利用無網格法求解二維及三維不可壓縮黏性流場之研究

Meshless Methods for 2D and 3D Incompressible Viscous Flows

詹淑菁(Shu-Jing Jan)

指導教授 ： 楊德良

國立臺灣大學/工學院/土木工程學研究所/博士(2004年)

https://doi.org/10.6342/NTU.2004.01873

摘要

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。

關鍵字

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

並列摘要

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.

並列關鍵字

velocity-vorticity formulation ； Stokes flows ； meshless methods ； incompressible Navier-Stokes equations ； multiquadrics method ； method of fundamental solutions

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