# Joukowsky Airfoil

Consider the uniform flow past a circular cylinder with circulation $C$ and speed $U>0$ given by the complex potential

\[
F(z)=Uz+\frac{Ua^2}{z}-\frac{i C}{2\pi}\log z.
\]

The streamfunction is just the imaginary part of this function, namely

\[
\psi = Uy\left(1-\frac{a^2}{x^2+y^2}\right)-\frac{C}{4\pi}\log\left(x^2+y^2\right).
\]

Hence the circle $x^2+y^2=a^2$ is still a streamline, with $\psi = -(C/2\pi)\log a$.

## Simulation

Consider now the uniform flow past the circular cylinder $c_0$ for $a=1$. We use the linear transformation $T(z)=-0.15+0.23i +r z$ to map the flow with circulation $C$ and speed $U$ around $|z|=1$ onto the flow around the circle $c_1$ with center $z_1=-0.15+0.23i$ and radius $r=0.23\sqrt{13\cdot 2}$. Then we use the mapping
\[
J(z)=z+\frac{1}{z}
\]
to map this flow around the Joukowsky airfoil.

The following simulation shows the uniform flow past the circular cylinder $c_1$ and its transformation to the Joukowsky airfoil. Drag the sliders to interact:

- Slider U = speed.
- Slider C = circulation.
- Slider T = apply transformation.
- Trace button = streamlines.

Sorry, the simulation is not supported for small screens.