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The thread: What is conserved — page 23

Page 23 of 33, continuing through the 292 essays this motif runs through.

292 essays carry this thread — page 23 of 33.

A threshold that is a curve, not a pressure. The tension a rectangular pulse has to reach to make a five-micron bubble run away, against how long the pulse lasts. At a tenth of a microsecond it takes forty bar; at a hundred microseconds it takes 1.05, which is within about one per cent of the static threshold. There is no such thing as the cavitation pressure of this bubble on its own.

A threshold that is also a duration

The cavitation number treats inception as a pressure: below it the liquid tears, above it does not. A five-micron bubble asked to grow in 0.3 microseconds needs 43 bar of tension and the same bubble given a hundred needs 1.05, because it has to make a journey and not merely respond.

A degree of temperature is worth 0.34 per cent of pressure. The error in the pressure a paint reports, against how much warmer the surface is at the test condition than at the reference, at four pressures. At 0.8 of the reference pressure the sensitivity is -0.34 per cent of pressure per kelvin, so 10 degrees is -3.45 per cent. That is not a small number against what the technique is used to measure, and a model's surface temperature is not uniform: it is warmer where the flow has been brought to rest and cooler where it has accelerated, which means the temperature error is largest exactly where the pressure gradients are. The standard remedy is a second, temperature-sensitive paint measured at the same time — an instrument added to correct an instrument.

What is taught wrongly

The paint that measures the wrong field

Dye, seeded particles and the optical methods all look through the flow or at something put in it. Pressure-sensitive paint looks at the surface, reports a scalar rather than a derivative, and turns a row of taps into a field. What quenches its luminescence is oxygen, which is what makes it a pressure gauge — and temperature, which is the other field a flow is guaranteed to produce.

The convergence exponent depends on the gas, which a dimensional exponent cannot. R ∝ (−t)^α for a converging shock, against the ratio of specific heats, for cylindrical and spherical symmetry. Guderley's exact values are marked and the agreement is to four figures. The Sedov blast's two-fifths is drawn beside them: it is the same for every gas, because it comes from dimensions and a conserved energy, and γ is dimensionless.

Regimes and numbers

An exponent dimensions cannot give

A blast wave's radius goes as the two-fifths power of time, and the two-fifths is arithmetic: count the dimensions and it falls out. A shock converging on a point goes as the 0.717 power, and no amount of counting will produce that number — because it depends on the gas, and γ is dimensionless.

How the weight is shared between the two surfaces, against the centre of gravity. The wing's load and the tail's, for an aeroplane in level flight, against the static margin. The two must sum to the weight and their moments must cancel, and those two equations decide the split. At a forward centre of gravity the tail carries down and the wing carries more than the weight; the crossing is where the tail carries nothing.

Circulation and lift

More lift than weight

An aeroplane in level flight is drawn with one arrow up and one arrow down, equal and opposite. That equation collapses the whole configuration onto one number, and it is not true of any aeroplane with a tail behind it: there are two surfaces, two equations, and the second decides the split.

The two long-wave speeds, and the swirl at which one of them stops. For uniform axial velocity the wave speeds follow from the criticality condition by a Galilean boost: c = W(1 ± 2S/j), with j the first zero of J1. The upstream-running root crosses zero exactly at S = j/2 = 1.9159, and above that swirl no disturbance can travel upstream — which is what subcritical and supercritical mean here and in an open channel.

Ideal flow

The swirl that holds a wave still

A swirling flow down a pipe carries waves, and above a certain swirl one of them stops moving. Below it, a disturbance downstream can send information upstream; above it, the flow has outrun its own waves. The words are open-channel flow's words, and they are the same words for the same reason.

The scalar spectrum, with its two ranges. A model scalar spectrum at a Schmidt number of two thousand — dye in water. Below the Kolmogorov wavenumber it is Obukhov and Corrsin's five-thirds, inherited from the velocity; above it there is no turbulence left and the spectrum is Batchelor's minus one, which contains no velocity spectrum at all.

Transition and turbulence

The scalar has its own cascade

Below the Kolmogorov scale there is no turbulence left, and a dye stirred into the flow goes on cascading anyway — on a spectrum whose exponent is minus one and whose amplitude contains no velocity spectrum at all. Resolving it costs the three-halves power of the Schmidt number, which for dye in water is a factor of ninety thousand.

Flows and fields

What a mean profile cannot tell anybody

Two flows are built here with mean velocity profiles that agree to four parts in 10¹⁷. One of them carries momentum across the shear and dissipates forty per cent more energy; the other carries nothing. Everything that distinguishes them is second order in a disturbance the mean cannot see at all.

The four waves, and the one that never goes away. The shock tube in space and time: a shock running right, an expansion fan running left, and the contact surface between them. The shock and the fan are travelling disturbances that leave; the contact is made of fluid, so it is carried along and is there for ever.

Compressible flow

A surface that remembers the diaphragm

Between the shock and the expansion in a shock tube there is a surface across which the pressure and the velocity are identical and the temperature differs by a factor of two. It is made of fluid, so it never goes away, and nothing in the pressure field says it is there.

$Spin against distance flown, not against time. The spin of a struck golf ball as a fraction of its launch spin, against how far it has flown. The integrated flight sits exactly on an exponential in distance, because the spin-down torque is proportional to speed times spin and the speed cancels. The constant is 1,202 metres and the drive is 200, so the ball arrives with five sixths of the spin it left with.$ Fluids at work

The ball that never forgets its spin

Commentary explains a late-swerving ball by saying the spin is dying away. The spin-down torque goes as speed times spin, so spin decays over a distance rather than a time — 1,202 metres for a golf ball against a 200-metre drive — and what dies away is the speed, which makes the swerve stronger.

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