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A low dissipation method to cure the grid-aligned shock instability. (English) Zbl 1453.76092
Summary: The grid-aligned shock instability prevents an accurate computation of high Mach number flows using low-dissipation shock-capturing methods. In particular one manifestation, the so-called carbuncle phenomenon, has been investigated by various different groups over the past decades. Nevertheless, the mechanism of this instability is still not fully understood and commonly is suppressed by the introduction of additional numerical dissipation. However, present approaches may either significantly deteriorate the resolution of complex flow evolutions or involve additional procedures to limit stabilization measures to the shock region. Instead of increasing the numerical dissipation, in this paper, we present an alternative approach that relates the problem to the low Mach number in transverse direction of the shock front. We show that the inadequate scaling of the acoustic dissipation in the low Mach number limit is the prime reason for the instability. Our approach is to increase the “numerical” Mach number locally whenever the advection dissipation is small compared to the acoustic dissipation. A very simple modification of the eigenvalue calculation in the Roe approximation leads to a scheme with less numerical dissipation than the original Roe flux which prevents the grid-aligned shock instability. The simplicity of the modification allows for a detailed investigation of multidimensional effects. By showing that modifications in flow direction affect the shock stability in the transverse directions we confirm the multidimensional nature of the instability. The efficiency and robustness of the modified scheme is demonstrated for a wide range of test cases that are known to be particularly prone to the shock instability. Moreover, the modified flux also is successfully applied to multi-phase flows.
##### MSC:
 76M12 Finite volume methods applied to problems in fluid mechanics 76L05 Shock waves and blast waves in fluid mechanics 65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs 76N15 Gas dynamics, general
##### Software:
AUSM; AUSMPW+; HE-E1GODF; HLLE
Full Text:
##### References:
 [1] Bagabir, A.; Drikakis, D., Mach number effects on shock-bubble interaction, Shock Waves, 11, 3, 209-218 (2001) · Zbl 0996.76038 [2] Haimovich, O.; Frankel, S. H., Numerical simulations of compressible multicomponent and multiphase flow using a high-order targeted ENO (TENO) finite-volume method, Comput. Fluids, 146, 105-116 (2017) · Zbl 1390.76439 [3] Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction (2013), Springer Science & Business Media [4] Godunov, S. K., A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Mat. Sb., 89, 3, 271-306 (1959) · Zbl 0171.46204 [5] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S. R., Uniformly high order accurate essentially non-oscillatory schemes, III, J. Comput. Phys., 71, 2, 231-303 (1987) · Zbl 0652.65067 [6] Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, 1, 202-228 (1996) · Zbl 0877.65065 [7] Harten, A.; Lax, P. D.; Leer, B.v., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25, 1, 35-61 (1983) · Zbl 0565.65051 [8] Einfeldt, B., On Godunov-type methods for gas dynamics, SIAM J. Numer. Anal., 25, 2, 294-318 (1988) · Zbl 0642.76088 [9] Toro, E. F.; Spruce, M.; Speares, W., Restoration of the contact surface in the HLL-Riemann solver, Shock Waves, 4, 1, 25-34 (1994) · Zbl 0811.76053 [10] Osher, S.; Solomon, F., Upwind difference schemes for hyperbolic systems of conservation laws, Math. Comput., 38, 158, 339-374 (1982) · Zbl 0483.65055 [11] Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43, 2, 357-372 (1981) · Zbl 0474.65066 [12] Quirk, J. J., A contribution to the great Riemann solver debate, (Upwind and High-Resolution Schemes (1997), Springer), 550-569 [13] Peery, K.; Imlay, S., Blunt-body flow simulations, (24th Joint Propulsion Conference (1988)), 2904 [14] Sanders, R.; Morano, E.; Druguet, M.-C., Multidimensional dissipation for upwind schemes: stability and applications to gas dynamics, J. Comput. Phys., 145, 2, 511-537 (1998) · Zbl 0924.76076 [15] Pandolfi, M.; D’Ambrosio, D., Numerical instabilities in upwind methods: analysis and cures for the “carbuncle” phenomenon, J. Comput. Phys., 166, 2, 271-301 (2001) · Zbl 0990.76051 [16] Gressier, J.; Moschetta, J.-M., Robustness versus accuracy in shock-wave computations, Int. J. Numer. Methods Fluids, 33, 3, 313-332 (2000) · Zbl 0980.76072 [17] Liou, M.-S., Mass flux schemes and connection to shock instability, J. Comput. Phys., 160, 2, 623-648 (2000) · Zbl 0967.76062 [18] Dumbser, M.; Moschetta, J.-M.; Gressier, J., A matrix stability analysis of the carbuncle phenomenon, J. Comput. Phys., 197, 2, 647-670 (2004) · Zbl 1079.76607 [19] Liou, M.-S., A sequel to AUSM: $$AUSM^+$$, J. Comput. Phys., 129, 2, 364-382 (1996) · Zbl 0870.76049 [20] Ren, Y.-X., A robust shock-capturing scheme based on rotated Riemann solvers, Comput. Fluids, 32, 10, 1379-1403 (2003) · Zbl 1034.76035 [21] Kim, S.-S.; Kim, C.; Rho, O.-H.; Hong, S. K., Cures for the shock instability: development of a shock-stable Roe scheme, J. Comput. Phys., 185, 2, 342-374 (2003) · Zbl 1062.76538 [22] Chen, S.-S.; Yan, C.; Lin, B.-X.; Li, Y.-S., A new robust carbuncle-free roe scheme for strong shock, J. Sci. Comput., 1-28 (2018) [23] Kim, S. D.; Lee, B. J.; Lee, H. J.; Jeung, I.-S., Robust HLLC Riemann solver with weighted average flux scheme for strong shock, J. Comput. Phys., 228, 20, 7634-7642 (2009) · Zbl 1391.76556 [24] Shen, Z.; Yan, W.; Yuan, G., A robust HLLC-type Riemann solver for strong shock, J. Comput. Phys., 309, 185-206 (2016) · Zbl 1351.76043 [25] Simon, S.; Mandal, J., A simple cure for numerical shock instability in the HLLC Riemann solver, J. Comput. Phys., 378, 477-496 (2019) · Zbl 1416.76160 [26] Kim, K. H.; Kim, C.; Rho, O.-H., Methods for the accurate computations of hypersonic flows: I. AUSMPW+scheme, J. Comput. Phys., 174, 1, 38-80 (2001) · Zbl 1106.76421 [27] Kitamura, K.; Shima, E., Towards shock-stable and accurate hypersonic heating computations: a new pressure flux for AUSM-family schemes, J. Comput. Phys., 245, 62-83 (2013) · Zbl 1349.76487 [28] Rodionov, A. V., Artificial viscosity in Godunov-type schemes to cure the carbuncle phenomenon, J. Comput. Phys., 345, 308-329 (2017) · Zbl 1378.76059 [29] Guillard, H.; Viozat, C., On the behaviour of upwind schemes in the low Mach number limit, Comput. Fluids, 28, 1, 63-86 (1999) · Zbl 0963.76062 [30] Guillard, H.; Murrone, A., On the behavior of upwind schemes in the low Mach number limit: II. Godunov type schemes, Comput. Fluids, 33, 4, 655-675 (2004) · Zbl 1049.76040 [31] Gottlieb, S.; Shu, C.-W.; Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM Rev., 43, 1, 89-112 (2001) · Zbl 0967.65098 [32] Li, X.-s.; Gu, C.-w., An all-speed Roe-type scheme and its asymptotic analysis of low Mach number behaviour, J. Comput. Phys., 227, 10, 5144-5159 (2008) · Zbl 1388.76207 [33] Kemm, F., Heuristical and numerical considerations for the carbuncle phenomenon, Appl. Math. Comput., 320, 596-613 (2018) · Zbl 1426.76200 [34] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54, 1, 115-173 (1984) · Zbl 0573.76057 [35] Elling, V., The carbuncle phenomenon is incurable, Acta Math. Sci., 29, 6, 1647-1656 (2009) · Zbl 1201.76126 [36] Hu, X. Y.; Khoo, B.; Adams, N. A.; Huang, F., A conservative interface method for compressible flows, J. Comput. Phys., 219, 2, 553-578 (2006) · Zbl 1102.76038 [37] Borges, R.; Carmona, M.; Costa, B.; Don, W. S., An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws, J. Comput. Phys., 227, 6, 3191-3211 (2008) · Zbl 1136.65076 [38] Bultelle, M.; Grassin, M.; Serre, D., Unstable Godunov discrete profiles for steady shock waves, SIAM J. Numer. Anal., 35, 6, 2272-2297 (1998) · Zbl 0929.76083 [39] Kalkhoran, I. M.; Smart, M. K., Aspects of shock wave-induced vortex breakdown, Prog. Aerosp. Sci., 36, 1, 63-95 (2000)
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