The thread: The exact theory is wrong — page 36
335 essays carry this thread — page 36 of 38.
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.
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.
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.
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.
Circulation and lift
A spar that must lift itself stops the span at a quarter of the weight
Held only to a lift and a bending moment, the least-drag wing never stops growing. An aeroplane's wing stops because its spar has weight, and the weight has to be lifted. Count it, and the best span is the one at which the spar is exactly a quarter of the lift when the wing's area is fixed, and a third when its chord is — whatever the loading. The best loading then sheds some of its tip load, but not all the way to Prandtl's bell; with the chord held and nothing else charged, it is the bell exactly, and the more of the drag the wing's own area makes, the nearer it comes back to the ellipse.
Compressible flow
The shock layer is Hill's vortex, and it heats the nose a tenth harder than Newton says
Behind a strong bow shock the gas is squeezed six-fold or more and turned round the nose in a thin layer, and if its density is taken as constant there the layer has an exact solution: the same family of rotational flows as Hill's spherical vortex, fitted to the shock's three conditions. It stands off further than a mass balance says, and its gas accelerates away from the stagnation point a fifth faster than Newton's pressure law implies. Since a nose's heating goes as the square root of that acceleration, the layer heats the nose about ten per cent harder than the estimate every first calculation of re-entry uses.
Viscosity
Two spheres settling at a slant keep their slant for ever, and three do not
Two equal spheres settling through a viscous liquid fall faster together than either alone — half as fast again if they touch in line, and still five per cent faster thirty radii apart. Slanted, the pair drifts sideways as it falls, by up to three and a half degrees, and keeps the slant, the separation and the drift for as long as it falls: the two spheres always move with exactly the same velocity. Release a third and the shape is lost at once, the middle sphere drawing ahead and the outer two closing in behind it. What a pair keeps and a trio does not is a statement about symmetry, and reciprocity is half of it.