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The thread: The exact theory is wrong — page 32

Page 32 of 32, continuing through the 288 essays this motif runs through.

288 essays carry this thread — page 32 of 32.

A slow bush leaves the bearing hunting, a sixth faster. The film's mean temperature excess over two cycles of a bearing hunting under a derating of 0.2 with a fifth of the housing's excess reaching the bush, for a bush that follows the housing instantly and one ten housing times slow. Both jump between a cool film and a hot one; the instant bush's period is 0.347 housing times, the slow bush's 0.291. The slow bush holds the walls at the cycle's mean, and the film's jumps are the same size.

A slow bush quickens the hunt it cannot start

A bearing's bush warms with its housing and makes the drive stiffer, and past a coupling of about a third it stops a derated bearing hunting. Give the bush a time of its own and the obvious fear is that the lag reopens the hunt. It cannot: whether a bearing hunts is decided by where its equilibrium sits, which no lag can move, and a lag on a positive link cannot make a settled state oscillate. What a slow bush does change is the hunt itself, which runs a sixth faster.

Each oblique shock added before the normal one buys less than the last. The best total-pressure recovery of a train of oblique shocks ending in a normal shock, against the number of oblique shocks, at Mach 2, 2.5, 3 and 4. At Mach 3 one oblique shock recovers 58.1%, two 74.4%, four 88.9% and eight 96.4%; a normal shock alone keeps 32.8%.

Compressible flow

The last shock should be the weakest

An intake slows a supersonic stream through a train of oblique shocks and one normal shock, and the rule for laying out the train is that its shocks should be equally strong. Solved exactly rather than quoted, the rule is right for the oblique shocks to four figures and wrong for the last one: the terminal normal shock should be about five per cent weaker than the others, and the turns that give equal strengths are not equal turns but grow along the train.

The meniscus shortens the fastest finger and its film lengthens it, and they cancel near Ca = 0.01. The fastest wavelength over the uncorrected π b/√Ca, against capillary number, with the in-plane tension reduced to π/4 alone, with the speed-dependent transverse jump alone, and with both. The reduced tension shortens every finger by √(π/4) = 0.8862; the film's jump lengthens it, more at higher capillary number; together they cancel at Ca = 0.0131, shorten the finger 5.8% at 10⁻⁴ and lengthen it 6.1% at 0.3.

Ideal flow

The meniscus slows the finger and hardly moves it

Air pushing oil through a narrow gap fingers at a wavelength the gap and one number set, on the assumption that surface tension acts only on the interface's curve in the plane. The interface is really a meniscus across the gap, leaving a film behind, and Park and Homsy's correction for it does two things that nearly cancel in the wavelength: it cuts the tension to π/4 and adds a drag that grows with speed. At bench speeds the fastest finger barely moves; it grows a quarter more slowly, and a channel goes unstable at four-fifths of the speed.

Between fixed ends the transparent-junction tree is the quiet one. The magnitude of the pulse reflected at the aortic root at the heart rate, 1.2 Hz, and its first four harmonics, for trees grown from the same aorta to the same 0.5 mm vessels by four branching rules. The Womersley-map tree reflects 0.028 at the heart rate and 0.0046 or less at the harmonics; the best single exponent 0.035; Murray's 0.083; and the area-preserving tree 0.21 at the heart rate though little at the harmonics.

Regimes and numbers

Between fixed ends the transparent tree is the quiet one

Grown from one aorta for six generations, an arterial tree whose every junction passes the pulse reflected more than one built to a single compromise exponent. Grown instead from the same aorta down to the same small vessels, however many generations each rule needs, the order turns over: the junction-by-junction tree is the quietest, and Murray's tree, which looked cheapest for the heart, becomes the dearest. The earlier ranking was a comparison between trees that fed different beds.

At fourth order a local slope reports less than a quarter of the anomaly. The anomaly each instrument reports at Reλ = 200 — Kolmogorov's n/3 less what it reads — against the order, for the log-normal field with μ = 0.25, with the departure its distribution carries. The local slope reports 0.026 of a true 0.111 at fourth order and 0.205 of 0.333 at sixth; extended self-similarity 0.245 of its own 0.261; the normalised moment reports the truth at every order.

Transition and turbulence

At higher orders the bottleneck hides intermittency

At second order the spectral bottleneck makes a field with no intermittency look intermittent. At higher orders it does the opposite. Every moment's local slope is steepened by the same finite-Reynolds-number excess, multiplied by half the order, while the intermittency correction pulls the other way. At a laboratory Reynolds number a fitted sixth-order exponent therefore shows less than two-thirds of the anomaly the field carries, and a fourth-order one less than a quarter.

Circulation and lift

A bending moment never chooses a span

Hold a wing's lift and its bending moment and ask for the least induced drag, and the answer does not stop: the span grows for ever and the tips push down. Both celebrated answers — Prandtl's bell at √(3/2) of the span for 8/9 of the drag, and the root-moment wing at 4/3 for 27/32 — are the span at which the least-drag loading's tips stop lifting. The structure sets how much is saved; a condition nobody states sets where the saving ends.

A free jet keeps its momentum; a wall jet keeps the flux of its momentum flux. Each quantity against distance, relative to its value at 2 cm, on logarithmic axes. The free jet's momentum flux is constant and its volume flux grows as x^(1/3). The wall jet's momentum flux falls as x^(−1/4) — the wall's shear takes it — while its volume flux grows as x^(1/4), and the one thing it conserves is Glauert's ∫u(∫u²)dy, the flux, through each height, of the momentum flux carried above it.

Viscosity

A jet along a wall keeps the flux of its momentum flux

A jet in open fluid keeps its momentum and gathers fluid as it goes. Blow the same jet along a wall and the wall takes its momentum away as drag, so it keeps neither. What it keeps is stranger: the flux, through each height, of the momentum flux carried above that height. That one invariant fixes everything else — the jet thickens as the three-quarter power of distance, slows as the inverse square root, and gathers fluid as the fourth root, more slowly than the free jet's cube root.

At Mach 3 the cowl caps external compression at 88 per cent. Best recovery at Mach 3 against the number of oblique shocks, free and with the total turn capped at the free stream's detachment angle, 34.07°, which is as far as the cowl's outer surface can turn. Free, six shocks reach 94.2%; capped, 86.7%, and no number of shocks, nor a continuous isentropic ramp turning the same angle, can pass 88.5%.

Compressible flow

The cowl caps what the ramps can buy

Every ramp in front of a supersonic intake turns the flow towards the cowl, and the cowl's outer surface has to turn the free stream at least as far. Past the free stream's own detachment angle it cannot, and that one angle caps every external intake: at Mach 3 no number of ramps, nor a smooth ramp, recovers more than 88 per cent, and the shock that is left at the throat is the strongest in the train. Above Mach 3 the cap falls fast, which is why fast intakes compress inside.

Out to a hundred neutral wavelengths the noisy flame is still speeding up. The front's speed in excess of a flat flame's against its width, on logarithmic axes, at noise strengths of a thousandth and a millionth, with the quiet pole front's and a power fitted from 20 to 100. At a thousandth the speed climbs from 1.05 at 12.5 to 2.71 at 100; at a millionth from 0.668 to 1.71. Both follow the 0.42 and 0.43 power of the width to within 4 per cent, and neither bends over; the quiet front stays at a half.

Flows and fields

A noisy flame is still speeding up a hundred wavelengths wide

A quiet wrinkled flame gains a fixed, bounded amount of speed over a flat one, however wide it is. A noisy flame keeps getting faster as it widens, and the open question was whether that ever stops. Out to a hundred neutral wavelengths it does not. The speed grows as about the 0.43 power of the width at two noise strengths a thousandfold apart. The noise sets only the prefactor, and the front's slope grows with the speed until it leaves the range the equation was derived for — which is where the growth stops being the equation's to decide.

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