The other branch of the same curve

The first essay in this anchor derived every ratio across a discontinuity from mass, momentum and energy alone, and the curve those three laws trace through the $(v, p)$ plane — the Hugoniot — passes through the initial state, because a jump of zero strength satisfies them.

Let the gas burn as it crosses, releasing a heat $q$ per unit mass, and it does not:

$$\frac{\gamma}{\gamma-1}\big(\tilde p\tilde v - 1\big) = \tfrac12(\tilde p - 1)(1 + \tilde v) + \tilde q,$$

with everything scaled on the initial state and $\tilde q = q/RT_1$. The curve lifts away from $(1,1)$, and the point that used to be the trivial solution is now in a gap that no wave can occupy.

Why the gap is forbidden #

The Rayleigh line — mass and momentum together — is the straight line from the initial state to the final one, and its slope is

$$\frac{\tilde p - 1}{1 - \tilde v} = \gamma M_1^2 > 0.$$

A slope that must be positive in these coordinates means the final state is either up-and-left of the initial one, which is a compression, or down-and-right, which is an expansion. The quadrant up-and-right — higher pressure and higher specific volume — needs a negative $\gamma M_1^2$, which is a negative square, and is therefore empty.

With heat release the Hugoniot passes through exactly that quadrant, from $\tilde v = 1$ to $\tilde v = 1 + \tilde q(\gamma-1)/\gamma$. Every point on that stretch satisfies all three conservation laws and no wave can reach it.

So a burning gas has no weak waves available to it. It must either run supersonically as a detonation — compression, up and to the left — or subsonically as a deflagration — expansion, down and to the right. There is nothing in between, and the width of the nothing is $1.71$ in specific volume for the heat release drawn.

The tangency #

On the detonation branch, draw Rayleigh lines from the initial state at increasing speed. A steep line cuts the branch twice. A shallow one misses it entirely. Between them there is exactly one that touches.

The tangency is the Chapman–Jouguet point, and it is the slowest self-sustaining detonation. Found here by a golden-section search on the wave Mach number along the branch — with nothing assumed about sound anywhere — it comes out at $M_1 = 3.1827$ for the heat release drawn, against a closed form of $\sqrt{1+H} + \sqrt H$ with $H = (\gamma^2-1)\tilde q/2\gamma$, agreeing to fifteen digits.

What the tangency turns out to mean #

The search was told to minimise a speed. What it found has a property nobody asked for:

$$M_2 = 1.0000000133.$$

The burnt gas leaves a Chapman–Jouguet detonation at exactly Mach one. That is a consequence of the tangency rather than an input to it, and it is the physical reason the CJ speed is the one nature picks. A sonic surface behind the wave means the expansion of the products cannot send a signal forward, so the wave is not slowed by anything happening behind it — which is the same argument the throat of a nozzle runs on, in a different setting.

What the same laws will not do #

Now the deflagration branch, and this is what the essay exists for.

Every point on it satisfies the same three conservation laws. The Chapman–Jouguet deflagration — the tangency on that side — is at $M_1 = 0.314$, which is $109$ metres a second in air at 300 K. Below it lies every slower wave, down to a crawl, and conservation cannot say which.

A real laminar flame runs at about half a metre a second.

That factor of two hundred is the whole difference between the two halves of this subject. The detonation speed is a conservation result. The flame speed is a transport result: what puts a flame at half a metre a second is the rate at which heat diffuses forward into unburnt gas and starts it reacting, which is a quantity appearing nowhere in the accounting.

The flame speed, and why it is a different calculation #

For completeness, the shape of the calculation that does give it. A laminar flame speed comes out as

$$S_L \sim \sqrt{\alpha\,\dot\omega/\rho},$$

a geometric mean of a thermal diffusivity and a reaction rate — the Mallard–Le Chatelier estimate, and every refinement of it has the same two ingredients. Neither appears in the Hugoniot. The heat release does, and it fixes the end states of the flame perfectly well; what it cannot fix is how fast the wave travels.

That division is worth carrying, because it is unusually clean. The jump conditions decide where a wave goes to. They do not decide how fast it gets there, unless something extra — a tangency, a sonic condition — selects a point on the branch. The detonation branch has such a selector and the deflagration branch does not.

Reading the diagram #

The $(v, p)$ plane repays a little practice, because almost every statement in this essay is a geometric one on it.

The initial state is a point, and every wave is a straight line from that point to somewhere on the Hugoniot. The line’s slope is minus the square of the mass flux, so a steeper line is a faster wave and a horizontal line is no wave at all.

A compression is up and to the left. Pressure rises, volume falls, the gas slows relative to the wave. Every shock in this collection is on that side.

An expansion is down and to the right. Pressure falls, volume rises. An ordinary gas cannot do that across a discontinuity — entropy forbids it — but a burning gas can, because the heat release supplies the entropy rise that the expansion would otherwise have to take from somewhere.

That last sentence is the one that makes deflagrations possible at all, and it is worth pausing on. A deflagration is an expansion discontinuity, which is exactly the thing the shock essays spend their words ruling out. What makes it legal is that it is not adiabatic: the second law is satisfied with room to spare because the reaction is generating entropy far faster than the expansion is consuming it.

The four waves a reactive gas has #

Putting the two branches together with the two tangencies gives four named waves, and the vocabulary is worth having straight.

A strong detonation sits above the CJ point on the compression branch, leaves the products subsonic, and needs a piston or a stronger wave behind it to keep it there.

The Chapman–Jouguet detonation is the tangency: the slowest self-sustaining compression wave, sonic behind, and the one a tube full of explosive gas settles on.

The Chapman–Jouguet deflagration is the tangency on the expansion branch, sonic behind, at Mach 0.314 here. It is a mathematical boundary rather than an observed wave.

And a weak deflagration is everything slower, which is every real flame ever measured.

The asymmetry is the point. Two of these four are picked out by a condition inside the conservation laws; the other two are picked out by whatever is driving the wave or by whatever transport process is carrying it, and the laws are silent about them.

What the heat release does to the numbers #

One sweep worth having, because it says which quantities are sensitive to the one imported number.

Doubling $\tilde q$ from 3 to 6 moves the CJ detonation Mach number from 2.51 to 3.18 — a 27 per cent change for a 100 per cent change in the heat — because the closed form goes as the square root of $H$ at large $H$. So the detonation speed is remarkably insensitive to how energetic the mixture is, which is why hydrocarbon–air detonations of very different chemistries all run at between 1,700 and 2,000 metres a second.

The forbidden gap, on the other hand, is directly proportional to $\tilde q$: it runs from $\tilde v = 1$ to $1 + \tilde q(\gamma-1)/\gamma$, so doubling the heat doubles its width. A more energetic mixture is not much faster and is much further from being able to burn gently.

And the CJ deflagration speed falls as the heat rises, because a larger expansion is harder to reach. That combination — a wider gap and a slower boundary — is why an energetic mixture is either burning quietly at half a metre a second or running at two kilometres a second, with the range between them wider than for a weak one.

What the model is worth on a real mixture #

Stoichiometric hydrogen in air releases about 3.4 MJ per kilogram. Feeding that into the closed form with $T_1 = 300$ K:

The measured value is between them, and it has to be: the gas on one side of the wave is cold hydrogen and air and the gas on the other is hot steam, and no single $\gamma$ is right for both.

The honest output of a single-γ model is a bracket, and the bracket spans forty per cent. Getting the number properly needs equilibrium chemistry — a composition that is solved for rather than assumed, with the products’ own thermodynamic data — which is a different calculation and not one this collection carries.

The overdriven case, and where the extra energy goes #

The strong branch above the CJ point is not a curiosity: it is what a detonation does when something is pushing it.

A wave driven by a piston, or by a stronger wave behind it, runs at whatever speed the driver imposes, sits on the strong branch, and leaves the products subsonic. Then the products can signal forward, and the wave’s speed is set by the driver rather than by the chemistry. Remove the driver and the wave decays to the CJ point, which is the slowest speed at which it can sustain itself, and stays there.

The weak branch — above CJ, leaving the products supersonic — is not reached by ordinary means at all. Getting to it would require passing through the CJ state, and the reaction cannot proceed through a sonic point in a steady wave. It is a solution of the algebra that the physics does not use, which is a situation this field has met before: the expansion shock satisfies mass, momentum and energy exactly and is ruled out by entropy, and this one is ruled out by an argument about how the wave is reached.

What a detonation actually is, inside #

The Chapman–Jouguet analysis treats the wave as a discontinuity, and it is not one. The Zel’dovich–von Neumann–Döring picture is that a detonation is a shock followed by a reaction zone: the gas is compressed by an ordinary non-reacting shock to a pressure well above the CJ value — the von Neumann spike — and then burns along the Rayleigh line down to the CJ state.

That structure is invisible to the accounting above, which is exactly the point of the accounting. The end states, the speed and the sonic condition are all correct without any of it. What the structure supplies is the thickness, the cell size, and whether the wave can propagate at all in a given tube — all of which are transport and chemistry, and none of which is conservation.

This is the same division the shock’s own structure has: the jump conditions are exact and contain no transport coefficient, and everything about the inside of the wave is transport.

The one length the accounting has no room for #

The Chapman–Jouguet speed contains no length, and neither does anything else above. A real detonation has one, it is measured rather than computed, and it decides every practical question about whether a detonation happens at all.

The front is not planar. It is a shifting network of transverse waves running across it, meeting each other and the leading shock at triple points, and each triple point is a Mach reflection — the three-shock structure of that essay, reproduced by the thousand and travelling with the wave. The triple points scratch their paths into a soot-coated foil on the tube wall, and what they leave is a startlingly regular pattern of diamonds. Its width is the cell size, $\lambda$.

That single length is the practical measure of a mixture’s detonability, and it varies far more between mixtures than any speed does. Acetylene with oxygen has cells around a millimetre; hydrogen in air, ten to fifteen millimetres; methane in air, of the order of thirty centimetres. The CJ speeds of those three differ by less than a factor of two, and their cell sizes by more than two orders of magnitude.

The consequences are all geometric, and they are the ones a safety case turns on.

A detonation cannot propagate in a passage much narrower than its own cell. Below roughly $\lambda/\pi$ in a round tube the transverse waves have nowhere to run and the wave fails — which is why a flame arrester is a stack of narrow channels and why it is specified against a cell size rather than against a speed.

And transmitting a detonation out of a tube into an open space needs a tube about thirteen cells across. Below that, the wave emerging into the unconfined mixture decays to a flame.

Initiating one directly costs an energy that scales as $\lambda^3$, so a mixture with cells thirty times larger is something like twenty thousand times harder to set off directly. That is the whole reason methane in air is treated as a much smaller detonation hazard than hydrogen, despite releasing a comparable amount of heat and running at a comparable speed.

None of it is in the Hugoniot. The accounting gives the speed and the end states exactly and cannot say whether the wave exists.

What is not in the velocity field #

The heat release, and the reaction rate — and they enter in different places.

The heat release enters the energy equation and therefore the Hugoniot. It fixes the branches, the forbidden gap, the CJ speed and every state either side of every wave. It is a single number and the whole of the conservation-law half of the subject follows from it.

The reaction rate enters nowhere in the accounting and decides the flame speed, the detonation’s thickness, its cell structure and whether it survives. It is not a number but a function of temperature and composition, and it is measured or computed by a different discipline.

So a flow with a chemical reaction in it needs two pieces of information that a velocity field does not contain, and the striking thing is how far the first one gets on its own.

Where this leaves the reader #

Three things worth taking away.

The gap is the result. A gas that can burn has no weak waves at all, which is why combustion is either a flame or a detonation and never something intermediate. The transition between them — deflagration-to-detonation — is not a continuous acceleration through the gap but a jump across it, and it is one of the genuinely difficult problems in the subject.

The CJ condition is a sonic condition. It joins the choked throat, the sonic point of a Fanno duct and the crossing of the Fanno and Rayleigh lines as another place where a flow arranges itself so that the downstream cannot signal upstream. That is not a coincidence: all four are the same statement about where a one-dimensional flow can be controlled from.

And the deflagration branch is the honest example of what conservation cannot do. Everywhere else in this collection, a control-volume argument that knows nothing about the interior gets the answer. Here it gets the branch and stops, and the missing physics is a rate.

The model limit #

Three, and they compound.

One γ. The gas changes composition and temperature across the wave and a single ratio of specific heats cannot describe both sides. The bracket above is the measurement of that.

One-step heat release. Real combustion is dozens of reactions with an induction period, and the induction period is what sets whether a detonation is stable or galloping. None of that is here.

And no dissociation. At CJ conditions the products are hot enough to be partly dissociated, which absorbs energy and lowers the effective heat release — so even the single number $q$ that the model does take is not a constant of the mixture but a function of the state it ends up in. The 3.4 MJ/kg used above is a nominal value and the equilibrium calculation gives less.