The bird at the point pays for the V

The lift beside a wing finds the ceiling for a flock. A row of $n$ wings tip to tip, loaded as a single optimum wing, pays in total what one wing pays alone, so each member pays $1/n$ of its lone induced drag. And Munk’s stagger theorem says the total cannot depend on how far ahead or behind any member flies, because induced drag depends only on the wake’s cross-section far downstream, and moving a wing along the stream moves nothing in that cross-section.

Both results are about a total. A flock is not one wing and it does not share a bank account. Each bird carries its own weight and trims its own wing, and what decides whether a position in a formation is worth flying is what that position costs the bird in it. That is a question the total cannot answer, and it has a sharp answer.

Nine lifting lines and every vortex between them #

Each member is a lifting line one span wide carrying the same elliptic loading, $\Gamma(y) = \Gamma_0 \sqrt{1 - (2y/b)^2}$, which is what a single wing trimmed for least induced drag carries. The loading is cut into horseshoe vortices — a bound segment along the wing and two trailing legs running downstream — and the vertical velocity each segment induces at each point of every other wing is evaluated from the exact Biot–Savart formula for a straight vortex segment. A member’s induced drag is its circulation times the downwash at its own bound line, integrated along its span, with the downwash summed from every vortex in the formation, its own included. It is reported as a fraction of what the same member pays flying alone.

Members sit one span apart across the stream, tips just touching, which is the closest a flat-wake model can place them without their wakes landing on one another. The sweep of the V is the angle each arm makes with a line abreast: at zero degrees the flock is a row, and at forty-five each member flies one span behind the one inboard of it.

Two wings, one saving, and a stagger that moves it #

The pair is where the division first shows, and it shows as a transfer.

Side by side, each wing saves 27.3 per cent of its lone induced drag. The optimally loaded pair of the essay before saves fifty. The difference is the price of each wing keeping its own elliptic loading rather than distorting it to suit its neighbour, and it is a large price, which a later section returns to.

Move the second wing behind the first and the saving does not change in total — the two shares sum to the same number at every stagger, to a part in $10^{15}$, which is Munk’s theorem verified by a calculation that knows nothing about it. But it changes hands almost completely. One span behind, the rear wing saves 50.9 per cent and the front wing 3.8. Four spans behind, the front wing gets less than one per cent.

Why the saving runs downstream #

The shares follow from what a lifting line’s field looks like to a neighbour ahead and to one behind.

The trailing legs are the source of the useful upwash beside a wing, and they run only downstream from the wing that sheds them. A neighbour behind flies alongside the fully developed trailing vortices and feels their whole upwash. A neighbour ahead sits beside legs that begin behind it; it feels less than half of what it would feel alongside them, falling towards nothing as the stagger grows.

The bound vortex adds a second contribution, and it is antisymmetric. A lifting line induces upwash ahead of it and downwash behind it, equally. So the rear wing’s bound vortex helps the front wing a little, and the front wing’s bound vortex costs the rear wing the same amount. The antisymmetric part cancels in the sum and adds to the transfer.

Munk’s theorem is the bookkeeping that says the two effects together move exactly as much benefit backwards as they take away from the front. Behind the arithmetic sits a physical asymmetry: a wing’s upwash is stored in its wake, and the wake lies downstream. A bird ahead can give; it can hardly receive.

A flock of nine, divided three ways #

In a line abreast every member is beside every other, at zero stagger, and the shares depend only on how many neighbours a member has and how near they are. The centre bird has four on each side and pays 0.340 of its lone drag. The two end birds have neighbours on one side only and pay 0.657. The saving is lopsided even with no sweep at all, and it is lopsided in favour of the middle.

Sweep the line back into a V and the transfer of the pair acts on every neighbour at once. Each member now has neighbours behind it on its inboard side and ahead of it on its outboard side, or the reverse, and it gains from the ones ahead and loses to the ones behind. The two ends of the line, which were the losers abreast, become the rearmost members of the V and collect from everyone ahead of them. The centre bird, which was the winner abreast, becomes the leader and has nobody ahead of it at all.

At a sweep of forty-five degrees the leader pays 0.894 of its lone drag — it saves under eleven per cent — while every other member pays between 0.346 and 0.390. The flock’s average is 0.4244, as it was abreast and as it is at every sweep.

The fairest V is nearly a straight line #

Two members compete for the worst position. Abreast it is the ends; in a V it is the point. The leader’s share rises with sweep and the ends’ falls, and they cross at a sweep of 7.5 degrees, where both pay 0.554 and nobody pays more. That is the V that minimises the worst member’s cost, and it is barely a V: each arm trails back one span for every seven and a half spans outboard.

The Vs seen in the sky are far sharper. An arm swept forty-five degrees makes a V whose opening is a right angle, and a skein of geese photographed from below rarely looks much flatter than that. On this calculation such formations are distinctly unfair to their leader, who at a sweep of sixty degrees pays 0.953 of its lone drag — within five per cent of flying alone — while it opens the air for everyone else.

A line abreast is also unfair, to its two ends, and the obvious question is why nobody flies the fair shallow V. The answer the essay before gives for why flocks fly Vs at all applies here too: a formation has to be held, and a bird holding station a span beside another needs to see it, needs room to correct, and needs not to drift into its downwash. A steep V puts every bird behind and outboard of the one it follows, where drifting inboard is resisted and the leader is in view. The shape is chosen by station-keeping, and the aerodynamics decides who pays for that choice.

Taking turns at the front #

If the leader pays most, the calculation predicts something about behaviour: a flock whose members are not related should not tolerate a permanent leader. The prediction has been tested. A 2015 study of northern bald ibises, a species reintroduced to migration by being trained to follow microlight aircraft, tracked individual birds through long formation flights and found them swapping positions, with pairs of birds matching closely the time each spent ahead of the other — a direct reciprocity in which the costly position is traded.

The numbers here say how much is being traded. In a V swept forty-five degrees the front position costs roughly two and a half times what a position behind it costs in induced drag, and induced drag is a large part of a large bird’s cruising power. A bird that led permanently would spend a substantially larger share of its energy than any follower for the whole of a migration. Nothing else about the formation creates so large a difference between its members, which is presumably why it is the thing they are seen to negotiate.

Two airliners, three kilometres apart #

The limit of large stagger is not a curiosity; it is the only formation airliners have flown in flight trials. Two aircraft cannot be flown a wingspan apart abreast, and the arrangement that has been tested puts the follower a few kilometres behind the leader and slightly outboard, positioned so that one wing sits in the upwash beside one of the leader’s rolled-up trailing cores. The reported savings for the follower were of the order of five per cent of its fuel.

The pair figure above says who gets that. At a stagger of four spans the front aircraft’s share of the saving has already fallen below one per cent, and at a stagger of fifty spans it is indistinguishable from zero. The follower collects the whole of the mutual saving and the leader collects nothing, so an airline pairing two of its own aircraft gains, while a pairing of two airlines is a transfer from one to the other unless they alternate the lead. It is the ibises’ negotiation again with a contract in place of a wingbeat.

The same limit sets how far behind the follower can usefully fly. Its benefit lives in the leader’s wake, and the leader’s wake does not last. In still air the rolled-up pair bends and pinches itself into rings within two or three minutes, which at cruising speed is thirty to forty kilometres; before that its cores have wandered, and the follower must track a moving target. The upwash a follower can find is the upwash of a wake still young enough to be the pair that the wing’s loading predicts, and the stagger at which a formation is flown is bounded on one side by the leader’s safety and on the other by the wake’s lifetime.

What the airliner trials and the birds share is the geometry of the benefit: beside the wake, never in it. Directly behind a wing is its downwash, which is the price of having ends paid again by anyone who flies there, and the useful region is a band outboard of each core whose width is a fraction of the span. Holding station in that band is the practical difficulty for an aircraft as it is for a bird, and the division computed here decides who is paid for managing it.

What trimming alone costs the flock #

The ceiling, one minus one over $n$, is reached only if the whole formation carries the loading of a single wing of $n$ spans, which would require each bird to unload its outboard wing near its neighbour’s tip and load it heavily elsewhere, differently in every position. A bird trimmed alone does not do that, and the calculation says what the omission costs.

A pair of trimmed wings saves 27.3 per cent of the induced drag where the optimum pair saves 50 — 55 per cent of the ceiling. Nine save 57.6 per cent against a ceiling of 88.9, and twenty-five save 65.5 against 96. The fraction kept rises slowly with the size of the flock, because a member in the middle of a long row has neighbours on both sides whose upwash fills in the dip its own elliptic loading leaves at each tip. The seventy-per-cent increase in range often quoted for a flock of twenty-five was computed for the ceiling, and two-thirds of that saving is the more honest expectation even before station-keeping errors are counted.

The wake, checked against its closed form #

The whole calculation rests on the downwash one wing induces at another’s bound line, and that has an independent check at zero stagger.

Far downstream, the downwash an elliptic wake induces outside its span is $(\Gamma_0/b)(1 - |y|/\sqrt{y^2 - b^2/4})$, a closed form, negative everywhere outside the tip, which is to say an upwash. At the wing’s own bound line the trailing legs are only half developed and the bound vortex induces nothing sideways, so the downwash there must be exactly half the far-field value. The horseshoe sum reproduces it to $4.8 \times 10^{-6}$ at points from just outside the tip to three spans away.

The lone wing’s drag is the other check. With 240 horseshoes it comes out half a per cent low, the error of a piecewise-constant loading, which falls as one over the number of horseshoes. Every formation number above is extrapolated from two resolutions to remove that error, and the same extrapolation brings the lone wing to $5 \times 10^{-6}$ of the closed form $L^2/\pi q b^2$.

What the picture cannot show #

Flat wakes. Every trailing vortex runs straight downstream from the wing that shed it, in one plane. Within a few spans a real wake rolls up into two cores at $\pi/4$ of the span, and a bird several spans behind is flying beside cores, not a sheet. The roll-up moves the upwash inboard, which is why real birds overlap their tips with the bird ahead rather than just touching them, and it changes the division between near and far members in a way this model cannot follow.

Steady wings. Birds flap, and a flapping wing’s wake is an alternating train of vortices whose useful part depends on the phase of the follower’s wingbeat; the essay before describes birds timing their beats to it. None of that is here.

Induced drag only, and no rolling moment. A bird at the end of an arm has upwash on its inboard wing and none on its outboard one, so it feels a rolling moment it must trim out with some cost of its own. The fractions are of induced drag alone; profile drag and the power of flapping are unchanged, so a bird’s total saving is smaller than its share here.

Identical birds. Every member has the same span and carries the same lift. A flock of mixed sizes redistributes the saving further, towards the birds that sit beside larger neighbours.

The convention the numbers depend on #

Sweep is measured from a line abreast, so zero is a row and ninety would be single file. Members are one span apart across the stream, centre to centre, so their tips touch; a V swept at angle $\Lambda$ puts each member one span times $\tan\Lambda$ behind the one inboard of it. Each share is the member’s induced drag as a fraction of its drag flying alone with the same loading. “The ceiling” is the formation’s saving if it were loaded as one optimum wing of its total span, which is $1 - 1/n$.

Who found it, and when #

Carl Wieselsberger computed the saving of a pair of wings flying beside each other in 1914, from the lifting-line theory Prandtl’s group had just produced. Max Munk proved in 1923 that stagger leaves the total unchanged. Lissaman and Shollenberger’s 1970 calculation of a twenty-five-bird formation gave the famous figure for range, for the optimum loading. Dietrich Hummel showed in 1983 that the shares in a straight V are unequal and that a formation in which every bird pays the same needs a V that is curved rather than straight. Portugal and colleagues measured ibises phasing their wingbeats in 2014, and Voelkl and colleagues found them trading the lead in 2015.

The same division problem appears whenever two lifting surfaces fly in tandem. A keel and a rudder share their induced drag in exactly this way, with the rudder, which flies in the keel’s wake, paying for the keel’s; and a biplane’s staggered wings have the same total at every stagger and a different split at each.

Still open: a V in which everybody pays the same #

A straight V has one fairest angle and it is not fair: at 7.5 degrees the leader and the two ends each pay 0.554 while the members between pay about 0.35. A formation in which every member pays exactly the same must give each member its own stagger and its own spacing, and Hummel’s answer was a curved V — the arms bending back more steeply towards their tips.

The calculation that follows makes each member’s position an unknown, asks for the positions at which every share is equal, and reports what that equal share is and how far it falls below the straight V’s average. It should be done with rolled-up wakes rather than flat ones, since the members several spans back fly beside cores — which also lets the tips overlap, as real birds’ do, and asks at last how much of the optimum pair’s half a flock of trimmed birds can recover by where it sits rather than by how it loads its wings.