The lift beside a wing

A wing leaves a velocity field behind it. Directly behind, between the two shed cores, that field is a downwash — that is the whole of why the wing has induced drag in the first place. Outboard of each tip it is an upwash, and the upwash is the useful part.

An aeroplane flying in the upwash beside another one meets its air at a slightly greater angle. Its own lift vector is tilted forward by that angle, and forward is thrust. It is the same arithmetic that makes induced drag exist, run with the sign the other way round.

The closed form at contact #

The tip-to-tip case is exact and worth doing in a line, because it is the whole result and everything else is a decay from it.

Two wings of span b, each carrying lift L, flying alone: each pays $L^2/(\pi q b^2)$, so together $2L^2/(\pi q b^2)$.

Now put their tips together and load the pair optimally. There is no longer any distinction between them in the Trefftz plane: the trace is a single line of length 2b carrying total lift 2L, and the optimum on it is elliptic, so the drag is

$$\frac{(2L)^2}{\pi q (2b)^2} = \frac{4L^2}{4\pi q b^2} = \frac{L^2}{\pi q b^2}.$$

Exactly half. The computation returns 0.4993 of the separate figure at forty segments per wing, and the gate requires it within two parts in a thousand.

That is a startlingly large number and it deserves to be doubted for the right reason, which is not the arithmetic. It is a ceiling: it assumes the pair is loaded as one wing would be, which means the inboard tips are carrying circulation and the loading is continuous across the join. Two aeroplanes flying at their own trim conditions are not doing that, and what they get is much less.

The decay, and the spacing that pays #

Pull the wings apart and the saving falls. The mechanism is the upwash field of a lifting wing, which outboard of the tip falls off roughly as the inverse of the distance, so the drag saving falls faster still.

By one span of separation between the tips there is very little left. By forty spans the gate requires there to be nothing: the saving must be under two per cent, and it is.

So the useful spacing is a fraction of a span, and the useful position is beside rather than behind. That is the geometric content of the whole subject, and it is the thing the drafting analogy gets wrong. A cyclist drafts by sitting in the wake. A wing gains by sitting beside it, and a wing sitting directly in the wake of another is in the downwash region and is worse off than it would be alone.

Where the upwash is, and how much of it there is #

It is worth having the field itself in view, because the whole subject is a statement about where a particular sign lives.

A lifting wing sheds vorticity into its wake and the wake induces a velocity field in the plane behind it. Between the two shed cores that velocity is downward — that is the induced angle every section of the wing itself feels, and it is why the wing has induced drag. Outboard of each core it is upward.

The magnitude falls off with distance from the core, and the useful quantity for a follower is the average upwash over its own span rather than the value at a point. A follower with a long span straddles more of the gradient and averages more of it away; a short-span follower sitting right in the upwash gets more of the peak. So the benefit depends on the ratio of the two spans as well as on the offset, which is a second design variable the tip-to-tip closed form conceals by having both spans equal.

Why a V and not a line abreast #

The arithmetic above says nothing about the longitudinal positions, and by Munk’s stagger theorem it cannot: the induced drag of the pair does not depend on how far ahead or behind one of them is.

So why is a formation a V rather than a row?

Because the theorem is about the drag of a given loading, and not about what loading a given geometry produces. Two aircraft abreast are each in the other’s near field, where the upwash is not yet developed and where the bound vortices interfere directly; two in echelon, one behind the other and offset, are in each other’s fully-developed wake where the Trefftz-plane arithmetic applies. The distance downstream needed for the wake to become the wake is a couple of spans, and that is the depth of the V.

And because a formation has to be flown. A bird or a pilot holding station beside another one at a tip overlap of a fraction of a span needs to see the reference aircraft, needs room to correct, and needs not to be in its downwash if it drifts. The V puts each participant behind and outboard, where the gradient of the upwash is such that drifting outboard loses benefit and drifting inboard is resisted — a position that is stable to fly rather than one that has to be held.

That is a control argument rather than an aerodynamic one, and it is the reason the observed spacing in bird formations is looser than the optimum this arithmetic gives.

What birds actually get #

The measured numbers are worth putting beside the ceiling, and the gap between them is the honest result.

The theoretical maximum for a pair at optimum spacing is a halving of induced drag. Induced drag is a large fraction of a bird’s total at cruise — birds fly at low speeds and high lift coefficients, which is exactly where induced drag dominates — so the ceiling on total drag saving is large.

Measured savings in flying formations of large birds are in the range of ten to fifteen per cent of total power, which is a great deal and is far below the ceiling. Both facts follow from the arithmetic. The ceiling assumes an optimum loading across the pair; birds are individually trimmed and hold station to a wingspan rather than to a tip overlap; and the gradient near the optimum is steep, so a bird half a span out of position is getting a fraction of what is available.

The mechanism is real, the ceiling is real, and holding station is the whole difficulty. That is also why the effect has been demonstrated repeatedly in aircraft — where the position can be held automatically — and has never been adopted operationally, because two airliners a wingspan apart is an arrangement nobody will certify.

The scaling with the number of participants #

The pair result generalises, and the generalisation is the reason a V has more than two birds in it.

Treat a formation of n wings, each of span b, each carrying lift L, arranged tip to tip. The Trefftz plane sees one line of length nb carrying nL, so the total drag is

$$\frac{(nL)^2}{\pi q (nb)^2} = \frac{L^2}{\pi q b^2},$$

which is the drag of one bird flying alone — shared between n of them. Each bird therefore pays 1/n of what it would pay alone, and the saving is (1 − 1/n).

Two birds: half. Five: eighty per cent. Twenty-five: ninety-six per cent. The returns diminish quickly, which is why the marginal value of the twentieth bird in a formation is very small and the marginal value of the second is enormous.

That is the calculation behind the seventy-per-cent range increase quoted for large flocks, and it carries the same caveat as everything else here: it is the ceiling for a formation loaded as a single optimum wing, and no assembly of independently trimmed birds achieves it.

The overlap that beats contact #

There is a detail in the curve that is easy to miss and is a genuine result rather than a numerical artefact: the optimum is not at tip-to-tip contact but at a small overlap.

The reason is in the loading rather than in the geometry. With the tips slightly overlapped, the combined trace is a line with a short doubled section in the middle, and the optimum loading on it puts a little negative circulation where they cross. That is a physically odd thing for two aeroplanes to do — one of them would have to be carrying down-load on its inner wing — and it is the arithmetic saying that the ideal single wing of that width would not have a step in the middle.

Which is worth stating as a caution rather than as advice. The optimum on a trace is the optimum for a lifting system free to load itself as it likes, and two aeroplanes are not free: each has to trim, each has to carry its own weight, and neither can carry negative lift on one wing to improve the pair’s total. The unconstrained optimum is a ceiling and the constrained one is lower.

What the model does not contain #

No wake roll-up. The trace is a straight line that stays where it is. A real wake rolls into two cores within a few spans, and a following aircraft at any realistic distance is flying beside a rolled-up vortex pair rather than beside a flat sheet. The induced drag of the leader is unchanged by the roll-up; what the follower experiences is not.

No decay, no atmosphere, no turbulence. The upwash a follower sits in is the leader’s undecayed wake. Atmospheric turbulence and the wake’s own instability destroy it in tens of seconds, and at airliner separations there is nothing left.

Each wing is loaded optimally as part of the pair, which no independently trimmed aircraft is. Every number here is a ceiling.

No trim, no control and no station-keeping. The whole practical subject is station-keeping and none of it is here.

And no vertical offset. Real formations are stepped in height as well as laterally, which changes the trace and which the same arithmetic would take. It is not asked here because the essay’s argument is about the lateral dimension.

There is a benefit behind, and it is not shelter either #

The essay refutes the drafting picture on the grounds that the region directly behind a wing is a downwash. That is right for a steady wake, and it leaves out a real mechanism that operates in the same position for an entirely different reason — one that arrives with the wingbeat phasing the ibis measurements found and that the steady theory cannot express.

A flapping wing does not shed a flat sheet. It sheds a street of discrete vortices, alternating in sign, convecting downstream at nearly the flight speed and carrying organised kinetic energy. A body sitting in that street is not in calmer air — it is in a strongly structured field whose sign reverses at the wingbeat frequency — and whether that field helps or hinders depends entirely on when the follower does what.

Get the timing right and the follower’s own motion is arranged so that it meets each vortex on the side where the induced velocity increases its effective incidence, and the energy comes out of the wake rather than out of the follower. Get it wrong by half a cycle and every one of those encounters costs. It is a resonance rather than a shelter, and the quantity that decides it is a phase, which does not appear anywhere in a steady calculation.

Two measurements make the point concretely.

Trout holding station behind a cylinder in a stream adopt a distinctive slalom — a Kármán gait, timed to the shedding — and their muscle activity, measured directly, falls well below what free-stream swimming at the same speed costs. They are not hiding in the wake’s velocity deficit; the deficit alone would be a poor bargain, since the fish must still hold station against the stream. They are extracting from the vortices, and the tell is that the gait’s frequency locks to the shedding frequency rather than to anything about the fish.

And the ibis flock that was instrumented in 2014 was found doing the same thing in the air. Birds in the outboard position phased their wingbeats to match the moving upwash of the bird ahead — and, more tellingly, a bird that found itself directly in trail flew in anti-phase, which is what a bird extracting from an alternating wake rather than sitting in a steady upwash would do.

So the honest statement is that a wake offers two quite different things and neither of them is shelter. Beside it there is a steady upwash, which the whole of this essay computes, and which needs only position. Inside it there is organised unsteady energy, which needs position and timing, which no steady theory contains, and which is available to a flapping animal and not to an aeroplane. That the birds turned out to be exploiting both is the reason the 1914 arithmetic got the geometry right and still understates what a flock achieves.

What an aircraft would have to do #

The effect has been demonstrated in flight more than once and has never entered service, and the reasons are worth listing because they are all outside the aerodynamics.

Station-keeping to a metre. The benefit lives in a band a fraction of a span wide with a steep gradient across it. A human pilot cannot hold that; an automatic system referenced to the leader’s position can, and the flight demonstrations have all used one.

The wake is not steady. A leader’s wake meanders with the atmosphere, so the band the follower has to sit in moves. The control problem is tracking a moving optimum whose position is not directly observable — it has to be inferred, usually from the follower’s own response.

And the follower is riding an unstable field. Just inboard of the useful position is the downwash region and the rolling moment that goes with it. A station-keeping failure that drifts the wrong way does not merely lose the benefit; it puts an aircraft into a wake encounter.

Set against that, the prize is a percentage of fuel on the longest and most expensive flights there are, and the arithmetic that says how much has been available since 1914. The obstacle has never been the mechanism, and it is not a small obstacle.

Who found it, and when #

Wieselsberger worked out the formation-flight saving in 1914, in the same Göttingen group and within a few years of the lifting line itself, and the paper was explicitly about birds. It is one of the earliest applications of the theory to something other than an aeroplane, and it got the mechanism right immediately: upwash beside the tip, not shelter behind.

Lissaman and Shollenberger’s 1970 paper in Science is the one usually cited, and it computed the optimum for a formation of many birds and found the saving scaling with the number of participants — a flock of twenty-five in a single V having, in principle, a range seventy per cent greater than one bird alone.

Measurement took another forty years. Portugal and colleagues instrumented a flock of northern bald ibises in 2014 and found the birds positioning themselves in the predicted upwash region and phasing their wingbeats to match the moving upwash of the bird ahead — which is a refinement nobody had asked for and which none of the arithmetic above contains, because all of it is steady.

That is the pattern worth taking away. The steady theory got the geometry right in 1914 and the mechanism right immediately, and the thing the birds turned out to be doing was an unsteady one that the steady theory has no vocabulary for at all.

Where the ladder goes next #

Every calculation in this essay treated the wake as a flat sheet lying where the wing left it. It does not stay there: within a few spans it rolls up into two concentrated cores, and where those cores end up is fixed before the roll-up starts — by an argument about what a roll-up can and cannot change.