Airfoil Flow Lab (2D) — streamlines, pressure and polar

• Raise the angle of attack past about 12° and watch C_L collapse (stall) in the polar.
• Pick High-lift: the zero-lift angle becomes negative, so lift exists at 0°.
• Look at the pressure plot: the area between the upper and lower curves is the lift.

Potential flow past a Joukowski aerofoil: a cylinder of radius R in the ζ-plane is mapped by z = ζ + 1/ζ. The Kutta condition (smooth flow off the trailing edge) fixes the circulation Γ = 4πR sin(α+β) (in units of U, with β set by camber).

Surface pressure coefficient C_p = 1 − (|V|/U)². Section lift c_l = 2Γ/(U c) (Kutta–Joukowski). Lift is also the integral ∮ C_p dx over the contour, so the two numbers must match; d’Alembert’s paradox says the pressure drag integral is zero.

Finite wing: lift slope is reduced by AR/(AR+2); induced drag C_Di = C_L²/(π e AR) with e = 0.85; profile drag grows with thickness; a smooth stall model switches to flat-plate-like C_L ≈ sin 2α after separation. Real stall depends on Reynolds number and surface roughness; this is a teaching model, not a design tool. Inviscid flow has no boundary layer, so the wake is not shown.

Lift L = ½ρU²S C_L, wing area S = c·b, chord c = 1.5 m, ρ = 1.225 kg/m³.