【作者】 王小军；

【导师】 祝家麟；

【作者基本信息】 重庆大学 ， 计算数学， 2010， 硕士

【摘要】 不可压缩粘性流体动力学方程组的Navier-Stokes方程在粘性很大,即雷诺数很小的情况下,可线性化为Stokes方程。考虑到边界的粘附条件,形成Stokes问题。鉴于边界元方法在处理Stokes问题中的连续性方程或者不可压缩条件时明显优于其它类型的方法,而且可以把计算速度场和计算压力分别开,因而是求Stokes问题数值解的理想方法。本文用边界元方法求解平面有界区域内带有非闭合直线段或曲线段的开边界的Stokes问题,即带有屏障的二维流体流动问题。求解这类问题的难点在于在数值逼近时需要妥善处理解在边界的端点附近存在的奇异性,本文对于含有开边界端点的边界单元,采用特别的插值函数,以模拟其固有的奇异性。我们利用对应于Stokes算子的Green公式和基本解所推导出来的第一类Fredholm向量积分方程。建立起与之等价的边界变分方程,采用Galerkin边界元求解,得出单层位势的向量密度,进而根据速度的边界积分表达式的离散形式计算流场中任意点的流速值。本文首先阐述了复连通区域二维Stokes问题的Galerkin边界元解法的思路,利用Fortran语言编制程序计算速度场,用Matlab画出相应的流线图,以检验程序的可靠性。然后主要针对有界区域内含有开边界的问题进行数值模拟。由于Galerkin边界元方法需要计算二重积分,本文推导出了带普通插值形函数和奇异形函数的积分公式,对存在奇异性的单元上的第一重积分中采用解析积分,针对第二重积分推导出了带奇异积分权函数的Gauss积分公式,对不存在奇异性的单元所有积分都采用数值积分。论文用若干数值算例模拟了复连通区域上以及含有开边界的有界区域上不可压缩粘性流体的绕流。更多还原

【Abstract】 Stokes equation is the linearization of Navier-Stokes equation, which is the governing equation of incompressible viscous flow, with small Reynolds number. It forms Stokes problem with the viscous boundary conditions attached. Among so many numerical methods to solve this problem, the boundary element method is an ideal method, since it is easy to handle the continuity equation, or the incompressibility condition,moreover, computing the velocity and computing the pressure can be separated in the solution procedure. In this paper,we solved the Stokes problem in a bounded plane region with non-closed line or curve segment as an open boundary,which is a two-Dimensional screen problem in fluid flow. The difficulty in solving this kind of problems is how to deal with the singularity near the endpoints on the open boundary. We have established the singular element with singular shape function while the element contains endpoint in order to simulate the inherent singularity.We establish the variational formulation based on the Fredholm vector integral equations of the first kind, which are derived by the Green’s formula and the fundamental solution corresponding to Stokes operator. Then, we solve the boundary variational equation by Galerkin Boundary Element Method, to get vector density function of the simple layer. The velocity at any point in the flow field is calculated by the discrete form of boundary integral expression.We first set forth the formulation of the Galerkin Boundary Element Method to solve the 2-D Stokes problem in a complex connected region, and devise the computing codes by Fortran program for calculate the velocity field of the flow. The streamlines are drawn by a Matlab program to verify the reliability of the program. Then we focus on the numerical simulation of the problem containing open boundary in a bounded domain. Because by Galerkin boundary element method we need to calculate the double integrals in computing matrix coefficients, we derived the analytical integral formula with shape functions contained singularity or not, and special Gauss integral formula with singular weight function. For the double integrations on singular boundary element, we carry out the first integration by analytical integral formula and the second integration by a Gauss integral formula with a special singular weight function, while the usual Gauss integral formulas are used when there is no singularity on the element. In this paper, several numerical tests simulated the flow of the viscous incompressible fluid in a complex connected region and a bounded region with an open boundary.更多还原