Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-23T15:40:25.846Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite Volume Lattice Boltzmann Method for Nearly Incompressible Flows on Arbitrary Unstructured Meshes

Part of: Basic methods in fluid mechanics Time-dependent statistical mechanics (dynamic and nonequilibrium) Incompressible viscous fluids

Published online by Cambridge University Press:  21 July 2016

Weidong Li  and
Li-Shi Luo
Weidong Li*
Affiliation:
Department of Mathematics & Statistics, Old Dominion University, Norfolk, Virginia 23529, USA Computational Science Research Center, Beijing 100193, China
Li-Shi Luo*
Affiliation:
Department of Mathematics & Statistics, Old Dominion University, Norfolk, Virginia 23529, USA Computational Science Research Center, Beijing 100193, China
*
*Corresponding author. Email addresses:[email protected] (W. D. Li), [email protected], [email protected] (L.-S. Luo)
*Corresponding author. Email addresses:[email protected] (W. D. Li), [email protected], [email protected] (L.-S. Luo)
Get access

Abstract

A genuine finite volume method based on the lattice Boltzmann equation (LBE) for nearly incompressible flows is developed. The proposed finite volume lattice Boltzmann method (FV-LBM) is grid-transparent, i.e., it requires no knowledge of cell topology, thus it can be implemented on arbitrary unstructured meshes for effective and efficient treatment of complex geometries. Due to the linear advection term in the LBE, it is easy to construct multi-dimensional schemes. In addition, inviscid and viscous fluxes are computed in one step in the LBE, as opposed to in two separate steps for the traditional finite-volume discretization of the Navier-Stokes equations. Because of its conservation constraints, the collision term of the kinetic equation can be treated implicitly without linearization or any other approximation, thus the computational efficiency is enhanced. The collision with multiple-relaxation-time (MRT) model is used in the LBE. The developed FV-LBM is of second-order convergence. The proposed FV-LBM is validated with three test cases in two-dimensions: (a) the Poiseuille flow driven by a constant body force; (b) the Blasius boundary layer; and (c) the steady flow past a cylinder at the Reynolds numbers Re=10, 20, and 40. The results verify the designed accuracy and efficacy of the proposed FV-LBM.

Keywords

Finite volume methodlattice Boltzmann equationarbitrary unstructured meshcomplex geometrymulti-dimensional scheme

MSC classification

Secondary: 76M12: Finite volume methods 76M28: Particle methods and lattice-gas methods 76D05: Navier-Stokes equations 82C40: Kinetic theory of gases
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 2 , August 2016 , pp. 301 - 324
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] He, X. and Luo, L.-S.. A priori derivation of the lattice Boltzmann equation. Phys. Rev. E, 55(6):R6333–R6336, 1997.CrossRefGoogle Scholar
[2] He, X. and Luo, L.-S.. Theory of lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation. Phys. Rev. E, 56(6):6811–6817, 1997.Google Scholar
[3] Yu, D.Z., Mei, R., Luo, L.-S., and Shyy, W.. Viscous flow computations with the method of lattice Boltzmann equation. Prog. Aerospace Sci., 39(5):329–367, 2003.CrossRefGoogle Scholar
[4] Luo, L.-S., Krafczyk, M., and Shyy, W.. Lattice Boltzmann method for computational fluid dynamics. In Blockley, R. and Shyy, W., editors, Encyclopedia of Arospace Engineering, chapter 56, pages 651–660. Wiley, New York, 2010.Google Scholar
[5] Yong, W.-A. and Luo, L.-S.. Accuracy of the viscous stress in the lattice Boltzmann equation with simple boundary conditions. Phys. Rev. E, 86:065701R, 2012.Google Scholar
[6] Dellar, P. J.. An interpretation and derivation of the lattice Boltzmann method using Strang splitting. Comput. Math. Appl., 65:129–141, 2013.Google Scholar
[7] Dellar, P. J. and Luo, L.-S.. Lattice Boltzmann methods. In Engquist, B., editor, Encyclopedia of Applied and Computational Mathematics, pages 774–778. Springer, Berlin, 2015.CrossRefGoogle Scholar
[8] Nannelli, F. and Succi, S.. The lattice Boltzmann-equation on irregular lattices. J. Stat. Phys., 68(3-4):401–407, 1992.Google Scholar
[9] Peng, G.W., Xi, H.W., Duncan, C., and Chou, S.H.. Lattice Boltzmann method on irregular meshes. Phys. Rev. E, 58(4):R4124–R4127, 1998.Google Scholar
[10] Peng, G.W., Xi, H.W., Duncan, C., and Chou, S.H.. Finite volume scheme for the lattice Boltzmann method on unstructured meshes. Phys. Rev. E, 59(4):4675–4682, 1999.CrossRefGoogle Scholar
[11] Xi, H.W., Peng, G.W., and Chou, S.H.. Finite-volume lattice Boltzmann method. Phys. Rev. E, 59(5B):6202–6205, 1999.CrossRefGoogle ScholarPubMed
[12] Xi, H.W., Peng, G.W., and Chou, S.H.. Finite-volume lattice Boltzmann schemes in two and three dimensions. Phys. Rev. E, 60(3):3380–3388, 1999.Google Scholar
[13] Stiebler, M., Tölke, J., and Krafczyk, M.. An upwind discretization scheme for the finite volume lattice Boltzmann method. Comput. Fluids, 35(8-9):814–819, 2006.Google Scholar
[14] Dubois, F. and Lallemand, P.. On lattice Boltzmann scheme, finite volumes and boundary conditions. Prog. Comput. Fluid Dym., 8(1-4):11–24, 2008.Google Scholar
[15] Ubertini, S. and Succi, S.. A generalised lattice Boltzmann equation on unstructured grids. Commun. Comput. Phys., 3(2):342–356, FEB 2008.Google Scholar
[16] Patil, D. V. and Lakshmisha, K. N.. Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh. J. Comput. Phys., 228(14):5262–5279, 2009.CrossRefGoogle Scholar
[17] Razavi, S. E., Ghasemi, J., and Farzadi, A.. Flux modelling in the finite-volume lattice Boltzmann approach. Int. J. Comput. Fluid Dyn., 23(1):69–77, 2009.Google Scholar
[18] Mavriplis, D. J.. Unstructured grid techniques. Annu. Rev. Fluid Mech., 29:473–514, 1997.Google Scholar
[19] Blazek, J.. Computational Fluid Dynamics: Principles and Applications. Elsevier, New York, 2nd edition, 2005.Google Scholar
[20] Li, W.D., Kaneda, M., and Suga, K.. A stable, low diffusion up-wind scheme for unstructured finite volume lattice Boltzmann method. In Proceedings of The 4th Asian Symposium on Computational Heat Transfer and Fluid Flow, 2013. June 3–6, 2013, Hong Kong, China.Google Scholar
[21] d’Humières, D.. Generalized lattice-Boltzmann equations. In Shizgal, B. D. and Weave, D. P., editors, Rarefied Gas Dynamics: Theory and Simulations, volume 159 of Prog. Astronaut. Aeronaut., pages 450–458, Washington, D.C., 1992. AIAA.Google Scholar
[22] Guo, Z.L., Zheng, C.G., and Shi, B.C.. An extrapolation method for boundary conditions in lattice Boltzmann method. Phys. Fluids, 14(6):2007–2010, 2002.Google Scholar
[23] Ginzbourg, I.. Boundary Conditions Problems in Lattice Gas Methods for Single and Multiple Phases. PhD thesis, Universite Paris VI, France, 1994.Google Scholar
[24] Ginzbourg, I. and Adler, P. M.. Boundary flow condition analysis for the three-dimensional lattice Boltzmann model. J. Phys. II, 4(2):191–214, 1994.Google Scholar
[25] Ginzburg, I. and d’Humières, D.. Multireflection boundary conditions for lattice Boltzmann models. Phys. Rev. E, 68(6):066614, 2003.Google Scholar
[26] Pan, C., Luo, L.-S., and Miller, C.T.. An evaluation of lattice Boltzmann schemes for porous medium flow simulation. Comput. Fluids, 35(8/9):898–909, 2006.Google Scholar
[27] Luo, L.-S., Liao, W., Chen, X., Peng, Y., and Zhang, W.. Numerics of the lattice Boltzmann method: Effects of collision models on the lattice Boltzmann simulations. Phys. Rev. E, 83(5):056710, 2011.CrossRefGoogle ScholarPubMed
[28] Schlichting, H. and Gersten, K.. Boundary Layer Theory. Springer, Berlin, 8th revised and enlarged edition, 2000.CrossRefGoogle Scholar
[29] White, F. M.. Viscous Fluid Flow. McGraw-Hill Higher Education, New York, 3rd edition, 2006.Google Scholar
[30] Zdravkovich, M.M.. Flow Around Circular Cylinders: Volume I: Fundamentals. Oxford University Press, Oxford, 1997.Google Scholar
[31] He, X. and Doolen, G. D.. Lattice Boltzmann method on curvilinear coordinates systems: Flow around a circular cylinder. J. Comput. Phys., 134:306–315, 1997.CrossRefGoogle Scholar
[32] Hejranfar, K. and Ezzatneshan, E.. Implementation of a high-order compact finite-difference lattice Boltzmann method in generalized curvilinear coordinates. J. Comput. Phys., 267:28–49, 2014.Google Scholar
[33] He, X., Luo, L.-S., and Dembo, M.. Some progress in lattice Boltzmann method: Part I. Nonuniform mesh grids. J. Comput. Phys., 129(2):357–363, 1996.Google Scholar
[34] He, X., Luo, L.-S., and Dembo, M.. Some progress in lattice Boltzmann method: Enhancement of Reynolds number in simulations. Physica A, 239(1–3):276–285, 1997.Google Scholar