Conformal Mapping



Any analytic function of a complex variable satisfies the equation for incompressible, irrotational flow:
Why?

We can, therefore, relate one flow field to another by setting:

where z' is related to z by an analytic function of z, z' = f(z). (Recall z = x + iy.)

The idea behind airfoil analysis by conformal mapping is to relate the flow field around one shape which is already known (by whatever means) to the flow field around an airfoil. Most often a circle is used as the first shape. The problem is to find an analytic function that relates every point on the circle to a corresponding point on the airfoil.


Joukowski found that the simple function: z' = z + 1/z
transforms a circle to a shape which looks a bit like an airfoil. By taking the origin of the circle at various points, different airfoil-like shapes are produced.

With just 2 degrees of freedom (the coordinates of the circle origin), the number of airfoil shapes that can be represented with the simple Joukowski transformation is limited. Furthermore, the thickness is greatest at 25% chord -- rather far forward.

This constraint has led to a number of generalizations of the Joukowski mapping that produce more practical shapes. But all of these methods are based on the same basic idea which is illustrated in the simple mapping.

We begin with a circle. The center of the circle is at Xc, Yc.

Every point z is mapped to the point z' by the relation: z' = z + 1/z.
If W(z) = F(z) + i G(z) is the complex potential function, then the velocity is given by:
w = u - iv = dW/dz.

We set W(z') = W(z) so that the velocity on the airfoil may be related to the velocity on the cylinder by:
w(z') = dW/dz' = (dW/dz) (dz/dz') = w(z) / ( 1-z-2).

Notice that this relates the velocity on the airfoil directly to that on the circle, but the relation blows up when z2 = 1.

We know that the velocity on the airfoil does not go to infinity anywhere. The reason that the mapping does not work at these points is that the mapping is not analytic here. This does not mean it cannot be used, it just means that we must make sure that such points are not in the flow: they must be inside or on the airfoil. Here we choose the circle so that it encloses the point -1,0, and we choose the circulation so that the velocity is 0 at the point 1,0. The point 1,0 maps to the airfoil trailing edge.

If we are to have a stagnation point on the cylinder at 1,0 we must have a certain amount of circulation. The origin of the circle thus determines the lift on the airfoil at a given angle of attack. (Also note that the point -1,0 must be enclosed.)

One of the troubles with conformal mapping methods is that parameters such as xc and yc are not so easily related to the airfoil shape. Thus, if we want to analyze a particular airfoil, we must iteratively find values that produce the desired section. A technique for doing this was developed by Theodorsen.

Another technique involves superposition of fundamental solutions of the governing differential equation. This method, discussed in subsequent sections, is called thin airfoil theory.