在本論文中，晶格波茲曼法與沉浸邊界法結合以模擬流經過靜止的或可移動的複雜幾何形狀的流場。此複雜幾何形狀是由拉格朗日標記點來表示，且外力被施加在拉格朗日標記點上以準確地滿足邊界上指定的速度。接著透過一個適當的離散 delta 函數，將拉格朗日標記點上的外力分配到尤拉網格點上。藉著尤拉網格點上已知的外力，晶格波茲曼法用以計算沉浸在卡氏計算範圍內，流經複雜幾何形狀的流場。目前提出的方法以計算以下的流場來檢驗：衰減渦旋流場，頂蓋拉動的方腔流場，旋轉圓柱流場與流經靜止的或移動的圓柱的流場。所有計算出來的結果與解析解或參考解有相當程度地吻合，並且滿足座標轉換的一致性。我們也檢驗拉格朗日標記點的間距對於計算準確度的影響；尤拉網格點上的誤差隨著拉格朗日標記點的間距的縮短而增加。目前計算出來的結果也顯示出壓力場在跨越沉浸邊界處會有不連續性，此現象在未來的研究中將被進一步地闡明。

In this thesis, the lattice Boltzmann method is combined with the immersed boundary technique to simulate complex geometry flows both with stationary and moving boundaries. The complex geometry is represented by Lagrangian markers and forces are exerted at the Lagrangian markers in order to satisfy exactly the prescribed velocity of the boundary. This force at the Lagrangian markers is then distributed to the Eulerian grid by a well-chosen discretized delta function. With the known force field in the Eulerian grid to mimic the boundary, the lattice Boltzmann method is used to compute the flow field where the complex geometry is immersed inside the Cartesian computational domain. The proposed method is examined by computing decaying vortex flow, lid driven cavity flow, rotating cylinder flow and flows over both stationary and moving cylinder. All the numerical results agree reasonably well with the analytical solution or the benchmark solution, and the Galilean invariance is satisfied. The influences of the Lagrangian marker spacing on the solution accuracy are also examined. It was observed that the error on the Eulerian grid increases when reducing the Lagrangian marker spacing. The predicted results also show a discontinuous pressure field across the immersed boundary, a phenomenon to be clarified in future study.

1. U. Frisch, B. Hasslacher, and Y. Pomeau, Phys. Rev. Lett. 56,1505 (1986); S. Wolfram, J. Stat. Phys. 45, 471 (1986).
2. F. J. Higuera, S. Succi, and R. Benzi, Europhys. Lett. 9, 345 (1989); F. J. Higuera and J. Jeme´nez, ibid. 9, 663 (1989).
3. P. L. Bhatnagar, E. P. Gross, and M. Krook, Phys. Rev. 94, 511 (1954).
4. S. Harris, “An Introduction to the Theory of the Boltzmann Equation”, Holt, Rinehart and Winston, New York, (1971).
5. R. L. Liboff, “Kinetic Theory”, Prentice Hall, Englewood Cliffs, NJ, (1990).
6. H. Chen, S. Chen, and W. H. Matthaeus, Phys. Rev. A 45, R5339 (1991); Y. H. Qian, D. d’Humie`res, and P. Lallemand, Europhys. Lett. 17, 479 (1992).
7. U. Frisch, D. d’Humières, 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. I. Ginzbourg, P.M. Alder, “Boundary flow condition analysis for the three-dimensional lattice Boltzmann model”, J. Phys. II (France) 4, 191 – 214 (1994)
10. I. Ginzbourg, D. d’Humières, “Local second-order boundary method for lattice Boltzmann models”, J. Statist. Phys. 84, 927 – 971 (1996)
11. I. Ginzbourg, D. d’Humières, “Local second-order boundary method for lattice Boltzmann models: Part II. Application to complex geometries”, Preprint (1996)
12. L.-S. Luo, “Analytic solutions of linearized lattice Boltzmann equation for simple flows”, J. Statist. Phys. 88, 913 –926 (1997)
13. X. He, Q. Zou, L.-S. Luo, M. Dembo, “Analytic solutions and analysis on non-slip boundary condition for the lattice Boltzmann BGK model”, J. Statist. Phys. 87, 115 – 136 (1997)
14. Q. Zou, X. He, “On pressure and velocity boundary conditions for the lattice Boltzmann BGK model”, Phys. Fluids 9, 1591 – 1598 (1997)
15. S. Chen, D. Martínez, R. Mei, “On boundary conditions in lattice Boltzmann methods”, Phys. Fluids 8, 2527 – 2536 (1996)
16. O. Filippova, D. Hänel, “Grid refinement for lattice-BGK models”, J. Comput. Phys. 147, 219 (1998)
17. R. Mei, L.-S. Luo, W. Shyy, “An Accurate Curved Boundary Treatment in the Lattice Boltzmann Method”, J. Comput. Phys. 155, 307 – 330 (1999)
18. P. Lallemand, L.-S. Luo, “Lattice Boltzmann method for moving boundaries”, J. Comput. Phys. 184, 406 – 421 (2003)
19. M. Bouzidi, M. Firdaouss, P. Lallemand, “Momentum transfer of a lattice-Boltzmann fluid with boundaries”, Phys. Fluids 13, 3452 – 3459 (2002)
20. J. M. Buick and C. A. Greated, “Gravity in a lattice Boltzmann model”, Physical Review E, 61, NUMBER 5, (MAY 2000)
21. Li-Shi Luo, “Theory of the lattice Boltzmann method: Lattice Boltzmann models for nonideal gases”, Physical Review E, 62, NUMBER 4, (OCTOBER 2000)
22. Dieter A. Wolf-Gladrow, “Lattice-Gas Cellular Automata and Lattice Boltzmann Models - An Introduction”, Springer, Berlin, Lecture Notes in Mathematics (2000)
23. C. S. Peskin, “Flow patterns around heart valves: a numerical method”, J. Comp. Phys. 10, 252 (1972)
24. M.-C. Lai and C. S. Peskin, “An immersed boundary method with formal second order accuracy and reduced numerical viscosity”, J. Comp. Phys. 160, 705 (2000)
25. D. Goldstein, R. Handler, and L. Sirovich, “Modeling a no-slip flow with an external force field”, J. Comp. Phys. 105, 354 (1993)
26. D. Goldstein, R. Hadler, L. Sirovich, “Direct numerical simulation of turbulent flow over a modeled riblet covered surface”, J. Fluid Mech. 302, 333. (1995)
27. E. M. Saiki, and S. Biringen, “Numerical simulation of a cylinder in uniform flow: application of a virtual boundary method”, J. Comp. Phys. 123, 450 (1996)
28. 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)
29. E. A. Fadlun, R. Verzicco, P. Orlandi, and J. Mohd-Yusof, “Combined immersed-boundary methods for three dimensional complex flow simulations”, J. Comp. Phys. 161, 30 (2000)
30. Tamas I. Gombosi, “Gaskinetic Theory”, Cambridge University Press, (1994)
31. X. He, 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).
32. Shen-Wei Su, Ming-Chih Lai, Chao-An Lin, “A simple immersed boundary technique for simulating complex flows with rigid boundary”, submitted to J. Comp. Phys.
33. U. Ghia, K. N. Ghia, and C. T. Shin, “High-Re solutions for incompressible flow using the Navier-Stokes Equations and a multi-grid method”, J. Comp. Phys. 48, 387 (1982)
34. Bengt Fornberg, “Steady incompressible flow past a row of circular cylinders”, J. Fluid Mech. 225, 655 (1991)
35. M. Schafer and S. Turek, in “Flow Simulation with High-Performance Computer II”, edited by E. H. Hirschel, Notes in Numerical Fluid Mechanics (Vieweg, Braunschweig, 1996), 52, 547-566