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

Spectrum — the series

9 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. One spectrum, two fields. Two one-dimensional fields built from the same amplitudes and different phases. Their energy spectra are identical to the last bit, their two-point correlations agree to 2e-15, their second-order structure functions are the same function — and no measurement of either kind can tell them apart. Everything that distinguishes them is in the phases, which is where the cascade lives.

    The moment a spectrum cannot hold

    A spectrum and a two-point correlation are the same object, transformed, so everything one of them records the other records too. Give a field the exact Kolmogorov amplitudes and independent phases and every second-order measurement comes out right — while the cascade, which lives in the phases, is not there at all.

    part 1 · turbulence
  2. Model spectra at four Reynolds numbers, compensated. The spectrum multiplied by k^(5/3) and divided by eps^(2/3), so that a true inertial range is a horizontal line at the Kolmogorov constant. What a finite Reynolds number has instead is a single maximum: it reaches 1.4996 at the highest and 1.49 at the lowest, and the band over which it is flat to one per cent goes from a third of a decade to two.

    The range a real Reynolds number does not have

    Kolmogorov's minus five thirds is a statement about a band of scales that has forgotten the forcing and does not feel the viscosity. Both conditions are about separation, and separation is exactly what a finite Reynolds number does not have much of.

    part 2 · turbulence
  3. 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.

    part 3 · turbulence
  4. On a smooth signal, sampling faster makes the lag-one scale longer. The average estimate of the integral scale from records three hundred scales long of the smooth process, against the number of samples per integral scale. The first zero and the exponential fit do not care how fast the record was sampled. The lag-one estimate grows in proportion to the sampling rate, because it is reading the curvature of the correlation at zero lag — the microscale — and dividing by the sample spacing.

    The best estimate of a scale assumes its shape

    There are four common ways to read an integral time scale off a turbulence record, and on the right signal the best of them is four times more precise than the usual one. On a signal whose correlation has a different shape it is off by a factor that no length of record reveals — and one of them turns out to be measuring the sampling rate rather than the flow.

    part 4 · turbulence
  5. A flow's window mean converges a power faster. The variance of the mean over an L × L window, as a fraction of the point variance, against the window's side in integral lengths, on logarithmic axes. The scalar field's falls as the inverse area. The plane cut through a three-dimensional flow falls the same way at half the level. The velocity of a two-dimensional incompressible flow falls as the inverse cube of the side, because its mean over any window is a streamfunction difference around the window's edge. Lines are closed forms; dots are Monte Carlo means over fifty generated fields.

    A snapshot counts areas, and a flow counts its edge

    A record at one point holds one independent value for every two integral scales it lasts. A snapshot of a field holds one for every integral area it covers, so at the same number of samples it holds far fewer, and sampling it more finely adds nothing. Except for a velocity in a two-dimensional incompressible flow, whose mean over a window is fixed by the window's edge alone: it converges a whole power faster than its area allows, and a large enough snapshot of it beats the record.

    part 5 · turbulence
  6. A frozen run counts the area it sweeps. Independent values in a run of frames while a frozen pattern is swept 100 integral lengths past a window 12.8 integral lengths square, against the distance the pattern moves between frames. For a scalar the run holds 353 values with a frame every half integral length, 362 with one every four and 388 with one every window width: the swept area in integral areas, whatever the frame rate. Only past a window width, when gaps open between frames, does the count fall. Treating frames an integral scale or two apart as independent, as a record would allow, counts 2075 at two integral lengths — 5.8 times too many.

    Frames that share their eddies count as one

    A particle-image run is a sequence of snapshots, and snapshots closer together than an integral time photograph the same eddies. When a mean flow carries a frozen pattern past the window, a run holds exactly the area it sweeps, counted in integral areas, and a faster camera adds nothing until frames stop overlapping — a threshold set by the window, not by the turbulence. In a flat flow one component escapes the rule: its run mean is fixed by the pattern at the two ends.

    part 6 · turbulence
  7. Small scales forget fast and large ones slowly, and the window only sees the large. Each scale's memory over a single probe's integral time against wavenumber in inverse samples, for one memory at every scale and for memories falling as the wavenumber to the minus two-thirds and minus one above the field's own wavenumber, 1/λ, and held there below it. All three give the same probe integral time; the window of 64 samples passes wavenumbers below about 2π/64 = 0.098, where the sweeping law's memory is 1.26 probe times and the single law's 1.

    The camera needs the window's memory, not the probe's

    A run of particle-image frames stops gaining independent values for its mean once frames are closer than the flow's memory — but the memory that counts is the window mean's, and when large eddies live longer than small ones it is longer than the integral time a single probe measures. Give each scale its own memory and the frame rate that stops buying moves by up to a quarter where the flow forgets before it is swept, by nothing where the sweep sets it, and by however much longer the largest eddies live. Fed the window's own memory, the single-memory rule gives it back to two per cent.

    part 7 · turbulence
  8. Zoomed into the inertial range, a camera needs frames faster than its window's memory says. The longest frame interval at which a run of window means still collects nine tenths of what continuous filming would, in probe integral times, against the window's diameter in integral scales, with no sweep, for one memory at every scale and for turnover and sweeping memories; dashed, the single-memory rule, 1.17 times the window mean's own memory, for the sweeping law. With the sweeping law the interval is 2.45 probe times for a window ten integral scales wide, 1.67 for one scale and 0.803 for a hundredth, where the rule says 1.18; with one memory it is 1.17 at every size.

    A camera zoomed into the cascade needs frames faster than its window remembers

    A wide camera window averages away the small eddies, so its mean remembers what the large eddies remember, and a run of frames is worth as many independent values as that memory allows. Zoom the window into the inertial range and the opposite is not what happens. A small window keeps nearly all the variance and almost exactly a single probe's memory — every eddy larger than itself still fills it — but much of what it keeps is the cascade's eddies, which forget far faster than the largest ones. Its frames then stop buying at an interval a third shorter than its own memory says, and a velocity gradient in the same window needs frames thirty times faster.

    part 8 · turbulence
  9. Noise in every frame pulls in the frame interval that stops buying, by the same share whatever the window sees. The longest frame interval at which a run still collects nine-tenths of continuous filming's independent values, over its value with no noise, against the noise's share of the window mean's variance, for a velocity window one integral scale wide, one a twentieth as wide, and a gradient in the small window; dashed, a single exponential memory in closed form. velocity, window ℓ: 0.782 at ν = 0.1, 0.199 at ν = 1; velocity, window ℓ/20: 0.804 at ν = 0.1, 0.243 at ν = 1; gradient, window ℓ/20: 0.819 at ν = 0.1, 0.242 at ν = 1.

    Noise in every frame is a memory of zero, and the run cannot measure it at its own pace

    A camera that adds an independent error to every frame adds a part of the signal with no memory at all. Infinitely many frames would still average it away, so the run's information still approaches continuous filming's — but the frame interval at which a faster camera stops buying is pulled in by the same share for a wide window, a zoomed one and a gradient, about half at a noise of three-tenths of the variance and a fifth at a noise equal to it. And the usual way to learn the noise, from the run's own correlation extrapolated to zero lag, works only at frames much faster than the run's information needs.

    part 9 · turbulence

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