kit – university of the state of baden-wuerttemberg and national research center of the helmholtz...

KIT – University of the State of Baden-Wuerttemberg and National Research Center of the Helmholtz Association

Institut für Technische Thermodynamik

www.kit.edu

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


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( )


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


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


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:


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


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


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


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


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)


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

KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH)

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)


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


benötigt.


Principle of the Evolution equation


benötigt.

∂ψ∂τ

= I −ψθψθ+

( )d ξ oψθθ oξ

mixing line

∂ψ∂τ

= I −ψθψθ+

( )F ψ θ( )( )

equilibrium curve


benötigt.



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


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


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


Estimation of the gradient



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)




Results: Stoichiometric Premixed Syngas Flame



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


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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benötigt.

For simpolicity: use For simpolicity: use visualization to monitor visualization to monitor the movement towards the movement towards the manifold.the manifold.


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


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.


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.

kit – university of the state of baden-wuerttemberg and national research center of the helmholtz...

Documents

transport slide

governing equation system

d ildm slide

detailed system solution

lineequilibrium curve

fast chemistry equation

gql application slide

line equilibrium curve