Mesh-Reduction Numerical Methods for Solving Low-Reynolds-Number Flow Fields

• Main Author: 李虹慧
• Other Authors: Chia-Ming Fan
• Format: Others
• Language: zh-TW
• Published: 2011
• Online Access: http://ndltd.ncl.edu.tw/handle/22132619018308696063

碩士 === 國立臺灣海洋大學 === 河海工程學系 === 99 === In this thesis, mesh-reduction numerical methods for solving low-Reynolds-number incompressible flow fields are proposed, and they can be described respectively as follows：1. Based on the Laplacian decomposition, the Stokes equations are recast as three Laplace equations. Using the modified collocation Trefftz method (MCTM), the numerical solutions of these three Laplace equations can be expressed by linear combination of the T-complete functions. The proposed meshless scheme is verified by analyzing direct and inverse Stokes problems. 2. By introducing the stream function, the Stokes equations are converted to a biharmonic equation, and then the MCTM is adopted to solve this equation. Using the characteristic length in MCTM can stabilize the numerical scheme and obtain highly accurate results. Inverse Stokes problems are stably analyzed by the proposed meshless method. 3. Utilizing the Laplacian decomposition, the Stokes equations are recast as three Laplace equations, by boundary knot method the numerical solutions of these three Laplace equations can be expressed as a linear combination of nonsingular fundamental solutions. In these three boundary-type meshless approaches, the unknown coefficients in the solution expressions are found by satisfying the boundary conditions at boundary collocation points. In addition, noises are added into the boundary conditions to demonstrate the outstanding stability of the proposed meshless schemes for dealing with the inverse Stokes problems. 4. The well-known Navier-Stokes equations are converted to a nonlinear biharmonic equation by using the stream function. The low-Reynolds-number flow fields problem are transformed to a system of algebraic equations by using the novel finite difference method. Then, by using the exponentially convergent scalar homotopy algorithm, the system of nonlinear algebraic equations is converted to a system of ordinary differential equations which will be solved by numerical integration method. In the novel finite difference method, only simple rectangular mesh is needed even for irregular physical domain. Hence, the task for generating mesh can be greatly simplified. On the other hand, the exponentially convergent scalar homotopy algorithm is exponential convergent and insensitive to initial guess. Therefore, it is very efficient to solve the system of nonlinear algebraic equations in novel finite difference method by the exponentially convergent scalar homotopy algorithm. In this thesis, several numerical examples are provided to validate the proposed numerical schemes for the numerical efficiency, stability and suitability.