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A Compact Third-Order Gas-Kinetic Scheme for Compressible Euler and Navier-Stokes Equations

Part of: Rarefied gas flows, Boltzmann equation Compressible fluids and gas dynamics, general

Published online by Cambridge University Press:  15 October 2015

Liang Pan  and
Kun Xu
Liang Pan
Affiliation:
Department of mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
Kun Xu*
Affiliation:
Department of mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
*
*Corresponding author. Email addresses: panliangjlu@sina.com (L. Pan), makxu@ust.hk (K. Xu)
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Abstract

In this paper, a compact third-order gas-kinetic scheme is proposed for the compressible Euler and Navier-Stokes equations. The main reason for the feasibility to develop such a high-order scheme with compact stencil, which involves only neighboring cells, is due to the use of a high-order gas evolution model. Besides the evaluation of the time-dependent flux function across a cell interface, the high-order gas evolution model also provides an accurate time-dependent solution of the flow variables at a cell interface. Therefore, the current scheme not only updates the cell averaged conservative flow variables inside each control volume, but also tracks the flow variables at the cell interface at the next time level. As a result, with both cell averaged and cell interface values, the high-order reconstruction in the current scheme can be done compactly. Different from using a weak formulation for high-order accuracy in the Discontinuous Galerkin method, the current scheme is based on the strong solution, where the flow evolution starting from a piecewise discontinuous high-order initial data is precisely followed. The cell interface time-dependent flow variables can be used for the initial data reconstruction at the beginning of next time step. Even with compact stencil, the current scheme has third-order accuracy in the smooth flow regions, and has favorable shock capturing property in the discontinuous regions. It can be faithfully used from the incompressible limit to the hypersonic flow computations, and many test cases are used to validate the current scheme. In comparison with many other high-order schemes, the current method avoids the use of Gaussian points for the flux evaluation along the cell interface and the multi-stage Runge-Kutta time stepping technique. Due to its multidimensional property of including both derivatives of flow variables in the normal and tangential directions of a cell interface, the viscous flow solution, especially those with vortex structure, can be accurately captured. With the same stencil of a second order scheme, numerical tests demonstrate that the current scheme is as robust as well-developed second-order shock capturing schemes, but provides more accurate numerical solutions than the second order counterparts.

Keywords

Third-order gas-kinetic schemecompact reconstructionNavier-Stokes solutions

MSC classification

Secondary: 76P05: Rarefied gas flows, Boltzmann equation 76N15: Gas dynamics, general
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 4 , October 2015 , pp. 985 - 1011
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Bhatnagar, P.L., Gross, E. P., Krook, M., A Model for Collision Processes in Gases I: Small Amplitude Processes in Charged and Neutral One-Component Systems, Phys. Rev. 94 (1954) 511–525.Google Scholar
[2]Cercignani, C., The Boltzmann Equation and its Applications, Springer-Verlag, (1988).Google Scholar
[3]Chapman, S., Cowling, T.G., The Mathematical theory of Non-Uniform Gases, third edition, Cambridge University Press, (1990).Google Scholar
[4]Cockburn, B., Shu, C. W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Mathematics of Computation, 52 (1989) 411–435.Google Scholar
[5]Cockburn, B., Shu, C. W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. Comput. Phys. 141 (1998), 199–224.Google Scholar
[6]Daru, V., Tenaud, C., High order one-step monotonicity-preserving schemes for unsteady compressible flow calculations, J. Comput. Phys. 193 (2004) 563–594.Google Scholar
[7]Ghia, U., Ghia, K. N., Shin, C. T, High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J. Comput. Phys. 48 (1982) 387–411.Google Scholar
[8]Godunov, S. K., A difference scheme for numerical computation of discontinuous solution of hyperbolic equation, Math. Sbornik. 47 (1959) 271–306.Google Scholar
[9]Harten, A., Engquist, B., Osher, S. and Chakravarthy, S. R.. Uniformly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys. 71 (1987) 231–303.Google Scholar
[10]Friedrich, O., Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids, J. Comput. Phys. 144 (1998) 194–212.CrossRefGoogle Scholar
[11]Fuchs, L., Tillmark, N., Numerical and experimental study of driven flow in a polar cavity, International Journal for Numerical Methods in Fluids. 5 (1985) 311–329.Google Scholar
[12]Gao, H. and Wang, Z.J., A high-order lifting collocation penalty formulation for the Navier-Stokes equations on 2D mixed grids, AIAA Paper 2009 3784.Google Scholar
[13]Guo, Z. L., Xu, K., and Wang, R.J., Discrete unified gas kinetic scheme for all Knudsen number flows: Low-speed isothermal case, Physical Review E 88, 033305 (2013).Google Scholar
[14]Huynh, H. T., A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods, AIAA Paper (2007) 4079.Google Scholar
[15]Huynh, H. T., A reconstruction approach to high-order schemes including discontinuous Galerkin for diffusion, AIAA Paper (2009) 403.Google Scholar
[16]Jiang, J., Qian, Y. H., Implicit gas-kinetic BGK scheme with multigrid for 3D stationary transonic high-Reynolds number flows, Computers & Fluids. 66 (2012) 21–28.CrossRefGoogle Scholar
[17]Jiang, G. S., Shu, C. W., Efficient implementation of Weighted ENO schemes, J. Comput. Phys. 126 (1996) 202–228.Google Scholar
[18]Jin, C., Xu, K., A unified moving grid gas-kinetic method in Eulerian space for viscous flow computation, J. Comput. Phys. 222 (2007) 155–175.Google Scholar
[19]Kumar, G., Girimaji, S. S., Kerimo, J., WENO-enhanced gas-kinetic scheme for direct simulations of compressible transition and turbulence, J. Comput. Phys. 234 (2013) 499–523.Google Scholar
[20]Li, Q., Xu, K., Fu, S., A high-order gas-kinetic Navier-Stokes flow solver, J. Comput. Phys. 229 (2010) 6715–6731.Google Scholar
[21]Liu, X. D., Lax, P. D., Solution of two-dimensional Riemann problems of gas dynamics by positive schemes, SIAM J. Sci. Comput. 19 (1998) 319–340.Google Scholar
[22]Liu, X. D., Osher, S., Chan, T., Weighted essentially non-oscillatory schemes, J. Comput. Phys. 115 (1994) 200–212.Google Scholar
[23]Liu, Y., Vinokur, M., Wang, Z.J., Spectral difference method for unstructured grids I: Basic formulation, J. Comput. Phys. 216 (2006) 780–801.Google Scholar
[24]Luo, J., Xuan, L. J., Xu, K., Comparison of fifth-order WENO scheme and WENO-gas-kinetic scheme for inviscid and viscous flow simulation, Commun. Comput. Phys. 14 (2013) 599–620.Google Scholar
[25]Luo, J., Xu, K., A high-order multidimensional gas-kinetic scheme for hydrodynamic equations, SCIENCE CHINA Technological Sciences, 56 (2013) 2370–2384.CrossRefGoogle Scholar
[26]Mieussens, L., On the asymptotic preserving property of the unified gas-kinetic scheme for the diffusion limit of linear kinetic models, J. Comput. Phys. 253 (2013) 138–156.Google Scholar
[27]Ohwada, T., Xu, K., The kinetic scheme for full Burnett equations, J. Comput. Phys. 201 (2004) 315–332.CrossRefGoogle Scholar
[28]Pan, L., Xu, K., Generalized coordinate transformation and gas-kinetic scheme, J. Comput. Phys. 287 (2015) 207–225.Google Scholar
[29]Reed, W. H., T.R. Hill, Triangular mesh methods for the neutron transport equation, Technical Report LA-UR-73-479, 1973, Los Alamos Scientific Laboratory, Los Alamos.Google Scholar
[30]Shu, C. W., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, Lecture Notes in Mathematics, Springer, 1998.Google Scholar
[31]Shu, C. W., Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes, J. Comput. Phys. 77 (1988) 439–471.Google Scholar
[32]Sjögreen, B., Yee, H.C., Grid convergence of high order methods for multiscale complex unsteady viscous compressible flows, J. Comput. Phys. 185 (2003) 1–26.Google Scholar
[33]Sod, G. A., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys. 27 (1978) 1–31.Google Scholar
[34]Toro, E., Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, (1997).Google Scholar
[35]Wang, Z. J., Spectral (finite) volume method for conservation laws on unstructured grids: basic formulation, J. Comput. Phys. 178 (2002) 210–251.Google Scholar
[36]Wang, Z. J., Gao, H., A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids, J. Comput. Phys. 228 (2009) 8161–8186Google Scholar
[37]Woodward, P., Colella, P., The numerical simulation of two dimensional fluids with strong shock, J. Comput. Phys. 54 (1984) 115–173.Google Scholar
[38]Xu, K., A Slope-update Scheme for Compressible Flow Simulation, J. Comput. Phys. 178 (2002) 252–259.Google Scholar
[39]Xu, K., Direct Modeling for Computational Fluid Dynamics: Construction and Application of Unified Gas-kinetic Schemes, World Scientific, (2015).Google Scholar
[40]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) 289–335.CrossRefGoogle Scholar
[41]Xu, K. and Huang, J., A unified gas-kinetic scheme for continuum and rarefied flows, J. Comput. Phys. 229 (2010) 7747–7764.Google Scholar
[42]Xu, K., Liu, H., Jiang, J., Multiple temperature kinetic model for continuum and near continuum flows, Physics of Fluids, 19 (2007) 016101.Google Scholar
[43]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 (2013) 540–557.Google Scholar