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

Vortex patch — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · inviscid
  2. 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.

    part 2 · inviscid
  3. Two shapes for each strain, and one of them holds. The strain rate, over the vorticity, at which an elliptical patch of aspect ratio λ stands still — Moore and Saffman's relation — rising to its maximum of 0.1501 at λ = 2.89 and falling again. Below the maximum there are two steady shapes for each strain: a rounder one, on which every disturbance computed here stays bounded, and an elongated one, which comes apart. Above it there is no steady shape at all.

    A strained vortex holds until it has no shape to hold

    A patch of vorticity in a strain has two steady shapes for every strain below 0.150 of its vorticity, a rounder one and a longer one, and none above. The longer one comes apart at the slightest nudge. The rounder one, computed with disturbances of two, three, four and five lobes, never does: it nods and holds right up to the strain at which it ceases to exist. So the existence limit is the real limit, and past it a vortex is not shattered but stretched — lingering first near the shape it has lost, for a time that grows as the fourth root of how close the strain is to the limit.

    part 3 · inviscid
  4. The ring is drawn off while the core keeps its shape. The edges of a vortex with a core of twice the ring's vorticity out to six-tenths of its radius, at four times, in a strain raised to 0.245 — just past the 0.227 that strips its ring. The strain stretches along the horizontal. Both edges lean into the ellipses of a strained vortex; then the ring's edge is pulled out at its two tips into arms, while the core inside stays close to an ellipse.

    A strained vortex loses its ring at the limit of what it holds

    A uniform patch of vorticity survives a strain up to 0.150 of its vorticity and no further. A real vortex is not uniform: its vorticity falls off outward, and in a strain its weak outer layers are drawn off while its core survives. The obvious way to predict how much survives is to apply the uniform limit layer by layer, each layer to its own vorticity. It is wrong by up to a factor of three. Built from a core inside a ring and followed by contour dynamics, the ring is stripped not at the limit of its own vorticity but at the limit of all the vorticity inside its edge, and a little after it.

    part 4 · inviscid

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