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The thread: One number decides the regime — page 39

Page 39 of 50, continuing through the 450 essays this motif runs through.

450 essays carry this thread — page 39 of 50.

The Earth turns a straight river into a very gentle bend. The transverse circulation through the depth of a straight river running at 1.2 m/s at latitude 50°N, in thousandths of the main flow, against height above the bed — and the circulation of the same river without rotation but bent to the left with a radius of 2V/f, 21 kilometres. The two nearly coincide: the Earth's rotation pushes the fast surface water right and the slow bed water left, exactly as a bend does, and for a river it is a bend of about twenty kilometres' radius everywhere at once.

The Earth bends every river a little

In 1860 Karl Ernst von Baer claimed that the Earth's rotation makes rivers in the northern hemisphere cut into their right banks. The Coriolis force on a river does drive a helix that throws surface water towards the right bank, the same helix a bend drives. For a mountain stream it is a ten-thousandth of what the stream's own bends do. For the great slow rivers of the far north it is a fifth to two-fifths, and there Baer's claim is not absurd.

An open rotor's slipstream shrinks; a duct holds it open. The radius of the slipstream behind a hovering rotor of radius R, against the distance behind the disc: for an open rotor, from the vortex-cylinder model, contracting towards R/√2 so that the wake ends with half the disc's area; and for a rotor in a straight duct, whose slipstream leaves at the full area, and in a duct that widens to 1.3 times it. The same thrust from a wider jet needs a slower one, and a slower jet wastes less energy.

Fluids at work

A duct is worth the square root of two

An open rotor squeezes its slipstream to half its own area and pays for the fast jet that results. Put the same rotor in a straight duct and the slipstream leaves at the rotor's full area, the jet is slower, and hovering costs 29 per cent less power. The duct is not a passive guard: it carries half the thrust itself, on the suction round its inlet lip. In cruise almost all of the advantage disappears.

A film just past its fold lingers where the fold was. The temperature excess at the middle of a sheared film, scaled by the temperature over which the viscosity falls by e, against time in thermal diffusion times, switched on from rest at five stresses either side of the fold. Below the fold the film settles. Just above it the film climbs to the fold's own centre temperature, 1.19, lingers there as if it had found a steady state, and only then runs away — the longer the closer it is to the fold.

Viscosity

A lost steady state still holds the film

Push a self-heating oil film one per cent past the stress at which it can no longer settle and it does not run away at once. It warms to the temperature the vanished steady state would have had, sits there as if nothing were wrong, and only then goes — after a delay that grows as one over the square root of the overload, and during nearly all of which the load could still be taken back.

On a smooth signal, sampling faster makes the lag-one scale longer. The average estimate of the integral scale from records three hundred scales long of the smooth process, against the number of samples per integral scale. The first zero and the exponential fit do not care how fast the record was sampled. The lag-one estimate grows in proportion to the sampling rate, because it is reading the curvature of the correlation at zero lag — the microscale — and dividing by the sample spacing.

Transition and turbulence

The best estimate of a scale assumes its shape

There are four common ways to read an integral time scale off a turbulence record, and on the right signal the best of them is four times more precise than the usual one. On a signal whose correlation has a different shape it is off by a factor that no length of record reveals — and one of them turns out to be measuring the sampling rate rather than the flow.

$Inside a strong blast: a shell of gas around a hot, nearly empty core. Left, the density, velocity and pressure inside a spherical strong blast wave at γ = 1.4, each as a fraction of its value just behind the shock, against the distance from the centre as a fraction of the shock radius. The density falls to nothing well inside the shock while the pressure levels off at 0.37 of its post-shock value. Right, the temperature, which is the pressure over the density and so rises without limit towards the centre.$ Compressible flow

The fireball is hollow

Inside a strong blast wave there is almost nothing. Half the air the shock has swept up lies in the outer four per cent of its radius; at half the radius the density is a hundredth of the air's; the centre is empty and, formally, infinitely hot. Integrating Sedov's equations to see this also shows that a published table of his constants was wrong at three of its four entries.

The slow bending wave, and where the cut-off stops describing it. The frequency of the slowest bending wave on a Rankine vortex — a helical wobble of the whole core that turns against the flow — against the wavenumber times the core radius, from Kelvin's exact relation and from his long-wave formula, which the cut-off method used for vortex pairs reproduces. They agree for long waves. Past ka = 1.44 the long-wave formula reverses sign and the exact wave does not, which is why a cut-off model's instabilities at such wavelengths are not real.

Ideal flow

A vortex is a waveguide

A spinning core is stiff in a way still fluid is not, and it carries waves along its length: an infinite family of them for every pattern round its axis, travelling at up to 0.83 of the swirl speed at its edge. The slowest is a helical bend that turns against the flow, and it is the wave every model of a bending vortex has been borrowing without saying so.

Time per litre rises in a straight line with the litres already filtered. The elapsed time divided by the filtrate collected, against the filtrate, for a constant-pressure filtration of a silica slurry through a cloth: integrated step by step, and Ruth's closed form. The line's slope is the cake's resistance, which grows with every litre, and its intercept is the cloth's, which does not. After a quarter of a cubic metre the cake is already the larger.

Fluids at work

A filter is slowed by what it has caught

Push a slurry through a cloth and within minutes the cloth no longer matters: the particles it has caught form a cake, the cake is a packed bed, and every litre filtered makes the bed thicker for the next. The filtrate grows only as the square root of time. And if the cake is soft, it packs tighter against the cloth the harder it is pushed, until more pressure buys almost nothing.

A slow current divides round the fluid over a hill (δ = 4). A uniform stream, left to right, over a Gaussian hill on the floor of a rotating layer, at δ = h₀/(H·Ro) = 4, where h₀/H is the hill's height against the depth and Ro the Rossby number U/fa; the onset for this shape is δ = 3.134. The dashed circle is the hill's e-folding radius. Water crossing the hill is squashed and, keeping its potential vorticity, spins clockwise; that anticyclone adds to the stream on one side and opposes it on the other. Here it stops the stream: the outlined region, 1.54 square radii, holds fluid that circles for ever and never leaves, and it sits beside the summit rather than on it, on the side where the swirl runs against the current.

Regimes and numbers

The hill a slow current will not climb

The Taylor–Proudman theorem says a rotating fluid goes round an obstacle rather than over it, as if a solid column stood above it. Conserving potential vorticity turns that limit into a threshold. A current is stopped over a hill once the hill's height against the depth exceeds a fixed multiple of the Rossby number — 2 for a flat-topped hill, 3.13 for a Gaussian one, 16/3 for a cone — and the fluid it holds sits beside the summit rather than on it.

Regimes and numbers

Hull speed is not the hump

The hull-speed rule says a displacement vessel meets a wall where its wave is as long as its hull, at a Froude number of 0.4. Michell's thin-ship integral, computed for a standard hull, puts that speed on the steep climb between the last hollow and the main hump, and the hump itself at 0.5. Past the hump the wave resistance grows more slowly than the speed. The hollows sit where the hull's own transverse wave switches off, and a bulb that cancels at one speed multiplies the resistance at another.

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