIMension)method.
Financial support by the Deutsche Forschungsgemeinschaft is gratefully
acknowledged.
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