The science
Smoke streaks here are exact streamlines of ideal (inviscid, incompressible) flow around a Joukowski aerofoil, which is the image of a circle under ζ = z + c²/z. The circle's centre offset sets the shape: the vertical offset is camber, the horizontal offset is thickness.
- Flow past a circle with circulation:
W(z) = U[(z−z0)e−iα + a²eiα/(z−z0)] − iΓ/(2π) ln(z−z0); velocity in the aerofoil plane is u − iv = W'(z) / (dζ/dz).
- Kutta condition: the velocity must stay finite at the sharp trailing edge, which fixes
Γ = −4π U a sin(α+β) (β is the camber angle). Lift per span is the Kutta–Joukowski law L = ρ U |Γ|.
- Pressure: Bernoulli gives
Cp = 1 − (V/U)². Integrating Cp around the profile gives lift and drag independently of Γ; the check button compares them.
- Thin-airfoil theory:
Cl ≈ 2π sin(α+β) (a little more for thick sections).
Honest simplifications: ideal flow has no boundary layer, so there is no stall and no drag (d'Alembert's paradox). Real wings separate near 12–16 degrees; the shaded zone and the status line only flag where this model stops being trustworthy. Lift per metre assumes a 1 m chord and sea-level air density 1.225 kg/m³. The 3D page's flow is a stylised visualisation; this page is the analytic version of it.