Joukowsky Transform

In applied mathematics, the Joukowsky transform, named after Nikolai Zhukovsky 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. 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 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 point = −1 (where the derivative is zero) and intersects the point = 1. This can be achieved for any allowable centre position by varying the radius of the circle.

Joukowsky airfoils have a cusp at their trailing edge. A closely related conformal mapping, the Kármán–Trefftz transform, generates the much 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.

Read more about Joukowsky Transform:  General Joukowsky Transform, Velocity Field and Circulation For The Joukowsky Airfoil, Kármán–Trefftz Transform

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