×

zbMATH — the first resource for mathematics

A three-dimensional explicit sphere function-based gas-kinetic flux solver for simulation of inviscid compressible flows. (English) Zbl 1349.76751
Summary: In this work, a truly three-dimensional (3D) flux solver is presented for simulation of inviscid compressible flows. Like the conventional multi-dimensional gas-kinetic scheme, in the present work, the local solution of 3D Boltzmann equation at the cell interface is used to evaluate the flux. On the other hand, different from most of the existing gas-kinetic schemes, which are constructed from Maxwellian distribution function, the present flux solver is derived from a simple distribution function defined on the spherical surface in the phase velocity space. As a result, the explicit expression of flux vector at the cell interface can be simply given. Since the simple distribution function is defined on the spherical surface, for simplicity, it is termed as sphere function hereafter. In addition, to simulate fluid flow problems with strong shock waves, the non-equilibrium part of the distribution function is regarded as numerical dissipation and involved in evaluating the inviscid flux at the cell interface. The weight of the non-equilibrium part is controlled by introducing a switch function which ranges from 0 to 1. In the smooth region, the switch function takes a value close to zero, while around the strong shock wave, it tends to one. To validate the proposed flux solver, several transonic, supersonic and hypersonic inviscid flows are simulated. Numerical results showed that the present solver can provide accurate numerical results for three-dimensional inviscid flows with strong shock waves.

MSC:
76M28 Particle methods and lattice-gas methods
76L05 Shock waves and blast waves in fluid mechanics
76H05 Transonic flows
76J20 Supersonic flows
76K05 Hypersonic flows
76N15 Gas dynamics, general
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anderson, J. D., Computational fluid dynamics: the basics with applications, (1995), McGraw-Hill
[2] Blazek, J., Computation fluid dynamics: principle and application, (2001), Elsevier · Zbl 0995.76001
[3] Toro, E. F., Riemann solvers and numerical methods for fluid dynamics, (2009), Springer · Zbl 1227.76006
[4] McDonald, P. W., The computation of transonic flow through tow-dimensional gas turbine cascades, (1971), ASME Paper 71-GT-89
[5] Pironneau, O., Finite element methods for fluids, (1989), Wiley · Zbl 0665.73059
[6] Patera, A. T., A spectral element method for fluid dynamics: laminar flow in a channel expansion, J. Comput. Phys., 54, 468-488, (1984) · Zbl 0535.76035
[7] Shu, C.; Richards, B. E., Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids, 15, 791-798, (1992) · Zbl 0762.76085
[8] Qiu, J. X.; Khoo, B. C.; Shu, C. W., A numerical study for the performance of the Runge-Kutta discontinuous Galerkin method based on different numerical fluxes, J. Comput. Phys., 212, 540-565, (2006) · Zbl 1083.65093
[9] Zhou, C. H.; Shu, C., A local domain-free discretization method to simulate three-dimensional compressible inviscid flows, Int. J. Numer. Methods Fluids, 61, 970-986, (2009) · Zbl 1252.76067
[10] Zhou, C. S.; Zhong, C. W., Filter-matrix lattice Boltzmann model for microchannel gas flows, Phys. Rev. E, 88, 053311, (2013)
[11] Liu, S.; Yu, P. B.; Xu, K.; Zhong, C. W., Unified gas-kinetic scheme for diatomic molecular simulations in all flow regimes, J. Comput. Phys., 259, 96-113, (2014) · Zbl 1349.76786
[12] Godunov, S. K., A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Mat. Sb., 47, 271-306, (1959) · Zbl 0171.46204
[13] Toro, E. F.; Spruce, M.; Speares, W., Restoration of the contact surface in the HLL-Riemann solver, Shock Waves, 4, 25-34, (1994) · Zbl 0811.76053
[14] Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43, 357-372, (1981) · Zbl 0474.65066
[15] van Leer, B., Towards the ultimate conservative difference scheme V. A second order sequel to Godunov’s method, J. Comput. Phys., 32, 101-136, (1979) · Zbl 1364.65223
[16] Liou, M. S.; Steffen, C. J., A new flux splitting scheme, J. Comput. Phys., 107, 23-39, (1993) · Zbl 0779.76056
[17] Ji, C. Z.; Shu, C.; Zhao, N., A lattice Boltzmann method-based flux solver and its application to solve shock tube problem, Mod. Phys. Lett. B, 23, 313-316, (2009) · Zbl 1419.76520
[18] Yang, L. M.; Shu, C.; Wu, J., Development and comparative studies of three non-free parameter lattice Boltzmann models for simulation of compressible flows, Adv. Appl. Math. Mech., 4, 454-472, (2012)
[19] Yang, L. M.; Shu, C.; Wu, J., A moment conservation-based non-free parameter compressible lattice Boltzmann model and its application for flux evaluation at cell interface, Comput. Fluids, 79, 190-199, (2013) · Zbl 1284.76313
[20] Shu, C.; Wang, Y.; Yang, L. M.; Wu, J., Lattice Boltzmann flux solver: an efficient approach for numerical simulation of fluid flows, Trans. Nanjing Univ. Aeronaut. Astronaut., 31, 1-15, (2014)
[21] Pullin, D. I., Direct simulation methods for compressible inviscid ideal-gas flow, J. Comput. Phys., 34, 231-244, (1980) · Zbl 0419.76049
[22] Mandal, J. C.; Deshpande, S. M., Kinetic flux vector splitting for Euler equations, Comput. Fluids, 23, 447-478, (1994) · Zbl 0811.76047
[23] Chou, S. Y.; Baganoff, D., Kinetic flux-vector splitting for the Navier-Stokes equations, J. Comput. Phys., 130, 217-230, (1997) · Zbl 0873.76057
[24] Xu, K., Gas-kinetic schemes for unsteady compressible flow simulations, VKI for Fluid Dynamics Lecture Series, vol. 1998-03, (1998)
[25] Xu, K.; Mao, M. L.; Tang, L., A multidimensional gas-kinetic BGK scheme for hypersonic viscous flow, J. Comput. Phys., 203, 405-421, (2005) · Zbl 1143.76553
[26] Yang, J. Y.; Hsieh, T. Y.; Shi, Y. H.; Xu, K., High-order kinetic flux vector splitting schemes in general coordinates for ideal quantum gas dynamics, J. Comput. Phys., 227, 967-982, (2007) · Zbl 1388.82005
[27] Yang, J. Y.; Muljadi, B. P.; Chen, S. Y.; Li, Z. H., Kinetic numerical methods for solving the semiclassical Boltzmann-BGK equation, Comput. Fluids, 85, 153-165, (2013) · Zbl 1290.76129
[28] Yang, L. M.; Shu, C.; Wu, J.; Zhao, N.; Lu, Z. L., Circular function-based gas-kinetic scheme for simulation of inviscid compressible flows, J. Comput. Phys., 255, 540-557, (2013) · Zbl 1349.76752
[29] Yang, L. M.; Shu, C.; Wu, J., A simple distribution function-based gas-kinetic scheme for simulation of viscous incompressible and compressible flows, J. Comput. Phys., 274, 611-632, (2014) · Zbl 1351.76257
[30] Zadehgol, A.; Ashrafizaadeh, M., Introducing a new kinetic model which admits an H-theorem for simulating the nearly incompressible flows, J. Comput. Phys., 274, 803-825, (2014) · Zbl 1351.76258
[31] Qu, K.; Shu, C.; Chew, Y. T., Alternative method to construct equilibrium distribution functions in lattice-Boltzmann method simulation of inviscid compressible flows at high Mach number, Phys. Rev. E, 75, 036706, (2007)
[32] Li, K.; Zhong, C. W., A lattice Boltzmann model for simulation of compressible flows, Int. J. Numer. Methods Fluids, 77, 334-357, (2015)
[33] Guo, Z. L.; Shu, C., Lattice Boltzmann method and its applications in engineering, (2013), World Scientific Publishing
[34] Barth, T. J.; Jespersen, D. C., The design and application of upwind schemes on unstructured meshes, (1989), AIAA Paper 89-0366
[35] Venkatakrishnan, V., Convergence to steady-state solutions of the Euler equations on unstructured grids with limiters, J. Comput. Phys., 118, 120-130, (1995) · Zbl 0858.76058
[36] Deiterding, R.
[37] Li, Q.; He, Y. L.; Wang, Y.; Tang, G. H., Three-dimensional non-free-parameter lattice-Boltzmann model and its application to inviscid compressible flows, Phys. Lett. A, 373, 2101-2108, (2009) · Zbl 1229.76090
[38] Deiterding, R.
[39] Kim, S. S.; Kim, C.; Rho, O. H.; Hong, S. K., Cure for shock instability: development of an improved roe scheme, (2002), AIAA-2002-0548
[40] Stolcis, L.; Johnston, L. J., Solution of the Euler equations on unstructured grids for two-dimensional compressible flow, Aeronaut. J., 94, 181-195, (1990)
[41] Slater, J. W., ONERA M6 wing · Zbl 0875.76256
[42] Schmitt, V.; Charpin, F., Pressure distributions on the ONERA-M6-wing at transonic Mach numbers, (1979), Experimental Data Base for Computer Program Assessment, Report of the Fluid Dynamics Panel Working Group 04, AGARD AR 138
[43] Lax, P. D.; Liu, X. D., Solution of two-dimensional Riemann problems of gas dynamics by positive schemes, SIAM J. Sci. Comput., 19, 319-340, (1998) · Zbl 0952.76060
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.