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On entropy generation and dissipation of kinetic energy in high-resolution shock-capturing schemes. (English) Zbl 1142.65073

Summary: This paper addresses entropy generation and the corresponding dissipation of kinetic energy associated with high-resolution, shock-capturing (Godunov) methods. Analytical formulae are derived for the rate of increase of entropy given arbitrary jumps in primitive variables at a cell interface. It is demonstrated that for general continuously varying flows the inherent numerical entropy increase of Godunov methods is not proportional to the velocity jump cubed as is commonly assumed, but it is proportional to the velocity jump squared.
Furthermore, the dissipation of kinetic energy is directly linked to temperature multiplied by change in entropy at low Mach numbers. The kinetic energy dissipation rate is shown to be proportional to the velocity jump squared and the speed of sound. The leading order dissipation rate associated with jumps in pressure, density and shear waves is detailed and further shown that at low Mach number it is the dissipation due to the perpendicular velocity jumps which dominates. This explains directly the poor performance of Godunov methods at low Mach numbers. The analysis is also applied to high-order accurate methods in space and time and all analytical results are validated with simple numerical experiments.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L45 Initial value problems for first-order hyperbolic systems
76N15 Gas dynamics (general theory)
76M12 Finite volume methods applied to problems in fluid mechanics
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[1] Guillard, H., Recent developments in the computation of compressible low Mach number flows, Flow turbul. combust., 76, 363-369, (2006) · Zbl 1125.76059
[2] 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
[3] Noh, W., Errors for calculations of strong shocks using and artificial viscosity and an artificial heat flux, J. comput. phys., 72, 78-120, (1987) · Zbl 0619.76091
[4] R. Christensen, Godunov methods on a staggered mesh – an improved artificial viscosity, Tech. Rep., Lawrence Livermore National Laboratory, 1990.
[5] Benson, D., Computational methods in Lagrangian and Eulerian hydrocodes, Comput. meth. appl. mech. eng., 235-394, (1992) · Zbl 0763.73052
[6] Volpe, G., Performance of compressible flow codes at low Mach number, Aiaa j., 31, 49-56, (1993) · Zbl 0775.76140
[7] R. Menikoff, Numerical anomalies mimicking physical effects, Tech. Rep., Los Alamos, 1995.
[8] Guillard, H.; Viozat, C., On the behaviour of upwind schemes in the low Mach number limit, Comput. fluids, 28, 63-86, (1999) · Zbl 0963.76062
[9] Guillard, H.; Murrone, A., On the behaviour of upwind schemes in the low Mach number limit: II. Godunov type schemes, Comput. fluids, 33, 655-675, (2004) · Zbl 1049.76040
[10] Boris, J.; Grinstein, F.; Oran, E.; Kolbe, R., New insights into large eddy simulation, Fluid dyn. res., 10, 199-228, (1992)
[11] D. Youngs, Application of MILES to Rayleigh-Taylor and Richtmyer-Meshkov mixing, AIAA-2003-4102.
[12] Kolmogorov, A., A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number, J. fluid mech., 13, 82-85, (1962) · Zbl 0112.42003
[13] Naterer, G., Heat transfer in single and multiphase systems, (2003), CRC Press
[14] Fureby, C.; Grinstein, F., Monotonically integrated large eddy simulation of free shear flows, Aiaa j., 37, 5, 544-556, (1999)
[15] Bejan, A., Entropy generation minimization: the method of thermodynamic optimization of finite-time systems and finite-time processes, (1996), CRC Press · Zbl 0864.76001
[16] Youngs, D., Three-dimensional numerical simulation of turbulent mixing by rayleigh – taylor instability, Phys. fluids A, 3, 5, 1312-1320, (1991)
[17] H. Bethe, On the theory of shock waves for an arbitrary equation of state, Tech. Rep., Office of Scientific Research and Development, 1942.
[18] ()
[19] Menikoff, R.; Plohr, B., The Riemann problem for fluid flow of real materials, Rev. mod. phys., 61, 1, 75-130, (1989) · Zbl 1129.35439
[20] Leveque, R., Finite volume methods for hyperbolic problems, (2002), Cambridge University Press · Zbl 1010.65040
[21] Toro, E., Riemann solvers and numerical methods for fluid dynamics, (1997), Springer-Verlag
[22] Klainerman, S.; Madja, A., Compressible and incompressible fluids, Commun. pure appl. math., 33, 399-440, (1982)
[23] 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
[24] Merriam, M., Smoothing and the second law, Comput. method appl. M, 64, 177-193, (1987) · Zbl 0607.76008
[25] Barth, T., Numerical methods for gasdynamic systems on unstructured meshes, () · Zbl 0969.76040
[26] Hugonoit, H., On the propagation of motion in bodies and in perfect gases in particular - II, J. de l’ecole polytech., 58, 1-125, (1889)
[27] Thornber, B.; Mosedale, A.; Drikakis, D., On the implicit large eddy simulation of homogeneous decaying turbulence, J. comput. phys., 226, 1902-1929, (2007) · Zbl 1219.76027
[28] Garnier, E.; Mossi, M.; Sagaut, P.; Comte, P.; Deville, M., On the use of shock-capturing schemes for large-eddy simulation, J. comput. phys., 153, 273-311, (1999) · Zbl 0949.76042
[29] Shu, C.-W., Total-variation-diminishing time discretizations, SIAM J. sci. stat. comp., 9, 1073-1084, (1988) · Zbl 0662.65081
[30] A. Jameson, Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings, AIAA 91-1596.
[31] Gottlieb, S.; Shu, C.-W., Total variation diminishing runge – kutta schemes, Math. comput., 67, 221, 73-85, (1998) · Zbl 0897.65058
[32] Spiteri, R.; Ruuth, S., A class of optimal high-order strong-stability preserving time discretization methods, SIAM J. sci. comput., 40, 2, 469-491, (2002) · Zbl 1020.65064
[33] van Leer, B., Towards the ultimate conservative difference scheme: IV. A new approach to numerical convection, J. comput. phys., 23, 276-299, (1977) · Zbl 0339.76056
[34] Drikakis, D.; Rider, W., High-resolution methods for incompressible and low-speed flows, (2004), Springer Verlag
[35] ()
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