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The thread: Taught wrongly, everywhere — page 25

Page 25 of 25, continuing through the 220 essays this motif runs through.

220 essays carry this thread — page 25 of 25.

$A flicker of a few per cent empties the suspension band and drains Stommel's traps more slowly. The share of a cloud released over one cell that is still within four cells of its start after the run, against the flicker's standard deviation as a fraction of the cells' mean strength, with a memory of three turnovers: inertialess particles falling at 0.3 of the cells' speed in still air, held in Stommel's traps, and particles of Stokes number 3 falling at 0.1, held in the suspension band. Stommel's traps hold 0.52, 0.41, 0.33, 0.16, 0.03, 0; the suspension band 0.22, 0.055, 0.015, 0.015, 0.005, 0, at flickers of 0, 0.05, 0.1, 0.2, 0.35, 0.5.$ Regimes and numbers

Flickering vortices cannot hold a settling particle up, and still make it fall faster

A steady lattice of vortices can do two things to a heavy particle settling through it: hold it up, on closed paths or on orbits of its own, and speed its fall, by flinging it into the down-draughts. Let the vortices strengthen and fade at random, as real eddies do, and the two come apart. A flicker of five per cent of the cells' strength empties three quarters of the band of particles held up by their inertia; a flicker of a third drains Stommel's traps almost entirely. The speed-up hardly notices a flicker of a tenth, and is halved only when the cells swing by half their strength and remember it for a few turnovers. Trapping needs a steady cell; the speed-up does not.

Through the transitional range Nikuradse's sand pipe runs below Colebrook's curve by up to a fifth. The friction factor against the Reynolds number for a pipe of relative roughness 1/120, with Nikuradse's roughness function and with Colebrook's, beside the smooth pipe. Nikuradse's falls furthest below Colebrook's, 21.9 per cent, at a Reynolds number of 11318, where the roughness is 6 wall units; both reach 0.0356 when fully rough.

Transition and turbulence

Sand and a commercial pipe part company at six wall units

A rough wall shifts the logarithmic velocity profile down by an amount that depends on the roughness height in wall units, and once the wall is fully rough every kind of roughness shifts it the same way. Before that they disagree. Nikuradse's uniform sand stays smooth until its grains reach four wall units and then rises steeply; Colebrook's commercial pipes feel grains a fifth of a wall unit high and rise smoothly. Put both into one log law and the friction factor of a pipe through its transitional range differs by up to a quarter, always at the same roughness in wall units, about six. The Moody chart, drawn from Colebrook, is the upper of the two.

Pressed with one load, a steeper blade leaves a thinner film, and without a land a steep one cannot float at all. The film a loaded scraper leaves on the surface, in micrometres, against the angle its face makes with the surface, at a load of 100 N per metre of span on an ink of 0.1 Pa·s moving at 5 m/s: with a flat land 0.1 mm long under the edge, and with none. With the land the film is 110 µm at 10°, 1.69 at 30° and 0.268 at 60°. Without one it is the whole gap, 145 µm at 8°, and above 44° the gap the load needs is below a ten-thousandth of a nanometre: the edge is in contact.

Viscosity

A pressed blade floats on its land and leaves half the gap behind

A scraper or a doctor blade is not held at a gap; it is pressed on with a load, and it rises until the liquid it lets through carries that load. Without a flat land under its edge the liquid can carry it only through the corner's logarithm, so the gap falls exponentially with the load, and a blade steeper than about forty-four degrees cannot float at a working load at all — its edge is in contact. Give it a land a tenth of a millimetre long and the film becomes half the gap, falls as one over the load, and is a quarter of a micrometre at sixty degrees. The land is also why a blade's film grows as its edge wears.

A five per cent distortion gives six blades a tone that no longer falls with the tip's speed, and two blades hardly notice. The sound power of the blade-passing fundamental against tip Mach number, thrust and shaft power held, for two and six blades: from the steady load, and from the thrust's fluctuation as each blade cuts an inflow distortion of 5 per cent rms. At tip Mach 0.4 six blades give 100 dB from the steady load and 113 from the distortion; two blades 127 and 109. The six-bladed steady tone overtakes its distortion tone at tip Mach 0.593.

Compressible flow

A distorted inflow gives back the tone more blades took away

A propeller's steady thrust sings through a Bessel function whose order is the number of blades times the harmonic, and that function is exponentially small at low tip speeds — which is why six blades are quieter than two. But a blade that cuts a non-uniform inflow carries a thrust that changes round the circle, and its s-th harmonic radiates through a Bessel function of order six minus s. The sixth needs no tip speed at all. A distortion of five per cent turns six blades' tone into one that hardly falls as the tips slow, fills the axis where the steady tone is silent, and cuts the six-blade advantage over two from twenty decibels to thirteen. Two blades never notice.

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