Joukowsky transform

Lore & Background

The transform is named after Nikolai Zhukovsky, who published it in 1910. In aerodynamics, it is used to solve for two-dimensional potential flow around a class of airfoils known as Joukowsky airfoils. These airfoils are generated in the complex z-plane by applying the Joukowsky transform to a circle in the ζ-plane. The circle encloses the point ζ = -1 (where the derivative is zero) and intersects the point ζ = 1. The coordinates of the circle's center 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; when a trailing edge angle of zero is specified, the Kármán–Trefftz transform reduces to the Joukowsky transform.

Reader's Guide

The Joukowsky transform is notable for its role in aerodynamics, where it enables the solution of two-dimensional potential flow around Joukowsky airfoils. These airfoils are generated by applying the transform to a circle in the ζ-plane, with the circle's center position and radius adjustable to vary the airfoil shape. The transform's right-inverse is not a global left-inverse because the mapping from ζ to z is 2-to-1, but a local left-inverse is always one of the right-inverse branches. The transform's legacy includes its relationship to the Kármán–Trefftz transform, which generalizes it by controlling the trailing edge angle; when that angle is zero, the Kármán–Trefftz transform reduces to the Joukowsky transform. This connection situates the Joukowsky transform as a foundational tool in the study of airfoil geometry and potential flow.

Did You Know?

• The transform maps a circle in the ζ-plane to a Joukowsky airfoil in the z-plane.
• The Kármán–Trefftz transform reduces to the Joukowsky transform when the trailing edge angle is zero.