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The thread: A smooth picture proves nothing — page 23

Page 23 of 31, continuing through the 272 essays this motif runs through.

272 essays carry this thread — page 23 of 31.

Two cylinders side by side pull together. The ideal stream past two cylinders side by side with a gap of a fifth of a radius between them. The stream squeezes through the gap faster than round the outsides, the pressure in the gap is lower, and each cylinder is pulled towards the other with a force of 3.6 ρU²a — while the pair together feels nothing at all.

The paradox is for the pair, not for each body

D'Alembert's paradox says a body in a steady ideal stream feels no force. Put two bodies in the stream and the theorem still holds — for the pair. Each cylinder on its own is pushed: side by side they pull together, with a force that grows without limit as the gap closes, and one behind the other they push apart, so that the front cylinder is driven upstream into the stream that is flowing past it. The forces are equal and opposite at every spacing and every angle, and their sum is exactly the zero the paradox promised.

In the pair's own frame, two eddies with no vorticity. The flow round two equal co-rotating vortices (orange dots), seen from the frame that turns with them, where it is steady; shading is the stream function. Round each vortex a lobe of fluid circulates; a band round both is bounded by a figure-eight through the saddle at the centre; and above and below, centred on the two points that make equilateral triangles with the vortices (blue dots), are two large eddies of fluid that circulate in this frame and carry no vorticity at all, bounded by the streamline through the two outer saddles at √5 half-separations.

Flows and fields

A vortex pair carries eddies no snapshot can see

Two equal vortices circling each other are, by every snapshot of the velocity gradient, two small vortices in a straining flow: outside their cores the flow is irrotational, and the gradient there is pure strain. Seen from the frame that turns with them, the same flow contains two large eddies, one on each side, centred where a third point would complete an equilateral triangle with the pair. Their fluid goes round with the pair for ever, and it carries no vorticity at all. They hold seventy times the area of the cores, and the only way to see them is to follow the fluid or to turn with it.

Where a motor's line crosses the film's own characteristic. The stress a self-heating film carries against the speed it lets one wall slide past the other, both scaled, for every steady state (solid): it rises while the film is cool, peaks at the fold and falls as the film heats and thins. A motor's torque falls along a straight line as its speed rises (dashed), and the film runs where the two cross. A stiff drive crosses once. A soft drive, shallower than the falling limb's steepest slope, 0.0738, can cross three times. A drive held at a fixed stress is a flat line, and at the fold's stress it only touches.

Viscosity

A motor turns the runaway into a jump

A self-heating oil film has a fold at fixed stress and none at fixed speed, and a real motor is neither: its torque falls along a line as its speed rises. Put that line across the film's own torque–speed curve and the runaway disappears for any motor at all. What replaces it depends on the line's slope. A drive softer than one fourteenth of the cold oil's resistance jumps to a hot state and keeps a memory of the load; a stiffer one does neither. And the boundary between the two falls exactly where the motor is turning at a quarter of its no-load speed, which no motor near its rated speed can reach.

A fuel-wetted plate sits far below the stream that cools it. The temperature of a throttle plate wetted by petrol, against the day's temperature at 80 per cent humidity, beside the temperature of the stream flowing over it — the mixture after half the fuel has evaporated into it, at the plate's recovery temperature. A plate kept wet with fresh fuel sits nine to twenty-three kelvin below the stream, more on warmer days, and below freezing on any day up to about 34 °C. A plate whose film has lost its light ends sits one to six kelvin below it. The dashed line is freezing.

What is taught wrongly

The throttle plate is a wet bulb for petrol

A carburettor's icing is usually worked out for the mixture — the fuel's latent heat spread through all the air — and that puts the warmest icing day near 15 °C, well short of the 30 °C the charts pilots use allow for. The ice does not grow in the mixture. It grows on the throttle plate, and a plate wet with evaporating petrol is a wet-bulb thermometer for fuel: nine to twenty-three kelvin colder than the stream flowing over it. Kept wet with fresh fuel it ices on a 30 °C day at full humidity and a 40 °C day at half — which is what the training handbook warns of. Once its light ends have gone it does not, and that difference is most of the story.

Whether a sloping ceiling drips in place is one ray's growth. How fast a disturbance grows as seen by an observer moving along the ceiling at speed v, for four speeds at which the film carries its disturbances downhill, in the film's own units. On a flat ceiling (V = 0) the growth peaks at a quarter for the observer standing still and falls to zero for one running at ±1.622. Tilting the ceiling slides the whole curve downhill. While the observer at the point disturbed, v = 0, still sees growth, the ceiling drips where the disturbance began; at V = 1.622 that observer sees none, and above it the disturbance grows only while it is carried away.

Transition and turbulence

A sloping ceiling drips downhill, or not at all

A film hanging from a level ceiling drips where its ripples form. Tilt the ceiling and the film flows downhill, carrying its ripples with it, and at some slope they are carried away faster than they spread and the ceiling stops dripping in place. That slope is set by one number, 1.622 — the speed at which a level ceiling's disturbance spreads — and for water it is tiny: a degree for a tenth of a millimetre of film, six for half a millimetre. Past it the ripples still grow, and still drip, but downhill, at a distance that grows with the slope: a ceiling shorter than that delivers its water to the edge.

Dry air makes a slow bang. The 10-to-90 per cent rise time of a steady shock against relative humidity, for jumps of 25, 50 and 90 pascals, all below the strength at which a discontinuity returns. Each falls roughly as the nitrogen relaxation time does, and doubles when the jump is halved. The rule marks the thermoviscous rise time of the 50-pascal shock, a few microseconds — two to three orders of magnitude below any of the curves.

Compressible flow

Oxygen makes a boom's crack, nitrogen its rise time

The shock at the front of a sonic boom is not the viscous shock of a textbook, a few microseconds thick. It is spread over a large fraction of a millisecond by the vibrational relaxation of the air's molecules, and the two gases do different jobs. Oxygen, fast, removes the discontinuity for any boom weaker than about ninety pascals and decides how much of the front is left at the frequencies the ear weighs most; nitrogen, slow, sets the rise time that is measured. Water vapour speeds both, so a dry day makes a softer bang.

Two shapes for each strain, and one of them holds. The strain rate, over the vorticity, at which an elliptical patch of aspect ratio λ stands still — Moore and Saffman's relation — rising to its maximum of 0.1501 at λ = 2.89 and falling again. Below the maximum there are two steady shapes for each strain: a rounder one, on which every disturbance computed here stays bounded, and an elongated one, which comes apart. Above it there is no steady shape at all.

Ideal flow

A strained vortex holds until it has no shape to hold

A patch of vorticity in a strain has two steady shapes for every strain below 0.150 of its vorticity, a rounder one and a longer one, and none above. The longer one comes apart at the slightest nudge. The rounder one, computed with disturbances of two, three, four and five lobes, never does: it nods and holds right up to the strain at which it ceases to exist. So the existence limit is the real limit, and past it a vortex is not shattered but stretched — lingering first near the shape it has lost, for a time that grows as the fourth root of how close the strain is to the limit.

A wrinkled flame settles into arcs meeting at a cusp. The steady front of a flame in a periodic domain 5, 10 and 20 neutral wavelengths wide, from the exact pole solution of the Michelson–Sivashinsky equation, with the burnt gas below and the flame advancing upwards; each is drawn across one period, scaled to the same width. Every one is a single smooth arc bulging into the fresh gas, meeting its neighbour in a sharp cusp pointing back into the burnt gas, and in these units the three arcs nearly coincide: only the cusp sharpens as the domain widens. In physical units the arc's depth grows in proportion to the domain's width, so the three flames are the same shape at three sizes.

Flows and fields

A wrinkled flame has one cusp and a speed limit

The linear theory of a flame says every long wrinkle grows and none is favoured. The weakly nonlinear theory — the Michelson–Sivashinsky equation — says where the growth goes: small wrinkles merge, the front settles into smooth arcs bulging into the fresh gas and meeting in sharp cusps, and in a domain of any width it ends with a single arc and a single cusp. That front is an exact solution made of poles in the complex plane, and its speed is a closed form that rises in steps as the domain admits more poles and then stops: beyond about five neutral wavelengths a wider flame is no faster, because it is the same shape at a larger size.

A film pulled apart asks for more tension than a liquid has. The pressure below ambient in the film under a sphere moving away from a wall, scaled by the tension the liquid can bear, against the distance from the axis in sphere radii, at the contact gap. The lubrication solution (dashed) asks for four times that tension on the axis. A liquid that cannot give it cavitates: a disc of vapour opens where the demand exceeds the floor, and outside it the pressure is exactly the solution it would have had. Here the disc reaches 0.12 sphere radii.

Viscosity

A torn film still pulls

A sphere bouncing off a wall under liquid has to climb back out through the film it squeezed, and the film pulls it back with a suction no real liquid can supply. Let the liquid cavitate and the obvious guess is that the sphere escapes the torn part of the film for free. It does not. The liquid round the vapour disc goes on pulling, and the disc itself holds the full tension over its area, so a film that tears at five times the contact gap saves a third of what the guess says — and the rebound threshold moves by five per cent where the guess said a third. At an atmosphere, in the liquids the threshold was measured in, it barely moves at all.

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