Concept

Isotropy — where it appears

The property of having no preferred direction: every statistic of the field is unchanged by rotating the frame. It is assumed in almost every turbulence result and holds exactly in none of them, which is why relations derived from it alone are the sharpest available test of how far a real flow is from it.

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

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 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

Named alongside it

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

Dissipation Inertial range Structure function Cascade Dimensionless Intermittency Kolmogorov's theory Model limit Spectrum Batchelor scale Conservation Correlation

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