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

Vortex patch — where it appears

A region of uniform vorticity bounded by a closed curve, which in two dimensions stays uniform because the vorticity of a material element cannot change. The edge is therefore the whole state, and an elliptical patch is an exact steady solution that rotates rigidly at a rate set by its shape alone.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

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 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.

inviscid · Vortex patch Three relaxed states with the same energy and enstrophy. The vorticity along the diagonal of the periodic square, through the centres of both vortices of the dipole, scaled by its rms value, for the three relations at the same ratio of enstrophy to energy, Z/E = 1.1 — except the linear state, which exists only at Z/E = 1. The sinh state concentrates its vorticity into sharp cores; the tanh state spreads it into flat-topped patches with steep edges; the linear state is a sine. All three carry the same two quadratic invariants in proportion.

The streamfunction says which relaxed state

Decaying two-dimensional turbulence ends in a large pair of vortices, and three theories say what that pair should look like: a sinh relation between vorticity and streamfunction, a tanh, or a straight line. Their scatter plots differ only in curvature, and a real flow's scatter hides curvature. Solve the three states in the same periodic box, at the same energy and enstrophy, and a statistic that separates them turns out to be one nobody looks at: the flatness of the streamfunction, which sits above the straight line's value for every sinh state and below it for every tanh state, and does not move when unrelaxed small eddies are added.

turbulence · Two-dimensional Four of six decayed states sit on the sinh branch; the other two lie above it. The streamfunction's flatness against the enstrophy-to-energy ratio Z/E: the sinh and tanh relaxed dipoles as curves, from the linear dipole at Z/E = 1 outwards, and the six decayed states at t = 600 as points. The tanh branch falls below 9/4 and stays near Z/E of one; the sinh branch rises. The two runs begun as two-level patches lie on the sinh branch at their own Z/E, 0.0049 to 0.055 from it, and so do the two begun on a k⁻³ spectrum, 0.04 to 0.072. The two random-phase runs lie above it, by 0.3 to 0.31: on the sinh side of 9/4, but not sinh dipoles.

Decaying flows end on the sinh side

Three theories predict the state a decaying two-dimensional flow relaxes to, and the streamfunction's flatness tells them apart: above 9/4 for the point-vortex theory's sinh, below for the two-level theory's tanh. Run the decay from three kinds of start and every run ends above 9/4 — including the one begun as two-level patches, the case the tanh theory was built for. Four of six land on the sinh branch itself; the other two are sinh-like without being sinh.

turbulence · Two-dimensional 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.

inviscid · Vortex patch

Named alongside it

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

Model limitContour dynamicsVorticityCirculationEquilibriumKirchhoff ellipseStabilityStrain rateEnstrophyFlatnessInverse cascadeNegative temperature

All concepts