×

zbMATH — the first resource for mathematics

An evaluation of lattice Boltzmann schemes for porous medium flow simulation. (English) Zbl 1177.76323
Summary: We quantitatively evaluate the capability and accuracy of the lattice Boltzmann equation (LBE) for modeling flow through porous media. In particular, we conduct a comparative study of the LBE models with the multiple-relaxation-time (MRT) and the Bhatnagar-Gross-Krook (BGK) single-relaxation-time (SRT) collision operators. We also investigate several fluid-solid boundary conditions including: (1) the standard bounce-back (SBB) scheme, (2) the linearly interpolated bounce-back (LIBB) scheme, (3) the quadratically interpolated bounce-back (QIBB) scheme, and (4) the multi-reflection (MR) scheme. Three-dimensional flow through two porous media-a body-centered cubic (BCC) array of spheres and a random-sized sphere-pack-are examined in this study. For flow past a BCC array of spheres, we validate the linear LBE model by comparing its results with the nonlinear LBE model. We investigate systematically the viscosity-dependence of the computed permeability, the discretization error, and effects due to the choice of relaxation parameters with the MRT and BGK schemes. Our results show unequivocally that the MRT-LBE model is superior to the BGK-LBE model, and interpolation significantly improves the accuracy of the fluid-solid boundary conditions.

MSC:
76M28 Particle methods and lattice-gas methods
76S05 Flows in porous media; filtration; seepage
PDF BibTeX Cite
Full Text: DOI
References:
[1] McNamara, G.R.; Zanetti, G., Use of the Boltzmann equation to simulate lattice-gas automata, Phys rev lett, 61, 20, 2332-2335, (1988)
[2] Higuera, F.J.; Jiménez, J., Boltzmann approach to lattice gas simulations, Europhys lett, 9, 663-668, (1989)
[3] Frisch, U.; Hasslacher, B.; Pomeau, Y., Lattice-gas automata for the navier – stokes equation, Phys rev lett, 56, 1505-1507, (1986)
[4] Frisch, U.; d’Humières, D.; Hasslacher, B.; Lallemand, P.; Pomeau, Y.; Rivet, J.-P., Lattice gas hydrodynamics in two and three dimensions, Complex syst, 1, 649-707, (1987) · Zbl 0662.76101
[5] He, X.; Luo, L.-S., A priori derivation of the lattice Boltzmann equation, Phys rev E, 55, R6333-R6336, (1997)
[6] He, X.; Luo, L.-S., Theory of lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation, Phys rev E, 56, 6811-6817, (1997)
[7] Junk, M.; Klar, A., Discretizations for the incompressible navier – stokes equations based on the lattice Boltzmann method, SIAM J sci comput, 22, 1-19, (2000) · Zbl 0972.76083
[8] Junk, M., A finite difference interpretation of the lattice Boltzmann method, Numer methods part diff equat, 17, 383-402, (2001) · Zbl 0987.76082
[9] Junk, M.; Yong, W.-A., Rigorous navier – stokes limit of the lattice Boltzmann equation, Asymptotic anal, 35, 165-185, (2003) · Zbl 1043.76003
[10] Junk, M.; Klar, A.; Luo, L.-S., Asymptotic analysis of the lattice Boltzmann equation, J computat phys, 210, 2, 676-704, (2005) · Zbl 1079.82013
[11] Chen, S.; Doolen, G.D., Lattice Boltzmann method for fluid flows, Annu rev fluid mech, 30, 329-364, (1998) · Zbl 1398.76180
[12] Yu, D.; Mei, R.; Luo, L.-S.; Shyy, W., Viscous flow computations with the method of lattice Boltzmann equation, Prog aerosp sci, 39, 329-367, (2003)
[13] Chen, H.; Chen, S.; Matthaeus, W.H., Recovery of the navier – stokes equations using a lattice-gas Boltzmann methods, Phys rev A, 45, 5339-5342, (1992)
[14] Qian, Y.H.; d’Humières, D.; Lallemand, P., Lattice BGK models for navier – stokes equation, Europhys lett, 17, 6, 479-484, (1992) · Zbl 1116.76419
[15] He, X.; Zou, Q.; Luo, L.-S.; Dembo, M., Analytic solutions and analysis on non-slip boundary condition for the lattice Boltzmann BGK model, J stat phys, 87, 115-136, (1997)
[16] d’Humières D. Generalized lattice-Boltzmann equations. In: Shizgal BD, Weave DP, editors. Rarefied gas dynamics: theory and simulations, volume 159 of Prog Astronaut Aeronaut, Washington (DC), 1992. AIAA, p. 450-8.
[17] Ginzbourg, I.; Adler, P.M., Boundary flow condition analysis for the three-dimensional lattice Boltzmann model, J phys II, 4, 191-214, (1994)
[18] Ginzburg, I.; d’Humières, D., Local second-order boundary methods for lattice Boltzmann models, J stat phys, 84, 5/6, 927-971, (1996) · Zbl 1081.82617
[19] Lallemand, P.; Luo, L.-S., Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance, and stability, Phys rev E, 61, 6, 6546-6562, (2000)
[20] d’Humières, D.; Ginzburg, I.; Krafczyk, M.; Lallemand, P.; Luo, L.-S., Multiple-relaxation-time lattice Boltzmann models in three dimensions, Philos tran roy soc lond A, 360, 437-451, (2002) · Zbl 1001.76081
[21] Lallemand, P.; Luo, L.-S., Theory of the lattice Boltzmann method: acoustic and thermal properties in two and three dimensions, Phys rev E, 68, 036706, (2003)
[22] Ginzburg, I.; d’Humières, D., Multireflection boundary conditions for lattice Boltzmann models, Phys rev E, 68, 066614, (2003)
[23] Bouzidi, M.; Firdaouss, M.; Lallemand, P., Momentum transfer of a Boltzmann-lattice fluid with boundaries, Phys fluids, 13, 11, 3452-3459, (2001) · Zbl 1184.76068
[24] Yu H, Luo L-S, Girimaji SS. LES of turbulent square jet flow using an MRT lattice Boltzmann model. Comput Fluids, this issue, doi:10.1016/j.compfluid.2005.04.009.
[25] Firdaouss, M.; Guermond, J.P.; Le Quéré, P., Nonlinear correction to darcy’s law at low Reynolds numbers, J fluid mech, 343, 331-350, (1997) · Zbl 0897.76091
[26] Lallemand, P.; Luo, L.-S., Lattice Boltzmann method for moving boundaries, J computat phys, 184, 406-421, (2003) · Zbl 1062.76555
[27] Ginzburg, I., Generic boundary conditions for lattice Boltzmann models and their application to advection and anisotropic-dispersion equations, Adv water res, 28, 1196-1216, (2005)
[28] Hasimoto, H., On the periodic fundamental solutions of the Stokes equation and their application to viscous flow past a cubic array of spheres, J fluid mech, 5, 317-328, (1959) · Zbl 0086.19901
[29] Sangani, A.S.; Acrivos, A., Slow flow through a periodic array of spheres, Int J multiphase flow, 8, 4, 343-360, (1982) · Zbl 0541.76041
[30] Pan, C.; Hilpert, M.; Miller, C.T., Pore-scale modeling of saturated permeabilities in random sphere packings, Phys rev E, 64, 6, 066702, (2001)
[31] Pan C, Luo L-S, Miller CT. An evaluation of lattice Boltzmann equation methods for simulating flow through porous media. In: Proceedings of the XVth International Conference on Computational Methods in Water Resources, Chapel Hill, 2004. Elsevier, p. 95-106.
[32] Ginzburg I, Carlier J-P, Kao C. Lattice Boltzmann approach to Richards’ equation. In: Proceedings of the XVth International Conference on Computational Methods in Water Resources, Chapel Hill, 2004. Elsevier, p. 15-23.
[33] Ginzburg I. Variably saturated flow described with the anisotropic lattice Boltzmann methods. Comput Fluids, this issue, doi:10.1016/j.compfluid.2005.11.001. · Zbl 1177.76314
[34] Pan, C.; Prins, J.F.; Miller, C.T., A high-performance lattice Boltzmann implementation to model flow in porous media, Comput phys commun, 158, 2, 89-105, (2004) · Zbl 1196.76069
[35] Hilpert, M.; McBride, J.F.; Miller, C.T., Investigation of the residual-funicular nonwetting-phase-saturation relation, Adv water res, 24, 2, 157-177, (2001)
[36] Hilpert, M.; Miller, C.T., Pore-morphology-based simulation of drainage in totally wetting porous media, Adv water res, 24, 3/4, 243-255, (2001)
[37] Yang, A.; Miller, C.T.; Turcoliver, L.D., Simulation of correlated and uncorrelated packing of random size spheres, Phys rev E, 53, 2, 1516-1524, (1996)
[38] Pan, C.; Hilpert, M.; Miller, C.T., Lattice-Boltzmann simulation of two-phase flow in porous media, Water resour res, 40, 1, W01501:1-W01501:14, (2004)
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.