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

Ensemble — where it appears

A population of realisations — of initial conditions, of starting points, of perturbed forecasts — whose statistics are the quantity being reported. A single number quoted for a flow is usually an average over one, and the spread is frequently larger than the mean.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

A millionth, doubling every three-quarters of a second. The separation of two trajectories started a millionth apart, on log axes against time. It grows as a straight line until it saturates at the size of the attractor, and the slope of that line is the largest Lyapunov exponent — measured here on the system by renormalising a nearby pair, not quoted. Sensitive dependence is what the straightness of the line means.

The randomness that is not in the equations

Turbulence is described in the language of statistics — means, variances, spectra, probability distributions — and none of that language appears in the equations it is a description of. The Navier–Stokes equations have no random term anywhere in them. What is random is the observer's ignorance of the initial data, and the flow's habit of amplifying it.

misconceptions · Randomness An hour for every tenfold, for ever. How long a forecast lasts, against how well the initial state is known. The relation is T = ln(tolerance/error)/lambda — exactly logarithmic — so improving the measurement by a factor of ten buys exactly the same extra time every time: ln(10)/lambda, which for this flow is 24.5 time units. It does not get harder and it does not get easier.

An hour for every tenfold

Turbulence is deterministic and unpredictable, and the exchange rate between those two is exact: measuring the initial state ten times better buys the same extra forecast time every time, for ever. A constant, and it belongs to the flow rather than to the instrument.

misconceptions · Randomness What a probe in a flame sees. Half the time hot light fluid at a hundred metres a second, half the time cold heavy fluid at twenty. That is what intermittency in a jet flame looks like at a point, and it is the simplest field in which the two averages of the velocity are different numbers.

Two averages of one flow

In a flow whose density varies there are two mean velocities, they are both correct, and across a flame they differ by a factor of two. One of them is what a hot wire returns; the other is what every compressible turbulence model is written in; and the mass flux is the single product they agree on.

kinematics · Averaging A record's length is counted in integral scales. The scatter of a record's mean, in units of the signal's own standard deviation, and the relative scatter of its variance, against the record's length in integral time scales: four hundred records at each of seven lengths, against the exact results. Both fall as the square root of the number of integral scales, √(2Tᵢ/T). A mean known to one per cent of σ needs twenty thousand integral scales. The exact variance curve is for the variance about the true mean; a short record can only measure it about its own mean, which is why the shortest records scatter less than the curve says.

A record is as long as its integral scales

A turbulence measurement of a million samples can hold less information than one of a thousand. What sets a record's worth is not how many numbers it contains but how many integral time scales it spans, and the integral scale — the unit every other error is counted in — is itself the hardest thing in the record to measure.

turbulence · Spectrum

Named alongside it

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

MeasurementAveragingChaosConvergenceCorrelationMixingPredictabilityStatisticsTurbulenceAdvectionClosureClosure problem

All concepts