Joukowsky transform

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 ${\displaystyle \tilde{W}={\tilde{u}}_x-i{\tilde{u}}_y,}$ around the circle in the ${\displaystyle \zeta }$-plane is $${\displaystyle \tilde{W}=V_{\mathrm{\infty }}{e}^{-i\alpha }+\frac{i\mathrm{\Gamma }}{2\pi (\zeta -\mu )}-\frac{V_{\mathrm{\infty }}{R}^2{e}^{i\alpha }}{(\zeta -\mu {)}^2},}$$

where

• ${\displaystyle \mu ={\mu }_x+i{\mu }_y}$ is the complex coordinate of the centre of the circle,
• ${\displaystyle V_{\mathrm{\infty }}}$ 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 $R=\sqrt{{\left(1-{\mu }_x\right)}^2+{\mu }_y^2}$,
• ${\displaystyle \mathrm{\Gamma }}$ is the circulation, found using the Kutta condition, which reduces in this case to $${\displaystyle \mathrm{\Gamma }=4\pi V_{\mathrm{\infty }}R\sin \left(\alpha +{\sin }^{-1}\frac{{\mu }_y}{R}\right).}$$

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

Here ${\displaystyle W=u_x-i{u}_y,}$ 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.

Kármán–Trefftz transform

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

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

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

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

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

Background

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

${\displaystyle \begin{array}{cc}z+2& =\zeta +2+\frac{1}{\zeta }=\frac{1}{\zeta }(\zeta +1{)}^2,\\ z-2& =\zeta -2+\frac{1}{\zeta }=\frac{1}{\zeta }(\zeta -1{)}^2.\end{array}}$

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

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

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

Symmetrical Joukowsky airfoils

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

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

The parameter ${\displaystyle ϵ}$ 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 ${\displaystyle a=1+ϵ}$.