Mixing — the series · Fluid Flow & Aerodynamics

Ballistic first, diffusive later. The mean square displacement of a parcel carried by a fluctuating velocity, from Taylor's 1921 integral with an exponential correlation of time scale 1. Below the correlation time it follows the straight-line law u²t² exactly — the parcel has not yet changed its mind — and above it the curve joins the diffusion law 2u²T·t, offset by the head start the ballistic phase gave it. The eddy diffusivity is u²T = 1.000, and it is a property of the correlation rather than of any equation of motion.

How far a parcel gets

Turbulent transport is one integral. A parcel carried by a fluctuating velocity goes in a straight line while it still remembers its own motion and performs a random walk once it has forgotten, and the crossover is the correlation time — so an eddy diffusivity is not a property a flow has, it is the limit of a measurement that has run for long enough.

part 1 · turbulence

The scalar spectrum, with its two ranges. A model scalar spectrum at a Schmidt number of two thousand — dye in water. Below the Kolmogorov wavenumber it is Obukhov and Corrsin's five-thirds, inherited from the velocity; above it there is no turbulence left and the spectrum is Batchelor's minus one, which contains no velocity spectrum at all.

The scalar has its own cascade

Below the Kolmogorov scale there is no turbulence left, and a dye stirred into the flow goes on cascading anyway — on a spectrum whose exponent is minus one and whose amplitude contains no velocity spectrum at all. Resolving it costs the three-halves power of the Schmidt number, which for dye in water is a factor of ninety thousand.

part 2 · turbulence

Three pairs released at different distances end on one line. The mean square separation of two parcels against time, both in Kolmogorov units, for pairs released one, thirty and a thousand Kolmogorov lengths apart in turbulence whose integral scale is a million of them. Each starts flat — the pair has not yet moved — and each joins the same line, g ε t³ with g = 0.5, after which nothing about where it started survives. Beyond the integral scale all three become ordinary diffusion, growing as t.

The further apart, the faster they part

Release two specks of smoke close together in turbulence and the distance between them does not diffuse the way a single speck wanders. It grows faster the larger it already is, because larger separations are pulled apart by larger eddies — as the cube of the time, forgetting where it started. The law is ninety years old, its constant is still argued about, and it needs a very large flow to be seen at all.

part 3 · turbulence

The longer a pair remembers, the less its cloud has tails. The kurtosis of the pairs' separation, ⟨r⁴⟩/⟨r²⟩², against how long each pair keeps its relative velocity, in units of the turnover time of eddies of its own size. Richardson's memoryless diffusion gives 3.76, dashed, with long tails of pairs that separate fast by chance; a Gaussian cloud gives 5/3. A memory of a tenth of a turnover time already takes the kurtosis to 2.55; one turnover time, to 1.82. The cloud's shape is a measurement of the memory.

A pair's memory shapes the cloud it spreads into

Richardson's diffusion of pair separations has no memory: each moment's push is independent of the last, and the cloud of separations grows as t³ with a peaked shape and long tails. Real pairs keep their relative velocity for about the turnover time of eddies their own size. Give them that memory and the cube law survives, because it is dimensional, but everything else about the cloud changes: its constant falls, its tails shrink, and its shape becomes a measure of how long a pair remembers. A memory of a tenth of a turnover time already takes the kurtosis from 3.76 to 2.5.

part 4 · turbulence

Bursts in the velocity put the tails back. The kurtosis of the pairs' separation against how long each pair remembers its velocity's direction, β, in turnover times. The lowest curve is a Gaussian velocity, as in the earlier calculation. The others give the velocity a flatness of 4, as measured in the inertial range, with its amplitude remembered for α turnover times. At the memory real pairs are estimated to have, β = 0.7, the Gaussian cloud's kurtosis is 1.89; with bursts remembered for three turnover times it is 3.41, and with the amplitude frozen 4.4 — either side of Richardson's 3.76, dashed.

Bursts and memory pull a cloud both ways

A pair of fluid particles that remembers its relative velocity spreads into a cloud with shorter tails than Richardson's, and the earlier calculation proposed reading the memory off the cloud's shape. Real relative velocities come in bursts, their amplitude set by a local dissipation that varies, and a burst that lasts pushes the tails back out — hard, because separation grows as the cube of diffusivity. At the memory real pairs are estimated to have, the two effects nearly cancel, and a cloud can have Richardson's exact shape for entirely the wrong reason.

part 5 · turbulence

A cloud's shape falls as it grows into calmer eddies. The kurtosis of a cloud of pairs released at a ten-thousandth of the integral scale, against its rms size, when the velocity's flatness follows the 1962 law, and for clouds whose velocity has one flatness, 3, 4 or 5, at every separation. The fixed-flatness clouds settle at a shape and keep it. The growing cloud rises to a kurtosis of 6.85 at 0.0022 L and then falls without settling, to 2.9 by 0.57 L, crossing all three.

A cloud grows out of its bursts and keeps their shape

Measured velocity differences are burstiest across the smallest separations and nearly Gaussian across the largest, so a cloud of particle pairs released close together starts in the fiercest intermittency and grows out of it. Its shape follows, but late: the kurtosis falls steadily as the cloud grows, always above what its present statistics would give, and the cube law's constant falls with it, so the growth exponent climbs towards three and never arrives.

part 6 · turbulence