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The thread: One number decides the regime — page 32

Page 32 of 34, continuing through the 305 essays this motif runs through.

305 essays carry this thread — page 32 of 34.

Three profiles that do not depend on the radius. The radial, azimuthal and axial velocities of the flow above a rotating disc, as functions of one similarity variable. The radial one is a jet: fluid thrown outward by the swirl it has picked up, peaking at 0.181 of the local disc speed a fifth of the way through the layer. The azimuthal one falls from the disc's own speed to nothing. And the axial one is the surprise — it does not vanish far from the disc but tends to a constant, so the disc draws fluid down onto itself at 0.8845 times the square root of the viscosity times the rotation rate, at every radius and for ever.

The solution that keeps its nonlinear term

Every exact solution before this one has been exact because the nonlinear term vanished. A rotating disc's does not vanish — at the wall it is the whole of the balance — and the reduction is exact anyway, because the radius divides out of all three momentum equations at once.

Six ways of reaching one speed. Six velocity histories, all starting from rest and all reaching exactly one at the same moment. Two are ramps, two are eased, one overshoots and comes back, and one goes backwards before it goes forwards.

Ideal flow

Everything about the start, except one vector

Six ways of accelerating a body from rest to the same speed produce six force histories with nothing in common — peaks spanning a factor of thirty-nine, two of them negative for part of the journey. The impulse left in the fluid is the same ten-figure number in every case, and so is the energy.

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.

Four over the dimension, and three dimensions is where the three comes from. The constant in front of the exact law, against the number of dimensions the flow lives in. Both exact results — Yaglom's for a scalar and the velocity's law for the mixed third moment — have this same constant, because both come from the same statement: an isotropic radial flux in separation space whose divergence is a constant sink. Integrating that divergence gives 4Q r/d and nothing else. The four-thirds everybody quotes is four over three, and the three is the space rather than anything about turbulence. The dots are the quadrature, which agrees with the closed form to 2·10⁻⁹.

Transition and turbulence

The fraction that is really four thirds

Turbulence has two exact results, not one. The second is about a scalar carried by the flow, its constant is four thirds rather than four fifths, and the difference between the two fractions has nothing in it about turbulence at all — it is the price of writing a three-component object in terms of one component.

Viscosity

One channel, one flux, two flows

Flow between two plane walls meeting at a line has an exact solution. Past a threshold that turns out to be a ratio of gamma functions, it has two — the same wedge carrying the same flux, once outward everywhere and once with the fluid running backwards along both walls, and nothing in the equations chooses.

The arrival map, and the place where it goes backwards. Where each boom ray lands on the ground, against the Mach number the aeroplane was doing when it launched it, for four accelerations from 15 km. In level flight this would be a straight line rising at the aeroplane's own speed. Here it falls before it rises: a ray launched at a higher Mach number is shorter and steeper, and near the cut-off it shortens faster than the aeroplane advances. Every minimum in these curves is a fold — two emission times delivering to one place — and on the fold itself neighbouring rays converge onto a single line.

Compressible flow

The carpet an accelerating aeroplane folds

Level flight launches every boom ray with the same invariant, so they run parallel and each place hears one boom. Accelerate, and each successive ray is shorter than the last — shorter, near the cut-off, than the aeroplane's own advance — so later rays overtake earlier ones and the arrival map folds onto a line.

Rolls turning side by side, with the fastest downwind water where they sink. The fastest-growing mode at a Langmuir number of 0.13, looking downwind, over two roll spacings of 2.89 decay depths and 4 decay depths down. The closed curves are streamlines of the overturning; the dashed curves are contours of the downwind velocity the rolls carry, positive under the lines where the water sinks. At the surface the cross-wind flow converges onto those lines, which is where floating foam and weed collect as windrows. The amplitude is arbitrary, as in any linear mode.

Flows and fields

The drift that turns a current into rolls

A current carrying a Stokes drift feels a force the drift makes out of the current's own vorticity, and under a wind that force is unstable. It turns the surface layer into rolls lined up downwind, with windrows where they sink. The rolls need both the current's shear and the drift's; their growth rate sees only the product; and the split between the two decides which motion gets the energy.

The throat holds the hammer back only while its cavity lasts. Left, pressure at the closing valve (red) and at the upstream face of the venturi (gold); right, the throat's cavity volume; after the valve shuts with a 5 mL cavity in the throat. The valve sees the full Joukowsky rise of 14.8 bar at once. The wave reaches the venturi 33.3 ms later, and for the next 7.9 ms the upstream pipe hears nothing: its pressure stays at 5 bar while the cavity is squeezed. When the cavity closes at 41.3 ms the surge passes into the upstream pipe at 13.8 bar above its steady pressure.

What is taught wrongly

A choked throat buys time, not silence

A venturi whose throat has reached vapour pressure passes a flow the downstream pressure cannot change, and it is tempting to read that as isolation: whatever happens downstream, the upstream pipe will not hear it. Slam a valve downstream and it hears it. The cavity at the throat holds the surge back only for as long as it takes to fill, and then lets 93 per cent of it through.

$A wave that travels and a wave that spreads. A harmonic pressure wave's amplitude and its instantaneous value along a tube, over two wavelengths of the inviscid wave, at four Womersley numbers. At α = 15 the wave marches on, a little weaker each wavelength. At α = 5 it is visibly damped. At α = 2 it is nearly gone within a wavelength. At α = 0.5 there is no wave to speak of: the disturbance falls away within a small fraction of the inviscid wavelength, as heat does into a wall.$ Regimes and numbers

The pulse that has to travel

In a rigid tube the Womersley number decides the shape of an oscillating flow. Make the wall elastic and the pressure pulse has to travel, a second number appears — the tube's length in wavelengths — and the first number turns out to decide something more basic than the profile: whether the tube carries a wave at all, or only a disturbance that spreads like heat.

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