×

A genuinely two-dimensional Riemann solver for compressible flows in curvilinear coordinates. (English) Zbl 1452.76131

Summary: A genuinely two-dimension Riemann solver for compressible flows in curvilinear coordinates is proposed. Following Balsara’s idea, this two-dimension solver considers not only the waves orthogonal to the cell interfaces, but also those transverse to the cell interfaces. By adopting the Toro-Vasquez splitting procedure, this solver constructs the two-dimensional convective flux and the two-dimensional pressure flux separately. Systematic numerical test cases are conducted. One dimensional Sod shock tube case and moving contact discontinuity case indicate that such two-dimensional solver is capable of capturing one-dimensional shocks, contact discontinuities, and expansion waves accurately. Two-dimensional double Mach reflection of a strong shock case shows that this scheme is with a high resolution in Cartesian coordinates. Also, it is robust against the unphysical shock anomaly phenomenon. Hypersonic viscous flows over the blunt cone and the two-dimensional Double-ellipsoid cases show that the two-dimensional solver proposed in this manuscript is with a high resolution in curvilinear coordinates. It is promising to be widely used in engineering areas to simulate compressible flows.

MSC:

76M12 Finite volume methods applied to problems in fluid mechanics
76N15 Gas dynamics (general theory)
76K05 Hypersonic flows
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
35Q31 Euler equations

Software:

RIEMANN; AUSM
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics (2009), Springer-Verlag · Zbl 1227.76006
[2] Chen, L.; Schaefer, L., Godunov-type upwind flux schemes of the two-dimensional finite volume discrete Boltzmann method, Comput. Math. Appl., 75, 3105-3126 (2018) · Zbl 1409.65061
[3] Jameson, A.; Schmidt, W.; Turkel, E., Numerical Solutions of the Euler Equations by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes (1981), AIAA Paper 1981-1259
[4] Swanson, R. C.; Turkel, E., Artificial Dissipation and Central Difference Schemes for the Euler and Navier-Stokes Equations (1987), AIAA Paper 1987-1107
[5] Roe, P. L., Approximate Riemann solvers, parameter vectors and difference schemes, J. Comput. Phys., 43, 357-372 (1981) · Zbl 0474.65066
[6] Qu, F.; Yan, C.; Sun, D., A new Roe-type scheme for all speeds, Comput. Fluids, 121, 11-25 (2015) · Zbl 1390.76686
[7] Steger, J. L.; Warming, R. F., Flux vector splitting of the inviscid gas-dynamics equations with application to finite difference methods, J. Comput. Phys., 40, 2 (1981) · Zbl 0468.76066
[8] van Leer, B., Flux vector splitting for the Euler equations, (Eighth International Conference of Numerical Methods in Fluid Dynamics. Eighth International Conference of Numerical Methods in Fluid Dynamics, Lecture Notes in Physics, vol. 170 (1982), Springer: Springer Berlin), 507-512
[9] Dumbser, M.; Balsara, D. S., A new efficient formulation of the HLLEM Riemann solver for general conservative and non-conservative hyperbolic systems, J. Comput. Phys., 304, 275-319 (2016) · Zbl 1349.76603
[10] Goetz, C. R.; Balsara, D. S.; Dunbser, M., A family of HLL-type solvers for the generalized Riemann problem, Comput. Fluids, 169, 201-213 (2018) · Zbl 1410.76228
[11] Qu, F.; Chen, J. J.; Sun, D.; Bai, J. Q., A new all-speed flux scheme for the Euler equations, Comput. Math. Appl., 77, 4, 1216-1231 (2019) · Zbl 1442.65204
[12] Liou, M. S., A sequel to AUSM: AUSM+, J. Comput. Phys., 129, 364-382 (1996) · Zbl 0870.76049
[13] 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
[14] Qu, F.; Yan, C.; Sun, D., A parameter-free upwind scheme for all speeds’ simulations, Sci. China, Technol. Sci., 58, 434-442 (2015)
[15] Qu, F.; Sun, D.; Yan, C., A new flux splitting scheme for the Euler equations II: E-AUSMPWAS for all speeds, Commun. Nonlinear Sci. Numer. Simul., 57, 58-79 (2018) · Zbl 1460.76640
[16] Liou, M. S., Open Problems in Numerical Fluxes: Proposed Resolutions (2011), AIAA Paper 2011-3055
[17] Kermani, M. J.; Plett, E. G., Modified Entropy Correction Formula for the Roe Scheme (2011), AIAA Paper 2011-0083
[18] Muller, B., Simple Improvements of an Upwind TVD Scheme for Hypersonic Flow (1989), AIAA Paper 1989-1977
[19] Kim, K. H.; Kim, C.; Rho, O. H., Cures for the shock instability, development of a shock-stable Roe scheme, J. Comput. Phys., 185, 342-374 (2003) · Zbl 1062.76538
[20] Einfeldt, B., On Godunov-type methods for gas dynamics, SIAM J. Numer. Anal., 25, 294-318 (1988) · Zbl 0642.76088
[21] Hu, L. J.; Yuan, L., A genuinely multidimensional Riemann solver based on the TV splitting, Appl. Math. Mech. Chin. Ed., 38, 243-264 (2017), (in Chinese)
[22] Roe, P. L., Discrete models for the numerical analysis of time-dependent multidimensional gas dynamics, J. Comput. Phys., 63, Article 458 pp. (1986) · Zbl 0587.76126
[23] Rumsey, C. B.; van Leer, B.; Roe, P. L., A multidimensional flux function with application to the Euler and Navier-Stokes equations, J. Comput. Phys., 105, Article 306 pp. (1993) · Zbl 0767.76039
[24] Colella, P., Multidimensional upwind methods for hyperbolic conservation laws, J. Comput. Phys., 87, 171-200 (1990) · Zbl 0694.65041
[25] LeVeque, R. J., Wave propagation algorithms for multidimensional hyperbolic systems, J. Comput. Phys., 131, 327-353 (1997) · Zbl 0872.76075
[26] Wendroff, B., A two-dimensional HLLE Riemann solver and associated Godunov-type difference scheme for gas dynamics, Comput. Math. Appl., 38, 175-185 (1999) · Zbl 0984.76064
[27] Fey, M., Multidimensional upwinding—part I: the method of transport for solving the Euler equations, J. Comput. Phys., 143, 159-180 (1998) · Zbl 0932.76050
[28] Fey, M., Multidimensional upwinding—part II: decomposition of the Euler equations into advection equations, J. Comput. Phys., 143, 181-203 (1998) · Zbl 0932.76051
[29] Brio, M.; Zakharian, A. R.; Webb, G. M., Two-dimensional Riemann solver for Euler equations of gas dynamics, J. Comput. Phys., 167, 177-195 (2001) · Zbl 1043.76042
[30] Abgrall, R., Approximation du problème de Riemann vraiment multidimensionnel des èquations d’Euler par une methode de type Roe, I: La linèarisation, C. R. Acad. Sci., Ser. I, 319, Article 499 pp. (1994) · Zbl 0813.76074
[31] Abgrall, R., Approximation du problème de Riemann vraiment multidimensionnel des èquations d’Euler par une methode de type Roe, I: solution du probleme de Riemann approchè, C. R. Acad. Sci., Ser. I, 319, Article 625 pp. (1994) · Zbl 0813.76075
[32] Balsara, D. S., Multidimensional HLLE Riemann solver: application to Euler and magnetohydrodynamic flows, J. Comput. Phys., 229, 1970-1993 (2010) · Zbl 1303.76140
[33] Balsara, D. S., Three dimensional HLL Riemann solver for conservation laws on structured meshes; application to Euler and magnetohydrodynamic flows, J. Comput. Phys., 295, 1-23 (2015) · Zbl 1349.76584
[34] Balsara, D. S., A two-dimensional HLLC Riemann solver for conservation laws: application to Euler and magnetohydrodynamic flows, J. Comput. Phys., 231, 7476-7503 (2012) · Zbl 1284.76261
[35] Balsara, D. S., Multidimensional Riemann problem with self – similar internal structure. Part II - application to hyperbolic conservation laws on unstructured meshes, J. Comput. Phys., 287, 269-292 (2015) · Zbl 1351.76091
[36] Mandal, J. C.; Sharma, V., A genuinely multidimensional convective pressure flux split Riemann solver for Euler equations, J. Comput. Phys., 297, 669-688 (2015) · Zbl 1349.76630
[37] Vides, J.; Nkonga, B.; Audit, E., A simple two-dimensional extension of the HLL Riemann solver for hyperbolic systems of conservation laws, J. Comput. Phys., 280, 643-675 (2015) · Zbl 1349.76403
[38] Balsara, D. S.; Nkonga, B., Multidimensional Riemann problem with self-similar internal structure. Part I - application to hyperbolic conservation laws on structured meshes, J. Comput. Phys., 227, 163-200 (2014) · Zbl 1349.76303
[39] Balsara, D. S.; Nkonga, B., Multidimensional Riemann problem with self-similar internal structure. Part III - a multidimensional analogue of the HLLI Riemann solver for conservative hyperbolic systems, J. Comput. Phys., 346, 25-48 (2017) · Zbl 1378.76056
[40] Chen, Y. X.; Toth, G.; Gombosi, T. I., A fifth-order finite difference scheme for hyperbolic equations on block-adaptive curvilinear grids, J. Comput. Phys., 305, 604-621 (2016) · Zbl 1349.65278
[41] Qu, F.; Yan, C.; Yu, J., A new flux splitting scheme for the Euler equations, Comput. Fluids, 102, 203-214 (2014) · Zbl 1391.76567
[42] Zhang, F.; Liu, J.; Chen, B. S., Modified multi-dimensional limiting process with enhanced shock stability on unstructured grids, Comput. Fluids, 161, 171-188 (2018) · Zbl 1390.76540
[43] Qu, F.; Sun, D.; Zuo, G., A study of upwind schemes on the laminar hypersonic heating predictions for the reusable space vehicle, Acta Astronaut., 147, 412-420 (2018)
[44] Toro, E. F.; Vazquez-Cendon, M. E., Flux splitting schemes for the Euler equations, Comput. Fluids, 70, 1-12 (2012) · Zbl 1365.76243
[45] Qu, F.; Yan, C.; Sun, D., Investigation into the influences of the low speed’s accuracy on the hypersonic heating computations, Int. Commun. Heat Mass Transf., 70, 53-58 (2016)
[46] Lung, T.; Roe, P., Towards a reduction of mesh imprinting, Int. J. Numer. Methods Fluids, 76, 450-470 (2014)
[47] Mandal, J. C.; Panwar, V., Robust HLL-type Riemann solver capable of resolving contact discontinuity, Comput. Fluids, 63, 148-164 (2012) · Zbl 1365.76164
[48] Borges, R.; Carmona, M.; Costa, B., An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws, J. Comput. Phys., 227, 3191-3211 (2008) · Zbl 1136.65076
[49] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54, 115-173 (1984) · Zbl 0573.76057
[50] Qu, F.; Sun, D.; Bai, J. Q.; Zuo, G., Numerical investigation of blunt body’s heating load reduction with combination of spike and opposing jet, Int. J. Heat Mass Transf., 127, 7-15 (2018)
[51] Qu, F.; Chen, J. J.; Sun, D.; Bai, J. Q.; Zuo, G., A grid strategy for predicting the space plane’s hypersonic aerodynamics heating loads, Aerosp. Sci. Technol., 86, 659-670 (2019)
[52] Kordulla, W.; Periaux, J., Attempt to evaluate the computations for Test Case 6.1: cold hypersonic flow past ellipsoidal shapes, (Hypersonic Flows for Reentry Problems, Vol. I & II. Hypersonic Flows for Reentry Problems, Vol. I & II, Antibes, France (1991), Springer-Verlag), 689-712
[53] Qu, F.; Sun, D., Investigation into the influences of the low-speed flows’ accuracy on RANS simulations, Aerosp. Sci. Technol., 70, 578-589 (2017)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.