The vorticity nothing decides

Steady two-dimensional Euler flow says that the vorticity is a function of the streamfunction: $\omega = F(\psi)$. On a streamline that comes from upstream, $F$ is a fact about the flow’s history — the vorticity that streamline was carrying when it arrived, and has been carrying ever since.

On a closed streamline there is no upstream. A recirculating bubble behind a body, an eddy in a cavity, the interior of a vortex ring: the fluid on those streamlines came from nowhere and has been going round for as long as the flow has existed. Nothing determines what vorticity it carries.

The ambiguity is therefore not one number per hole, which is what circulation is. It is a whole function, and there are as many exact steady solutions of the same problem, with the same boundary and the same conditions, as there are functions.

The two members, computed #

The left-hand flow is $\psi = A\sin kx\,\sin ky$, whose Laplacian is $-2k^2\psi$, so $\omega = 2k^2\psi$ and $F$ is linear. It is the flow everybody draws when they want a picture of an array of eddies, and it is exact.

The right-hand flow solves $\nabla^2\psi = -\omega_0$ with $\psi = 0$ on the walls — Prandtl’s torsion function, a Fourier series that is elementary if it is written as a particular solution plus a harmonic correction, and slowly convergent if it is written as a double sine series. Both were tried here and the second was discarded: its Laplacian converges as $1/mn$, so truncating it puts a six per cent ripple into the vorticity and makes an exact solution look like a poor one. In the corrected form every term of the correction is harmonic, so $\nabla^2\psi = -\omega_0$ holds at any truncation.

The residual $|\nabla^2\psi + F(\psi)|$ is below four parts in ten million for both, which is the check that both really are solutions.

The argument that chooses #

Prandtl gave it in 1904 and Batchelor made it a theorem in 1956, and it is one line.

Take the steady vorticity equation with viscosity kept, however small:

$$\mathbf{u}\cdot\nabla\omega = \nu\nabla^2\omega.$$

Integrate over the region enclosed by a closed streamline $C$. The left-hand side integrates to zero, because it is the divergence of $\omega\mathbf{u}$ and $\mathbf{u}\cdot\mathbf{n} = 0$ on $C$ — no vorticity is carried across a streamline. So

$$\nu\oint_C \frac{\partial\omega}{\partial n}\,dl = 0.$$

The $\nu$ cancels. With $\omega = F(\psi)$ the normal derivative is $F'(\psi)|\nabla\psi|$, and $F'$ is constant on the streamline, so

$$F'(\psi)\oint_C |\nabla\psi|\,dl = 0.$$

The remaining integral is a length times a speed. It is positive on any streamline the flow actually goes round. Therefore $F' = 0$: the vorticity inside a steady recirculating region is uniform.

What just happened #

Read the argument again for where the physics entered, because it is not where it looks.

Nothing was assumed about the size of $\nu$. The conclusion holds for $\nu = 10^{-30}$ as strongly as for $\nu = 1$. But it does not hold for $\nu = 0$, because at $\nu = 0$ the equation being integrated is $\mathbf{u}\cdot\nabla\omega = 0$, which is satisfied identically by any $F$ and gives no condition at all.

So: setting the viscosity to zero and letting it go to zero give different answers. The limit is singular, in the technical sense — the solution of the problem with a small parameter does not tend to the solution of the problem with the parameter set to zero. The inviscid theory has a family; the vanishing-viscosity limit has one member of it.

That is the strongest form this pattern takes anywhere in the collection. The missing information is not supplied by the equations, not by the boundary, and not by the history. It is supplied by a limit, and the limit remembers something the limiting equation has forgotten.

What a real cavity does about it #

The theorem describes a limit and a laboratory reaches it from below, so it is worth saying what the approach looks like.

Drive a cavity at a low Reynolds number and the interior vorticity is far from uniform: it is largest near the driven side and falls away with depth, because at low Reynolds number diffusion is fast enough to connect the interior to its boundaries directly and the whole region is a viscous flow. Raise the Reynolds number and diffusion becomes slower relative to circulation, the interior loses contact with its boundaries except through thin layers, and the core flattens.

The number that decides is the ratio of the circulation time to the diffusion time across the region, which is the Reynolds number itself. The core becomes uniform when a parcel goes round many times before diffusion can reach it, and that is a condition on the Reynolds number and not on the geometry — which is why the theorem, having taken $\nu \to 0$, can state a conclusion that mentions neither.

How long the limit takes to arrive #

The theorem states where a steady flow ends up and says nothing about how it gets there, and the missing quantity turns out to be a time rather than a Reynolds number — which is more useful, because it can be compared with how long an experiment or a computation was actually run.

The reasoning is the argument’s own, read forwards instead of backwards. The only thing that can change the vorticity on a closed streamline is diffusion across streamlines, since convection along one carries the vorticity round and returns it unchanged. So the core equilibrates on the diffusion time across the region, $L^2/\nu$, and the natural unit to measure that in is the turnover time $L/U$. Their ratio is

$$\frac{L^2/\nu}{L/U} = \mathrm{Re},$$

so a recirculating region reaches its uniform core after of order Re turnovers. At a Reynolds number of ten thousand that is ten thousand circuits of the eddy, while the outer flow has long since settled.

Two consequences follow, and both are practical.

A computation can look converged and not be. A steady solver’s residuals are dominated by the outer flow, which reaches its answer in a few flow-through times; the core vorticity inside a separation bubble is still drifting slowly towards its own limit long afterwards. A time-accurate calculation stopped after a few dozen turnovers has a bubble whose contents were essentially set by the initial condition — which is exactly the ambiguity this essay is about, arriving as a numerical artefact instead of a mathematical one. The tell is a core with vorticity gradients in it, and the right response is a longer run rather than a finer grid.

And an experiment has the same problem in reverse. A cavity started from rest takes an inconveniently long time to forget it, which is one reason clean measurements of the uniform core are made in steadily driven apparatus that has been running for a while rather than in a transient.

There is one arrangement that escapes the slow route, and it is worth naming because it is where the result was first worked out. A rotating container spins its contents up not by diffusion through the interior but by pumping: thin Ekman layers on the end walls suck fluid in, throw it outwards, and replace the whole interior in a time of order $\mathrm{Re}^{1/2}$ turnovers rather than $\mathrm{Re}$. Greenspan and Howard established that in 1963, and the square root is the difference between a tank that settles in a minute and one that would take an hour.

Which sharpens what the theorem is for. It is not a prediction to be checked against a snapshot; it is a statement about an end state, with a known and often prohibitive approach time. Used as a diagnostic it is excellent — a core that is not uniform has not finished — and used as a design formula it needs somebody to have waited.

Why the argument is not circular #

The suspicion to have about a one-line theorem is that it assumed its conclusion, and it is worth checking what would break it.

It needs the flow to be steady, in the sense of genuinely time-independent rather than statistically so. A recirculating bubble that is shedding, or breathing, or containing turbulence, does not satisfy the hypothesis, and most real separated regions do not.

It needs the streamlines to be closed, which means the region must be genuinely isolated from the outer flow rather than slowly exchanging fluid with it. Real bubbles entrain.

And it needs the limit to be taken at fixed geometry, which is the assumption that fails most quietly: as the Reynolds number rises the separated region usually changes shape, so the sequence of flows whose limit is being taken is not a sequence of solutions on one domain.

What the theorem does not need is any estimate of how small $\nu$ has to be, and that is its weakness as well as its elegance. It says the limit is uniform-vorticity and says nothing about the Reynolds number at which a real flow is close to it.

The axisymmetric version lands on the previous rung #

Run the same argument for an axisymmetric flow. The quantity carried along a streamline is $\omega_\varphi/\varpi$ rather than $\omega_\varphi$, so the conclusion is that $\omega_\varphi/\varpi$ is uniform inside a closed streamline.

That is exactly Hill’s spherical vortex, whose vorticity is $A\varpi$ with $A$ constant — checked here to five parts in a hundred million.

So the vortex that was chosen in 1894 because it was the one anybody could integrate turns out, sixty years later, to be the only spherical vortex a real fluid can approach. Every other member of Norbury’s family is an exact solution of the inviscid equations and none of them is a vanishing- viscosity limit at fixed geometry.

The energies are not close either #

It is worth having one more number, because “different flows” is a claim and forty-five per cent in the centre vorticity is a claim about one point.

Scale both members to the same peak streamfunction — the same maximum of the quantity that decides how much fluid is going round — and integrate the kinetic energy over the whole cell. The cellular flow comes to 2.467 and the uniform-vorticity flow to 3.238, in the same units: thirty-one per cent apart. The speed at the quarter point differs by twenty per cent.

So the two are not neighbours that a careful measurement would struggle to separate. They are different flows by every integral quantity anybody would think to compute, and the reason they are easily confused is that both are smooth, both are symmetric, and both draw as a set of nested closed curves — which is all a streamline picture of a recirculating region ever shows.

Where it shows up #

Three places, and the third is the one that decides whether a calculation of a separated flow is worth anything.

A cavity flow. Fluid driven across the mouth of a slot recirculates inside it, and the interior vorticity at high Reynolds number is measured to be roughly uniform over most of the core, with the gradients confined to thin layers against the walls. That is the theorem being visible in a measurement.

A separation bubble. The closed region behind a bluff body or under a leading-edge separation should, at high enough Reynolds number and if it were steady, have uniform vorticity in it. It is usually not steady, which is why the prediction is more useful as a diagnostic than as a design tool: a computed bubble with strong vorticity gradients in its core is either at a low Reynolds number or is not converged.

And Moffatt’s corner eddies, which are closed streamline regions at the other end of the Reynolds number entirely, where viscosity dominates and the theorem’s hypothesis of a vanishing $\nu$ is exactly wrong. The two results describe the same geometry at opposite limits and disagree completely about what is inside, which is the honest way to see how much of the answer the Reynolds number is carrying.

What the plane case would have to be, in a body’s wake #

Applying the result to a separation bubble makes a prediction with a number in it, and it is worth following through because the failure is instructive.

Take a steady separated region behind a body at high Reynolds number. Inside it, the theorem says $\omega$ is a constant $\omega_0$; and the value of that constant is not free, because the bubble’s boundary is a shear layer that arrives carrying vorticity from the body’s own boundary layer. Matching the two — the flux of vorticity into the region from the shear layer against the flux out through the rear — fixes $\omega_0$ in terms of the outer velocity and the bubble’s length.

Batchelor did that in 1956 and produced a model of a steady wake with a definite base pressure. It is one of the very few routes to a base pressure from theory rather than from a tapping, and it is the number free-streamline theory needs and cannot supply.

It does not agree with experiment, and the reason is the first hypothesis: real wakes at those Reynolds numbers are not steady. They shed. This site’s own solver cannot draw the street and neither can a theorem that begins by assuming a time derivative is zero.

What the picture cannot show #

A limit. Every figure here is a solution at one set of parameters, and the theorem is a statement about a sequence of solutions as a parameter goes to zero. Nothing in a picture of one flow can distinguish a member of the family from the member the limit selects; the distinguishing is done by an integral and reported as a number.

The thin layers. In a real high-Reynolds-number cavity the uniform core is surrounded by boundary layers on the walls and a shear layer across the mouth, and all of the vorticity gradients live in those. They are of thickness $\sqrt{\nu L/U}$, they are where the whole argument’s flux balance is struck, and at the Reynolds numbers a figure can be drawn at they are not thin.

And the difference between the two members is not visually striking. The centre vorticities differ by forty-five per cent and the streamline patterns look like near neighbours, because a streamfunction is a doubly integrated quantity and integration is a smoothing operation. That is the general reason this collection distrusts pictures of fields, and it is at its worst here.

One last remark on what kind of result this is. It is not a statement about fluids that happens to need a limit; it is a statement about limits that happens to be about fluids. A problem with a small parameter multiplying the highest derivative has more information in it than the problem with the parameter deleted, and the extra information does not disappear as the parameter shrinks — it condenses into a condition. Recognising that shape is worth more than the theorem, because the shape recurs and the theorem does not.

Who found it, and when #

Prandtl stated the result in his 1904 boundary-layer paper — the same eight pages that founded the subject — as a remark about the interiors of separated regions. Batchelor proved it properly in 1956 and generalised it to the axisymmetric case in the same year, in a pair of papers that also settled what happens in a region bounded by a free shear layer.

The surprising connection is with a completely different singular limit. The same structure appears whenever a small parameter multiplies the highest derivative: the equation with the parameter set to zero is of lower order, so it cannot satisfy all the boundary conditions, and the extra information the full problem carries survives in the limit as a solvability condition. A boundary layer is the standard example — that is what Prandtl’s eight pages were about — and this is the same phenomenon with the condition appearing not as a thin layer but as a constraint on a function in the interior. The two results are one paper apart in Prandtl’s own work and are usually taught fifty pages apart.

Where the ladder goes next #

Below this rung are the general rotational statement and the solution it selects.

Beside it are the eddies at the other end of the Reynolds number, where the same closed streamlines have entirely different contents, and where the flow lets go, which is where the closed region comes from in the first place.

And after it, the direction these rungs have been travelling: what happens when the missing information is supplied by the boundary layer itself.