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The thread: The exact theory is wrong — page 27

Page 27 of 31, continuing through the 276 essays this motif runs through.

276 essays carry this thread — page 27 of 31.

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.

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.

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.

An S-shaped set of states and a slow variable that crosses it. The film's steady mean temperature against the housing's, on the cool and hot branches of operating points, with the line on which the housing is in equilibrium with the film, at a derating of 0.22 per unit. Between housing temperatures 1.24 and 1.59 both branches exist. At the derating the essay uses, 0.22, neither meets the equilibrium line there: on the hot branch the housing is always warming, on the cool branch always cooling, so the film runs round the loop drawn over them, jumping at each fold.

Viscosity

A slow sensor makes the bearing cycle

Derate the motor driving a self-heating oil film on the temperature of its housing, which warms over minutes, and a protection meant to hold the bearing cool instead makes it cycle for ever: jump hot, warm the housing until the hot state is lost, drop cool, let the housing cool until the cool state is lost, jump again. The period belongs to the housing, the amplitude belongs to the film, and the derating strength decides only whether the cycle happens at all.

$The wall lets go long before the textbook's shock arrives. Where the flow leaves the nozzle's wall, as a fraction of the divergent section's length, against the ratio of chamber to ambient pressure. The inviscid picture runs full to the exit until the pressure ratio falls to 24.5, where a normal shock stands at the exit, and then moves the shock inside. The wall flow separates instead, when its pressure falls to Summerfield's 0.4 of ambient — at a pressure ratio of 192 — or to Schmucker's Mach-dependent value, at 138: six to eight times higher. Across that whole band the textbook's nozzle is full and the real one is not.$ Compressible flow

An overexpanded nozzle lets go before its shock arrives

The textbook's overexpanded nozzle runs full until its back pressure is high enough to hold a normal shock at the exit, and then draws the shock walking inside. A real nozzle never shows that sequence. Its wall's boundary layer cannot climb the pressure rise a normal shock imposes; it separates at a wall pressure of about four-tenths of ambient, which for a rocket nozzle happens at six to eight times the pressure ratio the textbook's shock needs. The separation is not a failure: it is what keeps the nozzle's thrust, and it is why a sea-level engine can be built twice the size it expands to.

Five gas films never meet their thermal crossing. Six air films placed by their squeeze number, which decides whether the gas leaves the gap or is trapped in it, and their thermal number, which decides whether the heat of compression leaves. The dashed diagonal is the estimate that the two differ by the Prandtl number. The devices lie between one and eleven decades below it, because heat leaves across the gap and the gas along the disc. Only the levitator at a twenty-micron gap comes within a decade of the thermal crossing at ωh²/κ ≈ 18.

Viscosity

The heat of a squeeze leaves across the gap

Squeeze a film of air and it warms, and whether that heat reaches the walls in time decides whether the gas is compressed isothermally or adiabatically. The natural estimate has the heat leaving the way the gas does, along the disc, and puts the thermal crossing beside the viscous one. It leaves across the gap instead, a distance hundreds of times shorter, so the crossing sits decades away — and when a film does reach it, what it gains is not stiffness but a second band of loss.

Ideal flow

A free pair turns, and forty-five degrees decides

Two cylinders in an ideal stream move to look bigger to it: side by side they close, in tandem they part. Free to turn as well, a staggered pair swings towards side by side — and does not stop there, because nothing in an ideal fluid damps the swing. Whether it then collides or flies apart is decided, far apart, by one angle and a theorem: the stream's pull on the pair falls as the inverse square of the spacing, and for such a force the sign of the energy alone decides, which changes at exactly forty-five degrees.

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