Joukowsky transform

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

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

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

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

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.

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 ${\textstyle R={\sqrt {\left(1-\mu _{x}\right)^{2}+\mu _{y}^{2}}}}$,
• ${\displaystyle \Gamma }$ is the circulation, found using the Kutta condition, which reduces in this case to
  $${\displaystyle \Gamma =4\pi V_{\infty }R\sin \left(\alpha +\sin ^{-1}{\frac {\mu _{y}}{R}}\right).}$$

  

The complex velocity ${\displaystyle W}$ around the airfoil in the ${\displaystyle z}$-plane is, according to the rules of conformal mapping and using the Joukowsky transformation,

$${\displaystyle W={\frac {\widetilde {W}}{\frac {dz}{d\zeta }}}={\frac {\widetilde {W}}{1-{\frac {1}{\zeta ^{2}}}}}.}$$

Here ${\displaystyle W=u_{x}-iu_{y},}$ with ${\displaystyle u_{x}}$ and ${\displaystyle u_{y}}$ the velocity components in the ${\displaystyle x}$ and ${\displaystyle y}$ directions respectively (${\displaystyle z=x+iy,}$ with ${\displaystyle x}$ and ${\displaystyle y}$ real-valued). From this velocity, other properties of interest of the flow, such as the coefficient of pressure and lift per unit of span can be calculated.

Example of a Kármán–Trefftz transform. The circle above in the ${\displaystyle \zeta }$-plane is transformed into the Kármán–Trefftz airfoil below, in the ${\displaystyle z}$-plane. The parameters used are: ${\displaystyle \mu _{x}=-0.08,}$ ${\displaystyle \mu _{y}=+0.08}$ and ${\displaystyle n=1.94.}$ Note that the airfoil in the ${\displaystyle z}$-plane has been normalised using the chord length.

The Kármán–Trefftz transform is a conformal map closely related to the Joukowsky transform. While a Joukowsky airfoil has a cusped trailing edge, a Kármán–Trefftz airfoil—which is the result of the transform of a circle in the ${\displaystyle \zeta }$-plane to the physical ${\displaystyle z}$-plane, analogue to the definition of the Joukowsky airfoil—has a non-zero angle at the trailing edge, between the upper and lower airfoil surface. The Kármán–Trefftz transform therefore requires an additional parameter: the trailing-edge angle ${\displaystyle \alpha .}$ This transform is[2][3]

${\displaystyle z=nb{\frac {(\zeta +b)^{n}+(\zeta -b)^{n}}{(\zeta +b)^{n}-(\zeta -b)^{n}}},}$

(A)

where ${\displaystyle b}$ is a real constant that determines the positions where ${\displaystyle dz/d\zeta =0}$, and ${\displaystyle n}$ is slightly smaller than 2. The angle ${\displaystyle \alpha }$ between the tangents of the upper and lower airfoil surfaces at the trailing edge is related to ${\displaystyle n}$ as[2]

${\displaystyle \alpha =2\pi -n\pi ,\quad n=2-{\frac {\alpha }{\pi }}.}$

The derivative ${\displaystyle dz/d\zeta }$, required to compute the velocity field, is

${\displaystyle {\frac {dz}{d\zeta }}={\frac {4n^{2}}{\zeta ^{2}-1}}{\frac {\left(1+{\frac {1}{\zeta }}\right)^{n}\left(1-{\frac {1}{\zeta }}\right)^{n}}{\left[\left(1+{\frac {1}{\zeta }}\right)^{n}-\left(1-{\frac {1}{\zeta }}\right)^{n}\right]^{2}}}.}$

In 1943 Hsue-shen Tsien published a transform of a circle of radius ${\displaystyle a}$ into a symmetrical airfoil that depends on parameter ${\displaystyle \epsilon }$ and angle of inclination ${\displaystyle \alpha }$:[4]

${\displaystyle z=e^{i\alpha }\left(\zeta -\epsilon +{\frac {1}{\zeta -\epsilon }}+{\frac {2\epsilon ^{2}}{a+\epsilon }}\right).}$

The parameter ${\displaystyle \epsilon }$ yields a flat plate when zero, and a circle when infinite; thus it corresponds to the thickness of the airfoil. Furthermore the radius of the cylinder ${\displaystyle a=1+\epsilon }$.

• Joukowski Transform NASA Applet
• Joukowsky Transform Interactive WebApp