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

Frames that share their eddies count as one counted the independent values a particle-image run holds for the mean of a field that a stream sweeps past the window while the field forgets itself. With one memory time $T$ for the whole field it found a clean rule: a faster camera stops buying once frames are closer than about 1.17 memory times, or once the stream carries the pattern less than a window’s width between frames, whichever comes first. It called the single memory the model’s weakest part. Large eddies carry most of a window mean’s variance and live longer than the integral time suggests; small ones live less; and the rule’s $T$ was a number a single probe in the flow would report, an average over all of them.

The question is practical. The integral time is easy to measure with a hot wire and impossible to measure from a PIV run at the frame rates where the rule matters. If the rule needs a different time, it needs a different measurement.

Each scale its own memory #

The fields are the earlier essay’s: Gaussian correlations of width four samples on the sampling lattice, a window of 64 samples a side — 12.8 integral lengths — and a stream that sweeps the pattern along one axis. Written in wavenumbers, a Gaussian field is a Gaussian spectrum, the window is a filter $|D_L(k_x)D_L(k_y)|^2$ with $D_L(k) = \sin(kL/2)/(L\sin(k/2))$, and the sweep is a phase $e^{ik_xUt}$ on each mode. The covariance between the window means of two frames a time $t$ apart is then

$$C(t) = \int \Phi(\mathbf{k})\,|D_L(k_x)D_L(k_y)|^2\cos(k_xUt)\,e^{-|t|/\tau(k)}\,\mathrm{d}^2k,$$

and a run of $N$ frames has a mean whose variance is the frame-pair sum of $C$. With $\tau(k) = T$ for every scale this is the earlier essay’s real-space double sum, computed by a code that shares nothing with it but the field, and the two agree to $10^{-13}$ at every lag tried and in the one-second PIV run, to the last digit of its 8,471 values.

The memory law is $\tau(k) = \tau_0\,(\max(k, k_c)/k_c)^{-p}$: falling as a power of wavenumber above the field’s own wavenumber $k_c = 1/\lambda$, and held at its largest value below it. The exponent $p = 1$ is the sweeping decorrelation Kraichnan and Tennekes argued for — a small eddy is carried past itself by the large ones in a time $1/(ku')$ — and $p = 2/3$ is an inertial-range eddy’s own turnover. Every law is calibrated so that a single probe’s integral time, the spectrum-weighted average of $\tau(k)$, is exactly $T$.

The figure says where the difference comes from. All three laws average to the same probe time, so a scale-dependent law must give the large scales more than $T$ and the small ones less. The window passes only wavenumbers below about 0.1 — a sixth of the field’s own — where the sweeping law’s memory is 1.26 probe times. A window mean is made almost entirely of eddies the probe’s average undercounts.

Why the window sees only the large scales is the same arithmetic the snapshot count rests on. A window 64 samples wide averages away every fluctuation shorter than itself, and of the field’s point variance it keeps about one per cent for a plane cut and two for a scalar: the part carried by scales comparable with the window or larger. Those are scales several times the integral length, out in the tail of the field’s correlation, where the spectrum is flat and nearly every eddy is bigger than the ones a probe’s average is dominated by. A probe sees every scale in proportion to its energy; a window sees the largest in proportion to their energy times the square of how little the window cancels them. Whatever memory law the field has, the window reads it at its low-wavenumber end.

The window remembers longer, by how much the large eddies do #

The window mean’s own memory — the integral of its autocorrelation, $\int C(t)\,\mathrm{d}t/C(0)$ — is 1.18 probe times for the turnover law and 1.26 for the sweeping one, for a scalar; for a plane cut through three-dimensional flow, whose streamwise component puts more of its variance at low wavenumber across the stream, 1.43 — the cross-stream correlation of a streamwise velocity being the transverse function that isotropy fixes from the longitudinal one, with a negative lobe that cancels the small scales in a window more thoroughly than a scalar’s correlation does. And it depends on a choice the measurement cannot see: where the largest scales’ memory stops growing. Held at the field’s own scale, as above, the window remembers 1.26 probe times. Let the sweeping law continue to twice the field’s scale before it saturates and the window remembers 1.93 probe times for a scalar and 2.35 for a plane cut; saturate it at half the scale and the window barely notices, 1.02.

That is the model’s real lesson, and it is uncomfortable. The window mean’s memory is set by the scales the probe’s integral time weights least, and how long those scales live is exactly what a single probe, averaging over all of them, cannot report. Two flows with the same probe integral time can have window memories that differ by a factor of two.

Where the frame rate moves, and where it does not #

The frame interval that stops buying follows. With no sweep — a window watching a flow that only forgets — it moves from 1.17 probe times with one memory to 1.47 with the sweeping law saturating at the field’s scale, a quarter longer; the dial shows 1.19 if the saturation is at half the scale and 2.2 if it is at twice. With a fast sweep nothing moves: the interval is 1.16 window crossings for every law, because a pattern carried clear of the window between frames gives a new frame whatever its memory. In between, where the pattern crosses the window in about a probe time, the interval moves by a few per cent.

The memory’s dependence on size matters only where memory, not sweep, decides the count — a slow tunnel, a quiescent tank, a recirculation zone with no mean flow through the window. There it moves the frame rate that stops buying by as much as the largest eddies outlive the average one.

The rule comes back if it is given the right time #

The surprising part is how little else changes. Take the single-memory rule — stop buying at 1.17 memory times with no sweep — and feed it the window mean’s own integral time instead of the probe’s. Across both memory laws and six saturation wavenumbers, from intervals of 1.19 to 2.85 probe times, the rule reproduces the computed interval to within 2.4 per cent. The window mean behaves like a field with one memory, its own; what the scale dependence changes is which memory that is.

That turns a modelling question into a measurement. The window mean’s autocorrelation can be read from the run itself: compute the window mean of each frame and the correlation between frames as a function of their separation in time. A run taken at a frame rate fast enough to resolve that correlation once — a few frames per memory time, for a few memory times — tells the experimenter the frame rate every later run needs, without a hot wire and without knowing how the large eddies forget. The probe’s integral time is the wrong number to bring to the camera, and the camera can measure the right one.

Measuring it costs little. The window mean’s autocorrelation needs a record several window memories long, sampled a few times per memory, and its integral converges as the square root of the record’s length in memories: a record of a few dozen window memories gives it to a few tens of per cent, which is all a frame-rate choice needs. In the still tank that is a few minutes of filming at a frame every half-second, once, before the runs that matter. Estimating the memory from the window means is also free of a trap that catches the probe: the best estimate of a scale assumes its shape, and a probe’s integral time read from a correlation that has not been followed to zero carries the assumption of an exponential tail, which a scale-dependent memory does not have. The window mean’s correlation is closer to exponential than the probe’s, because it is dominated by one band of scales, and its integral is the better-conditioned number of the two.

This is the connection to a record that is as long as its integral scales: a time series’ error is set by its own integral time, and a PIV run’s mean is a time series of window means whose integral time is not the point probe’s. The two were the same only because the earlier calculation gave every scale one memory.

A minute in a still tank #

Where no stream sweeps the pattern, the effect is not a tenth but a factor. Take a tank of water stirred and left with no mean flow through the window, an integral length of a centimetre, a probe memory of two seconds and the same field of view of 12.8 integral lengths. A one-minute run of a plane cut holds, at one frame a second, 1,301 independent values of the mean velocity if every scale forgets in two seconds, 932 if the memory follows the sweeping law and saturates at the field’s scale, and 592 if it saturates at twice the scale. The frame rate past which more frames stop buying is 0.43 Hz, 0.30 Hz or 0.19 Hz.

An experimenter who quotes a standard error from the first count, as the probe’s integral time invites, has under-stated it by up to a factor of 1.5 — the square root of 1,301 over 592 — and will find successive runs disagreeing by more than the error bars say they should. Nothing in the run’s own vector fields looks wrong. The window means simply remember longer than the number used to count them, and the only cure is to measure that memory from the window means.

A second of PIV #

The worked case is the earlier essay’s: a field of view 128 millimetres square in a 10 m/s stream, an integral length of 10 millimetres and a probe memory of 10 milliseconds, so the pattern crosses the window in 12.8 milliseconds — sweep and memory comparable. At 15 Hz every frame is independent and the second holds 1,285 values whatever the memory law. At 150 Hz it holds 8,471 with one memory, 8,024 with the turnover law and 7,864 with the sweeping one; at 5 kHz, 9,425, 8,708 and 8,462. The scale-dependent memory costs about a tenth of the values at the highest frame rates and nothing at the lowest, and the frame rate that reaches nine-tenths of the most falls from 150 Hz to 135. In a tunnel, where the sweep does most of the work, the earlier rule was nearly right for the wrong reason: the stream carried the pattern clear before the memory had a chance to matter.

Why two memory laws nearly agree #

The turnover law and the sweeping law disagree about every scale smaller than the field’s — one gives an eddy at ten times the field’s wavenumber a quarter of the probe time, the other an eighth — and their windows’ memories differ by only seven per cent. The reason is the saturation. Below the field’s wavenumber both laws hold the memory at its largest value, and the window lives almost entirely below that wavenumber, so both windows remember the saturated value; they differ only in how much of the probe’s average they spend on small scales, which sets how large the saturated value must be to keep the average at $T$. The steeper law starves the small scales more and gives the large ones correspondingly more, which is the seven per cent. The window’s memory is a fact about the largest scales and the calibration, not about the law’s exponent — one more reason that it has to be measured rather than derived from a model of the small scales.

The same reasoning says when the probe’s time is good enough. If the field’s energy sits mostly near its own scale, so that the probe’s average is dominated by the scales the window sees, the two times agree; the Gaussian fields here are of that kind, which is why the factor is a quarter and not a half. A turbulent flow with a long inertial range spreads its energy over more decades, and the range a real Reynolds number does not have is exactly the question of how many: the shorter the range, the closer a probe’s time is to the window’s. A camera in a high-Reynolds-number flow should expect the gap to be wider than any figure here.

What a faster camera is still for #

None of this says a fast camera is wasted. The frame rate that stops buying is a statement about the mean and about any statistic whose error is set by the largest scales. A time-resolved system pays for itself on what lives at short times — accelerations, the frequency spectrum’s high end, the correlation between consecutive frames itself — and on the third moment, the moment a spectrum cannot hold, whose estimate needs the small scales’ phases and not only their energy. What changes here is only the threshold for the mean, and it changes in the direction that makes slow cameras look worse than the probe’s integral time suggests, not better.

Three times, not one #

The probe’s integral time is an Eulerian time at a fixed point. A fluid parcel’s own memory, the Lagrangian integral time that sets how far a parcel gets by turbulent dispersion, is a second, usually shorter, because a parcel is carried through the field’s structure as it ages. The window mean’s memory is a third, set by the largest scales the window cannot average away. Experiments routinely quote one of these and use it for another. A camera’s frame rate should be set from the third, an estimate of dispersion from the second, and a hot wire’s record length from the first, and the calculation here is a reminder that in a flow whose eddies forget at rates depending on their size, the three can differ by factors of two.

The convention: whose integral time #

“Probe integral time” is the spectrum-weighted average memory, what a hot wire held in a flow with no mean velocity would measure, and every law is calibrated to it. “Window memory” is the integral of the window mean’s own autocorrelation with no sweep. Memories are exponential in time at every scale, and the frame interval that stops buying is the longest at which a long run collects nine-tenths of what continuous filming would. The window is 64 samples, 12.8 integral lengths; the field’s correlation width is four samples. Wavenumbers are per sample, and the window’s passband is quoted at its first zero, $2\pi/64$.

What the model leaves out #

Gaussian decorrelation. Kraichnan’s sweeping makes a scale’s correlation fall as a Gaussian in time, not an exponential; the integral times would change by a constant factor and the frame intervals with them, while the comparison between laws would not. A real spectrum. The fields are Gaussian-correlated, with no inertial range; a Kolmogorov spectrum has more energy at large scales, which would weight the window towards them even more. The saturation, which the model places by hand and which the figures show deciding the answer. Non-Gaussian statistics, which change the count of independent values for higher moments but not for the mean. And the sweep’s own variability: a turbulent stream does not sweep at one speed, and the large eddies that carry the memory also carry the pattern.

Where the question came from #

Taylor’s frozen-turbulence hypothesis of 1938 is the sweep. Kraichnan’s analysis of 1964 and Tennekes’s of 1975 argued that small scales decorrelate by being swept past an observer by the large ones, at a rate proportional to their wavenumber, which is the $p = 1$ law. The count of independent samples in a correlated record is Bendat and Piersol’s; its extension to a window swept past a camera is the snapshot’s area count carried into time. The calculation here asks the one question those pieces leave open, which time a camera’s run actually counts in.

Still open: a window that sees the inertial range #

The window here is wide — 12.8 integral lengths — so it averages away everything but the largest scales, and its memory is theirs. A PIV system zoomed in on a small region, or one measuring a quantity weighted towards small scales, like a velocity gradient, sees the other end of the memory law. The next calculation narrows the window through the integral scale into the inertial range and asks how the frame rate that stops buying changes when the window’s memory is the small eddies’ rather than the large ones’ — whether a zoomed-in camera needs a faster frame rate than a wide one looking at the same flow, and by how much the sweeping law’s short memories undercut the probe’s average.