The CJ theory of Lesson 3 never needed to know how fast the chemistry is: it treats the detonation as an infinitely thin sheet that turns fresh gas into burnt gas. But chemistry takes time. What happens in the half-millimetre behind the shock?
Shock first, fire second
During the Second World War, three people working independently — Yakov Zel'dovich in the Soviet Union, John von Neumann in the United States and Werner Döring in Germany — arrived at the same picture of a detonation's insides, now called the ZND modelZel’dovich, von Neumann and Döring’s picture of a detonation: an inert shock, then an induction zone where little happens, then a fast reaction zone that releases the heat.:
- A shock. The leading front is an ordinary, chemically inert shock wave, a fraction of a micrometre thick. It compresses and heats the fresh gas almost instantly — too fast for any chemistry. The state just behind it is the von Neumann spikeThe pressure peak just behind the leading shock, before any heat is released — roughly twice the CJ pressure.: roughly twice the CJ pressure, at a temperature of 1500–2000 K.
- An induction zone. Behind the spike, nothing much seems to happen. The gas drifts away from the shock while chain-branching reactions multiply radicals — H, O, OH — exponentially. Pressure and temperature stay almost flat. This Induction zoneThe region behind the shock where radicals build up but almost no heat is released yet. Its length Δ is extraordinarily sensitive to the post-shock temperature. has a length of a few tenths of a millimetre in hydrogen–air.
- A reaction zone. Once the radical pool is large enough, the fuel burns in a rush. The heat release accelerates the gas away from the shock; pressure falls and temperature rises, until the state reaches the CJ point — sonic relative to the front.
The heat released in the reaction zone pushes on the shock from behind, like a piston. The shock, in turn, heats the gas enough to start the reaction. That mutual support is what holds a detonation together.
The most sensitive number in detonation physics
The induction time follows an Arrhenius law: . For the activation energies of real fuels, that exponential is ferociously steep. A 5% drop in the shock speed lowers the von Neumann temperature by only about 8%, yet that more than doubles the induction length.
Going deeper
Going deeper: the sensitivity in one line
Write the induction time as . Differentiating, , the reduced activation energy. In the course's model, is about 8.6 for hydrogen–air and 8–11 for hydrocarbon–air mixtures (estimates from detailed chemistry put hydrogen–air lower, nearer 5–6). Since a strong shock has , a fractional change in the wave speed changes the induction time by a fraction of about — a 1% slowdown can lengthen the induction zone by 10–25%. The longer zone delays the heat release, which weakens the support for the shock, which slows it further: a feedback loop that a planar front cannot escape. That is why no real detonation is planar.
Watching one form
The ZND profile is a steady solution: it assumes the wave has been running forever. How does a real detonation get there? The simulation below uses the reactive version of the Euler solver from Lesson 2. A blast at the closed end of a tube drives a strong shock into the mixture. If the shock is hot enough, the gas behind it ignites after a short delay, and the reaction zone catches up with the shock and begins to push it. The wave starts out overdriven — faster than CJ, pushed by the initiator's pressure — then relaxes onto the Chapman–Jouguet speed.
Notice also the long, gentle decline in pressure behind the front: the Taylor expansion wave. The burnt gas has to come to rest against the closed end of the tube, and the expansion that slows it travels with the detonation, stretching as it goes. Nearly 40% of the CJ pressure (about three-eighths) remains on the closed end — the push a pulse detonation engine (Lesson 7) uses for thrust.
- The ZND model: an inert shock to the von Neumann spike, a quiet induction zone, then a reaction zone that releases the heat and relaxes to the CJ state.
- The heat release supports the shock; the shock ignites the gas. Their coupling makes a detonation.
- The induction length (a few tenths of a millimetre for hydrogen–air) is exponentially sensitive to the shock strength — the root of instability and failure.
- Behind the front, a Taylor expansion brings the burnt gas to rest against a closed end, leaving nearly 40% of the CJ pressure.
Next: the ZND picture is planar. Real fronts are anything but — and the pattern they make sets the size of every detonation engine.