Methods of Profile Theory

Since the Kutta-Joukowsky equation (Eq. 2-15), which forms the basis of lift theory, has been introduced, the computation of lift can now be discussed in more detail. First, the two-dimensional problem will be treated exclusively, that is, the airfoil of infinite span in incompressible flow. The theory of the airfoil of infinite span is also called profile theory. Comprehensive discussions of incompressible profile theory, taking into account nonlinear effects and friction, are given by Betz [5], von Karman and Burgers [70], Sears [59], Hess and Smith [23], Robinson and Laurmann [51], Woods [74], and Thwaites [67]. A comparison of results of profile theory with measurements was made by Hoerner and Borst [25], Riegels [50], and Abbott and von Doenhoff [1].

Profile theory can be treated in two different ways (compare [73]): first, by the method of conformal mapping, and second, by the so-called method of singularities. The first method is limited to two-dimensional problems. The flow about a given body is obtained by using conformal mapping to transform it into a known flow about another body (usually circular cylinder). In the method of singularities, the body in the flow field is substituted by sources, sinks, and vortices, the so-called singularities. The latter method can also be applied to three – dimensional flows, such as wings of finite span and fuselages. For practical purposes, the method of singularities is considerably simpler than conformal mapping. The method of singularities produces, in general, only approximate solutions, whereas conformal mapping leads to exact solutions, although these often require consider­able effort.