Hostname: page-component-848d4c4894-2xdlg Total loading time: 0 Render date: 2024-06-23T11:36:50.007Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  08 July 2016

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:mpeshuc@nus.edu.sg (C. Shu)
Get access

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.

MSC classification

Type
Research Article
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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. 856878.Google Scholar
[2]Pullin, D., Direct simulation methods for compressible inviscid ideal-gas flow, J. Comput. Phys., 34 (1980), pp. 231244.Google Scholar
[3]Mandal, J. and Deshpande, S., Kinetic flux vector splitting for Euler equations, Comput. Fluids, 23 (1994), pp. 447478.Google Scholar
[4]Chou, S. Y. and Baganoff, D., Kinetic flux-vector splitting for the Navier-Stokes equations, J. Comput. Phys., 130 (1997), pp. 217230.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. 611632.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. 5366.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. 127.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. 289335.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. 1739.Google Scholar
[13]Jiang, S. and Ni, G., A γ-model BGK scheme for compressible multifluids, Int. J. Numer. Methods Fluids, 46 (2004), pp. 163182.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. 708738.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. 405421.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. 49554976.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. 2128.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. 540557.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. 100106.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. 96113.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. 120130.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. 276299.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. 229243.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. 115173.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. 143149.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. 243251.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. 10251026.Google Scholar