Stable Runge-Kutta discontinuous Galerkin solver for hypersonic rarefied gaseous flow based on 2D Boltzmann kinetic model equations

doi:10.1007/s10483-017-2177-8

Applied Mathematics and Mechanics (English Edition) ›› 2017, Vol. 38 ›› Issue (3): 343-362.doi: https://doi.org/10.1007/s10483-017-2177-8

Stable Runge-Kutta discontinuous Galerkin solver for hypersonic rarefied gaseous flow based on 2D Boltzmann kinetic model equations

Wei SU¹, Zhenyu TANG², Bijiao HE¹, Guobiao CAI¹

1. School of Astronautics, Beihang University, Beijing 100191, China;
2. Beijing Institute of Spacecraft Environment Engineering, Beijing 100094, China

收稿日期:2015-12-06 修回日期:2016-07-22 出版日期:2017-03-01 发布日期:2017-03-01
通讯作者: Wei SU, E-mail:suwei@buaa.edu.cn E-mail:suwei@buaa.edu.cn
基金资助:
Project supported by the National Natural Science Foundation of China (No. 11302017)

Stable Runge-Kutta discontinuous Galerkin solver for hypersonic rarefied gaseous flow based on 2D Boltzmann kinetic model equations

Wei SU¹, Zhenyu TANG², Bijiao HE¹, Guobiao CAI¹

1. School of Astronautics, Beihang University, Beijing 100191, China;
2. Beijing Institute of Spacecraft Environment Engineering, Beijing 100094, China

Received:2015-12-06 Revised:2016-07-22 Online:2017-03-01 Published:2017-03-01
Contact: Wei SU E-mail:suwei@buaa.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (No. 11302017)

摘要/Abstract

摘要：

A stable high-order Runge-Kutta discontinuous Galerkin (RKDG) scheme that strictly preserves positivity of the solution is designed to solve the Boltzmann kinetic equation with model collision integrals. Stability is kept by accuracy of velocity discretization, conservative calculation of the discrete collision relaxation term, and a limiter. By keeping the time step smaller than the local mean collision time and forcing positivity values of velocity distribution functions on certain points, the limiter can preserve positivity of solutions to the cell average velocity distribution functions. Verification is performed with a normal shock wave at a Mach number 2.05, a hypersonic flow about a two-dimensional (2D) cylinder at Mach numbers 6.0 and 12.0, and an unsteady shock tube flow. The results show that, the scheme is stable and accurate to capture shock structures in steady and unsteady hypersonic rarefied gaseous flows. Compared with two widely used limiters, the current limiter has the advantage of easy implementation and ability of minimizing the influence of accuracy of the original RKDG method.

关键词: multiphase solid, equivalent homogeneous medium, constitutive resemblance, general normality rule, eqevalent elastoplastic constitutive equation, model equation, Courant-Friedrichs-Lewy (CFL) condition, hypersonic flow, conservative discretization, positivity-preserving limiter, discontinuous Galerkin (DG)

Abstract:

Key words: multiphase solid, equivalent homogeneous medium, constitutive resemblance, general normality rule, eqevalent elastoplastic constitutive equation, conservative discretization, positivity-preserving limiter, model equation, discontinuous Galerkin (DG), hypersonic flow, Courant-Friedrichs-Lewy (CFL) condition

中图分类号:

Wei SU, Zhenyu TANG, Bijiao HE, Guobiao CAI. Stable Runge-Kutta discontinuous Galerkin solver for hypersonic rarefied gaseous flow based on 2D Boltzmann kinetic model equations[J]. Applied Mathematics and Mechanics (English Edition), 2017, 38(3): 343-362.

参考文献

[1] Aristov, A. A. Direct Methods for Solving the Boltzmann Equation and Study of Non-equilibrium Flows, 1st ed., Springer, Dordrecht (2001)
[2] Alexeenko, A. A., Gimelshein, S. F., Muntz, E. P., and Ketsdever, A. D. Kinetic modeling of temperature driven flows in short microchannels. International Journal of Thermal Sciences, 45, 1045-1051(2006)
[3] Kolobov, V. I., Bayyuk, S. A., Arslanbekov, R. R., Aristov, V. V., Frolova, A., and Zabelok, S. Construction of a unified continuum/kinetic solver for aerodynamic problems. Journal of Spacecraft and Rockets, 42, 598-606(2005)
[4] Huang, A. B. and Giddens, D. P. The discrete ordinate method for the linearized boundary value problems in kinetic theory of gases. Proceedings of the 5th International Symposium on the Rarefied Gas Dynamics, Academic Press, New York (1967)
[5] Baranger, C., Claudel, J., Hérouard, N., and Mieussens, L. Locally refined discrete velocity grids for stationary rarefied flow simulations. Journal of Computational Physics, 257, 572-593(2014)
[6] Yang, J. Y. and Huang, J. C. Rarefied flow computations using nonlinear model Boltzmann equations. Journal of Computational Physics, 120, 323-339(1995)
[7] Mieussens, L. and Struchtrup, H. Numerical comparison of Bhatnagar-Gross-Krook models with proper Prandtl number. Physics of Fluids, 16, 2797-2813(2004)
[8] Li, Z. H. and Zhang, H. X. Numerical investigation from rarefied flow to continuum by solving the Boltzmann model equation. International Journal for Numerical Methods in Fluids, 42, 361-382(2003)
[9] Morinishi, K. Numerical simulation for gas microflows using Boltzmann equation. Computers and Fluids, 35, 978-985(2006)
[10] Titarev, V. A. Implicit numerical method for computing three-dimensional rarefied gas flows on unstructured meshes. Computational Mathematics and Mathematical Physics, 50, 1719-1733(2010)
[11] Cockburn, B. and Shu, C. W. The Runge-Kutta discontinuous Galerkin method for conservation laws V:multidimensional systems. Journal of Computational Physics, 141, 199-224(1998)
[12] Zhou, T., Li, Y., and Shu, C. W. Numerical comparison of WENO finite volume and Runge-Kutta discontinuous Galerkin methods. Journal of Scientific Computing, 16, 145-171(2001)
[13] Reed, W. H. and Hill, T. R. Triangular Mesh Methods for the Neutron Transport Equation, Los Almos Scientific Laboratory, Los Alamos (1973)
[14] Toulorge, T. and Desmet, W. CFL conditions for Runge-Kutta discontinuous Galerkin methods on triangular grids. Journal of Computational Physics, 230, 4657-4678(2011)
[15] Liu, H. and Xu, K. A Runge-Kutta discontinuous Galerkin method for viscous flow equations. Journal of Computational Physics, 224, 1223-1242(2007)
[16] Gobbert, M., Webster, S., and Cale, T. A Galerkin method for the simulation of the transient 2-D/2-D and 3-D/3-D linear Boltzmann equation. Journal of Scientific Computing, 30, 237-273(2007)
[17] Baker, L. L. and Hadjiconstantinou, N. G. Variance-reduced Monte Carlo solutions of the Boltzmann equation for low-speed gas flows:a discontinuous Galerkin formulation. International Journal for Numerical Methods in Fluids, 58, 381-402(2008)
[18] Alexeenko, A. A., Galitzine, C., and Alekseenko, A. M. High-order discontinuous Galerkin method for Boltzmann model equation. The 40th Thermophysics Conference, American Institute of Aeronautics and Astronautics, Seattle (2008)
[19] Alekseenko, A. M. Numerical properties of high order discrete velocity solutions to the BGK kinetic equation. Applied Numerical Mathematics, 61, 410-427(2011)
[20] Su, W., Alexeenko, A. A., and Cai, G. A parallel Runge-Kutta discontinuous Galerkin solver for rarefied gas flows based on 2D Boltzmann kinetic equations. Computers and Fluids, 109, 123-136(2015)
[21] Jiang, G. and Shu, C. W. On a cell entropy inequality for discontinuous Galerkin methods. Mathematics of Computation, 62, 531-538(1994)
[22] Arten, A., Lax, P. D., and Leer, B. On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Review, 25, 35-61(1983)
[23] Shu, C. W. TVB uniformly high-order schemes for conservation laws. Mathematics of Computation, 49, 105-121(1987)
[24] Biswas, R., Devine, K. D., and Flaherty, J. E. Parallel, adaptive finite element methods for conservation laws. Applied Numerical Mathematics, 14, 255-283(1994)
[25] Burbeau, A., Sagaut, P., and Bruneau, C. H. A problem-independent limiter for high-order RungeKutta discontinuous Galerkin methods. Journal of Computational Physics, 169, 111-150(2001)
[26] Zhu, J., Qiu, J., Shu, C. W., and Dumbser, M. Runge-Kutta discontinuous Galerkin method using WENO limiters II:unstructured meshes. Journal of Computational Physics, 227, 4330-4353(2008)
[27] Zhu, J., Zhong, X., Shu, C. W., and Qiu, J. Runge-Kutta discontinuous Galerkin method using a new type of WENO limiters on unstructured meshes. Journal of Computational Physics, 248, 200-220(2013)
[28] Zhang, X. and Shu, C. W. Positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms. Journal of Computational Physics, 230, 1238-1248(2011)
[29] Zhang, X., Xia, Y., and Shu, C. W. Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin schemes for conservation laws on triangular meshes. Journal of Scientific Computing, 50, 29-62(2012)
[30] 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. Physical Review, 94, 511-525(1954)
[31] Holway, L. H. New statistical models for kinetic theory:methods of construction. Physics of Fluids, 9, 1658-1673(1966)
[32] Bird, G. A. Molecular Gas Dynamics and the Direct Simulation, 2nd ed., Clarendon, Oxford (1994)
[33] Andries, P., Tallec, P. L., Perlat, J. P., and Perthame, B. The Gaussian-BGK model of Boltzmann equation with small Prandtl number. European Journal of Mechanics-B/Fluids, 19, 813-830(2000)
[34] Faires, R. L. and Burden, J. D. Numerical Analysis, 9th ed., Richard Stratton, Boston (2010)
[35] Shu, C. W. and Osher, S. Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of Computational Physics, 77, 439-471(1988)
[36] Mieussens, L. Discrete-velocity models and numerical schemes for the Boltzmann-BGK equation in plane and axisymmetric geometries. Journal of Computational Physics, 162, 429-466(2000)
[37] Rykov, V. A., Titarev, V. A., and Shakhov, E. M. Numerical study of the transverse supersonic flow of a diatomic rarefied gas past a plate. Computational Mathematics and Mathematical Physics, 47, 136-150(2007)
[38] Titarev, V. A. Conservative numerical methods for model kinetic equations. Computers and Fluids, 36, 1446-1459(2007)
[39] Zhang, X. and Shu, C. W. On maximum-principle-satisfying high order schemes for scalar conservation laws. Journal of Computational Physics, 229, 3091-3120(2010)
[40] Kolobov, V. I., Arslanbekov, R. R., Aristov, V. V., Frolova, V. V., and Zabelok, S. A. Unified solver for rarefied and continuum flows with adaptive mesh and algorithm refinement. Journal of Computational Physics, 223, 589-608(2007)
[41] Alsmeyer, H. Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam. Journal of Fluid Mechanics, 74, 497-513(1976)
[42] Yen, S. M. Temperature overshoot in shock waves. Physics of Fluids, 9, 1417-1418(1966)
[43] Shu, C. W. A brief survey on discontinuous Galerkin methods in computational fluid dynamics. Advances in Mechanics, 43, 541-554(2013)
[44] Krivodonova, L., Xin, J., Remacle, J. F., Chevaugeon, N., and Flaherty, J. E. Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Applied Numerical Mathematics, 48, 323-338(2004)
[45] Qiu, J. and Shu, C. W. A comparison of troubled-cell indicators for Runge-Kutta discontinuous Galerkin methods using weighted essentially nonscillatory limiters. SIAM Journal on Scientific Computing, 27, 995-1013(2005)
[46] Schreier, S. Compressible Flow, 2nd ed., Wiley, New York (1982)
[47] Laney, C. B. Computational Gas Dynamics, 1st ed., Cambridge University Press, New York (1998)
[48] Li, Z. H., Peng, A. P., Zhang, H. X., and Yang, J. Y. Rarefied gas flow simulations using high-order gas-kinetic unified algorithms for Boltzmann model equations. Progress in Aerospace Sciences, 74, 81-113(2015)

Stable Runge-Kutta discontinuous Galerkin solver for hypersonic rarefied gaseous flow based on 2D Boltzmann kinetic model equations

Stable Runge-Kutta discontinuous Galerkin solver for hypersonic rarefied gaseous flow based on 2D Boltzmann kinetic model equations

PDF

可视化

被引次数

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 6

编辑推荐

Metrics

本文评价

[1]	姜振华;阎超;于剑. High-order discontinuous Galerkin solver on hybrid anisotropic meshes for laminar and turbulent simulations[J]. Applied Mathematics and Mechanics (English Edition), 2014, 35(7): 799-812.
[2]	毛旭曹伟. Prediction of hypersonic boundary layer transition on sharp wedge flow considering variable specific heat[J]. Applied Mathematics and Mechanics (English Edition), 2014, 35(2): 143-154.
[3]	于剑;阎超;赵瑞. Assessment of shock capturing schemes for discontinuous Galerkin method[J]. Applied Mathematics and Mechanics (English Edition), 2014, 35(11): 1361-1374.
[4]	李邦明鲍麟童秉纲. Physical criterion study on forward stagnation point heat flux CFD computations at hypersonic speeds[J]. Applied Mathematics and Mechanics (English Edition), 2010, 31(7): 839-850.
[5]	李志辉毕林唐志共. Gas-kinetic numerical schemes for one- and two-dimensional inner flows[J]. Applied Mathematics and Mechanics (English Edition), 2009, 30(7): 889-904.
[6]	张培源;汤羽. THE CONSTITUTIVE EQUATION RESEMBLANCE OF ELASTIC-PLASTIC MULTIPHASE SOLID[J]. Applied Mathematics and Mechanics (English Edition), 1990, 11(10): 985-994.