In this thesis, three major issues are discussed. First, a consistent boundary condition for 2D and 3D lattice Boltzmann simulations is proposed. The unknown distribution
functions are obtained from the known distribution functions and the correction factors, where the correction factors at the boundary nodes are evaluated directly from the de‾nitions of density and momentum. This boundary condition is applied to two-dimensional Poiseuille flow, Couette flow with wall injection, and three-dimensional square duct flow. Numerical simulation indicate that the formulation is second order accurate. Then, we focus on the flow including an immersed boundary. Two kinds of immersed boundary treatment are proposed. One is the combination of direct forcing approach in the immersed boundary method and the lattice Boltzmann method(Method A), another is the curved boundary treatment in the lattice Boltzmann method(Method B). Three flow problems are simulated utilizing our immersed boundary treatments, i.e. decaying vortex, flow over an asymmetrically placed cylinder in a channel, and in-line oscillating cylinder in a fluid at rest. The results show that both methods can model the velocity field well,
but some inaccuracy of pressure occurs. This issue deserves further study.

[1] U. Frisch, B. Hasslacher, and Y. Pomeau, \Lattice-gas automata for the Navier-Stokes equation," Phys. Rev. Lett. 56, 1505, (1986).
[2] S. Wolfram, \Cellular automaton fluids 1: Basic theory," J. Stat. Phys. 45, 471, (1986).
[3] F. J. Higuera, S. Sussi, and R. Benzi, \3-dimensional flows in complex geometries with the lattice Boltzmann method," Europhys. Lett. 9, 345, (1989).
[4] F. J. Higuera, and J. Jemenez, \Boltzmann approach to lattice gas simulations," Europhys. Lett. 9, 663, (1989).
[5] 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," Phys. Rev. E 94, 511 (1954).
[6] S. Harris, \An introduction to the throry of the Boltzmann equation," Holt,Rinehart and Winston, New York, (1971).
[7] U. Frisch, D. d'Humiµeres, B. Hasslacher, P. Lallemand, Y. Pomeau, and J.P. Rivet, \Lattice gas hydrodynamics in two and three dimensions," Complex Syst. 1, 649, (1987).
[8] D. O. Martinez, W. H. Matthaeus, S. Chen, and D. C. Montgomery,\Comparison of spectral method and lattice Boltzmann simulations of two-dimensional hydrodynamics," Phys. Fluids 6, 1285, (1994).
[9] R. Mittal, and G.Iaccarino, \Immersed boundary methods," Annual review of Fluid Mechanics, 37, 239-261, (2005).
[10] 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," J. Stat. Phys. 87, 115 (1997).
[11] P. A. Skordos, \Initial and boundary conditions for the lattice Boltzmann method," Phys. Rev. E 48, 4823 (1993).
[12] T. Inamuro, M. Yoshino, and F. Ogino, \A non-slip boundary condition for lattice Boltzmann simulation," Phys. Fluids 7, 2928 (1995).
[13] S. Chen, D. O. Martinez, and R. Mei, \On boundary conditions in lattice Boltzmann methods," Phys. Fluids 8, 2527 (1996).
[14] Q. Zou, and X. He, \On pressure and velocity boundary conditions for the lattice Boltzmann BGK model," Phys. Fluids 9, 1591 (1997).
[15] C. F. Huo, \Development of boundary condition in lattice Boltzmann method," master thesis, National Tsing Hua University, Taiwan, (2006).
[16] A. J. C. Ladd, \Numerical simulation of particular suspensions via a discretized Boltzmann equation. Part II. Numerical results," J. Fluid Mech. 271, 311 (1994).
[17] O. Filippova, and D. HÄanel, \Grid refinement for lattice-BGK models," J. Comput. Phys, 147, 219 (1998).
[18] R. Mei, L. S. Luo, and W. Shyy, \An accurate curved boundary treatment in the lattice Boltzmann method," J. Comput. Phys, 155, 307 (1999).
[19] J. M. Buick, and C. A. Greated, \Gravity in a lattice Boltzmann model," Phys. Rev. E 61, 5307, (2000).
[20] L. S. Luo, \Theory of the lattice Boltzmann method: Lattice Boltzmann models for nonideal gases," Phys. Rev. E 62, 4982, (2000).
[21] L. S. Luo, \Analytic solutions of linearized lattice Boltzmann equation for simple flows," J. Stat. Phys. 88, 913 (1997).
[22] Z. L. Guo, C. G. Zheng, and B. C. Shi, \Discrete lattice effects on the forcing term in the lattice Boltzmann method," Phys. Rev. E, 65, 046308 (2002).
[23] R. Du and B. C. Shi, \A novel scheme for force term in the lattice BGK model," Journal of Modern Physics, 17, 945 (2006).
[24] J. Mohd-Yusof, \Combined immersed boundary/B-Spline method for simulations of flows in complex geometries," CTR Annual Research Briefs, NASA Ames/Stanford University, (1997).
[25] J. Kim, D. Kim, and H. Choi, \An immersed boundary finite-volume method for simulations of flow in complex geometries," J. Comput. Phys, 171, 132 (2001).
[26] E. Balaras, \Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations," Computer and Fluids, 33, 375 (2004).
[27] S. W. Su, M. C. Lai, and C. A. Lin, \An immersed boundary technique for simulating complex flows with rigid boundary," Computer and Fluids in press.
[28] Y. W. Chang, \Implementation of the immersed boundary method for flow in complex geometry," master thesis, National Tsing Hua University, Taiwan, (2006).
[29] D. J. Chen, K. H. Lin, and C. A. Lin, \Immersed boundary method based lattice Boltzmann method to simulate 2D and 3D complex geometry flows," International Journal of Modern Physics in press.
[30] Z. G. Feng, E. Efstathios, and Michaelides, \Proteus: a direct forcing method in the simulations of particle flows," J. Comput. Phys. 202, 20 (2005).
[31] Tamas I. Gombosi, \Gas kinetic theorym," Cambridge University Press, (1994).
[32] X. He, and L. S. Luo, \Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation," Phys. Rev. E 56, 6811-6817 (1997).
[33] D. A. Wolf-Gladrow, \Lattice-gas cellular automata and lattice Boltzmann models - an introduction," Springer, Lecture Notes in Mathematics, p.159, (2000).
[34] F. M. White, \Viscous Fluid Flow - 2nd ed.," McGraw-Hill, New York, (1991).
[35] D. J. Chen, \Immersed boundary method based LBM to simulate complex geometry flows," master thesis, National Tsing Hua University, Taiwan, (2004).
[36] M. schafer and S. Turek, \Flow simulation with High-Performance Computer II," edited by E. H. Hirschel, Notes in Numerical Fluid Mechanics (Vieweg, Braunschweig, 1996), 52, 547-566.
[37] R. Mei, D. Yu, W. Shyy, and L. S. Luo, \Force evaluation in the lattice Boltzmann method involving curved geometry," Phy. Rev. E 65 041203 (2002).
[38] D. Yu, R. Mei, L. S. Luo, and W, Shyy, \Viscous flow computations with the method of lattice Boltzmann equation," Progress in Aerospace Sciences 39, 329 (2003).
[39] K. H. Lin, \Immersed boundary based lattice Boltzmann method for 3-D complex flows," master thesis, National Tsing Hua University, Taiwan, (2005).
[40] H. DÄutsch, F. Durst, S. Becker, and H. Lienhart, \Low-Reynolds-number flow around an oscillating circular cylinder at low Keulegan-Carpenter numbers," J. Fluid Mech. 360, 249 (1998).
[41] J. Yang and E. Balaras, \An embedded-boundary formulation for large-eddy simulation of turbulent flows interacting with moving boundaries," J. Comput. Phys, 215, 20 (2006).
[42] P. Lallemand and L. S. Luo, \Lattice Boltzmann method for moving boundaries," J. Comput. Phys, 184, 406 (2003).