Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Formation — where it appears

A group of wings or birds flying so that each sits in the upwash of the others' trailing vortices and needs less power for its lift. The saving is the mutual induced drag, and how it is shared out depends on the arrangement rather than being equal.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Tip to tip, two wings cost exactly half of what they cost apart. The induced-drag saving of a pair of wings against the gap between their tips, as a fraction of what the two pay flying alone. With the tips touching the pair is one wing of twice the span carrying twice the lift, and the arithmetic of that is exact: the drag halves, and the computation returns 0.499257 of the separate figure. Pull them apart and the saving falls away with the square of the distance. Birds fly in a V because the tips are the part worth overlapping, and the spacing that pays is a small fraction of a span rather than any distance a formation could hold by eye.

The lift beside a wing

Fly two aeroplanes with their wingtips touching and the pair costs exactly half what the two cost apart. Not approximately half — the arithmetic is a closed form, because two wings tip to tip are one wing of twice the span, and induced drag goes as the square of it.

circulation · Formation A V seen from above, and what each member pays. Nine wings in a V swept 45 degrees, tips one span apart, each carrying the same lift, seen from above with the flight direction up the page. Beside each is its induced drag as a fraction of what it would pay flying alone. The leader at the point pays 0.89; every other member pays between 0.35 and 0.39. The saving has flowed backwards through the formation.

The bird at the point pays for the V

A flock's saving in a V is fixed by where its members sit across the stream, and the angle of the V cannot change it by a single per cent. What the angle changes is who gets the saving. It flows backwards through the formation, so that in a V swept forty-five degrees the leader pays nine-tenths of what it would pay alone while every bird behind it pays under four-tenths.

circulation · Formation The V in which every bird pays the same is curved. Nine birds one span apart, flying up the page, in plan: each short line is a wing, its height its distance behind the leader in spans. The straight V at 7.5° — the fairest straight V — and the V whose members' positions are solved so that every one pays exactly the same share of the induced drag. The equal-share arms leave the leader almost abreast and bend back ever more steeply: 1.6°, 7.6°, 17°, 32° from the apex to the tips.

A fair V is a curved V

In a straight V of birds somebody always pays more than somebody else: at the fairest angle the leader and the two birds at the tips pay half again what the birds between them pay. The V in which every bird pays exactly the same can be solved for, and it is not straight. Its arms leave the leader almost abreast and bend back ever more steeply, to thirty-two degrees at the tips of a flock of nine. The share every bird then pays is the flock's average, fixed by Munk's theorem before any position is chosen — so fairness costs nothing, and the only thing that can make the flock cheaper is flying closer together sideways.

circulation · Formation

Named alongside it

The objects these essays reach for when they reach for this one.

Induced dragModel limitMunk's stagger theoremOptimisationSpan loadingThe Trefftz planeUpwashThe Biot–Savart lawBound vortexCirculationDownwashInduced velocity

All concepts