The vortex a wing leaves behind

A wing lifts because there is circulation round it. That much is settled elsewhere on this site, and it raises an awkward question that the settling did not answer.

The wing was sitting still an hour ago. Sitting still, there was no circulation anywhere. Now there is. Circulation is not the kind of quantity that can simply appear, and it did.

The numbers in that figure are the essay. Round the wing, −2.400. Round the vortex left behind, +2.400. Round a circuit containing both, 0.000. Nothing was created.

What Kelvin’s theorem actually says #

The statement is about a material circuit — a closed loop drawn through the fluid and made of particular parcels of fluid, so that it deforms and moves with the flow rather than staying where it was drawn.

For such a circuit in an inviscid fluid of uniform density with only conservative body forces acting, the circulation round it does not change with time. DΓ/Dt = 0.

Two things about that are worth stressing before anything is built on it.

The circuit moves. This is not a statement about a fixed loop in space, and the distinction is exactly the Eulerian–Lagrangian one that runs through this subject. A fixed loop can perfectly well see its circulation change, because different fluid keeps arriving in it. A loop made of fluid cannot.

And the theorem is about the total, not about where it is. Vorticity can be moved, concentrated, stretched and rearranged inside the circuit however the flow likes. What it cannot do is change how much of it there is.

The argument for the starting vortex #

Take a large circuit, drawn in still air, well away from a wing that is not yet moving. Its circulation is zero, because everything is still.

Now start the wing. The circuit is made of fluid, so it deforms — it stretches out behind as the air it is made of is disturbed — but it is still the same circuit and it still contains the same fluid. Kelvin says its circulation is still zero.

The wing now has circulation round it, and the wing is inside the circuit. So something else inside the circuit must have the opposite circulation, in exactly the same amount. That something is the starting vortex, and it is left behind at the place where the wing began to move.

The argument has a satisfying quality that most arguments in this subject lack: it produces a prediction about something nobody was looking for. Nothing about lift suggests that an object should be left behind in the air, and the theorem insists on it.

It was duly photographed. Ludwig Prandtl’s laboratory took pictures of the vortex shedding from an aerofoil started impulsively in a water tank, and the vortex is exactly where the accounting says it should be, turning the way the accounting says it should turn.

It is worth being clear about what kind of argument that is. Nothing about the theorem explains the starting vortex in the sense of giving a mechanism; the mechanism is viscous and messy and happens at the trailing edge in the first few milliseconds. What the theorem does is forbid the alternative. A wing that acquired circulation with nothing shed would violate a conservation law, and conservation laws are the strongest instruments in physics precisely because they rule out possibilities without needing to know how anything works.

Why the wing keeps its circulation and the vortex does not follow #

An obvious objection: if the two are equal and opposite, why do they not cancel?

They do not cancel because they are in different places. Circulation is not a fluid property that can diffuse away on demand — it is vorticity, added up — and the vorticity of the two is concentrated in two separate regions with clean irrotational fluid in between. A circuit round one sees one; a circuit round both sees zero.

What actually happens to the starting vortex is that it is left behind. The wing flies away from it at flight speed, and the vortex sits in the air where it was made, spinning down slowly under viscosity over a matter of minutes.

That leads to a consequence worth stating plainly, because it is the sort of thing that sounds like a joke and is not. Every aircraft that has ever taken off has left a vortex on the runway. It is still there in some attenuated form for a while afterwards, and vortices of exactly this kind — the trailing pair rather than the starting one, but the same accounting — are the reason air traffic control imposes separation distances behind large aircraft.

The same reasoning runs in reverse and produces a second object nobody looks for. A wing that reduces its circulation must shed vorticity of the opposite sign, so every change of incidence, every control input, every gust leaves something behind. A wing in steady flight sheds nothing; a wing being flown sheds continuously, and the wake behind a manoeuvring aircraft is a record of what the pilot has been doing. Bringing an aircraft to a stop sheds a stopping vortex equal and opposite to the starting one, which finally balances the books for the whole flight.

What the solver computed, and how it was checked #

The field in the figures is two point vortices of equal and opposite strength in a light stream. It is exactly the kind of arrangement that could be asserted rather than measured, so it is measured.

Three circulations are computed by direct line integral of velocity around three closed loops: a small one round the wing, a small one round the shed vortex, and a large one containing both. Each loop is three thousand or six thousand segments, and at each segment midpoint the velocity is evaluated from the field and dotted into the segment.

The results are −2.400, +2.400 and 0.000 against a nominal strength of 2.4. That is the theorem measured on a field that was never told about it.

The check that makes this worth printing is the third number. The first two are, in a sense, guaranteed: a loop round an isolated vortex will return its strength because that is what a vortex is. The third is a cancellation between two contributions of size 2.4, and it comes out below 5 × 10⁻³ — which is the statement that the two vortices really are equal and opposite rather than approximately so.

Vorticity itself is checked as well. The site’s irrotationality assertion refuses solid-body rotation and passes a free vortex, and the pair drawn here is two free vortices: irrotational everywhere except at two points, with all the circulation living at those two points and none of it in between.

The pair is growing, and the growth is the lift #

The second figure shows the two vortices further apart than the first, and treats that as scene-setting. It is not: the separation is the lift, and the accounting closes in two lines.

A pair of equal and opposite vortices in a fluid carries a hydrodynamic impulse — the momentum that would have to be applied to create the arrangement from rest — of magnitude $\rho\Gamma d$, where $d$ is the distance between them, directed at right angles to the line joining them. That is a property of the pair rather than of either vortex, which is why neither figure’s individual line integral reveals it.

Now let the wing fly. The starting vortex stays where it was made and the wing goes on at speed $U$, so $d$ grows as $Ut$ and the impulse of the pair grows at the steady rate $\rho\Gamma U$, directed downwards through the fluid. Momentum is being handed to the air at exactly that rate, and the reaction on the wing is upward and equal:

$$L = \rho U \Gamma$$

which is Kutta–Joukowski, arriving here as a consequence of the book-keeping rather than as an independent result.

Two things are worth taking from that. The lift is not stored anywhere — it is the rate at which a separation grows, so a wing generating steady lift is an arrangement whose momentum increases without limit, which is only troubling until one remembers that the air is unbounded. And it reconciles the two accounts this collection keeps setting side by side: the momentum argument and the circulation argument are not two descriptions to choose between here. The circulation is fixed, the geometry does the growing, and their product is a force.

Where the circulation lives on the wing #

The wing’s own circulation is drawn above as a point vortex, which is a fiction that needs unpicking.

There is no vortex inside a wing. What there is, is a boundary layer — a film of strongly sheared fluid a millimetre or two thick wrapped round the surface — and that film is where the vorticity is. The upper and lower surfaces carry vorticity of opposite sign in unequal amounts, and the net is the circulation.

That is a genuinely surprising place for it to be. The circulation is the quantity that determines the whole lift of the wing, it is the quantity that makes the outer flow what it is for tens of metres in every direction, and every last bit of it lives inside a film thin enough that the ideal theory can ignore its existence and still describe the field correctly. A wing is, from the point of view of the air, an extremely thin sheet of spinning fluid that happens to have aluminium inside it.

Seen from a distance the distinction vanishes. Any circuit drawn in the clean air outside the layer encloses the whole of it and reports the same Γ, and that is why the wing can be replaced by a single bound vortex for any purpose that does not involve the surface itself.

The word bound is doing work. A free vortex moves with the fluid, as Helmholtz proved; a bound vortex does not, because it is being continuously regenerated at the surface. It is bound to the wing in the sense that it goes where the wing goes, and it is the one vortex in the subject that breaks Helmholtz’s rule — because the rule assumes an inviscid fluid and the layer generating it is the one place viscosity is not negligible.

What the picture cannot show #

The figures are drawn in the frame of the air rather than the frame of the wing, and the caption says so, because it matters. In the wing’s frame the starting vortex recedes at flight speed and the picture would be almost entirely empty.

More importantly, they are drawn as a steady field at one moment. The real process is irreducibly unsteady: over the first few chord-lengths of travel the circulation on the wing builds up from nothing, the shed vorticity rolls up into a concentrated core, and the flow near the trailing edge does something complicated involving a moving stagnation point. None of that is in these pictures, and this site’s solver cannot produce it.

There is also nothing here about how long it takes. The build-up is not instantaneous, and the relevant measure is chord-lengths travelled rather than seconds — the Wagner function, which describes the growth, reaches about half its final value in half a chord and 90% in around seven. A wing that has moved one chord-length has most of its circulation; a wing that has moved a hundredth of one has almost none.

And the point vortices are singular. At their centres the velocity is infinite, which is why both figures exclude a small disc round each. A real starting vortex has a core of finite size where the fluid rotates as a solid body, and that core is the only rotational part of the whole picture.

Where the model stops #

Kelvin’s theorem has four conditions and each is a way for the accounting to fail.

Inviscid. The one that matters most, because it is false everywhere it matters. Viscosity is precisely how the wing acquires circulation in the first place: the Kutta condition is a consequence of viscosity, and the starting vortex is shed from the boundary layer. The theorem is used here to reason about a process it forbids, and the justification is the usual one — viscosity acts in a thin layer, the circuits drawn are outside it, and the accounting between them is inviscid.

Uniform density, or at least density a function of pressure alone. Fails in a stratified atmosphere and behind a curved shock, both of which generate circulation from nothing.

Conservative body forces. Gravity qualifies; the Coriolis force in a rotating frame does not, which is why atmospheric flows have a source of vorticity this analysis has no room for.

A material circuit. The condition most often violated by accident, usually by someone applying the theorem to a fixed loop and concluding something false.

The first of those deserves one more sentence, because the position this essay occupies is genuinely awkward and glossing it would be dishonest. The theorem is being used to establish that a wing must shed a starting vortex, and the theorem holds only in a fluid where a wing could never acquire circulation in the first place. Both halves are needed and they contradict each other.

The resolution is the standard division of the flow into two regions, and it is Prandtl’s. Viscosity matters in a film next to the surface and in the wake that trails from it, and nowhere else. Every circuit drawn in these figures lies entirely outside that film, in fluid where the inviscid theorem applies exactly. The vorticity crosses from one side of the accounting to the other through the wake — which is precisely the one place the circuits are not allowed to be drawn, and precisely why the accounting works.

Who found it, and when #

Kelvin proved the theorem in 1869. Helmholtz’s vortex theorems, on which it builds, are from 1858. Both predate any suggestion that circulation had anything to do with flight by about forty years.

The application is Lanchester’s. Frederick Lanchester worked out in the 1890s that a wing must carry circulation and that the circulation must be shed at the tips as trailing vortices, and published it in 1907 in a book almost nobody read — partly because he wrote in a private vocabulary and partly because the Royal Society had declined the paper a decade earlier. Prandtl and his students arrived at the same picture independently and expressed it in a form the profession could use.

The starting vortex is the part of the picture that convinced people. A theory that says the flow must contain something nobody had noticed, followed by a photograph of that thing, is a very different sort of argument from a theory that explains what was already known.

There is a small injustice in how the credit settled, and it is worth recording. Lanchester had the whole physical picture — circulation, trailing vortices, the induced drag that comes with them — a decade before Prandtl’s group published anything on it, and got very little for it. The reasons are mostly about presentation rather than priority: he invented his own terms for everything, argued from physical reasoning where his readers wanted equations, and worked outside any institution. Prandtl supplied the equations and the students, and the students supplied the textbooks.

What Lanchester did get was the satisfaction of being right in print before anyone else, and a theorem named after neither of them.

Where the ladder goes next #

Next rungs on this anchor: the horseshoe vortex, which is what the bound vortex and the starting vortex become when the wing is given ends and the two are joined by trailing legs; the Wagner function, and how quickly circulation actually builds; the stopping vortex, shed when a wing decelerates and equal and opposite to the starting one; and wake turbulence, which is this accounting turned into an operational constraint on how closely aircraft may follow one another.

Then across to the sharp edge, which decides how much circulation is acquired, and to spin and circulation, where the difference between the two quantities this essay keeps switching between is set out properly.