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

Randomness — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · misconceptions
  2. 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.

    part 2 · misconceptions
  3. What the phase reaches, and what it does not. The difference between the two records, as a fraction, for five quantities. The variance and the autocorrelation are the same to machine precision because they are the spectrum. A narrow-band linear oscillator answers its own frequency and almost nothing else, so it is nearly phase-blind too. Everything extremal — the crest, the peak drag load, the range of the running integral — is not.

    The same statistics, and a different load

    A wind or wave specification is written as a spectrum, and a spectrum discards the phases. Two records built from one spectrum agree in variance to thirteen figures and in peak drag load by twenty per cent — and with the phases lined up, the same spectrum is a single impulse thirty-one times worse.

    part 3 · misconceptions

All series