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The thread: One number decides the regime — page 43

Page 43 of 50, continuing through the 450 essays this motif runs through.

450 essays carry this thread — page 43 of 50.

Forced at the integral scale, the third moment nearly reaches four-fifths. The Kármán–Howarth balance at a Taylor-scale Reynolds number of 200, on the same model spectrum, for a flow forced in a band at the spectrum's peak and for one decaying. Each term is divided by (4/5)εr and plotted against separation in Kolmogorov lengths. The viscous term is the same for both. The forcing term is negligible until the separation approaches the integral scale, and −Dₗₗₗ/((4/5)εr) for the forced flow peaks at 0.965 at 90 Kolmogorov lengths, where the decaying flow's peaks at 0.747 at 32.

Forcing at the integral scale leaves the cascade alone

A decaying flow's third moment falls a quarter short of the four-fifths law at the Reynolds numbers a grid reaches. A flow forced at its largest scales, on the same spectrum at the same Reynolds number, falls short by three and a half per cent. The shortfall is not what a finite Reynolds number does to every flow; it is what the source does, and how fast it closes is set by how far the source reaches into the inertial range.

Below the capillary length, the coat is set by the fibre, not the bath. The coat's thickness in units of ℓ_c Ca^(2/3) against the fibre's radius in capillary lengths, on logarithmic axes: the dynamic meniscus's 1.3376 over the curvature the static meniscus asks of it, 1/b + Z/ℓ_c². Thin fibres take Quéré's 1.3376 b Ca^(2/3), set by their own radius; thick ones take Landau–Levich's 0.9458 ℓ_c Ca^(2/3). The two laws cross at b = 0.71 ℓ_c, where the coat is 0.61 of either; at b = 0.1 ℓ_c it is 0.137 of what the plate's law would give.

Viscosity

A thin fibre coats by its own radius, and beads by it

A plate drawn out of a bath carries a film set by the capillary length, the size at which surface tension and gravity balance. A fibre thinner than that length carries a film set by its own radius instead, often a tenth of what the plate's law promises — and the same radius then decides how quickly that film gathers into beads. The faster the fibre is drawn, the thicker its coat and the shorter the length of it that comes out smooth.

Fluids at work

Air at a valve softens the hammer only in quantity

A vapour cavity at a closed valve holds the head at one value, and that is why the pressure after it collapses climbs a staircase of equal steps. Put a pocket of free air there instead and the valve becomes a spring. A millionth of the pipe's volume trims only the tallest, briefest pulses; a hundred-thousandth takes a third off them and moves them; a ten-thousandth can make the pulse taller than it was, even on a line that never cavitates. The air that reliably removes the hammer is a thousandth of the pipe, which is a deliberate air vessel rather than a little dissolved gas.

Circulation and lift

A canard pays for its stability in induced drag

The argument for a canard is that both of its surfaces lift upwards, while a tail pushes down and makes the wing carry the difference. Munk's stagger theorem turns the question into arithmetic: two surfaces' least induced drag depends only on how the lift is split between them, not on which is in front. Static margin sets the split. At a margin of a tenth of a chord a tail carries a small upload and costs under one per cent; a canard must carry a fifth of the weight on a third of the span and costs twelve, and the more stable it is made, the more it pays.

Next to the body the gas remembers the nose. The gas temperature, over the free stream's, against distance from the axis of a hemisphere-nosed cylinder at Mach 15, at 2, 20 and 200 diameters behind the nose; the body's surface is at a half. The gas next to the body crossed the nearly normal part of the bow shock and carries its entropy: 16.7 times the free stream's temperature at 2 diameters, 9.29 at 200, falling outwards to the gas that crossed the weaker, oblique shock. The blast-wave analogy's core at 2 diameters, dashed, runs off the top of the frame on its way to infinity.

Compressible flow

The gas beside a hypersonic body remembers its nose

The blast-wave analogy gets a blunt body's bow shock and afterbody pressure right and its temperature absurdly wrong: it puts the body in the empty core of an explosion, where the temperature has no bound. The real gas beside the body crossed the nearly normal shock at the nose and carries that crossing's entropy all the way down. Expanded to the afterbody's pressure, it is ten to twenty times the free stream's temperature at Mach 15, it never falls below a floor set by the nose alone, and the sheath that carries it is wider than the body.

Over a day the breeze turns right round. The tip of the surface wind vector over one day, onshore to the right and along the coast upwards, at 15°, 30° and 45° north, with a spin-down time of twelve hours, in units of the push over the daily frequency. Without rotation the wind would swing on and off shore along one line. The Coriolis force turns it clockwise through the day into an ellipse; at 30°, where the Earth's inertial period is exactly a day, the turning keeps pace with the push and the ellipse is nearly a circle, of radius 1.8 against the ellipses' 1.2 at 15° and 1.2 at 45°.

Ideal flow

At thirty degrees the Earth keeps time with the sea breeze

A sea breeze is Kelvin's circulation theorem failing: where warm air over land meets cool air over the sea, pressure and density surfaces cross and a circulation grows out of still air. Bjerknes' theorem prices the push, and the textbook's numbers give twenty-six metres a second in an hour — far more than any sea breeze blows. What stops it first is not friction but the evening: the push reverses before the wind can grow, and the day, not the drag, sets the breeze's size. Friction sets its hour, the Earth's rotation turns it through the day, and at thirty degrees, where the inertial period is a day, the turning keeps time with the push.

Squeezed past closing, the tube is held open by its own fluid. The channel's half-width along one wavelength of a travelling squeeze that would push an empty tube's wall 1.3 half-widths inward — past the centreline — and the wall the squeeze and the fluid's pressure make together, for three compliances. The empty squeeze would overlap the centreline across a sixth of the wavelength; with fluid in the tube, the pressure that builds ahead of the narrowest point pushes the wall back, and the gap stays open at a hundredth to a tenth of the half-width.

Flows and fields

A squeezed tube is held open by its own fluid

A peristaltic pump with a prescribed wall wave gets better the more its wave closes the channel, and at closure it becomes a piston. A real one squeezes an elastic wall and lets the fluid push back. Then the channel never closes: the pressure ahead of the narrowest point holds a gap open that grows as the square root of the wall's give, squeezing harder past closure pumps less, and the pump behaves as a displacement pump until a pressure of about one over its compliance blows the throat open — with an efficiency that stops short of one by an amount that grows with the wall's give.

Past its threshold a ridge slides at a speed its angles set. The steady speed of a ridge of liquid one square capillary length in cross-section, as a capillary number, against the plate's tilt, on three surfaces. Below each surface's threshold it is stuck. Past it the speed rises from zero, linearly at first, as far as the tilt allows. On clean glass the curve barely exists: between its threshold and the speed at which its uphill contact line fails there is a sliver of tilt, and a ridge pushed past that cannot slide steadily with a clean trailing edge.

Regimes and numbers

A sliding drop is held harder the faster it goes

A ridge of liquid on a tilted plate starts to slide when its weight beats the difference between its two contact angles' cosines. Once it moves, the angles move too: the front steepens and the back flattens, by a law set in the viscous corners at each edge. So the resistance rises with speed from exactly the static value, and a sliding drop has no kinetic friction lower than its static one — it stops at the tilt it started at. The back edge's angle falls to nothing at a finite speed, and past that no drop slides with a clean back.

Bursts in the velocity put the tails back. The kurtosis of the pairs' separation against how long each pair remembers its velocity's direction, β, in turnover times. The lowest curve is a Gaussian velocity, as in the earlier calculation. The others give the velocity a flatness of 4, as measured in the inertial range, with its amplitude remembered for α turnover times. At the memory real pairs are estimated to have, β = 0.7, the Gaussian cloud's kurtosis is 1.89; with bursts remembered for three turnover times it is 3.41, and with the amplitude frozen 4.4 — either side of Richardson's 3.76, dashed.

Transition and turbulence

Bursts and memory pull a cloud both ways

A pair of fluid particles that remembers its relative velocity spreads into a cloud with shorter tails than Richardson's, and the earlier calculation proposed reading the memory off the cloud's shape. Real relative velocities come in bursts, their amplitude set by a local dissipation that varies, and a burst that lasts pushes the tails back out — hard, because separation grows as the cube of diffusivity. At the memory real pairs are estimated to have, the two effects nearly cancel, and a cloud can have Richardson's exact shape for entirely the wrong reason.

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