A brief treatise on the Joukowski transformation

Abstract

The problem of predicting the flight characteristics of an arbitrary airfoil is very complicated, and does not readily lend itself to exact mathematical methods. However if certain simplifying assumptions are made the problem may be solved in certain cases, by the methods of conformal mapping of a complex variable. This solution is based upon the fact that the dynamics of fluid flow past a circular cross-section have been rather completely solved. Thus the problem reduces to one of finding a conformal transformation which will map the given airfoil into a circle. Unfortunately a general solution of this problem for any arbitrary shape airfoil has not been published. For certain special cases the problem has been solved by numerous writers, and the theoretical results have been checked by experimental data to a reasonable degree of accuracy. This paper will discuss primarily the transformation of a symmetrical airfoil into a circle by the Kutta-Joukowski transformation. The stream function for the airfoil will be transformed from that of a circle by the Joukowski transformation. The lift and moment for the airfoil will be discussed. Brief mention of other more complicated transformation will also be made.