以基本解方法求解反算及移動剛體之問題

Abstract

在本論文中，採用基本解法來分析反算及移動剛體之問題。在開始時，將基本解法結合矩陣的條件數來分析反算問題，包含二維拉普拉司方程式、柯西問題、遺失邊界條件和內部資料問題、散佈資料問題以及外型反算識別問題。再者，將基本解法結合穩態的史托克斯例來求解過度指定和不足指定部分邊界的反算史托克斯問題。史托克斯例的係數可以從任意兩個場域變數，例如速度、壓力、渦度或是流線函數來求得。將數值解和解析解加以比較均可以得到良好的結果。 接著，將基本解法與非穩態史托克斯例加以合併，加上對反算問題的數值經驗，可以直接分析半無窮域的非穩態史托克斯問題而不需任何的疊代或正規化處理。進一步的，基本解法結合非穩態史托克斯例可以成功模擬方形和圓形穴室流以及具有多種驅動邊界的非穩態史托克斯問題。模擬結果可清楚的呈現具有一段可動邊界、兩個轉動邊界以及兩個轉動的偏心圓流動現象。最後，將尤拉-拉格朗日基本解法與非穩態史托克斯例合併，用以求解有移動剛體的奈維爾-史托克斯方程式。首先，先驗證二維穴室流在雷諾數10和50的問題。接著，將此數值方法用以模擬有一移動圓柱的奈維爾-史托克斯方程式。尤拉-拉格朗日基本解法可以清楚且直接的描述移動剛體在流場中的現象。將數值方法與沈浸邊界有限元素法加以比較可得到良好的結果。

The method of fundamental solutions (MFS) is proposed to deal with the inverse and moving rigid body problems. Firstly, the MFS with condition number analysis is carried out for the inverse problems in 2D Laplace equation, Cauchy problems, problem with missing boundary condition and internal data, problem with scattered data, and shape identification problem. Then, the MFS based on the steady Stokeslets has been employed to solve the inverse Stokes problems with over- and under-specified boundary segments. The coefficients of the Stokeslets can be obtained from any two field variables among the u, v velocity, pressure, vorticity or stream function. The numerical results are almost identical with the analytical solutions and other numerical results. Furthermore, the unsteady Stokes flow in semi-infinite domain can be handled according to the experiences of dealing with the inverse problems. The MFS with unsteady Stokeslets can directly solve the semi-infinite domain problem without any iteration or regularization. In the next, the unsteady Stokes flow with various driven boundaries, square cavity and circular cavity will be solved. The variations also clearly demonstrate the phenomena of flow system with one moveable piece, two rotating belts and eccentric rotating cylinder. Finally, the Eulerian-Lagrangian method of fundamental solutions (ELMFS), which is a combination of the MFS and the Eulerian-Lagrangian method (ELM), is applied to solve the Navier-Stokes equations with moving rigid body. Further, the benchmark Navier-Stokes flow in lid-driven cavity is validated by the ELMFS based on the unsteady Stokelslets with Re=10 and Re=50. Finally, the phenomena of Navier-Stokes flow with a moving cylinder will be obtained and simulated. The ELMFS can be used clearly and directly to describe the moving rigid body phenomena in the fluid. The numerical results show good agreements with immersed-boundary finite element method (FEM).