The transform is
where z = x + iy is a complex variable in the new space and ζ = χ + iη is a complex variable in the original space. This transform is also called the Joukowsky transformation, the Joukowski transform, the Zhukovsky transform and other variations.
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 ζ 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.
A closely related transform is the Kármán–Trefftz transform, as used to specify the Kármán–Trefftz airfoil, another class of airfoils. The trailing edge of the Kármán–Trefftz airfoil does not have the cusp which the Joukowsky airfoil has.
The solution to potential flow around a circular cylinder is analytic and well known. It is the superposition of uniform flow, a doublet, and a vortex.
The complex velocity around the circle in the ζ plane is
The complex velocity W around the airfoil in the z plane is, according to the rules of conformal mapping and using the Joukowsky transformation:
Here W = ux − iuy, with ux and uy the velocity components in the x and y directions, respectively (z = x + iy, with x and y real-valued). From this velocity, other properties of interest of the flow, such as the coefficient of pressure or lift can be calculated.
A Joukowsky airfoil has a cusp at the trailing edge.
The Kármán–Trefftz transform is a conformal map closely related to the Joukowsky transform. While a Joukowsky airfoil has a cusped trailing edge, a Kármán–Trefftz airfoil — which is the result of the transform of a circle in the ς-plane to the physical z-plane, analogue to the definition of the Joukowsky airfoil — has a non-zero angle at the trailing edge, between the upper and lower airfoil surface. The Kármán–Trefftz transform therefore requires an additional parameter: the trailing-edge angle α. This transform is equal to:
The derivative dz / dζ, required to compute the velocity field, is equal to:
First, add and subtract two from the Joukowsky transform, as given above:
Dividing the left and right hand sides gives:
The right hand side contains (as a factor) the simple second-power law from potential flow theory, applied at the trailing edge near ζ = + 1. From conformal mapping theory this quadratic map is known to change a half plane in the ζ-space into potential flow around a semi-infinite straight line. Further, values of the power less than two will result in flow around a finite angle. So, by changing the power in the Joukowsky transform — to a value slightly less than two — the result is a finite angle instead of a cusp. Replacing 2 by n in the previous equation gives:
which is the Kármán–Trefftz transform. Solving for z gives it in the form of equation (A).