Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-22T14:08:18.932Z Has data issue: false hasContentIssue false
  • English
  • Français

A Unified Gas-Kinetic Scheme for Continuum and Rarefied Flows II: Multi-Dimensional Cases

Published online by Cambridge University Press:  20 August 2015

Juan-Chen Huang ,
Kun Xu  and
Pubing Yu
Juan-Chen Huang*
Affiliation:
Department of Merchant Marine, National Taiwan Ocean University, Keelung 20224, Taiwan
Kun Xu*
Affiliation:
Department of Merchant Marine, National Taiwan Ocean University, Keelung 20224, Taiwan
Pubing Yu*
Affiliation:
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
*
Email:[email protected]
Corresponding author.Email:[email protected]
Email:[email protected]
Get access

Abstract

With discretized particle velocity space, a multi-scale unified gas-kinetic scheme for entire Knudsen number flows has been constructed based on the kinetic model in one-dimensional case [J. Comput. Phys., vol. 229 (2010), pp. 7747-7764]. For the kinetic equation, to extend a one-dimensional scheme to multidimensional flow is not so straightforward. The major factor is that addition of one dimension in physical space causes the distribution function to become two-dimensional, rather than axially symmetric, in velocity space. In this paper, a unified gas-kinetic scheme based on the Shakhov model in two-dimensional space will be presented. Instead of particle-based modeling for the rarefied flow, such as the direct simulation Monte Carlo (DSMC) method, the philosophical principal underlying the current study is a partial-differential-equation (PDE)-based modeling. Since the valid scale of the kinetic equation and the scale of mesh size and time step may be significantly different, the gas evolution in a discretized space is modeled with the help of kinetic equation, instead of directly solving the partial differential equation. Due to the use of both hydrodynamic and kinetic scales flow physics in a gas evolution model at the cell interface, the unified scheme can basically present accurate solution in all flow regimes from the free molecule to the Navier-Stokes solutions. In comparison with the DSMC and Navier-Stokes flow solvers, the current method is much more efficient than DSMC in low speed transition and continuum flow regimes, and it has better capability than NS solver in capturing of non-equilibrium flow physics in the transition and rarefied flow regimes. As a result, the current method can be useful in the flow simulation where both continuum and rarefied flow physics needs to be resolved in a single computation. This paper will extensively evaluate the performance of the unified scheme from free molecule to continuum NS solutions, and from low speed micro-flow to high speed non-equilibrium aerodynamics. The test cases clearly demonstrate that the unified scheme is a reliable method for the rarefied flow computations, and the scheme provides an important tool in the study of non-equilibrium flow.

Keywords

76D0576P0582B4097M10Unified schemenon-equilibrium flowNavier-Stokes solution
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 3 , September 2012 , pp. 662 - 690
Copyright
Copyright © Global Science Press Limited 2012

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]Aristov, V. V., Direct Methods for Solving the Boltzmann Equation and Study of Nonequilib-rium Flows, Kluwer Academic Publishers (2001).CrossRefGoogle Scholar
[2]Belotserkovskii, O. M. and Khlopkov, Y. I., Monte Carlo Methods in Mechanics of Fluid and Gas, World Scientific (2010).CrossRefGoogle Scholar
[3]Bennoune, M., Lemou, M., and Mieussens, L., Uniformly Stable Numerical Schemes for the Boltzmann Equation Preserving the Compressible Navier-Stokes Asymptotics, J. Comput. Phys. 227 (2008), pp. 3781–3803.Google Scholar
[4]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. Rev. 94 (1954), pp. 511–525.Google Scholar
[5]Bird, G. A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford Science Publications (1994).Google Scholar
[6]Chapman, S. and Cowling, T. G., The Mathematical Theory of Non-uniform Gases, Cambridge University Press (1990).Google Scholar
[7]Chu, C. K., Kinetic-Theoretic Description of the Formation of a Shock Wave, Phys. Fluids 8 (1965), pp. 12–22.Google Scholar
[8]Degond, P., Liu, J. G., and Mieussens, L., Macroscopic Fluid Models with Localized Kinetic Upscale Effects, Multiscale Model. Simul. 5 (2006), pp. 940–979.Google Scholar
[9]Filbet, F. and Jin, S., A Class of Asymptotic Preserving Schemes for Kinetic Equations and related problems with stiff sources, J. Comput. Phys. 229 (2010), pp. 7625–7648.CrossRefGoogle Scholar
[10]Ghia, U., Ghia, K. N., and Shin, C. T., High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J. Comput. Phys. 48 (1982), pp. 387–411.Google Scholar
[11]Guo, Z. L., Liu, H. W., Luo, L. S., and Xu, K., A comparative study of the LBM and GKS methods for 2D near incompressible flows, J. Comput. Phys. 227 (2008), pp. 4955–4976.Google Scholar
[12]John, B., Gu, X. J., and Emerson, D. R., Effects of incomplete surface accommodation on non-equilibrium heat transfer in cavity flow: A parallel DSMC study, Comput. Fluids 45 (2011), pp. 197–201.CrossRefGoogle Scholar
[13]Kogan, M. N., Rarefied Gas Dynamics, Plenum Press, New York (1969).Google Scholar
[14]Kolobov, V. I., Arslanbekov, R. R., Aristov, V. V., Frolova, A. A., and Zabelok, S. A., Unified Solver for Rarefied and Continuum Flows with Adaptive Mesh and Algorithm Refinement, J. Comput. Phys. 223 (2007), pp. 589–608.Google Scholar
[15]Li, J. Q., Li, Q. B., and Xu, K., Comparison of the generalized Riemann solver and the gas-kinetic scheme for inviscid compressible flow simulations, J. Comput. Phys. 230 (2011), pp. 5080–5099.Google Scholar
[16]Li, Q. B. and Fu, S., On the Multidimensional Gas-Kinetic BGK Scheme, J. Comput. Phys. 220 (2006), pp. 532–548.Google Scholar
[17]Li, Q. B., Xu, K., and Fu, S., A High-Order Gas-Kinetic Navier-Stokes Solver, J. Comput. Phys. 229 (2010), pp. 6715–6731.CrossRefGoogle Scholar
[18]Li, Z. H. and Zhang, H. X., Gas-Kinetic Numerical Studies of Three-Dimensional Complex Flows on Spacecraft Re-Entry, J. Comput. Phys. 228 (2009), pp. 1116–1138.Google Scholar
[19]Maslach, G. J., and Schaaf, S. A., Cylinder Drag in the Transition from Continuum to Free-Molecule Flow, Phys. Fluids 6 (3) (1963), pp. 315–321.Google Scholar
[20]Metcalf, S. C., Berry, C. J. and Davis, B. M., An Investigation of the Flow About Circular Cylinders Placed Normal to a Low-Density, Supersonic Stream, Aeronautical Research Council, Reports and Memoranda No. 3416, Her Majesty’s Stationery Office, London (1965).Google Scholar
[21]Mieussens, L., Discrete-Velocity Models and Numerical Schemes for the Boltzmann-BGK Equtaion in Plane and Axisymmetric Geometries, J. Comput. Phys. 162 (2000), pp. 429–466.Google Scholar
[22]Pieraccini, S. and Puppo, G., Implicit-Explicit Schemes for BGK Kinetic Equations, J. Scientific Computing 32 (2007), pp. 1–28.Google Scholar
[23]Prendergast, K. H. and Xu, K., Numerical Hydrodynamics from Gas-Kinetic Theory, J. Comput. Phys. 109 (1993), pp. 53–66.Google Scholar
[24]Shakhov, E.M., Generalization of the Krook kinetic Equation, Fluid Dyn. 3, 95 (1968).Google Scholar
[25]Su, M. D., Xu, K., and Ghidaoui, M. S., Low-Speed Flow Simulation by the Gas-Kinetic Scheme, J. Comput. Phys. 150 (1999), pp. 17–39.Google Scholar
[26]Xu, K., Numerical Hydrodynamics from Gas-Kinetic Theory, Ph.D. thesis (1993), Columbia University.Google Scholar
[27]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
[28]Xu, K., and He, X. Y., Lattice Boltzmann Method and Gas-Kinetic BGK Scheme in the Low-Mach Number Viscous Flow Simulations, J. Comput. Phys. 190 (1) (2003), pp.100–117.Google Scholar
[29]Xu, K., Mao, M. L., and Tang, L., A Multidimensional Gas-Kinetic BGK Scheme for Hypersonic Viscous Flow, J. Comput. Phys. 203 (2005), pp. 405–421.Google Scholar
[30]Xu, K. and Huang, J. C., A Unified Gas-Kinetic Scheme for Continuum and Rarefied Flows J. Comput. Phys. 229 (2010), pp. 7747–7764.Google Scholar
[31]Xu, K. and Huang, J. C., An Improved Unified Gas-Kinetic Scheme and the Study of Shock Structures, IMA J. Appl. Math. 76 (2011), pp. 698–711.Google Scholar
[32]Xu, K., Martinelli, L., and Jameson, A., Gas-kinetic Finite Volume Methods, Flux-Vector Splitting and Artificial Diffusion, J. Comput. Phys. 120 (1995), pp. 48–65.CrossRefGoogle Scholar
[33]Xu, K. and Prendergast, K., Numerical Navier-Stokes Solutions from Gas-Kinetic Theory, J. Comput. Phys. 114 (1994), pp. 9–17.CrossRefGoogle Scholar
[34]Yang, J. Y. and Huang, J. C., Rarefied Flow Computations Using Nonlinear Model Boltzmann Equations, J. Comput. Phys. 120 (1995), pp. 323–339.CrossRefGoogle Scholar