Concept

Bound vortex — where it appears

The vorticity attached to a lifting wing, whose strength is the circulation round the section. It cannot end in the fluid, so it turns downstream at the tips and is closed by the starting vortex — which is what makes a horseshoe of it.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

The starting vortex, and the circulation it pays for. A wing that has just begun to lift, and the vortex it shed as it started. The circulation round the wing and the circulation round the shed vortex are equal and opposite, so a circuit large enough to contain both has no circulation at all — which is what Kelvin's theorem requires of a circuit that began at rest.

The vortex a wing leaves behind

A wing at rest has no circulation. A wing in flight has a great deal. Circulation round a circuit of fluid particles cannot change, so the difference had to come from somewhere — and it did, as an equal and opposite vortex dropped on the runway.

circulation · Lift

The wall that pushes back

A wing near the ground carries more lift than the same wing in free air, and the usual explanation — a cushion of compressed air underneath — is not what the equations say. The ground is a mirror, and what changes is not the pressure under the wing but the circulation the wing is forced to carry.

circulation · Images

The flap moves the curve and leaves its slope alone. Lift coefficient against incidence for several flap deflections. The lines are parallel: deflecting the flap gives the section lift at an incidence where it had none, and does not change how much lift each further degree of incidence buys.

What a flap does, and what it does not

Lowering a flap gives a wing lift at an incidence where it had none. It does not make the wing more responsive to being pitched — the lift curve moves sideways and its slope does not change, and those are two quite different things to buy.

circulation · Lift

The lift does not arrive when the incidence does. Bound circulation against distance travelled, in units of the settled value. The wing starts far short of its final lift and takes tens of chords to collect the rest, because every scrap of circulation it takes has to be paid for with an opposite vortex shed behind it, and that vortex's own downwash holds the wing back until it is far away. Wagner's exact 1925 answer is drawn beside the model in the colour this site keeps for a borrowed claim: the shapes agree and the model is slower, by about a fifth at its worst.

The lift that arrives late

A wing set into motion does not have its lift. Every scrap of circulation it takes has to be paid for by shedding an equal and opposite vortex behind it, and until that debt is far downstream its own downwash holds the wing back — for tens of chords, not for an instant.

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

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

Named alongside it

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

Circulation Downwash Kelvin's circulation theorem Kutta condition Lift coefficient Lift curve slope Starting vortex Thin-aerofoil theory The Biot–Savart law Boundary condition Camber Formation

All concepts