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

Inverse problem — where it appears

A problem that asks for the cause from the effect: the field that produced a recorded integral, the shape that produced a pressure. Such problems can have an exact answer and still amplify noise in the data without limit, so how the inversion treats errors matters as much as whether it exists.

Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

misconceptions · Visualisation What is left behind, and what it is made of. The two rolled-up vortices behind a large aircraft, drawn to scale against its span. They sit at 78.5 per cent of the span apart, each carries 508 square metres a second of circulation, and the pair descends at 1.72 metres a second under its own induction.

A wake that says what made it

Two vortices sitting behind an aeroplane carry its weight and its span in a form that can be read back out: circulation times separation times density times speed is the lift, exactly. The reading is exact for about a minute and worth nothing after four.

circulation · Tip vortex 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 layer read as slip gives a slip length that drifts. The slip length a drainage measurement would fit at each gap to the force of a 1-nanometre viscous layer, using the slip formula, for layers three and ten times as viscous and immobile. Far off it is the first-order value, −0.667 nm for κ = 3; closer, it shrinks towards zero — −0.597 nm at 10 nm and −0.525 at 5. A slip length that depends on the gap is the layer's signature, and the only one a drainage curve carries.

A viscous nanometre reads as slip until the gap is ten of it

A drainage experiment turns a molecular length at a wall into a measurable force, and reports it as a slip length. A layer of liquid a nanometre thick and several times as viscous as the bulk changes the same force by the same amount, to first order. Solved with the stratified viscosity, the two pictures part by a per cent only when the gap is eleven to seventeen layers wide, and the part of the difference a measurement can see is a slip length that drifts with the gap. A layer that does not move at all is exactly a wall in a different place.

misconceptions · The no-slip condition 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 Elasticity adds a spring at first order and a slip only at second. The damping's deficit, one minus the damping ratio, and the in-phase force against Λ on logarithmic axes. The in-phase force rises in proportion to Λ — a fitted slope of 1.0000 — and the damping's deficit as its square, 2.0000. A slip length changes the damping at first order and adds no spring at all, so a soft wall and a slipping one are distinct in kind: the soft wall announces itself first in the phase of the force.

A soft wall is read as slip only at second order

A drainage force smaller than Taylor's has been read as the liquid slipping at the wall. A wall that gives under the drainage pressure lowers the force too, and it can be mistaken for slip. Solved together, the flow and the elastic wall say how: the softness first adds a spring to the force, in phase with the motion, and only at second order takes anything from the damping a slip length is read from. When it does, the slip it imitates grows as the square of the frequency and falls as the square of the gap.

misconceptions · The no-slip condition The camber lines that carry them. The camber lines that carry the four loads at their ideal angles, heights in chords at a design lift coefficient of one. The uniform load's line is symmetric about mid-chord, 5.52% high; tapering the load over the last fifth moves the peak to 0.515 of the chord and raises it to 6.79%; tapering it over the whole chord, a = 0, puts the peak at 0.323.

A load carried to the trailing edge needs a hook

Thin-aerofoil theory runs backwards: ask for a chordwise load and it returns the camber line that carries it. Ask for the simplest load of all, the same everywhere, and the line comes back with a vertical tangent at both ends. The one at the nose is the price of a clean entry; the one at the tail is the Kutta condition being broken, and the hook it makes is what a Gurney flap is. Every NACA a-series line is a way of not paying it.

circulation · Thin-aerofoil A spread of sizes smears the stop band and pushes the fast band up. The phase speed of sound against frequency in water carrying air bubbles at a void fraction of 10⁻³: all of one millimetre radius, and spread from 10 µm to 1 mm as n ∝ R^(−q) with q = 0, 3/2 and 10/3. Every cloud starts at Wood's speed, 312 m/s, because at low frequency every bubble is driven below its resonance. The single size jumps through its stop band near 3 to 13 kHz into a fast band. The distributions climb through a band as wide as their resonances, 3.2 kHz for the largest to 301 kHz for the smallest, and the more small bubbles they carry the later they emerge faster than water.

The more small bubbles a cloud holds, the later it turns fast

Bubbles of one size give water three sound speeds: slow below their resonance, nothing in a stop band, faster than water above it. A cloud of many sizes has every resonance at once, so at any frequency its small bubbles soften it and its large ones stiffen it. The stop band becomes a wide band of attenuation, the fast band survives above it, and where the fast band starts is set by how much of the gas is in the smallest bubbles.

compressible · Speed of sound A mica sheet shields the glue's false slip only where the pressed region is narrower than the sheet bends. The slip length, in nanometres, that a dynamic drainage measurement at 100 Hz would read from a wall that does not slip, against the gap: a 3 mm sphere in water over bare glue of plane-strain modulus 1 GPa, and over mica sheets 1, 3 and 5 µm thick on the same glue. At 3 nm the bare glue reads 9.48 nm and the sheets 8.75, 2.68 and 1.06; at 30 nm, 0.1 against 0.099, 0.067 and 0.043.

A mica sheet hides its glue only at the smallest gaps, and changes how the false slip falls

A drainage measurement against a soft wall reads a slip that is not there, falling as the square of the gap. The wall in a surface-force apparatus is not a soft half-space but a stiff mica sheet on a soft glue, and the sheet spreads the drainage pressure over its own bending length before the glue feels it. Where the pressed region is wider than that length the sheet changes nothing; where it is narrower the sheet sinks as one piece, a spring in series with the liquid. That cuts the false slip only by the square of the bending length over the pressed width — threefold at three nanometres for three micrometres of mica — and it turns the slip's fall with the gap from the inverse square towards the inverse, which is the test an experimenter would have used to see it.

misconceptions · The no-slip condition 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.

Model limitMeasurementInstrumentLubricationThe no-slip conditionReynolds equationSlipSqueeze filmBoundary conditionElasticityFlow visualisationIll-posed

All concepts