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研究生： 徐偉烈
Wei-Lieh Hsu
論文名稱： Parallel Domain Decomposition Method for the Finite Element Approximation of Two-dimensional Navier-Stokes Equations with Slip Boundary Condition
指導教授： 黃楓南
Feng-Nan Hwang
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
論文出版年： 2014
畢業學年度： 102
語文別： 英文
論文頁數： 76
中文關鍵詞： 有限元素 、納維爾-史托克斯 、滑移邊界條件
外文關鍵詞： Slip Boundary Condition
相關次數： 點閱：11  下載：0 
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    • 一般流體力學在做數值模擬時，通常採用無滑移邊界條件，然而近來部分的實驗卻證實了在微小尺度或其他狀態下可能與事實違和。許多學者提出可以使用滑移邊界條件來取而代之，如此更加能夠更加真實模擬出事實的模樣。所以我們推測滑移邊界條件會改變典型流體的模樣，故本篇論文在假設已知有滑移的狀態下，使用滑移邊界條件來進行數值模擬，來檢視滑移對流體產生改變。
    在這篇論文中，我們先簡單介紹滑移邊界條件的背景以及我們所採用的模型，接著導出含邊界條件納維爾-史托克斯方程組的變分形式及使用牛頓-克雷洛夫-施瓦茨演算法解的大型稀稀疏非線性系統。我們使用一個具有解析解的例子來驗證我們的平行流體程式，並且我們將應用在頂部驅動穴流及突擴管流這兩個流體的基準問題上。我們藉由數值模擬來探究滑移對流體所影響的物理性質，例如發生分歧現象的雷諾數，以及分析解線性與非線性系統時的效能。

    In general, we usually impose the no-slip boundary condition when simulating the problem of fluid dynamics. But recently, some experimental evidences this condition is not applicable in small-scale system or other situations. Many researchers propose to use the slip boundary condition instead. Then the result would be consistent with real appearance. Thus, we speculate the typical appearance would change when we apply the slip boundary condition. Therefore, we assume there exist slip behavior. We simulate with slip boundary condition to observe the difference between no-slip.
    In this thesis, we first introduce the background of slip boundary condition and the model we used. Then we derive the variational formulation of the Navier-Stokes equation with the slip boundary condition and the resulting large, sparse nonlinear system of equations is solved by the parallel Newton-Krylov-Schwarz algorithm. We validate our parallel fluid code by considering a test case with an available analytical solution. We apply parallel Galerkin/least squares finite element flow code with the slip boundary condition to two benchmark problems -- lid-driven cavity flows and sudden expansion flows. We investigate numerically how the slip condition effects the physical behavior of the fluid flows, including the critical Reynolds number for the pitchfork bifurcation and the performance of the nonlinear and linear iterative methods for solving resulting linear sparse nonlinear system of equations.

    Tables ix Figures xi 1 Introduction 1 2 Navier-Stokes equations with slip boundary condition and their variational formulation 5 3 Solution algorithm 9 3.1 Galerkin/least-square finite element formulation 9 3.2 Basis functions of slip boundaries 10 3.3 Pseudo-transient Newton-Krylov-Schwarz method 13 3.4 Software development 14 4 Numerical results and discussion 15 4.1 Code validation 15 4.2 Applications 21 4.2.1 Lid-driven cavity flows 21 4.2.2 Sudden expansion flows 28 4.3 Implementation performance 42 5 Conclusions and future works 44 Bibliography 45 Appendix A 47 Appendix B 48 Appendix C 52

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