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

Vortex core — where it appears

The rotational centre of a vortex, where the swirl is fastest and the pressure lowest. Particles denser than the fluid are flung out of it and bubbles are drawn into it, so it concentrates its own cavitation nuclei and is where cavitation usually starts.

Named by 11 essays across 5 fields — each of them below, with the objects they name alongside it.

The core spreads and the outside never notices. The swirl velocity at four times a factor of four apart, with the free vortex Γ/2πr drawn behind them. Every curve leaves the free-vortex line at its own core radius and turns over into solid-body rotation inside it; outside the core all four are the same curve, to the precision of the plot. Viscosity has rounded off the singularity and changed nothing else. The peak swirl falls from 52.4 to 6.6 m/s across the four, and the circulation is identical for all of them.

What viscosity cannot take away

Leave a vortex alone in a viscous fluid and every local measure of it falls — the peak spin, the peak velocity, the enstrophy. The circulation round a large loop does not move at all, ever, and the far field is identical to the line vortex it started as.

viscous · Diffusion Q and the vorticity along a radius of one vortex. Two candidate measures of where the vortex is, along a radius of a Lamb–Oseen vortex. The vorticity is a Gaussian: positive at every radius, so a threshold on it puts the edge wherever the threshold is put. Q — the excess of rotation over strain — changes sign exactly once, at 1.121 core radii, and that radius is a property of the flow rather than of the person drawing it. Inside it, 71.5 per cent of the circulation.

Where a vortex stops

Four criteria decide where a vortex ends, and in two dimensions three of them are the same criterion. The fourth is a knob. And the one that is not a knob is not objective: a co-rotating pair of vortices occupies four per cent of a window to one observer and more than half of it to another.

kinematics · Coherent structures The relaxation time a particle actually has. Two quantities against the particle-to-fluid density ratio. β = 3ρ_f/(2ρ_p + ρ_f) is three for a bubble, one for a neutrally buoyant particle and nearly zero for anything heavy; it is the factor by which the fluid's own acceleration is felt. The other curve is the true relaxation time over the usual formula's, which is one for a heavy droplet, exactly three halves for a neutrally buoyant tracer, and unbounded for a bubble — the usual formula gives a bubble a relaxation time of zero, and therefore no dynamics at all.

The tracer that is not one

Every Stokes number is built on a relaxation time that counts the particle's own inertia and nothing else. Adding the two terms it leaves out gives a bubble a relaxation time where the usual formula gives zero, makes a neutrally buoyant tracer half as slow again as advertised, and sends bubbles into vortex cores that droplets are flung out of.

regimes · Particle A vortex has no energy. The kinetic energy of a Lamb–Oseen vortex inside a circle, per metre of its length, against the logarithm of that circle's radius. It is a straight line and it does not stop: outside the core the swirl is Γ/2πr, the energy density falls as 1/r², and the area grows as r², so every decade of radius adds the same amount. There is no such thing as the energy of a line vortex without a stated cutoff, and no cutoff is physical.

The energy a vortex cannot have

A line vortex has infinite kinetic energy. Not a large amount — infinite, growing without limit as the logarithm of however far out the counting stops. And it is losing that energy at a rate that is finite, exactly known, and contains no cutoff at all.

viscous · Diffusion 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 slow bending wave, and where the cut-off stops describing it. The frequency of the slowest bending wave on a Rankine vortex — a helical wobble of the whole core that turns against the flow — against the wavenumber times the core radius, from Kelvin's exact relation and from his long-wave formula, which the cut-off method used for vortex pairs reproduces. They agree for long waves. Past ka = 1.44 the long-wave formula reverses sign and the exact wave does not, which is why a cut-off model's instabilities at such wavelengths are not real.

A vortex is a waveguide

A spinning core is stiff in a way still fluid is not, and it carries waves along its length: an infinite family of them for every pattern round its axis, travelling at up to 0.83 of the swirl speed at its edge. The slowest is a helical bend that turns against the flow, and it is the wave every model of a bending vortex has been borrowing without saying so.

inviscid · Swirl 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 The wake a flapping bird leaves is a wave in the air. Side view, the air at rest: the path the leader's wingtip traced, where its tip vortex now lies, over two wingbeats; lengths along the flight path in wake wavelengths — the distance flown in one beat — and heights in spans, for a tip swinging a fifth of a span each way. A follower three-tenths of a wavelength behind that beats three-tenths of a beat later traces the same path and flies along the leader's vortex all the way; one that beats half a beat off that traces the mirror image and meets the vortex only twice a beat.

The follower beats in time with the wake, not the bird

A gliding bird can sit in its neighbour's upwash and stay there. A flapping bird's wake is a wave left in the air — the path its wingtip traced, rising and falling with every beat — and a bird behind gains only if its own wing is where that wave is when it arrives. The best timing is a rule with no aerodynamics in it: lag the bird ahead by the time the wake took to come, so that each wingtip retraces the path of the one before. Directly behind, the rule flips by half a beat, and it buys a smaller loss rather than a gain.

circulation · Formation 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 A follower can wander further fore and aft than sideways. A flapping follower's saving, averaged over a wander of its place with the standard deviation shown, as a fraction of its own induced drag: fore and aft, holding the phase that suits its average place, for tips swinging a tenth, a fifth and four-tenths of a span; and sideways, beating in phase. The saving in place is 0.837. Sideways it has halved at a wander of 0.185 spans; fore and aft, with a fifth-span swing, only at 0.804, a third of the wake's wavelength, and with the four-tenths a cruising bird swings, at 0.37. With a tenth-span swing it never halves.

A flapping follower can drift fore and aft, but not sideways

A bird in a flapping V has to be in the right place and beat at the right phase for that place, and no bird holds its place exactly. Drifting fore and aft costs it phase, at a full beat for every wavelength of the wake; drifting sideways takes it off the leader's tip vortex. The first is two to four times cheaper than the second, it can be bought back by re-timing the beat within about one beat of the drift, and the second cannot be bought back at all. So the precision a follower needs is sideways, and the attention it needs is on its timing.

circulation · Formation The two observers disagree twelvefold about where Q says the vortices are, and not at all about the LAVD edge. The Q = 0 contours and the outermost convex LAVD level over one orbit, in the laboratory and in the frame turning with the pair. Q > 0 covers 2.00 of the window in the laboratory and 24.94 in the turning frame; the LAVD edges enclose 6.29 in the laboratory and 6.29 in the turning frame, and the two LAVD fields differ nowhere by more than 8.0e-8 of their peak.

An edge no observer can move still depends on what it is averaged against

The Q-criterion puts a co-rotating vortex pair in four per cent of a window for one observer and in more than half of it for another. The cure is to average, along each fluid path, how far the vorticity departs from its mean — a number no observer's rotation can change — and take the outermost convex level of it round each core. Computed, it works: both observers find two discs of radius 1.00, twice the core radius, and no fluid inside them strays from its core in an orbit. But the edge has a setting of its own. The mean must be taken over some region, and as that region grows from a radius of 4 to 20 the edge moves from 0.90 to 1.13.

kinematics · Coherent structures

Named alongside it

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

Model limitVorticityWakeCirculationEigenvalueRotating frameVortex dynamicsThe Biot–Savart lawConservationDissipationEnstrophyFlapping wing

All concepts