# zbMATH — the first resource for mathematics

On the finite difference-based lattice Boltzmann method in curvilinear coordinates. (English) Zbl 0934.76074
The lattice Boltzmann method can be defined as a microscopic-based approach for solving the fluid flow problems at the macroscopic scales. The presently popular methods use regularly spaced lattices and cannot handle curved boundaries with desirable flexibility. To circumvent such difficulties, we present a finite difference-based lattice Boltzmann method (FDLBM) in curvilinear body-fitted coordinates on non-uniform grids. Several test cases, including the impulsively started cylindrical Couette flow, steady-state cylindrical Couette flow, steady flow over flat plates, and steady flow over a circular cylinder, are used to examine various issues related to the FDLBM. We also investigate the effect of boundary conditions for the distribution functions on the solution, the merits of second-order central difference and upwind schemes for advection terms, and the effect of the Reynolds number. $$\copyright$$ Academic Press.

##### MSC:
 76M28 Particle methods and lattice-gas methods 76M20 Finite difference methods applied to problems in fluid mechanics 76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
Full Text:
##### References:
 [1] Abe, T., Derivation of the lattice Boltzmann method by means of the discrete ordinate method for the Boltzmann equation, J. comput. phys., 131, 241, (1997) · Zbl 0877.76062 [2] 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 system, Phys. rev., 94, 511, (1954) · Zbl 0055.23609 [3] Cao, N.Z.; Chen, S.; Jin, S.; Martinez, D., Physical symmetry and lattice symmetry in lattice Boltzmann method, Phys. rev. E, 55, R21, (1997) [4] Chen, S.; Doolen, G.D., Lattice Boltzmann method for fluid flows, Ann. rev. fluid mech., 30, 329, (1998) · Zbl 1398.76180 [5] Chen, S.; Martinez, D.; Mei, R., On boundary conditions in lattice Boltzmann method, Phys. fluids, 8, 2527, (1996) · Zbl 1027.76630 [6] Chen, H.; Chen, S.; Matthaeus, W.H., Recovery of the navier – stokes equations using a lattice-gas Boltzmann method, Phys. rev. A, 45, R5339, (1992) [7] Fornberg, B., A numerical study of steady viscous flow past a circular cylinder, J. fluid mech., 98, 819, (1980) · Zbl 0428.76032 [8] He, X.; Luo, L.S., Apriori, Phys. rev. E, 55, R6333, (1997) [9] 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 [10] He, X.; Doolen, G., Lattice Boltzmann method on curvilinear coordinates system: flow around a circular cylinder, J. comput. phys., 134, 306, (1997) · Zbl 0886.76072 [11] Ladd, A.J.C., Numerical simulation of particular suspensions via a discretized Boltzmann equation. part 1. theoretical foundation, J. fluid mech., 271, 285, (1994) [12] Ladd, A.J.C., Numerical simulation of particular suspensions via a discretized Boltzmann equation. part 2. numerical results, J. fluid mech., 271, 311, (1994) [13] L. S. Luo, private communication, 1998 [14] 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, (1998) [15] Mei, R.; Plotkin, A., A finite difference scheme for the solution of the steady navier – stokes equation, Comput. & fluids, 14, 239, (1986) · Zbl 0599.76036 [16] Noble, D.R.; Chen, S.; Georgiadis, J.G.; Buckius, R.O., A consistent hydrodynamic boundary condition for the lattice Boltzmann method, Phys. fluid, 7, 203, (1995) · Zbl 0846.76086 [17] Schlichting, H., Boundary layer theory, (1979) [18] W. Shyy, Computational Modeling for Fluid Flow and Interfacial Transport, Elsevier, Amsterdam, 1994, 1997 [19] Shyy, W.; Thakar, S.T.; Ouyang, H.; Liu, J.; Blosch, E., Computational techniques for complex transport phenomena, (1997) [20] Succi, S., Lattice Boltzmann equation: failure or success?, Phys. A, 240, 221, (1997)
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.