Joukowsky transform

In applied mathematics, the Joukowsky transform (sometimes transliterated Joukovsky, Joukowski or Zhukovsky) is a conformal map historically used to understand some principles of airfoil design.

It is named after Nikolai Zhukovsky, who published it in 1910.

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 complex plane (

-plane) by applying the Joukowsky transform to a circle in the

The coordinates of the centre of the circle are variables, and varying them modifies the shape of the resulting airfoil.

Joukowsky airfoils have a cusp at their trailing edge.

A closely related conformal mapping, the Kármán–Trefftz transform, generates the broader class of Kármán–Trefftz airfoils by controlling the trailing edge angle.

The Joukowsky transform of any complex number

) components are: The transformation of all complex numbers on the unit circle is a special case.

Thus the complex unit circle maps to a flat plate on the real-number line from −2 to +2.

Transformations from other circles make a wide range of airfoil shapes.

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.

is the angle of attack of the airfoil with respect to the freestream flow, The complex velocity

-plane is, according to the rules of conformal mapping and using the Joukowsky transformation,

From this velocity, other properties of interest of the flow, such as the coefficient of pressure and lift per unit of span can be calculated.

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

-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

is a real constant that determines the positions where

between the tangents of the upper and lower airfoil surfaces at the trailing edge is related to

, required to compute the velocity field, is First, add and subtract 2 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

-space into potential flow around a semi-infinite straight line.

Further, values of the power less than 2 will result in flow around a finite angle.

So, by changing the power in the Joukowsky transform to a value slightly less than 2, the result is a finite angle instead of a cusp.

In 1943 Hsue-shen Tsien published a transform of a circle of radius

into a symmetrical airfoil that depends on parameter

yields a flat plate when zero, and a circle when infinite; thus it corresponds to the thickness of the airfoil.

Example of a Joukowsky transform. The circle above is transformed into the Joukowsky airfoil below.
Example of a Kármán–Trefftz transform. The circle above in the -plane is transformed into the Kármán–Trefftz airfoil below, in the -plane . The parameters used are: and Note that the airfoil in the -plane has been normalised using the chord length.