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

Flow visualisation — where it appears

The making of a flow's structure visible with smoke, dye, particles or refraction. Each technique renders a different curve or a different quantity, so what a photograph shows is a property of the instrument as much as of the flow.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

misconceptions · Visualisation The number on a streamline is a flow rate. Two streamlines and a crooked line drawn between them. The volume of fluid crossing that line every second, integrated from the velocity field, is the difference between the two streamfunction values at its ends — which is what makes a streamline a label rather than merely a curve.

The number on a streamline is a flow rate

Streamlines get drawn as decoration — curves the flow follows, spaced however the plotting looked best. Each one carries a number, the difference between two of those numbers is the volume of fluid passing between them every second, and it does not matter what route the measurement takes.

kinematics · Continuity Q and the vorticity along a radius of one vortex. Two candidate measures of where the vortex is, along a radius of a Lamb–Oseen vortex. The vorticity is a Gaussian: positive at every radius, so a threshold on it puts the edge wherever the threshold is put. Q — the excess of rotation over strain — changes sign exactly once, at 1.121 core radii, and that radius is a property of the flow rather than of the person drawing it. Inside it, 71.5 per cent of the circulation.

Where a vortex stops

Four criteria decide where a vortex ends, and in two dimensions three of them are the same criterion. The fourth is a knob. And the one that is not a knob is not objective: a co-rotating pair of vortices occupies four per cent of a window to one observer and more than half of it to another.

kinematics · Coherent structures Ideal flow past a sphere, drawn in a meridional plane. The Stokes stream function of ideal flow past a sphere, contoured at equal intervals. Its relations to the velocity carry factors of r sin θ that the plane stream function does not have, and differencing it reproduces the closed-form velocity to 10⁻¹¹. The surface speed at the equator is exactly one and a half times the free stream, against twice for a circular cylinder: a three-dimensional body lets the flow past in two directions rather than one.

The one number that runs out at three dimensions

A stream function is one function where the velocity is two, and four essays here are built on it. It exists because the divergence vanishes, it is single-valued only if nothing inside is making fluid, and in three dimensions it is not one function at all.

kinematics · Stream function 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 The same flow, from two different physics. Solid lines: the depth-averaged flow between two plates a small distance apart, with an obstacle standing between them. Dashed: the ideal-flow solution for the same obstacle. They are the same field to a part in ten billion, because averaging Stokes flow across a narrow gap gives a velocity that is the gradient of a harmonic potential. The cell has no inertia at all, which is the one hypothesis the ideal theory cannot do without.

The exact theory, drawn by viscosity

Two flat plates a millimetre apart, syrup between them, an obstacle in the gap. The Reynolds number is a hundredth, inertia is absent, and the streamline pattern is the potential flow past that obstacle — exactly, to a part in ten billion. The one hypothesis ideal flow cannot do without is the one this flow most conspicuously breaks.

inviscid · Hele-shaw 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.

misconceptions · Visualisation 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.

misconceptions · Visualisation

Named alongside it

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

MeasurementModel limitDivergenceInverse problemMass conservationModel validityNull spacePotential flowRegularisationStreamfunctionStreamlineAnalogy

All concepts