analysis of transient processes in the context of...

Institut für Technische Thermodynamik

Analysis of Transient Processes in the Context of REDIM

Viatcheslav Bykov, Alexander Neagos, Ulrich Maas

Institut für Technische Thermodynamik Karlsruher Institut für Technologie (KIT)


Overview

!   Reaction/Diffusion Manifolds

!   Analysis of the local time scales

!   Examples

!   Conclusions


!   Starting point: equation for the scalar field

!   Idea: use correlations introduced by the fast chemical kinetics (or by molecular transport)

!   Identify low-dimensional manifolds

!   Project governing equations onto LDM

Manifold Methods

0,000

0,002

0,004

0,006

0,008

0,010

0,1 0,2 0,3 0,4 0,5

wH2O

wCO2

fast

slow

convection chemistry transport

∂ψ∂t

= F ψ( ) +v ⋅gradψ + 1ρdivD gradψ = F ψ( ) +Ξ ψ ,∇ψ ,∇2ψ( )

ψ = h,p,w1,w2,…,wns( )

T

ψ =ψ θ( ), θ = θ1,…,θm( )

∂θ∂ t

= S θ( ) + v gradθ + 1ρ

P div D∗gradθ⎛ ⎝

⎞ ⎠


Decomposition of Motions

Decomposition into “very slow, intermediate and fast subspaces”

Problem: difficult to solve, dimensions change locally

Fψ = Zc Zs Zf( ) ⋅Nc

NsNf

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⋅

˜ Z c˜ Z s˜ Z f

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

convection chemistry transport

∂ψ∂t

= F ψ( ) +v ⋅gradψ + 1ρdivD gradψ = F ψ( ) +Ξ ψ ,∇ψ ,∇2ψ( )

λi Nc( ) < τcλireal Nf( ) < τs < λireal Ns( )

Zc∂ψ∂t

= ZcF ψ( ) − Zcv ⋅ gradψ + Zc1ρdivD gradψ

Zs∂ψ∂t

= ZsF ψ( ) − Zsv ⋅ gradψ + Zs1ρdivD gradψ

Zf∂ψ∂t

= ZfF ψ( ) − Zfv ⋅ gradψ + Zf1ρdivD gradψ

diffusion-convection equation for “quasi conserved” variables evolution along the LDM

ILDM-equations

Institut für Technische Thermodynamik KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH)

Evolution of a manifold according to reaction and diffusion

∂ψ∂t

= F ψ( ) −v ⋅gradψ + 1ρdivD gradψ

Reaction-Diffusion-Manifolds (REDIM)

(Bykov & Maas 2007)

∂ψ θ( )∂τ

= I −ψθψθ+( ) ⋅ F ψ θ( )( ) + 1ρD θ( )ψθσ +1ρξ D θ( )ψθ( )θ ξ

⎧⎨⎩

⎫⎬⎭

ξ=gradθ σ=div gradθ


!   Basic Procedure:

formulate initial guess specify boundary conditions estimate the gradient solve the evolution equation (PDE)

Stationary solution yields the REDIM

Numerical implementation


Tabulation strategy (REDIM)

ψ ,gradψ


!   1st limiting case: no diffusion slow invariant manifolds !   2nd limiting case: no reaction minimal surfaces !   special case: 1D, gradients from flames flamelet equation

!   Previous work: !   Application to premixed and non-premixed systems !   Implementation in laminar and turbulent flame calculations !   Extension to detailed transport models !   On the fly improvement of the gradient estimates

!   This Work: Analysis of the description of transient processes within the ILDM concept.

!   How do transient processes influence the REDIM? !   Do they only change the dynamics within the REDIM or do they

change the REDIM itself?


Counterflow methane air flame

Periodical perturbation of the fuel air ratio

König et al.

Dynamic Behavior in Physical Space


Counterflow methane air flame

Periodical perturbation of the fuel air ratio

Black: 20 Hz Red: 250 Hz Blue: 500 Hz Green: 100 Hz

König et al.

Dynamic Behavior in Composition Space


Consequences !   The quality of the description of transient processes will depend on the

dimension of the manifold. !   Chemistry and Transport must be analyzed in a coupled way


Counterflow configuration

Boundary layer approximation

Spatially 1D simulation

Detailed solution with INSFLA

Implementation of the reduced model in INSFLA

Test Case: Non-Premixed H2-Air Flame


Attracting Properties of the REDIM

!   2D REDIM (mesh) and convergence of an unsteady flame (cyan lines) towards the REDIM

!   For simplicity: use visualization to monitor the movement towards the manifold.


Source Term Eigenvalues

!   The invariance condition postulates that the dynamics of the detailed system will at any time be tangential to the REDIM.

!   In case of strong perturbations the detailed profile can leave the REDIM and then relax back to the REDIM.

!   Strong perturbations can lead to extinction.

!   How accurately can the REDIM capture the part of the dynamics tangential to the REDIM?

!   An answer can be obtained by investigation the eigenvalues of the Jacobian.


Source Term Eigenvalues

!   Jacobian of the chemical source term

!   Eigenvalues of a projected Jacobian

can be obtained via solving

Eigenvalues of the projected Jacobian are given by the eigenvalues of

Note: Degenerate cases can be handled by Schur-decomposition

Fψ ψ( ) =

∂F1∂ψ1

∂F1∂ψn

∂Fn∂ψ1

∂Fn∂ψn

⎛

⎝

⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

F̂ψ=ψθ

+Fψψθ F̂ψ⊥ = ψθ⊥( )+Fψψ θ⊥

Z Z +FψV =V N

Z +FψZ X = X N, V = Z X


Eigenvalues of tangential projection of the source term Jacobian

!   Projected Jacobian of the chemical source term

!   Plotted:

F̂ψ=ψθ

+Fψψθ

F̂ψX = X N

λmax N ( )


!   Projected Jacobian of the chemical source term

!   Plotted:

Eigenvalues of normal projection of the source term Jacobian

F̂ψ⊥ = ψθ⊥( )+Fψψ θ⊥

F̂ψ⊥X = X N⊥

λmax N⊥( )


Simulation of Flame Quenching !   Simulation quenching

!   Increase strain rate above the quenching limit

Reduced (white) and detailed (black) solution


REDIM – Detailed Transport Model and Pressure Dependence

!   investigation of the dependence of the REDIM on pressure


REDIM – Detailed Transport Model and Pressure Dependence

!   Simulation of the non-lineare dependence of the quenching limit on pressure

!   Comparison with detailed simulations

!   Test case for the ability of the reduction method to cope with changing reaction.

!   Test case for the ability of the reduction method to cope with detailed transport.

detailed reduced 1bar 560 1/s 600 1/s 2 bar 720 1/s 760 1/s 4 bar 580 1/s 680 1/s 8 bar 320 1/s 300 1/s


Conclusions

!   The concept of Reaction-Diffusion Manifolds (REDIM) is an efficient tool for model reduction.

!   Local time scale analyses reveal the dynamic behavior of movements along and towards the REDIM.

!   Transient system dynamics can be handled by the concept.

!   Future work: adaptive dimension hierarchical manifold concept

!   Financial support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged.

analysis of transient processes in the context of...

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