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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.

General Joukowsky transform

The Joukowsky transform of any complex number ${\displaystyle \zeta }$ to ${\displaystyle z}$ is as follows:

${\displaystyle {\begin{aligned}z&=x+iy=\zeta +{\frac {1}{\zeta }}\\&=\chi +i\eta +{\frac {1}{\chi +i\eta }}\\&=\chi +i\eta +{\frac {\chi -i\eta }{\chi ^{2}+\eta ^{2}}}\\&=\chi \left(1+{\frac {1}{\chi ^{2}+\eta ^{2}}}\right)+i\eta \left(1-{\frac {1}{\chi ^{2}+\eta ^{2}}}\right).\end{aligned}}}$

So the real (${\displaystyle x}$) and imaginary (${\displaystyle y}$) components are:

${\displaystyle {\begin{aligned}x&=\chi \left(1+{\frac {1}{\chi ^{2}+\eta ^{2}}}\right),\\y&=\eta \left(1-{\frac {1}{\chi ^{2}+\eta ^{2}}}\right).\end{aligned}}}$

Sample Joukowsky airfoil

The transformation of all complex numbers on the unit circle is a special case.

$${\displaystyle |\zeta |={\sqrt {\chi ^{2}+\eta ^{2}}}=1,}$$

which gives

$${\displaystyle \chi ^{2}+\eta ^{2}=1.}$$

So the real component becomes ${\textstyle x=\chi (1+1)=2\chi }$ and the imaginary component becomes ${\textstyle y=\eta (1-1)=0}$.

Thus the complex unit circle maps to a flat plate on the real-number line from −2 to +2.

Transformations from other circles make a wide range of airfoil shapes.

Velocity field and circulation for the Joukowsky airfoil

The solution to potential flow around a circular cylinder is analytic and well known. It is the superposition of uniform flow, a doublet, and a vortex.

The complex conjugate velocity ${\displaystyle {\widetilde {W}}={\widetilde {u}}_{x}-i{\widetilde {u}}_{y},}$ around the circle in the ${\displaystyle \zeta }$-plane is

$${\displaystyle {\widetilde {W}}=V_{\infty }e^{-i\alpha }+{\frac {i\Gamma }{2\pi (\zeta -\mu )}}-{\frac {V_{\infty }R^{2}e^{i\alpha }}{(\zeta -\mu )^{2}}},}$$

where

• ${\displaystyle \mu =\mu _{x}+i\mu _{y}}$ is the complex coordinate of the centre of the circle,
• ${\displaystyle V_{\infty }}$ is the freestream velocity of the fluid,

${\displaystyle \alpha }$ is the angle of attack of the airfoil with respect to the freestream flow,

• ${\displaystyle R}$ is the radius of the circle, calculated using ${\textstyle R={\sqrt {\left(1-\mu _{x}\right)^{2}+\mu _{y}^{2}}}}$,
• ${\displaystyle \mathrm{\Gamma <!--\; \Gamma \; -->}}$Zdroj:https://en.wikipedia.org?pojem=Joukowsky_transform
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