求解奈維爾-史托克思方程之向量勢能演算式：利用四階準確度局部微分積分法

Abstract

在本論文中為了模擬三維流場問題，我們選擇使用三維不可壓縮奈維爾-史托克思之向量勢能演算式為控制方程。向量勢能演算式為許多不需處理壓力項之解法中的一種，主要是藉著對動量方程取旋度，再以拉普拉斯運算子之向量勢能取代渦度傳輸方程中之渦度奈維爾-史托克思方程可以轉成只存在一種變數之向量勢能的四階偏微分方程(PDE)。與其他不需求解壓力項演算式比較，向量勢能演算法比較簡單，準確，更者，其計算求解比較有效率。對於向量勢能解題所需之邊界條件方面，除了應用已經被提出求解封閉流場問題的邊界條件，我們更進一步利用史托克思理論求解出非封閉流場所需之邊界條件。據我們所知，這是一個創新的突破。為了準確求解四階控制方程，本研究使用四階精度之局部微分積分法(LDQ)求解問題。通過非均勻格網的使用，我們可以有效率求得流場的數值解。為了驗證本數值算則求解四階控制方程的能力，我們利用兩個基準解的問題來檢驗本文所提演算式，包括二維穴室流場，以及二維後向階梯流場的模擬。結果確認本文所提演算式的正確性與可行性。藉著成功的利用本演算式求解二維流場問題，我們更進一步應用本演算式求解三維基準解問題，包括三維穴室流場之模擬，以及三維後向階梯流流場之模擬。結果與基準解吻合，這不但證明本演算法可以用來求解向量勢能演算式，也驗證了本文所提演算法的正確性。我們更進一步藉著數值模擬具體視覺化向量勢能之結構，而通過比較，也可以看出向量勢能與流線方程的不同。總結而言，我們成功的利用四階精確度區域微分積分法求解奈維爾-史托克思方程之向量勢能演算式，並可應用於求解非封閉流場的問題。從以前文獻看來，沒有任何一個文獻曾經提出過相似的想法，可以確信的是，本論文的創新結果對於三維流場的模擬提供了一個可行之方法。

To simulate three-dimensional flow problems in this thesis, the vector potential formulations of three-dimensional incompressible Navier-Stokes are chosen to govern the motion of fluid flow. The vector potential formulation belongs to one of the pressure-free algorithms which are obtained by taking curl to the momentum equations. By replacing the vorticity with Laplacian vector potential to the vorticity transport equations, the Navier-Stokes equations are transformed to fourth-order partial differential equations (PDEs) with one variable---vector potential. Comparing with other pressure-free algorithms, vector potential formulations are simpler and more accurate, and, moreover, the computation is more efficient. To the boundary conditions of vector potential, except the presented defined boundary conditions for confined flow, we further improved the algorithm to through-flow problem by introducing the concept of Stokes' theorem. To author's best knowledge, this improvement is groundbreaking. To accurately approximate these fourth-order governing equations, fourth-order-accuracy localized differential quadrature (LDQ) methods are employed. Through adopting the non-uniform mesh grids, the solutions can be obtained efficiently. To examine the ability of the proposed scheme to fourth-order governing equations, two benchmark problems are considered, including two-dimensional cavity flow problems and backward-facing step flow problems. The results show the accuracy and feasibility of the proposed scheme. By the successful implementation of the present scheme to two-dimensional flow problems, the proposed scheme is further employed to solve three-dimensional benchmark problems, including three-dimensional driven cavity flow problems and backward-facing step flow problems. The good performance not only demonstrates that the proposed scheme is able to be employed to solve the vector potential formulation, but also validates the correctness of the presented formulation. Furthermore, we specifically visualized the contour of vector potential by numerical simulation. The comparison between vector potential and stream functions show the difference of these two algorithms. Conclusively, the vector potential formulations of Navier-Stokes equations are successfully used to simulate the three-dimensional fluid motion, especially the fluid flow problems with through-flow. Through the application of the fourth-order-accuracy of localized differential quadrature method, the solutions can be accurately obtained, and the vector potential can be specifically visualized. From the previous literatures, no literature has ever presented the similar idea of this research. It is convinced that the groundbreaking findings in this thesis can provide a feasible way to simulate three-dimensional fluid motion.