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 40

Page 40 of 50, continuing through the 450 essays this motif runs through.

450 essays carry this thread — page 40 of 50.

A washed filter's best cycle is shorter, and still has one. The filtrate a batch filter produces per hour, averaged over filtering, washing and twenty minutes of emptying and refilling, against the filtrate collected in each cycle, for a slurry of 400 kilograms of silica per cubic metre and wash ratios of 0, 1, 2 and 5 pore volumes. Each curve has a best cycle, marked, and the wash both lowers it and moves it to a thinner cake: from 0.307 to 0.217 m³ per m² of cloth. Fluids at work

A washed filter works as long as it rests

A batch filter that has to be emptied between runs has a best cycle, and the rule for it is exact: the time the cake is responsible for equals the time the filter stands idle. Washing the cake does not break the rule, because a wash through a finished cake costs time in proportion to the square of its thickness, just as filtering it did. What a wash changes is the price, and the price is set by the slurry's concentration and by how far the wash front smears, not by the cake's softness.

 The V in which every bird pays the same is curved. Nine birds one span apart, flying up the page, in plan: each short line is a wing, its height its distance behind the leader in spans. The straight V at 7.5° — the fairest straight V — and the V whose members' positions are solved so that every one pays exactly the same share of the induced drag. The equal-share arms leave the leader almost abreast and bend back ever more steeply: 1.6°, 7.6°, 17°, 32° from the apex to the tips. Circulation and lift

A fair V is a curved V

In a straight V of birds somebody always pays more than somebody else: at the fairest angle the leader and the two birds at the tips pay half again what the birds between them pay. The V in which every bird pays exactly the same can be solved for, and it is not straight. Its arms leave the leader almost abreast and bend back ever more steeply, to thirty-two degrees at the tips of a flock of nine. The share every bird then pays is the flock's average, fixed by Munk's theorem before any position is chosen — so fairness costs nothing, and the only thing that can make the flock cheaper is flying closer together sideways.

 Where a motor's line crosses the film's own characteristic. The stress a self-heating film carries against the speed it lets one wall slide past the other, both scaled, for every steady state (solid): it rises while the film is cool, peaks at the fold and falls as the film heats and thins. A motor's torque falls along a straight line as its speed rises (dashed), and the film runs where the two cross. A stiff drive crosses once. A soft drive, shallower than the falling limb's steepest slope, 0.0738, can cross three times. A drive held at a fixed stress is a flat line, and at the fold's stress it only touches. Viscosity

A motor turns the runaway into a jump

A self-heating oil film has a fold at fixed stress and none at fixed speed, and a real motor is neither: its torque falls along a line as its speed rises. Put that line across the film's own torque–speed curve and the runaway disappears for any motor at all. What replaces it depends on the line's slope. A drive softer than one fourteenth of the cold oil's resistance jumps to a hot state and keeps a memory of the load; a stiffer one does neither. And the boundary between the two falls exactly where the motor is turning at a quarter of its no-load speed, which no motor near its rated speed can reach.

 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.

 Dry air makes a slow bang. The 10-to-90 per cent rise time of a steady shock against relative humidity, for jumps of 25, 50 and 90 pascals, all below the strength at which a discontinuity returns. Each falls roughly as the nitrogen relaxation time does, and doubles when the jump is halved. The rule marks the thermoviscous rise time of the 50-pascal shock, a few microseconds — two to three orders of magnitude below any of the curves. Compressible flow

Oxygen makes a boom's crack, nitrogen its rise time

The shock at the front of a sonic boom is not the viscous shock of a textbook, a few microseconds thick. It is spread over a large fraction of a millisecond by the vibrational relaxation of the air's molecules, and the two gases do different jobs. Oxygen, fast, removes the discontinuity for any boom weaker than about ninety pascals and decides how much of the front is left at the frequencies the ear weighs most; nitrogen, slow, sets the rise time that is measured. Water vapour speeds both, so a dry day makes a softer bang.

 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.

 Two shapes for each strain, and one of them holds. The strain rate, over the vorticity, at which an elliptical patch of aspect ratio λ stands still — Moore and Saffman's relation — rising to its maximum of 0.1501 at λ = 2.89 and falling again. Below the maximum there are two steady shapes for each strain: a rounder one, on which every disturbance computed here stays bounded, and an elongated one, which comes apart. Above it there is no steady shape at all. Ideal flow

A strained vortex holds until it has no shape to hold

A patch of vorticity in a strain has two steady shapes for every strain below 0.150 of its vorticity, a rounder one and a longer one, and none above. The longer one comes apart at the slightest nudge. The rounder one, computed with disturbances of two, three, four and five lobes, never does: it nods and holds right up to the strain at which it ceases to exist. So the existence limit is the real limit, and past it a vortex is not shattered but stretched — lingering first near the shape it has lost, for a time that grows as the fourth root of how close the strain is to the limit.

 The number of passes grows in proportion to Reynolds number. Passes the two rings make before their cores are close enough to merge, against the vortex Reynolds number, marched with the merging threshold at 0.24 of the core separation and at 0.30, and the closed form [(0.24 dₘᵢₙ)² − σ₀²] Re ÷ 4Tₚₐₛₛ, with the closest approach dₘᵢₙ and the time between passes Tₚₐₛₛ read from the inviscid pair. Both thresholds give a count in proportion to Re: one or two passes at a few thousand, ten or thirty at thirty thousand. Circulation and lift

Leapfrogging rings end by merging, not by parting

Two smoke rings that leapfrog in an ideal fluid do it for ever, because their energy binds them. A real fluid drains the energy, and the obvious guess is that the rings drift out of the bound state and part. They do not: diffusion drains the pair's energy and the energy of every possible pair of free rings together, and the margin that binds them never goes negative. What viscosity does instead is fatten the cores until, as one ring threads the other, the two are close enough to merge. That takes a number of passes in proportion to the Reynolds number — one to three at the few thousand of a laboratory smoke ring.

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