A wake told what to do

The frequency a wake chooses establishes the Strouhal number: a bluff body sheds vortices at a frequency proportional to the flow speed over its width, with a coefficient that is nearly constant over a wide range of Reynolds number.

That frequency is the wake’s own. This essay is about what happens when something else offers it a different one — and about the fact that a wake can be persuaded, which means it has a clock that can be reset.

What a shedding wake is, dynamically #

A wake behind a bluff body above the critical Reynolds number is a self-excited oscillator. It has a preferred frequency, it settles at a preferred amplitude, and it does both without being driven — which is what distinguishes it from a resonance.

The standard reduced description is a Stuart-Landau equation: an amplitude that grows exponentially when small, saturates through a cubic nonlinearity, and rotates in phase at the natural frequency. It is the same equation that describes every limit-cycle oscillator in physics, and its behaviour under forcing is correspondingly universal.

Capture #

Force the oscillator at a frequency near its own — by oscillating the body, by an upstream disturbance, by an acoustic field — and one of two things happens.

Inside a band, the wake abandons its own frequency and sheds at the forcing’s. The amplitude settles, the phase difference settles, and the wake’s own clock has been overwritten.

Outside it, the two frequencies coexist. The amplitude beats at their difference, and a spectrum of the signal shows both peaks and their combinations.

The width of the capture band is proportional to the forcing amplitude, from 0.093 at a forcing of 0.05 to 0.65 at 0.4 — which is the standard shape of a capture tongue and is the reason lock-in is described by a wedge in the plane of frequency and amplitude.

And what capture costs in time #

Capture is not instantaneous, and how long it takes depends on how far the forcing is from the wake’s own frequency.

Near resonance the wake settles in about fourteen time units. Near the edge of the band it takes eighty-six — a factor of six — and the time diverges as the boundary is approached.

The settling time diverges towards the edge: 14 periods at a detuning of 0.02, 18 at 0.06, 24 at 0.12, 28 at 0.18, 36 at 0.22, 54 at 0.245 and 86 at 0.252 — a factor of 6.1 across a twelve-fold change in detuning, with more than half of it in the last four per cent before the edge. That divergence is critical slowing down, and it has a clean interpretation in the terms used here. Near the edge the wake is neither following the forcing nor keeping its own time: it is spending many periods in a state that is a mixture of the two, and the mixture is a memory of both. The nearer the edge, the longer that memory persists.

What the band is measuring #

The capture band’s width has a reading worth extracting, because it turns the tongue into a measurement of something about the wake.

The band is wide when the oscillator is weakly attached to its own frequency, and narrow when it is strongly attached. Formally the width is the forcing divided by the oscillator’s amplitude and its sensitivity to detuning, so a wake with a large amplitude is harder to capture than one with a small one.

The width is exactly proportional to the forcing over the first three rows — 0.0929, 0.186, 0.371, each a doubling to three figures — and falls short only at the largest forcing, where 0.650 is 12 per cent under the 0.743 the proportionality would give. That is Adler’s linear width holding until the forcing stops being small.

That gives a practical statement. Near the onset of shedding the wake’s amplitude is small and it is easily captured — a tiny forcing locks it over a wide band. Well above onset the amplitude is large and capture needs proportionately more.

So a wake is most controllable exactly where it is weakest, which is the opposite of the intuition that a strong disturbance is easier to manipulate, and it is why active control of shedding is attempted near onset and abandoned at high Reynolds number.

The same statement in reverse explains why lock-in is such a hazard for a structure: a body whose natural frequency is close to the shedding frequency needs only a tiny motion to capture the wake, and once captured the wake drives the motion, which widens the band further.

What lock-in does to the amplitude #

Capturing a wake changes what it does as well as when. Near resonance the forcing reinforces the oscillation and the amplitude rises; near the edges of the band the forcing and the wake are working against each other and it falls.

That is why lock-in matters practically. A cylinder in a cross-flow that begins to vibrate at its structural frequency can capture its own wake, at which point the shedding is synchronised with the motion and feeds it — which is vortex-induced vibration, and it is the mechanism behind the failure of chimneys, risers, cables and heat-exchanger tubes — the same members whose loading two forces, and only one of them remembers computes in the absence of any motion at all.

What the solver computed, and how it was checked #

The wake is represented by a forced Stuart-Landau oscillator, integrated forwards. Lock-in is detected by measuring the drift of the phase difference between the oscillator and the forcing over the last quarter of the run: a locked state has essentially none and an unlocked one drifts at the beat frequency.

Three checks. That a wake forced near its own frequency locks — at a forcing of 0.3 and a detuning of 0.02 the phase drift over the last quarter of the run is 0.000000000 — and that one forced far from it does not, which at a forcing of 0.02 and a detuning of 0.5 gives a beat of 0.4994, within 0.2 per cent of the detuning itself. Those are the two ends of the claim. That the capture band grows with the forcing amplitude, monotonically across four amplitudes — 0.0929, 0.186, 0.371, 0.650. And that the settling time rises towards the edge, by at least a factor of four; it rises by 6.1, from 14 periods to 86.

One thing was looked for and not found, and it is recorded because the absence is informative. The capture band was swept upwards and downwards in forcing frequency to look for hysteresis — a band that depends on the direction of approach — and there is none in this model. The Stuart-Landau equation with a simple forcing has a symmetric tongue and no bistability; the hysteresis that real lock-in shows comes from a subcritical element that this description does not contain, and claiming it here would have been claiming a result the computation does not have.

What happens at the edge, in more detail #

The behaviour at the boundary of the band is worth one more paragraph, because it is where the two descriptions meet and it is genuinely peculiar.

Just inside the band the phase difference between the wake and the forcing sits at a fixed value near ninety degrees — the wake is locked, but only just, and it is holding on at the limit of what the forcing can supply.

Just outside, the phase does not sit anywhere. It advances quickly through most of a cycle and then lingers near the value it would have taken if it were locked, before slipping through. The signal therefore looks locked for long stretches interrupted by rapid phase slips, and the beat frequency is the rate of slipping rather than a smooth difference.

That intermittent character is the observable signature of being near the boundary, and it is easily mistaken for a noisy lock. Distinguishing them requires watching for long enough to see whether the slips accumulate, which is the same requirement the settling time imposes from the other side.

Why an oscillator and not a resonance #

The distinction is worth making because it changes what the response looks like and it is routinely blurred.

A resonance has no oscillation of its own. Force it and it responds; stop forcing it and it decays. Its response is largest at its natural frequency and falls away smoothly on both sides, and there is no band inside which anything qualitative changes.

A self-excited oscillator oscillates whether or not it is forced. Force it near its own frequency and it does not add the forcing to its own motion — it synchronises, discontinuously, at a threshold. The response is not a peak but a plateau with edges.

A wake is the second. So a measurement showing a large response near the shedding frequency is not by itself evidence of lock-in; what is evidence is the disappearance of the natural frequency from the spectrum, and that is what should be looked for.

Why the wake has a preferred amplitude at all #

The oscillator description carries an assumption worth unpacking, because it is what makes the wake capturable rather than merely forced.

The amplitude saturates because the equation is nonlinear: growth at small amplitude, damping at large, and a balance in between. That balance is what fixes the shedding’s strength, and it is why a vortex street looks the same at a given Reynolds number however it was started.

Without it, the wake would be a linear instability growing without bound and the question of capture would not arise: a linear system driven at one frequency simply responds at that frequency, and there is no band and no threshold.

So lock-in is a nonlinear phenomenon in an essential way. The band exists because the wake has a preferred amplitude to defend, and it is captured when the forcing can pull the phase faster than the detuning can push it away. That is Adler’s condition, and it is why the width is proportional to the forcing.

The physical origin of the saturation is the wake’s own feedback on the near-body flow — the street this site cannot draw is this collection’s account of what that flow is and of why the solver here cannot produce it.

Where else a flow is captured #

The hysteresis that is absent here is present, measured and load-bearing one field away, in two lifts at one incidence — which is worth knowing before reading its absence here as a general result.

Vortex-induced vibration. The standing example, described above, and the reason the subject is studied.

Combustion instability. A flame is an oscillator with a preferred frequency, and a chamber acoustic mode is a forcing; capture between them is what makes a combustor scream.

And a rotor in a duct. A blade row’s own unsteadiness can be captured by a duct mode, which is one of the mechanisms behind rotating stall — and the counting argument of a row that meets the row before it is what sets the frequencies on offer.

In all three the practical question is the same: is the forcing inside the band? And the band widens with the forcing, so a device that is quiet at low amplitude can capture itself at high.

Why lock-in is a memory question at all #

It is worth being explicit, because a synchronised state sounds like the opposite of a memory.

A wake shedding at its own frequency has an internal clock and its phase is a record of when it started: two identical wakes started a moment apart stay a moment apart for ever, because nothing in the equations couples them.

Forcing destroys that. Inside the band the phase is slaved to the forcing, so the wake’s memory of its own start-up is erased and replaced by a memory of the forcing’s. That is a real loss of information and it is the reason lock-in is used experimentally: a locked wake is reproducible where a free one is not, so phase-averaged measurements become possible.

Capture is a wake being given a new past, and the settling time is how long the substitution takes.

The other way a system’s state gets decided is by how it was started, which is the essay before this one and the same solver.

What a measurement should look for #

Since the signature is a disappearance rather than an appearance, it is worth listing what to record.

The spectrum, not the amplitude. A locked wake has one peak where a free one has two. An amplitude measurement alone cannot distinguish lock-in from a large forced response, and the two have completely different implications for a structure.

The phase, over many cycles. A locked phase is constant and an unlocked one drifts. Measuring the drift rate gives the beat frequency directly, and its vanishing is the sharpest available detector of the band’s edge.

And the approach, not just the state. The settling time diverges at the edge, so a measurement made before the wake has settled will report an unlocked state that is on its way to being locked. Runs near the boundary need to be long, and how long is exactly what this essay computes.

That last is the practical form of critical slowing down and it is a general hazard: a system near a threshold takes longest to reveal which side of it is on, which is the same warning a puff that does not know how old it is gives about measuring a transition in a pipe of finite length.

What the picture cannot show #

The oscillator here is one complex number and a wake is a flow field. Everything about the spatial structure — where the vortices form, how far downstream the synchronisation extends, whether the near wake and the far wake lock together — is outside a model with one degree of freedom.

Nothing here shows a spectrum either, and a spectrum is what an experiment measures: the disappearance of one peak and the growth of another is the observable signature, and this model produces the phase drift instead.

The same lag, one more time #

The settling time near the band edge is the last of this collection’s response times and it is worth putting beside the others, because it is produced by a different mechanism and behaves the same way.

A wake’s lift deficiency falls with reduced frequency because shed vorticity has not convected away — the lag that makes flutter possible.

A turbulence’s stress lags a change in strain by a turnover — a closure with no memory at all.

A polymer’s stress relaxes over its own time — the fluid that has not finished its last deformation.

And a wake being captured settles over a time that diverges at the edge of the band.

The first three are first-order relaxations with a fixed time constant. This one is not: its time constant depends on how far the forcing is from resonance, and diverges. That makes it the one case here where the memory’s own length is a function of the question being asked — which is what a threshold does to a relaxation, and is why critical slowing down is a signature of a bifurcation rather than of a material property.

What lock-in does to a fatigue calculation #

The engineering consequence is not the amplitude but the count, and the two are usually confused.

An unlocked wake produces a broad-band load whose energy is spread over a range of frequencies, so a structure sees many partial cycles at its own frequency and few large ones.

A locked wake produces a narrow-band load at exactly one frequency, and if that frequency is the structure’s own the cycles accumulate coherently. The amplitude may be no larger; the number of full-amplitude cycles per hour is very much larger.

So damage rises faster than load does, because fatigue accumulates as a high power of the range and lock-in converts a distribution of ranges into a single large one. That is why a riser’s design case is the lock-in condition rather than the extreme current, and why the width of the band matters more than its depth.

Who found it, and when #

Synchronisation of oscillators is Huygens’ observation of 1665, about pendulum clocks on a shared beam, and its modern theory is Adler’s from 1946 in electronics. Lock-in of a vortex wake was established experimentally by Bishop and Hassan in 1964 and by Koopmann in 1967.

The reduced description as a Stuart-Landau oscillator is Landau’s amplitude equation applied to the wake, and its use for the vortex street is Provansal, Mathis and Boyer’s from 1987 — which is what put the subject on the same footing as every other synchronisation problem.

Limits recorded rather than smoothed over #

One degree of freedom. A wake has many, and the reduced description holds near the onset of shedding rather than at high Reynolds number. Its coefficients would have to be fitted to a particular wake.

No hysteresis, and real wakes have some. Stated above at length. The model’s tongue is symmetric, real lock-in boundaries are not, and the difference is a nonlinearity this equation does not carry.

Forcing amplitude in arbitrary units. The forcing here is a term added to the amplitude equation, and relating it to a physical body displacement requires a calibration that is not attempted.

Forcing at one frequency. A real disturbance has a spectrum, and a nonlinear oscillator forced at several frequencies at once can lock to a combination of them rather than to any one — which is where the subject becomes considerably richer and is not attempted here.

And nothing is a flow. No cylinder, no Reynolds number, no vortices — which is a considerable distance from this collection’s usual standard, and is stated so that the essay is read as the dynamics of an oscillator rather than as a computation of a wake.