Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: Taught wrongly, everywhere — page 24

Page 24 of 25, continuing through the 220 essays this motif runs through.

220 essays carry this thread — page 24 of 25.

Followed along their streamlines, deep parcels at 4° arrive nearly together, and at 0° the upper leads by more than outside. The arrival lag at 95% chord, lower parcel minus upper, as a multiple of the lag outside both layers, against the streamline both parcels follow, as a share of the upper layer's mass flux at the trailing edge. At 4° it is 0.46 for the deepest streamline drawn, −0.16 at a twentieth, 0.47 at a fifth and 1 at the layer's edge; at 0°, 1.47, 1.68, 1.33 and 1. What is taught wrongly

Followed rather than held, the parcels near a wing nearly tie

Parcels held at a fixed fraction of a boundary layer's thickness made the side that turns turbulent first decide which arrives first near the wall, and at a wing's ordinary incidence they gave the upper parcel a lead ten times the one outside the layer. Real smoke is not held at a depth: it follows a streamline. Near the attachment point the layer is thinner than any filament's height, so every filament starts outside it and is swallowed later, and on the upper surface, whose turbulent layer grows faster, it is swallowed deeper. Followed this way, the deep parcels at 4° arrive nearly together, and at a Reynolds number of a million the lower one can come first.

 Sheared between plates, a gas transmits its full viscous stress down to a few free paths across, and then a share proportional to its pressure. The shear stress a gas transmits between two plates, over Navier–Stokes's μU/H, against the rarefaction δ = √π/(2Kn), from the linearised kinetic equation; with Navier–Stokes and first-order slip, 1/(1 + 2·1.016/δ), and the free-molecular δ/√π. The kinetic stress is 0.908 of μU/H at δ = 20, 0.501 at 2, 0.0979 at 0.2; the slip formula is within 2.7 per cent at δ = 1. Flows and fields

A gas's viscosity ignores its pressure until the free path reaches the walls

Maxwell predicted in 1860 that a gas's viscosity does not depend on its density — halve the pressure and half as many molecules carry momentum across the flow, each twice as far — and measured it with swinging discs in 1866: unchanged from an atmosphere to a sixtieth of one. Solved across every pressure with the kinetic equation, the shear a gas transmits between two plates obeys him over five decades and then, once the free path reaches the gap, falls in proportion to the pressure. A millimetre gap reads half the true viscosity at fifteen pascals, and a vacuum gauge reads pressure from exactly that fall.

 A strike cannot choose its spin and launch apart: each angle of attack is one line across the plane, and only an upward one reaches low spin with height. Backspin and launch angle of the ball as the dynamic loft is changed from 1° to 24° above the angle of attack, for a 45 m/s head striking at angles of attack of −4°, 0°, +4° and +8°; on each line, the loft that gives the longest total. At −4° the best is 2413 rpm at 4.43°; at 0° the best is 2129 rpm at 7.43°; at +4° the best is 1636 rpm at 9.71°; at +8° the best is 1221 rpm at 12.3°. The free plane's best for a 66.6 m/s ball is 1349 rpm at 12.5°. Fluids at work

A driver reaches the best drive only by striking upward

The longest drive asks for a launch near twelve degrees with little more than a thousand revolutions a minute of backspin, and a club cannot ask for those separately. Its strike sets the spin by the angle between the head's path and its face, and the launch by the face's loft, so a player who swings level can trade spin for launch only along one line — and that line misses the best drive. Striking upward moves the line. At a 45 m/s head speed the longest total comes at an attack of nine degrees upward with fourteen of loft, 4.6 metres past the best level strike, and it still lands near thirty-five degrees.

 A mica sheet shields the glue's false slip only where the pressed region is narrower than the sheet bends. The slip length, in nanometres, that a dynamic drainage measurement at 100 Hz would read from a wall that does not slip, against the gap: a 3 mm sphere in water over bare glue of plane-strain modulus 1 GPa, and over mica sheets 1, 3 and 5 µm thick on the same glue. At 3 nm the bare glue reads 9.48 nm and the sheets 8.75, 2.68 and 1.06; at 30 nm, 0.1 against 0.099, 0.067 and 0.043. What is taught wrongly

A mica sheet hides its glue only at the smallest gaps, and changes how the false slip falls

A drainage measurement against a soft wall reads a slip that is not there, falling as the square of the gap. The wall in a surface-force apparatus is not a soft half-space but a stiff mica sheet on a soft glue, and the sheet spreads the drainage pressure over its own bending length before the glue feels it. Where the pressed region is wider than that length the sheet changes nothing; where it is narrower the sheet sinks as one piece, a spring in series with the liquid. That cuts the false slip only by the square of the bending length over the pressed width — threefold at three nanometres for three micrometres of mica — and it turns the slip's fall with the gap from the inverse square towards the inverse, which is the test an experimenter would have used to see it.

 Two spheres settle faster together than apart, faster in line than side by side, and still faster at thirty radii than either alone. The settling speed of a pair of equal spheres over a single sphere's, against the distance between their centres in radii: along the line of centres, exactly (the bispherical solution) and by Batchelor's far-field series; across it, by the series; dashed, each sphere's Stokeslet alone, 1 + 3/(2ρ) and 1 + 3/(4ρ). Touching in line the pair falls 1.55 times as fast; at 5 radii, 1.286 in line and 1.154 side by side; at 30, 1.05 and 1.025. Viscosity

Two spheres settling at a slant keep their slant for ever, and three do not

Two equal spheres settling through a viscous liquid fall faster together than either alone — half as fast again if they touch in line, and still five per cent faster thirty radii apart. Slanted, the pair drifts sideways as it falls, by up to three and a half degrees, and keeps the slant, the separation and the drift for as long as it falls: the two spheres always move with exactly the same velocity. Release a third and the shape is lost at once, the middle sphere drawing ahead and the outer two closing in behind it. What a pair keeps and a trio does not is a statement about symmetry, and reciprocity is half of it.

 With the weight chosen from the data, non-negativity still saves views, and the saving shrinks as the noise grows. The fewest views that bring each prior's reconstruction of a field of two puffs within fifteen per cent of the truth, against the noise on the rays as a share of the rms ray: solid with each prior's weight chosen by the discrepancy principle, from the data alone; dashed with the weight that knows the answer. Points drawn at 25 never get there. At one per cent noise no prior needs 8 views and non-negativity 4; at three per cent, 8 and 6, against 4 with the answer known; at ten per cent, more than 24 and 10, against 7. What is taught wrongly

A prior chosen from noisy pictures keeps most of its gain, and a wrong one loses its alibi

With exact rays, a reconstruction's prior could be judged by its error, and a constraint that was wrong gave itself away by missing its own rays. With noise every prior needs a weight — how far to trust the rays against the assumption — and the weight has to be chosen without the answer. Chosen where the misfit falls to the noise, it costs the smooth and unassuming priors almost nothing and non-negativity two or three views, so the ranking survives. What does not survive is the refutation: a dictionary that misses a field's rays by four per cent is caught at three per cent noise and invisible at seven, where it fits the noise as well as any prior and is about as accurate as they are.

 On a NACA section thickness multiplies the lift by more than on a Joukowski section, by nearly one plus its thickness. The lift of a NACA four-digit section by panels over thin-aerofoil theory's, against its thickness: the 2412-type at 4°, and the symmetric section's lift slope over 2π; beside the Joukowski law 1 + 0.77 t/c. At 12 per cent the cambered section's factor is 1.115 and the symmetric slope's 1.101, against Joukowski's 1.092; at 18 per cent, 1.174 and 1.152. Ideal flow

A NACA section multiplies its lift by nearly one plus its thickness, and a nose correction cannot supply it

On Joukowski's exact sections, thickness — which lifts nothing alone — multiplies the lift of camber and incidence by 1 + 0.77 t/c. Solved by panels, the NACA four-digit sections that real wings use do more: their lift slope grows as about 1 + 0.84 t/c, and a cambered 2412 lifts 11.5 per cent more than thin-aerofoil theory says. The more of the thickness sits near the nose, the more it multiplies. Added as thin-aerofoil theory adds them, the parts lose a quarter of the suction peak and put the centre of pressure two hundredths of the chord too far aft. Riegels's nose factor, the classical repair, makes the nose speed finite and moves the peak, and adds no lift at all: the missing lift is not a nose effect.

 A sphere two per cent larger overtakes its neighbour round a path that is its own mirror image, and leaves at the offset it came in at. The centre of a sphere two per cent larger than its neighbour, relative to the smaller sphere, as it overtakes it in a viscous liquid, from four horizontal offsets; the circle is the distance at which they would touch. Each path bulges outward as it passes and is symmetric about the smaller sphere's level: from 0.85, 1.5, 2.5, 4 radii the larger sphere leaves at 0.8499, 1.5, 2.5, 4, after closest approaches of 2.27, 2.83, 3.74, 5.15 radii. Viscosity

A larger sphere overtakes its neighbour and leaves on the line it came in on

Two equal spheres settling together never change their shape. Make one of them two per cent larger and the pair becomes an encounter: the larger sphere sinks past the smaller round a path that bulges out and comes back, slowed sixteenfold at contact but never stopped, and it leaves at exactly the offset it arrived with. The smaller sphere swings seven radii sideways while the larger is above it and gives every radius back while the larger is below. In Stokes flow, meeting a neighbour displaces nothing — unless the spheres touch, and a hundredth of a radius of roughness is enough to make every touching path leave on the same line.

 Kicked at ninety per cent of its peak flow, the compressor is held only when both modes have an actuator. The flow through the compressor, as a fraction of the peak flow, after a stall cell of amplitude 0.05 is started at 90 per cent of the peak flow, at B = 2, under four controllers: a tip injector of gain 0.7 and capacity 2 per cent of the peak flow with the throttle fixed; the injector with a pressure-driven throttle; the pressure-and-flow throttle with no injector; and both. injector, throttle fixed: surges or runs away; injector, pressure throttle: surges or runs away; throttle pair, no injector: stalls; injector and throttle pair: recovers. Fluids at work

A compressor needs one actuator for each way it can fail

A throttle alone cannot hold a compressor left of its peak in a large plenum: it has one degree of freedom and the machine has two ways to fail, and the throttle's own surge gain runs into a ceiling that the plenum's size does not move. Give the stall cell its own actuator — air injected at the rotor tips against the cell — and give the throttle a second signal, the flow, and a machine at B = 2 is held at ninety per cent of its peak flow, recovering from a kicked cell nearly three times the injector's capacity. Let the injection lag the cell by a radian and the line still holds, but almost no disturbance is survived.

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