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  • 學位論文

利用修正有限配點法對二維自由液面勢流流場之數值模擬

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

指導教授 : 蔡丁貴

摘要


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。

並列摘要


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.

參考文獻


黃翊恆 (2011),”以局部化徑向基底函數微分積分法求解二維自由液面問題”國立台灣大學土木工程學研究所碩士論文
吳南靖 (2008),”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”國立台灣大學土木工程學研究所博士論文
楊琛灃 (2009),”以完全非線性無網格方法數值模擬三維邊界移動造波”國立台灣大學土木工程學研究所碩士論文
張裕弦 (2008),”孤立波溯升之研究”國立成功大學水利及海洋工程學研究所博士論文
王冠閔 (2004),”斜坡地形變化對孤立波溯升高度的影響”國立成功大學機械工程學研究所博士論文

被引用紀錄


陳 捷(2015)。以權重最小二乘局部近似法應用於不恆定淺水流水理數值模擬之研究〔碩士論文,國立臺灣大學〕。華藝線上圖書館。https://doi.org/10.6342%2fNTU.2015.10792

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