The thread: One number decides the regime — page 58
521 essays carry this thread — page 58 of 58.
Transition and turbulence
Transition along a plate is a count of spots, and one number gives its length
A boundary layer does not turn turbulent everywhere at some station. Arrowhead patches of turbulence appear at random and sweep downstream, spreading as they go, and a probe at a fixed place reads a square wave. Treat those patches as independent, add one rate of production and one spot shape, and the entire transition zone is a line of arithmetic: the same formula Kolmogorov wrote down in 1937 for crystals growing in a cooling melt. It has only one free number in it, and that one number states the zone's length.
Viscosity
A Leidenfrost drop floats on a bearing it feeds itself
A bead of water on a hot pan vanishes in seconds, unless the pan is much hotter, when it skates about for minutes. Between the drop and the plate is a film of its own steam a tenth of a millimetre thick, and that film is a lubricated bearing with no shaft and no pump: its supply is the drop's own evaporation, which keeps it thin where it is thick and thick where it is thin. Its thickness obeys a fourth root of everything. On a plate too hot to touch, the drop's lifetime is a hundred times what it is on a plate that is merely hot.
Regimes and numbers
A flapping plate has no best Strouhal number until it pays for drag
Fish, birds and bats of very different sizes all cruise with a tail or wing beat at a Strouhal number between 0.2 and 0.4, and the usual story credits the wake: a reverse vortex street that has one most-amplified frequency. Garrick's exact inviscid theory of 1936 gives something else. A plunging plate's efficiency has no peak: it falls from one at no flapping at all to a half at frantic flapping, for every amplitude. A best Strouhal number exists only once the plate has a body's drag to pay for, and it then lands in the right band with no wake instability in the argument.
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.
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.
Fluids at work
A coil's friction is one number, and the coil holds its flow laminar
Bend a pipe into a loose coil and the flow inside is not the same flow bent. The fast core is thrown outward, slow fluid returns round the walls, and a pair of counter-rotating cells appears, about a tenth of the mean speed in an ordinary coil. Dean showed in 1927 that one number governs all of it: Reynolds times the square root of the tightness. Below about 12 nothing changes. Above it the friction rises by a curve the coil's geometry never enters again. And the same secondary flow does something unexpected to transition. A coil stays laminar at four or five times the Reynolds number that breaks a straight pipe.
What is taught wrongly
Two connected balloons part over a range of stretch, and even out on either side
Air flows from high pressure to low until the pressures match, and everyone concludes that two connected balloons end up the same size. Soap bubbles famously do not. Rubber does something stranger than either: a pair evens out at low stretch, parts into one large and one small balloon over a wide middle range, and evens out again at high stretch. Equal pressure is all the air ever asks for, and one rubber curve lets the same pressure be held at three different sizes. What the pair does also depends on how it was filled.
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.