Numerical Simulation of 2D Free-surface Potential Flows Using Modified Finite Point Method

本文以垂直二維水槽為研究對象，探討當矩形水槽受到水平強制振動時，內部液體之非線性沖激(sloshing)行為；以及藉由數值水槽的造波，模擬孤立波(solitary wave)於等水深以及變水深二維水槽之流體運動情形。本文假設槽內液體滿足勢能流場，以無網格法中的修正有限配點法(modified finite point method, MFPM)，藉由局部多項式(local polynomial method)來近似連續函數，求解拉普拉斯方程式(Laplace equation)，以得到計算格點上之速度（velocity）與速度勢（velocity potential）的分布，再配合二階中央插分法處理時間微分項，及Lagrangian座標描述計算出自由液面上質點的位置。 本研究模擬計算了水槽受到水平簡諧外力時，水槽內部自由液面的震盪情況。模擬結果顯示，本模式在模擬自由振盪問題與文獻上之資料（Lin & Liu, 2008）相當地吻合，模擬的計算效率也更高。 等水深時，本模式模擬孤立波之前進，波形不會變形或減衰，驗證本模式的準確性。在變水深的情況下，孤立波的波形會隨著水深變淺而尖銳化，最後形成不穩定的翻轉（overturning）。 本文數值模擬的結果與文獻(Grilli and Subramanya,1996)的資料作比對，檢驗其波形，在波形的比較上，本研究與Grilli and Subramanya （1996)的結果呈現良好的吻合，足以說明本模式可以描述捲浪型碎波。This study focuses on applicability of a numerical method in two problems, first, nonlinear liquid sloshing inside a two-dimensional tank that is subjected to horizontal forced oscillations, and second, a solitary wave propagating over a constant depth and over water with a gentle slope. The liquid flow is assumed to satisfy potential flow theory. In this study, a meshless numerical method which is named modified finite point method (MFPM) is employed. Based on collocation, this method uses polynomials as the local solution form needed in the collocation approach. The MFPM of Laplace equation is applied to solve the potential and velocity distributions at the grids of the computational domain and on the boundaries of the tank. An explicit time marching technique is developed by utilizing the leap-frog second-order central difference scheme. Lagrangian description are used to compute the position and the relative unknowns of the liquid particles on the free surface. In the present study, the free surface displacements in a tank due to horizontally harmonic forced oscillation of two-dimensional rectangular tank are computed. It is shown that present numerical results agree very well with other research results (Lin & Liu, 2008). Present numerical model also demonstrated better computational efficiency. For a solitary wave propagating over a water of constant depth, accuracy of present model is verified by observing no deformation and decay of the numerical wave form. When present model is applied to the case of a solitary wave over a water of a gentle slope, the shape of the wave steepens as it propagates over the slope and then turns over to break. The calculated wave shapes are compared with numerical results of Grilli and Subramanya (1996). Present model has demonstrated its capability in simulating a plunging solitary wave