When a shock cannot bounce

A wedge in a supersonic stream makes an oblique shock that turns the flow through the wedge angle. When that shock reaches a wall — or a symmetry plane, which is the same thing — the flow has to be turned back, because the wall is a streamline. A second shock does it, and the two meet at a point.

That is regular reflection, and the whole of it is two applications of the θ–β–M relation in series: the reflected shock has to deflect the flow behind the incident one by exactly $\theta_w$, back the other way.

Where it runs out #

The incident shock has already slowed the flow. So the reflected shock is working with a smaller Mach number, and a smaller Mach number can turn a flow through a smaller angle.

At Mach 4 the free stream itself can be turned through $38.77°$. The flow behind a $25.6°$ wedge shock is at Mach $2.32$, which can be turned through $25.61°$ — exactly the wedge angle. Above that there is no attached reflected shock at any angle.

Reflection therefore fails at $25.61°$, thirteen degrees below the free stream’s own limit.

That is the detachment criterion, and it is the first of the two boundaries this essay is about.

What replaces it #

What appears instead is a triple point. The intersection lifts off the wall, and three shocks meet at it: the incident shock, a reflected shock, and a nearly normal Mach stem standing on the surface. A slip line trails downstream from the junction, across which the velocity jumps and the pressure does not.

The configuration is fixed by two statements:

$$\theta_1 - \theta_2 = \theta_3\quad\text{(the two sides of the slip line are parallel)},$$

$$p_2 = p_3\quad\text{(and at the same pressure)}.$$

Solved here as a root find on the Mach stem’s own deflection — parametrised that way rather than by the stem’s angle, because below the maximum-deflection angle the θ–β–M relation returns the weak root and asking afterwards for the strong shock at that deflection builds the residual out of two configurations that are not the same one.

The second boundary #

Now the question that makes this essay more than a catalogue: at what wedge angle does the three-shock solution first exist?

Not at the detachment angle. It appears earlier, at the point where its root sits at the very end of its range — a Mach stem that is exactly normal and deflects nothing. The residual there is

$$p_2(\text{regular reflection}) - p(\text{normal shock at } M_\infty),$$

so the condition is that the pressure behind a regular reflection has risen to the pressure behind a normal shock in the free stream. That is the von Neumann criterion, and at Mach 4 it is at $20.854°$.

Checked two ways: by that closed characterisation, and by simply scanning upwards for the first wedge angle at which the three-shock root exists. The two share no arithmetic and agree to two parts in a thousand million.

Both, at once #

So between $20.85°$ and $25.61°$ at Mach 4, both configurations exist. Each satisfies mass, momentum and energy across every one of its shocks. Each is stable. And the conservation laws have two answers.

The domain is $0.38°$ wide at Mach 2.5, $4.75°$ at Mach 4 and $9.50°$ at Mach 7. Below about Mach 2.2 the two criteria cross and the ambiguity disappears: there, the three-shock solution appears only after the two-shock one has gone.

What decides, when the equations do not #

The answer is the history of the experiment.

Raise the wedge angle slowly through the domain and the regular reflection persists — it exists, it is stable, and there is no reason for it to change — all the way to detachment at $25.61°$. Lower it again and the Mach reflection persists all the way down to von Neumann at $20.85°$.

That is a hysteresis loop, and its width is the dual-solution domain. It is the one place in this collection where the missing piece of information is not a property of the gas, of the observer, or of the model, but of what happened a minute ago.

Reading the two criteria against each other #

The two boundaries are conditions of different kinds, and the difference is worth being explicit about because it decides which one a flow meets first.

Detachment is an existence condition on the two-shock solution. It says the reflected shock has run out of turning capacity: there is no attached solution at any angle, so the configuration cannot continue. It is the same statement oblique-shock makes about a blunt wedge, applied to the second shock rather than the first.

Von Neumann is an existence condition on the three-shock solution. It says a Mach stem in equilibrium with the reflected shock has become possible — the pressures can now be matched with a stem of non-zero strength.

There is no reason those two should coincide, and they do not. Which comes first depends on the Mach number: above about 2.2 the three-shock solution becomes possible before the two-shock one becomes impossible, which is the domain; below it, the order reverses and there is a range in which neither steady configuration exists. That last case is the von Neumann paradox, and what happens there is a configuration with a curved reflected shock that the three-shock theory does not describe at all.

What the triple point is made of #

The three-shock solution deserves unpacking, because it is the first configuration in this collection with a genuinely two-dimensional shock structure in it.

Upstream is the free stream. Crossing the incident shock gives state 1, deflected towards the wall by the wedge angle. Crossing the reflected shock gives state 2, turned back by $\theta_2$. Crossing the Mach stem directly from the free stream gives state 3, turned by $\theta_3$.

States 2 and 3 are adjacent, and they must agree about two things and are free to disagree about everything else. They agree on pressure, because a pressure difference across a surface with nothing to hold it would accelerate the fluid. They agree on direction, because otherwise the surface between them would not be a surface. They disagree on speed, on density, on temperature and on entropy — state 3 has crossed one nearly normal shock and state 2 has crossed two oblique ones, so state 3 has much the larger entropy rise.

A surface across which the pressure is continuous and the velocity is not is a vortex sheet, and this one is the same object thin-aerofoil theory puts along a wing’s camber line and a wing’s wake trails behind it. Here it is made by two shocks of different strengths meeting, and it is one of the cleanest examples in the subject of vorticity appearing in a flow with no viscosity anywhere.

The numbers at one condition #

To make the configuration concrete, here is Mach 4 on a thirty-degree wedge — inside the Mach reflection regime, since thirty degrees is above detachment.

The incident shock sits at $\beta_1 = 37.76°$ and raises the pressure by a factor of $9.16$. The Mach stem takes the free stream through $\theta_3 = 16.51°$ on the strong branch, at $\beta_3 = 85.09°$ — nearly normal, as the name says — and raises the pressure by $18.36$. The reflected shock turns state 1 back through $\theta_2 = 13.49°$ and multiplies its pressure by $2.005$, bringing it to $18.36$ as well.

The two deflections sum to $30.000000°$ and the two pressures agree to $2.4\times10^{-13}$. Neither of those was arranged: the root find adjusted one number, the stem’s deflection, and both conditions closed together because the configuration has exactly the right number of freedoms.

That is worth noticing as a check on the whole scheme. A three-shock configuration has two unknowns and two equations, so it is determined — and the fact that solving one of them satisfies the other is the statement that the two are not independent, which is what the geometry says.

Whether it is real #

The prediction is old — von Neumann in the 1940s — and the experimental confirmation is not, because the hysteresis is fragile.

Wind-tunnel free-stream turbulence, or a small disturbance from a support, can knock the flow out of the regular reflection and into the Mach one before the detachment angle is reached, so early experiments found the transition at von Neumann and reported no hysteresis. The loop was observed cleanly only when tunnels quiet enough became available, in the 1990s, and it is now routinely reproduced in both wind tunnels and computations.

This is worth saying plainly because it is an unusual epistemic situation. The theory says the answer is not determined; the experiment agrees, and what the experiment measures is how quiet the tunnel is. Two facilities disagreeing about the transition angle are both right about their own flows.

The paradox in the corner of the same diagram #

The crossing near Mach 2.2 is mentioned above as the place where the ordering of the two criteria swaps, and what happens below it deserves more than a clause, because it stood as an open problem for over fifty years.

For a weak incident shock at a small wedge angle, experiments show what looks unmistakably like a Mach reflection: an incident shock, a reflected shock, a stem on the wall, and a shear layer trailing from the junction. And the three-shock theory of this essay has no solution there at all. The two conditions — parallel flow and matched pressure across the slip line — cannot be met by any combination of three shocks. The picture exists and the theory says it cannot.

That is the von Neumann paradox proper, and the resolutions offered for decades were all of the form perhaps it is not really a triple point. They were right, and the reason took until the 2000s to establish, because it is a matter of resolution rather than of principle.

What is there instead of a triple point is a four-wave structure. Behind the junction sits a tiny region in which the flow is supersonic, terminated by an expansion fan — so the reflected shock does not meet the stem at a point but is joined to it through a small patch with its own internal wave system, sometimes a sequence of them with weak shocklets between. The patch supplies the extra freedom the three-shock conditions were short of, and everything closes.

It is small. A few per cent of the Mach stem’s own height, which puts it below what the photographs of the 1940s could show and below what the computations of the 1980s could resolve — so for half a century every measurement of it returned a triple point because a triple point was the finest thing the instrument could report. It was found by computations fine enough to see it and then looked for, and found, in experiments.

That is a good caution to carry away from an essay whose whole content is a diagram of criteria. The diagram’s boundaries are exact statements about a model with three shocks in it, and one corner of it describes a flow that has four waves — which no amount of care with the three-shock algebra could ever have revealed.

Where the ambiguity comes from #

It is worth naming the structure, because it recurs.

The two-shock solution and the three-shock solution are two branches of one algebraic system, and the domain is the region in which both branches are real. That is the same shape as the two roots of the θ–β–M relation — a weak and a strong shock at the same deflection — and as the two Mach numbers at one area ratio in a duct.

In each of those cases the collection has had to say what picks the branch, and the answer has been different every time: the weak oblique shock is picked by the downstream pressure being free, the duct’s branch is picked by whether the throat is choked, and here it is picked by history. The equations routinely admit more than one answer, and the selector is never in the equations.

What the Mach stem does that a regular reflection does not #

Three consequences worth having, because the two configurations are not merely different pictures.

A subsonic pocket. The Mach stem is nearly normal, so the flow behind it is subsonic — $M_3 = 0.47$ in the case computed. That subsonic region sits on the wall and is in communication with everything downstream of it, so what happens far downstream can reach back and change the triple point’s position. A regular reflection has supersonic flow everywhere and no such channel.

A slip line. The two streams either side of the junction have the same pressure and different velocities, so there is a shear layer trailing downstream from the triple point. It rolls up, it is unstable, and it is a source of vorticity in a flow with no viscosity in the model.

And a total-pressure penalty. Gas that crosses the near-normal stem loses far more total pressure than gas that crosses two oblique shocks. For a supersonic intake, which is the practical setting for all of this, the difference between the two configurations is the difference between an acceptable pressure recovery and an unacceptable one.

Why an intake designer cares #

The whole subject exists because of intakes. A supersonic intake decelerates the flow through a series of oblique shocks before a final normal one, and the point of the series is that several weak shocks lose much less total pressure than one strong one — which is what a shock costs, applied as a design principle.

Those oblique shocks reflect off the intake walls. If a reflection goes over to a Mach configuration, a near-normal stem appears in the middle of the duct, the pressure recovery collapses, and the subsonic pocket behind it can interact with the downstream conditions and drive the whole intake unstable — the phenomenon called buzz.

So the boundaries in this essay are design limits, and the dual-solution domain is a region an intake is kept out of rather than a curiosity. A configuration inside it will work perfectly until something disturbs it.

What is not in the flow #

The history. The Mach number, the wedge angle, the gas and the geometry are all specified, and inside the domain they do not determine the answer. What determines it is the path taken to reach those conditions.

This is the only essay among those gathered here whose missing quantity is of that kind. Elsewhere the answer needs something about the gas, or about the observer, or about a boundary — all of which are properties of the situation. Here the situation is fully specified and the answer is still not, because the system has two stable states and no rule for choosing.

That is a bifurcation with hysteresis, and it belongs to the same family as stall, as the laminar–turbulent transition’s dependence on disturbances, and as every other place in this collection where the phrase it depends what happened before has had to be used.

The model limit #

Three, and the third is the one that matters for comparison with experiment.

Steady, two-dimensional, inviscid, perfect gas. The whole analysis is algebra on the θ–β–M relation and inherits every hypothesis behind it.

A straight Mach stem. The three-shock theory treats the stem as locally straight at the triple point. A real stem is curved, which means it makes vorticity and its own shape is part of the solution.

And no boundary layer. A real reflection happens on a wall with a boundary layer on it, and the adverse pressure gradient through the shock can separate that layer — producing a lambda foot at the reflection point which changes the effective wall shape and hence the effective wedge angle. In a practical intake that interaction is usually larger than the difference between the two configurations this essay computes, which is why the criteria are design boundaries rather than predictions.

There is a fourth, quieter limit. Everything here is steady, and the transition between the two configurations is not: crossing a boundary means one structure collapsing and another forming, over a time set by how fast the flow sweeps the region. Nothing in a steady theory says how long that takes, or whether the flow overshoots on the way, and in an intake the answer to both is what decides whether the transition is a nuisance or a failure.