以高斯幅狀基底函數之無網格方法數值模擬完全非線性水面波

Abstract

本論文利用無網格數值方法，建立一個能模擬完全非線性水面波的數值模式。在空間方面，以拉普拉斯方程式(Laplace Equation)基本解(Fundamental Solution) 之線性組合擬合流體速度勢(Velocity Potential)，將前述基本解之中心(Source Point)置於計算邊界以外的位置，僅需由適當少數邊界點上的邊界條件即可進行求解，不需要積分，也不需要處理任何奇異點(Singularity)。在時間方面，利用二階中央差分，將自由液面邊界條件離散成為顯式的差分式，不需要迭代，也不需在時間上做積分，即可推測下個時間步（Time Step）求解所需之自由液面邊界條件。此種離散方式亦稱為跳蛙法（Leap Frog Scheme）。而推算下個時間步自由液面邊界條件需要自由液面梯度，則透過高斯幅狀基底函數(Gaussian Radial Basis Function)來擬合自由液面的水位高程，進而得到求解所需之條件。 本論文先模擬孤立波在三維渠道內之傳遞，並檢查質量與能量之守恒，來證明本數值方法之正確性。然後，再分別模擬二維及三維的問題，來說明本數值方法之適用性。各項數值模擬，均與前人研究之解析解、數值解或實驗結果比較，以說明模擬結果之準確可靠。

A numerical model for fully nonlinear free surface waves is developed in present study, by applying a boundary-type meshless approach with leap frog time marching scheme. Adopting Gaussian Radial Basis Functions to fit the free surface, a non-iterative approach to discretize the nonlinear free surface boundary conditions is formulated. Using the fundamental solutions of the Laplace equation as the solution form of the velocity potential, free-surface wave problems can be solved by collocations at only a few boundary points since the governing equation is automatically satisfied. The accuracy of present method is verified by comparing the simulated propagation of a solitary wave in a three dimensional flume with an exact solution. The applicability of present model is illustrated by employing it to simulate both two dimensional and three dimensional problems. Good agreement can be found in comparing the results of present numerical model with other schemes, as well as with analytic solutions and available experimental data.