納維爾-斯托克斯方程式一直是探討流場問題時重要的統御方程式，藉由此方程式，我們可以探討在不同幾何形狀、流動速度、流體密度及流體黏滯係數下，流場的流動行為。而科學家們在求解此方程式時，經常會利用到有限元素法(Finite element method, FEM)、有限體積法(Finite volume method, FVM)、有限差分法(Finite difference method, FDM)……等方法。上述的方法各有優缺點，但有限元素法對幾何邊界的自由性是其餘方法無法比擬的。吾人考慮到往後欲求解複雜幾何形狀的流場，故本研究將使用有限元素法來求解納維爾-斯托克斯方程式。
在空間的離散上，吾人將利用有限元素法對統御方程式進行離散，但在該方法中仍存在不同階數的離散手法，吾人欲使系統穩定，故本研究將採用P2-P1泰勒-胡德(Taylor-Hood)有限元素法[1]，以錯開流場中的速度及壓力。在時間的離散上，吾人將採用隱式的尤拉法(Implicit Euler method)。接著，吾人將搭配合適的初始條件(Initial condition)及邊界條件(Boundary condition)，並透過自行撰寫的程式以獲得流場的流動行為、阻力係數(Drag coefficient)、升力係數(Lift coefficient)及壓力差(Pressure difference)。最後，吾人將透過與各個文獻比較模擬數據，以驗證本研究的可靠性及準確性。
最終的模擬結果顯示，在雷諾數為100的流動中，本研究所獲得之阻力係數、升力係數及壓力差皆與文獻相近。同時，吾人亦找出本研究方法對雷諾數的限制為400，若欲求流場之雷諾數高於此數值，則該流場將不適用本研究方法。

The Navier-Stokes equations have always been a crucial governing equation for the analysis of flow distribution. We can explore the flow behavior of the flow field under different geometry shape, flow rate, fluid density, and fluid viscosity. In practical use, as the complexity of the involved situation increases, the solving difficulties rises. To deal with these equations properly, scientists take advantages of finite element method, finite volume method, finite difference method, and etc. to solve these equations. Each method has its pros and cons, and we should choose the one that best suits the problem needs and situations. In this research, given that we inclined to compute the flow field across complex geometric shapes, we decide to utilize FEM to solve Navier-Stokes equations since the FEM has very low restrictions on the geometric boundary, which is incomparable to any other method.
To solve the equations with the use of FEM, we must obtain spatial discretization to the governing equations. We need to choose the most suitable method for this study among various different order discretization schemes. To ensure the system’s stability, we adapt P2-P1 Taylor -Hood finite element method [1]to stagger velocity and pressure of the flow field. And in terms of time discretization, we employ implicit Euler method. Furthermore, we introduce appropriate initial conditions and boundary conditions, and obtain the flow behavior, drag coefficient, lift coefficient and pressure difference through our hand craft code. Lastly, the simulated data will be compared with various literatures to verify the reliability and accuracy of this research.
The final results indicate that the calculated drag coefficient, lift coefficient and pressure difference are similar to the results of the literatures when the flow has a Reynolds number equals 100. Meanwhile, we have discovered that this study has a limitation on Reynolds number over 400. If you want to solve the flow field with Reynolds number higher than the restriction, this research method will not be suitable for this situation.

1. Taylor, C. and P. Hood, A numerical solution of the Navier-Stokes equations using the finite element technique. Computers & Fluids, 1973. 1(1): p. 73-100.
2. Chorin, A.J., Numerical solution of the Navier-Stokes equations. Mathematics of computation, 1968. 22(104): p. 745-762.
3. Dennis, S.C.R. and G.-Z. Chang, Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100. Journal of Fluid Mechanics, 1970. 42(3): p. 471-489.
4. Bell, J.B., P. Colella, and H.M. Glaz, A second-order projection method for the incompressible navier-stokes equations. Journal of Computational Physics, 1989. 85(2): p. 257-283.
5. Weinan, E. and J.-G. Liu, Projection method with spatial discretizations. 1992, manuscript.
6. Weinan, E. and J.-G. Liu, Projection method I: convergence and numerical boundary layers. SIAM journal on numerical analysis, 1995: p. 1017-1057.
7. Weinan, E. and J.-G. Liu, Projection method ii: Godunov-ryabenki analysis. SIAM journal on numerical analysis, 1996: p. 1597-1621.
8. Brooks, A.N. and T.J. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Computer methods in applied mechanics and engineering, 1982. 32(1-3): p. 199-259.
9. Glowinski, R. and O. Pironneau, Finite element methods for Navier-Stokes equations. Annual review of fluid mechanics, 1992. 24(1): p. 167-204.
10. Hashiguchi, M. and K. Kuwahara, Two-Dimensional Study of Flow past a Circular Cylinder. 数理解析研究所講究録, 1996. 974: p. 164-169.
11. du Toit, C.G., Finite element solution of the Navier-Stokes equations for incompressible flow using a segregated algorithm. Computer Methods in Applied Mechanics and Engineering, 1998. 151(1): p. 131-141.
12. Rajani, B., A. Kandasamy, and S. Majumdar, Numerical simulation of laminar flow past a circular cylinder. Applied Mathematical Modelling, 2009. 33(3): p. 1228-1247.
13. Zienkiewicz, O.C., R.L. Taylor, and J.Z. Zhu, The finite element method: its basis and fundamentals. 2005: Elsevier.
14. Courant, R., K. Friedrichs, and H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik. Mathematische annalen, 1928. 100(1): p. 32-74.
15. Rannacher, R., Finite element methods for the incompressible Navier-Stokes equations, in Fundamental directions in mathematical fluid mechanics. 2000, Springer. p. 191-293.
16. Wendland, W.L. and M. Efendiev, Analysis and simulation of multifield problems. Vol. 12. 2012: Springer Science & Business Media.
17. Sch‰fer, M., et al. Benchmark Computations of Laminar Flow Around a Cylinder. 1996.