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 and its right-inverse are z = ζ + 1 ζ , ζ = 1 2 z ± ( 1 2 z ) 2 − 1 = 1 1 2 z ∓ ( 1 2 z ) 2 − 1 , {\displaystyle z=\zeta +{\frac {1}{\zeta }},\qquad \zeta ={\tfrac {1}{2}}z\pm {\sqrt {{\bigl (}{\tfrac {1}{2}}z{\bigr )}^{2}-1}}={\frac {1}{{\tfrac {1}{2}}z\mp {\sqrt {{\bigl (}{\tfrac {1}{2}}z{\bigr )}^{2}-1}}}},} where z = x + i y {\displaystyle z=x+iy} is a complex variable in the new space and ζ = χ + i η {\displaystyle \zeta =\chi +i\eta } is a complex variable in the original space. The right-inverse is not a global left-inverse because ζ ↦ z {\displaystyle \zeta \mapsto z} is 2-to-1; but a local left-inverse is always one of the right-inverse branches. 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 ( z {\displaystyle z} -plane) by applying the Joukowsky transform to a circle in the ζ {\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 ζ = − 1 {\displaystyle \zeta =-1} (where the derivative is zero) and intersects the point ζ = 1. {\displaystyle \zeta =1.} This can be achieved for any allowable centre position μ x + i μ y {\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...