The collection

Every essay — page 33

Page 33 of 61, continuing through the fields in the same order.

Flows and fields Ideal flow Circulation and lift Viscosity Regimes and numbers Compressible flow Transition and turbulence Fluids at work What is taught wrongly Series Concepts Regimes Refutations Search

Flows and fields

Streamlines, particle paths and the field that carries them. What is conserved, what a picture of a flow can show, and what it cannot.

Squeezed past closing, the tube is held open by its own fluid. The channel's half-width along one wavelength of a travelling squeeze that would push an empty tube's wall 1.3 half-widths inward — past the centreline — and the wall the squeeze and the fluid's pressure make together, for three compliances. The empty squeeze would overlap the centreline across a sixth of the wavelength; with fluid in the tube, the pressure that builds ahead of the narrowest point pushes the wall back, and the gap stays open at a hundredth to a tenth of the half-width.

A squeezed tube is held open by its own fluid

A peristaltic pump with a prescribed wall wave gets better the more its wave closes the channel, and at closure it becomes a piston. A real one squeezes an elastic wall and lets the fluid push back. Then the channel never closes: the pressure ahead of the narrowest point holds a gap open that grows as the square root of the wall's give, squeezing harder past closure pumps less, and the pump behaves as a displacement pump until a pressure of about one over its compliance blows the throat open — with an efficiency that stops short of one by an amount that grows with the wall's give.

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Every parcel goes straight while the streamlines wave. A uniform stream across the page carrying a pattern of transverse velocity with it, v = 0.6 sin(x − t): the streamlines at one instant, which wave, and the paths of five parcels over the next six time units, which are straight lines. Each parcel keeps the transverse velocity it started with, because the pattern moves with it; the streamlines are a snapshot of a pattern that is sliding past, and no parcel ever follows one.

A parcel goes straight while the streamlines curve

In a steady flow every parcel accelerates while the picture never changes. The opposite also happens: a flow whose picture changes all the time while no parcel accelerates at all, because the local and convective halves of the acceleration cancel exactly. Such a flow has no pressure gradient anywhere, and that is a severe demand. An incompressible flow can meet it only as a pure shear at every point; a cloud that can compress meets it freely and pays later, when its straight paths cross in a caustic.

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Moving air needs a quarter of the inertia still air does. The first crossing against Stokes number for a cloud thrown into still air — time in units of the cloud's fastest convergence rate — and for a cloud carried by air converging on a line, in units of the air's strain rate. Still air only brakes: its threshold is one. Converging air keeps pushing: its threshold is a quarter, and heavy particles in it cross in about one strain time. Dots: particles integrated in the converging sine flow; the curve is the linearised oscillator.

Drag decides whether a cloud folds, and the air decides at what

A cloud of free particles keeps its velocities, and wherever it converges its paths cross in finite time: a caustic, the density infinite along a sheet. Give each particle a drag on the air and the answer depends on what the air is doing. In still air the drag only brakes, and a cloud folds only if its Stokes number is above one. In air that is itself converging the drag keeps pushing, and a quarter is enough — the same quarter that decides whether a droplet hits a wing.

7 figures $At a coast the swell's water goes back, or goes along. The Lagrangian mean current — the Stokes drift plus the Eulerian current — under an 8-second swell arriving straight at a coast, in seas 8, 25 and 60 metres deep, each as a fraction of its own surface drift against depth as a fraction of the water's. Left, the part across the shore, positive onshore; right, the part along it. The cross-shore parts carry no net water. At 8 metres, inside the Ekman layer, the whole exchange is across the shore; at 25 the return is shared with a current along the coast; at 60 the open shelf's own rotation has already taken the transport back, and the coast has little left to do.$

A coast sends the drift back, or sends it along

A shelf sea under a swell keeps part of the Stokes drift's transport, turned to the right by the rotating Earth, and near a coast some of it points at the land. The sea answers with a slope in its surface. In water shallower than an Ekman depth the slope drives the water straight back as an undertow, as it would without rotation; a few Ekman depths down the coast turns the transport into a current along the shore instead, and there the sea can slope down towards the land it faces.

6 figures $Posts slip as one over the root of their fraction. The slip length of a surface of posts, over the period of their square array, against the fraction of the surface that is solid, beside Philip's stripes along and across and the dilute limit in which each post is a lone disc dragged edgewise, (3/16)√(π/φ). At a hundredth solid the posts slip 2.87 periods against the stripes' 1.32; at a tenth 0.597 against 0.591, nearly equal; at a half 0.0663 against 0.11 along the stripes and 0.0552 across them.$

A post holds liquid back by its radius, a stripe by its length

A surface of no-slip posts standing in a gas-filled texture lets liquid slip over it, and the slip length grows as one over the square root of the solid fraction, where a surface of stripes grows only as its logarithm. The reason is the oldest contrast in slow flow: a disc dragged through a viscous liquid has a drag proportional to its radius, which is Stokes' law, and a line has no finite drag at all, which is Stokes' paradox. But sparse posts beat stripes only while they are sparse.

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Below a quarter the cloud settles on the cell walls; above it, it crosses them. Paths of particles released with the air's velocity inside one cell of the vortex lattice, at Stokes numbers of 0.15 and 0.4, over 25 time units; the cell's walls, the separatrices joining the saddles, are drawn. At 0.15 the particles are flung outwards by the rotation and approach the walls ever more closely without crossing, so the cloud is compressed onto lines. At 0.4 they reach a wall with enough speed to overshoot it into the next cell, where they meet particles coming the other way: the cloud folds.

A lattice of vortices folds a cloud at the saddles' quarter

A cloud of heavy particles folds in still air above a Stokes number of one, and in converging air above a quarter. Air that turns as well as converges might have settled between them. In a lattice of vortices it settles exactly on the quarter: the vortices fling the particles to the cell walls, but only the saddles where the walls meet can make them cross. Below the quarter nothing folds, and the cloud is gathered onto the walls without limit instead.

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The current along the shore runs fastest three Ekman depths down. The depth-averaged Eulerian current along the shore, positive to the left looking shoreward, against the local depth in Ekman depths, for a swell arriving 20° off the shore-normal at the shelf edge and for one arriving head-on. Both make a jet to the right, fastest at 2.9 Ekman depths for the oblique swell (-1.42 mm/s at 37.6 m) and 2.6 for the head-on one (-0.948 mm/s), and a weak current the other way in water shallower than one Ekman depth. Rotation turns the transport the shelf returns, so even a swell with no alongshore component drives one.

A sloping shelf puts the swell's current three Ekman depths down

A swell's Stokes transport arrives at a coast and the sea must send it back or turn it aside. On a flat shelf one depth decided which. On a real shelf, deepening from the beach to its edge, every depth is present at once, and the sea does both: it sets up a few centimetres against the beach, dips a tenth of a millimetre where the depth passes two Ekman depths, and runs a current along the shore that is fastest about three Ekman depths down — wherever the friction puts it.

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Moving the saddles never lowers the quarter; fast large wobbles raise it. The smallest Stokes number at which a cloud of particles in the oscillating lattice folds within sixty time units, against the oscillation's amplitude ε, at frequencies of 1, 3 and 10 times the saddles' strain rate. With ε = 0 it is the steady lattice's 0.251. Moving the lattice never takes it below a quarter; at ε = 1 it is 0.283, 0.311 and 0.324 at the three frequencies.

Wobbling saddles keep the quarter, and raise it

Heavy particles in a steady lattice of vortices fold onto the cell walls only above a Stokes number of a quarter, the converging saddles' own threshold. Random flows fold at every Stokes number, so it was natural to suspect the quarter of belonging to steady saddles. Set the lattice wobbling and the particles cross from cell to cell, seventy per cent of them in forty time units, and still none folds below a quarter. The threshold belongs to the strongest strain the flow has; moving the saddles only moves particles out of reach of it, and raises the threshold.

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A swept wall makes vorticity along its isobars, not across its flow

A still wall in a pressure gradient puts vorticity into the fluid at a rate set by the gradient, and the source is a vector: it lies in the wall, along the isobars. On an unswept body the isobars run across the flow and so does the vorticity, which is the ordinary boundary layer. On a swept one the isobars run along the span and the flow does not, so part of every new vortex line points along the flow — half of it, over the accelerating front of a cylinder swept 35°. That part is where a swept wing's crossflow comes from.

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A stronger strain folds particles below a quarter only if it lasts. The Stokes number above which particles in a lattice with pulsing strain 1 + a sin ωt fold onto its walls, against the pulse's frequency, for amplitudes of 0.15, 0.3 and 0.5, with the steady lattice's quarter dashed. A slow pulse folds them at the peak strain's quarter, 1/(4(1 + a)): 0.194 at a = 0.3 and ω = 0.05, against 0.192. A fast one folds them only at the mean's: 0.249 at ω = 5.

A stronger strain folds a cloud only if it lasts

Heavy particles in a lattice of vortices fold onto the walls above a Stokes number of a quarter of the saddles' strain. Let the strain pulse and the threshold moves — down to the quarter of the peak strain when the pulse is slow, back up to the quarter of the mean when it is fast — because a particle responds to the strain it has been in for about four strain times, not to the strain at the instant. In a random strain the same rule makes the folds arrive in bursts, at the excursions that are strong and long at once.

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A nose at incidence makes its vorticity along the flow

A wall makes vorticity along its isobars, and on a swept wing that sends part of it along the flow. A fuselage at a few degrees of incidence has no sweep anywhere, and still most of the vorticity its wall makes points along the flow — a fifth for a body twice as long as it is wide, four-fifths for one eight times — because at incidence its pressure varies round the body faster than along it. A sphere, at any incidence, makes none. The lee vortices begin at the wall.

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Out to a hundred neutral wavelengths the noisy flame is still speeding up. The front's speed in excess of a flat flame's against its width, on logarithmic axes, at noise strengths of a thousandth and a millionth, with the quiet pole front's and a power fitted from 20 to 100. At a thousandth the speed climbs from 1.05 at 12.5 to 2.71 at 100; at a millionth from 0.668 to 1.71. Both follow the 0.42 and 0.43 power of the width to within 4 per cent, and neither bends over; the quiet front stays at a half.

A noisy flame is still speeding up a hundred wavelengths wide

A quiet wrinkled flame gains a fixed, bounded amount of speed over a flat one, however wide it is. A noisy flame keeps getting faster as it widens, and the open question was whether that ever stops. Out to a hundred neutral wavelengths it does not. The speed grows as about the 0.43 power of the width at two noise strengths a thousandfold apart. The noise sets only the prefactor, and the front's slope grows with the speed until it leaves the range the equation was derived for — which is where the growth stops being the equation's to decide.

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