Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: Everything happens in a thin layer — page 8

Page 8 of 10, continuing through the 84 essays this motif runs through.

84 essays carry this thread — page 8 of 10.

A particle released in still fluid, with the history and without. The velocity of a hundred-micron particle released at one metre a second in air, computed with the unsteady Stokes force and with the quasi-steady drag alone. After five particle time constants the one that remembers is still moving nearly four times as fast. Viscosity

The drag that integrates a whole history

A particle released in still air does not stop exponentially. The unsteady drag on it carries a term that is an integral over everything the particle has already done, weighted by the inverse square root of how long ago — so there is no time constant, and five particle time constants later it is still moving four times faster than the quasi-steady answer allows.

 In clean water a bubble rises nearly three times as fast as the same bubble in tap water. The terminal rise speed of an air bubble in water at 20 °C against its radius, from buoyancy balanced against drag: with a clean, shear-free surface using Moore's law, and with a surface immobilised by contamination using the rigid-sphere correlation. At 0.3 mm the clean bubble rises at 13.0 cm/s against 6.7; at 0.5 mm at 31.0 against 11.2, a factor of 2.76. Beyond a radius of 0.47 mm the clean bubble's Weber number passes one, its shape flattens, and a spherical calculation stops describing it; that region is shaded. Nothing about the bubble's size, gas or liquid changes between the two curves — only whether its surface can move. Flows and fields

The vorticity a clean surface cannot refuse

A clean bubble's surface cannot hold a shear stress, and it is easy to conclude that it makes no vorticity. On a curved surface it must carry exactly 2κu — three times the speed over the radius at a sphere's equator, whatever the Reynolds number. That is so much weaker than a rigid wall's that the flow stays irrotational to leading order, and the bubble's drag is the dissipation of that irrotational flow: 48/Re, three to ten times below a rigid sphere's.

 What the fluid at one height is listening to. The weight the fluid two millimetres above a moving wall gives to the wall's velocity a given delay earlier, in water. It peaks at two thirds of a second and has a tail that falls as the delay to the power minus three halves — so the fluid is responding to a broad stretch of the wall's past rather than to a moment of it. Viscosity

The wall the fluid is listening to

Water two millimetres above a moving wall is responding to what the wall did two thirds of a second ago — most likely. Half of its response is older than four and a half seconds, a tenth is older than two minutes, and the average age of what it is responding to does not exist at all.

 Slip follows the stripes more closely than the shear does. Plan views of a striped surface, with the stripes running across each panel, for a shear at 0°, 30°, 54.7° and 90° to them. The faint arrow is the direction of the shear; the dark one is the slip velocity it produces, whose component along the stripes is the along-stripe slip length times the shear and whose component across them is half that. The slip is turned towards the stripes by 0.0°, 13.9°, 19.5° and 0.0°. It is largest, 19.47°, for a shear at 54.74°, where tan θ = √2. A surface with a tensor for a boundary condition can push a flow sideways, which a scalar slip length never can. Flows and fields

Twice as slippery along as across

A surface of alternating gas and solid stripes lets a liquid slip, and a flow far above it sees one number in place of the pattern — but the number depends on which way the flow goes. Along the stripes it is Philip's logarithm; across them it is exactly half, for a reason that takes one substitution to show. And the logarithm means that the slip is bought by the pattern's period rather than by how much of it is gas.

 The flow stops and the stress does not. Shear stress against time for a fluid sheared at a constant rate for two seconds and then left alone. Nothing is moving after the second mark and the fluid is still stressed: 37 per cent of the peak one relaxation time later, and 0.7 per cent after five. Viscosity

The fluid that has not finished its last deformation

Shear a polymer solution for two seconds and stop. Nothing is moving and the fluid is still stressed — 37 per cent of the peak one relaxation time later, and still measurably stressed after five. Two histories imposing exactly the same total strain leave it in states differing by a factor of 3.7.

 A spun cylinder carries its whole circulation at once, and hides it until the vorticity has left. The circulation round circles of radius r about a cylinder of radius a started spinning at once, as a share of the circulation of its own surface, 2πa²Ω, against r/a on a logarithmic axis, at νt/a² = 0.01, 0.1, 1, 10 and 100. At the surface it is the whole of it from the first instant, because no slip makes the fluid there turn with the cylinder. Just outside, the spin-up has laid down an equal and opposite ring of vorticity, so the circulation round a larger circle is only what has diffused past it: at two radii 0.000, 0.035, 0.611, 0.936 and 0.993 of the surface's at the same five times. The circulation a Magnus rotor needs is in the fluid the moment it spins; the far field learns of it only as fast as the counter-vorticity moves out. Flows and fields

A wall puts in exactly its own speed

A wall sliding in its own plane makes vorticity at a rate equal to its acceleration, with no viscosity in the rate. So however a wall is started, the vorticity it has put into the fluid is its speed, to the last digit; a wall that stops takes all of it back and leaves the fluid moving; and a spinning cylinder carries its whole circulation from the first instant, hidden behind an equal and opposite ring until viscosity carries the ring away.

 Three modes carry heat up to a ceiling of three; the rolls they were cut from keep going. The Nusselt number — heat carried across the layer over what conduction alone would carry — against r, the Rayleigh number over its critical value, for Lorenz's three-mode truncation, 1 + 2(r − 1)/r, and for steady rolls at the same wavenumber computed with 6 and with 42 temperature modes. At r = 2, r = 5 and r = 30 Lorenz gives 2.000, 2.600 and 2.933; the 42-mode rolls give 2.143, 3.323 and 5.970. Lorenz's value can never exceed three whatever the Rayleigh number; the rolls' keeps rising. The ceiling is not in the convection. It is in the three modes, which have only one way to thin the thermal layers, and half of it is used by the time the layer is twice past onset. Transition and turbulence

The truncation that cannot carry three times the heat

Lorenz's three modes give a convecting layer's heat flux in one line — one plus twice (r − 1) over r — and it can never reach three times what conduction carries. The same rolls computed with forty-two modes agree with that line exactly at onset, carry 7.1 per cent more heat at twice the critical Rayleigh number, and twice as much at thirty times it. The ceiling is not in the convection; it is in having one sine to draw the temperature with.

 Two external flows that agree where it matters and nowhere else. The velocity just outside the boundary layer, for two pressure distributions, against distance along the surface. They cross at the half-way station with the same value and the same gradient, and they have nothing else in common: one accelerates steadily and the other does most of its accelerating at once. Viscosity

A layer that is an integral of everything upstream

Two surfaces are given external velocity distributions that agree exactly at one station — the same speed and the same gradient. The boundary layers there differ by 38 per cent in momentum thickness, and the two surfaces separate five per cent of their length apart.

 The gas that does not stop at the wall. Channel flow profiles with and without slip, at a Knudsen number of a twentieth. The slipping profile does not reach zero at the wall: the gas there is moving, by an amount proportional to the mean free path times the velocity gradient. Regimes and numbers

Slip is a memory of one mean free path

A molecule arriving at a wall last collided about a mean free path away and carries the velocity from there. Averaged over arrivals and departures, that leaves the gas at the wall moving — by two per cent of the centreline speed at a Knudsen number of a hundredth, and sixty per cent more flow through a microchannel at a tenth.

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