以修正有限配點法模擬水波港池共振問題

Abstract

本文主要是以有限配點無網格數值計算方法，探討水波造成港池內部共振的現象。由於水波入射港池造成的共振現象，會嚴重影響港埠設施及港內船舶的安全，因此，港池共振分析在港灣規劃設計中，便成為相當重要需考慮避免發生情況。本研究之目的即藉由數值模擬分析含防波堤港池之共振效應，並以解析解來驗證本模式的準確性，希能建立計算數值模式作為港灣工程設計時的分析應用工具。 本研究的控制方程式為緩坡方程式(mild-slop equation)，在等水深的情況下，故原方程式可簡化為赫姆霍茲方程式(Helmholtz equation)。數值計算方法上則採用無網格法中(Wu & Tsay,2011)的修正有限配點法(Modified finite point method, MFPM)，其乃利用局部多項式(Local polynomial method)來近似所欲求解之函數。在先前的研究中，修正有限配點法已可準確計算計算點上的函數值及其偏導數值。本文以計算區域區隔的方法，提升防波堤存在時波場計算的準確性。無網格法佈點容易，對於不規則形狀更能顯現其效用。 在模式驗證上，以三種港池類型，其中包含有防波堤之港池，探討港池區域的波高放大係數（R）與入射波週波數（k）乘上港池半徑（a）或乘上港池長度（l）的關係，本模式之正確性，由模擬計算結果與Mei and Petroni (1973)所推導得出之解析解進行比較，得到驗證。本模式更進一步將模擬計算結果以港池振盪之二維平面等振幅線圖及流速向量分布圖展示，可以提供港灣設計的依據。

The objective of this study focuses on extending the modified finite point mesh-less numerical model to the analysis of the harbor resonance induced by water waves. Harbor resonance is a phenomenon caused by ocean waves intruding into the harbor on coastal areas. When it occurs, it would seriously affect the safety of the ships in the harbor. Therefore, analysis of harbor resonance is an important matter in the process of harbor planning and design. The purpose of this study is to verify present model’s accuracy and applicability by comparing with available analytical solutions. Present numerical model after verifications can apply to harbor engineering practices. In this study, the governing equation is the mild-slop equation. It can be simplified to the Helmholtz equation when water depths remain constant in the computational domain. A special mesh-less numerical methods, namely, modified finite point method (MFPM) (Wu & Tsay, 2011) is employed in present study. Based on collocation, this method uses polynomials as the local solution form needed in the collocation approach. In previously research, it has been shown that MFPM can efficiently calculate the solutions and the partial derivatives of the unknown function. When breakwaters appear in the computational domain, a concept of subdomains is designed to obtained accurate solutions. Present mesh-less numerical method is easy to generate computational points, especially in irregular regions. To verify accuracy of present numerical model, examples of three types of harbors, with or without breakwaters, are calculated to obtain amplification factor at a specific point for different parameters, which are products of incident wave numbers and radius of circular harbor or length of a rectangular one. Present numerical results are compared with the analytical solutions by Mei and Petroni(1973),. Very good agreements are observed. Two-dimensional contours of wave amplitude and graphs of velocity vectors are demonstrated in these examples.