Mass flux schemes and connection to shock instability. (English) Zbl 0967.76062
From the summary: We analyze numerical mass fluxes with an emphasis on their capability for accurately capturing shock and contact discontinuities. We examine several prominent numerical flux schemes and analyze the structure of numerical diffusivity. This leads to a detailed investigation into the cause of certain catastrophic breakdowns by some numerical flux schemes. In particular, we identify the dissipative terms that are responsible for shock instabilities, such as the odd-even decoupling and the so-called “carbuncle phenomenon.” As a result, we propose a conjecture stating the connection of the pressure difference term to these multidimensional shock instabilities, and hence a cure to those difficulties. The validity of this conjecture has been confirmed by examining a wide class of upwind schemes.
MSC:
| 76M12 | Finite volume methods applied to problems in fluid mechanics |
| 76L05 | Shock waves and blast waves in fluid mechanics |
Keywords:
carbuncle phenomenon; shock instability; AUSM schemes; shock-stable schemes; numerical mass fluxes; contact discontinuities; numerical diffusivity; dissipative terms; upwind schemesSoftware:
AUSMReferences:
| [1] | S. K. Godunov, A difference method for the numerical calculation of discontinuous solutions of hydrodynamic equations, Mat. Sb. 47, 271, 1959, (translation, US JPRS: 7225, November 1960, ).; S. K. Godunov, A difference method for the numerical calculation of discontinuous solutions of hydrodynamic equations, Mat. Sb. 47, 271, 1959, (translation, US JPRS: 7225, November 1960, ). · Zbl 0171.46204 |
| [2] | Wada, Y.; Liou, M.-S., An accurate and robust flux splitting scheme for shock and contact discontinuities, SIAM J. Sci. Comput., 18, 633 (1997) · Zbl 0879.76064 |
| [3] | M.-S. Liou, Probing numerical fluxes: Mass flux, positivity, and entropy-satisfying property, AIAA paper 97-2035-CP, 1997.; M.-S. Liou, Probing numerical fluxes: Mass flux, positivity, and entropy-satisfying property, AIAA paper 97-2035-CP, 1997. |
| [4] | Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43, 357 (1981) · Zbl 0474.65066 |
| [5] | Osher, S.; Solomon, F., Upwind difference schemes for hyperbolic systems of conservation laws, Math. Comp., 38, 339 (1982) · Zbl 0483.65055 |
| [6] | van Leer, B., Flux-vector splitting for the Euler equations, in Lecture Notes in Physics (1982), Springer-Verlag: Springer-Verlag Berlin, p. 507 |
| [7] | D. Hänel, R. Schwane, and, G. Seider, On the accuracy of upwind schemes for the solution of the Navier-Stokes equations, AIAA paper 87-1105-CP, 1987.; D. Hänel, R. Schwane, and, G. Seider, On the accuracy of upwind schemes for the solution of the Navier-Stokes equations, AIAA paper 87-1105-CP, 1987. |
| [8] | Einfeldt, B., On Godunov-type methods for gas-dynamics, SIAM J. Numer. Anal., 25, 294 (1988) · Zbl 0642.76088 |
| [9] | Einfeldt, B.; Munz, C. D.; Roe, P. L.; Sjögreen, B., On Godunov-type methods near low densities, J. Comput. Phys., 92, 273 (1991) · Zbl 0709.76102 |
| [10] | Obayashi, S.; Wada, Y., Practical formulation of a positively conservative scheme, AIAA J., 32, 1093 (1994) · Zbl 0800.76269 |
| [11] | Liou, M.-S., A sequel to AUSM: \(AUSM^+\), J. Comput. Phys., 129, 364 (1996) · Zbl 0870.76049 |
| [12] | Liou, M.-S.; Steffen, C. J., A new flux splitting scheme, J. Comput. Phys., 107, 23 (1993) · Zbl 0779.76056 |
| [13] | Quirk, J. J., A contribution to the great Riemann solver debate, Internat. J. Numer. Methods Fluids, 18, 555 (1994) · Zbl 0794.76061 |
| [14] | F. Coquel, and, M.-S. Liou, Hybrid upwind splitting (HUS) by a field-by-field decomposition, NASA TM 106843, 1995.; F. Coquel, and, M.-S. Liou, Hybrid upwind splitting (HUS) by a field-by-field decomposition, NASA TM 106843, 1995. |
| [15] | E. Shima, and, T. Jounouchi, AUSM type upwind schemes, preprint, 1996.; E. Shima, and, T. Jounouchi, AUSM type upwind schemes, preprint, 1996. |
| [16] | Swanson, R. C.; Radespiel, R.; Turkel, E., On some numerical dissipation schemes, J. Comput. Phys., 147, 518 (1998) · Zbl 0934.76050 |
| [17] | K. M. Peery, and, S. T. Imlay, Blunt-body flow simulations, in, AIAA/SAE/ASME/ASEE 24th Joint Propulsion Conference, 1988; AIAA paper 88-2904.; K. M. Peery, and, S. T. Imlay, Blunt-body flow simulations, in, AIAA/SAE/ASME/ASEE 24th Joint Propulsion Conference, 1988; AIAA paper 88-2904. |
| [18] | H.-C. Lin, Dissipation additions to flux-difference splitting, in, 10th AIAA Computational Fluid Dynamics Conference, 1991; AIAA paper 91-1544.; H.-C. Lin, Dissipation additions to flux-difference splitting, in, 10th AIAA Computational Fluid Dynamics Conference, 1991; AIAA paper 91-1544. |
| [19] | Edwards, J. R., A low-diffusion flux-splitting scheme for Navier-Stokes calculations, Comput. & Fluids, 26, 635 (1997) · Zbl 0911.76055 |
| [20] | Pandolfi, M.; D’Ambrosio, D., Upwind methods and carbuncle phenomenon, Proceedings of the Fourth European Computational Fluid Dynamics Conference, 126 (1998) |
| [21] | Jameson, A., Analysis and design of numerical schemes for gas dynamics. II. Artificial diffusion and discrete shock structure, Internat. J. Comput. Fluid Dynamics, 5, 1 (1995) |
| [22] | Sanders, R.; Morano, E.; Druguet, M., Multidimensional dissipation for upwind schemes: Stability and applications to gas dynamics, J. Comput. Phys., 145, 511 (1998) · Zbl 0924.76076 |
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