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The thread: A smooth picture proves nothing — page 29

Page 29 of 31, continuing through the 272 essays this motif runs through.

272 essays carry this thread — page 29 of 31.

A wing averages a gust through its own loading

A finite span averages a turbulent gust and so tames the load it causes, but it does not average uniformly. The reverse-flow theorem says exactly how it weights the span: by the loading the wing makes at a uniform incidence. An elliptic wing therefore filters the gust the way a round aperture diffracts light, passes eight per cent more of its short scales than a uniform strip, and crosses its mean load up to four per cent more often.

Four of six decayed states sit on the sinh branch; the other two lie above it. The streamfunction's flatness against the enstrophy-to-energy ratio Z/E: the sinh and tanh relaxed dipoles as curves, from the linear dipole at Z/E = 1 outwards, and the six decayed states at t = 600 as points. The tanh branch falls below 9/4 and stays near Z/E of one; the sinh branch rises. The two runs begun as two-level patches lie on the sinh branch at their own Z/E, 0.0049 to 0.055 from it, and so do the two begun on a k⁻³ spectrum, 0.04 to 0.072. The two random-phase runs lie above it, by 0.3 to 0.31: on the sinh side of 9/4, but not sinh dipoles.

Transition and turbulence

Decaying flows end on the sinh side

Three theories predict the state a decaying two-dimensional flow relaxes to, and the streamfunction's flatness tells them apart: above 9/4 for the point-vortex theory's sinh, below for the two-level theory's tanh. Run the decay from three kinds of start and every run ends above 9/4 — including the one begun as two-level patches, the case the tanh theory was built for. Four of six land on the sinh branch itself; the other two are sinh-like without being sinh.

The current along the shore runs fastest three Ekman depths down. The depth-averaged Eulerian current along the shore, positive to the left looking shoreward, against the local depth in Ekman depths, for a swell arriving 20° off the shore-normal at the shelf edge and for one arriving head-on. Both make a jet to the right, fastest at 2.9 Ekman depths for the oblique swell (-1.42 mm/s at 37.6 m) and 2.6 for the head-on one (-0.948 mm/s), and a weak current the other way in water shallower than one Ekman depth. Rotation turns the transport the shelf returns, so even a swell with no alongshore component drives one.

Flows and fields

A sloping shelf puts the swell's current three Ekman depths down

A swell's Stokes transport arrives at a coast and the sea must send it back or turn it aside. On a flat shelf one depth decided which. On a real shelf, deepening from the beach to its edge, every depth is present at once, and the sea does both: it sets up a few centimetres against the beach, dips a tenth of a millimetre where the depth passes two Ekman depths, and runs a current along the shore that is fastest about three Ekman depths down — wherever the friction puts it.

The pocket escapes at the least pressure its states can stand. The liquid pressure each equilibrium of the pocket needs, against the pocket's volume, for a crevice with a one-micrometre mouth in a wall the water wets at 40° and one it does not, at 110°. As the crown's pressure falls the pocket follows its curve to the right: the rim slides out of the cone, pins at the mouth, and the meniscus bulges into a cap. The curve's minimum is the escape threshold — -106 kPa for the wetting wall and -138 kPa for the non-wetting one — and below it no state exists: the pocket becomes a cavity.

What is taught wrongly

A crevice keeps the nucleus a free bubble loses

A free bubble at a siphon's crown dissolves in milliseconds, so it cannot be what breaks a siphon that has run for an hour. A pocket of gas in a crack of the hose wall can be, because its meniscus is curved by the wall and need not dissolve. Followed through its equilibria, the pocket escapes at a tension set almost entirely by the crack's mouth. What the wall's wettability decides is slower and more consequential: whether the crack fills, or keeps drawing gas in until the siphon breaks.

Heading into the sea, follow the long waves and fly level through the short. The dinghy's mean drag against the crossover encounter frequency below which it follows the surface, in units of the sea's peak frequency, heading into seas of significant height 0.2, 0.4 and 0.6 m. Following everything — the right-hand end — costs the heave's lift, and flying level through everything — the left — costs the foil's swinging depth and, in the roughest sea, breaches. Between them each sea has a best crossover: for 0.4 m at 2.92 times the peak frequency, 67.6 N against 67.9 flying level and 199 following.

Fluids at work

A foil in a random sea follows it up to its peak

In a regular wave a foiling boat chooses between holding its height and following the surface. A real sea is a spectrum, and a wand that filters its signal can do both at once: follow the long waves and fly level through the short. The best place to divide them is close to the frequency of the sea's own peak, the divided control beats either pure strategy, and in a rough sea it is the only one of the three that keeps the foil a safe distance under the surface and the drag near its calm-water value.

$A wake held at the stern moves the volume aft a third as far. The least-resistance hull's centre of volume, in per cent of the half-length from midships, negative aft, against the design Froude number, holding the Wigley hull's length, draught, depth profile and displacement. With the linear wake it sits 2.8 per cent aft at Fr 0.30; with the shaped wake of the same propeller-disc fraction, 0.86 if uniform in depth, 1.5 if deepest at the waterline, and 0.11 if deepest at the keel, where the sources make the fewest waves.$ Regimes and numbers

A wake held at the stern keeps the bulb and loses the lean

A wake that slows the water steadily from bow to stern makes Michell's least-resistance hull fuller aft and puts its bulb at the bow. A real ship's wake is nearly nothing along most of the hull and strong only in its last few metres, deepest at the keel. Held there, with the same wake at the propeller, it moves the hull's volume aft by a twentieth to a half as much, and it keeps a third to two-thirds of the bulb's preference for the bow. The lean was a property of the water along the run, and the bulb a property of the water at the stern.

A lagged re-timing passes no error down the V. The variance of each bird's phase error, in radians squared, against its place in one arm of a V, for fore-and-aft wander of half a span correlated over four beats. Holding a fixed phase, every bird's error is its offset from the bird ahead, 3.16. Re-timing with a one-beat lag, every bird's error is 0.632 — the first follower's and the thirtieth's alike, to the last digit. Re-timing as smoothly but through two half-beat lags, the error grows from 0.819 at the first follower to 1.05 at the tenth and 1.08 at the thirtieth, and is still growing slowly there.

Circulation and lift

A wandering flock passes no error down the V

In a flapping V every bird re-times its beat to the wake of the bird ahead, whose own beat is imperfectly timed, so the errors ought to pile up along the arm. With the simplest way of re-timing they do not, at all: the thirtieth bird is off its phase by exactly as much as the first. A one-beat lag and its complement add to one at every frequency, and that identity telescopes the whole arm. Re-time more smoothly and the errors do accumulate — by about a third, and then they stop.

$Below the bubbles' resonance the mixture is slow; above its stop band it is faster than water. The phase speed of sound in water carrying millimetre air bubbles, against frequency, at void fractions of 10⁻⁴, 10⁻³ and 10⁻². At low frequency each is Wood's mixture speed with the gas isothermal — 312 m/s at 10⁻³, not the 366 an adiabatic gas would give. Approaching the bubbles' resonance near 3.2 kHz the speed falls further, then jumps through a stop band in which the wave hardly propagates, and above it the phase speed exceeds water's — at 10⁻³ 1640 m/s at 30 kHz — before returning to 1481 m/s from above.$ Compressible flow

Above their resonance, bubbles make water faster

Wood's formula says a pinch of air makes water's sound slower than air's, and it is right at low frequency. Each bubble, though, is a spring with water for its mass, and the wave drives it. Below the bubbles' resonance the mixture is slow; through a band above it no sound propagates at all; and above that band the same bubbly water carries sound faster than pure water does. Even the slow end is not quite Wood's, because a slowly squeezed bubble keeps its heat.

The water climbs the body, and the wetted width outruns the drawing. The wetted half-width against penetration for a wedge of 10° deadrise and a circular cylinder of unit radius: where the body crosses the undisturbed level, von Kármán's width, and where the risen water meets it, Wagner's. For the wedge Wagner's width is π/2 = 1.571 times the geometric one at every depth; for the circle it is √2 = 1.414 times, at small penetration.

Ideal flow

Water climbs a falling wedge, and the load comes from the climbing

A hull that strikes the sea sets the water under it moving, and the force is the rate at which it does so. The obvious estimate measures the wetted width where the hull crosses the undisturbed surface. But the water does not wait: pushed aside, it rises up the hull and wets it sooner, a factor π/2 wider for a wedge and √2 for a round bottom. The force carries that factor twice, and the peak pressure, where a thin jet leaves the hull, carries it squared on a cotangent that grows without bound as the bottom flattens.

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