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

Intermittency — where it appears

The property of being present only part of the time, whether at the edge of a shear layer or in the patchiness of the dissipation. It makes a conventional average an average over two states, and the switching contributes a variance that is not turbulence.

Named by 12 essays across 2 fields — each of them below, with the objects they name alongside it.

The inertial range, and the slope read back off it. The model energy spectrum at Re = 1e+6, with production rolling off below the integral scale and dissipation cutting it off above the Kolmogorov scale. The straight middle is the inertial range, and the number printed beside it is the slope least-squares fitted to the drawn points over the middle of that range — not the −5/3 that went in.

Where the energy goes

Energy enters a turbulent flow at the largest scale and leaves it at the smallest, and in between there is nothing for it to depend on but the rate at which it is passing through. Two quantities and one dimensional argument fix the shape of the spectrum, and the exponent is −5/3.

turbulence · Cascade The one exact law, and where viscosity takes it back. The two terms of the Kármán–Howarth relation against separation, measured in Kolmogorov lengths. The four-fifths term rises linearly with r and is the whole of the law at large separation; the viscous term, 6ν dS₂/dr, falls as r^(−1/3) and takes over below 5.64η. Nothing in the calculation was told what η is: the crossing is at the same multiple of it at every viscosity tried, which is what makes the dissipation scale the lower end of the inertial range rather than a separate assumption.

The one exact result

Almost nothing in turbulence follows from the equations without a model in it. One thing does — Kolmogorov's four-fifths law, which fixes the third moment of the velocity differences at −(4/5)εr with no adjustable constant anywhere in it. The companion two-thirds law is not exact, and the 4.02 everybody quotes in it turns out to be a gamma function.

turbulence · Structure function Most of the fluctuation is the switching. What a hot wire at the edge of a shear layer records, taken apart. The conditional intensity — the fluctuation inside the turbulent fluid — is flat at 0.16 by construction. The measured intensity peaks 20 per cent above it, and the excess is the third curve: the variance of a signal that keeps switching between two mean velocities, which peaks where the switching is most even and is not turbulence at all. A model calibrated against the measured curve is being fitted to an artefact of averaging.

Turbulent some of the time

At the edge of a jet or a wake a probe is inside turbulent fluid for part of the time and in perfectly smooth flow for the rest, and an ordinary time average mixes the two. Most of the fluctuation it records there is not turbulence at all — it is the switching between two states, and it peaks where the switching is most even rather than where the turbulence is strongest.

turbulence · Intermittency Four sets of scaling exponents, all of them exact at the third moment. zeta_p against p for K41, the beta-model, the log-normal model and She–Leveque. Every one of them passes through zeta_3 = 1 exactly, because the four-fifths law is a consequence of the equations and a model that missed it would be wrong about the one thing that is known. What they disagree about is every other moment.

The exponents that stop being thirds

Kolmogorov's 1941 theory says every moment of the velocity difference scales with the same exponent, p over three, so the distribution keeps its shape at every scale. It does not. The exponents fall below the line, by more the higher the moment, and what the departure measures is a dimension.

turbulence · Intermittency The mean flux, flat across the inertial shells and equal to the dissipation. The time-averaged transfer out of the first n shells, computed as the rate at which the nonlinearity changes their energy rather than from a remembered formula. It is constant to a tenth across the middle of the ladder and equal to the dissipation, which is the cascade — and it is an average.

A flux that runs both ways

The cascade is a statement about a mean. Kolmogorov's four-fifths law fixes an average and the constant flux through the inertial range is an average, and neither says anything about what the transfer is doing at any instant — which turns out to be running backwards a substantial part of the time.

turbulence · Cascade What a probe in a flame sees. Half the time hot light fluid at a hundred metres a second, half the time cold heavy fluid at twenty. That is what intermittency in a jet flame looks like at a point, and it is the simplest field in which the two averages of the velocity are different numbers.

Two averages of one flow

In a flow whose density varies there are two mean velocities, they are both correct, and across a flame they differ by a factor of two. One of them is what a hot wire returns; the other is what every compressible turbulence model is written in; and the mass flux is the single product they agree on.

kinematics · Averaging Two lifetimes, crossing. The mean time for a turbulent puff to decay and the mean time for it to split into two, against Reynolds number, on a logarithmic axis spanning thirty orders of magnitude. Below the crossing puffs die faster than they multiply; above it they multiply faster than they die, and the flow stays turbulent.

A puff that does not know how old it is

A patch of turbulence in a pipe below the critical Reynolds number dies at random, and its chance of dying in the next second does not depend on how long it has already lasted. The flow that contains it has a memory anyway, because the patches multiply — and where multiplying overtakes dying is a Reynolds number.

turbulence · Transition How far apart two points can be and still be correlated. The correlation between the logarithm of the dissipation at two points, against how far apart they are in units of the smallest scale, over four decades. It falls as a ratio of logarithms — so it is still a quarter at a thousand smallest scales, and reaches a half only at a hundred.

A dissipation correlated across every scale

The dissipation is supposed to be the most local quantity in turbulence — a thing happening at the smallest eddies, everywhere and independently. A multiplicative cascade makes its logarithm correlated over a distance that is a ratio of logarithms, so two points a thousand smallest scales apart still agree a quarter of the time.

turbulence · Intermittency One curve from two to one, with four thirds somewhere in the middle. The ratio of the transverse second-order structure function to the longitudinal one, against separation, at three Reynolds numbers. Every value on every curve follows from the longitudinal function alone by a relation with no dynamics in it. It is exactly 2 where the field is smooth, exactly 1 beyond the correlation length, and it passes through four thirds on the way — but it passes through rather than resting there, and how nearly it rests is the whole of what a Reynolds number buys.

A relation with no turbulence in it

Isotropy and incompressibility alone fix the transverse structure function from the longitudinal one. Divide the relation through and it says the ratio of the two is one plus half the local slope — so the exponent everybody measures as 0.70 and the ratio everybody measures as 1.35 are one measurement, and a model spectrum with no intermittency in it produces both.

turbulence · Structure function Remembered pressure turns the runaway into a teardrop. The velocity gradient's two invariants, Q against R, both scaled by the mean enstrophy, for an ensemble of parcels whose pressure remembers one Kolmogorov time of deformation, at a memory of a tenth of a large-eddy time. The restricted Euler equation sends every parcel off to infinity along the right-hand branch of Vieillefosse's curve, dashed. With the remembered pressure the parcels stay: they crowd along that branch in the strain-dominated quadrant and above the axis on the left where vortices are being stretched — the teardrop measured in turbulence.

The pressure a parcel remembers keeps it finite

The restricted Euler equation follows a parcel's velocity gradient with the pressure's shape thrown away, and every gradient it follows blows up. Give the parcel back a pressure that remembers how its neighbourhood was deformed over the last Kolmogorov time — nothing more — and no gradient blows up at all. The ensemble settles into the teardrop measured in turbulence, its vorticity lines up with the middle strain axis, and its intermittency grows as the memory shortens. What the memory cannot do is keep the one identity homogeneity demands, and that miss says where the rest of the pressure lives.

kinematics · Deformation 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 At fourth order a local slope reports less than a quarter of the anomaly. The anomaly each instrument reports at Reλ = 200 — Kolmogorov's n/3 less what it reads — against the order, for the log-normal field with μ = 0.25, with the departure its distribution carries. The local slope reports 0.026 of a true 0.111 at fourth order and 0.205 of 0.333 at sixth; extended self-similarity 0.245 of its own 0.261; the normalised moment reports the truth at every order.

At higher orders the bottleneck hides intermittency

At second order the spectral bottleneck makes a field with no intermittency look intermittent. At higher orders it does the opposite. Every moment's local slope is steepened by the same finite-Reynolds-number excess, multiplied by half the order, while the intermittency correction pulls the other way. At a laboratory Reynolds number a fitted sixth-order exponent therefore shows less than two-thirds of the anomaly the field carries, and a fourth-order one less than a quarter.

turbulence · Structure function

Named alongside it

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

DissipationMeasurementModel limitInertial rangeSpectrumStructure functionThe Kolmogorov scaleScaling exponentTurbulenceCorrelationReynolds numberAveraging

All concepts