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

The camera needs the window’s memory, not the probe’s asked how often a particle-image camera should take frames of a turbulent field, if each frame’s window mean is to add information to the run. Frames that share their eddies count as one had answered it with one memory for every scale: frames closer than about 1.17 integral times apart share their eddies and add little. The camera essay gave each scale its own memory, the large eddies living longest, and found that a wide window — 12.8 integral lengths — averages away the small scales, keeps the large ones and so remembers longer than a probe does, 1.26 probe times under the sweeping law. The rule came back if it was given the window’s own memory instead of the probe’s.

It ended on the other end of the memory law. A camera zoomed in on a small region, or one measuring a quantity weighted towards small scales such as a velocity gradient, sees the eddies that forget fastest. The question was how the frame interval that stops buying changes when a window looks into the inertial range.

The answer is not the mirror image of the wide window. A small window does not take on the small eddies’ memory: the large eddies still fill it, and its mean remembers almost exactly what a probe does. But the window’s correlation is no longer a single decay. It drops quickly at first, as the small eddies it holds forget, and slowly afterwards, and a run of frames has to sample that quick drop. The frame interval that stops buying falls a third below what the window’s own memory predicts, and for a gradient, whose variance lives in the small eddies, zooming in is everything.

A field with a cascade in it #

The earlier essays’ fields were Gaussian-correlated: their spectra fell off steeply above the integral wavenumber, with no inertial range to zoom into. This field has one. Its two-dimensional spectrum is $\Phi(k) \propto (1 + (k\ell)^2)^{-4/3}\, e^{-(k\eta)^2}$, so its one-dimensional spectrum falls as $k^{-5/3}$ between the integral scale $\ell$ and a dissipative scale a thousand times smaller — the local slope between $k\ell = 20$ and $40$ is $-5/3$ to five parts in a thousand. A second field in the same flow is the first weighted by $k^2$: a velocity gradient or a vorticity, whose variance sits at the small end of the cascade. A spectrum written this way has no trouble with the moment a spectrum cannot hold: the dissipative cut-off keeps the gradient’s variance finite.

The memory is the camera essay’s. Each wavenumber forgets exponentially over its own time, $\tau(k) \propto k^{-p}$ above the integral wavenumber and the largest eddies’ memory below it, and the whole is scaled so that a single probe, which averages over every scale, reports an integral time of $T$. The exponent $p = 0$ is one memory for every scale; $p = 2/3$ is an eddy’s own turnover in the inertial range; $p = 1$ is the sweeping of small eddies past a point by the large ones, which is what a probe in a real flow mostly measures. The gradient shares the flow’s memory law, since it is the same eddies seen through a derivative.

The window is a disc of diameter $L$. That is a small departure from the camera essay’s square window: a disc’s passband is isotropic, so the window mean’s covariance becomes a single integral over the magnitude of the wavenumber,

$$C(t) = \int \Phi(k)\,\left[\frac{2J_1(kL/2)}{kL/2}\right]^2 J_0(kUt)\, e^{-t/\tau(k)}\, 2\pi k\,\mathrm{d}k,$$

with the sweep $U$ averaged over its direction. A logarithmic grid of five thousand wavenumbers resolves the window’s lobes from a window ten integral scales wide to one three thousandths of one.

The quantity computed is the earlier essays’: the share of continuous filming’s independent values a run of frames collects when the frames are $\Delta t$ apart, and the longest $\Delta t$ at which that share is still nine tenths — the frame interval past which a faster camera buys little.

The interval falls as the window shrinks #

The first figure is the result, for a window watching a flow that only forgets — no mean flow carrying the pattern across it. With one memory at every scale the interval is 1.17 probe times whatever the window, which is the earlier rule. With the sweeping law it depends strongly on the window. Ten integral scales across, 2.45 probe times: the wide window keeps only the long-lived large scales, as the camera essay found. One integral scale across, 1.67. A tenth, 0.97. A hundredth, 0.80.

The dashed curve is the single-memory rule fed the window’s own memory: 1.17 times the window mean’s integral time. For a wide window it is right, 2.46 against 2.45. For a zoomed window it is wrong, 1.18 against 0.80 at a hundredth of an integral scale. The rule that came back for the wide window fails for the narrow one, and it fails in the direction that costs data: a camera set to the rule’s frame rate would take frames half again too far apart and lose a tenth or more of what the run could hold.

The turnover law, $p = 2/3$, gives the same shape, less steeply: from 2.09 probe times at ten integral scales to 0.94 at a hundredth.

A small window keeps the probe’s memory #

The second figure shows why the rule fails. A window a hundredth of an integral scale across keeps 98.7 per cent of the point variance: it averages over a region in which almost everything varies together, because almost all the variance is in scales larger than the window. Its memory is 1.01 probe times — almost exactly a probe’s. One integral scale across, the window keeps only 52 per cent of the variance, the large scales, and remembers 1.56 probe times; ten across, four per cent and 2.11.

So the zoomed window’s mean is, to a first approximation, a probe’s reading, with the same variance and the same integral time. If one memory described it, the rule would hold, and the interval would be 1.17 probe times. It is 0.80 because the window’s mean is not a probe’s reading in the one respect the integral time hides: how that memory is distributed.

A fast fall, then a slow one #

The third figure draws the window mean’s autocorrelation, for a window a hundredth of an integral scale across under the sweeping law, beside the single exponential that has the same integral. At a fiftieth of a probe time the window’s correlation has already fallen to 0.924, where the exponential is at 0.980; at one probe time it is 0.308 against 0.373. The window’s correlation drops faster at first, as the cascade’s eddies it holds — those larger than the window and smaller than the integral scale, a fifth of the variance lying above ten integral wavenumbers, each living a small fraction of a probe time — forget, and then follows the largest eddies’ slower decay. Integrated, the two curves have the same area. Sampled, they do not behave alike.

A run of frames takes its independent values from the correlation’s shape near the origin as well as from its area. Frames spaced half a probe time apart have correlations that are lower than the single exponential says, so each frame holds a little more new information — but frames spaced more widely than the fast part’s own time miss that fast part altogether, and the run’s efficiency falls once the spacing passes it. The fast part sets the frame interval, and the slow part sets the memory: the rule, which reads only the memory, sees the slow part.

The dial widens the window. At a tenth of an integral scale the fast drop is smaller, at one scale smaller still, and at ten scales the window has averaged the cascade’s eddies away, keeping only scales larger than the integral scale, which all share the largest eddies’ memory. Its correlation is then a single exponential — which is why the rule holds there, and why it held for the camera essay’s wide window.

A gradient lives at the small end #

The fourth figure measures a velocity gradient through the same window, in the same flow, with the same memory law. A gradient’s variance is dominated by the small eddies, and its point integral time is only 0.0076 probe times — the dissipative eddies’ memory, under sweeping. For a gradient the window matters as it never did for the velocity: the interval is 0.028 probe times for a window a hundredth of an integral scale across, 0.13 for a tenth and 0.85 for one. A wide window averages the gradient’s small scales away and leaves only its weak large-scale part, which remembers long; a narrow one keeps them.

At a hundredth of an integral scale the gradient needs frames thirty times closer than the velocity does in the same window. A camera that measures vorticity or strain from the same images it measures velocity from is therefore sampling two quantities with very different memories, and a frame rate chosen for the velocity is far too slow for the gradient’s own statistics — a point the best estimate of a scale assumes its shape made in another form, about how far a statistic’s estimate depends on what the statistic weights.

A mean flow makes the crossing decide #

The fifth figure carries the pattern past the window at one integral scale per probe time, the situation of a camera looking at a flow with a mean speed through its window. The earlier essays found that a fast sweep makes the interval a fixed number of window crossings whatever the memory, since a pattern carried clear of the window between frames gives a new frame. The zoomed window reaches that state sooner. At a hundredth of an integral scale the interval is 0.43 probe times with one memory and 0.47 with the sweeping law — the memory law hardly matters. At ten integral scales it is 1.13 and 2.21, the memory law nearly doubling it.

The two limits are the two ways a frame becomes new: the pattern has moved, or it has forgotten. A small window’s pattern moves out of it quickly, so for any mean flow worth the name a zoomed camera’s frame rate is set by the crossing; only in a still tank or a recirculation zone does the fast-forgetting part of its variance decide it.

A tank, a camera and two frame rates #

Put numbers on it. A water tank stirred so that its integral scale is twenty millimetres and a probe’s integral time fifty milliseconds is watched by a camera zoomed onto a window two millimetres across, a tenth of the integral scale, with no mean flow through it. Under the sweeping law the window mean remembers 1.11 probe times, so the single-memory rule sets the frame interval at 1.30 probe times, 65 milliseconds. The computed interval is 0.97 probe times, 48 milliseconds: a camera run at the rule’s rate would collect 85 per cent of the independent values continuous filming would give, where the computed rate collects 90. If the same images are used for the vorticity, the interval is 0.13 probe times — six and a half milliseconds, a frame rate seven times the velocity’s — and a run at the velocity’s computed rate collects only 29 per cent of the vorticity’s.

For the dissipation, whose variance is weighted further towards the small end than the gradient’s and whose spatial structure is correlated across every scale, the requirement would be stricter still. The direction of the correction is the same in every case: the more a measured quantity is weighted towards small scales, the more its frame rate is set by the cascade’s fast memories rather than by its own integral time, and the less the integral time is a guide.

Why a small window is a probe #

The result that surprised is the second figure’s: a small window does not take on the small eddies’ memory. The reason is in where the variance of a turbulent field lives. In a cascade with a $k^{-5/3}$ spectrum the variance in scales smaller than $L$ grows only as $L^{2/3}$ — the second-order structure function’s law, a neighbour of the one exact result — so a window a hundredth of an integral scale across has very little variance finer than itself to average away: in the model it loses 1.3 per cent. Every eddy larger than the window is nearly uniform across it and passes into its mean whole. A small window’s mean is a probe’s reading with its finest grain averaged off, and a probe’s reading contains the whole cascade.

So the zoomed window’s failure of the rule is a probe’s failure too. A single probe in this field has an integral time of $T$ by construction, and four fifths of its variance lies in eddies smaller than the integral scale whose memories, under sweeping, are shorter than $T$ in proportion to their size; it too stops buying at about 0.78 probe times, not 1.17. The rule needs a single memory, and only a window wide enough to keep nothing but the largest eddies has one. The camera essay’s wide window restored it by averaging the cascade away; a zoomed window cannot.

That is also why a camera’s window and a probe agree on the integral time a measurement reports — the number a record is as long as its integral scales uses for a single probe’s record length — and still disagree with the rule on how often they must sample. An integral time adds the memories, weighted by variance. Sampling depends on how widely they are spread.

The convention: probe times and integral scales #

Times are in units of the probe integral time $T$ of the velocity — the integral of a single point’s autocorrelation. Window sizes are diameters, in units of the integral scale $\ell$ of the spectrum’s form. “Stops buying” means the run’s efficiency, its independent values per unit time over continuous filming’s, has fallen to nine tenths. The memory laws are exponents of $\tau(k)$ above the integral wavenumber; below it every scale has the largest eddies’ memory. The sweep, where there is one, is in integral scales per probe time, averaged over its direction relative to the window, which a disc’s symmetry permits.

What the model leaves out #

The spectrum is stated. A von Kármán–Kolmogorov form with a thousandfold inertial range is a model of a moderately high Reynolds number; a laboratory flow with a shorter range would put less of its variance in fast-forgetting scales and soften every effect here. The range a real Reynolds number does not have is the reason the range was made long rather than realistic.

The memories are exponential. Kraichnan’s sweeping gives each scale a Gaussian decay in time rather than an exponential one, which changes the correlation’s shape near the origin — exactly where the frame interval is decided. The direction of the result should survive; the numbers depend on it.

The window is a disc. A square window’s sharper corners pass a little more of the small scales; the camera essay’s square window and this disc agree in scale but not in detail.

Noise and resolution. A real particle-image system resolves only scales several times its interrogation spacing and adds measurement noise that has no memory at all — an extra fast part in every correlation, which would lower every interval here further.

Who asked it before #

The sampling of correlated records goes back to the integral-time rules that Taylor and Lumley’s textbook arguments made standard for single probes; the sweeping decorrelation of small scales is Tennekes’s of 1975, after Kraichnan; the question of how particle-image fields should be sampled in time was raised as soon as high-speed systems made the frame rate a choice. The finding here — that a window inside the inertial range keeps a probe’s memory but needs frames a third faster, and that the single-memory rule fails there while holding for wide windows — is this calculation’s, for the field and memory laws stated.

How it was checked #

The Bessel functions $J_0$ and $J_1$ match five tabulated values, a zero of $J_0$ among them, to three parts in a billion. The field’s one-dimensional spectrum has the inertial slope to five parts in a thousand. Every memory law is calibrated to the probe’s integral time to rounding error. And with one memory for every scale the window mean’s correlation must be the exponential $e^{-t/T}$ exactly, whatever the window, since every scale decays alike; the computed correlation matches it to $10^{-13}$, and the interval comes out at the earlier essays’ 1.17 for every window.

Still open: noise that forgets at once #

Every interval here assumes the camera records the field exactly. A real particle-image measurement adds noise to every vector, independent from frame to frame — a part of the variance with no memory at all, which the window mean averages but does not remove. In the language of the third figure, noise is the fastest possible fall at the origin. The next calculation would add a stated noise level per vector, averaged over the window’s vectors, and ask how much noise it takes before the frame interval that stops buying is set by the noise rather than by the flow — and whether, for a gradient, which differences neighbouring vectors and so amplifies their noise, that point is reached at the noise levels real systems have.