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A Switch Function-Based Gas-Kinetic Scheme for Simulation of Inviscid and Viscous Compressible Flows

Part of: Permutation groups

Published online by Cambridge University Press:  08 July 2016

Yu Sun ,
Chang Shu ,
Liming Yang  and
C. J. Teo
Yu Sun
Affiliation:
Department of Mechanical Engineering, National University of Singapore 10 Kent Ridge Crescent , Singapore 119260, Singapore
Chang Shu*
Affiliation:
Department of Mechanical Engineering, National University of Singapore 10 Kent Ridge Crescent , Singapore 119260, Singapore
Liming Yang
Affiliation:
Department of Aerodynamics, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Yudao Street, Nanjing 210016, China
C. J. Teo
Affiliation:
Department of Mechanical Engineering, National University of Singapore 10 Kent Ridge Crescent , Singapore 119260, Singapore
*
*Corresponding author. Email:[email protected] (C. Shu)
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Abstract

In this paper, a switch function-based gas-kinetic scheme (SF-GKS) is presented for the simulation of inviscid and viscous compressible flows. With the finite volume discretization, Euler and Navier-Stokes equations are solved and the SF-GKS is applied to evaluate the inviscid flux at cell interface. The viscous flux is obtained by the conventional smooth function approximation. Unlike the traditional gas-kinetic scheme in the calculation of inviscid flux such as Kinetic Flux Vector Splitting (KFVS), the numerical dissipation is controlled with a switch function in the present scheme. That is, the numerical dissipation is only introduced in the region around strong shock waves. As a consequence, the present SF-GKS can well capture strong shock waves and thin boundary layers simultaneously. The present SF-GKS is firstly validated by its application to the inviscid flow problems, including 1-D Euler shock tube, regular shock reflection and double Mach reflection. Then, SF-GKS is extended to solve viscous transonic and hypersonic flow problems. Good agreement between the present results and those in the literature verifies the accuracy and robustness of SF-GKS.

Keywords

Switch functiongas-kinetic schemenumerical dissipationinviscid and viscous flows

MSC classification

Secondary: 20B40: Computational methods
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 5 , October 2016 , pp. 703 - 721
Copyright
Copyright © Global-Science Press 2016 

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References

[1]May, G., Srinivasan, B. and Jameson, A., An improved gas-kinetic BGK finite-volume method for three-dimensional transonic flow, J. Comput. Phys., 220 (2007), pp. 856–878.Google Scholar
[2]Pullin, D., Direct simulation methods for compressible inviscid ideal-gas flow, J. Comput. Phys., 34 (1980), pp. 231–244.Google Scholar
[3]Mandal, J. and Deshpande, S., Kinetic flux vector splitting for Euler equations, Comput. Fluids, 23 (1994), pp. 447–478.Google Scholar
[4]Chou, S. Y. and Baganoff, D., Kinetic flux-vector splitting for the Navier-Stokes equations, J. Comput. Phys., 130 (1997), pp. 217–230.Google Scholar
[5]Yang, L., Shu, C. and Wu, J., A simple distribution function-based gas-kinetic scheme for simulation of viscous incompressible and compressible flows, J. Comput. Phys., 274 (2014), pp. 611–632.CrossRefGoogle Scholar
[6]Xu, K., Gas-kinetic schemes for the unsteady compressible flow simulations, VKI for Fluid Dynamics Lecture Series, (1998), 1998-03.Google Scholar
[7]Xu, K., Numerical Hydrodynamics from Gas-Kinetic Theory, Ph.D. Thesis, Columbia University, New York, 1993.Google Scholar
[8]Prendergast, K. H. and Xu, K., Numerical hydrodynamics from gas-kinetic theory, J. Comput. Phys., 109 (1993), pp. 53–66.CrossRefGoogle Scholar
[9]Chae, D., Kim, C. and Rho, O.-H., Development of an improved gas-kinetic BGK scheme for inviscid and viscous flows, J. Comput. Phys., 158 (2000), pp. 1–27.Google Scholar
[10]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 (2001), pp. 289–335.Google Scholar
[11]Bhatnagar, P. L., Gross, E. P. and Krook, M., A model for collision processes in gases I. small amplitude processes in charged and neutral one-component systems, Phys. Review, 94 (1954), 511.Google Scholar
[12]Su, M., Xu, K. and Ghidaoui, M., Low-speed flow simulation by the gas-kinetic scheme, J. Comput. Phys., 150 (1999), pp. 17–39.Google Scholar
[13]Jiang, S. and Ni, G., A γ-model BGK scheme for compressible multifluids, Int. J. Numer. Methods Fluids, 46 (2004), pp. 163–182.Google Scholar
[14]Li, Z.-H. and Zhang, H.-X., Study on gas kinetic unified algorithm for flows from rarefied transition to continuum, J. Comput. Phys., 193 (2004), pp. 708–738.Google Scholar
[15]Xu, K., Mao, M. and Tang, L., A multidimensional gas-kinetic BGK scheme for hypersonic viscous flow, J. Comput. Phys., 203 (2005), pp. 405–421.Google Scholar
[16]Guo, Z., Liu, H., Luo, L.-S. and Xu, K., A comparative study of the LBE and GKS methods for 2D near incompressible laminar flows, J. Comput. Phys., 227 (2008), pp. 4955–4976.Google Scholar
[17]Jiang, J. and Qian, Y., Implicit gas-kinetic BGK scheme with multigrid for 3D stationary transonic high-Reynolds number flows, Comput. Fluids, 66 (2012), pp. 21–28.Google Scholar
[18]Yang, L., Shu, C., Wu, J., Zhao, N. and Lu, Z., Circular function-based gas-kinetic scheme for simulation of inviscid compressible flows, J. Comput. Phys., 255 (2013), pp. 540–557.Google Scholar
[19]Li, W., Kaneda, M. and Suga, K., An implicit gas kinetic BGK scheme for high temperature equilibrium gas flows on unstructured meshes, Comput. Fluids, 93 (2014), pp. 100–106.CrossRefGoogle Scholar
[20]Liu, S., Yu, P., Xu, K. and Zhong, C., Unified gas-kinetic scheme for diatomic molecular simulations in all flow regimes, J. Comput. Phys., 259 (2014), pp. 96–113.Google Scholar
[21]Venkatakrishnan, V., Convergence to steady state solutions of the Euler equations on unstructured grids with limiters, J. Comput. Phys., 118 (1995), pp. 120–130.CrossRefGoogle Scholar
[22]Van Leer, B., Towards the ultimate conservative difference scheme IV. A new approach to numerical convection, J. Comput. Phys., 23 (1977), pp. 276–299.CrossRefGoogle Scholar
[23]Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics: a Practical Introduction, Springer Science & Business Media, 2009.Google Scholar
[24]Mittal, Sanjay and Tezduyar, T., A unified finite element formulation for compressible and incompressible flows using augmented conservation variables, Comput. Methods Appl. Mech. Eng., 161 (1998), pp. 229–243.Google Scholar
[25]Woodward, P. and Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54 (1984), pp. 115–173.CrossRefGoogle Scholar
[26]Cook, P. H., McDonald, M. A. and Firmin, M. C. P., Aerofoil RAE2822-pressure distributions, and boundary layer and wake measurements, AGARD Report AR 138, (1979).Google Scholar
[27]Lee, C. B. and Wang, S., Study of the shock motion in a hypersonic shock system/turbulent boundary layer interaction, Exp. Fluids, 19 (1995), pp. 143–149.Google Scholar
[28]Lee, C. B., Hong, Z. X., Kachanov, Y. S., Borodulin, V. I. and Gaponenko, V. V., A study in transitional flat plate boundary layers: measurement and visualization, Exp. Fluids, 28 (2000), pp. 243–251.Google Scholar
[29]Lee, C. B. and Wu, J. Z., Transition in wall-bounded flows, Appl. Mech. Rev., 61 (2008), 0802.CrossRefGoogle Scholar
[30]Kim, K., Kim, C., Rho, O. H. and Hong, S., Cure for shock instability: development of an improved Roe scheme, AIAA Paper, 548 (2002), 2002.Google Scholar
[31]Wieting, A. and Holden, M., Experimental study of shock wave interference heating on a cylinder leading edge, AIAA Paper, (1988).Google Scholar
[32]Yoon, S. and Jameson, A., Lower-upper symmetric-Gauss-Seidel method for the Euler and Navier-Stokes equations, AIAA J., 26 (1988), pp. 1025–1026.Google Scholar