Turning the other way is free

A supersonic stream that meets a corner turning into it shocks. A supersonic stream that meets a corner turning away from it does something entirely different, and the difference is not a matter of degree.

Why one converges and the other spreads #

The mechanism is entirely geometric and it is worth having before the algebra.

A Mach wave sits at $\arcsin(1/M)$ to the local flow. Turning the flow away from itself accelerates it, so the Mach number rises and the Mach angle falls — and each successive wave is more swept back than the last. Waves that lean progressively further back can never overtake one another, so the family spreads apart and stays spread apart.

Turning the flow into itself does the reverse. Each successive wave is less swept back than the last, so the later waves gain on the earlier ones, and the family converges. Where they meet, the compression has become a discontinuity, which is a shock.

The asymmetry is in that one sentence, and every other property in this essay follows from it. There is no such thing as an expansion shock because the waves that would have to make one are running apart.

Isentropic in the strongest sense #

A shock is a finite jump across which entropy rises. An expansion is a continuum of infinitesimal turns, and the entropy each one produces goes as the cube of its own angle.

Sum the cubes of $n$ turns of size $\theta/n$ and the total goes as $\theta^3/n^2$, which goes to zero as $n \to \infty$. The fan is that limit taken exactly, so the entropy produced is exactly zero.

Not small. Zero.

That is the whole content of the Prandtl–Meyer expansion, and it makes the fan far easier to compute than the shock it mirrors: since the process is isentropic, all the isentropic relations apply, and the total pressure and total temperature both come through untouched.

The function that counts how far a flow has turned #

The bookkeeping is a single function. $\nu(M)$ is the angle through which a stream at Mach 1 would have to turn to reach Mach $M$:

$$\nu(M) = \sqrt{\frac{\gamma+1}{\gamma-1}}\arctan\sqrt{\frac{\gamma-1}{\gamma+1}(M^2-1)} - \arctan\sqrt{M^2-1}$$

and turns simply add. A flow at Mach 2 has $\nu = 26.38°$; turn it a further 10° and it has $\nu = 36.38°$, which corresponds to Mach 2.385.

That additivity is what makes supersonic geometry so tractable. There is no path dependence, no history, no need to know how the flow got to Mach 2 — only its Prandtl–Meyer angle, which is a state variable in the same way the Mach number is. That is a stronger property than it sounds: it means a supersonic flow’s whole compressible history is carried in one number, where an incompressible boundary layer carries its history in a profile shape that no single quantity summarises.

How far the state actually moves #

The numbers across a fan are larger than the modest turning angles suggest, and it is worth putting them beside the compression case to see how asymmetric the pair is.

At Mach 2, turning away by 10°: the Mach number goes to 2.385, the pressure falls to 0.548 of what it was, the density to 0.646, the temperature to 0.848. The pressure coefficient on the surface is −0.161.

At Mach 2, turning into the flow by the same 10°: the Mach number falls to 1.641, the pressure rises by a factor of 1.707, and the pressure coefficient is +0.252.

So the compression moves the pressure further than the expansion does — a factor of 1.71 up against a factor of 1.82 down, which is comparable — but it is the cost that differs completely, and the cost is what a designer is buying or selling.

The limit at which the gas runs out #

$\nu(M)$ does not grow without bound. As $M \to \infty$ it approaches

$$\nu_{\max} = \frac{\pi}{2}\left(\sqrt{\frac{\gamma+1}{\gamma-1}} - 1\right) = 130.45°$$

so a stream at Mach 1 can be turned through 130.45° and no further. Beyond that there is no solution, because the gas would have to expand to zero pressure, zero density and zero temperature — it would have reached vacuum, and there is nothing left to turn.

This is a genuine limit rather than a failure of the theory, and the solver treats it as one: machFromNu refuses an angle at or beyond $\nu_{\max}$ with a message saying the flow would have to expand past vacuum. The rejection test asks for exactly that, and requires the refusal.

Physically, a corner turning through more than the available angle produces a region of vacuum next to the wall, with the flow separating from the surface and leaving it. That is what a rocket plume does when it exits into space: it turns until it runs out of Prandtl–Meyer angle, and then stops following anything.

What the solver computes, and how it is checked #

expansionFan takes a Mach number and a turning angle, adds the angle to the Prandtl–Meyer function, inverts by bisection, and assembles the state ratios from the isentropic relations on both sides.

assertExpansionIsIsentropic then checks the result three ways, and the first is the one that matters. It computes $p/\rho^\gamma$ across the fan from the returned ratios and requires it to be unchanged to $10^{-12}$ — which is a genuine test, because the pressure and density ratios were assembled from separate isentropic expressions and a slip in either exponent would break it.

It also requires the flow to accelerate and cool, which is the direction an expansion goes, and requires the trailing Mach wave to lie behind the leading one. That last check is the geometric statement that the fan opens: a fan whose waves converge is not a fan, and the rejection test hands it exactly that case.

Three further refusals guard the edges: an expansion claimed to lose total pressure, a Prandtl–Meyer angle requested below Mach one where the function is undefined, and a turn past vacuum.

The corner does not have to be a corner #

Nothing in the derivation required the turn to happen at a single point. A centred fan — all the waves radiating from one corner — is the sharp-corner case, and a surface that curves gradually produces the same total turn with the waves originating at successive points along it.

The states are identical. The Prandtl–Meyer function depends only on the total angle turned, so a flow that has been turned 10° over a rounded shoulder arrives at exactly the same Mach number, pressure and temperature as one turned 10° at a corner.

What differs is where the waves go, and that matters for anything downstream. A centred fan’s characteristics all cross the same region and interact with whatever is there; a distributed one’s are spread out. In a supersonic nozzle this is the whole of the contouring problem — the expansion section is shaped so that the waves it generates are cancelled by the wall further along, delivering parallel uniform flow at the exit rather than a diverging jet.

That cancellation is why a good supersonic nozzle is a specific curve rather than a cone, and it is computed by the method of characteristics, marching the same wave bookkeeping through a two-dimensional field. This site does not implement it, and the nozzle figures elsewhere in this field are quasi-one-dimensional for that reason: they get the states right along the axis and say nothing about whether the exit flow is parallel.

Why the aerofoil problem becomes elementary #

Put the two turns together and a supersonic section can be solved exactly, face by face, with no inversion and no iteration.

That is a complete solution of a lifting aerofoil in three lines of reasoning, and it should be compared with what the same problem costs subsonically: a conformal map, or a panel method inverting a matrix over the whole surface, plus a Kutta condition to pick the circulation, none of which is needed here.

The reason is the change of type: supersonic flow carries no information upstream, so each face’s state depends only on the faces before it, and the problem marches rather than inverting. Supersonic aerofoil theory is easier than subsonic aerofoil theory, which is one of the least expected facts in the subject.

The reflection that has no reflection #

One more consequence, and it is the sort of thing that only makes sense once the fan is understood as a family of waves rather than as an event.

A shock striking a solid wall reflects as a shock, because the wall requires the flow to be parallel to it and a second compression is what achieves that. An expansion fan striking a wall reflects as an expansion fan, for the same reason.

But an expansion fan striking a free boundary — the edge of a jet, where the pressure rather than the direction is fixed — reflects as a compression, because holding the pressure constant requires undoing what the fan did. That is what produces the regular diamond pattern visible in a rocket or jet plume: alternate reflections between the free boundary and the axis, each converting one kind of wave into the other, running for many cells downstream until dissipation flattens them — a wave system with the same reflection bookkeeping as an image system in ideal flow, and a rare case where the two halves of this site use the same trick.

The plume structure of an over- or underexpanded nozzle is that process, and each bright cell is one round trip.

What the waves do after they leave the surface #

The face-by-face solution above stops at the body, and the waves do not. Following them outward qualifies the claim that the method is exact, and explains why the flow a long way from a supersonic aeroplane has forgotten almost everything about it.

The first qualification is a reflection. Every Mach wave generated at the surface runs outward until it reaches the bow shock, and a shock is not a wall — it is a discontinuity whose angle depends on the flow behind it, so a wave arriving at it changes it and something is reflected back towards the body. The marching solution takes no account of that returning wave. For a wedge or a flat plate the bow shock is straight and the reflections are absent, and the method is genuinely exact; for a curved surface they exist, they are second order in the turning angles, and shock–expansion theory is an excellent approximation rather than a closed solution. The essay’s “each face depends only on the faces before it” is exactly true only when the shock is straight.

The second is a merger, and it decides what anybody on the ground hears. An expansion fan travels slightly faster than the shock ahead of it, because a shock moves at more than the local sound speed and a Mach wave moves at exactly it — so over distance the fan catches the shock and weakens it, wave by wave. The compressions from the body’s aft end do the reverse and coalesce into a rear shock.

Run that far enough and the whole structure collapses into two discontinuities with a linear pressure ramp between them: a front shock, a fall through ambient, and a rear shock — the N-wave. Nothing else survives. The individual faces, the fans, the reflections and the entire near-field detail have merged into three numbers: the front overpressure, the duration, and the rear rise.

The decay is slow and its exponents are worth carrying. For an axisymmetric body the shock overpressure falls as $r^{-3/4}$ and the signature stretches as $r^{1/4}$ — so a boom does not fade in the way a spherical wave would, and it lasts longer the further it has come.

Which is the honest reading of this essay’s own tidiness. Face by face, the answer is exact and elementary. Kilometre by kilometre, the answer becomes an N-wave whose shape depends on the body only through one integral of it, and everything the marching solution computed so carefully has been averaged away.

The asymmetry is thermodynamics, not geometry #

The wave-convergence argument at the top of this essay is a mechanism, and mechanisms explain how rather than whether. It is worth checking that the deeper reason agrees with it, because on this site the mechanical story and the thermodynamic one have already parted company once.

They agree here. The second law permits a process only if it does not lower entropy. A compression discontinuity raises entropy, so it is permitted; an expansion discontinuity would lower it, so it is not. The wave geometry is the way nature implements that restriction — waves that would have to form a forbidden discontinuity are exactly the waves that run apart — but the restriction does not depend on the implementation.

That distinction has a test. In a BZT fluid, where the isentrope curves the other way near the critical point, the thermodynamics reverses: expansion shocks are permitted and compression ones are not. And the wave behaviour reverses with it, because the sign of the same second derivative controls both. The mechanism follows the thermodynamics rather than the other way round, and the two cannot be made to disagree.

Where the model stops #

Two-dimensional and steady. Prandtl–Meyer is a planar result. Axisymmetric expansions — round a cone’s shoulder, or out of a circular nozzle — are governed by the method of characteristics and are not this closed form.

Supersonic throughout. The function is defined from Mach one upwards and the solver refuses below it, because there are no Mach waves to make a fan out of.

No viscosity. A real fan striking a boundary layer thickens it, and a strong enough expansion can relaminarise a turbulent layer — an effect with no representation here at all.

Perfect gas, constant $\gamma$. A very large expansion cools the gas enough that condensation becomes possible, and in a wind tunnel expanding from a humid reservoir the fan can be made visible by exactly that: the flow condenses, and the theory that predicted the temperature no longer applies to the fluid it produced.

What the picture cannot show #

The fan is drawn with eleven lines and it has infinitely many.

The lines are a sampling, and the number of them carries no meaning — drawing twenty changes nothing about the boundaries or the states. The genuinely continuous nature of the turn is the property the figure most needs to convey and is the one it can only gesture at, since any drawing of a continuum is a set of discrete marks.

The figure is also silent about strength. Every line in the fan is drawn identically, and in reality each carries an equal share of the turning, which means the pressure gradient through the fan is steepest where the waves are closest. Nothing in the drawing shows that.

And the states between the waves are not drawn at all. The figure gives the state before and the state after; the continuum of intermediate states, each at its own Mach number with its own Mach angle, is what makes the fan open, and it appears only as the changing angle of the lines.

Who found it, and when #

Theodor Meyer’s 1908 dissertation at Göttingen, supervised by Prandtl, contains both the oblique shock relations and this expansion. It is one of the most productive theses in the subject: two of the three elementary supersonic waves, worked out in one document, at a moment when nothing flew fast enough to need either.

Prandtl had produced the essential idea a few years earlier while studying steam nozzles, which is the industrial context that drove most early compressible work. The application to aerofoils waited until Ackeret’s linear theory in 1925, and the application to actual aircraft until the 1940s.

There is a pattern in this field worth noting as it closes: the mathematics of supersonic flow was essentially complete thirty years before anything flew supersonically, and the thing that had to arrive was not theory but engines.

Where the ladder goes next #

Compression and expansion are now both in hand, and between them they solve any supersonic section exactly.

What comes out of that solution is a drag — in a steady, inviscid flow, on a closed body, which is precisely the situation d’Alembert’s paradox says has no drag at all. Every hypothesis of that proof still holds except one, and the consequence is a whole category of drag that has nothing to rub against.