The follower beats in time with the wake, not the bird

The lift beside a wing found that two wings flying tip to tip cost exactly half what they cost apart, because together they are one wing of twice the span. The bird at the point pays spread that saving along a V and found it shared unequally, and a fair V is a curved V bent the V’s arms until every member paid the same. All three had a quiet assumption in common. The birds glided. Each one’s wake was a steady pattern hanging in the air behind it, and each follower could place its wing in the upwash beside that pattern and leave it there.

Birds of the size that fly in V’s do not glide on migration; they flap, and a flapping wing’s wake is not a steady pattern. Its tip vortex is shed from the wingtip, and the wingtip is going up and down. What is left in the air is a wave: a vortex line laid along the path the tip traced, high where the tip was high and low where it was low, and stronger where the wing was working harder. The curved V’s last section asked what that does to a follower, and this essay answers it. The answer is a timing rule, and it is the rule a flock of ibises was measured following in 2014.

A wave left in the air #

The first figure is the geometry, drawn in the air rather than on the birds. The leader flies from right to left and its wingtip swings up and down; the air it leaves behind is at rest, so the path its tip traced stays where it was drawn, a sine wave whose wavelength is the distance the bird flies in one wingbeat. For a bird beating its wings a few times a second at ten or so metres a second, that is a few metres — two or three spans.

A follower some distance behind flies along the same line and meets, at every moment, the part of the wave the leader drew a little earlier: the part it drew when it was where the follower now is. If the follower flaps with a phase lag equal to the time the leader took to cover that distance, measured in beats — three-tenths of a beat for a follower three-tenths of a wavelength behind — its own wingtip traces exactly the same path, and it flies along the leader’s tip vortex all beat long. If it flaps half a beat off that, its tip traces the mirror image, crossing the leader’s vortex only twice a beat and spending the rest of it above or below.

The model: a wake frozen where it was drawn #

The calculation is the one the gliding essays used, cut down to what the flapping changes. Each bird is a horseshoe: a lifting line whose two trailing vortices sit $\pi b/4$ apart, the spacing that gives an elliptically loaded wing of span $b$ its lift and induced drag. Each tip vortex has a core, a twentieth of a span across unless stated, inside which its velocity stops growing, as a real vortex’s does. By Munk’s stagger theorem, the place of a wing along the stream does not change the pair’s total, but it does decide who is paid; the follower’s share of the interaction is computed in the Trefftz plane at its own position: the leader’s two tip vortices, at the height the leader’s tip had when it shed them, and the leader’s circulation at that moment, act on the follower’s wing at its present height and circulation. The integral of the downwash across the follower’s span has a closed form in logarithms.

Flapping enters twice. The leader’s tip height swings as $a\cos\omega t$, and its circulation swells on the downstroke and slackens on the upstroke, as $\Gamma_0(1 + \beta\sin\omega t)$ with $\beta$ a half. The follower does both with a phase lag $\varphi$. The wake is frozen — it does not sink, and it does not roll up further between the birds — and each instant is treated as steady, which asks that a beat be long compared with the time the air takes to cross a wing — the condition slow enough to be steady puts a number on. The saving is averaged over a beat and expressed as a fraction of what the follower would pay flying alone and flapping the same way.

In the upwash, beat with the wake #

The second figure is the answer. A follower placed with its inboard tip vortex on the leader’s outboard one — the placement the gliding essays found best — saves 84 per cent of its own induced drag when its phase matches the wake’s: exactly what a gliding follower saves in the same place. Flapping costs nothing, provided it is timed. Half a beat off, the same follower with a tip swinging a tenth of a span each way saves 35 per cent; with a swing of a fifth, 19. Most of the benefit of the formation is in the timing.

The saving is largest exactly in phase with the wake for a reason that has nothing to do with the details. Two things are being correlated: the follower’s height with the height of the vortex it meets, and its circulation with the strength of that vortex. Both are best matched when the follower does what the leader did at the place it now occupies, and both are matched exactly then. The search that found the best phase — a scan and a golden-section refinement, with no knowledge of the answer — lands on the wake’s phase to a hundred-thousandth of a radian at every spacing tried.

The saving is a fraction of induced drag, and induced drag is only part of what a flapping bird pays. A migrating bird flies somewhere between the speed for least power, where a parabolic drag polar makes induced drag three-quarters of the whole, and the speed for greatest range, where it is half. A follower in phase that saves 84 per cent of its induced drag therefore saves between two-fifths and three-fifths of its aerodynamic power; one half a beat off, with a swing of a fifth of a span, a tenth to a seventh. Measured on the bird rather than the air, the gains are smaller still, since the muscles’ efficiency and the body’s resting costs do not fall with the drag: Weimerskirch and colleagues found great white pelicans flying in formation in 2001 with heart rates about an eighth lower than when flying alone.

The rule has no aerodynamics in it #

The third figure turns the result into a rule. In the upwash, the best lag is 360 degrees for every wavelength behind: the follower should be as many beats behind the leader as the wake took to reach it. It is a statement about kinematics — a speed, a distance and a frequency — and the aerodynamics decides only its sign: in the upwash the follower wants to meet the vortex, so it copies the leader’s path.

Directly behind the leader the sign flips. There the follower sits in the leader’s downwash, which is a cost, and the best it can do is keep its wing as far from the leader’s wake as possible — to trace the mirror image of the leader’s path, half a beat off the wake. The best lag directly behind is the same rule plus 180 degrees.

Both halves are what Steven Portugal and his colleagues found when they tracked a flock of northern bald ibises on migration in 2014, with loggers fast enough to follow each wingbeat. Birds flying in the V’s upwash positions timed their beats so that each wingtip followed the path of the tip ahead — “spatial phase coherence”, in their words — and birds flying directly behind another flapped in antiphase with that path. The calculation here derives both from nothing but a frozen wake and a correlation, and does not know what a bird is.

The upwash is a narrow thing #

How much the timing matters depends on how narrow the upwash is, and the fourth figure shows it for a gliding follower moved up and down. With a core a twentieth of a span across, the saving is 84 per cent level with the leader’s vortex, 52 per cent a tenth of a span above or below it, and a third at two-tenths. The upwash beside a vortex falls off as the inverse of the distance from it, and a follower’s wing collects it over a span; move the wing off the vortex’s height and the part of the span nearest the vortex, where the upwash is strongest, is the part it loses first.

A fatter core spreads the upwash and lowers its peak: a core of a tenth of a span gives 56 per cent at the peak, and one of a fifth, 30. The core’s size is the least certain number in the calculation — a flapping bird’s tip vortex is not a steady line vortex with a sharp core, and its size depends on how the wing sheds its circulation — and it is the one the next figure shows mattering most.

How far out of phase a follower can drift #

The fifth figure asks how precise the timing has to be. For each tip swing it finds the phase error at which the follower is down to half its best saving. Below a swing of about twice the core size, the follower never loses half, whatever its phase: its wing never gets far enough from the vortex for the timing to matter much. Above that the window closes. With a core of a twentieth of a span, a swing of a fifth of a span leaves a window of 72 degrees either side of the right phase, and three-tenths of a span leaves 46. With a core of a fifth, the same swings leave 143 and 78.

So the tolerance is set by the swing against the core, and the core is the thing a bird cannot see. A large bird’s wingtip swings a large fraction of its span; whether that makes the timing critical or merely helpful turns on a vortex structure that has not been measured behind flying birds in formation. What the calculation can say is that the timing is worth most to the birds whose tips swing furthest, and that the window is tens of degrees — a fraction of a beat — rather than a few.

The swing a cruising bird chooses #

Whether a real bird sits in the narrow part of that figure is not a free question, because the swing and the wavelength are tied together by how birds fly. Flying animals of every size cruise with a Strouhal number — the wingbeat frequency times the tip’s peak-to-peak excursion, divided by the flight speed — between about 0.2 and 0.4, the band in which a flapping foil’s wake is most efficient at making thrust, and Taylor, Nudds and Thomas found birds, bats and insects clustered in it in 2003. The Strouhal number is also exactly the ratio of the tip’s peak-to-peak swing to the wake’s wavelength, since the wavelength is the speed over the frequency.

So a bird cruising at a Strouhal number of 0.3 with a wake two or three spans long swings its tip about 0.3 to 0.45 of a span each way. That is on the right of the fifth figure, where the window has closed to a few tens of degrees for any core thinner than a fifth of a span. The efficient way to flap and the need to time the flapping come together: a bird that beats at the Strouhal number its own propulsion prefers has made its tip swing large enough that the formation’s saving depends on the phase.

A wave that runs back through the flock #

Applied down a whole V, the rule has a striking consequence. Each follower lags the bird ahead of it by the time the wake takes to arrive, so the second bird lags the leader by its distance behind, the third lags the second by its distance behind that, and the phases add. Every bird is therefore at the phase set by its own distance behind the leader, whichever route through the flock is counted, and every wingtip traces the same path in the air. Seen from the ground, the flock’s wingbeats form a wave that runs from the point of the V to the ends of its arms at exactly the flight speed — so that in the air, where the wake is, nothing is travelling at all: each bird simply puts its wing where the one in front put its own.

That is a rule a bird can follow with information it has. It needs neither to see the leader nor to know the spacing, only to feel the upwash at its own wing and keep it at its best, and the phase that does that is the one in the rule. The same wake that carries the saving carries the instruction.

Directly behind, a smaller loss #

The sixth figure is the follower directly behind. Level with the leader’s wake, in its downwash, it pays 2.23 times its own induced drag on top of flying alone — more than three times the cost of flying solo, a position no bird would choose. In phase with the wake it meets that downwash all beat long, whatever its swing. Half a beat off, its wing passes above and below the leader’s wake and the penalty falls: to 1.13 of its own drag at a swing of a tenth of a span, 0.79 at a fifth and 0.48 at four-tenths. No phase and no swing turns the place into a gain.

That is what makes the ibises’ antiphase flapping in line interesting. It is not a way of using the position; it is a way of surviving it — of paying less for a place that a bird may occupy while changing position, or while the flock is reforming. The same rule that places a bird in the upwash tells a bird that has ended up behind another how to limit the damage.

What was checked #

The seventh figure is the ledger. The closed-form integral of a cored vortex’s downwash across a span agrees with direct quadrature of the vortex’s velocity to four parts in $10^{8}$. The searched best phase matches the wake’s own phase in the upwash, and the wake’s phase plus half a beat directly behind, at three spacings. With no flapping at all the saving does not depend on phase to fifteen figures, which is the gliding formation recovered. And for a small swing, the whole curve of saving against phase is predicted by a second-order expansion — the static saving’s curvature in height for the swing, the correlation of the two circulations for the lift — to a tenth of a per cent.

What a frozen wake cannot show #

The wake moves. A real wake sinks at the induced downwash and rolls up further between the birds; the follower meets the wave lower than it was drawn and a little changed. A steady descent shifts the best height, not the best phase, as long as it is uniform.

The wing is not rigid. Birds flex and fold their wings on the upstroke, shortening the span, so the tip path is not a sine wave and the wake’s two tip vortices are not always $\pi b/4$ apart. The rule survives any periodic path, since it only asks the follower to copy it; the numbers do not.

A beat is not slow. A wake wavelength of two or three spans makes the flapping quick enough that the lift arrives late on each stroke, and the circulation’s swing lags the motion. That changes $\beta$’s phase relative to the height, and the two correlated quantities then prefer slightly different lags.

The core is assumed. Every saving here depends on it, and the fourth figure is there to show by how much.

The convention: phase against the wake #

Phases are measured against the wake’s own phase at the follower’s position, $2\pi\Delta x/\lambda$, where $\Delta x$ is the streamwise distance behind the leader and $\lambda = U/f$ is the distance flown in one beat. A positive lag means the follower’s beat comes later. Savings are fractions of the follower’s own induced drag flying alone with the same flapping, so that zero is flying alone and one would be flying for free; the lift swing $\beta$ is a half and the tip swing $a$ is given in spans each way. Lengths are in spans.

Who found it, and when #

Lissaman and Shollenberger in 1970 put horseshoe vortices on a V of gliding birds and started the modern argument about formation savings. The possibility that flapping birds would have to time their beats to use the upwash was raised in the decades after, including by Willis, Peraire and Breuer, who simulated flapping formations; the measurement that settled what birds actually do is Portugal and colleagues’ of 2014, on northern bald ibises trained to follow a microlight, which found spatial phase coherence in the upwash and antiphase in line.

Still open: a follower that can hold only one of the two #

The rule asks a follower to do two things at once: to be at the right place in the V and to beat at the right phase for that place. A bird that changes its position within the flock changes its right phase with it, at 360 degrees for every wavelength it drops back. The next calculation lets the follower’s streamwise position wander — as measured birds’ positions do, by a fraction of a span over a few beats — and asks what a bird should do if it can adjust its phase only slowly: whether it is better to hold the phase that suits its average position and accept the loss as it wanders, or to hold its position more tightly at the cost of the extra power that takes, and how that trade depends on the swing and the core.