What Is a Detonation?

A self-sustaining supersonic combustion wave — fundamentally different from a flame. The shock wave itself triggers the reaction, and the reaction sustains the shock.

Flame vs Explosion

Light a candle. The flame spreads slowly through the wax vapor — a few centimeters per second. The hot gas expands gently. The unburnt fuel ahead of the flame doesn't know it's coming until the heat reaches it by conduction. This is a deflagration: a subsonic combustion wave.

Now detonate dynamite. The explosion rips through the material at thousands of meters per second — faster than sound in the explosive. The unburnt material ahead is hit by a shock wave so violent that it compresses and heats the fuel enough to ignite it instantly. The shock and the reaction are locked together, each sustaining the other. This is a detonation.

The key differences

Deflagration (flame): Subsonic. Heat diffuses forward into the fuel. Gentle pressure rise. The fuel "knows" the flame is coming.
Detonation: Supersonic. A shock wave slams into the fuel, compressing and igniting it instantly. Enormous pressure spike. The fuel has no warning.

Think of it this way: in a deflagration, the fire chases the fuel. In a detonation, the shock wave catches the fuel by surprise.

The UFC model proposes that the Big Bang was not an explosion into space — it was a detonation wave propagating through a pre-existing superfluid, converting ground-state fluid into the hot plasma we call the universe.

Chapman-Jouguet Theory

In the 1890s, Chapman and Jouguet independently worked out the theory of steady-state detonation waves. Their key insight: a detonation has a minimum velocity at which it can sustain itself.

The Hugoniot Curve

Consider a shock wave converting reactants (state 1) to products (state 2). Conservation of mass, momentum, and energy constrain the possible end states. Plot all possible product states in the $(P, V)$ plane (pressure vs specific volume) — you get the Hugoniot curve.

The Hugoniot for detonation products sits above the inert shock Hugoniot because chemical energy has been released — the products are hotter and at higher pressure for the same compression.

The Rayleigh Line

Conservation of mass and momentum across the shock front gives a straight line in $(P, V)$ space called the Rayleigh line. Its slope is proportional to $-D^2$, where $D$ is the detonation velocity. Every point on this line satisfies the jump conditions for a shock at speed $D$.

The CJ Point

The detonation state must lie on both the Hugoniot curve and a Rayleigh line. The Rayleigh line with the smallest slope (lowest $D$) that still touches the Hugoniot is tangent to it — this tangent point is the Chapman-Jouguet (CJ) point.

At the CJ point:

The detonation velocity $D_\text{CJ}$ is the minimum self-sustaining detonation speed
The flow behind the wave is exactly sonic relative to the wave front — information from behind cannot catch up to the shock
The detonation is self-sustaining — it needs no external energy input

This is a thermodynamic inevitability: any overdriven detonation (faster than CJ) will relax back toward the CJ state. The CJ detonation is an attractor.

Rankine-Hugoniot Jump Conditions

Setup

A planar shock wave moves at velocity $D$ into undisturbed fuel (state 1: $\rho_1, P_1, u_1 = 0$). Behind the shock, the products are at state 2: $\rho_2, P_2, u_2$. In the shock frame, the fuel enters at $w_1 = D$ and products leave at $w_2 = D - u_2$.

Mass conservation

$$\rho_1 w_1 = \rho_2 w_2 \quad \Longrightarrow \quad \rho_1 D = \rho_2 (D - u_2)$$

Whatever mass enters the shock per unit time must leave it.

Momentum conservation

$$P_1 + \rho_1 w_1^2 = P_2 + \rho_2 w_2^2$$

Pressure force plus momentum flux is continuous across the shock.

Energy conservation

$$e_1 + \frac{P_1}{\rho_1} + \frac{w_1^2}{2} = e_2 + \frac{P_2}{\rho_2} + \frac{w_2^2}{2} + q$$

$e$ is specific internal energy, $q$ is the specific energy released by the chemical reaction (negative = exothermic, so often written with opposite sign convention).

Hugoniot relation

Eliminating velocities, we get the Hugoniot in terms of thermodynamic quantities:

$$e_2 - e_1 = \frac{1}{2}(P_1 + P_2)(v_1 - v_2) - q$$

where $v = 1/\rho$ is specific volume. This defines the locus of all possible end states.

Rayleigh line

From mass and momentum conservation:

$$P_2 - P_1 = \rho_1^2 D^2 (v_1 - v_2)$$

This is a straight line in $(v, P)$ space with slope $-\rho_1^2 D^2$. Steeper = faster detonation.

CJ condition

The CJ point is where the Rayleigh line is tangent to the Hugoniot:

$$\left.\frac{dP}{dv}\right|_{\text{Hugoniot}} = \frac{P_2 - P_1}{v_2 - v_1}$$

At this point, the flow behind the detonation is sonic: $w_2 = c_2$ (local sound speed). This gives:

$$D_\text{CJ}^2 = v_1^2 \frac{P_2 - P_1}{v_1 - v_2}$$

ZND Model

Structure

Zel'dovich, von Neumann, and Doering independently proposed the internal structure of a detonation wave:

Von Neumann spike: An inert shock compresses the fuel to the highest pressure point on the inert Hugoniot. No reaction has occurred yet — just compression.
Reaction zone: The compressed, superheated fuel reacts. As energy is released, the state moves along the Rayleigh line from the von Neumann point toward the CJ point.
CJ plane: Reaction is complete. The products are at the CJ state. Flow is sonic relative to the wave.

Pressure profile

Pressure peaks at the von Neumann spike (inert shock compression), then decreases through the reaction zone as chemical energy converts compression energy into kinetic energy, reaching the CJ pressure at the end of the reaction zone.

The von Neumann spike pressure can be 2-3x the CJ pressure. This is the most destructive part of the wave.

Summary

A detonation is a coupled shock-reaction wave. The CJ velocity $D_\text{CJ}$ is the minimum self-sustaining speed, determined entirely by thermodynamics. The ZND model describes the internal structure: inert shock $\to$ reaction zone $\to$ CJ plane.

Expert Notes

Cellular Instability

The ZND model assumes a perfectly planar, steady detonation. In reality, all detonations are cellular. Erpenbeck (1964) showed that the planar ZND solution is linearly unstable: small transverse perturbations grow and form a complex pattern of interacting shock waves.

The detonation front develops a cellular structure visible on smoked foils: a network of diamond-shaped cells traced by triple points where three shock waves meet (the leading shock, a transverse wave, and a Mach stem).

Triple Points and Cellular Structure

Triple points travel transversely across the detonation front, tracing out the cell boundaries. The cell width $\lambda$ is a fundamental length scale of the detonation — typically 10-100 times the ZND reaction zone length. Regular cell patterns indicate marginally unstable detonations; highly irregular patterns indicate strongly unstable ones.

The cell size $\lambda$ determines:

Critical diameter for detonation propagation: $d_{\text{crit}} \approx 13\lambda$ (round tubes)
Critical energy for direct initiation: $E_c \propto \lambda^3$
Detonation limits: detonation fails when the channel is too narrow to sustain at least one cell

Why This Matters for UFC

The UFC detonation is cellular too. The transverse wave pattern from the cosmological detonation imprints a characteristic scale on the universe — and the cell boundaries, where triple-point collisions create extreme local conditions, are where structure formation is seeded. This is the same instability that creates cells in chemical detonations, scaled up to cosmological dimensions.