Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: One number decides the regime — page 34

Page 34 of 34, continuing through the 305 essays this motif runs through.

305 essays carry this thread — page 34 of 34.

Three answers to one question: what happens after a body is jerked into motion. The force following a step change in a body's velocity, for three models. The ideal one is a spike at the instant and nothing afterwards. The viscous one falls as the inverse square root of time and never reaches zero. The compressible one holds while the signal is still crossing the body and then settles. Ideal flow

The theory with no memory in it

Laplace's equation has no time in it, so an ideal flow's response to a body being jerked into motion is instantaneous and complete. Its indicial kernel is a spike and nothing afterwards. Beside it sit the two kernels that are not, and the comparison says which ingredient every memory in this collection came in through.

 Flat until the back pressure reaches a value, then gone. Entrainment ratio against back pressure, as a multiple of the suction pressure, for three mixing-section sizes of one steam ejector driven from a motive supply a hundred times its suction pressure. Each is exactly flat while the entrained stream is choked beside the jet, up to its critical back pressure. The dashed lines join that point to the back pressure at which the entrainment has fallen to nothing, 1.4 per cent higher for the middle machine. The model fixes those two ends and not the path between them. A larger mixing section entrains more and breaks at a lower back pressure. Fluids at work

A choke that belongs to two streams

A steam ejector entrains a fixed amount of gas whatever its back pressure, up to a pressure where it stops. The flat part is a choke, and the entrained gas is not at Mach one when it happens: it is at 0.886, because the supersonic jet beside it is part of the same throat. One area ratio then trades that entrainment for compression, and the trade decides how a vacuum train is built.

 The best efficiency runs from nine-eighths of φ² to one. The best efficiency a peristaltic pump can reach, against the fraction of the channel its wave closes, with its two limits. For a shallow wave it is 9φ²/8, which is small — a wave closing a fifth of the channel is at best 4.5 per cent efficient. As the wave closes the channel the best efficiency tends to one, and its shortfall shrinks in proportion to the remaining gap: about 1.9(1 − φ). Nothing in between is independent of the amplitude. Flows and fields

The pump that is better the more it squeezes

A waving sheet swims at a cost per metre with no amplitude in it. A waving wall pumping fluid is the same mechanism turned round, and its efficiency is nothing like that: it starts at nine-eighths of the amplitude ratio squared, is exactly 2 − √3 at half closure, and rises towards one as the wave closes the tube — where the pump stops being a wave and becomes a piston.

 The part of a field 4 cameras cannot see. Middle, a field with no symmetry on a 16 × 16 grid. Left, the part of it that projects to exactly nothing in every one of 4 views, shaded one way above zero and the other below: a pattern of streaks and hollows that cancels along every ray. Right, the field with that part taken away. The middle and right fields give identical pictures in all 4 views, to 2e-13 of the largest ray, and no reconstruction from those views can tell them apart. The invisible part is one combination of 177 independent patterns the views cannot see. What is taught wrongly

Four cameras and a field they cannot see

An axisymmetric flow can be rebuilt from one photograph because its symmetry supplies every other view. A flow without an axis has to be photographed from several directions, and a few directions do not merely give a noisy answer — they leave whole patterns of density that every camera records as nothing. Four views of a 16 × 16 field see 79 of its 256 independent patterns and are exactly blind to the rest.

 Where the eddies outconduct the molecules. The ratio of turbulent to molecular heat diffusivity across the pipe at Reτ = 2000, for five fluids, on logarithmic axes. Wherever it is above one the eddies carry more heat than conduction does. For air it passes one inside the buffer layer and reaches 135; for water, earlier and higher. For liquid sodium it never reaches one anywhere: at its peak, halfway to the axis, the eddies carry just over half what conduction carries, and the temperature profile is set by conduction across the whole pipe. Regimes and numbers

The heat the eddies do not carry

In a laminar layer the temperature and the velocity have different thicknesses in every fluid but one. In a turbulent pipe the eddies carry both, and the difference nearly vanishes — for air, water and oil alike. It does not vanish for a liquid metal, whose molecules conduct heat faster than the eddies can, and the boundary between the two behaviours is a Péclet number of about four hundred at every Reynolds number.

 A boom gathers most of its age in the thin air near the aeroplane. The share of the total age gathered above each height, for the ray under the track and the last ray computed near the carpet's edge, with the share of the path length travelled above each height for comparison. Under the track 66 per cent of the age is gathered above the tropopause in 26 per cent of the path. The edge ray spends most of its path in the lowest few kilometres and gathers only 11 per cent of its age below 3 km, because the same pressure distorts dense air far more slowly than thin air. Compressible flow

A boom is aged in the thin air it starts in

The rays that reach the edge of a sonic-boom carpet travel two and a half times as far as the one under the track, and it is natural to expect their signatures to have aged accordingly. They have not. A pressure wave distorts thin air far faster than dense air, so two-thirds of a boom's ageing is done in the stratosphere near the aeroplane, and the extra kilometres near the ground add little — which decides how far out a boom shaped to be quiet stays quiet.

 The plateau everybody looks for is a summit, and a low one. −Dₗₗₗ/((4/5)εr) against separation in decaying turbulence at five Taylor-scale Reynolds numbers. None has a plateau at one. Each rises through the viscous range and turns over, peaking at 0.49, 0.63, 0.75, 0.85, 0.90 for Reλ = 50, 100, 200, 500, 1000. A measurement of ε that takes the largest value of this curve as four-fifths reads each of those shortfalls as a smaller dissipation. Transition and turbulence

The decay inside the four-fifths law

The four-fifths law gives the dissipation of turbulence from one measured moment with no constant in it, which makes it the obvious way to measure how the dissipation coefficient travels during a decay. But the law is exact only in a limit, and a decaying flow is not in it. The decay itself takes a quarter off the moment at the Reynolds numbers grids reach — and the bias moves as the flow decays, by a sixth, in the direction opposite to the effect being looked for.

 Three lobes, then a filament. An ellipse of aspect ratio 4 with a three-lobed bump of three thousandths, as contour dynamics carries it, drawn in the frame turning with the undisturbed ellipse at t = 0, 30 and 42. By t = 30 the bump has grown to a visible three-fold asymmetry — one end fattened, the other thinned — and by t = 42, about a turn and a tenth of the ellipse, the thinned end is being drawn out into a filament. The march is stopped there, while the area is still conserved to a few parts in a thousand; resolving the filament needs a contour that adds nodes, which this one does not. Ideal flow

Past three, an ellipse is a shear layer

Kirchhoff's elliptical vortex turns for ever without changing shape, and Love showed in 1893 that it stops being stable at an aspect ratio of exactly three. Computed, that threshold turns out to be the first of a sequence — a new way of coming apart every one and a half aspect ratios — and the sequence ends somewhere recognisable. A long enough ellipse is a strip of vorticity, and it comes apart the way a shear layer does, at a rate Rayleigh found for the strip.

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