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The thread: What is conserved — page 35

Page 35 of 43, continuing through the 384 essays this motif runs through.

384 essays carry this thread — page 35 of 43.

Every gradient in one plane, and where each one goes. The two invariants of a trace-free velocity gradient, R across and Q up. Under the restricted Euler equation the combination 27R²/4 + Q³ never changes, so every trajectory is one of these curves, and R never decreases, so every curve is travelled from left to right. Vieillefosse's line, where the combination is zero, has two branches: the left one runs into the origin and is the only way to reach it, and the right one is where every other trajectory ends, at infinity, in finite time.

A gradient left to itself

Follow a parcel's velocity gradient with nothing acting on it but its own square and the part of the pressure a single parcel can know about. Every starting gradient but a set of measure zero reaches infinity in a finite time, and it gets there as a sheet with its spin lying in the sheet.

A gust, the aeroplane's response, and the load it is left with. A one-minus-cosine gust twenty-five chords long, as an angle of attack, against distance flown, and the lift it produces on two aeroplanes free to rise — a light single with a mass parameter of 16 and an airliner at eleven kilometres with 149. Both loads lag the gust, because lift takes several chords to build. The light aeroplane's also peaks lower and falls away early: it has started to rise with the gust, which lowers its own angle of attack.

Circulation and lift

The aeroplane rises out of its own gust

A gust pushes a wing up, and the wing, being attached to an aeroplane that is free to move, starts rising — which lowers its angle of attack and takes away part of the load the gust was delivering. How much depends on one number, and the formula every airworthiness code used for forty years is a fit to the calculation this essay does again.

Three pairs released at different distances end on one line. The mean square separation of two parcels against time, both in Kolmogorov units, for pairs released one, thirty and a thousand Kolmogorov lengths apart in turbulence whose integral scale is a million of them. Each starts flat — the pair has not yet moved — and each joins the same line, g ε t³ with g = 0.5, after which nothing about where it started survives. Beyond the integral scale all three become ordinary diffusion, growing as t.

Transition and turbulence

The further apart, the faster they part

Release two specks of smoke close together in turbulence and the distance between them does not diffuse the way a single speck wanders. It grows faster the larger it already is, because larger separations are pulled apart by larger eddies — as the cube of the time, forgetting where it started. The law is ninety years old, its constant is still argued about, and it needs a very large flow to be seen at all.

Four tubes to scale, and the level Jurin's law gives each. Water rising in tubes of radius half a capillary length to four, drawn to one scale, with the level of the liquid far outside the tubes at the bottom. The red line in each tube is Jurin's height, 2ℓ²/R. In the narrow tube the surface is nearly flat and sits on it. In the wide ones the liquid gathers in a rim at the wall, the centre hardly rises, and Jurin's level runs through the middle of a surface that is nowhere near it — while still being its exact average.

Regimes and numbers

A law that is exact as an average

Jurin's law gives the height water climbs in a tube as twice the square of the capillary length over the radius. As a statement about the height it is an approximation for narrow tubes. As a statement about the mean height of everything lifted it is exact for every tube — and a wide tube shows what the average was hiding: a rim of water at the wall and almost nothing in the middle.

A gust spectrum at a point, across a span, and as a load. The spectrum of vertical gust velocity at a single point — von Kármán's, falling as the minus five-thirds power — and the same gust averaged across a wing whose span is a tenth of the turbulence scale and across one as wide as the scale. Averaging removes the wavenumbers shorter than the span, one power steeper. Through Sears' filter for a chord of a five-hundredth of the scale, the tenth-span load falls as the minus eleven-thirds power, steep enough to converge.

Circulation and lift

The span takes away the infinity, not the gust

A two-dimensional wing in turbulence reverses its load infinitely often, because the gust spectrum never falls fast enough. A real wing averages the gust across its span, and the average keeps ninety-five per cent of the gust's strength and removes the infinity. The load's crossing rate settles at about two per turbulence scale flown and stops depending on where anyone cut the spectrum off.

Where the reflected shock meets the driver gas. Distance against time in a helium-driven air tube, the incident shock in red running from the diaphragm to the end wall, the contact surface behind it in green, and the reflected shock coming back. Where it meets the contact, an under-tailored tube sends an expansion back to the wall — in ochre — which lowers the reservoir's pressure, and the contact drifts back towards the diaphragm; an over-tailored one sends a shock, which raises it. At the tailored Mach number, 3.41, nothing is sent back, the contact stops dead, and the reservoir at the end wall holds until something slower arrives.

Compressible flow

The shock that passes without an echo

A shock tunnel's reservoir lasts until a wave comes back from the driver gas to spoil it. For each pair of gases there is one incident shock strength at which nothing comes back at all, and it is found by asking the reflected shock to stop two different gases at the same pressure.

Viscosity

A flow pinned between two guesses

The minimum-dissipation principle says the true flow is the cheapest one the walls allow, so any guessed velocity carries too little. It has a twin that nobody teaches: any guessed stress in balance with the pressure carries too much, and it needs no wall condition at all. Between the two, the flow through a duct with no formula is pinned down to as many figures as anyone wants.

The gas flows in, against the heat. The radial velocity of the gas, in units of α/a, at the same four times — from the motion of the gas shells themselves — with the heat flux, reversed in sign and in the same units, drawn as dots. The two coincide: at constant pressure the gas moves at (γ − 1)/(γp) times the heat flux, against it. Heat leaves the spot outward and the gas comes in, everywhere.

Flows and fields

The air flows in against the heat

A gas moving at a ten-thousandth of the speed of sound is as incompressible as anything gets, by the usual arithmetic. Heat one spot of it and let the spot cool: the gas changes its volume threefold, and it flows towards the hot spot while the heat flows out, at a speed fixed by the heat flux alone. At constant pressure, a joule is a volume.

The first three ways a drop can ring. A drop's first three modes of oscillation, each drawn at the two ends of its swing: the shaded outline and the red one are half a period apart. The fundamental, l = 2, alternates between a lemon and a lentil. The next two ring at 1.94 and 3.00 times its pitch, with three and four lobes. The swings are drawn at a fifth of the radius so they can be seen; the frequencies belong to swings much smaller than that.

Regimes and numbers

A drop rings like a bell

Disturb a drop and it swings between a lemon and a lentil at a pitch set by its surface tension and its size, and it keeps swinging for dozens of cycles because water is nearly frictionless at that scale. The usual story has a falling raindrop's wake ringing it. The wake strikes several times too fast for the note the drop plays.

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