Joukowsky transform

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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.^[1]

The transform is

${\displaystyle z=\zeta +{\frac {1}{\zeta }},}$

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.

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.

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.