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The thread: One number decides the regime — page 37

Page 37 of 58, continuing through the 521 essays this motif runs through.

521 essays carry this thread — page 37 of 58.

$The vapour fraction that would stop a cavity, against temperature. How much vapour a cavity would have to make, as a multiple of the liquid volume it cools, before the cooling had used up the whole driving tension. In cold water it is six hundred and the effect is unreachable; at the boiling point it is below one and the effect is the dominant thing in the problem. Four decades across the range a pump might see, out of a vapour density that rises by a factor of two hundred.$ Fluids at work

A cavity that cools the water it came from

The usual account takes the vapour pressure as a property of the liquid, looked up once. It is a property of the liquid at whatever temperature the cavity has left it — and making vapour costs latent heat, so a cavity suppresses itself. In cold water that is unmeasurable and in liquid hydrogen it is the dominant term.

A drop fed into a tube, to scale, up to the largest one it can hold. Six members of the family of static drops a tube of radius half a capillary length holds, drawn to scale and hung from the rim, with their volumes in cubic capillary lengths. The drop grows from a shallow cap through a hemisphere to a bulb with a neck. The last is the fold: no static drop on this tube holds more, and it holds 76 per cent of what Tate's law says the tube can carry.

Regimes and numbers

The drop falls at a fold

Tate's law says a drop leaves a tube when its weight equals the tension round the rim. No force balance decides it. A tube holds a family of static drops, the family has a largest member, and the drop falls because there is no static shape with more liquid in it — which Tate's balance overestimates by a quarter on a millimetre tube and underestimates on a wide one.

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.

The water a submerged circle carries depends on how fast it is shaken. The added mass of a horizontal circular cylinder heaving under deep water, as a multiple of ρπa², against the wavenumber of the waves its frequency makes, Ka = ω²a/g, for five depths of its centre, with the depth set drawn dark: 1.1 radii, least added mass -0.182. Slowly shaken, every depth borrows more than its free-space mass; quickly shaken, less. Between, each curve rises to a peak and then falls through a trough, and for the shallowest circle the trough goes below zero — to -0.182 at Ka = 0.76.

What is taught wrongly

The borrowed mass that goes negative

A body's added mass is taught as the water it carries with it, fixed by its shape. Put the body under a surface that can make waves and the amount depends on how fast it is shaken, runs from a wall's value to a free boundary's and past both, and for a circle close enough to the surface falls below zero.

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.

The same thickness, drawn as a body of revolution and as a wing section. The surface speed along three prolate spheroids of thickness ratio a half, a quarter and a tenth, and along the plane ellipses of the same ratios, in units of the stream speed. Each plane section speeds the flow over its middle by exactly its own thickness ratio. The body of revolution of the same ratio does so by much less — at a tenth, by two per cent where the ellipse manages ten.

Ideal flow

Thickness costs a fuselage far less than a wing

A wing section a tenth as thick as it is long speeds the air over its middle by ten per cent. A body of revolution with the same proportions speeds it by two. The difference is not a detail of shape: a plane section pays for its thickness linearly and a body of revolution pays quadratically, and that one exponent is why a fuselage reaches the speed of sound on its surface long after its wing.

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.

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