Joukowski Airfoils

(Book 18.11)

The flat plate airfoil does not work well in practice, since the flow does not really want to bend about the sharp leading edge, like it does not want to bend around the sharp trailing edge:

To get a decent airfoil, we need to round the leading edge.

Joukowski airfoils achieve a rounded leading edge as follows: after the mapping :

they add an additional mapping .This additional mapping leaves the trailing edge alone, but makes the cylinder slightly larger so that the leading edge is no longer on the circle that maps onto the plate:

(The smaller yellow circle is the unit circle, the same size as the circle in the plane.) The mapping is

Now we map the (yellow) unit circle again onto a plate using the Joukowki transormation:

By giving C also a small imaginary part, we can also move the circle vertically, giving asymmetric airfoils.

Exercise:

Set and use your plotting software to plot the shape of the Joukowski airfoil for various small values of and . Use lots of points to plot.

Joukowski airfoils have a trailing edge that ends in a cusp.

You should now be able to do the exercise above.