Concept

Scaling exponent — where it appears

The power to which a separation is raised in a structure function's dependence on it. Dimensional reasoning gives one third of the order for every moment; measurements give less at every order above the third, and the shortfall is what intermittency means.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

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

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

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.

Intermittency Model limit Inertial range Structure function Dissipation The Kolmogorov scale Measurement Probability distribution Reynolds number Spectrum Correlation Flatness

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