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

Kurtosis — where it appears

The fourth moment of a distribution divided by the square of its variance, a measure of how heavy its tails are. A Gaussian has three; a larger value means rare large departures matter more than the variance alone suggests.

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

Also named here as richardson dispersion — the same set of essays touches all of them, so they are one junction rather than several.

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.

turbulence · Mixing 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.

turbulence · Mixing 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.

turbulence · Mixing

Named alongside it

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

Inertial rangeLagrangianModel limitRichardson dispersionStochastic modelMemoryPair dispersionProbability distributionDiffusionIntegral scaleIntermittencyOrnstein uhlenbeck

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