zbMATH — the first resource for mathematics

Benchmark computations based on lattice-Boltzmann, finite element and finite volume methods for laminar flows. (English) Zbl 1177.76313
Summary: The goal of this article is to contribute to the discussion of the efficiency of lattice-Boltzmann (LB) methods as CFD solvers. After a short review of the basic model and extensions, we compare the accuracy and computational efficiency of two research simulation codes based on the LB and the finite-element method (FEM) for two-dimensional incompressible laminar flow problems with complex geometries. We also study the influence of the Mach number on the solution, since LB methods are weakly compressible by nature, by comparing compressible and incompressible results obtained from the LB code and the commercial code CFX. Our results indicate, that for the quantities studied (lift, drag, pressure drop) our LB prototype is competitive for incompressible transient problems, but asymptotically slower for steady-state Stokes flow because the asymptotic algorithmic complexity of the classical LB-method is not optimal compared to the multigrid solvers incorporated in the FEM and CFX code. For the weakly compressible case, the LB approach has a significant wall clock time advantage as compared to CFX. In addition, we demonstrate that the influence of the finite Mach number in LB simulations of incompressible flow is easily underestimated.

76M28 Particle methods and lattice-gas methods
76M10 Finite element methods applied to problems in fluid mechanics
76M12 Finite volume methods applied to problems in fluid mechanics
PDF BibTeX Cite
Full Text: DOI
[1] Wolf-Gladrow, DA, Lattice-gas cellular automata and lattice-Boltzmann models, Lecture notes in mathematics, (2000), Springer Berlin · Zbl 0999.82054
[2] Succi, S., The lattice Boltzmann equation, (2001), Oxford Science Publications Oxford
[3] Hänel, D., Molekulare gasdynamik [in german], (2004), Springer-Verlag
[4] Bhatnagar, P.L.; Gross, E.P.; Krook, M., A model for collision processes in gases, Phys rev, 94, 511, (1954) · Zbl 0055.23609
[5] Qian, Y.H.; d’Humiéres, D.; Lallemand, P., Lattice BGK models for navier – stokes equation, Europhys lett, 17, 479-484, (1992) · Zbl 1116.76419
[6] He, X.; Luo, L.-S., Theory of lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation, Phys rev E, 56, 6, 6811-6817, (1997)
[7] d’Humières, D., Generalized lattice Boltzmann equations, (), 450-458
[8] 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)
[9] d’Humières, D.; Ginzburg, I.; Krafczyk, M.; Lallemand, P.; Luo, L.-S., Multiple-relaxation-time lattice Boltzmann models in three-dimensions, Philo trans R soc lond A, 360, 1792, 437-451, (2002) · Zbl 1001.76081
[10] Ginzburg, I.; d’Humières, D., Multi-reflection boundary conditions for lattice-Boltzmann models, Phys rev E, 68, 066614, (2003)
[11] Lockard, D.P.; Luo, L.-S.; Milder, S.D.; Singer, B.A., Evaluation of powerflow for aerodynamic applications, J stat phys, 107, 1/2, 423-478, (2002) · Zbl 1126.76355
[12] Schäfer, M.; Turek, S., Benchmark computations of laminar flow over a cylinder, Notes in numerical fluid mechanics, 52, (1996), Vieweg Verlag Braunschweig, p. 547-66 · Zbl 0874.76070
[13] Noble, D.R.; Georgiadis, J.G.; Buckius, R.O., Comparison of accuracy and performance for lattice Boltzmann and finite difference simulations of steady viscous flow, Int J numer meth fluids, 23, 1-18, (1996) · Zbl 0866.76069
[14] Bernsdorf, J.; Durst, F.; Schäfer, M., Comparison of cellular automata and finite volume techniques for simulation of incompressible flow in complex geometries, Int J numer meth fluids, 29, 251-264, (1999) · Zbl 0940.76067
[15] Kandhai, D.; Vidal, D.J.-E.; Hoekstra, A.G.; Hoefsloot, H.; Iedema, P.; Sloot, P.M.A., Lattice-Boltzmann and finite element simulation of fluid flow in a SMRX static mixer reactor, Int J numer meth fluids, 31, 1019-1033, (1999) · Zbl 0964.76072
[16] Junk, M., A finite difference interpretation of the lattice Boltzmann method, Numer meth part differen eqn, 17, 383-402, (2001) · Zbl 0987.76082
[17] Junk, M.; Klar, A.; Luo, L.-S., Asymptotic analysis of the lattice Boltzmann equation, J comput phys, 676-704, (2005) · Zbl 1079.82013
[18] Lai, Y.G.; Lin, C.-L.; Huang, J., Accuracy and efficiency study of lattice Boltzmann method for steady flow simulations, Numer heat transfer J B, 39, 21-43, (2001)
[19] www.exa.com.
[20] Tölke, J.; Krafczyk, M.; Rank, E.; Berrioz, R., Discretization of the Boltzmann equation in velocity space using a Galerkin approach, Comput phys commun, 129, 91-99, (2000) · Zbl 0976.76062
[21] Tölke, J.; Krafczyk, M.; Schulz, M.; Rank, E., Implicit discretization and non-uniform mesh refinement approaches for FD discretizations of LBGK models, Int J mod phys C, 9, 8, 1143-1157, (1998)
[22] Tölke, J.; Krafczyk, M.; Rank, E., A mutligrid-solver for the discrete Boltzmann-equation, J stat phys, 107, 573-591, (2002) · Zbl 1007.82004
[23] Lee, T.; Lin, C.-L., A characteristic Galerkin method for discrete Boltzmann equation, J comput phys, 171, 1, 336-356, (2001) · Zbl 1017.76043
[24] Shi, X.; Lin, J.; Yu, Z., Discontinuous Galerkin spectral element lattice Boltzmann method on triangular element, Int J numer meth fluids, 42, 1249-1261, (2003) · Zbl 1033.76046
[25] Nanelli, F.; Succi, S., The lattice Boltzmann equation on irregular lattices, J stat phys, 68, 3/4, 401, (1992) · Zbl 0925.82036
[26] Filippova, O.; Hänel, D., Boundary-Fitting and local grid refinement for LBGK models, Int J mod phys C, 8, 1271, (1998)
[27] Yu, D.; Mei, R.; Shyy, W., A multi-block lattice Boltzmann method for viscous fluid flows, Int J numer meth fluids, 39, 2, 99-120, (2002) · Zbl 1036.76051
[28] Mei, R.; Yu, D.; Shyy, W.; Luo, L.-S., Force evaluation in the lattice Boltzmann method involving vurved geometry, Phys rev E, 65, 041203, (2002)
[29] Schulz, M.; Krafczyk, M.; Tölke, J.; Rank, E., Parallelization strategies and efficiency of CFD computations in complex geometries using lattice Boltzmann methods on high-performance computers, (), 115-122
[30] Freudiger S. Effiziente Datenstrukturen für Lattice-Boltzmann-Simulationen in der computergestützten Strömungsmechanik. Diploma thesis, TU München, 2001.
[31] Crouse, B.; Rank, E.; Krafczyk, M.; Tölke, J., A LB-based approach for adaptive flow simulations, Int J mod phys B, 17, 109-112, (2002)
[32] Crouse B. Lattice-Boltzmann Strömungssimulationen auf Baumdaten-strukturen. dissertation [in German]. TU München; 2002.
[33] Turek, S., Efficient solvers for incompressible flow problems: an algorithmic and computational approach, (1999), Springer Berlin · Zbl 0930.76002
[34] Turek, S.; Rannacher, R., A simple nonconforming quadrilateral Stokes element, Numer meth part differen eqn, 8, 97-111, (1992) · Zbl 0742.76051
[35] Turek, S., On discrete projection methods for the incompressible navier – stokes equations: an algorithmical approach, Comput meth appl mech eng, 143, 271-288, (1997) · Zbl 0898.76069
[36] Yu D. Viscous flow computations with the lattice Boltzmann equation method. PhD thesis. University of Florida; 2002.
[37] 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
[38] Lallemand, P.; Luo, L.-S., Lattice Boltzmann method for moving boundaries, J comput phys, 184, 406-421, (2003) · Zbl 1062.76555
[39] Thürey N. A single-phase free-surface lattice-Boltzmann method. Diploma thesis. IMMD10. University of Erlangen-Nuremberg; 2003.
[40] http://www-waterloo.ansys.com/cfx/.
[41] Tölke J, Freudiger S, Krafczyk M. An adaptive scheme using hierarchical grids for lattice Boltzmann multiphase flow simulations. Comp. Fluids, in press. doi:10.1016/j.compfluid.2005.08.010. · Zbl 1177.76332
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.