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Coarse- and fine-grid numerical behavior of MRT/TRT lattice-Boltzmann schemes in regular and random sphere packings. (English) Zbl 1351.76238

Summary: We analyze the intrinsic impact of free-tunable combinations of the relaxation rates controlling viscosity-independent accuracy of the multiple-relaxation-times (MRT) lattice-Boltzmann models. Preserving all MRT degrees of freedom, we formulate the parametrization conditions which enable the MRT schemes to provide viscosity-independent truncation errors for steady state solutions, and support them with the second- and third-order accurate (“linear” and “parabolic”, respectively) boundary schemes. The parabolic schemes demonstrate the advanced accuracy with weak dependency on the relaxation rates, as confirmed by the simulations with the D3Q15 model in three regular arrays (SC, BCC, FCC) of touching spheres. Yet, the low-order, bounce-back boundary rule remains appealing for pore-scale simulations where the precise distance to the boundaries is undetermined. However, the effective accuracy of the bounce-back crucially depends on the free-tunable combinations of the relaxation rates. We find that the combinations of the kinematic viscosity rate with the available “ghost” antisymmetric collision mode rates mainly impact the accuracy of the bounce-back scheme. As the first step, we reduce them to the one combination (presented by so-called “magic” parameter \(\Lambda\) in the frame of the two-relaxation-times (TRT) model), and study its impact on the accuracy of the drag force/permeability computations with the D3Q19 velocity set in two different, dense, random packings of 8000 spheres each. We also run the simulations in the regular (BCC and FCC) packings of the same porosity for the broad range of the discretization resolutions, ranging from 5 to 750 lattice nodes per sphere diameter. A special attention is given to the discretization procedure resulting in significantly reduced scatter of the data obtained at low resolutions. The results reveal the identical \(\Lambda\)-dependency versus the discretization resolution in all four packings, regular and random. While very small \(\Lambda\) values overestimate the drag measurements several-fold on the coarse grids, \(\Lambda > 1\) may overestimate the permeability at the same extent. In low resolution region we provide practical guidelines, extending previously known solutions for the straight/diagonal Poiseuille flow. Analysis of the high-resolution region reveals the collapse of the solutions obtained with all the considered \(\Lambda\) values with the average rate of -1.3, followed by their common, smooth, first-order convergence with the rate of -1.0 as the best, towards the reference solutions provided by the “parabolic” schemes. High-quality power-law fits estimate that the bounce-back would reach their accuracy (obtained at about 200 nodes per sphere) for two-order magnitude higher grid resolution.

MSC:

76M28 Particle methods and lattice-gas methods
65N75 Probabilistic methods, particle methods, etc. for boundary value problems involving PDEs
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[1] Adhikary, R.; Suchi, S., Duality in matrix lattice Boltzman models, Phys. Rev. E, 78, 066701 (2008)
[2] Aidun, C. K.; Clausen, J. R., Lattice-Boltzmann method for complex flows, Annu. Rev. Fluid Mech., 42, 439-472 (2010) · Zbl 1345.76087
[3] Ahrenholz, B.; Tölke, J.; Krafczyk, M., Lattice-Boltzmann simulations in reconstructed parametrized porous media, Int. J. Comput. Fluid Dyn., 20, 369-377 (2006) · Zbl 1370.76141
[4] Bijeljic, B.; Mostaghimi, P.; Blunt, M. J., Insights into non-Fickian solute transport in carbonates, Water Resour. Res., 49, 2714-2728 (2013)
[5] Boek, E.; Venturoli, M., Lattice-Boltzmann studies of fluid flow in porous media with realistic rock geometries, Comput. Math. Appl., 59, 2305-2314 (2010) · Zbl 1193.76104
[6] Bouzidi, M.; Firdaouss, M.; Lallemand, P., Momentum transfer of a Boltzmann-lattice fluid with boundaries, Phys. Fluids, 13, 3452-3459 (2001) · Zbl 1184.76068
[7] Cho, H.; Jeong, N.; Sung, H. J., Permeability of microscale fibrous porous media using the lattice Boltzmann method, Int. J. Heat Fluid Flow, 44, 435-443 (2013)
[8] Chun, B.; Ladd, A. J.C., Interpolated boundary conditions for lattice Boltzmann simulations of flows in narrow gaps, Phys. Rev. E, 75, 066705 (2007)
[9] Contrino, D.; Lallemand, P.; Asinari, P.; Luo, Li-Shi, Lattice Boltzmann simulations of the thermally driven square 2D cavity at high Rayleigh numbers, J. Comput. Phys., 275, 257-272 (2014) · Zbl 1349.76671
[10] Dellar, P. J., Incompressible limits of lattice Boltzmann equations using multiple relaxation times, J. Comput. Phys., 191, 351-370 (2003) · Zbl 1076.76063
[11] Dubois, F.; Lallemand, P., Towards higher order lattice Boltzmann schemes, J. Stat. Mech., 2009, P06006 (2009) · Zbl 1459.76097
[12] Dubois, F.; Lallemand, P.; Tekitek, M., On a superconvergent lattice Boltzmann boundary scheme, Comput. Math. Appl., 59, 2141-2149 (2010) · Zbl 1193.76106
[13] Freund, H.; Bauer, J.; Zeiser, T.; Emig, G., Detailed simulation of transport processes in fixed beds, Ind. Eng. Chem. Res., 44, 6423-6434 (2005)
[14] Gallivan, M. A.; Noble, D. R.; Georgiadis, J. G.; Buckius, R. O., An evaluation of the bounce-back boundary condition for lattice Boltzmann simulations, Int. J. Numer. Methods Fluids, 25, 249-263 (1997) · Zbl 0889.76061
[15] Geller, S.; Krafczyk, M.; Tölke, J.; Turek, S.; Hron, J., Benchmark computations based on lattice Boltzmann, finite element and finite volume method for laminar flows, Comput. Fluids, 35, 888-897 (2006) · Zbl 1177.76313
[16] Ginzbourg, I.; Adler, P. M., Boundary flow condition analysis for the three-dimensional lattice Boltzmann model, J. Phys. II Fr., 4, 191-214 (1994)
[17] Ginzbourg, I.; d’Humières, D., Local second-order boundary method for lattice Boltzmann models, J. Stat. Phys., 84, 927-971 (1996) · Zbl 1081.82617
[18] Ginzburg, I.; d’Humières, D., Multi-reflection boundary conditions for lattice Boltzmann models, Phys. Rev. E, 68, 066614 (2003)
[19] Ginzburg, I., Equilibrium-type and Link-type lattice Boltzmann models for generic advection and anisotropic-dispersion equation, Adv. Water Resour., 28, 1171-1195 (2005)
[20] Ginzburg, I., Lattice Boltzmann modeling with discontinuous collision components, hydrodynamic and advection-diffusion equations, J. Stat. Phys., 126, 157-203 (2007) · Zbl 1198.82039
[21] Ginzburg, I., Consistent lattice Boltzmann schemes for the Brinkman model of porous flow and infinite Chapman-Enskog expansion, Phys. Rev. E, 77, 0666704 (2008)
[22] Ginzburg, I.; Verhaeghe, F.; d’Humières, D., Two-relaxation-time lattice Boltzmann scheme: about parametrization, velocity, pressure and mixed boundary conditions, Commun. Comput. Phys., 3, 427-478 (2008)
[23] Ginzburg, I.; Verhaeghe, F.; d’Humières, D., Study of simple hydrodynamic solutions with the two-relaxation-times lattice Boltzmann scheme, Commun. Comput. Phys., 3, 519-581 (2008)
[24] Ginzburg, I.; d’Humières, D.; Kuzmin, A., Optimal stability of advection-diffusion lattice Boltzmann models with two relaxation times for positive/negative equilibrium, J. Stat. Phys., 139, 1090-1143 (2010) · Zbl 1205.82049
[25] Ginzburg, I., Truncation errors, exact and heuristic stability analysis of two-relaxation-times lattice Boltzmann schemes for anisotropic advection-diffusion equation, Commun. Comput. Phys., 11, 1439-1502 (2012) · Zbl 1373.76241
[26] Ginzburg, I., Multiple anisotropic collisions for advection-diffusion lattice Boltzmann schemes, Adv. Water Resour., 51, 381-404 (2013)
[27] Giraud, L., Fluides visco-élastiques par la méthode de Boltzmann sur réseau (1997), PhD, Université Paris VI
[28] Guo, Z.; Zhao, T. S.; Shi, Y., Preconditionned lattice-Boltzmann method for steady flows, Phys. Rev. E, 70, 066706 (2004)
[29] Hammou, H.; Ginzburg, I.; Boulerhcha, M., Two-relaxation-times Lattice Boltzmann schemes for solute transport in unsaturated water flow, with a focus on stability, Adv. Water Resour., 34, 779-793 (2011)
[30] Higuera, F.; Succi, S.; Benzi, R., Lattice gas dynamics with enhanced collisions, Europhys. Lett., 9, 345-349 (1989)
[31] van der Hoef, M. A.; Beetstra, R.; Kuipers, J. A.M., Lattice-Boltzmann simulations of low-Reynolds-number flow past mono- and bidisperse arrays of spheres: results for the permeability and drag force, J. Fluid Mech., 528, 233-254 (2005) · Zbl 1165.76369
[32] Hubert, C.; Schafei, B.; Parmegiani, A., A new pore-scale model for linear and non-linear heterogeneous dissolution and precipitation, Geochim. Cosmochim. Acta, 124, 109-130 (2014)
[33] d’Humières, D., Generalized lattice-Boltzmann equations, Prog. Astronaut. Aeronaut., 59, 450-548 (1992)
[34] d’Humières, D.; Ginzburg, I.; Krafczyk, M.; Lallemand, P.; Luo, L.-S., Multiple-relaxation-time lattice Boltzmann models in three dimensions, Philos. Trans. R. Soc. Lond. A, 360, 437-451 (2002) · Zbl 1001.76081
[35] d’Humières, D.; Ginzburg, I., Viscosity independent numerical errors for lattice Boltzmann models: from recurrence equations to “magic” collision numbers, Comput. Math. Appl., 58, 823-840 (2009) · Zbl 1189.76405
[36] Junk, M.; Yang, Z., Convergence of lattice Boltzmann methods for Stokes flows in periodic and bounded domains, Comput. Math. Appl., 55, 1481-1491 (2008) · Zbl 1142.76451
[37] Jodrey, W. S.; Tory, E. M., Computer simulation of close random packing of equal spheres, Phys. Rev. A, 32, 2347-2351 (1985)
[38] Kandhai, D.; Koponen, A.; Hoekstra, A. G.; Kataja, M.; Timonen, J.; Sloot, P. M.A., Implementation aspects of 3d lattice-bgk: boundaries, accuracy and a new fast relaxation method, J. Comput. Phys., 150, 482-501 (1999) · Zbl 0937.76066
[39] Khirevich, S.; Daneyko, A.; Höltzel, A.; Seidel-Morgenstern, A.; Tallarek, U., Statistical analysis of packed beds, the origin of short-range disorder, its impact on eddy dispersion, J. Chromatogr. A, 1217, 4713-4722 (2010)
[40] Khirevich, S.; Höltzel, A.; Daneyko, A.; Seidel-Morgenstern, A.; Tallarek, U., Structure-transport correlation for the diffusive tortuosity of bulk, monodisperse, random sphere packings, J. Chromatogr. A, 1218, 6489-6497 (2011)
[41] Khirevich, S.; Höltzel, A.; Tallarek, U., Validation of pore-scale simulations of hydrodynamic dispersion in random sphere packings, Commun. Comput. Phys., 13, 801-822 (2013)
[42] Khirevich, S.; Höltzel, A.; Hlushkou, D.; Seidel-Morgenstern, A.; Tallarek, U., Structure-transport analysis for particulate packings in trapezoidal microchip separation channels, Lab Chip, 8, 1801-1808 (2008)
[44] Komrakova, A. E.; Eskin, D.; Derksen, J. J., Lattice Boltzmann simulations of a single n-butanol drop rising in water, Phys. Fluids, 25, 042102 (2013)
[45] Koivu, V.; Decain, M.; Geindreau, C.; Mattila, K.; Bloch, J.-F.; Kataja, M., Transport properties of heterogeneous materials. Combining computerised X-ray micro-tomography and direct numerical simulations, Int. J. Comput. Fluid Dyn., 23, 713-721 (2009) · Zbl 1278.76110
[46] Kuzmin, A.; Ginzburg, I.; Mohamad, A. A., A role of the kinetic parameter on the stability of two-relaxation-times advection-diffusion lattice Boltzmann scheme, Comput. Math. Appl., 61, 3417-3442 (2011) · Zbl 1225.76233
[47] Ladd, A. J.C., Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation, J. Fluid Mech., 271, 285-309 (1994) · Zbl 0815.76085
[48] Ladd, A. J.C., Moderate Reynolds number flows through periodic and random arrays of aligned cylinders, J. Fluid Mech., 349, 31-66 (1997) · Zbl 0912.76014
[49] Ladd, A. J.C.; Verberg, R., Lattice-Boltzmann simulations of particle-fluid suspensions, J. Stat. Phys., 104, 1191-1251 (2001) · Zbl 1046.76037
[50] Lallemand, P.; Luo, L.-S., Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance and stability, Phys. Rev. E, 61, 6546-6562 (2000)
[51] Larson, R. E.; Hidgon, J. J.L., A periodic grain consolidation model of porous media, Phys. Fluids A, 1, 38-46 (1989) · Zbl 0656.76079
[52] Leriche, E.; Lallemand, P.; Labrosse, G., Stokes eigenmodes in cubic domain: primitive variable and lattice Boltzmann formulations, Appl. Numer. Math., 58, 935-945 (2008) · Zbl 1143.65090
[53] Luo, L.-S.; Liao, W.; Chen, X.; Peng, Y.; Zhang, W., Numerics of the lattice Boltzmann method: effects of collision models on the lattice Boltzmann simulations, Phys. Rev. E, 83, 056710 (2011)
[54] Maier, R. S.; Kroll, D. M.; Kutsovsky, Y. E.; Davis, H. T.; Bernard, R. S., Simulation of flow through bead packs using the lattice Boltzmann method, Phys. Fluids, 10, 60-74 (1998)
[55] Maier, R. S.; Kroll, D. M.; Bernard, R. S.; Howington, S. E.; Peters, J. F.; Davis, H. T., Pore-scale simulation of dispersion, Phys. Fluids, 12, 2065-2079 (2000) · Zbl 1184.76340
[56] Maier, R. S.; Schure, M. R.; Gage, J. P.; Seymour, J. D., Sensitivity of pore-scale dispersion to the construction of random bead packs, Water Resour. Res., 44, W06S03 (2008)
[57] Maier, R. S.; Bernard, R. S., Lattice-Boltzmann accuracy in pore-scale flow simulation, J. Comput. Phys., 229, 233-255 (2010) · Zbl 1213.76203
[58] Manwart, C.; Aaltosalmi, U.; Koponen, A.; Hilfer, R.; Timonen, J., Lattice-Boltzmann and finite-difference simulations for the permeability for three-dimensional porous media, Phys. Rev. E, 66, 016702 (2002)
[59] Narváez, A.; Zauner, T.; Raischel, F.; Hilfer, R.; Harting, J., Quantitative analysis of numerical estimates for a the permeability of porous media from lattice-Boltzmann simulations, J. Stat. Mech., P11026 (2010)
[60] Narváez, A.; Yazdchi, K.; Luding, S.; Harting, J., From creeping to inertial flow in porous media: a lattice Boltzmann-finite element study, J. Stat. Mech., P02038 (2013) · Zbl 1456.76127
[61] Onoda, G. Y.; Liniger, E. G., Random loose packings of uniform spheres the dilatancy onset, Phys. Rev. Lett., 64, 2727-2730 (1990)
[62] Pan, C.; Luo, L.-S.; Miller, C. T., An evaluation of lattice Boltzmann schemes for porous media simulation, Comput. Fluids, 35, 898-909 (2006) · Zbl 1177.76323
[63] Qian, Y.; d’Humières, D.; Lallemand, P., Lattice BGK models for Navier-Stokes equation, Europhys. Lett., 17, 479-484 (1992) · Zbl 1116.76419
[64] Reis, T.; Phillips, T. N., Alternative approach to the solution of the dispersion relation for a generalized lattice Boltzmann equation, Phys. Rev. E, 77, 026702 (2008)
[65] Rothman, D. H., Cellular-automaton fluids: a model for flow in porous media, Geophysics, 53, 509-518 (1988)
[66] Sangani, A. S.; Acrivos, A., Slow flow through a periodic array of spheres, Int. J. Multiph. Flow, 8, 343-360 (1982) · Zbl 0541.76041
[67] Sengupta, A.; Hammond, P. S.; Frenkel, D.; Boek, E. S., Error analysis and correction for lattice Boltzmann simulated flow conductance in capillaries of different shapes and alignments, J. Comput. Phys., 231, 2634-2640 (2012) · Zbl 1426.76617
[68] Silbert, L., Jamming of frictional spheres random loose packing, Soft Matter, 6, 2918-2924 (2010)
[69] van der Smann, R. G.M., Galilean invariant lattice Boltzmann scheme for natural convection on square and rectangular lattices, Phys. Rev. E, 74, 026705 (2006)
[70] Song, C.; Wang, P.; Makse, H. A., A phase diagram for jammed matter, Nature, 453, 629-632 (2008)
[71] Stewart, M. L.; Ward, A. L.; Rector, D. R., A study of pore geometry effects on anisotropy in hydraulic permeability using the lattice-Boltzmann method, Adv. Water Resour., 29, 1328-1340 (2006)
[72] Talon, L.; Bauer, D.; Gland, N.; Youssef, S.; Auradou, H.; Ginzburg, I., Assessment of the two relaxation time lattice-Boltzmann scheme to simulate Stokes flow in porous media, Water Resour. Res., 48, W04526 (2012)
[73] Talon, L.; Bauer, D., On the determination of a generalized Darcy equation for yield stress fluid in porous media using a lattice Boltzmann TRT scheme, Eur. Phys. J. E, 36, 139-149 (2013)
[74] Verberg, R.; Ladd, A. J.C., Simulation of low-Reynolds-number flow via a time-independent lattice-Boltzmann method, Phys. Rev. E, 60, 3366-3373 (1999)
[75] Vikhansky, A.; Ginzburg, I., Taylor dispersion in heterogeneous porous media: extended method of moments, theory, and modelling with two-relaxation-times lattice Boltzmann scheme, Phys. Fluids, 26, 022104 (2014)
[76] Wang, J.; Wang, D. H.; Lallemand, P.; Luo, L.-S., Lattice Boltzmann simulations of thermal convective flows in two dimensions, Comput. Math. Appl., 65, 262-286 (2013) · Zbl 1268.76050
[77] Zick, A. A.; Homsy, G. M., Stokes flow through periodic arrays of spheres, J. Fluid Mech., 115, 13-26 (1982) · Zbl 0515.76039
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