本文採用流線座標法搭配擴充型von Mises轉換，即以x=x(ξ)、ψ=ψ(η)透過疊代聯立求解渦度傳輸方程式與Laplace方程式，以獲得到流線位置 y(ξ,η)與渦度 ω(ξ,η)，此種數值模式可稱之為y-ω模式，來分析二維低雷諾數黏性流流過物體的流場問題。以下為本文分析與討論相關流場之要點。
第一，勢流場的解析解與y-ω模式所得數值解之比較有助於誤差分析。除了兩種流況本質之差異外，當選擇「近似方程式與其變數導數之解析解」作為最佳離散形式時，使用物體邊界幾何貼合格網系統，必須適當控制格網尺度，以減小數值誤差。第二，由於流線位置y的變動加上幾何參數f^2中含有ω的因素，求解變數y與ω形成彼此互制的非線性關係，容易導致求解程序的不安定性，同時又因線性化處理需要反覆疊代，因此必須小心地遵循合理的疊代次序來安排計算模式的演算法則。第三，在特定雷諾數的流況，須藉由調整格網尺度，以測試ψ-ω模式的結果，再由追蹤近壁流函數ψ與渦度ω的數值，以評估y-ω模式是否合理處理固壁邊界條件；並另由測試複變勢流格網的幾何係數解析性，佐以比對前人實驗照片，分析了y-ω模式在不同格網系統的數值結果，希望藉此提升計算結果的精度。最後，為延伸y-ω模式應用在雷諾數大於6之圓柱體流，由延長柱體尾端形狀與壁面吸流等不同控制分離流的方式，尋求控制條件用以分析這些控制分離流的有效性，裨供後續相關研究之參考。

The thesis discusses how a viscous flow model is applied to a low Reynolds number flow past a two-dimensional object; such a model, based on the streamline-coordinate method associated with the extended von Mises transformation, x=x(ξ) and ψ=ψ(η), is called the y-ω model which solves the Laplace equation and the vorticity transport equation through iteration process to obtain the positions y(ξ,η) of a streamline along η=const. and the vorticity ω(ξ,η) on it. The following are the brief of the studied flow analyzed and discussed in this thesis.
First, the comparison between the analytical solution in potential flow and the numerical solution obtained from the y-ω model is done in error analysis. Except for their intrinsic deviation, one finds that the numerical error can be reduced by suitable grid-scale control in a boundary-fitted grid system when the best discretized form in the model is taken as the analytic form of the approximated equations and the derivatives. Secondly, due to the streamline locus y and the invocation of vorticity ω varied in a nonlinear manner with the geometric parameter f^2, the interaction of unknowns y and ω might badly cause the instability within solution scheme. Meanwhile, since the nonlinearity always requests iterative treatment, one must cautiously arrange a reasonable order in numerical algorithm during iterations. Thirdly, for the flow at a specific Re, one must justify the grid scale to test the solution accuracy of the y-ω model, and then track the wall values of stream function ψ and vorticity ω to assess whether the solid-boundary conditions is reasonable modeled. In addition, I also tested the analytic expression of geometric coefficients from the complex grid system in potential flow, referred to the laboratory photos investigated previously by others, and analyzed my results from various grid configurations of the y-ω model. Indeed, this effort could improve the numerical accuracy of the model. At last, to extend the application of the y-ω model for a circular cylinder flow without separation beyond Re=6, I tried to analyze two devices to avoid the flow separation, namely, elongation of the tail length and employment of flow suction by porous trailing surface. Some conditions to control the flow separation are proposed accordingly to give some useful information for future research.

1. 張志華，「孤立波與結構物在黏性流體中互制作用之研究」國立成功大學水利及海洋工程研究所博士學位論文，1997
2. 唐啟釗、韓若怡，「波動對孔隙柱體之質動量擴散作用」第30屆海洋工程研討會論文集，2008
3. 唐啟釗，「流線座標應用於非穩態黏性流場之計算」第 36 屆海洋工程研討會論文集，2014
4. Boadway, J.D.,“Transformation of elliptic partial differential equations for solving two-dimensional boundary value problems in fluid flow,”Int. J. Numer Meth Eng., 10(3), pp. 527-533, 1976.
5. Barron, R.M.,“Computation of incompressible potential flow using Von Mises Coordinates,”Math. Comput. Simulation, 31, pp. 177-188, 1989.
6. Barron, R.M., Hamdan, M.H.,“The double Von Mises transformation in the boundaries: Theory and analysis study of two-phase fluid flow over curved,”Int. J. Numer Methods Fluids., 14, pp. 883-905, 1992.
7. Bilus, I., Ternikb, P., Zunic, Z.,“Further contributions on the flow past a stationary and confined cylinder: Creeping and slowly moving flow of power law fluids,”J. Fluids Struct., Vol. 27, pp. 1278–1295, 2011.
8. Chen, C.J., Chen, H.C., The finite analytic method. IIHR, U. Iowa, Iowa City, IA, 1982.
9. Chen, C.J., Choi, S.K., The finite analytic method and its applications : laminar and turbulent flows past two dimensional and axisymmetric bodies. IIHR, U. Iowa, Iowa City, IA, U.S., 1990.
10. Dyke, M.V., An Album of Fluid Motion. The Parabolic Press, U.S., 1982.
11. Finn, R. K.,“Determination of the drag on a cylinder at low Reynolds number,”J. Appl. Phys. vol 24, pp. 771-773, 1953.
12. Huang, C.Y., Dulikravich, G.S.,“Stream function and stram-function-coordinate (sfc) formulation for inviscid flow field calculations,”Texas Inst. Computational Mechanics, Department of Aerospace Engineering and Engineering Mechanics, U. Texas, Austin, TX, U.S., 1986.
13. Jaeger, C.,“Engineering Fluid Mechanics,”Blackie and Son, Edinburgh, 1956.
14. Jayaweera, K. O. L. F., Mason, B. J.,“The behaviour of freely falling cylinders and cones in a viscous fluid,”J. Fluid Mech. vol 22, pp. 709-720, 1965.
15. Kaplun, S.,“Low Reynolds number flow past a circular cylinder,”J. Math. Mech., vol 6, pp. 595-603, 1957.
16. Khalili, A., Liu, B.,“Stokes’paradox: creeping flow past a two-dimentional cylinder in an infinite domain,”J. Fluid Mech., vol 817, pp. 374-387, 2017.
17. Kida, T., Take, T.,“Asymptotic expansions for low Reynolds number flow past a cylindrical body,”JSME Intl. J. vol 35, pp. 144-150, 1992.
18. Lamb, H., Hydrodynamics. Cambridge Uni. Press. Cambridge, 1932.
19. Majda, A.J., Bertozzi, A.L., Vorticity and incompressible flow. Cambridge Univ. Press. Cambridge, 2002.
20. Patankar, S.V., Numerical Heat Transfer and Fluid Flow. Hemisphere Publishing Corp. U.S., 1980.
21. Skinner, L. A.,“Generalized expansions for slow flow past a cylinder,”Q. J. Mech. Appl. Maths, vol 28, pp. 333-340, 1975.
22. Thom, A.,“The flow past circular cylinders at low speeds,”Proceedings of the Royal Society of London. Series A, Vol. 141, No. 845, pp. 651-669, 1993.
23. Tayfour E.-B.,“Fluid flow at small reynolds number: Numerical applications,” Department of mathematics and statistics. College of Science Sultan Qaboos Univ. P.O. Box 36, Al-Khod 123 Sultanate of Oman, 2006.
24. Veysey, J. & Goldenfeld, N.,“Singular perturbations in simple low Reynolds number flows: from boundary layers to the renormalization group,”Rev. Mod. Phys. vol 79, pp. 883-927, 2007.
25. White, F.M., Viscous Fluid Flow. McGraw-Hill, Inc. New York, 1992.