Bykov, V.

Maas, U.

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Viatcheslav V. Bykov Reduction of reacting flow models by the REDIM methodReduction of reacting flow models by the REDIMmethodViatcheslav V. Bykov, Ulrich MaasInstitue of Technical Thermodinamics, Karlsruhe University (TH),Kaiserstrasse 12, 76128 Karlsruhe, Germany1 IntroductionIt is known that typical kinetic models used in recting flows are extremely large, stiff and non-linear[1]. Hence methods of dimension reduction of large detailed chemical kinetic models are very importantand have been developed intensively in the last years. Although significant progress is made in boththeoretical [2, 3, 4] and numerical aspects (see [5, 6, 7, 8] for more references and details about existingmethods of system reduction) there are fundamentally weak points in many existing methods [9]. Amain problem of application of reduction methods or reduced kinetics models is the fact that the reducedchemistry modelling is very often based purely on the analysis of the source term, neglecting the potentialimpact of the transport processes on the so called ‘slow manifold’. Additionally, applying the sametransport term for reduced model as for the detailed one without a proper projection, while the sourceterm is replaced by a reduced chemistry model, can lead to inaccurate results as well (see e.g. [9] formore details).In the present work the method of Reaction Diffusion Manifolds (REDIM) for automatic reductionof chemical kinetics models depending on transport properties of the media and configuration of thereacting flow is discussed. The approach allows to incorporate the effect of coupling of reaction anddiffusion processes. The reduced kinetic mechanism is constructed as a table of a slow manifold meshin the composition or state space. The manifold table contains all necessary information about thereduced kinetics as well as about the projection of the original system of governing equations on thislow dimensional manifold. Thus, the actual reduction is realized by a reformulation of the system ofgoverning equations on the low dimensional manifold that approximates the full system dynamics in thestate space.The proposed method follows the general framework of invariant system manifolds (see e.g. [10, 11]).The main assumption which allows application of this concept is the existence of differences in time scalesof subprocesses in reacting flows leading to a decomposition of the system dynamics. This assumptionis evidently valid for typical cases of dynamical systems describing recting flows where large differencesin time scales stem from the natural properties of combustion kinetics.The method is applied to a premixed laminar flame of relatively simple structure, but neverthelesscontaining main features of recting flows. The comparison of the full system behavior and reduced oneshows the potential of the suggested approach to approximate the system dynamics even in regions ofthe state space far from the equilibrium point of the source term.21st ICDERS - July 23-27, 2007 - Poitiers 1EVA-STAR (Elektronisches Volltextarchiv – Scientific Articles Repository) http://digbib.ubka.uni-karlsruhe.de/volltexte/1000010856 First published in:Viatcheslav V. Bykov Reduction of reacting flow models by the REDIM method2 Mathematical backgroundWe start with a formulation of a typical system of PDEs of a low Mach number reacting flow and use adetailed equation system in general symbolic vector notations (an extension to high Mach number flowsis possible without any principal difficulties)∂Ψ∂t= F (Ψ) − v grad (Ψ) −1ρdiv (D · grad (Ψ)) ≡ Φ (Ψ) , (1)where v represents the velocity field, ρ the density and D the (n by n)-dimensional matrix of thetransport/diffusion coefficients [1], in the present work equal diffusivities is assumed. The state vectorΨ is the (n = ns + 2)-dimensional state space vector Ψ =(h, p, w1M1, ...,wnsMns)T, where h denotes theenthalpy, p the pressure, w1, ..., wns the species mass fractions and M1, ..., Mns the molar masses. F (Ψ)is the n-dimensional vector of the thermo-chemical source term and t denotes the time.According to our assumption, the system solution in the state space is ”close” or belongs to an ms-dimensional invariant manifold defined by an explicit function Ψ (θ): M = {Ψ = Ψ (θ) , Ψ : Rms → Rn},where θ is a ms-dimensional parameter on the manifold, which represents local coordinates. The slowmanifold is defined by using a projection operator onto the normal space (TM)⊥: P(TM)⊥ = I −ΨθΨ+θof M and by the invariance condition [10])(I − Ψθ Ψ+θ)· Φ (Ψ) = 0. (2)Here Ψ+θ is the Moore-Penrose pseudo-inverse of Ψθ [12]. To solve the manifold equation (2) areformulation in terms of a multi-dimensional parabolic PDEs system for Ψ = Ψ (θ, t) is suggested∂Ψ (θ)∂t=(I − Ψθ (θ) Ψ+θ (θ))· Φ (Ψ (θ)) , (3)such that the stationary solution Ψ (θ,∞) defines the desired manifold. The system (3) is integratedstarting from an initial guess for the invariant manifold given by the extended ILDM approach [13] untilthe solution converges. Thus, according to the basic assumption of the invariant manifolds method, thestate vector Ψ of (1), in any point of the flow and at any time, belongs to the manifold i.e. the systemdynamics in the state space is completely described as a movement within this manifold.3 Reaction Diffusion Manifolds and implementationNote that the assumption about the dependence of the stationary set of the system (3) on local coor-dinates only, namely Ψ (t, x) = Ψ (θ (t, x)), is not generally valid and used here as an approximation.There is also a problem with such formulation because the last term in the transport part of Φ (Ψ) (see(1) and (4) depends on the gradient of the local parameter [11]Φ (Ψ (θ)) = F (Ψ) − v Ψθ grad (θ)−− 1ρ(D · Ψθ div (grad (θ)) + (D · Ψθ)θ ◦ grad (θ) ◦ grad (θ)) .(4)To remove the dependence on the gradient of local parameter (4) the following modification of theevolution equation (3) is suggested (see [11] for more details)∂Ψ (θ)∂t=(I − ΨθΨ+θ)·(F (Ψ) −1ρD∥∥ΨTθ grad (Ψ)∥∥2 1msTr (A)), (5)where(Tr (A))i =∑jΨiθjθj(ΨTθjΨθj)−2, ΨTθjΨθj =∑k(∂Ψk∂θj)2.21st ICDERS - July 23-27, 2007 - Poitiers 2Viatcheslav V. Bykov Reduction of reacting flow models by the REDIM methodThe modified invariance equation (5) defines the REDIM and incorporates the impact of the diffusionthrough the modified diffusion term and the gradient of the local parameter grad (θ), which is nowapproximated by the norm of the original system gradient -∥∥ΨTθ grad (Ψ)∥∥. Therefore, if one knowsroughly the norm of the gradient, then it is sufficient to approximate the reduced manifold with anacceptable level of accuracy.CO0123456H2HO2OHH2CHOOC0O12304011223344H0C0O012E1EE0050554E2E20E50650E5035E3E05054E4E05055E5E05056E6E05057E7E0505Figure 1: Projections of the state space. Left figure - 2D manifolds: red+blue is the extended ILDMused as an initial guess, green is the 2D REDIM; black curve with filled symbols is the full systemstationary solution. Right figure - full (black) and reduced (colored) system stationary solutions fordifferent approximations of the system gradient, the upper figure is based on a 1D model, the lowerfigure shows the result of a 2D reduced model.Figure 1 (left) shows a 2D REDIM for a syngas/air system and the detailed system stationary solutionin a projection onto the H2O−CO2−HCO subspace. One can see that the system solution deviates fromthe standard ILDM, which is used as an initial solution for (5), in the domain where the temperature islow (close to the origin), but the REDIM captures rather well the system solution.The right figures illustrate how the stationary solution of the reduced model depends on the approx-imations of the system gradient of the syngas/air laminar flat flame example. In this case the systemgradient approximations were varied within two orders of magnitude, but nevertheless the reduced modelproduces reasonable results even for minor species like HCO and in the so called low temperature do-main. Moreover, increasing of the manifold dimension by one it becomes less sensitive to the estimationof the system gradient. Although two orders were used for the gradient estimation the deviation of thereduced solutions are in a small range of several percent.4 ConclusionsA new approach for automatic mechanism reduction - the REDIM method is discussed in this work. Themethod is developed as an attempt to approximate the PDEs system behavior in the state space. The21st ICDERS - July 23-27, 2007 - Poitiers 3Viatcheslav V. Bykov Reduction of reacting flow models by the REDIM methodREDIM manifold is defined as a stationary set of a multidimensional parabolic PDEs system defined inthe state space. The dependence on the spatial coordinates is removed due to the suggested modificationof the so called invariance condition, which overcomes the problem of an infinite dimensional structureof the PDE system’s slow manifold.The constructed manifold approximates well enough the stationary system solutions even in the low-temperature domain, which previously was a problematic point of several reduction methods. It allows areasonable description of the minor species evolution through the laminar flame front. It was found thatthe sensitivity to the system gradient estimation decreases with the increase of the manifold dimension.Numerical simulations show that the reduced and detailed models are in a good agreement.References[1] Warnatz J et al. (2000). Combustion. Springer Verlag.[2] Zagaris A et al. (2004). Fast and slow dynamics for the computational singular perturbation Method.Multiscale Modelling and Simulation. 2: 613-638.[3] Bykov V et al. (2005). Novel numerical decomposition approaches for multiscale combustion andkinetic models. Journal of Physics: Conference Series. 22: 1-29.[4] Davis MJ (2006). Low-dimensional manifolds in reaction-diffusion equations. 1. Fundamental Aspects& 2. Numerical Analysis and Method Development. Journal of Physical Chemistry A. 110: 5235-5272.[5] Nafe J and Maas U (2003). Hierarchical Generation of ILDMs of Higher Hydrocarbons. Comb. Flame,135:1726.[6] Bykov V et al. (2006). On a modified version of ILDM approach: asymptotic analysis based onintegral manifolds. IMA J. Applied Mathematics, 71:359382.[7] Borok S et al. (2006). Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena.Springer, BerlinHeidelbergNew York, 55-81.[8] Valorani M et al. (2006). An automatic procedure for the simplification of chemical kinetic mecha-nisms based on CSP. Combustion and Flame. 146: 29-51.[9] Lam SH (2006). Reduced chemistry-diffusion coupling, accepted to appear in Combustion Scienceand Technology.[10] Gorban A et al. (2004). Constructive methods of invariant manifolds for kinetic problems. PhysicsReports. 396:197-403.[11] Bykov V and Maas U (2006). The extension of the ILDM concept to reaction-diffusion manifolds,Combustion Theory and Modelling, in print.[12] Golub GH and C.F. van Loan (1989). Matrix Computation. The Hopkins University Press. Balti-more, London.[13] Bykov V and Maas U (2007). Extension of the ILDM method to the domain of slow chemistry.Proc. Comb. Inst. 31, 465-472.21st ICDERS - July 23-27, 2007 - Poitiers 4

Reduction of reacting flow models by the REDIM method

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Reduction of reacting flow models by the REDIM method

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