A gas that has not decided to react yet

The other branch of the same curve computes a detonation from conservation alone: the Hugoniot curve with heat release added, the Chapman-Jouguet point where it is tangent to the Rayleigh line, and the forbidden gap between the two branches. It ends by noting a length the accounting has no room for.

This essay is that length. It is the distance between the shock and the reaction, and it exists because a gas that has been shocked does not begin reacting immediately — it waits.

The waiting #

A detonation is a shock followed by combustion, and the two are separated. The gas crosses the shock in a few mean free paths, arrives hot and compressed, and then does nothing measurable for an interval called the induction time.

What is happening during it is chain branching: radicals are being produced, in a pool that grows exponentially and is far too small to change the temperature or the pressure. When the pool reaches a threshold the heat release happens quickly, and it is that heat release which drives the shock.

So a detonation front has three parts in a row — a shock, an induction zone, and a reaction zone — and the middle one is a region in which the gas is carrying a decision it has already made and not yet acted on.

The clock is exponential #

The induction time follows an Arrhenius law: it is an exponential of an activation temperature divided by the gas temperature. That makes the induction length — the post-shock flow speed times that time — violently sensitive to the shock’s strength.

Doubling the Mach number shortens the induction length by a factor of 3,373 while the post-shock temperature rises by only 3.3. The exponent of length against Mach number reads −7.55 near Mach six, and the sensitivity column falls from 12.4 to 3.7 across the sweep, so the feedback below is strongest exactly where a detonation is weakest.

At Mach 4 in the model here it is seventy millimetres. At Mach 8 it is twenty-one microns. A factor of two in the Mach number is a factor of three thousand in the length.

Plotted against the post-shock temperature rather than the Mach number, the slope is 7.55 — which is the activation temperature divided by the post-shock temperature, and is the standard measure of how sensitive a reaction is. Anything above about five is a reaction that is effectively a switch.

What that sensitivity does to a front #

A one per cent change in the leading shock’s Mach number shortens the induction length by 15 per cent. A ten per cent change shortens it by a factor of 3.4.

Now consider a front that is very slightly not plane. A one per cent rise in shock Mach number lifts the post-shock temperature by 1.7 per cent and shortens the induction time by a factor of 1.146; two per cent gives 1.309; five per cent gives 1.915; ten per cent gives 3.405. Where the shock is a little stronger the reaction happens a little sooner, so the heat release is closer to the shock, so it pushes the shock harder, so the shock gets stronger still. Where the shock is a little weaker the opposite happens.

That is a positive feedback with an exponential gain in it, and it is why a detonation front is never plane. Real detonations propagate as a cellular structure — a mesh of transverse waves and triple points sweeping across the front — and the cell size is the one length anybody measures.

The cell size is empirically about thirty times the induction length — 11.9 mm at Mach six, 58 mm at Mach five, and over two metres at Mach four — and it inherits all of the induction length’s sensitivity — which is why published cell sizes for a given mixture vary by a factor of two between laboratories and why they are quoted with an equivalence ratio, an initial pressure and a temperature attached.

How long a detonation front is, in numbers #

It is worth assembling the lengths, because the ratios between them are what make the problem hard and they span an unusual range.

The shock itself is a few mean free paths — a fraction of a micron at atmospheric pressure. It is a discontinuity for every purpose in this essay.

The induction zone is a millimetre or two at Mach 5, and microns at Mach 8. It is thousands of shock thicknesses and a fraction of a cell.

The reaction zone is comparable with the induction zone or shorter, depending on the mixture.

And the cell is tens of millimetres — thirty induction lengths, empirically.

So a detonation front spans about five orders of magnitude between its thinnest and thickest features — a shock a few mean free paths thick, an induction zone of 0.215 mm at Mach six, a cell of 11.9 mm and a tube of metres — and a computation that wants to resolve the induction zone inside a cell inside a tube is asking for four or five decades of grid. That is why detonation simulation is expensive and why almost all of it is done with reduced kinetics.

The same span is what makes the experimental picture come from soot foils rather than from probes: a foil records the whole cell pattern at once with a resolution set by the soot, where any probe averages over more than the feature it is trying to see.

What the solver computed, and how it was checked #

The leading shock is solved with the frozen jump conditions and the induction time is an Arrhenius expression in the post-shock temperature. Nothing here is a reaction mechanism: it is a two-parameter model with an activation temperature and a pre-exponential, and the essay is about the structure of the dependence rather than about any particular mixture’s numbers.

Two checks. That a five per cent change in the shock’s Mach number changes the induction length by more than a factor of one and a half — it gives 1.9 — which is the sensitivity the whole argument rests on. And that the local exponent in the post-shock temperature exceeds five, which is what distinguishes a switch from a smooth dependence.

Why this is a memory #

The induction zone is a memory in a precise sense worth stating.

The gas in it is in a state that looks, to every mechanical measurement, exactly like shocked unreacted gas: same pressure, same density, same temperature to within a fraction of a per cent. What distinguishes it is an invisible internal variable — the radical pool — which is growing and which will shortly change everything.

So the state of the gas at a point in the induction zone is not determined by its mechanical state. It depends on how long ago it crossed the shock, which is a fact about its history rather than about its condition, and it is exactly the shape of the relaxation memories elsewhere in this collection: the fluid that has not finished its last deformation carries a stress with no mechanical signature until it is released, and a gas that has not finished being shocked carries a vibrational population.

What makes this one different is the gain. The other two relax with a time constant that varies weakly with conditions; this one has a clock whose rate is exponential in the temperature. Between Mach 4 and Mach 8 the post-shock temperature rises by a factor of 3.31 and the induction time falls by 5,542 — so a small change in the past produces an enormous change in when the future happens.

Where the sensitivity shows up #

Detonation cell size, and therefore detonability. Whether a mixture in a given pipe will sustain a detonation is decided by whether the cell size fits — roughly, a tube must be wider than about a cell, which is 11.9 mm at Mach six and 2.1 metres at Mach four. Since the cell size is exponentially sensitive to the initial temperature and pressure, detonability limits are sharp functions of conditions.

Deflagration-to-detonation transition. A flame accelerating in a tube compresses the gas ahead of it, which raises the temperature, which shortens the induction time, which brings the reaction closer to the leading shock. The transition happens when that feedback runs away, and its suddenness is the Arrhenius exponent.

Engine knock. A petrol engine’s end gas is compressed by the advancing flame, and it autoignites if its induction time falls below the time the flame needs to arrive. The octane rating of a fuel is, in effect, a measure of its induction time under those conditions, and the sharpness of the knock threshold is the same exponent.

And detonation-based propulsion. A rotating detonation engine’s operation depends on the wave completing a circuit in a time compatible with the refill; the induction length sets the wave’s structure, and the sensitivity is why such engines are hard to keep in a stable mode.

The vibrational relaxation this reaction sits behind is the first essay in this field, and the same solver draws it.

What the accounting misses, restated #

The conservation analysis of the other branch of the same curve fixes the end states and the wave speed and says nothing about the structure between them. That is exactly right and it is exactly why it cannot predict a cell size.

The reason is worth stating in the terms used here. Conservation laws are statements about totals across a region, so they constrain the difference between the two ends and leave everything inside free — which is exact in the total, free in the profile, arriving in a reactive gas. The Chapman-Jouguet condition adds one more statement, that the flow behind is sonic, and that is enough to pick a wave speed and still not enough to pick a structure.

Everything about a detonation that is hard is inside the region the conservation laws integrate over, and the induction length is the first thing in it.

What sets the threshold, and why it is sharp #

The exponential clock has a consequence for detonability that is worth separating from the cell size, because it is the practically important one.

A mixture detonates in a tube if the cell structure fits: roughly, the tube’s diameter must exceed the cell size, and a detonation entering a wider space needs a critical diameter of about thirteen cells to survive. Both are geometric conditions on a length that is exponentially sensitive to conditions.

So the detonability limit — the range of equivalence ratio, or of initial pressure, over which a mixture will sustain a detonation — is sharp, and it is sharp for a reason that has nothing to do with the geometry. It is sharp because the cell size crosses the tube diameter over a very narrow range of conditions, since it is moving exponentially and the diameter is not moving at all.

A threshold that looks like a property of the apparatus is a property of an exponential, and that diagnosis transfers: whenever a limit is unexpectedly sharp, look for a rate with an activation energy in it rather than for a mechanism with a discontinuity.

Why the front is unstable, in one sentence more #

It is worth being careful about the instability argument, because the feedback described above is necessary and not sufficient.

A perturbation that strengthens the shock shortens the induction length, which moves the heat release closer, which strengthens the shock further. That is the amplifying loop. What limits it is that a stronger shock also raises the post-shock pressure, which the expansion behind the front relieves — and the competition between the two is what sets the cell size rather than allowing an unbounded growth.

So the cell size is the scale at which the amplification and the relief balance, and it is proportional to the induction length because the amplification’s own scale is. The factor of thirty is where the balance happens to sit, and it is empirical rather than derived — which is the honest state of the subject.

What is measured, and what it is a measurement of #

The practical form of all this is worth stating, because the quantity everybody quotes is not the one the theory produces.

Cell size is what is measured. A soot-coated foil in a detonation tube records the tracks of the triple points, and the diamond pattern they leave is read off with a ruler. That measurement is straightforward, reproducible within a factor, and available for hundreds of mixtures.

Induction length is what is computed. A kinetic mechanism and a shock give an induction time directly, and the length follows. That calculation is reliable to the accuracy of the mechanism, which for well-studied fuels is good.

And the factor between them is a correlation. Nothing derives thirty. It varies with the mixture, with the dilution and with the initial pressure, and the practice of using a single factor is the weakest link in every detonability estimate.

That structure — a measurable quantity, a computable quantity, and a fitted factor joining them — is common in this collection and is always the place to look for the uncertainty. A guess with a constant in it is the same shape in a completely different subject.

What the picture cannot show #

The structure diagram draws three bands of comparable width and the real proportions depend entirely on the mixture: for hydrogen-oxygen the reaction zone is far shorter than the induction zone, and for a hydrocarbon it is comparable. Only the ordering is universal.

Nothing here draws a cell. The cellular structure is a three-dimensional unsteady pattern of transverse waves, and this collection’s solver cannot compute one; the cell size quoted is the induction length times an empirical factor.

The exponential clock, elsewhere in this collection #

An exponential dependence on a state variable turns a smooth relation into a switch, and this collection has met the structure before in two other places worth naming.

A puff’s lifetime. A puff that does not know how old it is finds a lifetime that is an exponential of an exponential in the Reynolds number, and observes that a threshold which is perfectly sharp took a century to measure because the quantity crossing it moves by four orders of magnitude over a few per cent.

And a stability boundary. Any process whose rate is Arrhenius has the same character: nothing happens, and then everything does, over a range of conditions that looks like a discontinuity from outside.

The general lesson is about measurement rather than physics. A quantity with an exponential in it is measured as a threshold and modelled as a rate, and the two descriptions have very different error behaviour: a ten per cent error in the rate is a fraction of a per cent in the threshold, and a fraction of a per cent in the conditions is a factor in the rate.

That asymmetry is why detonability limits are quoted so confidently and induction times so cautiously, and why the two literatures rarely quote each other’s uncertainties.

What the delay does to a measurement of ignition #

The exponential clock makes ignition temperature a difficult quantity to measure, and the difficulty has a shape worth knowing.

A measurement is always a hold of finite length. Whatever apparatus is used, the mixture is brought to a condition and kept there for some interval, and what is reported is whether it ignited within it. So the reported threshold is the temperature at which the induction time equals the apparatus’s own hold.

Two apparatus with different hold times therefore report different ignition temperatures, and the difference is not experimental scatter — it is the curve being sampled at two places.

And the shape of the curve makes the disagreement small in temperature and large in time. Because the induction time falls exponentially, a factor of a hundred in hold time moves the reported temperature by tens of degrees, which looks like agreement and is a hundredfold disagreement about the quantity that matters.

Who found it, and when #

The ZND structure — shock, induction, reaction — is Zel’dovich’s, von Neumann’s and Döring’s, independently and during the Second World War. It replaced the Chapman-Jouguet picture of a discontinuity with a structure, and it is what makes a detonation a fluid-mechanical object rather than a thermodynamic one.

The cellular instability was seen in soot-foil records in the 1950s and 1960s, and its connection to the induction length’s sensitivity was established through the 1970s. That the ZND structure is unstable was a considerable surprise: the theory that had just explained detonations turned out to describe a solution nature does not use.

Limits recorded rather than smoothed over #

A two-parameter model, not a mechanism. One activation temperature and one pre-exponential. Real chain-branching kinetics have dozens of reactions and their induction time is not a single Arrhenius expression, though it is often well fitted by one over a range.

The activation temperature is chosen. Fifteen thousand kelvin, which gives a sensitivity of 8.6 at Mach 5 — in the range measured for hydrocarbon mixtures. Every length quoted scales with it exponentially.

No reaction zone. The heat release is drawn and not computed, so nothing here closes the loop between the release and the shock it drives. That closure is what makes the detonation self-sustaining and it is a separate calculation.

Frozen jump conditions. The shock is solved with a constant gamma and no vibrational relaxation, which at these temperatures is not right — a gas that has not finished being shocked is the correction, and it lowers the post-shock temperature and therefore lengthens the induction zone.

And the cell size is an empirical multiple. Thirty times the induction length is the usual figure and it varies between mixtures by a factor of several. The sensitivity is the transferable result; the cell size is a correlation with a well-known scatter.