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

Probability distribution — where it appears

The full statistical description of a fluctuating quantity, of which the mean and the variance are two numbers. In turbulence and in chaotic stretching the distribution is broad enough that its moments are dominated by rare events, so an average is a measurement of a tail rather than of a typical case.

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

One rate per moment, and none of them the same. lambda_p = ln<l^p>/(p t) against p. As p goes to zero it is the Lyapunov exponent, the rate of the typical element; at p = 1 it is the rate of the average length, which is nearly twice as large. If ln l were exactly Gaussian this would be a straight line with the Lyapunov exponent as its intercept, and the departure from that line is the same multifractality the velocity increments have.

The stretching rate that is not one number

A material line in a flow gets longer, and there is a theorem saying its length grows at a definite exponential rate. There is also a rate at which the average length grows, and it is nearly twice as large — and a different rate for every moment of the distribution.

kinematics · Material lines 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 Two different moments of one distribution. The permeability and the specific surface of a log-normal bundle, against the width of its pore-size distribution at a fixed median. The permeability rises by five orders because it is a fourth moment and the widest tubes dominate it; the surface falls because it is a first moment and the narrowest tubes dominate that.

A permeability that is only the geometry

Kozeny–Carman says that a porous medium's permeability follows from its porosity and its specific surface. Both are real, both are exactly measurable, and they do not determine the answer: forty-nine tube bundles built with identical values of each span a factor of eighty-two in permeability.

applied · Porous

Named alongside it

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

MomentsModel limitABC flowAveragingChaotic advectionCorrelationDarcy's lawDeformation gradientErgun's equationFlatnessFour-fifths lawFractal dimension

All concepts