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

Enstrophy — where it appears

The integral of the square of the vorticity, which two-dimensional flow conserves in addition to energy. That second conservation law is what forces energy to travel to large scales rather than small ones in two dimensions.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

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 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 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 The condensate, and the limit of no friction which does not exist. In a steady state the friction must remove everything the forcing puts in, so the energy is eps/(2 alpha) and the coherent velocity is sqrt(eps/alpha) — exactly a minus one half power, checked to 10⁻¹². As the friction is weakened the condensate grows without bound: the limit alpha to zero is not a flow with a weak condensate, it is a flow with no steady state at all.

Where the inverse cascade stops

Two-dimensional turbulence sends its energy upward in scale, and the upward direction has an end: the box. Without something to remove the energy before it arrives, it accumulates there in a pair of vortices filling the domain, and the limit of no friction has no steady state at all.

turbulence · Two-dimensional The two dissipations, side by side. The strain form on the left and the enstrophy form on the right, for the same field, on the same scale. They have their maxima in different places — 0.46 apart on a box of side 2 pi — and neither is a smoothed version of the other. One says the dissipation is in the strained regions and the other says it is in the rotating ones, which is nearly a complete disagreement about what a turbulent flow is doing.

Equal on average, and nothing else

The rate at which a fluid turns motion into heat can be written two ways, and every textbook says the two are equivalent. Their averages are equal to fourteen decimal places. Point by point they are uncorrelated, and their maxima are in different places.

turbulence · Dissipation

Named alongside it

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

DissipationConservationModel limitVorticityCirculationInverse cascadeKinetic energySimilarity solutionViscosityVortex coreVorticity diffusionAveraging

All concepts