Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: Taught wrongly, everywhere — page 21

Page 21 of 22, continuing through the 190 essays this motif runs through.

190 essays carry this thread — page 21 of 22.

The drag follows the surfactant load, and a surface pressure of μU pays nearly all of it. How far the drag has climbed from the clean bubble's to the rigid sphere's, against the mean surfactant load over the whole bubble as a surface pressure in units of μU. A load of 0.2 μU makes a 60° cap and a third of the climb; 0.56 a 90° cap and 70 per cent of it; 1 a 120° cap and 94 per cent. The dashed line is the cap's share of the surface for the same caps: the drag runs ahead of the area covered. Viscosity

A thousandth of a monolayer holds a bubble still

A clean bubble rising slowly through water feels two-thirds of a rigid sphere's drag, and real bubbles almost never do, because surfactant swept to the rear holds the surface still over a cap there. Solving the flow with the cap in it shows how little that takes. The drag runs ahead of the area covered — half-way to rigid with a third of the surface held — and the surfactant needed is set by the viscous stress, not by the surface tension. For a bubble a tenth of a millimetre across, a thousandth of a monolayer, spread as a cap, makes it rise within a few per cent of a solid ball.

 The ring is drawn off while the core keeps its shape. The edges of a vortex with a core of twice the ring's vorticity out to six-tenths of its radius, at four times, in a strain raised to 0.245 — just past the 0.227 that strips its ring. The strain stretches along the horizontal. Both edges lean into the ellipses of a strained vortex; then the ring's edge is pulled out at its two tips into arms, while the core inside stays close to an ellipse. Ideal flow

A strained vortex loses its ring at the limit of what it holds

A uniform patch of vorticity survives a strain up to 0.150 of its vorticity and no further. A real vortex is not uniform: its vorticity falls off outward, and in a strain its weak outer layers are drawn off while its core survives. The obvious way to predict how much survives is to apply the uniform limit layer by layer, each layer to its own vorticity. It is wrong by up to a factor of three. Built from a core inside a ring and followed by contour dynamics, the ring is stripped not at the limit of its own vorticity but at the limit of all the vorticity inside its edge, and a little after it.

 Every ball's best angle is on one curve. The launch angle that carries a ball furthest over level ground, against β, the square of its launch speed measured in terminal speeds. At small β the air does nothing and the answer is 45°. It falls slowly as β grows: 41° at β = 1, 32.6° at 10 and 20.3° at 1000. A shot put sits at β = 0.009 and 44.9°, a golf ball struck without spin at β = 2.39 and 38.1°, a shuttlecock hit at 30 m/s at β = 19.5 and 30.2°. Fluids at work

The terminal speed sets the best launch angle

In a vacuum every ball goes furthest at forty-five degrees. In air the best angle is lower, and it is lower by an amount that depends on one number only: how many times its own terminal speed the ball is launched at. A shot put, a baseball and a shuttlecock are three points on one curve, which falls from 45° to 41° at once and then needs a thousandfold more speed to reach 20°.

 Viscosity slows every ripple, and the short ones most. The growth rate of a varicose ripple on a liquid jet against its wavenumber times the jet's radius, at Ohnesorge numbers of 0, 0.1, 1 and 10, in units of the capillary time. Every ripple longer than the circumference grows. Viscosity damps each one in proportion to the square of its wavenumber, so the short ones lose most: the fastest moves from ka = 0.697 inviscid to 0.344 at Oh = 1 and 0.123 at Oh = 10, and its rate falls from 0.343 to 0.0114. Regimes and numbers

Viscosity lets a jet break, later and into bigger drops

A thread of honey falls for metres before it breaks, and a thread of water for centimetres, which suggests that viscosity holds a jet together. It does not: it cannot stop any ripple longer than the jet's circumference from growing. It slows them, the short ones most, so the ripple that wins is longer, it takes the viscous time rather than the capillary one to win, and each drop it makes is bigger. The wavelength grows as the square root of the Ohnesorge number and the drop as its sixth root.

 The waves a wing makes on the sea cost it almost nothing. The wave drag of a wing skimming deep water, as a fraction of its lift, against speed, at heights of half a metre, one and two. Each curve peaks at the speed √(2gh) — 4.43 m/s at one metre, whatever the chord — and there the wave drag is 4.4·10⁻⁴ of the lift, the density ratio of air to water times Cₗ c/4eh. At a cruise of 50 m/s it is 9.31·10⁻⁶ of the lift. What is taught wrongly

A wing over the sea barely touches it

A wing skimming the water presses its whole weight onto the surface beneath it, spread over a width about its own height. The sea is a deformable ground, and one might expect it to give way: a trough under the wing, waves behind, a drag to pay. It barely notices. Air is eight hundred times lighter than water, and that one ratio sets the dent at millimetres, the waves at a few millimetres high, and the wave drag at less than a two-thousandth of the lift even at the worst speed.

 Lift takes the launch angle down far faster than drag does. The best launch angle against β for a ball whose lift is a fixed fraction ℓ of its drag, from ℓ = 0 — the drag-only curve — to ℓ = 1. At β = 10 drag alone puts the best angle at 32.6°; a lift-to-drag ratio of 0.4 takes it to 15.9° and 0.8 makes a level launch best. Once the curve reaches zero it stays there: the lift lifts the ball, and any angle given at launch is height the lift would have bought for less drag. Fluids at work

A spinning ball should leave low

Drag alone takes a ball's best launch angle from 45° to the high thirties at the speeds sport is played at. Backspin takes it into the teens, because a ball with lift is a glider and a glider buys its height with lift rather than with angle. Every tenth of lift-to-drag ratio takes a fixed slice off the angle, and above a line that depends on the launch speed the best flight leaves the ground level.

 The parcel over the top arrives first, and never waits for the other. Two parcels released together far upstream, a whisker above and below the streamline that divides at the nose of a Joukowski section 11.8% thick at 4°. Both crawl past the nose; then the upper one is carried over the top faster and reaches the far line 1.13 time units ahead — 0.279 chord-transit times — against the 1.59 that the circulation divided by the speed squared predicts. Behind the section they travel on side by side at the stream's speed and the gap between them never closes. What is taught wrongly

The upper parcel leads by the circulation

Transit time looked like a quantity that could not be measured: a parcel released near the dividing streamline crawls past the nose for as long as one likes. The difference between two parcels' transit times, one either side of that streamline, does not crawl. It converges, and on a thin section it is the circulation divided by the square of the speed — the lift, measured in seconds. The upper parcel arrives first by that much, and behind the wing the two never close the gap.

 Below 9.8 µm in silicone oil, a gradient of a degree per millimetre outruns gravity. The speed of a clean air bubble in 10 cSt silicone oil against its radius, on logarithmic axes: migrating in a gradient of 1 K/mm, which grows in proportion to the radius, and rising under its weight, which grows with its square. They cross at 9.81 µm and 31 µm/s. The crossing radius, 3|dσ/dT|G/(2ρg), has no viscosity in it: both speeds are set against the same viscous drag. Viscosity

The warm side pulls a bubble with no force on it

A bubble in a liquid whose temperature varies from place to place moves towards the warm side, with no gravity and nothing pushing it. The surface tension is lower where the liquid is warmer, the surface is pulled towards the cold pole, and the bubble goes the other way. Young, Goldstein and Block's speed comes out of four interface conditions and one more: that the total force on the bubble is zero. That one condition removes the point force a sinking or rising body always carries, so the bubble's disturbance dies a hundred times faster with distance, and the radius at which it balances its own buoyancy has no viscosity in it.

 A washed-out rectangular wing is at its best at one lift coefficient. The span efficiency of a rectangular wing of aspect ratio 8 against lift coefficient, untwisted and with 2°, 4° and 6° of washout. Untwisted, it is 0.9367 at every lift coefficient. Each washout has one lift coefficient where it is best — 0.254, 0.509 and 0.763 — proportional to the twist, and each reaches the same peak, 0.9911, there. Below the peak the efficiency falls fast. Circulation and lift

Washout is right at one lift coefficient

A twisted wing's loading is the sum of two shapes in a proportion that changes with speed, so its induced drag is a parabola in lift coefficient with its least at one design point, and that point moves in proportion to the twist. On a rectangular wing four degrees of washout puts it at a lift coefficient of 0.51 and recovers almost all the span efficiency the planform lost. On a well-tapered wing the same four degrees puts it at nineteen, and the washout is a drag paid at every speed for the stall alone.

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