A wing flies because the air loops around it. The lift per unit span is nothing but the circulation, scaled: L′ = ρ v∞ Γ. Double the loop and you double the lift; kill the loop and you kill the flight — that last fact is the resolution of d'Alembert's paradox. Down the center, data flows: density, speed and circulation go in, the theorem computes, the lift comes out. The blue team builds and defends it; the red team tries to break it.
source M. W. Kutta, "Auftriebskräfte in strömenden Flüssigkeiten," Illustrierte Aeronautische Mitteilungen 6 (1902) — showed circulation ↔ lift; N. E. Joukowski, "De la chute dans l'air de corps légers…," Bull. Inst. Aérodynamique de Koutchino (1906) — first stated L′ = ρV Γ. No stable digital original; cited by author/title/year. Rendered, not quoted.
Three facts, and the wing is a theorem, not a mystery:
1 the lift per span is L′ = ρ v∞ Γ — strictly proportional to the circulation Γ. 2 Γ = ∮ v·dl, the loop integral of velocity around any curve enclosing the wing — and for potential flow it is the same for every such loop. 3 the force is perpendicular to the free stream v∞. No circulation, no lift.
For the current settings, the machine measures Γ on two different enclosing loops and reports the lift:
| quantity | value |
|---|
A wing carries bound vorticity: the airfoil is wrapped in a sheet of spin. Integrate that spin over the section and you get exactly the circulation Γ — this is the vorticity sphere's ∮v·dl = ∬(∇×v)·dA made into flight.
Kutta (1902) and Joukowski (1906) closed the loop: the bound spin around a wing is the lift, L′ = ρ v Γ. Where vorticity asks "how much does the field curl?", Kutta-Joukowski answers "then this much you can fly on." Each sphere is the next one's premise.
The blue team's live check: re-derive the four invariants — linearity in Γ, zero-Γ→zero-lift, perpendicularity, loop-independence — from the pure functions and confirm them. If red tampers, this badge is where it shows.
Kutta-Joukowski needs three numbers about a 2-D section of wing:
| symbol | meaning | units |
|---|---|---|
| ρ | fluid density | kg·m⁻³ |
| v∞ | free-stream speed | m·s⁻¹ |
| Γ | circulation ∮ v·dl | m²·s⁻¹ |
Γ is the whole game: it is the net swirl the wing binds into the flow. Feed these three into the panel below and the lift per unit span falls straight out.
The lift is recomputed from L′ = ρ v∞ Γ on the spot — never looked up. Γ is also re-measured live by integrating v·dl around two different loops.
Streamlines: uniform flow + a bound vortex Γ around the cylinder. With Γ≠0 the flow crowds on one side (faster = lower pressure) and the stagnation points slide — that asymmetry is the lift.
Slide Γ toward zero: the lift falls with it, hitting exactly 0 at Γ=0 — d'Alembert's paradox, live.
What the machine produces, proven: the lift per unit span L′ = ρ v∞ Γ — linear in Γ (so doubling Γ doubles L′, checked to 1e-12), zero when Γ=0 (no circulation, no flight), perpendicular to v∞, and driven by a Γ that is the same on any enclosing loop (two loops agree to 1e-9). The current lift is above; the invariants are the output.
The blue team's witness (left) confirms these live; the red team (right) tries to make lift go as Γ².
It also predicts zero drag (d'Alembert) — false for any real wing. Real lift needs a starting vortex to establish Γ; real wings are finite, so 3-D induced drag and lifting-line effects appear that no 2-D circulation captures. K-J is the exact skeleton, not the whole body.
"Air over the longer top surface must rejoin its partner, so it goes faster (equal-transit)." Cut. The top air arrives sooner, not together; equal-transit is a myth. The speed-up is the circulation, and its size is L′ = ρvΓ.
"Bernoulli alone explains lift." Kept, corrected. Bernoulli turns a speed difference into a pressure difference — but why one side is faster is the bound Γ. K-J is what ties speed, pressure and force together.
"d'Alembert proved wings cannot have drag." Cut. The paradox is an artifact of inviscid theory; real drag comes from viscosity, separation and (in 3-D) trailing vortices.
The red team's move: rewrite the theorem so lift goes as Γ² instead of Γ. Now doubling the circulation quadruples the lift and Γ=0… still gives 0, so it hides — but linearity is gone. The blue team's witness (window 7) is watching.
Flip lift to Γ² and the "double Γ ⇒ double lift" invariant breaks — the witness recomputes, disagrees with the known linear law, and turns red. Nothing is faked; the attack is real and it is caught.