Comparison of Conventional and Low-Dimensional Manifold Methods to Reduce Reaction Mechanisms

Eggels, Rlgm; Louis, J.J.J.; Kok, Jacobus B.W.; Goey, de Lph Philip

doi:10.1080/00102209708935634

Cited by 11 publications

(7 citation statements)

References 6 publications

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“…Refinements, applications, and evaluations of the ILDM method against direct numerical simulations can be found in Refs. [1][2][3][4]11,12,37,39,42,47,58,[68][69][70]73,[86][87][88].…”

Section: Many Chemical Reaction Mechanisms In Combustionmentioning

confidence: 99%

Asymptotic analysis of two reduction methods for systems of chemical reactions

Kaper

2002

Physica D: Nonlinear Phenomena

132

127

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This paper concerns two methods for reducing large systems of chemical kinetics equations, namely, the method of intrinsic low-dimensional manifolds (ILDMs) due to Maas and Pope [Combust. Flame 88 (1992) Chem. Phys. 93 (1990) 1072. Both methods exploit the separation of fast and slow reaction time scales to find low-dimensional manifolds in the space of species concentrations where the long-term dynamics are played out. The asymptotic expansions of these manifolds (ε ↓ 0, where ε measures the ratio of the reaction time scales) are compared with the asymptotic expansion of M ε , the slow manifold given by geometric singular perturbation theory. It is shown that the expansions of the ILDM and M ε agree up to and including terms of O(ε); the former has an error at O(ε 2 ) that is proportional to the local curvature of M 0 . The error vanishes if and only if the curvature is zero everywhere. The iterative method generates, term by term, the asymptotic expansion of M ε . Starting from M 0 , the ith application of the algorithm yields the correct expansion coefficient at O(ε i ), while leaving the lower-order coefficients invariant. Thus, after applications, the expansion is accurate up to and including the terms of O(ε ). The analytical results are illustrated on a planar system from enzyme kinetics (Michaelis-Menten-Henri) and a model planar system due to Davis and Skodje.

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“…Refinements, applications, and evaluations of the ILDM method against direct numerical simulations can be found in Refs. [1][2][3][4]11,12,37,39,42,47,58,[68][69][70]73,[86][87][88].…”

Section: Many Chemical Reaction Mechanisms In Combustionmentioning

confidence: 99%

Asymptotic analysis of two reduction methods for systems of chemical reactions

Kaper

2002

Physica D: Nonlinear Phenomena

132

127

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“…This state is consistent with Rankine-Hugoniot jump conditions for the inviscid equivalent of Eqs. (1)(2)(3)(4)(5)(6)(7)(8)(9)(10). Knowledge of these parameters allows determination of all other dependent variables at t = 0 s through the use of the governing equations.…”

Section: Governing Equations Initial and Boundary Conditionsmentioning

confidence: 99%

“…Equation (5) is a constitutive equation for the evolution of species which is the law of mass action with Arrhenius kinetics. Equations (6)(7)(8) give constitutive relations for mass, momentum, and energy diffusion which are the ordinary Fick's law, Newtonian stress-strain rate relation, and Fourier's law, all with assumed constant diffusion coefficients. The contribution to the heat flux due to species diffusion has been neglected here, though this is easily remedied.…”

Section: Governing Equations Initial and Boundary Conditionsmentioning

confidence: 99%

“…(1)(2)(3)(4)(5)(6)(7)(8)(9)(10) are M i , a j , β j , E j , , ν ij , D ij , µ, k, and h o i . They represent the molecular mass of species i, the kinetic rate constant of reaction j, the temperature dependency exponent of reaction j, the activation energy of reaction j, the universal gas constant ( = 8.31441 J/mol/K), the stoichiometric coefficient of the ith species of reaction j, the mass diffusivity of the ith species due to the jth species, the dynamic viscosity, the thermal conductivity, and the standard enthalpy of formation of species i.…”

Section: Governing Equations Initial and Boundary Conditionsmentioning

confidence: 99%

“…A number of studies have appeared in recent years advancing the technique and some variants, cf. Blansenbrey, et al 5 ; Eggels, et al 6 ; Hamiroune, et al 7 ; Lam 8 ; Schmidt, et al 9 ; Norris 10 ; and Yang and Pope 11,12 . In this study we have used the method of Maas and Pope, which systematically equilibrates the fast time scales and resolves the slow time scales of the reaction mechanism, thereby eliminating the associated stiffness.…”

Section: Introductionmentioning

confidence: 99%

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Novel modeling of hydrogen/oxygen detonation

Rastigejev

Singh

Bowman

et al. 2000

38th Aerospace Sciences Meeting and Exhibit

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A standard ignition delay problem for a mixture of hydrogen, oxygen, and argon in a shock tube is extended to the viscous regime and solved using the method of Intrinsic Low Dimensional Manifolds (ILDM) coupled with a Wavelet Adaptive Multilevel Representation (WAMR) spatial discretization technique. An operator splitting method is used to describe the reactions as a system of ordinary differential equations at each spatial point. The ILDM method is used to eliminate the stiffness associated with the chemistry by decoupling processes which evolve on fast and slow time scales. The fast time scale processes are systematically equilibrated, thereby reducing the dimension of the phase space required to describe the reactive system. The WAMR captures the detailed spatial structures automatically with a small number of basis functions thereby further reducing the number of variables required to describe the system. Using a maximum of only 300 collocation points and 15 scale levels allows results with striking resolution of fine scale viscous and induction zones to be obtained. Additionally, the resolution of physical diffusion processes minimizes the effects of potentially reaction-inducing artificial entropy layers associated with numerical diffusion.

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