Numerical Conformal Mapping and Its Application to the Two-Dimensional Potential Flow Analysis

共形映射在科学和工程中很常见。但是，除了特殊的区域，确切的映射函数是未知的。特别地，无界多连通域的共形映射作为二维势流分析的一种方法是众所周知的。封闭Jordan曲线外的区域到平行缝域、圆形缝域和径向缝域的数值共形映射分别与绕障碍物的均匀流、涡流和点源流问题直接相关，本文给出了多连通区域到平行缝域、圆形缝域和径向缝域的数值共形映射的一种简单方法。我们将映射问题归结为一对共轭调和函数的Dirichlet问题，并采用电荷模拟方法，其中共轭调和函数对由复对数势的线性组合近似。该方法具有以下特点：1.平行狭缝域上的共形映射，狭缝平行于虚轴和平行于真实的轴，构成一个对偶问题，可以用对偶方法计算.圆形和径向狭缝域上的共形映射构成了一个对偶问题，可以用对偶方法计算.他们被重新制定成一个更完整的形式，其中要解决的线性方程组具有相同的系数矩阵。

Conformal mappings are familiar in science and engineering. But, exact mapping functions are not known except for special domains. Particularly, conformal mappings of unbounded multiply-connected domains are well known as a method of two-dimensional potential flow analysis. The mappings of a domain exterior to closed Jordan curves onto parallel, circular and radial slit domains are directly related to the problems of uniform, vortex and point-souce flows around obstacles, respectively.We presented a simple method of numerical conformal mappings of the multiply-connected domains onto the parallel, circular and radial slit domains. We reduced the mapping problems to the Dirichlet problem with a pair of conjugate harmonic functions and employed the charge simulation method, where the pair of conjugate harmonic functions are appoximated by a linear combination of complex logarithmic potentials. The method has the following features :1. Conformal mappings onto the parallel slit domains with slits parallel to the imaginary axis and parallel to the real axis constitute a dual problem and can be computed in a dual way.2. Conformal mappings onto the circular and radial slit domains constitute a dual problem and can be computed in a dual way.3. They are reformulated into a more integrated form in which the linear equations to be solved have the same coefficient matrix.