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The thread: What is conserved — page 27

Page 27 of 33, continuing through the 292 essays this motif runs through.

292 essays carry this thread — page 27 of 33.

The stress a closure predicts, and the stress there is. The Reynolds-stress anisotropy through a step change in the strain rate, against what an eddy viscosity gives — which is the equilibrium value at every instant. The real stress takes about a turnover to get there, and during that turnover the closure is wrong by up to sixty per cent.

A closure with no memory at all

An eddy viscosity says the Reynolds stress is the mean strain rate times a number, now. The stress it is standing in for takes a turnover to arrive, so the closure is the zero-frequency limit of a response that has a lag in it — and the curve it is the limit of is the same shape as an aerofoil's lift deficiency.

The same plate borrows more near a wall and less near a free surface. The added mass of a plate closing broadside on a boundary, as a multiple of its free-air value, against the gap in chords on a logarithmic axis, for a solid wall and for a boundary held at constant pressure — a free surface struck quickly, or the edge of an open jet. At a tenth of a chord the wall gives 1.966 and the free boundary 0.677; at 0.035 chords 3.97 and 0.584. The wall's value grows without limit as the gap closes, because the fluid in the gap has to be squeezed out. The free boundary's falls towards exactly one half, because a plate lying on a free surface sets in motion only the half-space below it. Same plate, same fluid, same speed — the boundary decides the sign, through the one thing it is allowed to tell the flow.

What is taught wrongly

The borrowed mass the boundary decides

A body accelerating near a solid wall has to squeeze out the fluid between them, and borrows more mass than it would in the open. The same body accelerating near a free surface, or inside an open-jet wind tunnel, borrows less. The fluid, the body and the speed are identical, and what reverses the answer is the one thing each boundary is allowed to tell the flow.

A skin lags the air by a time its own thickness sets. A skin held at one flight condition, relaxing towards the adiabatic wall temperature. The approach is exponential with a time constant ρc τ / h — 8.37 s at 2 mm, 25.1 s at 6 mm, 50.2 s at 12 mm — so the temperature a steady recovery calculation gives is reached after several minutes rather than at once. The fluid supplies one number to this calculation, the adiabatic wall temperature, and the structure supplies everything else.

Compressible flow

The skin that lags the flight

A wall can be told its temperature or told nothing, and both are solved problems. A real skin is told neither. It has heat capacity, so its temperature is a transient whose time constant is its own thickness divided by what the layer delivers — and the number the steady calculation returns is an upper bound a short exposure never collects.

A 204 m hammer traded for a 8.57 m swing over 299 seconds. The water level in a 10 m surge tank at the end of a 2 km tunnel 3 m across, carrying 2 m/s, after the turbine is shut off at once, with the level measured from the reservoir's. Without friction it rises to V₀√(L Aₜ/g Aₛ) = 8.57 m and swings with a period 2π√(L Aₛ/g Aₜ) = 299.1 s, the integration agreeing with both closed forms. With the tunnel's 5 m of friction the level starts 5 m below the reservoir, peaks at 5.61 m after 98 s, falls to −3.70 m, and decays. The same tunnel shut at its end with no tank would take the Joukowsky rise of 204 m. The tank does not remove the column's momentum; it gives it a free surface to push against, slowly.

Fluids at work

A tank that turns a hammer into a swing

Shut a turbine at the end of a two-kilometre tunnel in two seconds and the valve takes a rise of 256 metres of head. Put a shaft open to the air beside it and the rise is 51, the tunnel never carries the closure as a wave at all, and its water slows instead against a level that climbs for a minute and a half — to a height that is a closed form with the tank's area under a square root.

The excess energy is the energy of the difference, exactly. Add any admissible perturbation to the potential flow and its kinetic energy rises by precisely the energy of the perturbation itself — not approximately, and not to leading order. The measured excess and the perturbation's own energy lie on one another to two parts in 10¹¹ across a sixty-fold range of amplitude.

Ideal flow

How much more than the least

Of all the flows that conserve mass and stay inside the walls, the ideal one carries the least energy. That is a theorem, and the useful half of it is the part nobody quotes: it says by exactly how much every other flow misses, and the answer is the square of how wrong it looks.

How flat the optimum is. The same curve near its maximum, with the bands within a tenth and a half of one per cent of the best shaded. Every taper ratio from 0.31 to 0.42 is within a tenth of a per cent of the optimum, and the whole band from 0.25 to 0.51 is within half a per cent. The optimum is exact, and choosing it rather than its neighbour buys nothing a wing can measure.

Circulation and lift

The optimum that does not matter

Elliptic loading gives the least induced drag, exactly, and the proof is one line: the penalty is a sum of squares. Which is also why the optimum is flat enough that a quarter of the design space sits inside a tenth of a per cent of it.

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.

How far apart two points can be and still be correlated. The correlation between the logarithm of the dissipation at two points, against how far apart they are in units of the smallest scale, over four decades. It falls as a ratio of logarithms — so it is still a quarter at a thousand smallest scales, and reaches a half only at a hundred.

Transition and turbulence

A dissipation correlated across every scale

The dissipation is supposed to be the most local quantity in turbulence — a thing happening at the smallest eddies, everywhere and independently. A multiplicative cascade makes its logarithm correlated over a distance that is a ratio of logarithms, so two points a thousand smallest scales apart still agree a quarter of the time.

Two forces on one cylinder, a quarter of a cycle apart. The inertia and drag terms of Morison's equation over one wave period, at a Keulegan-Carpenter number of ten. The inertia term follows the acceleration and peaks where the velocity is zero; the drag term follows the velocity and peaks where the acceleration is. They are a quarter of a cycle apart and they are different kinds of quantity.

Regimes and numbers

Two forces, and only one of them remembers

Morison's equation adds an inertia term to a drag term and is usually presented as an empirical patch. It is not: the two terms are the two kinds of memory this collection has been separating, one a function of the present acceleration and one a function of the wake left by the previous half cycle.

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