Vorticity is the local spin of a fluid — how fast a tiny paddle wheel set adrift would turn. The engine computes ω = curl v straight from the velocity field, and lands on the surprise: a free vortex has circular streamlines yet no spin at all off its axis. Curved motion and local rotation are two different things. Rendered, not quoted.
source H. von Helmholtz, “Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen,” Journal für die reine und angewandte Mathematik (Crelle) 55 (1858) 25–55 — doi:10.1515/crll.1858.55.25 (AMBER: paywalled reprint; Tait’s 1867 English translation is the accessible witness)
In 2D, vorticity is the scalar curl of the velocity field:
ω = ∂vy/∂x − ∂vx/∂y
Three canonical fields test it:
rigid rotation v = Ω×r = (−Ωy, Ωx) → uniform ω = 2Ω.
simple shear v = (y, 0) → ω = −1.
free vortex vθ = Γ/(2πr) → ω = 0 for every point off the axis.
Partials are taken by central finite difference; linear fields come out exact, the vortex leaves a residual bounded below.
The local spin of flow — vorticity; ω = curl v, twice the angular velocity, zero for the free vortex off-axis. Follow that same circulation downstream and it becomes the thing that holds an aircraft up: the bound circulation Γ whose lift is L = ρvΓ in the-kutta-joukowski — no circulation, no lift.
Live re-check of the load-bearing invariant — rigid rotation must read ω = 2Ω under the current engine. If the Red tamper swaps curl for divergence, this flips.
A velocity field v(x,y) and a probe point. Left of the canvas: rigid rotation, Ω=0.5. Right: a free vortex, Γ=2π. Each little cross is a fluid element carried by the flow.
Both fields carry elements on circles. Watch the crosses: on the left every element spins as it orbits (that spin is the vorticity, 2Ω). On the right the crosses orbit but keep their orientation — the free vortex is irrotational off-axis.
“Curved streamlines mean the fluid is rotating. The free vortex whirls in circles — obviously it spins.”
Wrong, and it is the whole point. Circulation around a loop can be non-zero while the vorticity inside is zero, because all the spin is concentrated on the singular axis. A paddle wheel dropped off-axis orbits the center without turning on its own axle. Curvature of a path ≠ rotation of an element.
“Vorticity equals the angular velocity of the fluid.”
→ It is twice the local angular velocity: ω = 2Ω. Off by a factor of 2 forever.
“Irrotational just means the fluid is at rest or moving in straight lines.”
→ The free vortex is irrotational yet everywhere in motion on circular streamlines. Irrotational is about local spin, not global path.
“Divergence and curl both measure how a field turns.”
→ Divergence measures spreading; curl measures rotation. Confusing them is exactly the tamper below.
The disclosed planted void: replace curl v with div v in the “vorticity” operator. A rigid rotation has div v = 0, so it would be reported as having zero spin — the ω = 2Ω invariant breaks and the Witness (7) catches it live.