Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: Taught wrongly, everywhere — page 17

Page 17 of 22, continuing through the 190 essays this motif runs through.

190 essays carry this thread — page 17 of 22.

A slow current divides round the fluid over a hill (δ = 4). A uniform stream, left to right, over a Gaussian hill on the floor of a rotating layer, at δ = h₀/(H·Ro) = 4, where h₀/H is the hill's height against the depth and Ro the Rossby number U/fa; the onset for this shape is δ = 3.134. The dashed circle is the hill's e-folding radius. Water crossing the hill is squashed and, keeping its potential vorticity, spins clockwise; that anticyclone adds to the stream on one side and opposes it on the other. Here it stops the stream: the outlined region, 1.54 square radii, holds fluid that circles for ever and never leaves, and it sits beside the summit rather than on it, on the side where the swirl runs against the current. Regimes and numbers

The hill a slow current will not climb

The Taylor–Proudman theorem says a rotating fluid goes round an obstacle rather than over it, as if a solid column stood above it. Conserving potential vorticity turns that limit into a threshold. A current is stopped over a hill once the hill's height against the depth exceeds a fixed multiple of the Rossby number — 2 for a flat-topped hill, 3.13 for a Gaussian one, 16/3 for a cone — and the fluid it holds sits beside the summit rather than on it.

 Hull speed is not the hump. The wave-resistance coefficient of the Wigley hull, R/(½ρU²S) in thousandths, against the Froude number U/√(gL), from Michell's thin-ship integral. The curve rises through a series of humps and hollows and peaks at Fr = 0.499. The hull-speed rule's Froude number, 1/√(2π) = 0.399, where the transverse wave is as long as the hull, is neither: it lies on the steep climb between the last hollow, at 0.346, and the main hump. Regimes and numbers

Hull speed is not the hump

The hull-speed rule says a displacement vessel meets a wall where its wave is as long as its hull, at a Froude number of 0.4. Michell's thin-ship integral, computed for a standard hull, puts that speed on the steep climb between the last hollow and the main hump, and the hump itself at 0.5. Past the hump the wave resistance grows more slowly than the speed. The hollows sit where the hull's own transverse wave switches off, and a bulb that cancels at one speed multiplies the resistance at another.

 A fuel-wetted plate sits far below the stream that cools it. The temperature of a throttle plate wetted by petrol, against the day's temperature at 80 per cent humidity, beside the temperature of the stream flowing over it — the mixture after half the fuel has evaporated into it, at the plate's recovery temperature. A plate kept wet with fresh fuel sits nine to twenty-three kelvin below the stream, more on warmer days, and below freezing on any day up to about 34 °C. A plate whose film has lost its light ends sits one to six kelvin below it. The dashed line is freezing. What is taught wrongly

The throttle plate is a wet bulb for petrol

A carburettor's icing is usually worked out for the mixture — the fuel's latent heat spread through all the air — and that puts the warmest icing day near 15 °C, well short of the 30 °C the charts pilots use allow for. The ice does not grow in the mixture. It grows on the throttle plate, and a plate wet with evaporating petrol is a wet-bulb thermometer for fuel: nine to twenty-three kelvin colder than the stream flowing over it. Kept wet with fresh fuel it ices on a 30 °C day at full humidity and a 40 °C day at half — which is what the training handbook warns of. Once its light ends have gone it does not, and that difference is most of the story.

 Whether a sloping ceiling drips in place is one ray's growth. How fast a disturbance grows as seen by an observer moving along the ceiling at speed v, for four speeds at which the film carries its disturbances downhill, in the film's own units. On a flat ceiling (V = 0) the growth peaks at a quarter for the observer standing still and falls to zero for one running at ±1.622. Tilting the ceiling slides the whole curve downhill. While the observer at the point disturbed, v = 0, still sees growth, the ceiling drips where the disturbance began; at V = 1.622 that observer sees none, and above it the disturbance grows only while it is carried away. Transition and turbulence

A sloping ceiling drips downhill, or not at all

A film hanging from a level ceiling drips where its ripples form. Tilt the ceiling and the film flows downhill, carrying its ripples with it, and at some slope they are carried away faster than they spread and the ceiling stops dripping in place. That slope is set by one number, 1.622 — the speed at which a level ceiling's disturbance spreads — and for water it is tiny: a degree for a tenth of a millimetre of film, six for half a millimetre. Past it the ripples still grow, and still drip, but downhill, at a distance that grows with the slope: a ceiling shorter than that delivers its water to the edge.

 One fast pump or two slow ones. Pump efficiency against the flow asked for, for two pumps into a system with 60 per cent static lift: one pump under speed control, up to the flow at which it reaches full speed; both under speed control; and the best number running with the pumps held at full speed and a valve taking up the difference. Below 0.601 of the design flow one pump is better than two, and above it two are better — before the single pump has run out of speed at 0.65. The throttle is far worse at every flow. Fluids at work

One fast pump or two slow ones

A pumping station with several identical pumps in parallel has a control that a single pump does not: how many of them are running. With no static lift the answer is all of them, always, slowed together. With lift it is a number that falls with the flow and is almost never whole — the flow times the number of pumps, over the square root of the head the system asks. Staging rounds it, and the right moment to round up comes before the running pumps have run out of speed.

 The exponent a pulsing tree wants is between two and three. The tree's reflection at its root, averaged over the pulse's ten harmonics with the pulse's own weights, against the branching exponent k in r_parent^k = 2 r_daughter^k — two is area-preserving, three is Murray's law. With a wave speed that does not change with radius the best exponent is 2.15, just above the inviscid match of two; with one that rises as smaller arteries stiffen, 2.58, just above 2.5. Viscosity in the smallest branches pushes the best exponent a little towards Murray's. Regimes and numbers

Murray's law is not the rule for a pulse

Murray's law sizes a branching vessel for the cheapest steady flow, and at every junction built to it a pressure pulse is partly reflected. The rule that makes a junction transparent to a pulse is a different exponent, set by how the wave speed changes with radius. A tree is not the sum of its junctions, either: a Murray tree six generations deep reflects a third of the pulse at the heart rate, three times what one of its junctions does, and viscosity in the smallest branches means no area rule can make it transparent at every frequency. The best a pulsing tree can do lies between area-preserving and Murray.

 A slip law implies a slip length that changes with the rate. The slip length — slip velocity over the true wall shear rate — against the wall shear rate, for a melt whose slip follows a velocity law in stress with exponent 1.5, 2 or 3, all matched to slip at 2 mm/s at a stress of 100 kPa, and for a melt with a constant slip length, dashed. With the exponent of 2 the implied length falls from 0.22 mm to 0.08 mm across the range of rates; only a law whose exponent is exactly 1/n, 2.5 here, implies a constant length. What is taught wrongly

A melt's slip length has an exponent it cannot choose

A polymer melt slips at the wall, and there are two ways to say by how much: a slip length, fixed, so the slip velocity follows the wall's shear rate; or a slip law, the slip velocity as a power of the wall's stress. The literature moves between them as though they were the same, and for a melt they are not. A constant slip length forces the slip law's exponent to be exactly one over the melt's flow index — 2.5 for a typical melt — so a measured exponent of 2 is a slip length that changes with the rate. And temperature separates them outright: a tenfold change in viscosity moves the slip velocity tenfold under one description and three-hundredfold under the other.

 The suction a diffuser needs to beat Betz on its own exit. The best power per unit of exit area against the suction held behind the exit, as a pressure coefficient −c. The curve is (2/3√3)(1 + c)^(3/2) up to c = ½ and the dots are a direct search. It starts at 0.385, sixty-five per cent of Betz's limit, and crosses 16/27 at c = 1/3 exactly — the pressure on the back face of Betz's open disc. A diffuser with no suction at its exit is worse than a bare rotor as large as its exit; one that beats the limit on its exit area is holding more suction there than the open disc holds behind itself. Fluids at work

A duct beats Betz only on the area it chooses

Put a wind turbine inside a flaring duct and it can take more than 16/27 of the wind's power through its rotor, which is the claim. Measure the same power against the duct's exit, the area the device actually fills in the wind, and a plain diffuser takes less than a bare rotor of that size would. It crosses the limit only when the duct holds a suction behind its exit, and the suction it needs is exactly the pressure on the back face of Betz's own disc.

 The largest nucleus decides how tall a siphon can be. The tallest crown a siphon of degassed water can run over, above the upper reservoir's surface, against the radius of the largest gas nucleus the water carries, for outlets 0.2, 1 and 3 m below the upper reservoir. Nuclei of ten microns and more leave the ordinary limit of about ten metres, where the crown reaches the vapour pressure. A largest nucleus of one micron lets a slow siphon stand 14.4 m tall; of three-tenths of a micron, 27.3 m. The water's own strength is never the limit: its dirt is. What is taught wrongly

A degassed siphon is as tall as its largest nucleus allows

The textbook says a siphon cannot lift water more than about ten metres, because above that the pressure at its crown would fall below zero. Water can hold a pressure below zero — a tension — and siphons of degassed water have run over crowns taller than the barometric height. How much taller is not set by the water's strength, which is enormous, but by the largest speck of gas it carries. A nucleus of a micron lets a slow siphon stand fourteen metres tall; three-tenths of a micron, twenty-seven. And the flow's own speed takes metres off every limit.

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