Frames that share their eddies count as one

A snapshot counts areas, and a flow counts its edge counted the independent values in a single frame of a turbulent field: one for every integral area the frame covers, for a scalar — about 45 in a frame 64 vectors square at five vectors to an integral length — twice that for a plane cut through three-dimensional flow, and ten times more for a two-dimensional incompressible flow, whose window mean is set by the streamfunction round the window’s edge. It ended where every real measurement begins. Nobody takes one frame. A particle-image velocimetry run is a sequence of frames, a simulation is saved as a sequence of snapshots, and frames closer together than an integral time photograph eddies they share.

A record is as long as its integral scales gave the rule for a time series at a point: one independent value for every two integral time scales, however finely it is sampled. The question here is what the rule becomes for a sequence of fields, and the answer turns out to depend on something a probe never sees — how the pattern moves relative to the window, rather than how fast it changes.

A field that moves and a field that forgets #

A turbulent field changes in a fixed window for two reasons. The mean flow carries the pattern through the window, which moves eddies in at one side and out at the other without changing them; and the pattern itself evolves, eddies straining and breaking and forgetting their earlier shape. Taylor’s frozen-turbulence hypothesis keeps only the first, and it is the reason a probe in a wind tunnel can be read as a spatial record at all. The second is what a probe riding with the mean flow would see.

The simplest space-time field with both is a frozen pattern swept at a speed U and forgetting at a rate 1/T:

$$\rho(x, y, t) = \rho_s(x - Ut,\; y)\, e^{-|t|/T},$$

where $\rho_s$ is one of the snapshot essay’s three spatial correlations. Two numbers then set everything: how far the pattern moves between frames, in integral lengths, and how many memory times pass between them. Their ratio is $UT$, the distance the mean flow carries an eddy in one memory time. In a wind tunnel with a 10 m/s stream and eddies 10 mm across that forget themselves in about the time a velocity fluctuation of 1 m/s takes to cross them, $UT$ is 100 mm — about ten integral lengths. Turbulence in a moving stream is swept much faster than it forgets, which is why the frozen hypothesis works as well as it does.

Every spatial correlation here is a sum of products of a function of x and a function of y, and the sweep acts only on x. The variance of the mean over an L × L window and N frames is therefore an exact double sum over pairs of frames: each pair contributes its time factor $e^{-|k|\Delta t/T}$ times the window sum along x shifted by the distance the pattern moved between them, $Uk\Delta t$, times the window sum across. No simulation is needed to compute it; one is needed to check it, and was.

A frozen run counts its swept area #

Start with the frozen limit, where the eddies never change and only the sweep matters. The figure fixes the run by how much pattern passes the window — 100 integral lengths — and varies the interval between frames. With a frame every half integral length the scalar run holds 353 independent values; with one every four integral lengths, 362; with one every window width, 388. Over a factor of twenty-five in frame rate the count barely moves. It is the area of the strip the window has swept — 100 plus 12.8 integral lengths long and 12.8 wide — divided by the four square integral lengths that one independent value of a scalar field occupies, which comes to about 360. A frozen run photographs a strip of pattern, and the strip holds what it holds however many times any part of it is photographed.

The count falls only when the interval passes a window width, because then gaps open between frames and pattern goes past unphotographed; at twenty integral lengths between frames it is down to 271. The plane cut behaves identically at about twice the count, 670 against 353, as it did for a single frame.

The dashed line is the count a record’s rule would give. A record counts one independent value per two integral time scales, and a frame interval of two integral lengths of sweep is two integral times of a probe fixed in the stream. Treat those 51 frames as independent, each worth one frame’s 45 values, and the run holds 2,075 — 5.8 times the truth. The error is a factor of the window width over twice the integral length, and it is the factor by which frames overlap. A frame is not a probe: it samples the pattern over a width, and a pattern swept one integral length has moved only a thirteenth of the way through the window.

With no mean flow, frames count like a record #

The other limit has no sweep at all: a window fixed in a flow with no mean, as in a stirred tank, a convection cell or a decaying box of turbulence photographed from its centre. Now the only way frames become independent is for the pattern to forget itself. The figure takes a run of 40 memory times and varies the interval. The count rises as frames are added until they are about a memory time apart and then stops: 873 at one memory time, 927 at a tenth. The record’s rule — one independent frame for every two memory times, each worth one frame’s count — gives 902, and the run’s count sits on it. A field that does not move is a record of fields, and the record’s rule carries over exactly, with the frame’s own count as the unit.

The frame rate that stops buying #

Real flows sit between the two limits, and the practical question is where a faster camera stops buying independent values. The figure computes it: the longest interval between frames at which a long run still collects nine-tenths of the values continuous filming would give, as the sweep in one memory time rises from a hundredth of a window width to a hundred.

With no sweep the interval is 1.17 memory times: frames closer than about a memory time are the same eddies photographed again. With a fast sweep it is 1.16 window crossings — the time for the mean flow to carry the pattern across the window once and a little more — and it has nothing to do with the turbulence’s own scales at all. The change between the two happens where the pattern crosses the window in about one memory time, $UT \approx L$.

That puts a number on something PIV practice knows qualitatively. For the mean and for anything whose error is set by the integral scales, a frame rate faster than about one frame per window crossing, or one per memory time if that is shorter, buys nothing. The gain from a time-resolved system is real for quantities that live at short times — accelerations, the spectrum at high frequency, the moment a spectrum cannot hold — and absent for the statistic most runs are used to report.

A flat flow is fixed by its ends #

The snapshot essay’s most surprising result was that a two-dimensional incompressible flow’s window mean converges a power faster than its area allows, because the mean of $u = \partial\psi/\partial y$ over a window is a streamfunction difference between its top and bottom edges, divided by its height. A frozen run carries that argument into time, and the sweep direction splits the two components.

The streamwise component u is a derivative across the sweep. A frozen run’s mean of it is the streamfunction difference between the top and bottom edges of the swept strip, averaged along the strip’s length — and the strip’s edges are long, so their average converges as the inverse of the strip’s length, like an area. Its variance falls as the inverse of the run, slope −0.98 in the figure, starting from the single frame’s advantage and keeping it.

The cross-stream component $v = -\partial\psi/\partial x$ is a derivative along the sweep. Its run mean is the streamfunction’s difference between the strip’s two ends, divided by the strip’s length, and the ends do not grow as the run does. Its variance falls as the inverse square of the run, slope exactly −2: a run twice as long gives a mean four times as precise, where any area average gives twice. At 160 integral lengths of sweep its variance is a hundredth of the streamwise component’s.

That advantage belongs to a pattern that does not change. Let the pattern forget itself over 25 integral lengths of sweep and the two ends of the strip are no longer ends of one frozen streamfunction; each stretch of the run carries ends of its own, and past a few memory times the cross-stream variance bends back to the inverse of the run. The inverse square is a statement about a frozen pattern, and it holds for runs shorter than the pattern’s memory. One function instead of two is the reason a flat flow has a streamfunction at all; this is the second place in two essays that the streamfunction’s existence turns a counting problem into a boundary problem.

A second of PIV #

A worked case makes the scales concrete. A planar PIV system looks at a field of view 128 mm square in a wind tunnel running at 10 m/s, with an integral length of 10 mm and eddies that forget in 10 ms; the measurement is a plane cut through three-dimensional turbulence, so each frame holds about 86 independent values of the mean streamwise velocity. The flow crosses the window in 12.8 ms and forgets in 10, so the sweep and the memory are comparable, $UT/L$ about 0.8.

At 15 Hz — a common double-frame PIV system — frames are 67 ms apart, far more than either time, and every frame is independent: one second holds 1,285 values, 15 frames times 86. At 150 Hz the second holds 8,471, which is nine-tenths of all it can ever hold. At 5 kHz, a time-resolved system taking thirty-three times as many frames as 150 Hz, it holds 9,425. For the mean, the last factor of thirty in frame rate buys eleven per cent. The 15 Hz system run for seven and a half seconds holds as much as the 5 kHz system run for one.

The count also says how to state an uncertainty honestly. The standard error of a run’s mean is the point standard deviation divided by the square root of the count in the figure, not of the number of frames and not of the number of vectors. A 5 kHz second has 5,000 frames and about twenty million vectors, and holds under ten thousand independent values of the mean.

A periodic box sweeps nothing #

The sweep rule has a consequence for simulations that is easy to miss, because a simulation’s window is usually the whole domain. A direct simulation of a channel or a boundary layer is run in a box that is periodic in the streamwise direction, and statistics are averaged over the box and over saved snapshots. The mean flow sweeps the pattern through the box — and out of the far end, and back in at the near one. A frozen pattern swept through a periodic box is the same pattern translated, and its average over the whole box is unchanged by translation. For a whole-box mean, the sweep buys exactly nothing, however many flow-through times pass.

So a periodic simulation’s snapshots become independent only by memory, never by sweeping, and the record’s rule applies with the largest eddies’ memory time — which for the long structures of a wall flow can be comparable with, or longer than, the time the mean flow takes to cross a typical box. Snapshots saved every flow-through time, a common choice, can share their largest eddies for many saves in a row. The same argument does not apply to a subwindow of the box, which is swept like a PIV frame, nor to a wind tunnel, whose pattern never comes back. The constants of the one exact result and the correlations behind a dissipation correlated across every scale are usually reported from exactly such boxes, and an error bar on them has to be counted in memory times.

There is a physical counterpart. A wing flying through a frozen gust pattern is a window of one dimension being swept, and the span takes away the infinity found the same geometry doing a different job: there the span’s average removes the small scales from the load, where here the window’s average removes them from the count. And a flat flow whose integral length keeps growing, as in the cascade that runs backwards, has no fixed window width in integral lengths at all; its count per frame falls as its eddies grow, and a run in it must be counted against the integral length of each moment rather than of the run.

What was checked #

Two limits are identities: a single frame must be the snapshot essay’s window variance, and ten frames that neither overlap nor remember must divide it by ten. Both hold to rounding error. The double sum was then checked against Monte Carlo runs: fields generated spectrally on a periodic 256 × 256 lattice, each frame the previous one shifted by the sweep and mixed with a fresh field in the proportion that makes consecutive frames correlate by exactly $e^{-\Delta t/T}$, with the mean taken over four 32-sample windows per field far enough apart to be independent. For the frozen scalar, both components of the frozen flat flow and the forgetting scalar, the Monte Carlo variances are 1.02, 1.04, 1.07 and 0.89 of the double sum: within the sampling error of 240 to 480 run means, which is about nine per cent at one standard deviation. The tests also refuse a run of no frames, a sweep that runs backwards against a fixed window and a velocity component the model does not have.

What the model leaves out #

A single memory time. The space-time correlation multiplies a frozen pattern by one exponential. Real eddies forget at rates that depend on their size — small ones quickly, large ones slowly — so the large scales that dominate a mean are frozen for longer than the small ones, and a scale-dependent memory would move the threshold towards the sweep-dominated side for the mean and away from it for gradients.

Uniform sweeping. Every eddy moves at the mean speed. In a shear flow the sweeping speed varies across the window and the large eddies move at their own convection velocity rather than the local mean; the frozen strip becomes a sheared one, and a sheared strip mixes the two components’ behaviours.

Gaussian statistics and one correlation shape. As in the snapshot essay, the variance of a mean depends only on the correlation, and the exponents here — the swept area, the record’s rule, the inverse square — are set by geometry rather than by the Gaussian’s form.

Point samples in time. A PIV frame is an average over the two exposures of a pair, which is short enough to ignore for a mean; a time-resolved system’s own exposure is a filter in time of the kind the window every vector is averaged over is in space.

A camera is not a probe #

The picture this run of calculations has built up is that a measurement’s independent values are counted in the currency of whatever it integrates over. A probe integrates over time, and counts integral times. A frame integrates over area, and counts integral areas — unless the flow has a streamfunction, when it counts the edge. A run of frames in a moving stream integrates over the strip it sweeps, and counts that strip’s area, which is why the frame interval that matters is the window crossing time rather than the integral time a probe would suggest. The error in borrowing a probe’s rule for a camera is not a small correction; in the case drawn it is a factor of six, and it grows with the field of view. The best estimate of a scale assumes its shape found that the integral scale itself is hard to measure; the reassuring part here is that the threshold for a camera in a fast stream does not need it.

Who worked it out #

Taylor proposed the frozen-turbulence hypothesis in 1938, in the paper on the spectrum of turbulence that made a probe’s time record a spatial one. The space-time correlation as a swept pattern multiplied by a decay is the simplest of a family of models, with Kraichnan’s 1964 sweeping analysis and He and Zhang’s 2006 elliptic model among the more refined. The effective sample size of a correlated sequence is Bayley and Hammersley’s, from 1946, and the uncertainty of PIV statistics, including the reduction from correlated samples, was systematised by Sciacchitano and Wieneke in 2016.

Still open: a memory that depends on size #

The single memory time is the model’s weakest part, and it matters in exactly the regime a PIV run most often sits in, where the sweep and the memory are comparable. Large eddies, which carry most of a mean’s variance, are frozen for longer than the integral time suggests, and small ones for less. The next calculation replaces the one exponential with a memory that scales as each wavenumber’s own turnover time — the sweeping-decorrelation form Kraichnan’s analysis gives — and asks how far the frame rate that stops buying moves, and whether the count for a frame rate in the crossover is then set by the large eddies’ memory or by the window. The answer decides whether the rule drawn here can be applied with the integral time a probe measures, or needs a separate measurement of how long the largest eddies live.