# Joukowsky Airfoil

> Air flowing past a wing, found by mapping the flow around a circle with Z = z + 1/z; the angle of attack animates.

In 1910 Nikolai Joukowsky found a simple way to compute the airflow around a wing. The flow of an ideal fluid around a circle is known exactly, and the map $Z = z + 1/z$ sends a circle through the point $z = 1$ to a wing-shaped curve with a sharp trailing edge. Because the map preserves angles, it carries the circle's flow to the flow around the wing.

In this document the circle has its centre at $(c_x, c_y)$ and passes through $z = 1$, so its radius is $R$. The stream function $\psi$ of the flow around the circle combines a uniform wind at the angle of attack $a$, its reflection in the circle, and a circulation $\Gamma$. To find $\psi$ at a point of the wing plane, the document inverts the map: $u$, $v$, $m$, $s$ and $w$ compute the complex square root of $Z^2 - 4$ in real numbers, and $k$ keeps the root outside the circle. The shading shows $\psi$, and the curves $\sin(\pi \psi/0.25) = 0$ are the streamlines, the paths that the air follows.

The circulation is set by the Kutta condition: the air must leave the sharp trailing edge smoothly. That fixes $\Gamma = 4\pi R\sin(a + \arcsin(c_y/R))$, and the lift on the wing is proportional to $\Gamma$. As $a$ animates, watch the streamlines crowd together above the wing, where the air moves faster and the pressure is lower. Edit $c_x$ to make the wing thicker or thinner, and $c_y$ to change its curvature.

After a Desmos graph by Fabrice Neyret.

- [Open the interactive version](https://graph-paper.io/showcase/en/joukowsky-airfoil)
- [Learn more on Wikipedia](https://en.wikipedia.org/wiki/Joukowsky_transform)