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The thread: A smooth picture proves nothing — page 22

Page 22 of 31, continuing through the 272 essays this motif runs through.

272 essays carry this thread — page 22 of 31.

Two eddies that stand still behind a cylinder in ideal flow. A uniform stream past a circular cylinder with a pair of opposite vortices standing 2 radii behind its centre and 0.859 either side of the axis, each of strength 10.31 in units of the stream speed times the radius — Föppl's equilibrium. The shading is the stream function: the fluid between the vortices and the cylinder circulates in two closed eddies and never leaves, while the stream divides round the whole assembly as if it were one longer body.

Two eddies can stand behind a cylinder, but not for long

Ideal flow, which has no viscosity and no wake, can still hold a pair of eddies standing behind a cylinder — at any distance behind it, with a strength fixed by the distance. They cost the cylinder nothing. And they cannot stay: nudged sideways by a thousandth of a radius, the pair grows its displacement exponentially and leaves, which is the first step of shedding a wake.

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.

What is taught wrongly

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.

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.

Every particle leaves a flow that is a vortex at every instant. Eight fluid particles starting on a small circle in Haller's rotating-saddle flow, followed for 2.4 time units. At every instant the velocity gradient has complex eigenvalues and a positive Q — the flow is a vortex by every criterion that reads a snapshot — and every particle spirals outward, its distance growing as e to the time. The fluid is not held; it is flung.

Flows and fields

Every snapshot says vortex, and every particle leaves

There is a flow in which the velocity gradient has complex eigenvalues at every point and every instant — a vortex by every criterion that reads a snapshot, in the frame the flow is measured in — and in which every fluid particle is flung away exponentially. The snapshot is not wrong about the gradient. It is wrong about the fluid, and it is wrong exactly when the strain's axes turn.

Where a carburettor's cold comes from. The temperature changes in a carburettor's throat, in kelvin: the cooling of the air as it expands to a hundred metres a second, the far smaller cooling of the metal it flows over, the cooling as petrol evaporates into it, and the warming as the water in humid air condenses back out. The fuel does five times what the venturi does to the air and fifty times what it does to the metal.

What is taught wrongly

Carburettor ice is made by the fuel, not the venturi

Pilots are taught that a carburettor ices because air cools as it expands through the venturi. The expansion does cool the air, by five kelvin at a hundred metres a second, but the metal the ice grows on barely feels it. The cold comes from the petrol evaporating, and whether it makes ice is decided by the water in the air, which warms the mixture as it condenses. The worst day is not a cold one.

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 washed filter's best cycle is shorter, and still has one. The filtrate a batch filter produces per hour, averaged over filtering, washing and twenty minutes of emptying and refilling, against the filtrate collected in each cycle, for a slurry of 400 kilograms of silica per cubic metre and wash ratios of 0, 1, 2 and 5 pore volumes. Each curve has a best cycle, marked, and the wash both lowers it and moves it to a thinner cake: from 0.307 to 0.217 m³ per m² of cloth.

Fluids at work

A washed filter works as long as it rests

A batch filter that has to be emptied between runs has a best cycle, and the rule for it is exact: the time the cake is responsible for equals the time the filter stands idle. Washing the cake does not break the rule, because a wash through a finished cake costs time in proportion to the square of its thickness, just as filtering it did. What a wash changes is the price, and the price is set by the slurry's concentration and by how far the wash front smears, not by the cake's softness.

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