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The thread: A smooth picture proves nothing — page 26

Page 26 of 31, continuing through the 272 essays this motif runs through.

272 essays carry this thread — page 26 of 31.

$An edge is worth its sharpness against the capillary length. The extra height a rounded edge holds over a flat plate, as a fraction of what a sharp edge holds, against the rounding radius in capillary lengths, for Young angles of 30°, 60° and 90°. An edge rounded to a thirtieth of a capillary length, about a tenth of a millimetre for water, keeps about ninety-five per cent. One rounded to a third of a millimetre keeps between eight and nine tenths; one rounded to a whole capillary length, about a third; and past that the worth falls as the inverse of the radius.$ What is taught wrongly

A rounded edge spills before its line goes round

Gibbs's band says a sharp edge lets a liquid stand at any angle across a range as wide as the edge's turn. No real edge is sharp, and on a rounded one the line does not stop; it slides round the curve, meeting it at the Young angle everywhere. The band survives the sliding. What the rounding costs is height, because the line drops as it goes round, and past a point the drop outweighs the steeper angle and the liquid spills before the line has reached the far face. An edge is worth its sharpness measured against the capillary length, and nothing smaller.

Stratification makes a hill easier to block. The strength at which a current over a Gaussian hill first stops and traps fluid, δc = (h₀/H)/Ro at onset, against the stratification B = NH/fa — the depth over the height fa/N to which a rotating stratified flow feels the hill. Homogeneous, it is the 3.134 of the unstratified layer. As B grows the hill's anticyclone gathers at the bottom, where it is stronger, and the threshold falls; in deep water it falls as 2.243/B, which is a Froude number: the current is blocked when N h₀/U exceeds 2.243, and the rotation has dropped out.

Regimes and numbers

A stratified sea blocks a current sooner and traps less

A slow current in a rotating layer stops over a hill once the hill's height over the depth, divided by the Rossby number, passes 3.134, and a column of water over the hill is trapped from floor to surface. In a stratified sea the hill's anticyclone gathers near the floor and fades upward over a height of fa/N. It is stronger there, so the current stops sooner; and it is confined there, so what it traps is a cap, not a column. In deep water the threshold loses the rotation altogether and becomes a Froude number: a current is blocked when N h₀/U exceeds 2.24.

Every parcel goes straight while the streamlines wave. A uniform stream across the page carrying a pattern of transverse velocity with it, v = 0.6 sin(x − t): the streamlines at one instant, which wave, and the paths of five parcels over the next six time units, which are straight lines. Each parcel keeps the transverse velocity it started with, because the pattern moves with it; the streamlines are a snapshot of a pattern that is sliding past, and no parcel ever follows one.

Flows and fields

A parcel goes straight while the streamlines curve

In a steady flow every parcel accelerates while the picture never changes. The opposite also happens: a flow whose picture changes all the time while no parcel accelerates at all, because the local and convective halves of the acceleration cancel exactly. Such a flow has no pressure gradient anywhere, and that is a severe demand. An incompressible flow can meet it only as a pure shear at every point; a cloud that can compress meets it freely and pays later, when its straight paths cross in a caustic.

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.

Four vortices bend twice as fast, on shorter waves. Growth rate of the fastest bending wave against its wavelength, both in the units of the equivalent single pair — one unit of growth is one e-fold in the time the wake sinks by its own spacing. Crow's pair grows at most at 0.827, at 8.54 spacings. A flap vortex of 0.3 of the tip's strength at 0.4 of its station keeps a Crow band just below that and adds a second, at 1.57 and 1.8 spacings; one of 0.5 at 0.3 merges the two into one band peaking at 1.64 and 4.4 spacings.

Circulation and lift

Four vortices bend faster, and bend the wrong one

With its flaps down a wing sheds four vortices, not two: a strong one at each tip and a weaker one at each flap's edge, and on each side the two circle one another as the wake sinks. The orbit makes the wake unstable at waves a fifth of Crow's length, growing twice as fast. But the wave that grows is the weak flap vortex being bent round the strong tip vortex, which hardly moves, while the long bend that pinches the tips together grows no faster than before. Only a counter-rotating inner vortex makes the whole wake unstable, and it does so at every wavelength.

Tailored, the reservoir waits for the driver's own expansion. Distance against time in a helium-driven air tube at the tailored Mach number, 3.41, with a driver half as long as the driven tube: the incident shock, the contact surface, the reflected shock and the shock it transmits into the driver gas, and the driver's expansion — its head running back to the driver's closed end, and the head reflected from there. Nothing returns from the contact, and the reservoir at the end wall holds from 0.293 until the reflected head arrives at 0.555 L/a₁: a test time of 0.261.

Compressible flow

A tailored tube buys its test time with its driver

Tailoring a shock tube removes the wave the contact surface would send back to the reservoir. What ends the reservoir then is slower: the driver's own expansion, which runs back to the driver's closed end, reflects, and has to cross the whole tube to reach the end wall. Its arrival is exact in one dimension, because the reflected head crosses the incident fan as a simple wave, and the answer is that test time is bought with driver length — about seven-tenths of a driven-tube crossing time per driver length for helium — and that tailoring is worth nothing with a driver shorter than a quarter of the tube.

$A waist moves the fuselage's overspeed off the root chord. The fuselage's axial perturbation velocity at its own surface, as a fraction of flight speed, along its length at Mach 0.8, with the wing's root chord between the two rules: the plain fuselage, which adds 0.0251 everywhere along the chord; a narrow waist 6.2 per cent deep that cancels it at mid-chord and raises it at the chord's ends; and a wide one 31 per cent deep that leaves none of it positive along the chord and lifts it on the body ahead of and behind the wing.$ Ideal flow

A waist moves the overspeed and cannot remove it

A fuselage adds two per cent of overspeed at the wing root, and a waist — a local narrowing of the body where the wing joins it — is the obvious cure. Slender-body theory says exactly what a waist can do. Its sources and sinks sum to nothing, so the velocity it adds along the body integrates to zero: it cannot remove the overspeed, only move it. Moved off the whole root chord it needs a third of the fuselage's cross-section at the wing, a sixth of what the transonic area rule would take, and it lands on the fuselage just ahead of and behind the wing, where there is room for it.

A cloud's shape falls as it grows into calmer eddies. The kurtosis of a cloud of pairs released at a ten-thousandth of the integral scale, against its rms size, when the velocity's flatness follows the 1962 law, and for clouds whose velocity has one flatness, 3, 4 or 5, at every separation. The fixed-flatness clouds settle at a shape and keep it. The growing cloud rises to a kurtosis of 6.85 at 0.0022 L and then falls without settling, to 2.9 by 0.57 L, crossing all three.

Transition and turbulence

A cloud grows out of its bursts and keeps their shape

Measured velocity differences are burstiest across the smallest separations and nearly Gaussian across the largest, so a cloud of particle pairs released close together starts in the fiercest intermittency and grows out of it. Its shape follows, but late: the kurtosis falls steadily as the cloud grows, always above what its present statistics would give, and the cube law's constant falls with it, so the growth exponent climbs towards three and never arrives.

Moving air needs a quarter of the inertia still air does. The first crossing against Stokes number for a cloud thrown into still air — time in units of the cloud's fastest convergence rate — and for a cloud carried by air converging on a line, in units of the air's strain rate. Still air only brakes: its threshold is one. Converging air keeps pushing: its threshold is a quarter, and heavy particles in it cross in about one strain time. Dots: particles integrated in the converging sine flow; the curve is the linearised oscillator.

Flows and fields

Drag decides whether a cloud folds, and the air decides at what

A cloud of free particles keeps its velocities, and wherever it converges its paths cross in finite time: a caustic, the density infinite along a sheet. Give each particle a drag on the air and the answer depends on what the air is doing. In still air the drag only brakes, and a cloud folds only if its Stokes number is above one. In air that is itself converging the drag keeps pushing, and a quarter is enough — the same quarter that decides whether a droplet hits a wing.

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