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 38

Page 38 of 50, continuing through the 450 essays this motif runs through.

450 essays carry this thread — page 38 of 50.

The rudder should carry an eighth of the side force. The drag of keel and rudder together, as a percentage of the keel carrying everything, against the rudder's share of the side force, for a rudder 0.809 of the keel's depth. The induced drag is least with the keel alone — a shallower rudder makes the combined loading less elliptic for any load it takes. The profile drag is least near equal lift coefficients. Their sum is least at 12.4 per cent, 0.97 per cent below the keel alone; for the yacht drawn here, a rudder 0.809 of the keel's depth, that is 12.4 per cent. Fluids at work

The rudder pays for the keel's wake

A rudder angled to carry side force feels drag out of all proportion to its share, and that is the sensation behind the rule that weather helm is slow. The pair's drag does not care who is billed. It is least with the rudder carrying an eighth of the load, at under a degree of helm, and that share does not change with how hard the boat is pressed.

 Every velocity guesses low and every stress guesses high. Two bounds on the flow rate of a duct, in units of G a⁴/μ, against the number of mesh cells per half-side: from above, the best stress field in equilibrium with the pressure gradient (red); from below, the best velocity that vanishes on the walls (green). For the square the series value 0.562308 (grey) lies between them at every mesh. For the L-shaped duct, which has no closed form, they close on 0.21399 to 0.21415. Viscosity

A flow pinned between two guesses

The minimum-dissipation principle says the true flow is the cheapest one the walls allow, so any guessed velocity carries too little. It has a twin that nobody teaches: any guessed stress in balance with the pressure carries too much, and it needs no wall condition at all. Between the two, the flow through a duct with no formula is pinned down to as many figures as anyone wants.

 The gas flows in, against the heat. The radial velocity of the gas, in units of α/a, at the same four times — from the motion of the gas shells themselves — with the heat flux, reversed in sign and in the same units, drawn as dots. The two coincide: at constant pressure the gas moves at (γ − 1)/(γp) times the heat flux, against it. Heat leaves the spot outward and the gas comes in, everywhere. Flows and fields

The air flows in against the heat

A gas moving at a ten-thousandth of the speed of sound is as incompressible as anything gets, by the usual arithmetic. Heat one spot of it and let the spot cool: the gas changes its volume threefold, and it flows towards the hot spot while the heat flows out, at a speed fixed by the heat flux alone. At constant pressure, a joule is a volume.

 A record's length is counted in integral scales. The scatter of a record's mean, in units of the signal's own standard deviation, and the relative scatter of its variance, against the record's length in integral time scales: four hundred records at each of seven lengths, against the exact results. Both fall as the square root of the number of integral scales, √(2Tᵢ/T). A mean known to one per cent of σ needs twenty thousand integral scales. The exact variance curve is for the variance about the true mean; a short record can only measure it about its own mean, which is why the shortest records scatter less than the curve says. Transition and turbulence

A record is as long as its integral scales

A turbulence measurement of a million samples can hold less information than one of a thousand. What sets a record's worth is not how many numbers it contains but how many integral time scales it spans, and the integral scale — the unit every other error is counted in — is itself the hardest thing in the record to measure.

 One rim, one drop, every angle in the band. Water on a disc the size of a US cent, 19.1 mm across, pinned at its rim. The faint outlines are the drop at apparent angles of 60°, 90°, 120° and 150°; the dark one is the drop at the angle set, 120°, holding 1.13 mL. The rim stays put while the angle climbs, and every shape is an equilibrium: 0.52 mL at 60°, 1.44 mL at 150°. They are flattened by gravity into puddles with rounded edges, not spherical caps. What is taught wrongly

An edge holds any angle it is given

A liquid's contact angle is taught as a property of the liquid and the solid. On a smooth face it is. On a sharp edge the line stops, the angle is free to take any value in a band as wide as the edge is sharp, and that freedom is how a coin carries a dome of water and a glass stands full above its brim.

 The first three ways a drop can ring. A drop's first three modes of oscillation, each drawn at the two ends of its swing: the shaded outline and the red one are half a period apart. The fundamental, l = 2, alternates between a lemon and a lentil. The next two ring at 1.94 and 3.00 times its pitch, with three and four lobes. The swings are drawn at a fifth of the radius so they can be seen; the frequencies belong to swings much smaller than that. Regimes and numbers

A drop rings like a bell

Disturb a drop and it swings between a lemon and a lentil at a pitch set by its surface tension and its size, and it keeps swinging for dozens of cycles because water is nearly frictionless at that scale. The usual story has a falling raindrop's wake ringing it. The wake strikes several times too fast for the note the drop plays.

 Which bends of a vortex pair grow, and how fast. The square of the growth rate of a sinusoidal bend of a trailing vortex pair, in units of one e-fold per descent time, against the bend's wavenumber times the spacing, for a core of elliptic loading's size. Where the square is positive the bend grows. The symmetric mode, in which the two vortices bend as mirror images, grows over a long-wave band and fastest at a wavelength of 8.54 spacings, by 0.827 e-folds in the time the pair takes to sink one spacing. The antisymmetric mode, bending in step, is stable across the same band. Circulation and lift

A wake ends by bending, not by fading

The two vortices behind an aeroplane do not simply weaken until they are gone. In still air each one bends in the other's strain, the bend grows by a factor of e every thirty-three seconds behind an airliner, and after a couple of minutes the pair has pinched itself into a chain of rings. The wavelength it chooses is eight and a half times the spacing, and the vortex cores hardly enter.

 The drip spacing is set by the depth of the layer. The spacing of the fastest-growing wave, in units of 2π capillary lengths, against the depth of the hanging layer in capillary lengths, for water, glycerol and honey. A thin film of any of them drips at √2. Deep water drips at √3. Deep glycerol and honey keep growing past √3, their fastest wave as long as the layer is deep, because viscosity slows short waves more than long ones. Transition and turbulence

How far apart a ceiling drips

A layer of liquid hanging from a ceiling is heavy fluid over light, and every ripple on it longer than about seventeen millimetres grows. Which ripple grows fastest, and so how far apart the drips form, is usually given as one number. It is at least three, and what chooses between them is not the liquid's surface tension but the depth of the layer.

 The unburnt gas bends round a wrinkle, and the bending makes it grow. Streamlines of the unburnt gas flowing up towards a flame with a small sinusoidal wrinkle, one wavelength across, for a density ratio of seven, with the wrinkle exaggerated. The expansion behind the flame pushes back on the gas ahead of it where the flame bulges forward, so the streamlines spread there and the gas arrives slower, and they crowd together where the flame lags, so the gas arrives faster. A flame that burns into the gas at a fixed speed therefore advances further where it was already ahead. Flows and fields

A flat flame is unstable at every size

A flame expands the gas it burns, and the expansion pushes back on the fresh gas ahead. Where the flame bulges forward the fresh gas slows, so the bulge burns further forward; where it lags, the gas speeds up and it falls further back. Every wrinkle grows, the shorter ones faster, and what finally gives a real flame a size is how its burning speed responds to its own curvature.

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