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

Trailing vortex — where it appears

The concentrated vortex shed from the tip of a lifting wing, carrying away the circulation that the wing's lift requires to end somewhere. Its downwash over the wing tilts the lift backwards into induced drag, and the pair behind a large aircraft can persist for minutes.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

Two of opposite sign go somewhere. Two vortices of equal and opposite strength. Each is carried by the other's field, both are carried the same way, and the pair travels in a straight line at Γ/2πd forever, keeping its separation exactly. The speed is a consequence of one vortex's field evaluated at the other, and nothing else.

Vortices move each other

A vortex alone in an infinite fluid sits exactly still, forever — its own field is antisymmetric about it and there is nothing at its centre to be carried by. Everything a vortex does, another vortex did, and two of them already exhaust what can be written down.

inviscid · Vortex dynamics The wake a flapping bird leaves is a wave in the air. Side view, the air at rest: the path the leader's wingtip traced, where its tip vortex now lies, over two wingbeats; lengths along the flight path in wake wavelengths — the distance flown in one beat — and heights in spans, for a tip swinging a fifth of a span each way. A follower three-tenths of a wavelength behind that beats three-tenths of a beat later traces the same path and flies along the leader's vortex all the way; one that beats half a beat off that traces the mirror image and meets the vortex only twice a beat.

The follower beats in time with the wake, not the bird

A gliding bird can sit in its neighbour's upwash and stay there. A flapping bird's wake is a wave left in the air — the path its wingtip traced, rising and falling with every beat — and a bird behind gains only if its own wing is where that wave is when it arrives. The best timing is a rule with no aerodynamics in it: lag the bird ahead by the time the wake took to come, so that each wingtip retraces the path of the one before. Directly behind, the rule flips by half a beat, and it buys a smaller loss rather than a gain.

circulation · Formation A phantom biplane partner doubles the induced drag. Induced drag at a given lift, as a multiple of its deep-water value, against the foil's depth in spans. The surface's image of the foil is an identical wing, equally loaded, twice the depth above it, so the real foil pays its own induced drag plus the mutual drag of a biplane with a gap of twice its depth: 1 + σ, with σ Prandtl's biplane factor. The elliptic biplane's 1 + σ and the aspect-ratio-9 foil's lattice both run to two as the depth closes.

A foil under the surface flies with a phantom

At foiling speed the water's surface cannot hold a pressure, and the image that condition requires is not the reversed one a wall makes. For a horizontal foil it is an identical wing, equally loaded, above the surface — a biplane partner that takes lift and gives nothing back, doubling the induced drag as the foil rises. For the board that pierces the surface it is a reversed copy, which makes the board pay two and a half times what a keel under a hull pays. It is the board, not the foil, that decides how deep a foiling boat rides.

applied · Sailing A follower can wander further fore and aft than sideways. A flapping follower's saving, averaged over a wander of its place with the standard deviation shown, as a fraction of its own induced drag: fore and aft, holding the phase that suits its average place, for tips swinging a tenth, a fifth and four-tenths of a span; and sideways, beating in phase. The saving in place is 0.837. Sideways it has halved at a wander of 0.185 spans; fore and aft, with a fifth-span swing, only at 0.804, a third of the wake's wavelength, and with the four-tenths a cruising bird swings, at 0.37. With a tenth-span swing it never halves.

A flapping follower can drift fore and aft, but not sideways

A bird in a flapping V has to be in the right place and beat at the right phase for that place, and no bird holds its place exactly. Drifting fore and aft costs it phase, at a full beat for every wavelength of the wake; drifting sideways takes it off the leader's tip vortex. The first is two to four times cheaper than the second, it can be bought back by re-timing the beat within about one beat of the drift, and the second cannot be bought back at all. So the precision a follower needs is sideways, and the attention it needs is on its timing.

circulation · Formation

Named alongside it

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

Induced dragModel limitFlapping wingFormation flightPhaseThe Trefftz planeUpwashVortex coreWakeBiplaneCirculationFree surface

All concepts