The Joukowsky transform, also called the Joukowsky transformation, the Joukowski transform, the Zhukovsky transform and other variations, is a conformal map historically used to understand some principles of airfoil design.
The transform is
where is a complex variable in the new space and is a complex variable in the original space.
In aerodynamics, the transform is used to solve for the two-dimensional potential flow around a class of airfoils known as Joukowsky airfoils. A Joukowsky airfoil is generated in the z plane by applying the Joukowsky transform to a circle in the plane. The coordinates of the centre of the circle are variables, and varying them modifies the shape of the resulting airfoil. The circle encloses the origin (where the conformal map has a singularity) and intersects the point z=1. This can be achieved for any allowable centre position by varying the radius of the circle.
The complex velocity around the circle in the plane is
The complex velocity W around the airfoil in the z plane is, according to the rules of conformal mapping,
A Joukowsky airfoil has a cusp at the trailing edge.
The Karman-Trefftz transform is a conformal map derived from the Joukowsky transform. While a Joukowsky airfoil has a cusped trailing edge, a Karman-Trefftz airfoil has a finite trailing edge. The Karman-Trefftz transform therefore requires an additional parameter: the trailing edge angle.
When the Joukowsky transform is written as a composition of three transformations, one of these can be modified independently. The Joukowsky transform is , where S3, S2, S1 are:
Reducing the exponent in by a small amount increases the thickness of the trailing edge to a finite amount.