A streakline rolls up where nothing is rolling

The line the dye actually draws made the point that a streakline is the curve most photographs really show, and that it depends on the entire history of the flow and not on the velocity at the moment of the picture. It showed that a streakline can cross itself, which a streamline never can, and it ended with the warning that this makes it a treacherous witness. The shutter is part of the answer cited, in passing, Hama’s note of 1962 which said a streakline picture of a wave can look like a row of vortices.

That sentence should be followed up, because of how much of the history of fluid mechanics has been read from pictures of rolled-up smoke. The canonical illustrations of transition, of a mixing layer, of a wake, are nearly all streaklines, and every one of them shows spirals. This essay computes the smallest flow in which the spirals appear, so that what they are, and what they are not, can be stated in numbers.

The flow is as plain as a flow can be. Uniform shear, plus a sine wave of normal velocity travelling along it:

$$u = 1 + y, \qquad v = a\sin(x - t).$$

Its divergence is zero exactly, since $u$ has no $x$ in it and $v$ no $y$. It satisfies no equation of motion in particular, and it does not need to, because everything here is kinematic: a stated velocity field, and particles following it. The wave’s phase speed is one, so the level $y=0$ moves with the wave. Release dye at the origin, and follow it.

Anyone shown that curve without its caption would say that the flow contains a row of vortices, and very probably count them. It does not contain vortices in any sense that the word could support.

What the vorticity is doing #

The vorticity is

$$\omega = a\cos(x-t) - 1,$$

so at $a=0.05$ it runs between $-1.05$ and $-0.95$. It is a uniform field with a five-per-cent ripple. Compare a row that really is a row of vortices.

The exact solution of Stuart’s at $A=0.5$ has its vorticity vary by a factor of nine between its core and braid, and a real roll-up would do the same. Yet a streakline in a five-per-cent ripple shows spirals at least as tight as the ones a Stuart’s row would make. There is no vorticity peak for the eye to be seeing.

The clue to what is being seen comes from viewing the flow in the wave’s own frame, where the wave stands still. There the streamfunction is

$$\Psi = \tfrac12 y^2 + a\cos X, \qquad X = x-t,$$

and that does have closed curves. It has the separatrix of a pendulum, with eyes of half-height $2\sqrt{a}$, which is 0.447. So the flow has a kinematic eye: a region in which fluid, seen from a frame moving at the wave’s speed, travels round closed loops. Particles at the wave’s level oscillate about the wave’s phase like a pendulum with a small amplitude, with angular frequency $\sqrt{ak}$ times the square root of the shear. They are “trapped” in the sense of a surfer. The dye at the level is held in the eye, so the streakline stays within $\pm0.447$, and within the eye the differential rotation of a pendulum whose period depends on amplitude winds any line into a spiral.

So the spirals are real, and they do mark something real. They mark the place where a fluid particle moves with a wave. This is not a vortex. It is a kinematic trapping, and it has no connection to whether the vorticity is concentrated. The same picture appears for plasma electrons trapped by a wave, where nobody suggests that the spirals in phase space are whirlpools.

The time it takes depends on the eye #

So the rate at which the dye winds up should be the rate at which the eye turns over, and it is. Measuring the time at which the streakline first overturns gives the following.

That formula has no threshold. It has no critical amplitude and no minimum strength of wave at which the dye begins to roll. A wave half a per cent of the shear rolls the dye, by time 26.5, into a row of spirals, smaller in height, and just as convincing in a picture. The factor in time for a tenfold weaker wave is only three: $\sqrt{10}$. So a photographer does not have to be anywhere near the instability to get spirals. A disturbance too weak to be detected by a hot wire, carried in a flow that is stable and decaying, can still fill a smoke visualisation with structure.

That result has a hard edge. The argument behind Hama’s note is that a marker in a nominally laminar layer with a small travelling wave gives beautiful pictures of what look like the first vortices, which are easily misread as the beginning of breakdown. Those pictures can appear while the wave’s own dynamics are still linear. Everything happening in the picture is geometry applied to a wave.

Why a pendulum winds a line #

The mechanism inside the eye is simple enough to state completely. In the wave’s frame a particle obeys

$$\ddot X = -aS\sin X,$$

which is exactly a pendulum with $X$ as its angle, measured from the eye’s centre: the shear turns a vertical displacement into a horizontal speed, and the wave turns a horizontal position into a vertical speed. The centre of the eye is the pendulum hanging at rest, the separatrix is the swing that just reaches the top, and small oscillations have an angular frequency $\sqrt{a}$ in units of the shear. At $a=0.05$ the period at the centre is $2\pi/\sqrt{0.05}=28.1$.

The winding comes from one property of the pendulum, the one every clockmaker knows: its period depends on its swing. Small swings take 28.1 and large ones longer, without limit at the separatrix. So particles entering the eye at the same moment but different heights fall out of step, the outer ones lagging, and any line of particles that was once straight is sheared along its own oscillation. That is a spiral. The first fold, at time 10, comes at about a third of a period, when the first arrivals have swung over and are moving back beneath the later ones. After that, the number of turns grows in proportion to the time, without bound and without any further mechanism, and the spiral tightens as far as a dye’s diffusion allows. A phase-space diagram of electrons trapped by a plasma wave looks exactly the same for exactly the same reason, and there the curves are not even in physical space.

It also explains the sensitivity to height. A particle released outside the eye is a pendulum going over the top, always circulating in the one direction, and it is not trapped. The dye’s height decides whether it starts inside the separatrix, and the eye’s height of $2\sqrt{a}$ is a very modest one, 0.447 in units where the wavelength is $2\pi$. A dye port one eye-height too high, at $y=0.5$, is a pendulum just over the top, moving slowly near the separatrix and folding late; at $y=2$ it passes each crest in a fraction of a period and is merely deflected.

What this would be in a wind tunnel #

Put numbers on it. Take a mixing layer with a mean speed of 10 metres a second, a vorticity thickness of a centimetre, and so a shear of about 1000 per second. A natural wave there is about seven thicknesses long, so $k$ is about 90 per metre, and the speed of the wave at the mean, $U$, is 10 metres a second, so that one unit of time in this essay’s scaling is about a millisecond and the wave’s scaled shear is about one. A disturbance of half a per cent is a normal velocity of 5 centimetres a second, an amount far below what most tunnels would call a strong wave. It takes 26.5 units of time to fold the dye, that is, about 30 milliseconds, in which the flow has gone 30 centimetres. So a smoke wire in such a tunnel would show spirals about thirty centimetres downstream, from a wave too weak to have changed the mean flow or the layer’s thickness. A photographer standing there would see an unmistakable row of vortices at a station where a velocity probe says the layer is laminar and gently wavy.

Those are rough numbers from a flow with a very simple form, and a real layer would be growing while they applied. Yet the arithmetic is not subtle. The time to fold scales as the inverse square root of the wave’s size, and a wave has to be extraordinarily weak before the distance needed to fold it exceeds the length of a working section.

The same flow from a different port #

It gets worse, or better, depending on the purpose. The pattern depends strongly on where the dye goes in.

So two photographs, taken at the same moment of the same flow by two dye ports a small height apart, show a train of vortices and a wavy ribbon. Nothing in the flow differs between the two. The vortices were not in the flow, and in a flow with real vortices no such sensitivity would be possible. The same applies to a smoke wire moved through a shear layer: anyone who has adjusted one has seen its picture change character across a few millimetres. A mistake here is hard to detect by looking at the result: each photograph is coherent, sharp and plausible, and neither carries a warning.

A related fact comes from the marker’s own history. Dye shows where fluid has been since its release, and two flows with the same instantaneous velocity field can have utterly different streaklines. A timeline in the same flow at the same moment says something else again.

The pathline shows nothing resembling a vortex, since each particle only traces a pendulum’s arc in the wave’s frame. The timeline is folding too, but differently, with its folds tied to the moving eye rather than to a succession of releases. Only the streakline piles its different fluid particles into a spiral, because only the streakline is a line joining particles with different histories. Between them, the three curves differ by as much as they could.

A record of a flow that has gone #

The final consequence is the most useful in practice. Dye has no dynamics. Once it has been folded into a spiral, it stays there, carried along, however the flow changes afterwards. So a streakline is a record of something that happened upstream, and it can far outlive the event.

Let the wave die with distance: replace $a$ by $a\,e^{-x/L}$, with a decay length of 15, and start at $a=0.08$. The field remains divergence-free, and the shear is unchanged.

At $x=40$ the wave is 7 per cent of what it was, its eye has shrunk, and a streakline released there would barely stir. The dye arriving from upstream carries on showing the rolls it was given, as if the eye were still there at full height, for as long as the picture lasts.

That is what Cimbala, Nagib and Roshko found behind bluff bodies in 1988. The smoke picture in a far wake showed a well-formed vortex street at distances where hot-wire surveys showed that the street had decayed and been replaced by a new, unrelated instability. The smoke was still drawing the street well downstream of where the street had stopped being a flow. The photograph was entirely faithful, and it was a photograph of history. The opposite trouble exists as well. A faint dye pattern need not mean a weak flow: it might mean the dye has not had long enough.

What a vortex would need to show #

The reasonable question is how anyone tells them apart. One thing that does not work is seeing the flow in another frame, though it is often a useful step: as an edge no observer can move discusses, closed streamlines exist in a chosen frame and fail to exist in another, so a kinematic eye only shows that a frame exists. Local eigenvalue criteria don’t entirely settle it either. Here the velocity gradient has eigenvalues $\pm\sqrt{a\cos(x-t)}$ in units of the shear, imaginary where the cosine is negative, so that the Okubo–Weiss criterion marks half the layer as vortical. It is a real fact about the local kinematics that the rate of rotation of nearby fluid is $\sqrt{a}$, the turnover rate of the eye. But for a flow at $a=0.005$, a detector is calling something a vortex whose vorticity differs from the surroundings by half a per cent.

What a vortex in the ordinary sense needs is an excess of vorticity over its surroundings, by a factor large enough that it induces the motion it sits in. That can be read from a measurement of velocity; it cannot come from a picture of a marker. A boundary that only exists over a window has the same message from another direction. A velocimetry measurement of this flow would, in the wave’s frame, show closed streamlines with no vorticity peak. And that is what to look for: a closed curve in the wave’s frame and a near-uniform colour in a map of vorticity.

What the picture cannot show #

Whether the wave is the right one. A flow of this form is not a solution of the Navier–Stokes equations in any strict sense, although uniform shear plus a neutral wave comes close to a local description of a shear layer’s critical level. Real flows have a wave whose amplitude grows, whose vorticity responds, and which has real vorticity concentration when it saturates. So the picture is not an argument that mixing layers do not contain vortices. They do, and a sheet that rolls up concentrates its vorticity genuinely. What the figures say is that seeing a spiral is no evidence of that.

Diffusion and buoyancy of the dye. The dye is a mathematical curve with no thickness and no diffusion. A real dye filament has a width that blurs tight rolls, and that would make them look even more like a blob of something. Smoke has an inertia that lets it stray, usually negligibly. A real picture’s spirals are less tight than a curve’s, and more vortex-like.

The third dimension. Everything is two-dimensional, and a real sheet of smoke is a thin slice through a flow where dye can enter and leave the plane of the light.

The convention #

Lengths are in units of $1/k$ and times in units of $1/(kU)$, where $U$ is the wave’s phase speed and also the shear’s speed at unit height, so $S=k=c=1$. The amplitude $a$ is therefore $v/U$, and also the wave’s share of the shear in the vorticity. A fold is counted when a vertical line cuts the curve three times or more, and first-fold times are found on a half-unit grid in time.

How it was checked #

Each dye particle is integrated by fourth-order Runge–Kutta at a step of 0.03 from its own release to the picture’s time, so no streakline depends on any other. With the amplitude set to zero the curve is exactly straight, and no folds are counted. The dye stays within the kinematic eye’s half-height to three figures at this amplitude: a check with content, because it compares a marching result to a closed form derived from the wave-frame streamfunction. The model refuses a negative amplitude, a zero wavenumber and a time of zero.

Who warned about it, and when #

Hama’s short paper, Streaklines in a perturbed shear flow, appeared in Physics of Fluids in 1962. Williams and Hama extended it to two waves in 1980, and Kurosaka and Sundaram’s article of 1986 gave a set of examples. Cimbala, Nagib and Roshko’s work on far wakes dates from 1988. None of this has stopped the pictures being used, and they are not misused for being pretty: dye remains a superb way of finding where a flow is doing something. It answers where, and not what.

Still open: what a two-wave flow draws #

One wave makes a row of identical spirals. Williams and Hama’s two-wave calculation produced pairing-like patterns from two linear waves with no pairing in them, which was startling in the 1980s, when the visible pairing of a mixing layer’s spirals was the main evidence for the growth mechanism. The next calculation would add a subharmonic of stated amplitude and phase to this wave, draw the dye through both, and find at what subharmonic amplitude a picture looks like a pairing. Then it would march a real layer with an inviscid solver through a genuine pairing, draw its dye with the same procedure, and ask whether any measure computed from the two pictures alone can tell them apart.