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 48

Page 48 of 50, continuing through the 450 essays this motif runs through.

450 essays carry this thread — page 48 of 50.

Helium wastes least for most stiffnesses; xenon reaches furthest. The film's loss tangent — damping over stiffness — against its stiffness, each gas traced from a tenth of an atmosphere up to the pressure of its greatest stiffness. Raising the pressure buys stiffness and costs loss for every gas. Up to about a hundred newtons per unit relative amplitude helium's film loses least at any stiffness — 0.29 at 50 N against air's 0.4 and xenon's 0.46. Beyond, helium is near its peak of 127 N and the heavy monatomic gases take over: xenon reaches 149 N, and at 120 N loses 0.73 against helium's 0.78. Viscosity

The gas sets a squeeze film's loss, not its stiffness

A squeeze-film levitator wastes part of every cycle as heat that crosses the gap, and the gas and its pressure decide how much. Fill one levitator with six gases at one atmosphere and its stiffness hardly changes while its loss nearly doubles from helium to xenon. Raise the pressure and every gas stiffens towards a peak near ten atmospheres. Helium wastes least at any stiffness up to a hundred newtons; above that, only the heavy monatomic gases get there.

 With a small plenum, stall feedback closes the hysteresis loop. The stall amplitude against the throttle as it is closed slowly past the peak and opened again, at B = 0.2, with no stall feedback and with feedback at twice the critical gain; closing solid, opening dashed. Without feedback the cell appears suddenly near γ₀ = 0.6094 and vanishes only at 0.6428, and the loop between the two legs has an area of 0.078. With feedback the cell grows gradually as the throttle closes and shrinks along the same curve as it opens: the area is 0.0014. Fluids at work

A throttle can hold the peak only in a small plenum

A compressor makes its most pressure at the peak of its characteristic, and at the peak it either surges or stalls. A throttle moved by feedback can fight both. Driven by the plenum's pressure it damps surge, with a gain that grows as the square of Greitzer's B; driven by the flow, it cannot. Driven by the stall cell's amplitude it turns the sudden jump into deep stall into a gradual one and closes the hysteresis loop — but only while the plenum is small. Above B of about a third the same feedback turns the jump into a cycle.

 The entrance grows with the Péclet number, peaks, and then shrinks. The distance from a wall-temperature step at which the local Nusselt number has come within 5 per cent of its developed value, in diameters, against the Péclet number, for turbulent pipes at Reτ = 500, 1000, 2000 and 5000, with the slug's laminar line L/D = 0.029·Pe. The liquid metals sit on the line; the entrance peaks at 11.6 diameters at Pe ≈ 4700 for Reτ = 1000 and falls to about three and a half diameters for water. Regimes and numbers

The longest thermal entrance belongs to neither metal nor gas

A liquid metal carries its heat across a turbulent pipe by conduction, as a laminar flow would, and so its thermal entrance should be long. It is long only when its Péclet number is large. Below a few hundred the entrance grows in proportion to the Péclet number with the constant of a fluid moving as a solid block, and it leaves that scaling close to the threshold at which the eddies first match conduction. The longest entrance in a turbulent pipe belongs to the fluids between the metals and the gases, and how long it is depends on a number the measurements have never pinned down.

 An elliptic wing filters the gust like a round aperture. How much of a gust component with spanwise wavenumber k₂ survives the span average, against κ = k₂b/2, for three weights. A uniform strip is a slit and its filter is sinc², falling as κ⁻² between its zeros. An elliptic loading is a circular aperture and its filter is the Airy pattern (2J₁(κ)/κ)², whose first zero is at 3.83 and whose envelope falls as κ⁻³, but which stays near one to larger κ: the ellipse's weight is narrower, so its filter is wider. A rectangular wing of aspect ratio nine weights the span by its own loading and lies between, with the ellipse's κ⁻³ fall because its loading too goes to zero at the tips as a square root. Circulation and lift

A wing averages a gust through its own loading

A finite span averages a turbulent gust and so tames the load it causes, but it does not average uniformly. The reverse-flow theorem says exactly how it weights the span: by the loading the wing makes at a uniform incidence. An elliptic wing therefore filters the gust the way a round aperture diffracts light, passes eight per cent more of its short scales than a uniform strip, and crosses its mean load up to four per cent more often.

 A cold wall holds its layer on longer; a hot one lets it go sooner. The pressure-gradient parameter β at which the wall shear vanishes, against the wall's total enthalpy over the edge's. With the velocity deaf to the temperature it is −0.1988 for every wall. Coupled, it is -0.3264 for a wall at absolute zero, -0.2623 for a wall at half the edge's total enthalpy, -0.1988 at one, -0.1573 at one and a half and -0.1295 at two. Cooling delays separation by more than the same heating hastens it. Compressible flow

A cold wall holds its layer on longer

A compressible boundary layer's pressure gradient pushes on its density, and its density is its temperature. Solve the velocity and the enthalpy together and the wall's temperature reaches the velocity: a cold wall holds its layer on to an adverse gradient two-thirds steeper, a hot one lets it go a third sooner, the stagnation point's heat-to-friction ratio more than doubles between them, and the cold band an uncoupled layer put above a cooled nose disappears.

 The current along the shore runs fastest three Ekman depths down. The depth-averaged Eulerian current along the shore, positive to the left looking shoreward, against the local depth in Ekman depths, for a swell arriving 20° off the shore-normal at the shelf edge and for one arriving head-on. Both make a jet to the right, fastest at 2.9 Ekman depths for the oblique swell (-1.42 mm/s at 37.6 m) and 2.6 for the head-on one (-0.948 mm/s), and a weak current the other way in water shallower than one Ekman depth. Rotation turns the transport the shelf returns, so even a swell with no alongshore component drives one. Flows and fields

A sloping shelf puts the swell's current three Ekman depths down

A swell's Stokes transport arrives at a coast and the sea must send it back or turn it aside. On a flat shelf one depth decided which. On a real shelf, deepening from the beach to its edge, every depth is present at once, and the sea does both: it sets up a few centimetres against the beach, dips a tenth of a millimetre where the depth passes two Ekman depths, and runs a current along the shore that is fastest about three Ekman depths down — wherever the friction puts it.

 The pocket escapes at the least pressure its states can stand. The liquid pressure each equilibrium of the pocket needs, against the pocket's volume, for a crevice with a one-micrometre mouth in a wall the water wets at 40° and one it does not, at 110°. As the crown's pressure falls the pocket follows its curve to the right: the rim slides out of the cone, pins at the mouth, and the meniscus bulges into a cap. The curve's minimum is the escape threshold — -106 kPa for the wetting wall and -138 kPa for the non-wetting one — and below it no state exists: the pocket becomes a cavity. What is taught wrongly

A crevice keeps the nucleus a free bubble loses

A free bubble at a siphon's crown dissolves in milliseconds, so it cannot be what breaks a siphon that has run for an hour. A pocket of gas in a crack of the hose wall can be, because its meniscus is curved by the wall and need not dissolve. Followed through its equilibria, the pocket escapes at a tension set almost entirely by the crack's mouth. What the wall's wettability decides is slower and more consequential: whether the crack fills, or keeps drawing gas in until the siphon breaks.

 Heading into the sea, follow the long waves and fly level through the short. The dinghy's mean drag against the crossover encounter frequency below which it follows the surface, in units of the sea's peak frequency, heading into seas of significant height 0.2, 0.4 and 0.6 m. Following everything — the right-hand end — costs the heave's lift, and flying level through everything — the left — costs the foil's swinging depth and, in the roughest sea, breaches. Between them each sea has a best crossover: for 0.4 m at 2.92 times the peak frequency, 67.6 N against 67.9 flying level and 199 following. Fluids at work

A foil in a random sea follows it up to its peak

In a regular wave a foiling boat chooses between holding its height and following the surface. A real sea is a spectrum, and a wand that filters its signal can do both at once: follow the long waves and fly level through the short. The best place to divide them is close to the frequency of the sea's own peak, the divided control beats either pure strategy, and in a rough sea it is the only one of the three that keeps the foil a safe distance under the surface and the drag near its calm-water value.

 A wake held at the stern moves the volume aft a third as far. The least-resistance hull's centre of volume, in per cent of the half-length from midships, negative aft, against the design Froude number, holding the Wigley hull's length, draught, depth profile and displacement. With the linear wake it sits 2.8 per cent aft at Fr 0.30; with the shaped wake of the same propeller-disc fraction, 0.86 if uniform in depth, 1.5 if deepest at the waterline, and 0.11 if deepest at the keel, where the sources make the fewest waves. Regimes and numbers

A wake held at the stern keeps the bulb and loses the lean

A wake that slows the water steadily from bow to stern makes Michell's least-resistance hull fuller aft and puts its bulb at the bow. A real ship's wake is nearly nothing along most of the hull and strong only in its last few metres, deepest at the keel. Held there, with the same wake at the propeller, it moves the hull's volume aft by a twentieth to a half as much, and it keeps a third to two-thirds of the bulb's preference for the bow. The lean was a property of the water along the run, and the bulb a property of the water at the stern.

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