×

zbMATH — the first resource for mathematics

Progress in gas-kinetic upwind schemes for the solution of Euler/Navier-Stokes equations. I: Overview. (English) Zbl 1365.76279
Summary: Three gas-kinetic upwind schemes for the solution of the Euler/Navier-Stokes equations are reviewed. They are Kinetic Flux-Vector Splitting (KFVS), Kinetic Wave/Particle Splitting (KWPS), and Bhatnagar-Gross-Krook (BGK) methods. For the Euler equations, the most sophisticated BGK scheme can be interpreted as a relaxation scheme between the two KFVS schemes with different moments, KFVS and KFVS_\(u^0\). It improves the accuracy over the KFVS scheme and the robustness over the KFVS_\(u^0\) scheme. The direct generalization of this relaxation approach to the Navier-Stokes equations leads to a much simpler BGK scheme than the one in the literature. In this simplified BGK scheme, there exist two types of particle collision time. The one in the BGK model acts as a relaxation parameter. Its role is to add some numerical dissipation from the KFVS scheme to the KFVS_\(u^0\) scheme. On the other hand, the one in the Chapman-Enskog expansion of the gas distribution function is related to physical dissipation. Following the same approach, another type of BGK schemes is further developed, which is a relaxation scheme between KWPS and KWPS_\(u^0\). In spite of the fact that the KWPS scheme is more diffusive than the KFVS scheme, a BGK scheme based on KWPS and KWPS_\(u^0\) is found not only computationally more efficient but also less diffusive than a BGK scheme based on KFVS and KFVS_\(u^0\). However, this issue needs further and more rigorous investigation by performing the numerical analysis of a model 1-D convection-diffusion equation.

MSC:
76N15 Gas dynamics (general theory)
76M28 Particle methods and lattice-gas methods
Software:
HLLE
PDF BibTeX Cite
Full Text: DOI
References:
[1] Harten, A.; Lax, P.D.; van Leer, B., Upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM rev, 25, 35, (1983) · Zbl 0565.65051
[2] Chen, H.; Kandasamy, S.; Orszag, S.; Shock, R.; Succi, S.; Yakhot, V., Extended Boltzmann kinetic equation for turbulent flows, Science, 301, 633, (2003)
[3] Tang L, Yang J. Effect of higher-order hydrodynamics on separated flow simulation. AIAA paper 2009-3562; 2009.
[4] Pullin, D.I., Direct simulation methods for compressible inviscid ideal gas flow, J comput phys, 34, 231, (1980) · Zbl 0419.76049
[5] Mandal, J.C.; Deshpande, S.M., Kinetic flux vector splitting for Euler equations, Comput fluids, 23, 447, (1994) · Zbl 0811.76047
[6] Chou, S.Y.; Baganoff, D., Kinetic flux-vector splitting for the navier – stokes equations, J comput phys, 130, 217, (1997) · Zbl 0873.76057
[7] Roe, P.L., Approximate Riemann solvers, parameter vectors and difference schemes, J comput phys, 43, 357, (1981) · Zbl 0474.65066
[8] Rao, S.V.R.; Deshpande, S.M., Kinetic theory based wave-particle splitting scheme for Euler equations, Aero soc India J, 44, 329, (1992)
[9] Agarwal RK, Acheson KE. A kinetic theory based wave/particle flux splitting scheme for the Euler equations. AIAA paper 95-2178; 1995.
[10] Prendergast, K.H.; Xu, K., Numerical hydrodynamics from gas-kinetic theory, J comput phys, 109, 53, (1993) · Zbl 0791.76059
[11] Xu, K., A gas-kinetic BGK scheme for the navier – stokes equations and its connection with artificial dissipation and Godunov method, J comput phys, 171, 289, (2001) · Zbl 1058.76056
[12] May, G.; Jameson, A., An improved gas-kinetic BGK finite-volume method for three-dimensional transonic flow, J comput phys, 220, 856, (2007) · Zbl 1370.76096
[13] Tang L. Improved Euler simulation of helicopter vortical flows. Ph.D. thesis. University of Maryland, College Park; 1998.
[14] Tang, L.; Baeder, J.D., Improving Godunov-type reconstructions for simulation of vortex-dominated flows, J comput phys, 213, 659, (2006) · Zbl 1088.76039
[15] Xu K. Numerical hydrodynamics from gas-kinetic theory. Ph.D. thesis. Columbia University; 1993. · Zbl 0791.76059
[16] Tang L. Progresses in gas-kinetic upwind schemes for the solution of Euler/Navier-Stokes equations II. Generalized KFVS/KWPS schemes, in preparation.
[17] Tang L. Progresses in gas-kinetic upwind schemes for the solution of Euler/Navier-Stokes equations III. Kinetic Flow-Variable Splitting (KFVS) schemes, in preparation.
[18] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J comput phys, 54, 115, (1984) · Zbl 0573.76057
[19] Cook PH, McDonald MA, Firmin MCP. Aerofoil RAE 2822 - pressure distributions, and boundary layer and wake measurements. AGARD Report AR 138; 1979.
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.