A shock that will not stand still

Relaxing the transonic equation to convergence gives numbers to read off the result: a pressure distribution, a supersonic run, a shock at a stated fraction of the chord, a drag. Every one of those is a property of a steady solution, and the equation being solved has no time in it.

Put the same section in a tunnel at the same conditions and, over a band of Mach numbers and incidences, there is no steady solution to be found. The shock sweeps forward and back across something like a fifth of the chord, the lift oscillates with it at a few tens of hertz, and an aeroplane doing this shakes hard enough that the condition is a certified flight limit rather than a matter of comfort — and unlike flutter it needs no structural degree of freedom at all. That is buffet, and it is the most important thing about the transonic range that a steady calculation cannot produce.

It is not a resonance #

The first thing to establish is what the frequency is not, because a structure that shakes at a definite frequency invites one particular explanation and it is the wrong one here.

Buffet is not the structure ringing. The frequency does not move when the wing’s stiffness is changed, a rigid model in a tunnel does it just as a flexible one does, and it is not flutter, which is an instability of the coupled structure and flow and disappears if the structure is made rigid. It is also not vortex shedding, whose Strouhal number is about 0.2 and whose mechanism needs a bluff body.

What the frequency belongs to is the flow, and specifically to a journey. Something has to travel somewhere and come back, and the period is how long the round trip takes.

That distinction is worth more than a paragraph of definitions, because it is the difference between two completely different engineering situations. A resonance can be tuned away: stiffen the structure, move the natural frequency, and the problem goes elsewhere. A feedback loop cannot, because its period belongs to the flow and the flow does not care what the structure is made of. Buffet is therefore a limit on the flight envelope rather than on the airframe, and an aircraft that buffets at Mach 0.89 and a certain lift coefficient will still buffet there after every structural modification anyone can make to it.

It also means the frequency carries information. A resonance tells an observer about the structure; a loop tells them about the journey. A measured buffet frequency is, read backwards, a measurement of the distance from the shock to the trailing edge and of the speed at which waves make their way back along it — which is the use this essay eventually puts it to.

A loop with two legs #

Lee’s account, which is the one that has survived thirty years of being tested, is a feedback loop with no resonant element in it at all:

1. The shock foot separates the boundary layer. Disturbances shed at the foot are carried downstream through the separated shear layer to the trailing edge, at a fraction $k_p$ of the free-stream speed.
2. Arriving at the trailing edge, they generate pressure waves that travel upstream through the subsonic flow outside the wake and reach the shock.
3. Arriving at the shock, they move it — which changes where the disturbances are shed, and the loop closes.

Nothing in that description has a natural frequency. The period is a sum of two travel times, so

$$T = \frac{c - x_s}{k_p U_\infty} + \frac{c - x_s}{a_u}, \qquad \mathrm{St} = \frac{f c}{U_\infty} = \frac{1}{\left(1 - x_s/c\right)\left(1/k_p + U_\infty/a_u\right)},$$

with $a_u$ the speed at which the upstream-running wave actually makes progress against the flow.

The leg that crawls #

That upstream speed is where all the interest is, and it is a small difference of two large quantities.

A pressure wave travels at the local speed of sound relative to the fluid. The fluid behind the shock is moving downstream at $u_2$, so the wave’s progress against the ground is $a_2 - u_2$, which can be written

$$a_u = u_2\left(\frac{1}{M_2} - 1\right).$$

Just behind a transonic shock the flow is barely subsonic. At $M_2 = 0.90$ the wave comes back at a tenth of the free-stream speed; at $0.94$ at a sixteenth; at $0.97$ at a thirty-second. It is a wave running up an escalator that is very nearly going as fast as it is.

So the period is essentially the return trip. At $M_2 = 0.97$ the upstream leg is ninety-three per cent of the total, and the first leg — the one involving the separated shear layer, the boundary layer, and all the physics a calculation is least confident about — contributes seven per cent.

That inversion is the most useful thing in the whole model, and it is worth stating as a principle. The part of a feedback loop that sets its period is the slowest leg, and the slowest leg here is the one made of the cleanest physics: an acoustic wave in an inviscid subsonic stream. The messy leg is fast and therefore nearly irrelevant.

The magnitude is worth dwelling on because it is genuinely extreme. Three per cent of the free-stream speed, on an aeroplane at Mach 0.85 and thirty-five thousand feet, is about eight metres a second — walking pace, in air that is moving past at two hundred and fifty. A pressure disturbance made at the trailing edge of a two-metre chord takes a quarter of a second to reach the shock, against seven thousandths of a second for a sound wave in still air to cover the same distance. The flow has slowed the messenger by a factor of thirty, and the frequency of the whole phenomenon is that delay.

It also explains why the frequency is low. Buffet is felt at tens of hertz, not the hundreds that a chord-length divided by a flight speed would suggest, and the usual account — that the structure responds at its own low frequencies — has the causality backwards. The forcing is low-frequency because the loop is slow, and the structure happens to have modes there because wings are large.

The number that comes out, and the number that is measured #

Putting numbers in gives an immediate test, and the test is not generous: the loop has to produce a Strouhal number near 0.07, and it has a factor of thirty available to it depending on one input.

With the shock at 0.6 of the chord and disturbances convected at 0.4 of the free stream, the loop gives a Strouhal number of 0.21 at a post-shock Mach number of 0.90, 0.14 at 0.94, and 0.073 at 0.97. The measured band is 0.06 to 0.08. So the arithmetic reproduces the measurement at a post-shock Mach number of about 0.97, and the interesting question is whether that is a plausible value or a fudge.

It is plausible, and it is what a transonic shock on a lifting section actually produces. The pressure rise across a shock whose upstream Mach number is 1.15 — a typical value on a wing near the buffet boundary — leaves the flow behind it at about 0.87; one at 1.05 leaves it at 0.95; one at 1.02 at 0.98. The shocks that cause buffet are weak ones, because a strong shock separates the layer so completely that the flow does not reattach and the oscillation gives way to a steady stall. So a post-shock Mach number in the high nineties is exactly the range the phenomenon lives in, and the loop meeting the measurement there is a check rather than a coincidence.

Read the other way, the measurement becomes an instrument. A measured Strouhal number of 0.07, with the shock position known from a pressure survey, says the flow behind the shock is at Mach 0.971 and that the returning wave is making three per cent of the free-stream speed. Neither of those is easy to measure directly, and both come out of a frequency and a ruler.

Three inputs, and only one of them is worth arguing about #

The convection fraction $k_p$ is the parameter a sceptic would attack first. It is not derived from anything; it is a number near 0.4 taken from measurements of separated shear layers, and a reader is entitled to ask what the model is worth if it contains a fitted constant.

The answer is that it barely contains it. A two per cent change in $k_p$ moves the frequency by 0.23 per cent, because $k_p$ only enters the fast leg and the fast leg is seven per cent of the period. A two per cent change in $M_2$ moves the frequency by thirty-six per cent. The model’s weakest ingredient is the one it hardly depends on, and its strongest sensitivity is to a quantity that follows from the steady flow field.

That is a good position for a model to be in and a bad position for a prediction to be in, which is the tension the rest of this essay is about.

What a steady solver can supply, and what it cannot #

The post-shock Mach number is exactly the sort of thing the relaxation two essays below computes, so the loop can be fed with its own numbers rather than with assumptions.

The sweep runs from a Strouhal number of 0.012 to one of 0.85 across three steps in the similarity parameter — which is to say across about five hundredths in Mach number — and the measured 0.06 to 0.08 falls between the first two of them.

That is a bracket rather than a prediction, and the reason is stated rather than hidden: a symmetric section at zero lift does not buffet. Buffet needs lift, because it needs the shock strong enough and far enough forward to separate the boundary layer beneath it, and a symmetric section at zero incidence puts its shock near the trailing edge where there is nothing left to separate. The three cases above have their shocks at 0.58, 0.70 and 0.83 of the chord; a lifting section in buffet has its shock nearer 0.4 to 0.5.

Why the boundary is a correlation #

The sensitivity is the practical conclusion, and it explains something about how transonic aerodynamics is actually practised.

A buffet boundary is quoted, in every aircraft’s flight manual, as a curve of Mach number against lift coefficient. It is established by flight test and by tunnel test, and it is not computed from first principles even now. The usual explanation is that separation is hard to predict, which is true and is not the whole of it.

The other half is the arithmetic above. Predicting the frequency to ten per cent requires the post-shock Mach number to better than a per cent, which requires the shock’s strength to better than a per cent, which requires the pressure distribution ahead of it to better than a per cent — on a flow that has a separated boundary layer under it, whose displacement changes the shock’s position, which changes the separation. A quantity that is a small difference of two numbers of order one cannot be computed to a per cent by a method that gets each of them to two.

So the buffet boundary is measured for the same reason the drag-divergence Mach number is measured: not because the physics is unknown, but because the physics is arranged so that the answer depends on a subtraction.

It is worth separating two things that the phrase “hard to predict” runs together. The onset — does this condition buffet at all — is hard for the usual reason: it depends on whether a boundary layer separates, and how much uphill a layer can take under a shock is at the edge of what any turbulence model does well. The frequency, given that it buffets, is hard for a quite different reason, which is the one this essay is about: it is a well-posed question about an inviscid wave speed whose answer is a small difference, and no amount of turbulence modelling helps with a subtraction.

The two difficulties have different remedies, and that is the practical point. A better turbulence model would move the onset prediction and would barely touch the frequency; a better resolution of the flow immediately behind the shock would do the reverse. A reader looking at a computation that gets the boundary right and the frequency wrong, or the reverse, now has a way of telling which half of the problem it has solved.

What the picture cannot show #

There is no boundary layer anywhere in this calculation. The loop’s first leg exists because the shock separates one, and the model takes the separation as given rather than predicting it. Whether a particular condition buffets at all is precisely the question this cannot answer.

The shock is taken to be at one place while it is in fact sweeping. The model computes a period from a mean state, which is legitimate if the excursion is small and is a stretch at a fifth of the chord — the post-shock Mach number varies through the cycle, and the loop’s own sensitivity to it means the instantaneous period does too.

The upstream wave travels outside the wake and the calculation places it inside the inviscid flow. The real path is along the edge of the separated region, where the speed is not the inviscid one, and that is a correction to the very quantity the answer is most sensitive to.

And the downstream leg’s constant is not derived. It is a measured convection fraction for a separated shear layer, applied here without a shear-layer calculation. The essay’s defence of it is that it hardly matters, which is a defence of the frequency and not of the picture.

Nothing here says the oscillation happens. Every number above is conditional on there being one, and a loop’s period is a property it has whether or not its gain exceeds one. That is the largest gap in the account and it is taken up at the end.

And the loop is drawn on one surface. A real wing’s shock is swept, its buffet is three-dimensional, and the disturbances travel along the span as well as along the chord — which is why a swept wing’s buffet has a broader spectrum than a two-dimensional section’s, and why a two-dimensional Strouhal number transfers to an aircraft only approximately.

Who found it, and when #

Buffet was a flight problem from the first aircraft fast enough to have one, and was well described by the mid-1950s. Lee proposed the feedback model in 1990, after measurements on a supercritical aerofoil showed pressure disturbances propagating upstream from the trailing edge at speeds that matched the observed period; the model has since been tested against unsteady computations and against experiments on several sections, and it accounts for the frequency without accounting for the onset.

The surprising connection is with something with no shock in it at all. The same arithmetic — a disturbance carried downstream at one speed, a wave returning upstream at the local sound speed minus the local flow speed, and a period that is the sum — is the edge tone, the hole tone and the cavity’s own oscillation. Every one of those is a feedback loop rather than a resonance, every one has a period that is a round trip, and every one produces a Strouhal number of order a tenth for the same reason: the return leg is slow because it is travelling against the flow that carried the outward leg. What is unusual about buffet is only how slow: an ordinary subsonic cavity’s return wave makes half the free-stream speed, and a transonic shock’s makes three per cent of it.

Still open: whether the loop closes at the same place it opens #

The model computes a period and says nothing about an amplitude, and the missing piece is not a detail — it is the whole of what a buffet boundary is.

A feedback loop oscillates when its round-trip gain exceeds one, and the period the loop computes is the period it would oscillate at, not a statement that it will. Every ingredient of the gain is missing here: how large a pressure disturbance a given shock displacement sheds at the foot, how much of it survives the trip to the trailing edge, how much of that is converted into an upstream-running wave, and how far the shock moves per unit of pressure perturbation arriving at it. All four are computable in principle; the last is the most interesting, because a shock’s displacement for a given downstream pressure change is a property of the steady solution and could be read off the relaxation directly, by perturbing the trailing-edge pressure and watching where the shock settles.

That sensitivity — chords of shock movement per unit of pressure coefficient at the trailing edge — is a number the steady solver can produce and nobody appears to have tabulated against the similarity parameter. Whether it grows fast enough, as the parameter falls, to take the loop gain through one at the place the buffet boundary is measured would turn Lee’s model from an account of the frequency into an account of the onset. It would also say something the correlations cannot: whether the boundary is set by the loop’s gain or by the separation appearing at all.