Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Theme

The thread: Everything happens in a thin layer — page 10

Page 10 of 14, continuing through the 125 essays this motif runs through.

125 essays carry this thread — page 10 of 14.

What a downstream blade sees going past. The axial velocity a blade in the second row meets, over one revolution, as it passes through the wakes of thirty upstream blades. Each dip is one wake, and the blade meets all thirty of them every time it goes round. Circulation and lift

A row that meets the row before it

A compressor blade is loaded and unloaded thirty times a revolution by geometry it does not have. The forcing sits at the blade count of the row in front of it and at multiples of that, and how far up the harmonics it reaches is decided by how far apart the two rows are.

 What the rays say the edge is: nearly as loud, and then nothing. The overpressure across the carpet relative to the value under the track, from ray-tube spreading alone, at three Mach numbers from 15 km. It falls gently and then stops: at Mach 1.8 it is 0.92 ten kilometres out, 0.77 at twenty-three and 0.65 on the last ray, 36 km out, with silence beyond. Geometrical acoustics says the carpet ends at a cliff, and the boom at the edge of a real carpet fades as a rumble. The cliff is what the model says; it is also where the model stops. Compressible flow

The edge is a rumble, not a quieter bang

The rays that reach the outer half of a sonic-boom carpet arrive nearly horizontally, having travelled almost three times as far as the one under the track. Ray theory says they still carry two-thirds of the overpressure, right up to a line beyond which there is nothing. Neither half of that is what is heard — which is the useful result, because it says the edge's loudness is not a ray quantity at all.

 A boom gathers most of its age in the thin air near the aeroplane. The share of the total age gathered above each height, for the ray under the track and the last ray computed near the carpet's edge, with the share of the path length travelled above each height for comparison. Under the track 66 per cent of the age is gathered above the tropopause in 26 per cent of the path. The edge ray spends most of its path in the lowest few kilometres and gathers only 11 per cent of its age below 3 km, because the same pressure distorts dense air far more slowly than thin air. Compressible flow

A boom is aged in the thin air it starts in

The rays that reach the edge of a sonic-boom carpet travel two and a half times as far as the one under the track, and it is natural to expect their signatures to have aged accordingly. They have not. A pressure wave distorts thin air far faster than dense air, so two-thirds of a boom's ageing is done in the stratosphere near the aeroplane, and the extra kilometres near the ground add little — which decides how far out a boom shaped to be quiet stays quiet.

 Where a film stops being a damper and starts being a spring. The stiffness and the damping of a circular gas film against its squeeze number, both in units of ambient pressure times disc area. At the left the stiffness is nothing and the damping is Stefan's incompressible law; at the right the damping has gone and the stiffness has reached π, which is an isothermal gas trapped in the gap with no way out. They cross at σ = 6.23, and neither limit was put in by hand — both fall out of a Bessel function of a complex argument. Viscosity

A damper that turns into a spring

A film of oil squeezed between two plates resists motion and stores nothing, and that is a property of the oil rather than of the film. Fill the same gap with air and the same equation gives a film that stores and resists nothing — above a squeeze number of six, with nothing changed but the frequency and the gap.

 Below a Stokes number of ten a ball does not come back. The restitution of an impact through a liquid film, as a fraction of the same impact's dry restitution, against Stokes number. Nothing rebounds below the threshold and the recovery above it is a hyperbola: the restitution, as a fraction of its dry value, is one minus the critical Stokes number over the Stokes number, with that critical value 10.18 computed from the film and the dry restitution alone. The measured curve, drawn beside it, is the same expression with ten in place of that number — and ten is what four decades of viscosity and four materials all give. Viscosity

A ball that bounces in water and not in oil

A squeeze film cannot be closed with any finite energy, so nothing should ever touch anything. A sphere dropped into a tank nevertheless rebounds, and whether it does is decided by a number near ten that four materials and four decades of viscosity all agree on.

 Where the inverse cube stops being an inverse cube. The force resisting a sphere's approach to a plane, against the gap, for four slip lengths. With no slip the force goes as one over the gap and has no limit. Any slip at all bends the curve over: below the slip length the force grows only as a logarithm. The curves separate where the gap falls through the slip length, which is exactly where a measurement has to be made and is why the apparatus has to close to nanometres rather than microns. What is taught wrongly

A length you can only measure by squeezing

No slip is a boundary condition rather than a law, and nothing in the equations requires it. The evidence that it is right for an ordinary liquid to within a nanometre comes from the one experiment whose small length is chosen rather than given — a squeeze film closed to tens of nanometres.

 The current that runs across the river. The transverse velocity through the depth in a bend, beside the downstream velocity that causes it. The surface water is thrown outward and the bed water is driven inward, the two cross at about half the depth, and the whole profile carries no net discharge — which it cannot, because the cross-section has to pass what it is given. What it does carry is the bed load, inward. Fluids at work

The current that runs across the river

The standard channel calculation is one-dimensional, and a river is not. Put a bend in it and the surface tilts by a few centimetres, which is invisible — and a helix forms, a few per cent of the main flow, which decides where the river will be in a hundred years.

 The speed past the flank, in the cell and in the ideal flow it is meant to show. The gap-averaged speed along the top of a cylinder, against the distance from its surface in cylinder radii, for three cells — the cylinder's radius 10, 35 and 100 times the Brinkman length h/√12 — and for ideal flow. Ideal flow slides along the wall at twice the stream speed. The cell's flow must stop at the wall and does so in a layer a few Brinkman lengths thick, outside which it rejoins the ideal curve. Ideal flow

The cell draws a larger cylinder

A Hele-Shaw cell draws the streamlines of ideal flow past an obstacle and cannot obey the one rule ideal flow breaks: the fluid must stop at the obstacle's wall. It does stop there, in a layer a third of the gap thick, and far away the whole correction amounts to one thing — the cell is drawing ideal flow past a cylinder one layer-thickness too big, and the obstacle has exactly that cylinder's drag.

 A film just past its fold lingers where the fold was. The temperature excess at the middle of a sheared film, scaled by the temperature over which the viscosity falls by e, against time in thermal diffusion times, switched on from rest at five stresses either side of the fold. Below the fold the film settles. Just above it the film climbs to the fold's own centre temperature, 1.19, lingers there as if it had found a steady state, and only then runs away — the longer the closer it is to the fold. Viscosity

A lost steady state still holds the film

Push a self-heating oil film one per cent past the stress at which it can no longer settle and it does not run away at once. It warms to the temperature the vanished steady state would have had, sits there as if nothing were wrong, and only then goes — after a delay that grows as one over the square root of the overload, and during nearly all of which the load could still be taken back.

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