Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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 2 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

Named alongside it

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

IntermittencyModel limitStructure functionCorrelationDissipationFlatnessFour-fifths lawFractal dimensionInertial rangeIsotropyThe Kolmogorov scaleLog-normal

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