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:

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

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.

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

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

First, add and subtract 2 from the Joukowsky transform, as given above:

${\displaystyle {\begin{aligned}z+2&=\zeta +2+{\frac {1}{\zeta }}={\frac {1}{\zeta }}(\zeta +1)^{2},\\[3pt]z-2&=\zeta -2+{\frac {1}{\zeta }}={\frac {1}{\zeta }}(\zeta -1)^{2}.\end{aligned}}}$

Dividing the left and right hand sides gives

${\displaystyle {\frac {z-2}{z+2}}=\left({\frac {\zeta -1}{\zeta +1}}\right)^{2}.}$

The right hand side contains (as a factor) the simple second-power law from potential flow theory, applied at the trailing edge near ${\displaystyle \zeta =+1.}$ From conformal mapping theory, this quadratic map is known to change a half plane in the ${\displaystyle \zeta }$-space into potential flow around a semi-infinite straight line. Further, values of the power less than 2 will result in flow around a finite angle. So, by changing the power in the Joukowsky transform to a value slightly less than 2, the result is a finite angle instead of a cusp. Replacing 2 by ${\displaystyle n}$ in the previous equation gives^[2]

${\displaystyle {\frac {z-n}{z+n}}=\left({\frac {\zeta -1}{\zeta +1}}\right)^{n},}$

which is the Kármán–Trefftz transform. Solving for ${\displaystyle z}$ gives it in the form of equation A.

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 ${\displaystyle a=1+\epsilon }$.