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

Visualisation — the series

9 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Four ways of photographing one flow, and what each of them records. The same solved flow, rendered as four different laboratory techniques would record it. Smoke from a port gives a streakline; tufts give direction with no speed in it at all; an oil film gives the direction of the friction on the surface rather than the flow above it; pressure taps give a scalar with no direction in it. None of the four is the velocity field, and only the first happens to coincide with a streamline, because this flow is steady.

    What a photograph of a flow shows

    Wind-tunnel pictures are the evidence this whole subject is argued from, and hardly anybody says which quantity a given technique records. They are not interchangeable — smoke, tufts, oil and pressure taps measure four different things, and none of them is the velocity field.

    part 1 · misconceptions
  2. 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.

    part 2 · misconceptions
  3. Three instruments, three derivatives, one field. A density field — a shock, smoothed to its own thickness — and what each of the three optical techniques records across it, each scaled to its own peak so the shapes can be compared. Interferometry follows the density itself; schlieren follows its first derivative and peaks where the density is changing fastest; shadowgraph follows the second and is a light-and-dark pair straddling the same place. None of them is looking at the flow: the refractive index of a gas is linear in its density, so every one of them is a densitometer and the differences between them are differences of calculus rather than of apparatus.

    An instrument that takes a derivative

    Dye, smoke and seeded particles mark the fluid and read the marks. The optical ones mark nothing — light passes through and is bent — and what a plate records is not the flow but a derivative of its density. Two of the three are therefore exactly blind to a uniform stream, at any speed it happens to have.

    part 3 · misconceptions
  4. A degree of temperature is worth 0.34 per cent of pressure. The error in the pressure a paint reports, against how much warmer the surface is at the test condition than at the reference, at four pressures. At 0.8 of the reference pressure the sensitivity is -0.34 per cent of pressure per kelvin, so 10 degrees is -3.45 per cent. That is not a small number against what the technique is used to measure, and a model's surface temperature is not uniform: it is warmer where the flow has been brought to rest and cooler where it has accelerated, which means the temperature error is largest exactly where the pressure gradients are. The standard remedy is a second, temperature-sensitive paint measured at the same time — an instrument added to correct an instrument.

    The paint that measures the wrong field

    Dye, seeded particles and the optical methods all look through the flow or at something put in it. Pressure-sensitive paint looks at the surface, reports a scalar rather than a derivative, and turns a row of taps into a field. What quenches its luminescence is oxygen, which is what makes it a pressure gauge — and temperature, which is the other field a flow is guaranteed to produce.

    part 4 · misconceptions
  5. 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.

    part 5 · misconceptions
  6. One per cent of noise on the image, 1.43 on the axis. The field recovered by onion peeling on 50 rings from one view carrying noise of one per cent of the instrument's own peak reading, from an interferometer's projection and from a schlieren system's deflection, against the true field. On the axis, where the truth is 1, the projection gives 1.431 and the deflection 0.876; over the whole radius their root-mean-square errors are 0.0742 and 0.0219, so the deflection is the quieter route here. Near the edge both are clean, and the error gathers towards the axis — where the field is largest and the flow usually most interesting.

    One view is enough, and the axis pays for it

    An axisymmetric flow — a jet, a plume, a flame — can be reconstructed from a single optical view, because Abel's integral inverts exactly. The inversion runs from the outside in, every error made on the way reaches the axis, and whether it arrives multiplied depends on which instrument took the picture.

    part 6 · misconceptions
  7. 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.

    part 7 · misconceptions
  8. Four views, four guesses at what they missed. The earlier essay's field — two puffs and an oblique streak — and its reconstructions from the same four views, each choosing the part of the field the views cannot see by a different assumption. With none it is set to zero: an error of 42.8 per cent. The smoothest field the views allow: 39.8. A non-negative field: 24.7. A sum of Gaussian puffs of one width: 37.6. The first three meet every one of the 96 rays; the dictionary's does not.

    A prior the pictures can refute is the one worth having

    A few views of a flow with no symmetry leave most of the field unseen, and a reconstruction has to fill that part in by assuming something. Three common assumptions were tried on the same field and the same views. Non-negativity saves three views of ten, and five on a field of puffs; smoothness saves one; a dictionary of known shapes saves none unless it is exactly right, and when it is nearly right it can be wrong by a factor of a hundred. The prior that helps most is also the one the data can catch being wrong.

    part 8 · misconceptions
  9. With the weight chosen from the data, non-negativity still saves views, and the saving shrinks as the noise grows. The fewest views that bring each prior's reconstruction of a field of two puffs within fifteen per cent of the truth, against the noise on the rays as a share of the rms ray: solid with each prior's weight chosen by the discrepancy principle, from the data alone; dashed with the weight that knows the answer. Points drawn at 25 never get there. At one per cent noise no prior needs 8 views and non-negativity 4; at three per cent, 8 and 6, against 4 with the answer known; at ten per cent, more than 24 and 10, against 7.

    A prior chosen from noisy pictures keeps most of its gain, and a wrong one loses its alibi

    With exact rays, a reconstruction's prior could be judged by its error, and a constraint that was wrong gave itself away by missing its own rays. With noise every prior needs a weight — how far to trust the rays against the assumption — and the weight has to be chosen without the answer. Chosen where the misfit falls to the noise, it costs the smooth and unassuming priors almost nothing and non-negativity two or three views, so the ranking survives. What does not survive is the refutation: a dictionary that misses a field's rays by four per cent is caught at three per cent noise and invisible at seven, where it fits the noise as well as any prior and is about as accurate as they are.

    part 9 · misconceptions

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