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A New Multiple-relaxation-time Lattice Boltzmann Method for Natural Convection

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Abstract

This article is devoted to the study of multiple-relaxation-time (MRT) lattice Boltzmann method with eight-by-eight collision matrix for natural convection flow. In the velocity space, eight speed directions are used and the corresponding incompressible multiple-relaxation-time model with force term is presented. D2Q4 model is for temperature field. The coupled double distribution functions (DDF) overcome artificial compressible effect corresponding to the standard MRT model. The simulations of natural convection flows with Pr=0.71 for air and Ra=103–109 are carried out and excellent agreements are obtained to demonstrate the numerical accuracy and stability of the proposed model.

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References

  1. McNamara, G.R., Zanetti, G.: Use of the Boltzmann equation to simulate lattice gas automata. Phys. Rev. Lett. 61, 2332–2335 (1988)

    Article  Google Scholar 

  2. Higuera, F.J., Succi, S., Benzi, R.: Lattice gas dynamics with enhanced collisions. Europhys. Lett. 9(4), 345–349 (1989)

    Article  Google Scholar 

  3. Boghosian, B.M., Taylor, W.: Quantum lattice-gas model for the many-particle Schrodinger equation in d dimensions. Phys. Rev. E 57(1), 54–66 (1998)

    Article  MathSciNet  Google Scholar 

  4. Hasslacher, B., Pomeau, Y.: Lattice gas automata for the Navier Stokes equations. Phys. Rev. Lett. 56, 1505–1508 (1986)

    Article  Google Scholar 

  5. Shan, X.W., He, X.Y.: Discretization of the velocity space in the solution of the Boltzmann equation. Phys. Rev. Lett. 80, 65–68 (1998)

    Article  Google Scholar 

  6. Palpacelli, S., Succi, S.: The quantum Lattice Boltzmann equation: recent developments. Commun. Comput. Phys. 4(5), 980–1007 (2008)

    Google Scholar 

  7. Kuznik, F., Obrecht, C., Rusaouen, G.: LBM based flow simulation using GPU computing processor. Comput. Math. Appl. 59, 2380–2392 (2010)

    Article  MATH  Google Scholar 

  8. Benzi, R., Succi, S., Vergassola, M.: The lattice Boltzmann equation: theory and applications. Phys. Rep. 222, 145–197 (1992)

    Article  Google Scholar 

  9. Ladd, A.J.C.: Sedimentation of homogeneous suspensions of non-Brownian spheres. Phys. Fluids 9, 491–499 (1997)

    Article  Google Scholar 

  10. Nestler, B., Aksi, A., Selzer, M.: Combined lattice Boltzmann and phase-field simulations for incompressible fluid flow in porous media. Math. Comput. Simul. 80, 1458–1468 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Succi, S., Filippova, O., Smith, G., Kaxiras, E.: Applying the lattice Boltzmann equation to multiscale fluid problems. Comput. Sci. Eng. 3(6), 26–37 (2001)

    Article  Google Scholar 

  12. Prasianakis, N.I., Chikatamarla Shyam, S., Karlin, I.V., et al.: Entropic lattice Boltzmann method for simulation of thermal flows. Math. Comput. Simul. 72(2–6), 179–183 (2006)

    Article  MATH  Google Scholar 

  13. Palpacelli, S., Succi, S.: The quantum lattice Boltzmann equation: recent developments. Commun. Comput. Phys. 4, 980C1007 (2008)

    Google Scholar 

  14. Shi, B.C., Guo, Z.L.: Lattice Boltzmann model for nonlinear convection diffusion equations. Phys. Rev. E 79, 016701 (2009)

    Article  Google Scholar 

  15. Shi, B.C., Guo, Z.L.: Lattice Boltzmann model for the one dimensional nonlinear Dirac equation. Phys. Rev. E 79, 066704 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Borja, S.C., Tsai, F.T.C.: Non-negativity and stability analyses of lattice Boltzmann method for advection diffusion equation. J. Comput. Phys. 228(1), 236C56 (2009)

    Google Scholar 

  17. Blaak, R., Sloot, P.M.A.: Lattice dependence of reaction-diffusion in lattice Boltzmann modeling. Comput. Phys. Commun. 129, 256C66 (2000)

    Article  MathSciNet  Google Scholar 

  18. Karlin, I.V., Gorban, A.N., Succi, S., Boffi, V.: Exact equilibria for lattice kinetic equations. Phys. Rev. Lett. 81, 1–6 (1998)

    Article  Google Scholar 

  19. Karlin, I.V., Ferrante, A., Ottinger, H.C.: Perfect entropy functions of the Lattice Boltzmann method. Europhys. Lett. 47, 182–188 (1999)

    Article  Google Scholar 

  20. Ansumali, S., Karlin, I.V.: Entropy function approach to the lattice Boltzmann method. J. Stat. Phys. 107, 291–308 (2002)

    Article  MATH  Google Scholar 

  21. Ansumali, S., Karlin, I.V., Ottinger, H.C.: Minimal entropic kinetic models for hydrodynamics. Europhys. Lett. 63, 798–804 (2003)

    Article  Google Scholar 

  22. Lallemand, P., Luo, L.S.: Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance and stability. Phys. Rev. E 61(6), 6546–6562 (2000)

    Article  MathSciNet  Google Scholar 

  23. Ginzburg, I.: Equilibrium-type and link-type lattice Boltzmann models for generic advection and anisotropic-dispersion equation. Adv. Water Resour. 28(11), 1171–1195 (2005)

    Article  Google Scholar 

  24. Ginzburg, I.: Generic boundary conditions for lattice Boltzmann models and their application to advection and anisotropic dispersion equations. Adv. Water Resour. 28(11), 1196–1216 (2005)

    Article  Google Scholar 

  25. D’humières, D., Bouzidi, M., Lallemand, P.: Thirteen-velocity three-dimensional lattice Boltzmann model. Phys. Rev. E 63(6), 066702 (2001)

    Article  Google Scholar 

  26. D’humières, D., Ginzburg, I., Krafczyk, M., et al.: Multiple-relation-time lattice Boltzmann models in three dimensions. Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci. 360(1792), 437–451 (2002)

    Article  MATH  Google Scholar 

  27. Mccracken, M.E., Abaham, J.: Multiplerelaxation-time lattice-Boltzmann model for multiphase flow. Phys. Rev. E 71(3), 036701 (2005)

    Article  Google Scholar 

  28. Hiroaki, Y., Makoto, N.: Multiple-relaxation-time lattice Boltzmann model for the convection and anisotropic diffusion equation. J. Comput. Phys. 229(20), 7774–7795 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Du, R., Shi, B.C., Chen, X.W.: Multi-relaxation-time lattice Boltzmann model for incompressible flow. Phys. Lett. A 359(6), 564–572 (2006)

    Article  MATH  Google Scholar 

  30. Du, R., Shi, B.C.: Incompressible MRT lattice Boltzmann model with eight velocities in 2D space. Int. J. Mod. Phys. C 20(7), 1023–1037 (2009)

    Article  MATH  Google Scholar 

  31. Mezrhab, A., Moussaoui, M.A., et al.: Double MRT thermal lattice Boltzmann method for simulating convective flows. Phys. Lett. A 347, 3499–3507 (2010)

    Article  Google Scholar 

  32. Guo, Z.L., Shi, B.C., Zheng, C.G.: A coupled lattice BGK model for the Boussinesq equation. Int. J. Numer. Methods Fluids 39(4), 325–342 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  33. Guo, Z.L., Zheng, C.G., Shi, B.C.: An extrapolation method for boundary conditions in lattice Boltzmann method. Phys. Fluids 14(6), 2007–2010 (2002)

    Article  Google Scholar 

  34. Dixit, H.N., Babu, V.: Simulation of high Rayleigh number natural convection in a square cavity using the lattice Boltzmann method. Int. J. Heat Mass Transf. 49, 727–739 (2006)

    Article  MATH  Google Scholar 

  35. Massaioli, F., Benzi, R., Succi, S.: Exponential tails in two-dimensional Rayleigh-Bénard convection. Europhys. Lett. 21(3), 305–310 (1993)

    Article  Google Scholar 

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Acknowledgements

Authors would like to thank Dr. Zhenhua Chai and Lin Zheng for helpful discussions. This work is supported by the National Natural Science Foundation of China (Grant No: 11026181).

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Authors and Affiliations

  1. Department of Mathematics, Southeast University, Nanjing, 210096, P.R. China

    Rui Du

  2. Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, P.R. China

    Wenwen Liu

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  1. Rui Du
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  2. Wenwen Liu
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Correspondence to Rui Du.

Additional information

This research is supported by National Natural Science Foundation of China (No. 10871044).

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Du, R., Liu, W. A New Multiple-relaxation-time Lattice Boltzmann Method for Natural Convection. J Sci Comput 56, 122–130 (2013). https://doi.org/10.1007/s10915-012-9665-9

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  • DOI: https://doi.org/10.1007/s10915-012-9665-9

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