In this thesis, the validity of the adopted MRT lattice Boltzmann model [14] is examined by computing two-dimensional Poiseuille flow, lid-driven cavity flow, and three-dimensional Poiseuille flow in a square duct. We also use the density fluctuation and assume the mean density to reduce effects due to compressibility [48]. The present simulation indicates that SRT method fails to simulate higher Reynolds number flows due to the limitation of tau value. In contrast to the SRT method, the tau value for MRT method can be close to 0.5 for 2-D and 3-D simulations owing to the different relaxation rates. It should be noted that 3D MRT computations are very sensitive to boundaries, especially the treatment of the corners and edges. As adopting smaller value of tau combined with MRT method, higher Reynolds number flow can be realized. Finally, parallel-MRT and parallel-SRT models are implemented for large grid density simulations, and simulated results are presented.

Bibliography
[1] S. Chen, H. Chen, D. O. Martinez, and W. H. Matthaeus, "Lattice Boltzmann model for simulation of magnetohydrodynamics", Physics Review Letters 67,3776, (1991).
[2] Y. H. Qian, D. d'Humiµeres, and P. Lallemand, "Lattice BGK model for Navier-Stokes equation", Europhysics Letters 17, 479, (1992).
[3] S. Chen and G. D. Doolen, "Lattice Boltzmann method for fluid flow", Annual Reviews of Fluid Mechanics 30 329, (1998).
[4] U. Frisch, B. Hasslacher, and Y. Pomeau, "Lattice-gas automata for the Navier-Stokes equation", Physics Review Letters 56, 1505, (1986).
[5] S. Wolfram, "Cellular automaton fluids 1: Basic theory", Journal Statistic Physics 45, 471, (1986).
[6] F. J. Higuera, S. Sussi, and R. Benzi, "3-dimensional flows in complex geometries with the lattice Boltzmann method", Europhysics Letters 9, 345,
(1989).
[7] F. J. Higuera, and J. Jem¶enez, "Boltzmann approach to lattice gas simulations", Europhysics Letters 9, 663, (1989).
[8] P. L. Bhatnagar, E. P. Gross, and M. Grook, "A model for collision processes in gases. I. small amplitude processes in charged and neutral one-component systems", Physics Reviews E 94, 511, (1954).
[9] S. Harris, "An introduction to the throry of the Boltzmann equation", Holt, Rinehart and Winston, New York, (1971).
[10] Y. H. Qian, D. d'Humi¶eres, and P. Lallemand, "Lattice BGK models for Navier-Stokes equation", Europhysics Letters 17, 479, (1992).
[11] H. Chen, S. Chen, and W. H. Matthaeus, "Recovery of the Navier-Stokes quations using a lattice gas Boltzmann method", Physics Reviews A 45, R5339, (1992).
[12] U. Frisch, D. d'Humiµeres, B. Hasslacher, P. Lallemand, Y. Pomeau, and J. P. Rivet, "Lattice gas hydrodynamics in two and three dimensions", Complex
System 1, 649, (1987).
[13] D. O. Martinez, W. H. Matthaeus, S. Chen, and D. C. Montgomery, "Comparison of spectral method and lattice Boltzmann simulations of two-dimensional hydrodynamics", Physics Fluids 6, 1285, (1994).
[14] P. Lallemand, L. Luo, "Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance, and stability", Physics Review E 61, 6546-6562, (2000).
[15] F. J. Higuera, S. Succi, R. Benzi, "Lattice gas-dynamics with enhanced collisions", Europhysics Letters 9, 345 -349, (1989).
[16] S. Succi S, G. Amati, R. Benzi. "Challenges in lattice Boltzmann computing", Journal of Statistical Physics 81, 5 -16, (1995).
[17] Y. H. Qian, S. Succi, S.A. Orszag. "Recent advances in lattice Boltzmann computing", Annual Review of Computational Physics, vol. III, World Scientific: Singapore, 195-242.(1995)
[18] D. Kandhai, A. Koponen, A. Hoekstra, M. Kataja, J. Timonen, PMA. Sloot. "Lattice Boltzmann hydrodynamics on parallel systems", Computer Physics Communications 111, 14 -26, (1998).
[19] A. Dieter, Wolf-Gladrow. "Lattice Gas Cellular Automata and Lattice Boltzmann Models". Springer: Berlin.(2000)
[20] M. Krafczyk, M. Schulz, E. Rank. "Lattice-gas simulations of two-phase flow in porous media", Communications in Numerical Methods in Engineering 14,
7 -12 (1998)
[21] J. Bernsdorf, G. Brenner, F. Durst. "Numerical analysis of the pressure drop in porous media flow with lattice Boltzmann (BGK) automata", Computer
Physics Communications 129, 247-255, (2001)
[22] D. M. Freed. "Lattice-Boltzmann method for macroscopic porous media modeling", International Journal of Modern Physics 9, 1491-1503, (1998).
[23] K. Kono, T. Ishizuka, H. Tsuba. "Application of lattice Boltzmann model to multiphase flows with phase transition", Computer Physics Communications 129, 110 -120, (2001).
[24] S. Hou, X. Shan, Q. Zou, G. D. Doolen, W. E. Soll. "Evaluation of two lattice Boltzmann models of multiphase flows", Journal of Computation Physics 138,
695 -713, (1997)
[25] X. He, S. Chen, R. Zhang. "A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh Taylor instability", Journal of Computation Physics 152, 642- 663, (1999).
[26] Y. Hashimoto, Y. Ohashi. "Droplet dynamics using the lattice-gas method", International Journal of Modern Physics C, 8, 977-983, (1997).
[27] H. Xi, C. Duncan. "Lattice Boltzmann simulations of three-dimensional single droplet deformation and breakup under simple shear flow", Physical Review E, 59, 3022-3026, (1999).
[28] S. Hou. "Lattice Boltzmann Method for Incompressible, Viscous Flow", Ph.D. Thesis, Department of Mechanical Engineering, Kansas State University.(1995).
[29] D. D'Humieres. "Generalized lattice Boltzmann equation", In Rarefied Gas Dynamics: Theory and Simulations, Progress in Astronautics and Aeronautics,
vol. 159, Shizgal BD, Weaver DP (eds). AIAA: Washington, DC, 45-458.(1992)
[30] X. He, Q. Zou, L.-S. Luo, and M. Dembo, "Analytic solution of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model", Journal Statistic Physics 87, 115, (1997).
[31] P. A. Skordos, "Initial and boundary conditions for the lattice Boltzmann method", Physics Reviews E 48, 4823, (1993).
[32] D. R. Noble, S. Chen, J. G. Georgiadis, and R. O. Buckius, "A consistent hydrodynamic boundary condition for the lattice Boltzmann method", Physics Fluids 7, 203, (1995).
[33] T. Inamuro, M. Yoshino, and F. Ogino, "A non-slip boundary condition for lattice Boltzmann simulation", Physics Fluids 7, 2928, (1995).
[34] S. Chen, D. O. Martinez, and R. Mei, "On boundary conditions in lattice Boltzmann methods", Physics Fluids 8, 2527, (1996).
[35] Q. Zou and X. He, "On pressure and velocity boundary conditions for the lattice Boltzmann BGK model", Physics Fluids 9, 1591, (1997).
[36] C. F. Ho, C. Chang, K. H. Lin and C. A. Lin, "Consistent boundary conditions for 2D and 3D lattice Boltzmann simulations", CMES, Vol. 44, no.2, pp.137-155. (2009).
[37] M. Krafczyk, J. Tolke, L. Luo, "Large-eddy simulation with a multiple-relaxation -time LBE model", Journal of Modern Physics B Vol. 17, Nos. 1& 2 (2003).
[38] O. Metais and J. Ferziger, "New Tools in Turbulence Modelling". Les Houches School, 21-31 May (1996), XVII, 298 pp.. Springer-Verlag Berlin Heidelberg New York.
[39] S. Hou et al., "Pattern Formation and Lattice Gas Automata", Fields Institute Communications 6, ed. A.T. Lawniczak and R. Kapral (AMS, Providence, 1996), p. 151.
[40] J. Smagorinsky, Monthly Weather Rev. 91, 99 (1963).
[41] Tamas I. Gombosi, "Gas kinetic theorym", Cambridge University Press,(1994).
[42] X. He and L. Luo, "Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation", Physics Reviews E 56, 6811, (1997).
[43] Y. Shao and J. Wu, "Simulation of lid-driven cavity °ows by parallel lattice Boltzmann method using multi- relaxation-time scheme", International Journal for Numerical Methods in Fluids, 46:921-937. (2004)
[44] D. D'Humieres, I. Ginzburg, M. Krafczyk, P. Lallemand and L.S.Luo, "Multiple-relaxation-time lattice Boltzmann models in three dimensions", The Royal Society, 360, 437-451. (2002)
[45] U. Ghia, K. N. Ghia, C. T. Shin, "High-Resolutions for incompressible flow using the Navier-Stokes equations and a multigrid method", J. Comput. Phys.48, 387-411. (1982)
[46] O. Botella, R. Peyret, "Benchmark spectral results on the lid-driven cavity flow", Computers Fluids 27, 421-433. (1998)
[47] F. M. White, "Viscous Fluid Flow - 2nd ed.", McGraw-Hill, New York, (1991).
[48] X. He and L. Luo. "Lattice Boltzmann model for the incompressible Navier-Stokes equation", Journal of statistic physics, Vol.88, Nos.3/4, (1997).
[49] I. Ginzburg, F. Verhaeghe, and D. d'Humieres, "Two-relaxation-time lattice Boltzmann scheme: about parametrization,velocity, pressure and mixed boundary conditions", Communication in computational physics Vol. 3, No.2, pp.427-478, (2007).
[50] L. Zheng, B. Shi, and Z. Guo, "Multiple-relaxation-time model for the correct thermohydrodynamic equations", Physics Review E 78, 026705, (2008).