Where vorticity comes from

Air arriving at an aeroplane has no vorticity in it. Air leaving has a great deal — a boundary layer full of it, a wake made of it, and a pair of trailing vortices that will still be turning over minutes later. Somewhere between the two it was created, and the equations of motion have no term that creates it: vorticity is stretched, tilted, transported and diffused by them, and never made.

It is made at the surface. And the rate at which it is made contains no viscosity at all.

Four lines at the wall #

Take the momentum equation along a stationary, impermeable wall and evaluate it on the wall, where both velocity components vanish:

$$0 = -\frac{1}{\rho}\frac{\partial p}{\partial x} + \nu\frac{\partial^2 u}{\partial y^2}.$$

The convective terms are gone because the velocity is zero. What is left says that the curvature of the velocity profile at the wall is fixed by the pressure gradient.

Now notice what that curvature is. In two dimensions the vorticity is $\omega = -\partial u/\partial y$ near a wall, so $\partial^2 u/\partial y^2 = -\partial \omega/\partial y$, which is the diffusive flux of vorticity out of the surface — the rate at which the wall is handing vorticity to the fluid. Therefore

$$\sigma \;=\; -\nu\frac{\partial \omega}{\partial y}\bigg|_{w} \;=\; \frac{1}{\rho}\frac{\partial p}{\partial x}.$$

The viscosity has cancelled. It made the no-slip condition necessary — an inviscid fluid may not be told anything about its tangential velocity, which is the whole of the counting argument — and it does not appear in what the condition produces. A wall generates vorticity at a rate set by the pressure it is under, and the fluid’s stickiness decides only how thickly that vorticity is spread once it is out.

Checked against a solver that knows nothing about it #

The identity is tested here on the Falkner–Skan family, where the edge velocity is $U_e = Cx^m$ and the pressure gradient is therefore known in closed form.

Those profiles are produced by bisecting on the wall slope until the shot reaches the free stream — a procedure with no reference to pressure gradients, wall fluxes or vorticity anywhere in it. The third derivative at the wall is then read off the tabulated second derivative by a fourth-order one-sided difference, and the equation says it must equal $-\beta$.

It does, at every $\beta$ tried, and the vorticity flux that follows equals the pressure gradient to $2\times10^{-6}$ of $U_e^2/x$ — with the factor $(m+1)/2$ from the similarity variable folded in, which is where an error would have shown.

A flat plate makes no vorticity at all #

Set $\beta = 0$ and the flux is zero. The computed value is $2\times10^{-14}$.

That is a stronger statement than it first sounds. A flat plate at zero incidence has a boundary layer full of vorticity, growing thicker with distance, and none of it is being created along the plate. All of it entered at the leading edge, where the pressure gradient is not zero and where the boundary-layer approximation fails; downstream of that the layer is redistributing what it already has by diffusion, spreading it thinner as it goes.

So the growth of a flat plate’s boundary layer is a diffusion problem with a fixed amount of vorticity in it, not a generation problem. That reframing is worth keeping, because it explains why the integrated vorticity in the layer is constant along the plate — a fact the momentum integral states in other language, and which follows here in one line.

And a closed body makes exactly as much of each sign #

Integrate the flux round a closed body. The flux is $(1/\rho)\,\partial p/\partial s$ and the pressure is single valued, so going once round returns to where it started:

$$\oint \sigma\,ds = \frac{1}{\rho}\oint \frac{\partial p}{\partial s}\,ds = 0.$$

Computed on a cylinder’s surface pressure, it comes to $5\times10^{-16}$.

A body cannot change the total vorticity of the fluid around it. It can only separate positive from negative — send one into a wake on one side and the other into a wake on the other side, or shed one downstream and keep the other in a boundary layer.

That is Kelvin’s theorem arriving from the wall rather than from a material loop, and it is the reason a wing that acquires circulation must leave an equal and opposite starting vortex behind it. The circulation was not created. It was separated, and the other half of it is in the air a mile back.

The result also settles a question that sounds like a puzzle. If a wing at a fixed angle in a steady flow is not generating any net vorticity, what keeps its boundary layer supplied? Nothing needs to: the layer is not consuming vorticity, it is carrying it downstream and diffusing it. A steady flow is a steady balance — vorticity in through the surface on one side, out through the wake, and the total in the field constant. The generation is not zero anywhere except a flat plate; it is equal and opposite, which is a different statement and a stronger one.

The sign of the source is the sign of the trouble #

The flux is positive where the pressure rises along the surface, which is exactly where a boundary layer is in danger.

An adverse gradient injects vorticity of the opposite sign to what the layer already carries. That opposing vorticity accumulates near the wall, the profile’s curvature there goes negative, an inflection point appears in it, and if the injection continues the wall slope reaches zero — which is what separation is.

So separation has an account in this language that is a little sharper than the usual one about running out of momentum. A layer separates because the wall has been feeding it vorticity of the wrong sign for too long, and the amount fed is the integral of the pressure rise. That is why the criterion for separation is a property of the pressure distribution rather than of the fluid, and why a laminar layer separates at a definite Falkner–Skan β regardless of Reynolds number.

There is a numerical consequence of the flux picture that is worth recording, because it is how several classes of solver are built. If the total vorticity a body makes is decided by its surface pressure, then an inviscid calculation is enough to say how much vorticity a viscous flow will have — which is the basis of every vortex method: solve the potential problem, read the surface slip velocity, and shed exactly enough vorticity to cancel it. The viscosity enters only in how the shed vorticity then spreads.

The same reading explains why a favourable gradient is stabilising in a way that is more than “there is more momentum”. A falling pressure feeds vorticity of the same sign the layer already has, into the region nearest the wall, which steepens the wall slope and makes the profile fuller. A fuller profile has no inflection point, and a profile with no inflection point is stable to the disturbances that matter. Accelerating a boundary layer does not merely postpone separation: it removes the ingredient instability needs.

What the picture cannot show #

It is two-dimensional and the wall is stationary and impermeable. A moving wall adds a term — the surface acceleration is a vorticity source too, which is how a spinning cylinder generates circulation — and a porous one adds another. In three dimensions the flux is a vector and there is a second contribution from the surface’s own motion.

The identity is exact and the picture of a “source” is not. Vorticity is not a substance and does not obey a conservation law with a flux in the way mass does; what is exact is the equation for $\partial\omega/\partial y$ at the wall, and calling it a flux is an interpretation that works beautifully in two dimensions and needs care in three.

And the leading edge is missing from every figure here. That is where a flat plate’s vorticity actually comes from, it is a singular point of the boundary-layer approximation, and nothing in this collection resolves it. The statement “a flat plate makes no vorticity” is a statement about the part of the plate where the approximation holds, which is all of it except the part that matters for this question.

One further consequence is worth drawing out because it is unexpected. The flux depends on the pressure gradient and not on the wall shear — so the place on a body where the most vorticity is being made is not the place where the friction is highest. On an aerofoil the friction peaks near the leading edge, where the layer is thinnest; the vorticity source peaks where the pressure is changing fastest, which is a little further back on the suction side and, on the pressure side, at a different station again. Two maps of the same surface, both correct, showing different things.

The one interior source, and why it was left out #

The opening sentence of this essay says the equations have no term that creates vorticity. That is true of the equations this collection has been solving, and the qualification is worth making explicit, because the term that has been dropped is the source of a great deal of the vorticity in the world.

Written without assuming constant density, the vorticity equation carries a term the constant-density version does not:

$$\frac{D\boldsymbol{\omega}}{Dt} = (\boldsymbol{\omega}\cdot\nabla)\mathbf{u} + \frac{\nabla\rho \times \nabla p}{\rho^2} + \nu\nabla^2\boldsymbol{\omega}.$$

The middle term is the baroclinic one, and it is a genuine interior source: it makes vorticity at a point in the fluid, with no wall anywhere near, out of nothing but two gradients that are not parallel. If the density is uniform, $\nabla\rho$ is zero and it disappears. If the density depends on the pressure alone — a barotropic fluid, which is what an isentropic compressible flow is — the two gradients are parallel by construction and it disappears again. Every flow in this essay is one or the other, which is why the wall was the only source in sight.

The mechanism is a torque, and it is easier to see than the algebra suggests. A pressure gradient pushes on a parcel through its centre of pressure; the parcel’s inertia resists through its centre of mass; and if the density is not uniform across the parcel those two points do not coincide. A force applied off the centre of mass is a couple, and a couple applied to a fluid element is vorticity.

Three of the most-photographed flows in the subject are made this way. A shock wave passing over an interface between two gases deposits vorticity on it — the pressure gradient is normal to the shock, the density gradient is normal to the interface, and where they cross the interface rolls up into the spikes and bubbles of the Richtmyer–Meshkov instability. A cold sea and a warm land under one atmosphere have horizontal density gradients under vertical pressure gradients, and the circulation that results is the sea breeze — a wind generated in the interior of a fluid with no surface driving it. And a curved shock on a blunt body puts different amounts of entropy into different streamlines, which is a density gradient across a pressure gradient, and the flow behind it is rotational for ever afterwards. That last one is why an inviscid flow need not be irrotational, and it is the reason a potential-flow method cannot be used anywhere behind a bow shock.

None of that weakens the wall argument; it bounds it. The clean statement is that in a constant-density or barotropic fluid, all vorticity enters through surfaces, at a rate the surface pressure gradient fixes, and the total round a closed body is zero. Admit a density that varies independently of the pressure and there is a second source, distributed through the volume, and the neat conservation statement above no longer holds — a stratified flow can gain net circulation without any boundary being involved at all.

Which is the more interesting way round to state it. The reason a wall is the only source in most of aerodynamics is not a property of walls. It is that air at low speed is very nearly a fluid whose density does not vary independently of its pressure, and the moment that stops being true — behind a shock, in a flame, in an atmosphere with a temperature front in it — the fluid starts making its own.

Why an inviscid fluid cannot start anything #

The wall-flux argument settles a question that ideal flow leaves hanging, and it settles it in a sentence.

A body in an inviscid fluid may carry circulation: the mathematics permits any value of it, and the whole family of flows round a cylinder with circulation are legitimate solutions of the equations. What ideal flow cannot do is say how a body acquired one, and it cannot, because circulation is the integral of vorticity and there is no mechanism in Euler’s equations for making any. Kelvin’s theorem says as much from the other end: the circulation round a material circuit is constant for ever.

So a wing starting from rest in a genuinely inviscid fluid would never begin to lift. It is not that lift would be small, or would take a long time to develop; there is no term in the equations that could produce the required vorticity, and the flow would remain the circulation-free solution with its rear stagnation point on the upper surface, for as long as anybody cared to wait.

What breaks that deadlock is the one condition an inviscid fluid may not be told. No slip forces a shear at the surface, the shear is vorticity, and the pressure gradient round the sharp trailing edge drives it into the fluid until the flow leaves the edge smoothly and the driving gradient collapses. The Kutta condition is the outcome of that process rather than an axiom, and the starting vortex is the receipt.

Who noticed, and why it is not better known #

Lighthill set the argument out properly in 1963, in a chapter of Laminar Boundary Layers that is still the clearest statement of it, and the identity itself is implicit in Prandtl’s original 1904 paper — which is exactly the observation that the boundary-layer equations evaluated at the wall give the curvature in terms of the pressure gradient.

It is not better known because it is not needed for calculation. Nobody computes a boundary layer by tracking vorticity flux; the momentum integral is easier and gives the same answers. What the flux picture buys is explanation: why an inviscid fluid can carry circulation but cannot acquire it, why a starting vortex has to exist, why a flat plate’s layer grows without being fed, and why an adverse pressure gradient is dangerous in a way that a favourable one of the same size is not.

The surprising connection is with the counting argument that opens this field. The no-slip condition was described there as the price of viscosity’s extra order — something a wall is allowed to say because the equation has room for it. This essay is the other half: what the wall does by saying it. A boundary condition is not only a constraint. It is a source, and the strength of the source is fixed by a quantity — the surface pressure gradient — that ideal flow computes perfectly well without any viscosity in it at all.

Where the ladder goes next #

The rung above is the three-dimensional version, where the flux is a vector, the surface’s own curvature enters, and the vorticity that leaves a swept wing’s surface has a spanwise component that the two-dimensional argument cannot produce.

The one beside it is the moving wall: a surface that accelerates generates vorticity at a rate its own acceleration sets, which is how an impulsively started plate produces the Stokes layer this collection already solves, and how a rotating cylinder acquires the circulation that makes the Magnus effect.