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The thread: The exact theory is wrong — page 23

Page 23 of 23, continuing through the 206 essays this motif runs through.

206 essays carry this thread — page 23 of 23.

Three speeds, and the Froude number at which two of them cross. The kinematic wave speed and the two dynamic wave speeds, all divided by the speed of a shallow-water wave on still water, against the Froude number. The kinematic wave is three halves of the water speed and the downstream dynamic wave is the water speed plus one, so they cross at Froude two — and beyond that crossing the news of a change in flux arrives before the news of a change in depth, which uniform flow cannot survive.

When the flux outruns the pressure

A kinematic wave is derived by throwing the momentum equation away, and the derivation never says when that is allowed. It is allowed until the kinematic wave, which travels faster than the water, overtakes the downstream dynamic wave as well — at which point uniform flow ceases to exist and a concrete chute carries a train of surges instead of a sheet.

A count of states that has a maximum in it. How many configurations of thirty point vortices have each energy, sampled from the measure their own Hamiltonian defines — which is the area measure, because a vortex's coordinates are its own conjugate pair. The count peaks at an energy of -0.054 and falls away on both sides, which no ordinary system's does. Above the peak, adding energy reduces the number of ways of arranging the fluid.

Transition and turbulence

An equilibrium a three-dimensional flow cannot have

The condensate an inverse cascade ends in is treated as what is left over when the energy has nowhere further to go. It is not a remainder. A two-dimensional fluid's phase space is its own region and therefore has finite volume, so its entropy has a maximum, its temperature changes sign, and the clustered state above that point is an equilibrium.

$The depth at the edge, against how much pressure is left there. The brink depth as a fraction of the critical depth, against the share of the hydrostatic pressure the brink is taken to retain. A free jet — no pressure at all — gives exactly two thirds, to 1e-16. The measured ratio of 0.715 is met at 40 per cent of hydrostatic, which is more residual pressure than a falling nappe looks as though it should have.$ Fluids at work

The depth at the edge is not the critical one

Every open-channel calculation assumes the pressure is hydrostatic, and usually says so nowhere. At a free overfall it is not, the assumption's failure can be measured, and the measurement is one number — a brink depth that is 0.715 of the critical depth where a hydrostatic reading would make it one.

Three aircraft, and the only thing the drag can see. The cross-sectional area against station, for a clean body, the same body with a wing added, and the body cut away where the wing is. The wave drag depends on this curve and on nothing else about the three shapes — so the middle one pays for its lump and the third does not, although the third has the same wing.

Ideal flow

The drag that can only see one curve

A slender body's far field is a line of sources of strength A′(x), and nothing else about its shape reaches it. So its supersonic wave drag depends on the cross-sectional area distribution alone — and a fuselage cut away where the wing joins it can carry the wing for nothing.

Three dies, one melt, three flow curves. The apparent flow curve a capillary rheometer reports, for three dies whose radii differ by a factor of four. A material property cannot depend on the instrument, so the fact that these three do not lie on one another is a measurement of something other than a viscosity. The single curve drawn beneath them is what the same melt gives with the wall condition set to no slip.

What is taught wrongly

The wall a melt really does slip on

For a simple liquid the slip length is under two nanometres, which is why no-slip is safe everywhere. For an entangled polymer melt it is a millimetre, which is larger than the die it is being pushed through — so the melt moves as a plug and the viscosity a rheometer reports is a property of the instrument.

The only term that turns spin into thrust, and what it is made of. The coupling term of the propulsion matrix against the drag anisotropy, with the geometry held fixed. It is exactly proportional to the difference of the two drag coefficients, so it is zero when they are equal — not small, zero — and no helix of any pitch turned at any rate would move. A real filament sits at 1.66, which is a third of the way from useless to the unreachable limit.

Viscosity

Two drags, or nothing swims

A bacterium turns a corkscrew and goes forward, and the reason is not the corkscrew. It is that a thin filament dragged broadside resists more than the same filament dragged end-on. Make the two resistances equal and the thrust is not small but exactly zero, for every pitch and every rate.

A frequency set almost entirely by how nearly sonic the flow behind the shock is. The Strouhal number of the loop against the local Mach number behind the shock. The band drawn across it is what is measured on aerofoils in buffet, 0.06 to 0.08, and the arithmetic meets it at a post-shock Mach number of 0.971. Two per cent either side of that number changes the frequency by a third, which is why buffet onset is predicted by correlation rather than by calculation.

Compressible flow

A shock that will not stand still

A converged transonic solution puts the shock in a definite place. Over a band of conditions the real one sweeps back and forth across a fifth of the chord at a fixed frequency, and the period is set almost entirely by how nearly sonic the flow just behind it is — a quantity that is small, hard to compute, and multiplies everything.

$The vapour fraction that would stop a cavity, against temperature. How much vapour a cavity would have to make, as a multiple of the liquid volume it cools, before the cooling had used up the whole driving tension. In cold water it is six hundred and the effect is unreachable; at the boiling point it is below one and the effect is the dominant thing in the problem. Four decades across the range a pump might see, out of a vapour density that rises by a factor of two hundred.$ Fluids at work

A cavity that cools the water it came from

The usual account takes the vapour pressure as a property of the liquid, looked up once. It is a property of the liquid at whatever temperature the cavity has left it — and making vapour costs latent heat, so a cavity suppresses itself. In cold water that is unmeasurable and in liquid hydrogen it is the dominant term.

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