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

Vortex pair — where it appears

Two line vortices close together, which move each other: equal and opposite ones travel together in a straight line, equal and like-signed ones orbit their midpoint. A pair is the simplest wake, the far field of a lifting wing and the unit whose instabilities end most trailing wakes.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Elliptic loading rolls up to πb/4, and it is π that puts it there. Three loadings on the same span carrying the same lift, with the rolled-up core positions marked below each. The core sits at the centroid of the shed vorticity, which is a quadrature over the loading and needs nothing about the roll-up itself. Elliptic loading gives πb/8 from the centreline, so the pair ends up 0.7854 of the span apart — the number every wake-separation rule is written against, and one of the few places in this subject where π turns up in an answer an engineer uses directly. The bell rolls up to 0.586 and a nearly rectangular loading to 0.978, because it sheds at the tips.

Where the wake ends up

The sheet a wing sheds rolls up into two cores within a few spans, and nobody can compute the roll-up cheaply. Nobody has to: what the cores conserve is fixed before they form, and for an elliptically loaded wing the answer contains π and comes out at 78.5 per cent of the span.

circulation · Tip vortex In the pair's own frame, two eddies with no vorticity. The flow round two equal co-rotating vortices (orange dots), seen from the frame that turns with them, where it is steady; shading is the stream function. Round each vortex a lobe of fluid circulates; a band round both is bounded by a figure-eight through the saddle at the centre; and above and below, centred on the two points that make equilateral triangles with the vortices (blue dots), are two large eddies of fluid that circulate in this frame and carry no vorticity at all, bounded by the streamline through the two outer saddles at √5 half-separations.

A vortex pair carries eddies no snapshot can see

Two equal vortices circling each other are, by every snapshot of the velocity gradient, two small vortices in a straining flow: outside their cores the flow is irrotational, and the gradient there is pure strain. Seen from the frame that turns with them, the same flow contains two large eddies, one on each side, centred where a third point would complete an equilateral triangle with the pair. Their fluid goes round with the pair for ever, and it carries no vorticity at all. They hold seventy times the area of the cores, and the only way to see them is to follow the fluid or to turn with it.

kinematics · Frames Four vortices bend twice as fast, on shorter waves. Growth rate of the fastest bending wave against its wavelength, both in the units of the equivalent single pair — one unit of growth is one e-fold in the time the wake sinks by its own spacing. Crow's pair grows at most at 0.827, at 8.54 spacings. A flap vortex of 0.3 of the tip's strength at 0.4 of its station keeps a Crow band just below that and adds a second, at 1.57 and 1.8 spacings; one of 0.5 at 0.3 merges the two into one band peaking at 1.64 and 4.4 spacings.

Four vortices bend faster, and bend the wrong one

With its flaps down a wing sheds four vortices, not two: a strong one at each tip and a weaker one at each flap's edge, and on each side the two circle one another as the wake sinks. The orbit makes the wake unstable at waves a fifth of Crow's length, growing twice as fast. But the wave that grows is the weak flap vortex being bent round the strong tip vortex, which hardly moves, while the long bend that pinches the tips together grows no faster than before. Only a counter-rotating inner vortex makes the whole wake unstable, and it does so at every wavelength.

circulation · Tip vortex

Named alongside it

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

Model limitWakeThe Biot–Savart lawCentroidCirculationCoherent structuresConserved quantityEigenvalueFrame dependenceInduced dragInduced velocityLinear stability

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