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 2 essays across 2 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

Named alongside it

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

VorticityBurgers vortexCascadeCirculationConservationDimensionlessDissipationEnergy cascadeEnergy spectrumEnstrophyInverse cascadeKelvin's circulation theorem

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