# 臺灣博碩士論文加值系統

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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 ixFigures xi1 Introduction 12 Navier-Stokes equations with slip boundary condition and theirvariational formulation 53 Solution algorithm 93.1 Galerkin/least-square finite element formulation 93.2 Basis functions of slip boundaries 103.3 Pseudo-transient Newton-Krylov-Schwarz method 133.4 Software development 144 Numerical results and discussion 154.1 Code validation 154.2 Applications 214.2.1 Lid-driven cavity flows 214.2.2 Sudden expansion flows 284.3 Implementation performance 425 Conclusions and future works 44Bibliography 45Appendix A 47Appendix B 48Appendix C 52
 [1] ParaView homepage. http://www.paraview.org/.[2] CUBIT homepage. https://cubit.sandia.gov/, 2008.[3] M. P. Brenner E. Lauga and H. A. Stone. In Microfluidics: The No-Slip Boundary Condition, chapter 15. J. Foss, C. Tropea and A. Yarin and Springer, 2005.[4] L.P. Franca and S.L. Frey. Stabilized finite element methods. II: The incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering, 99(2-3):209–233, 1992.[5] Q. He and X.-P. Wang. Numerical study of the effect of Navier slip on the driven cavity flow. Zeitschrift für Angewandte Mathematik and Mechanik, 68:856–871, 2012.[6] G. Pineau J.-L. Guermond, C. Migeon and L. Quartapelle. Start-up flows in a three-dimensional rectangular driven cavity of aspect ratio 1 : 1 : 2 at Re = 1000. Journal of Fluid Mechanics, 450:169–199, 2002.[7] V. John. Slip with friction and penetration with resistance boundary conditions for the Navier-Stokes equations – numerical tests and aspects of the implementation. Journal of Computational and Applied Mathematics, 147:287–300, 2002.[8] V. John and A. Liakos. Time-dependent flow across a step: the slip with friction boundary condition. International Journal for Numerical Methods in Fluids, 50:713–731, 2006.[9] G. Karypis. METIS web page. http://glaros.dtc.umn.edu/gkhome/metis/parmetis/overview.[10] A. Kundt and E. Warburg. On friction and thermal conductivity in rarefied gases. Philosophical Magazine, 50:53, 1875.[11] J. C. Maxwell. On the condition to be satisfied by a gas at the surface of a solid body. Scientific Papers, 2:704, 1879.[12] C.L. Navier. Mémoire sur les lois du mouvement des fluides. Mem. Acad. R. Sci. Paris., 6:389–416, 1823.[13] T. Mullin R.M. Fearn and K.A. Cliffe. Nonlinear flow phenomena in a symmetric sudden expansion. Journal of Fluid Mechanics, 211:595–608, 1990.[14] W. D. Gropp D. Kaushik M. G. Knepley L. C. McInnes B. F. Smith S. Balay, K. Buschelman and H. Zhang. PETSc users manual. Technical Report ANL- 95/11 - Revision 3.5, Argonne National Laboratory, 2014.[15] W. D. Gropp D. Kaushik M. G. Knepley L. C. McInnes B. F. Smith S. Balay, K. Buschelman and H. Zhang. PETSc Web page. http://www.mcs.anl.gov/petsc/, 2014.[16] D. Keyes R. Melvin X.-C. Cai, W. Gropp and D. Young. Parallel Newton-Krylov-Schwarz algorithms for the transonic full potential equation. SIAM Journal on Scientific Computing, 19:246–265, 1998.
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 1 發展求解不可壓縮Navier-Stokes及相關組成方程式之有限元素模型

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