Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: The exact theory is wrong — page 39

Page 39 of 39, continuing through the 345 essays this motif runs through.

345 essays carry this thread — page 39 of 39.

Under twenty degrees of aft sweep removes divergence altogether, and forward sweep brings it down to half. The dynamic pressure at which the wing diverges, in units of EI over the lift slope, chord and cube of the span, against sweep, positive aft. It is 24.7 for the straight wing, 15.3 at 10° forward and 11.8 at 30° forward. It is 38.6 at 5° aft, 692 at 10°, and past about 18.7° no divergence is left. Circulation and lift

A bent wing twists itself: aft sweep out of trouble, forward sweep into it

A wing bends upward under its own lift. Unswept, that does nothing to its incidence. Swept, it does, because a section cut along the wind rotates as the beam bends: aft sweep turns that section nose-down and forward sweep turns it nose-up. Aft sweep has been working as an automatic gust relief since the 1940s, and it removes a straight wing's divergence entirely at under twenty degrees. Forward sweep halves the speed at which the wing destroys itself. No forward-swept fighter flew until composite skins could be laid to twist against the bending.

 Every polymer leaves the Newtonian line at its own point, runs along its own branch, and stops at the same asymptote. The friction factor against Reynolds number. Newtonian Prandtl–Kármán is 0.0045 at 10⁵; Virk's asymptote 8.86·10⁻⁴. The three-layer profile with an elastic layer out to y⁺ = 30, 100, 300 gives 0.00239, 0.00137, 9.97·10⁻⁴ at that Reynolds number. A laminar pipe at the same Reynolds number would be at 1.6·10⁻⁴: the asymptote lies between the two and is still turbulent. Transition and turbulence

Polymers stop at an asymptote that no polymer chooses

A few parts per million of a long-chain polymer can cut a pipe's turbulent friction by four fifths. That is astonishing enough, but the stranger fact was measured by Virk around 1970. The reduction has a limit, and the limit is a single friction law, 19 log Re √f − 32.4, that does not depend on which polymer, what weight, which solvent or how much of it. Every polymer solution leaves the ordinary curve at its own point and arrives on the same line. An asymptote with no properties of the additive in it is a statement about turbulence, and half a century on, nobody can derive it.

 A bubble in water speeds up, slows as it flattens, and bottoms out at the speed of the slowest ripple. Rise speed against equivalent diameter in water. A clean surface reaches 0.298 m/s at 1.54 mm; a contaminated one has no such peak. Both then join the wave law, whose minimum is 0.236 m/s at 5.61 mm. Kelvin's slowest capillary–gravity wave is 0.231 m/s. Regimes and numbers

A bubble in water bottoms out at the speed of the slowest ripple

A bigger bubble should rise faster, and up to a couple of millimetres it does. Then it flattens, starts to wobble and slows down, and between 2 and 10 millimetres across it rises at nearly the same speed whatever its size, a little over 23 centimetres a second. That number is familiar from somewhere else. It is the speed of the slowest possible ripple on a water surface, the same capillary–gravity minimum that sets Kelvin's wind. Mendelson's wave analogy of 1967 makes the link a formula, and the two speeds agree to two per cent in every liquid.

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