Tokyo Journal of Mathematics

Numerical Methods for Chemically Reacting Fluid Flow Computation under Low-Mach Number Approximation

Toshiyuki Arima

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A mathematical model of environmental fluid is presented to describe fluid flow motions with large density variations. Moreover the associated numerical methods are discussed. The model of environmental fluid is formulated as an unsteady low-Mach number flow based on the compressible Navier-Stokes equations. For low-Mach number flows, the acoustic effects are assumed to be weak relative to the advection effects. Under this assumption, detailed acoustic effects can be removed from governing equations. The low-Mach number formulation thus enables numerical flow analysis with a projection methodology that uses high-order accurate upwind difference of the convection terms with a time step restricted solely by an advection Courant-Friedrichs-Lewy (CFL) condition. The algorithm presented here is based on an iterative implicit time evolution of second order accuracy and a high-accurate spatial discretization with TVD properties for unsteady low-Mach number flows. It is seen from the results on the verification for test cases of flows with a wide range of density variations that our numerical method is validated.

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Tokyo J. of Math. Volume 29, Number 1 (2006), 167-198.

First available in Project Euclid: 20 December 2006

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Arima, Toshiyuki. Numerical Methods for Chemically Reacting Fluid Flow Computation under Low-Mach Number Approximation. Tokyo J. of Math. 29 (2006), no. 1, 167--198. doi:10.3836/tjm/1166661873.

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  • R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena, John Wiley & Sons, Inc. (1960).
  • S. R. Chakravarthy and S. Osher, A new class of high accuracy TVD scheme for hyperbolic conservation laws, AIAA Paper 85 -0363, (1985).
  • U. Ghia, K. N. Ghia and C. T. Shin, High-Re solutions for incompressible flow using the Navier-Stokes equations and a multi-grid method, Journal Computational Physics, 48 (1982), 387–411.
  • A. Harten, On a class of high resolution Total-Variation-Stable finite-difference schemes, SIAM Journal of Numerical Analysis, 21 (1) (1984), 1–12.
  • R.Klein, Semi-implicit extension of Gudonov-type scheme based on low Mach number asymptotic I: One-dimensional flow, Journal Computational Physics, 121 (1995), 213–237.
  • I. P. Jones, A comparison problem for numerical methods in fluid dynamics, The 'Double-Glazing' problem, in Numerical Methods in Thermal Problems, Proceedings of the First International Conference (1979), 338–348.
  • B. van Leer, Toward the ultimate conservative difference scheme. 4, A new approach to numerical convection, Journal of Computational Physics, 23 (1977), 276–299.
  • A. Majda, Compressible fluid flow and systems of conservation laws in several space variable, Springer, New York (1984).
  • H. N. Najm, P. S. Wyckoff and O. M. Knio, A Semi-implicit Numerical Scheme for Reacting Flow, Journal of Computational Physics, 143 (1998), 381–402.
  • C. M. Rhie and W. L. Chow, A numerical study of the turbulent flow past an isolated airfoil with trailing edge separation, AIAA Journal, 21 (11) (1983), 1525–1532.
  • D. B. Spalding and P. L. Stephenson, Laminar flame propagation in hydrogen+bromine mixtures, Proceedings of Royal Society of London, A324, 315 (1971).
  • G. de Vahl Daivis and I. P. Jones, Natural Convection in a Square Cavity. A Comparison Exercise, International Journal for Numerical Method in Fluids, 3, 1983, 227–248.
  • C. R. Wilke, Journal of Chemical Physics, 18, 517 (1950).