: Mathematics: Joukowsky transform

mathematical physics > conformal map > Joukowsky transform

geometry > differential geometry > conformal geometry > conformal map > Joukowsky transform

geometry > geometric transformation > conformal map > Joukowsky transform

mathematical analysis > complex analysis > complex function > conformal map > Joukowsky transform

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. The transform is
${\\displaystyle z=\\zeta +{\ rac {1}{\\zeta }},}$
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.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Joukowsky_transform)