zbMATH — the first resource for mathematics

Numerics of the lattice Boltzmann method on nonuniform grids: standard LBM and finite-difference LBM. (English) Zbl 1390.76710
Summary: The present study is focused on the comparison between the standard “collide-and-stream” lattice Boltzmann method (LBM) and the Lax-Wendroff-based finite-difference LBM (FDLBM) on block-structured nonuniform grids with an adaptive mesh refinement (AMR) strategy. While the standard LBM (SLBM) is found to be slightly faster than the FDLBM, the latter is shown to be more stable at higher Reynolds numbers. Although both approaches are as accurate in simulation of fluid flow problems, the SLBM has a more complicated algorithm and its implementation is more involved; this is mainly because, in applying SLBM, the AMR blocks at different refinement levels do not advance in time simultaneously. On the other hand, the underlying differences between the cell-center and cell-vertex data structures are explained and their advantages and disadvantages are highlighted. In general, the cell-center data structure is favorable because it is more efficient in terms of computational time and memory. The effect of the interpolation schemes on the order of accuracy of the LBM is also investigated. It is reestablished that the popular linear interpolation degrades the order of accuracy of LBM to first order. A variety of benchmark studies, including Taylor-Green decaying vortex, gravity-driven Poiseuille flow, thin shear layer instability, and unsteady flow past a square cylinder, are carried out to assess SLMB and FDLBM with a multiple-relaxation-time collision operator.

76M28 Particle methods and lattice-gas methods
76M20 Finite difference methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
PDF BibTeX Cite
Full Text: DOI
[1] Samet, H., The design and analysis of spatial data structures, (1990), Addison-Wesley Reading, MA
[2] Rantakokko, J.; Thuné, M., Parallel computing, (2009), Springer London, [chapter 5]
[3] Chen, S.; Doolen, G., Lattice Boltzmann method for fluid flows, Annu Rev Fluid Mech, 30, 329, (1998)
[4] Yu, D.; Mei, R.; Luo, L.-S.; Shyy, W., Viscous flow computations with the method of lattice Boltzmann equation, Prog Aerosp Sci, 39, 329, (2003)
[5] He, X.; Luo, L.-S.; Dembo, M., Some progress in lattice Boltzmann method. part i. nonuniform mesh grids, J Comput Phys, 129, 357, (1996) · Zbl 0868.76068
[6] Yu, D.; Mei, R.; Shyy, W., A multi-block lattice Boltzmann method for viscous fluid flows, Int J Numer Methods Fluids, 39, 99, (2002) · Zbl 1036.76051
[7] Kandhai, D.; Soll, W.; Chen, S.; Hoekstra, A.; Sloot, P., Finite-difference lattice-BGK methods on nested grids, Comput Phys Commun, 129, 100, (2000) · Zbl 0976.76067
[8] Sofonea, V.; Sekerka, R. F., Viscosity of finite difference lattice Boltzmann models, J Comput Phys, 184, 422, (2003) · Zbl 1062.76556
[9] Lee, T.; Lin, C.-L., An Eulerian description of the streaming process in the lattice Boltzmann equation, J Comput Phys, 185, 445, (2003) · Zbl 1047.76106
[10] Fakhari, A.; Lee, T., Finite-difference lattice Boltzmann method with a block-structured adaptive-mesh-refinement technique, Phys Rev E, 89, 033310, (2014)
[11] Eitel-Amor, G.; Meinke, M.; Schroder, W., A lattice-Boltzmann method with hierarchically refined meshes, Comput Fluids, 75, 127, (2013) · Zbl 1277.76082
[12] Nannelli, F.; Succi, S., The finite volume lattice Boltzmann equation, J Stat Phys, 68, 401, (1992) · Zbl 0925.82036
[13] Chen, H., Volumetric formulation of the lattice Boltzmann method for fluid dynamics: basic concept, Phys Rev E, 58, 3955, (1998)
[14] Chen, H.; Filippova, O.; Hoch, J.; Molvig, K.; Shock, R.; Teixeira, C., Grid refinement in lattice Boltzmann methods based on volumetric formulation, Phys A, 362, 158, (2006)
[15] Rohde, M.; Kandhai, D.; Derksen, J. J.; van den Akker, H. E.A., A generic, mass conservative local grid refinement technique for lattice-Boltzmann schemes, Int J Numer Methods Fluids, 51, 439, (2006) · Zbl 1276.76060
[16] Lee, T.; Lin, C.-L., A characteristic Galerkin method for discrete Boltzmann equation, J Comput Phys, 171, 336, (2001) · Zbl 1017.76043
[17] Filippova, O.; Hänel, D., Grid refinement for lattice-BGK models, J Comput Phys, 147, 219, (1998) · Zbl 0917.76061
[18] Lin, C.-L.; Lai, Y.-G., Lattice Boltzmann method on composite grids, Phys Rev E, 62, 2219, (2000)
[19] Lagrava, D.; Malaspinas, O.; Latt, J.; Chopard, B., Advances in multi-domain lattice Boltzmann grid refinement, J Comput Phys, 231, 4808, (2012) · Zbl 1246.76131
[20] Crouse, B.; Rank, E.; Krafczyk, M.; Tölke, J., A LB-based approach for adaptive flow simulations, Int J Mod Phys B, 17, 109, (2003)
[21] Tölke, J.; Freudiger, S.; Krafczyk, M., An adaptive scheme using hierarchical grids for lattice Boltzmann multi-phase flow simulations, Comput Fluids, 35, 820, (2006) · Zbl 1177.76332
[22] Wu, J.; Shu, C., A solution-adaptive lattice Boltzmann method for two-dimensional incompressible viscous flows, J Comput Phys, 230, 2246, (2011) · Zbl 1391.76643
[23] Chen, Y.; Kang, Q.; Cai, Q.; Zhang, D., Lattice Boltzmann method on quadtree grids, Phys Rev E, 83, 026707, (2011)
[24] Matsumoto, T., Self-gravitational magnetohydrodynamics with adaptive mesh refinement for protostellar collapse, Publ Astron Soc Jpn, 59, 905, (2007)
[25] He, X.; Luo, L.-S., Lattice Boltzmann model for the incompressible Navier-Stokes equation, J Stat Phys, 88, 927, (1997) · Zbl 0939.82042
[26] Lallemand, P.; Luo, L.-S., Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance, and stability, Phys Rev E, 61, 6546, (2000)
[27] Dupuis, A.; Chopard, B., Theory and application of an alternative lattice Boltzmann grid refinement algorithm, Phys Rev E, 67, 066707, (2003)
[28] Guo, Z.; Zheng, C.; Zhao, T. S., A lattice BGK scheme with general propagation, J Sci Comput, 16, 569, (2001) · Zbl 1039.76054
[29] Guzik, S. M.; Weisgraber, T. H.; Colella, P.; Alder, B. J., Interpolation methods and the accuracy of lattice-Boltzmann mesh refinement, J Comput Phys, 259, 461, (2014) · Zbl 1349.76687
[30] Mei, R.; Luo, L.-S.; Lallemand, P.; d’Humiéres, D., Consistent initial conditions for lattice Boltzmann simulations, Comput Fluids, 35, 855, (2006) · Zbl 1177.76319
[31] Tölke, J.; Krafczyk, M., Second order interpolation of the flow field in the lattice Boltzmann method, Comput Math Appl, 58, 898, (2009) · Zbl 1189.76416
[32] Buick, J. M.; Greated, C. A., Gravity in a lattice Boltzmann model, Phys Rev E, 61, 5307, (2000)
[33] Guo, Z.; Zheng, C.; Shi, B., Discrete lattice effects on the forcing term in the lattice Boltzmann method, Phys Rev E, 65, 046308, (2002) · Zbl 1244.76102
[34] He, X.; Zou, Q.; Luo, L.-S.; Dembo, M., Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model, J Stat Phys, 87, 115, (1997) · Zbl 0937.82043
[35] Fakhari, A.; Lee, T., Multiple-relaxation-time lattice Boltzmann method for immiscible fluids at high Reynolds numbers, Phys Rev E, 87, 023304, (2013)
[36] Michalke, A., On the inviscid instability of the hyperbolic-tangent velocity profile, J Fluid Mech, 19, 543, (1964) · Zbl 0129.20302
[37] Fontane, J.; Joly, L., The stability of the variable-density Kelvin-Helmholtz billow, J Fluid Mech, 612, 237, (2008) · Zbl 1151.76475
[38] Davis, R. W.; Moore, E. F., A numerical study of vortex shedding from rectangles, J Fluid Mech, 116, 475, (1982) · Zbl 0491.76042
[39] Okajima, A., Strouhal numbers of rectangular cylinders, J Fluid Mech, 123, 379, (1982)
[40] Sohankar, A.; Norberg, C.; Davidson, L., Low-Reynolds-number flow around a square cylinder at incidence: study of blockage, onset of vortex shedding and outlet boundary condition, Int J Numer Methods Fluids, 26, 39, (1998) · Zbl 0910.76067
[41] Saha, A. K.; Muralidhar, K.; Biswas, G., Transition and chaos in 2D flow past a square cylinder, J Eng Mech, 126, 523, (2000)
[42] Pavlov, A. N.; Sazhin, S. S.; Fedorenko, R. P.; Heikal, M. R., A conservative finite difference method and its application for the analysis of a transient flow around a square prism, Int J Numer Methods H, 10, 6, (2000) · Zbl 0966.76061
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.