Wind tunnel
Change the airfoil shape, the angle of attack and the speed. Watch smoke flow around the wing, see where pressure rises and falls — and find out when the wing stops lifting.
↑↓ angle of attack ←→ speed 1–5 presets Shift finer steps
CL(α) for the current airfoil; the dot is the current setting.
−Cp along the chord: up = suction. The area between the curves ≈ lift.
The two explanations — pressure and “Newtonian” — are not competing theories, just two sides of the same phenomenon.
An airfoil set at an angle of attack (and cambered) turns the air flowing around it. Ahead of the wing the stream rises slightly, and behind the trailing edge it heads down — follow the streamlines. Since the wing pushes air down, by Newton's third law the air pushes the wing up.
A cruising airliner deflects tens of tons of air downward every second. The change in momentum of that mass is exactly the force that keeps it aloft.
For the stream to curve, the pressure in the air has to change. Above the wing there is low pressure, and below it and at the leading edge — high pressure. Where the air speeds up, the pressure drops (Bernoulli's principle along a streamline). The sum of these pushes over the whole surface is the lift.
p + 12ρV2 = const along a streamline
Cp = p − p∞½ ρ v2 = 1 − (V / v)2 V — local speed, v — flight speed
L = 12 ρ v2 S CL ρ — air density, S — wing area
The coefficient CL depends on the airfoil shape and the angle of attack. For a thin airfoil, theory gives
CL ≈ 2π (α − α0) α in radians, i.e. about 0.11 per degree
α0 is the angle at which lift vanishes — for cambered airfoils it is negative, so they lift even at α = 0°. Twice the speed means four times the force.
L′ = ρ v Γ, CL = 2Γv c Kutta–Joukowski theorem (force per meter of span)
Γ is the circulation — a measure of how strongly the flow “circles” around the airfoil. Its value is set by the Kutta condition: air must leave the sharp trailing edge smoothly instead of wrapping around it.
Many books say: air particles split at the leading edge must meet again at the trailing edge, and since the path over the top is longer, the air on top moves faster. That's not true.
Nothing forces the particles to meet. Air above the wing reaches the trailing edge well before the air from underneath and never meets it again. Besides, the top path is only 1–3% longer — that would give far too little lift. The myth also can't explain how a flat plate lifts, or how a plane flies upside down.
At too high an angle of attack, the air near the upper surface can't follow its curve: the boundary layer separates, a turbulent wake forms behind the wing, lift drops sharply and drag rises. For typical airfoils this happens at about 12–16°. Camber and flaps shift this threshold, but don't remove it.
Note: the potential flow we compute here has no viscosity, so it never separates. The stall, the gray wake behind the wing and the drop in CL come from a simple empirical model fitted to typical airfoils, not from a flow calculation.
D = 12 ρ v2 S CD, CD ≈ CD0 + k (CL − CL,opt)2 + CL2π e λ
The first two terms are the profile drag (friction and pressure), the last is the induced drag, which arises at the tips of a wing with a finite span b; λ = b2/S is the aspect ratio. That is why gliders have long, narrow wings. The ratio L/D is the glide ratio: it tells you how many meters a glider travels horizontally while losing 1 m of height.
In pure potential flow the drag comes out as zero (d’Alembert's paradox) — so we estimate drag with an empirical formula: CD0 ≈ 0.0045 + 0.011 t + 0.018 |m|, k = 0.005, e = 0.9.
We compute the flow as potential flow — inviscid, incompressible and two-dimensional — using the vortex panel method: the airfoil outline is split into 100 segments carrying a linearly varying vortex, we require that no air passes through the surface, and we add the Kutta condition at the trailing edge. The solution gives the velocity at every point, the streamlines, the Cp distribution and the circulation Γ, and from it CL. The smoke is thousands of particles carried by this velocity field.
Simplifications: we compute lift for the airfoil section, as if the wing were very long; induced drag is added with the formula for a wing of aspect ratio λ. Drag and stall come from empirical models, and the flaps in the takeoff preset are replaced by a more strongly cambered airfoil. The model ignores air compressibility (important above about Mach 0.3, i.e. about 370 km/h at sea level) and Reynolds number effects. Air density comes from the International Standard Atmosphere (ISA).
A “NACA m p tt” airfoil is a camber line with maximum camber m (% of chord) at position p (tenths of the chord), wrapped in a symmetric thickness t (% of chord). For x from 0 to 1:
yt = 5t (0.2969 √x − 0.1260 x − 0.3516 x2 + 0.2843 x3 − 0.1036 x4)
yc = mp2(2px − x2) for x < p, yc = m(1 − p)2(1 − 2p + 2px − x2) for x ≥ p
The last coefficient, −0.1036 (instead of the classic −0.1015), closes the trailing edge into a sharp wedge, as the Kutta condition requires. The minimum thickness is 1% — the panel method can't describe a truly flat plate.