Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Particle-image velocimetry — where it appears

A measurement of a velocity field from two photographs of seeded fluid, each vector the displacement that best matches the particle pattern in a small window. Each vector is therefore an average over its window, and the window acts as a spatial filter on everything derived from the field.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

The exposure lengths that give the true mean, and the ones that do not. How far a finite exposure of a shedding wake lands from the long-exposure mean, against how many shedding periods the shutter was open for. It is exactly zero at every whole number of periods — an average over a complete cycle is the mean, with no error at all — and between the zeros it falls as one over the exposure. Nothing about the picture tells the reader which of these they are looking at.

The shutter is part of the answer

Flow photographs are compared as though the exposure were a detail of the camera. It is a term in the measurement: an average over exactly one shedding period returns the true mean to fourteen figures, any other length carries a residue that falls only as one over the exposure, and a two-pulse velocity reading is short by exactly the sinc of the swept angle.

misconceptions · Visualisation A window one core wide reads the vortex 7.5 per cent slow. The tangential velocity across a Lamb–Oseen vortex, in units of its core radius and of Γ/2π divided by it, as it is and as particle image velocimetry reports it with square interrogation windows of three widths — the average of the velocity over each window, which is what a correlation over the window returns to first order. The true peak is 0.6382 at 1.1209 core radii. A window 0.5 core radii wide reports 98.0 per cent of it, 1.021 times as far out, a window 1 core radii wide reports 92.5 per cent of it, 1.084 times as far out and a window 2 core radii wide reports 77.1 per cent of it, 1.334 times as far out. The instrument that measures velocity directly still reports a slower, fatter vortex than the one there, by an amount set entirely by the window against the core.

The window every vector is averaged over

Particle image velocimetry is the one flow-visualisation technique that reports the velocity itself, and it still applies an operator: every vector is an average over an interrogation window. A window is a filter with a transfer function, and it makes a vortex slower and fatter, a thin shear layer exactly as thick as the window, and some features smaller than the window point the wrong way.

misconceptions · Visualisation The part of a field 4 cameras cannot see. Middle, a field with no symmetry on a 16 × 16 grid. Left, the part of it that projects to exactly nothing in every one of 4 views, shaded one way above zero and the other below: a pattern of streaks and hollows that cancels along every ray. Right, the field with that part taken away. The middle and right fields give identical pictures in all 4 views, to 2e-13 of the largest ray, and no reconstruction from those views can tell them apart. The invisible part is one combination of 177 independent patterns the views cannot see.

Four cameras and a field they cannot see

An axisymmetric flow can be rebuilt from one photograph because its symmetry supplies every other view. A flow without an axis has to be photographed from several directions, and a few directions do not merely give a noisy answer — they leave whole patterns of density that every camera records as nothing. Four views of a 16 × 16 field see 79 of its 256 independent patterns and are exactly blind to the rest.

misconceptions · Visualisation A frozen run counts the area it sweeps. Independent values in a run of frames while a frozen pattern is swept 100 integral lengths past a window 12.8 integral lengths square, against the distance the pattern moves between frames. For a scalar the run holds 353 values with a frame every half integral length, 362 with one every four and 388 with one every window width: the swept area in integral areas, whatever the frame rate. Only past a window width, when gaps open between frames, does the count fall. Treating frames an integral scale or two apart as independent, as a record would allow, counts 2075 at two integral lengths — 5.8 times too many.

Frames that share their eddies count as one

A particle-image run is a sequence of snapshots, and snapshots closer together than an integral time photograph the same eddies. When a mean flow carries a frozen pattern past the window, a run holds exactly the area it sweeps, counted in integral areas, and a faster camera adds nothing until frames stop overlapping — a threshold set by the window, not by the turbulence. In a flat flow one component escapes the rule: its run mean is fixed by the pattern at the two ends.

turbulence · Spectrum Small scales forget fast and large ones slowly, and the window only sees the large. Each scale's memory over a single probe's integral time against wavenumber in inverse samples, for one memory at every scale and for memories falling as the wavenumber to the minus two-thirds and minus one above the field's own wavenumber, 1/λ, and held there below it. All three give the same probe integral time; the window of 64 samples passes wavenumbers below about 2π/64 = 0.098, where the sweeping law's memory is 1.26 probe times and the single law's 1.

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

A run of particle-image frames stops gaining independent values for its mean once frames are closer than the flow's memory — but the memory that counts is the window mean's, and when large eddies live longer than small ones it is longer than the integral time a single probe measures. Give each scale its own memory and the frame rate that stops buying moves by up to a quarter where the flow forgets before it is swept, by nothing where the sweep sets it, and by however much longer the largest eddies live. Fed the window's own memory, the single-memory rule gives it back to two per cent.

turbulence · Spectrum Zoomed into the inertial range, a camera needs frames faster than its window's memory says. The longest frame interval at which a run of window means still collects nine tenths of what continuous filming would, in probe integral times, against the window's diameter in integral scales, with no sweep, for one memory at every scale and for turnover and sweeping memories; dashed, the single-memory rule, 1.17 times the window mean's own memory, for the sweeping law. With the sweeping law the interval is 2.45 probe times for a window ten integral scales wide, 1.67 for one scale and 0.803 for a hundredth, where the rule says 1.18; with one memory it is 1.17 at every size.

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

A wide camera window averages away the small eddies, so its mean remembers what the large eddies remember, and a run of frames is worth as many independent values as that memory allows. Zoom the window into the inertial range and the opposite is not what happens. A small window keeps nearly all the variance and almost exactly a single probe's memory — every eddy larger than itself still fills it — but much of what it keeps is the cascade's eddies, which forget far faster than the largest ones. Its frames then stop buying at an interval a third shorter than its own memory says, and a velocity gradient in the same window needs frames thirty times faster.

turbulence · Spectrum Noise in every frame pulls in the frame interval that stops buying, by the same share whatever the window sees. The longest frame interval at which a run still collects nine-tenths of continuous filming's independent values, over its value with no noise, against the noise's share of the window mean's variance, for a velocity window one integral scale wide, one a twentieth as wide, and a gradient in the small window; dashed, a single exponential memory in closed form. velocity, window ℓ: 0.782 at ν = 0.1, 0.199 at ν = 1; velocity, window ℓ/20: 0.804 at ν = 0.1, 0.243 at ν = 1; gradient, window ℓ/20: 0.819 at ν = 0.1, 0.242 at ν = 1.

Noise in every frame is a memory of zero, and the run cannot measure it at its own pace

A camera that adds an independent error to every frame adds a part of the signal with no memory at all. Infinitely many frames would still average it away, so the run's information still approaches continuous filming's — but the frame interval at which a faster camera stops buying is pulled in by the same share for a wide window, a zoomed one and a gradient, about half at a noise of three-tenths of the variance and a fifth at a noise equal to it. And the usual way to learn the noise, from the run's own correlation extrapolated to zero lag, works only at frames much faster than the run's information needs.

turbulence · Spectrum

Named alongside it

The objects these essays reach for when they reach for this one.

MeasurementAveragingSamplingIntegral scaleModel limitMemory kernelCorrelationInertial rangeInstrumentMisconceptionModel validitySpectrum

All concepts