The thread: What is conserved — page 46
424 essays carry this thread — page 46 of 48.
Regimes and numbers
Below separation a cylinder's drag is a Stokes layer, and KC stops deciding the force
Morison's equation splits the force on a cylinder in an oscillating flow into inertia and drag, and the Keulegan–Carpenter number says which dominates — once the flow separates and leaves a wake. Below that, nothing separates, and the drag term describes the thin oscillating boundary layer on the cylinder's surface. Its coefficient then grows without limit as the motion shrinks, as one over the Keulegan–Carpenter number, and that exactly cancels the number's power to decide anything: the drag's share of the force is fixed by the frequency parameter alone, a few per cent at most, until Honji's instability ends the regime.
Circulation and lift
Sweep slows a wing root's gust cycles and leaves their excess over the lift
On a straight wing every station meets a gust at the same instant, and one wing root's bending moment crosses its mean nearly a fifth more often than the lift, because it reads the rolling half of the gust. Sweep the wing back and the outer stations meet the gust later, by their distance aft. That delay averages the gust along the flight path as a second, softer chord would: at 35° every load crosses its mean about a tenth less often, the root slightly more than the lift since its weight sits further out, and the variance hardly changes at all. The rolling excess survives almost whole.
Compressible flow
More blades hush a propeller only while its tips are slow
A propeller's thrust and torque are steady in the blade's own frame, and a steady force held still makes no sound. Carried round a circle it sings at the blade-passing frequency, and Gutin's formula says how loudly: through a Bessel function whose order is the harmonic's and whose argument is that order times the tip's speed towards the listener over the speed of sound. Such a function is exponentially small until its argument nears its order. So spreading the same thrust over more blades, which raises the order of the fundamental, silences it by twenty-five decibels at a tip Mach number of 0.4 and by one at 0.95.
Regimes and numbers
Inertia empties the vortices that would hold a settling particle, and then speeds its fall
A particle that settles slower than a lattice of vortices turns can be carried round a cell for ever: without inertia, its path is a closed curve, and Stommel showed in 1949 that half a cell can be such a trap. Give the particle the slightest inertia and every trap leaks, in a time that grows as one over its Stokes number; the freed particle is flung to the cells' edges, rides their downdraughts, and falls on average several times faster than it would through still air. More inertia undoes some of that, and a narrow band of it can hold a slow particle up indefinitely on an orbit of its own.
Fluids at work
The layer every airway restarts bends Murray's cube towards the turbulent pipe's
A laminar fork does not lose energy the way a turbulent one does, in a burst of mixing at the split. It loses it all the way down each daughter, because the boundary layer starts again at every flow divider and an airway a few diameters long ends before the layer has grown across it. Pedley measured that extra dissipation in 1970, and in the form he wrote it the radius cancels: it depends on the flow and the length alone. Price it into the cheapest tree and Murray's cube law gives way to a flow growing as the radius to the power twelve-fifths or thirteen-fifths, the band the turbulent pipe's exponent sits in, across the first eleven generations of a resting lung.
Viscosity
A levitator lifts by its amplitude, not its gas
A squeeze-film levitator holds its load up by the mean of a pressure that swings symmetrically about nothing, and the gas in the gap was expected to decide how much: a gas that cannot shed its heat of compression in time is stiffer, so it should lift more. Solved to every order in the amplitude, with the heat crossing the gap, it barely does. At a levitator's own thermal numbers every common gas lifts within a few per cent of the isothermal film while its losses differ several-fold, and even a gas far too slow to shed its heat lifts only a fifth more, because the walls quietly pump heat out of it and cool its middle.
Compressible flow
A flap that closes the mouth starts Busemann's biplane below its design point
Busemann's biplane cancels its own waves at one Mach number and cannot get there: its channel narrows to a throat that will not swallow the shock standing in its mouth until the aeroplane is flying faster than the design point. Hinge the front of each element's inner ramp and turn it into the channel, and the mouth closes towards the throat while the throat stays put. A flap a fifth of the chord long, turned in three and a half degrees, starts the channel at Mach 1.72 instead of 2.15; once started, the flaps can shut at any speed above 1.60 and the zero at the design point returns exactly.
Viscosity
A round jet hands its wall jet a strength proportional to its throw
A round jet striking a plate spreads into a radial wall jet, and like the plane jet before it, the wall jet keeps not the jet's momentum but Glauert's flux of momentum flux — the momentum the jet brought times the fluid it gathered on the way. A round jet gathers fluid at a rate set by the viscosity alone, eight π ν per metre of travel, whatever its strength. So the invariant it hands on grows in exact proportion to the distance it was thrown, where a plane jet's grew as the cube root: hold the nozzle four times further away and the wall jet is four times as strong and runs twice as fast.
Ideal flow
A step's face is pushed at its foot and pulled at its lip, equally
Ideal flow running up a step has two corners in it, and the local theory of corners says what each does: the inside corner at the foot brings the flow to rest, the outside corner at the lip sends it round at infinite speed. Solved exactly, the step's face feels both — a push of 2/π of a dynamic pressure times its height spread over its lower four-fifths, and a pull concentrated at the lip — and they cancel exactly, at every ramp angle, because one reflection of the mapping exchanges the two corners. A step has no drag in ideal flow for the same reason a flat plate's lift has no drag: a finite force carried by an infinite speed.