2.1 Constitutive equations

Introducing the differential operator

(2.1) $$\begin{eqnarray}\displaystyle \frac{\text{D}}{\text{D}t}\equiv \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+u\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}, & & \displaystyle\end{eqnarray}$$

and neglecting the viscosity and the conductivity, the equations governing the one-dimensional flow of a reacting gaseous mixture, considered as an ideal gas $p/\unicode[STIX]{x1D70C}=(c_{p}-c_{v})T$ with constant specific heats per unit of mass $c_{p}$ and $c_{v}$ , $\unicode[STIX]{x1D6FE}\equiv c_{p}/c_{v}=\text{cst.}$ , take the form

(2.2a,b ) $$\begin{eqnarray}\displaystyle \frac{1}{\unicode[STIX]{x1D70C}}\frac{\text{D}}{\text{D}t}\unicode[STIX]{x1D70C}+\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}=0,\quad \frac{\text{D}}{\text{D}t}u=-\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x}, & & \displaystyle\end{eqnarray}$$

(2.3a,b ) $$\begin{eqnarray}\displaystyle \frac{\text{D}}{\text{D}t}\left[\ln T-\frac{(\unicode[STIX]{x1D6FE}-1)}{\unicode[STIX]{x1D6FE}}\ln p\right]=\frac{q_{m}}{c_{p}T}\frac{{\dot{w}}}{t_{r}},\quad \frac{\text{D}}{\text{D}t}Y=\frac{{\dot{w}}}{t_{r}}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}$ , $u$ , $p$ , $T$ , $Y$ , $q_{m}$ , $t_{r}$ and ${\dot{w}}$ are respectively the density, the flow velocity in the laboratory frame, the pressure, the temperature, the progress variable ( $Y=0$ in the initial mixture and $Y=1$ in the burnt gas), the chemical heat release per unit mass of mixture, the reaction time and the non-dimensional reaction rate. The first equation in (2.3) is the equation for the conservation of energy, written in the form of a production of entropy by the rate of heat release. The problem is closed by the chemical kinetics controlling the heat release. In the simplest case of a one-step irreversible reaction sensitive to the temperature, neglecting the pressure effect for simplicity, the reaction rate ${\dot{w}}$ is a function of $Y$ and $T$ , ${\dot{w}}(Y,T)>0$ , where $1-Y$ is the reduced mass fraction of the reactive species. At the initial temperature $T_{u}$ the system is frozen, ${\dot{w}}(Y,T_{u})=0$ , in a composition $Y=0$ , far from the chemical equilibrium in the burnt gas, $Y=1,T=T_{b}$ , ${\dot{w}}(1,T_{b})=0$ . The formulation is easily extended to a multiple-step chemical scheme.

Introducing the sound speed $a=\sqrt{\unicode[STIX]{x1D6FE}p/\unicode[STIX]{x1D70C}}$ and eliminating the temperature in the left-hand side of (2.3) using $(c_{p}-c_{v})T=p/\unicode[STIX]{x1D70C}$ , the density can be eliminated from the Euler equations (2.2) to take the form of two nonlinear hyperbolic equations for $p$ and $u$ , extending the Riemann characteristic equations to reacting mixtures,

(2.4) $$\begin{eqnarray}\displaystyle \frac{1}{\unicode[STIX]{x1D6FE}}\frac{\text{D}^{\pm }}{\text{D}t}\ln p\pm \frac{1}{a}\frac{\text{D}^{\pm }}{\text{D}t}u=\frac{q_{m}}{c_{p}T}\frac{{\dot{w}}(Y,T)}{t_{r}}, & & \displaystyle\end{eqnarray}$$

where the two following differential operators have been introduced

(2.5) $$\begin{eqnarray}\displaystyle \frac{\text{D}^{\pm }}{\text{D}t}\equiv \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+(u\pm a)\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}, & & \displaystyle\end{eqnarray}$$

see Clavin & Searby (Reference Clavin and Searby2016). In an inert gas ( ${\dot{w}}=0$ ) the linearized version of (2.4) reduces to acoustic waves propagating in two opposite directions.

2.2 Boundary conditions and multiple scale analysis

Because of the difference of time scales between the elastic collisions and the inelastic collisions that are responsible for the chemical heat release, a detonation wave is constituted of an inert shock front followed by a reaction zone (Zeldovich–von Neumann–Döring (ZND) structure). Considering the lead shock as a discontinuity in the flow of inert gas, the conditions in the compressed gas at the shock front, called the Neumann state and denoted by $N$ , take the form

(2.6a-c ) $$\begin{eqnarray}\displaystyle \frac{p_{N}}{p_{u}}=1+\frac{2\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D6FE}+1}(M_{u}^{2}-1),\quad \frac{\unicode[STIX]{x1D70C}_{N}}{\unicode[STIX]{x1D70C}_{u}}=\frac{1+(M_{u}^{2}-1)}{1+\displaystyle \frac{\unicode[STIX]{x1D6FE}-1}{\unicode[STIX]{x1D6FE}+1}(M_{u}^{2}-1)},\quad \frac{u_{N}}{a_{u}}=\left(1-\frac{\unicode[STIX]{x1D70C}_{u}}{\unicode[STIX]{x1D70C}_{N}}\right)M_{u}, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where the subscript $u$ denotes the fresh mixture at rest, $M_{u}(t)\equiv {\mathcal{D}}/a_{u}$ . These well-known Rankine–Hugoniot conditions are obtained by the conservation of mass, $\unicode[STIX]{x1D70C}_{N}({\mathcal{D}}-u_{N})=\unicode[STIX]{x1D70C}_{u}{\mathcal{D}}$ , momentum and total energy across the discontinuity. The propagation is supersonic, $M_{u}>1$ , with an increase of entropy, $s_{N}>s_{u}$ , imposed by the dissipative mechanisms inside the inner structure of the shock, see text books.

In addition to the boundary conditions at the Neumann state (2.6), $Y=0$ , ${\dot{w}}(0,T_{N})\neq 0$ , $p=p_{N},\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{N}$ and $u=u_{N}$ , a boundary condition is required in the burnt gas for solving the detonation dynamics. Generally the latter condition corresponds to the state of the burnt gas at rest nearby the piston. The problem simplifies when the length scale $l_{ext}$ of the flow in the burnt gas (approximately the thickness of the rarefaction wave) is larger than the detonation thickness $l_{deto}\approx a_{N}t_{r}$ , $l_{ext}\gg l_{deto}$ , so that the problem is addressed by a multiple scale analysis. The solution can be decomposed into two regions, a thin zone for the internal structure of the detonation wave ( ${\dot{w}}\neq 0$ ) and a large zone for the external flow $u_{ext}(x,t)$ of the rarefaction wave in the burnt gas ( ${\dot{w}}=0$ ). The problem is then solved by matching the inner solution with the external flow. Introducing the equation for the trajectory of the shock front $x=x_{f}(t)$ , the study of the inner structure of the detonation is performed using a non-dimensional coordinate of order unity $\unicode[STIX]{x1D709}=(x-x_{f}(t))/l_{deto}$ . For a detonation propagating from left to right, ${\dot{x}}_{f}\equiv \text{d}x_{f}/\text{d}t>0$ , $u\geqslant 0$ , as in figure 1, the initial mixture and the compressed gas correspond respectively to $\unicode[STIX]{x1D709}>0$ and $\unicode[STIX]{x1D709}<0$ . The matching region corresponds to $-l_{ext}\ll x-x_{f}(t)\ll -l_{deto}$ . In terms of the inner variable $\unicode[STIX]{x1D709}$ , the boundary condition in the burnt gas to be used in the analysis of the inner structure corresponds to

(2.7) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D709}\rightarrow -\infty :\quad u=u_{b} & & \displaystyle\end{eqnarray}$$

where $u_{b}$ is determined by the matching condition with the external flow $u_{ext}(x_{f}(t)+l_{deto}\unicode[STIX]{x1D709},t)$ . To leading order in the limit $l_{ext}\gg l_{deto}$ , the variations of the external flow for variations of $\unicode[STIX]{x1D709}$ of order unity are negligible,

(2.8) $$\begin{eqnarray}\displaystyle u_{b}(t)\approx u_{ext}(x_{f}(t),t). & & \displaystyle\end{eqnarray}$$

Notice that $u_{b}(t)$ is an outcome of the analysis even when the external flow field $u_{ext}(x_{,}t)$ is prescribed. To the leading order of the multiple length scale analysis, $l_{ext}/l_{deto}\gg 1$ , the instantaneous velocity of the lead shock ${\mathcal{D}}(t)=\text{d}x_{f}/\text{d}t$ is obtained by the inner solution of (2.2)–(2.3) with the boundary conditions (2.6) at $\unicode[STIX]{x1D709}=0$ (Neumann state) and (2.7)–(2.8) at $\unicode[STIX]{x1D709}\rightarrow -\infty$ (burnt gas).

3.1 Quasi-isentropic condition in the burnt gas

Consider an overdriven detonation close to the CJ regime, characterized by the small parameter $\unicode[STIX]{x1D700}$

(3.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D700}\equiv \frac{(M_{u}^{2}-M_{u_{CJ}}^{2})}{(M_{u_{CJ}}^{2}-1)}\ll 1. & & \displaystyle\end{eqnarray}$$

The pressure and the density in the burnt gas, expressed in terms of the propagation Mach number $M_{u}$ , can be written in the form of an expansion in powers of $\unicode[STIX]{x1D700}^{1/2}$ , see § A.1. Introducing the notations

(3.2a,b ) $$\begin{eqnarray}\displaystyle A_{1}\equiv \frac{\unicode[STIX]{x1D6FE}(M_{u_{CJ}}^{2}-1)}{1+\unicode[STIX]{x1D6FE}M_{u_{CJ}}^{2}},\quad \text{and}\quad A_{2}\equiv \sqrt{\frac{M_{u_{CJ}}^{2}+1}{M_{u_{CJ}}^{2}}}, & & \displaystyle\end{eqnarray}$$

equation (A 7) leads to the departure of $p_{b}$ and $\unicode[STIX]{x1D708}_{b}\equiv 1/\unicode[STIX]{x1D70C}_{b}$ from their CJ values in the form

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{(p_{b}-p_{b_{CJ}})}{p_{b_{CJ}}}=A_{1}[A_{2}\sqrt{\unicode[STIX]{x1D700}}+\unicode[STIX]{x1D700}]+O(\unicode[STIX]{x1D700}^{3/2}), & \displaystyle\end{eqnarray}$$

(3.4) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FE}\frac{(\unicode[STIX]{x1D708}_{b}-\unicode[STIX]{x1D708}_{b_{CJ}})}{\unicode[STIX]{x1D708}_{b_{CJ}}}=-\left[\frac{(p_{b}-p_{b_{CJ}})}{p_{b_{CJ}}}-A_{1}\frac{(M_{u}^{2}-M_{u_{CJ}}^{2})}{M_{u_{CJ}}^{2}}\right]\frac{M_{u_{CJ}}^{2}}{M_{u}^{2}}, & \displaystyle\end{eqnarray}$$

where the second term in the bracket on the right-hand side of (3.4) is of order $\unicode[STIX]{x1D700}$ . According to the thermodynamic relation $\unicode[STIX]{x1D6FF}s/c_{v}=\unicode[STIX]{x1D6FF}p/p+\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D708}/\unicode[STIX]{x1D708}$ , equations (3.3)–(3.4) show that the isentropic condition is verified at the leading order in the limit $\unicode[STIX]{x1D700}\ll 1$ , $\unicode[STIX]{x1D6FF}s/c_{v}\ll \unicode[STIX]{x1D6FF}p/p$ ,

(3.5a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D700}\ll 1:\quad \unicode[STIX]{x1D6FF}p_{b}/p_{b_{CJ}}=O(\sqrt{\unicode[STIX]{x1D700}}),\quad \unicode[STIX]{x1D6FF}s_{b}/c_{v}=(\unicode[STIX]{x1D6FF}p_{b}/p_{b_{CJ}})+\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D708}_{b}/\unicode[STIX]{x1D708}_{b_{CJ}})=O(\unicode[STIX]{x1D700}), & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}p_{b}=p_{b}-p_{b_{CJ}}$ and $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D708}_{b}=\unicode[STIX]{x1D708}_{b}-\unicode[STIX]{x1D708}_{b_{CJ}}$ .

The condition in (3.1) for a quasi-isentropic approximation, $(M_{u}^{2}-M_{u_{CJ}}^{2})\ll (M_{u_{CJ}}^{2}-1)$ , includes the limit of small heat release $(M_{u_{CJ}}-1)\ll 1$ which is used in § 4,

(3.6) $$\begin{eqnarray}\displaystyle (M_{u}^{2}-M_{u_{CJ}}^{2})\ll (M_{u_{CJ}}^{2}-1)\ll 1. & & \displaystyle\end{eqnarray}$$

3.2 The rarefaction wave behind a slightly overdriven detonation

Consider a slightly overdriven detonation propagating initially with a constant velocity ${\mathcal{D}}_{o}>{\mathcal{D}}_{CJ}$ satisfying the condition $\unicode[STIX]{x1D700}\ll 1$ . From now on, $\unicode[STIX]{x1D700}$ is defined by (3.1) when $M_{u}$ is replaced by the initial propagation Mach number ${\mathcal{D}}_{o}/a_{u}$ ,

(3.7) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D700}\equiv ({\mathcal{D}}_{o}^{2}-{\mathcal{D}}_{CJ}^{2})/({\mathcal{D}}_{CJ}^{2}-a_{u}^{2})\ll 1. & & \displaystyle\end{eqnarray}$$

When the velocity of the detonation front ${\mathcal{D}}(t)$ is decreasing, ${\mathcal{D}}_{u_{CJ}}\leqslant {\mathcal{D}}<{\mathcal{D}}_{o}$ the quantity $({\mathcal{D}}^{2}-{\mathcal{D}}_{CJ}^{2})/({\mathcal{D}}_{CJ}^{2}-a_{u}^{2})$ is smaller than $\unicode[STIX]{x1D700}$ but of the same order as $\unicode[STIX]{x1D700}$ . According to (A 12)–(A 14), the flow $u_{o}(x,t)$ in the centred rarefaction wave, generated in the burnt gas as soon as the piston is suddenly stopped at $t=0$ , takes the form of a self-similar solution of the isentropic Euler equations recalled in § A.2, see also figure 1,

(3.8a,b ) $$\begin{eqnarray}\displaystyle 0<t<t_{i}:\quad 0\leqslant x\leqslant a_{po}t:\,u_{o}=0,\quad (u_{bo}+a_{bo})t<x<{\mathcal{D}}_{o}t+x_{o}:\,u_{o}=u_{bo},\qquad & & \displaystyle\end{eqnarray}$$

(3.9) $$\begin{eqnarray}\displaystyle a_{po}t\leqslant x\leqslant (u_{bo}+a_{bo})t:\quad u_{o}(x,t)=\frac{2}{(\unicode[STIX]{x1D6FE}+1)}\left[\frac{x}{t}-a_{po}\right], & & \displaystyle\end{eqnarray}$$

where $u_{bo}$ and $a_{bo}$ are respectively the flow velocity and the sound speed in the burnt gas adjacent to the detonation front propagating with the constant velocity ${\mathcal{D}}_{o}$ and $a_{po}=a_{bo}-(\unicode[STIX]{x1D6FE}-1)u_{bo}/2$ is the sound speed in the quiescent gas adjacent to the piston at rest. The subscript $o$ means that this flow is associated with the period for which the detonation still propagates with its initial constant velocity ${\mathcal{D}}_{o}$ , the instantaneous position of the detonation front $x_{f}(t)$ being $t<t_{i}:\,x_{f}={\mathcal{D}}_{o}t+x_{o}$ where $x_{o}$ denotes the position of the detonation at $t=0$ i.e. when the piston is suddenly stopped, see figure 1. The flow behind the detonation, $x\leqslant x_{f}(t)$ , is usually no longer given by (3.9) as soon as the leading edge of the rarefaction wave interacts with the detonation front, i.e. for $t>t_{i}$ . Generally, during the decrease of the detonation velocity $\text{d}x_{f}/\text{d}t\equiv {\mathcal{D}}(t)$ from ${\mathcal{D}}_{o}$ down to ${\mathcal{D}}_{CJ}$ ( $t>t_{i}:D_{CJ}\leqslant {\mathcal{D}}(t)<{\mathcal{D}}_{o}$ ), both the entropy and the Riemann invariant $J_{-}\equiv u_{b}-2a_{b}/(\unicode[STIX]{x1D6FE}-1)$ start to vary with ${\mathcal{D}}(t)$ . However, when the initial detonation velocity ${\mathcal{D}}_{o}$ is close to ${\mathcal{D}}_{CJ}$ , these variations are negligible at the leading order in the limit (3.7), $\unicode[STIX]{x1D700}\ll 1$ , as shown now. The ideal gas law $a_{b}/a_{b_{CJ}}=\sqrt{(p_{b}/p_{b_{CJ}})(\unicode[STIX]{x1D70C}_{b_{CJ}}/\unicode[STIX]{x1D70C}_{b})}$ and the mass conservation $\unicode[STIX]{x1D70C}_{b}({\mathcal{D}}-u_{b})=\unicode[STIX]{x1D70C}_{u}{\mathcal{D}}$ yield

(3.10) $$\begin{eqnarray}\displaystyle -\frac{2}{\unicode[STIX]{x1D6FE}-1}\frac{\unicode[STIX]{x1D6FF}a_{b}}{a_{b_{CJ}}} & = & \displaystyle -\frac{1}{\unicode[STIX]{x1D6FE}-1}\left[\frac{\unicode[STIX]{x1D6FF}p_{b}}{p_{b_{CJ}}}+\unicode[STIX]{x1D6FF}\left(\frac{\unicode[STIX]{x1D70C}_{b_{CJ}}}{\unicode[STIX]{x1D70C}_{b}}\right)\right],\end{eqnarray}$$

(3.11) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D6FF}u_{b}}{a_{b_{CJ}}} & = & \displaystyle -\frac{\unicode[STIX]{x1D70C}_{u}{\mathcal{D}}}{\unicode[STIX]{x1D70C}_{b_{CJ}}a_{b_{CJ}}}\unicode[STIX]{x1D6FF}\left(\frac{\unicode[STIX]{x1D70C}_{b_{CJ}}}{\unicode[STIX]{x1D70C}_{b}}\right)+\left(1-\frac{\unicode[STIX]{x1D70C}_{u}}{\unicode[STIX]{x1D70C}_{b}}\right)\frac{{\mathcal{D}}_{CJ}}{a_{b_{CJ}}}\frac{\unicode[STIX]{x1D6FF}{\mathcal{D}}}{{\mathcal{D}}_{CJ}}.\end{eqnarray}$$

Using the sonic condition of the CJ wave, ${\mathcal{D}}_{CJ}-u_{b_{CJ}}=a_{b_{CJ}}$ , $\unicode[STIX]{x1D70C}_{b_{CJ}}a_{b_{CJ}}=\unicode[STIX]{x1D70C}_{u}D_{CJ}$ and neglecting terms of order smaller than $\unicode[STIX]{x1D700}$ , equation (3.11) yields

(3.12) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D6FF}u_{b}}{a_{b_{CJ}}}=-\unicode[STIX]{x1D6FF}\left(\frac{\unicode[STIX]{x1D70C}_{b_{CJ}}}{\unicode[STIX]{x1D70C}_{b}}\right)+\frac{\unicode[STIX]{x1D6FF}{\mathcal{D}}}{{\mathcal{D}}_{CJ}}. & & \displaystyle\end{eqnarray}$$

Adding (3.10) and (3.12) then shows

(3.13) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D6FF}J_{-}}{a_{b_{CJ}}}=-\frac{1}{\unicode[STIX]{x1D6FE}-1}\frac{\unicode[STIX]{x1D6FF}s}{c_{v}}+\frac{\unicode[STIX]{x1D6FF}{\mathcal{D}}}{{\mathcal{D}}_{CJ}}+\text{terms smaller than }\unicode[STIX]{x1D700}. & & \displaystyle\end{eqnarray}$$

When the detonation velocity decreases, the order of magnitude of the variation $\unicode[STIX]{x1D6FF}J_{-}/a_{b_{CJ}}$ is, according to (3.13), the same as for $\unicode[STIX]{x1D6FF}s/c_{v}$ . Then, according to (3.5), $\unicode[STIX]{x1D6FF}s/c_{v}=O(\unicode[STIX]{x1D700})$ , the variation of $J_{-}$ , $\unicode[STIX]{x1D6FF}J_{-}/a_{b_{CJ}}=O(\unicode[STIX]{x1D700})$ , is smaller than the variation of the flow velocity, $\unicode[STIX]{x1D6FF}u_{b}/a_{b_{CJ}}=O(\sqrt{\unicode[STIX]{x1D700}})$ , see (3.3), (3.4). The order of magnitude for the sound speed in the quiescent gas adjacent to the piston at rest is, according to (A 9) and (A 14)–(A 16),

(3.14) $$\begin{eqnarray}\displaystyle (a_{po}-a_{p_{CJ}})/a_{b_{CJ}}=O(\unicode[STIX]{x1D700}). & & \displaystyle\end{eqnarray}$$

Neglecting terms of order $\unicode[STIX]{x1D700}$ , the situation is somehow similar to the case of weak shocks described by Friedrichs (Reference Friedrichs1948); the flow behind the detonation has a constant entropy and a constant Riemann invariant $J_{-}$ . For such flows the $C_{+}$ characteristics (along which the Riemann invariant $J_{+}$ is constant) are straight lines, see Courant & Friedrichs (Reference Courant and Friedrichs1948) and § A.2. Therefore, the rarefaction wave for $t>t_{i}$ is just the continuation of that for $t<t_{i}$ . In other words, restricting attention to the leading order in the limit $\unicode[STIX]{x1D700}\rightarrow 0$ , the decelerating detonation does not influence the burnt gas flow, which continues to be part of the centred rarefaction for $t<t_{i}$ when the detonation is moving at the constant velocity ${\mathcal{D}}_{o}$ , see figure 1.

3.3 Sonic condition during the decay

When the modification to the centred rarefaction wave is neglected, the trajectory of the detonation front $x_{f}(t)$ is simply determined by requiring that the flow velocity (3.9) at $x=x_{f}(t)$ satisfies the jump conditions across a detonation (hydrodynamic discontinuity) propagating at the velocity ${\mathcal{D}}=\text{d}x_{f}/\text{d}t$ , $u_{o}(x_{f},t)=u_{b}(t)$ ,

(3.15) $$\begin{eqnarray}\displaystyle u_{b}(t)=\frac{2}{(\unicode[STIX]{x1D6FE}+1)}\left[\frac{x_{f}}{t}-a_{po}\right]. & & \displaystyle\end{eqnarray}$$

Subtracting the first equation in (A 16) from (3.15) and neglecting terms of order $\unicode[STIX]{x1D700}$ yields

(3.16) $$\begin{eqnarray}\displaystyle u_{b}-u_{b_{CJ}}=\frac{2}{\unicode[STIX]{x1D6FE}+1}\left[\frac{x_{f}}{t}-{\mathcal{D}}_{CJ}\right], & & \displaystyle\end{eqnarray}$$

where, according to (3.14), the quantity $(a_{po}-a_{p_{CJ}})/a_{b_{CJ}}$ has been neglected in front of $(u_{b}-u_{b_{CJ}})/a_{b_{CJ}}=O(\sqrt{\unicode[STIX]{x1D700}})$ . The expression of the left-hand side of (3.16) in terms of $M_{u}=(\text{d}x_{f}/\text{d}t)/a_{u}$ is obtained from (3.3), (3.4) using mass conservation, $u_{b}({\mathcal{D}})=(1-\unicode[STIX]{x1D70C}_{u}/\unicode[STIX]{x1D70C}_{b}){\mathcal{D}}$ . Retaining only the leading-order term in the limit (3.1) yields

(3.17) $$\begin{eqnarray}\displaystyle \frac{u_{b}-u_{b_{CJ}}}{a_{u}} & = & \displaystyle \frac{2}{\unicode[STIX]{x1D6FE}+1}\frac{\left(M_{u_{CJ}}^{2}-1\right)}{2M_{u_{CJ}}}A_{2}\sqrt{\unicode[STIX]{x1D700}}+\text{terms of order }\unicode[STIX]{x1D700}\end{eqnarray}$$

(3.18) $$\begin{eqnarray}\displaystyle & {\approx} & \displaystyle \frac{2}{\unicode[STIX]{x1D6FE}+1}\frac{1}{A_{3}}\sqrt{\frac{1}{a_{u}}\frac{\text{d}}{\text{d}t}(x_{f}-{\mathcal{D}}_{CJ}t)},\quad \text{where}~A_{3}\equiv \frac{M_{u_{CJ}}^{3/2}}{\sqrt{(M_{u_{CJ}}^{4}-1)/2}}\end{eqnarray}$$

and where the expression $M_{u}-M_{u_{CJ}}=a_{u}^{-1}\text{d}(x_{f}-{\mathcal{D}}_{CJ}t)/\text{d}t$ has been used. Putting together (3.16) and (3.18) yields a differential equation for the front position $x_{f}(t)$ ,

(3.19) $$\begin{eqnarray}\displaystyle \sqrt{\frac{1}{a_{u}}\frac{\text{d}(x_{f}-{\mathcal{D}}_{CJ}t)}{\text{d}t}}=A_{3}\frac{(x_{f}-{\mathcal{D}}_{CJ}t)}{a_{u}t}. & & \displaystyle\end{eqnarray}$$

For future consideration it is worth writing this equation in non-dimensional form by introducing a reference time scale $t_{ref}$ which is arbitrary at this level

(3.20) $$\begin{eqnarray}\displaystyle \sqrt{\frac{\text{d}\unicode[STIX]{x1D6FC}}{\text{d}\unicode[STIX]{x1D70F}}}=\frac{1}{\sqrt{2}}\frac{\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D70F}}\quad \text{where}~\unicode[STIX]{x1D70F}\equiv \frac{t}{t_{ref}},\quad \unicode[STIX]{x1D6FC}\equiv 2A_{3}^{2}\frac{(x_{f}-{\mathcal{D}}_{CJ}t)}{a_{u}t_{ref}}>0. & & \displaystyle\end{eqnarray}$$

Using the beginning of the interaction as the initial condition, $t=t_{i}:x_{f}={\mathcal{D}}_{o}t_{i}+x_{o}$ (see figure 1), written in non-dimensional form, $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}_{i}:\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{i}$ where $\unicode[STIX]{x1D6FC}_{i}\equiv 2A_{3}^{2}[({\mathcal{D}}_{o}-{\mathcal{D}}_{CJ})t_{i}+x_{o}]/a_{u}t_{ref}$ , integrating (3.20), yields

(3.21) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70F}\geqslant \unicode[STIX]{x1D70F}_{i}:\quad \frac{1}{\unicode[STIX]{x1D6FC}}=\frac{1}{2}\left(\frac{1}{\unicode[STIX]{x1D70F}}-\frac{1}{\unicode[STIX]{x1D70F}_{i}}\right)+\frac{1}{\unicode[STIX]{x1D6FC}_{i}}, & & \displaystyle\end{eqnarray}$$

and in the long-time limit

(3.22) $$\begin{eqnarray}\displaystyle \lim _{\unicode[STIX]{x1D70F}\rightarrow \infty }\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{\infty }\quad \text{where}~\frac{1}{\unicode[STIX]{x1D6FC}_{\infty }}\equiv \frac{1}{\unicode[STIX]{x1D6FC}_{i}}-\frac{1}{2\unicode[STIX]{x1D70F}_{i}}. & & \displaystyle\end{eqnarray}$$

According to the definition of $\unicode[STIX]{x1D6FC}$ in (3.20), the asymptotic value $\unicode[STIX]{x1D6FC}_{\infty }$ can also be expressed in terms of the length $\unicode[STIX]{x0394}x_{f\infty }=\lim _{t\rightarrow \infty }(x_{f}-{\mathcal{D}}_{CJ}t)$ in figure 1, $\unicode[STIX]{x1D6FC}_{\infty }=2A_{3}^{2}\unicode[STIX]{x0394}x_{f\infty }/(a_{u}t_{ref})$ . The initial condition $\unicode[STIX]{x1D6FC}_{i}$ can be expressed in terms of the initial distance from the detonation and the sonic point $\unicode[STIX]{x0394}x_{fi}\equiv ({\mathcal{D}}_{o}-{\mathcal{D}}_{CJ})t_{i}+x_{o}$ , $\unicode[STIX]{x1D6FC}_{i}=2A_{3}^{2}\unicode[STIX]{x0394}x_{fi}/a_{u}t_{ref}$ so that $\unicode[STIX]{x0394}x_{f\infty }$ and $\unicode[STIX]{x0394}x_{fi}$ are linked by the relation

(3.23) $$\begin{eqnarray}\displaystyle \frac{1}{\unicode[STIX]{x0394}x_{f\infty }}\equiv \frac{1}{\unicode[STIX]{x0394}x_{fi}}-\frac{A_{3}^{2}}{a_{u}t_{i}}. & & \displaystyle\end{eqnarray}$$

According to (3.21), the asymptotic behaviour in the long-time limit takes the form

(3.24) $$\begin{eqnarray}\displaystyle & \displaystyle \lim _{\unicode[STIX]{x1D70F}\rightarrow \infty }(\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D6FC}_{\infty })\approx -\unicode[STIX]{x1D6FC}_{\infty }^{2}/(2\unicode[STIX]{x1D70F}), & \displaystyle\end{eqnarray}$$

(3.25) $$\begin{eqnarray}\displaystyle & \displaystyle t\gg t_{i}:\quad 2A_{3}^{2}\frac{x_{f}-{\mathcal{D}}_{CJ}t}{a_{u}t_{ref}}\approx \unicode[STIX]{x1D6FC}_{\infty }-\frac{\unicode[STIX]{x1D6FC}_{\infty }^{2}}{2}\frac{t_{ref}}{t}~\Rightarrow ~\frac{{\mathcal{D}}-{\mathcal{D}}_{CJ}}{a_{u}}\approx A_{3}^{2}\left(\frac{\unicode[STIX]{x0394}x_{f\infty }}{a_{u}t}\right)^{2}, & \displaystyle\end{eqnarray}$$

showing that the detonation velocity decreases toward the velocity of the self-propagating front like the inverse of the time squared. A similar long-time behaviour was obtained by Levin & Chernyi (Reference Levin and Chernyi1967) and by Liñan et al. (Reference Liñan, Kurdyumov and Sanchez2012) for $M_{u}\gg 1$ .

According to (3.21)–(3.25), the distance $\unicode[STIX]{x0394}x_{f}$ from the detonation front to the point where the flow velocity is sonic (relatively to the front), ${\mathcal{D}}-u=a$ , increases with the time from $\unicode[STIX]{x0394}x_{fi}$ to $\unicode[STIX]{x0394}x_{f\infty }$ , $\unicode[STIX]{x0394}x_{fi}<\unicode[STIX]{x0394}x_{f\infty }$ , see (3.23). The distance $\unicode[STIX]{x0394}x_{f}$ thus never becomes shorter than $\unicode[STIX]{x0394}x_{fi}$ . In other words, during the decay, the sonic condition $a+u={\mathcal{D}}$ is satisfied at a finite distance from the front and the distance $\unicode[STIX]{x0394}x_{f}$ never shrinks to zero. The sonic condition ${\mathcal{D}}_{CJ}-u_{b_{CJ}}=a_{b_{CJ}}$ is satisfied at the detonation front only in the limit $t\rightarrow \infty$ when the slope of the flow velocity in the burnt gas becomes zero, see the self-explanatory figure 1. This result, obtained at the leading order in the limit $\unicode[STIX]{x1D700}\rightarrow 0$ and not mentioned before, will be useful in the following.

4.1 Detonation model

The analysis is performed using the distinguished limit,

(4.1a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D716}^{2}\equiv q_{m}/(c_{p}T_{u})\ll 1,\quad (\unicode[STIX]{x1D6FE}-1)/\unicode[STIX]{x1D716}=h~\Rightarrow ~(M_{u_{CJ}}-1)=\unicode[STIX]{x1D716}+O(\unicode[STIX]{x1D716}^{2}), & & \displaystyle\end{eqnarray}$$

where $h$ is of order unity or smaller in the limit $\unicode[STIX]{x1D716}\rightarrow 0$ , the ordering of $(M_{u_{CJ}}-1)$ being obtained from (A 5). The first relation indicates that the relative variation of temperature is of order $\unicode[STIX]{x1D716}^{2}$ so that, assuming a polytropic gas, the relative variation of the sound speed is also of order $\unicode[STIX]{x1D716}^{2}$ , $a/a_{u}=1+O(\unicode[STIX]{x1D716}^{2})$ . In the limit (4.1) the flow is transonic across the whole detonation structure leading to two main simplifications: firstly the variation of the sound speed can be neglected and secondly the dynamical problem is one of two time scales, as discussed now. The propagation velocity (in the reference frame of the lead shock) of the two downstream running waves (acoustic and entropy) are both of the order of the sound speed which is larger by a factor $1/\unicode[STIX]{x1D716}$ than the propagation velocity of the upstream running acoustic wave. Therefore, these waves introduce a time lag of the order of the transit time $t_{r}\approx l_{deto}/a_{u}$ which is smaller than the time lag introduced by the upstream running acoustic wave of order $t_{r}/\unicode[STIX]{x1D716}$ . At the leading order in the limit (4.1), the effects of the downstream running waves are considered as instantaneous. The time scale of the dynamics of the inner structure is the longest time delay in the loops so that the period of oscillations of weakly unstable detonations is

(4.2) $$\begin{eqnarray}\displaystyle t_{ref}=t_{r}/\unicode[STIX]{x1D716}, & & \displaystyle\end{eqnarray}$$

see the discussion below (B 4)–(B 6) in § B.1 for more details. For stable detonations the time $t_{ref}$ is the typical delay involved by the linear response of the inner structure to small external disturbances as the unsteady rarefaction wave in the burnt gas of an overdriven detonation.

Then it is convenient to introduce the following reduced coordinates of order unity; the non-dimensional time $\unicode[STIX]{x1D70F}$ reduced by $t_{ref}$ and the non-dimensional distance from the lead shock $\unicode[STIX]{x1D709}$ reduced by $a_{u}t_{r}$

(4.3a-d ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70F}\equiv \frac{t}{t_{ref}}=\unicode[STIX]{x1D716}\frac{t}{t_{r}},\quad \unicode[STIX]{x1D709}\equiv \frac{x-x_{f}(t)}{a_{u}t_{r}},\quad \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}=\frac{1}{t_{r}}\left(\unicode[STIX]{x1D716}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}-\frac{1}{a_{u}}\frac{\text{d}x_{f}}{\text{d}t}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right),\quad \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}=\frac{1}{a_{u}t_{r}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where the $x$ -coordinate is orientated as in figure 1, corresponding to $\unicode[STIX]{x1D709}<0$ in the shocked gas and $\text{d}x_{f}/\text{d}t={\mathcal{D}}(t)>0$ . To avoid confusion, notice that this orientation of the $x$ -coordinate, which was used in the pioneering works of Courant & Friedrichs (Reference Courant and Friedrichs1948) and Levin & Chernyi (Reference Levin and Chernyi1967), is the opposite to the one in Clavin & Williams (Reference Clavin and Williams2002) and Clavin & Williams (Reference Clavin and Williams2009). It is also convenient to introduce reduced unknowns; the non-dimensional distance of the lead shock from the sonic point $\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70F})$ , the non-dimensional difference of velocity between the lead shock and the CJ velocity $\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70F})\equiv \text{d}\unicode[STIX]{x1D6FC}/\text{d}\unicode[STIX]{x1D70F}>0$ and the non-dimensional flow velocity shifted by the constant term $({\mathcal{D}}_{CJ}-a_{u})>0$ , $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})$ ,

(4.4a-c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FC}\equiv \frac{x_{f}(t)-{\mathcal{D}}_{CJ}t}{a_{u}t_{r}}>0,\quad \dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}\equiv \frac{\displaystyle \frac{\text{d}x_{f}}{\text{d}t}-{\mathcal{D}}_{CJ}}{\unicode[STIX]{x1D716}a_{u}},\quad \unicode[STIX]{x1D707}\equiv \frac{1}{\unicode[STIX]{x1D716}a_{u}}[u-({\mathcal{D}}_{CJ}-a_{u})]>0. & & \displaystyle\end{eqnarray}$$

The definition of $\unicode[STIX]{x1D6FC}$ in (4.4) is the same as in (3.20), $2A_{3}^{2}\approx 1/\unicode[STIX]{x1D716}$ in the limit (4.1).

Introducing the instantaneous distribution of reaction rate $w(\unicode[STIX]{x1D709},\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}})={\dot{w}}(Y,T)$ , Clavin & Williams (Reference Clavin and Williams2002) showed that, to leading order in the limit (4.1), the intrinsic dynamics of the inner structure is reduced to solving a single partial differential equation for the non-dimensional flow velocity $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})$ , see § B.1,

(4.5) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D707}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+(\unicode[STIX]{x1D707}-\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}})\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D707}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}=\frac{1}{2}w(\unicode[STIX]{x1D709},\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}). & & \displaystyle\end{eqnarray}$$

The usual nonlinear term of the Burgers type characterizes transonic flows as in Bdzil & Stewart (Reference Bdzil and Stewart1986). The unknown velocity of the lead shock $\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70F})$ is obtained by using the boundary conditions (2.6) at the lead shock in (B 3) and (2.7), (2.8) in the burnt gas,

(4.6a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D709}=0:\,\unicode[STIX]{x1D707}=1+2\,\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70F}),\quad \unicode[STIX]{x1D709}\rightarrow -\infty :\,\unicode[STIX]{x1D707}=\unicode[STIX]{x1D707}_{b}. & & \displaystyle\end{eqnarray}$$

Here only the first-order correction to $\unicode[STIX]{x1D707}$ is retained at the Neumann state ( $\unicode[STIX]{x1D709}=0$ ), anticipating that $\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}$ is a small quantity when the propagation regime is sufficiently close to the CJ regime, see (4.14). The pulsations of the detonation are controlled by the thermal sensitivity of the rate of the chemical heat release. The instantaneous distribution of the rate of heat release $w(\unicode[STIX]{x1D709},\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}})$ , to be introduced into the right-hand side of (4.5), is obtained in § B.1 by the solution of (2.3) with the boundary conditions (2.6) at $\unicode[STIX]{x1D709}=0$ when the compressional heating is negligible. In the following we will use the approximate expression in (B 13), obtained by a scaling law,

(4.7) $$\begin{eqnarray}\displaystyle w(\unicode[STIX]{x1D709},\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}})\approx \text{e}^{b\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}}\overline{w}_{CJ}(\unicode[STIX]{x1D709}\text{e}^{b\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}}), & & \displaystyle\end{eqnarray}$$

where $\overline{w}_{CJ}(\unicode[STIX]{x1D709})$ is the distribution of heat release rate in the unperturbed CJ wave ( $\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}=0$ ) and, according to (B 12), $\text{e}^{-b\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}}$ is the non-dimensional induction length, the coefficient $b$ measuring the thermal sensitivity. The boundary condition $\unicode[STIX]{x1D707}_{b}$ in (4.6) is determined in § 4.5 by matching the inner solution with the external flow in the burnt gas. It is worth looking first to the steady state.

4.2 Steady states

Consider first the steady case of a detonation moving at constant velocity $\overline{{\mathcal{D}}}$ , $\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}=\overline{m}$ where, according to (4.4), $\overline{m}$ is a positive constant,

(4.8) $$\begin{eqnarray}\displaystyle \dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}=\overline{m}\equiv \frac{\overline{{\mathcal{D}}}-{\mathcal{D}}_{CJ}}{\unicode[STIX]{x1D716}a_{u}}>0, & & \displaystyle\end{eqnarray}$$

the overbar identifying the steady states. The expression of the flow velocity in the burnt gas $\overline{\unicode[STIX]{x1D707}}_{b}$ in terms of the detonation velocity $\overline{m}$ is obtained by integrating the steady version of (4.5) ( $\unicode[STIX]{x2202}\unicode[STIX]{x1D707}/\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}=0$ ), $\text{d}(\overline{\unicode[STIX]{x1D707}}-\overline{m})^{2}/\text{d}\unicode[STIX]{x1D709}=\overline{w}$ , from the burnt gas at $\unicode[STIX]{x1D709}=-\infty$ to the lead shock $\unicode[STIX]{x1D709}=0$ , using the normalized condition $\int _{-\infty }^{0}\overline{w}(\unicode[STIX]{x1D709},\overline{m})\,\text{d}\unicode[STIX]{x1D709}=1$ , valid to the leading order in the limit $\unicode[STIX]{x1D716}\ll 1$ , since $\overline{w}(\unicode[STIX]{x1D709},\overline{m})={\dot{w}}(\overline{Y},\overline{T})$ and $\text{d}\overline{Y}/\text{d}\unicode[STIX]{x1D709}=-{\dot{w}}(\overline{Y},\overline{T})+O(\unicode[STIX]{x1D716})$ with $\overline{Y}|_{\unicode[STIX]{x1D709}=0}=0$ and $\overline{Y}|_{\unicode[STIX]{x1D709}=-\infty }=1$ . This yields $(\overline{\unicode[STIX]{x1D707}}-\overline{m})_{\unicode[STIX]{x1D709}=0}^{2}-(\overline{\unicode[STIX]{x1D707}}-\overline{m})_{\unicode[STIX]{x1D709}=-\infty }^{2}=1$ . Assuming that the departure from the CJ velocity is sufficiently small $\overline{m}\ll 1$ , $\unicode[STIX]{x1D709}=0:\,\overline{\unicode[STIX]{x1D707}}=1+2\,\overline{m}$ , the integration across the detonation structure then leads to

(4.9) $$\begin{eqnarray}\displaystyle (1+\overline{m})^{2}-(\overline{\unicode[STIX]{x1D707}}_{b}-\overline{m})^{2}=1~\Rightarrow ~\overline{\unicode[STIX]{x1D707}}_{b}=\overline{m}+\sqrt{(1+\overline{m})^{2}-1}, & & \displaystyle\end{eqnarray}$$

where the $+$ sign is chosen according to the ZND solution. According to (4.9) the flow velocity in the burnt gas takes the form

(4.10a,b ) $$\begin{eqnarray}\displaystyle \overline{m}\ll 1~\Rightarrow ~\overline{\unicode[STIX]{x1D707}}_{b}^{2}\approx 2\overline{m},\quad \overline{\unicode[STIX]{x1D707}}_{b}\approx \sqrt{2\overline{m}}, & & \displaystyle\end{eqnarray}$$

which is in agreement with (3.3), (3.4) for small $\unicode[STIX]{x1D700}$ . The square root in the expression of $\overline{\unicode[STIX]{x1D707}}_{b}$ illustrates the marginal character of the CJ regime.

4.3 Flow velocity in the burnt gas

For the decay of detonation, the unknown function $\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70F})\geqslant 0$ is obtained from (4.5), (4.6) when the boundary condition in the burnt gas $\unicode[STIX]{x1D707}_{b}$ is expressed in terms of $\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70F})$ by matching the inner solution with the external flow. For an inert gas ( $w=0$ ) there is a particular solution of (4.5) of the form

(4.11) $$\begin{eqnarray}\displaystyle w=0:\quad \unicode[STIX]{x1D707}_{ext}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})=\frac{\unicode[STIX]{x1D709}+\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70F})}{\unicode[STIX]{x1D70F}}. & & \displaystyle\end{eqnarray}$$

In the limit (4.1), the expression (4.11) is precisely that of the centred rarefaction wave (3.9) in the transonic region where $\unicode[STIX]{x1D709}$ is of order unity, as shown by using the approximations $a_{po}/a_{u}\approx 1$ and $\unicode[STIX]{x1D6FE}\approx 1$ and the definition of $\unicode[STIX]{x1D70F}$ in (4.3),

(4.12) $$\begin{eqnarray}\displaystyle \frac{u}{a_{u}}+1\approx \frac{(x-x_{f})/(a_{u}t_{r})}{t/t_{r}}+\frac{x_{f}/(a_{u}t_{r})}{t/t_{r}}=\frac{\unicode[STIX]{x1D716}}{\unicode[STIX]{x1D70F}}\left[\unicode[STIX]{x1D709}+\frac{x_{f}-{\mathcal{D}}_{CJ}t}{a_{u}t_{r}}\right]+\frac{{\mathcal{D}}_{CJ}}{a_{u}} & & \displaystyle\end{eqnarray}$$

leading to (4.11) because, according to (4.4), one has $u/a+1={\mathcal{D}}_{CJ}/a_{u}+\unicode[STIX]{x1D716}\unicode[STIX]{x1D707}$ .

The external flow in the burnt gas therefore takes the form (4.11) if the entropy emitted at $\unicode[STIX]{x1D709}=0$ is constant so that the centred rarefaction wave is not disturbed by the dynamics of the detonation. According to the discussion in § 3, this assumption is verified when the propagation velocity is sufficient close to the CJ velocity, $\unicode[STIX]{x1D700}\ll 1$ .

4.4 Double limit

One is thus led to analyse the detonation decay near the CJ regime $\unicode[STIX]{x1D700}\ll 1$ , for small heat release $\unicode[STIX]{x1D716}\equiv \sqrt{q_{m}/(c_{p}T_{u})}\approx (M_{u_{CJ}}-1)\ll 1$ , see (3.7) and (4.1). The problem is thus investigated in the double limit $\unicode[STIX]{x1D700}\rightarrow 0$ and $\unicode[STIX]{x1D716}\rightarrow 0$ . Considering an initial overdriven detonation (velocity ${\mathcal{D}}_{o}$ ) verifying (3.7), the propagation Mach number of the decaying detonation satisfies the ordering

(4.13) $$\begin{eqnarray}\displaystyle (M_{u}-M_{u_{CJ}})/(M_{u_{CJ}}-1)=O(\unicode[STIX]{x1D700})~\Rightarrow ~(M_{u}-M_{u_{CJ}})=({\mathcal{D}}-{\mathcal{D}}_{CJ})/a_{u}=O(\unicode[STIX]{x1D700}\unicode[STIX]{x1D716}). & & \displaystyle\end{eqnarray}$$

The regimes under consideration correspond to the double limit (3.6) and, according to (4.4), the reduced velocity of the shock front $\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}$ is a small quantity of order $\unicode[STIX]{x1D700}$

(4.14) $$\begin{eqnarray}\displaystyle (M_{u}-M_{u_{CJ}})\ll (M_{u_{CJ}}-1)\ll 1~\Rightarrow ~\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}=O(\unicode[STIX]{x1D700}), & & \displaystyle\end{eqnarray}$$

with, according to (4.8)–(4.10), $\overline{\unicode[STIX]{x1D707}}_{b}=O(\sqrt{\unicode[STIX]{x1D700}})$ , in agreement with (3.3), (3.4). The terms of order $\unicode[STIX]{x1D716}^{2}$ at $\unicode[STIX]{x1D709}=0$ being neglected in (4.6), the relation $\unicode[STIX]{x1D716}^{2}\ll \unicode[STIX]{x1D716}\unicode[STIX]{x1D700}$ should be used in (4.13) for consistency. Then, the double limit to be considered in the asymptotic analysis is

(4.15a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D700}\rightarrow 0\quad \text{and}\quad \unicode[STIX]{x1D716}\ll \unicode[STIX]{x1D700}. & & \displaystyle\end{eqnarray}$$

Retaining the terms of order $\unicode[STIX]{x1D716}^{2}$ would have introduced more tedious calculations but no new physical insights.

To summarize, the decay of the detonation is obtained in the limit (4.15) by the solution of (4.5) satisfying the boundary conditions (4.6), $\unicode[STIX]{x1D707}_{b}$ being given by the matching condition with the external flow in (4.11). Anticipating, on the basis of the results in § 3.3, that $\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70F})$ goes to a constant when $\unicode[STIX]{x1D70F}$ increases, the flow velocity in (4.11) goes to zero when $\unicode[STIX]{x1D70F}\rightarrow \infty$ for any value of $\unicode[STIX]{x1D709}$ of order unity. The value $\unicode[STIX]{x1D707}=0$ corresponds effectively to the sonic condition of CJ waves ${\mathcal{D}}_{CJ}-u_{b_{CJ}}=a_{b_{CJ}}$ , $\lim _{\unicode[STIX]{x1D716}\rightarrow 0}\unicode[STIX]{x1D707}_{b_{CJ}}=0$ , as shown from the expression of $\unicode[STIX]{x1D707}$ in (4.4) when using the relation $a_{u}\approx a_{b_{CJ}}$ , valid in the limit (4.15).

4.5 Simplification coming from the initial condition

Matching the inner flow with (4.11) becomes simpler when $\unicode[STIX]{x1D6FC}\gg 1$ , $x_{f}(t)-{\mathcal{D}}_{CJ}t\gg a_{u}t_{r}$ ( $\unicode[STIX]{x0394}x_{f}\gg a_{u}t_{r}$ in the notation of figure 1), namely when the distance between the lead shock and the point where the sonic condition holds is larger than the detonation thickness. This is automatically satisfied for typical initial conditions, namely when the initial distance between the detonation and the piston at $t=0$ (time at which the piston is suddenly arrested) is larger than the detonation thickness, $x_{o}\gg a_{u}t_{r}$ , see figure 1. Under such conditions the term $\unicode[STIX]{x1D709}$ (of order unity in the internal structure) is negligible in front of $\unicode[STIX]{x1D6FC}$ in (4.11). The external flow close to the detonation is nearly uniform and takes the form $\unicode[STIX]{x1D707}_{ext}\approx \unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70F})/\unicode[STIX]{x1D70F}$ . This approximation is equivalent to neglecting the gradient of the reduced velocity of the flow compared to the inverse of the detonation thickness, $\unicode[STIX]{x2202}\unicode[STIX]{x1D707}_{ext}/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}=1/\unicode[STIX]{x1D70F}\ll 1$ . Matching then leads to a boundary condition (4.6) in the burnt gas similar to (2.7), (2.8) with

(4.16) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{b}(\unicode[STIX]{x1D70F})=\frac{\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70F})}{\unicode[STIX]{x1D70F}}=O(\sqrt{\unicode[STIX]{x1D700}}), & & \displaystyle\end{eqnarray}$$

where the same order of magnitude of the flow velocity in the burnt gas as in the steady state near the CJ regime, has been used, $\unicode[STIX]{x1D707}_{b}=O(\sqrt{\unicode[STIX]{x1D700}})$ . This condition is compatible with $\unicode[STIX]{x1D6FC}\gg 1$ when considering instants of time sufficiently large, $\unicode[STIX]{x1D70F}\gg 1/\sqrt{\unicode[STIX]{x1D700}}$ . This condition is automatically satisfied for $t>t_{i}$ if $t_{i}\gg t_{r}/(\unicode[STIX]{x1D716}\sqrt{\unicode[STIX]{x1D700}})$ that corresponds to an initial distance $x_{o}$ that is large compared to the detonation thickness $l_{deto}\approx a_{u}t_{r}$ ,

(4.17) $$\begin{eqnarray}\displaystyle x_{o}\gg a_{u}t_{r}~\Leftrightarrow ~t_{i}\gg t_{r}/(\unicode[STIX]{x1D716}\sqrt{\unicode[STIX]{x1D700}})\quad \text{i.e.}~\unicode[STIX]{x1D70F}_{i}\gg 1/\sqrt{\unicode[STIX]{x1D700}}~\Rightarrow ~\unicode[STIX]{x1D6FC}\gg 1, & & \displaystyle\end{eqnarray}$$

as shown from the relation $x_{o}/(a_{u}t_{i})\approx \unicode[STIX]{x1D716}\sqrt{2\unicode[STIX]{x1D700}}$ , which is obtained from the relation $x_{o}=[a_{bo}-({\mathcal{D}}_{o}-u_{bo})]t_{i}$ in figure 1 when using (3.3), (3.4) with (A 4), (A 5) and (3.7), neglecting the terms of order $\unicode[STIX]{x1D716}^{2}$ in (4.1), $(M_{uo}^{2}-1)\approx \unicode[STIX]{x1D716}(1+\unicode[STIX]{x1D700})$ . To summarize, for ordinary initial conditions, $x_{o}\gg a_{u}t_{r}$ , the problem is reduced to solving (4.5) using the boundary conditions (4.6) with (4.16).

The solution for a detonation considered as a hydrodynamic discontinuity in § 3 is recovered by the quasi-steady state approximation, namely when the unsteady term is neglected in (4.5). Integrating (4.5) across the detonation structure, from $-\infty$ to 0, using the condition of normalization, $\int _{-\infty }^{0}w(\unicode[STIX]{x1D709},\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}})\,\text{d}\unicode[STIX]{x1D709}=1$ , yields

(4.18) $$\begin{eqnarray}\displaystyle {\textstyle \frac{1}{2}}(\unicode[STIX]{x1D707}-\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}})^{2}|_{\unicode[STIX]{x1D709}=-\infty }^{\unicode[STIX]{x1D709}=0}={\textstyle \frac{1}{2}}~\Rightarrow ~{\textstyle \frac{1}{2}}(\unicode[STIX]{x1D707}^{2}|_{\unicode[STIX]{x1D709}=0}-\unicode[STIX]{x1D707}_{b}^{2})-(\unicode[STIX]{x1D707}|_{\unicode[STIX]{x1D709}=0}-\unicode[STIX]{x1D707}_{b})\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}={\textstyle \frac{1}{2}}, & & \displaystyle\end{eqnarray}$$

and equation (3.20) is effectively recovered by the boundary conditions (4.6) (4.16) when the terms smaller than $\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}=O(\unicode[STIX]{x1D700})$ are neglected, $\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}\equiv \text{d}\unicode[STIX]{x1D6FC}/\text{d}\unicode[STIX]{x1D70F}=(1/2)(\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D70F})^{2}$ .

4.6 Instantaneous distribution of the rate of heat release

Another simplification concerns the instantaneous distribution of the heat release rate $w(\unicode[STIX]{x1D709},\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}})$ on the right-hand side of (4.5). Its determination becomes more easy if the compressional heating in the thermal equation (2.3) can be neglected, which is the cases if $(\unicode[STIX]{x1D6FE}-1)$ is sufficiently small, $h\ll 1$ in (4.1) as will be assumed from now on,

(4.19) $$\begin{eqnarray}\displaystyle (\unicode[STIX]{x1D6FE}-1)/\unicode[STIX]{x1D716}\ll 1. & & \displaystyle\end{eqnarray}$$

Following the work of Clavin & He (Reference Clavin and He1996), the analysis, recalled in § B.1, leads to (4.7). According to (4.6) and (4.16), $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})$ is of order unity close to the lead shock and of order $\sqrt{\unicode[STIX]{x1D700}}$ at the end of the reaction zone. Therefore, according to (4.14), the non-dimensional velocity $\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70F})=O(\unicode[STIX]{x1D700})$ is smaller than $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})$ . For stable detonations against planar disturbances only the linear term in the expansion of the instantaneous reaction rate (4.7) in powers of $\unicode[STIX]{x1D6FF}\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70F})$ is retained at the leading order in the limit (4.15). The corresponding expression of the variation $\unicode[STIX]{x1D6FF}w(\unicode[STIX]{x1D709},\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}})$ is similar to that in the stability analysis of Clavin & Williams (Reference Clavin and Williams2002). Neglecting the terms smaller than $\unicode[STIX]{x1D700}$ , equation (4.5) then takes the form

(4.20) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D707}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+[\unicode[STIX]{x1D707}-\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70F})]\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D707}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}=\frac{1}{2}\overline{\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D709})+\frac{b}{2}\overline{\unicode[STIX]{x1D6FA}}(\unicode[STIX]{x1D709})\unicode[STIX]{x1D6FF}\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70F})\quad \text{where}~\overline{\unicode[STIX]{x1D6FA}}(\unicode[STIX]{x1D709})\equiv \frac{\text{d}}{\text{d}\unicode[STIX]{x1D709}}[\unicode[STIX]{x1D709}\overline{\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D709})], & & \displaystyle\end{eqnarray}$$

(4.21a,b ) $$\begin{eqnarray}\displaystyle \int _{-\infty }^{0}\overline{\unicode[STIX]{x1D6FA}}(\unicode[STIX]{x1D709})\,\text{d}\unicode[STIX]{x1D709}=0,\quad \int _{-\infty }^{0}\overline{\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D709})\,\text{d}\unicode[STIX]{x1D709}=1, & & \displaystyle\end{eqnarray}$$

where the coefficient $b$ in front of $\overline{\unicode[STIX]{x1D6FA}}(\unicode[STIX]{x1D709})$ in (4.20) is a measure of the sensitivity to the Neumann temperature, see (B 12), (B 13). The function $\overline{\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D709})$ is the distribution of heat release rate in the unperturbed overdriven detonation, and $\overline{\unicode[STIX]{x1D6FA}}(\unicode[STIX]{x1D709})$ characterizes the deformation of the distribution of the overdriven detonation in steady state when the Neumann temperature is modified from its unperturbed value, see (B 10), (B 11).

An implicit relation between the non-dimensional shock position (relative to the sonic point), $\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70F})\gg 1$ , and its velocity (relative to that of the CJ wave), $\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70F})\equiv \text{d}\unicode[STIX]{x1D6FC}/\text{d}\unicode[STIX]{x1D70F}=O(\unicode[STIX]{x1D700})$ , is then obtained during the decay ( $\unicode[STIX]{x1D70F}\geqslant \unicode[STIX]{x1D70F}_{i}$ , $\unicode[STIX]{x1D70F}_{i}\equiv \unicode[STIX]{x1D716}t_{i}/t_{r}\gg 1/\sqrt{\unicode[STIX]{x1D700}}$ ) by solving the partial differential equation (4.20) for the flow velocity field $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})$ with the two boundary conditions (4.6) using (4.16). When equation (4.20) is fully linearized and when the boundary condition (4.16) is replaced by an unperturbed uniform flow, $\unicode[STIX]{x1D709}\rightarrow -\infty :\unicode[STIX]{x1D707}=\overline{\unicode[STIX]{x1D707}}_{b}=$ cst., the stability analysis of Clavin & Williams (Reference Clavin and Williams2002) against planar disturbances is recovered, see § B.2.

4.7 Linear dynamics at the beginning of the decay

The initial condition of the detonation decay is a steady weakly overdriven detonation for small heat release in the limit (4.15). Assume to begin with that this initial detonation is stable. Unstable detonations will be considered in § 4.8. Denoting $\unicode[STIX]{x1D6FC}_{i}$ the position of the lead shock at time $\unicode[STIX]{x1D70F}_{i}$ when the rarefaction wave catches the unperturbed overdriven detonation, the initial conditions are, according to (4.8)–(4.10),

(4.22a-d ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}_{i}:\quad \unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{i},\quad {\mathcal{D}}={\mathcal{D}}_{o},\quad \text{d}\unicode[STIX]{x1D6FC}/\text{d}\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D700},\quad \unicode[STIX]{x1D6FC}_{i}/\unicode[STIX]{x1D70F}_{i}=\sqrt{2\unicode[STIX]{x1D700}}, & & \displaystyle\end{eqnarray}$$

where the last relation comes from (4.10) and (4.16). During the interaction between the rarefaction wave and the overdriven detonation the flow velocity at the end of the reaction zone, $\unicode[STIX]{x1D707}_{b}(\unicode[STIX]{x1D70F})$ , is, according to (4.16), $\unicode[STIX]{x1D70F}>\unicode[STIX]{x1D70F}_{i}:\unicode[STIX]{x1D707}_{b}(\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D70F}<\unicode[STIX]{x1D6FC}_{i}/\unicode[STIX]{x1D70F}_{i}$ . During a short period of time after $\unicode[STIX]{x1D70F}_{i}$ this flow takes the form $\unicode[STIX]{x1D707}_{b}=\overline{\unicode[STIX]{x1D707}}_{b}+\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D707}_{b}$ , with $\overline{\unicode[STIX]{x1D707}}_{b}=\unicode[STIX]{x1D6FC}_{i}/\unicode[STIX]{x1D70F}_{i}$ and

(4.23) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}\unicode[STIX]{x1D707}_{b}(\unicode[STIX]{x1D70F})=\frac{\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70F})}{\unicode[STIX]{x1D70F}}-\frac{\unicode[STIX]{x1D6FC}_{i}}{\unicode[STIX]{x1D70F}_{i}}\approx \frac{\unicode[STIX]{x1D6FC}_{i}}{\unicode[STIX]{x1D70F}_{i}}\left[\frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D6FC}_{i}}-\frac{(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{i})}{\unicode[STIX]{x1D70F}_{i}}\right], & & \displaystyle\end{eqnarray}$$

where the term $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D6FC}_{i}\equiv (\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D6FC}_{i})/\unicode[STIX]{x1D6FC}_{i}$ is smaller than $(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{i})/\unicode[STIX]{x1D70F}_{i}$ by a factor $\sqrt{\unicode[STIX]{x1D700}}$ because $\text{d}\unicode[STIX]{x1D6FC}/\text{d}\unicode[STIX]{x1D70F}=O(\unicode[STIX]{x1D700})$ , $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}=O(\unicode[STIX]{x1D700}(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{i}))$ , and, according to (4.22), $\unicode[STIX]{x1D6FC}_{i}/\unicode[STIX]{x1D70F}_{i}=O(\sqrt{\unicode[STIX]{x1D700}})$ , so that $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D6FC}_{i}=O(\sqrt{\unicode[STIX]{x1D700}}(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{i})/\unicode[STIX]{x1D70F}_{i})$ . Therefore, for a short period of time, the small variation of the non-dimensional flow at the end of the reaction zone takes the form

(4.24) $$\begin{eqnarray}\displaystyle 0<\frac{\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{i}}{\unicode[STIX]{x1D70F}_{i}}\ll 1,~\unicode[STIX]{x1D709}\rightarrow -\infty :\quad \unicode[STIX]{x1D6FF}\unicode[STIX]{x1D707}=\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D707}_{b}(\unicode[STIX]{x1D70F})=-\sqrt{2\unicode[STIX]{x1D700}}\left(\frac{\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{i}}{\unicode[STIX]{x1D70F}_{i}}\right), & & \displaystyle\end{eqnarray}$$

and the linear dynamics around the initial state at $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}_{i}$ is the solution of the fully linearized version of (4.20). This linear equation is given in (B 19) when the distribution $\overline{\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D709})$ is replaced by the steady distribution $\overline{\unicode[STIX]{x1D714}}_{o}(\unicode[STIX]{x1D709})$ of the unperturbed overdriven detonation before its interaction with the rarefaction wave. The initial dynamics is the linear response to the disturbance (4.24) in the burnt gas. It is controlled by (B 32) in § B.3, obtained as a straightforward extension of the linear dynamics of Clavin & Williams (Reference Clavin and Williams2002) for the stability analysis that is briefly recalled in § B.2. As explained in § B.3, equation (B 32) for the linear response is obtained from $y(z,\unicode[STIX]{x1D70F})\equiv \overline{\unicode[STIX]{x1D707}}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D707}(z,\unicode[STIX]{x1D70F})$ after having introduced the linear disturbances $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D707}$ , $\unicode[STIX]{x1D707}=\overline{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D709})+\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})$ , the linear perturbation $\unicode[STIX]{x1D6FF}\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70F})\equiv \text{d}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70F})/\text{d}\unicode[STIX]{x1D70F}$ , $\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}=\unicode[STIX]{x1D700}+\unicode[STIX]{x1D6FF}\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70F})$ , the change of variable $z=\int _{0}^{\unicode[STIX]{x1D709}}\text{d}\unicode[STIX]{x1D709}/\overline{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D709})$ , $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}z=\overline{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D709})\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}$ and the function $g(z)$ in (B 21), $\int _{-\infty }^{0}g(z)\,\text{d}z=\overline{\unicode[STIX]{x1D707}}(0)-\overline{\unicode[STIX]{x1D707}}_{b}$ . The initial dynamics is thus described by (B 32) in which, according to (4.10), $\overline{\unicode[STIX]{x1D707}}_{b}=\sqrt{2\unicode[STIX]{x1D700}}$ , $\overline{\unicode[STIX]{x1D707}}_{b}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D707}_{b}=-2\unicode[STIX]{x1D700}(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{i})/\unicode[STIX]{x1D70F}_{i}$ ,

(4.25) $$\begin{eqnarray}\displaystyle 0<\frac{(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{i})}{\unicode[STIX]{x1D70F}_{i}}\ll 1:\quad 2\unicode[STIX]{x1D6FF}\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70F})=\int _{-\infty }^{0}g(z^{\prime })\unicode[STIX]{x1D6FF}\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70F}+z^{\prime })\,\text{d}z^{\prime }-2\unicode[STIX]{x1D700}\,\frac{(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{i})}{\unicode[STIX]{x1D70F}_{i}}. & & \displaystyle\end{eqnarray}$$

The last term (forcing term) in (4.25) is of order $\unicode[STIX]{x1D700}$ so that the solution of the linear equation (4.25) is also of order $\unicode[STIX]{x1D700}$ , $\unicode[STIX]{x1D6FF}\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70F})=O(\unicode[STIX]{x1D700})$ , as it should be in the double limit (4.14). Considering stable detonations to planar disturbances, the attention is limited to small sensitivity to temperature, $b<b_{c}$ where $b_{c}$ is the critical value denoting the instability threshold, recalled below (B 27). For $\unicode[STIX]{x1D70F}\leqslant \unicode[STIX]{x1D70F}_{i}$ , the shock velocity ${\mathcal{D}}_{o}$ is constant, $\unicode[STIX]{x1D6FF}\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70F})=0$ , and there is no forcing term, so that the lower bound of the integral can be replaced by $-(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{i})$

(4.26) $$\begin{eqnarray}\displaystyle 0<\frac{(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{i})}{\unicode[STIX]{x1D70F}_{i}}\ll 1:\quad 2\unicode[STIX]{x1D6FF}\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70F})=\int _{-(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{i})}^{0}g(z^{\prime })\unicode[STIX]{x1D6FF}\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70F}+z^{\prime })\,\text{d}z^{\prime }-2\unicode[STIX]{x1D700}\frac{(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{i})}{\unicode[STIX]{x1D70F}_{i}}. & & \displaystyle\end{eqnarray}$$

The decrease of the velocity for $\unicode[STIX]{x1D70F}>\unicode[STIX]{x1D70F}_{i}$ is driven by the last term and is modulated by the integral describing the response of the internal structure.

4.8 Nonlinear decay towards the CJ regime

The main outcome of the linear analysis is to transform the partial differential equation (4.20) for the flow velocity field $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})$ into an integral equation (4.25) for the detonation velocity $\unicode[STIX]{x1D6FF}\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70F})$ . As explained at the end of § B.2, the linear approximmation (B 32) cannot be used during the final stage of the decay because, according to the sonic condition in the CJ wave, one gets $\overline{\unicode[STIX]{x1D707}}_{b_{CJ}}=0$ , so that the linear forcing term is zero. The nonlinear character is also illustrated by the nonlinear form of the self-similar solution (3.20). Before considering unstable detonations, consider first stable CJ detonations. Close to the CJ regime, $\overline{\dot{\unicode[STIX]{x1D6FC}}}_{\unicode[STIX]{x1D70F}}=0$ , an extension of the integral equation can be anticipated by replacing the forcing term in the burnt gas $\overline{\unicode[STIX]{x1D707}}_{b}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D707}_{b}$ in (B 32) by its nonlinear version $\unicode[STIX]{x1D707}_{b}^{2}/2$ with, according to (4.16), $\unicode[STIX]{x1D707}_{b}(\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70F})/\unicode[STIX]{x1D70F}$ ,

(4.27) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70F}\gg \unicode[STIX]{x1D70F}_{i}:\quad 2\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70F})=\int _{-\infty }^{0}g(z^{\prime })\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70F}+z^{\prime })\,\text{d}z^{\prime }+(1/2)[\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70F})/\unicode[STIX]{x1D70F}]^{2}. & & \displaystyle\end{eqnarray}$$

The physical interpretation of (4.27) is simple. The result (3.20) for a detonation considered as a hydrodynamic discontinuity, $\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}=(1/2)(\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D70F})^{2}$ , is recovered from the quasi-steady-state approximation of (4.27). When the variation of $\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70F})$ on the time scale of the acoustic wave across the detonation structure is neglected in the integral term $\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70F}+z^{\prime })\rightarrow \dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70F})$ , the normalization condition $\int _{-\infty }^{0}g(z^{\prime })\,\text{d}z^{\prime }=1+O(\unicode[STIX]{x1D700})$ ( $\overline{\unicode[STIX]{x1D707}}_{N_{CJ}}=1+O(\unicode[STIX]{x1D700})$ , $\overline{\unicode[STIX]{x1D707}}_{b_{CJ}}=0$ ) yields effectively (3.20). For a stable CJ detonation, the integral term represents the linear response of the inner structure to the unsteady rarefaction wave in the external flow of burnt gas.

Recalling the orders of magnitudes of $\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}=O(\unicode[STIX]{x1D700})$ in (4.14) and $\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D70F}=O(\sqrt{\unicode[STIX]{x1D700}})$ in (4.16), the derivative with respect to $\unicode[STIX]{x1D70F}$ of the nonlinear term $(\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D70F})^{2}$ in (4.27) is of order $\unicode[STIX]{x1D700}/\unicode[STIX]{x1D70F}$ which is smaller than $\unicode[STIX]{x1D700}$ for ordinary initial conditions, $\unicode[STIX]{x1D70F}_{i}\gg 1/\sqrt{\unicode[STIX]{x1D700}}$ , see (4.17). Therefore, retaining time delays in the nonlinear term $(\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D70F})^{2}$ of the same order of magnitude as the response time, similar to $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D707}_{b}(\unicode[STIX]{x1D70F}-|\overline{z}_{b}|)$ in (B 31), would introduce corrections that are negligible to leading order in the asymptotic analysis in the limit (4.15).

Let us show now that (4.27) is effectively obtained at the leading order in the asymptotic analysis from (4.5), (4.6) using the boundary condition (4.16) in the burnt gas $\unicode[STIX]{x1D709}\rightarrow -\infty$ . Introducing the decomposition $\unicode[STIX]{x1D707}=\overline{\unicode[STIX]{x1D707}}_{CJ}(\unicode[STIX]{x1D709})+\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})$ into (4.20) and retaining the nonlinear term $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D707}\unicode[STIX]{x2202}(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D707})/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}$ yields

(4.28) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D707}\unicode[STIX]{x2202}\unicode[STIX]{x1D707}/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}=\overline{\unicode[STIX]{x1D707}}_{CJ}\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D707}}_{CJ}/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}+\unicode[STIX]{x2202}(\overline{\unicode[STIX]{x1D707}}_{CJ}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D707})/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}+\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D707}\unicode[STIX]{x2202}(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D707})/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}, & \displaystyle\end{eqnarray}$$

(4.29) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{\unicode[STIX]{x1D707}}_{CJ}\left[\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D707}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D707}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right]=\left[\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\right](\overline{\unicode[STIX]{x1D707}}_{CJ}\unicode[STIX]{x1D707})+\overline{\unicode[STIX]{x1D707}}_{CJ}\frac{\text{d}\overline{\unicode[STIX]{x1D707}}_{CJ}}{\text{d}z}+\frac{1}{2}\frac{\unicode[STIX]{x2202}(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D707})^{2}}{\unicode[STIX]{x2202}z}, & \displaystyle\end{eqnarray}$$

where the change of variable $\unicode[STIX]{x1D709}\rightarrow z$ , introduced in (B 20), has been used. It is worth mentioning in passing that no singularity of the wave breaking type can be produced in the shocked gas by the kinematic term $\unicode[STIX]{x2202}\unicode[STIX]{x1D707}/\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}+\unicode[STIX]{x1D707}\unicode[STIX]{x2202}\unicode[STIX]{x1D707}/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}$ because the flow velocity $\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})>0$ is an increasing function of $\unicode[STIX]{x1D709}$ . Introducing the quantity

(4.30) $$\begin{eqnarray}\displaystyle y(z,\unicode[STIX]{x1D70F})\equiv \overline{\unicode[STIX]{x1D707}}_{CJ}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D707}+(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D707})^{2}/2, & & \displaystyle\end{eqnarray}$$

equation (4.29) takes the form

(4.31) $$\begin{eqnarray}\displaystyle \overline{\unicode[STIX]{x1D707}}_{CJ}\left[\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D707}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+(\unicode[STIX]{x1D707}-\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}})\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D707}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right] & = & \displaystyle \left[\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\right]y+\frac{1}{2}\overline{\unicode[STIX]{x1D707}}_{CJ}\overline{w}_{CJ}\nonumber\\ \displaystyle & & \displaystyle -\,\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}\frac{\text{d}\overline{\unicode[STIX]{x1D707}}_{CJ}}{\text{d}z}-\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x2202}(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D707})}{\unicode[STIX]{x2202}z}-\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D707})}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}.\end{eqnarray}$$

Anticipating that the leading-order terms are of order $\unicode[STIX]{x1D700}$ , $\unicode[STIX]{x2202}\unicode[STIX]{x1D707}/\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}=O(\unicode[STIX]{x1D700})$ , the last two terms on the right-hand side of (4.31) are negligible because they are smaller than $\unicode[STIX]{x1D700}$ , as shown by using the following results. (i) The CJ regime is characterized by $\overline{\dot{\unicode[STIX]{x1D6FC}}}_{\unicode[STIX]{x1D70F}}=0$ , $\overline{\unicode[STIX]{x1D707}}_{CJ}(\unicode[STIX]{x1D709})=O(1)$ close to the Neumann state and $\overline{\unicode[STIX]{x1D707}}_{b_{CJ}}=0$ in the burnt gas. (ii) According to (4.14), $\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}=O(\unicode[STIX]{x1D700})$ . (iii) According to (4.6) and (4.16), $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D707}$ is of order $\unicode[STIX]{x1D700}$ close to the Neumann state ( $\unicode[STIX]{x1D709}=0$ ) and of order $\sqrt{\unicode[STIX]{x1D700}}$ in the burnt gas. (iv) According to (4.16), the derivative of $\unicode[STIX]{x1D707}$ with respect to time in the burnt gas, $\unicode[STIX]{x2202}\unicode[STIX]{x1D707}_{b}/\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}=O(\sqrt{\unicode[STIX]{x1D700}}/\unicode[STIX]{x1D70F})$ , is smaller than $\sqrt{\unicode[STIX]{x1D700}}$ for ordinary initial conditions, $\unicode[STIX]{x1D70F}_{i}\gg 1/\sqrt{\unicode[STIX]{x1D700}}$ , see (4.17). Equation (4.31) then yields

(4.32) $$\begin{eqnarray}\displaystyle \overline{\unicode[STIX]{x1D707}}_{CJ}\left[\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D707}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+(\unicode[STIX]{x1D707}-\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}})\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D707}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right] & = & \displaystyle \frac{1}{2}\overline{\unicode[STIX]{x1D707}}_{CJ}w(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})\nonumber\\ \displaystyle & = & \displaystyle \left[\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z}\right]y+\frac{1}{2}\overline{\unicode[STIX]{x1D707}}_{CJ}\overline{w}_{CJ}-\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}\overline{\unicode[STIX]{x1D707}}_{CJ}\frac{\text{d}\overline{\unicode[STIX]{x1D707}}_{CJ}}{\text{d}\unicode[STIX]{x1D709}}.\end{eqnarray}$$

As in (4.20), and for the same reason, the reaction rate $w(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})$ can be linearized around the steady CJ solution. Therefore, introducing the function $g(z)$ in (B 21), defined here for the unperturbed CJ wave, the equation for $y(z,\unicode[STIX]{x1D70F})$ , defined in (4.30), takes a form similar to the linear analysis

(4.33) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}y}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+\frac{\unicode[STIX]{x2202}y}{\unicode[STIX]{x2202}z}=g_{CJ}(z)\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70F}), & & \displaystyle\end{eqnarray}$$

but with a different boundary condition in the burnt gas. According to (4.30), the boundary condition (4.6), $\overline{\unicode[STIX]{x1D707}}_{b_{CJ}}=0$ , $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D707}_{b}=\unicode[STIX]{x1D707}_{b}$ , leads to $z\rightarrow -\infty :\,y\rightarrow (\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D707}_{b})^{2}/2=\unicode[STIX]{x1D707}_{b}^{2}/2$ , so that, according to (4.16), the solution of (4.33) should satisfy the boundary conditions

(4.34a,b ) $$\begin{eqnarray}\displaystyle z=0:\,y=2\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70F})=2\unicode[STIX]{x1D700},\quad z\rightarrow -\infty :\,y=(1/2)[\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70F})/\unicode[STIX]{x1D70F}]^{2}=O(\unicode[STIX]{x1D700}), & & \displaystyle\end{eqnarray}$$

where the contribution of the nonlinear term $(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D707})^{2}$ in $y$ has been neglected at the Neumann state $\unicode[STIX]{x1D709}=0$ because it is there of order $\unicode[STIX]{x1D700}^{2}$ . To summarize, in the detonation decay toward a stable CJ wave in the limit (4.15) and (4.19), the nonlinear effects come only from the flow in the burnt gas. Following the same development as in the linear response in § B.2, equation (4.27) is then directly obtained from (4.33), (4.34).

The difference between the initial distribution $g(z)$ and $g_{CJ}(z)$ is of order $\sqrt{\unicode[STIX]{x1D700}}$ and can be neglected to leading order. An equation for $\text{d}\unicode[STIX]{x1D6FC}/\text{d}\unicode[STIX]{x1D70F}\equiv ({\mathcal{D}}(\unicode[STIX]{x1D70F})-{\mathcal{D}}_{CJ})/\unicode[STIX]{x1D716}a_{u}$ , including the long-time limit (4.27) and the linear evolution (4.25) just after the initial condition, $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}_{i}:$ ${\mathcal{D}}={\mathcal{D}}_{o}$ , $\text{d}\unicode[STIX]{x1D6FC}/\text{d}\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D700}$ , $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{i}$ , $\unicode[STIX]{x1D6FC}_{i}/\unicode[STIX]{x1D70F}_{i}=\sqrt{2\unicode[STIX]{x1D700}}$ , takes the form

(4.35) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70F}\geqslant \unicode[STIX]{x1D70F}_{i}:\quad 2\frac{\text{d}\unicode[STIX]{x1D6FC}}{\text{d}\unicode[STIX]{x1D70F}}=\int _{-(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{i})}^{0}g(z^{\prime })\frac{\text{d}\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70F}+z^{\prime })}{\text{d}\unicode[STIX]{x1D70F}}\,\text{d}z^{\prime }+\frac{1}{2}\frac{\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70F})^{2}}{\unicode[STIX]{x1D70F}^{2}}+\unicode[STIX]{x1D700}\int _{-\infty }^{-(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{i})}g(z^{\prime })\,\text{d}z^{\prime }, & & \displaystyle\end{eqnarray}$$

which corresponds to (4.25) in which the last term on the right-hand side and $\unicode[STIX]{x1D6FF}\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}=({\mathcal{D}}-{\mathcal{D}}_{o})/\unicode[STIX]{x1D716}a_{u}$ are replaced by the nonlinear term $(1/2)[(\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D70F})^{2}-(\unicode[STIX]{x1D6FC}_{i}/\unicode[STIX]{x1D70F}_{i})^{2}]$ and $(\text{d}\unicode[STIX]{x1D6FC}/\text{d}\unicode[STIX]{x1D70F})-\unicode[STIX]{x1D700}$ respectively. The initial condition, $\text{d}\unicode[STIX]{x1D6FC}/\text{d}\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D700}$ , is effectively recovered at $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}_{i}$ in (4.35), since $(\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D70F})^{2}/2=\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D700}\int _{-\infty }^{0}g(z)\,\text{d}z=\unicode[STIX]{x1D700}$ while the first term on the right-hand side vanishes. In the long-time limit, $\unicode[STIX]{x1D70F}\gg \unicode[STIX]{x1D70F}_{i}$ , the last term on the right-hand side of (4.35) vanishes as soon as $\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{i}$ is larger than a value of order unity since $g(z^{\prime })$ is zero outside the detonation structure, so that (4.27) is recovered. Introducing a shift of origin and the unknown function $a(t)$ , $\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}=\unicode[STIX]{x1D700}\text{d}a/\text{d}t$ , $\text{d}a/\text{d}t=O(1)$ and using (4.22),

(4.36a-c ) $$\begin{eqnarray}\displaystyle t\equiv \unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{i},\quad a(t)\equiv \frac{\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70F})-\unicode[STIX]{x1D6FC}_{i}}{\unicode[STIX]{x1D700}},\quad \unicode[STIX]{x1D6FC}_{i}/\unicode[STIX]{x1D70F}_{i}=\sqrt{2\unicode[STIX]{x1D700}}, & & \displaystyle\end{eqnarray}$$

equation (4.35) takes the form

(4.37) $$\begin{eqnarray}\displaystyle t\geqslant 0:\quad 2\frac{\text{d}a(t)}{\text{d}t}=\int _{-t}^{0}g(z^{\prime })\frac{\text{d}a(t+z^{\prime })}{\text{d}t}\,\text{d}z^{\prime }+\left[\frac{1+\displaystyle \frac{\sqrt{\unicode[STIX]{x1D700}/2}}{\unicode[STIX]{x1D70F}_{i}}a(t)}{1+t/\unicode[STIX]{x1D70F}_{i}}\right]^{2}+\int _{-\infty }^{-t}g(z^{\prime })\,\text{d}z^{\prime }, & & \displaystyle\end{eqnarray}$$

satisfying the initial conditions $t=0:a=0,\text{d}a/\text{d}t=1$ . Here $g(z)$ is defined in (B 21) for the steady solution of the CJ wave,

(4.38) $$\begin{eqnarray}\displaystyle g(z)\equiv \left[\frac{b}{2}\frac{\text{d}}{\text{d}\unicode[STIX]{x1D709}}[\unicode[STIX]{x1D709}\overline{w}_{CJ}(\unicode[STIX]{x1D709})]+\frac{\text{d}}{\text{d}\unicode[STIX]{x1D709}}\overline{\unicode[STIX]{x1D707}}_{CJ}(\unicode[STIX]{x1D709})\right]\overline{\unicode[STIX]{x1D707}}_{CJ}(\unicode[STIX]{x1D709}), & & \displaystyle\end{eqnarray}$$

where the variable $z$ is defined in (B 20), $z=\int _{0}^{\unicode[STIX]{x1D709}}\text{d}\unicode[STIX]{x1D709}^{\prime }/\overline{\unicode[STIX]{x1D707}}_{CJ}(\unicode[STIX]{x1D709}^{\prime })$ . Equation (4.37) is valid for a stable CJ detonation and, according to (4.17), for $\unicode[STIX]{x1D70F}_{i}\gg 1/\sqrt{\unicode[STIX]{x1D700}}$ .

A straightforward extension of (4.37) to unstable CJ detonations ( $b>b_{c}$ ) is obtained from (4.32) when the distribution of reaction rate $w(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})$ in (4.7) is no longer linearized,

(4.39) $$\begin{eqnarray}\displaystyle t\geqslant 0:\,2\dot{a}_{t}(t)=\int _{-t}^{0}G(z^{\prime },\dot{a}_{t}(t+z^{\prime }))\,\text{d}z^{\prime }+\left[\frac{1+\displaystyle \frac{\sqrt{\unicode[STIX]{x1D700}/2}}{\unicode[STIX]{x1D70F}_{i}}a(t)}{1+t/\unicode[STIX]{x1D70F}_{i}}\right]^{2}+\int _{-\infty }^{-t}G(z^{\prime },1)\,\text{d}z^{\prime }, & & \displaystyle\end{eqnarray}$$

where the notation $\dot{a}_{t}\equiv \text{d}a/\text{d}t$ ( $\unicode[STIX]{x1D700}\dot{a}_{t}=\dot{\unicode[STIX]{x1D6FC}}_{\unicode[STIX]{x1D70F}}$ ) has been introduced,

(4.40) $$\begin{eqnarray}\displaystyle G(z,\dot{a}_{t})\equiv \frac{\overline{\unicode[STIX]{x1D707}}_{CJ}(\unicode[STIX]{x1D709})}{2\unicode[STIX]{x1D700}}[\text{e}^{\unicode[STIX]{x1D700}b\dot{a}_{t}}\overline{\unicode[STIX]{x1D714}}_{CJ}(\unicode[STIX]{x1D709}\text{e}^{\unicode[STIX]{x1D700}b\dot{a}_{t}})-\overline{\unicode[STIX]{x1D714}}_{CJ}(\unicode[STIX]{x1D709})]+\frac{\text{d}\overline{\unicode[STIX]{x1D707}}_{CJ}(\unicode[STIX]{x1D709})}{\text{d}z}\dot{a}_{t}. & & \displaystyle\end{eqnarray}$$

Here also the quasi-steady-state approximation is recovered when the delay in the first integral term is negligible, $\int _{-\infty }^{0}G(z^{\prime },\dot{a}_{t}(t))\,\text{d}z^{\prime }=\dot{a}_{t}(t)$ . For a given initial time $\unicode[STIX]{x1D70F}_{i}$ , for example $\unicode[STIX]{x1D70F}_{i}=1/\unicode[STIX]{x1D700}$ , equation (4.39) involves two scalar parameters: $b$ for the thermal sensitivity and $\unicode[STIX]{x1D700}\ll 1$ for the initial proximity of the CJ regime. The distribution of the rate of heat release in the steady-state solution of the CJ wave $\overline{\unicode[STIX]{x1D714}}_{CJ}(\unicode[STIX]{x1D709})$ , $\text{d}\overline{\unicode[STIX]{x1D707}}_{CJ}/\text{d}z=\overline{\unicode[STIX]{x1D714}}_{CJ}/2$ , is also involved in this integral equation describing the decay of detonation toward the CJ regime, $\dot{a}_{t}(t)=O(1)$ , $\dot{a}_{t}(0)=1$ , $\lim _{t\rightarrow \infty }\dot{a}_{t}=0$ . Anticipating that the quasi-steady-state approximation (3.22) provides a correct order of magnitude, $\lim _{t\rightarrow \infty }\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{\infty }\approx (1+\sqrt{\unicode[STIX]{x1D700}/2})\unicode[STIX]{x1D6FC}_{i}$ , $\lim _{t\rightarrow \infty }a\approx \unicode[STIX]{x1D6FC}_{i}/\sqrt{2\unicode[STIX]{x1D700}}=\unicode[STIX]{x1D70F}_{i}$ , the term $\sqrt{\unicode[STIX]{x1D700}/2}\,a(t)/\unicode[STIX]{x1D70F}_{i}\approx \sqrt{\unicode[STIX]{x1D700}/2}$ is expected to be negligible in the numerator of the bracket in (4.39). Therefore, to the leading order of the asymptotic analysis in the limits (4.14) and (4.19), the small parameter $\unicode[STIX]{x1D700}$ disappears from (4.37),

(4.41) $$\begin{eqnarray}\displaystyle t\geqslant 0:\quad 2\dot{a}_{t}(t)=\int _{-t}^{0}g(z^{\prime })\dot{a}_{t}(t+z^{\prime })\,\text{d}z^{\prime }+(1+t/\unicode[STIX]{x1D70F}_{i})^{-2}+\int _{-\infty }^{-t}g(z^{\prime })\,\text{d}z^{\prime }. & & \displaystyle\end{eqnarray}$$

For unstable detonations, which are typically the case in real situations, the small parameter $\unicode[STIX]{x1D700}$ appears only in the nonlinear term $G(z,\dot{a}_{t})$ which controls the amplitude of the nonlinear oscillations,

(4.42) $$\begin{eqnarray}\displaystyle t\geqslant 0:\quad 2\dot{a}_{t}(t)=\int _{-t}^{0}G(z^{\prime },\dot{a}_{t}(t+z^{\prime }))\,\text{d}z^{\prime }+(1+t/\unicode[STIX]{x1D70F}_{i})^{-2}+\int _{-\infty }^{-t}G(z^{\prime },1)\,\text{d}z^{\prime }. & & \displaystyle\end{eqnarray}$$

When the instability threshold is characterized by a critical value of $b$ of order unity, namely when the distribution $\overline{\unicode[STIX]{x1D714}}_{CJ}(\unicode[STIX]{x1D709})$ is sufficiently smooth, the amplitude of the nonlinear oscillations $|\dot{a}_{t}|$ becomes large for $b=O(1/\unicode[STIX]{x1D700})$ . In the spirit of asymptotic analyses, the solutions are limited to the unstable region in which the amplitude of oscillations are such that the term $\int _{-\infty }^{0}G(z^{\prime },\dot{a}_{t}(t+z^{\prime }))\,\text{d}z^{\prime }$ is of order unity.