×

zbMATH — the first resource for mathematics

Development of a three dimensional meshless numerical method for the solution of the Boltzmann equation on complex geometries. (English) Zbl 1410.76371
Summary: We propose a three dimensional meshless numerical scheme based on the Boltzmann equation for the simulation of fluid flows in domains of complex geometries. The velocity space discretization in our method is the same as the standard lattice Boltzmann method, i.e., the same lattices are used to discretize microscopic velocities. The velocity-discrete Boltzmann equation with the BGK collision approximation is discretized in time using the Lax-Wendroff scheme, and subsequently in space using the meshless local Petrov-Galerkin method based on local radial basis functions augmented with polynomials. The method is first validated and tuned by solving two benchmark problems: the pressure-driven flow in an elliptical tube, and the three-dimensional lid-driven cavity flow. The error analysis for the first case shows the second order accuracy of the proposed method. The flow in a packed bed is then solved in order to illustrate the efficiency of the proposed method in the simulation of geometrically complex flows. The results show that both the computational time and the memory usage of our method are lower than those of the standard lattice Boltzmann method for the geometrically complex flows.

MSC:
76M28 Particle methods and lattice-gas methods
65M75 Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs
Software:
Mfree2D
PDF BibTeX Cite
Full Text: DOI
References:
[1] Natsui, S.; Sawada, A.; Terui, K.; Kashihara, Y.; Kikuchi, T.; Suzuki, R. O., Dem-sph study of molten slag trickle flow in coke bed, Chem Eng Sci, 175, 25-39, (2018)
[2] Das, S.; Deen, N.; Kuipers, J., Multiscale modeling of fixed-bed reactors with porous (open-cell foam) non-spherical particles: hydrodynamics, Chem Eng J, 334, 741-759, (2018)
[3] Jourak, A.; Hellström, J. G.I.; Lundström, T. S.; Frishfelds, V., Numerical derivation of dispersion coefficients for flow through three-dimensional randomly packed beds of monodisperse spheres, AlChE J, 60, 2, 749-761, (2014)
[4] Dixon, A. G.; Taskin, M. E.; Stitt, E. H.; Nijemeisland, M., 3D cfd simulations of steam reforming with resolved intraparticle reaction and gradients, Chem Eng Sci, 62, 18, 4963-4966, (2007)
[5] Gunjal, P. R.; Ranade, V. V.; Chaudhari, R. V., Computational study of a single-phase flow in packed beds of spheres, AlChE J, 51, 2, 365-378, (2005)
[6] Guardo, A.; Coussirat, M.; Recasens, F.; Larrayoz, M.; Escaler, X., Cfd study on particle-to-fluid heat transfer in fixed bed reactors: convective heat transfer at low and high pressure, Chem Eng Sci, 61, 13, 4341-4353, (2006)
[7] Kuroki, M.; Ookawara, S.; Ogawa, K., A high-fidelity cfd model of methane steam reforming in a packed bed reactor, J Chem Eng Jpn, 42, Supplement, S73-S78, (2009)
[8] Magnico, P., Hydrodynamic and transport properties of packed beds in small tube-to-sphere diameter ratio: pore scale simulation using an Eulerian and a Lagrangian approach, Chem Eng Sci, 58, 22, 5005-5024, (2003)
[9] Maier, R. S.; Kroll, D.; Kutsovsky, Y.; Davis, H.; Bernard, R. S., Simulation of flow through bead packs using the lattice Boltzmann method, Phys Fluids, 10, 1, 60-74, (1998)
[10] Sullivan, S.; Sani, F.; Johns, M.; Gladden, L., Simulation of packed bed reactors using lattice Boltzmann methods, Chem Eng Sci, 60, 12, 3405-3418, (2005)
[11] Van der Hoef, M.; Beetstra, R.; Kuipers, J., 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)
[12] Pan, C.; Luo, L.-S.; Miller, C. T., An evaluation of lattice Boltzmann schemes for porous medium flow simulation, ComputFluids, 35, 8, 898-909, (2006)
[13] Maier, R.; Bernard, R., Lattice-Boltzmann accuracy in pore-scale flow simulation, J Comput Phys, 229, 2, 233-255, (2010)
[14] Aidun, C. K.; Clausen, J. R., Lattice-Boltzmann method for complex flows, Annu Rev Fluid Mech, 42, 439-472, (2010)
[15] Chen, S.; Doolen, G. D., Lattice Boltzmann method for fluid flows, Annu Rev Fluid Mech, 30, 1, 329-364, (1998)
[16] Succi, S., The lattice Boltzmann equation for fluid dynamics and beyond, (2001), Oxford university press
[17] Wolf-Gladrow, D. A., Lattice-gas cellular automata and lattice Boltzmann models: an introduction, 1725, (2000), Springer Verlag
[18] Chun, B.; Ladd, A., Interpolated boundary condition for lattice Boltzmann simulations of flows in narrow gaps, PhysRev E, 75, 6, 066705, (2007)
[19] McNamara, G. R.; Zanetti, G., Use of the Boltzmann equation to simulate lattice-gas automata, Phys Rev Lett, 61, 20, 2332-2335, (1988)
[20] McNamara, G. R.; Garcia, A. L.; Alder, B. J., Stabilization of thermal lattice Boltzmann models, J Stat Phys, 81, 1-2, 395-408, (1995)
[21] Sterling, J. D.; Chen, S., Stability analysis of lattice Boltzmann methods, J Comput Phys, 123, 1, 196-206, (1996)
[22] He, X.; Luo, L.-S., Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation, Phys Rev E, 56, 6, 6811, (1997)
[23] Bardow, A.; Karlin, I.; Gusev, A., General characteristic-based algorithm for off-lattice Boltzmann simulations, EPL (Europhysics Letters), 75, 3, 434, (2006)
[24] Cao, N.; Chen, S.; Jin, S.; Martinez, D., Physical symmetry and lattice symmetry in the lattice Boltzmann method, Phys Rev E, 55, 1, R21-R24, (1997)
[25] Mei, R.; Shyy, W., On the finite difference-based lattice Boltzmann method in curvilinear coordinates, J Comput Phys, 143, 2, 426-448, (1998)
[26] Xi, H.; Peng, G.; Chou, S.-H., Finite-volume lattice Boltzmann method, Phys Rev E, 59, 5, 6202, (1999)
[27] Ubertini, S.; Bella, G.; Succi, S., Lattice Boltzmann method on unstructured grids: further developments, Phys Rev E, 68, 1, 016701, (2003)
[28] Stiebler, M.; Tölke, J.; Krafczyk, M., An upwind discretization scheme for the finite volume lattice Boltzmann method, Comput Fluids, 35, 8, 814-819, (2006)
[29] Patil, D. V.; Lakshmisha, K., Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh, J Comput Phys, 228, 14, 5262-5279, (2009)
[30] Lee, T.; Lin, C.-L., An Eulerian description of the streaming process in the lattice Boltzmann equation, J Comput Phys, 185, 2, 445-471, (2003)
[31] Li, Y.; LeBoeuf, E. J.; Basu, P., Least-squares finite-element scheme for the lattice Boltzmann method on an unstructured mesh, Phys Rev E, 72, 4, 046711, (2005)
[32] Shi, X.; Lin, J.; Yu, Z., Discontinuous Galerkin spectral element lattice Boltzmann method on triangular element, Int J Numer Methods Fluids, 42, 11, 1249-1261, (2003)
[33] Düster, A.; Demkowicz, L.; Rank, E., High-order finite elements applied to the discrete Boltzmann equation, Int J Numer Methods Eng, 67, 8, 1094-1121, (2006)
[34] Min, M.; Lee, T., A spectral-element discontinuous Galerkin lattice Boltzmann method for nearly incompressible flows, J Comput Phys, 230, 1, 245-259, (2011)
[35] Zadehgol, A.; Ashrafizaadeh, M.; Musavi, S. H., A nodal discontinuous Galerkin lattice Boltzmannmethod for fluid flow problems, Comput Fluids, 105, 58-65, (2014)
[36] Musavi, S. H.; Ashrafizaadeh, M., Meshless lattice Boltzmann method for the simulation of fluid flows, Phys Rev E, 91, 2, 023310, (2015)
[37] Bhatnagar, P. L.; Gross, E. P.; Krook, M., A model for collision processes in gases. i. small amplitude processes in charged and neutral one-component systems, Phys Rev, 94, 3, 511, (1954)
[38] Lax, P.; Wendroff, B., Systems of conservation laws, Selected Papers Volume I, 263-283, (2005)
[39] Atluri, S. N., The meshless method (MLPG) for domain & BIE discretizations, 677, (2004), Tech Science Press Forsyth
[40] Liu, G.-R., Mesh free methods: moving beyond the finite element method, 712, (2009), CRC Press Boca Raton
[41] Liu, G.-R.; Gu, Y.-T., An introduction to meshfree methods and their programming, (2005), Springer
[42] He, X.; Chen, S.; Doolen, G. D., A novel thermal model for the lattice Boltzmann method in incompressible limit, J Comput Phys, 146, 1, 282-300, (1998)
[43] Reddy, J. N., An introduction to the finite element method, 2, (2006), McGraw-Hill New York
[44] Panton, R. L., Incompressible flow, (2013), John Wiley & Sons
[45] Shankar, P.; Deshpande, M., Fluid mechanics in the driven cavity, Annu Rev Fluid Mech, 32, 1, 93-136, (2000)
[46] Ku, H. C.; Hirsh, R. S.; Taylor, T. D., A pseudospectral method for solution of the three-dimensional incompressible Navier-Stokes equations, J Comput Phys, 70, 2, 439-462, (1987)
[47] Chen, S.; Martinez, D.; Mei, R., On boundary conditions in lattice Boltzmann methods, Phys Fluids, 8, 9, 2527-2536, (1996)
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.