The gas a reflected shock refuses is inside the layer

A tailored tube buys its test time with its driver followed the waves in a shock tube after its interface was tailored to send nothing back, and found the reservoir at the end wall lasting until the driver’s own expansion, reflected from the driver’s closed end, crossed the whole tube: about seven-tenths of a driven-tube crossing time per driver length for helium. It ended on the limit that usually arrives sooner. The reflected shock, on its way back from the end wall, meets the boundary layer the incident shock left on the wall, and there it does not stay a plane. Its foot lifts off the wall into a λ, a bubble of stalled gas collects beneath it, and when the λ reaches the driver gas, cold driver gas runs along the wall under it towards the end wall and spoils the reservoir.

The first question in that chain is whether the foot forms at all. It has a classical answer, Mark’s criterion of 1958, which asks one thing of the gas at the wall. This essay asks the same thing of the whole boundary layer, and the answer changes where it matters most — at the high Mach numbers a hydrogen-driven tailored tube runs at.

What the reflected shock asks of the gas it meets #

The tube is the one diaphragm that makes every wave, with air at 300 K in the driven section. The incident shock sets the air behind it moving towards the end wall at a speed $u_2$ — 901 metres a second at the helium-tailored Mach number of 3.41. At the end wall that air must stop, and the reflected shock is what stops it: it runs back from the wall at 462 metres a second, and the air it passes is brought to rest at the reservoir pressure $p_5$, 73 times the fill pressure.

Seen from the reflected shock, the core air arrives at $u_2$ plus the shock’s own speed and is brought to the shock’s speed behind it, and the jump the conservation laws allow puts it at $p_5$. The gas in the boundary layer is different. At the wall it is at rest in the laboratory, so it arrives at the reflected shock at the shock’s speed alone, and it is cold, at the wall’s temperature rather than the 957 K of the shocked core. To be brought into the reservoir it must reach $p_5$ too, and the most pressure any gas can reach by being brought to rest in the shock’s frame is its stagnation pressure. If that is below $p_5$, the gas cannot enter the reservoir. It piles up in front of the shock’s foot, pushes the foot forward and lifts it off the wall, and the core gas above it passes through an oblique leading shock to the bubble’s pressure before reaching the reservoir through a second. That is the bifurcation, and Mark’s criterion is the test: the wall gas’s stagnation pressure against $p_5$.

Mark’s answer for air #

The first figure’s upper curve is Mark’s. Below an incident Mach number of 1.32 the wall gas’s stagnation pressure exceeds $p_5$ and the shock stays plane. Above it the ratio falls, to half at Mach 2.5 to 3.4 — at the helium-tailored point it is 0.51 — and the reflected shock is strongly bifurcated. Then it rises again, and above Mach 6.5 it is above one: by Mark’s test the bifurcation stops.

The rise has a plain cause. The reflected shock gets faster as the incident one does, and the wall gas, cold and so with a low sound speed, meets it at a Mach number that grows without limit: 1.33 at the helium point, 2.1 at Mach 6, 3.4 at Mach 10. Its pitot pressure grows as the square of that Mach number, faster than $p_5$ does. A strong enough shock meets cold wall gas so fast that the gas has pressure to spare.

On that reading a hydrogen-driven tube, tailored at Mach 6.0, runs with the wall gas at 0.89 of $p_5$ — barely refused, a weak bifurcation, close to the edge of none. The rest of this essay is about why that reading is wrong.

The gas that fails is inside the layer #

The wall gas is one layer of the boundary layer, and not a representative one. Moving outward from the wall, the gas speeds up from rest to $u_2$ and its temperature does something less simple: it rises first, because friction dissipates the core’s kinetic energy in the layer, and then falls to the core’s own temperature at the edge. The gas at the wall is kept at the wall’s temperature by conduction; the gas a little way out is heated towards the recovery temperature, the temperature a gas reaches when brought to rest by friction rather than isentropically — the same temperature that a pipe cannot hold its gas at the wall’s temperature turned on. The Crocco–Busemann relation, in Walz’s form with a turbulent layer’s recovery factor of 0.89, gives the temperature as a quadratic in the gas’s velocity, and the second figure uses it to compute each layer’s stagnation pressure in the reflected shock’s frame.

At Mach 2 and 3.41 the wall gas is the worst and the curve rises from it: the gas further out is moving towards the reflected shock and arrives faster. At Mach 6 and 10 the curve dips first. Gas a fifth to a quarter of the way into the layer’s velocity range has been heated to three and eight and a half times the wall’s temperature. Its sound speed is higher in proportion to the square root, while its extra velocity towards the shock is only a fraction of the core’s, so it meets the shock at a lower Mach number than the wall gas does — and its stagnation pressure is the lowest in the layer: 0.64 of $p_5$ at Mach 6, where the wall gas has 0.89, and 0.69 at Mach 10, where the wall gas has twice $p_5$.

The hydrogen point’s numbers show the mechanism plainly. The wall gas meets the reflected shock at 733 metres a second with a sound speed of 347: Mach 2.11. The core gas meets it at 2,421 metres a second with a sound speed of 979: Mach 2.47, and passes into the reservoir with a fifth of $p_5$ to spare. Between them, gas moving at 325 metres a second in the laboratory — a fifth of the core’s speed — has been heated by friction to 897 K on the way to the layer’s recovery temperature of 3,645 K. It meets the shock at 1,058 metres a second with a sound speed of 600: Mach 1.76, lower than either neighbour. Its speed towards the shock has grown by less than half from the wall’s, while its temperature has tripled. That Mach number, not the wall’s, is the one that fails.

That is the first figure’s lower curve. Across the layer, gas is refused at every incident Mach number above 1.32, and the lowest ratio levels off near 0.69 rather than rising through one. Mark’s test is exact for the gas it tests; it tests the wrong gas once the shock is strong enough for friction to heat the layer.

How much of the layer is refused #

Whether the foot forms is one question; how much gas it has to hold is another, and the third figure answers it by weighting each layer by its mass. A turbulent layer’s velocity is taken to follow the one-seventh power law, and each layer’s density goes as one over its temperature at the layer’s uniform pressure. The refused share is about a tenth of a per cent at Mach 1.5, 6 per cent at Mach 2, 19 at the helium-tailored point, 26 at hydrogen’s and 28 at Mach 10.

The share keeps growing where Mark’s test says the refusal stops. The gas that collects under the foot is not a sliver at the wall that disappears above Mach 6.5; it is a quarter of the whole layer, the part near the wall where the gas is slow relative to the shock for its temperature, and it grows with the shock.

The foot’s leading shock #

The pressure of the stalled gas sets the λ’s geometry. The core gas, arriving at the reflected shock at its relative Mach number, passes first through the foot’s leading leg, an oblique shock that raises it to the bubble’s pressure, and its angle follows from the oblique-shock relation — the same relation whose limit a wedge too blunt for its shock to stay attached reaches. The fourth figure takes the bubble’s pressure as Mark’s wall gas’s and as the layer’s lowest. At the helium-tailored point both give a leading shock leaning at 47 degrees to the wall. Mark’s steepens towards the normal as its refusal weakens, to 71 degrees at Mach 6, and vanishes at 6.5. The layer’s settles near 54 to 56 degrees at high Mach numbers: a foot that stays well formed.

The angle is where a measurement could discriminate between the two tests. A photograph or a shadowgraph of the λ at an incident Mach number of 6 to 8 in air would show either a nearly normal foot fading out — Mark’s picture — or a foot still leaning at about fifty-five degrees, which is what the layer’s gas predicts.

How many atoms a molecule has #

The fifth figure varies the test gas. The lower end of the refusal barely moves, from Mach 1.23 at a ratio of specific heats of 1.15 to 1.57 for a monatomic gas. The upper end by Mark’s test falls steeply as the ratio rises — 9.9 at 1.3, 6.5 at 1.4, 4.6 at 1.5, 2.8 at 5/3 — because a gas with fewer internal degrees of freedom heats more when shocked, and its reservoir pressure grows more slowly. Across the layer, every gas with a ratio of 1.5 or less refuses gas up to the scan’s Mach 12, and a monatomic gas up to 5.3.

The monatomic gas is the exception in depth as well as range. Its refusal is shallow everywhere: the lowest stagnation pressure in its layer is never below 0.895 of $p_5$, against 0.49 for air. That is consistent with argon being the gas in which reflected shocks are seen to bifurcate least, and it is the physical reason a facility that can choose its test gas has one that is kind to its reservoir.

Why the wall was the natural place to look #

Mark’s choice of the wall gas was not careless. At the incident Mach numbers the shock tubes of the 1950s ran at, two to four, the wall gas really is the worst, as the second figure’s first two curves show, and the criterion predicted the bifurcation those tubes saw. The layer’s interior only takes over above Mach 4, where the friction heating grows with the square of the core’s Mach number relative to the wall, and the reflected-shock tunnels that run there — the high-enthalpy facilities driven by hydrogen or by free pistons — came later. It is in exactly those facilities that driver-gas contamination is the binding limit on test time, and a test that lets the wall gas through there, as Mark’s does, predicts a weak or vanishing bifurcation at just the conditions where the gas carried along the wall does the most damage. The layer’s gas removes that contradiction without changing anything in Mark’s reasoning except the gas it is applied to.

What that means for a tailored tube #

For the tube of the previous essays the answer is now definite. At the helium-tailored point, Mach 3.41, both tests agree: the wall gas is the worst, it reaches half of $p_5$, and a fifth of the boundary layer cannot enter the reservoir. At hydrogen’s, Mach 6.0, Mark’s test says the bifurcation is weak and nearly gone; the layer says it is strong, with a quarter of the layer refused. In no tailored condition of an air tube is the reflected shock plane. The mechanism that the shock that passes without an echo tailored away one wave to protect — the reservoir’s steadiness — is attacked by this one at every condition the tailoring can reach.

The wave-limited test times of the previous essay are therefore upper bounds at every point, as it said, and the gap between them and the contamination-limited times depends on how fast the refused gas and the driver gas behind it are carried to the end wall. That is the remaining step, and it is a different kind of calculation: the refused gas is a mass, collected under the foot, and what matters is where it goes.

What was checked #

The shock tube’s states come from the previous essays’ calculation, and the reflected shock’s Mach number derived from them matches the textbook closed form to rounding at three ratios of specific heats and three incident Mach numbers. The isentropic and Rayleigh stagnation-pressure formulas agree exactly at Mach one, where the one hands over to the other, and the layer calculation’s wall gas is Mark’s criterion to the last bit. The foot’s leading-shock angle, put back through the oblique-shock relation, gives the pressure it was built to reach. The checks refuse a negative Mach number and a station outside the layer.

What the calculation leaves out #

The temperature profile is a model. The Crocco–Busemann–Walz relation is exact for a laminar layer at unit Prandtl number with no pressure gradient and a good approximation for turbulent layers; the layer behind a moving shock is neither exactly. The minimum inside the layer depends on the profile’s shape, and a measured profile would move the numbers, not the fact that the hottest slow gas is inside.

Real gas. At Mach 6 the reservoir is at 5,000 kelvin, where air dissociates and the ratio of specific heats stops being a number. The reservoir pressure and the core’s temperature both move, and so does the criterion.

The criterion is a threshold. It says which gas cannot enter the reservoir; it does not say how the foot grows, how big the bubble becomes or how the gas leaves it. Those are the subject of the next step. And a surface that remembers the diaphragm is a reminder that the contact the reflected shock later meets is not a plane either.

The wall’s temperature. The wall is held at the fill temperature, which is right for a shock tube’s short run; a heated or cooled wall changes the wall gas and, through the profile, the gas inside the layer.

Who worked it out #

Mark’s analysis of the reflected shock’s interaction with the boundary layer is from 1958, and it has been the standard criterion since; Davies and Wilson in 1969 followed the refused gas into the jet of driver gas it feeds. The temperature–velocity relation is Crocco’s and Busemann’s from the 1930s, with Walz’s extension to a recovery factor below one. Applying the criterion across the layer rather than at the wall is the calculation here.

Still open: where the refused gas goes #

The refused gas collects under the foot, and when the reflected shock reaches the contact surface the gas being refused is driver gas. The next calculation follows that mass: it grows the incident shock’s boundary layer along the tube, collects the refused share under the foot as the reflected shock sweeps back, and, once the foot is in the driver gas, lets the gas that passed the λ’s two legs — which arrives at the reservoir pressure with more total pressure than gas that passed one normal shock — run along the wall as a jet towards the end wall. The question is the jet’s arrival time against the previous essay’s wave-limited time, for helium and hydrogen drivers, and whether the tube’s diameter, through the boundary layer’s thickness, decides which comes first.