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The thread: One number decides the regime — page 51

Page 51 of 52, continuing through the 463 essays this motif runs through.

463 essays carry this thread — page 51 of 52.

A resistive wall lengthens a slug's thermal entrance by half, and no more. The 5 per cent thermal entrance length of a slug flow, as a multiple of its length with the wall held at a fixed temperature, against the wall Biot number hw·D/kf. A wall that conducts well changes nothing; as the wall's resistance grows the entrance lengthens to a peak of 1.501 times at a Biot number of 2.29, and then shortens slightly to 1.457 times, the fixed-flux value, for a wall that conducts hardly at all.

A steel wall lengthens a sodium entrance by half

A liquid metal's thermal entrance is long because its heat crosses the pipe by conduction, and a steel tube conducts a tenth as well as sodium, so the wall is part of the path. Put it in series with the fluid and the entrance does grow — but by half at the very most, at a wall Biot number near two, which is where sodium in a steel tube happens to sit. A wall that resists more changes the boundary condition from a fixed temperature to a fixed flux, not the length.

A free interface's thickness ratio heads to the root of its Prandtl number. The ratio of the shear layer's thickness to the density interface's against time, for an interface starting four times thinner than its shear, heat and salt in water. A free shear layer thickens by viscosity faster than the interface by conduction, so the ratio heads to √Pr: 2.64 for heat, 25.8 for salt. Held at its thickness, the shear lets the ratio fall through 2 — at about 50 s for heat and 4725 s for salt.

Transition and turbulence

What holds the shear decides whether the waves last

Holmboe's travelling waves need a density interface about half as thick as the shear around it, and diffusion thickens both. If the shear layer is held at its thickness, the interface catches up — in under a minute for heat in a laboratory tank, over an hour for salt — and the waves get two e-folds or a hundred and seventy. If the shear layer is free it thickens faster than the interface, the ratio heads to the root of the Prandtl number, and the waves last until the Richardson number ends them instead.

A stronger strain folds particles below a quarter only if it lasts. The Stokes number above which particles in a lattice with pulsing strain 1 + a sin ωt fold onto its walls, against the pulse's frequency, for amplitudes of 0.15, 0.3 and 0.5, with the steady lattice's quarter dashed. A slow pulse folds them at the peak strain's quarter, 1/(4(1 + a)): 0.194 at a = 0.3 and ω = 0.05, against 0.192. A fast one folds them only at the mean's: 0.249 at ω = 5.

Flows and fields

A stronger strain folds a cloud only if it lasts

Heavy particles in a lattice of vortices fold onto the walls above a Stokes number of a quarter of the saddles' strain. Let the strain pulse and the threshold moves — down to the quarter of the peak strain when the pulse is slow, back up to the quarter of the mean when it is fast — because a particle responds to the strain it has been in for about four strain times, not to the strain at the instant. In a random strain the same rule makes the folds arrive in bursts, at the excursions that are strong and long at once.

A trainer's sortie wants the washout of its cruise, not of its average. The span efficiency of a rectangular wing of aspect ratio 8 averaged over a training sortie — eight minutes' climb, thirty of cruise, ten of slow flight and twelve of circuits — against its linear washout. Untwisted it is 0.9367. The best washout for the sortie is 3.43°, reaching 0.9774: the optimum for a lift coefficient of 0.436, the distance-weighted harmonic mean of the sortie's. The cruise alone would want 2.6°; the time-averaged lift coefficient, 0.671, would want 5.28° and lose 1.2% of the efficiency.

Circulation and lift

One twist is chosen for the speed flown per metre

A twisted wing is best at one lift coefficient and an aeroplane flies at many. Average its induced drag over a flight the way fuel and height are actually spent, per metre, and the twist that wins is the one for the root-mean-square speed of the flight — so the cruise outvotes the slow flight, a glider's circling barely counts, and a trainer's usual three or four degrees of washout turn out to be right for reasons nobody fitted them for.

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.

Viscosity

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.

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