Spin is not the same as going round

Two words in this subject sound like synonyms and are not. Circulation is about a flow going round something. Vorticity is about the fluid itself spinning. A flow can have plenty of the first with none of the second, and it is precisely the flows that do which make aeroplanes work.

The test that separates them is one of the oldest pictures in fluid mechanics: put a tiny paddle wheel in the flow, let it be carried along, and watch whether it turns.

The middle panel is the one nobody argues with. The other two are the essay.

The wheel is not measuring what it looks like #

The paddle wheel gets described as though it detects “going round”, and it does not. It has two arms and it responds to the average of their two turning rates, which is a different quantity from either.

Consider the arm pointing along the flow direction. It turns at a rate set by how the transverse velocity changes as one moves along it — ∂v/∂x. Now the arm pointing across the flow. It turns at a rate set by how the streamwise velocity changes across the stream, and with a minus sign, because turning the top of that arm forwards rotates it the other way: −∂u/∂y.

Vorticity is the sum of those two:

$$\omega = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}$$

and the wheel turns at half of it. The figure prints both terms under each panel, and the whole argument is visible in those two numbers. In the free vortex they are −1.56 and +1.56, which sum to zero. In solid-body rotation they are +0.90 and +0.90, which sum to 1.80. In the shear they are 0.00 and −0.90.

So the wheel in the first panel is being sheared into a diamond without being rotated, and the wheel in the third is being rotated without anything in the picture going round at all.

The two cases that offend intuition #

The free vortex is worth dwelling on because it is the first of them.

Its velocity falls off as one over the distance from the centre: fast near the middle, slow far out. The streamlines are circles. Everything about the picture says this is spinning.

What actually happens to a parcel is this. As it goes round, the side of it nearer the centre is moving faster than the side further out, so the parcel is stretched around the circle and squeezed towards it. That is a deformation, and it is a large one. But the two effects on its orientation — the outer side lagging, the parcel’s own progress around the circle carrying it — cancel exactly, and the parcel arrives back where it started facing the same way it set off.

Bathwater is the everyday example and it is a good one. A leaf floating in a draining bath travels in circles and does not spin about its own centre; it goes round the drain always pointing the same way, right up until it reaches the core.

The exception is that core. At the centre the model says the velocity is infinite, which is nonsense, and a real vortex has a small region there where the fluid rotates like a solid body. All the vorticity of a real vortex lives in that core. Outside it there is none, which is why the whole outer flow can be treated with a theory that assumes there is none.

The third panel is the same surprise in reverse. A simple shear — a fluid where the speed varies linearly across the stream and every streamline is a straight parallel line — is rotating everywhere. Not going round anything, since there is nothing to go round, but rotating: a paddle wheel dropped anywhere in it turns steadily.

The reason is that only one arm of the cross has anything happening to it. The arm across the stream has its top end in faster fluid and its bottom end in slower, so it tips over. The arm along the stream sits in fluid that is all moving the same way and does not tip at all. The average of a turning arm and a stationary arm is not zero.

This is the case that matters for the layer next to a surface, where the flow is almost exactly a shear. The boundary layer is the most rotational region in any ordinary flow, and it looks the calmest.

What the solver computed, and how it was checked #

The three fields in the first figure are given by formula, and the two turning rates at the marked point are computed by central differences on those formulas rather than worked out on paper. That is deliberate: an essay whose figure printed the analytic answer would be showing its own algebra back to itself.

The check that makes this family safe is a pair, not a single test.

assertIrrotational sweeps a domain and refuses any field whose vorticity exceeds a tolerance anywhere. Applied to solid-body rotation, it refuses: worst vorticity 1.400, which is 2Ω exactly. Applied to a free vortex, over the same grid with the singular core excluded, it passes at a worst value of 1.6 × 10⁻⁷ — which is round-off, not physics.

Those two results together are what makes the check worth having. A test that only ever refused things could be refusing everything; a test that only ever passed could be blind. The two flows in question have identical streamline pictures, and the check separates them.

The second half is subtler and lives in the acceleration essay: the Bernoulli check there deliberately walks a general path rather than a streamline, because along a streamline the identity it tests holds for rotational flows too. A check that only walked streamlines would pass on solid-body rotation. Handed a radial path through it, it fails, and that failure is asserted in the gate.

The two quantities, related properly #

Circulation and vorticity are not unrelated, and pretending they are would replace one confusion with another. Stokes’s theorem ties them together exactly:

$$\Gamma = \oint_C \mathbf{u}\cdot d\mathbf{l} = \iint_S \omega \, dA$$

The circulation round a closed circuit equals the vorticity added up over any surface that circuit bounds. So circulation is not an independent thing; it is vorticity, counted.

Read carefully, that is what makes the free vortex possible rather than what makes it impossible. Take a circuit that does not enclose the centre: the vorticity inside it is zero everywhere, so the circulation round it is zero, and a paddle wheel carried round that circuit comes back unturned. Take a circuit that does enclose the centre and the surface it bounds must contain the core, where all the vorticity is. The circulation is then whatever that core contains, and it is the same for every circuit that encloses it — which is why the strength of a vortex is a single number rather than a function of how far away one stands.

The same accounting is what lets a wing be treated as a point vortex from a distance. All the vorticity is in the boundary layer wrapped round it; any circuit drawn in the clean air outside encloses the whole of it and reports the same Γ.

The two panels are one object #

The first and second panels of the opening figure are presented as opposites, and in a real vortex they are the inside and the outside of the same thing.

Rankine’s model is the simplest statement of it: the fluid rotates as a solid body out to a core radius, and outside that the swirl falls off as one over the distance. The speed therefore rises linearly from the axis, peaks exactly at the core’s edge, and decays beyond it. The vorticity is uniform at twice the core’s angular rate inside and exactly zero outside — so all of the vorticity a vortex possesses is inside its core, and the entire outer field is irrotational however violent it looks.

Read the circulation and the accounting closes. Inside the core it grows as the square of the radius, because a growing circuit is enclosing more vorticity; outside it is constant, because there is no more to enclose. That constant is the vortex’s strength, and it is the same number measured at one core radius or at a kilometre — which is the property Stokes’ theorem guarantees and the reason a distant vortex can be treated as a point.

The pressure has a tidy consequence too. Integrating the centripetal balance inwards gives a deficit at the core’s edge, and integrating on through the solid-body core adds exactly as much again — so the pressure drop from far away to the axis is precisely twice the drop from far away to the core boundary. Half the suction of a tornado, a bath vortex or a wingtip vortex is inside the visible core.

So the paddle wheel that does not turn and the paddle wheel that does are the same wheel at two radii of one vortex, and the boundary between them is the only place in the flow where viscosity has ever acted.

Why it decides where Bernoulli may be used #

The distinction is not a curiosity. It sets the boundary of the most-used equation in the subject.

Bernoulli’s relation between speed and pressure holds along a streamline in any steady inviscid flow. That is the weak form and it is nearly always what is meant. But in an irrotational flow it holds between any two points whatever, on the same streamline or not, with a single constant for the whole field.

That upgrade is what makes the ideal theory usable. It is why the pressure coefficient round an aerofoil can be computed from the local speed alone, without tracing where each parcel came from. Take irrotationality away and every streamline gets its own constant, and the constants are only knowable by following each streamline back to somewhere the pressure is known.

That figure carries the whole reconciliation. The potential exists, so the flow is irrotational. The potential does not close up — walking once round the centre leaves it larger by Γ than it started — so the circulation is not zero. Irrotational everywhere and circulating anyway: the two statements sit in one picture, and the dashed line is where they are forced to meet.

Where the vorticity in a real flow actually is #

All of the above is about where vorticity is not. It is worth seeing where it is, because the answer is startlingly concentrated.

Two thin sheets against the surface, of opposite sign, trailing off into the wake. That is the whole of it. The rest of the field is irrotational to the accuracy of the grid, which is why the ideal solution describes it as well as it does.

This is the shape of the subject in one image. Viscosity acts only where there are large velocity gradients; large velocity gradients occur only next to surfaces, because that is where the fluid is forced from the free-stream speed to zero; and vorticity is what those gradients are. So viscosity, gradients and vorticity are three names for one thin region, and everything outside it can be handled by a theory that has none of the three in it.

It also explains a fact about the picture that looks like an artefact and is not. The bands are strongest right at the surface and weakest in the middle of the wake, even though the wake is where the flow looks most disturbed. The wake is slow, which is not the same as sheared, and vorticity responds only to the second.

What the picture cannot show #

The paddle-wheel figure is drawn at one instant and it is drawn at one point per panel.

The one instant matters because the cross is shown after a short interval that the author chose. Draw it after a longer one and the free-vortex cross is sheared into something so elongated that reading its orientation becomes a matter of squinting. The claim is about the rate at time zero, and any finite interval is an approximation to it.

The one point matters more. Vorticity is a field: it has a value everywhere, and in a general flow it varies from place to place. Three panels with one marked point each cannot show that. For the two flows in the figure the value happens to be uniform — zero everywhere in the vortex except the core, and 2Ω everywhere in solid-body rotation — so the single point is representative. In the flow round a real wing it would not be, because the vorticity is essentially all concentrated in a film a millimetre thick.

And the figure cannot show the core of the free vortex, because there is nothing there to draw. The model has an infinity at the centre and the panel simply stops short of it. Everything this essay says about the vortex is a statement about the outside.

Where the model stops #

Vorticity is created at solid surfaces and essentially nowhere else in the flows this site draws. That single sentence is the whole reason ideal theory works as well as it does: a flow that starts irrotational stays irrotational, so long as it never touches anything.

Kelvin’s theorem is the formal version, and the consequences run right through the subject. It is why a wing that begins to move has to leave a starting vortex behind it, since the circulation it acquires has to be balanced. It is why the vorticity in a real flow is confined to the boundary layer and the wake, and irrotational everywhere else. It is why the exact theory can throw away viscosity and still describe most of the field correctly.

The theorem has conditions, and each one is a way for vorticity to appear where there was none. A fluid whose density depends on more than its pressure can generate it — which is how a compressible flow behind a curved shock ends up rotational. A rotating reference frame supplies it, which is most of atmospheric dynamics. And a body force that is not the gradient of anything supplies it, which is most of magnetohydrodynamics.

None of that is drawn on this site, and none of it is claimed here.

Who found it, and when #

The distinction is Helmholtz’s, in an 1858 paper on vortex motion that is one of the few papers in this subject a modern reader can pick up unaided. He proved that vortex lines move with the fluid, that their strength is conserved, and that in an ideal fluid a vortex can be neither created nor destroyed — results so striking that Kelvin spent two decades trying to build atoms out of knotted vortex rings.

The atomic theory was wrong and the mathematics was not. Knot theory as a subject exists largely because Peter Guthrie Tait started tabulating knots in order to tabulate Kelvin’s atoms, which is about as clean an example as there is of a wrong physical idea leaving a permanent mathematical inheritance.

The paddle-wheel picture itself is a teaching device of much later vintage, and it has been quietly doing damage as well as good: it is drawn so often as a wheel, with a rim, that readers take it to be reporting on the flow going round it. The version above draws it as two independent arms, because two arms is what the mathematics has.

Where the ladder goes next #

Next rungs on this anchor: Kelvin’s circulation theorem, proved rather than quoted, and the exact conditions under which it can fail; the vorticity equation, which is what the momentum equation becomes when the pressure is eliminated from it and which explains why vortex tubes stretch and spin up; vortex stretching, which is the mechanism by which turbulence takes energy from large scales and hands it to small ones; and the Biot–Savart law, which reconstructs the whole velocity field from the vorticity alone.

Then across to circulation and lift, where the distinction drawn here becomes the difference between a wing that flies and a picture of one, and to Bernoulli’s hypotheses, where it decides which of two very different equations a reader is entitled to use.