On the behavior of upwind schemes in the low Mach number limit. II: Godunov type schemes. (English) Zbl 1049.76040

Guillard, Hervé; Murrone, Angelo

Comput. Fluids 33, No. 4, 655-675 (2004).

Summary: This paper presents an analysis of Godunov scheme in the low Mach number regime [for part I see the first author and C. Viozat, ibid. 28, No. 1, 63–86 (1999; Zbl 0963.76062)]. We study the Riemann problem and show that the interface pressure contains acoustic waves of order \(\mathcal O(M_{\ast})\), where \(M_{\ast}\) is the reference Mach number, even if the initial data are well-prepared and contain only pressure fluctuations of order \(\mathcal O(M_{\ast}^2)\). We then propose to modify the fluxes computed by Godunov type schemes by solving a preconditioned Riemann problem instead of the original one. We show that this strategy allows to recover a correct scaling of pressure fluctuations. Numerical experiments confirm these theoretical results.

MSC:

76M12	Finite volume methods applied to problems in fluid mechanics
76M20	Finite difference methods applied to problems in fluid mechanics
76N15	Gas dynamics (general theory)

Keywords:

Riemann problem; interface pressure; acoustic waves

Citations:

Zbl 0963.76062

Software:

HE-E1GODF

References:

[1]	Bijl, H.; Wesseling, P., A unified method for computing incompressible and compressible flows in boundary fitted coordinate, J. comput. phys, 141, 153-173, (1998) · Zbl 0918.76054
[2]	Clerc, S., Numerical simulation of the homogeneous equilibrium for two-phase flows, J. comput. phys, 161, 354-375, (2000) · Zbl 0965.76051
[3]	Dervieux, A.; Guezengar, D.; Guillard, H.; Viozat, C., Analysis of low Mach simulations with compressible upwind codes, ()
[4]	Radespiel, R.; Turkel, E.; Kroll, N., Assessment of preconditioning methods for multidimensional aerodynamics, Comput. fluids, 26, 613-634, (1997) · Zbl 0893.76063
[5]	Grenier, E., Oscillatory perturbations of the navier – stokes equations, J. math. pures. appl, 76, 477-498, (1997) · Zbl 0885.35090
[6]	Guillard, H.; Abgrall, R., Modélisation des fluides numériques compressibles, Series in applied mathematics, vol. 5, (2001), Gauthier-Villars/North Holland Paris · Zbl 0988.76003
[7]	Guillard, H.; Viozat, C., On the behavior of upwind schemes in the low Mach number limit, Comput. fluids, 28, 63-96, (1999) · Zbl 0963.76062
[8]	Klainerman, S.; Majda, A., Compressible and incompressible fluids, Commun. pure appl. math, 35, 629-653, (1982) · Zbl 0478.76091
[9]	Klein, R., Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics I: one-dimensional flow, J. comput. phys, 121, 213-237, (1995) · Zbl 0842.76053
[10]	Kreiss, H.-O., Problems with different time scales for partial differential equations, Commun. pure appl. math, 33, 399-440, (1980)
[11]	Masella, J.M.; Faille, I.; Galouët, T., On an approximate Godunov scheme, Int. J. comput. fluid dyn, 12, 133-149, (1999) · Zbl 0944.76041
[12]	Metivier, G.; Schochet, S., The incompressible limit of the nonisentropic Euler equations, Arch rat mech anal, 158, 61-90, (2001) · Zbl 0974.76072
[13]	Metivier G, Schochet S. Limite incompressible des équations d’Euler non-isentropiques. Séminaire équations aux dérivées partielles de l’école Polytechnique, 2001. Available from: http://www.maths.univ-rennes1.fr/metivier/preprints.html
[14]	Paillère, H.; Clerc, S.; Viozat, C.; Toumi, I.; Magnaud, J.-P., Numerical methods for low Mach number thermal-hydraulic flows, ()
[15]	Paillère H, Viozat C, Kumbaro A, Toumi I. Comparison of low Mach number models for natural convection problems. In: Eurotherm Seminar No 63. Genoa, Italy, 6-8 September 1999
[16]	Schall E, Viozat C, Koobus B, Dervieux A. On the computation of unsteady ans steady low Mach flows with implicit upwind methods. In: ECCOMAS 2001, 2001 · Zbl 1043.76050
[17]	Schochet, S., Asymptotics for symmetric hyperbolic systems with a large parameter, J. differen. equat, 75, 1-27, (1988) · Zbl 0685.35014
[18]	Schochet, S., Fast singular limits of hyperbolic PDE’s, J. differen. equat, 114, 476-512, (1994) · Zbl 0838.35071
[19]	Toro, E.F., Riemann solvers and numerical methods for fluid dynamics, (1997), Springer-Verlag Berlin · Zbl 0888.76001
[20]	Turkel, E., Preconditioned methods for solving the incompressible and low speed compressible equations, J. comput. phys, 72, 277-298, (1987) · Zbl 0633.76069
[21]	Turkel, E., Review of preconditioning methods for fluid dynamics, Appl. numer. math, 12, 257-284, (1993) · Zbl 0770.76048
[22]	Turkel, E., Preconditional techniques in computational fluid dynamics, Annu. rev. fluid mech, 31-385, (1999)
[23]	Turkel, E.; Fiterman, A.; van Leer, B., Preconditioning and the limit of the compressible to the incompressible flow equations for finite difference schemes, (1994), John Wiley and Sons Chichester, p. 215-34
[24]	Ukai, S., The incompressible limit and the initial layer of the compressible Euler equations, J. math. Kyoto univ, 26, 323-331, (1986) · Zbl 0618.76074
[25]	van Leer, B.; Darmofal, D.-L., Steady Euler solutions in O(n) operations, (), 24-33 · Zbl 0986.76075
[26]	Vierendeels, J.; Riemslagh, K.; Dick, E., A multigrid semi-implicit line method for viscous incompressible and low-Mach number flows on high aspect ratio grid, J. comput. phys, 154, 310-341, (1999) · Zbl 0955.76067
[27]	Viozat C. Calcul d’écoulements stationnaires et instationnaires à petit nombre de Mach et en maillages étirés. PhD thesis, University of Nice, Sophia, Antipolis, 1998
[28]	Viozat C. Calcul d’écoulements diphasiques dans une tuyère: Influence de la renormalisation du schéma de flux. Technical Report SYSCO/LGLS/RT/00-014, CEA, 2000
[29]	Volpe, G., Performance of compressible flow codes at low Mach number, Aiaa j, 31, 49-56, (1993) · Zbl 0775.76140
[30]	Zienkiewicz, O.C.; Szmelter, J.; Perraire, J., Compressible and incompressible flows: an algorithm for all seasons, Comput. meth. appl. mech. eng, 78, 1, 105-121, (1990) · Zbl 0708.76099