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研究生:詹淑菁
研究生(外文):Shu-Jing Jan
論文名稱:利用無網格法求解二維及三維不可壓縮黏性流場之研究
論文名稱(外文):Meshless Methods for 2D and 3D Incompressible Viscous Flows
指導教授:楊德良楊德良引用關係
指導教授(外文):Der-Liang Young
學位類別:博士
校院名稱:國立臺灣大學
系所名稱:土木工程學研究所
學門:工程學門
學類:土木工程學類
論文種類:學術論文
論文出版年:2004
畢業學年度:92
語文別:英文
論文頁數:159
中文關鍵詞:MFS納維爾史托克方程式圓形穴室流方形穴室流無網格法MQ三維不可壓縮黏性流場速度-渦度法
外文關鍵詞:velocity-vorticity formulationStokes flowsmeshless methodsincompressible Navier-Stokes equationsmultiquadrics methodmethod of fundamental solutions
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本論文以速度-渦度表示式為主體的Navier-Stokes方程式,作為主要研究主題,並以近年來才開始發展的無網格數值方法,取代以往所使用的傳統有網格數值方法。根據式中雷諾數為極小值的Stokes flow,到屬於低雷諾數時的流況,探討在這些流況下,以無網格數值方法所模擬的結果,是否合乎以往的理論、實驗或其他數值方法所求得的研究成果,並驗證新的數值方法在計算流體領域使用的可行性。
本研究採用二維圓形、方形及三維方形穴室流作為驗證的流場,分別以無網格法中的method of fundamental solutions(MFS)數值方法去模擬Stokes方程式,及利用multiquadrics(MQ)數值方法求解不可壓縮黏性流場,包括Stokes流場和Navier-Stokes流場。
相較固有的數值方法而言,無網格數值方法的本質上是採用佈點方式模擬。因此,在驗證例子裡,以邊界佈點形式的MFS數值方法對二維的圓形穴室流、方形穴室流各以200點及80點模擬;三維方形穴室流則以1176點佈點求解。另一方面,以MQ數值方法模擬二維及三維方形穴室流,分別以雷諾數趨近無窮小、二維Re=100、400及三維Re=10,50,100三大部分為模擬流況。依據不同的流況、及電腦設備,所採取的點數也不盡相同,而點數分佈範圍為441點到3375點。
就模擬結果比較而言,發現儘管是以少數點佈點方式模擬,所得到的結果不僅和其他數值方法求得的結果一致,也大致符合實際的物理現象。
依目前的研究結果顯示,若要模擬更為複雜的流況,除了增加佈點會造成對電腦軟硬體的需求增加外,MFS的距離係數及MQ的空間參數的決定,會是另一個需詳細考量的問題。因此,在本研究裡,僅就部分例子之結果加以討論。
The development of robust and efficient numerical algorithms for both steady-state and transient simulations of the Stokes equations and Navier-Stokes equations for two-dimensional and three-dimensional incompressible viscous flow is an active research field. A meshless method based on the multiquadrics (MQ) method has been developed to solve the 2D and 3D Stokes flows and Navier-Stokes equations in velocity-vorticity formulation. Numerical results are also reported using the method of fundamental solutions (MFS) in order to compare its performance with the MQ method. The method of fundamental solutions (MFS) based on the Stokeslet is successfully implemented for the numerical solution of Stokes flow problems. The MFS does not require a discretized interior domain and boundary integration to obtain the solutions for the flow variables. We implemented the MQ method to solve the Stokes equations and Navier-Stokes equations for two and three-dimensional flow problems. The method employed a coupled numerical solution algorithm by combining the boundary equations along with the governing equations to form a single global matrix for all the field variables. The computation of the velocity and the vorticity variables are completed by satisfying the continuity equation for the velocity field and the solenoidal constraint for the vorticity field. The multiquadrics method is found to be an efficient scheme for low Reynolds number flows, which has been demonstrated in this study. Two-dimensional and three-dimensional flow solutions for Stokes equations in a circular cavity, square cavity and cubic cavity are established and compared with available benchmark solutions by using both the MFS and MQ algorithms.
誌謝 i
摘要 ii
Abstract iii
Table of Contents iv
List of Tables vii
List of Figures viii
Symbols xv
Chapter 1 Introduction 1
1.1 Mathematical modeling 1
1.2 Numerical methods 2
1.3 Meshless methods 4
1.4 Objectives of the present work 6
1.5 Organization of the dissertation 7
Chapter 2 Literature Review 8
2.1 Method of fundamental solutions 8
2.2 Multiquadrics method 10
2.2.1 Stokes flow 10
2.2.2 Navier –Stokes equations 13
Chapter 3 Governing equations 18
3.1 Navier-Stokes equations in the primitive variable form 18
3.2 Navier-Stokes equations in the velocity-vorticity form 20
3.3 Stokes flow equations in the velocity-vorticity form 22
3.4 Stokeslet equations in the primitive variables form 24
3.5 Initial and boundary conditions 28
Chapter 4 Numerical solutions using multiquadrics method and method of fundamental solutions 30
4.1 Numerical solutions of 2D and 3D Stokes flow using MFS 30
4.2 Numerical solutions of Stokes flow using MQ 33
4.2.1 2D Stokes flow using MQ 33
4.2.2 3D Stokes flow using MQ 35
4.3 Numerical solutions of Navier-Stokes equations using MQ 39
4.3.1 2D Navier-Stokes equations using MQ 39
4.3.2 3D Navier-Stokes equations using MQ 43
Chapter 5 Results and discussions of method of fundamental solutions for Stokes flows 55
5.1 MFS and Stokeslet solutions 55
5.2 2D Stokes flow in a circular cavity 55
5.3 2D Stokes flow in a square cavity 63
5.4 2D Stokes flow in a cavity with wave-shaped bottom 69
5.5 3D Stokes flow in a cubical cavity 73
Chapter 6 Results and discussions of multiquadrics method for Stokes flows and Navier-Stokes equations 80
6.1 MQ solutions for 2D Stokes flows 80
6.1.1 2D results of Stokes flow in a square cavity using MQ 80
6.1.2 2D results of Stokes flow in a circular cavity using MQ 82
6.2 MQ solutions for 3D Stokes flow in a cubic cavity 85
6.3 MQ solutions for Navier-Stokes equations 94
6.3.1 MQ solutions for 2D Navier-Stokes equations 94
6.3.2 MQ solutions for 3D Navier-Stokes equations 96
6.4 Effect of Space parameter c 109
Chapter 7 Conclusions 122
7.1 Main conclusions 122
7.2 Limitations and scope for further research126
References 129
Appendix 1:Multiquadrics with radial basis function 135
A.1.1 Meshless numerical methods 135
A.1.2 Radial basis function 137
A.1.3 Illustration of the MQ method 141
Appendix 2: Fundamental solutions using Stokeslet 152
Appendix 3: List of publications 158
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