2.1. Two-time-scale analysis of the rarefaction wave

In the limit of small heat release, the flow of burnt gas close to the detonation front is controlled by a single equation, which corresponds to (6.6) of Clavin & Denet (Reference Clavin and Denet2020) by setting the reaction rate equal to zero. In this section, we show that this equation can be extended throughout the rarefaction wave. For that purpose we study the problem of fluid mechanics behind a supersonic discontinuity in the limit of a propagation velocity larger than the sound speed by a small amount. Denoting $\gamma$, $a$, $p$ and $u$ the ratio of specific heat, the isentropic speed of sound, the pressure and the gas velocity (relative to the laboratory frame where the uncompressed gas is at rest), an inert compressible flow in a spherical geometry is governed by two hyperbolic equations

(2.1)\begin{equation} \frac{1}{\gamma p}\left[\frac{\partial}{\partial t}+(u\pm a)\frac{\partial}{\partial r}\right ]p \pm\frac{1}{a}\left[\frac{\partial}{\partial t}+(u\pm a)\frac{\partial}{\partial r}\right ]u={-} 2\frac{u}{r}, \end{equation}

where $r$ is the radius and $t$ the time, see Liñán et al. (Reference Li nan, Kurdyumov and Sanchez2012) and Clavin & Denet (Reference Clavin and Denet2020). For a detonation treated as a discontinuity, the pressure $p$ and the gas velocity $u$ on the detonation front $r=r_f(t)$ are given in terms of the propagation velocity $\mathcal {D}(t)$ by the conservation equations recalled in Appendix A, see (A1)–(A6),

(2.2a,b)\begin{equation} r=r_f(t): \quad \frac{1}{\gamma}\frac{p-p_u}{p_u}=M(t) \frac{u}{a_u}, \end{equation}

where $M(t)\equiv \mathcal {D}(t)/a$ is the Mach number and the subscript $u$ identifies the constant properties upstream from the front. Written in the reference frame attached to the front equations (2.1) read

(2.3)\begin{equation} \left. \begin{gathered} x\equiv r-r_f(t)\leqslant0, \quad \textrm{d}r_f(t)/\textrm{d}t=\mathcal{D}(t)\\ \frac{1}{\gamma p}\left[\frac{\partial}{\partial t}+(u\pm a-\mathcal{D})\frac{\partial}{\partial x}\right ]p \pm\frac{1}{a}\left[\frac{\partial}{\partial t}+(u\pm a-\mathcal{D})\frac{\partial}{\partial x}\right ]u={-} 2\frac{u}{r_f(t)+x}. \end{gathered} \right\} \end{equation}

In the limit of small heat release $q_m/c_p T_u \equiv \epsilon ^2 \ll 1$, where $q_m $ and $c_p $ are respectively the heat release and the specific heat per unit mass and $T_u$ is the initial temperature of the unburnt gaseous mixture, the velocity of the planar CJ wave in steady state is slightly higher than the sound speed, see (A2),

(2.4a,b)\begin{equation} 0< (\mathcal{D}_{o_{CJ}}-a)/a \approx \epsilon \ll 1, \quad \mathcal{D}_{o_{CJ}}\approx (1+\epsilon)a, \end{equation}

and attention is focused on detonations sufficiently close to the CJ regime

(2.5)\begin{equation} 0\leqslant(\mathcal{D}-\mathcal{D}_{o_{CJ}})/\epsilon a =O(1) \quad \text{or smaller}. \end{equation}

In the limit $\epsilon \ll 1$, $0<(M-1)=O(\epsilon )$, according to (A1)–(A6), the gas velocity is smaller than the speed of sound $u/a=O(\epsilon )$ at the front. The condition for which a weak shock wave $0<(M-1)\ll 1$ can be considered as a discontinuity is discussed in Clavin & Williams (Reference Clavin and Williams2002). According to (2.2a,b), the relative pressure variation is as small as $u/a$ and the variation of $(1/\gamma )\delta p/ p-u/a$ is even smaller, of order $(M-1)u/a$, $(1/\gamma )\delta p/ p-u/a=O(\epsilon ^2)$. The variation of the speed of sound is also of order $\epsilon ^2$ $\delta a/a=O(\epsilon ^2)$. Neglecting terms of order $\epsilon ^2$, the variation of $a$ is negligible, and the isentropic relation $(1/\gamma )\delta p/ p\approx u/a$ holds at the front, up to the second order in the perturbation analysis for $\epsilon \ll 1$ under the conditions (2.4a,b) and (2.5) for which the relative jump of entropy across the detonation front is of order $\epsilon ^2$. Then, the rarefaction wave is a quasi-transonic flow which can be analysed by a two-time-scale analysis.

It was known long ago that the rarefaction wave $u(r,t)$ behind a spherical detonation is delimited by a weak discontinuity at the radius $r_0(t)$ of a core of stagnant gas ($r_0<r_f$) which grows with the speed of sound

(2.6a–c)\begin{align} \text{d} r_0(t)/\text{d} t=a, \quad r\leqslant r_0(t): u=0 \Rightarrow \,\text{d} u \left(r_0(t),t\right)/\text{d} t=0; \quad r> r_0(t): u>0. \end{align}

The flow $u$ is oriented in the same direction as the propagation $\mathcal {D}>0$, $u\geqslant 0$, and is increasing monotonically from zero at $r=r_0(t)$ to a value (at the detonation front $r_f$) smaller than the speed of sound by a factor $\epsilon$, so that the ordering $u/a=O(\epsilon )$ holds throughout the rarefaction wave. The equations of the two characteristics $\mathcal {C}_{\pm }$ in (2.3) involve two differential operators

(2.7a,b)\begin{equation} {\partial }/{\partial t}+V_{{\pm}}{\partial }/{\partial x}, \quad V_{{\pm}}\equiv (u\pm a)-\mathcal{D}. \end{equation}

The scalars $I_+\equiv (1/\gamma )\delta p/p + u/a$ and $I_-\equiv (1/\gamma )\delta p/p - u/a$ that are transported by upstream running characteristic $\mathcal {C}_{+}$ and downstream running characteristic $\mathcal {C}_{-}$ respectively, are modified by the flow divergence on the right-hand side of (2.3) whose order of magnitude is $\epsilon a/r$,

(2.8)\begin{equation} \frac{\partial I_\pm}{\partial t}+V_{{\pm}}\frac{\partial I_\pm }{\partial x}={-}2\frac{u}{r}=O(\epsilon a/r). \end{equation}

Under the conditions (2.4a,b) and (2.5), the flow with respect to the front is quasi-transonic, $\mathcal {D}-u=O(\epsilon a)$. The propagation velocity $V_-\equiv u-a-\mathcal {D} < 0$ of the downstream-running characteristics $\mathcal {C}_-$ is approximately $-a$ throughout the rarefaction wave. (We use here the same convention as in Clavin & Williams Reference Clavin and Williams2002; upstream-running (downstream-running) characterizes a propagation towards the shock (the core of stagnation gas) in the frame attached to the front.) Near the detonation front where the flow relative to the front is subsonic $\mathcal {D}-u \leqslant a$ (overdriven regime), the characteristic $\mathcal {C}_+$ is upstream running $V_+=a-(\mathcal {D}-u)>0$. But a sonic point appears when $u$ decreases since $V_+=0$ when $\mathcal {D}-u=a$. Therefore the characteristic $\mathcal {C}_+$ becomes downstream running ($V_+<0$) behind the sonic point, which stands close to the front because we consider propagation regimes of the detonation that are close to the CJ regime which is characterized by the sonic condition at the detonation front. Then, the transport of $I_-$ by $\mathcal {C}_-$ from the sonic point to $r_0$ is quasi-instantaneous compared to the slow transport of $I_+$ by $\mathcal {C}_+$, $\vert V_+ \vert /a = O(\epsilon )$, $\vert V_- \vert /a \approx 1$. Anticipating that the thickness of the rarefaction wave $\Delta r\equiv (r_f-r_0 )$ is smaller than the detonation radius $\Delta r/r_f=O(\epsilon )$, see § 2.2, the modification of $I_-$ by the geometrical effect during the transit time $\Delta r/a$ of $\mathcal {C}_-$ is of order $\epsilon \Delta r/r_f =O(\epsilon ^2)$ and can be neglected. Then, to leading order in the limit $\epsilon \ll 1$, the isentropic relation of acoustics $(1/\gamma )\delta p/p=u/a$, which is valid inside the detonation structure in the limit of small heat release, holds throughout the rarefaction wave $u\ll a$, $u/\epsilon a=O(1)$. Then, neglecting the short time delay and the small flow modification (of order $\epsilon ^2 a$) introduced by the fastest downstream-running mode and retaining only the slow time scale, the leading order of the flow $u( r,t)$ is controlled by a single equation corresponding to the simple wave associated with $\mathcal {C}_+$,

(2.9)\begin{gather} \left[\frac{\partial}{\partial t}+(u +a)\frac{\partial}{\partial r}\right ]u={-} a\frac{u}{r}, \end{gather}

(2.10a–c)\begin{gather}x\equiv r-r_f(t)\leqslant0, \quad \left[\frac{\partial}{\partial t}+(u +a-\mathcal{D})\frac{\partial}{\partial x}\right ]u={-} a\frac{u}{r_f(t)+x}, \quad \frac{\text{d} r_f}{\text{d} t}= \mathcal{D}(t). \end{gather}

2.2. Self-similar rarefaction wave behind a spherical CJ detonation

Generally speaking, when the modification to the inner structure of the leading front is ignored (zero detonation thickness), a self-similar solution exists when a finite amount of energy is deposited quasi-instantaneously by a quasi-punctual external source at the centre because there are no length and time scales in the problem. This is the case for the blast wave of Sedov (Reference Sedov1946) and Taylor (Reference Taylor1950b) and the rarefaction wave of Zeldovich (Reference Zeldovich1942) and Taylor (Reference Taylor1950a) behind a CJ detonation, obtained in the limit of large Mach number $M\gg 1$, see, for example, Clavin & Searby (Reference Clavin and Searby2016). We show below that, in the opposite limit of small heat release $M_{o_{CJ}}-1\ll 1$, there is also a self-similar solution for the rarefaction wave behind a CJ detonation, which is qualitatively similar to the self-similar solution of Zeldovich (Reference Zeldovich1942) and Taylor (Reference Taylor1950a). The difference concerns mainly the extension of the self-similar rarefaction wave which is small compared to the radius of the detonation in the limit $M_{o_{CJ}}- 1\ll 1$.

2.2.1. Formulation

Consider the spherical detonation propagating with the constant CJ velocity $\mathcal {D}_{o_{CJ}}\approx (1+\epsilon )a$ in the limit (2.4a,b) $\epsilon \ll 1$. According to the conservation equations (A1)–(A6), the boundary condition at the front takes the form

(2.11a,b)\begin{equation} x=0: \quad u\approx\epsilon a. \end{equation}

Dividing by $\epsilon a$ and introducing the non-dimensional flow of order unity in the limit $\epsilon \ll 1$, $v (x, t)\equiv u/(\epsilon a)$, $v\in [0, 1]$, (2.10a–c) yields

(2.12a,b)\begin{gather} \left[\frac{\partial}{\partial t}+(u+ a-\mathcal{D}_{o_{CJ}})\frac{\partial}{\partial x}\right ]v={-} a\frac{v}{r_f(t)+x}, \quad v \equiv \frac{u}{\epsilon a}, \end{gather}

(2.13)\begin{gather} \left[\frac{1}{a}\frac{\partial}{\partial t}+(v-1)\epsilon\frac{\partial}{\partial x}\right ]v={-} \frac{v}{r_f(t)+x}, \end{gather}

where (2.4a,b) has been used. This suggests rescaling the distance from the front by using the new space variable $\eta \equiv x/\epsilon$, so that (2.11a,b) and (2.13) read

(2.14a,b)\begin{align} \eta \equiv [r-r_f(t)]/\epsilon, \quad \left[\frac{1}{a}\frac{\partial}{\partial t}+(v-1)\frac{\partial}{\partial \eta}\right ]v={-} \frac{v}{r_f(t)+\epsilon \eta}; \quad \eta =0: v=1. \end{align}

The downstream relation (2.6a–c) is automatically fulfilled at $\eta =\eta _0(t) \equiv [r_0(t)-r_f(t)]/\epsilon$ since the relations $v(\eta _0(t), t)=0$ and $\text {d} v(\eta _0(t), t)/\text {d} t =0$ are verified by (2.14a,b), $v=0:\partial v/\partial t- a\partial v/\partial \eta =0$, yielding $\text {d}\eta _0(t)/\text {d} t= -a$, so that, using (2.4a,b) $\text {d}r_f/\text {d} t=(1+\epsilon )a$, one gets $\text {d} r_0/\text {d} t=a$.

Introducing the two lengths $r_{fi}$ and $r_{0i}$ characterizing the initial condition $t=0$,

(2.15a–c)\begin{gather} r_f(t)=\mathcal{D}_{o_{CJ}}t+r_{fi}\approx (1+\epsilon)at+ r_{fi}, \quad r_0(t)=at+ r_{0i}, \quad r_{fi}>r_{0i}, \end{gather}

(2.16a,b)\begin{gather}\eta_0(t)\equiv{-}[r_f(t)-r_0(t)]/\epsilon={-}at+\eta_{0i}, \quad \eta_{0i}\equiv{-}(r_{fi}-r_{0i})/\epsilon<0, \end{gather}

with $\epsilon \vert \eta _{0i} \vert$ being the initial thickness of the rarefaction wave, we will show below that a self-similar solution of (2.14a,b) exists in the limit $\epsilon \ll 1$ if the initial thickness of the rarefaction wave $r_{fi}-r_{0i}$ is smaller than the initial radius of the detonation $r_{fi}$ by a factor $\epsilon$, as it is the case in the linear solution presented in Appendix B.1,

(2.17a–c)\begin{equation} (r_{fi}-r_{0i})/r_{0i} =O(\epsilon ) ,\quad r_{f}(t)-r_{0}(t)=\epsilon (at-\eta_{0i}), \quad \vert \eta_{0i}\vert =O(r_{0i}). \end{equation}

To leading order, the denominator $r=r_f+\epsilon \eta$ on the right-hand side of (2.14a,b) can be replaced by $r_0(t)$ throughout the rarefaction wave $r_0(t)\leqslant r\leqslant r_f(t)$ since, according to (2.17a–c), $r_f(t)=r_0(t)+\epsilon at-\epsilon \eta _{0i}$ so that $r_f(t)=(1+\epsilon )r_0(t)+O(\epsilon r_{0i})$. Therefore, replacing the time variable $t$ by a time-like variable $\nu \equiv r_0(t)=at+r_{0i}$ whose dimension is a length, (2.14a,b) for the flow $v( \eta , \nu )$ takes the form of a Burgers-like equation, free from parameter, in which the local viscous dissipation is replaced by a global (linear) damping rate on the right-hand side,

(2.18)\begin{equation} \epsilon \to 0: \quad \left[\frac{\partial}{\partial \nu}+(v-1)\frac{\partial}{\partial \eta}\right ]v={-} \frac{v}{\nu}, \quad \eta =0: \ v=1, \end{equation}

where

(2.19a–c)\begin{equation} \nu\equiv at+r_{0i}, \quad \eta \equiv \frac{r-r_f(t)}{\epsilon}, \quad v (\eta, \nu)\equiv \frac{u(x, t)}{\epsilon a}. \end{equation}

The parameter $r_{0i}$ in the definition of the time-like variable $\nu$ can be eliminated by a change of time origin.

2.2.2. Infinite gradient of the flow on the front. Self-similar solution

As in the self-similar solution of Zeldovich (Reference Zeldovich1942) and Taylor (Reference Taylor1950a) obtained in the opposite limit of large Mach number $\mathcal {D}_{o_{CJ}}/a \gg 1$, the solution of (2.18) is singular on the detonation front where the gradient of the flow becomes infinite. The flow being constant on the front $v=1$, the unsteady term $\partial v/\partial \nu$ in (2.18) is negligible around $\eta =0^-$ so that the steady-state approximation holds near the detonation front $(v-1){\partial v}/{\partial \eta }\approx - {1}/{\nu }$

(2.20a,b)\begin{equation} 1-v= \sqrt {\frac{2}{\nu}(-\eta)}, \quad 1-\frac{u}{\epsilon a}= \sqrt {\frac{2[r_f(t)-r]}{\epsilon(at+r_{0i})}}, \end{equation}

the time derivative $\partial v /\partial \nu \vert _{\eta =0^-} =\sqrt {-2\eta }/(2\nu ^{3/2})$ being negligible in a boundary layer at the front $\vert \partial v /\partial \nu \vert _{\eta =0^-} \ll 1/\nu$,

(2.21a,b)\begin{equation} \vert \eta \vert \ll {2 \nu}, \quad r_f(t)-r\ll 2\epsilon(at+r_{0i}). \end{equation}

The divergence of the flow gradient on the front of a spherical CJ detonation $\partial v /\partial \eta \vert _{\eta =0^-} \propto 1/(-\eta )^{1/2}$ is a consequence of the sonic condition $\eta =0: \,v-1=0,\ \forall t$.

The self-similar solution of (2.18) is obtained by looking for a solution in the form

(2.22)\begin{gather} v(\eta, \nu)=\text V(z),\quad \textrm{with}\ z\equiv \eta/\nu \Rightarrow \frac{u}{\epsilon a} =V\left(\frac{r-r_{f}(t))}{\epsilon r_{0}(t)}\right), \end{gather}

(2.23)\begin{gather} {[}-(1+z)+V]\frac{\text{d}V}{\textrm{d}z}={-}V, \quad z=0:\ V=1. \end{gather}

After multiplication by $1/V^2$, (2.23) takes the form $\text {d} [(1+z)/V]/\text {d} z+(1/V)\text {d} V/\text {d} z =0$, then, the solution $V(z)$ satisfying the boundary condition at $z=0$ is the root of a transcendental equation

(2.24)\begin{equation} V\ln V-V+(z+1)=0. \end{equation}

According to this equation, the radius $r=r_0(t)$ of the spherical core of stagnant gas $V=0$ corresponds to $z=-1$. Therefore, $r_0(t)$ and $r_f(t)$ are linked by the relation

(2.25)\begin{equation} z\equiv[{r_0(t)-r_{f}(t)}]/{\epsilon r_{0}(t)}={-}1 \Leftrightarrow (1+\epsilon) r_0(t)= r_f(t), \end{equation}

in agreement with the assumption (2.17a–c) in the limit of small heat release (2.4a,b), which finally takes the more restrictive form

(2.26)\begin{equation} (r_{fi}-r_{0i}) =\epsilon r_{0i}, \end{equation}

in a consistent way with $r_f(t)\approx (1+\epsilon ) at+r_{fi}$ and $r_0(t)=at+r_{0i}$ yielding (2.26) for $z=-1$. The velocity profile of the self-similar rarefaction wave is plotted in figure 1. Close to the detonation front $z\approx 0$, namely for $V= 1+\delta V$ with $\vert \delta V\vert \ll 1$, $\ln V\approx \delta V-\delta V^2/2+\cdots$, $V\ln V \approx -1+(\delta V)^2/2+\cdots$ so that (2.24) yields $(\delta V)^2/2 +z\approx 0$ and the relation (2.20a,b) is recovered for $\vert z\vert \ll 1$, $V\approx 1-\sqrt {-2z}$. At the radius of the stagnant core, the root of (2.24) goes to zero $\lim _{z=-1} V=0^+$ with a zero gradient $\text {d} V/\text {d} z\vert _{z=-1} =0$, as shown by taking the limit $V\to 0^+$ of the derivative of (2.24) $\text {d} (V\ln V)/\text {d} z-\text {d}V/\text {d} z+1=0$ leading to $\lim _{V\to 0^+}(\text {d}V/\text {d} z)=-2/\ln V\to 0^+$.

Figure 1. Solution of (2.23)–(2.24) representing the self-similar rarefaction flow behind the front of a spherical CJ detonation considered as a discontinuity for small heat release $(\mathcal {D}_{o_{CJ}}-a)/a\approx \epsilon \ll 1$. The reduced flow $v= u(r,t)/\epsilon a$ is plotted vs the reduced distance from the front $z= (r-r_f(t) )/\epsilon r_{0}(t)$ with $r_f(t)=\mathcal {D}_{o_{CI}}t+r_{fi}$, $r_0(t)=at+r_{0i}$ and, according to (2.26), $r_{fi} = (1+\epsilon )r_{0i}$.

To conclude the rarefaction wave behind a spherical CJ detonation sustained by a small heat release ($M_{o_{CJ}}-1\approx \epsilon \ll 1$) is similar to the self-similar solution for $M_{o_{CJ}}\gg 1$. The only difference is quantitative; the extension of the rarefaction wave is smaller than the detonation radius $r_f(t)$ by a factor $M_{o_{CJ}}-1$, $(r_f-r_0)/r_f\approx (M_{o_{CJ}}-1)/M_{o_{CJ}}$ for small heat release while $(r_f-r_0)/r_f=1/2$ for $M_{o_{CJ}}\gg 1$.

2.3. Rarefaction wave behind an overdriven detonation considered as a discontinuity

Within the framework of the discontinuous model the decay of the propagation velocity of an overdriven detonation $\mathcal {D}(\tau )$ in a spherical geometry to the planar Chapmann–Jouguet velocity $\mathcal {D}_{o_{CJ}}$ occurs systematically after a finite time and at a finite radius. In this section, we derive an analytical expression for the rarefaction wave behind an overdriven detonation treated as a discontinuity approaching the CJ velocity, in the limit $\epsilon \ll 1$. The subsequent relaxation to the self-similar solution is discussed in § 3.

2.3.1. Formulation

As already mentioned, there is no length (or time) scale in the direct-initiation problem with the discontinuous model. However, in view of bridging the gap with the study in the second part of this manuscript, it is useful to introduce the non-dimensional space and time variables $\xi$ and $\tau$ as well as the reduced flow field $\mu (\xi ,\tau )$ that are of order unity inside the unsteady (and curved) inner structure of the detonation in the limit (2.4a,b)–(2.5)

(2.27a–c)\begin{gather} \xi \equiv \frac{(r-r_f(t))}{l}, \quad \tau\equiv \epsilon \frac{a\,t}{l}, \quad \mathcal{D}=\frac{\text{d} r_f}{\text{d} t}, \end{gather}

(2.28)\begin{gather} \mu(\xi,\tau)\equiv v-1 = {u}/{\epsilon a}-1 \Leftrightarrow u/{a}=\epsilon(1+\mu), \end{gather}

where $l$ is the detonation thickness and $1/t_r=a/l$ the reaction rate at the Neumann state,

(2.29a,b)\begin{equation} \frac{\partial }{\partial r}=\frac{1}{l}\frac{\partial }{\partial \xi},\quad \frac{\partial }{\partial t}=\frac{\epsilon}{t_r}\frac{\partial }{\partial \tau}-\frac{\mathcal{D}}{l}\frac{\partial }{\partial \xi}. \end{equation}

For clarity, the results will be written with the two types of variables. Introducing the reduced radius $\tilde {r}_f(\tau )$ and propagation velocity $\dot {\alpha }_\tau (\tau )$

(2.30a,b)\begin{equation} \tilde{r}_f(\tau)\equiv \epsilon r_f(\tau)/l, \quad \dot{\alpha}_\tau(\tau) \equiv [\mathcal{D}(\tau)-\mathcal{D}_{o_{CJ}}]/(\epsilon a), \end{equation}

(2.10a–c) reduces to (6.6) in Clavin & Denet (Reference Clavin and Denet2020) without the reaction term

(2.31)\begin{equation} \frac{\partial \mu}{\partial \tau} + ( \mu -\dot{\alpha}_\tau)\frac {\partial \mu}{\partial \xi} ={-}\frac{1}{\tilde{r}_f(\tau) +\epsilon \xi}\ (1+\mu ),\end{equation}

where

(2.32a,b)\begin{equation} \frac{\text{d} \tilde{r}_f(\tau)}{\text{d} \tau}=\frac{\mathcal{D}}{a}=(1+\epsilon)+ \epsilon \dot{\alpha}_\tau(\tau),\quad \tilde{r}_f(\tau)=\tilde{r}_{fi}+(1+\epsilon)\tau+\epsilon \int_0^\tau\dot{\alpha}_\tau \,\text{d} \tau, \end{equation}

see Appendix C.1. In (2.31), $\tilde {r}_{fi}\equiv \epsilon r_{fi}/l$ is associated with the initial position of the front $r_{fi} \equiv r_f(0)$. The sonic condition with respect to the front $(\mathcal {D}-u)= a$ corresponds to $(\mu -\dot {\alpha }_\tau )=0$, and a subsonic condition $(\mathcal {D}-u) < a$ corresponds to $(\mu -\dot {\alpha }_\tau )>0$ while $(\mu -\dot {\alpha }_\tau )<0$ characterizes a supersonic flow relative to the detonation front. The decay of the detonation velocity to the CJ velocity corresponds to $\dot {\alpha }_\tau \to 0^+$ and $\mu _f \to 0^+$.

As already mentioned, the radius $r=r_0(\tau )$ of the spherical core of stagnant gas $u(r_0(\tau ), \tau )=0$

(2.33a–c)\begin{equation} r < r_0(\tau): u=0, \quad r > r_0(\tau): u > 0, \quad\partial u/\partial r\vert_{r=r^+_0}\geqslant 0 \end{equation}

is a weak discontinuity of the flow moving at the speed of sound $\text {d} r_0(t)/\text {d} t=a$. Introducing the reduced thickness of the rarefaction wave $\vert \xi _0(\tau ) \vert$

(2.34)\begin{equation} \xi_0(\tau)\equiv \frac{[r_0(\tau)-r_f(\tau)]}{l}<0; \quad \xi=\xi_0(\tau): \mu={-}1\quad \text{i.e.} \quad \mu\left(\xi_0(\tau),\tau \right)={-}1, \end{equation}

the weak discontinuity is recovered in (2.31) at $\xi =\xi _0(\tau )$ where ${\partial \mu }/{\partial \tau }\vert _{\xi =\xi _0^+} -(1+\dot {\alpha }_\tau ) {\partial \mu }/{\partial \xi }\vert _{\xi =\xi _0^+}=0$ and $\mu =-1$ by differentiating the last expression in (2.34), $\text {d} \mu (\xi _0(\tau ),\tau )/\text {d} \tau =\partial \mu /\partial \tau \vert _{\xi =\xi _0^+}+(\text {d} \xi _0/\text {d}\tau )\partial \mu /\partial \xi \vert _{\xi =\xi _0^+}$ $=0$ yielding

(2.35a,b)\begin{equation} \text{d} \xi_0/\text{d}\tau={-}(1+\dot{\alpha}_\tau), \quad \xi_0(\tau)={-}\tau-\int_0^\tau \dot{\alpha}_\tau(\tau')\,\text{d}\tau'+\xi_{0i}, \end{equation}

which corresponds effectively to $\text {d} r_0/\text {d} t=a$ that is $\text {d} (r_0- r_f)/\text {d} t=a-\mathcal {D}$ with, according to (2.4a,b) and (2.30a,b), $(a-\mathcal {D})=-\epsilon a(1+\dot {\alpha }_\tau )$. When the order of magnitude of $\dot {\alpha }_\tau$ is not larger than unity in the limit $\epsilon \to 0$, $(\mathcal {D}-a)/a =O( \epsilon )$, (2.35a,b) shows that the thickness of the rarefaction wave $r_f(t)-r_0(t)$ increases with a velocity of order $\epsilon a$ smaller than the speed of sound by a factor $\epsilon$. Considering $\tau =O(1)$ in the limit of small heat release, $\lim _{\epsilon \to 0}\epsilon \xi _0=0$, (2.32a,b) yields $\lim _{\epsilon \to 0} \tilde {r}_{f}=\tau +\tilde {r}_{fi}$, so that, to leading order, (2.31) reduces to

(2.36)\begin{equation} \epsilon \to 0, \quad \frac{\partial \mu}{\partial \tau} + \left[ \mu -\dot{\alpha}_\tau(\tau)\right]\frac {\partial \mu}{\partial \xi} ={-}\frac{1+\mu}{\tau+\tilde{r}_{fi}},\quad\textrm{where}\ \tilde{r}_{fi}\equiv \epsilon \frac{r_{fi}}{l}=O(1). \end{equation}

Rescaling the non-dimensional length $\xi$ and time $\tau$ with $\tilde {r}_{fi}\equiv \epsilon r_{fi}/l$,

(2.37a–c)\begin{equation} \tilde{\tau} \equiv \frac{\tau}{\tilde{r}_{fi}}=\frac{a t}{r_{fi}}, \quad\tilde{\xi} \equiv\frac{\xi}{\tilde{r}_{fi}}=\frac{r-r_f(t)}{\epsilon \,r_{fi}}, \quad \tilde{\xi}_0(\tilde{\tau})\equiv\frac{\xi_0(\tau)}{\tilde{r}_{fi}}=\frac{r_0(t)-r_f(t)}{\epsilon \,r_{fi}} \end{equation}

(2.35a,b)–(2.36) take a form free from parameter

(2.38a,b)\begin{equation} \frac{\partial \mu}{\partial \tilde{\tau}} + [ \mu -\dot{\alpha}_\tau(\tilde{\tau})]\frac {\partial \mu}{\partial \tilde{\xi}} ={-}\frac{(1+\mu)}{\tilde{\tau}+1}, \quad \frac{\text{d} \tilde{\xi}_0}{\text{d}\tilde{\tau}}={-}[1+\dot{\alpha}_\tau(\tilde{\tau})]. \end{equation}

Once a general solution of (2.38a,b) is known, the dynamics of the front $\dot {\alpha }_\tau (\tilde {\tau })$ is obtained by a boundary condition at the front. For the discontinuous model, the instantaneous flow of burned gas at the front, denoted by $\mu _f(\tilde {\tau })\equiv \mu (\tilde {\xi }=0,\tilde {\tau })$, is given by the conservation of mass, momentum and energy in Appendix A, leading to express $\mu _f(\tilde {\tau })$ in terms of $\dot {\alpha }_\tau (\tilde {\tau })$

(2.39)\begin{equation} \tilde{\xi}=0: \quad \mu\equiv \mu_f(\tilde{\tau})=\mathcal{M}_f(\dot{\alpha}_\tau(\tilde{\tau})). \end{equation}

Equations (2.38a,b) and (2.39) represent an eigenvalue problem in which the unknown function $\dot {\alpha }_{\tau }(\tilde {\tau })$ appears in the boundary condition (2.39) and in (2.38a,b).

2.3.2. Simplified formulation near the CJ velocity

The formulation gets simpler when attention is focused on the end of the detonation decay when the velocity is close to the CJ velocity,

(2.40a,b)\begin{equation} 0<\frac{(\mathcal{D}_i-\mathcal{D}_{o_{CJ}})}{\epsilon a}\ll 1, \quad 0\leqslant\dot{\alpha}_\tau \equiv \frac{(\mathcal{D}-\mathcal{D}_{o_{CJ}})}{\epsilon a}\ll 1. \end{equation}

Expanding (A1)–(A2) for small values of $( \mathcal {D}- \mathcal {D}_{o_{CJ}} )/\epsilon a$ yields the well-known square root relation between the flow of burnt gas at the front ${u_b}(t)$ and the detonation velocity $\mathcal {D}(\tau )$ near the CJ regime ${u_b}/{\epsilon a}\approx 1+\sqrt {2( \mathcal {D}- \mathcal {D}_{o_{CJ}} )/\epsilon a}$, obtained by the relation $( {M_{o_{CJ}}-M_{o_{CJ}}^{-1}})^2/({M-M^{-1}})^2 \approx 1\text {--}2( \mathcal {D}- \mathcal {D}_{o_{CJ}} )/\epsilon a$ in (A1). Then the boundary condition at the front simplifies to

(2.41a,b)\begin{gather} {u_f(\tau)}/{\epsilon a}\approx1+\sqrt{2\left( \mathcal{D}(\tau)- \mathcal{D}_{o_{CJ}}\right )/\epsilon a}, \quad \mu_f(\tau)\approx \sqrt{2 \dot{\alpha}_\tau(\tau)}\ll 1, \end{gather}

(2.42)\begin{gather} 0<\mu_{fi}\equiv {u_{fi}}/{\epsilon a}-1\approx \sqrt {{2(\mathcal{D}_i-\mathcal{D}_{o_{CJ}})}/{\epsilon a}}\ll 1, \end{gather}

where $u_{fi}$ and $\mu _{fi}$ denote the initial value of $u_{f}(\tau )$ and $\mu _{f}(\tau )\equiv {u_{fi}}/{\epsilon a}-1$ with $u_{fi}>\epsilon a$. The term $[\mu -\dot {\alpha }_\tau (\tilde {\tau })]$ can then be replaced by $\mu$ in (2.36) because the flow field in the rarefaction wave increases monotonically with the radius, from $u=0$ at $r=r_0$ to a positive value $u_f$ on the front ($r=r_f$), $\mu \in [-1, \mu _f]$, with, according to (2.40a,b)–(2.41a,b), $0\leqslant \dot {\alpha }_\tau \ll \mu _f \ll 1$ so that $\dot {\alpha }_\tau ( \tau ) \ll \vert \mu (\xi , \tau ) \vert$ $\forall \xi \in [\xi _0,\, 0]$. Under the condition (2.40a,b), the unknown velocity of the front $\dot {\alpha }_\tau (\tilde {\tau })$ does not appear explicitly anymore on the left-hand side of (2.38a,b) which reduces to the Burgers-like equation in (2.18),

(2.43)\begin{equation} \epsilon \to 0, \quad 0<\dot{\alpha}_\tau \ll 1 \Rightarrow \frac{\partial \mu}{\partial \tilde{\tau}} + \mu \frac {\partial \mu}{\partial \tilde{\xi}} ={-}\frac{(1+\mu)}{\tilde{\tau}+1}, \quad \tilde{\xi}_{0}(\tilde{\tau})={-}\tilde{\tau}+\tilde{\xi}_{0i}, \end{equation}

the unknown function being the flow at the front $\mu _f(\tilde {\tau }) \equiv u_f(\tilde {\tau })/\epsilon a -1$.

As we shall see in § 3.1, the solution $\mu (\tilde {\xi }, \tilde {\tau })$ of (2.43) reaches the self-similar CJ solution (2.23)–(2.24) in the long-time limit if the initial thickness of the rarefaction wave scales as (2.17a–c), $\vert \tilde {\xi }_{0i}\vert =O(1)$ in the limit $\epsilon \to 0$. Notice that (2.43), divided by $r_{fi}$, yields (2.18) for $v=\mu +1$, the difference with the CJ problem being the boundary condition on the front ($\tilde {\xi }=0$) which now involves an unknown flow $\mu _f(\tilde {\tau })\ne 0$, $v(\eta =0, \nu ) \ne 1$.

2.3.3. Analytical solutions

The rarefaction wave behind a spherical detonation is a nonlinear solution of the Euler equations, which cannot be described by a linearized approximation, even if the flow velocity is smaller than the speed of sound as it is the case in the limit of small heat release. By comparison, the linear solution is briefly recalled in Appendix B. Equations (2.38a,b) and (2.43) have analytical solutions $\mu (\tilde {\xi }, \tilde {\tau })$ which provide us with an expression of the unsteady flow on the front $\mu _f(\tilde {\tau })= \mu (0, \tilde {\tau })$ in terms of the unknown function $\dot {\alpha }_\tau (\tilde {\tau })$ which, according to (2.30a,b), represents the propagation velocity of the front $\mathcal {D}(t)$. The dynamics of the front $\dot {\alpha }_\tau (\tilde {\tau })$ is then obtained in a second step through the boundary condition on the front. For example, the Rankine–Hugoniot condition yields the relaxation of a pure shock freely propagating in a spherical geometry which is derived in Appendix D. The end of the decay of an overdriven detonation, treated as a discontinuity, is obtained from (2.43) by using the boundary condition (2.41a,b).

Analytical solutions of (2.38a,b) are obtained in the form of separated variables,

(2.44a–d)\begin{align} \mu(\tilde{\xi}, \tilde{\tau})= A(\tilde{\tau}) B(\tilde{\eta})-1,\quad \tilde{\eta}\equiv \tilde{\xi}-\tilde{\xi}_0(\tilde{\tau}), \quad \mu(\tilde{\xi}_0(\tilde{\tau}), \tau )={-}1, \quad \frac{\text{d} \tilde{\xi}_0}{\text{d}\tilde{\tau}}={-}(1+\dot{\alpha}_\tau). \end{align}

Introducing the notation $B'\equiv \text {d} B/\text {d} \tilde {\eta }$ and $\dot {A}\equiv \text {d} A/\text {d} \tilde {\tau }$, ${\partial \mu }/{\partial \tilde {\tau }}=\dot {A}+(1+\dot {\alpha }_\tau )AB'$, $\mu {\partial \mu }/{\partial \tilde {\xi }}=(AB-\dot {\alpha }_\tau )AB'$, the unknown function $\dot {\alpha }(\tilde {\tau })$ disappears from the equations for $A(\tilde {\tau })$ and $B(\tilde {\eta })$, yielding

(2.45)\begin{equation} \dot{A} B+A^2BB'={-}AB/(\tilde{\tau}+1) \Rightarrow -\text{d} A^{{-}1}/\text{d} \tilde{\tau}+ A^{{-}1}/(\tilde{\tau}+1)={-}B', \end{equation}

the second equation being obtained after division by $A^2 B$. The left-hand side of the second equation in (2.45) is a function of $\tilde {\tau }$ only, while the right-hand side is a function of $\tilde {\eta }$. Therefore, the two sides should be equal to the same constant yielding an ordinary differential equation for $A(\tilde {\tau })$ and $B(\tilde {\eta })$ respectively

(2.46a,b)\begin{equation} \text{d} A^{{-}1}/\text{d} \tilde{\tau}- A^{{-}1}/(\tilde{\tau}+1)=k, \quad \text{d} B/\text{d} \tilde{\eta}= k. \end{equation}

The constant $k$ has to be obtained by an initial condition. Integration of the second equation is straightforward leading to a uniform gradient (straight profile of the flow). Introducing the initial value $\tau =0: A=A_i$, the solution of the first equation is,

(2.47)\begin{equation} A^{{-}1}=(1+\tilde{\tau})A_i^{{-}1}+k(1+\tilde{\tau})\ln(1+\tilde{\tau}), \end{equation}

the first term on the right-hand side being the general solution of the homogeneous equation and the second term a particular solution. Solutions of (2.43) then take the form

(2.48)\begin{gather} \mu(\tilde{\xi}, \tilde{\tau})= \frac{k[\tilde{\xi}-\tilde{\xi}_0(\tilde{\tau})]}{(1+\tilde{\tau})[A_i^{{-}1}+k\ln(1+\tilde{\tau})]}-1 \end{gather}

(2.49)\begin{gather}\tilde{\xi}=0:\quad \mu_f( \tilde{\tau})= \frac{-k\tilde{\xi}_0(\tilde{\tau})}{(1+\tilde{\tau})[A_i^{{-}1}+k\ln(1+\tilde{\tau})]}-1, \quad \tilde{\tau}=0: 1+\mu_{fi}={-}k\tilde{\xi}_{0i}A_i, \end{gather}

where the initial values $\mu _{fi}$ and $\tilde {\xi }_{0i}$ have been used, $\tilde {\tau }=0: \mu _f=\mu _{fi},\,\tilde {\xi }_0=\tilde {\xi }_{0i}$. Eliminating $A_i$ in favour of $\mu _{fi}$, $A_i^{-1}=-k\tilde {\xi }_{0i}/(1+\mu _{fi})$, the constant $k$ is also eliminated from the expressions of $\mu (\tilde {\xi }, \tilde {\tau })$ and $\mu _f( \tilde {\tau })$, leading to a two-parameter family of solutions involving the parameters $\mu _{fi}$ and $\tilde {\xi }_{0i}$,

(2.50a,b)\begin{gather} \mu(\tilde{\xi}, \tilde{\tau})= \frac{[\tilde{\xi}-\tilde{\xi}_0(\tilde{\tau})]}{(1+\tilde{\tau})[\theta_i+\ln(1+\tilde{\tau})]}-1, \quad \theta_i\equiv \frac{-\tilde{\xi}_{0i}}{(1+\mu_{fi})}>0, \end{gather}

(2.51a,b)\begin{gather}\mu_f( \tilde{\tau})= \frac{- \tilde{\xi}_{0}(\tilde{\tau})}{(1+\tilde{\tau})[\theta_i+\ln(1+\tilde{\tau})]}-1, \quad \tilde{\xi}_0(\tilde{\tau})={-} \left[\tilde{\tau}+\int_0^{\tilde{\tau}}\dot{\alpha}_\tau(\tilde{\tau}')\text{d} \tilde{\tau}'\right]+\tilde{\xi}_{0i}. \end{gather}

These expressions for the flow field are solutions to (2.38a,b) and also to (2.43) when $\dot {\alpha }_\tau (\tilde {\tau })$ is small. Notice that the unknown function $\dot {\alpha }_\tau (\tilde {\tau })$ appears only through $\tilde {\xi }_0(\tilde {\tau })$. Written with the original variables, denoting $u_f(t)$ the flow on the front, $u_{fi}$ its initial value ($t=0$), $r_{fi}$ the initial radius of the front and $\mathcal {D}_i>\mathcal {D}(t)$ the initial detonation velocity, (2.50a,b)–(2.51a,b), using the definition in (2.28) $1+\mu =u/\epsilon a$, take the form

(2.52a–c)\begin{gather} 0< \frac{u_{fi}}{ \epsilon a}-1\ll 1, \quad r_{0}(t)\leqslant r\leqslant r_f(t): \quad \frac{u(r,t)}{u_{f}(t)}= \frac{r-r_0(t)}{r_f(t)-r_0(t)}, \end{gather}

(2.53a,b)\begin{gather}r_0(t)=at +r_{0i},\quad\frac{u_f(t)}{u_{fi}}= \frac{r_f(t)-r_0(t)}{r_{fi}+at}{\left [\frac{r_{fi}-r_{0i}}{r_{fi}}+\frac{u_{fi}}{a}\ln\left (1+\frac{at}{r_{fi}}\right)\right ]^{{-}1}}, \end{gather}

where $u_{fi}/a=(r_{fi}-r_{0i})/(r_{0i}\theta _i)$. Self-consistency of the asymptotic analysis in the limit $\epsilon \to 0$ $u_{fi}/a=O(\epsilon )$ is ensured by the scaling

(2.54)\begin{equation} \epsilon \to 0:\quad \frac{r_{f}(t)-r_{0}(t)}{\epsilon\,r_{0}(t)}={-} \frac{\tilde{\xi}_0(\tilde{\tau})}{\tilde{\tau} +1}=O(1). \end{equation}

Under the condition (2.42), the integral term can be neglected in the expression (2.51a,b) of $\tilde {\xi }_0$, so that (2.54) reduces to (2.25) $(r_{f}(t)-r_{0}(t))/\epsilon \,r_{0}(t)=1$ for $\tilde {\xi }_{0i}=-1$.

Notice also that $\mu (\tilde {\xi }, \tilde {\tau })$ in (2.50a,b)–(2.53a,b) can be written in the same form as the self-similar solution of (2.18), $v=[z+\mathcal {A}(\nu )]/\mathcal {B}(\nu )$ with $z=\eta /\nu$, $\mathcal {B}(\nu )=(\theta _i+\ln \nu )$ and $\mathcal {A}(\nu )=[r_f(t)-r_0(t)]/\epsilon r_0(t)$ which reduces to $\mathcal {A}(\nu )=1-(1+\tilde {\xi }_{0i})/\nu$ for $\dot {\alpha }_\tau \ll 1$ and $\mathcal {A}(\nu )=1$ for $\tilde {\xi }_{0i}=-1$.

2.3.4. Simplified expression of the rarefaction flow near the CJ regime

According to (2.50a,b) and/or (2.52a–c), the gradient of the flow is uniform and decreases monotonically with the time. According to (2.51a,b) and/or (2.53a,b) the flow velocity on the front also decreases and the planar CJ velocity is reached at finite time $t_t$

(2.55a–c)\begin{equation} t= t_t: u_f/\epsilon a =1, \quad \mathcal{D}=\mathcal{D}_{o_{CJ}} \Leftrightarrow \tilde{\tau} =\tilde{\tau}_t: \mu_f=0, \quad \tilde{\tau}_t \equiv a t_t/r_{fi}=\tau_t/\tilde{r}_{fi}. \end{equation}

However, at $t-t_t=0^-$, the flow field (2.50a,b)–(2.53a,b) is different from the self-similar rarefaction wave behind a CJ wave (2.22)–(2.24) plotted in figure 1. We will show in § 3.1 that the relaxation of the flow towards the CJ rarefaction wave occurs progressively for $t>t_t$ after a sudden and sharp transition of the flow gradient on the front at $t=t_t$.

At the end of the decay, $\vert \dot {\alpha }_{\tau }\vert \ll 1$, (2.51a,b) where the integral term $\int _0^{\tilde {\tau }}\dot {\alpha }_\tau (\tilde {\tau }')\,\text {d} \tilde {\tau }'$ is neglected (initial condition close to the CJ velocity), introducing $\mu _f(\tilde {\tau }_t)=0$, leads to a transcendental equation for $\tilde {\tau }_t$ in terms of $\tilde {\xi }_{0i}$ and $\mu _{fi}$

(2.56a,b)\begin{align} \int_0^{\tau_t}\dot{\alpha}_\tau( \tau')\text{d} \tau'\ll \tau_t,\quad \mu_f(\tilde{\tau}_t)=0 \Rightarrow \tilde{\tau}_t - \tilde{\xi}_{0i}= (1+\tilde{\tau}_t)\left[ -\frac{\tilde{\xi}_{0i}}{(1+\mu_{fi})}+\ln(1+\tilde{\tau}_t)\right ], \end{align}

which has a single positive root which is small for $\mu _{fi} \ll 1$ whatever $\tilde {\xi }_{0i}<0$, see Appendix C.2. This root yields a simple expression of $\tilde {\tau }_t$ (and/or $t_t$) in terms of the initial value of the flow at the front

(2.57)\begin{equation} \mu_{fi} \ll 1: \quad \tilde{\tau}_t\approx \mu_{fi}\ll 1 \Rightarrow \tilde{\tau}_t\equiv \frac{a t_t}{r_{fi}}\approx \left(\frac{u_{fi}}{ \epsilon a}-1\right)\ll 1,\quad \forall\ \tilde{\xi}_{0i}<0 \end{equation}

obtained from the Taylor expansion of (2.56a,b) when the quadratic terms $\tilde {\tau }_t^2$ are neglected. Limiting our attention to $0\leqslant t \leqslant t_t$, $0\leqslant \tilde {\tau }=O(\tilde {\tau }_t)$, quadratic terms $\tilde {\tau }^2=(a t/r_{fi})^2$ are negligible, and a Taylor expansion of (2.50a,b)–(2.51a,b) in powers of $\tilde {\tau }_t\approx \mu _{fi}\ll 1$, limited to first order, yields

(2.58)\begin{equation} 0\leqslant \tilde{\tau}\leqslant \tilde{\tau}_t\ll 1: \quad\mu \approx \frac{\tilde{\xi}}{(-\tilde{\xi}_{0i})}\left[1+\tilde{\tau}_t-\left (1+\frac{1}{-\tilde{\xi}_{0i}}\right )\tilde{\tau} \right ]+ \tilde{\tau}_t-\tilde{\tau}, \quad \vert\tilde{\xi}_{0i}\vert=O(1). \end{equation}

The downstream condition at the weak discontinuity, $\xi =\xi _0(\tilde {\tau }) : \mu =-1$ with $\tilde {\xi }_0=-\tilde {\tau }+\tilde {\xi }_{0i}$ in (2.43), is recovered at the first order of the Taylor expansion of (2.58) in the form $\tilde {\xi }_0/(-\tilde {\xi }_{0i})=-1-\tilde {\tau }/(-\tilde {\xi }_{0i})$. According to (2.42) and (2.57), there is a boundary layer near the detonation front where the flow (2.58) takes an even simpler form at the end of the decay $0\leqslant \tilde {\tau }\leqslant \tilde {\tau }_t\ll 1$

(2.59a,b)\begin{gather} -\tilde{\xi}\equiv \frac{(r_f-r)}{\epsilon r_{fi}}=O(\tilde{\tau}_t), \quad \tilde{\tau}_t\approx \mu_{fi}= \sqrt{\frac{2(\mathcal{D}_i-\mathcal{D}_{o_{CI}})}{\epsilon a}}\ll 1, \end{gather}

(2.60a,b)\begin{gather}\mu(\tilde{\xi}, \tilde{\tau})\approx \frac{\tilde{\xi}}{-\tilde{\xi}_{0i}}-(\tilde{\tau}-\tilde{\tau}_t), \quad \mu_f(\tilde{\tau})\approx (\tilde{\tau}_t-\tilde{\tau}), \end{gather}

(2.61a,b)\begin{gather}\frac{u(r,t)}{\epsilon a}-1\approx \frac{r-r_f(t)}{ \epsilon r_{fi}}+\frac{a\,(t_t-t)}{ r_{fi}}, \quad \frac{u_f(t)}{\epsilon a}-1\approx \frac{a(t_t-t)}{ r_{fi}}, \end{gather}

obtained when terms of second order $\tilde {\tau }_t^2$ are neglected. This leads to

(2.62a–c)\begin{align} \frac{\partial \mu(\tilde{\xi}, \tilde{\tau})}{\partial \tilde{\tau}} \approx{-}1, \quad \left.\frac{\partial u(r, t)}{\partial r}\right\vert_{r=r_f} \approx \frac{a}{r_f},\quad \frac{1}{\epsilon a}\frac{ \text{d} u_f(t)}{\text{d} t}\approx{-}\frac{a}{ r_{f}} \Rightarrow \frac{\text{d} u_f(t)}{\text{d} t}\approx{-}a\frac{u_f}{ r_{f}}, \end{align}

where $r_{fi}$ has been replaced by $r_f$ in the denominators since $r_f-r_{fi}=O(\epsilon r_{f})$. The terms of order $\epsilon$ being neglected in (2.43), retaining the small term $(\tilde {\tau }_t-\tilde {\tau })$ which is of order $\tilde {\tau }_t\approx \mu _{fi}\equiv (u_{fi}/ \epsilon a)-1$, is meaningful in (2.60a,b) in an intermediate asymptotic limit $\epsilon \ll \sqrt {2(\mathcal {D}_i-\mathcal {D}_{o_{CJ}})/\epsilon a}\ll 1$.

The last equation in (2.62a–c) corresponds to the closure assumption used in Clavin & Denet (Reference Clavin and Denet2020) who neglected the small gradient of the flow at the exit of the reaction zone. This equation is indeed quite general close to the CJ regime and was derived previously in the opposite limit $M_{o_{CJ}}\gg 1$ by Liñán et al. (Reference Li nan, Kurdyumov and Sanchez2012). This can be seen in (2.9)–(2.10a–c); because of the transonic character of the burnt gas flow near the detonation, $u+a-\mathcal {D} \ll 1$, the term involving the gradient of the flow on the left-hand side becomes negligible near the detonation front, the unsteady term being balanced by the divergence of the flow $-a u/r$. More precisely in the limit of small heat release, using $\mathcal {D}_{o_{CJ}}-a\approx \epsilon a$, $\mathcal {D} -a=\epsilon a (1+\dot {\alpha }_\tau )\approx \epsilon a$, $u_f=O(\epsilon a)$, $\mu =u/\epsilon a-1$, see (2.28), and, according to (2.62a–c), $\partial u/\partial r\vert _{r=r_f}\approx a/r_f$ with (2.57) $\mu _f<\mu _{fi}\ll 1$, the gradient term on the left-hand side of (2.10a–c) is shown to be smaller than the curvature term $au/r$ on the right-hand side by a factor $\epsilon$,

(2.63)\begin{equation} (u_f+a-\mathcal{D})\left.\frac{\partial u}{\partial x}\right\vert_{x=0} \approx \epsilon \mu_f a \frac{a}{r_f}\ll a \frac{\epsilon a}{r_f}=O\left( \frac{ a u_f}{r_f}\right). \end{equation}

As we shall see, this small gradient of the rarefaction flow cannot be ignored in the critical dynamics studied in § 4 because it controls the instantaneous position of the sonic point inside the rarefaction wave.

3.1. Abrupt transition of the flow on the front

Both the flows (2.50a,b) and (2.22)–(2.23) vanish at the radius of the spherical core of stagnant gas but the gradient of the flow is uniform and finite in the former while it is infinite on the detonation front in the latter, see figure 1. The sonic condition of (2.50a,b) (relative to the detonation front) which is located at finite distance behind the detonation front for $t<t_t$, reaches the front at $t=t_t$. Within the framework of the discontinuous model, the velocity of the burnt gas relative to the lead shock cannot become smaller than the sound speed on the detonation front. According to (2.41a,b)–(2.42), neither $\dot {\alpha }_t(t)$ nor $\mu _f(t)$ can become negative; they should vanish simultaneously at $t=t_t$ and stay equal to zero at later times $t>t_t$

(3.1)\begin{equation} t>t_t: \quad u_f=\epsilon a,\quad\mu_f=0, \quad \mathcal{D}=\mathcal{D}_{o_{CJ}}, \quad r_f(t)=\mathcal{D}_{o_{CJ}}(t-t_t)+r_{f}(t_t). \end{equation}

Therefore, the decrease of $u_f(t)$ with a quasi-constant deceleration rate (2.61a,b) for $t\leqslant t_t$ stops suddenly at $t=t_t$ since $u_f$ stays constant after $t_t$. A jump of deceleration of the flow is thus produced at $t=t_t$ on the detonation front

(3.2a,b)\begin{equation} \left.\frac{1}{\epsilon a}\frac{\text{d} u_f}{\text{d} t}\right\vert^{t_t^+}_{t_t^-}=\frac{a}{r_f(t_t)}, \quad \left.\frac{\text{d} \mu_f}{\text{d} \tilde{\tau} }\right\vert^{\tilde{\tau}_t^+}_{\tilde{\tau}_t^-}=\frac{r_{fi}}{r_f(t_t)}\approx 1. \end{equation}

However, the trajectory in the phase space of velocity–radius “$\mathcal {D}$-$r_f$” is tangent to the axis $\mathcal {D}=\mathcal {D}_{o_{CJ}}$ at $t=t_t$, as shown by (2.41a,b) $\dot {\alpha }_\tau = \mu ^2_f(\tau )/2$ by using (2.61a,b) $\mu _f \approx {a(t_t-t)}{ r_{fi}}$

(3.3)\begin{align} &\epsilon\ll1, (t-t_t) \to 0^-: \quad [\mathcal{D}(t)-\mathcal{D}_{o_{CJ}}]/a \to (\epsilon /2)[(t-t_t )a/r_{fi}]^2, \nonumber\\ &\text{d} r_f/\text{d} t\approx \mathcal{D}_{o_{CJ}}=a(1+\epsilon), \quad (\mathcal{D}-\mathcal{D}_{o_{CJ}})/a \to (\epsilon /2)(r_f-r_{f}(t_t) )^2/r_{fi}^2, \end{align}

the last relation being valid near the CJ regime $0\leqslant \mathcal {D}-\mathcal {D}_{o_{CJ}}\ll \mathcal {D}_{o_{CJ}}$ in the limit of small heat release $(\mathcal {D}_{o_{CJ}}-a)/a= \epsilon \ll 1$. The jumps in (3.2a,b) and the tangency of the trajectories at $t=t_t$ are consequences of the discontinuous model and are no longer valid when small modification of the inner structure of the detonation is taken into account.

3.2. Transitory regime. An analytical study

The transitory flow between the transition from the straight profile (2.50a,b) for $\tilde {\tau }\leqslant \tilde {\tau }_t$ to the self-similar solution (2.22)–(2.24), denoted $\mu ^{(tr)}(\tilde {\xi }, \tilde {\tau })$ in the following, is solution of (2.43) for a flow velocity at the front kept equal to its CJ value after $t_t$

(3.4)\begin{equation} \tilde{\tau} \geqslant \tilde{\tau}_t:\quad \frac{\partial \mu^{(tr)}}{\partial \tilde{\tau}} + \mu^{(tr)} \frac {\partial \mu^{(tr)}}{\partial \tilde{\xi}} ={-}\frac{1+\mu^{(tr)}}{\tilde{\tau}+1}, \quad \tilde{\xi}=0: \mu^{(tr)}=0. \end{equation}

This flow is equal to (2.50a,b) (with $\mu _{fi}>0$) when $t < t_t$,

(3.5)\begin{equation} \tilde{\tau} \approx \tilde{\tau}_t^{-}, \quad \tilde{\xi}_{0}(\tilde{\tau}_t)\leqslant \tilde{\xi}\leqslant0:\quad \mu^{(tr)}(\tilde{\xi}, \tilde{\tau}_t)= \frac{\tilde{\xi}}{- \tilde{\xi}_{0i}+\tilde{\tau}_t}=\frac{\tilde{\xi}}{-\tilde{\xi}_0(\tilde{\tau}_t)}, \end{equation}

where the simplified expression (2.58) has been used by introducing (2.56a,b)–(2.57) into (2.50a,b) neglecting the quadratic terms $\tilde {\tau }^2_t$. Equation (3.5) says simply that the transitory flow at $\tilde {\tau }=\tilde {\tau }_t$ has a straight profile corresponding to the reduced thickness $\vert \tilde {\xi }_0(\tilde {\tau }_t)\vert$ of the rarefaction wave. Simultaneously with the jump of deceleration (3.2a,b), the flow gradient on the detonation front, which is finite for $\tilde {\tau }<\tilde {\tau }_t$, jumps at $\tilde {\tau }=\tilde {\tau }_t$ to become infinite at $\tilde {\tau }=\tilde {\tau }_t^+$ and stays infinite afterwards $\tilde {\tau }>\tilde {\tau }_t$

(3.6a,b)\begin{equation} \tilde{\tau}=\tilde{\tau}_t^-{\ll} 1:\left. \frac{\partial \mu}{\partial \tilde{\xi}}\right\vert_{\tilde{\xi}=0^-}=\frac{1}{-\tilde{\xi}_0(\tilde{\tau}_t)}, \quad \tilde{\tau}>\tilde{\tau}_t: \lim_{\tilde{\xi}\to 0^-}\frac{\partial \mu^{(tr)}}{\partial \tilde{\xi}}\approx \frac{1}{\sqrt{-2\tilde{\xi}}}. \end{equation}

These expressions are obtained by the same method as in § 2.2.2 for the self-similar CJ solution (2.22)–(2.23). Recalling that no boundary condition is used on the front to derive (2.50a,b)–(2.51a,b), this flow is still solution of (2.43) for $\tilde {\tau }>\tilde {\tau }_t$ when $\mu _f(\tilde {\tau })$ decreases below zero. The corresponding flow, denoted $\mu ^{(wd)}(\tilde {\xi }, \tilde {\tau })$ from now on, takes the form

(3.7a–c)\begin{align} \tilde{\tau}_t\approx \mu_{fi}\ll 1,\quad \tilde{\tau}\geqslant\tilde{\tau}_t,\quad \mu^{(wd)}(\tilde{\xi}, \tilde{\tau})\approx \frac{\tilde{\xi}+\tilde{\tau}-\tilde{\xi}_{0i}}{(1+\tilde{\tau})[-\tilde{\xi}_{0i}/(1+\tilde{\tau}_t)+\ln(1+\tilde{\tau})]}-1, \end{align}

where, neglecting the quadratic terms $\tilde {\tau }^2_t$, the flow at $\tau =\tau _t$ is given in (3.5), $\mu ^{(wd)}(\tilde {\xi }, \tilde {\tau }_t)=\mu ^{(tr)}(\tilde {\xi }, \tilde {\tau }_t)$. The instantaneous position of the weak discontinuity corresponding to (3.7a–c), $\tilde {\xi }=\tilde {\xi }_0(\tilde {\tau })$: $1+\mu ^{(wd)}(\tilde {\xi }_0, \tilde {\tau }_t)=0$, takes the form

(3.8)\begin{equation} \tilde{\xi}_0(\tilde{\tau})=\tilde{\xi}_0(\tilde{\tau}_t)-(\tilde{\tau}-\tilde{\tau}_t). \end{equation}

Equation (3.7a–c) cannot represent the transitory flow nearby the detonation front for $\tilde {\tau } > \tilde {\tau }_t:$ $\mu ^{(wd)}(\tilde {\xi }, \tilde {\tau })\ne \mu ^{(tr)}(\tilde {\xi }, \tilde {\tau })$ since $\mu ^{(tr)}(0,\tilde {\tau })=0$ while $\mu ^{(wd)}(0,\tilde {\tau })<0\ \forall \,\tilde {\tau } > \tilde {\tau }_t$. However, (3.7a–c) still represents the transitory flow at a distance from the front large enough, as explained now by the method of characteristics.

The singular perturbation (3.6a,b), which is generated instantaneously at $\tilde {\tau }=\tilde {\tau }_t$ on the front, induces a disturbance which is transported downstream with the velocity $\mu ^{(tr)}$ by the mode $\mathcal {C}_+$ (downstream running for $\tilde {\tau }>\tilde {\tau }_t$ since $\mu ^{(tr)}\leqslant 0$). According to (3.4), the characteristic curves $\tilde {\xi }=\tilde {\xi }_{_{\,\mathcal {C}_+}}(\tilde {\tau })$

(3.9)\begin{equation} \text{d} \tilde{\xi}_{_{\,\mathcal{C}_+}}/\text{d} \tilde{\tau}= \mu^{(tr)}(\tilde{\xi}_{_{\,\mathcal{C}_+}}(\tilde{\tau}), \tilde{\tau}) \end{equation}

corresponds to the following equation for the flow velocity transported by $\mathcal {C}^+$

(3.10a,b)\begin{equation} \mu^{(tr)}_{_{\,\mathcal{C}_+}}(\tilde{\tau})\equiv \mu^{(tr)}(\tilde{\xi}_{_{\,\mathcal{C}_+}}(\tilde{\tau}), \tilde{\tau} ),\quad \frac{\text{d} \mu^{(tr)}_{_{\,\mathcal{C}_+}}(\tilde{\tau})}{\text{d} \tilde{\tau}}={-}\frac{1+\mu^{(tr)}_{_{\,\mathcal{C}_+}}(\tilde{\tau})}{\tilde{\tau} +1}. \end{equation}

Denoting $\tilde {\xi }=\tilde {\xi }_{_{\,\mathcal {C}_+}}(\tilde {\tau }, \tilde {\tau }')$ the characteristic curve leaving the front ($\tilde {\xi }=0$) at $\tilde {\tau }=\tilde {\tau }'\geqslant \tilde {\tau }_t$, integration of (3.10a,b) from $\tilde {\tau } =\tilde {\tau }': \mu ^{(tr)}_{_{\,\mathcal {C}_+}}(\tilde {\tau }')=0$ yields the flow velocity transported by this characteristic in the form

(3.11)\begin{equation} \tilde{\tau}\geqslant\tilde{\tau}'\geqslant \tilde{\tau}_t:\quad 1+ \mu^{(tr)}(\tilde{\xi}_{_{\,\mathcal{C}_+}}(\tilde{\tau}, \tilde{\tau}'), \tilde{\tau} ) =\frac{\tilde{\tau}'+1}{\tilde{\tau} +1}. \end{equation}

Integrating (3.9) from $\tau =\tau ':$ $\tilde {\xi }_{_{\,\mathcal {C}_+}}(\tilde {\tau }', \tilde {\tau }')=0$ using (3.11), then determines the characteristic curves $\tilde {\xi }=\tilde {\xi }_{_{\,\mathcal {C}_+}}(\tilde {\tau }, \tilde {\tau }')$

(3.12)\begin{gather} \tilde{\tau}\geqslant\tilde{\tau}'\geqslant \tilde{\tau}_t:\quad \frac{\partial \tilde{\xi}_{_{\,\mathcal{C}_+}}(\tilde{\tau}, \tilde{\tau}')}{\partial \tilde{\tau}}= \frac{\tilde{\tau}'+1}{\tilde{\tau}+1}-1\leqslant0 \end{gather}

(3.13)\begin{gather} \tilde{\xi}_{_{\,\mathcal{C}_+}}(\tilde{\tau}, \tilde{\tau}')= (\tilde{\tau}'+1)\ln\frac{\tilde{\tau}+1}{\tilde{\tau}'+1}-(\tilde{\tau}-\tilde{\tau}'). \end{gather}

All these characteristics leave the front slowly since their velocity is zero at the front ($\xi =0$) $\mu ^{(tr)}(0,\tilde {\tau }')=0$, $\partial \tilde {\xi }_{_{\,\mathcal {C}_+}}(\tilde {\tau }, \tilde {\tau }')/\partial \tilde {\tau }\vert _{\tilde {\tau }=\tilde {\tau }'}=0$ $\forall \tilde {\tau }'\geqslant \tilde {\tau }_t$. They all reach the same velocity in the long-time limit, $\lim _{\tilde {\tau } \to \infty } \partial \tilde {\xi }_{_{\,\mathcal {C}_+}}(\tilde {\tau }, \tilde {\tau }')/\partial \tilde {\tau }=-1$. Moreover, in agreement with the fact that no shock can be created when the flow $u(r,t)$ increases in space in the direction of the flow, $\partial u/\partial r >0$, these characteristic curves do not cross each other. This is checked by noticing that the modulus of their velocity satisfies the following relation $\tilde {\tau }_1' >\tilde {\tau }_2' \Rightarrow \vert \partial \tilde {\xi }_{_{\,\mathcal {C}_+}}(\tilde {\tau }, \tilde {\tau }_1')/\partial \tilde {\tau }\vert <\vert \partial \tilde {\xi }_{_{\,\mathcal {C}_+}}(\tilde {\tau }, \tilde {\tau }_2')/\partial \tilde {\tau }\vert$, see figure 2. Therefore, the trajectory $\tilde {\xi }=\tilde {\xi }_{_{\,\mathcal {C}_+}}(\tilde {\tau }, \tilde {\tau }_t)$ associated with the characteristic curve leaving the front at the earliest time $\tilde {\tau }'=\tilde {\tau }_t$ is, according to (3.13),

(3.14)\begin{gather} \tilde{\tau}\geqslant\tilde{\tau}_t,\quad\tilde{\xi}_{_{\,\mathcal{C}_+}}(\tilde{\tau}, \tilde{\tau}_t)= (\tilde{\tau}_t+1)\ln\frac{\tilde{\tau}+1}{\tilde{\tau}_t+1}-(\tilde{\tau}-\tilde{\tau}_t), \end{gather}

(3.15)\begin{gather} = (\tilde{\tau}_t+1)\ln(\tilde{\tau}+1)-\tilde{\tau}+O(\tilde{\tau}_t^2), \end{gather}

with, according to (3.11), the following expression of the flow which is transported

(3.16)\begin{equation} \mu^{(tr)}(\tilde{\xi}_{_{\,\mathcal{C}_+}}(\tilde{\tau}, \tilde{\tau}_t), \tilde{\tau} )= \frac{\tilde{\tau}_t+1}{\tilde{\tau}+1}-1. \end{equation}

In other words, $\tilde {\xi }=\tilde {\xi }_{_{\,\mathcal {C}_+}}(\tilde {\tau }, \tilde {\tau }_t)$ is the equation of the leading edge of the disturbance resulting from keeping equal to zero the front velocity $\tilde {\xi }=0:$ $\mu ^{(tr)}=0$ $\forall \tilde {\tau }>\tilde {\tau }_t$. Ahead of this leading edge, the flow $\mu ^{(tr)}(\tilde {\xi }, \tilde {\tau })$ is equal to the ‘unperturbed’ flow (3.7a–c),

(3.17)\begin{equation} \tilde{\xi} \leqslant \tilde{\xi}_{_{\,\mathcal{C}_+}}(\tilde{\tau}, \tilde{\tau}_t):\quad \mu^{(tr)}(\tilde{\xi}, \tilde{\tau})=\mu^{(wd)}(\tilde{\xi}, \tilde{\tau}). \end{equation}

Figure 2. Example of three characteristic curves $\tilde {\xi }=\tilde {\xi }_{_{\,\mathcal {C}_+}}(\tilde {\tau }, \tilde {\tau }')$ for $\tilde {\tau }'=\tilde {\tau }_t$, $\tilde {\tau }'=\tilde {\tau }'_2$ and $\tilde {\tau }'=\tilde {\tau }'_1$ with $\tilde {\tau }_t<\tau '_2<\tau '_1$. In the dashed region $\tilde {\xi }\leqslant \tilde {\xi }_{_{\,\mathcal {C}_+}}(\tilde {\tau }, \tilde {\tau }_t)$, the transitory flow is equal to the unperturbed flow (3.7a–c) (straight profile whose slope is decreasing to zero) $\mu ^{(tr)}(\tilde {\xi }, \tilde {\tau })=\mu ^{(wd)}(\tilde {\xi }, \tilde {\tau })$.

The junction point at which $\mu ^{(wd)}(\tilde {\xi }, \tilde {\tau })$ is equal to (3.16)

(3.18)\begin{equation} \frac{\tilde{\xi}(\tilde{\tau})+\tilde{\tau}-\tilde{\xi}_{0i}}{-\tilde{\xi}_{0i}/(1+\tilde{\tau}_t)+\ln(1+\tilde{\tau})}=\tilde{\tau}_t+1 \Rightarrow \tilde{\xi}(\tilde{\tau})={-}\tilde{\tau} +(\tilde{\tau}_t+1)\ln(1+\tilde{\tau})\quad \forall\, \tilde{\xi}_{0i}, \end{equation}

corresponds effectively to the characteristics (3.14) when the quadratic terms $\tilde {\tau }^2_t$ are neglected as it should be.

Strictly speaking, this point does not reach the core of stagnant gas (3.8) $\tilde {\xi }_0(\tilde {\tau })$ in the long-time limit, $\lim _{\tilde {\tau }\to \infty }\tilde {\xi }_{_{\,\mathcal {C}_+}}(\tilde {\tau }, \tilde {\tau }_t)- \tilde {\xi }_0(\tilde {\tau })\approx \ln \tilde {\tau }$. However, this is true in the sense of the self-similar analysis when using the self-similar variable $z_{_{\,\mathcal {C}_+}}(\tilde {\tau })\equiv [r_{_{\,\mathcal {C}_+}}(t)-r_f(t)]/\epsilon r_0(t) \approx \tilde {\xi }_{_{\,\mathcal {C}_+}}(\tilde {\tau }, \tilde {\tau }_t)/(-\tilde {\tau })$, yielding, according to (3.15)

(3.19a,b)\begin{equation} \lim_{\tilde{\tau} \to \infty} \tilde{\xi}_{_{\,\mathcal{C}_+}}(\tilde{\tau}, \tilde{\tau}_t)/\tilde{\xi}_0(\tilde{\tau})=1, \quad\lim_{\tilde{\tau} \to \infty} z_{_{\,\mathcal{C}_+}}(\tilde{\tau}) ={-}1. \end{equation}

To conclude, the transitory flow $\mu ^{(tr)}(\tilde {\xi }, \tilde {\tau })$ is composed of two parts. The part corresponding to $\tilde {\xi } \leqslant \tilde {\xi }_{_{\,\mathcal {C}_+}}(\tilde {\tau }, \tilde {\tau }_t)$ is the straight profile $\mu ^{(wd)}(\tilde {\xi }, \tilde {\tau })$ in (3.7a–c) whose slope is uniform and decreases to zero $\lim _{\tilde {\tau }\to \infty }(1+\tilde {\tau })^{-1}[-\tilde {\xi }_{0i}/(1+\tilde {\tau }_t)+\ln (1+\tilde {\tau })]^{-1}=0$. The transitory flow has a curved shape in the range $\tilde {\xi }_{_{\,\mathcal {C}_+}}(\tilde {\tau }, \tilde {\tau }_t)\leqslant \tilde {\xi }\leqslant 0$, joining the straight solution $\mu ^{(wd)}(\tilde {\xi }, \tilde {\tau })$ at $\tilde {\xi } = \tilde {\xi }_{_{\,\mathcal {C}_+}}(\tilde {\tau }, \tilde {\tau }_t)$, while the slope is infinite at the front ($\tilde {\xi }=0$). The evolution of the rarefaction wave is sketched in figure 3 where the theoretical expressions for the coordinates (position and flow velocity at time $t$) of the point of junction with the straight profile are given.

Figure 3. Sketch of the transitory flow, plotted in a universal form for $\tilde {\xi }_{0i}=-1$ and $\tilde {\tau }_t\approx \mu _{fi}\ll 1$, $\tilde {\tau }_t$ being the cross-over time (2.56a,b). The bold curve in black is the rarefaction wave $u^{(tr)}/\epsilon a= 1+\mu ^{(tr)}$ behind a detonation treated as a discontinuity, after the cross-over time $\tilde {\tau }>\tilde {\tau }_t$. The slope is infinite at the front and the profile is straight away from the front where $\mu ^{(tr)}=\mu ^{(wd)}$. Expressed in terms of the space variable $\tilde {\xi }/(-\tilde {\xi }_0)=[r-r_f(t)]/[r_f(t)-r_0(t)]$, $-\tilde {\xi }_0(\tilde {\tau })=\tilde {\tau }+1$, the profile of the transitory flow is parameter free, provided the time is measured by $(1+\tilde {\tau })/(1+\tilde {\tau }_t)$. The bold points are the time dependent radius and flow velocity of the junction point where the transitory flow $\mu ^{(tr)}$ matches $\mu ^{wd}$ in (3.7a–c). The coloured straight lines represent the rarefaction wave (2.50a,b) of the overdriven detonations before the velocity reaches the planar CJ velocity $\tilde {\tau } < \tilde {\tau }_t$, the red one being just before the transition to CJ, $\tilde {\tau } -\tilde {\tau }_t=0^-$.

These theoretical results are fully confirmed by the numerical solution of (3.4)–(3.5). An example of numerical result is shown in figure 4. When the result is plotted with a space coordinate reduced by the thickness of the CJ rarefaction wave, the numerical analysis shows that the transitory flow and the self-similar CJ solution are quasi-identical in the range between the detonation front and the junction point travelling with the characteristic $\mathcal {C}_+$ issued from the front at $t=t_t$ at which the front reaches the CJ velocity. Outside this range, the numerical result shows a straight profile down to the stagnant core, in full agreement with (3.7a–c). The trajectory of the junction point between the two parts of the transitory flow corresponds exactly to the theoretical result plotted in figure 3.

Figure 4. Numerical solution (solid line) of (3.4)–(3.5), describing the transitory flow $u^{(tr)}/\epsilon a = 1+\mu ^{(tr)}$ for $\tilde {\tau }-\tilde {\tau }_t=3$, using the non-dimensional radius, reduced by the thickness of the rarefaction wave. The dotted line is the self-similar rarefaction flow (2.23)–(2.25) behind the spherical CJ detonation, which is reached in the long-time limit $\tilde {\tau }-\tilde {\tau }_t\to \infty$. The dashed straight line is (3.7a–c) for $\tilde {\tau }-\tilde {\tau }_t=3$. The trajectory of the point at the junction of the numerical solution of (3.4)–(3.5) and the straight profile (3.7a–c) is in full agreement with the theoretical prediction (3.12)–(3.18). To summarize, the numerical solution of (3.4)–(3.5) is found to be exactly the straight profile (3.7a–c) below the junction while, in the range between the front and the junction point, there is no noticeable difference between the numerical solution of (3.4)–(3.5) and the self-similar solution (2.23)–(2.25).

4.1. Detonation model and method of solution

The reduced variables of order unity governing the structure and the dynamics of the inner structure in the limit of small heat release $\epsilon \to 0$ are $\xi =O(1)$ and $\tau =O(1)$ in (2.27a–c) where $r_{fi}$ is the initial radius of the detonation at $\tau =0$. According to (2.36)–(2.37a–c), the link of $\xi$, $\tau$ with $\tilde {\xi }\equiv \xi /\tilde {r}_{fi}$ and $\tilde {\tau }=\tau /\tilde {r}_{fi}$ is through the parameter $\tilde {r}_{fi}\equiv \epsilon r_{fi}/l$ of order unity in the limit $\epsilon \to 0$.

As in Clavin & Denet (Reference Clavin and Denet2020), we will use the detonation model of Clavin & Williams (Reference Clavin and Williams2002), obtained by an asymptotic analysis in the limit of small heat release, coupled to the Newtonian approximation neglecting the compressional heating in the reaction zone where the chemical energy is released (after the induction period),

(4.1)\begin{equation} \epsilon \equiv \sqrt {q_m/c_pT_u} \approx M_{o_{CJ}}-1 \ll 1, \quad (\gamma -1) < \epsilon. \end{equation}

The Rankine–Hugoniot conditions (A3)–(A5) in the limit (4.1) take the form,

(4.2a,b)\begin{gather} T_N/T_u\approx 1+2(\gamma -1)(M-1), \quad b\equiv 2\epsilon (\gamma-1)\mathcal{E}/k_BT_u, \end{gather}

(4.3a,b)\begin{gather}T_{No_{CJ}}/T_u\approx 1+2(\gamma -1)\epsilon, \quad (\mathcal{E}/k_BT_u^2)(T_N-T_{No_{CJ}})= b\dot{\alpha}_{\tau}, \end{gather}

where $\mathcal {E}$ is the activation energy governing the induction length, and $b=O(1)$ is its reduced form. A first simplification concerns the temperature and the mass fraction of species which, to leading order in the limit (4.1), are solutions of the steady-state version of the conservation equations of energy and mass, satisfying the Rankine–Hugoniot conditions at the lead shock, see the original article Clavin & Williams (Reference Clavin and Williams2002) recalled in § B.1 of Clavin & Denet (Reference Clavin and Denet2018). The unsteady distribution of the rate of heat release $\omega (\xi , \tau )$ can then be expressed in terms of the unsteady detonation velocity and the distribution in steady state, $\omega (\xi , \tau )= \bar \omega (\xi , \dot {\alpha }_\tau (\tau ))$. Focusing our attention on propagation velocity close enough to the CJ velocity, $y(\tau )\equiv b\dot {\alpha }_\tau (\tau )=O(1)$, the unsteady distribution of reaction rate takes the form $\omega (\xi , \tau )= \bar \omega (\xi , y(\tau ))$ which will be denoted simply $\omega (\xi ,y)$ in the following. For the sake of simplicity we will use here the scaling law assuming that the distribution of reaction rate is fully governed by the induction length. Introducing the distribution of the planar CJ wave $\omega _{o_{CJ}}(\xi )$, the reaction-rate distribution $\omega (\xi , y)$ then takes the form

(4.4a,b)\begin{equation} \omega(\xi, y)= \text{e}^{y(\tau)}\omega_{o_{CJ}}(\xi \text{e}^{y(\tau)}),\quad y(\tau)\equiv b\dot{\alpha}_{\tau}(\tau)=\frac{b}{\epsilon}\frac{(\mathcal{D}(\tau)-\mathcal{D}_{o_{CJ}})}{a}=O(1). \end{equation}

The chemical kinetics thus appears only through the activation energy $b$ and the CJ distribution $\omega _{o_{CJ}}(\xi )$.

In order to further simplify the presentation, the same bounded-thickness model for the inner structure of the planar CJ detonation as in (3.15)–(3.17b) of Clavin & Denet (Reference Clavin and Denet2020) will be used,

(4.5a,b)\begin{gather} \xi\leqslant-1: \omega_{o_{CJ}}(\xi)=0, \quad \xi>{-}1: \omega_{o_{CJ}}(\xi) >0, \end{gather}

(4.6a,b)\begin{gather}\int_{{-}1}^0 \omega_{o_{CJ}}(\xi')\,\text{d} \xi'=1, \quad \text{d} \omega_{o_{CJ}}/\text{d} \xi\vert_{\xi+1=0^+}\geqslant0, \end{gather}

so that, in the limit of small heat release, the structure of the flow in the planar CJ wave ($y=0$) $\mu _{o_{CJ}}(\xi )$ is solution to

(4.7a–c)\begin{equation} \mu_{o_{CJ}}\frac{\text{d} \mu_{o_{CJ}}}{\text{d} \xi}= \frac{1}{2}\omega_{o_{CJ}}(\xi), \quad \xi\leqslant-1: \mu_{o_{CJ}}=0, \quad \xi=0:\mu_{o_{CJ}}=1. \end{equation}

According to (4.4a,b) and (4.5a,b) the exit of the reaction zone $\xi =\xi _b(\tau )$ of the unsteady inner structure of the spherical combustion wave is

(4.8)\begin{equation} \xi_b(\tau)={-}\text{e}^{{-}y(\tau)}, \end{equation}

describing a substantial increase of the reaction-wave thickness when the velocity decreases below the CJ velocity $y<0$. However, for a reduced detonation velocity of order unity $\vert y\vert =O(1)$, the thickness of the detonation is of the same order of magnitude as in the planar CJ wave, $\text {e}^{-y}=O(1)$ in the limit $\epsilon \to 0$. The variation of $y$ is bounded from below since a supersonic velocity $\mathcal {D} \geqslant a$ implies $\dot {\alpha }_\tau \equiv (\mathcal {D}-\mathcal {D}_{o_{CJ}})/\epsilon a =(\mathcal {D}-a)/\epsilon a-1\geqslant -1$ so that $y\equiv b \dot {\alpha }_\tau \geqslant -b$. In fact, according to the well-known chemical-kinetics quenching in combustion, occurring at a cross-over temperature for which the recombination reactions become faster than the chain-branching reactions, the rate of heat release vanishes earlier. In real combustion, this cross-over temperature $T_c$ at which the reaction rate decreases sharply and the induction time increases strongly is typically $T_c \approx 1000\ \textrm {K}$ in ordinary condition while the Neumann temperature of a CJ wave is approximately $T_{No{CJ}}\approx 2000\ \textrm {K}$, see Sanchez & Williams (Reference Sanchez and Williams2014) and Clavin & Searby (Reference Clavin and Searby2016). Considering that ordinary combustion cannot proceed below $T_c$ with $T_{No{CJ}} -T_c \approx (T_{No{CJ}} -T_u)/2$, a chemical quenching is assumed to be produced in the limit of small heat release for $y=y_c$ with typically $y_c= - b/2$. Then, the following condition is added to (4.4a,b)–(4.5a,b)

(4.9)\begin{equation} y \leqslant y_c\approx{-}b/2: \quad \omega(\xi, y) =0,\quad \forall \,\xi. \end{equation}

The reaction being quenched below $y\leqslant y_c$, detonation fails systematically as soon as the detonation velocity decreases below the lower bound $y_c$.

Another key simplification of the limit (4.1) concerns the two-time-scale nature of the dynamics which, to a lesser extent, characterizes also real detonations. To leading order in the limit (4.1), following the reasoning used in § 2.1, the unsteady flow inside the inner structure of a spherical detonation is governed by a single equation corresponding to (2.36) plus an additional term on the right-hand side associated with the reaction rate

(4.10)\begin{gather} \epsilon \to 0: \frac{\partial \mu}{\partial \tau} + \left( \mu -\frac{y}{b}\right)\frac {\partial \mu}{\partial \xi} =\frac{1}{2}\omega(\xi, y(\tau)) - \frac {(1+\mu)}{\tilde{r}_{f}(\tau)}, \end{gather}

(4.11)\begin{gather}\xi=0: \mu=1+2y(\tau)/b, \end{gather}

with, according to (2.30a,b)–(2.32a,b),

(4.12)\begin{equation} \epsilon \to 0: \quad \tilde{r}_f(\tau)\equiv \epsilon {r_f(\tau)}/{l}\approx \tau+ \tilde{r}_{fi}. \end{equation}

The key point which is used to obtain (4.10) is that the variations of pressure and flow velocity across the inner structure of the reaction wave are identical to leading order in the limit (4.1), ${\rm \pi} =\mu +1$ where ${\rm \pi} \equiv (1/\epsilon \mu )$, $\ln (p/p_u) \approx u/a$, see Clavin & Williams (Reference Clavin and Williams2002) and also § B.1 of Clavin & Denet (Reference Clavin and Denet2018) and § 3.1 of Clavin & Denet (Reference Clavin and Denet2020) for a spherical geometry. The boundary condition at the lead shock (4.11) is the Rankine–Hugoniot relation (4.2a,b)–(4.3a,b) obtained from (A5)–(A6) for $M-1=\epsilon (1+y/b)$. The last term on the right-hand side of (4.10) is the damping rate (2.36) $(l/a)(u/r)$ due to the divergence of the burnt-gas flow near the front for $r_f\gg l$ in a spherical geometry, $u/r\approx u/r_f$.

When attention is focused on the inner structure, $\xi =O(1)$, the time-dependent velocity of the lead shock $y(\tau )$ is obtained as an eigenfunction of the system (4.10)–(4.11) plus a boundary condition at the exit of the reaction zone. In contrast to Clavin & Denet (Reference Clavin and Denet2020) where, considering the burnt-gas flow as uniform, the decelerating flow $\mu _b(\tau )$, solution to $\text {d} \mu _b/\text {d} \tau \approx -(1+\mu _b)/\tilde {r}_f$, was imposed in the burnt gas $\xi =-\text {e}^{-y}$, the solution of (4.10)–(4.11) will be matched now with the non-uniform flow of the rarefaction wave, solution to the inert version (2.36) of (4.10), denoted $\mu ^{ext}( \xi , \tau )$ from now on. In overdriven regimes, this external flow field $\mu ^{ext}( \xi , \tau )$ is given by the analytical expressions (2.50a,b)–(2.51a,b), as discussed now.

4.2. Overdriven regimes. Splitting and matching

The burnt-gas flow at the exit of the reaction zone of a supersonic reaction wave in the overdriven regime is, by definition, subsonic relative to the lead shock, $\mathcal {D}-u_b < a \Leftrightarrow \mu ^{ext}(-\text {e}^{-y(\tau )}, \tau )>y(\tau )/b$. The sonic point $\xi =\xi _s(\tau )$, $\mu ^{ext}(\xi _s(\tau ), \tau )=y(\tau )/b$, is located inside the rarefaction flow, behind the inner structure of the combustion wave. Overdriven regimes are then characterized by the relations

(4.13a–c)\begin{equation} \xi_s(\tau)<{-}\text{e}^{{-}y(\tau)}, \quad \mu^{ext}(\xi_s(\tau), \tau)=y(\tau)/b, \quad \mu^{ext}(-\text{e}^{{-}y(\tau)}, \tau)>y(\tau)/b, , \end{equation}

the last inequality being in agreement with a rarefaction flow of inert gas which is an increasing function of the radius $\partial \mu ^{ext}/\partial \xi >0$. The larger the overdrive factor $\mu ^{ext}(-\text {e}^{-y(\tau )}, \tau )-y(\tau )/b>0$ is, the larger is the distance between the sonic point and the exit of the reaction zone.The characteristic $\mathcal {C}_+$ of (2.36) is downstream running (towards the weak discontinuity $\xi =\xi _0<0$) behind the sonic point ($\xi <\xi _s$, $\mu ^{ext} < y/b)$) while it is upstream running (towards the lead shock $\xi =0$) for $\xi >\xi _s$, $\mu ^{ext}>y/b$, see figure 5. Then, under the condition (4.13a–c), modifications of the inner structure of the combustion wave cannot influence the rarefaction flow in the burnt gas, which is still given by (2.50a,b)–(2.51a,b),

(4.14)\begin{gather} \xi\leqslant-\text{e}^{{-}y(\tau)} : \mu(\xi,\tau)=\mu^{ext}(\xi, \tau), \end{gather}

(4.15)\begin{gather}\mu^{ext}( \xi, \tau)=\frac{ \xi}{\tilde{r}_{f}(\tau)[(-\tilde{\xi}_{0i})/(1+\mu^{ext}_{fi})+\ln(\tilde{r}_f/\tilde{r}_{fi})]}+ \mu_f^{ext}( \tau), \end{gather}

where $\tilde {r}_f(\tau )$ is the reduced radius (4.12). Introducing the notation

(4.16)\begin{equation} Y(\tau)\equiv (1/\tau)\int_0^{ \tau}\dot{\alpha}_{\tau}( \tau')\,\text{d} \tau'=O(y/b), \end{equation}

the function $\mu _f^{ext}(\tau )$ is, according to (2.51a,b),

(4.17)\begin{equation} \mu^{ext}_f(\tau) = \frac{\tau \left[1+Y(\tau)\right ]+(-\tilde{\xi}_{0i})\tilde{r}_{fi}}{ \tilde{r}_f(\tau)[(-\tilde{\xi}_{0i})/(1+\mu^{ext}_{fi})+\ln(\tilde{r}_f/\tilde{r}_{fi})]}-1. \end{equation}

The external flow $\mu ^{ext}(\xi , \tau )$ is a linear function of the radius (straight shape) with a slope decreasing monotonically to zero, regardless of the time-dependent velocity of the front $y(\tau )$. The dependence of $\mu _f^{ext}(\tau )$ on the past of the unknown solution $y(\tau )$ through the integral $Y(\tau )$ comes from the increase of the thickness of the rarefaction wave with the time, $-\xi _0(\tau )\equiv {[r_f(\tau )-r_0(\tau )]}/{l}$, $-\text {d}\xi _{0}/\text {d} \tau = 1+y/b$, see (2.34)–(2.35a,b). At large radius and close to the CJ velocity the external flow takes the simplified form (C12), reminiscent of (2.61a,b). To summarize, for overdriven regimes, the boundary condition in the burnt gas of the solution of (4.10) is (4.14)

(4.18)\begin{gather} \xi_s(\tau)<{-}\text{e}^{{-}y(\tau)}; \quad \xi\leqslant -\text{e}^{{-}y(\tau)}: \mu(\xi,\tau)= \mu^{ext}(\xi,\tau), \end{gather}

(4.19)\begin{gather} \text{and, in particular} \quad \xi={-}\text{e}^{{-}y(\tau)}: \mu(\xi,\tau)= \mu^{ext}(-\text{e}^{{-}y(\tau)},\tau). \end{gather}

Figure 5. Sketch of the flow in an overdriven regime with $0 < y/b < \mu ^{ext}_{f}$, $\mu ^{ext}(\xi _{s},0)=y/b$ $<\mu ^{ext}_{f}$, $\xi _{s}<-\text {e}^{-y}$. The distances plotted on the horizontal and vertical axis correspond to the simplified expressions (C12), $\tau _t$ corresponding to the time at which $\mu ^{ext}_{f}=0$.

Introducing the decomposition

(4.20)\begin{equation} \mu(\xi, \tau)=\mu^{ext}(\xi, \tau)+\hat \mu(\xi, \tau), \end{equation}

and subtracting (2.36) from (4.10)

(4.21)\begin{equation} \frac{\partial \hat \mu}{\partial \tau} + \left( \mu -\frac{y}{b}\right)\frac {\partial \mu}{\partial \xi}- \left( \mu^{ext} -\frac{y}{b}\right)\frac {\partial \mu^{ext}}{\partial \xi}= \frac{1}{2}\text{e}^{y(\tau)} \omega_{o_{CJ}}(\xi\,\text{e}^{y(\tau)})-\frac{\hat{\mu}}{\tilde{r}_f}, \end{equation}

the dynamics of the lead shock $y(\tau )$ during the decay of a combustion wave in the overdriven regime corresponds to the eigenfunction of the following problem

(4.22)\begin{gather} \tau\leqslant \tau_s:\quad\frac{\partial \hat \mu}{\partial \tau} + \left( \hat \mu -\frac{y}{b}\right)\frac {\partial \hat \mu}{\partial \xi} = \frac{1}{2}\text{e}^{y(\tau)} \omega_{o_{CJ}}(\xi \,\text{e}^{y(\tau)})-\frac{\hat \mu}{\tilde{r}_{f}}-\frac{\partial}{\partial \xi}[ \mu^{ext}{\hat \mu}], \end{gather}

(4.23)\begin{gather}\xi=0: \quad \hat \mu=1+2y(\tau)/b-\mu_f^{ext}(\tau); \quad \xi \leqslant -\text{e}^{{-}y(\tau)}: \hat \mu=0, \end{gather}

where $\mu ^{ext}(\xi , \tau )$ is given in (4.15) and $\tau _s$ denotes the time at which the overdriven regime is no longer verified. The boundary conditions at $\xi =0$ and at $\xi = -\text {e}^{-y(\tau )}$ are respectively the Rankine–Hugoniot relation (4.11) and (4.18).

Typically, $\xi _s(\tau )$ increases when $y(\tau )$ decreases, see § 5.1, so that the sonic point approaches the end of the reaction $\xi _b(\tau )=-\text {e}^{-y}$. As soon as the sonic point crosses the exit of the reaction zone, $\tau \geqslant \tau _s$, $\xi _s(\tau _s)=-\text {e}^{-y(\tau _s)}$, $\mu ^{ext}(\xi _s,\tau )=y/b$, the propagation regime is no more overdriven; the external flow which is still solution of (2.38a,b) in the burnt gas $\xi <-\text {e}^{-y(\tau )}$, is influenced by the disturbances emitted from the reactive transonic flow and transported downstream (towards the weak discontinuity $\xi =\xi _0<0$) by the characteristic $\mathcal {C}_+$. Then, for $\tau > \tau _s$, $\xi _s(\tau )>-\text {e}^{-y(\tau )}$, the boundary condition (4.19) is no longer valid and the rarefaction wave is no longer represented accurately by the analytical expression (4.15) everywhere in the burnt gas. In other words, soon after $\tau _s$, the rarefaction flow behind the end of the reaction ($\xi <-\text {e}^{-y(\tau )}$) is modified by the heat which is released in the region delimited by the end of the reaction and the sonic point ($-\text {e}^{-y}\leqslant \xi \leqslant \xi _s$). However, as in the discontinuous model in § 3.2, see figures 3 and 4, the solution (4.15) is recovered downstream until the arrival of the disturbance transported by the characteristic $\mathcal {C}_+$. The subsequent dynamics for $\tau \geqslant \tau _s$ should then be analysed by the numerical solution of (4.10)–(4.11) using an initial condition at $\tau = \tau _s$ given by the solution of the eigenvalue problem (4.22)–(4.23).

4.3. Initial condition

In a real initiation process the trajectory of the lead shock is fully determined by the strong blast wave (inert flow) which is generated initially by a quasi-punctual and quasi-instantaneous deposit of external energy at the centre. According to the self-similar solution of Sedov (Reference Sedov1946) and Taylor (Reference Taylor1950b), this strong blast wave depends only on the amount of energy which is deposited, see the discussion of Liñán et al. (Reference Li nan, Kurdyumov and Sanchez2012) for more details. Such an initial stage corresponds to a large Mach number which is beyond the scope of the asymptotic analysis in the limit $0 < M-1\ll 1$. The critical condition of initiation will be investigated in the present article by a parametric study of the initial conditions of the solution of (4.10)–(4.17). Three parameters of order unity in the limit $\epsilon \to 0$ are involved at $\tau =0$; the initial radius $\tilde {r}_{fi}\equiv \epsilon {r_{fi}}/{l}$, the initial thickness of the rarefaction wave $-\tilde {\xi }_{0i}\equiv (r_{fi}-r_{0i})/\epsilon r_{fi}>0$ and a positive parameter $\mu ^{ext}_{fi}$. The initial propagation velocity $y_i=b(\mathcal {D}_i-\mathcal {D}_{o_{CJ}})/\epsilon a$ is related to the above-mentioned parameters through the inner structure of the detonation. Our attention will be focused on the initial velocity of weakly overdriven detonations close to the CJ velocity, typically $y_i/b=0.15\text {--}0.5$ and on an initial extension of the rarefaction wave much larger than the detonation thickness, as in real initiation processes near criticality

(4.24)\begin{equation} r_{fi} -r_{0i} \gg l \Rightarrow (-\tilde{\xi}_{0i})\tilde{r}_{fi}\gg 1. \end{equation}

The numerical integration of (4.10)–(4.11) is initialized at $\tau =0$ by igniting the exothermal reaction in the inert flow (4.15)

(4.25)\begin{equation} \tau=0: \quad \mu^{ext}( \xi)=\frac{ (1+\mu^{ext}_{fi})}{(-\tilde{\xi}_{0i})\tilde{r}_{fi}}\,\xi+ \mu_{fi}^{ext}\quad \text{with} \ \mu_{fi}^{ext}>0, \end{equation}

leading quickly, on a time scale shorter than unity, to the structure (4.13a–c) of an overdriven regime sketched in figure 5, $\mathcal {D}-u_b < a \Leftrightarrow \mu _{fi}^{ext}(-\text {e}^{-y_i})-y_i/b > 0$. For $\mu _{fi}^{ext}$ large enough, the initial detonation velocity is larger than the CJ velocity $y_i>0$. As explained at the beginning of § 4.2, the rarefaction flow of burnt gas (4.15)–(4.17) is not perturbed by the heat release since the regime is overdriven.