A wall puts in exactly its own speed

Where vorticity comes from found that every scrap of vorticity in the flow past a body enters through its surface, and that the rate at which it enters contains no viscosity at all: evaluate the momentum equation on a stationary wall, where the velocity vanishes, and the diffusive flux of vorticity out of the wall is the pressure gradient along it. That essay ended by naming the wall it had left out — the one that moves.

The same four lines settle that one. Take a wall sliding in its own plane at a speed $U(t)$ that it chooses, with the fluid stuck to it. On the wall the fluid’s velocity is the wall’s, so its acceleration there is $dU/dt$, and the momentum equation evaluated at the surface keeps that one term:

$$\frac{dU}{dt} = -\frac{1}{\rho}\frac{\partial p}{\partial x} + \nu\frac{\partial^2 u}{\partial y^2}\bigg|_{w}.$$

With $\omega = -\partial u/\partial y$, the last term is the diffusive flux of vorticity out of the wall, so

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

A wall is a vorticity source whose strength is its own acceleration, added to the pressure term, and the viscosity has cancelled from the acceleration term for exactly the reason it cancelled from the pressure term. Everything below takes the pressure gradient to be zero and looks only at what acceleration does.

Three starts, one total #

Integrate the source over time and the budget is immediate. The rate at which vorticity crosses the wall is $dU/dt$, so the total that has crossed it by any instant is the wall’s speed at that instant, and nothing has left the fluid by any other route. A wall moving at 0.1 metres a second has put exactly 0.1 metres a second of vorticity, per unit of its area, into the fluid above it.

The figure shows what that does and does not fix. A wall started at once puts its whole budget in at the first instant, in a sheet of zero thickness that viscosity immediately begins to spread; three seconds later that vorticity has diffused a millimetre or two, with its peak still at the surface. A wall ramped up over a second has made its vorticity at a steady rate throughout that second, so some of it is three seconds old and some is two, and the profile is broader and lower. A smooth start differs again. The areas under the three curves are the same to six figures, because the area is the wall’s speed and all three walls are moving at the same speed.

That is a much stronger statement than it looks. The wall’s history is completely different in the three cases, and the viscosity could be anything: nothing about the total depends on either. The history decides where the vorticity is, and the viscosity decides how fast it spreads once it is there. How much of it there is was decided by the wall’s speed alone.

A calculation that knows only the flux #

The source law was derived in two lines from an equation evaluated at a surface, and a derivation that short deserves a check that does not repeat it.

The check is two calculations with nothing in common but the wall. The first solves for the velocity. It knows the wall’s speed at every instant and imposes no slip, the ordinary way, and its vorticity is read off afterwards as the slope of the velocity. The second solves for the vorticity directly, as a diffusing quantity in cells. It contains no velocity at all, it does not know the wall is moving, and its only information about the wall is a single number on the bottom face of the bottom cell: the flux $dU/dt$.

If the source law were wrong — a factor of two, a missing term, a sign — the second calculation would put the wrong amount of vorticity in at the wrong rate, and the dots would drift off the lines. For a smooth start they agree to eight parts in ten million of the peak. Across two wall histories and two viscosities a hundred times apart the worst disagreement is a thousandth of the peak, and it sits at the corners of the steady ramp, where the acceleration jumps and a finite time step cannot follow it exactly.

The budget closes on the speed #

The same second calculation gives the budget directly. Its dots are the vorticity summed over every cell, and its lines are the wall’s speed. A fast start makes vorticity quickly and a slow one slowly, and at every instant of every run the sum sits on the speed.

That is Kelvin’s theorem arriving from the wall rather than from a material loop. Circulation is vorticity added up, and for this flow the circulation round a tall rectangle — up through the fluid far away, along the top where nothing moves, down, and back along the wall — is the wall’s speed times the rectangle’s length. The wall is the one side of that loop whose velocity changes, and the vorticity inside the loop changes to match it. Nothing is created in the interior; nothing is destroyed; it enters at the bottom, at the acceleration’s rate, and spreads.

What viscosity decides #

If viscosity is absent from the amount, it must be present somewhere, and it is: in the distance.

A wall started at once under water, air and glycerol holds the same vorticity above it at every instant. After one second it is concentrated within about a millimetre of the surface in water, spread over four in air and over three and a half centimetres in glycerol, and the centroid of each moves out as the square root of the viscosity times the time — the diffusion length that the wall that shakes found for an oscillating wall and that sets how thick every boundary layer is.

So the division of labour is exact. The wall’s speed decides how much vorticity there is, and the viscosity decides where it is. A fluid with a thousand times less viscosity does not have less vorticity above a started wall; it has the same amount packed thirty times more tightly, with a peak thirty times higher — which is why the shear stress at a started wall, the peak vorticity times the viscosity, falls as the square root of the viscosity rather than as the viscosity itself.

A wall that stops #

A wall that stops is a wall that accelerates the other way, and the source law applies unchanged.

Stopping is an impulsive start in reverse, so it injects a sheet of vorticity equal and opposite to the one the start put in, at the surface, beneath the old vorticity that has had a second to diffuse outwards. From then on the two sheets spread together, and the profile has a positive layer above and a negative layer below. The total is zero, as it must be for a wall with no speed.

The fluid is still moving. Its momentum per unit area is 0.058 millimetre-metres per second half a second after the stop and 0.021 after seven seconds, falling because the stopped wall now holds the fluid back. And that momentum is exactly the first moment of the vorticity: integrating the velocity by parts gives $\int u\,dy = \int y\,\omega\,dy$, so a profile whose vorticity sums to zero but whose positive part sits further from the wall than its negative part carries momentum.

That separates two things the word “vorticity” tends to run together. The net vorticity above a wall records whether the wall is moving now. The distribution of vorticity records what the fluid is doing, and a fluid can be in motion with no net vorticity above it at all. An oscillating wall is the extreme case: its net vorticity follows its velocity and averages to zero over a cycle, while the wave of alternating layers it sends upwards carries energy away into the fluid for as long as it shakes.

The shaken wall, read as a source #

The wall that shakes solved Stokes’s oscillating wall as a problem of velocity, and found that the shear at the wall leads the wall’s own velocity by an eighth of a cycle. The source law reads the same flow as a budget, and adds a second phase to it.

A wall moving at $U_0\cos\omega t$ holds, at every instant, a net vorticity of $U_0\cos\omega t$ above it: the budget closes on the speed here as everywhere. The rate at which vorticity is entering is the acceleration, $-U_0\omega\sin\omega t$, which leads the velocity by a quarter of a cycle. And the vorticity sitting at the surface — which is the shear, and which is what the wall feels — leads by an eighth, half-way between the two, because it is the part of the budget that has entered most recently and has not yet diffused away.

So the three phases are one statement. The source runs a quarter of a cycle ahead because it is a rate; the net runs with the velocity because it is a total; and the surface value runs between them because it is a total weighted towards the recent past, with a memory whose kernel is the diffusion equation’s own. Averaged over a cycle the net vorticity above a shaken wall is zero, and the energy it sends into the fluid is not.

A floor that moves with the stream #

The source law has one practical consequence that engineers arrived at without it. A model tested for ground effect sits above a floor, and a stationary floor under a moving stream grows a boundary layer that a real road or runway does not have: the air there is at rest relative to the ground, and the ground is at rest.

The fix is a belt running under the model at the stream’s speed. Read as a source, what the belt does is exact rather than approximate. Where the floor moves at the free-stream speed and the pressure along it is uniform, neither term in $\sigma = dU/dt + (1/\rho)\,\partial p/\partial x$ is non-zero — the belt is not accelerating, and there is no gradient — so the floor makes no vorticity at all. What remains is made where the model’s own pressure field reaches the floor, which is the ground effect the test exists to measure, and the belt is the only floor that makes that and nothing else.

A stationary floor under the same stream is not accelerating either. It makes its vorticity at the leading edge of the floor, where the stream first meets a surface at a different speed — a sheet put in impulsively, as a started wall puts in its speed — and then spreads it downstream as a layer. The belt removes the difference in speed, and with it the sheet.

A cylinder spun up #

The flat wall has a curved relative in which the budget has a visible consequence for lift.

A cylinder of radius $a$ starts spinning at $\Omega$. No slip makes the fluid touching it turn with it at once, so the circulation round a circle drawn just on its surface is $2\pi a \cdot \Omega a = 2\pi a^2\Omega$ from the first instant, in any fluid.

The fluid a little further out has not moved. So the circulation round a slightly larger circle is still zero, and the difference between the two circles is the vorticity between them: the spin-up has put a thin ring of vorticity, of total strength exactly $-2\pi a^2\Omega$, into the fluid next to the surface. The cylinder’s own rotation and that ring add to nothing, and from far away there is no circulation at all.

Then the ring diffuses outwards. Every circle it crosses stops enclosing it, and the circulation round that circle rises towards the surface’s value. Lift with no wing at all is the Magnus effect computed from exactly that circulation — so the figure answers a question the ideal-flow calculation cannot ask, which is where a spinning cylinder’s circulation comes from. It is not built up by viscosity. It is present at the surface at once, and viscosity carries away the counter-vorticity that hides it.

The arrival is a diffusion clock. Half the circulation reaches two radii in a time of $0.665\,a^2/\nu$ and twenty radii in $143\,a^2/\nu$, and the ratio of the time to the square of the radius settles towards a constant as the cylinder’s own size stops mattering. For a cylinder of ten centimetres’ radius in water that is nearly two hours to reach two radii; in a stream the counter-vorticity is carried downstream far faster than it diffuses, which is why a real rotor’s lift arrives on the flow’s time scale and not on this one. The amount it arrives at is the same either way.

What the circulation costs #

Circulation is not energy, and the figure that shows the difference is the torque.

The torque is large while the layer is thin and falls as it thickens, as a started wall’s drag does. It does not fall to zero. It settles at $4\pi\mu\Omega a^2$, the torque of the flow a spinning cylinder ends with — a potential vortex, whose speed falls as one over the radius, which has no vorticity anywhere and is still shearing everywhere, and so still dissipates. That is the same fact that gives a clean bubble its drag from an irrotational flow: vorticity and dissipation are not the same thing, and a flow can carry either without the other.

The calculation balances its own books. The time integral of the torque the cylinder exerts equals the angular momentum of the fluid at the end, to a hundred-thousandth, and with the torque settled at a constant the angular momentum grows in proportion to time. The circulation arrives for nothing; the angular momentum does not, and a spinning cylinder in an unbounded fluid goes on setting more distant fluid turning, and paying for it, for ever.

What the moving-wall model leaves out #

Pressure gradients. Every wall here moves through fluid at rest with no pressure gradient, so the only source is the acceleration. A body moving through a fluid has both, and the two terms add; the pressure term is the one that decides separation, and the acceleration term is the one that decides what an unsteady body puts in on top of it.

Curvature and three dimensions. A flat wall and a circular cylinder keep the source a scalar. On a curved surface in three dimensions it is a vector, the surface’s own curvature adds terms, and a surface can shed vorticity pointing in directions the flat argument cannot produce.

A stream. The cylinder spins in still fluid, so its counter-vorticity can leave only by diffusion. In a stream it is convected away, and the lift a started wing takes time to reach is set by how far downstream its shed vorticity has travelled, not by how far it has diffused.

Stability. Every flow here is laminar and stays so. A started wall’s layer is stable for a long time; a cylinder spinning fast enough inside an outer boundary is not, and breaks into Taylor’s cells.

A budget with a long history #

Rayleigh solved the impulsively started plate in 1911, and Stokes the oscillating one sixty years earlier, both as problems of velocity. Reading the wall as a source of vorticity, with a strength fixed by the momentum equation evaluated on the surface, is Lighthill’s framing of 1963, and the moving-wall term in it was set out explicitly by Morton in the 1980s in an account of how vorticity is generated and destroyed. The framing adds nothing to the velocity solutions. What it adds is a budget that holds for every wall motion at once, and a way of saying where a body’s circulation came from that ideal flow, which has no source at all, cannot.

Still open: the source as a vector #

The flat wall and the cylinder make vorticity pointing one way only, parallel to the surface and across the flow. A surface curved in two directions — a swept wing, the hull of a ship, a bubble — makes vorticity whose direction depends on the surface’s curvature as well as on the pressure gradient and the acceleration, and part of it points along the flow. That is where the spanwise vorticity of a swept wing’s boundary layer comes from, and it is the version of this budget that three dimensions require.

Beside it is what the vorticity does once it has left the wall. Two sheets of it, or two rings, or a ring and its image in a surface, move one another at speeds fixed by their circulation alone — vortices move each other — and a shear-free surface, which cannot hold a stress, turns out to make vorticity too: a clean surface cannot refuse it wherever it is curved.