本研究中發展了兩種可在卡氏網格中計算具有浸沈物體的不可壓縮流場之計算方法。第一種方法是將物體所在的區域視為同樣是被流體所佔據，但此部份的流體其速度和溫度是已知或可由其它的結構方程式和溫度方程式求得，而且其速度滿足無散(Divergence-free)的要求，並藉由定義物體體積比(Volume-of-Body, VOB)函數來描述流場中的物體。對於同時存在有流體和物體的界面網格，可經由流體部份和物體部份的速度和溫度利用體積加權平均後得到網格形心上的速度和溫度。然而，經由體積加權平均的處理手段之後，會將物體和流體之間的界面模糊化，使其存有一個網格寬度的不準度，因此限制了在浸沈邊界附近的流場解僅能達到一階的準確度。為了增加浸沈邊界附近解的準確度，則發展了局部的網格加密技術來有效地提升在浸沈邊界附近的網格解析度。
第二種方法為虛擬網格法(Ghost Cell Method)，此方法將位於物體表面內的第一層網格定義為虛擬網格(Ghost Cell)，在虛擬網格上的變數值則利用鄰近流場解與物體表面上的邊界值以外插方式求得，所以在虛擬網格中即隱含了所需的邊界條件。在本研究中，對於求取虛擬網格上的值，提出了新的外插求值的方式，此外插方式是穩定且容易處理的，並且可得到二階準確的流場解。
本研究中使用隱式的壓力修正法(Implicit Fractional Step Pressure-Correction Method)來求解不可壓縮的Navier-Stokes方程式。以有限體積法於非結構性卡氏網格(Unstructured Cartesian Mesh)上進行方程式的數值離散，並使用多重網格法(Multigrid Method)來增加求解過程中的穩定性及收斂速率。本研究並使用平行運算的方式來提升三維計算的效率，及結合動態的局部網格加密(Dynamic Local Refinement)技巧以有效控制計算所需的格點數量。在不同的計算中驗證了兩種計算方法的準確度，以及對於具有靜止或移動的浸沈物體時的流場計算能力。

In this dissertation, two non-body-fitted grid methods to treat immersed bodies in incompressible flows are developed on Cartesian meshes with local refinement. The first method is the Volume of Body (VOB) method, in which the body is treated as being made of the same fluid as outside with a prescribed divergence-free velocity field and temperature field. In this view the volume fraction in a cell occupied by the body, or Volume of Body function, can be used to track the presence of the immersed body. The body-fluid interface is similar to a fluid-fluid interface encountered in the Volume of Fluid (VOF) method for the two-fluid flow problems. For the grid cells containing the body-fluid interface, the solution is a volume fraction-weighted average of fluid solution and body solution. However, such a volume-averaged procedure smears the sharp fluid-body interface into a width of one cell size and limits the accuracy of the solution near the immersed boundary to only first order. To improve the solution near the boundary, the local refinement capability around the immersed boundary is developed.
The second method is a new Ghost Cell method. Ghost cells are finite volume cells inside the body surface with at least one neighboring fluid cells. In this method the ghost cell values are determined not by the flow equations, but by a flow reconstruction in the vicinity of the body surface in such a way that the no-slip boundary conditions are implicitly satisfied at the immersed boundary. In this work a new reconstruction model for the ghost cells is developed for flows with low to moderate high Reynolds number. It is shown that the new Ghost Cell method is simple and stable, and achieves overall second order accurate flow solutions.
Both VOB method and Ghost Cell method are implemented on unstructured Cartesian flow solver based on an implicit fractional step pressure-correction method. To enhance stability and convergence of the solution process, multigrid methods are developed to solve the difference equations for pressure, velocity and temperature field. To improve the efficiency of 3D computations, parallel computing based on Message Passing Interface (MPI) protocol on a PC cluster is implemented in this work. The dynamic local refinement strategy is adopted in treating moving immersed bodies to reduce the grid size. The spatial accuracy of both methods is verified in the 2D steady flow with a stationary immersed body. Various steady and unsteady flows over 2D and 3D bodies are computed to validate the proposed methods. The capability of the proposed methods to treat moving bodies is also demonstrated.

1. Yerry, M. A. and Shephard, M. S., “Automatic Three-Dimensional Mesh Generation by the Modified-Octree Technique,” Int. J. Numer. Methods Eng., Vol. 20, pp. 1965-1990, 1984.
2. Lohner, R. and Parikh, P., “Generation of Three-Dimensional Unstructured Grids by the Advancing-Front Method,” Int. J. Numer. Meth. Fluids, Vol. 8, pp. 1135-1149, 1988.
3. Chao, M. J., “Unstructured Grid Generation and Agglomeration Multigrid Method for the Euler Solutions Over Complex Aircraft Configurations,” Ph. D. Thesis, Institute of Aeronautics and Astronautics, National Cheng Kung University, Tainan,Taiwan, R.O.C., June 1997.
4. Batina, J. T., “Unsteady Euler Airfoil Solutions Using Unstructured Dynamic Meshes,” AIAA J., Vol. 28, no. 8, pp. 1381-1388, 1990.
5. Benek, J. A., Buning, P. G., and Steger, J. L., “A 3-D Chimera Grid Embedding Technique,” AIAA Paper 85-1523, 1985.
6. Wang, Z. J., Parthasarathy, V., “A Fully Automated Chimera Methodology for Multiple Moving Body Problems,” Int. J. Numer. Meth. Fluids, Vol. 33, pp. 919-938, 2000.
7. Zeeuw, D. D., and Powell, K. G., “An Adaptively Refined Cartesian Mesh Solver for the Euler Equations,” J. Comput. Phys., Vol. 104, pp. 56-68, 1993.
8. Pember, R. B., Bell, J. B. and Colella, P., “An Adaptive Cartesian Grid Method for Unsteady Compressible Flow in Irregular Regions,” J. Comput. Phys., Vol. 120, pp. 278-304, 1995.
9. Ye T., Mittal R., Udaykumar H.S. and Shyy W., “An Accurate Cartesian Grid Method for Viscous Incompressible Flows with Complex Immersed Boundaries,” J. Comput. Phys., Vol. 156, pp. 209-240, 1999.
10. Udaykumar, H. S., Mittal, R., Rampunggoon, P. and Khanna, A., “A Sharp Interface Cartesian Grid Method for Simulating Flows with Complex Moving Boundaries,” J. Comput. Phys., Vol. 174, pp. 345-380, 2001.
11. Peskin C.S., “The Immersed Boundary method,” Acta Numerica, Vol. 11, pp. 479-517, 2002.
12. Peskin, C. S., and McQueen, D. M., “A Three-Dimensional Computational Method for Blood Flow in the Heart I. Immersed Elastic Fibers in a Viscous Incompressible Fluid,” J. Comput. Phys., Vol. 81, pp. 372-405, 1989.
13. Lai, M. C., and Peskin, C. S., “An Immersed Boundary Method with Formal Second-Order Accuracy and Reduced Numerical Viscosity,” J. Comput. Phys., Vol. 160, pp. 705-719, 2000.
14. Goldstein D., Handler R. and Sirovich L., “Modeling a No-slip Flow Boundary with an External Force Field,” J. Comput. Phys., Vol. 105, pp. 354-366, 1993.
15. Saiki, E. M. and Biringen, S., “Numerical Simulation of a Cylinder in Uniform Flow: Application of a Virtual Boundary Method,” J. Comput. Phys., Vol. 123, pp. 450-465, 1996.
16. Mohd-Yusof, J., “Combined Immersed-Boundaries/B-spline Methods fir Simulations of Flow in Complex Geometries,” CTR Annual Research Briefs, NASA Ames/Stanford University, 1997.
17. Fadlun E.A., Verzicco R., Orlandi P. and Mohd-Yuosof J., “Combined Immersed-Boundary Finite-Difference Methods for Three-Dimensional Complex Flow Simulations,” J. Comput. Phys., Vol. 161, pp. 35-60, 2000.
18. Iaccarino, G., and Verzicco, R., “Immersed Boundary Technique for Turbulent Flow Simulations,” Appl. Mech. Rev., Vol. 56, no. 3, pp. 331-347, 2003.
19. Balaras, E., “Modeling Complex Boundaries Using an External Force Field on Fixed Cartesian Grids in Large-Eddy Simulations,” Comput. Fluids, Vol. 33, pp. 375-404, 2004.
20. Sheu, T. W. H., Ting, H. F. and Lin, R. K., “An Immersed Boundary Method for the Incompressible Navier-Stokes Equations in Complex Geometry,” Int. J. Numer. Meth. Fluids, Vol. 56, pp. 877-898, 2008.
21. Tseng, Y. H. and Ferziger, J. H., “A Ghost-cell Immersed Boundary Method for Flow in Complex Geometry,” J. Comput. Phys., Vol. 192, pp. 593-623, 2003.
22. Pacheco, J. R., Pacheco-Vega, A., Rodic, T. and Peck R. E., “Numerical Simulations of Heat Transfer and Fluid Flow Problems Using an Immersed-Boundary Finite-Volume Method on Nonstaggered Grids,” Numer. heat transf., B Fundam., Vol. 48, pp. 1-24, 2005.
23. Pan, D., “An Immersed Boundary Method for Incompressible Flows Using Volume of Body Function,” Int. J. Numer. Meth. Fluids, Vol. 50, pp. 733-750, 2006.
24. Pan, D., “An Immersed Boundary Method on Unstructured Cartesian Meshes for Incompressible Flows with Heat Transfer,” Numer. heat transf., B Fundam., Vol. 49, pp. 277-297, 2006.
25. Silva, A.L.F. Lima E, Silveira-Neto, A. and Damasceno, J.J.R., “Numerical Simulation of Two-dimensional Flows over a Circular Cylinder Using the Immersed Boundary Method,” J. Comput. Phys., Vol. 189, pp. 351-370, 2003.
26. Kim, J., Kim, D. and Choi, H., “An Immersed-Boundary Finite-Volume Method for Simulations of Flow in Complex Geometry,” J. Comput. Phys., Vol. 171, pp. 132-150, 2001.
27. Rhie, C. M. and Chow W. L., “A Numerical Study of the Turbulent Flow past an Airfoil with Trailing Edge Seperation,” AIAA J., Vol. 21, no. 11, pp. 1525-1532, 1983.
28. Sheu T.W.H. and Tsai S.F., “Flow Topology in a Steady Three-Dimensional Lid-Driven Cavity,” Comput Fluids, Vol. 31, pp. 911-934, 2002.
29. Jiang Bo-nan, Lin T.L. and Povinelli L.A., “Large-Scale Computation of Incompressible Viscous Flow by Least-Squares Finite Element Method,” Comput. Methods Appl. Mech. Eng., Vol. 114, pp. 213-231, 1994.
30. Bennett, B. A. V. and Hsueh, J., “Natural Convection in a Cubic Cavity: Implicit Numerical Solution of Two Benchmark Problems,” Numer. Heat Transf. A Appl., Vol. 50, pp. 99-123, 2006.
31. Sophy, T., Sadat, H. and Prax, C., “A Meshless Formulation for Three-Dimensional Laminar Natural Convection,” Numer. Heat Transf. B Fundam., Vol. 41, pp. 433-445, 2002.
32. Lo, D. C., Young, D. L. and Murugesan, K., “GDQ Method for Natural Convection in a Cubic Cavity Using Velocity-Vorticity Formulation,” Numer. Heat Transf. B Fundam., Vol. 48, pp. 363-386, 2005.
33. Fornberg, B., “Steady Viscous Flow Past a Sphere at High Reynolds Numbers,” J. Fluid Mech., Vol. 190, pp. 471-489, 1988.
34. Johnson, T. A. and Patel, V. C., “Flow past a Sphere up to a Reynolds Number of 300,” J. Fluid Mech., Vol. 378, pp. 19-70, 1999.
35. Marella, S., Krishnan, S., Liu, H. and Hdaykumar H. S., “Sharp Interface Cartesian Grid Method I: An Easily Implemented Technique for 3D Moving Boundary Computations,” J. Comput. Phys., Vol. 210, pp.1-31, 2005.
36. Bagchi, P., Ha, M. Y. and Balachandar, S., “Direct Numerical Simulation of Flow and Heat Transfer from a Sphere in a Uniform Cross-Flow,” J. Fluids Eng., Vol. 123, pp. 347-358, 2001.
37. Bouard, R. and Coutanceau, M., “The Early State of Development of the Wake behind an Impulsively Started Cylinder for ,” J. Fluid Mech., Vol. 101, pp. 583-607, 1980.
38. Berger, M. J. and Oliger, J., “Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations,” J. Comput. Phys., Vol. 53, pp. 484-512, 1984.
39. Berger, M. J. and Colella, P., “Local Adaptive Mesh Refinement for Shock Hydrodynamics,” J. Comput. Phys., Vol. 82, pp. 64-84, 1989.
40. Bell, J., Berger, M., Saltzman, J. and Welcome, M., “Three-Dimensional Adaptive Mesh Refinement for Hyperbolic Conservation Laws,” SIAM J. Sci. Comput., Vol. 15, no. 1, pp. 127-138, 1994.
41. Mei, R., “Flow due to an Oscillating Sphere and an Expression for Unsteady Drag on the Sphere at Finite Reynolds Number,” J. Fluid Mech., Vol. 270, pp. 133-174, 1994.
42. Wang, Z. J., “Two Dimensional Mechanism for Insect Hovering,” Phys. Rev. Lett., Vol. 85, no. 10, pp. 2216-2219, 2000.