Concept

Cascade — where it appears

The passage of energy from the large scales of a turbulent flow to the small ones, at a rate set by the largest. It is a flux rather than a shape, so it is carried by the third moment of the velocity differences and is invisible to any spectrum.

Named by 9 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

One spectrum, two fields. Two one-dimensional fields built from the same amplitudes and different phases. Their energy spectra are identical to the last bit, their two-point correlations agree to 2e-15, their second-order structure functions are the same function — and no measurement of either kind can tell them apart. Everything that distinguishes them is in the phases, which is where the cascade lives.

The moment a spectrum cannot hold

A spectrum and a two-point correlation are the same object, transformed, so everything one of them records the other records too. Give a field the exact Kolmogorov amplitudes and independent phases and every second-order measurement comes out right — while the cascade, which lives in the phases, is not there at all.

turbulence · Spectrum

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

Two things moving by a million, and their product standing still. Viscosity, the squared velocity gradient at the dissipation scale, and their product, across six decades of Reynolds number at a fixed large-scale flow. The viscosity falls by a factor of 1e+6; the squared gradient rises by exactly the same factor, because η falls as Re^(−3/4) and u_η as Re^(−1/4); and the dissipation ν(u_η/η)² does not move at all. That is the dissipation anomaly stated as arithmetic: the limit of the dissipation as viscosity vanishes is not the value it takes when viscosity is zero.

The limit that is not the value

Dissipation is viscosity times the square of a velocity gradient, so it ought to vanish as the viscosity does. It does not. The gradient rises by exactly the factor the viscosity falls by, the product stands still, and a fluid with no viscosity at all would dissipate nothing — which is why the limit and the value are different numbers.

turbulence · Dissipation

Deviation: how far the flow leaves from the blade angle, against solidity. The angle between the outlet flow and the blade, for a row of flat plates at 30° stagger meeting a flow at 45°. An open row barely turns the flow at all and the deviation is nearly the whole of the intended turning; a tight row guides it to within a thousandth of a degree of the blade angle. Nothing about the blade changed between the two ends of this curve.

A row is not a set of aerofoils

An isolated aerofoil's incidence is measured from the free stream. A compressor blade's cannot be, because an infinite row of identical blades above and below it has a circulation that is part of its own free stream — and the velocity the theorem uses is a vector mean that exists nowhere in the machine.

circulation · Cascade

The production jumps and the dissipation does not. Production and dissipation against time, through a step change in the strain rate. The production follows the strain immediately — it is the strain squared times an eddy viscosity — and the dissipation moves by less than one per cent at the instant of the step, because it is set by a cascade that has not been told yet.

A dissipation that lags its production

Change the strain rate on a patch of turbulence and the production of energy follows instantly — it is the strain squared. The dissipation moves by less than one per cent, because it is the far end of a cascade that has not been told yet, and the two are out of balance by a factor of six for the next turnover.

turbulence · Dissipation

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

What a downstream blade sees going past. The axial velocity a blade in the second row meets, over one revolution, as it passes through the wakes of thirty upstream blades. Each dip is one wake, and the blade meets all thirty of them every time it goes round.

A row that meets the row before it

A compressor blade is loaded and unloaded thirty times a revolution by geometry it does not have. The forcing sits at the blade count of the row in front of it and at multiples of that, and how far up the harmonics it reaches is decided by how far apart the two rows are.

circulation · Cascade

Four over the dimension, and three dimensions is where the three comes from. The constant in front of the exact law, against the number of dimensions the flow lives in. Both exact results — Yaglom's for a scalar and the velocity's law for the mixed third moment — have this same constant, because both come from the same statement: an isotropic radial flux in separation space whose divergence is a constant sink. Integrating that divergence gives 4Q r/d and nothing else. The four-thirds everybody quotes is four over three, and the three is the space rather than anything about turbulence. The dots are the quadrature, which agrees with the closed form to 2·10⁻⁹.

The fraction that is really four thirds

Turbulence has two exact results, not one. The second is about a scalar carried by the flow, its constant is four thirds rather than four fifths, and the difference between the two fractions has nothing in it about turbulence at all — it is the price of writing a three-component object in terms of one component.

turbulence · Structure function

Named alongside it

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

Dissipation Turbulence Inertial range Kolmogorov's theory Spectrum Structure function Conservation Dimensionless Measurement Memory kernel Model limit Model validity

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