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

Structure function — where it appears

The average of a power of the velocity difference across a separation, which is the real-space counterpart of a spectrum. The second order carries the same information as the spectrum; the third carries the energy flux and is fixed exactly by Kolmogorov's law.

Named by 11 essays across one field — each of them below, with the objects they name alongside it.

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 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 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 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 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 The plateau everybody looks for is a summit, and a low one. −Dₗₗₗ/((4/5)εr) against separation in decaying turbulence at five Taylor-scale Reynolds numbers. None has a plateau at one. Each rises through the viscous range and turns over, peaking at 0.49, 0.63, 0.75, 0.85, 0.90 for Reλ = 50, 100, 200, 500, 1000. A measurement of ε that takes the largest value of this curve as four-fifths reads each of those shortfalls as a smaller dissipation.

The decay inside the four-fifths law

The four-fifths law gives the dissipation of turbulence from one measured moment with no constant in it, which makes it the obvious way to measure how the dissipation coefficient travels during a decay. But the law is exact only in a limit, and a decaying flow is not in it. The decay itself takes a quarter off the moment at the Reynolds numbers grids reach — and the bias moves as the flow decays, by a sixth, in the direction opposite to the effect being looked for.

turbulence · Decay Forced at the integral scale, the third moment nearly reaches four-fifths. The Kármán–Howarth balance at a Taylor-scale Reynolds number of 200, on the same model spectrum, for a flow forced in a band at the spectrum's peak and for one decaying. Each term is divided by (4/5)εr and plotted against separation in Kolmogorov lengths. The viscous term is the same for both. The forcing term is negligible until the separation approaches the integral scale, and −Dₗₗₗ/((4/5)εr) for the forced flow peaks at 0.965 at 90 Kolmogorov lengths, where the decaying flow's peaks at 0.747 at 32.

Forcing at the integral scale leaves the cascade alone

A decaying flow's third moment falls a quarter short of the four-fifths law at the Reynolds numbers a grid reaches. A flow forced at its largest scales, on the same spectrum at the same Reynolds number, falls short by three and a half per cent. The shortfall is not what a finite Reynolds number does to every flow; it is what the source does, and how fast it closes is set by how far the source reaches into the inertial range.

turbulence · Decay Half the forced summit is gone in a fifth of an eddy time. The third moment of the velocity differences as a fraction of four-fifths εr, against separation in Kolmogorov lengths of the forced flow, at the instant the forcing is switched off and 0.1, 0.2, 0.5, 1 and 4 large-eddy times later. Forced, the curve peaks at 0.94 near 67 η. A tenth of an eddy time later the summit is 0.899, at a fifth 0.838, at a half 0.759, and after that it hardly moves: 0.742 at one and 0.72 at four, where the flow is simply decaying. The large separations fall first and farthest.

The summit forgets the forcing before the dissipation does

A flow forced at its largest scales carries the four-fifths law closer to exact than a decaying one. Switch the forcing off and the third moment has to pass from one value to the other. It does not wait for the cascade to drain: half the forced summit is gone in a fifth of an eddy turnover, while the dissipation has hardly moved, and the largest separations overshoot the decaying value before they settle on it.

turbulence · Decay 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 However long its forcing remembers, a stationary flow's averaged four-fifths summit stays at 0.94. The summit of −Dₗₗₗ/((4/5)εr) computed from the time-averaged spectrum and source — the third moment averaged, then divided by the averaged dissipation — against the forcing's correlation time in large-eddy times, for a forcing amplitude whose logarithm wanders with standard deviation 0.5 and 1, over runs of 200 to 400 eddy times. The steady forcing's summit is 0.94; across correlation times from 0.03 to 10 eddy times and both spreads the averaged summit lies between 0.9371 and 0.9438, within its batch errors of the steady value.

A wandering forcing leaves the averaged four-fifths law alone

A forced flow's third moment loses half its excess within a fifth of an eddy time of the forcing stopping, so a forcing that wanders ought to drag a simulation's four-fifths ratio about with it. Instantaneously it does: the summit swings from 0.8 to 1.4. Averaged it does not — the third moment averaged and then divided stays at the steady forcing's 0.94 for every correlation time from a thirtieth of an eddy time to ten. The reason is that the balance is linear and stationarity fixes its source; what the forcing's memory changes is only the answer to a different question, the average of a ratio.

turbulence · Decay

Named alongside it

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

DissipationInertial rangeSpectrumCascadeFour-fifths lawModel limitIntermittencyMeasurementReynolds numberClosureCorrelationDecay

All concepts