A sheet that cannot stay a sheet

Every separated flow drawn on this site has a vortex sheet in it. The wake behind a bluff body, the shear layer at the edge of a jet, the sheet trailing from a wing — all of them are regions where the vorticity is large and the thickness is small, and all of them are idealised the same way: take the thickness to zero at fixed circulation per unit length, and what is left is a surface across which the tangential velocity jumps.

The idealisation is not a convenience. It is what makes the exact theory able to say anything at all about a separated flow, because a sheet is a boundary condition and a thick layer is a partial differential equation.

Every wavelength at once is the first thing that goes wrong with it, and that essay is careful about what the failure means: the growth rate is proportional to the wavenumber without bound, so the initial-value problem is ill-posed, and an ill-posed answer is a model reporting a scale it left out. This essay is about what happens after the linear stage, and the answer is that the sheet does not merely grow. It produces a singularity of its own, in finite time, from a start as smooth as anything in this collection.

The equation, in the only coordinate that works #

Parameterise the sheet by circulation rather than by arclength. That is the whole trick, and it works because circulation is conserved on each material element: the amount of vorticity between two marked points cannot change, however violently the distance between them does.

With $\Gamma$ as the parameter and $z = x + iy$ the position, the self-induced motion of a periodic sheet of period one is

$$\frac{\partial \bar z}{\partial t}(\Gamma, t) = \frac{1}{2i}\,\mathrm{PV}\!\int \cot\big(\pi(z(\Gamma) - z(\Gamma'))\big)\,d\Gamma'.$$

That is the Birkhoff–Rott equation. Discretised at $N$ points it is a system of point vortices with the exact periodic kernel, and the principal value is taken automatically by leaving out the singular self-term.

Krasny’s regularisation replaces the kernel by a smoothed one with a length $\delta$ in it, and at $\delta = 0$ the smoothed kernel is the exact one. So the same code computes the idealisation and its repair, and the difference between them is a single number.

A flat sheet is an equilibrium, so everything below is the perturbation’s doing #

The check that has to come first is that the flat sheet does nothing. A uniform sheet on $y = 0$ induces no velocity on itself: every element’s contribution is cancelled by its mirror across the evaluation point, and the sum is zero. Computed over 256 points it is $6.4\times10^{-16}$, which is round-off and not a number.

That matters because it means the whole of what follows is produced by the perturbation, and nothing by the representation.

The growth rate is not the one in the textbook, and the difference is exact #

The linear stage cannot be measured off a run, and the reason is the subject of this essay. Instead the operator is linearised directly: perturb the flat sheet by one Fourier mode, project the induced velocity back onto the same mode, and take the eigenvalues of the resulting four-by-four.

What comes out is not $\pi m$. It is

$$\sigma_m = \pi m\left(1 - \frac{m}{N}\right),$$

to 1.8 parts in $10^{10}$, over every $N$ and $m$ tested. The form was not looked up; it came out of the operator, and it is the sharpest available statement of what a discretisation does to this problem. At long wavelengths it is Kelvin–Helmholtz’s rate. At the shortest wave a grid of $N$ points can carry it is half of it.

And the continuum limit is recovered: at fixed $m$, refining from 64 points to 1024 brings the rate monotonically up to $\pi m$, reaching it to a part in a thousand at the last.

The fastest-growing mode is the grid #

Read the parabola $\pi m(1 - m/N)$ for what it says. It is maximised at $m = N/2$, and its value there is $\pi N/4$. The fastest-growing disturbance a discretised vortex sheet has is at the grid scale, and its growth rate rises without bound as the computation is refined.

At $N = 128$ the mode is $m = 63$ of a possible 64 and it grows at 100.5, which is $\pi N / 4$ exactly. Nothing in the physics is at that wavelength. It is the representation’s own instability, and it is faster than everything the calculation is for.

Smoothing the kernel over a length $\delta$ cuts it off. At $\delta = 0.2$ the fastest mode is $m = 5$, growing at 5.6 — a wavelength of a fifth of the period, which is a length rather than a grid spacing.

What that does to a computation, measured #

Round-off is $10^{-16}$. A perturbation of amplitude $10^{-2}$ grows at $\pi$. The grid mode grows at $\pi N/4$, so it reaches the size of the perturbation after a time

$$t \approx \frac{\ln(10^{14})}{\pi N/4},$$

which at $N = 128$ is 0.32 — before the singularity the computation exists to find.

The consequence is the refutation this essay carries. Refining the unregularised computation from 64 points to 256 multiplies the reported peak curvature by four hundred: 1.3, then 218, then 533. A refinement study whose answer runs away from its neighbours as the grid is halved is not converging on anything, and it is not obvious from the pictures, which are smooth at every resolution.

Krasny’s filter removes it. Set every Fourier coefficient below $10^{-13}$ to zero at each step and the seed of the grid mode is gone; the physics, which is far above that threshold, is untouched. The same refinement then gives 1.333, 1.367, 1.384 — four per cent across a factor of four in the grid.

This is a filter on the representation and not a change to the equation, and it stops helping exactly when the physics reaches the threshold — which is the run whose spectrum has stopped decaying, and which is the next section.

The measurement that says a singularity is forming #

A finite computation cannot show a curvature reaching infinity. What it can show is the width of the strip of analyticity closing.

A curve that is analytic in a strip of half-width $\rho$ about the real axis has Fourier coefficients decaying as $e^{-k\rho}$; that is a theorem about analytic functions rather than a fitted form. A singularity reaching the real axis is $\rho = 0$. So fit $\ln|a_k|$ against $k$ over an intermediate band at each time — low $k$ carries the initial condition and high $k$ carries round-off — and watch the exponent.

It falls, smoothly, and the linear extrapolation of its last dozen samples reaches zero at

$$t_c = 0.3829,$$

against Moore’s asymptotic value of 0.375 for this initial condition — 2.1 per cent. The curvature has reached 6.1 by the last station computed and is rising steeply; the extrapolation is what says where it is going.

What the limit threw away, and what it produced instead #

The idealisation removed the smallest scale in the problem. The sheet then produced one of its own, of size zero, at a time that is finite and computable.

That is the shape of every essay in this collection that asks what a limit leaves behind, and it is worth stating plainly because it is not the usual moral. The usual moral about vortex sheets is that they are unstable and that viscosity would smooth them. Both are true and neither is this. The nonlinear evolution of the idealised sheet has a singularity, and it arrives whether or not anything is smoothing it, because the equation being solved has no length in it at all.

Why the linear stage is not the story #

It is worth saying why an essay is needed at all, given that the linear instability is already established.

A growth rate proportional to the wavenumber without bound is an alarming result and it is often taken to be the whole difficulty: the model is ill-posed, so it is unusable, and the discussion ends. That reading is too quick in both directions.

It is too pessimistic because ill-posedness of a linearised problem does not by itself prevent the nonlinear one from having a smooth solution for a while, and this one does — the sheet is analytic up to a finite time, and the analyticity strip’s width is what measures it.

And it is too optimistic because it suggests the trouble is confined to the shortest wavelengths, which a filter or a smoothing would remove. The singularity at $t_c$ is not at a short wavelength. It forms at the scale of the initial perturbation, out of an initial condition containing one Fourier mode, and no amount of high-wavenumber filtering prevents it.

So the linear result says the model has no smallest scale, and the nonlinear result says it makes one.

What is conserved while all this is happening #

A computation that is producing a singularity is a computation whose numbers should be doubted, so it is worth saying what it is holding on to.

The sheet’s total circulation is conserved by construction: each point carries a fixed $\Gamma_j$ and the sum never changes. That is not evidence of anything, because it cannot fail.

The linear impulse $\int y\,d\Gamma$ can fail, and it does not. Over a full run it moves by $1.1\times10^{-17}$ against a starting value of $8\times10^{-19}$ — which is to say the two are the same zero. The impulse is the sheet’s version of the quantity the momentum with no value is about: the momentum of an unbounded two-dimensional flow is conditionally convergent and depends on the shape of the region it is summed over, and the impulse is the well-defined thing that replaces it. It is conserved here for the same reason Kelvin’s theorem holds, and what survives being wound up is the essay that tests that directly by advecting a material loop until nothing about its shape is recognisable.

So the singularity is not the integration losing something. Every invariant the equation has is still there when the curvature is six and rising.

And then the answer depends on how it is put back #

Set $\delta > 0$ and the sheet does not develop a singularity at all. It rolls up into a spiral with a finite core, and it keeps going for as long as the integration runs.

The core’s size is $\delta$, and so is everything about the answer. Peak curvature at $t = 1$ falls from 1093 at $\delta = 0.05$ to 6.85 at $\delta = 0.4$, a clean power law over a factor of eight. There is no physical principle in the computation that fixes $\delta$: it is the thickness that was removed in the first place, being put back by hand.

That is the honest position and it is not a scandal. A shear layer has a thickness, the sheet model threw it away, and the model cannot be integrated past a finite time without it. What is not honest is a computation that takes $\delta = 0$, refines until the pictures look sharp, and reports the result: that computation is measuring $\pi N/4$.

The two pictures together are the choice the subject offers. Take $\delta$ small and the computation resolves more of the physics and less of it reliably; take $\delta$ large and it resolves a spiral whose core is a number somebody chose. There is no setting at which both are true, and the parameter sweep is the honest form of the result.

Where the limit is legitimate, which is most of this collection #

None of this makes the sheet model wrong where the collection uses it.

The street this site cannot draw builds the Kármán vortex street from point vortices and says so; what it computes is a spacing ratio, which is a property of an equilibrium and not of an evolution. Vortices move each other and three is the most that can be predicted are about the dynamics of a finite number of vortices, where there is no grid scale to run away with. And a trailing sheet’s roll-up over a few chords, with $\delta$ chosen as a real core radius, is a model with a stated length in it.

The failure is specific: it is the limit $\delta \to 0$ at fixed time, or equivalently the limit $t \to t_c$ at fixed $\delta = 0$. Those are the two ways of asking the sheet what it does at scales it does not have.

The number to carry #

$t_c \approx 0.38$ in the units used here is not a portable number, but the statement behind it is.

The critical time scales with the initial amplitude: a smaller perturbation takes longer to reach the singularity, and the scaling is logarithmic because the growth is exponential. So a sheet perturbed at $10^{-4}$ rather than $10^{-2}$ takes about $\ln(100)/\pi = 1.5$ time units longer, which is four times the whole of the run above.

In practical terms, a vortex sheet reaches its singularity a few e-folding times of the Kelvin–Helmholtz instability after the perturbation becomes visible. Not many; not few; and always finite.

That is the useful form of the result. A computation using a sheet model has a budget measured in e-folding times, and it can be spent on the linear stage or on the roll-up but not on both.

What is not computed here #

The singularity’s structure is not resolved. Moore’s analysis says the curvature diverges and gives the exponent; this computation measures the analyticity strip closing and extrapolates. Those are different statements, and the second is the weaker one.

The critical time is sensitive to the fitting band and to the resolution. At 256 points with the band from $k = 5$ to $k = 25$ it is 0.383; at 384 points with a wider band it is 0.42. The scatter is larger than the 2 per cent quoted against Moore’s value, and the number to trust is “a little under four tenths” rather than the four figures the fit prints.

The filter is not free. It works because the physical spectrum at the fitting band is far above $10^{-13}$ for the whole of the run used. Near $t_c$ the spectrum flattens and the two meet, and beyond that point the filtered computation is as untrustworthy as the unfiltered one — which is the reason the extrapolation stops where it does.

Why it matters two essays away #

The sheet is the cleanest case of a pattern this collection meets repeatedly: an idealisation that removes a length, an answer that then depends on the length being put back, and a computation that will happily produce a number without it.

What a point vortex is not is the same story for a single vortex rather than a sheet — a patch shrunk to a point loses a shape, and the shape’s own rotation rate diverges. The part of the flow inside the body is the same story for a body: thin it, and a singularity that was safely inside arrives on the surface. And the limit that is not the value is the same story for viscosity itself, where the dissipation does not vanish with the thing that causes it.

In each case the limit is easy to write down and the residue is the subject.

There is one more thing the sheet is good for, and it is a warning about pictures rather than about models. Every frame of the roll-up above is smooth, closed and plausible, at every resolution and at every smoothing length, including the ones that are computing round-off. The site’s own rule is that a smooth picture proves nothing — a wrong flow field is beautiful — and this is the sharpest case of it in the collection: the difference between a converged answer and a four-hundredfold artefact is not visible in the drawing, and it took a refinement study and a spectrum to find. Neither of those is a figure a reader would have asked for.