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 56

Page 56 of 57, continuing through the 509 essays this motif runs through.

509 essays carry this thread — page 56 of 57.

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.

 Losing the leading-edge suction doubles a slender wing's drag due to lift and multiplies a long wing's by about half its aspect ratio. The drag due to lift with no leading-edge suction, Cₗ²/Cₗα, over the induced drag a full suction leaves, Cₗ²/(πAe), against aspect ratio, from a vortex lattice for rectangular, delta and elliptic flat wings; with the closed form 1 + √(1 + A²/4) that Helmbold's lift slope gives an elliptic wing. Rectangular: 2.15 at A = 1, 3.46 at 4, 5.33 at 8, 7.11 at 12; delta: 2.22 at 0.5 and 2.83 at 2. The slender limit is 2. Circulation and lift

A wing that cannot hold its suction gains little from its span

The leading-edge suction is a small force concentrated at one edge, and on a finite wing it decides what the aspect ratio is worth. With it, the drag due to lift is the induced drag, which falls as the span grows. Without it, the drag due to lift is the lift tilted back by the incidence, and that falls only as fast as the lift slope rises — which stops at the section's 2π. A slender delta that loses its suction pays exactly twice its induced drag; a wing of aspect ratio eight pays more than five times it. Without the suction no span can lift a wing's best lift-to-drag ratio past a ceiling set by the section alone.

 Noise in every frame pulls in the frame interval that stops buying, by the same share whatever the window sees. The longest frame interval at which a run still collects nine-tenths of continuous filming's independent values, over its value with no noise, against the noise's share of the window mean's variance, for a velocity window one integral scale wide, one a twentieth as wide, and a gradient in the small window; dashed, a single exponential memory in closed form. velocity, window ℓ: 0.782 at ν = 0.1, 0.199 at ν = 1; velocity, window ℓ/20: 0.804 at ν = 0.1, 0.243 at ν = 1; gradient, window ℓ/20: 0.819 at ν = 0.1, 0.242 at ν = 1. Transition and turbulence

Noise in every frame is a memory of zero, and the run cannot measure it at its own pace

A camera that adds an independent error to every frame adds a part of the signal with no memory at all. Infinitely many frames would still average it away, so the run's information still approaches continuous filming's — but the frame interval at which a faster camera stops buying is pulled in by the same share for a wide window, a zoomed one and a gradient, about half at a noise of three-tenths of the variance and a fifth at a noise equal to it. And the usual way to learn the noise, from the run's own correlation extrapolated to zero lag, works only at frames much faster than the run's information needs.

 A stagnation point can leave the flow through the wall, and the wall's point changes kind to pay for it. The heights of the interior stagnation points on the axis of a swirling stagnation flow onto a plate, against the skin friction's spreading rate σ: the lower point, of index +1, and the upper, of index −1; along the wall, the skin-friction focus, a separation of index +1 where σ < 0 and an attachment of index −1 where σ > 0. As σ falls to zero the lower point descends into the wall as z ≈ σ and the wall's focus turns from attachment to separation; as σ rises to one half the two interior points meet at a height of one and vanish. Interior plus half the wall is −½ throughout. Flows and fields

A stagnation point can leave the flow through the wall, and the count is kept by halves

A no-slip wall is a sheet of zeros, so the index count that holds for a flow in open space has nothing to count there. Divide the velocity by the height and the wall's zeros shrink to the skin friction's nodes, foci and saddles; reflect the result through the wall and each of them is counted once while every point in the fluid is counted twice. That is why the points on a wall are halves, and an exact swirling flow onto a plate shows the halves doing their job: a stagnation point in the fluid can disappear without a partner by sinking into the wall, where the skin friction's focus changes from attachment to separation to pay for it. A section through the flow sees none of this, because in the section every one of the points is a saddle.

 Shaken at the right frequency, a liquid with one relaxation time moves sixteen hundred times the flow its viscosity allows. The amplitude of the oscillating flow rate over the steady Poiseuille flow the same pressure-gradient amplitude drives through a Newtonian liquid of the same zero-shear viscosity, against the frequency, in a tube 1 cm in radius. A Maxwell liquid with λ = 1 s and η₀ = 30 Pa·s peaks at 1660 times at 6.63 Hz and again at 317 times near 15.2 Hz; a Rouse spectrum with the same longest time and viscosity reaches 20.2 at 19.8 Hz; the Newtonian liquid only falls, to 0.99 at 30 Hz. Regimes and numbers

A liquid with one relaxation time rings in a tube, and a spectrum of times silences it

Push a viscoelastic liquid back and forth along a rigid tube and, if its relaxation time is long against the cycle, the tube becomes a cavity for shear waves in the liquid itself. At the cavity's resonance a wormlike-micelle solution in a tube one centimetre across moves sixteen hundred times the flow a Newtonian liquid of the same viscosity would. The Deborah number says when the liquid is elastic enough to ring; it cannot say whether it will, because a polymer with a spectrum of relaxation times and the same longest one damps the same resonance to a factor of twenty.

 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.

 A travelling bubble's B-factor grows as the square root of its age, and the bubble turns heat-limited when it reaches B*. The vapour a growing bubble has made over the liquid it has cooled, against its age, in water at 80 °C under a tension 10 kPa below the vapour pressure, while inertia limits its growth; the dashed line is B, at which the cooling would use up the whole tension. B is 4.3 at a microsecond and 136 at a millisecond, and meets B = 31.6 at 53.8 µs, when its radius is 0.141 mm. Fluids at work

A bubble is throttled by its age, and a sheet by its length

A cavity cools the liquid it boils, and the cooling lowers the vapour pressure it needed. Whether that matters was a question about one number, the B-factor — vapour made over liquid cooled — and no real cavity's was known. Computed, the two kinds a pump has are nothing alike. A travelling bubble cools a thin shell that thins further as it stretches, so its B grows as the square root of its age and reaches the critical value at Brennen's thermal time: a millisecond in water at 60 °C, four microseconds at the boiling point. An attached sheet keeps renewing the liquid it cools, so its B is a few, fixed by its length and speed. In warm water the bubbles are suppressed forty degrees before the sheet.

 The second gradient separates the layer soonest after a short release, latest after a dozen thicknesses, and sooner again after a long one. The distance past the second gradient's start at which the layer separates, in its thickness there, against the release's length in the thickness at the first release: Green's lagged layer, the same layer restarted at the second gradient with no memory, and the shape-only layer. Lagged: 2.91 after no release, 5.07 after 12 thicknesses, 3.95 after 60; with no memory, 3.16, 4.81 and 3.91. Transition and turbulence

A layer's memory of one gradient helps it through the next, unless the next comes too soon

A turbulent layer released from an adverse gradient keeps the gradient's eddies for a dozen thicknesses, entraining more than its shape asks. Decelerate it again before they fade — behind the next compressor stage, or past a flap — and the question is whether that memory helps or hurts. Both, depending on when. A second gradient that arrives within about three thicknesses of the release meets a layer whose entrainment is still catching up, and it separates eight per cent sooner than a layer of the same shape with no history. One that arrives later meets eddies still working from the first gradient, and the layer holds on up to five per cent longer. The crossing sits at three thicknesses for every pair of gradients tried.

 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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