The window every vector is averaged over

The paint that measures the wrong field ends by naming the technique the others are proxies for. Smoke reports where fluid went, schlieren a derivative of density, paint a pressure contaminated by temperature; particle image velocimetry reports the velocity itself, and it is the only member of the family that does. That essay also refuses it the exemption its position invites. PIV has its own operator — a correlation window, a particle response time, a pulse separation — and the questions asked of every other instrument apply to it unchanged.

Two of those three have been computed. The shutter is part of the answer prices the pulse separation, which is a window in time, and finds a two-pulse reading of a spinning flow over half a turn returning exactly 2/π of the speed. The tracer that is not one prices the particle, whose response time is not the one usually written down. What is left is the window in space — the interrogation window every vector is the average of — and it turns out to be the one that changes the picture most.

A vector is an average #

A PIV measurement photographs a sheet of flow twice, a short interval apart, with the fluid seeded by particles. Each photograph is divided into interrogation windows, and in each window the displacement that best matches the first image’s particle pattern to the second’s is found by cross-correlation. That displacement, divided by the interval, is the vector drawn at the window’s centre.

When the velocity varies across the window the particles in it do not all move by the same amount, and the correlation peak is where the pattern as a whole matches best. To first order that is the average displacement of the particles in the window, and with the particles spread uniformly it is the average of the velocity over the window’s area. The convention every figure here uses is exactly that: a square window, uniform weight, the first-order average — no weighting by particle brightness, no broadening of the correlation peak, no pulse separation, no noise. It is the best case, and it is the case in which the window is the only operator left.

The flow used to test it is a Lamb–Oseen vortex, a Gaussian patch of vorticity whose velocity is known in closed form. Its tangential velocity rises from zero at the centre to a peak of 0.638, in units of the circulation over 2π times the core radius, at 1.121 core radii, and falls as the inverse of the radius outside. Each windowed velocity is the average of that field over a square, computed by quadrature.

A window half a core radius wide reports 98.0 per cent of the peak. A window one core radius wide reports 92.5 per cent, and a window two core radii wide reports 77.1 per cent. Each also moves the peak outwards — to 1.021, 1.084 and 1.334 times its true radius — because averaging across the peak pulls in the lower velocities on both sides and the smaller velocities inside the core outweigh the larger ones outside. The vortex a PIV system reports is slower and fatter than the one in the flow, and it is slower and fatter by an amount set entirely by the window against the core.

The window as a transfer function #

An average over a window is a linear operation, and a linear operation on a field is best described by what it does to each wave the field is made of. Average a sinusoid of wavenumber $k$ over a window of width $W$ centred at its crest, and the result is the crest’s value multiplied by

$$H(k) = \frac{\sin(kW/2)}{kW/2}.$$

That is the window’s transfer function, and it is the same sinc that describes a finite exposure in time.

The dots are sinusoids averaged over the window numerically and they agree with the closed form to a part in a billion, which is the check that the quadrature behind every other figure is doing what it says.

Three features of the curve are worth reading off it. Long waves pass almost untouched: a wave ten windows long keeps 98 per cent of its amplitude. A wave exactly one window long reads zero, because the window contains one whole crest and one whole trough and they cancel; a flow feature of that size is simply absent from the field. And waves between one half and one window long are reported with their sign reversed — the curve goes negative — deepest at $kW = 8.99$, where the window returns −0.217 of the amplitude. The band repeats with alternating signs and shrinking size for ever.

The time-domain version of the zero is the one the shutter essay finds: an exposure of exactly one shedding period returns the mean exactly, because a whole period cancels. The space version has the same cancellation and the same negative lobes, and it has one property the time version is spared. A shutter’s time axis has a direction and a reader rarely looks at a flow at frequencies above the frame rate. A vector field’s spatial axis has none, and the features a window reverses are drawn right beside the ones it reports correctly.

A feature the window reports backwards #

A field varying with a wavelength of 0.75 windows sits in the first negative lobe, at $kW = 8.38$, and the window reports it at −0.2067 of its amplitude. Where the flow runs one way, the vectors point the other.

Nothing in the reported field is inconsistent. It is smooth, it has exactly the wavelength of the real feature, and its amplitude of a fifth is plausible for a weak secondary flow. A reversed feature is indistinguishable, in a single vector field, from a real weak feature of the opposite sign — and in a flow full of structures near the window size, a turbulent boundary layer or the cores of a vortex street at modest resolution, some of what is drawn is of that kind.

The test is not in the field. It is in the method: process the same images with a different window. A real feature changes in amplitude a little as the window changes and keeps its sign. A reversed feature moves through the transfer function’s lobes as $kW$ changes, and changes size sharply, vanishes at $kW = 2\pi$, and reappears with the correct sign once the window is shorter than half its wavelength. That is the practical reason for processing a PIV data set at two window sizes rather than one, and it is rarely stated as a reason.

Three numbers a vortex is reported by #

A vortex is usually summarised by three numbers: the peak velocity, the radius of the core, and the vorticity at the centre. The window moves all three, and it does not move them equally.

The central vorticity is computed the way a PIV analysis computes it: central differences of the windowed vectors on a grid with half-window spacing. At a window one core radius wide it is 76.7 per cent of the truth, against 92.5 for the peak velocity; at three core radii it is 25.4 per cent against 62.9.

The vorticity loses first and loses most for a reason the transfer function makes exact. Vorticity is a derivative of velocity, and differentiating multiplies each wave in the field by its wavenumber — which weights the field towards the short waves, where the window’s response is smallest. A derivative of a filtered field is filtered harder than the field, and the vorticity of a vortex is more concentrated in space than its velocity, so it has more of its content where the window hurts. The second derivative, needed for strain gradients or for the viscous term of a pressure reconstruction, is worse again.

What the window does not change is the total. Circulation is vorticity added up, an average moves vorticity about without creating or destroying it, and so the circulation round a loop well outside both the core and the window survives the filter: at five core radii with a window one core wide it is kept to two parts in ten billion. The one number a PIV field reports about a vortex without correction is the integral, and it is the number that says least about the core’s structure.

A shear layer as thick as the window #

A shear layer is the other feature a velocity field is read for — the edge of a jet, a mixing layer, the outer part of a boundary layer — and it is summarised by its thickness, the velocity difference across it over its steepest gradient.

For a layer shaped as a hyperbolic tangent the answer is closed. The slope of a box average is the difference of the field at the box’s two ends divided by its width, so the steepest reported gradient is $2\tanh(W/2\delta)/W$ against a true $1/\delta$, and the reported thickness is

$$\frac{\delta_\text{reported}}{\delta_\text{true}} = \frac{W}{2\delta\tanh(W/2\delta)}.$$

A window as wide as the layer’s scale reports it 8.2 per cent too thick; twice as wide, 31 per cent; four times as wide, 2.08 times as thick. And as the window grows the ratio tends to $W/2\delta$ — the reported thickness becomes the window’s own width, whatever the layer’s real thickness is. A shear layer thinner than the window does not appear thin in a PIV field; it appears exactly as thick as the window, and it looks like a perfectly ordinary layer.

That is the measurement version of a warning from stability theory. A layer with a kink in it makes the instability of a shear layer depend on the shape of its profile, and a sheet of zero thickness is unstable at every wavelength. A PIV field of a thin layer has had its profile replaced by a box of the window’s width, so its most unstable wavelength and growth rate, estimated from the measured profile, belong to a layer the window made.

What overlapping the windows buys #

The standard response to the window’s cost is to overlap neighbouring windows, so that vectors are reported on a grid finer than the window. It helps, and the amount it helps is exactly bounded.

More overlap puts the grid points closer together, and a central difference over a shorter distance is a better estimate of the derivative. At half overlap the three windows give 93.1, 76.7 and 43.8 per cent of the central vorticity; at seven-eighths overlap they come close to their ceilings. The ceiling is the vorticity averaged over a single window — 96.0, 85.1 and 55.8 per cent — which is what a perfect derivative of window-averaged velocities would return, since differentiating and averaging commute.

So overlap buys back the error of the difference and nothing of the error of the window. It multiplies the number of vectors, and every one of them has passed through the same filter; a field four times as dense has four times as many samples of the same smoothed flow. The only way past the ceiling is a smaller window, which costs particles per window and so costs noise, and that trade between resolution and noise is the design problem of every PIV experiment.

What a smaller window costs #

The obvious answer to every figure above is a smaller window, and the reason it is not simply chosen is that a window has to contain enough particles to correlate. A common guideline is about ten particle images a window; with fewer, the true correlation peak is lost among chance matches between unrelated particles. Halving a window’s width quarters its area and the particles in it, so a window half the size needs four times the seeding density.

Seeding more densely is not free either. Particles scatter and shadow one another, so a densely seeded light sheet is dimmer and noisier; and eventually the particles change the flow they are carried by, since a heavily seeded fluid is a suspension with its own viscosity and its own inertia. The resolution of a PIV measurement is set in the end by how much dust the flow will carry, not by the camera, and every figure above is a statement about that choice: a vortex core, a shear layer or a wave smaller than the spacing between particles cannot be recovered by correlation at all, whatever the pixels do.

That is also why the answer to the window is not a better camera. A sensor with more pixels makes each particle image larger or each window cover less of the flow, and neither puts more particles in the window. The one direction that genuinely helps is fewer windows with more particles each — which is a larger window — or more images of the same flow, averaged, which helps only a flow that holds still.

The field a turbulence calculation also has #

There is a connection here that makes the window less of an instrumental nuisance and more of a familiar object.

A large-eddy simulation of turbulence does not compute the velocity field. It computes the velocity field averaged over a spatial filter the width of its grid, and the equations it solves for that filtered field contain a term it cannot compute — the stresses carried by the motion smaller than the filter, which have to be supplied by a model. That term is the whole difficulty of the method, and it is the same accounting that makes any averaged description of turbulence need a closure.

A PIV field of a turbulent flow is a filtered field in exactly that sense, with the interrogation window as the filter. Its velocities are the resolved velocities, its gradients are the resolved gradients, and every quantity built from gradients — the dissipation rate most of all, which is dominated by the smallest scales — is the resolved part of that quantity. A dissipation rate computed directly from PIV gradients is too small, and it is too small by the amount a sub-filter model would have to add. The literature that estimates dissipation from PIV does exactly what a large-eddy simulation does: it models what the window removed.

What the first-order window leaves out #

The average is weighted by nothing. A real correlation weights particles by the brightness of their images, so a window with a few bright particles at one side reports their velocity rather than the window’s mean. That is a random error on top of the systematic one computed here.

The correlation peak is taken to shift, not to broaden. Strong velocity gradients across a window smear the correlation peak as well as moving it, and beyond a displacement difference of a few pixels across the window the peak splits or is lost. The first-order average is the limit of weak gradients, which is the limit in which the window’s filtering is also mildest.

The particles are perfect. Particles heavier than the fluid are thrown out of vortex cores, and the tracer that is not one shows that even a neutrally buoyant particle does not respond on the time scale usually quoted. A core emptied of particles has no vectors in it at all.

The pulse separation is zero. A finite interval between the two images is a second window, in time, and its effect compounds with this one’s; the shutter essay computes it separately.

And the flow is planar. A light sheet has a thickness, particles cross it between exposures, and a vector is an average through the sheet’s thickness as well as across the window — a third direction of the same filter.

The two results the figures rest on are computed twice. The Lamb–Oseen peak lands where its closed form puts it, 1.12091 core radii at 0.63817, to four and five figures; the window’s measured transfer matches sin(kW/2)/(kW/2) at five wavenumbers including the negative lobe, to a part in a billion; and the shear layer’s reported thickness matches its closed form at six window widths.

Where the method came from #

Particle image velocimetry grew out of laser speckle measurements in the late 1970s and early 1980s, and became a routine instrument around 1990, when the photographs could be taken on digital cameras and correlated by computer rather than analysed optically. The effect of the interrogation window on spatial resolution was analysed from the start by the people who built the method, and adaptive schemes that deform and shrink the windows iteratively were developed through the 1990s precisely to push the window below the scale of the features being measured.

What those schemes do not do is remove the window. Every refinement moves the filter to a smaller scale; none turns an average into a point value, because a displacement cannot be measured from a single particle without knowing which particle it is. The instrument that measures velocity directly is, in the end, an instrument that measures an averaged velocity directly, and the transfer function is the price of the word.

Still open: one view of a flow of revolution #

Every instrument in this family so far has reported a field in the plane it looked at. The optical ones do something stranger: schlieren and interferometry integrate along the line of sight, so what they record at a point in the image is a sum over every point on the ray behind it. For a flow with no symmetry that sum cannot be undone from one picture.

For a flow of revolution — a jet, a plume, the wake of a projectile — it can, exactly, and the undoing has a cost that is concentrated where the flow is most interesting. That is one view is enough, and the axis pays for it, and it finds that the older of the two optical instruments is the better one for the job.