A Three-dimensional Multiple-Relaxation-Time Lattice Boltzmann Method for Whole-Speed-Range

doi:10.19596/j.cnki.1001-246x.7680

Abstract

Abstract: A three-dimensional (3D) free-parameter multiple-relaxation-time lattice Boltzmann method for high speed compressible and low speed incompressible flows is presented. In the approach transformation matrix is constructed according to irreducible representation basis functions of SO(3) group. Equilibria of nonconserved moments are chosen so as to recover compressible Navier-Stokes equations through Chapman-Enskog analysis. Sizes of discrete velocities are flexible. Influence of model parameters on numerical stability is analyzed. Reference values of parameters are suggested. To validate performance of the model, several well-known benchmark problems ranging from 1D to 3D are simulated. Numerical results are in good agreement with analytical solutions and/or other numerical results.

Key words: lattice Boltzmann method, compressible flows, multiple-relaxation-time

CLC Number:

CHEN Feng, XU Aiguo, ZHANG Guangcai, JIAO Peigang. A Three-dimensional Multiple-Relaxation-Time Lattice Boltzmann Method for Whole-Speed-Range[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2018, 35(4): 379-387.

References

[1] SUCCI S. The lattice Boltzmann equation for fluid dynamics and beyond[M]. Oxford:Clarendon Press, 2001:77-176.
[2] 郭照立,郑楚光,李青,等. 流体动力学的格子Boltzmann方法[M]. 武汉:湖北科学技术出版社, 2002:1-166.
[3] 何雅玲,王勇,李庆. 格子Boltzmann方法的理论及应用[M]. 北京:科学出版社, 2009:1-237.
[4] BENZI R, SUCCI S, VERGASSOLA M. The lattice Boltzmann equation:Theory and applications[J]. Phys Rep, 1992, 222(3):145-197.
[5] XU A G,ZHANG G C,GAN Y B,et al. Lattice Boltzmann modeling and simulation of compressible flows[J]. Front Phys, 2012, 7(5):582-600.
[6] 许爱国,张广财,李英骏,等. 非平衡与多相复杂系统模拟研究-Lattice Boltzmann动理学理论与应用[J].物理学进展, 2014, 34(3):136-167.
[7] ALEXANDER F J, CHEN H, CHEN S, DOOLEN G D. Lattice Boltzmann model for compressible fluids[J]. Phys Rev A, 1992, 46(4):1967-1970.
[8] YU H D, ZHAO K H. Lattice Boltzmann method for compressible flows with high Mach numbers[J]. Phys Rev E, 2000, 61(4):3867-3870.
[9] YAN G W, ZHANG J Y, LIU Y H, DONG Y F. A multi-energy-level lattice Boltzmann model for the compressible Navier-Stokes equations[J]. Int J Numer Meth Fluids, 2007, 55(1):41-56.
[10] SUN C H, HSU A T. Three-dimensional lattice Boltzmann model for compressible flows[J]. Phys Rev E, 2003, 68(1):016303.
[11] SHAN X W, YUAN X F, CHEN H D. Kinetic theory representation of hydrodynamics:A way beyond the Navier-Stokes equation[J]. J Fluid Mech, 2006, 550:413-441.
[12] QU K, SHU C, CHEW Y T. Alternative method to construct equilibrium distribution functions in lattice-Boltzmann method simulation of inviscid compressible flows at high Mach number[J]. Phys Rev E, 2007, 75(3):036706.
[13] LI Q, HE Y L, WANG Y, TANG G H. Three-dimensional non-free-parameter lattice-Boltzmann model and its application to inviscid compressible flows[J]. Phys Lett A, 2009, 373(25):2101-2108.
[14] RAN Z. Note on invariance of one-dimensional lattice-Boltzmann equation[J]. Chinese Phys Lett, 2007, 24(12):3332-3335.
[15] RAN Z. Thermal equation of state for lattice Boltzmann gases[J]. Chinese Phys B, 2009, 18(6):2159-2167.
[16] WATARI M,TSUTAHARA M. Possibility of constructing a multispeed Bhatnagar-Gross-Krook thermal model of the lattice Boltzmann method[J]. Phys Rev E, 2004, 70(1):016703.
[17] WATARI M,TSUTAHARA M. Supersonic flow simulations by a three-dimensional multispeed thermal model of the finite difference lattice Boltzmann method[J]. Phys A, 2006, 364:129-144.
[18] TSUTAHARA M,KATAOKA T,SHIKATA K,TAKADA N. New model and scheme for compressible fluids of the finite difference lattice Boltzmann method and direct simulations of aerodynamic sound[J]. Comput Fluids, 2008, 37(1):79-89.
[19] PAN X F, XU A G, ZHANG G C, JIANG S. Lattice Boltzmann approach to high-speed compressible flows[J]. Int J Mod Phys C, 2007, 18(11):1747-1764.
[20] GAN Y B, Xu A G, Zhang G C, et al. Two-dimensional lattice Boltzmann model for compressible flows with high Mach number[J]. Phys A, 2008, 387(8):1721-1732.
[21] CHEN F, XU A G, ZHANG G C, et al. Highly efficient lattice Boltzmann model for compressible fluids:Two-dimensional case[J]. Commun Theor Phys, 2009, 52(4):681-693.
[22] CHEN F, XU A G, ZHANG G C, LI Y J. Three-dimensional lattice Boltzmann model for high-speed compressible flows[J]. Commun Theor Phys, 2010, 54(6):1121-1128.
[23] GAN Y B, XU A G, ZHANG G C, YANG Y. Lattice BGK kinetic model for high-speed compressible flows:Hydrodynamic and nonequilibrium behaviors[J]. Europhys Lett, 2013, 103(2):24003.
[24] ZHANG Y D, XU A G, ZHANG G C, ZHU C M. A highly efficient discrete Boltzmann model for detonation[J]. Chinese J Comp Phys, 2016, 33(5):505-515.
[25] HUMIERES D. Generalized lattice-Boltzmann equations, in rarefied gas dynamics:Theory and simulations[M]. Washington:AIAA Press, 1992:450-458.
[26] LAⅡEMAND P, LUO L S. Theory of the lattice Boltzmann method:Dispersion, dissipation, isotropy, Galilean invariance, and stability[J]. Phys Rev E, 2000, 61(6):6546.
[27] PREMNATH K N, ABRAHAM J. Three-dimensional multi-relaxation lattice Boltzmann models for multiphase flows[J]. J Comp Phys, 2007, 224(2):539-559.
[28] YOSHIDA H, NAGAOKA M. Multiple-relaxation-time lattice Boltzmann model for the convection and anisotropic diffusion equation[J]. J Comp Phys, 2010, 229(20):7774-7795.
[29] HUANG J J, HUANG H B, SHU C, et al. Hybrid multiple-relaxation-time lattice-Boltzmann finite-difference method for axisymmetric multiphase flows[J]. J Phys A:Math Theor, 2013, 46(5):055501.
[30] CHEN F,XU A G, ZHANG G C, et al. Multiple-relaxation-time lattice Boltzmann approach to compressible flows with flexible specific-heat ratio and Prandtl number[J]. Europhys Lett,2010, 90(5):54003.
[31] CHEN F,XU A G, ZHANG G C, et al. Multiple-relaxation-time lattice Boltzmann model for compressible fluids[J]. Phys Lett A, 2011, 375:2129-2139.
[32] STERLING J D,CHEN S Y. Stability analysis of lattice Boltzmann methods[J]. J Comput Phys, 1996, 123(1):196-206.
[33] SETA T,TAKAHASHI R. Numerical stability analysis of FDLBM[J]. J Stat Phys, 2002, 107(1):557-572.
[34] LATINI M,SCHILLING O,DON W S. Effects of order of WENO flux reconstruction and spatial resolution on reshocked two-dimensional Richtmyer-Meshkov instability[J]. J Comput Phys,2007,221(2):805-836.
[35] 柏劲松,李平,王涛,等. 可压缩多介质粘性流体的数值计算[J]. 爆炸与冲击, 2007, 27(6):515-521.