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Detonation capturing for stiff combustion chemistry. (English) Zbl 0936.80005

Summary: This paper contributes to the topic of unphysical one-cell-per-time-step travelling combustion wave solutions in numerical computations of detonation waves in the presence of stiff chemical source terms. These false weak detonation solutions appear when a gas dynamics-chemistry operator-splitting technique is used in conjunction with modern shock-capturing schemes for compressible flow simulations.
A detailed analysis of piecewise constant three-state weak solutions of the Fickett-Majda detonation model equations is carried out. These structures are idealized analogues of the fake numerical solutions observed in computations. The analysis suggests that the problem can be cured by introducing a suitable ignition temperature below which the chemistry is frozen. It is found that the threshold temperatures needed to effectively suppress the undesired numerical artefacts are considerably lower than any temperature actually found in the reaction zone of a resolved detonation. This is in contrast to earlier suggestions along the same lines in the literature and it allows us to propose the introduction of such a low and otherwise irrelevant ignition temperature threshold as a routine measure for overcoming the problem of artificial weak detonations.
The criterion for choosing the ignition temperature is then extended to the reactive Euler equations and extensive computational tests for both the model and the full equations demonstrate the effectiveness of our strategy. We consider the behaviour of a first-order Godunov-type scheme as well as its second-order extension in space and time using van Leer’s MUSCL approach and Strang splitting.

MSC:

80A25 Combustion
80M20 Finite difference methods applied to problems in thermodynamics and heat transfer
80A30 Chemical kinetics in thermodynamics and heat transfer
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References:

[1] Fickett W, Detonation (1979)
[2] Taki S, 18th Symp. on Combustion pp 1671– (1981)
[3] Schoeffel, S U and Ebert, F. A numerical investigation of a reestablishment of a quenched gaseous detonation. Proc. 16th Int. Conf. on Shocktubes and Waves. Weinheim: VCH.
[4] Boris J P, Numerical Simulation of Reactive Flow (1987) · Zbl 0875.76678
[5] DOI: 10.1016/0010-2180(92)90084-3 · doi:10.1016/0010-2180(92)90084-3
[6] Quirk, J J and Short, M. Numerical investigation of pulsating detonations. Proc. 20th Int. Symp. on Shock Waves. Pasadena, CA. Vol. 2, pp.1125–30. Singapore: World Scientific.
[7] Williams F A, Combustion Theory, The Fundamental Theory of Chemically Reacting Flow Systems (1985)
[8] DOI: 10.1137/0705041 · Zbl 0184.38503 · doi:10.1137/0705041
[9] DOI: 10.1137/0907073 · Zbl 0633.76060 · doi:10.1137/0907073
[10] DOI: 10.1137/0141006 · Zbl 0472.76075 · doi:10.1137/0141006
[11] DOI: 10.1119/1.11973 · doi:10.1119/1.11973
[12] DOI: 10.1137/0153062 · Zbl 0787.65062 · doi:10.1137/0153062
[13] DOI: 10.1016/0021-9991(90)90097-K · Zbl 0682.76053 · doi:10.1016/0021-9991(90)90097-K
[14] DOI: 10.1137/0729074 · Zbl 0759.65060 · doi:10.1137/0729074
[15] DOI: 10.1137/0151018 · Zbl 0734.76082 · doi:10.1137/0151018
[16] DOI: 10.1016/0010-2180(94)00232-H · doi:10.1016/0010-2180(94)00232-H
[17] Nikiforakis N, Proc. R. Soc. A 452 (1997)
[18] Lax P D, SIAM Regional Conference Series in Applied Mathematics 11, in: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (1973) · doi:10.1137/1.9781611970562
[19] Strehlow R A, Combustion Fundamentals (1984)
[20] Courant R, Supersonic Flow and Shock Waves (1948)
[21] DOI: 10.1137/0151016 · Zbl 0731.76076 · doi:10.1137/0151016
[22] DOI: 10.1137/0143071 · Zbl 0572.76062 · doi:10.1137/0143071
[23] Klein R, IMA Volumes in Mathematics and its Applications 35, in: Dynamical Issues in Combustion Theory (1990)
[24] Fernandez G, Nonlinear Hyperbolic Equations–Theory, Numerical Methods, and Applications 24 pp 128– (1989)
[25] DOI: 10.1137/0725021 · Zbl 0642.76088 · doi:10.1137/0725021
[26] DOI: 10.1137/0721062 · Zbl 0565.65048 · doi:10.1137/0721062
[27] Munz C D, Notes on Numerical Fluid Mechanics 14 (1986)
[28] LeVeque R J, Numerical Methods for Conservation Laws (1990)
[29] DOI: 10.1137/0153066 · Zbl 0782.76054 · doi:10.1137/0153066
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