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

Vortex dynamics — where it appears

The study of how concentrated regions of vorticity move and interact, each carried by the velocity the others induce. It reduces a field problem to the motion of a few objects, which is why a ring, a pair and a street can all be understood without a grid.

Named by 9 essays across 2 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 speed depends on the core; the slope does not. A ring's self-induced speed against the logarithm of the cutoff used to compute it. The line is a quadrature of the reduced integral; the dots are an independent sum of cross products over a hundred and twenty thousand straight segments. Both diverge as the cutoff is thinned — a filament of zero thickness would move infinitely fast — so no answer here is a ring's speed until a core model is chosen. The slope, Γ/4πR, is the same whatever is chosen, and is what the check measures.

A ring moves because it is bent

A straight vortex filament induces exactly no velocity on itself — every element's direction is parallel to the line joining it to the point being evaluated, and the cross product is zero. Bend it into a ring and it drives itself forward, at a speed that diverges logarithmically as the core is thinned.

circulation · Vortex ring Wound up, and worth exactly what it started with. A material loop in a steady cellular flow — an exact solution of Euler's equations — drawn at four times. Each streamline in the cell has its own period, so the loop is stretched steadily into a spiral: by the last frame its perimeter is 5.7 times what it started as. The circulation round it is 0.903741 at the start and 0.903666 at the end. Nothing about the curve survives except the number.

What survives being wound up

Draw a loop of marked fluid particles and let the flow carry it. It will be stretched, folded and wound into a spiral until nothing about its shape is recognisable, and the circulation round it will not have moved at all — provided three conditions hold, each of which can be broken on purpose.

inviscid · Kelvin A lift curve that keeps climbing to 49 degrees. The lift of a slender delta of aspect ratio 1, split into the potential term that an attached flow would give and the vortex term the separation adds, with a conventional wing's curve behind them. The delta's is nonlinear from the start and reaches 1.68 at 49 degrees, where a conventional wing stalled at fifteen. That is the whole design case for the shape: not that it is efficient — it is not — but that it still has lift at incidences where an ordinary wing has none, which is what a delta-winged aircraft needs on approach and in a turn.

Lift out of a failure

Separation is what ends an ordinary wing's lift curve. A slender delta with sharp leading edges separates on purpose, rolls the shed sheet into a pair of vortices above its upper surface, and takes most of its lift from the suction they induce — with a curve that climbs to forty-nine degrees.

circulation · Vortex lift Four vortices, and the end of prediction. The same three vortices as before with a fourth, weaker one added near the middle. Three point vortices have three independent invariants for three degrees of freedom and cannot be chaotic; four have the same three invariants and one more degree of freedom, and generically are. The energy and the impulses are conserved here to fourteen digits over the whole run, which is what makes the tangle a property of the system rather than of the arithmetic.

Three is the most that can be predicted

Point vortices are the simplest dynamical system fluid mechanics has — no cores, no viscosity, no approximations, four exactly conserved quantities. Three of them are integrable and cannot be chaotic. Add a fourth and the same equations, conserving the same quantities to fourteen digits, stop being predictable at all.

inviscid · Vortex dynamics At a fixed impulse every orbit is a curve of constant energy. Axial separation of two coaxial rings against the radius of the ring that started smaller, for six pairs with the same total impulse, all starting in one plane. Pairs starting nearer equal (inner curves) trace closed loops: the separation swings from one sign to the other as the rings take turns, and the radius swings with it. Pairs starting further apart run off to one side and never return. The pale curves are the separatrix, the energy of two equal free rings — it passes through the plane of the start at a ratio of 0.340 and its arms reach out to infinite separation.

Two rings leapfrog only if they start alike

Two coaxial smoke rings passing through each other in turn is the most famous thing vortices do, and it is not what two rings do in general. Set them off from one plane with different radii and below a definite ratio the smaller one draws ahead and never comes back. Which of the two happens is decided before either ring moves, by whether two separate rings could hold the pair's energy.

circulation · Vortex ring Which bends of a vortex pair grow, and how fast. The square of the growth rate of a sinusoidal bend of a trailing vortex pair, in units of one e-fold per descent time, against the bend's wavenumber times the spacing, for a core of elliptic loading's size. Where the square is positive the bend grows. The symmetric mode, in which the two vortices bend as mirror images, grows over a long-wave band and fastest at a wavelength of 8.54 spacings, by 0.827 e-folds in the time the pair takes to sink one spacing. The antisymmetric mode, bending in step, is stable across the same band.

A wake ends by bending, not by fading

The two vortices behind an aeroplane do not simply weaken until they are gone. In still air each one bends in the other's strain, the bend grows by a factor of e every thirty-three seconds behind an airliner, and after a couple of minutes the pair has pinched itself into a chain of rings. The wavelength it chooses is eight and a half times the spacing, and the vortex cores hardly enter.

circulation · Tip vortex The number of passes grows in proportion to Reynolds number. Passes the two rings make before their cores are close enough to merge, against the vortex Reynolds number, marched with the merging threshold at 0.24 of the core separation and at 0.30, and the closed form [(0.24 dₘᵢₙ)² − σ₀²] Re ÷ 4Tₚₐₛₛ, with the closest approach dₘᵢₙ and the time between passes Tₚₐₛₛ read from the inviscid pair. Both thresholds give a count in proportion to Re: one or two passes at a few thousand, ten or thirty at thirty thousand.

Leapfrogging rings end by merging, not by parting

Two smoke rings that leapfrog in an ideal fluid do it for ever, because their energy binds them. A real fluid drains the energy, and the obvious guess is that the rings drift out of the bound state and part. They do not: diffusion drains the pair's energy and the energy of every possible pair of free rings together, and the margin that binds them never goes negative. What viscosity does instead is fatten the cores until, as one ring threads the other, the two are close enough to merge. That takes a number of passes in proportion to the Reynolds number — one to three at the few thousand of a laboratory smoke ring.

circulation · Vortex ring 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 limitInduced velocityCirculationThe Biot–Savart lawHamiltonianKinetic energyVortex coreVortex ringConservationCore modelEigenvalueImpulse

All concepts