Circulation is vorticity, added up

Two quantities on this site are constantly mistaken for one another, and the confusion is expensive because lift is one of them.

Circulation is a property of a loop: pick a closed curve in the fluid, walk round it, and add up the component of the velocity along the direction of travel. Vorticity is a property of a point: it is twice the local rate of rotation of the fluid there, and a paddle wheel dropped into the flow measures it.

They have different units, different arguments and different meanings, and a theorem says that one is the integral of the other.

The theorem, and why it is worth testing #

Stokes’ theorem, in the flat two-dimensional case, says

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

for any closed curve $C$ bounding a region $A$ on which the field is defined. The left is circulation, the right is total vorticity, and the theorem makes them equal.

On paper that settles the confusion. Circulation is not spin, it is accumulated spin — the amount of rotation inside the loop, wherever inside the loop it happens to be, added together with a sign.

The reason to compute both anyway is that the theorem is a statement about exact fields and the things drawn on this site are numerical ones. A vorticity–streamfunction solver keeps vorticity as a stored array and produces velocity by differencing a streamfunction that was obtained by solving a Poisson equation from that same array. Those two objects can drift apart — a boundary condition applied to one and not the other is enough — and nothing else the solver reports would show it. Stokes’ theorem is therefore a check on the field, not on the mathematics.

What the solver computed, and how it was checked #

The loop is a rectangle in the wake, from six tenths of a diameter behind the body to three diameters, and from just above the centreline to one and a half diameters up. It sits entirely in fluid, which turns out to matter.

The circulation is computed by walking the four sides in 1,200 steps each, evaluating the velocity at the midpoint of every step and accumulating $\mathbf{u}\cdot d\mathbf{l}$. The vorticity total is computed on a 420 × 420 grid of sample points inside the rectangle, each contributing $\omega \, dA$. One walks the boundary and never looks inside; the other samples the inside and never touches the boundary.

The results, at four Reynolds numbers:

Nothing in the solver arranges this. The agreement is at the level of the discretisation error of two different quadratures on a coarse grid, and it is the strongest available evidence that the stored vorticity really is the curl of the velocity the figures are drawn with.

The rejection test is the other half. Handing the check a vorticity field multiplied by 1.4 — a perfectly smooth, plausible field, wrong by an amount no picture would reveal — makes it refuse.

The loop that appears to break it #

Move the rectangle so that it straddles the cylinder and the agreement falls apart: −2.8802 round the boundary against −2.6337 inside, which is 8.6% and far outside anything the numerics can explain.

The theorem has not failed. The rectangle now contains 27,600 sample points that are inside the solid body, where the field does not exist and the solver stores nothing, and those cells are skipped. What is being compared is a line integral over a complete boundary against an area integral over a region with a hole in it.

This is not a defect to be tidied away. It is the mechanism behind the single most important fact in the subject: a loop drawn around a wing can have circulation while the flow everywhere on and inside that loop is irrotational, because the loop is not enclosing a simply connected patch of fluid. The starting vortex is the bookkeeping entry that keeps the total honest.

Worth noticing in passing: a rectangle drawn symmetrically about the body gives −0.0001 round the loop and −0.0005 inside it. Both are zero. The wake of a symmetric body at these Reynolds numbers carries equal and opposite vorticity above and below the centreline, so the total is nothing at all — which is why this comparison is made on a one-sided loop, and why a check written only against the symmetric case would have compared two zeros and passed no matter what.

Circulation with nothing spinning #

The cleanest case is the one with no viscosity in it at all.

Sampling the vorticity of that field at points scattered through it returns $3 \times 10^{-8}$, $-1.7 \times 10^{-9}$, $-5.7 \times 10^{-9}$ — round-off, on a field whose analytic vorticity is exactly zero away from the origin. And yet the line integral round a loop containing the origin is 2, to six decimal places.

Both are true because the origin is not in the domain. The velocity goes as $1/r$ and is undefined at the centre, so the region bounded by the loop is not one on which the theorem’s hypotheses hold. Move the loop to one side, so that it encloses no singular point, and the circulation drops to $-6 \times 10^{-10}$.

That pair — 2 for a loop around the centre, zero for a loop beside it — is the whole content of the distinction. Circulation is not a local property being reported at a distance. It is an integral, and it is sensitive to what is inside the curve, including things that are not fluid.

The local measurement, made three times #

The theorem relates an integral to a sum of local values, so it is worth being sure the local value means what it is supposed to. The measurement is the one a paddle wheel makes: drop a cross of two short fluid lines into the flow, let it go, and watch whether the average of the two arms’ turning rates is anything other than zero.

Those three cases are the reason a picture of streamlines cannot answer the question. The first two have identical streamline patterns — concentric circles — and opposite answers. The third has straight parallel streamlines and a non-zero answer. Circulation round a large loop would separate the first two, and the theorem is what connects that global measurement to the local one.

How a wing acquires circulation from nowhere #

The most useful application of the multiply-connected loophole is the one that makes flight describable.

A wing sitting in an irrotational stream is surrounded by fluid with no vorticity in it anywhere. A loop drawn around the wing nonetheless has circulation, and by Kutta–Joukowski that circulation is the lift. Both statements hold at once because the loop encircles the wing, so the region it bounds is not a patch of fluid at all: it has a hole in it, and the theorem’s hypotheses fail exactly there.

If instead the loop is drawn as a material circuit — a circle of fluid particles, far from the wing, drawn before the wing ever moved — then it does enclose only fluid, its circulation was zero at rest, and Kelvin’s theorem says it must stay zero. When the wing starts and acquires circulation, that material loop still encloses zero, so something with the opposite circulation must be inside it. It is, and it is the starting vortex left on the runway.

The whole of that argument is bookkeeping on this one theorem, and none of it works without the distinction between a loop enclosing fluid and a loop enclosing a body.

Where the vorticity actually is #

In a viscous flow the theorem earns its keep in a second way: it says where to look.

Vorticity in the flow past a body is not spread through it. It is manufactured at the surface, by the no-slip condition, and then diffuses and convects away in a thin layer and a wake. Everywhere else — the entire outer flow — is irrotational to a very good approximation, which is what makes ideal theory useful at all despite being wrong about drag.

That concentration is the reason a loop drawn anywhere in the outer flow, not enclosing the body, returns essentially zero — and the reason a loop that does enclose the body returns the bound circulation, which is the lift divided by $\rho U$.

Why the wake’s circulation grows with Reynolds number #

The table above has a pattern in it worth reading rather than passing over. The circulation in the one-sided wake box is −0.24 at Reynolds number 1, −0.50 at 10, −1.66 at 40 and −2.73 at 100. It grows by more than a factor of ten across the range, on a body of the same size in a stream of the same speed.

The vorticity is made at the wall, at a rate set by the velocity gradient there, and it is removed by two processes: it is carried downstream, and it diffuses. Diffusion is what cancels the two signs against each other, since the upper half of the flow makes vorticity of one sign and the lower half the other. Raising the Reynolds number weakens diffusion relative to convection, so less of the vorticity is annihilated on the way and more of it survives into the box.

That is also the sense in which a wake is the vorticity: the region behind a bluff body that looks different from the rest of the picture is precisely the region into which the surface’s vorticity has been swept. Everything outside it can be treated as irrotational, and the whole of the boundary-layer decomposition rests on that being true.

The number the site cannot get from this is a drag coefficient. Reading drag out of the wake is how a wind tunnel does it, and it does not work on this grid: the momentum deficit surveyed at different stations disagrees by more than its own size, and the site’s own machinery refuses it. Circulation survives that limitation because it is an integral of a stored quantity rather than a difference of two nearly equal ones.

What the wall is actually doing #

“Manufactured at the surface by the no-slip condition” is the standard sentence and it names a requirement rather than a rate. The rate is knowable, and knowing it turns the vorticity account into something that can be followed round a body.

Write the momentum equation at a stationary wall, where the velocity is zero and every convective term vanishes with it, and what is left is a balance between two things: the pressure gradient along the wall, and the diffusion of vorticity away from it. So the flux of vorticity into the fluid is proportional to the tangential pressure gradient, and to nothing else:

$$\nu\,\frac{\partial \omega}{\partial n} \;=\; -\frac{1}{\rho}\,\frac{\partial p}{\partial s}$$

A wall is not a source of vorticity because it is sticky. It is a source because there is a pressure gradient along it, and a wall with no pressure gradient on it injects none at all.

Follow that round a cylinder and the whole story of the flow reads off. Over the front the pressure falls, so vorticity of one sign is injected steadily and swept downstream into a thin layer. Past the shoulder the pressure rises, the sign of the injection reverses, and vorticity of the opposite sense begins to accumulate next to the wall underneath the layer already there. Separation is what happens when enough of it has arrived to reverse the shear at the surface, which is why an adverse gradient is the villain of every essay in the viscous field: it is the term that runs the source backwards.

And the bookkeeping closes globally. A body started from rest in still fluid injects equal and opposite totals — everything the layer carries is matched by what was left behind, which is Kelvin’s theorem read as a statement about a manufacturing process rather than about a material circuit.

What the picture cannot show #

The area integral is only as good as the resolution of the field it sums. On this grid the layer next to the body is a few cells thick, so the vorticity generated at the wall is smeared over a distance the physics would put it in a much thinner strip. The total survives that smearing — it is an integral, and integrals are forgiving — but no statement about the vorticity profile across the layer should be read off these pictures.

A second thing the pictures cannot show is the sign convention doing its work. Vorticity is a signed quantity and the wake carries both signs, so a region containing equal amounts of each has none. A picture of vorticity magnitude, which is what a great many published visualisations show, destroys exactly the information the theorem uses.

Where the model stops #

Everything here is two-dimensional. In three dimensions the theorem still holds with the vorticity vector projected onto the surface normal, and the immediate consequence is Helmholtz’s result that vortex lines cannot end in the fluid — they close on themselves or terminate on a boundary. That is what forces the trailing vortices of a finite wing to exist at all, and it is why a wing with ends pays a drag that an infinite one does not.

The solver’s own limit is the one recorded elsewhere: it does not shed a periodic vortex street at these Reynolds numbers, so the wake here is a standing pair rather than a train, and nothing on this page should be read as a statement about shedding.

Why this is the check to run on a new solver #

Of all the assertions on this site, this is the one that would be worth running first on a field whose provenance was unknown.

Most checks compare a computed quantity against a value from somewhere else — a closed form, a tabulated constant, another solver. That works only where such a value exists, which rules out exactly the interesting cases: a viscous wake at Reynolds number 40 has no closed form to be checked against. Stokes’ theorem needs no external value at all. It compares the field against itself, along two routes that a broken field has no way to reconcile.

It is also cheap and unforgiving in the right proportions. A quadrature over a boundary and a quadrature over an area both converge as the mesh is refined, so agreement to a fraction of a per cent on a coarse grid is a strong statement; and the two integrals weight the field completely differently, so an error concentrated anywhere in the interior shows up in one and not the other.

The general form of that idea is the habit the whole site runs on: compute a thing twice by routes that share nothing, and treat the agreement as the evidence. The lift is computed by circulation and by surface pressure; the force on a wing is computed on its surface and on a box in the fluid; and here the rotation in a region is computed on its edge and in its middle.

Who found it, and when #

The theorem is Kelvin’s, communicated to Stokes in a letter in 1850, and Stokes set it as an examination question in Cambridge in 1854, which is how it acquired the name. Its use in fluid mechanics is older than either in spirit: Helmholtz’s vortex theorems of 1858 are the same observation applied to a fluid that conserves its circulation.

The distinction it settles — that a flow can circulate without rotating — is the one that took the subject longest to absorb, and the one this site’s paddle-wheel essay exists to demonstrate. It is still routinely got backwards in print.

Where the ladder goes next #

Below this rung, spin is not the same as going round makes the local distinction with three flows that have identical streamline pictures. Above it, the same accounting turns into Kelvin’s theorem and the starting vortex: if circulation round a material circuit cannot change, and a wing acquires some, then an equal and opposite amount had to be left behind.

The other direction leads to the boundary layer, where the vorticity is manufactured, and to how thick that layer is — a question that is really about how far the vorticity has diffused from the wall that made it.