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

Contour dynamics — where it appears

The method of advecting a region of uniform vorticity by tracking its boundary alone, since the interior needs no representation. The velocity is a contour integral of a logarithm, the area is conserved exactly, and the approximation is only in the curve — which makes it one of the few exact methods in the subject.

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

Also named here as vortex patch — the same set of essays touches all of them, so they are one junction rather than several.

Kirchhoff's rotation rate, which a point vortex does not have. A patch of uniform vorticity bounded by an ellipse turns rigidly at omega a b/(a+b)², a rate that depends on the shape alone. It is largest for a circle, where it is unobservable, and falls away as the patch is drawn out. A point vortex has no shape and therefore no entry on this axis at all.

The shape a vortex keeps

Outside a circular patch of uniform vorticity the flow is exactly the point vortex's — not nearly, exactly — so replacing one by the other looks free. It is not. The patch has a shape, the shape has a rotation rate of its own, and there is a strain above which no shape exists at all.

inviscid · Vortex patch The patches' centroids, against the point-vortex circle. Two circular patches of uniform vorticity, advected by nothing but the velocity their own boundaries induce, over one full co-rotation. Their centroids stay within one per cent of a separation of the exact point-vortex orbit — which they must, because the exterior field of a circular patch is the point vortex's and a harmonic function's area average over a disc is its value at the centre.

What a point vortex is not

Two circular patches of vorticity move exactly as two point vortices do — the centroid velocity is the point model's with no correction of any order, and that is a theorem rather than an approximation. The trouble starts the moment they stop being circular, which is immediately.

inviscid · Vortex dynamics Three lobes, then a filament. An ellipse of aspect ratio 4 with a three-lobed bump of three thousandths, as contour dynamics carries it, drawn in the frame turning with the undisturbed ellipse at t = 0, 30 and 42. By t = 30 the bump has grown to a visible three-fold asymmetry — one end fattened, the other thinned — and by t = 42, about a turn and a tenth of the ellipse, the thinned end is being drawn out into a filament. The march is stopped there, while the area is still conserved to a few parts in a thousand; resolving the filament needs a contour that adds nodes, which this one does not.

Past three, an ellipse is a shear layer

Kirchhoff's elliptical vortex turns for ever without changing shape, and Love showed in 1893 that it stops being stable at an aspect ratio of exactly three. Computed, that threshold turns out to be the first of a sequence — a new way of coming apart every one and a half aspect ratios — and the sequence ends somewhere recognisable. A long enough ellipse is a strip of vorticity, and it comes apart the way a shear layer does, at a rate Rayleigh found for the strip.

inviscid · Vortex patch

Named alongside it

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

EquilibriumModel limitVortex patchCirculationKirchhoff ellipsePoint vortexRegularisationStrain rateTwo-dimensional flowVorticityConserved quantityDiscretisation

All concepts