The discontinuity that has a thickness

Every ratio across a shock on this site comes from mass, momentum and energy, and not one of those three equations mentions viscosity. That is why the jump conditions work: the answer either side is independent of how the gas gets from one state to the other, so the same relations serve a shock in air, in a shock tube, in a supernova remnant and in a numerical scheme.

The thickness is the opposite. It is entirely viscosity, and the shock is a discontinuity only in a theory that has thrown viscosity away.

The reduction #

Putting it back gives a steady one-dimensional profile, and at Prandtl number $3/4$ the whole problem collapses to a first-order equation.

Mass gives $\rho u = m$. Momentum gives $p + mu - \tfrac43\mu u' = P$. And at $\mathrm{Pr} = 3/4$ the viscous and conductive fluxes combine so that the stagnation enthalpy is exactly constant, $h + \tfrac12u^2 = H$ — which is the whole reason that value is special. Eliminating $p$ and $h$ leaves

$$\frac{4\gamma\mu}{3m}\,u\,\frac{\mathrm du}{\mathrm dx} = \tfrac12(\gamma+1)(u-u_1)(u-u_2),$$

whose right side vanishes at the two end states.

That is the jump conditions reappearing as the equilibria of a differential equation. They were not put in; they are where the right-hand side is zero.

The profile #

Integrating outwards from the steepest point in both directions gives a monotone profile running from $u_1$ to $u_2$, with its measured maximum gradient matching the closed form to one part in ten thousand.

The gradient is steepest at $u = \sqrt{u_1u_2}$ — the geometric mean, which comes out of maximising $(u-u_1)(u-u_2)/u$ and is a small pleasure of the algebra. Defining the thickness as the velocity drop over that maximum gradient gives

$$\delta = \frac{8\gamma\mu(\sqrt{u_1}+\sqrt{u_2})}{3\rho_1u_1(\gamma+1)(\sqrt{u_1}-\sqrt{u_2})}.$$

The numbers #

For air at sea level, with $\mu = 1.79\times10^{-5}$ and a mean free path of 64 nanometres:

Below the line the continuum description has produced a structure thinner than the distance a molecule travels between collisions. That is not a small error; it is the theory being used outside the hypothesis that produced it, and measured shock thicknesses at those Mach numbers are several times larger than this.

The end states, meanwhile, remain exact. The jump conditions contain no transport coefficient, so nothing about the failure of the interior model touches them.

The two halves of a shock calculation #

Putting that division plainly, because it is the essay’s point:

Set by the conservation laws, and containing no viscosity: the two states, every ratio between them, the entropy rise, the total-pressure loss, and everything in this collection’s other three shock essays. Exact at any Mach number.

Set by the transport, and containing nothing else: the thickness, the shape of the profile, and the requirement of $\mathrm{Pr} = 3/4$ for a closed form. Outside its own range above Mach 3.

The measurement that separates them #

The cleanest demonstration is a sweep in which only the viscosity changes.

Scale $\mu$ from a quarter of air’s to four times it, at Mach 2. The thickness follows exactly — the ratios are the scale factors to nine decimal places, a range of sixteen — and the entropy rise does not move at all, at $\Delta s/R = 0.327291106$ in every row.

The irreversibility is decided by the two end states and the distance over which it happens is decided by the transport. Those are different questions and the word shock runs them together.

That is a strange thing to be true. The entropy production is caused by viscosity and conduction — there is no other mechanism — and its total is independent of how much of either there is. Halving the viscosity halves the gradients, quarters the dissipation rate per unit volume, and doubles the distance over which it acts, and the product is unchanged.

Reading the phase plane #

The first-order equation deserves a little more attention than a figure caption, because the way it is solved is unusually clean and the structure it reveals is general.

$u' = k(u-u_1)(u-u_2)/u$ is an autonomous first-order equation, so its behaviour is entirely determined by the sign of the right-hand side. Between the two roots the product is negative and $u$ decreases with $x$: a profile released anywhere in the interval runs monotonically from $u_1$ to $u_2$ and cannot do anything else. There is no oscillation, no overshoot, and no possibility of a non-monotone solution.

Outside the interval the product is positive and the solution runs away, which is the equation saying that a shock cannot connect any other pair of states. Together those two facts are a proof of existence and uniqueness for the profile, obtained by looking at a quadratic.

And the two equilibria are approached exponentially, at rates $k(u_1-u_2)/u_1$ and $k(u_1-u_2)/u_2$ — so the profile has infinite extent in principle and a well-defined thickness in practice, which is why the thickness has to be defined rather than measured off the ends. The maximum-slope definition used here is the usual one; a definition based on where the velocity has fallen to within one per cent of $u_2$ gives a number two or three times larger, and papers quoting shock thicknesses do not always say which they mean.

Where the entropy is actually made #

The constancy of the entropy rise under a change of viscosity is worth unpacking, because it is the kind of result that is easy to state and hard to believe.

Entropy is produced inside the shock at a rate $\Phi/T$ where $\Phi$ is the viscous dissipation, which goes as $\mu(\mathrm du/\mathrm dx)^2$. Halving $\mu$ halves that prefactor. It also doubles the thickness, which halves the gradient, which quarters the square. So the production rate per unit volume falls by a factor of eight — and the volume over which it acts doubles, and the time a parcel spends inside doubles as well.

Working through: the entropy produced per unit mass is $\int(\Phi/T)\,\mathrm dt$ along a particle path, and every factor of $\mu$ cancels between the dissipation and the residence time. The answer is whatever the end states demand.

The physical reading is that the second law is a constraint on the states and the viscosity is a mechanism for satisfying it. A gas crossing a shock must produce a definite amount of entropy, because the states either side are fixed by conservation; the transport coefficients decide only how quickly and over what distance. A gas with less viscosity spreads the same production over a longer distance, and a gas with none at all — the inviscid limit — produces it in zero distance, which is where the discontinuity comes from.

That is why the entropy rise across a shock can be computed with no transport properties at all, and it is one of the more surprising things in the subject.

Where the thickness comes from, physically #

The competition is worth naming. A shock is a compression steepening under the mechanism characteristics describes, and viscosity spreading it out. The thickness is where the two balance.

That gives the weak-shock scaling directly: the steepening rate goes as the amplitude and the spreading rate as the viscosity over the width squared, so $\delta\propto\mu/(M-1)$ for a weak shock. At Mach 1.05 the thickness is fifty mean free paths and at Mach 8 it is a third of one, which is that scaling with the strong-shock saturation on the end of it.

The same balance is what sets the thickness of every other internal structure in this collection: a boundary layer is diffusion against advection, a vortex core is diffusion against rotation. A shock is the one where the opposing process is nonlinear steepening rather than a mean flow, which is why it is the only one that gets thinner as the flow gets stronger.

Why the continuum model fails so early #

A boundary layer at ordinary conditions is millimetres thick and comfortably a continuum. A shock at Mach 5 is not, and the reason is that a shock is the only structure in gas dynamics whose thickness is set against a molecular scale rather than against a macroscopic one.

Both the viscosity and the mean free path come from the same kinetic theory: $\mu\sim\rho\bar c\lambda$. So the ratio $\delta/\lambda$ contains no macroscopic length at all, and depends only on the Mach number. A shock does not become more continuum-like in a bigger apparatus, at higher pressure or in a larger vehicle. There is no size of experiment in which a Mach 5 shock is thick.

That puts shock structure in the same category as rarefied flow, which this collection reaches by lowering the density rather than by raising the Mach number, and both arrive at the same failure of the same hypothesis.

What a numerical scheme does instead #

There is a practical consequence that this collection should record, because every computed shock in it was produced this way.

A finite-volume scheme solving the Euler equations has no viscosity in its equations and a great deal of it in its arithmetic. Upwinding, artificial dissipation and flux limiters all add a numerical viscosity whose size is set by the mesh spacing rather than by the gas, and the shocks such a scheme produces are several cells thick for exactly the reason the shocks here are several mean free paths thick — a balance between steepening and diffusion.

The consequence is that a computed shock’s thickness is a property of the mesh and carries no physical information whatever. Refining the mesh makes it thinner without limit, and the end states converge to the right ones because the scheme is conservative and the jump conditions do not care how the transition happened. That is the whole reason shock-capturing works: the arithmetic is allowed to be wrong about the inside because the conservation laws make it right about the outside.

The kinematic-wave essay’s front is the same situation in its simplest setting — a first-order scheme with numerical diffusion, producing a front at the correct chord speed and the wrong width — and reading the two together is worth doing, because the discipline they share is one of the most useful in computational fluid dynamics: trust a computed shock’s position and states, never its profile.

What is measured, and how #

Since the model fails above Mach 3, it is worth saying what the measurements are.

Shock thickness is measured by electron-beam absorption or by laser interferometry across a shock in a low-density wind tunnel, where lowering the density raises the mean free path until the structure is millimetres rather than nanometres. That is the same trick rarefied-flow work uses throughout, and it is legitimate because the Knudsen-number dependence is what is being measured.

The results are consistent and unflattering to the Navier–Stokes description. Measured thicknesses exceed the Navier–Stokes prediction by a factor of two or more above Mach 2, and the density profile is asymmetric in a way the model cannot produce. Solutions of the Boltzmann equation, or direct simulation of the molecules, reproduce both.

So this is one of the small number of places in fluid mechanics where the continuum equations are known to be quantitatively wrong and the reason is understood exactly. It is worth having in a collection that spends most of its length on those equations being right.

The Prandtl number, and what happens without it #

$\mathrm{Pr} = 3/4$ is close to air’s actual 0.71 and it is not air’s actual value, and the exactness above depends on it.

At that value the viscous and conductive fluxes in the energy equation combine into a total derivative of the stagnation enthalpy, so $h_0$ is exactly constant through the shock and the system reduces to one equation. At any other Prandtl number it does not, the energy equation stays independent, and the structure is a two-dimensional phase-plane problem with no closed form.

The consequences are qualitative as well as quantitative. At $\mathrm{Pr} < 3/4$ the temperature profile overshoots — there is a region inside the shock hotter than the downstream temperature — and at $\mathrm{Pr} > 3/4$ it does not. That overshoot is real and measured, and it is invisible in the model here.

So the closed form is bought with a coincidence about a dimensionless number, and it is worth knowing which results survive dropping it. The thickness’s scaling does; its constant does not; and the temperature profile’s shape is entirely a casualty.

What is not in the flow #

The viscosity — which is the quantity the jump conditions were remarkable for not containing.

That is an unusual entry on the list these essays keep. Everywhere else the missing quantity is something a model has abstracted away and which the reader has to be told. Here it is a quantity the rest of the field is proud of not needing, and the pride is justified: not needing it is what makes the jump conditions exact.

The price is that the same theory has nothing to say about the inside of the wave, and the moment a question is asked about the inside — how thick, what shape, does the temperature overshoot, how much radiation is emitted — a transport coefficient has to be supplied and the answer stops being universal.

The deeper objection, which is not about length #

The table above condemns the model by comparing two lengths, and there is a stronger objection that does not depend on measuring anything. It is about whether the quantities in the equations are defined at all inside the wave.

The Navier–Stokes constitutive relations — stress proportional to strain rate, heat flux proportional to temperature gradient — are not fundamental. They are the first term of an expansion of the Boltzmann equation in the Knudsen number, valid when the molecular velocity distribution is a small perturbation of a local Maxwellian. Inside a strong shock the Knudsen number is of order one, so the expansion has no small parameter, and the failure is in the constitutive law rather than in the conservation equations it is substituted into.

What the distribution actually looks like there is not a perturbed Maxwellian at all. It is bimodal — recognisably a mixture of the upstream population and the downstream one, with the fast cold molecules and the slow hot ones coexisting in the same volume and not yet having collided enough to merge. Mott-Smith built a model of the shock on exactly that picture in 1951, treating the profile as a changing weighted sum of the two end-state Maxwellians, and it gives thicknesses far closer to measurement than the continuum solution does.

And with a bimodal distribution, “the temperature” is not one number. The mean square molecular velocity along the flow direction and across it are different, substantially so — the compression acts on one component and not the others — so a shock has a parallel translational temperature and a transverse one that disagree by a large factor at its steepest point. A local thermodynamic temperature, and therefore a local entropy and a local pressure, are not defined inside a strong shock.

Which sharpens the essay’s division a last time. The end states are exact because they are equilibrium states, connected by conservation laws that hold whatever happens between. Between them lies a region in which the gas is not in equilibrium, its temperature is not a scalar, and the equations that drew the profile above were never entitled to be there.

The model limit #

Four.

Prandtl number 3/4, discussed above.

Constant viscosity. Air’s rises roughly as $T^{0.7}$, and the temperature inside a Mach 5 shock rises by a factor of six, so the viscosity is not constant across the very structure being computed. That alone changes the thickness by tens of per cent.

No caloric imperfection. At Mach 5 the temperature behind the shock is well into the vibrational range, so $\gamma$ is not 1.4 — and the relaxation time for vibration is longer than the shock’s own transit time, so the structure has a second, much thicker layer behind it in which the vibrational mode fills. Measured shock “thicknesses” at high Mach number are often that relaxation zone rather than the viscous one.

And the continuum hypothesis itself, which the essay’s central table shows failing at Mach 3. Below that the model is describing something real; above it, the honest statement is that the end states are exact and the profile between them is a picture of a structure that the equations drawing it cannot support.