Abstract
The usual lattice-Boltzmann schemes for fluid flow simulations operate with square and cubic lattices. Instead of relying on square lattices it is possible to use rectangular and orthorombic lattices as well. Schemes using rectangular lattices can be constructed in several ways. Here we construct a rectangular scheme, with the BGK collision operator, by introducing 2 additional discrete velocities into the standard D2Q9 stencil and show how the same procedure can be applied in three dimensions by extending the D3Q19 stencil. The weights and scaling factors for the new stencils are found as the solutions of the well-known Hermite quadrature problem, assuring isotropy of the lattice tensors up to rank four (Philippi et al., Phys. Rev. E 73(5):056702, 2006) This isotropy is a necessary and sufficient condition for assuring the same second order accuracy of lattice-Boltzmann equation with respect to the Navier–Stokes hydrodynamic equations that is found with the standard D2Q9 and D3Q19 stencils. The numerical validation is done, in the two-dimensional case, by using the new rectangular scheme with D2R11 stencil for simulating the Taylor–Green vortex decay. The D3R23 stencil is numerically validated with three-dimensional simulations of cylindrical sound waves propagating from a point source.
Similar content being viewed by others
References
Benzi, R., Succi, S., Vergassola, M.: The lattice Boltzmann equation: theory and applications. Phys. Rep. 222(3), 145 (1992)
Aidun, C., Clausen, J.: Lattice-Boltzmann method for complex flows. Annu. Rev. Fluid Mech. 42, 439 (2010)
Inamuro, T., Sturtevant, B.: Numerical study of discrete-velocity gases. Phys. Fluids A 2(12), 2196 (1990)
McNamara, G., Garcia, A., Alder, B.: Stabilization of thermal lattice Boltzmann models. J. Stat. Phys. 81(1–2), 395 (1995)
Reider, M., Sterling, J.: Accuracy of discrete-velocity BGK models for the simulation of the incompressible Navier-Stokes equations. Comput. Fluids 24(4), 459 (1995)
Cao, N., Chen, S., Jin, S., Martínez, D.: Physical symmetry and lattice symmetry in the lattice Boltzmann method. Phys. Rev. E 55(1), R21 (1997)
Nannelli, F., Succi, S.: The lattice Boltzmann equation on irregular lattices. J. Stat. Phys. 68(3–4), 401 (1992)
Amati, G., Succi, S., Benzi, R.: Turbulent channel flow simulations using a coarse-grained extension of the lattice Boltzmann method. Fluid Dyn. Res. 19(5), 289 (1997)
Chen, H.: Volumetric formulation of the lattice Boltzmann method for fluid dynamics: basic concept. Phys. Rev. E 58(3), 3955 (1998)
Lee, T., Lin, C.-L.: A characteristic Galerkin method for discrete boltzmann equation. J. Comput. Phys. 171(1), 336 (2001)
Shi, X., Lin, J., Yu, Z.: Discontinuous Galerkin spectral element lattice Boltzmann method on triangular element. Int. J. Numer. Meth. Fluids 42(11), 1249 (2003)
Li, Y., LeBoeuf, E., Basu, P.: Least-squares finite-element lattice Boltzmann method. Phys. Rev. E 69(6), 065701(R) (2004)
Abe, T.: Derivation of the Lattice Boltzmann method by means of the discrete ordinate method for the Boltzmann equation. J. Comput. Phys. 131(1), 241 (1997)
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)
Shan, X., He, X.: Discretization of the velocity space in the solution of the Boltzmann equation. Phys. Rev. Lett. 80(1), 65 (1998)
Philippi, P., Hegele Jr, L.: From the continuous to the lattice Boltzmann equation: the discretization problem and thermal models. Phys. Rev. E 73(5), 056702 (2006)
Shan, X., Yuan, X.-F., Chen, H.: Kinetic theory representation of hydrodynamics: a way beyond the Navier-Stokes equation. J. Fluid Mech. 550, 413 (2006)
Shan, X.: General solution of lattices for Cartesian lattice Bhatanagar-Gross-Krook models. Phys. Rev. E 81(3), 036702 (2010)
Zhou, J.: Rectangular lattice Boltzmann method. Phys. Rev. E 81(2), 026705 (2010)
Bhatnagar, P., Gross, E., 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)
Bouzidi, M., d’Humières, D., Lallemand, P., Luo, L.-S.: Lattice Boltzmann equation on a two-dimensional rectangular grid. J. Comput. Phys. 172(2), 704 (2001)
Chikatamarla, S., Karlin, I.: Comment on “Rectangular lattice Boltzmann method”. Phys. Rev. E 83(4), 048701 (2011)
Koelman, J.: A simple lattice Boltzmann scheme for Navier-Stokes fluid flow. Europhys. Lett. 15(6), 603 (1991)
van der Sman, R.: Galilean invariant lattice Boltzmann scheme for natural convection on square and rectangular lattices. Phys. Rev. E 74(2), 026705 (2006)
He, X., Luo, L.-S., Dembo, M.: Some progress in lattice Boltzmann method. Part I. nonuniform mesh grids. J. Comput. Phys. 129(2), 357 (1996)
Higuera, F., Succi, S., Benzi, R.: Lattice gas dynamics with enhanced collisions. Europhys. Lett. 9(4), 345 (1989)
d’Humières, D.: Generalized lattice-Boltzmann equations. In Shizgal, B., Weave, D. (eds.), Rarefied Gas Dynamics: Theory and Simulations Prog. Astronaut. Aeronaut, vol. 159, p. 450. AIAA Press, Washington (1992)
Ginzburg, I.: Equilibrium-type and link-type lattice Boltzmann models for generic advection and anisotropic-dispersion equation. Adv. Water Resour. 28(11), 1171 (2005)
Philippi, P., Hegele Jr, L., Surmas, R., Siebert, D., Dos Santos, L.: From the Boltzmann to the lattice-Boltzmann equation: beyond BGK collision models. Internat. J. Modern Phys. C 18(4), 556 (2007)
Shan, X., Chen, H.: A general multiple-relaxation-time Boltzmann collision model. Internat. J. Modern Phys. C 18(4), 635 (2007)
d’Humières, D., Bouzidi, M., Lallemand, P.: Thirteen-velocity three-dimensional lattice Boltzmann model. Phys. Rev. E 63(6), 066702 (2001)
Alim, U., Entezari, A., Möller, T.: The Lattice-Boltzmann method on optimal sampling lattices. IEEE Trans. Vis. Comput. Graphics 15(4), 630 (2009)
Qian, Y., d’Humières, D., Lallemand, P.: Lattice BGK models for Navier-Stokes equation. Europhys. Lett. 17(6), 479 (1992)
Petkov, K., Qiu, F., Fan, Z., Kaufman, A., Mueller, K.: Efficient LBM visual simulation on face-centered cubic lattices. IEEE Trans. Vis. Comput. Graphics 15(5), 802 (2009)
Krüger, T., Varnik, F., Raabe, D., Second-order convergence of the deviatoric stress tensor in the standard Bhatnagar-Gross-Krook lattice Boltzmann method. Phys. Rev. E 82(2), 025701(R) (2010)
Ladd, A., Verberg, R.: Lattice-Boltzmann simulations of particle-fluid suspensions. J. Stat. Phys. 104(5–6), 1191 (2001)
Latt, J., Chopard, B.: Lattice Boltzmann method with regularized pre-collision distribution functions. Math. Comput. Simul. 72(2–6), 165 (2006)
Kinsler, L., Frey, A., Coppens, A., Sanders, J.: Fundamentals of Acoustics, 4th edn. Wiley, New York (2000)
Viggen, E.: The lattice Boltzmann method in acoustics. In Proceedings of 33rd Scandinavian Sympousium on Physical Acoustics, Geilo, Norway (2010)
Viggen, E.: Viscously damped acoustic waves with the lattice Boltzmann method. Phil. Trans. R. Soc. A 369(1944), 2246 (2011)
Acknowledgments
Petrobras funding is gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hegele Jr, L.A., Mattila, K. & Philippi, P.C. Rectangular Lattice-Boltzmann Schemes with BGK-Collision Operator. J Sci Comput 56, 230–242 (2013). https://doi.org/10.1007/s10915-012-9672-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-012-9672-x