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The thread: A smooth picture proves nothing — page 21

Page 21 of 31, continuing through the 272 essays this motif runs through.

272 essays carry this thread — page 21 of 31.

The speed past the flank, in the cell and in the ideal flow it is meant to show. The gap-averaged speed along the top of a cylinder, against the distance from its surface in cylinder radii, for three cells — the cylinder's radius 10, 35 and 100 times the Brinkman length h/√12 — and for ideal flow. Ideal flow slides along the wall at twice the stream speed. The cell's flow must stop at the wall and does so in a layer a few Brinkman lengths thick, outside which it rejoins the ideal curve.

The cell draws a larger cylinder

A Hele-Shaw cell draws the streamlines of ideal flow past an obstacle and cannot obey the one rule ideal flow breaks: the fluid must stop at the obstacle's wall. It does stop there, in a layer a third of the gap thick, and far away the whole correction amounts to one thing — the cell is drawing ideal flow past a cylinder one layer-thickness too big, and the obstacle has exactly that cylinder's drag.

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.

A record's length is counted in integral scales. The scatter of a record's mean, in units of the signal's own standard deviation, and the relative scatter of its variance, against the record's length in integral time scales: four hundred records at each of seven lengths, against the exact results. Both fall as the square root of the number of integral scales, √(2Tᵢ/T). A mean known to one per cent of σ needs twenty thousand integral scales. The exact variance curve is for the variance about the true mean; a short record can only measure it about its own mean, which is why the shortest records scatter less than the curve says.

Transition and turbulence

A record is as long as its integral scales

A turbulence measurement of a million samples can hold less information than one of a thousand. What sets a record's worth is not how many numbers it contains but how many integral time scales it spans, and the integral scale — the unit every other error is counted in — is itself the hardest thing in the record to measure.

One rim, one drop, every angle in the band. Water on a disc the size of a US cent, 19.1 mm across, pinned at its rim. The faint outlines are the drop at apparent angles of 60°, 90°, 120° and 150°; the dark one is the drop at the angle set, 120°, holding 1.13 mL. The rim stays put while the angle climbs, and every shape is an equilibrium: 0.52 mL at 60°, 1.44 mL at 150°. They are flattened by gravity into puddles with rounded edges, not spherical caps.

What is taught wrongly

An edge holds any angle it is given

A liquid's contact angle is taught as a property of the liquid and the solid. On a smooth face it is. On a sharp edge the line stops, the angle is free to take any value in a band as wide as the edge is sharp, and that freedom is how a coin carries a dome of water and a glass stands full above its brim.

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.

Circulation and lift

A wake ends by bending, not by fading

The two vortices behind an aeroplane do not simply weaken until they are gone. In still air each one bends in the other's strain, the bend grows by a factor of e every thirty-three seconds behind an airliner, and after a couple of minutes the pair has pinched itself into a chain of rings. The wavelength it chooses is eight and a half times the spacing, and the vortex cores hardly enter.

The drip spacing is set by the depth of the layer. The spacing of the fastest-growing wave, in units of 2π capillary lengths, against the depth of the hanging layer in capillary lengths, for water, glycerol and honey. A thin film of any of them drips at √2. Deep water drips at √3. Deep glycerol and honey keep growing past √3, their fastest wave as long as the layer is deep, because viscosity slows short waves more than long ones.

Transition and turbulence

How far apart a ceiling drips

A layer of liquid hanging from a ceiling is heavy fluid over light, and every ripple on it longer than about seventeen millimetres grows. Which ripple grows fastest, and so how far apart the drips form, is usually given as one number. It is at least three, and what chooses between them is not the liquid's surface tension but the depth of the layer.

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