Every wavelength at once

A vortex sheet is the idealisation of a shear layer taken to its limit: two streams of the same fluid at different speeds, meeting across a surface of zero thickness, with all of the vorticity concentrated on that surface as a delta function.

It is a useful object. Thin-aerofoil theory is built out of one, the trailing wake of a wing is modelled as one, and the lifting line is a statement about a sheet’s roll-up. The model earns its place many times over on this site.

Asked what happens when it is disturbed, it gives an answer that cannot be right, and the way in which it cannot be right is the most instructive thing about it.

The answer, in one line #

For two streams of equal density separated by a sheet, with a velocity difference ΔU across it, the linear analysis gives a disturbance of wavenumber k a growth rate

$$\sigma = \frac{k\,\Delta U}{2}$$

Every wavenumber grows. There is no neutral wavenumber, no band, and no maximum. Halve the wavelength and the growth rate doubles, without limit.

The derivation is short and worth having in outline, because the structure of the answer is where the trouble is. Potential flow on each side, a disturbance of the interface proportional to e^(ikx − iωt), and two matching conditions at the interface: the interface moves with the fluid on both sides, and the pressure is continuous across it. Two conditions, two unknown amplitudes, and a dispersion relation that is a quadratic in ω. Its roots are complex conjugates for every real k, and the imaginary part is kΔU/2.

Complex conjugates means one root grows and the other decays at the same rate. There is no stable band anywhere.

Why unbounded growth is a statement about the model #

An initial-value problem whose growth rate rises without bound with wavenumber is ill-posed in the sense of Hadamard: the solution does not depend continuously on the initial data. Perturb the initial condition by an arbitrarily small amount at a sufficiently high wavenumber, and after any fixed time the solution has changed by an arbitrarily large amount.

That is not a defect that careful numerics can work around. It is a statement that the mathematical problem as posed does not have the property a physical problem must have.

The useful reading is the one this site takes throughout: a model that produces an unbounded answer is naming the quantity it does not contain. The vortex sheet has a velocity difference and a density, and from those two no length can be built. Growth rate has dimensions of inverse time, ΔU of length over time, and the only way to make one from the other is to multiply by a wavenumber — so σ ∝ kΔU was forced before any analysis began, and the constant ½ is all the analysis supplied.

The missing quantity is a length. Supply one and the problem is well posed.

Supplying the length #

Replace the discontinuity by U(y) = ΔU tanh(y/δ)/2 and everything changes character.

The profile still has an inflection point, at the centre, so Rayleigh’s criterion still permits instability. But now the disturbance has something to compare its wavelength with. A wave much longer than δ cannot tell the layer from a sheet and grows at nearly the sheet’s rate. A wave much shorter than δ sees a locally uniform shear, which has no inflection point in the region it occupies, and does not grow at all.

The crossover is exact: kδ = 1, with the neutral eigenfunction sech(y/δ) and phase speed equal to the mean of the two streams. The figure above finds it by residual rather than by recollection, and the residual at the neutral mode comes out below 10⁻⁸.

The fastest-growing wave sits at about kδ = 0.44 — a wavelength of roughly fourteen layer thicknesses, or about seven times the thickness by the more common definition of δ. That is the wavelength a photograph of a Kelvin–Helmholtz billow shows, and it is why those photographs show a regular spacing rather than the finest structure the resolution can capture.

Why the growth rate is what it is #

The dispersion relation can be got at without the algebra, and the argument is worth having because it makes the k in σ = kΔU/2 inevitable rather than surprising.

Think of the sheet as a row of vortices rather than as a surface. A sinusoidal displacement of the sheet does two things at once. It concentrates vorticity at the points where the sheet has been tilted most steeply against the shear, because the sheet is being locally stretched or compressed along its own length; and the concentrations then induce velocities at each other’s positions.

The induced velocity at a distance r from a line vortex of strength Γ goes as Γ/2πr. Concentrations a wavelength apart therefore induce velocities proportional to 1/λ, which is to say proportional to k. The amount of vorticity concentrated per unit length is proportional to ΔU. Multiply and the growth rate is proportional to kΔU.

Nothing in that argument required an analysis, and nothing in it can be avoided by doing the analysis more carefully. The unbounded growth is in the geometry of the model, and specifically in the fact that a vortex’s induced velocity has no cutoff. A real layer’s vorticity is spread over a thickness, and spreading it is exactly what supplies the cutoff.

This is also the reason the numerical treatment of vortex sheets is difficult in a way that looks like bad luck and is not. A point-vortex discretisation of a sheet inherits the ill-posedness of the sheet and produces chaotic point motion at the grid scale, which refining the grid makes worse rather than better. Every practical method — vortex blobs, regularised kernels, a filtered sheet — works by putting a length back in, and the length is the method’s, not the physics’.

Where it is seen #

The instability is one of the most visible in the subject, and the visibility is a consequence of the regularity the finite thickness supplies.

Billow clouds. A layer of air sliding over another with a temperature difference between them, and enough moisture in the upper one to condense: the roll-up is drawn in cloud, at the wavelength the layer thickness selects, and the row of billows is nearly evenly spaced.

Wind over water. The initial generation of capillary waves is the same instability with surface tension supplying an additional length, which cuts off the short end more sharply and sets the minimum phase speed of water waves at 0.23 m/s.

The shear layer behind any separation. When a boundary layer separates, the layer of vorticity it leaves behind is a free shear layer, and it is unstable by this mechanism from the moment it detaches. This is the ordinary route by which a separated flow becomes unsteady, and it is much faster than anything happening inside the attached layer upstream.

A jet’s edge. The annular shear layer at the lip of a jet rolls up into vortex rings at the frequency this analysis picks, which is why a jet’s noise spectrum has a peak in it and why acoustic forcing at that frequency changes a jet’s spreading rate so dramatically.

Growing, and being carried away while it grows #

There is a question the analysis above cannot ask, because it is posed in a frame moving with the layer: where does the growing disturbance end up? A wave that doubles in amplitude while being swept three metres downstream is a different object from one that doubles while staying put, and the difference decides what kind of thing the flow is.

Watch a disturbance released at one point and follow the whole wave packet rather than a single wavenumber. If the packet grows but is convected away, so that the fluid at the release point returns to quiet, the flow is convectively unstable. If the packet spreads upstream as well as downstream, so that the release point itself never recovers, it is absolutely unstable.

The distinction is not academic; it is the difference between two kinds of machine.

A convectively unstable flow is an amplifier. It has no frequency of its own — it takes whatever disturbances arrive at its origin and magnifies the ones inside its unstable band. So its behaviour depends on the noise environment, which is why two nominally identical mixing layers in two laboratories do not behave alike, and why a jet’s shear layer can be locked to an imposed frequency: force it acoustically anywhere in the band and it will roll up on cue. That is exactly the sensitivity the jet paragraph above describes, and it is the signature of an amplifier rather than a curiosity about jets.

An absolutely unstable flow is an oscillator. It selects its own frequency, sustains it without any input, and cannot easily be persuaded to do anything else. That is why the wake in the figure below has a Strouhal number — a definite dimensionless frequency belonging to the body and the Reynolds number, not to whatever was happening upstream.

And the criterion separates the essay’s own two examples. A mixing layer between two streams moving the same way is convectively unstable, always: the mean advection outruns the packet’s upstream edge. It becomes absolutely unstable only when the velocity ratio $\Delta U/(U_1+U_2)$ exceeds about 1.315 — which requires the slow stream to be moving backwards. A free shear layer between co-flowing streams never qualifies; a wake with a recirculating region behind the body does, because reversed flow is exactly what a recirculation is.

So the transition from a shear layer that merely amplifies to a wake that sings is a statement about whether there is return flow, and the essay’s separated-layer example sits on one side of it while its vortex street sits on the other.

What the roll-up conserves, which is all that survives the linearisation #

The linear analysis stops being valid at an amplitude comparable with the layer thickness, which on a growing disturbance is a matter of a few e-foldings. Almost nothing about the subsequent motion is computable. One thing is, and it is a conservation statement rather than a solution.

Circulation round a material circuit is conserved in an inviscid flow — Kelvin’s theorem, which this site derives and checks elsewhere. A billow that has rolled up out of one wavelength of sheet therefore carries exactly the circulation that wavelength of sheet contained, which is ΔU·λ.

That is enough to pin the strength of the resulting vortices without any knowledge of how the roll-up proceeded, and it is why a model made of point vortices can be a useful description of a rolled-up layer even though the roll-up itself was never computed. The circulation is a bookkeeping quantity, the bookkeeping is exact, and the bookkeeping survives every nonlinear thing that happens in between.

The same argument sets the spacing. If a layer rolls up at the wavelength linear theory selects, the vortices are that wavelength apart, and their strength follows from the circulation in one wavelength. Two of the three quantities describing a row of vortices are therefore fixed by the linear problem plus a conservation law, and only the third — what the row does subsequently — is out of reach.

The density case, and one number worth carrying #

If the two streams have different densities, gravity enters and the dispersion relation acquires two more terms — a stabilising one from the density difference in a gravitational field, and a stabilising one from surface tension if there is an interface.

The result for air over water is worth stating because it is quantitative and because it disagrees with the observed answer in an interesting direction. The analysis says that wind over water is unstable above a wind speed of about 6.6 m/s, and that the wave first generated has a wavelength of 1.7 cm.

Water is observably disturbed by winds well below 6.6 m/s. The discrepancy is not in the algebra; it is that the model has a discontinuous wind profile and the real one is a turbulent boundary layer with its own structure, and the mechanism that generates the first ripples — Miles’ critical-layer instability, from 1957 — is a different mechanism that this model does not contain.

That is the same lesson twice: the sheet named the length it lacked, and the constant-profile analysis names the profile structure it lacks.

The same shape, in three other places #

The vortex sheet is not the only model on this site that answers a question with an infinity, and collecting the cases makes the diagnosis look less like a special pleading for this one.

A point vortex has infinite velocity at its centre, and the missing quantity is a core radius. Every practical use of one — the lifting line, the image system in ground effect, the street model above — either keeps its evaluation points away from the centre or puts a core in by hand.

The leading edge of a thin aerofoil has infinite velocity in thin-aerofoil theory, and the missing quantity is the nose radius. The theory is excellent about lift, which is an integral, and useless about the suction peak, which is a local value at exactly the place the model has none.

A sharp trailing edge in inviscid flow permits infinite velocity round it, and what supplies the missing constraint is not a length but a physical requirement — the Kutta condition — which is the same repair by a different route: an assumption is added because the model as posed does not determine the answer.

In every case the infinity is diagnostic. It appears exactly where a real fluid has a small scale that the idealisation removed, and it points at the scale rather than at a mistake. A model that answered these questions with a large finite number instead would be harder to correct, because nothing about a large number says which quantity is missing.

Where the model stops #

Linear analysis predicts the wavelength that grows fastest and the rate at which it grows. Both are statements about a disturbance small enough that its square may be neglected.

By the time a billow is visible in a cloud, its amplitude is comparable with the layer thickness and the linearisation has been invalid for some time. The roll-up, the pairing of adjacent billows, the secondary instabilities on the braids between them and the eventual breakdown to three-dimensional turbulence are all nonlinear and none of them is computed anywhere on this site.

The one further thing that can be said cheaply is a conservation statement rather than a solution: whatever the layer does, the circulation in it is conserved, so the total vorticity in a rolled-up billow equals the vorticity that was in the length of sheet it came from. That is the accounting the vortex-street model in the figure above obeys, and it is why a model with no body in it can still get a spacing ratio right.

Who found it, and when #

Helmholtz posed the problem in 1868 in a paper on discontinuous fluid motion, and observed that the surface of separation was unstable.

Kelvin analysed it in 1871, including gravity and surface tension, and obtained the wind-over-water threshold of about 6.6 m/s — which he knew disagreed with observation and said so.

Rayleigh’s finite-thickness analysis followed in 1880 and 1887 as part of the work that produced the inflection criterion, and the tanh layer’s exact neutral solution has been the standard example ever since.

Taylor in 1931 and Goldstein in 1931 independently established the stratified generalisation, which introduces the Richardson number and the quarter that has been the criterion for stratified shear instability ever since.

Michalke computed the growth-rate curve for the tanh profile properly in 1964, which is when the fastest-growing wavelength became a number rather than an estimate.

Where the ladder goes next #

This anchor’s two rungs are the two ends of one problem — a profile with structure, and a profile with none — and the next step is to stop asking which flows go unstable and start asking what the unstable state is like.

That begins with an accounting exercise rather than a solution: what averaging costs, which is the discovery that writing down the equations for the mean flow produces more unknowns than equations, exactly and for ever.