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The thread: A smooth picture proves nothing — page 19

Page 19 of 20, continuing through the 172 essays this motif runs through.

172 essays carry this thread — page 19 of 20.

$A gust's lift keeps falling where a pitching wing's stops at a half. The magnitude of Sears' function, the lift a wing gets flying through a sinusoidal gust as a fraction of the quasi-steady value, against the reduced frequency on a logarithmic axis, beside Theodorsen's function for a wing that pitches or heaves. The two agree at low frequency. Above a reduced frequency of about a tenth they part: Theodorsen's levels off at one half, because a moving wing changes its whole boundary condition at once, while Sears' keeps falling as one over the square root of 2πk, because several wavelengths of gust lie along the chord and cancel.$ Regimes and numbers

The gusts that cancel along the chord

A wing that pitches keeps half its circulatory lift however fast it moves. A wing flying through a gust does not: once the gust is a few chords long, its ups and downs lie along the chord together and cancel, and the lift falls without limit. For an airliner that barely touches the root-mean-square gust load, and cuts the load spectrum at the wing's own torsion frequency to a quarter of the quasi-steady value.

The arrival map, and the place where it goes backwards. Where each boom ray lands on the ground, against the Mach number the aeroplane was doing when it launched it, for four accelerations from 15 km. In level flight this would be a straight line rising at the aeroplane's own speed. Here it falls before it rises: a ray launched at a higher Mach number is shorter and steeper, and near the cut-off it shortens faster than the aeroplane advances. Every minimum in these curves is a fold — two emission times delivering to one place — and on the fold itself neighbouring rays converge onto a single line.

Compressible flow

The carpet an accelerating aeroplane folds

Level flight launches every boom ray with the same invariant, so they run parallel and each place hears one boom. Accelerate, and each successive ray is shorter than the last — shorter, near the cut-off, than the aeroplane's own advance — so later rays overtake earlier ones and the arrival map folds onto a line.

Rolls turning side by side, with the fastest downwind water where they sink. The fastest-growing mode at a Langmuir number of 0.13, looking downwind, over two roll spacings of 2.89 decay depths and 4 decay depths down. The closed curves are streamlines of the overturning; the dashed curves are contours of the downwind velocity the rolls carry, positive under the lines where the water sinks. At the surface the cross-wind flow converges onto those lines, which is where floating foam and weed collect as windrows. The amplitude is arbitrary, as in any linear mode.

Flows and fields

The drift that turns a current into rolls

A current carrying a Stokes drift feels a force the drift makes out of the current's own vorticity, and under a wind that force is unstable. It turns the surface layer into rolls lined up downwind, with windrows where they sink. The rolls need both the current's shear and the drift's; their growth rate sees only the product; and the split between the two decides which motion gets the energy.

The throat holds the hammer back only while its cavity lasts. Left, pressure at the closing valve (red) and at the upstream face of the venturi (gold); right, the throat's cavity volume; after the valve shuts with a 5 mL cavity in the throat. The valve sees the full Joukowsky rise of 14.8 bar at once. The wave reaches the venturi 33.3 ms later, and for the next 7.9 ms the upstream pipe hears nothing: its pressure stays at 5 bar while the cavity is squeezed. When the cavity closes at 41.3 ms the surge passes into the upstream pipe at 13.8 bar above its steady pressure.

What is taught wrongly

A choked throat buys time, not silence

A venturi whose throat has reached vapour pressure passes a flow the downstream pressure cannot change, and it is tempting to read that as isolation: whatever happens downstream, the upstream pipe will not hear it. Slam a valve downstream and it hears it. The cavity at the throat holds the surge back only for as long as it takes to fill, and then lets 93 per cent of it through.

One curve from two to one, with four thirds somewhere in the middle. The ratio of the transverse second-order structure function to the longitudinal one, against separation, at three Reynolds numbers. Every value on every curve follows from the longitudinal function alone by a relation with no dynamics in it. It is exactly 2 where the field is smooth, exactly 1 beyond the correlation length, and it passes through four thirds on the way — but it passes through rather than resting there, and how nearly it rests is the whole of what a Reynolds number buys.

Transition and turbulence

A relation with no turbulence in it

Isotropy and incompressibility alone fix the transverse structure function from the longitudinal one. Divide the relation through and it says the ratio of the two is one plus half the local slope — so the exponent everybody measures as 0.70 and the ratio everybody measures as 1.35 are one measurement, and a model spectrum with no intermittency in it produces both.

The kept transport spirals into nothing as the sea deepens. The net Lagrangian transport as a vector, scaled on the Stokes transport, traced as the water depth increases from a quarter of an Ekman depth to eight, for an 8-second swell with an eddy viscosity of 0.01 m²/s. Shallow water keeps the whole transport pointing with the waves, at the right-hand end. As the sea deepens the vector shortens and swings to the right, crosses the across-wave axis near two Ekman depths, and winds into the origin, which is the open ocean's exact cancellation.

Flows and fields

The floor that gives the drift back

In the open ocean the Coriolis force drives a current that cancels a swell's Stokes transport exactly. Over a continental shelf the sea floor holds a stress, and whatever it holds is transport the rotation does not take back. How much survives depends almost only on the depth in Ekman depths; which way it points depends on the wave.

One number decides which pulse grows. The pressure pulse and the flow pulse at the far end of the tube, each as a multiple of its value at the entrance, against the load's reflection coefficient. A load that reflects pressure with the same sign — a stiffer or narrower continuation — amplifies the pressure pulse and damps the flow pulse. One that reflects it inverted — a wider continuation, or many branches — does the opposite. With no reflection both fall slightly, by the wave's own attenuation. The two curves cross near Γ = 0 and pull apart on either side.

Regimes and numbers

The pulse that grows as it leaves the heart

The pressure pulse measured at the wrist is larger than the pulse in the aorta that drives it, and the flow pulse is smaller. Nothing downstream is pumping. A wave reflected from the end of an elastic tube arrives back in step with the outgoing wave near the end and out of step near the start, and a single number — the reflection coefficient — decides whether it is the pressure or the flow that grows.

What the rays say the edge is: nearly as loud, and then nothing. The overpressure across the carpet relative to the value under the track, from ray-tube spreading alone, at three Mach numbers from 15 km. It falls gently and then stops: at Mach 1.8 it is 0.92 ten kilometres out, 0.77 at twenty-three and 0.65 on the last ray, 36 km out, with silence beyond. Geometrical acoustics says the carpet ends at a cliff, and the boom at the edge of a real carpet fades as a rumble. The cliff is what the model says; it is also where the model stops.

Compressible flow

The edge is a rumble, not a quieter bang

The rays that reach the outer half of a sonic-boom carpet arrive nearly horizontally, having travelled almost three times as far as the one under the track. Ray theory says they still carry two-thirds of the overpressure, right up to a line beyond which there is nothing. Neither half of that is what is heard — which is the useful result, because it says the edge's loudness is not a ray quantity at all.

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.

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