Wing Lab

Angle of attack 🔊

α is the angle between the wing's chord line and the oncoming air. Thin-airfoil theory predicts C_L ≈ 2π(α − α₀), about 0.11 per degree, where α₀ is the zero-lift angle (negative for a cambered wing). Right now α = 5°, α₀ = −2°, and this slice gives C_L = 0.

Pressure: Bernoulli 🔊

Along a streamline in steady flow, p + ½ρv² is constant, so where air speeds up its pressure drops. The back wall and the wing's paint show the pressure coefficient C_p = 1 − (v/V)², scaled by the dynamic pressure ½ρV². The smoke is colored by speed: warm where the air slows toward the stagnation point, icy blue where it races over the top. Lift is the net push of the pressure difference over the whole surface.

Newton: downwash 🔊

The green arrows show the wing bending the airflow down behind it (and up ahead of it). Accelerating air downward takes a downward force, so the air pushes the wing up: Newton's third law. Pressure and downwash are not rival theories. They are two descriptions of one flow: the pressure field is what turns the air.

The lift equation, live 🔊

ρ is sea-level air density, S is a 747-size wing area, and C_L,wing is 83% of the tunnel slice's value (wingtip and friction losses).

The math inside the tunnel 🔊

Every smoke point follows an exact ideal-flow solution. Flow around a spinning cylinder is bent into a wing shape by the Joukowski map z = ζ + 1/ζ. The spin, or circulation Γ, is fixed by the Kutta condition: the only value that lets air leave the sharp trailing edge smoothly. Then the Kutta–Joukowski theorem gives lift per unit span: L′ = ρVΓ, with Γ = 4πVR·sin(α + β). The wing spans the tunnel wall to wall, so this 2D solution is the same at every spanwise slice: the smoke rake releases it at several stations.

The equal-transit myth 🔊

Many books claim the top air must go farther, so it must speed up to meet the bottom air at the trailing edge. Watch the yellow smoke: the top half arrives first and never meets its old partner. Nothing forces them to meet. The top air is faster because of the pressure field that circulation creates, not because of a meeting.

Stall and flow separation 🔊

Near the surface, a thin boundary layer of slowed air must climb from the low-pressure peak toward higher pressure at the back (an adverse pressure gradient). Past the stall angle it can't, so it separates, rolls into vortices and sheds a turbulent wake. Suction collapses and drag jumps. Ideal flow has no friction, so it can't predict this. The stalled picture here is an illustration: a vortex grows over the upper surface and sheds, with a counter-vortex from the trailing edge, at a Strouhal number of about 0.2; the Kutta condition is re-applied with those vortices present. In stall the colors come from the ideal flow that carries the stalled lift, plus low pressure in the vortex cores, and the lift number follows a typical post-stall curve.

Flaps and stall speed 🔊

In level flight L = W, so the slowest speed is V_s = √(2W / ρSC_L,max). Flaps raise C_L,max, so V_s drops, which is why pilots extend them to land. They also add drag and lower the stall angle a little. For a Cessna 172 (2,400 lb, 174 ft²) our model gives . The handbook lists about 48 kt clean and 40 kt with full flaps.