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 ${\displaystyle z=x+iy}$ is a complex variable in the new space and ${\displaystyle \zeta =\chi +i\eta }$ 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 complex plane (${\displaystyle z}$-plane by applying the Joukowsky transform to a circle in the ${\displaystyle \zeta }$-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 ${\displaystyle \zeta =-1}$ (where the derivative is zero) and intersects the point ${\displaystyle \zeta =1.}$ This can be achieved for any allowable centre position ${\displaystyle \mu _{x}+i\mu _{y}}$ by varying the radius of the circle.

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