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The thread: What is conserved — page 40

Page 40 of 43, continuing through the 384 essays this motif runs through.

384 essays carry this thread — page 40 of 43.

The exit area is a condition, not a second choice. Critical entrainment and critical compression ratio for a steam ejector with a mixing tube of 60 nozzle throats, against the nozzle's exit area as a multiple of the exit that delivers the jet at exactly the pressure it shares with the entrained gas. Both peak at the matched exit, at every suction drawn, so no exit area buys entrainment at the cost of compression or the other way. Halving the exit costs 2.4 per cent of the entrainment at a hundredth of the motive pressure, doubling it 3.2 per cent.

An ejector's nozzle exit is a condition, not a choice

A steam ejector has two areas a designer can pick, the mixing tube's and the nozzle exit's, and one of them looked like a second way round the trade between entrainment and compression. It is not. Both are largest with the exit matched to the pressure the jet meets, as a rocket's thrust is, and every other exit draws a curve inside the matched one. What moves the machine is loss, and the losses sort themselves: only the nozzle's reaches the entrainment, and the sharp edge of the characteristic belongs to the ideal machine alone.

$A flow's window mean converges a power faster. The variance of the mean over an L × L window, as a fraction of the point variance, against the window's side in integral lengths, on logarithmic axes. The scalar field's falls as the inverse area. The plane cut through a three-dimensional flow falls the same way at half the level. The velocity of a two-dimensional incompressible flow falls as the inverse cube of the side, because its mean over any window is a streamfunction difference around the window's edge. Lines are closed forms; dots are Monte Carlo means over fifty generated fields.$ Transition and turbulence

A snapshot counts areas, and a flow counts its edge

A record at one point holds one independent value for every two integral scales it lasts. A snapshot of a field holds one for every integral area it covers, so at the same number of samples it holds far fewer, and sampling it more finely adds nothing. Except for a velocity in a two-dimensional incompressible flow, whose mean over a window is fixed by the window's edge alone: it converges a whole power faster than its area allows, and a large enough snapshot of it beats the record.

Stratification makes a hill easier to block. The strength at which a current over a Gaussian hill first stops and traps fluid, δc = (h₀/H)/Ro at onset, against the stratification B = NH/fa — the depth over the height fa/N to which a rotating stratified flow feels the hill. Homogeneous, it is the 3.134 of the unstratified layer. As B grows the hill's anticyclone gathers at the bottom, where it is stronger, and the threshold falls; in deep water it falls as 2.243/B, which is a Froude number: the current is blocked when N h₀/U exceeds 2.243, and the rotation has dropped out.

Regimes and numbers

A stratified sea blocks a current sooner and traps less

A slow current in a rotating layer stops over a hill once the hill's height over the depth, divided by the Rossby number, passes 3.134, and a column of water over the hill is trapped from floor to surface. In a stratified sea the hill's anticyclone gathers near the floor and fades upward over a height of fa/N. It is stronger there, so the current stops sooner; and it is confined there, so what it traps is a cap, not a column. In deep water the threshold loses the rotation altogether and becomes a Froude number: a current is blocked when N h₀/U exceeds 2.24.

Every parcel goes straight while the streamlines wave. A uniform stream across the page carrying a pattern of transverse velocity with it, v = 0.6 sin(x − t): the streamlines at one instant, which wave, and the paths of five parcels over the next six time units, which are straight lines. Each parcel keeps the transverse velocity it started with, because the pattern moves with it; the streamlines are a snapshot of a pattern that is sliding past, and no parcel ever follows one.

Flows and fields

A parcel goes straight while the streamlines curve

In a steady flow every parcel accelerates while the picture never changes. The opposite also happens: a flow whose picture changes all the time while no parcel accelerates at all, because the local and convective halves of the acceleration cancel exactly. Such a flow has no pressure gradient anywhere, and that is a severe demand. An incompressible flow can meet it only as a pure shear at every point; a cloud that can compress meets it freely and pays later, when its straight paths cross in a caustic.

Ideal flow

A free pair turns, and forty-five degrees decides

Two cylinders in an ideal stream move to look bigger to it: side by side they close, in tandem they part. Free to turn as well, a staggered pair swings towards side by side — and does not stop there, because nothing in an ideal fluid damps the swing. Whether it then collides or flies apart is decided, far apart, by one angle and a theorem: the stream's pull on the pair falls as the inverse square of the spacing, and for such a force the sign of the energy alone decides, which changes at exactly forty-five degrees.

One bowl, and a line across it for every way of flying. The least induced drag of a wing, a canard and a tail together, as a function of the canard's and the tail's shares of the lift, drawn as rings of equal drag about the bowl's foot, where each carries 1.9 per cent. A static margin of a tenth of a chord and a wing pitching moment draw a straight line of trimmed splits across the bowl. The tail aircraft trims where its line crosses the axis of zero canard load; the three-surface aircraft slides along the same line to the point nearest the foot.

Circulation and lift

A third surface is worth a square

A canard and a tail each leave an aircraft one free share of its lift, and the static margin spends it. Give an aircraft both and one share stays free after trim, so the split can slide along a line to the point of least induced drag. What that slide is worth turns out to be a square: half the bowl's curvature times the square of how far the tail aircraft already trims from twice the bowl's foot. For a cruising wing the saving is under one per cent and less than the canard's own skin friction; it pays only for a wing whose pitching moment is as large as a flapped section's.

Four vortices bend twice as fast, on shorter waves. Growth rate of the fastest bending wave against its wavelength, both in the units of the equivalent single pair — one unit of growth is one e-fold in the time the wake sinks by its own spacing. Crow's pair grows at most at 0.827, at 8.54 spacings. A flap vortex of 0.3 of the tip's strength at 0.4 of its station keeps a Crow band just below that and adds a second, at 1.57 and 1.8 spacings; one of 0.5 at 0.3 merges the two into one band peaking at 1.64 and 4.4 spacings.

Circulation and lift

Four vortices bend faster, and bend the wrong one

With its flaps down a wing sheds four vortices, not two: a strong one at each tip and a weaker one at each flap's edge, and on each side the two circle one another as the wake sinks. The orbit makes the wake unstable at waves a fifth of Crow's length, growing twice as fast. But the wave that grows is the weak flap vortex being bent round the strong tip vortex, which hardly moves, while the long bend that pinches the tips together grows no faster than before. Only a counter-rotating inner vortex makes the whole wake unstable, and it does so at every wavelength.

Tailored, the reservoir waits for the driver's own expansion. Distance against time in a helium-driven air tube at the tailored Mach number, 3.41, with a driver half as long as the driven tube: the incident shock, the contact surface, the reflected shock and the shock it transmits into the driver gas, and the driver's expansion — its head running back to the driver's closed end, and the head reflected from there. Nothing returns from the contact, and the reservoir at the end wall holds from 0.293 until the reflected head arrives at 0.555 L/a₁: a test time of 0.261.

Compressible flow

A tailored tube buys its test time with its driver

Tailoring a shock tube removes the wave the contact surface would send back to the reservoir. What ends the reservoir then is slower: the driver's own expansion, which runs back to the driver's closed end, reflects, and has to cross the whole tube to reach the end wall. Its arrival is exact in one dimension, because the reflected head crosses the incident fan as a simple wave, and the answer is that test time is bought with driver length — about seven-tenths of a driven-tube crossing time per driver length for helium — and that tailoring is worth nothing with a driver shorter than a quarter of the tube.

$A waist moves the fuselage's overspeed off the root chord. The fuselage's axial perturbation velocity at its own surface, as a fraction of flight speed, along its length at Mach 0.8, with the wing's root chord between the two rules: the plain fuselage, which adds 0.0251 everywhere along the chord; a narrow waist 6.2 per cent deep that cancels it at mid-chord and raises it at the chord's ends; and a wide one 31 per cent deep that leaves none of it positive along the chord and lifts it on the body ahead of and behind the wing.$ Ideal flow

A waist moves the overspeed and cannot remove it

A fuselage adds two per cent of overspeed at the wing root, and a waist — a local narrowing of the body where the wing joins it — is the obvious cure. Slender-body theory says exactly what a waist can do. Its sources and sinks sum to nothing, so the velocity it adds along the body integrates to zero: it cannot remove the overspeed, only move it. Moved off the whole root chord it needs a third of the fuselage's cross-section at the wing, a sixth of what the transonic area rule would take, and it lands on the fuselage just ahead of and behind the wing, where there is room for it.

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