Conformal Mapping: the Joukowski Airfoil

What you're actually seeing

The left panel is solved with nothing more than the classic flow-around-a-cylinder result, plus one circulation term added to satisfy the Kutta condition — the physical requirement that flow leaves a sharp trailing edge smoothly instead of whipping around it. That circulation is computed in closed form: Γ = 4πU∞R sin(α − θ_TE), where θ_TE is just the angle of the one point on the circle that the mapping will turn into the sharp trailing edge.

Once Γ is fixed, every point on every streamline in the z-plane gets pushed through w = z + b²/z — not just the boundary. The right panel isn't a separate calculation; it's the literal image of the left panel under one transformation. Solve once, map everywhere.

Why one point on the circle becomes a sharp corner

The mapping's derivative is dw/dz = 1 − b²/z², which is exactly zero at z=±b. A conformal map is angle-preserving everywhere except where its derivative vanishes — at those two points, angles get doubled. The circle is built to pass through z=b exactly, so that one point becomes the airfoil's sharp trailing edge, while z=−b is tucked safely inside the circle, where its corner-doubling effect never shows up on the boundary at all. That's the entire trick behind the airfoil's shape: one carefully placed circle, one fixed singularity, one cusp.

The bigger pattern

This is one instance of a much more general move: when a problem is hard in its natural domain, transform it into a domain where it's easy, solve the easy version, then transform the answer back. Logarithms turn multiplication into addition. The Laplace transform turns a differential equation into an algebraic one. An electrical-circuit analogy turns an unfamiliar diffusion or vibration problem into a familiar resistor-and-capacitor network. Conformal mapping turns a hard-shaped flow domain into a circle. The mechanics differ every time — the strategy doesn't.