Where the wake ends up

Behind every lifting wing there is a sheet of vorticity, shed continuously along the span, and it does not stay a sheet. It is unstable to its own induced velocity: the edges curl inward, the curl draws in more of the sheet, and within a few spans what was a flat ribbon has become two concentrated cores.

Following that process is expensive. It is an unsteady, three-dimensional, strongly nonlinear calculation, and it is exactly the kind of thing this site’s honesty rules forbid it to claim it can do on a coarse grid.

It does not have to. The answer is fixed before the process starts.

What a roll-up cannot change #

Betz’s argument, from 1932, is a conservation argument and its shape is the one this site keeps meeting: identify what the process cannot alter, and the answer follows without following the process.

Two things are conserved as the sheet rolls up.

The total circulation of each half. Vorticity in an inviscid flow is neither created nor destroyed; it moves. Whatever each half of the sheet shed is what each core ends up with, and it is the peak circulation on the wing, Γ₀.

And the centroid of the vorticity in each half. This is the vortex-dynamics analogue of a centre of mass: for a two-dimensional distribution of vorticity in an unbounded fluid, the centroid moves only under the influence of external vorticity, so within one half of a symmetric wake it stays put. The roll-up shuffles the vorticity about; it does not move its centroid.

So the two cores end up at the centroids of the two halves’ shed vorticity, and those are integrals over the loading:

$$\bar y = \frac{1}{\Gamma_0}\int_0^{b/2} y\left(-\frac{d\Gamma}{dy}\right) dy.$$

Nothing in that expression knows about the roll-up.

The number, and where π comes from #

For elliptic loading $\Gamma = \Gamma_0\sqrt{1-(2y/b)^2}$, and the integral is elementary:

$$\bar y = \frac{\pi b}{8},\qquad\text{so the cores are}\qquad b' = \frac{\pi b}{4} = 0.7854\,b.$$

Seventy-eight and a half per cent of the span. The solver takes the same quantity by quadrature over four hundred segments and requires it within five parts in a thousand of the closed form; it comes out at 0.7834, the discrepancy being entirely the tip singularity, where an elliptic loading sheds an infinite vorticity density over an infinitesimal length.

That π is worth pausing on. It is one of very few places in this subject where a transcendental number appears in a quantity an engineer uses directly — the separation standards behind every large aircraft are written against a core spacing, and the core spacing has a π in it, and the π is there because the loading is a semicircle.

Why it is not the tips #

That figure is the answer to the common picture, and it is worth stating flatly.

Vorticity is shed everywhere the circulation changes, in proportion to $-d\Gamma/dy$. For an elliptic wing the shedding is concentrated near the tips because that is where the loading falls fastest, but it is not at the tips, and it is not only at the tips.

The centroid of that distribution is therefore inboard of the tip, always, for every loading. How far inboard depends on the loading and on nothing else:

• elliptic — 0.783 of the span, the classical value;
• bell — 0.586, because that loading is deliberately concentrated inboard and sheds over a much wider band;
• nearly rectangular — 0.978, because a loading that stays flat to the tip and then drops sheds essentially everything there.

The solver requires that ordering: a flat loading must roll up wider than an elliptic one, and a bell narrower. The wake’s geometry is a property of the span loading, which is a property of the planform and the twist, and a tip device changes it only insofar as it changes the loading.

What the arithmetic does not touch #

The conservation argument gives the positions. It says nothing at all about two other things a wake has, and being clear about the division is most of the practical content.

The core size. Betz’s construction gives a radial distribution of circulation in the rolled-up core as well as its position, by matching the circulation contained within a radius to the circulation shed outboard of the corresponding station. That is a genuine result and it is more delicate than the centroid, because it assumes the roll-up is orderly and that vorticity from one station ends up at one radius. Real cores are turbulent and their size is set by that turbulence.

And the decay. A wake’s cores weaken over minutes, and none of the mechanisms is in this arithmetic. Atmospheric turbulence tears them apart; stratification generates baroclinic vorticity that opposes them; and the pair has a long-wave instability of its own — the Crow instability — in which the two cores develop a sinuous distortion, approach each other, and link into a train of rings. Which of those dominates decides how long a wake lasts, and separation minima are set from measurements of it rather than from any theory here.

So the honest summary is that the geometry is a closed form and the strength is somebody’s measurement, and every figure in this essay says which of the two it is drawing.

The invariant argument, in general #

The shape of Betz’s reasoning is worth extracting from the wake, because it is one of the most useful patterns in this subject and this site has now used it several times.

A process is intractable. Rather than compute it, ask what it cannot change, and see whether the answer is determined by those quantities alone.

• The wake’s roll-up is intractable; circulation and vorticity centroid are conserved; the core positions follow.
• A control volume containing a machine is intractable; mass, momentum and energy crossing its boundary are conserved; the limit on what the machine can extract follows.
• A hydraulic jump is intractable; mass and momentum across it are conserved and energy is not; the conjugate depths follow.
• A shock is intractable; three conservation laws hold across it; the jump conditions follow.

In every case the reward is the same: an exact answer to a particular question about a process nobody can follow, at the price of learning nothing about the process itself. Betz’s argument gives the positions of the cores and nothing whatever about how they got there — not the time it takes, not the shape of the sheet on the way, not the core size.

Knowing which questions a conservation argument can answer is most of the skill in using one, and the failure mode is asking it for the thing it cannot give. A wake’s strength over time is such a question, and the essay’s own figure marks the curve for it as an illustration rather than a result.

The hazard, and why it is a matrix rather than a distance #

Wake separation rules are not written as one distance. They are a matrix: heavy behind heavy, medium behind heavy, light behind heavy, and so on. The reason is geometric and it is in that figure.

A following aircraft spanning both cores has one wing in each core’s upwash region and one in each downwash region, and much of the rolling moment cancels. A follower small enough to fit between them, or to sit inside one, gets the whole of one core’s field across its span and is rolled by all of it.

And the small aircraft has less to answer with. Its ailerons are shorter, its roll authority is smaller in absolute terms, and its own roll inertia is much smaller — so it responds faster to a disturbance it can do less about. Both halves of the mismatch scale the wrong way, which is why the rules are asymmetric and why the dangerous case is a light aircraft behind a heavy one rather than the reverse.

The leader’s weight comes in because the circulation does. Γ₀ scales with lift over span and speed — a heavy aircraft, slow, with flaps down and a short span, is the strongest wake generator there is, which is precisely the configuration on approach.

Three practical consequences #

Wakes descend. The pair induces a downward velocity on itself of $\Gamma_0/(2\pi b')$, so the whole wake sinks at a few hundred feet a minute and levels off a few hundred feet below the flight path. That is a consequence of the two positions this essay computes and it is why the advice for following an aircraft is to stay above its flight path.

And near the ground they stop descending and move apart. The ground is a wall, so the image system applies: each core sees its own reflection below the runway, and the pair’s descent turns into a lateral drift outwards. In a light crosswind one core can be held stationary over the runway centreline, which is the specific meteorological condition wake encounters on approach happen in.

And the spacing is not the wingspan. An aircraft avoiding a wake by staying a wingspan to one side of the flight path is avoiding the wrong place, because the cores are at 0.39 of the span each side of the centreline and their fields extend well beyond that.

Reading a photograph of a wake #

Condensation makes a wake visible under the right conditions and the pictures are among the most familiar in the subject. Three things are worth knowing before reading one.

The visible core is not the vortex. What condenses is the region where the pressure — and with it the temperature — has fallen far enough for the air to reach saturation, which is a small region near the centre where the swirl is fastest. The circulation extends far outside it. A photograph of a wake shows the coldest part of it, not the extent of the field, and the field a following aircraft meets is many times wider than the white tube.

Two cores, not two tips. Every visible wake is a pair, and the pair is inboard of the wingtips by the amount this essay computes. On an aircraft with flaps deployed there is often a second pair from the flap edges, which merges with the outboard pair over a few spans — a real effect and one this arithmetic can handle, because a flapped wing’s loading has two places where the circulation changes fast and the centroid integral sees both.

And the sinuous distortion is the instability. A wake photographed a long way behind an aircraft often shows the two tubes waving in a long wavelength and eventually joining into a chain of rings. That is the Crow instability, its wavelength is about eight core spacings, and it is the mechanism that destroys a wake in still air. It is not turbulence and it is not decay; it is a linear instability of a vortex pair, and it is one of the few places in this subject where the end of a flow structure is as well understood as its beginning.

The two pairs a flapped wing makes #

There is a case worth spelling out because it is the one an air traffic controller is actually dealing with, and because it shows the centroid argument doing work a picture cannot.

A wing with inboard flaps deployed has a loading with a step in it: high inboard where the flaps are, lower outboard where they are not, and falling to zero at the tip. The circulation therefore changes fast in two places — at the flap edge and at the tip — and vorticity is shed strongly at both.

So the wake starts as four cores rather than two, and the outboard pair and the flap pair are counter-rotating relative to each other on each side. They orbit, merge over some tens of spans, and end up as a single pair whose position is the centroid of the whole half-span’s shedding — which is what the integral in this essay computes, and which it computes correctly without knowing that there was ever an intermediate stage with four cores in it.

That is the argument’s strength stated as sharply as it can be. The number of intermediate stages is irrelevant. Anything the vorticity does among itself leaves the centroid where it was, so the end state is the same whether the wake rolled up from a smooth sheet, from two pairs, or from four. What it does change is the time the process takes, and an aircraft following at a shorter distance may meet the intermediate arrangement rather than the final one — which is a real operational difference and is completely outside this arithmetic.

What ends a wake in still air #

The Crow instability has been named three times here as the thing that finally destroys a pair, and its mechanism is worth having, because it is the one part of a wake’s life that is neither a conservation law nor a measurement.

Each core sits in the strain field of its partner — a two-dimensional straining flow whose stretching and compressing directions lie at forty-five degrees. Give one filament a small sideways wobble. The part of the wobble lying along the stretching direction is pulled further out, which is growth; meanwhile the filament’s own curvature makes the perturbation rotate, carrying it out of the favourable orientation and into the unfavourable one, which is not. The two effects compete, and they balance only for perturbations long enough that the self-induced rotation is slow.

So the instability is a long-wave one, and its preferred wavelength comes out at roughly eight times the core spacing — several hundred metres behind an airliner, which is why the waving is visible in a photograph at all. The two cores approach at the crests, touch, and reconnect into a train of rings, and the pair as such has ceased to exist.

Which is why the deliberate version has been studied: oscillate a control surface at the right frequency and the wake can be persuaded to destroy itself sooner.

What the model does not contain #

No roll-up. The whole point is that none is computed. What is computed is where a roll-up must end, given that circulation and centroid are conserved.

The centroid argument assumes the wake is isolated. It is exact for one half of a symmetric wake in an unbounded inviscid fluid. Near the ground, near another aircraft, or in a stratified atmosphere it is not.

No core structure and no core size. The cores here are points. Everything about the velocity inside a core — which is what a following aircraft actually encounters — is outside this arithmetic, and the hazard figure puts in a cut-off radius by hand and says so.

No decay, no instability, no turbulence. Named above and computed nowhere.

And no viscosity. Which is the interesting one, because viscosity cannot change the answer anyway: circulation round a material loop enclosing a decaying vortex is conserved even as every local measure of it falls. The wake gets weaker in the sense that its peak velocity falls, and it does not get weaker in the sense that its circulation goes anywhere.

Who found it, and when #

Albert Betz published the roll-up invariants in 1932, and the argument is a good example of a kind of reasoning that was characteristic of that group: rather than attempt an intractable dynamical calculation, find the quantities the dynamics cannot change and read the answer off them. Kaden had computed the self-similar roll-up of a semi-infinite sheet a few years earlier, and Betz’s step was to notice that the end state did not require it.

The πb/4 result then sat in the literature as a piece of theory for about forty years, and became operationally important very suddenly in the early 1970s — when wide-body aircraft entered service and a series of accidents to smaller aircraft following them made wake turbulence a regulatory problem. The separation categories in use today date from that period, and they are a rare case of a piece of 1930s vortex dynamics being converted directly into air traffic control procedure.

The theory did not change when it became urgent. What changed was that the aircraft got heavier and the traffic got denser, and a number that had been an elegant consequence of a conservation law became the thing that sets how many aeroplanes an hour a runway can accept.

Where the ladder goes next #

Every essay so far has treated a wing as a rigid object in a flow. It is not rigid: it bends, it twists, and the aerodynamic forces depend on how much it has bent and twisted. When those two motions couple, the wing can take energy out of the airstream — and the failure that follows is the fastest and most destructive in the subject, and is routinely explained by naming the wrong mechanism entirely.