A fair V is a curved V

The bird at the point pays for the V puts nine trimmed birds — each carrying its own weight on its own elliptically loaded wing — into a straight V, and divides the flock’s induced drag among them. Munk’s stagger theorem fixes the total: however far ahead or behind any bird flies, the flock together pays 0.424 of what nine birds would pay flying alone, a number set entirely by their spacing across the flight path. The sweep of the V only moves the cost around. Swept back steeply, the leader pays most; nearly abreast, the leader and the birds at the tips pay most. At the fairest straight angle, 7.5 degrees, the leader and the two tip birds pay 0.554 and the birds between them about 0.35.

That essay ends on Dietrich Hummel’s answer, from 1983: if every bird is to pay the same, the V must be curved, with the arms bending back more steeply towards their tips. This essay solves for that V — every position an unknown, every share required to be equal — and asks what shape it is, what each bird then pays, and whether fairness has a price.

The problem, and why it has a solution #

Each bird is a lifting line one span wide, carrying the same elliptic loading as every other, built from 60 horseshoe vortices whose velocities come from the exact Biot–Savart formula for a straight segment, with wakes that trail straight downstream. The birds sit one span apart across the flight path, wingtip to wingtip. What is free is how far behind the leader each flies. A symmetric V of nine has four such distances, one for each rank on an arm, and four conditions: each rank’s share of the induced drag must equal the leader’s.

Four equations in four unknowns need not have a solution, but here there is a reason to expect one. Munk’s theorem says the total does not depend on the fore-and-aft positions, so moving a bird backwards can only transfer drag between birds, never create or destroy it. Every rank has a lever — its own position — and the levers act on the shares in different proportions. Newton’s method, started from the straight V at 7.5 degrees and using a Jacobian built by finite differences, converges in five steps, to share differences below $10^{-10}$.

What each bird pays #

The first figure is the answer by rank. In a straight V swept 30 degrees the leader pays most and the share falls towards the tips. In a straight V swept 7.5 degrees the leader and the tips pay most and the middle of each arm least. In the solved V every rank pays the same, and the number it pays, with the panels extrapolated to fine resolution, is 0.424 of flying alone — exactly the flock’s average in both straight Vs.

That equality is the check the calculation passes and did not impose. Newton’s method was asked only to make the shares equal to one another, not to any particular number; that the common value comes out as the straight Vs’ average, to six parts in $10^{17}$, is Munk’s theorem appearing unbidden. The total over the flock was also computed directly for straight Vs at zero, 7.5 and 30 degrees and for the solved curved V, and the four agree to rounding.

So fairness costs nothing. The flock that shares its work equally pays exactly what the flock that makes its leader pay half as much again pays, because the fore-and-aft positions are a way of dividing the saving and not of making it.

The shape of a fair V #

The second figure draws it. The first birds behind the leader fly almost abreast of it, 0.028 of a span back. The next rank is 0.16 spans back, the next 0.47 and the tip birds 1.09. Measured as the angle of the line joining each bird to its neighbour nearer the apex, the arm is swept 1.6, 7.6, 17.2 and 31.9 degrees: it leaves the leader nearly straight across and bends back ever more steeply. That is Hummel’s curved V, recovered without being asked for.

The curve has a plain reason. A bird’s share depends on how much of its neighbours’ upwash it flies in, and on which side. The leader has a neighbour on each side but is ahead of both, and in a flat-wake model a bird ahead of another gains less from it than a bird beside it does; the leader is helped most by neighbours abreast of it. A tip bird has a neighbour on one side only and so gains least, and it can be brought down to the average only by giving it more of that one neighbour’s upwash than the neighbour gets back — which a bird well behind its neighbour receives, by the stagger half of Munk’s theorem: a bird behind its neighbour gains what the neighbour loses. The ranks between need progressively more of that transfer the closer they are to the tips, and each additional increment of stagger buys less of it, so the arm curves.

Two birds, and what stagger does #

The whole mechanism is visible with two birds, and the numbers make it concrete. Side by side, wingtip to wingtip, each flies in the other’s upwash and each pays 0.727 of what it would pay alone. Move one of them half a span behind the other and the pair’s total does not change, but its division does: the bird in front pays 0.920 and the bird behind 0.535. A full span back, 0.962 and 0.492; two spans back, 0.987 and 0.467. The sum is 1.454 throughout, to rounding.

The bird in front loses because it has moved ahead of the upwash its partner’s bound vortex makes, which is strongest beside the partner and weaker ahead of it; the bird behind gains because it has moved into the full upwash of its partner’s trailing vortex, which has developed by the time it arrives. Munk’s theorem is the statement that the two changes are equal and opposite, and it holds because both are the same interaction seen from either end: the drag one lifting line induces on another plus the drag the other induces back depends only on their lateral separation.

A V is nine of these pairings at once. The leader is in front of everybody, so every pairing it takes part in shifts drag onto it; the tip birds are behind everybody on their side, so every pairing shifts drag off them. That is why a straight V swept back steeply makes the leader pay most. The fair V uses the only freedom there is — how far behind its neighbour each bird sits — to balance those transfers, and since each transfer’s size depends on the stagger in a way that saturates, the arms have to bend back further and further to keep finding more.

The bigger the flock, the harder the arms bend #

The third figure repeats the calculation for flocks of five, seven, nine and eleven. Every one leaves the apex nearly abreast; the last segment of the arm is swept 16.6 degrees in a flock of five and 36.8 in a flock of eleven. A larger flock has more birds in the middle of each arm gaining from neighbours on both sides, so the tip birds have further to fall behind to be brought up to the average.

This is also a test the shape could fail: if the equal-share solution were an artefact of nine birds, the family would not be smooth. It is. The curves nest, and the sweep at each segment grows steadily with the number of birds behind it.

What an equal share is worth #

The fourth figure gives the equal share as the flock grows, extrapolated to fine panels: 0.613 of flying alone for three birds, 0.507 for five, 0.455 for seven, 0.424 for nine and 0.404 for eleven. Each bird in a larger flock pays less, because a larger fraction of the flock flies with neighbours on both sides, and the gains fall more slowly the bigger the flock is. The leader of a straight V at 7.5 degrees pays 0.652 in a flock of three and 0.545 in a flock of eleven; the gap between it and the fair share widens from 0.04 to 0.14 as the flock grows, because the straight V’s leader has neighbours abreast but a straight V never places them where the leader gains most.

Only the spacing across the path matters #

The fifth figure changes the one thing Munk’s theorem says does matter: the spacing across the flight path. With wingtips touching, nine birds pay 0.424 each. Open a gap of a tenth of a span between neighbours and the share rises to 0.630: the gap gives away 36 per cent of the saving, because the upwash just outside an elliptically loaded wing is strongest at its tip and falls off steeply outward. At a gap of half a span each bird pays 0.84, and most of the benefit of flying in company is gone.

The steepness is the practical content of the whole calculation. A bird that gets its fore-and-aft position slightly wrong changes who pays but not what the flock pays; a bird that drifts sideways by a tenth of its span changes what everyone pays. Munk’s theorem says where the precision has to go.

Overlapping wingtips cannot be computed this way. With flat wakes a neighbour’s trailing vortex would cross a wing’s bound vortex, and the model becomes singular; real wakes roll up into cores within a span or two, which is what lets real birds overlap, and which is the calculation the lead of the earlier essay asked for and this one does not do.

How precisely a bird must hold its place #

The solved positions are exact for the model, and a real bird can hold them only approximately. The calculation can say which errors matter. Move the first bird behind the leader back by a tenth of a span from its fair position, and its share falls from 0.411 to 0.281 while the leader’s rises to 0.540: a tenth of a span of drift near the apex moves an eighth of a lone bird’s drag onto its neighbour. Move a tip bird back by the same tenth and its share changes by 0.012. Near the apex, where the birds fly almost abreast, stagger is a sensitive lever; at the tips, where they are already well staggered, it is a weak one. The flock total does not move in either case — Munk again.

Sideways is a different matter. Open the lateral spacing of the fair V by five per cent of a span, leaving every fore-and-aft position alone, and every share rises, to between 0.52 and 0.56 at this panel count, against a flock average of 0.41 before. A small lateral error costs the whole flock, fore-and-aft errors only move the cost from one bird to another. A bird that has to choose what to be precise about should be precise about how far it is from its neighbour’s wingtip, and it can afford to be sloppy about how far behind it is — except next to the leader, where sloppiness is paid for by the leader.

Why the fore-and-aft positions cannot matter #

It is worth saying once why Munk’s theorem holds, since the whole essay leans on it. The induced drag of any system of lifting surfaces is the kinetic energy the system leaves behind in the air per unit distance flown, and that energy can be computed far downstream, in a plane across the wake — the Trefftz plane. There, the wake of every bird is a sheet of trailing vorticity, and the energy depends only on where those sheets are in the plane and how strong they are. How far ahead or behind a bird flew when it laid its sheet down does not appear, because by the time the sheets reach the plane they are all simply side by side. A wing that leaves the plane uses the same argument to show that a winglet works by reshaping that cross-section, and the argument here is its opposite: the fore-and-aft positions are invisible in the one place the drag is decided.

The argument depends on the wakes being flat and fixed. Real wakes roll up and descend, and a bird several spans behind flies beside a rolled-up core rather than a sheet, which is where the flat-wake shares stop being exact. A rolled-up pair of cores also wanders, and bends until it links, though over distances of hundreds of spans that a flock never spans.

What the birds do #

Measurements now exist to set against this. In 2014 Steven Portugal and his colleagues fitted a flock of northern bald ibises, trained to follow a microlight on migration, with GPS loggers and accelerometers, and found them flying at positions close to those that theory places in the upwash of the bird ahead, and timing their wingbeats so that each wingtip followed the path of the one in front — flapping, in a sense, in phase with the neighbour’s wake. Flocks of large birds are commonly seen with arms that bend back towards their tips, and flocks change leaders; neither observation says, on its own, which of the explanations is at work.

What the calculation says is that a flock does not need to rotate its leader to share the work, since a curve can share it for free; and that a flock that does rotate its leader, and holds a straight V, pays exactly as much as one that curves. The choice between them is about something the lifting-line model does not contain — how precisely a bird can hold a position that is optimal only to within a tenth of a span, in a rolled-up, flapping wake.

What was checked #

The sixth figure lists them: every share against the flock’s average, which Munk’s theorem requires and Newton’s method did not impose; the total for the curved V against straight Vs at three sweeps; Newton’s residual; the positions. The lifting-line method itself was checked in the essay that built it, against the lone elliptic wing’s closed-form drag, against the Trefftz-plane downwash, and against Munk’s theorem pair by pair.

What the picture cannot show #

Flat wakes and fixed wings. The wakes trail straight back and never roll up, and the wings do not flap; the positions are for gliding birds with the lifting-line wake, which is exact only for small lift coefficients.

Trimmed birds with one loading. Each bird carries its own weight on the same elliptic loading, which is not what a bird’s wing carries: a flapping wing, with its bending constrained, may be closer to the bell-shaped loading, whose tips carry less and so leave a weaker upwash for a neighbour. Loading a wing for the flock rather than for itself would do better still, and a bird cannot.

One lateral spacing. Every neighbour is the same distance across the path. Letting the lateral spacing vary by rank as well gives more freedom and a cheaper flock only by closing the gaps, which the flat-wake model cannot follow past tips touching.

The convention the numbers depend on #

Shares are each bird’s induced drag as a fraction of the same bird flying alone. Distances are in spans. Sweep is the angle of the line from a bird to its neighbour nearer the apex, measured back from the line across the flight path. The fine-panel shares are extrapolated from 40 and 80 horseshoes per wing by Richardson’s rule, the discretisation’s error falling as one over the panel count; the positions change by a tenth of a per cent between 40 and 120 panels and are quoted at 60.

Who found it, and when #

Max Munk proved the stagger theorem in 1919, working on biplanes, and two wings and it does not matter where follows it through. Lissaman and Shollenberger’s paper of 1970 estimated the saving of a V of birds, and Dietrich Hummel’s of 1983 computed the forces on each member and proposed the curved V for equal shares. The ibis measurements are Portugal and colleagues’, published in 2014.

Still open: the flock that flaps #

A gliding flock can place its birds in each other’s upwash and leave them there. A flapping flock cannot: each bird’s wake is a wave, rising and falling with its wingbeat, and a bird behind it gains only if it flaps in the right phase with the wave arriving at its wings. The next calculation gives each member a wingbeat — a lift that varies sinusoidally in time along a wake that carries the variation downstream at the flight speed — and asks, for each fore-and-aft spacing, which phase of flapping minimises the follower’s drag, and whether the phase the ibises chose is the one that falls out.