2.1. Conservative form

The equations for mass and momentum are

(2.1a,b)\begin{equation} \frac{\partial \rho}{\partial t}={-}\frac{\partial(\rho u)}{ \partial x}, \quad \frac{\partial (\rho u)}{\partial t}={-} \frac{\partial\left[p+\rho u^2- \mu\partial u \partial {x} \right]} {\partial x} , \end{equation}

where $\rho$, $p$ and $u$ are respectively the density, the pressure and the velocity of the flow in the laboratory frame and $\mu$ is the viscosity. For the sake of simplicity we will consider an irreversible one-step exothermal reaction. This simplification can be removed and the fundamental result does not depend qualitatively on a detailed chemical scheme provided the laminar flame velocity is highly sensitive to the temperature. Introducing the progress variable $Y$, namely the mass fraction of products, $Y=0$ in the unburned mixture and $Y=1$ in the burned gas, the chemical heat release per unit mass $q_m$, the heat conductivity $\lambda$ and the diffusion coefficient $D$, the equation for energy, written in conservative form, is

(2.2)\begin{align} &\frac{\partial \left [\rho (c_vT +u^2/2-q_m Y)\right]}{\partial t}\nonumber\\ &\quad ={-}\frac{\partial \left[ \rho u\left( c_vT+p/\rho+u^2/2-q_m Y\right)-\lambda \partial T/\partial x-\mu u\partial u/\partial x+q_m\rho D\partial Y/\partial x\right]}{\partial x}, \end{align}

with the perfect gas law, $p=c_v(\gamma -1)\rho T=nk_BT$ in which $k_B$ is the Boltzmann constant and $n$ the molecular density. The ratio of specific heats $\gamma \equiv c_p/c_v$ is assumed constant for simplicity. The conservation equation of species for a one-step reaction, written in terms of the progress variable, $Y\in [0, 1]$ reads

(2.3)\begin{equation} \frac{\partial (\rho Y)}{\partial t} ={-}\frac{\partial\left[ \rho u Y-\rho D\partial Y/\partial x\right]}{\partial x} +\rho W(Y,T), \end{equation}

with a reaction rate in the form of an Arrhenius law for a large activation energy $\mathcal {E} \gg k_BT$, proportional to the frequency of binary collisions $1/t_{coll}$

(2.4)\begin{equation} W(Y,T)= \frac{B}{t_{coll}}(1-Y)^{2}\exp({-\mathcal{E}/k_BT}). \end{equation}

These macroscopic equations are solutions of the Boltzmann equation in the hydrodynamic limit (macroscopic length scales larger than the mean free path and time larger than $t_{coll}$, respectively). The solution shows how the frequency of binary collisions $1/t_{coll}$, the diffusion coefficient $D$ and the sound speed $a=\sqrt {\gamma p/\rho }$ are related in a perfect gas $p=nk_BT$

(2.5a,b)\begin{equation} D=\frac{1}{6\sqrt \gamma nr_o^2}a, \quad \frac{D}{t_{coll}}= \frac{8\sqrt {\rm \pi}}{3\gamma }a^2, \end{equation}

$r_o$ being the radius of the molecules. The parameter $B$, usually called the pre-factor, is the reduced activation energy times the initial molecular dilution of reactant $y_i$

(2.6)\begin{equation} B=y_i\frac{\mathcal{E}}{k_BT} \quad \Rightarrow \quad B\frac{D}{t_{coll}}=\frac{8\sqrt {\rm \pi}}{3}y_i\frac{\mathcal{E}}{m}. \end{equation}

Equations (2.5a,b) and (2.6) are useful to express the laminar flame velocity in terms of the flame temperature. For the sake of simplicity the molecules of reactants, products and diluent are assumed to have the same mass $m$ ($\rho =nm$) and the same radius $r_o$.

According to the asymptotic analysis of Zeldovich–Frank–Kamenetskii (ZFK), in the limit of large activation energy $\mathcal {E}/k_BT_b\gg 1$, the laminar flame velocity $U_b(T_b)$ relative to the burned gas denoted by the subscript $b$ ($T_b$ is the flame temperature) takes the form

(2.7)\begin{equation} U_b(T_b)=\frac{1}{\tilde{\beta}_b^{3/2}}\sqrt{8B_b\frac{D_{b}}{t_{coll b}}\exp({-\mathcal{E}/k_BT_b})},\quad \text{where} \ \tilde{\beta}_b \equiv \frac{q_m}{c_pT_b}\frac{\mathcal{E}}{k_BT_b}, \end{equation}

showing how small is the Mach number of the laminar flame velocity

(2.8)\begin{equation} \varepsilon\equiv \frac{U_b}{a_b}=\sqrt{\frac{ 2!\,16{\rm \pi}^{1/2}}{3\gamma} y_i}\,\frac{1}{\tilde{\beta}_b^{3/2}}\sqrt{\frac{\mathcal{E}}{ k_BT_b}} \exp({-\mathcal{E}/2k_BT_b});\end{equation}

see a didactic presentation in Clavin & Searby (Reference Clavin and Searby2016). Typical orders of magnitude in real flames are ${U_b}/{a_b}\approx 10^{-3}\unicode{x2013}10^{-2}$. Equation (2.8) leads to similar values $\varepsilon \approx 10^{-3}$ for ${\mathcal {E}}/{ k_BT_b}\approx 10$, $q_m/c_pT_b\approx 0.8$ and $\varepsilon \approx 10^{-2}$ for smaller values of ${\mathcal {E}}/{ k_BT_b}$ in energetic mixtures. According to (2.5a,b)–(2.8), the ratio of the laminar flame velocity for two flames with different temperatures takes the form

(2.9)\begin{equation} \frac{U_b(T_{b1})}{U_b(T_{b2})}= \left (\frac{T_{b1}}{T_{b2}}\right )^{3}\exp{-\frac{\mathcal{E}}{2k_B}\left( \frac{1}{T_{b1}}-\frac{1}{T_{b2}}\right )}. \end{equation}

The main effect of a complex network of chemical kinetics is to modify the power law in the Arrhenius pre-factor and to introduce a temperature cutoff $T_c\in [850\unicode{x2013}1200\ {\rm K}]$ below which the combustion cannot proceed. In addition, the activation energy $\mathcal {E}$ varies with the temperature for $T>T_c$. For a large activation energy $\mathcal {E}/(k_BT_b)\gg 1$ and far from the chemical quenching $T_b>T_c$, the temperature variation of $\mathcal {E}$ can be neglected in a limited range of flame temperature $\Delta T_b\ll \mathcal {E}/(\text {d}\mathcal {E}/\text {d} T_b)$. The upper bound of $\Delta T_b/T_b$ for the validity of this inequality is not small when the relative variation of the activation energy is smaller than the reduced activation energy $(T_b/\mathcal {E})\,\text {d}\mathcal {E}/\text {d}T_b< \mathcal {E}/k_BT_b$. In such conditions, (2.9) is reduced to an Arrhenius law

(2.10)\begin{equation} {\beta}\gg 1, \quad \frac{\mathcal{E}}{k_BT_{b2}}\left (\frac{T_{b1}}{T_{b2}}-1 \right )=O(1): \ \frac{ U_b(T_{b1})}{U_b(T_{b2})}\approx \exp\left[\frac{\mathcal{E}/2}{k_BT_{b2}}\left (\frac{T_{b1}}{T_{b2}}-1 \right )\right ]. \end{equation}

However in highly reactive mixtures (stoichiometric ${\rm H}_2$ or ${\rm C}_2{\rm H}_4$ mixtures in pure oxygen) the flame temperature is large and the reduced activation energy is of order unity so that the power law $({T_{b1}}/{T_{b2}} )^{3}$ in (2.9) cannot be ignored.

2.2. Non-dimensional equations in the Lagrangian form

From now on, the reference state used in the non-dimensional equations is the burned gas of the steady flame at the initial condition (labelled $i$). The latter is a self-similar solution with the lead shock at infinity. Denoting by $T_{ui}$ and $\rho _{bi}$ respectively the temperature of the unburned gas and the density of the burned gas in the initial state, the reference temperature, velocity, density and pressure are

(2.11a–d)\begin{align} T_{ref}=T_{bi}=T_{ui}+q_m/c_p, \quad U_{ref}=U_b(T_{bi}),\quad \rho_{ref}= \rho_{bi}, \quad p_{ref}=(c_p-c_v)\rho_{ref}T_{ref}. \end{align}

Using the corresponding flame thickness and transit time $d_{ref}=D_{ref}/U_{ref}$ and $t_{ref}=d_{ref}/U_{ref}=D_{ref}/U^2_{ref}$, where $D_{ref}$ is the molecular diffusion coefficient in the reference state, the following non-dimensional variables are introduced:

(2.12)\begin{gather} \tau\equiv t/t_{ref}, \quad \xi\equiv (x-X_P)/d_{ref}, \quad \text{where} \ t_{ref}\equiv d_{ref}/U_{ref}=D_{ref}/U^2_{ref}, \end{gather}

(2.13a–e)\begin{gather} r\equiv \frac{\rho}{\rho_{ref}}, \quad v\equiv \frac{u}{U_{ref}}, \quad {\rm \pi}\equiv\frac{p}{p_{ref}}, \quad \theta\equiv \frac{T}{T_{ref}}, \quad v_P\equiv \frac{U_P}{U_{ref}},\end{gather}

where $x=X_P(t)$ is the instantaneous position of the flame (for example, the maximum of reaction rate), $u(x,t)$ and $U_P(t)\equiv \text {d} X_P/\text {d} t$ are respectively the flow velocity and the propagation speed of the flame in the laboratory frame (not to be confused with the laminar flame velocity $U_b$). The non-dimensional mass flux across the flame $m \equiv \rho (U_P-u)/\rho _{ref}U_{ref}$ takes the form

(2.14)\begin{equation} m(\xi,\tau)= r\left[v_P(\tau)-v(\xi,\tau)\right ]>0, \quad r\equiv {\rm \pi}/{\theta}. \end{equation}

Introducing the Mach number of the laminar flame, the reduced heat release and the reduced activation energy

(2.15a–c)\begin{equation} \varepsilon\equiv \frac{U_{ref}}{{a_{ref}}}\approx 10^{{-}2}, \quad q\equiv \frac{q_m}{c_pT_{ref}}\approx 0.7, \quad \beta\equiv \frac{\mathcal {E}}{k_B T_{ref}}=4\unicode{x2013}8, \end{equation}

and using the relations $1/(\rho _{ref} c_p T_{ref})=(\gamma -1)/(\gamma p_{ref})=(\gamma -1)/(\rho _{ref} a_{ref}^2)$ and $D_{ref}=\lambda /(\rho _{ref} c_p)=\mu /\rho _{ref}$ ($Le=1$ for simplicity), the non-dimensional Lagrangian form of (2.1a,b)–(2.4), written in the frame moving with the flame $t_{ref}\partial /\partial t \to \partial /\partial \tau -v_P\partial /\partial \xi$, reads

(2.16)$$\begin{gather} \frac{\partial r}{\partial \tau}=\frac{\partial m}{\partial \xi}, \quad r={\rm \pi}/\theta \end{gather}$$

(2.17)$$\begin{gather}\frac{\partial v}{\partial \tau} -m \frac{\partial v}{\partial \xi}={-}\frac{1}{\gamma \varepsilon^2 }\frac{\partial {\rm \pi}}{\partial \xi}+\frac{\partial^2 v}{\partial \xi^2} \end{gather}$$

(2.18)$$\begin{gather}\left[ r\frac{\partial }{\partial \tau}-m\frac{\partial }{\partial \xi}\right]Y-\frac{\partial^2 Y}{\partial \xi^2}=w(\theta, Y) \end{gather}$$

(2.19)$$\begin{gather}\left[ r\frac{\partial }{\partial \tau}-m\frac{\partial }{\partial \xi}\right ]\theta- \frac{(\gamma-1)}{\gamma}\left[ \frac{\partial }{\partial \tau}+[v-v_P]\frac{\partial }{\partial \xi}\right ]{\rm \pi} =\frac{\partial^2 \theta }{\partial \xi^2}+ (\gamma-1)\varepsilon^2\left(\frac{\partial v}{\partial \xi}\right )^2+qw. \end{gather}$$

For a large activation energy, the non-dimensional reaction rate takes the form

(2.20a,b)\begin{align} w(\theta, Y)\equiv t_{ref}w=\frac{\beta_{ref}^{3}}{8}\frac{{\rm \pi}_b}{\theta_b} (1-Y)^{2} \exp\left[\frac{\mathcal{E}}{k_B T_{ref}}\left({\theta} -1\right )\right], \quad \beta_{ref} \equiv \frac{q_m}{c_pT_{ref}}\frac{\mathcal{E}}{k_BT_{ref}}, \end{align}

the subscript $b$ denoting the burned gas. Eliminating the density by using the perfect gas law $r={\rm \pi} /\theta$, the four equations (2.16)–(2.19) concern four fields $v(\xi,\tau )$, ${\rm \pi} (\xi,\tau )$, $Y(\xi,\tau )$ and $\theta (\xi,\tau )$ plus an unknown function $v_P(\tau )$ appearing in the mass flux $m(\xi,\tau )$ (2.14).

2.3. Mass-weighted coordinate

The analysis of the unsteady flame structure is more easily performed using the mass-weighted coordinate $z$ and the reduced mass flux at the origin ($z=0$) $m(\tau )\equiv m(z=0,\tau )$ with, according to (2.14)

(2.21)\begin{equation} m(\tau)= r(0,\tau) \left[v_P(\tau)-v(0,\tau) \right]=\frac{{\rm \pi}(0,\tau)}{\theta(0,\tau)} \left[v_P(\tau)-v(0,\tau) \right]>0. \end{equation}

Introducing the change of variables $(\xi,\tau )\to (z,\tau )$

(2.22a,b)$$\begin{gather} z\equiv\int_0^\xi r(\xi',\tau)\,\text{d}\xi', \quad \frac{\partial }{\partial \xi}=r \frac{\partial }{\partial z}=\frac{\rm \pi}{\theta} \frac{\partial }{\partial z}, \end{gather}$$

(2.23)$$\begin{gather}\left.\frac{\partial }{\partial \tau}\right\vert_\xi=\left.\frac{\partial }{\partial \tau}\right\vert_z+ \left[\int_0^\xi \frac{\partial r(\xi',\tau)}{\partial \tau}\text{d}\xi'\right ] \frac{\partial }{\partial z}=\left.\frac{\partial }{\partial \tau}\right\vert_z+\left [m(\xi, \tau)-m(\tau)\right ] \frac{\partial }{\partial z}, \end{gather}$$

(2.24)$$\begin{gather}\left(v-v_P(\tau)\right) \frac{\partial }{\partial \xi}={-}m(\xi,\tau) \frac{\partial }{\partial z}\Rightarrow \left.\frac{\partial }{\partial \tau}\right\vert_\xi+\left( v-v_P(\tau)\right) \frac{\partial }{\partial \xi}= \left.\frac{\partial }{\partial \tau}\right\vert_z-m(\tau) \frac{\partial }{\partial z}, \end{gather}$$

where the function $m(\tau )$ in front of the derivative with respect to $z$ depends only on the time. Continuity (2.16) written with the variables ($z,\tau$)

(2.25)\begin{equation} r\frac{\partial m(z,\tau)}{\partial z}= \frac{\partial r(z,\tau)}{\partial \tau}+\left[m(z,\tau)-m(\tau)\right ]\frac{\partial r}{\partial z}, \end{equation}

yields after multiplication by $1/r^2$

(2.26)\begin{equation} \frac{1}{r}\frac{\partial m(z,\tau)}{\partial z}= \left.-\frac{\partial (1/r)}{\partial \tau}\right\vert_{z}-\left[ m(z,\tau)-m(\tau)\right ]\frac{\partial (1/r)}{\partial z}, \end{equation}

to give, using (2.14) $m(z,\tau )= -r(z,\tau ) [v(z,\tau )-v_P(\tau ) ]>0$,

(2.27)\begin{align} \frac{\partial m(z,\tau)}{\partial z}={-}\left[v(z,\tau)-v_P(\tau)\right]\frac{\partial r}{\partial z}-r \frac{\partial v}{\partial z} \quad \Rightarrow \quad \frac{\partial v}{\partial z}= \left.\frac{\partial (1/r)}{\partial \tau}\right\vert_{z}-m(\tau)\frac{\partial (1/r)}{\partial z}, \end{align}

yielding the gradient of the flow in terms of $1/r=\theta (z,\tau )/{\rm \pi} (z,\tau )$. For simplicity, the diffusion coefficient $D$ is assumed to verify $\rho ^2D$= constant, $\partial (\rho D \partial /\partial x)/\partial x\to (\rho ^2 D/\rho ^2_{ref}D_{ref}) r\partial ^2/\partial z^2 =r\partial ^2/\partial z^2$. Then, (2.16)–(2.19) yield

(2.28)\begin{gather} \frac{\partial v}{\partial z}=\left[\frac{\partial }{\partial \tau}-m(\tau)\frac{\partial }{\partial z} \right]\frac{\theta}{\rm \pi}, \nonumber\\ =\frac{1}{\rm \pi}\left[\frac{\partial }{\partial \tau}-m(\tau)\frac{\partial }{\partial z} \right]\theta-\frac{\theta}{{\rm \pi}^2}\left[\frac{\partial }{\partial \tau}-m(\tau)\frac{\partial }{\partial z} \right]{\rm \pi} \end{gather}

(2.29)$$\begin{gather} \left[\frac{\partial v}{\partial \tau}-m (\tau) \frac{\partial v}{\partial z} -\frac{\partial^2 v}{\partial z^2}\right]={-}\frac{1}{\gamma \varepsilon^2}\frac{\partial {\rm \pi}}{\partial z} \end{gather}$$

(2.30)$$\begin{gather}\left[\frac{\partial Y}{\partial \tau}-m ( \tau)\frac{\partial Y}{\partial z} -\frac{\partial^2 Y}{\partial z^2}\right ]=w(\theta, Y),\quad Y(z, \tau)\in [0, 1] \end{gather}$$

(2.31)$$\begin{gather}\left[\frac{\partial \theta}{\partial \tau}-m ( \tau)\frac{\partial \theta}{\partial z} -\frac{\partial^2 \theta}{\partial z^2}\right ]= q w(\theta, Y)+\frac{(\gamma-1)}{\gamma}\frac{\theta}{\rm \pi}\left[\frac{\partial {\rm \pi}}{\partial\tau} -m(\tau) \frac{\partial {\rm \pi}}{\partial z}\right] \nonumber\\ + (\gamma-1)\varepsilon^2 \left(\frac{\partial v}{ \partial z}\right)^{\!2} , \end{gather}$$

where the perfect gas law $r={\rm \pi} /\theta$ has been used to eliminate the density. When the dissipative terms (heat conduction, viscosity and reaction rate) are neglected, equation for energy (2.31) takes the form of the entropy wave in an inert gas

(2.32)\begin{equation} \frac{1}{\theta}\left[\frac{\partial \theta}{\partial \tau}-m ( \tau)\frac{\partial \theta}{\partial z} \right ] -\frac{(\gamma-1)}{\gamma}\frac{1}{\rm \pi}\left[\frac{\partial {\rm \pi}}{\partial\tau} -m(\tau) \frac{\partial {\rm \pi}}{\partial z}\right]=0. \end{equation}

3.1. Backflow models

The flame on the tip is treated as planar and orthogonal to the propagation axis. Following Clanet & Searby (Reference Clanet and Searby1996), the longitudinal gradient of burned-gas flow on the tube axis $u(x, t)$ is roughly modelled by a source term of mass whose origin is the burning of the lateral flame parallel to the wall. Denoting the laminar flame velocity (relative to the burned gas) of this lateral flame $U_{bw}(x,t)$, the gradient of the flow on the tube axis $u(x, t)$ is approximated by a one-dimensional mass conservation in an incompressible flow

(3.2)\begin{equation} \frac{\partial u}{\partial x}= \frac{2}{R}U_{bw}(x,t) , \end{equation}

where $R$ is the radius of the tube. The longitudinal backflow $u_b(t)$ impinging the tip from behind is obtained by integration along the axis of the finger flame. For a closed-end tube on the burned-gas side, assuming that the incompressible flow of burned gas is at rest behind the elongated flame, one gets

(3.3)\begin{equation} u_b(t)\equiv u(x=X_p(t),t)=\frac{2}{R}\int_{X_P-L}^{X_P}U_{bw}(x,t)\,\text{d}\,x, \end{equation}

where $X_p(t)$ is the position of the tip and $L(t)$ the length of the elongated flame. Neglecting both the heat loss on the wall and the unsteadiness of the burned-gas flow treated as incompressible, $U_{bw}$ is uniform along the lateral flame and equal to the laminar flame speed on the tip at the same time $U_{bw}=U_b(t)$. Moreover, if the unsteadiness of the inner flame structure is negligible, $U_b(t)$ is given by (2.7)–(2.10) $U_b(t)=U_b(T_b(t))$. Under these approximations, the backflow takes the form introduced by Clavin & Tofaili (Reference Clavin and Tofaili2021)

(3.4)\begin{equation} \text{instantaneous backflow model:}\quad u_b(t)=S(t)\,U_b(t), \quad U_b(t)\equiv U_b\left(T_b(t)\right), \end{equation}

where $S$ is the elongation of the finger flame ($S=2L/R$ in cylindrical geometry) and $T_b(t)=T_u(t) + Q/c_p$ where $T_u(t)$ is the temperature of the fresh mixture just ahead of the flame.

Unsteadiness of the flow of burned gas in an elongated flame is too complicated to be described analytically. Hopefully a detailed study is not useful in the following. This unsteady effect will be roughly modelled by a delay $\Delta t(X_P-x)$ for transferring to the tip the flow of burned gas issued from the lateral flame at a distance $X_p-x$ from the tip

(3.5)\begin{equation} u_b(t)\approx\frac{2}{R}\int_{X_P-L}^{X_P}U_{b} \left(t-\Delta t(X_P-x) \right)\text{d}\,x. \end{equation}

Assuming a slow evolution of the laminar flame velocity $U_b/(\text {d}U_b/\text {d}t) \gg \Delta t(L)\geqslant \Delta t(X_P-x)$, a Taylor expansion yields

(3.6)\begin{equation} u_b(t)\approx\frac{2L(t)}{R}U_b(t) -\frac{2}{R}\frac{\text{d} U_b}{\text{d} t}\int_{X_P-L}^{X_P}\Delta t(X_P-x)\,\text{d}\,x. \end{equation}

Assuming that the variation of the radial burned gas out of the lateral flames is propagated by the downstream-running compression waves with a quasi-constant sound speed $a$, $\Delta t(X_P-x)\approx (X_P-x)/a$,

(3.7)\begin{equation} \int_{X_P-L}^{X_P}\Delta t(X_P-x)\,\text{d}\,x\approx L^2/2a, \end{equation}

equation (3.6) yields, after introducing the overall delay $\Delta t_w\approx (1/2){L}/{a}$,

(3.8)\begin{align} \text{delayed model of backflow: } \quad u_b(t)\approx S(t)\left[ U_b(t)-\Delta t_w\frac{\text{d}U_b}{\text{d}t}\right]\approx S(t)\,U_b(t-\Delta t_w), \end{align}

in which the variation with the time of $\Delta t_w$ is a negligible second-order effect.

3.2. Limit of large activation energy

The ZFK analysis in the limit of large activation energy has been extended more than forty years ago to unsteady structure of flames, see Clavin & Searby (Reference Clavin and Searby2016) for a didactic presentation. Choosing the instantaneous position of the reaction sheet as the origin on the $z$-axis and introducing the notation

(3.9a,b)\begin{equation} \beta\equiv \mathcal{E}/k_BT_{ref}, \quad \theta_b(\tau)\equiv T_b(t)/T_{ref}, \end{equation}

for the reduced activation energy and the reduced flame temperature, the jump conditions on the reaction sheet take the form

(3.10)\begin{gather} \left.\begin{gathered} z\leqslant 0:\quad Y=1,\\ z=0:\quad Y=1, \quad \theta=\theta_b(\tau), \quad \left(\theta_b-1\right )=O(1/\beta), \end{gathered}\right\}\end{gather}

(3.11)$$\begin{gather} \beta\gg 1, \quad \beta(\theta_b-1)=O(1): \quad \left. \frac{\partial \theta}{\partial z}\right\vert_{z=0^+}={-} q \exp\left[\frac{\beta}{2}\left(\theta_b-1\right )\right] +O(1/\beta), \end{gather}$$

(3.12)$$\begin{gather}\left.\frac{\partial \theta}{\partial z}\right\vert_{z=0^-}=\left.\frac{\partial \theta}{\partial z}\right \vert_{z=0^+}-q\left.\frac{\partial Y}{\partial z}\right\vert_{z=0^+}+O(1/\beta^2), \end{gather}$$

with $z=0^+$ and $z=0^-$ denoting the preheated zone side of the reaction zone and the exit on the burned-gas side, respectively. Equation (3.11) is valid to order unity while (3.12) is valid up to first order $1/\beta \ll 1$ (included). The backflow of burned gas is applied on the reaction sheet so that a boundary condition concerning the flow velocity is added

(3.13)$$\begin{gather} z=0\quad \tau>0: \quad v=v_b(\tau), \end{gather}$$

(3.14)$$\begin{gather}\tau\leqslant0: \quad v= v_b(0)=S_i. \end{gather}$$

To leading order in the limit $\beta \gg 1$, the variation of flame temperature is retained in the Arrhenius factor of (2.10) only. Therefore, when the inner flame structure is in steady state (denoted by an overbar), the laminar flame velocity reads

(3.15)\begin{align} {\beta}\gg 1, \quad\beta(\bar{\theta}_b-1)=O(1): \ {\bar{U}_b(T_b)}/{U_{ref}}=\bar{m} =\exp\left[{\beta}\left(\bar{\theta}_b-1\right )/2\right]+O(1/\beta), \end{align}

where $\bar {\theta }_b(\tau )=\theta _u(\tau )+q$ is the flame temperature and $\theta _u(\tau )=T_u(\tau )/T_{ref}$ the instantaneous temperature of unburned gas just ahead of the flame. The instantaneous backflow (3.4), written in non-dimensional form, then yields

(3.16)\begin{equation} \bar{v}_b(\tau)= S(\tau)\bar{m}=[1+\epsilon \tau]S_i\exp\left ({\beta}\left [ \theta_u(\tau)+q-1\right ]/2\right ), \end{equation}

and the delayed backflow model (3.8) reads

(3.17a–c)\begin{equation} {\beta}\gg 1,\quad \tau\geqslant0:\quad v_b(\tau)=\bar{v}_b(\tau)\left[1- (\beta/2)\frac{\Delta t_w}{t_{ref}} \frac{\text{d}\theta_u}{\text{d} \tau}\right ],\quad \Delta t_w\approx \frac{L}{2a}. \end{equation}

When the unsteadiness of the inner structure of the flame (studied in § 7) is taken into account, the mass flux $\bar {m}$ in (3.17a–c) is replaced by its unsteady version $m(\tau )=\bar {m}+\delta m$, $\bar {v}_b=S(\tau )\bar {m} \to S(\tau )m(\tau )$, the reduced backflow including the two unsteady effects (in the burned gas and in the inner flame structure) takes the form,

(3.18a,b)\begin{gather}{\beta}\gg 1,\ \tau\geqslant0:\ v_b(\tau)=S(\tau)\exp({{\beta}[\bar{\theta}_b(\tau)-1]/2})\left [1+ \frac{\delta m(\tau)}{\bar{m}(\tau)} \right ]\left[1- (\beta/2)\frac{\Delta t_w}{t_{ref}}\frac{\text{d}\theta_u}{\text{d}\tau}\right ]. \end{gather}

Here, $\delta m(\tau )$ is computed by the unsteady analysis of the inner structure.

3.3. Initial condition

Before the elongation starts to increase, the initial condition is a self-similar solution (constant elongation, steady flame structure, $\theta _b(0)=1$) with a constant backflow (3.14) $v_b(0)=S_i$ and a uniform flow ahead of the flame $\theta =1-q$, $v=v_b(0)+q$, the lead shock far ahead from the flame front being considered at infinity. The problem being hyperbolic in the unburned gas outside the flame structure, the downstream boundary condition far away from the flame for solving the unsteady problem is given by the initial solution

(3.19)\begin{equation} \left.\begin{gathered} z \to \infty:\quad {\rm \pi}\approx 1+O(\varepsilon^2),\quad Y=0, \quad \theta=1-q+O(\varepsilon^2),\\ v=S_i+q+O(\varepsilon^2). \end{gathered}\right\}\end{equation}

The neglected terms are of the same order of magnitude as the pressure jump across a laminar flame $\delta p/p=O(\varepsilon ^2)$, according to the steady-state version of (2.29).

4.1. Preliminary insights into the unburned-gas flow ahead of the flame

When the flame accelerates the flame acts a semi-transparent piston so that simple compression waves are sent in the unburned gas. The dissipative mechanisms being negligible in this external flow, the entropy wave (2.32) propagates from right to left in the reference frame attached to the flame since the flame runs from left to right faster than the flow in the laboratory flame, $m(\tau )>0$. Therefore, as long as no new shock wave is formed on the leading edge of the compression wave, the entropy is constant ahead of the flame and equal to the downstream entropy ($z\to \infty$). The isentropic condition ${\rm \pi} =\theta ^{\,\gamma /(\gamma -1)}$ for small pressure variations, $\theta =\theta _i(z)+\delta \theta$, ${\rm \pi} =1+\delta {\rm \pi}$, $\lim _{z\to \infty } \theta _i=1-q$, yields

(4.1)\begin{equation} {\delta {\rm \pi}}\ll 1:\quad {\delta \theta}/{\theta_i(z)}=[(\gamma-1)/{\gamma}]{\delta {\rm \pi}}, \end{equation}

the subscript $i$ denoting the initial unperturbed flow $\tau =0: {\rm \pi}_i=1$. Anyway, in the limit of small Mach number of the laminar flame $\varepsilon \ll 1$, the shocks that could be produced by the accelerating flame are weak so that the small entropy jump across the shock is of order of magnitude $\varepsilon ^3$ and thus is negligible when the analysis is limited to $\varepsilon$-term.Then, according to continuity, (2.28),

(4.2a,b)\begin{align} {\delta {\rm \pi}}=O(\varepsilon),\quad z\gg 1: \frac{\partial v}{\partial z}=\left[\frac{\partial }{\partial \tau}-m(\tau)\frac{\partial }{\partial z} \right]\left [\delta \theta-\theta_i\delta {\rm \pi}\right ]={-}\theta_i\frac{1}{\gamma}\left[\frac{\partial }{\partial \tau}-m(\tau)\frac{\partial }{\partial z} \right]\delta {\rm \pi}.\end{align}

The viscosity being negligible ahead of the flame, (2.29) yields

(4.3)\begin{equation} {\delta {\rm \pi}}\ll 1:\quad \left[\frac{\partial }{\partial \tau}-m (\tau) \frac{\partial}{\partial z}\right ]v={-}\frac{1}{\gamma \varepsilon^2}\frac{\partial \delta{\rm \pi}}{\partial z}. \end{equation}

The flow $v(z,\tau )$ can be eliminated from (4.2a,b) and (4.3) to give in the linear approximation

(4.4)\begin{equation} \frac{(1-q)}{ {\rm \pi}^2}\left[\frac{\partial }{\partial \tau}-m(\tau)\frac{\partial }{\partial z} \right]^2\delta {\rm \pi}=\frac{1}{\varepsilon^2}\frac{\partial^2 }{\partial z^2}\delta {\rm \pi}. \end{equation}

Therefore, considering a time scale of the order of the transit time of fluid particles across the flame, $\partial /\partial \tau =O(1)$, the pressure varies in space with a length scale larger than the flame thickness by a factor of order $1/\varepsilon$

(4.5)\begin{equation} \frac {\partial \delta {\rm \pi}/\partial z}{\partial \delta {\rm \pi}/\partial \tau}=O(\varepsilon). \end{equation}

The unsteady term of (4.4) balances the second derivative with respect to space on the right-hand side since the term $[m{\partial {\rm \pi}}/{\partial z}]$ is negligible in front of $[(1/\varepsilon ){\partial {\rm \pi}}/{\partial z}]$ for $m=O(1)$. Therefore, the pressure fluctuation satisfies the linear wave equation

(4.6a,b)\begin{equation} \varepsilon \ll 1:\quad \frac{\partial^2 \delta {\rm \pi}}{\partial \tau^2}=\frac{(1-q)}{\varepsilon^2}\frac{\partial^2 \delta {\rm \pi}}{\partial z^2}; \quad a^2=\theta a^2_{ref}\Rightarrow \frac{\partial^2 \delta {\rm \pi}}{\partial t^2}\approx a_i^2 \frac{\partial^2 \delta {\rm \pi}}{\partial x^2}, \end{equation}

written in the original variables using (2.12) and (2.22a,b). Therefore, in the limit $\varepsilon \ll 1$, the external flow of unburned gas ahead of the flame is governed by the linear acoustics with a negligible Doppler effect $m(\partial \delta {\rm \pi}/\partial z)/(\partial \delta {\rm \pi}/\partial \tau )=O(\varepsilon )$ leading to the scalings

(4.7)\begin{equation} \tau=O(1) \quad \Rightarrow \quad \varepsilon z =O(1). \end{equation}

According to the continuity equation (4.2a,b), ${\partial v}/{\partial z}$ is of order $\partial \delta {\rm \pi}/\partial \tau$. Anticipating that the change in flow velocity is of order $S_i$ times the laminar flame velocity $\delta v=O(S_i)$, $\delta v$ and the pressure vary on the large length scale $\partial v/\partial z=O(\varepsilon S_i)$, the variation of the non-dimensional pressure $\delta {\rm \pi}$ in the external zone is of order $\varepsilon S_i$. In the external zone ahead of the flame, the pressure takes the form

(4.8a,b)\begin{equation} {\rm \pi}=1+\varepsilon {\rm \pi}_1(\varepsilon z,\tau), \quad {\rm \pi}_1=O(S_i). \end{equation}

The nonlinear solution of the compression wave obtained by Clavin & Champion (Reference Clavin and Champion2022) confirms that the limit $\varepsilon \ll 1$ leads to the linear wave (4.6a,b)–(4.8a,b).

4.2. Distinguished limit

According to (4.8a,b), the spatial variation of pressure in the external flow $\delta {\rm \pi}=\varepsilon {\rm \pi}_1=O(\varepsilon S_i)$, is larger than in the inner structure of the flame by a factor $1/\varepsilon$ since ${\rm \pi} _u-{\rm \pi} _b=\gamma \varepsilon ^2 (v_u-v_b)\,m$ where the subscript $u$ denotes the unburned state just ahead of the flame. Neglecting terms of order $\varepsilon ^2$, the pressure is treated as uniform inside the preheated zone of the flame structure $z=O(1): 1-{\rm \pi} =O(\varepsilon ^2)$. This is also the case in the thin reaction zone $z=O(1/\beta )$ across which the gradient of the flow velocity varies of order unity so that, according to (2.29), $\partial ^2 v/\partial z^2\approx (\partial {\rm \pi}/\partial z)/\gamma \varepsilon ^2$ $\Rightarrow$ $\partial v/\partial z\vert ^{0+}_{0-}\approx {\rm \pi}\vert ^{0+}_{0-}/\gamma \varepsilon ^2=O(1) \Rightarrow {\rm \pi}\vert ^{0+}_{0-}=O(\varepsilon ^2)$. According to (3.15) in the limit of large activation energy $\beta \gg 1$, the compressional heating (4.1) influences the laminar flame velocity and the flame structure as soon as the compression-induced increase of the flame temperature is of the same order of magnitude as the inverse of the activation energy $(\gamma -1)\,\delta {\rm \pi}=O(1/\beta )$ $\Rightarrow$ $(\gamma -1)\,\varepsilon \delta S_i=O(1/\beta )$. Therefore, as in the previous analysis of Clavin (Reference Clavin2022), the distinguished limit to be considered in the DDT study is similar to that in Deshaies & Joulin (Reference Deshaies and Joulin1989)

(4.9)\begin{equation} \varepsilon \to 0,\quad \beta\to \infty:\quad (\gamma-1)\,\beta \varepsilon\,S_i=O(1). \end{equation}

The comparison with the ZFK expression (2.8) of $\varepsilon$ yields the order of magnitude of $S_i$, typically between $5$ and $10$.

4.3. Equations in the limit $\varepsilon \ll 1$

According to (4.8a,b), the pressure disturbance is small and varies in space on the rescaled coordinate $z_1\equiv \varepsilon z=O(1)$

(4.10a–d)\begin{equation} \partial {\rm \pi}/\partial z=O(\varepsilon^2S_i), \quad {\rm \pi}=1+\varepsilon {\rm \pi}_1(z_1, \tau), \quad {\rm \pi}_1=O(S_i),\quad z_1\equiv \varepsilon z. \end{equation}

Outside the thin reaction sheet, neglecting second-order terms $O(\varepsilon ^2)$, (2.28)–(2.31) take the form

(4.11)\begin{gather} \left.\begin{gathered} {\rm \pi}=1+\varepsilon {\rm \pi}_1:\quad \frac{\partial v}{\partial z}=\left[\frac{\partial }{\partial \tau}-m(\tau)\frac{\partial }{\partial z} \right] \frac{\theta}{1+\varepsilon {\rm \pi}_1},\\ \varepsilon\ll 1: \quad\frac{\partial v}{\partial z}=[1-\varepsilon {\rm \pi}_1]\left[\frac{\partial }{\partial \tau}- m(\tau)\frac{\partial }{\partial z} \right]\theta -\varepsilon \theta\left[\frac{\partial {\rm \pi}_1}{\partial\tau} -m(\tau) \frac{\partial {\rm \pi}_1}{\partial z}\right]+O(\varepsilon^2), \end{gathered}\right\}\end{gather}

(4.12)$$\begin{gather} \left[\frac{\partial v}{\partial \tau}-m (\tau) \frac{\partial v}{\partial z} -\frac{\partial^2 v}{\partial z^2}\right]={-}\frac{1}{\gamma }\frac{1}{\varepsilon}\frac{\partial {\rm \pi}_1}{\partial z} \end{gather}$$

(4.13)$$\begin{gather}\left[\frac{\partial Y}{\partial \tau}-m ( \tau)\frac{\partial Y}{\partial z} -\frac{\partial^2 Y}{\partial z^2}\right ]=0,\quad Y(z, \tau)\in [0, 1] \end{gather}$$

(4.14)$$\begin{gather}\left[\frac{\partial \theta}{\partial \tau}-m ( \tau)\frac{\partial \theta}{\partial z} -\frac{\partial^2 \theta}{\partial z^2}\right ]= \varepsilon \frac{(\gamma-1)}{\gamma}{\theta}\left[\frac{\partial {\rm \pi}_1}{\partial\tau} -m(\tau) \frac{\partial {\rm \pi}_1}{\partial z}\right]+O(\varepsilon^2). \end{gather}$$

Equation (4.14) shows how the effect of compressional heating in the unburned gas outside the flame thickness ($z\gg 1$: $[1/\theta ]{\partial \theta }/{\partial \tau }=\varepsilon [(\gamma -1)/{\gamma }]{\partial {\rm \pi}_1}/{\partial \tau }$) is transmitted to the reaction sheet ($z=0$) by the entropy wave, as it is modified by the heat conduction inside the preheated zone (second derivative on the left-hand side). According to (4.10a–d), the terms involving $\partial {\rm \pi}_1/\partial z=O(\varepsilon )$ in (4.11) and (4.14) are negligible in the inner structure of the flame (of order $\varepsilon ^2$)

(4.15)$$\begin{gather} \frac{\partial v}{\partial z}=[1-\varepsilon {\rm \pi}_1]\left[\frac{\partial }{\partial \tau}-m(\tau)\frac{\partial }{\partial z} \right]\theta -\varepsilon \theta\frac{\partial {\rm \pi}_1}{\partial\tau} +O(\varepsilon^2), \end{gather}$$

(4.16)$$\begin{gather}\left[\frac{\partial \theta}{\partial \tau}-m ( \tau)\frac{\partial \theta}{\partial z} -\frac{\partial^2 \theta}{\partial z^2}\right ]=\varepsilon \frac{(\gamma-1)}{\gamma}{\theta}\frac{\partial {\rm \pi}_1}{\partial\tau}+O(\varepsilon^2). \end{gather}$$

Introducing (4.16) into (4.15), the flow gradient inside the flame structure is expressed in terms of the heat flux and the time derivative of the pressure

(4.17)\begin{equation} \frac{\partial v}{\partial z}= [1-\varepsilon {\rm \pi}_1]\frac{\partial^2 \theta}{\partial z^2}-\varepsilon \frac{1}{\gamma}{\theta}\frac{\partial {\rm \pi}_1}{\partial\tau}+O(\varepsilon^2). \end{equation}

To summarize, in the distinguished limit (4.9), the problem consists in solving (4.12), (4.13), (4.16) and (4.17) with the jump conditions (3.10)–(3.16) on the reaction sheet and the boundary conditions (3.19) at infinity.

Here the problem is solved analytically in the frame attached to the reaction sheet by an asymptotic method. The corresponding one-dimensional numerical analysis to be performed later for the purpose of comparison is not straightforward. Solving the basic equations for a reaction rate given $W(Y,T)$ as an initial value problem would require to apply a boundary condition at the exit of the moving reaction zone, which is not so usual, see the text below (7.16).

5.1. Back to the external flow ahead of the flame

Denoting the external flow ahead of the flame by the subscript $ext_+$ the initial condition takes the form

(5.1a–d)\begin{equation} \tau=0:\quad \theta_{ext_+}=1-q, \quad v_{ext_+}= S_i+q, \quad {\rm \pi}=1, \quad ({\rm \pi}_1=0). \end{equation}

Using the rescaled coordinate $z_1=\varepsilon z$ in (4.10a–d), (4.1) and (4.8a,b)

(5.2)\begin{equation} \theta_{ext_+}(z_1,\tau)= (1-q)\left[ 1+\varepsilon\frac{\gamma-1}{\gamma}{\rm \pi}_1(z_1,\tau)\right]+O(\varepsilon^2), \end{equation}

equation (4.17) takes the form

(5.3)\begin{equation} \left.\begin{gathered} \frac{\partial v_{ext_+}}{\partial z}=[1-\varepsilon {\rm \pi}_1]\left[\frac{\partial }{\partial \tau}-m(\tau)\frac{\partial }{\partial z} \right]\theta_{ext_+} -\varepsilon \theta_{ext_+}\frac{\partial {\rm \pi}_1}{\partial\tau} +O(\varepsilon^2)\\ \text{yielding}\quad \frac{\partial v_{ext_+}(z_1,\tau)}{\partial z_1}={-} (1-q)\frac{1}{\gamma}\frac{\partial {\rm \pi}_1(z_1,\tau)}{\partial \tau}+O(\varepsilon) \end{gathered}\right\}. \end{equation}

Combined with the leading order of (4.12) in the external flow

(5.4)\begin{equation} \frac{\partial v_{ext_+}(z,\tau)}{\partial \tau}={-}\frac{1}{\gamma }\frac{\partial {\rm \pi}_1(z_1,\tau)}{\partial z_1} +O(\varepsilon), \end{equation}

the derivative of (5.3) with respect to $\tau$, after elimination of $v_{ext_+}$, leads to the wave equation (4.6a,b) for the pressure

(5.5)\begin{equation} \varepsilon \ll 1:\quad \frac{\partial^2 {\rm \pi}_1(z_1,\tau)}{\partial \tau^2}=\frac{1}{1-q}\frac{\partial^2 {\rm \pi}_1(z_1,\tau)}{\partial z_1^2}+O(\varepsilon) ,\end{equation}

where, in the mass-weighted coordinates, the non-dimensional sound speed in the external zone is $1/\sqrt {1-q}$. This is easily confirmed as follows:

(5.6)\begin{equation} z_1=\varepsilon z=[U_{ref}/a_{ref}][\rho/\rho_{ref}]x /[U_{ref}t_{ref}], \end{equation}

using $\tau =t/t_{ref}$, the ratio $z_1/\tau$ takes the form $z_1/\tau =[x/a t] \rho a/[\rho _{ref}a_{ref}]$ to give, using $\rho a/[\rho _{ref}a_{ref}]=[p/p_{ref}]\sqrt {T_{ref}/T}$, $(z_1/\tau )=[(x/a t)/\sqrt {\theta }][1+O(\varepsilon )]$ with, according to (5.2), $\theta =1-q+O(\varepsilon )$.

The flow of unburned gas being uniform and steady far ahead from the flame, the external flow is a downstream-running compression wave (propagating in the same direction as the flame) with a leading edge in the form of a weak singularity propagating with the sound speed relative to the flow. Then, according to (5.5), ${\rm \pi} _1(z_1, \tau )$ and $v_{ext_+}(z_1,\tau )=S_i+q+\delta v_{ext_+}(z_1,\tau )$ are functions of a single variable $\tau -\sqrt {1-q}\,z_1$

(5.7)\begin{equation} \left.\begin{gathered} z_1\leqslant \tau/\sqrt{1-q}: \quad {\rm \pi}_1={\rm \pi}_u(\tau-\sqrt{1-q}\,z_1), \quad \delta v_{ext_+}=\varPhi(\tau-\sqrt{1-q}\,z_1),\\ \text{with} \quad {\rm \pi}_u(\tau)\equiv {\rm \pi}_1\vert_{z_1=0},\quad {\rm \pi}_u(0)=0,\quad\varPhi(\tau)\equiv \delta v_{ext_+}\vert_{z_1=0},\quad \varPhi(0)=0 \end{gathered}\right\},\end{equation}

so that the quasi-uniform pressure in the flame structure is $1+\varepsilon {\rm \pi}_u(\tau )$. Using (5.7) in the form $\partial v_{ext_+}/\partial z_1=-\sqrt {1-q}\,\partial v_{ext_+}/\partial \tau$, (5.4) yields

(5.8)\begin{equation} \frac{\partial v_{ext_+}}{\partial z_1}=\frac{\sqrt{1-q}}{\gamma }\frac{\partial {\rm \pi}_1}{\partial z_1} +O(\varepsilon), \end{equation}

in agreement with the linear relation $\delta p\approx \bar {\rho }\,\bar {a}\,\delta u$ of an acoustic wave propagating from left to right. The flow field $v_{ext_+}(z_1,\tau )$ is obtained by integrating (5.8) from the leading edge of the compression wave where, to leading order, the boundary conditions (3.19) $v_{ext_+}= S_i+q$ and ${\rm \pi} _1=0$ hold

(5.9a,b)\begin{equation} v_{ext_+}(z_1,\tau)-(S_i+q)=\frac{\sqrt{1-q}}{\gamma}{\rm \pi}_1(z_1, \tau) + O(\varepsilon),\quad \delta v_{ext_+}\approx \frac{\sqrt{1-q}}{\gamma }{\rm \pi}_1(z_1, \tau). \end{equation}

According to (5.7) and (5.9a,b), the relation linking the flow to the pressure just ahead of the flame ($z_1=0$) takes the form

(5.10)$$\begin{gather} v_{ext_+}(z_1=0,\tau)=S_i+q+\frac{\sqrt{1-q}}{\gamma }{\rm \pi}_u(\tau)+O(\varepsilon) \end{gather}$$

(5.11)$$\begin{gather}\left.\frac{\partial {\rm \pi}_1}{\partial z_1}\right\vert_{z_1=0}={-}\sqrt{1-q}\frac{\text{d} {\rm \pi}_u(\tau)}{\text{d} \tau} \Rightarrow\left.\frac{\partial v_{ext_+}}{\partial z_1}\right\vert_{z_1=0}={-}\frac{(1-q)}{\gamma }\frac{\text{d}{\rm \pi}_u(\tau)}{\text{d} \tau} +O(\varepsilon), \end{gather}$$

in agreement with (5.3). These relations are useful for matching the flow of the inner structure with the external flow of unburned gas in § 5.3.

5.2. Matching the temperature

From now on, the superscript (i) denotes the solution in the preheated zone of the flame structure ($z>0$). Inside the flame structure, the spatial variation of pressure introduces a negligible term of order $\varepsilon ^2$ so that $\partial {\rm \pi}_1/\partial \tau$ in (4.16) and (4.17) can be replaced by the function $\textrm {d} {\rm \pi}_u(\tau )/\textrm {d}\tau \equiv \partial {\rm \pi}_1/\partial \tau \vert _{z_1=0}$ describing the coupling of the flame structure with the external solution on the cold gas. Therefore, (4.16) can be written as

(5.12)\begin{equation} z\geqslant0:\quad \left[\frac{\partial \theta^{(i)}}{\partial \tau}-m (\tau)\frac{\partial \theta^{(i)}}{\partial z} -\frac{\partial^2 \theta^{(i)}}{\partial z^2}\right ]=\varepsilon \frac{(\gamma-1)}{\gamma}\theta^{(i)}(z,\tau)\frac{\text{d} {\rm \pi}_u(\tau)}{\text{d}\tau}+O(\varepsilon^2) ,\end{equation}

where ${\rm \pi} _u(\tau )\equiv {\rm \pi}_1\vert _{z_1=0}$. The boundary condition at infinity on the cold gas side of the preheated zone ($z=O(1)$, $z\to \infty$) is obtained by matching the preheated zone $\theta ^{(i)}(z,\tau )$ and the external flow $\theta _{ext_+}(z_1,\tau )$

(5.13a,b)\begin{align} \lim_{z\to \infty} \theta^{(i)}(z,\tau) =\theta_{ext_+}(z_1,\tau)\vert_{z_1=0}, \quad \lim_{z\to \infty} \partial \theta^{(i)}/\partial z=\varepsilon \partial \theta_{ext_+}/\partial z_1\vert_{z_1=0} =O(\varepsilon^2), \end{align}

where, according to (5.2) $\theta _{ext_+}=(1-q)+\varepsilon (1-q)[{(\gamma -1)}/{\gamma }]{\rm \pi} _1(z_1,\tau )+O( \varepsilon ^2)$, $\partial \theta _{ext_+}/\partial z_1=O(\varepsilon )$ so that $\lim _{z\to \infty } \partial \theta ^{(i)}/\partial z=O(\varepsilon ^2)$ is negligible to first order in a perturbation analysis for small $\varepsilon$

(5.14a,b)\begin{equation} \lim_{z\to \infty} \theta^{(i)}(z,\tau)- (1-q)\approx\varepsilon(1-q)\frac{\gamma-1}{\gamma}{\rm \pi}_u(\tau), \quad\lim_{z\to \infty} \partial \theta^{(i)}/\partial z \approx 0.\end{equation}

Equations (4.13) and (5.12) have to be solved using (5.14a,b) and the boundary conditions (3.10)–(3.12) on the reaction sheet ($z=0$) in the distinguished limit (4.9). Equation (3.11) involves the flame temperature $\theta _b(\tau )-1\equiv \theta ^{(i)}(z,\tau )\vert _{z=0}-1=O(1/\beta )$ which is a time-dependent eigenvalue of the problem, obtained by the jump condition (3.12). In the fully unsteady problem, the solution in the burned-gas side of the reaction sheet $z<0$ is required in (3.12). We will come back to this question later.

5.3. Matching the flow velocity

Matching the flow velocity in the preheated zone with the external flow of cold gas yields

(5.15)$$\begin{gather} \lim_{z\to \infty}v^{(i)}(z, \tau)=v_{ext_+}(z_1,\tau)\vert_{z_1=0}=S_i+q+\frac{\sqrt{1-q}}{\gamma }{\rm \pi}_u(\tau)+O(\varepsilon) , \end{gather}$$

(5.16)$$\begin{gather}\lim_{z\to \infty}\frac{ \partial v^{(i)}}{\partial z}=\left.\varepsilon \frac{\partial v_{ext_+}}{\partial z_1}\right\vert_{z_1=0}={-}\varepsilon \frac{(1-q)}{\gamma }\frac{\text{d} {\rm \pi}_u(\tau)}{\text{d}\tau} +O(\varepsilon^2) \end{gather}$$

where (5.10) and (5.11) have been used. Equations (5.15) and (5.16) yield

(5.17)\begin{equation} \lim_{z\to \infty}v^{(i)}(z,\tau)\to S_i+q+\frac{\sqrt{1-q}}{\gamma }{\rm \pi}_u(\tau)-\varepsilon z\frac{(1-q)}{\gamma }\frac{\text{d}{\rm \pi}_u(\tau)}{\text{d} \tau} +O(\varepsilon^2). \end{equation}

Integration of (4.17), written in the preheated zone in the form

(5.18)\begin{align} \frac{\partial v^{(i)}}{\partial z}&= \left[1-\varepsilon {\rm \pi}_u(\tau) \right]\frac{\partial^2 \theta^{(i)}}{\partial z^2}-\varepsilon \frac{1}{\gamma}\left[\theta^{(i)}-(1-q) \right]\frac{\text{d}{\rm \pi}_u(\tau)}{\text{d}\tau}\nonumber\\ &\quad - \varepsilon (1-q)\frac{1}{\gamma}\frac{\text{d} {\rm \pi}_u(\tau)}{\text{d}\tau}+O(\varepsilon^2), \end{align}

from the reaction sheet $z=0: v^{(i)}=v_b(\tau )$ yields

(5.19)\begin{equation} \left.\begin{gathered} z=O(1):\quad v^{(i)}(z,\tau)-v_b(\tau) =\left [1-\varepsilon {\rm \pi}_u(\tau) \right]\left(\frac{\partial \theta^{(i)}}{\partial z}-\left.\frac{\partial \theta^{(i)}}{\partial z}\right\vert_{z=0^+}\right) \\ -\varepsilon \frac{1}{\gamma}\frac{\text{d} {\rm \pi}_u(\tau)}{\text{d} \tau}\int^{z}_0\left[{\theta}^{(i)}-(1-q)\right]\text{d} z-\varepsilon z (1-q)\frac{1}{\gamma}\frac{\text{d}{\rm \pi}_u(\tau)}{\text{d} \tau}+O(\varepsilon^2). \end{gathered}\right\}\end{equation}

Thanks to (5.14a,b) $\lim _{z\to \infty } \theta ^{(i)}(z,\tau )= (1-q)+O(\varepsilon )$, the leading order of the integral on the right-hand side of (5.19) is well defined in the limit $z\to \infty$ and is of order unity. Then, using (5.17), the limit $z\to \infty$ of (5.19) yields

(5.20)\begin{align} &v_b(\tau)-\left[ S_i+q+\frac{\sqrt{1-q}}{\gamma }{\rm \pi}_u(\tau)\right ] \nonumber\\ &\quad =\left [1-\varepsilon {\rm \pi}_u(\tau) \right ]\left.\frac{\partial \theta^{(i)}}{\partial z}\right\vert_{z=0^+}+\varepsilon \frac{1}{\gamma}\frac{\text{d}{\rm \pi}_u(\tau)}{\text{d} \tau}\int^{+\infty}_0[{\theta^{(i)}}-(1-q)]\,\text{d}z+O(\varepsilon^2), \end{align}

where the thermal flux out of the reaction sheet ${\partial \theta ^{(i)}}/{\partial z}\vert _{z=0^+}$ is obtained in terms of the flame temperature $\theta _b$ by the jump condition (3.11). The integral term on the right-hand side of (5.20) is meaningful as soon as ${\textrm {d}{\rm \pi} _u(\tau )}/{\textrm {d} \tau }<\varepsilon$.

5.4. Master equation

Using the jump relation (3.11), (5.20) yields

(5.21)\begin{align} &v_b(\tau)-\left[ S_i+q+\frac{\sqrt{1-q}}{\gamma }{\rm \pi}_u(\tau) \right]\nonumber\\ &\quad={-} q \left [1-\varepsilon {\rm \pi}_u(\tau)\right ] \exp\left[\frac{\beta}{2}\left(\theta_b-1\right )\right]+\varepsilon \frac{1}{\gamma}\frac{\text{d}{\rm \pi}_u(\tau)}{\text{d}\tau}\int^{+\infty}_0[{\theta^{(i)}}-(1-q)]\,\text{d}z+O(\varepsilon^2). \end{align}

Anticipating that $\beta (\theta _b-1)$ is of order unity in the distinguished limit (4.9) and neglecting $\varepsilon$ terms, (5.21) gives a general equation of order unity, called the master equation

(5.22)\begin{equation} v_b(\tau)-\left[ S_i+q+\frac{\sqrt{1-q}}{\gamma }{\rm \pi}_u(\tau) \right] ={-} q \exp\left(\frac{\beta}{2}\left[\theta_b(\tau)-1\right ]\right)+O(\varepsilon). \end{equation}

This equation is valid for the ZFK model of flames even if the inner structure is unsteady. Under the quasi-steady approximation, (5.22) can be obtained more directly by the conservation of mass across the flame $\bar {U}_b/\bar {U}_L=\bar {T}_b/T_u$ combined with the relation between the laminar flame velocities $\bar {U}_b$ and $\bar {U}_L$ and the flows in the unburned mixture ahead of the flame $u_u$ and in the burned gas $\bar {u}_b$, $u_u-\bar {u}_b=\bar {U}_b-\bar {U}_L=[(\bar {T}_b-T_u)/\bar {T}_b]\bar {U}_b$. The latter expression takes the form $u_u-\bar {u}_b= q\,\bar {U}_b[1+O(1/\beta )]$ in the limit of large activation energy, the relative variation of the flame temperature $\bar {T}_b(\tau )/\bar {T}_b(0)-1$ being of order $1/\beta$. Using (5.15) $u_u(\tau )=S_i+q+[\sqrt {1-q}/{\gamma }]{\rm \pi} _u(\tau )$ and (2.9) $\bar {U}_b\approx U_b(0)\exp ({\beta (\bar {\theta }_b-1)/2})$, (5.22) is recovered. Although the laminar flame velocity $\bar {U}_b$ (2.9) is no longer valid for an unsteady flame structure, (5.22) is still valid when the unsteady flame temperature $\theta _b(\tau )$ computed from the unsteady flame structure is used on the right-hand side.

6.1. Quasi-steady inner structure

If the inner structure of the flame is in steady state (denoted by an overbar), the terms $\partial \theta ^{(i)}/\partial \tau$ and $\varepsilon \,{\textrm {d}{\rm \pi} _u(\tau )}/{\textrm {d}\tau }$ are neglected in (5.12) leading to

(6.1)\begin{equation} \left.\begin{gathered} z\geqslant0:\ \bar{Y} =\text{e}^{-\bar{m}z},\quad \bar{\theta}^{(i)}(z,\tau) = \left[\bar{\theta}_b- \theta_{u}\right ]\text{e}^{-\bar{m}\,z}+\theta_u,\\ z\leqslant 0:\ \bar{Y} =1,\quad \bar{\theta}^{(i)}=\bar{\theta}_b(\tau) \end{gathered}\right\},\end{equation}

where the short notation

(6.2)\begin{equation} \theta_u(\tau)\equiv(1-q)\left[ 1+\varepsilon\frac{\gamma-1}{\gamma}{\rm \pi}_u(\tau)\right], \end{equation}

has been introduced for the gas temperature just ahead of the flame as it is modified by the downstream-running acoustic wave in the unburned gas, see (5.14a,b). Introducing the parameter $b$ of order unity in the distinguished limit (4.9)

(6.3)\begin{equation} b\equiv \frac{\beta \varepsilon}{2}(1-q)\frac{(\gamma-1)}{\gamma}, \quad \varepsilon \to 0,\quad \beta\to \infty:\quad bS_i=O(1), \end{equation}

the jump conditions acrosss the reaction sheet (3.11) and (3.12) yield

(6.4a,b)$$\begin{gather} \bar{\theta}_b(\tau)=\theta_u+q, \quad \bar{\theta}_b-1= \varepsilon(1-q) \frac{\gamma-1}{\gamma}{\rm \pi}_u+O(\varepsilon^2), \end{gather}$$

(6.5a,b)$$\begin{gather}\beta(\bar{\theta}_b-1)/2={\beta}\left [\theta_u+q-1\right ]/2=b{\rm \pi}_u,\quad \bar{m}=\text{e}^{b{\rm \pi}_u}+O(1/\beta), \end{gather}$$

(6.6)$$\begin{gather}\bar{\theta}^{(i)}(z,\tau)=q\text{e}^{-\bar{m}\,z}+(1-q)\left[ 1+\varepsilon\frac{\gamma-1}{\gamma}{\rm \pi}_u(\tau)\right] . \end{gather}$$

6.2. Backflow

In the quasi-steady approximation, the backflow reduces to the instantaneous model (3.16)

(6.7)\begin{equation} v_b(\tau)=S\,\bar{m}=S(\tau)\exp({b{\rm \pi}_u(\tau)}). \end{equation}

Introducing (6.7) into the master equation (5.22) yields the same transcendental equation for the flame pressure ${\rm \pi} _u={\rm \pi} _1\vert _{z_1=0}=[p/p_{ref}\vert _{z_1=0}-1]/\varepsilon =O(1)$ (or the flame temperature) as obtained when the flame is considered as a discontinuity, see Clavin (Reference Clavin2022)

(6.8)\begin{equation} (S+q)\text{e}^{b{\rm \pi}_u}= S_i+q+(\sqrt{1-q}/{\gamma })\,{\rm \pi}_u. \end{equation}

6.2.1. Turning point

Introducing the notation

(6.9a–c)$$\begin{gather} \vartheta\equiv b{\rm \pi}_u=O(1), \quad \zeta\equiv S+q, \quad \zeta(\tau)=(1+\epsilon \tau)S_i+q, \end{gather}$$

(6.10)$$\begin{gather}\tilde{b}\equiv b\gamma/\sqrt{1-q}=(\beta \varepsilon/2)(\gamma-1)\sqrt{1-q}=O(1/S_i) \end{gather}$$

(6.8) takes the reduced form

(6.11a–c)\begin{equation} \zeta \text{e}^\vartheta-\zeta_i-\vartheta /\tilde{b}=0;\quad \tau=0: \quad\zeta=\zeta_i\equiv S_i+q, \quad \vartheta=0. \end{equation}

The solution yields the pressure in terms of the elongation $\vartheta (\zeta )$. The solution depends on the initial elongation $\zeta _i$ and involves a single parameter $\tilde b$. Using the elongation vs the time in (3.1) $S( \tau )=[1+\epsilon \tau ]S_i$, the solution $\vartheta (\zeta )$ provides us with the dynamics of the pressure and/or the flame temperature. The graph of the inverse function $\zeta (\vartheta )$ is a bell-shaped curve sketched in figure 2, the maximum of which corresponds to $\zeta =\zeta ^*$ and $\vartheta =\vartheta ^*$

(6.12a–c)\begin{equation} \left.\frac{\text{d}\zeta}{\text{d}\vartheta}\right\vert_{\vartheta=\vartheta^*}=0: \quad \zeta^* \text{e}^{\vartheta^*}=\frac{1}{\tilde{b}}, \quad \vartheta^*=1-\zeta_i\,\tilde{b}>0, \quad \frac{\zeta^*}{\zeta_i}=\frac{\text{e}^{[\tilde{b}\zeta_i-1]}}{\tilde{b}\zeta_i}\geqslant1, \end{equation}

the inequality ${\zeta ^*}/{\zeta _i}\geqslant 1$ being valid for all the reactive gaseous mixtures ($0<\tilde {b}\zeta _i\leqslant 1$), see Clavin (Reference Clavin2022). The dynamics of the flame is represented by the C-shaped curve $\vartheta (\zeta )$ with a turning point at the critical elongation $S^*$, $\zeta ^*=S^*+q$, $\textrm {d} \vartheta /\textrm {d}\zeta \vert _{\zeta =\zeta ^*}=\infty$. There is no more solution to (6.11a–c) for $\zeta >\zeta ^*$, For $\zeta <\zeta ^*$, there are two branches of solutions $\bar {\vartheta }_\pm =\vartheta ^*\pm \sqrt {2(\zeta ^*-\zeta )/\zeta ^*}$, $\bar {\vartheta }_--\vartheta ^*<0<\bar {\vartheta }_+- \vartheta ^*$, ${\textrm {d}\bar {\vartheta }_-}/{\textrm {d}\zeta }>0$ and ${\textrm {d}\bar {\vartheta }_+}/{\textrm {d}\zeta }<0$ for the other, see figure 2. According to the thermodynamics law, the temperature increases during an adiabatic compression so that the physical branch of solutions is $\bar {\vartheta }_-(\zeta )$.

Figure 2.

Sketch of the solutions ‘elongation $\zeta$ vs pressure $\vartheta$’ (not to scale). The two branches $\bar {\vartheta }_\pm (\zeta )$ of solutions of the quasi-steady equation (6.11a–c) in thick line show the critical elongation $\zeta ^*$, above which no quasi-steady solutions ($\bar {\vartheta }_-$ is the physical solution) exist. The horizontal arrows in broken line indicate the direction of the trajectories of (6.26) for $\tilde {\epsilon }_w>0$ showing the stability of the branch of physical solutions of (6.11a–c), see Strogatz (Reference Strogatz1994). They are in the opposite direction for $\tilde \epsilon _w<0$. The solution of the dynamical equation (6.26) $\vartheta (\zeta )$ in thin red line shows the finite-time divergence of the pressure at the elongation $\zeta _c>\zeta ^*$, $\lim _{(\zeta -\zeta _c)\to 0^-}\vartheta =\infty$. The red arrows indicate the direction of increasing time for an elongation increasing with the time. According to (6.30a,b), the relative difference in critical elongations is small for a small elongation rate (3.1) $\epsilon \ll 1$: $(\zeta _c-\zeta ^*)/\zeta ^*=O( (\epsilon S_i /\zeta ^*)^{2/3} )$.

As noticed in Clavin (Reference Clavin2022), the limiting case $\zeta _i=\zeta ^*$, $S=S_{max}^*$ corresponds to a universal critical Mach number $u^*_u/a^*_u={2}/[{\beta (\gamma -1)}]$ characterizing the pre-conditioned flow of unburned gas just before the DDT onset. This critical Mach number of the cold flow is typically $u^*_u/a^*_u\approx$ 0.65 for ordinary flames ($\beta \approx 8$) and becomes slightly supersonic $u^*_u/a^*_u>1$ for a very energetic mixture ($\beta \lesssim 4$) while the laminar flame velocity remains very subsonic $U^*_b/a^*_b\approx 0.05$, in agreement with the experiments of Kuznetsov et al. (Reference Kuznetsov, Liberman and Matsukov2010) and the numerics of Liberman et al. (Reference Liberman, Ivanov, Kiverin, Kuznetsov, Chukalovsky and Rakimova2010) and Ivanov et al. (Reference Ivanov, Kiverin and Liberman2011).

6.2.2. Finite-time singularity of the flow gradient

According to (6.12a–c) $\textrm {d} \zeta /\textrm {d} \tau =\epsilon S_i$, the elongation and the flame velocity $\bar {m}=\textrm {e}^{\vartheta }$ ($\vartheta \equiv b{\rm \pi} _u$) increase first slowly with the time $\textrm {d}\bar {m}/\textrm {d} \tau =O(\epsilon S_i)$, and, according to (6.12a–c), the flame acceleration diverges abruptly ${\textrm {d}\vartheta }/{\textrm {d} \zeta }\vert _{\zeta =\zeta ^*}= \infty$, $\textrm {d} \bar {m}/\textrm {d} \tau \vert _{\tau =\tau ^*}=\epsilon S_i\bar {m}^* {\textrm {d} \vartheta }/{\textrm {d} \zeta }\vert _{\zeta =\zeta ^*}=\infty$ when the elongation reaches $S^*$, namely when the flame velocity reaches the critical value $\bar {m}^*=\textrm {e}^{\vartheta ^*}$ which is finite ($\vartheta ^*< 1$). As in the piston problem considered in Clavin & Tofaili (Reference Clavin and Tofaili2021), (6.11a–c) takes a generic form near the critical point $\textrm {d} \zeta /\textrm {d} \vartheta \vert _{\vartheta =\vartheta ^*}=0$,

(6.13)$$\begin{gather} \left.\frac{\text{d}^2 \zeta}{\text{d}\vartheta^2}\right\vert_{\vartheta=\vartheta^*}={-}\zeta^*\quad \Rightarrow \quad \frac{\zeta^*-\zeta}{\zeta^*}\ll 1: \frac{\zeta^*-\zeta}{\zeta^*}\approx \frac{1}{2}(\vartheta^*-\vartheta)^2 \end{gather}$$

(6.14a,b)$$\begin{gather}\vartheta^*-\vartheta\approx \sqrt 2\sqrt{\frac{\zeta^*-\zeta}{\zeta^*}},\quad b({\rm \pi}_u^*-{\rm \pi}_u)\approx \sqrt{\frac{(S^*-S)}{(S^*+q)/2}}, \end{gather}$$

obtained by a Taylor expansion. The dynamics of flame pressure and flame temperature near the critical condition at $\tau =\tau ^*$ takes the form

(6.15)\begin{equation} \vartheta^*-\vartheta(\tau)\approx\kappa\sqrt{\tau^*-\tau} \quad\text{where}\ \kappa\equiv \sqrt{2\epsilon\frac{S_i}{S^*+q}}=O(\sqrt{\epsilon}), \end{equation}

exhibiting the finite-time singularity of the flame acceleration

(6.16a–c)$$\begin{gather} \tau/ \tau^*-1\to 0_-: \ \vartheta \to \vartheta^*=1-\zeta_i\,\tilde{b}>0,\quad {\rm \pi}_u\to {\rm \pi}_u^*=\vartheta^*/b,\quad\bar{m}\to\bar{m}^*=\text{e}^{\vartheta^*} \end{gather}$$

(6.17a–c)$$\begin{gather}\frac{\text{d} \vartheta}{\text{d} \tau}\approx \frac{\kappa/2}{\sqrt{\tau^*-\tau}}, \quad \frac{\text{d} {\rm \pi}_u}{\text{d} \tau}\approx \frac{\kappa/2b}{\sqrt{\tau^*-\tau}}, \quad \frac{1}{\bar{m}}\frac{\text{d} \bar{m}}{\text{d} \tau}\approx \frac{\kappa/2}{\sqrt{\tau^*-\tau}}. \end{gather}$$

According to (5.3)–(5.11) and (6.15)–(6.17a–c)

(6.18)\begin{equation} \left.\begin{gathered} \tau/ \tau^*-1\to 0_-:\quad \frac{\partial v_{ext_+}(z_1,\tau)}{\partial z_1}\approx{-} \frac{(1-q)}{\gamma}\frac{\kappa/2}{b}\frac{1}{\sqrt{\tau^*-\tau+z_1\sqrt{1-q}}},\\ \frac{\partial v_{ext_+}(z_1,\tau)}{\partial \tau}\approx\frac{\sqrt{1-q}}{\gamma}\frac{\kappa/2}{b}\frac{1}{\sqrt{\tau^*-\tau+z_1\sqrt{1-q}}}, \end{gathered}\right\} \end{equation}

the gradient and acceleration of the external unburned flow diverge on the flame

(6.19a,b)\begin{align} \tau\to \tau^*:\quad\left.\frac{\partial v_{ext_+}}{\partial z_1}\right\vert_{z_1=0}&\approx \frac{(1-q)}{\gamma}\frac{\kappa/2}{b}\frac{1}{\sqrt{\tau^*-\tau}},\nonumber\\ \quad \left.\frac{\partial v_{ext_+}}{\partial \tau}\right\vert_{z_1=0}&\approx \frac{\sqrt{1-q}}{\gamma}\frac{\kappa/2}{b}\frac{1}{\sqrt{\tau^*-\tau}}. \end{align}

This suggests a finite-time singularity of the flow gradient leading to the formation of a shock inside the quasi-isobaric flame structure. The DDT mechanism is associated with an even more violent phenomenon: a catastrophic behaviour of the flame structure is predicted below by the delayed backflow model.

6.3. Delayed backflow model. Catastrophic dynamics

The unsteady flow behind the tip of the elongated flame is a too complex problem for an analytical study. Equation (3.17a–c) is a simplified model for studying the main consequence of this unsteadiness, a detailed expression of the delay $\Delta t_w$ is not useful in the following. Only the order of magnitude of $\Delta t_w$ matters for a clear understanding of the phenomenon. Still assuming the inner structure of the flame in steady state $\bar {m}=\textrm {e}^{b{\rm \pi} _u(\tau )}$, $\textrm {d}\bar {m}/\textrm {d} \tau \approx \bar {m} \,b\,\textrm {d} {\rm \pi}_u/\textrm {d}\tau$ the delayed backflow model (3.17a–c) reads

(6.20)$$\begin{gather} v_b =S_i(1+\epsilon \tau)\,\bar{m}\left[ 1-\frac{\Delta t_w}{t_{ref}} b\frac{\text{d} {\rm \pi}_u}{\text{d}\tau}+\dots\right ] \end{gather}$$

(6.21)$$\begin{gather}v_b \approx S_i\text{e}^{b{\rm \pi}_u(\tau)}+\epsilon \tau\,S_i \text{e}^{b{\rm \pi}_u^*}-S_i\,\text{e} ^{b{\rm \pi}_u^*}\frac{\Delta t_w}{t_{ref}}b\frac{\text{d}{\rm \pi}_u}{\text{d} \tau}+\dots, \end{gather}$$

where, considering $\Delta t_w/t_{ref}$ of order unity and ${\textrm {d}{\rm \pi} _u}/{\textrm {d}\tau }$ of order $\epsilon < 1$, the $\textrm {e}^{b{\rm \pi} _u}$ term in the second and third terms on the right-hand side has been considered as constant nearby the turning point for simplicity.

6.3.1. Dynamical equation for the pressure and the flame temperature

Introducing (6.5a,b) and (6.21) into the master equation (5.22) yields an ordinary differential equation (ODE) of first order for ${\rm \pi} _u(\tau )$

(6.22)\begin{equation} S_i\text{e}^{b{\rm \pi}_u(\tau)}+\epsilon \tau S_i \text{e}^{b{\rm \pi}^*_u}-S_i\text{e}^{b{\rm \pi}_u^*}\frac{\Delta t_w}{t_{ref}}b\frac{\text{d}{\rm \pi}_u}{\text{d}\tau} -\left[ S_i+q+\frac{\sqrt{1-q}}{\gamma }{\rm \pi}_u(\tau) \right]={-} q \text{e}^{b{\rm \pi}_u(\tau)}, \end{equation}

which can be written in the form

(6.23)$$\begin{gather} \left [S_i(1+\epsilon \tau)+q\right ]\text{e}^{b{\rm \pi}_u(\tau)}-\left[ S_i+q+\frac{\sqrt{1-q}}{\gamma }{\rm \pi}_u(\tau) \right] =K_wb\frac{\text{d}{\rm \pi}_u(\tau)}{\text{d}\tau}. \end{gather}$$

(6.24)$$\begin{gather}\text{where}\quad K_w\equiv S_i\text{e}^{b{\rm \pi}_u^*}\frac{\Delta t_w}{t_{ref}}>0. \end{gather}$$

Written with the notation (6.10) and (6.11a–c), a nonlinear ODE for $\vartheta (\zeta )=b{\rm \pi} _u(\zeta )$ is obtained

(6.25)\begin{equation} \zeta\exp\vartheta-\zeta_i -{\vartheta }/{\tilde{b}}= \tilde K_w\frac{\text{d} \vartheta}{\text{d}\zeta}, \quad \text{where}\ \tilde{K}_w\equiv\epsilon S_i\,K_w \quad \text{and}\quad \tilde{b}\equiv b\gamma/\sqrt{1-q}. \end{equation}

The roots of the left hand-side of the first equation (6.25) are the quasi-steady solutions (6.11a–c) of the instantaneous backflow model $\overline \vartheta _\pm (\zeta )$ shown in figure 2. Focusing our attention on the vicinity of the turning point, following (6.15), a power expansion in $(\vartheta -\vartheta ^*)$ limited to the quadratic terms yields

(6.26)\begin{equation} \frac{(\zeta-\zeta^*) }{ \zeta^*}+\frac{(\vartheta^*-\vartheta)^2 }{2 }= \tilde \epsilon_w \frac{\text{d}\vartheta}{\text{d}\zeta}, \quad \text{where}\ \tilde{\epsilon}_w\equiv \tilde{b}\tilde{K}_w=\epsilon \frac{S_i^2}{\zeta^*}\frac{\Delta t_w}{t_{ref}}. \end{equation}

For $\zeta <\zeta ^*$, the sign of the left-hand side of (6.26) is positive for $\vartheta < \bar {\vartheta }_-$ and for $\vartheta > \bar {\vartheta }_+$ (negative for $\bar {\vartheta }_-<\vartheta <\bar {\vartheta }_+$). The trajectories in the phase space of (6.26) show that the physical branch $\bar {\vartheta }_-(\zeta )$ is stable since $\tilde {\epsilon }_w>0$. The other branch $\bar {\vartheta }_+(\zeta )$ is unstable. A finite-time singularity of the solution of (6.26) occurs around the turning point, as shown now.

6.3.2. Dynamical saddle-node bifurcation

Equation (6.26) describes the dynamics nearby a saddle-node bifurcation. Such an equation was extensively used for sharp transitions in different problems of physics or biophysics. The theory of catastrophic events based on this equation has been recently revisited and extended by Peters, Le Berre & Pomeau (Reference Peters, Le Berre and Pomeau2012). Conveniently rescaled

(6.27a,b)\begin{equation} (1/2^{2/3})(\zeta^*/\tilde{\epsilon}_w)^{1/3}(\vartheta-\vartheta^*) \to y', \quad (1/2^{1/3})(\zeta^*/\tilde{\epsilon}_w)^{2/3}(\zeta-\zeta^*)/\zeta^* \to t', \end{equation}

equation (6.26), after multiplication by $(\zeta ^*/\sqrt 2\tilde {\epsilon }_w)^{2/3}$, takes a generic normal form

(6.28)\begin{equation} \frac{\text{{d}y}'(t')}{\text{d} t' }=t' + y'^{2} ,\end{equation}

with two fixed points for $t'<0$, the stable one corresponding to the negative root $y'=-\sqrt {-t'}$ (physical branch of solutions). The fixed points collapse at $t'=0$ and there is no more fixed point for $t'>0$ (saddle-node bifurcation). Considering an initial condition on the stable branch $t'=t'_i<0: y'=-\sqrt {-t_i'}$ for $-t_i'/t'_c =y'^2_i/t'_c$ larger than unity, the asymptotic solution of (6.28) is obtained in terms of the Airy function to give

(6.29)\begin{equation} \lim_{t\to t'_c}y'(t')=\frac{1}{t'_c-t'}-\frac{t'_c}{3}(t'_c-t')+\dots\quad \text{where}\ t'_c\approx 2.338\ldots; \end{equation}

see the references in Peters et al. (Reference Peters, Le Berre and Pomeau2012). The finite-time singularity (6.29) is of the same type as the solution of the Riccati equation ${{\textrm {d}y}'}/{\textrm {d} t' }=y'^{2}$. According to (6.29), the pressure and the flame temperature $\vartheta \equiv b{\rm \pi} _u(\tau )$ blow up at time $\tau =\tau _c$ for a finite elongation $\zeta _c=S_i(1+\epsilon \tau _c)+q$, $(\zeta _c-\zeta ^*)=(\tau _c-\tau ^*)\epsilon S_i$

(6.30a,b)\begin{align} \frac{\zeta_c-\zeta^*}{( 2\zeta^*\tilde{\epsilon}_w^2)^{1/3}}= 2.338\ldots, \quad b({\rm \pi}_u-{\rm \pi}_u^*)\equiv \vartheta-\vartheta^*\approx \frac{2\tilde \epsilon}{\zeta_c-\zeta}=\frac{ 2b\gamma }{\sqrt{1-q}}\frac{K_w}{\tau_c-\tau}. \end{align}

The solution to (6.26) increases above the critical elongation $\vartheta (\tau )>\vartheta ^*$ and diverges like $K_w/(\tau _c-\tau )$ where $\tau _c-\tau ^*\propto K_w^{2/3}/(\epsilon S_i)^{1/3}$, see figure 2. The smaller the elongation rate $\epsilon < 1$, the closer to the turning point $S^*$ the pressure blowup, $\lim _{\epsilon \to 0}(S_c-S^*)/S_i\propto (\epsilon S_iK_w)^{2/3}$.

To summarize, unsteadiness of the burned-gas flow which is modelled by a delay in the backflow in § 6.3 has a drastic effect on the dynamics, more drastic than the instantaneous backflow in § 6.2: the flame structure is blown off as a whole in finite time when the elongation increases slowly. A strong increase in pressure and flame temperature occurs abruptly with a sudden shrinking of the flame thickness for an elongation slightly larger than the critical elongation of the instantaneous backflow model.

7.1. Unsteady inner structure of the flame

Introducing the decomposition $w=\bar {w}+\delta w$ where $\overline w$ denotes the quasi-steady approximation of the inner structure, a perturbation analysis of (4.13) and (4.16) is performed in the distinguished limit (4.9)

(7.1a–d)\begin{equation} \varepsilon \ll\epsilon \ll 1,\quad \text{d}{\rm \pi}_u/\text{d} \tau=O(\epsilon),\quad \beta \gg 1, \quad (\gamma-1)\beta \varepsilon S_i=O(1), \end{equation}

retaining unsteady terms of order $\varepsilon \,\textrm {d} {\rm \pi}_u/\textrm {d} \tau =O(\varepsilon \epsilon )$ in these equation and neglecting smaller terms, namely those of order $\varepsilon (\textrm {d} {\rm \pi}_u/\textrm {d} \tau )^2$ and $\varepsilon ^2$. First-order terms are sufficient to draw a final conclusion concerning the finite-time singularity.

7.1.1. Preheated zone ($z\geqslant 0$)

Anticipating that $\beta ({\theta _b-\bar {\theta }_b})\propto \textrm {d}{\rm \pi} _u/\textrm {d} \tau$ and $\theta ^{(i)}-\bar {\theta }^{(i)}\propto \bar {\theta }^{(i)}\,\textrm {d}{\rm \pi} _u/\textrm {d} \tau$, the temperature $\theta ^{(i)}$ can be replaced by $\bar {\theta }^{(i)}$ in front of the pressure term on the right-hand side of (5.12)

(7.2)$$\begin{gather} z\geqslant0: \ \left[\frac{\partial \theta^{(i)}}{\partial \tau}-m (\tau)\frac{\partial \theta^{(i)}}{\partial z} -\frac{\partial^2 \theta^{(i)}}{\partial z^2}\right ]\approx \frac{(\gamma-1)}{\gamma}\left [ q\text{e}^{-\bar{m}\,z}+(1-q)\right]\varepsilon\,\frac{\text{d} {\rm \pi}_u}{\text{d}\tau}, \end{gather}$$

(7.3a,b)$$\begin{gather}\lim_{z\to \infty} \theta^{(i)}(z,\tau) =(1-q)\left[1+\varepsilon\frac{\gamma-1}{\gamma}{\rm \pi}_u(\tau)\right], \quad \lim_{z\to \infty} \frac{\partial \theta^{(i)}}{\partial z}=0, \end{gather}$$

where the $\varepsilon ^2$-terms have been neglected in the boundary conditions (5.14a,b). Introducing the decomposition

(7.4a–c)\begin{equation} \theta^{(i)}(z,\tau)=\bar{\theta}^{(i)}+\delta \theta^{(i)},\quad m(\tau)=\bar{m}+\delta m, \quad \bar{m}=\text{e}^{b{\rm \pi}_u}, \end{equation}

with, according to (6.6),

(7.5a,b)\begin{equation} z\geqslant0:\quad\frac{\partial \bar{\theta}^{(i)}}{\partial \tau}={-} qz \text{e}^{-\bar{m}\,z}\frac{\text{d}\bar{m}}{\text{d}\tau}+\varepsilon(1-q) \frac{(\gamma-1)}{\gamma}\frac{\text{d}{\rm \pi}_u}{\text{d}\tau}, \quad \frac{1}{\bar{m}}\frac{\text{d}\bar{m}}{\text{d}\tau}=b\frac{\text{d}{\rm \pi}_u}{\text{d}\tau}, \end{equation}

equation (7.2) reads after subtracting ${\partial \bar {\theta }^{(i)}}/{\partial \tau }$ and $-\delta m{\partial \bar {\theta }^{(i)}}/{\partial z}=q\bar {m}\textrm {e}^{-\bar {m}z}\delta m$

(7.6)\begin{align} z\geqslant0: \left[\frac{\partial \delta \theta^{(i)}}{\partial \tau}-\bar{m}\,\frac{\partial \delta \theta^{(i)}}{\partial z} -\frac{\partial^2\delta \theta^{(i)}}{\partial z^2}\right ]&\approx{-}q\,\bar{m}\,\text{e}^{-\bar{m}\,z}\delta m\nonumber\\ &\quad + q\left[b\,\bar{m}\,z\,\text{e}^{-\bar{m}z}+\varepsilon \frac{(\gamma-1)}{\gamma} \text{e}^{-\bar{m}\,z}\right] \frac{\text{d}{\rm \pi}_u}{\text{d}\tau}, \end{align}

with, according to the boundary conditions (7.3a,b),

(7.7a,b)\begin{equation} \lim_{z\to \infty}\delta \theta^{(i)}=0, \quad \lim_{z\to \infty}\text{d}\delta \theta^{(i)}/\text{d}z=O(\varepsilon^2). \end{equation}

Introducing the decomposition $Y=\bar {Y}+\delta$Y into (4.13) using $\bar {Y}=\textrm {e}^{-\bar {m} z}$, $-\delta m \,\partial \bar {Y}/\partial z=\bar {m}\textrm {e}^{-\bar {m}z}\delta m$ and $\partial \bar {Y}/\partial \tau =-\bar {m}\,z\,\textrm {e}^{-\bar {m}z}b\,\textrm {d}{\rm \pi} _u/\textrm {d} \tau$ yields

(7.8a,b)\begin{equation} \left[\frac{\partial \delta Y}{\partial \tau}-\bar{m}\,\frac{\partial \delta Y}{\partial z} -\frac{\partial^2\delta Y}{\partial z^2}\right ]\approx{-}\bar{m}\text{e}^{-\bar{m}\,z}\delta m+\bar{m}\,z\,\text{e}^{-\bar{m}z}b\,\frac{\text{d}{\rm \pi}_u}{\text{d}\tau}, \quad \lim_{z\to \infty}\delta Y=0. \end{equation}

The equation for $\delta Z\equiv \delta \theta ^{(i)}-q\delta Y$ is free from the term $\delta m(\tau )$

(7.9a,b)\begin{equation} z\geqslant0:\quad\left[\frac{\partial \delta Z}{\partial \tau}-\bar{m}\,\frac{\partial \delta Z}{\partial z} -\frac{\partial^2\delta Z}{\partial z^2}\right ]\approx \varepsilon \,q\frac{(\gamma-1)}{\gamma}\text{e}^{-\bar{m}z} \frac{\text{d}{\rm \pi}_u}{\text{d}\tau}, \quad \lim_{z\to \infty}\delta Z=0. \end{equation}

Anticipating that $\delta Z$ is of order $\varepsilon \, {\textrm {d}{\rm \pi} _u}/{\textrm {d}\tau }$, neglecting terms of order $\varepsilon \, ({\textrm {d}{\rm \pi} _u}/{\textrm {d}\tau })^2$, the reduced laminar flame speed $\bar {m}=\textrm {e}^{b{\rm \pi} _u}+O(1/\beta )$ is treated as constant in (7.9a,b). The unsteady term $\partial \delta Z/\partial \tau <\delta Z$ can be neglected in front of $\partial \delta Z/\partial z$ and $\partial ^2 \delta Z/\partial z^2$ since $\partial /\partial z=O(1)$. After integration $\int _z^\infty \textrm {d}z$, this yields

(7.10)\begin{equation} \left.\begin{gathered} z\geqslant0:\quad \bar{m} \,\delta Z+\frac{\partial \delta Z}{\partial z}= \varepsilon q \frac{(\gamma-1)}{\gamma}\frac{1}{\bar{m}}\text{e}^{-\bar{m}\,z} \frac{\text{d}{\rm \pi}_u}{\text{d}\tau}\\ \left.\Rightarrow \quad \bar{m} \delta \theta_b+\frac{\partial \delta Z}{\partial z}\right\vert_{z=0^+}= \varepsilon q\frac{(\gamma-1)}{\gamma}\frac{1}{\bar{m}} \frac{\text{d}{\rm \pi}_u}{\text{d}\tau}, \end{gathered}\right\}\end{equation}

where the boundary conditions $z=0^+: \ \delta Y=0$, $\delta Z=\delta \theta _b(\tau )\equiv \delta \theta ^{(i)}(z=0,\tau )$ have been used. Equation (7.10) can be checked by the small frequency limit of the Fourier transform of (7.9a,b). Using the relations ${\partial \bar {\theta }^{(i)}}/{\partial z} \vert _{z=0^-}=0$ and $Y\vert _{z=0^-}=0$ on the burned-gas side, the jump condition (3.12) takes the form

(7.11)\begin{equation} \left.\frac{\partial \theta^{(i)}}{\partial z}\right\vert_{z=0^-}=\left.\frac{\partial \theta^{(i)}}{\partial z}\right\vert_{z=0^+}-q\left.\frac{\partial Y}{\partial z}\right\vert_{z=0^+}\quad \Rightarrow\quad \left.\frac{\partial \delta \theta^{(i)}}{\partial z}\right\vert_{z=0^-}=\left.\frac{\partial \delta Z}{\partial z}\right\vert_{z=0^+}. \end{equation}

According to (7.10) and (7.11), the unsteadiness-induced modification of flame temperature $\delta \theta _b$, $\theta _b=\bar {\theta }_b+\delta \theta _b$, is expressed in terms of the temperature gradient on the burned-gas side of the reaction sheet

(7.12)\begin{equation} \bar{m} \,\delta \theta_b+\left.\frac{\partial \delta \theta^{(i)}}{\partial z}\right\vert_{z=0^-}= \varepsilon q\frac{(\gamma-1)}{\gamma}\frac{1}{\bar{m}} \frac{\text{d}{\rm \pi}_u}{\text{d}\tau}. \end{equation}

The right-hand side of (7.12) is part of the perturbation of the flame temperature. The first-order correction to $\beta (\theta _b-1)$ requires the investigation of the temperature in the burned-gas flow ($z<0$) for computing $\partial \delta \theta ^{(i)}/\partial z \vert _{z=0^-}$.

7.1.2. Burned gas $z\leqslant 0$

For the analysis of the burned gas, one has to be back to (4.14). According to (6.1) and (6.5a,b), $\bar {\theta }^{(i)}-1=O(\varepsilon {\rm \pi}_u)$ can be neglected in the factor of the pressure term on the right-hand side of (4.14)

(7.13)$$\begin{gather} z\leqslant 0:\ \left[\frac{\partial \theta^{(i)}}{\partial \tau}-m (\tau)\frac{\partial \theta^{(i)}}{\partial z} -\frac{\partial^2 \theta^{(i)}}{\partial z^2}\right ]=\varepsilon \frac{(\gamma-1)}{\gamma}\left[\frac{\partial {\rm \pi}_1}{\partial\tau}-m(\tau) \frac{\partial {\rm \pi}_1}{\partial z}\right]\left[1+O(\varepsilon) \right ] \end{gather}$$

(7.14a,b)$$\begin{gather}\lim_{z \to - \infty}\theta^{(i)}(z,\tau)= 1, \quad \lim_{z \to - \infty}{\rm \pi}_1(z,\tau)=0, \end{gather}$$

where the boundary conditions (7.14a,b) at infinity on the burned-gas side are given by the initial condition (hyperbolic problem). Neglecting $\varepsilon \partial ^2{\rm \pi} _1/\partial z^2$ in the burned gas, the energy equation (7.13) and (7.14a,b) can be written as an entropy equation

(7.15a–c)\begin{align} z\leqslant0:\left[\frac{\partial }{\partial \tau}-\bar{m} \frac{\partial }{\partial z}-\frac{\partial^2 }{\partial z^2}\right]s\approx 0, \quad s\equiv \left[ \theta^{(i)}-\frac{(\gamma-1)}{\gamma}\varepsilon{\rm \pi}_1\right], \quad \lim_{z\to -\infty}s=1. \end{align}

Anticipating that the unsteadiness-induced disturbances are of order $\varepsilon \,\textrm {d}{\rm \pi} _u/\textrm {d} \tau$, the mass flux $m$ has been replaced by its unperturbed expression $\bar {m}$ on the left-hand side of (7.15a–c) since the attention is limited to the leading order. Equation (7.15a–c) shows how the entropy which is generated across the flame structure during the flame acceleration

(7.16)\begin{equation} s_b(\tau)\equiv s(z=0, \tau)=\bar{\theta}_b(\tau)+\delta \theta_b(\tau)-(\gamma-1)\varepsilon {\rm \pi}_u(\tau)/\gamma ,\end{equation}

escapes the reaction zone from the hot side ($m>0$). This leakage of entropy is the main difference between a flame pushed from behind by a flow of burned gas and an adiabatic (and impermeable) piston. The upstream boundary condition $\lim _{z\to -\infty }s=1$ can also be viewed as resulting from the damping of the transient variation of entropy by the heat conduction in the (inert) burned gas. The solution to (7.15a–c) is easily obtained using the Fourier transform $s-1=\textrm {e}^{\textrm {i}\omega \tau }\tilde s_{\omega }(z)$ and the two relations (6.4a,b) $\bar {\theta }_b-1=\varepsilon (1-q)(\gamma -1){\rm \pi} _u/{\gamma }$ and (7.16), $s_b-1=\delta \theta _b-q(\gamma -1){\rm \pi} _u/{\gamma }$

(7.17)$$\begin{gather} z\leqslant0:\quad\tilde s_{\omega}(z)=\left[\widetilde{\delta \theta_b}-q\frac{\gamma-1}{\gamma}\varepsilon \tilde{\rm \pi}_u\right]\exp{\tilde{k}\,z}, \end{gather}$$

(7.18)$$\begin{gather}\tilde k(\omega) \equiv \frac{1}{2}\left[-\bar{m}+\sqrt{\bar{m}^2+4{\rm i}\omega} \right ]\approx \frac{{\rm i}\omega}{\bar{m}}+\frac{\omega^2}{\bar{m}^3}+\dots, \end{gather}$$

to give on the reaction sheet, using $\partial \tilde {s}_{\omega }/\partial z\vert _{z=0_ -}\to \tilde {k} [\widetilde {\delta \theta _b}-q({\gamma -1}/{\gamma })\varepsilon \tilde {\rm \pi}_u]$ and, in the low frequency limit, $\tilde {k}\to (1/\bar {m})\,\textrm {d}/\textrm {d}\tau$

(7.19)\begin{align} \left.\frac{\partial s}{\partial z}\right\vert_{z=0^-}=\left.\frac{\partial \delta \theta^{(i)}}{\partial z}\right\vert_{z=0^-}-q\frac{(\gamma-1)}{\gamma}\varepsilon\left.\frac{\partial {\rm \pi}_1}{\partial z}\right\vert_{z=0^-}\approx \frac{1}{\bar{m}}\frac{\text{d}\delta \theta_b}{\text{d}\tau}-q\frac{(\gamma-1)}{\gamma}\frac{1}{\bar{m}}\varepsilon \frac{\text{d}{\rm \pi}_u}{\text{d} \tau}.\end{align}

This low frequency result corresponds to the undamped transport by the entropy wave $\partial s/{\partial \tau }-\bar {m} {\partial s}/{\partial z}\approx 0$. The conduction-induced damping rate is of next order in the limit of small frequency. This is similar to freely propagating acoustic waves in planar geometry. In the limit $\varepsilon \ll 1$, the pressure gradient in the burned gas is negligible, the dominant effect being through the increase rate in pressure (time derivative). To avoid any cumulative effect that could induce acoustical instabilities reviewed in Clavin & Searby (Reference Clavin and Searby2016), the ends of the tube have been assumed sufficiently far away from the flame. Therefore, neglecting $\varepsilon {\partial {\rm \pi}_1}/{\partial z} \vert _{z=0^-}$, (7.19) reads

(7.20)\begin{equation} \text{burned gas:}\quad\left.\frac{\partial \delta \theta^{(i)}}{\partial z}\right\vert_{z=0^-}\approx \frac{1}{\bar{m}}\frac{\text{d}\delta \theta_b}{\text{d}\tau}-q\frac{(\gamma-1)}{\gamma}\frac{1}{\bar{m}}\varepsilon \frac{\text{d}{\rm \pi}_u}{\text{d}\tau}. \end{equation}

7.1.3. Unsteady modification to the flame temperature

Introducing (7.20) into (7.12) leads to

(7.21)\begin{equation} \bar{m} \,\delta \theta_b+\frac{1}{\bar{m}}\frac{\text{d}\delta \theta_b}{\text{d} \tau}\approx2 q\frac{(\gamma-1)}{\gamma}\frac{1}{\bar{m}} \varepsilon \frac{\text{d}{\rm \pi}_u}{\text{d}\tau}. \end{equation}

To first order, the time derivative on the left-hand side of (7.21) is negligible

(7.22)\begin{equation} \delta \theta_b\approx 2 q\frac{(\gamma-1)}{\gamma}\frac{1}{\bar{m}^2} \varepsilon \frac{\text{d}{\rm \pi}_u}{\text{d}\tau}=2 q\frac{(\gamma-1)}{\gamma} \exp[{-2b{\rm \pi}_u}]\varepsilon \frac{\text{d}{\rm \pi}_u}{\text{d}\tau}. \end{equation}

This can be checked in the low frequency limit of the Fourier transform of (7.21)

(7.23)\begin{align} \widetilde{\delta \theta_b}\approx 2 q\frac{(\gamma-1)}{\gamma}\frac{1}{\bar{m}^2}\, \frac{{\rm i}\omega}{1+{\rm i}\omega/\bar{m}^2} \varepsilon\tilde{\rm \pi}_u\approx 2 q\frac{(\gamma-1)}{\gamma}\frac{1}{\bar{m}^2} \varepsilon {\rm i}\omega \,\tilde{\rm \pi}_u\left[ 1-\frac{1}{\bar{m}^2}{\rm i}\omega+O(\omega^2) \right].\end{align}

Thanks to the minus sign in front of $\textrm {i}\omega$ in the bracket, (7.23) describes a causal response of the flame temperature to the time derivative of pressure. However, the causality link between the flame temperature $\theta _b(\tau )=\bar {\theta }_b(\tau )+\delta \theta _b(\tau )$ and the pressure ${\rm \pi} _u(\tau )$ is the reverse

(7.24)$$\begin{gather} \theta_b(\tau)=1+\varepsilon(1-q)\frac{\gamma-1}{\gamma}{\rm \pi}_u(\tau)+2\varepsilon ,q\frac{(\gamma-1)}{\gamma} \exp[{-2b{\rm \pi}_u}] \frac{\text{d}{\rm \pi}_u}{\text{d}\tau}+O(\varepsilon^2 ), \end{gather}$$

(7.25)$$\begin{gather}\beta(\theta_b-1)/2=b\left[{\rm \pi}_u(\tau)+(\Delta \tau_{\theta}) \,\text{d}{\rm \pi}_u(\tau)/\text{d} \tau+\dots\right]\approx b{\rm \pi}_u(\tau+\Delta \tau_{\theta}), \end{gather}$$

(7.26)\begin{gather} \left.\begin{gathered} \exp({\beta(\theta_b-1)/2})\approx \exp({b{\rm \pi}_u(\tau)}) \left[1+(\Delta \tau_{\theta})b\frac{\text{d}{\rm \pi}_u(\tau)}{\text{d} \tau}+\dots\right],\\ \Delta \tau_{\theta} =\frac{2q}{1-q}\exp({-2b{\rm \pi}_u})>0, \end{gathered}\right\}\end{gather}

where, according to (6.3), $b\equiv {\beta \varepsilon }(1-q){(\gamma -1)}/2{\gamma }=O(1)$ and the $\tau$-variation of ${\rm \pi} _u$ in the coefficient $\exp ({-2b{\rm \pi} _u})$ in front of $\textrm {d} {\rm \pi}_u/\textrm {d} \tau$ on the right-hand side of (7.24) is negligible since it introduces a correction of following order. Focusing attention near the turning point, the time delay is quasi-constant

(7.27)\begin{equation} \Delta \tau_{\theta}\approx \frac{2q}{1-q}\text{e}^{{-}2b{\rm \pi}_u^*}=O(1). \end{equation}

Equations (7.24)–(7.26) show that the flame temperature and the reaction rate at time $\tau$ are related to the pressure at a later time $\tau +\Delta \tau _{\theta }$, $\Delta \tau _{\theta }>0$. As we shall see later, this promotes an instability of the physical branch of the C-shaped curve ‘flame velocity vs elongation’.

7.1.4. Unsteady modification to the mass flux

Anticipating that ${\delta Y}$ is of order $\varepsilon \,\textrm {d} {\rm \pi}_u/\textrm {d}\tau \ll 1$, the unsteady term on the left-hand side of (7.8a,b) can be neglected to first order

(7.28)\begin{equation} -\bar{m}\,\frac{ \partial \delta Y}{\partial z} -\frac{\partial^2\delta Y}{\partial z^2}\approx{-}\bar{m}\text{e}^{-\bar{m}\,z} {\delta m}+\bar{m}\,z\,\text{e}^{-\bar{m}z}b\,\frac{\text{d} {\rm \pi}_u}{\text{d}\tau}. \end{equation}

Integration with the two boundary conditions $z=0: {\delta Y}=0$ and $\lim _{z\to \infty } {\delta Y}=0$ yields

(7.29)$$\begin{gather} \delta Y={-}\text{e}^{-\bar{m} z}\,z\delta m+\left[\text{e}^{-\bar{m} z}\frac{z^2}{2} +\frac{1}{\bar{m}}\text{e}^{-\bar{m} z}\,z\right]b\frac {\text{d} {\rm \pi}_u}{\text{d}\tau}, \end{gather}$$

(7.30)$$\begin{gather}\left.\frac{\text{d} \delta Y}{\text{d} z}\right\vert_{z=0^+}={-}\delta m+\frac{1}{\bar{m}}b\,\frac {\text{d} {\rm \pi}_u}{\text{d}\tau}. \end{gather}$$

According to the jump (3.11) and (3.12), the gradients on the flame sheet take the form

(7.31a,b)$$\begin{gather} \left.\frac{\partial \theta^{(i)}}{\partial z}\right\vert_{z=0^+}={-} q \text{e}^{b{\rm \pi}_u}\left[1+\frac{\beta}{2}\delta\theta_b+\dots\right],\quad \left.\frac{\partial \delta \theta^{(i)}}{\partial z}\right\vert_{z=0^+}\approx{-} q \text{e}^{b{\rm \pi}_u}\frac{\beta}{2}\delta\theta_b \end{gather}$$

(7.32)$$\begin{gather}\left.\frac{\partial \delta \theta^{(i)}}{\partial z}\right\vert_{z=0^+}=q\left.\frac{\partial \delta Y}{\partial z}\right\vert_{z=0^+}+O(1/\beta). \end{gather}$$

After simplification by $q$, (7.31a,b) and (7.32) read

(7.33)\begin{equation} - \text{e}^{b{\rm \pi}_u}\frac{\beta}{2}\delta\theta_b=\left.\frac{\partial \delta Y}{\partial z}\right\vert_{z=0^+}+O(1/\beta). \end{equation}

Introducing (7.25) into (7.33) and using (7.30) leads to the unsteady modification to the mass flux across the reaction sheet defining the instantaneous laminar flame velocity relative to the burned gas $U_b(\tau )$

(7.34)\begin{equation} - \text{e}^{b{\rm \pi}_u}\Delta \tau_{\theta}\, b\frac{\text{d} {\rm \pi}_u}{\text{d} \tau} \approx{-}\delta m+\frac{1}{\bar{m}}b\,\frac {\text{d} {\rm \pi}_u}{\text{d}\tau} \quad \Rightarrow \quad \delta m \approx \text{e}^{{-}b{\rm \pi}_u}\frac{1+q}{1-q}b\,\frac {\text{d} {\rm \pi}_u}{\text{d}\tau}. \end{equation}

The modification of mass flux can be used to compute the variation of the speed of the reaction sheet $U_P(t)$ when the backflow $u_b(t)$ and the flame temperature are known.

7.2. Dynamical effect of the unsteadiness of the inner structure

The dynamics is governed by the ODE for ${\rm \pi} _u(\tau )$ when the expressions of $v_b$ and $\beta (\theta _b-1)$ in terms of ${\rm \pi} _u$ are introduced into the master equation (5.22). The main difference with the previous analysis of § 6 is that the unsteady flame temperature on the right-hand side of (5.22) is no longer the temperature of the unburned gas simply shifted by the heat release as in a steady laminar flame.

7.2.1. Instantaneous backflow model

In a first step, assume for simplicity that the lateral flame (quasi-parallel to the lateral wall) are quasi-steady, the unsteadiness being limited to the flame structure on the tip of the elongated front. Neglecting heat loss at the wall, the temperature in the tongues of unburned gas engulfed near the wall is assumed to be the same as in the flame on the tip of the elongated front, this temperature being modified by the longitudinal compression wave propagating ahead of the tip. This is an accurate approximation when the elongation is larger than the acoustic wavelength. Then, the instantaneous backflow (3.4) reads

(7.35)\begin{equation} v_b(\tau)/S_i=(1+\epsilon\tau)\bar{m}(\tau)=(1+\epsilon\tau)\text{e}^{b{\rm \pi}_u(\tau)}\approx \text{e}^{b{\rm \pi}_u(\tau)}+\epsilon \tau\,\text{e}^{b{\rm \pi}^*_u}+\dots .\end{equation}

Introducing (7.26) and (7.35) into (5.22) yields

(7.36)\begin{equation} S_i\text{e}^{b{\rm \pi}_u(\tau)}+\epsilon \tau S_i \text{e}^{b{\rm \pi}^*_u}-\left[ S_i+q+\frac{\sqrt{1-q}}{\gamma }{\rm \pi}_u(\tau) \right]\approx{-} q \text{e}^{b{\rm \pi}_u(\tau)}- \text{e}^{{-}b{\rm \pi}_u^*}\frac{2q}{1-q}\,b\,\frac{\text{d}{\rm \pi}_u(\tau)}{\text{d} \tau}, \end{equation}

which can be written

(7.37)\begin{equation} \left [S_i(1+\epsilon \tau)+q\right ]\text{e}^{b{\rm \pi}_u(\tau)}-\left[ S_i+q+\frac{\sqrt{1-q}}{\gamma }{\rm \pi}_u(\tau) \right] ={-} \text{e}^{{-}b{\rm \pi}_u^*}\frac{2q}{1-q}\,b\,\frac{\text{d}{\rm \pi}_u(\tau)}{\text{d} \tau}, \end{equation}

and, using the notation (6.10) and (6.11a–c),

(7.38)$$\begin{gather} \zeta\exp\vartheta-\zeta_i -\vartheta /\tilde{b}={-}\tilde{K}_{\theta}\frac{\text{d} \vartheta}{\text{d}\zeta}, \quad \text{where}\ \tilde{K}_{\theta}\equiv\epsilon S_i\,\text{e}^{{-}b{\rm \pi}_u^*}\frac{2q}{1-q}>0 \end{gather}$$

(7.39)$$\begin{gather}\frac{(\zeta-\zeta^*) }{ \zeta^*}+\frac{(\vartheta^*-\vartheta)^2 }{2 }={-}\tilde{\epsilon}_\theta \frac{\text{d} \vartheta}{\text{d}\zeta}, \quad \text{where}\ \tilde{\epsilon}_\theta\equiv \tilde{b}\tilde{K}_\theta. \end{gather}$$

The difference with (6.25) and (6.26) is the sign on the right-hand side. The negative sign obtained for the instantaneous backflow shows that, according to the trajectories in the phase space of (7.38) and (7.39), the unsteadiness of the inner flame structure promotes an instability of the physical branch $\bar {\vartheta }_-(\zeta )$ of the quasi-steady solutions discussed in § 6.2.1. This would be unfortunate for the study of the DDT when the elongation increases since the physical branch of quasi-steady solutions could not be followed up to the vicinity of the turning point. Hopefully, the delay in the backflow restores the stability as shown now.

7.2.2. Delayed backflow model

Still assuming that the lateral flames are in steady state, the non-dimensional expression of the delayed backflow is the same as (6.21). Introducing (7.26) and (6.21) into the master equation (5.22) yields a non-dimensional ODE similar to (6.26)

(7.40)\begin{equation} \frac{(\zeta-\zeta^*) }{ \zeta^*}+\frac{(\vartheta^*-\vartheta)^2 }{2 }= \tilde{\epsilon} \frac{\text{d} \vartheta}{\text{d}\zeta}, \quad \text{where}\ \tilde{\epsilon}\equiv \tilde{b}(\tilde{K}_w-\tilde{K}_\theta), \end{equation}

so that the phenomenology is the same as in § 6.3.2 provided $\tilde {K}_w>\tilde {K}_\theta$ which is typically the case, see the text below (7.43).

7.2.3. Unsteady structure of the lateral flames

If the inner flame structure of the lateral flames is not in steady state, a delay is involved in the radial flow of burned gas $U_b$ feeding the longitudinal backflow on the axis of the elongated flame front. The unsteady laminar flame velocity $U_b$ of the lateral flames is computed with the first-order disturbance of the mass rate across the reaction sheet $m(\tau )$ in (7.34). The additional delay in the backflow pushing the flame tip takes the form

(7.41)\begin{equation} v_b/S_i\approx \text{e}^{b{\rm \pi}_u(\tau)}+\epsilon \tau \,\text{e}^{b{\rm \pi}^*_u}+\text{e}^{{-}b{\rm \pi}_u^*}\frac{1+q}{1-q}b\frac{\text{d} {\rm \pi}_u}{\text{d} \tau}+\dots. \end{equation}

Introducing (7.26) and (7.41) into the master equation (5.22) yields an ODE for $\vartheta =b{\rm \pi} _u$, namely for the pressure and/or the flame temperature similar to (7.40) but involving an additional destabilizing term $\tilde {K}_{f}$

(7.42a,b)\begin{equation} \tilde{\epsilon}\equiv\tilde{b}(\tilde{K}_w-\tilde{K}_\theta-\tilde{K}_{f}), \quad \tilde{K}_{f}\equiv \epsilon S_i^2\text{e}^{{-}b{\rm \pi}_u}\frac{1+q}{1-q}. \end{equation}

The same equation as (6.26) is obtained in which $K_w$ is replaced by $K_w-(K_\theta +K_f)>0$. The same finite-time singularity as that described at the end of § 6.3.2 is obtained, provided the delays satisfy the following condition:

(7.43)\begin{equation} \tilde{K}_w>\tilde{K}_\theta+\tilde{K}_{f} \quad \Rightarrow \quad \frac{L/a_{ref}}{d_{ref}/U_{ref}}>\text{e}^{{-}2b{\rm \pi}_u^*}\left[ \frac{1}{S_i}\frac{2q}{1-q}+\frac{1+q}{1-q}\right] ,\end{equation}

which is verified for a length of the finger-like flame sufficiently elongated compared to the flame thickness $L/d_{ref}> \textrm {e}^{-2b{\rm \pi} _u^*}/\varepsilon$. This is already the case for $\varepsilon \approx 10^{-2}$ and a flame elongation larger than a cell size of a few centimetres (tube diameter).