Joukowski Conformal Map & Airfoil

Joukowski Transform: The map w = z + c²/z (with c=1) is a conformal mapping — it preserves angles locally. Applied to a circle in the complex plane that passes through z=1, it produces a cusped teardrop (symmetric airfoil). Offsetting the circle center changes camber and thickness. Potential flow around a circle (with circulation Γ) is analytically solvable; the conformal map carries those streamlines to exact inviscid flow around the airfoil. The Kutta-Joukowski theorem gives lift L = ρUΓ per unit span. This was the foundation of theoretical aerodynamics (Zhukovsky, 1906) and still underlies modern thin-airfoil theory.