A snapshot counts areas, and a flow counts its edge

A record is as long as its integral scales established the rule for a time series at one point: its mean is as good as a mean of one independent value for every two integral time scales the record lasts, however finely it is sampled. The best estimate of a scale assumes its shape then compared the ways of measuring that integral scale from the record itself. Both worked with one probe and a clock.

A great deal of modern turbulence data is not a record. A particle-image velocimetry frame is a field of vectors on a grid; a simulation is saved as snapshots, each a field in two or three dimensions. The question that essay left is how a snapshot compares: does a frame of 64 × 64 vectors hold more or fewer independent values than a record of 4,096 samples? It sounds like a question of bookkeeping, and the obvious answer — count integral scales in both — is wrong in two different directions depending on what the field is.

Three fields with one integral length #

To compare like with like the fields need the same integral length, and the cleanest way is to give them the same longitudinal correlation: the correlation of a velocity component with itself at a separation along its own direction, which is the one a probe in a stream measures through Taylor’s hypothesis. Here it is Gaussian, $e^{-r^2/2\lambda^2}$, so the integral length is $\lambda\sqrt{\pi/2}$, about $1.25\lambda$, in every case. What differs is the transverse correlation — the same component at a separation across its direction — and that difference is the subject.

The first field is a scalar: the same Gaussian correlation in every direction. A temperature or a dye concentration behaves like this, and so does a velocity component whenever anyone treats it as if it were a scalar, which is most of the time.

The second is a plane cut through a three-dimensional incompressible field, the situation of a planar PIV measurement in real turbulence. The in-plane component $u$ is $\partial A_3/\partial y - \partial A_2/\partial z$ for a vector potential A; the first term is a derivative within the plane, the second a derivative out of it, and the second looks from inside the plane like an ordinary scalar.

The third is a two-dimensional incompressible flow, $u = \partial\psi/\partial y$ for a streamfunction $\psi$ — a soap film, a stratified layer, a large-scale atmospheric flow, a two-dimensional simulation. One function instead of two is the reason such a flow can be written this way at all, and it turns out to be the reason for everything below.

Along the component the three correlations are one curve. Across it they part. The scalar’s transverse correlation is the same Gaussian. The plane cut’s falls to zero at $\sqrt2\,\lambda$ and dips slightly below, the average of a Gaussian and a curve that swings negative. The two-dimensional flow’s swings negative so far that its integral out to infinity is exactly zero — a consequence of incompressibility and nothing else, since its transverse correlation is the second derivative of the streamfunction’s, and a second derivative integrates to a difference of first derivatives that both vanish. The relation between longitudinal and transverse correlations in three dimensions is the one with no turbulence in it; this is its two-dimensional cousin.

A probe cannot tell them apart #

One consequence of choosing the fields this way is worth stating before any window is drawn. A probe held in a stream, reading one component as the flow carries the pattern past it, records the correlation along the direction of the mean flow — the longitudinal one, if the component is the streamwise velocity. That correlation is identical for the three fields. So is everything a record measures: its variance, its integral time scale, its spectrum, and the scatter of its mean over any duration. A hot wire in the scalar field, in the plane cut and in the two-dimensional flow produces three statistically indistinguishable signals.

The difference is entirely in the direction the probe never samples. That is why the record essays could say nothing about it, and why a rule for snapshots cannot be borrowed from a rule for records by counting integral scales along one axis and squaring. The transverse correlation carries information about the field that no single-point time series holds, and incompressibility is what shapes it.

The mean over a window, exactly #

The error of a spatial mean is set by the correlation integrated over every pair of points in the window. For these three fields the correlations separate into a product of a function of x and a function of y, so the double sum over an L × L window is a product of two single sums: V(L), the average of the Gaussian over pairs in a line of L points, and W(L), the average of the transverse flow correlation over the same pairs. The scalar’s variance is $V^2$, the plane cut’s is $\tfrac12(VW + V^2)$, the flow’s is $VW$.

W is where the flow’s peculiarity lives. Its integrand is $-\lambda^2$ times the second derivative of the Gaussian, and summing a second derivative over pairs leaves only what happens at the ends: $W$ is $2\lambda^2/L^2$ times a factor that tends to one. The sum over the window’s interior cancels. Physically, the mean of $\partial\psi/\partial y$ over a window is the difference of ψ along its top and bottom edges divided by the window’s height, and ψ at the edges knows nothing about the interior.

The figure draws the three variances against window size in integral lengths, as closed forms, with Monte Carlo means over fifty generated fields on a 256 × 256 lattice as dots; the two agree within six per cent at every window checked. For large windows the scalar’s variance tends to $4\ell^2/L^2$ — one independent value for every square two integral lengths on a side. The plane cut’s tends to half that, $2\ell^2/L^2$, because half its variance comes from the out-of-plane derivative, which behaves as a scalar, and the other half from the in-plane derivative, which cancels. The flow’s tends to $4\ell\lambda^2/L^3$, and by a window thirteen integral lengths wide it is a nineteenth of the scalar’s.

Area and perimeter #

The local exponent — how much a doubling of the window improves the mean — shows the two regimes directly. Past a few integral lengths the scalar and the plane cut settle on −2: every new integral area in the window adds an independent value, and doubling the side quadruples them. The flow settles on −3: its mean is set by the window’s edge, which doubles when the side does, while the edge’s contribution is divided by an area that quadruples, so each doubling improves the mean eightfold. The approach is not quick: the flow’s exponent is still −2.8 at three integral lengths and −2.9 at six, because the window’s edge has to be long compared with the correlation before the corners and the partial cancellation near them stop mattering. Below an integral length or two all three exponents are shallower, because the window sits inside a single correlated patch and averaging it barely helps.

The same argument runs in three dimensions, one power higher. The mean velocity over a cube of a three-dimensional incompressible field is a surface integral of the vector potential, so its variance falls as the inverse fourth power of the side, not the inverse cube that the volume would give. A simulation’s volume average of a velocity component converges a power faster than a volume average of its temperature. A plane cut throws that away: the out-of-plane derivative is invisible to any in-plane boundary, and it restores the area law, at half strength.

Records against snapshots #

Counting independent values — the point variance over the variance of the mean — against the number of samples puts a record and the three snapshots on one footing. At five samples per integral length the record holds one value for every ten samples. A 64 × 64 snapshot, 4,096 samples, holds 45 if it is a scalar, 86 if it is a plane cut through three-dimensional flow, and 855 if it is a two-dimensional flow; the record of 4,096 samples holds 409. The scalar snapshot holds about a ninth of the record’s count, the plane cut about a fifth, and the flow about twice.

The reason for the scalar’s deficit is the geometry of counting. A record needs $2\ell/\Delta x$ samples for each independent value; a field needs $(2\ell/\Delta x)^2$, because the value is an area and it must be resolved in both directions. At five samples per integral length that is ten samples against a hundred, and the gap is a factor of ten in the count. The flow’s snapshot starts below the record at small windows, where its edge is not yet long compared with the correlation, and overtakes it once its side passes 5.8 integral lengths.

Oversampling a snapshot #

The factor 2ℓ/Δx has a consequence that the usual practice runs straight into. A record’s count grows with its sampling rate only until the samples are about an integral scale apart — beyond that, the record essay showed, extra samples add nothing. The same is true of a snapshot, and there the extra samples are paid for in two directions. The figure fixes a window 12.8 integral lengths wide and samples it from once to twenty times per integral length. The snapshot’s count does not move: 45 values for a scalar, 86 for a plane cut, about 860 for a flow, whether the vectors are a tenth of an integral length apart or half. A record with as many samples as the window gains in proportion.

That is the whole difference between sampling in time and in space for a mean. Finer PIV spacing is worth having for gradients, dissipation and the window every vector is averaged over, all of which are small-scale quantities. For the mean flow, and for anything whose error is governed by the integral scale, a frame holds as many independent values as it holds integral areas, and the only way to get more is a larger field of view or more frames.

What this means for a measurement #

The practical rules follow directly. A mean velocity from a planar PIV frame of real turbulence should be credited with about $L^2/(2\ell^2)$ independent values — half the area in integral squares, doubled because the plane cut’s in-plane cancellation halves the variance — and a frame is typically a few integral lengths across, so a single frame holds a few tens. Converged statistics come from frames, and frames separated in time by less than an integral time share their largest eddies and are not independent either. For a scalar field — a concentration in a mixing layer, a temperature in a convection cell — the count is half that again.

For a genuinely two-dimensional flow the correction runs the other way, and it is large. A soap-film or a two-dimensional simulation’s mean velocity over a window is far better determined than an area count suggests, and an error bar computed by treating the velocity as a scalar is too wide by the square root of the ratio, about four at a window thirteen integral lengths across. That is also the setting of the cascade that runs backwards, where energy accumulates at the largest scales and the integral length grows in time; there the window must be measured against the current integral length, and the advantage shrinks as the flow’s own scale approaches the window.

A worked case makes the size of the effect concrete. A planar PIV system in a wind-tunnel boundary layer with a field of view of 100 mm and an integral length of 20 mm sees five integral lengths across its frame. On the plane-cut rule that frame holds about twelve independent values for the mean of the streamwise velocity, however many vectors the processing returns — 128 × 128 or 32 × 32 makes no difference to that number. An analysis that credits the frame with its 16,384 vectors, or even with its 25 integral-length squares treated as independent, reports an error bar on the frame mean several times too narrow. The same frame’s vorticity, which lives at small scales, is a different matter: its count is set by the vorticity’s own much shorter correlation, and there the fine grid is exactly what buys the independent values.

None of this changes for higher-order statistics in the same simple way. The structure functions that the one exact result constrains are averages of products of velocity differences, not of velocities, and their error depends on fourth-order correlations that incompressibility does not make cancel. The boundary law is a property of means of a component, and of anything linear in it.

What was checked #

Each closed form was checked against fields generated spectrally — white noise filtered to the Gaussian spectrum, and differentiated in Fourier space for the flow and the plane cut — on a periodic 256 × 256 lattice, at windows of 8, 16 and 32 samples: the nine window variances agree with the closed forms to within 5.6 per cent, the sampling error of forty fields. The generated fields’ correlations along and across the component match the closed forms to about a hundredth. The flow’s window factor reaches its continuum boundary form to two parts in ten thousand at a window of ten correlation lengths. The tests also refuse a window of zero width, a field kind the model does not define and a tolerance of zero.

What the model leaves out #

Gaussian statistics. Every field here is Gaussian, and the variance of a mean depends only on the correlation, so the results hold for any field with these correlations. The bursts that make pair dispersion sensitive to the fourth moment do not reach a mean.

One correlation shape. The Gaussian correlation has no tail. A correlation with an exponential or power-law tail changes the constants, and a transverse correlation whose negative lobe is slow to decay approaches the perimeter law more slowly; the exponents are set by incompressibility and dimension, and do not change.

Point samples. A PIV vector is an average over an interrogation window, which smooths the field at scales below the window. That lowers the point variance a little and leaves the variance of a large-window mean untouched, so it raises the apparent count slightly without changing any of the laws; the correction is the one the window every vector is averaged over computes, and it matters for small-scale statistics rather than for means.

Periodic, homogeneous fields. A mixing layer’s statistics vary across it, and the integral length with them. A window in an inhomogeneous flow is not one sample of one field, and the count has to be done piece by piece.

Who worked it out #

The variance of a spatial average in terms of the correlation’s integral is Taylor’s, from 1921, extended to fields by the statistics of spatial processes in the decades after. That the mean of a solenoidal field over a region is a boundary term is Stokes’ theorem; its consequence for the convergence of spatial averages in incompressible turbulence is noted in the literature on turbulence statistics, where the vanishing of the transverse correlation’s integral in two dimensions is standard. Estimates of how many independent samples a PIV frame holds are part of the uncertainty analysis developed for the method from the 1990s onwards.

Still open: frames that share their eddies #

Everything here is a single snapshot. A PIV run is a sequence of frames, and frames closer together than an integral time share their largest eddies, so a run’s count is not the frames’ counts added. For a record the conversion was two integral times per value; for a sequence of snapshots it is a space-time integral of the correlation, and a frozen field swept past the window by a mean velocity makes consecutive frames overlap in space as well as time. The next calculation generates sequences of fields advected and decorrelating at chosen rates, and asks how many independent values a run of N frames holds — and at what frame rate a PIV system stops buying anything for the mean.