Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-22T13:15:00.522Z Has data issue: false hasContentIssue false
  • English
  • Français

Improving the Stability of the Multiple-Relaxation-Time Lattice Boltzmann Method by a Viscosity Counteracting Approach

Published online by Cambridge University Press:  21 December 2015

Chunze Zhang ,
Yongguang Cheng ,
Shan Huang  and
Jiayang Wu
Chunze Zhang
Affiliation:
State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China
Yongguang Cheng*
Affiliation:
State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China
Shan Huang
Affiliation:
State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China
Jiayang Wu
Affiliation:
State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China
*
*Corresponding author. Email:[email protected] (Y. G. Cheng)
Get access

Abstract

Numerical instability may occur when simulating high Reynolds number flows by the lattice Boltzmann method (LBM). The multiple-relaxation-time (MRT) model of the LBM can improve the accuracy and stability, but is still subject to numerical instability when simulating flows with large single-grid Reynolds number (Reynolds number/grid number). The viscosity counteracting approach proposed recently is a method of enhancing the stability of the LBM. However, its effectiveness was only verified in the single-relaxation-time model of the LBM (SRT-LBM). This paper aims to propose the viscosity counteracting approach for the multiple-relaxation-time model (MRT-LBM) and analyze its numerical characteristics. The verification is conducted by simulating some benchmark cases: the two-dimensional (2D) lid-driven cavity flow, Poiseuille flow, Taylor-Green vortex flow and Couette flow, and three-dimensional (3D) rectangular jet. Qualitative and Quantitative comparisons show that the viscosity counteracting approach for the MRT-LBM has better accuracy and stability than that for the SRT-LBM.

Keywords

65M1078A48Multiple-relaxation-time lattice Boltzmann methodviscosity counteractinghigh Reynolds number flowPoiseuille flowCouette flowTaylor-Green vortex flowlid-driven cavity flow
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 1 , February 2016 , pp. 37 - 51
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]Chen, S. Y. and Doolen, G. D., Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid. Mech., 30 (1998), pp. 329–364.CrossRefGoogle Scholar
[2]Qian, Y. H., D’humières, D. and Lallemand, P., Lattice BGK models for Navier-Stokes equation, EPL-Europhys. Lett., 17 (1992), pp. 479.Google Scholar
[3]Sterling, J. D. and Chen, S. Y., Stability analysis of lattice Boltzmann methods, J. Comput. Phys., 123 (1996), pp. 196–206.CrossRefGoogle Scholar
[4]Hou, S., Sterling, J. and Chen, S., A lattice Boltzmann subgrid model for high Reynolds number flows, Pattern Formation and Lattice Gas Automata, 6 (1996), pp. 151–166.Google Scholar
[5]He, X. Y. and Doolen, G. D., Lattice Boltzmann method on curvilinear coordinates system: flow around a circular cylinder, J. Comput. Phys., 134 (1997), pp. 306–315.CrossRefGoogle Scholar
[6]Zhang, R. Y., Chen, H. D. and Qian, Y. H., Effective volumetric lattice Boltzmann scheme, Phys. Rev. E, 63 (2001), 56705.Google Scholar
[7]Shu, C., Niu, X. D. and Chew, Y. T., A fractional step lattice Boltzmann method for simulating high Reynolds number flows, Math. Comput. Simulat., 72 (2006), pp. 201–205.CrossRefGoogle Scholar
[8]Humieres, D., Generalized lattice Boltzmann equations, Rarefied Gas Dynamics-Theory and Simulations, (1994), pp. 450–458.Google Scholar
[9]Lallemand, P. and Luo, L. S., Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability, Phys. Rev. E, 61 (2000), 6546.Google Scholar
[10]Luo, L. S., Liao, W. and Chen, X. W., Numerics of the lattice Boltzmann method: effects of collision models on the lattice Boltzmann simulations, Phys. Rev. E, 83 (2011), 056710.Google Scholar
[11]Humières, D., Multiple-relaxation-time lattice Boltzmann models in three dimensions, Philos. T. R. Soc. A, 360 (2002), pp. 437–451.Google Scholar
[12]Krafczyk, M., Tölke, J. and Luo, L. S., Large-eddy simulations with a multiple-relaxation-time LBE model, Int. J. Mod. Phys. B, 17 (2003), pp. 33–39.CrossRefGoogle Scholar
[13]Luo, L. S., Krafczyk, M. and Shyy, W., Lattice Boltzmann method for computational fluid dynamics, Encyclopedia of Aerospace Engineering, 2010.Google Scholar
[14]Yu, H. D., Luo, L. S. and Girimaji, S. S., LES of turbulent square jet flow using an MRT lattice Boltzmann model, Comput. Fluids, 35 (2006), pp. 957–965.Google Scholar
[15]Premnath, K. N., Pattison, M. J. and Banerjee, S., Generalized lattice Boltzmann equation with forcing term for computation of wall-bounded turbulent flows, Phys. Rev. E, 79 (2009), 026703.Google Scholar
[16]Cheng, Y. G. and Zhang, H., A viscosity counteracting approach in the lattice Boltzmann BGK model for low viscosity flow: Preliminary verification, Comput. Math. Appl., 61 (2011), pp. 3690–3702.Google Scholar
[17]Cheng, Y. G. and Li, J. P., Introducing unsteady non-uniform source terms into the lattice Boltzmann model, Int. J. Number. Meth. Fluid, 56 (2008), pp. 629–641.CrossRefGoogle Scholar
[18]He, X. Y., Zou, Q. S. and Luo, L. S., Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model, J. Stat. Phys., 87 (1997), pp. 115–136.Google Scholar
[19]Pozrikidis, C. and Ferziger, J. H., Introduction to theoretical and computational fluid dynamics, Phys. Today, 50 (1997), 72.Google Scholar
[20]Erturk, E., Corke, T. C. and Gökçöl, C., Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers, Int. J. Numer. Meth. Fluid., 48 (2005), pp. 747–774.Google Scholar
[21]Yu, H. D., Sharath, S. and Girimaji, , Near-field turbulent simulations of rectangular jets using lattice Boltzmann method, Phys. Fluids, 17 (2005), 125106.Google Scholar
[22]Tsuchiya, Y. and Horikoshi, C., On the spread of rectangular jets, Exp. Fluids, 4 (1986), pp. 197–204.Google Scholar