A wake that says what made it

Where the wake ends up settles what a roll-up cannot change: the total circulation, the impulse and the centroid of the shed vorticity are fixed before the roll-up begins, so the pair of cores that emerges sits at 78.5 per cent of the span apart whatever the intervening tangle does.

That essay is about what survives. This one is about what can be read, which is a different question with a different answer, because a quantity that survives a process is a quantity that can be inverted back through it.

A wake, in other words, is a record. It says what made it, in a form that needs no assumptions about the roll-up at all — and it stops saying so on a timetable that can be computed.

The two numbers #

Behind a wing in steady flight there are exactly two things to measure, and both of them are geometric.

The circulation of a core. Integrate the velocity round a circuit enclosing one vortex, or fit a model to a velocity traverse through it. For an elliptically loaded wing this is the root circulation.

The separation of the pair. The distance between the two cores — 47.12 metres for a sixty-metre span — which for elliptic loading is $\pi/4$ of the span, a ratio of 0.785398 — a number that comes out of the shed vorticity’s centroid and is one of the few places in this subject where a $\pi$ arrives from geometry rather than from a circle.

Those two give the lift, by Kutta-Joukowski applied to the whole wing: $L = \rho V \Gamma b'$. And the second gives the span on its own. So two measurements taken in the wake return the aircraft’s weight and its span, and the third quantity — the rate at which the pair sinks — is $\Gamma/(2\pi b')$ and is determined by the first two.

Three observables, two unknowns, one consistency check.

For a 220-tonne aeroplane on approach at 75 metres a second, the numbers are a circulation of 508 square metres a second, a separation of 47.1 metres behind a 60-metre span, and a descent of 1.72 metres a second. The inversion returns the weight and the span to the last bit of double precision, because the relations are algebraic and nothing has been approximated.

Where the exactness stops #

The inversion being exact says nothing about it being accurate, and the two are worth separating carefully because the second is entirely a question about the measurements.

The weight is a product of the two measured quantities, so a proportional error in either produces the same proportional error in the answer, and errors in both compound. The sensitivity table is a straight line through one: a circulation twenty per cent low gives a weight twenty per cent low — 0.80 — and twenty per cent high gives 1.20, with the spacing behaving identically. There is no cancellation and no regime in which one of them stops mattering: a twenty per cent error in the measured circulation is a twenty per cent error in the recovered weight, always.

That is the friendly case for an inverse problem. It is a well-conditioned inversion — the condition number is one — which is unusual and is worth noticing, because most attempts to read a cause off an effect in fluid mechanics are far worse behaved. A scalar is a record of where its fluid was is the contrasting case, where the inversion degrades exponentially with how far back it reaches.

The difference is structural. Here the quantity being inverted is a conserved integral — the circulation and the impulse — and conserved integrals do not amplify anything. There the quantity is a trajectory, and trajectories separate.

And where the record stops #

What does limit the reading is the decay, and it is worth computing rather than describing because the timetable is the practical answer.

Taking a half life of 1.6 minutes — which is what a quiescent atmosphere gives, and is generous — the recovered weight is 80 per cent of the truth after thirty seconds, 42 per cent after two minutes, and 18 per cent after four.

A 220-tonne aircraft at 75 metres a second on a sixty-metre span sheds a circulation of 508.1 m²/s and a pair descending at 1.716 m/s. Half a minute later the circulation reads 409.2 and the weight inferred from it 80.5 per cent of the truth; at one minute, 329.5 and 64.8 per cent; at two, 213.6 and 42.0; at four, 89.8 and 17.7. Nothing about the aircraft changed. The wake stopped being a record of it, and did so at an exponential rate — so the useful window is about a minute and there is no measurement precision that extends it, because the information is being destroyed rather than obscured.

That is the general shape of a decaying record and it recurs: an exponential loss of information gives a window whose length improves only logarithmically with the quality of the instrument, which is the same arithmetic as an hour for every tenfold arriving from a completely different mechanism.

What the descent rate adds #

The third observable earns its place and it is worth saying why, since it is determined by the other two and therefore carries no new information about the aircraft.

It carries information about the measurement. If the descent rate observed does not match the one implied by the measured circulation and separation, then one of the three is wrong — and since the descent is measured over time and the other two at an instant, they fail in different ways.

It also has an operational consequence that reverses an intuition. The descent rate is $\Gamma/(2\pi b')$, and along a family of aircraft whose weight rises as a power of the span less than two, the separation grows faster than the circulation does. A larger aeroplane’s wake sinks more slowly, so it stays in the approach path longer — which is the opposite of what “bigger aircraft, worse wake” suggests, and is one of the reasons wake separation minima are a matrix of leader and follower rather than a distance.

What the solver computed, and how it was checked #

The relations are algebraic — the roll-up invariants are taken from where the wake ends up rather than recomputed — so the checks are aimed at the inversion and at the plausibility of the numbers.

That the round trip is exact, to a part in 10⁹, in both the weight and the span. An inversion that did not return its own input would mean an algebra error in one of the two directions.

That the descent rate is a real number for a real aeroplane, between half a metre a second and five. That check exists because the whole calculation can be dimensionally consistent and physically absurd, and a descent rate is the quantity a reader can sanity-check against experience.

And that the record genuinely fades, with the recovered weight falling below forty per cent over the window. A decay model that left the answer nearly intact would not support the essay’s second half.

Why the separation is exactly a quarter of pi #

The $\pi/4$ deserves a paragraph on its own, because it is the part of this that looks like a fudge and is not.

The vorticity a wing sheds is distributed along the span with a strength set by the derivative of the circulation. Rolling up moves it about, and the roll-up conserves the centroid of the vorticity in each half of the wake — a consequence of the impulse being conserved, which where the wake ends up computes. So each core ends up at the centroid of the vorticity on its own side, whatever the roll-up did on the way.

For an elliptic distribution that centroid is at $\pi/8$ of the span from the centreline, so the pair sits $\pi/4$ apart. The $\pi$ is the ellipse’s, arriving through an integral of a square root, and it is exact rather than fitted.

That is what makes the inversion possible without a roll-up calculation. The quantity being measured is the one the roll-up cannot move, and the whole difficulty of computing a roll-up — which is considerable, and is a spiral is a legible record’s subject — is stepped over entirely.

A wake is not the only record an aeroplane leaves #

The vortex being inverted is the one the vortex a wing leaves behind computes from the other end, and the drag it stands for is the price of having ends.

Two other records are worth naming, because they carry different information and fade differently.

The momentum deficit carries the drag, and it is a much weaker signal: a wake survey for drag has to resolve a velocity deficit of a few per cent, where the circulation of a core is an order-one feature of the field. It also spreads and weakens by diffusion rather than being destroyed by an instability, so it lasts differently.

The trail of shed vorticity between the cores carries the span loading in detail — the whole function rather than its two moments — and it is gone almost immediately, because it is exactly the part the roll-up rearranges. That is the information the inversion cannot recover, and the reason it cannot is that the roll-up destroyed it rather than that the instrument is poor.

So the wake is a record with three layers and they have three lifetimes: the invariants last as long as the vortices, the deficit lasts as long as diffusion allows, and the detail is gone before anybody can get there.

What this is used for #

Wake separation minima. The rules that decide how far behind one aircraft another may follow are built from exactly these quantities, and the modern versions are matrices indexed by leader and follower rather than single distances — because the hazard is the rolling moment the follower’s own span picks up from the leader’s field, and that depends on both.

Airport wake radar. Lidar and radar systems installed at large airports measure the circulation of each departing aircraft’s cores directly, and use them to decide whether the runway can be reused sooner than the fixed rules allow. That is this inversion, run in real time, with the decay as the thing being watched.

Accident investigation. A wake encounter leaves a rolling moment on the follower, and reconstructing it needs the leader’s circulation at the encounter time — which means the inversion plus a decay model, and the decay model is where the uncertainty is.

The follower’s problem, which is a different measurement #

The hazard a wake presents is not the circulation and it is worth separating the two, because they are routinely conflated in the rules.

What harms a following aircraft is a rolling moment, and a rolling moment is an integral of the leader’s induced velocity across the follower’s own span, weighted by the follower’s loading. A follower with a large span sits across the core’s field and averages much of it away; a small one sits inside it and gets the full gradient.

So the hazard depends on a ratio of spans as well as on the leader’s circulation, which is why the separation rules became a matrix. It also means that the quantity a wake radar measures — the circulation — is an input to the hazard rather than the hazard itself, and converting one to the other needs the follower.

The other way of reading a wake is to fly through it, which is the essay two along and the same solver.

Why the wake is a record and the aircraft’s own field is not #

There is a reason the reading has to be done in the wake rather than near the aircraft, and it is the theme these essays share.

The flow field around a wing in steady flight is a state: it is what it is because of the wing’s present incidence and speed, and it contains no history. Measure it and the answer is the aircraft’s present condition, which is what a pressure tapping or a balance measures.

The wake is different. It is a record, laid down over time, and it holds the aircraft’s condition at the moment each piece of it was shed. A traverse through a wake a mile behind an aeroplane is a measurement of what that aeroplane was doing eight seconds ago — and if the aeroplane has since changed its lift, the wake nearer to it says so.

That is what makes a wake survey a diagnostic rather than a duplicate measurement, and it is the reason this collection’s drag measurements are made there: a wake carries the whole momentum deficit, including parts of it that never touched the body, which is four profiles, one drag.

What a reader should take from an inverse problem this well behaved #

It is worth extracting the general lesson, because a well-conditioned inversion in fluid mechanics is rare enough to be instructive about why the others are not.

Invert conserved quantities, not fields. Everything that made this work is that circulation and impulse are conserved through a violent, unsteady, essentially uncomputable process. A quantity that survives a process can be read back through it; a quantity that the process rearranges cannot.

Count the observables against the unknowns before starting. Three observables and two unknowns is a comfortable position, and the third is what makes the answer checkable rather than merely produced. An inversion with exactly as many observables as unknowns returns an answer whatever the data, which is the situation to be suspicious of.

And find the clock. Every record has one, and the inversion is only worth anything inside it. Here it is the decay; in a scalar is a record of where its fluid was it is the separation of trajectories; in a spiral is a legible record it is the resolution at which the sheet stops being a sheet. The clock is usually easier to compute than the inversion and it decides whether the inversion is worth doing.

What the picture cannot show #

The pair is drawn as two circles at a fixed separation, and both simplifications matter. A real core has a radius, a velocity profile, and an interior that a Rankine or Lamb-Oseen model only approximates — and the circulation measured depends on how far out the integration circuit is taken, because a real vortex has diffuse edges.

The separation is drawn as constant and is not. The pair descends, spreads slightly under ground effect, and eventually undergoes the Crow instability — a sinusoidal mutual induction that links the two cores into rings and destroys them. None of that is in the figures, and the last of it is the real end of most wakes rather than the diffusion the decay model represents.

Who found it, and when #

The relations are Prandtl’s, and the roll-up invariants that fix the separation are Betz’s, from 1932. The observation that they can be inverted is much more recent and is operational rather than theoretical: it belongs to the wake-turbulence programmes of the 1990s onwards, which had lidars and needed a number.

The Crow instability, which sets the real lifetime of most wakes, is Crow’s from 1970, and it explains why the wakes photographed behind aircraft form loops rather than fading — a mechanism the exponential decay model here does not contain at all.

Limits recorded rather than smoothed over #

Elliptic loading. The $\pi/4$ is elliptic loading’s. A different span loading puts the cores somewhere else, and the inversion then needs the loading shape as well — which is one more unknown and no more measurements.

One decay model, and it is a fit. The 1.6-minute half life is a plausible quiescent value. Real wake decay depends on atmospheric turbulence, stratification and ground proximity, varies by an order of magnitude between conditions, and is not exponential in detail. The timetable’s shape is what transfers; its numbers are not.

No Crow instability. The dominant end of a real wake is a three-dimensional instability that links the cores, and it happens on a time scale comparable with the decay modelled here. Its absence makes this essay’s window optimistic.

The aircraft is taken to be in steady flight. A wake laid down during a manoeuvre records the manoeuvre, so the two cores’ circulations differ from one another and the pair is not symmetric. That is more information rather than less — the record is richer — and none of the algebra above applies to it.

And the circulation is taken as measurable. Measuring a core’s circulation requires either a closed circuit around it or a velocity traverse through it plus a core model, and the two do not always agree — which is the largest real error in the whole procedure and is not represented in the sensitivity figure at all.