A spiral is a legible record

A sheet that cannot stay a sheet is this collection’s account of the vortex sheet as a model: what the zero-thickness limit throws away, why the resulting object is unstable at every wavelength, and how the answer after the singularity depends on how the sheet is put back together.

This essay is about the thing the sheet becomes, and about a property of it that is easy to miss because it is so simple. A sheet is a one-dimensional object carried by the flow. Its points are fluid particles strung along a line, and a line’s points cannot change their order without passing through one another.

That single fact turns a rolled-up sheet from a picture into a record.

The order is never lost #

Two hundred material points are placed along one period of an almost-flat sheet, perturbed by one per cent of the period, and carried by the velocity the sheet itself induces over 1.2 time units in 240 steps, with the kernel smoothed over 0.15 of a period. The perturbation grows, the sheet steepens, and the ends wind into a spiral.

Throughout, the computation counts how many adjacent pairs ever swap places. Over 240 steps and 199 adjacent pairs that is 47,760 order checks, and the answer is zero. It is not a small number and it is not a tolerance: two points on a material line cannot cross, because crossing means occupying the same place at the same instant, and the velocity field is single-valued.

So the parameter along the sheet — where each point started — is a label the fluid keeps for ever, and the spiral inherits it. Colouring the sheet by initial position shows the arms of the spiral in order, each arm a known interval of the original line.

What that makes the picture #

A photograph of a rolled-up shear layer — behind a splitter plate, at the edge of a jet, along the top of a breaking wave — is usually read as a state: this is what the flow looks like now. It is that, and it is also a time-ordered record, and the second reading carries information the first does not.

The outermost arm is the part of the sheet that has been winding longest. The innermost is the newest, and the label they carry runs from 0.000 to 0.995 along the sheet in 200 equal steps. The number of turns is a clock: a spiral with three turns has been rolling for longer than one with two, in a way that is measurable off the picture without any knowledge of the velocity.

That is why smoke and dye pictures of shear layers are as useful as they are, and why they mislead when they are read as instantaneous. It is the same distinction streamlines are not the paths particles take makes for curves: what a camera records in an unsteady flow is an accumulation, and the accumulation is the interesting part.

Why the order survives when almost nothing else does #

The protection is worth spelling out, because it is stronger than a numerical statement and it is the reason the record is trustworthy at all.

The velocity field is single-valued: at any instant, each point of space has one velocity. Two fluid particles at different places therefore move differently, and two at the same place move identically — so particles can approach one another arbitrarily closely and can never swap. That is a consequence of the flow being a flow rather than a collection of independent motions, and it does not depend on the equations being Euler’s, or on the fluid being inviscid, or on anything else.

What it does depend on is the sheet being material. A vortex sheet in an ideal fluid is: it is made of fluid particles and it moves with the average of the velocities on its two sides. In a viscous fluid the layer diffuses, so the object being tracked is a vorticity distribution rather than a set of particles, and vorticity is not conserved along particle paths in the presence of diffusion. The ordering argument then applies to the fluid and not to the vorticity, which are different things at the scale of the layer.

So the reading below is a reading of an ideal sheet, and its transfer to a real shear layer is by the usual argument — that at high Reynolds number the layer is thin and diffusion has not had time to matter — which is exactly the argument that fails once the layer becomes turbulent.

The strength per unit length, which halves #

There is a second quantity the roll-up changes, and it explains why a rolled-up sheet is a weaker object than the sheet it came from.

The circulation distributed along the sheet is conserved: it is Kelvin’s theorem applied to the sheet itself, and it is the reason the drift was the instrument can treat a circulation as an exactly fixed number. The sheet’s length, though, is not conserved at all. Over the run its arclength grows by a factor of 2.13.

Circulation fixed and length multiplied by 2.13 means the strength per unit length has fallen to 47.0 per cent of what the flat sheet carried. The sheet is being drawn out and diluted, in the same way and for the same reason a material line is stretched in two strainings, and the order they came in.

The consequence for the flow is that the sheet becomes progressively less able to do the thing that made it unstable. Its growth rate at a given wavelength is proportional to its local strength, which has fallen to 47.0 per cent by the end of the run and to 37.3 in the sharpest of the three smoothings, so a rolled and stretched sheet is more stable than a flat one — which is part of why a roll-up settles into a recognisable core rather than continuing to fragment without limit.

How much of the record belongs to the model #

The uncomfortable part, and the part it would be dishonest to leave to a footnote.

The Birkhoff-Rott integral that governs a sheet has a kernel that is singular where two parts of the sheet touch, and an unsmoothed sheet develops a curvature singularity in finite time — the result a sheet that cannot stay a sheet computes. Past that moment the equation has no solution at all, and every computation of a roll-up is a computation of a regularised sheet: the kernel is smoothed over a length, and the length is a statement about how thin the shear layer is really being taken to be.

Three smoothing lengths spanning a factor of 2.5 give final arclengths of 2.68, 2.13 and 1.52 — a spread of 76 per cent. The sharpest winds further and packs more sheet into its core; the bluntest has barely begun. The shapes are qualitatively the same and no number taken off the core is.

So the record is legible and part of what it records is the model. The order of the arms is a fact about the fluid; the number of turns is a fact about the fluid and the smoothing together; and anything read off the core is mostly the smoothing.

Which parts of the record are safe to read #

That split is worth making explicit, because it is what the essay is for.

Safe: the order. Which arm came from which part of the sheet is a topological statement and it does not depend on the smoothing at all. Every one of the three computations has the same arms in the same order.

Fairly safe: the number of turns, as a comparison. Two pictures of the same apparatus at different times can be compared by counting turns, because whatever the effective smoothing is, it is the same in both.

Not safe: anything about the core. The spacing of the inner arms, the peak vorticity, the core radius and the rate at which the innermost turn winds are all set by the regularisation. A measurement of a core radius from a computation is a measurement of the computation.

And not safe: the time at which anything happened. The roll-up rate depends on the smoothing, so converting turns into seconds needs a calibration that the model does not supply.

What the solver computed, and how it was checked #

One period of sheet, two hundred material points, the periodic Birkhoff-Rott kernel smoothed over a stated length, stepped with the same fourth-order rule the rest of this collection uses.

Two things are asserted and one is unusual. That no pair of points ever changes order, which is the essay’s whole claim and is checked at every step rather than at the end — a crossing that happened and was undone would be invisible in a final count. That the arclength really grew, by at least a factor of two, so the dilution being described is happening. And that the smoothing changes the answer, by at least five per cent, because a claim that a parameter matters should be refused if it does not.

That last check is the one worth pointing at. It is written so that it fails if the smoothing turns out not to matter — which would be the good news, and would mean the roll-up was converged. It reads 76 per cent, so the news is the other one, and the essay says so rather than quoting one computation.

Why a sheet keeps its order and a fluid does not #

There is an apparent tension with the rest of this collection and resolving it is instructive.

A boundary that only exists over a window describes fluid being pulled apart and interleaved until neighbouring parcels have unrelated histories. Here a set of material points keeps a strict order for ever. Both are true, and the difference is dimensional.

A line in a plane has an inside and an outside and its points have a cyclic or linear order that no continuous motion can change. That is a topological constraint and it is absolute. A region has no such structure: two parcels in a two-dimensional patch can be exchanged by a continuous volume-preserving motion, and a chaotic flow does exactly that, repeatedly.

So the memory a sheet keeps is a one-dimensional memory — an ordering — and it survives everything. The memory a patch of fluid keeps is a map, and it is destroyed by the same motion that preserves the ordering. The dimension of the object decides how much of its past is protected, which is a statement worth carrying out of this collection.

Counting turns, and what a count is worth #

Since the number of turns is the clock, it is worth asking how good a clock it is.

The winding is not uniform in time, and the arclength says by how much. At a fifth of the run the sheet has stretched by 0.2 per cent; at two fifths, 1.2 per cent; at three fifths, 7.8; at four fifths, 50.5; and it ends at 112.7. Ninety-three per cent of the stretching happens in the last two fifths of the run. Early on the sheet steepens without winding at all — the perturbation is growing, the ends are being drawn towards one another, and no arm has yet formed. Then the first turn happens quickly, and each subsequent turn takes longer than the one before, because the inner arms are wound by a smaller effective circulation and are closer together.

So turns against time is a decelerating curve rather than a straight line, and reading a time off a turn count requires the curve, which requires the smoothing, which is exactly the parameter the model does not fix. A turn count compares two pictures well and dates one picture badly.

That is a common shape for a measurement in this subject: a quantity that is reliable as a difference and unreliable as an absolute. The wave-drag numbers in the drag that is made of waves have the same character, and so does anything read off a visualisation whose tracer was introduced at an unknown time.

What legibility costs is the essay after this one, computed by the same machinery.

Where the same reading applies #

Smoke visualisation in a shear layer. The classic photographs of a mixing layer show a row of rolled-up cores, and the smoke in each one is arranged in arms in the order it entered. Counting arms gives the number of pairings the layer has been through.

A starting vortex. The vortex a wing leaves behind is a rolled sheet of exactly this kind, and the same reading applies: its arms record the order in which the sheet left the trailing edge, so the outside of the core is the earliest part of the start-up.

And a tip vortex, further downstream. Where the wake ends up follows the trailing sheet as it rolls into two cores. The order is preserved there too, so the material at the outside of a tip vortex came from the outboard end of the wing’s trailing edge.

What a record like this is good for #

The practical value of reading a spiral as a record is not aesthetic, and three uses are worth naming.

Identifying the pairing history of a mixing layer. A layer’s cores merge as it grows downstream, and each merger doubles the amount of sheet in a core. Counting arms in a core says how many mergers it has been through, which is a quantity that no measurement of the mean profile contains — for the reason what a mean profile cannot tell anybody gives in general.

Deciding whether a computation has lost material. A code that inserts points into a sheet to keep it resolved has stopped tracking the fluid, and the tell is that the arms lose their labelling. Carrying the initial parameter as a passive quantity and checking that it remains monotone along the sheet is a one-line diagnostic that catches it.

And separating an initial condition from a mechanism. The arms record where the sheet came from, so an experiment whose spiral shows arms from an unexpected part of the sheet is an experiment whose inlet condition is not what it was thought to be. That is the sort of question a mean measurement cannot raise, because it has already averaged over the arms.

What the picture cannot show #

Three things, and all three are the same limit seen from different sides.

The drawn sheet is a polygon through two hundred points, and since the arclength has more than doubled while the point count has not, the spacing along the sheet has grown by the same 2.13 — by the end of the run adjacent arms are closer together than the spacing along the sheet. The picture therefore draws arms that are, in the model, closer than the resolution can support — which is the same failure the drift was the instrument describes for a material loop.

The smoothing length is not drawn anywhere. It is a parameter of the kernel rather than a thickness on the page, and a figure that showed it as a band would be the more honest picture and would obscure the spiral entirely.

And nothing here shows the vorticity of the sheet, only its position. A sheet’s strength varies along it, and the diluted parts contribute less to the flow than the picture’s uniform line suggests.

Who found it, and when #

The Birkhoff-Rott equation is from the 1930s and 1960s in its two forms, and the desingularised computation that made roll-up calculations reliable is Krasny’s, from 1986 — which is the method used here. The result that a sheet forms a singularity in finite time is Moore’s, from 1979.

The reading of a rolled sheet as a record is older than any of them and belongs to experiment: the smoke pictures of the 1970s mixing-layer work were read that way immediately, because the alternative — that the arms were somehow independent structures — is not available once anybody has watched the sequence rather than a single frame.

Limits recorded rather than smoothed over #

One period, two dimensions, no viscosity. A real shear layer is three-dimensional, and its secondary instabilities break the arms up in ways nothing here contains.

The order argument needs the sheet to stay a sheet. Once the layer becomes turbulent it is no longer a one-dimensional object carried by the flow, and the protection the ordering enjoyed is gone with it. The reading applies to the laminar roll-up and to the early cores, not to a developed mixing layer.

The smoothing is not a thickness. It regularises the kernel; it does not represent a viscous core with a profile. A comparison between the smoothing length and a measured layer thickness is a calibration rather than a correspondence.

And the run stops before the arms touch. Where two arms of the spiral come within a smoothing length of each other the computation is representing them as one object, and everything after that is the regularisation talking.