KIT – University of the State of Baden-Wuerttemberg and National Research Center of the Helmholtz Association
Institut für Technische Thermodynamik
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Coping with complexity: Model Reduction for Coping with complexity: Model Reduction for the Simulation of Turbulent Reacting flowsthe Simulation of Turbulent Reacting flows
V. Bykov, U. Maas (Karlsruhe Institute of Technology)V. Bykov, U. Maas (Karlsruhe Institute of Technology)
V. Goldsh‘tein (Ben Gurion University)V. Goldsh‘tein (Ben Gurion University)
Institut für Technische Thermodynamik2
Overview
IntroductionIntroduction
Manifold-Based Concepts for Model ReductionManifold-Based Concepts for Model Reduction
Dimension reduction for reaction/diffusion systemsDimension reduction for reaction/diffusion systems
ImplementationImplementation
ConclusionsConclusions
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equation for the scalar fieldequation for the scalar field
filtered or averagedfiltered or averaged
Problems: 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?On which level of accuracy does this equation system have to be solved?Reduce the dimension of the governing equation system!Reduce the dimension of the governing equation system!Note: Chemistry has to be analyzed in the context of a reacting flow!Note: Chemistry has to be analyzed in the context of a reacting flow!
convectionchemistry transport
∂ψ∂t
= F ψ( ) −v ⋅gradψ −1
ρdivD gradψ = F ψ( ) + Ξ ψ,∇ψ,∇2ψ( )
Conservation Equations
ψ = h,p,w1,w2,K ,wns( )
T
∂ψ∂t
= F ψ( ) −v ⋅gradψ +1
ρdivD gradψ = F ψ( ) + Ξ ψ,∇ψ,∇2ψ( )
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describe temporal evolution of the species concentrations in chemical describe temporal evolution of the species concentrations in chemical reactionsreactions
needed for modeling reacting flowsneeded for modeling reacting flows
species conservation equations
averaged species conservation equations
FDF/PDF-transport equation
source terms are functions of the thermokinetic statesource terms are functions of the thermokinetic state
concept of elementary reactionsconcept of elementary reactions
Q = M iω i
Q =∂
∂Ψαρ Ψ( )Sα Ψ( )f[ ]
ω i = ωi T , p,c1,c2 ,K ,cns( ) ωi = ωi h, ρ,w1,w 2,K ,wns( )
ρ∂w i
∂t+ ρ
r v grad w i( ) + div j i = Q = M iωi
rl =Al Tβl exp −Ea,l / RT( ) c j
a j ,l
j=1
ns∏ ω i = rl ˜ a i,l − ai,l( )
l =1
n r∑
Chemical Source Terms
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Points of View
detailed chemistrydetailed chemistry
equation for the scalar field comprises equation for the scalar field comprises nnss + 2 equations + 2 equations
Warnatz, Maas, Dibble: Combustion 2001Warnatz, Maas, Dibble: Combustion 2001
detailed and accurate, but enormous detailed and accurate, but enormous computational effortcomputational effort
enormous amount of unimportant enormous amount of unimportant informationinformation
infinitely fast chemistryinfinitely fast chemistry
equation for the scalar field reduces equation for the scalar field reduces to an equation system for to an equation system for hh, , pp, c, cii
all species concentrations and the all species concentrations and the temperature are known as funcions of temperature are known as funcions of these variablesthese variables
COOxidation
H2
Oxidation
CH4/C2H6Oxidation
CH3OH
Oxidation
CnH2n+2
Oxidation
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Stiff chemical kinetics as well as molecular transport processes cause Stiff chemical kinetics as well as molecular transport processes cause the existence of attractors in composition spacethe existence of attractors in composition space
ILDMs of higher hydrocarbons(Maas & Pope 1992, Blasenbrey & Maas 2000)
Correlation analysis of DNS-Data (Maas & Thevenin 1998)
Observation:
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Decomposition of Motions
Decomposition into “very slow, intermediate and fast subspaces”Decomposition into “very slow, intermediate and fast subspaces”
Fψ = Zc Zs Zf( )⋅
Nc
Ns
Nf
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
⋅
˜ Z c˜ Z s˜ Z f
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
convectionchemistry transport
∂ψ∂t
= F ψ( ) +v ⋅gradψ +1ρ
divD gradψ = F ψ( ) +Ξ ψ,∇ψ,∇2ψ( )
λi Nc( ) <τc
λireal Nf( ) <τs <λi
real Ns( )
%Zc∂ψ∂t
= %ZcF ψ( ) −%Zcv ⋅gradψ + %Zc1ρ
divD gradψ
%Zs∂ψ∂t
= %ZsF ψ( ) −%Zsv ⋅gradψ + %Zs1ρ
divD gradψ
%Zf∂ψ∂t
= %ZfF ψ( ) −%Zfv ⋅gradψ + %Zf1ρ
divD gradψ
diffusion-convection equation
for “quasi conserved” variables
evolution along the LDM
ILDM-equations
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Low-Dimensional Manifold Concepts
QSSA (Bodenstein 1913)QSSA (Bodenstein 1913)
Set right hand side for qss species to zero
ILDM (Maas & Pope 1992)ILDM (Maas & Pope 1992)
Use eigenspace decomposition of Jacobian
GQL (Bykov et al. 2007)GQL (Bykov et al. 2007)
Use eigenspace decomposition of global quasilinearization matrix
Fψ = Zs Zf( ) ⋅Ns 00 Nf
⎛ ⎝ ⎜ ⎞
⎠ ⎟⋅˜ Zs˜ Zf
⎛
⎝ ⎜
⎞
⎠ ⎟
˜ Z f ψ( )F ψ( ) =0∂ψ∂t
= F ψ( )
system equation manifold equation
%Zf =
0 1 00 0 1
⎛⎝⎜
⎞⎠⎟
T =F ψ( )| |ψ1 L ψ1
| |
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
−1
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the system is transformed into fast/slow subsystemsthe system is transformed into fast/slow subsystems
( )⎪⎩
⎪⎨
⎧
=
=
0ss
ff
zQ~
zQ~
zFQ~
Qdt
dz ( )
( )⎪⎩
⎪⎨
⎧
=
=
0zFQ~
zFQ~
Qdt
dz
f
ss
fast subsystem:fast subsystem: slow subsystem: slow subsystem:
Projection of the state space of the CO-H2-O2 system
Reduction - decomposition of motions
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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 2007Bykov, Goldshtein, Maas 2007
GQL application
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Red curve: detailed solutiongreen mesh: 2D GQL manifold red cubes: reference set, Spheres: reduced solution
GQL for an Ignition Problem
Temperature dependence of the ignition delay timeCircles: reduced model (ms = 14)red dashed curve: detailed model (md=31)
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REDIMs for Le = 1 and equal diffusivitiesREDIMs for Le = 1 and equal diffusivities
Bykov and Maas 1997
Stationary solution gives the invariant manifold, Stationary solution gives the invariant manifold, is an estimate for grad is an estimate for grad
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Evolution of a manifold according to reaction and diffusionEvolution of a manifold according to reaction and diffusion
∂ψ∂t
= F ψ( ) −v ⋅gradψ +1ρ
divD gradψ
Reaction-Diffusion-Manifolds (REDIM)
∂ψ∂τ
= I −ψθψθ+
( ) ⋅ F ψ( ) +dρ
ξ oψθθ oξ⎧⎨⎩
⎫⎬⎭
(Bykov & Maas 2007)
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Principle of the Evolution equation
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∂ψ∂τ
= I −ψθψθ+
( )d ξ oψθθ oξ
mixing line
∂ψ∂τ
= I −ψθψθ+
( )F ψ θ( )( )
equilibrium curve
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Principle of the Evolution equation
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∂ψ∂τ
= I −ψθψθ+
( )d ξ oψθθ oξ
mixing line
∂ψ∂τ
= I −ψθψθ+
( )F ψ θ( )( )
equilibrium curve
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KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH)
Evolution equation for the manifoldEvolution equation for the manifold
Basic Procedure: Basic Procedure:
• formulate initial guess• specify boundary conditions• estimate the gradient• solve the evolution equation (PDE)
∂ψ θ( )∂τ
= I −ψθψθ+
( ) ⋅ F ψ θ( )( ) +1ρ
D θ( )ψθσ +1ρ
ξ o D θ( )ψθ( )θ oξ⎧⎨⎩
⎫⎬⎭
ξ=gradθ σ=div gradθ
Extension to detailed transport
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Comparison ILDM-REDIM
Premixed syngas/air systemPremixed syngas/air systemLeft: red mesh: ILDM, green mesh: REDIMLeft: red mesh: ILDM, green mesh: REDIMRight: reaction rate of CO2, mesh: domain of existence of the 2D ILDMRight: reaction rate of CO2, mesh: domain of existence of the 2D ILDM
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It has been shown (Bykov & Maas 2007) that a good estimate gets more It has been shown (Bykov & Maas 2007) that a good estimate gets more and more unimportant for increasing dimensionand more unimportant for increasing dimension
In this work: use gradients from typical flameletsIn this work: use gradients from typical flamelets
KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH)
Estimation of the gradient
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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 solutiongreen: Le=1, equal diffusivities blue: detailed transport, no thermal diffusionred: detailed transportvery 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)
symbols: reduced solution; curves: detailed solutiongreen: Le=1, equal diffusivities blue: detailed transport, no thermal diffusionred: detailed transportvery good gradient estimates used from flamelets (cf. Bykov & Maas 2008)
Results: Stoichiometric Premixed Syngas Flame
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KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH)
Results: Stoichiometric Premixed Syngas Flame
symbols: reduced solution; curves: detailed solutiongreen: Le=1, equal diffusivities blue: detailed transport, no thermal diffusionred: detailed transportvery 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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Attracting Properties of the REDIM
2D REDIM (mesh) and convergence of an unsteady flame (cyan lines) towards 2D REDIM (mesh) and convergence of an unsteady flame (cyan lines) towards the REDIMthe REDIM
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For simpolicity: use For simpolicity: use visualization to monitor visualization to monitor the movement towards the movement towards the manifold.the manifold.
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Implementation
ILDM
GQL
REDIM
interpolation
mass
momentum
energy
∂ρ∂ t
=K
∂ρr v
∂ t=K
∂ρu∂ t
=K
reduced variables
reaction transport
∂ θ∂ t
=S θ( ) + PΞ ψ θ( ),∇ψ θ( ),∇2ψ θ( )( )
S θ( ),ψ θ( ),T θ( ), ρ θ( ),P θ( )
θ,h,p,∇θ,∇h,τ
CFD-code
reduced states
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Example: LES of a premixed flame
Large eddy simulation and experimental studies of turbulent premixed combustion near extinctionP. Wang, F. Zieker, R. Schießl, N. Platova, J. Fröhlich, U. MaasEuropean Combustion Meeting 2011
Scatter plot of temperature vs. hydrogen mass fraction. = 0.71 at one time step, calculated from LES resolved values.
Instantaneous contours of temperature, red line: ZH=0.7. An event of local extinction is seen around x/R=8, r/R=1.
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Conclusions
Efficient methods for kinetic model reduction and its subsequent Efficient methods for kinetic model reduction and its subsequent implementation in reacting flow calculations have been presented.implementation in reacting flow calculations have been presented.
GQLGQL and and ILDMILDM allow an efficient decoupling of fast chemical processes allow an efficient decoupling of fast chemical processes
The slow chemistry domain can be treated efficiently by the REDIM The slow chemistry domain can be treated efficiently by the REDIM ((REREaction-action-DIDIffusion-ffusion-MManifold, anifold, REREduction of the duction of the DIMDIMension)-ension)-method.method.
Financial support by the Financial support by the Deutsche Forschungsgemeinschaft Deutsche Forschungsgemeinschaft is gratefully is gratefully acknowledged.acknowledged.