以區域微分積分法求解不規則計算域問題及其在流體力學及熱傳播上的應用

Abstract

本研究提出一個改良的區域微分積分（local differential quadrature, LDQ）數值模式，以求解不規則計算域問題並應用於流場及熱傳播問題的模擬。此處提出的區域微分積分模式乃改良自微分積分法，是一個具備高階精確度及良好計算效率的數值方法。然而傳統的微分積分法受限於計算矩陣非常條件不良（ill-conditioned）、對計算網格型式過於限制以及不易處理不規則邊界等缺點，讓這個方法無法廣泛推廣於數值模擬的領域。本研究利用區域化的數值公式及網格截切的特殊技術，順利的解決了以上幾個微分積分法的重大缺陷。在求解二維及三維的不規則幾何形狀計算域的卜易松及赫姆霍茲等偏微分方程式的數值測試中，可以觀察到此改良過的數值模型成功的保留了傳統微分積分法的高階精確度，並加強了良好計算效率的優點。在求解許多經典的二維及三維計算流體力學（computational fluid dynamics, CFD）問題中，證實了本研究所提出的區域微分積分方法的良好計算效能。例如二維及三維的矩形穴室流（cavity flow）問題，流經圓柱的流場計算及後向階梯（backward facing step）計算域的強制對流流場和溫度場的數值分析。本數值模式在這個研究中，更被成功地推廣應用於二維及三維流體力學及熱傳播在不規則計算域中的模擬。例如求解流經加熱波浪狀底床的二維渠流流場及溫度場問題，還有內嵌球形障礙物的三維穴室流場模擬。另外，與本數值模式有關的無網格的加密計算技術，以及極具發展潛力的移動邊界處理技術，也都在這個研究中被提出。藉由一些數值試驗，證實本數值模式可優異的用來解決需利用無網格加密計算技術及移動邊界技術之流場與熱流場之問題。

This study proposes a local differential quadrature (LDQ) algorithm to handle the irregular domain problems and applies it to solve the flow and heat transfer problems. The proposed LDQ method is developed from the classic differential quadrature (DQ) method, a numerical method with high order accuracy and economic efficiency. However the ill-conditioned matrix, sensitive computational mesh and restricted geometry have limited the extensive applications of the conventional DQ method. This study successfully overcomes those above mentioned disadvantages by adopting the localized numerical schemes and a grid cutting technique. The numerical tests of solving the two- and three-dimensional Poisson and Helmholtz equations have verified that the proposed LDQ model maintains the high order accuracy and enhances the economical efficiency of the DQ method. The present model is also extended to apply to the two- and three-dimensional fluid mechanics and heat transfer problems. The good performance of the LDQ method is demonstrated by solving many classical computational fluid dynamics (CFD) problems such as the two- and three-dimensional lid-driven cavity flows, flow around fixed circular cylinder and forced convection problem in backward facing step channel. The applications of two-dimensional channel flow over heating ripple beds and three-dimensional cavity flows with a fixed sphere have also revealed the excellent capability of the proposed LDQ for simulating the flow and heat transfer problems with irregular domains. This investigation also undertakes the techniques of the refining process of meshless computational nodes, and the treatment of moving boundaries related to the proposed model. The proposed LDQ algorithm has also demonstrated the excellent performance to deal with the meshless refining process and moving boundary problems in irregular domains by some numerical experiments.