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 7 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 Three relaxed states with the same energy and enstrophy. The vorticity along the diagonal of the periodic square, through the centres of both vortices of the dipole, scaled by its rms value, for the three relations at the same ratio of enstrophy to energy, Z/E = 1.1 — except the linear state, which exists only at Z/E = 1. The sinh state concentrates its vorticity into sharp cores; the tanh state spreads it into flat-topped patches with steep edges; the linear state is a sine. All three carry the same two quadratic invariants in proportion.

The streamfunction says which relaxed state

Decaying two-dimensional turbulence ends in a large pair of vortices, and three theories say what that pair should look like: a sinh relation between vorticity and streamfunction, a tanh, or a straight line. Their scatter plots differ only in curvature, and a real flow's scatter hides curvature. Solve the three states in the same periodic box, at the same energy and enstrophy, and a statistic that separates them turns out to be one nobody looks at: the flatness of the streamfunction, which sits above the straight line's value for every sinh state and below it for every tanh state, and does not move when unrelaxed small eddies are added.

turbulence · Two-dimensional Four of six decayed states sit on the sinh branch; the other two lie above it. The streamfunction's flatness against the enstrophy-to-energy ratio Z/E: the sinh and tanh relaxed dipoles as curves, from the linear dipole at Z/E = 1 outwards, and the six decayed states at t = 600 as points. The tanh branch falls below 9/4 and stays near Z/E of one; the sinh branch rises. The two runs begun as two-level patches lie on the sinh branch at their own Z/E, 0.0049 to 0.055 from it, and so do the two begun on a k⁻³ spectrum, 0.04 to 0.072. The two random-phase runs lie above it, by 0.3 to 0.31: on the sinh side of 9/4, but not sinh dipoles.

Decaying flows end on the sinh side

Three theories predict the state a decaying two-dimensional flow relaxes to, and the streamfunction's flatness tells them apart: above 9/4 for the point-vortex theory's sinh, below for the two-level theory's tanh. Run the decay from three kinds of start and every run ends above 9/4 — including the one begun as two-level patches, the case the tanh theory was built for. Four of six land on the sinh branch itself; the other two are sinh-like without being sinh.

turbulence · Two-dimensional

Named alongside it

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

Model limitVorticityDissipationInverse cascadeConservationTwo-dimensional turbulenceCirculationFlatnessKinetic energyNegative temperatureSimilarity solutionStatistical mechanics

All concepts