Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-27T02:14:47.463Z Has data issue: false hasContentIssue false
  • English
  • Français

Comparison of Simulations of Convective Flows

Published online by Cambridge University Press:  03 June 2015

Pierre Lallemand  and
François Dubois
Pierre Lallemand
Affiliation:
Beijing Computational Science Research Center, Beijing Run Ze Jia Ye, China
François Dubois*
Affiliation:
Structural Mechanics and Coupled Systems Laboratory, Conservatoire National des Arts et Métiers, Paris, France Department of Mathematics, University Paris-Sud, Bât. 425, F-91405 Orsay Cedex, France
*
*Corresponding author. Email addresses: pierre.lallemand1@free.fr (P. Lallemand), francois.dubois@math.u-psud.fr (F. Dubois)
Get access

Abstract

We show that a single particle distribution for the “energy-conserving” D2Q13 lattice Boltzmann scheme can simulate coupled effects involving advection and diffusion of velocity and temperature. We consider various test cases: non-linear waves with periodic boundary conditions, a test case with buoyancy, propagation of transverse waves, Couette and Poiseuille flows. We test various boundary conditions and propose to mix bounce-back and anti-bounce-back numerical boundary conditions to take into account velocity and temperature Dirichlet conditions. We present also first results for the de Vahl Davis heated cavity. Our results are compared with the coupled D2Q9-D2Q5 lattice Boltzmann approach for the Boussinesq system and with an elementary finite differences solver for the compressible Navier-Stokes equations. Our main experimental result is the loss of symmetry in the de Vahl Davis cavity computed with the single D2Q13 lattice Boltzmann model without the Boussinesq hypothesis. This result is confirmed by a direct Navier Stokes simulation with finite differences.

Keywords

650576N1580A2082C20Bounce-backnatural convectionadiabatic wallde Vahl Davis
Type
Research Article
Information
Communications in Computational Physics , Volume 17 , Issue 5 , May 2015 , pp. 1169 - 1184
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Batchelor, G.K.. An Introduction to Fluid Dynamics, Cambridge University Press, 1967.Google Scholar
[2]Benzi, R., Succi, S., Vergassola “The lattice Boltzmann equation: theory and applications”, Physics Reports, vol. 222, pp. 145–197, 1992.Google Scholar
[3]Dubois, F.. “Stable lattice Boltzmann schemes with a dual entropy approach for monodimensional nonlinear waves”, Computers and Mathematics with Applications, vol. 65, pp. 142–159, 2013.Google Scholar
[4]Dubois, F., Lallemand, P.. “Towards higher order lattice Boltzmann schemes”, Journal of Statistical Mechanics: Theory and Experiment, P06006, 2009.CrossRefGoogle Scholar
[5]Eggels, J., Somers, J.. “Numerical-simulation of free convective flow using the lattice-Boltzmann scheme”, International Journal of Heat and Fluid Flow, vol. 16, pp. 357–364, 1995.Google Scholar
[6]Ginzburg, I.. “Generic boundary conditions for lattice Boltzmann models and their application to advection and anisotropic dispersion equations”, Advances in Water Resources, vol. 28, pp. 1196–1216, 2005.Google Scholar
[7]Hénon, M.. “Viscosity of a Lattice Gas”, Complex Systems, vol. 1, pp. 763–789, 1987.Google Scholar
[8]d’Humiéres, D.. “Generalized Lattice-Boltzmann Equations”, in Rarefied Gas Dynamics: Theory and Simulations (Eds Shizgal, B.D. and Weave, D.P.), vol. 159 of AIAA Progress in Astronautics and Astronautics, pp. 450–458, 1992.Google Scholar
[9]Junk, M., Klar, A., Luo, L.S.. “Asymptotic analysis of the lattice Boltzmann equation”, Journal of Computational Physics, vol. 210, pp. 676–704, 2005.Google Scholar
[10]Khobalatte, B., Perthame, B.. “Maximum principle on the entropy and second-order kinetic schemes”, Mathematics of Computation, vol. 62, pp. 119–131, 1994.CrossRefGoogle Scholar
[11]Lallemand, P., Luo, L.-S.. “Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability”, Physical Review E, vol. 61, pp. 6546–6562, June 2000.CrossRefGoogle ScholarPubMed
[12]Lallemand, P., Luo, L.-S.. “Theory of the lattice Boltzmann method: Acoustic and thermal properties in two and three dimensions”, Physical Review E, vol. 68, pp. 036706, 2003. vol. 68, no. 3, 2003Google Scholar
[13]Lallemand, P., Dubois, F.. “Some results on energy-conserving lattice Boltzmann models”, Computers and Mathematics with Applications, vol. 65, pp. 831–844, 2013.Google Scholar
[14]Landau, L.D., Lifshitz, E.M.. Fluid Mechanics, Pergamon Press, 1959.Google Scholar
[15]Quéré, P. Le. “Accurate solutions to the square thermally driven cavity at high Rayleigh number”, Computers and Fluids, vol. 20, pp. 29–41, 1991.Google Scholar
[16]Mezrhab, A., Moussaoui, A., Jami, M., Naji, H.. “Double MRT thermal lattice Boltzmann method for simulating convective flows”, Physics Letters A vol. 374, pp. 3499–3507, 2010.CrossRefGoogle Scholar
[17]Nie, X., Shan, X., Chen, H.. “Thermal lattice Boltzmann model for gases with internal degrees of freedom”, Physical Review E, vol. 77, pp. 035701(R), 2008.Google Scholar
[18]Qian, Y., Zhou, Y.. “Complete Galilean-invariant lattice BGK models for the Navier-Stokes equation”, Europhysics Letters, vol. 42, pp. 359–364, 1998.Google Scholar
[19]Roache, P.J.. Computational Fluid Dynamics, 446 pages, Hermosa Publishers, Albuquerque, 1976.Google Scholar
[20]Davis, G. De Vahl. “Natural convection of air in a square cavity: A benchmark numerical solution”, Int. J. of Num. Meth. in fluids, vol. 3, pp. 249–264, 1983.Google Scholar
[21]Wang, J., Wang, D., Lallemand, P., Luo, L-S.. “Lattice Boltzmann simulations of thermal convective flows in two dimensions”, Computers and Mathematics with Applications, vol. 65, pp. 262–286, 2013.Google Scholar