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The thread: A smooth picture proves nothing — page 27

Page 27 of 31, continuing through the 272 essays this motif runs through.

272 essays carry this thread — page 27 of 31.

Dissolved gas takes height from the largest nuclei first. The tallest crown a siphon runs over, against the radius of its largest nucleus at the reservoir, when a nucleus at the crown must neither break the water at once nor grow to break it by diffusion, for water at 0, 10, 30, 60 and 100 per cent of saturation. Fully degassed water gives the static limit. For a one-micron nucleus, gas below about a fifth of saturation changes nothing — 14.41 m against 14.41 — and saturated water brings it to 10.7 m. A five-micron nucleus in saturated water breaks the siphon at 2.4 m.

A crown dissolves its nuclei or breaks on them, and fast

A siphon of degassed water runs until its crown's tension reaches the breaking threshold of its largest nucleus. A nucleus lodged at the crown does not stay the size it arrived: it gains or loses gas by diffusion, and across a micron diffusion takes milliseconds, so the crown gives its verdict almost at once. Every stable nucleus loses gas unless the water holds more than half the crown's tension, and past a knee in gas content, dissolved air takes height from a siphon's crown that no amount of further degassing gives back.

A cone hands the terminal shock a spread of Mach numbers. At Mach 2, the Mach number across the annulus a cowl on the conical shock captures, from the cone's surface to the shock, for cones of 15°, 22° and 28°, the last the best for recovery. Behind a wedge the flow is uniform; behind a cone it is fastest just behind the shock and slowest at the surface, 1.43 to 1.32 for the best cone. The terminal shock acts on all of it.

Compressible flow

A cone intake's shock sees the whole capture

A cone's shock keeps far more total pressure than a wedge's at the same angle, because the cone finishes its turn in an isentropic compression after the shock. An intake then needs a terminal normal shock, and that shock meets not one Mach number but the whole spread the cone leaves across the captured annulus — fastest by the shock, where most of the air is, and slowest at the surface, where little is. Averaged over the mass, the cone's advantage over a wedge at the best angle for each is under two points of recovery, and the usual estimate from the surface Mach number nearly doubles it.

A frozen run counts the area it sweeps. Independent values in a run of frames while a frozen pattern is swept 100 integral lengths past a window 12.8 integral lengths square, against the distance the pattern moves between frames. For a scalar the run holds 353 values with a frame every half integral length, 362 with one every four and 388 with one every window width: the swept area in integral areas, whatever the frame rate. Only past a window width, when gaps open between frames, does the count fall. Treating frames an integral scale or two apart as independent, as a record would allow, counts 2075 at two integral lengths — 5.8 times too many.

Transition and turbulence

Frames that share their eddies count as one

A particle-image run is a sequence of snapshots, and snapshots closer together than an integral time photograph the same eddies. When a mean flow carries a frozen pattern past the window, a run holds exactly the area it sweeps, counted in integral areas, and a faster camera adds nothing until frames stop overlapping — a threshold set by the window, not by the turbulence. In a flat flow one component escapes the rule: its run mean is fixed by the pattern at the two ends.

Ideal flow

Drag makes every free pair meet

Two cylinders set free in an ideal stream swing towards side by side like a pendulum, and whether they collide or part is fixed, far apart, by forty-five degrees. Give each a drag on its motion relative to the stream and the pendulum stops swinging — but it does not stop the pair. A drag coefficient of order one turns the swing into a creep, lets no pair get far from where it started, and brings every one of them round to meet near side by side.

$At a coast the swell's water goes back, or goes along. The Lagrangian mean current — the Stokes drift plus the Eulerian current — under an 8-second swell arriving straight at a coast, in seas 8, 25 and 60 metres deep, each as a fraction of its own surface drift against depth as a fraction of the water's. Left, the part across the shore, positive onshore; right, the part along it. The cross-shore parts carry no net water. At 8 metres, inside the Ekman layer, the whole exchange is across the shore; at 25 the return is shared with a current along the coast; at 60 the open shelf's own rotation has already taken the transport back, and the coast has little left to do.$ Flows and fields

A coast sends the drift back, or sends it along

A shelf sea under a swell keeps part of the Stokes drift's transport, turned to the right by the rotating Earth, and near a coast some of it points at the land. The sea answers with a slope in its surface. In water shallower than an Ekman depth the slope drives the water straight back as an undertow, as it would without rotation; a few Ekman depths down the coast turns the transport into a current along the shore instead, and there the sea can slope down towards the land it faces.

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.

What is taught wrongly

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.

Fly level through short waves, follow long ones. The mean drag of the two strategies against wavelength, for a wave 0.4 m high at 8 m/s: flying level at the best mean depth for that sea, and following the surface at the calm boat's best depth, with the flat-water drag as the faint line. Level flight costs about the same whatever the wavelength; following costs the square of the heave it demands and falls away as the waves lengthen. Into the waves the two cost the same at 15.6 m; running with them, where the boat meets each wave slowly, at 5.9 m.

Fluids at work

A foil flies level through a short sea and follows a long one

A foiling boat in waves has two ways to fly. It can hold its height and let the surface rise and fall over its foil, or it can follow the surface and heave with every wave. Flying level costs a deeper ride and a few per cent of drag whatever the wavelength; following costs the square of the heave, which grows with the frequency the boat meets the waves at. Into a head sea the two cost the same at a wavelength of about sixteen metres, running with the waves at about six, and the wave's height hardly moves either.

$A follower can wander further fore and aft than sideways. A flapping follower's saving, averaged over a wander of its place with the standard deviation shown, as a fraction of its own induced drag: fore and aft, holding the phase that suits its average place, for tips swinging a tenth, a fifth and four-tenths of a span; and sideways, beating in phase. The saving in place is 0.837. Sideways it has halved at a wander of 0.185 spans; fore and aft, with a fifth-span swing, only at 0.804, a third of the wake's wavelength, and with the four-tenths a cruising bird swings, at 0.37. With a tenth-span swing it never halves.$ Circulation and lift

A flapping follower can drift fore and aft, but not sideways

A bird in a flapping V has to be in the right place and beat at the right phase for that place, and no bird holds its place exactly. Drifting fore and aft costs it phase, at a full beat for every wavelength of the wake; drifting sideways takes it off the leader's tip vortex. The first is two to four times cheaper than the second, it can be bought back by re-timing the beat within about one beat of the drift, and the second cannot be bought back at all. So the precision a follower needs is sideways, and the attention it needs is on its timing.

Viscosity

A warm bush stops the bearing hunting

A bearing whose motor is derated on its housing's temperature can hunt, jumping between a cool film and a hot one on the housing's clock. Let the bush warm with the housing, and a second feedback joins the first, of the kind that usually makes things run away. It does the opposite: a warm wall is a stiffer drive, the stiffening closes the S the cycle runs round, and past a bush coupling of about a third the bearing settles, warm and derated, however the protection is set.

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