Conformal Mapping of a Circle

Abstract

Mapping techniques are mathematical methods which are frequently applied for solving fluid flow problems in the interior and about bodies of nonregular shape. Since the advent of supercomputers such techniques have become quite important in the context of numerical grid generation [1]. In introductory courses in fluid dynamics students learn how to calculate the circulation of an incompressible potential flow about a so-called “Joukowski airfoil” [3] which represent the simplest airfoils of any technical relevance. The physical plane where flow about the airfoil takes place is in a complex p ═ u + iυ plane where i ═ √–1. The advantage of a Joukowski transform consists in providing a conformal mapping of the p plane on a z ═ x + iy plane such that calculating the flow about the airfoil gets reduced to the much simpler problem of calculating the flow about a displaced circular cylinder. A special form of the mapping function p = f(z) = u(z) + iv(z) of the Joukowski transform reads

$${\rm{p = }}\frac{1}{2}\left( {z + \frac{{a^2 }}{z}} \right)$$

((11.1))

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