A gas that has not finished being shocked

When gamma stops being a number is this collection’s account of what happens to the jump conditions when the gas’s heat capacity varies: the ratio of specific heats is a count of the modes carrying energy, and at high temperature there are more of them.

That essay computes the equilibrium answer. This one is about the fact that equilibrium takes time to arrive, and about the region behind a shock in which the gas is not the gas the jump conditions describe.

Two thicknesses, a thousand apart #

A shock is thin. The discontinuity that has a thickness computes how thin: a few mean free paths, which is a fraction of a micron at ordinary densities.

Vibrational relaxation is slow. A molecule needs thousands of collisions before its vibrational mode comes into equilibrium with the translational one, because the energy quantum is large and the coupling is weak.

So the gas passes through the shock in a few collisions and leaves it with its vibrational modes exactly as they were: cold, in their pre-shock population. At Mach six on 300 K air that means leaving the shock at 2,382 K and arriving at 2,059 a third of a millimetre later, having travelled at 395 m/s while the vibrational modes filled. All of the shock’s energy has gone into translation, which is what a thermometer would read, and the temperature is therefore higher than the equilibrium calculation gives.

The two temperatures #

At Mach 6 in air at 300 K, the frozen answer is 2,382 K and the equilibrium answer is 2,059 — 13.6 per cent apart. The gas leaves the shock at the first and arrives at the second about four tenths of a millimetre later.

The energy book-keeping is exact and worth following. The vibrational energy per unit mass rises from its pre-shock value towards the value appropriate to the local temperature, and every joule that goes into vibration comes out of translation. The temperature falls as the vibrational levels fill up.

That is a relaxation memory of exactly the kind the fluid that has not finished its last deformation computes for a polymer: one internal state variable, an exponential kernel, and a definite time constant past which the past is gone.

How the gap grows #

At Mach 2 the two answers differ by three tenths of a per cent and the distinction is not worth making. At Mach 10 they differ by 22 per cent, and 1,300 kelvin.

The reason is that vibrational modes are frozen out at low temperature and excited at high: the population at 500 K is negligible and at 4,000 K it is a substantial fraction of the internal energy. So the amount of energy the relaxation can move grows faster than the temperature does, and the effect is a hypersonic one rather than a supersonic one.

And how thick the region is #

The relaxation length is the flow speed behind the shock times the relaxation time — a millimetre in these conditions, growing with Mach number because a stronger shock leaves faster gas.

Seven Mach numbers, one gas:

The drop grows by a factor of 67 across the sweep while the length grows by only 2.3 — so the zone stays about a third of a millimetre wherever it is and the error in ignoring it does not. A third of a millimetre is a strange length to find in this problem. It is thousands of times the shock’s own thickness, so on the scale of the shock the relaxation zone is enormous; and it is a small fraction of any body, so on the scale of a vehicle it is thin. It sits exactly between the two scales the problem otherwise has, which is why it is easy to overlook in a calculation that has only those two.

What the solver computed, and how it was checked #

The jump conditions are solved with the vibrational mode frozen, giving the state the gas leaves the shock in. The mode is then relaxed by the Landau-Teller equation along the flow, with the temperature falling as energy leaves translation.

Three checks. That the temperature falls by at least five per cent, so there is an effect. That the fall is essentially complete within the computed relaxation length, with the measured one-over-e distance between a fifth and five times the velocity-times-relaxation-time estimate: at Mach six it reads 0.298 mm against an estimate of 0.395, a ratio of 0.754. And that the gap grows monotonically with the Mach number, which would catch a sign error in the energy book-keeping.

One failure is recorded because it was immediate and total. The relaxation time is a microsecond and the flow crosses the region in a few of them, so a step chosen to draw a picture is far longer than the relaxation time — which makes the explicit difference stiff. The first version produced a temperature of minus 3,060 K on its second step and not-a-number by its third. The relaxation is now integrated exactly over each step, which is possible because it is linear, and the step size is a drawing decision again.

What the jump conditions actually claim #

It is worth restating what the jump conditions do and do not say, because the essay is not a criticism of them.

They are conservation statements across a control volume that straddles the shock: mass, momentum and energy in equals out. They are exact, they assume nothing about what happens inside, and they are the reason the jump does not ask what made it is true.

What they need is a thermodynamic state on each side, and a state requires equilibrium. The downstream side of the control volume therefore has to be drawn far enough back for the gas to have equilibrated — which is far enough back to include the relaxation zone.

So the jump conditions are right about the gas at the far side of the relaxation and silent about everything in between. Applying them at the shock itself is applying an equilibrium relation to a gas that is not in equilibrium, which is not an approximation but a category error — and it happens to give the frozen answer if the frozen gamma is used, which is exactly what is done here.

That is the honest formulation: two sets of jump conditions, one at each end of the relaxation, with a first-order relaxation joining them.

Where this changes an answer #

Heat transfer to a hypersonic vehicle. The gas arriving at a surface behind a bow shock has a temperature that depends on how far it has come and on how fast it is going, because those decide how much of the relaxation has happened. A calculation using the equilibrium temperature underestimates the convective heating in the region near the shock and overestimates it further back.

Optical measurements. Emission and absorption spectroscopy in a shock tube reads the population of particular states, so it reads the relaxation directly — which is how relaxation times are measured, and also why a spectroscopic temperature and a pitot-derived one disagree in the relaxation zone.

And the shock stand-off distance. A shock that lies on the body computes the stand-off from the density ratio, and the density ratio depends on which of the two temperatures the gas is at. A frozen shock layer is thicker than an equilibrium one, and on a small body at high altitude the layer can be frozen throughout.

Which regime a body is in #

The comparison that decides everything is between the relaxation length and the body, and it produces three regimes worth naming.

Equilibrium. The body is much larger than the relaxation length — a metre against 0.395 mm is a ratio of 2,530 — so the gas equilibrates immediately on the scale of the flow and the equilibrium jump conditions apply everywhere except in a thin sheet. That is the case for a large vehicle at low altitude.

Frozen. The body is much smaller than the relaxation length, so the gas passes it before anything happens and the frozen conditions apply throughout — the gas leaves the shock at 2,382 K and the body sees that rather than the equilibrium 2,059, a difference of 323 K in the temperature that drives the heating. That is the case for a small body at high altitude, where the density is low and the relaxation time is correspondingly long.

And non-equilibrium, which is neither. The body and the relaxation length are comparable, the state varies through the shock layer, and there is no single set of jump conditions that applies. That is the case for most real re-entry, and it is why hypersonic computation carries species and mode equations rather than a gamma.

The parameter that separates them is a Damköhler number — a flow time divided by a relaxation time — and it is the same construction as the Deborah number of a solid if it is not given time with a chemical relaxation in place of a mechanical one.

The same shock read as a pair of totals is the last essay in this field, on the same machinery.

Why the relaxation time is the hard number #

Everything above depends on one quantity that is not computed here, and it is worth saying how badly.

The vibrational relaxation time is a strong function of temperature and pressure — it falls roughly exponentially with the cube root of the reciprocal temperature, which is the Millikan-White correlation — and it varies between species by orders of magnitude. Nitrogen relaxes slowly, oxygen faster, and mixtures with water vapour faster still by a large factor.

So the millimetre quoted here is an order of magnitude rather than a number, and the shape of the result is what transfers: a frozen state at the shock, an exponential approach, and a length that is a velocity times a time. Getting the time right for a particular gas is an experimental question, and shock tubes exist largely to answer it.

The same shape, in three neighbouring essays #

The fourth is not a gas at all: the drag that integrates a whole history is the same structure on a particle, where the state that has not finished changing is the fluid around it.

The fourth is not a gas at all: the drag that integrates a whole history is the same structure on a particle, where the state that has not finished changing is the fluid around it.

The structure — a state that is imposed instantly and an internal variable that takes time to follow — is the commonest one in this collection, and three neighbours are worth naming.

A viscoelastic fluid has a stress that relaxes towards its equilibrium value over a relaxation time, which is the fluid that has not finished its last deformation. The mathematics is identical: one internal variable, one exponential, one time constant.

A turbulence stepped in strain has a dissipation that takes a turnover to catch up, which is a dissipation that lags its production.

And a rotor’s inflow takes a fraction of a wake convection time, which is the inflow that takes time to arrive.

What distinguishes this case is that the internal variable is a molecular one, so the relaxation time comes from kinetic theory rather than from a fitted constant — and that the imposed change is a genuine discontinuity rather than a rapid one, so the separation between the two time scales is as clean as it ever gets.

Where else a gas has not caught up #

Vibration is the slowest of several relaxations and the others are worth naming, because they happen in sequence and each has its own length.

Translation and rotation equilibrate in a few collisions, so they are part of the shock rather than behind it — which is why the shock has a thickness at all.

Vibration takes thousands of collisions, and is what this essay computes.

Dissociation takes longer still and is a chemical rather than a modal relaxation, so behind a strong enough shock there is a further zone in which the composition is changing.

And ionisation is slower again.

A strong shock in air therefore has a sequence of zones behind it, each with its own length, each with a different effective gamma, and the ordering is fixed by how many collisions each process needs. That sequence is the reason a hypersonic flow field cannot be described by any single equation of state.

What a measurement in the zone reads #

Since the region is a millimetre thick and the state varies through it, it is worth asking what any particular instrument returns, because the answers differ.

A pitot probe measures a total pressure, which is a mechanical quantity and does not care which modes hold the energy. It reads the same thing at either end of the relaxation, to within the small correction the changing gamma makes.

A thermocouple reads a translational temperature, if it reads anything at all in a millimetre — which it does not, since the probe is far larger than the zone. Any contact measurement here is averaging over the whole relaxation.

Emission spectroscopy reads the population of the states it is looking at, so it reads the vibrational temperature directly and is the only one of the three that can resolve the zone at all.

And an interferometer reads density, which is set by the pressure and the translational temperature — so it sees the relaxation as a density gradient behind the shock, which is what the classic shock-tube photographs show.

The general point is the one this collection keeps arriving at: an instrument reports the quantity it couples to, and in a gas out of equilibrium those quantities have stopped agreeing about what the temperature is.

What the picture cannot show #

The relaxation is drawn as a single temperature falling, and there are really two temperatures: a translational one, which falls, and a vibrational one, which rises from its pre-shock value to meet it. Drawing both is the honest picture and it needs the vibrational temperature to be defined, which requires the mode to be in a Boltzmann distribution among its own levels — an assumption that is itself only approximately true early in the relaxation.

Nothing here draws the shock. It is a line at the left-hand edge of every figure, thousands of times thinner than the region being drawn, and the discontinuity that has a thickness is where it is resolved.

The number to carry #

One dimensionless group decides whether any of this matters, and it is worth stating on its own.

Form the flow time — a body length divided by the flow speed behind the shock — and divide it by the relaxation time. Above about ten, the gas equilibrates and the equilibrium jump conditions apply. Below about a tenth, the gas is frozen and the frozen ones do. In between, neither does.

At Mach 6 in air at sea level that ratio is enormous for anything bigger than a millimetre, so the equilibrium answer is right. At a hundred kilometres altitude the density is a millionth of sea level’s, the relaxation time is correspondingly a million times longer, and a metre-scale body is frozen.

So the same vehicle passes through all three regimes on the way down, which is the reason re-entry aerothermodynamics is a subject rather than a calculation.

Who found it, and when #

Vibrational relaxation was measured in the 1930s by ultrasonic dispersion — sound absorption in a gas depends on it — and the shock-tube measurements that made it a practical subject are from the 1950s and 1960s, driven by re-entry.

Landau and Teller’s 1936 theory gives the relaxation as a first-order approach with a rate that depends exponentially on the temperature to the minus one third, and it remains the standard form. Millikan and White’s 1963 correlation is the fit to it that everybody uses.

Limits recorded rather than smoothed over #

A single relaxation time, and a harmonic oscillator. The vibrational mode is treated as a single harmonic oscillator with one relaxation time. Air has two diatomic species with different characteristic temperatures and different rates, and real oscillators are anharmonic.

No dissociation, no chemistry. At the Mach numbers where the effect is largest here, real air is dissociating, and the dissociation absorbs far more energy than the vibration. The temperatures quoted above Mach 8 are therefore too high even as frozen values.

The relaxation time is an input. One microsecond, chosen to be plausible. Every length in this essay scales linearly with it and none of the temperature differences do.

One relaxing mode, in one direction. The gas here only ever relaxes towards a hotter vibrational state. Behind an expansion the reverse happens — the vibrational modes are left over-populated and relax downwards — and in a nozzle the flow can leave before they do, which is vibrational freezing and is why rocket-nozzle performance calculations carry a frozen and an equilibrium bound.

And the shock is a discontinuity. The jump conditions are applied across a surface with no structure, which is exactly right for the vibrational problem — the shock is thin compared with the relaxation — and exactly wrong for the shock structure itself.