Method of Fundamental Solutions for Solving the Velocity-Vorticity Formulation

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Project title

Code/計畫編號

NSC98-2218-E019-001

Translated Name/計畫中文名

以基本解數值方法求解速度渦度方程式

Project Coordinator/計畫主持人

Chia-Ming Fan

Funding Organization/主管機關

National Science and Technology Council

Department/Unit

Department of Harbor and River Engineering

Website

https://www.grb.gov.tw/search/planDetail?id=1817095

Year

2009

Start date/計畫起

01-02-2009

Expected Completion/計畫迄

31-07-2009

Bugetid/研究經費

126千元

ResearchField/研究領域

土木水利工程

"本計畫將測試目前最新的無網格數值計算方法，也就是基本解數值計算方法，求解速度渦度方程式。直接求解不可壓縮黏性流體的流動控制方程式會遇到壓力條件無法明確給定的問題，因此，使用渦度的觀念可以將控制方程式轉換成速度渦度方程式，此一方程式系統可以避免直接求解壓力的問題，因此本計畫將求解速度渦度方程式來分析不可壓縮黏性流體的流動。基本解數值方法為目前新發展的無網格數值計算方法之一，由於在計算過程中不需要記錄點與點的關係，因此大大的提高了計算的效率與降低電腦資源的需求量。本計畫預計以有限差分法先將時間軸離散，將每一時刻的渦度傳輸方程式轉換成對流擴散反應方程式，再採用基本解數值方法求解每一時刻的渦度分佈。一旦渦度求出來之後就可以當作速度卜松方程式的非齊性項，並用基本解數值方法再將速度分佈求解出來。此一數值方法具有高精準度且快收斂的特性，因此在求解流場上不僅避免記錄網格的關係，還可以利用非常少的計算點數求出相當精確的答案。所以本計畫預計以基本解數值方法求解速度渦度方程式，分析此一無網格數值計算方法在模擬不可壓縮流體流動的可行性與效率。""The objective of this project is to analyze the viscous incompressible fluid flow by the method of fundamental solutions. In most practical situation, the conditions for pressure can not be specified clearly when the Navier-Stokes equations are considered. Therefore, by introducing the vorticity, the Navier-Stokes equations can be transformed to the velocity-vorticity formulation. Hence, the problems of pressure can be avoided. In this project, we will test the efficacy of the method of fundamental solutions for solving the velocity-vorticity formulation. The method of fundamental solutions is one of the promising meshless methods, since the nodal connectivity is no more needed in the simulation. It can greatly simplify the computation resource required and increase the efficiency of the simulation. First, the finite difference method will be used for discretizing time axis. Hence, the time-dependent vorticity transport equation will be transformed to the convection-diffusion-reaction equations for every time step. The method of fundamental solutions will be used for solving the vorticity. Then the Poisson equations can be solved by using the same meshless method. From previous researches, the method of fundamental solutions can obtain very accurate numerical results even very few nodes are used. Therefore, the objective of this project is to develop a numerical scheme, based on the method of fundamental solutions, to analyze the velocity-vorticity formulation."

Keyword(s)

無網格方法
基本解方法
速度渦度方程式
高精準度
快收斂
meshless method
method of fundamental solutions
velocity-vorticity formulation
high accuracy
fast convergence

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Method of Fundamental Solutions for Solving the Velocity-Vorticity Formulation

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