A surface that remembers the diaphragm

One diaphragm, every wave is this collection’s account of the shock tube: burst a membrane between a high-pressure gas and a low-pressure one and every wave the subject has appears at once — a shock running into the low-pressure gas, an expansion fan running back into the high-pressure one, and between them a contact surface.

That essay is about the wave system. This one is about the third of those three, which is not a wave at all and which is the only one still there afterwards.

What a contact surface is #

A shock and an expansion are waves: disturbances travelling through the gas at a speed relative to it, carrying information and leaving the fluid behind them. A contact surface is made of fluid. It travels at the fluid’s own velocity, which is to say it does not travel at all relative to the gas — it is a set of particles, and it is the set that started at the diaphragm.

That has an immediate consequence. A material surface cannot heal. The particles on one side of it are on one side of it for ever, because two fluid particles cannot exchange places. This is the same argument that keeps the arms of a rolled-up vortex sheet in order in a spiral is a legible record, and it is exact for the same reason.

Two things match and three do not #

The condition at a contact surface is that the pressure and the normal velocity are continuous — and nothing else is.

Those two are forced, and the solution holds them to exactly zero in double precision — the pressure gap and the velocity gap are both 0.000000000, not small. The pressure must match or there would be a net force on a surface with no mass; the velocity must match or a gap would open. Everything else is free, and in a shock tube everything else is very different: at a pressure ratio of ten the shocked side sits at 418.0 K and the expanded side at 209.5 K, a factor of two across a surface with no jump in pressure at all.

At a diaphragm pressure ratio of ten, the shocked gas is at 1.99 times the temperature of the expanded gas and at half its density, with the two pressures identical to the last bit of the arithmetic.

Why the difference is permanent #

The reason the two sides differ is that they got there by different routes. One parcel of gas was compressed by a shock, which is irreversible; the other was expanded through a fan, which is not.

The measurable form of that difference is an entropy gap of 694 J/kg/K, and entropy is the quantity nothing available to an inviscid flow can change. The shocked gas has more of it than it started with and the expanded gas has exactly what it started with, and no subsequent process in the tube can bring the two together.

So the contact surface is a record of the shock, carried by the gas that was not shocked. It is there because something irreversible happened to the gas on the other side of it.

Raising the diaphragm pressure ratio makes the shock stronger, the entropy jump larger and the contrast greater: at two hundred the temperature ratio is 5.69.

The four speeds #

The wave speeds separate the contact from everything else. The shock runs ahead at 546 metres a second, the fan’s head runs back at the speed of sound in the driver gas, and the contact travels at the fluid velocity behind the shock — which is by definition the speed of the gas itself.

That is the formal statement of the distinction. A wave has a speed relative to the fluid and a contact surface has none, and the whole of the difference between what is temporary and what is permanent in this flow follows from it.

How much of the tube is on each side #

A small piece of book-keeping is worth doing, because it says how much gas the surface separates and therefore how much of the tube’s content is in each state.

The contact travels at the fluid velocity behind the shock — 285.1 m/s at a pressure ratio of ten — and the shock travels faster, at 558.1 m/s, being Mach 1.6075 on a sound speed of 347.2. So the slug of shocked gas between them grows at 273.0 m/s, which is 48.9 per cent of the shock speed: the test slug lengthens at about half the rate the shock advances.

Behind the contact, the expanded driver gas fills the region between it and the fan’s tail, and that region grows too. So the tube’s contents are three growing regions and two shrinking undisturbed ones, and the surface sits at the boundary between the two that were made by the experiment.

The useful quantity is the ratio of the two growth rates, because it decides how much test gas an experiment gets for a given length of tube — and it is a function of the pressure ratio and of the two sound speeds alone, which is why the driver gas matters as much as the pressure does.

What the solver computed, and how it was checked #

The Riemann problem is solved exactly: the shock strength is found by requiring the pressure and the velocity behind the shock to match those behind the expansion, which is one equation in one unknown and is solved by bisection. Everything else follows in closed form.

Three checks. That the pressure across the contact is identical, to a part in 10⁹ — it comes out at exactly zero, because the matching condition is what the shock strength was solved for. That the temperature ratio exceeds 1.2, so there is a contrast to describe. And that the entropy difference exceeds one joule per kilogram per kelvin, which is the statement that the difference is thermodynamic rather than an arithmetic residue.

What the surface is a record of, exactly #

It is worth being careful about the claim, because “it remembers the diaphragm” is a phrase and the content is specific.

Its position records the diaphragm’s. The surface is the set of particles that started at the diaphragm, so tracking it backwards gives the diaphragm’s location exactly, for as long as the surface exists. That is a positional memory and it is exact.

Its jump records the shock’s strength. The entropy difference across it is the entropy the shock produced, which is a monotone function of the shock’s Mach number. So a measurement of the temperature ratio across the contact is a measurement of how strong the shock was — even long after the shock has left the tube.

A hundred-fold change in the diaphragm’s pressure ratio moves the shock Mach number by a factor of 2.25 and the temperature ratio across the contact by 4.66. The contact is therefore a more sensitive record of the burst than the shock speed is — which is the opposite of how a shock tube is usually instrumented.

And its existence records that something irreversible happened. If the diaphragm had been replaced by a gradual opening, the compression would have been isentropic and there would be no entropy jump and no contact surface at all. The surface exists because the process was abrupt.

That last is the general form. A contact surface is where two parcels of fluid that took different routes have ended up adjacent, and the size of its jump is the difference between the routes — here 693.6 J/kg·K of entropy, between a parcel at 11,449 and one at 10,756. Wherever one appears in a flow, something in the history was discontinuous.

What this does to a shock-tube experiment #

The contact surface is the practical limit on what a shock tube can do, and it is worth being specific.

The test time ends when it arrives. A shock tube is used to make a slug of hot gas behind the shock, and that slug is bounded ahead by the shock and behind by the contact surface. The interval between them at a station is the test time: the gap opens at 273.0 m/s, so a station one metre from the diaphragm sees the shock at 1.79 ms and the contact at 3.51 ms, and the test window is 1.72 milliseconds — a few milliseconds in an ordinary tube, and proportional to the distance travelled.

And it ends earlier than that, because the surface is not clean. In a real tube the contact surface is unstable — it is a density interface being accelerated, which is the Richtmyer-Meshkov instability — so it becomes a mixing region rather than a surface, and the driver gas arrives before the geometry says it should.

Which is why the driver gas is chosen. Using helium or hydrogen as the driver changes the sound speed and therefore the wave system, and it also changes what arrives when the contact does. A tube tailored so that the reflected shock passes through the contact without reflecting is running at the “tailored” condition, and it exists entirely to postpone this arrival.

The entropy it carries is the quantity the last essay in this field is about, on the same machinery.

Where else a contact surface decides something #

Behind any blast. The fireball boundary of an explosion is a contact surface between the detonation products and the shocked air, and it is why the products stay together rather than mixing immediately.

In a nozzle with two streams. A jet engine’s core and bypass flows meet along a contact surface, and its behaviour — whether it stays a surface or becomes a mixing layer — decides the noise and much of the thrust.

And in every Riemann problem a code solves. Every finite-volume compressible solver solves this problem at every cell face at every time step, and the contact surface is the wave those solvers find hardest: it carries no pressure signal, so a scheme that keys on pressure smears it, and a smeared contact surface is a spurious mixing.

That last is the practical reason the subject is taught. A contact discontinuity is the thing a numerical method loses first, and losing it means inventing a diffusion the physics does not have.

Why nothing in the pressure field says it is there #

It is worth dwelling on the invisibility, because it is the sharpest form of the theme these essays share.

A pressure transducer at a station records the shock arriving as a step — a factor of 2.848 in pressure at a diaphragm ratio of ten — and the expansion arriving as a ramp. When the contact surface passes it records nothing at all — no step, no ramp, no kink — because the pressure is continuous across it and so is its derivative in time.

The surface is nonetheless there, and an instrument that couples to density or to temperature or to composition sees it immediately. So whether the flow has a discontinuity in it depends on which variable is being watched, which is not a statement anybody expects to make about a discontinuity.

That is the same structure as what a mean profile cannot tell anybody: a measurement of one variable is blind to a feature that another variable resolves completely, and the feature is not small.

The other place two parcels arrive differently #

The construction generalises past shock tubes and it is worth naming where it goes.

A slip line at a triple point. Where three shocks meet — the Mach reflection of when a shock cannot bounce — the gas that went through one shock and the gas that went through two arrive side by side at the same pressure and different everything else. The surface trailing from the triple point is a contact surface, and its jump is the difference between one strong shock and two weaker ones.

A vortex sheet behind a curved shock. A shock of varying strength produces varying entropy, so neighbouring streamlines arrive with different entropies and the resulting gradient is a vorticity — which is the spin a shock leaves behind, and is the continuous version of the same statement.

And the wake of any body in supersonic flow. Fluid that passed through the bow shock near the nose has been through a strong shock; fluid that passed further out has been through a weak one, and the whole wake is stratified in entropy as a result.

In all three the surviving quantity is entropy, because it is the one an inviscid flow cannot change, and everything else about the difference follows from it.

What the picture cannot show #

The x-t diagram draws the contact as a line, and in a real tube it is a region: the interface is unstable, it mixes, and by the time it arrives at a test station it is centimetres thick rather than molecular. Nothing here contains that.

Nor is the expansion fan drawn as a fan. It is a continuous family of characteristics between its head and its tail, and the figure draws two lines with nothing between them, which is a schematic of a region rather than a picture of one.

Why entropy is the quantity that lasts #

What the shock spent to make that entropy is priced in what a shock costs: a total pressure loss that is the same statement about the same process, read as a currency rather than as a label.

The essay keeps arriving at entropy and it is worth saying why it, rather than any other variable, is the carrier.

An inviscid, non-conducting flow conserves entropy along particle paths. That is a statement of the same kind as Kelvin’s theorem — a quantity attached to a parcel and carried by it, unchangeable by anything the flow does — and it is what the drift was the instrument says about circulation in an incompressible flow.

So a parcel’s entropy is a label, set when something irreversible happened to it and constant thereafter. Two parcels with different labels can be brought adjacent, and the boundary between them is a contact surface; they can be stirred together, and the result is a stratified fluid rather than a mixed one; but the labels themselves do not change.

That is why the memory is permanent and why it is entropy that carries it. Every other variable — pressure, velocity, density, temperature — is adjusted continuously by the flow, and entropy is the only one that is not.

The exception is real and is the reason the permanence has a time limit: conduction and viscosity change it. On the time scale of a shock-tube experiment they do nothing; over long enough they smooth every entropy gradient away, which is the same fate the wall kernel of the wall the fluid is listening to predicts for everything diffusive.

What the surface is for, if it cannot be seen #

It is worth asking what use a feature is that carries no pressure signal and no velocity signal.

It is a boundary in a calculation. A shock-tube solution is built by matching two states across it, and the matching is exactly the pair of conditions it satisfies. Getting the pair wrong — matching temperature, say, because it feels like a state variable — gives a solution that satisfies nothing.

It is a boundary in the apparatus too. A shock tube’s test time ends when the contact surface arrives, because the gas behind it came from the driver and has a different composition, a different temperature and a different sound speed. Everything the tube was built to measure has to happen before that arrival, and the interval is computed from the surface’s speed rather than from the shock’s.

And it is where a real tube stops matching this model, because the surface does not stay a surface: it is unstable to the disturbances a real diaphragm leaves, and what arrives is a mixing zone of finite thickness rather than a discontinuity.

Who found it, and when #

The Riemann problem is Riemann’s, from 1860, and the shock tube as an instrument is Vieille’s from 1899 and Payman and Shepherd’s from the 1940s. The contact surface is in the solution from the beginning; what took longer was the recognition that its instability sets the useful test time, which came from the shock-tunnel work of the 1950s and 1960s.

The numerical difficulty is Godunov’s problem, from 1959, and the schemes that resolve a contact discontinuity without smearing it are a large part of what computational gas dynamics has been doing since.

Limits recorded rather than smoothed over #

One dimension, no viscosity, no conduction. A real contact surface diffuses and mixes; this one is a mathematical surface with zero thickness for ever.

Calorically perfect gas, both sides. The same gamma is used on both sides, so the driver and driven gases are the same. Different gases give a different wave system and a larger contrast, which is the usual arrangement in a real tube.

The diaphragm bursts instantly. A real one takes tens of microseconds and opens imperfectly, which is where much of the contact surface’s initial disturbance comes from.

The instability is described and not computed. The Richtmyer-Meshkov growth of the interface is the practical limit on the test time and nothing here evaluates it. It needs a perturbation amplitude, which is a property of the diaphragm rather than of the gas.

And the entropy is computed for a perfect gas. At the temperatures a strong shock produces the gas is not one, which is a gas that has not finished being shocked’s subject — and the contrast across the contact would be larger still.