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

Vortex stretching — where it appears

The intensification of vorticity when a vortex tube is drawn out along its own axis, conserving its circulation while its cross-section shrinks. It is the mechanism usually drawn for the cascade, and it is absent in two dimensions.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

The one place the stretching argument closes. Burgers' vortex: an axisymmetric strain carrying vorticity inwards at exactly the rate viscosity spreads it outwards. The vorticity profile is a Gaussian of radius √(4ν/α), the swirl velocity peaks at 1.12 core radii rather than at the core radius itself, and the circulation reaches its full value by about two. The steady vorticity equation is evaluated on this profile by differencing it, not by re-deriving it.

The spin that feeds itself

Stretch a vortex tube and its spin rises in exact proportion, because the circulation round it cannot change and its area has fallen. Nothing in that argument sets a limit — and the one flow where the limit can be written down exactly puts it at a length of √(4ν/α).

kinematics · Vorticity 80 per cent of the energy goes to the larger scale. One unit of energy is taken out of the middle wavenumber and shared between its two neighbours. Two conservation laws decide the split completely: the energy must add up, and so must the enstrophy, which weights each wavenumber by k². For (1, 2, 4) the answer is that 80.0 per cent of the energy goes up in scale and 80.0 per cent of the enstrophy goes down. There is no model of turbulence anywhere in that: it is two linear equations in two unknowns, and it is why a two-dimensional flow organises itself into large vortices while its gradients get finer.

The cascade that runs backwards

Three-dimensional turbulence carries energy from large scales to small ones and dissipates it. Take away one dimension and the term that does it vanishes identically, a second quantity becomes conserved, and two conservation laws between them force the energy to go the other way — up in scale, into ever larger vortices.

turbulence · Two-dimensional Every gradient in one plane, and where each one goes. The two invariants of a trace-free velocity gradient, R across and Q up. Under the restricted Euler equation the combination 27R²/4 + Q³ never changes, so every trajectory is one of these curves, and R never decreases, so every curve is travelled from left to right. Vieillefosse's line, where the combination is zero, has two branches: the left one runs into the origin and is the only way to reach it, and the right one is where every other trajectory ends, at infinity, in finite time.

A gradient left to itself

Follow a parcel's velocity gradient with nothing acting on it but its own square and the part of the pressure a single parcel can know about. Every starting gradient but a set of measure zero reaches infinity in a finite time, and it gets there as a sheet with its spin lying in the sheet.

kinematics · Deformation A slow current divides round the fluid over a hill (δ = 4). A uniform stream, left to right, over a Gaussian hill on the floor of a rotating layer, at δ = h₀/(H·Ro) = 4, where h₀/H is the hill's height against the depth and Ro the Rossby number U/fa; the onset for this shape is δ = 3.134. The dashed circle is the hill's e-folding radius. Water crossing the hill is squashed and, keeping its potential vorticity, spins clockwise; that anticyclone adds to the stream on one side and opposes it on the other. Here it stops the stream: the outlined region, 1.54 square radii, holds fluid that circles for ever and never leaves, and it sits beside the summit rather than on it, on the side where the swirl runs against the current.

The hill a slow current will not climb

The Taylor–Proudman theorem says a rotating fluid goes round an obstacle rather than over it, as if a solid column stood above it. Conserving potential vorticity turns that limit into a threshold. A current is stopped over a hill once the hill's height against the depth exceeds a fixed multiple of the Rossby number — 2 for a flat-topped hill, 3.13 for a Gaussian one, 16/3 for a cone — and the fluid it holds sits beside the summit rather than on it.

regimes · Geostrophy

Named alongside it

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

Model limitVorticityStrain rateBurgers vortexCascadeCirculationClosure problemConservationCoriolisDimensionlessDiscriminantDissipation

All concepts