5.1. The flame speed

The burning rate $m^*$, as defined in (3.18), is obtained by matching the hydrodynamic solution $m_{hyd} = m^{(0)}_{hyd} + \delta m^{(1)}_{hyd}$ at $n=0^-$ with the corresponding solution in the flame zone, namely $m \sim m^{(0)} + \delta m^{(1)}$ as $\eta \to -\infty$. Evidently, to leading order,

(5.1)\begin{equation} m_{hyd} \sim m^{(0)}_{hyd}|_{n=0^-} = \lim_{\eta \to -\infty} m^{(0)} = M . \end{equation}

To proceed to the next order, we note that as $\eta \to -\infty$,

(5.2)\begin{equation} \left.\begin{gathered} \mathcal{J}_2 \sim \frac{J_1}{MT_{u}} + \frac{\mathcal{J}_1 (\eta)}{T_{u}}, \\ T_{u} \left( \frac{\mathcal{J}_3} {\vartheta(\eta)} - \mathcal{J}_4 \right) \sim{-} \frac{\dot J_1}{M \dot T_{u}} + \frac{J_1}{M T_{u}} + \dfrac{\mathcal{J}_1 (\eta)}{T_{u}}, \end{gathered}\right\} \end{equation}

such that

(5.3)\begin{align} m \sim M + \delta \left\{ A - \frac{P}{MT_{u}} \left(\left({\mathbb{K}} + \frac{1}{\gamma} \frac{\dot P}{P} - \frac{\dot M}{M} \right) J_1 + \dot J_1 \right) + \left( M \kappa - \frac{P\mathbb{K}}{T_{u}} - \frac{\dot P}{\gamma T_{u}} \right) \mathcal{J}_1(\eta)\right\}. \end{align}

The hydrodynamic solution to $O(\delta )$ at $n = 0^-$ takes the form

(5.4)\begin{equation} m_{hyd} \sim M + \delta \left\{ m^{(1)}_{hyd}|_{n=0^-} + \left.\frac{\partial m^{(0)}_{hyd}}{\partial n}\right|_{n=0^-} \mathcal{J}_1(\eta) \right\}, \end{equation}

where, here and below, it is understood that $M$ must be expanded to $O(\delta )$, namely replaced by $M + \delta (\partial {M}/\partial {T_{f}})T_b^{(1)}$ with the $O(\delta )$ retained when necessary. Matching then yields an expression for $m^*$ which, after dividing by $\rho _u$, gives the expression

(5.5) \begin{equation} S_{f} = \frac{M}{\rho_u} - \frac{\delta}{M} \left\{ \mathcal{M}_1 \left( \mathbb{K} - \frac{\dot M}{M}\right) + \mathcal{M}_2 \frac{\dot P}{P} \right\} + \unicode{x2134}(\delta). \end{equation}

The flame speed $S_{f}$ is modulated by two time-dependent coefficients

(5.6)\begin{equation} \left.\begin{gathered} \mathcal{M}_1 = \frac{ T_{f}}{q} J_1+ \left( \frac{T_a}{T_{f} } \right)^{2} \frac{\beta(Le_{eff} - 1)}{2 q} T_{u} J_2, \\ \mathcal{M}_2 = \mathcal{M}_1- \frac{\gamma -1}{\gamma} \left\{J_1+ \left( \frac{\lambda_u}{T_{u}} - \frac{\lambda_b}{T_{f}}\right) \frac{T_{u}T_{f}}{q}+ \left( \frac{T_a}{T_{f} } \right)^{2} \frac{\beta(Le_{eff} - 1)}{2 q}T_{u}^2 J_3 \right\}, \end{gathered}\right\} \end{equation}

which results from (5.3) after expressing the scaled parameters ${le_{eff}}$ and $\varphi$ in terms of the effective Lewis number $Le_{eff}$ and the excess-to-deficient reactant mass ratio $\varPhi$ while using the Zel'dovich number $\beta$ in lieu of $\beta _0$. Matching the gradients does not reveal new information because the gradient of the mass flux to leading order can also be deduced from the continuity equation (3.2) when evaluated at $n=0^-$. The $O(\delta )$ correction to the flame speed consists of two terms; the first will be recognized below as the voluminal stretch and the second is the proportionate time rate of change of the pressure; the coefficients $\mathcal {M}_1$ and $\mathcal {M}_2$, which can arguably be referred to as Markstein numbers, will be further discussed below.

The flame speed, as defined in (3.17), is based on the prevalent convention that it is measured relative to the velocity of the fresh unburned gas. It is often convenient to relate the propagation speed to the gas velocity of the burned gas and define the flame speed as $S_{f}^{b} \equiv m^{b}/\rho _b$, where the superscript $^{b}$ here and below signifies that the quantity is measured relative to the burned gas. This requires evaluating the mass flux $m^{b}$ at the burned edge of the flame, i.e. at $n=0^+$, which is directly obtained by matching with the solution (4.26) as $\eta \to +\infty$. The flame speed is then given by

(5.7)\begin{equation} \frac{\rho_b}{\rho_u} S_{f}^{b} = \frac{M}{\rho_u} - \frac{\delta}{M}\left\{\mathcal{M}_1^{b} \left( \mathbb{K} - \frac{\dot M}{M} \right) + \mathcal{M}_2^{b} \frac{\dot P}{P} \right\} + \unicode{x2134}(\delta), \end{equation}

with time-dependent coefficients

(5.8)\begin{equation} \left.\begin{gathered} \mathcal{M}_1^{b} = \frac{T_{u}}{q} J_1 + \left( \frac{T_a}{T_{f}}\right)^{2} \frac{\beta(Le_{eff} - 1)}{2 q} T_{u} J_2,\\ \mathcal{M}_2^{b} = \mathcal{M}_1^{b}- \frac{\gamma -1}{\gamma} T_{u}^2 \left\{\frac{1}{q} \left( \frac{\lambda_u}{T_{u}} - \frac{\lambda_b}{T_{f}}\right) + \left( \frac{T_a}{T_{f} } \right)^{2} \frac{\beta(Le_{eff} - 1)}{2 q}J_3 \right\}, \end{gathered}\right\} \end{equation}

which may be referred to as the burned Markstein numbers. Here too, matching the gradients does not reveal new information; the gradient of the mass flux to leading order can be also deduced from the continuity equation (3.2) when evaluated at $n=0^+$.

From the above relations, it can be verified that there is an $O(\delta )$ jump in the normal mass flux across the entire flame given by

(5.9)\begin{equation} [\![ m ]\!] = \frac{\delta}{M} \left\{\rho_u J_1 \left(\mathbb{K} - \frac{\dot M}{M}\right) + \frac{\mathrm{d}}{\mathrm{d}t}(\rho_u J_1)\right\}. \end{equation}

Although the mass flux is conserved across an infinitesimally thin flame, when accounting for the finite (albeit small) thickness of the flame zone, the normal mass flux leaving the flame differs from that into the flame due to transverse transport and/or mass accumulation. One also notes that the jump in the gas velocity across the flame given by

(5.10)\begin{equation} [\![ \boldsymbol v \boldsymbol{\cdot} \boldsymbol n]\!] = \frac{T_{f} - T_u}{P} m^* + \frac{T_{f}}{P} [\![ m ]\!]= \frac{qM}{P} + O(\delta) \end{equation}

always increases in time, a direct result of the rise in flame temperature.

Although the density and temperature of the unburned gas $\rho _u$ and $T_u$ remain spatially uniform to all orders in $\delta$, the state of the burned gas is modified to $O(\delta )$ by corrections arising from variations in the flame temperature. In particular, the temperature is determined by solving

(5.11)\begin{equation} \rho \frac{\mathrm{D}T}{\mathrm{D}t} - \frac{\gamma - 1}{\gamma} \frac{\mathrm{d}P}{\mathrm{d}t} = \delta \boldsymbol{\nabla} \boldsymbol{\cdot} (\lambda \boldsymbol{\nabla} T), \end{equation}

subject to

(5.12)\begin{align} T_{f} &= T_{u} + q - \frac{\delta P}{M^2} \left\{ \frac{\gamma-1}{\gamma} \left( J_1 + \frac{q\lambda_b }{ T_{u} + q} \right) \frac{\dot P}{P} \right\} \nonumber\\ &\quad - ({Le}_D -1) \frac{\delta }{M} \left\{\left( \frac{P J_2}{M} \right) \mathbb{K} + \frac{\mathrm{d}}{\mathrm{d}t} \left(\frac{P J_2}{M} \right)\right\} \end{align}

at $n=0^+$, with appropriate boundary conditions to $O(\delta )$ at the vessel's walls. The expression for $T_{f}$ results from matching with the solution in the flame zone as $\eta \to + \infty$. Although the flame temperature $T_{f}$ varies primarily in time as a result of the pressure rise, there are small changes in temperature associated with flame stretch that depend on the Lewis number of the deficient reactant, i.e. the fuel in a lean mixture and the oxidizer in a rich mixture. The dependence on flame stretch was corroborated with direct numerical simulations for freely expanding spherical flames  (Giannakopoulos et al. Reference Giannakopoulos, Gatzoulis, Frouzakis, Matalon and Tomboulides2015) where it was shown that, depending on the equivalence ratio that determines whether ${Le}_D$ is above/below one, the flame temperature is lower/higher than its adiabatic value and increases/decreases towards $T_a$ when ${\mathbb {K}} \to 0$.

The equations describing the flow field and state variables in the hydrodynamic zones were presented in § 3 to leading order. To extend the results to $O(\delta )$, (2.9)–(2.13) must be considered with the reaction term set to zero and diffusion added as correction terms. For the determination of the flow field, the $O(\delta )$ corrections to the RH conditions (3.9) must be derived, as carried out in Matalon & Matkowsky (Reference Matalon and Matkowsky1982) for freely propagating flames. Not being the focus of this work, the derivation will not be pursued here. For configurations that depend on a single spatial coordinate, such as planar or spherical flames, the velocity field is directly determined from the continuity equation (3.7) which, when extended to $O(\delta )$ takes the form

(5.13)\begin{equation} \boldsymbol\nabla\boldsymbol{\cdot} \boldsymbol v + \frac{1}{\gamma P} \frac{\mathrm{d}P}{\mathrm{d}t} = \delta \frac{1}{P}\boldsymbol{\nabla} \boldsymbol{\cdot} (\lambda \boldsymbol{\nabla} T), \end{equation}

and needs to be solved subject to the jump condition (5.9) across the flame surface. The momentum equation is used a posteriori to evaluate the dynamic pressure.

As a final remark, we note that the expressions for the flame speed and Markstein lengths (relative to burned or unburned gas), and the flame temperature simplify to the corresponding expressions for freely propagating flames. With $P= T_{u} =1$, the flame temperature, to leading order, is the adiabatic flame temperature $T_a = 1+q$ and the density is a piecewise constant function with $\rho _u = 1$ and $\rho _b = 1/T_a$. Then, the flame speed depends on a single Markstein number $\mathcal {M}_1$ that takes the form

(5.14)\begin{equation} \mathcal{M}_1 = \frac{\sigma}{\sigma - 1} \int_{1}^{\sigma} \frac{\lambda(x) }{x} \mathrm{d}\kern 0.06em x + \frac{\beta(Le_{eff} - 1)}{2 (\sigma - 1)} \int_{1}^{\sigma} \frac{\lambda(x) }{x} \ln {\left(\frac{\sigma - 1}{x - 1}\right)} \mathrm{d}\kern 0.06em x, \end{equation}

where $\sigma = \rho _u / \rho _b$ is the unburned-to-burned density ratio. The stretch-corrected flame speed and temperature reduce to

(5.15)\begin{equation} \left.\begin{gathered} S_f = 1 - \delta \mathcal{M}_1 {\mathbb{K}}, \\ T_{f} = T_{a} - \delta \left\{({Le}_D -1) \int_{1}^{\sigma} \frac{\lambda(x) }{x} \ln {\left(\frac{\sigma - 1}{x - 1}\right)} \mathrm{d}\kern 0.06em x\right\} \mathbb{K}, \end{gathered}\right\} \end{equation}

as in Matalon et al. (Reference Matalon, Cui and Bechtold2003).

5.2. Pressure rise

The pressure rise in the vessel is determined from (2.17) where the integral on the right-hand side is to be carried out over the entire volume. In the hydrodynamic limit it simplifies to

(5.16)\begin{equation} \frac{\mathrm{d}P}{\mathrm{d}t} = \frac{ \gamma q}{{\mathbb{V}}} \int \omega \, \mathrm{d} n \, \mathrm{d} \mathcal{A} = \frac{\gamma q}{{\mathbb{V}}} \int_{\mathcal{A}_{f}} \int_{-\infty}^{\infty} \omega \delta \lambda \mathrm{d} \eta \, \mathrm{d} \mathcal{A}, \end{equation}

where the integration is carried out across the flame zone and over the entire flame surface $\mathcal {A}_{f}$. Since the chemical reaction rate $\omega$ is replaced by a Dirac delta function, the latter simplifies to

(5.17)\begin{equation} \frac{\mathrm{d}P}{\mathrm{d}t} ={-} \frac{\gamma}{{\mathbb{V}}} \int_{\mathcal{A}_{f}} \left[\frac{\partial T}{\partial\eta}\right] \mathrm{d} \mathcal{A}. \end{equation}

The integrand consists of the jump in temperature gradient across the reaction sheet, namely the fraction of the total heat generated by the chemical reaction and conducted in different proportions to the burned and unburned gas. Under adiabatic conditions, more heat is transported to the fresh mixture justifying the assertion that the pressure always rises. The jump in temperature gradient across $\eta =0$ is obtained directly from the solution constructed in the previous section, such that

(5.18)\begin{equation} \frac{\mathrm{d}P}{\mathrm{d}t} = \frac{\gamma q {\mathbb{A}}_{f}}{{\mathbb{V}}} \left\{ M - \delta \left(\frac{T_a}{T_{f} } \right)^{2} \frac{\beta(Le_{eff} - 1)}{2q} \left(\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{P J_2}{M} \right)+ \frac{PJ_2}{M} \frac{1}{{\mathbb{A}}_{f}} \int_{\mathcal{A}_{f}} \mathbb{K} \, \mathrm{d} \mathcal{A}\right)\right\}, \end{equation}

where ${\mathbb {A}}_{f}$ is the flame surface area. For consistency, $O(\delta ^2)$ terms in the curly bracketed expression must be discarded. It may be verified when using (4.22) that the factor multiplying the last term in this equation decreases in time implying that the influence of flame stretch diminishes as the pressure rises. Equation (5.18) can be also expressed as

(5.19)\begin{equation} \frac{\mathrm{d}P}{\mathrm{d}t} = \frac{\gamma}{{\mathbb{V}}}\int_{\mathcal{A}_{f}} (q m^* + T_{f} [\![ m]\!]) \,\mathrm{d} \mathcal{A} + \delta \left(\frac{{\mathbb{A}}_{f}}{\mathbb{V}} \right)^{2} \gamma(\gamma-1) q J_1, \end{equation}

a form found useful in the next section.

5.3. Flame stretch

Flame stretch, first introduced by Karlovitz et al. (Reference Karlovitz, Denniston, Knapschaefer and Wells1953), is a measure of the deformation of the flame surface resulting from its motion and from the underlying flow field. It is defined as the proportionate rate of change of the area of an infinitesimal surface element $A$ of the flame front, namely ${\mathbb {K}} \equiv A^{-1} \,\mathrm {d}{A}/\mathrm {d}{t}$. The stretch rate appeared naturally in the mass conservation equation (4.24) when expressed in terms of intrinsic surface coordinates, and may be expressed in the coordinate-free form as

(5.20)\begin{equation} {\mathbb{K}} ={-} V_{f} \kappa - \underbrace{\boldsymbol n \boldsymbol{\cdot} \boldsymbol\nabla \times (\boldsymbol v \times \boldsymbol n)}_{- \boldsymbol\nabla_{s} \boldsymbol{\cdot} \boldsymbol v_{{\perp}}}, \end{equation}

obtained from the basic definition (Matalon Reference Matalon1983). The first term corresponds to the surface dilatation resulting from the motion of the flame front, and the second term is the surface extension/compression resulting from the underlying velocity gradient on the surface of the flame front. The curvature can be evaluated from $\kappa = - \boldsymbol \nabla \boldsymbol {\cdot } {\boldsymbol n}$, evaluated at the flame front. Since $\boldsymbol v_{\perp }$ is continuous across the flame (to leading order), the stretch rate is uniquely defined on the flame surface and can be evaluated from (5.20) on either side of the interface.

It is often stated that flame stretch is a combination of curvature and strain, a statement that requires careful consideration. When expanding the right-hand side of (5.20), one finds that

(5.21)\begin{equation} {\mathbb{K}} = (\boldsymbol v \boldsymbol{\cdot} \boldsymbol n - V_{f}) \kappa + K_S + \boldsymbol\nabla\boldsymbol{\cdot} \boldsymbol v, \end{equation}

where $K_S = - \boldsymbol n \boldsymbol {\cdot } \boldsymbol E \boldsymbol {\cdot } \boldsymbol n$ is the local strain rate. It must be noted that the first two terms on the right-hand side admit different values when evaluated on the unburned and burned sides of the flame front. Thus, using (5.21) to determine the stretch rate requires specifying on which side of the flame front this expression is to be evaluated. For a freely propagating flame in the hydrodynamic limit, $\boldsymbol \nabla \boldsymbol {\cdot } \boldsymbol v = 0$ on either side of the flame and the stretch rate is indeed a combination of curvature and strain, but each of these components have different expressions when evaluated on the unburned and burned sides. On the unburned side of the flame, $\boldsymbol v \boldsymbol {\cdot } \boldsymbol n - V_{f} \sim 1$ and ${\mathbb {K}} = \kappa + K_S^{u}$; on the burned side of the flame ${\boldsymbol v \boldsymbol {\cdot } \boldsymbol n - V_{f} \sim \sigma}$, where $\sigma$ is the unburned-to-burned density ratio and ${\mathbb {K}} = \sigma \kappa + K_S^{b}$. Both yield the same value for the stretch rate because $[\![ K_S ]\!] = - (\sigma - 1) \kappa$, a relation derived in Matalon et al. (Reference Matalon, Cui and Bechtold2003). For a flame propagating in a closed vessel, the stretch rate is not just a combination of curvature and strain because the volume dilatation rate $\boldsymbol \nabla \boldsymbol {\cdot } \boldsymbol v \ne 0$ due to gas compression, but may be expressed in the form

(5.22)\begin{equation} \mathbb{K} = \frac{M}{\rho} \kappa + K_S - \frac{1}{\gamma P} \frac{\mathrm{d}P}{\mathrm{d}t},\quad \text{at}\ \psi={0^\pm}.\end{equation}

A planar flame, for example, is not stretched, but in a closed vessel it experiences normal strain that balances the volume dilatation rate, thus ensuring that ${\mathbb {K}} = 0$. Similarly, a spherical flame experiences normal straining and a non-zero dilatation rate such that (5.22) simplifies to ${\mathbb {K}} = - V_{f} \kappa$ as it should. These examples will be further discussed in § 6.

Although the flame zone expressed in units of the diffusion length $l_d$ is of infinite extent, the distance required for the temperature to rise from the cold value $T_{u}$ to the flame temperature $T_f$ occurs over a relatively small multiple of this length. The flame thickness may therefore be defined as the distance from the reaction zone to the point where the temperature falls within $1\,\%$ of $T_{u}$. For a freely propagating flame ($M=1$), this distance is constant; typically $l_{f} \approx 7 l_d$ with an accuracy of $10^{-4}$. Under confinement, however, $l_{f} \sim 1/M$ and, since $M$ increases in time due to the pressure rise, the flame becomes consistently thinner when propagating towards the end of the vessel, as shown in figure 2(a). The graph exhibits the dependence of $l_{f}$ on time, represented by $P(t)$, for various values of $q$, with $\varphi =0, \lambda = T^{0.7}$ and $\beta = 10$. Changes in equivalence ratio and/or the dependence $\lambda (T)$ show no significant quantitative difference in the behaviour of $l_{f}$.

Figure 2.

The dependence of (a) the flame thickness $l_f$ and (b) the flame temperature $T_{f}$ scaled by the adiabatic flame temperature $T_a$ on pressure $P(t)$ representing time, for various values of the heat release parameter $q$; the curves end at $P = P_{e}$ marked on the axis.

Due to the significant decrease in the flame thickness with time, the correction to the flame speed resulting from the physicochemical processes occurring inside the flame zone has been expressed in (5.5) as a multiple of $\delta /M$, which being proportional to $l_{f}/l$, is a more accurate measure of its thickness. The proportionate time rate of change of the flame thickness ${l_{f}}^{-1} \,\mathrm {d}{l_{f}}/\mathrm {d}{t} = - \dot M/M$ combines with the proportionate time rate of change of the area of a flame surface element to define a voluminal stretch rate

(5.23)\begin{equation} \frac{1}{V} \frac{\mathrm{d}V}{\mathrm{d}t} = \frac{1}{A} \frac{\mathrm{d}A}{\mathrm{d}t} + \frac{1}{l_{f}} \frac{\mathrm{d}l_{f}}{\mathrm{d}t} = {\mathbb{K}} - \frac{\dot M}{M}, \end{equation}

where $V = l_{f} A$ is an infinitesimal volume element of the flame zone. Accordingly, the flame speed (5.5) was found to depend on the voluminal stretch rate that measures the local deformation of a volume element of the flame zone. The voluminal stretch concept was first introduced by Buckmaster (Reference Buckmaster1979) in the context of slowly varying flames, as noted in the introduction.

5.4. The effective Lewis number

Although distinct Lewis numbers were assumed for the two reactants, the burning rate was found to depend on a stoichiometry-weighted combination of the two, or an effective Lewis number of the form

(5.24)\begin{equation} \left.\begin{gathered} Le_{eff} = \frac{{Le}_E + {Le}_D (1 + \varDelta)}{2+ \varDelta},\\ \varDelta = \left(\frac{T_a}{{T_{f}}} \right)^{2} \beta(\varPhi - 1),\quad \varPhi \sim \left\{\begin{array}{@{}ll@{}} 1/\phi & \text{lean mixtures}, \\ \phi & \text{rich mixtures}, \end{array}\right. \end{gathered}\right\} \end{equation}

where $\phi$ is the equivalence ratio and $\varDelta$ measures the deviation from stoichiometry. It is an average of the Lewis numbers of fuel and oxidizer, with a heavier weight on the deficient component in the mixture. The effective Lewis number is the mean value of ${Le}_D$ and ${Le}_E$ for a stoichiometric mixture ($\varDelta = 0$) and tends towards ${Le}_D$ for conditions sufficiently far from stoichiometry ($\varDelta \gg 1$). It also varies in time due to the dependence of $T_{f}/T_a$ on pressure shown in figure 2(b). The figure displays the dependence of the (leading-order) flame temperature scaled by the adiabatic temperature $T_a$ on pressure, which varies in time from $P=1$ initially to the end pressure $P_{e}=1 + \gamma q$; the latter is shown on the axis for each value of $q$. Given that $T_{f}/T_a$ does not increase considerably, and the uncertainty in accurately specifying the individual Lewis numbers, a time-averaged value can serve as the representative $Le_{eff}$.

Figure 3 shows the dependence of $Le_{eff}$, averaged over the total propagation time, on the equivalence ratio $\phi$ of the mixture, for several fuels burning in air. They are plotted for the representative value $q=6$, which characterizes mixtures of various heat releases $Q$ and mass fractions $Y_{D_{{0}}}$. The determination of the effective $Le_{eff}$ requires specifying the limiting Lewis numbers; the fuel Lewis number ${Le}_F$ for lean mixtures and the oxygen Lewis number ${Le}_O$ for rich mixtures. These values are defined as the ratio of the thermal diffusivity of the mixture to the binary diffusivity of the fuel/oxygen into N$_2$, as per the dilute-mixture assumption used when adopting Fick's law in the governing equations; they were estimated near the lean/rich flammability limits and are marked by a dashed line in the figure. The formula (5.24) then provides the distribution of $Le_{eff}$ as a function of the equivalence ratio $\phi$, approaching the respective limiting Lewis numbers for lean and rich mixtures. A monotonically decreasing curve results for heavy hydrocarbon fuels with ${Le}_F > 1$, and a monotonically increasing curve results for light fuels with ${Le}_F < 1$, such as hydrogen and methane. The transitional curves display a weak dependence on the Zel'dovich number, which has been specified in this figure as $\beta = 10$.

Figure 3.

The dependence of the effective Lewis number $Le_{eff}$, averaged over the total propagation time, on the mixture equivalence ratio for several fuels burning in air. The dashed lines mark the limiting Lewis numbers; the fuel Lewis number ${Le}_F$ for lean mixtures and the oxygen Lewis number ${Le}_O$ for rich mixtures.

5.5. The Markstein numbers

When viewed on the hydrodynamic scale, the flame is effectively an interface embedded in the flow field; the preheat zone of the flame collapses to $\psi = 0^-$ and the post-reaction zone of the flame to $\psi = 0^+$. The propagation speed can be expressed relative to either the unburned or burned gas. When expressed relative to the unburned gas, the flame speed $S_{f}$ exhibits a linear dependence on the voluminal stretch rate, modulated by a parameter $\mathcal {M}_1$, and on the rate of pressure rise, modulated by a parameter $\mathcal {M}_2$. Similarly, when expressed relative to the burned gas, the flame speed $S_{f}^{b}$ exhibits a linear dependence on the voluminal stretch rate, modulated by a parameter $\mathcal {M}^{b}_1$, and on the rate of pressure rise, modulated by a parameter $\mathcal {M}^{b}_2$. We note that unlike $S_{f}$, which mimics the diffusion and reaction processes occurring in the entire flame zone and is not evaluated at any specific isotherm, $S_{f}^{b}$ is effectively evaluated at the flame temperature $T = T_{f}$. This has practical implications when seeking comparison with experimental and/or numerical simulations aimed at quantifying the dependence of the flame speed on stretch and pressure rise. Measurements and simulations that resolve the flame structure require selecting an isotherm to represent the flame surface, and the burned Markstein numbers in that regard are the most relevant ones. Indeed, it has been established that the most appropriate isotherm for the evaluation of the flame speed, which is well conditioned and displays the best agreement with the theory, must be sufficiently close to the burned side of the flame (Giannakopoulos et al. Reference Giannakopoulos, Gatzoulis, Frouzakis, Matalon and Tomboulides2015). The flame speed $S_{f}$ and Markstein numbers $\mathcal {M}_1$ and $\mathcal {M}_2$, on the other hand, are the appropriate expressions to use for the evolving flame front when determining the flow field in the hydrodynamic regions.

Although the Markstein numbers (5.6) and (5.8) are pressure dependent, for a given mixture they are mostly affected by the effective Lewis number $Le_{eff}$. Given that their dependence on time for a wide range of $Le_{eff}$ is quite moderate, they can be represented by an average value. In figure 4 we show the variation of the Markstein numbers (unburned and burned, averaged over the total propagation time) of selected fuels burning in air across the entire range of the stoichiometric spectrum, which evidently mirror the variation of $Le_{eff}$ on the equivalence ratio. We note that the second member on the right-hand side of (5.8) is identically zero for $\lambda (T)=T$ and negligibly small otherwise, implying that $\mathcal {M}_2^{b} \approx \mathcal {M}_1^{b} = \mathcal {M}^{b}$. For the unburned coefficients, $\mathcal {M}_2 < \mathcal {M}_1$, but the difference is typically small and does not exceed $20\,\%$.

Figure 4.

The dependence of the Markstein numbers (averaged over the total propagation time) on the mixture equivalence ratio $\phi$ for several fuels burning in air.

There are circumstances where despite the moderate variation of the Markstein numbers with time they may change sign during the propagation. This becomes particularly relevant when attempting to extract the burned Markstein number from experimental measurements. Since the integrals $J_1$ and $J_2$ are both positive, $\mathcal {M}^{b}$ is strictly positive for $Le_{eff} > {Le}^{cr}_{eff}$ and negative otherwise, where

(5.25)\begin{equation} {Le}^{cr}_{eff}\sim 1 - \frac{2}{\beta} \left(\frac{T_{f}}{T_a}\right)^{2} \frac{J_1}{J_2}. \end{equation}

In most cases, $\mathcal {M}^{b}$ is either positive or negative, but in a limited range of sub-unity Lewis number, estimated as $0.75 \lessapprox Le_{eff} \lessapprox 0.82$, it may change sign during the propagation. The implication being that in such cases, the location where flame speed measurements are taken to estimate the Markstein numbers becomes important. Two examples illustrating this effect are shown in figure 5, where the dependence of $\mathcal {M}^{b}$ on time is shown for several values of the equivalence ratio. For a sufficiently lean hydrogen–air mixture, such as $\phi \le 0.775$, the Markstein number is strictly negative and becomes strictly positive for $\phi \ge 0.845$; it changes sign during the propagation when $\phi \approx 0.8$. The opposite trend is observed for a rich octane–air mixture when diluted with CO$_2$ to increase the effective Lewis number; the Markstein number is strictly positive for $\phi \le 1.98$, negative for $\phi \ge 2.92$ and changes sign during the propagation when $\phi \approx 2.4$.

Figure 5.

Two cases illustrating circumstances where the ‘burned’ Markstein number $\mathcal {M}^{b}$ may change sign during the propagation. Results are shown for the (a) H$_2$–air mixture and (b) C$_8$H$_{18}$–air–CO$_2$ mixture.

6.1. Planar flame

We consider first the propagation of a planar flame travelling from left to right in a rectangular channel, as shown in figure 6(a). If the channel's width (in the $y$ and $z$ directions) is chosen as the hydrodynamic length scale $l$, the flame surface area ${\mathbb {A}}_f =1$ and the ratio ${\mathbb {A}}_f /{\mathbb {V}} = 1/L$. The flame front is described by $\psi (x,t) \equiv x_f (t) - x = 0$, such that $0 < x < x_f$ corresponds to the burned gas and $x_f < x < L$ to the unburned gas region. The propagation speed is then given by $V_{f} = - \dot x_f$.

To leading order, the gas velocity determined from (5.13) and satisfying $u(0,t) = u(L,t)=0$ is given by

(6.1)\begin{equation} u(x,t) = \left\{\begin{array}{@{}ll@{}} \dfrac{\dot{P}}{\gamma P}(L - x), & x_f < x < L, \\ -\dfrac{\dot{P}}{\gamma P} x, & 0 < x < x_f, \end{array}\right. \end{equation}

with the pressure obtained by solving $\dot P = \gamma q M/L$ subject to $P(0)=1$, and $M$ given by (4.22). The RH relation $[\![ \rho (\dot {x}_f - u)]\!] =0$ then yields an expression for the flame position,

(6.2)\begin{equation} x_f (t) = \left( 1 - \frac{P_{e} - P}{P_{e} - 1} P^{{-}1/\gamma} \right) L, \end{equation}

where $P_{e} = 1+ \gamma q$ is the end pressure. In a sufficiently long channel ($L \gg 1$), the pressure increases linearly in time, namely $P \sim 1 + \gamma qt/L$, and the velocity simplifies to

(6.3)\begin{equation} u(x,t) \sim \left\{\begin{array}{@{}llc@{}} q - q(x + \gamma q t)/L + O(1/L), & x_f < x < L, \\ - q x /L, & 0 < x < x_f, \end{array}\right. \end{equation}

where the additional $O(1/L)$ contribution in the unburned gas comes from the expansion of $M$. When $L \to \infty$, the solution tends to that of a planar flame propagating from the closed end of a semi-infinite channel towards an open end at a speed $\dot x_f \sim 1+q$. Determining the state of the burned gas requires first solving the advection equation

(6.4)\begin{equation} \frac{\partial\mathcal{E}}{\partial t} + \frac{\dot{P}}{\gamma P} x \frac{\partial\mathcal{E}}{\partial x} = 0 \end{equation}

for the entropy function, subject to $\mathcal {E} = \mathcal {E}_b$ at $x = x_f$. The general solution is $\mathcal {E} = f(xP^{1/\gamma })$, where the function $f(\zeta )$ is determined from the relation

(6.5)\begin{equation} f(\zeta) = 1 + \frac{q}{P^{(\gamma - 1)/\gamma}} \quad \text{for}\ \zeta = \left( P^{1/\gamma} - \frac{P_{e} - P}{P_{e} - 1} \right) L . \end{equation}

Evidently, $f(0) = 1+q$ and $f(\zeta ) \sim 1$ as $\zeta \to \infty$, but the values of interest are limited to $\zeta$ corresponding to $1 \le P < P_{e}$, with the end point determined by the parameters $\gamma$ and $q$.

Next we examine the influences arising from the finite thickness of the flame. Evidently, the planar flame surface is not stretched; although it experiences normal strain, the strain rate $K_S = - \partial u / \partial x$ balances the dilatation rate resulting from gas compression such that ${\mathbb {K}} = K_S + \boldsymbol \nabla \boldsymbol {\cdot } \boldsymbol v = 0$. The flame thickness however decreases as the pressure rises and the influence of the internal structure on the flame speed, given by (5.5), diminishes in time. Equation (5.18) for the pressure simplifies to

(6.6)\begin{equation} \frac{\mathrm{d}P}{\mathrm{d}t} = \frac {\gamma q}{L} \left\{ M - \delta\left( \frac{T_a}{T_{f} } \right)^{2} \frac{\beta(Le_{eff} - 1)}{2q} \frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{P J_2}{M} \right) \right\}, \end{equation}

with the only changes at $O(\delta )$ resulting from non-unity Lewis number effects. Their contributions however are generally small and amount to less than $5\,\%$ for typical values of $\delta$ and a wide range of $Le_{eff}$. The RH condition (5.9) combined with (5.19) yields

(6.7)\begin{equation} x_{f} = \left(1 - \frac{P_{e} - P}{P_{e}-1} P^{{-}1/\gamma} \right) L + \frac{\delta}{q } \left(\frac{T_f J_1}{M} - \frac{T_a \widehat{J_1}}{ P^{1/\gamma}} \right), \end{equation}

relating flame location to pressure. Although the gas velocity is affected to $O(\delta )$ by the corresponding variations in pressure, the velocity in the burned gas includes an additional correction resulting from the right-hand side of (5.13), such that

(6.8)\begin{equation} u ={-} \left.\frac{\dot{P}}{\gamma P} x + \delta \lambda f'(x P^{1/\gamma})\right|^x_{x=0}, \quad 0 < x < x_f, \end{equation}

where the prime in $f^{\prime }$ denotes differentiation with respect to the function's argument and $\lambda$ at the wall is evaluated at $T(0) = P^{(\gamma - 1)/\gamma } T_a$. Overall, the contributions arising from the internal flame structure on the propagation of a planar flame are small.

The results are illustrated in figures 7 and 8 for a representative value $\delta = 0.05$, starting with the solution of an unconfined planar flame at $x = 0$. Although the computations account for the $O(\delta )$ corrections, the solutions appear indistinguishable from their leading-order counterparts.

Figure 7.

(a) The pressure $P$ and (b) the mass burning rate $m^*$ as a function of time, for a planar flame propagating in a closed channel, for various values of the heat release parameter $q$.

Figure 8.

(a) Density and (b) temperature profiles of a planar flame in a closed channel at equal time intervals, for a representative value $q=6$.

Figure 7 shows the dependence of the pressure $P$ and burning rate $m^*$ on time, for several values of the heat release parameter $q$. The pressure and burning rate, which at first grow linearly, increase rapidly after the flame has travelled beyond the middle of the vessel. Although the transition appears to occur earlier for larger values of $q$, the duration of flame propagation is also shortened. Note that the overall pressure increases by nearly ten fold, approaching $P_{e} = 1 + \gamma q$ when the flame reaches the end of the channel.

Density and temperature profiles at equal time intervals are shown in figure 8. The gas on both sides of the flame is compressed continuously as the pressure rises. In the unburned gas region the density and temperature remain spatially uniform but vary in time. The larger pressure translates primarily in increasing the density of the fresh mixture, which reaches a value of approximately five times its initial value, near the end of propagation. The temporally increasing temperature and density of the moving flame front are advected with the flow upstream such that their values in the burned gas region are no longer spatially uniform. A temperature gradient is established in the burned gas region and is responsible for the re-illumination observed in photographic records of flames propagating in closed spherical bombs at the centre of the vessel, a phenomenon known as the Flamm-Mache effect (Flamm & Mache Reference Flamm and Mache1917; Lewis & von Elbe Reference Lewis and von Elbe1987). The finite temperature gradient observed at $x=0$ can be smoothed out in a thin thermal boundary layer to ensure the adiabaticity condition at the wall.

Velocity profiles are plotted in figure 9(a) for the representative value $q = 6$. Initially the burned gas trapped between the flame and the left wall is at rest, the flame advances at a speed $\dot x_{f} \sim 1+q$ and the fresh mixture is forced to the right at a constant velocity $u^* \sim q$. As the flame propagates forward, the fresh mixture is being forced against a denser gas causing it to slow down, gradually coming to rest at the end of the channel. The jump in velocity across the flame increases due to the higher flame temperature and the burned gas is swiftly advected away from the flame at an increasing speed. The flame propagation speed $\dot x_{f} = S_{f} + u^*$ results from a balance between the increasing flame speed and the gradual reduction in gas velocity ahead of the flame front. The flame speed $S_{f}$ and propagation speed $\dot x_{f}$ are shown in figure 9(b) for increasing values of $q$, starting respectively from $1$ and $1+q$ at time $t=0$ to a common value when the flame reaches the end of the vessel and $u^*=0$. Being an intrinsic property of the combustion process, the flame speed (dashed curves) increases continuously in time and is higher for larger values of heat release. The propagation speed (solid curves) on the other hand depends on the flow conditions, such that the flame accelerates throughout the vessel for low values of $q$ and decelerates for large values of $q$. Indeed, the velocity profiles shown in figure 9(a) at equal time intervals demonstrate that the flame for $q=6$ decelerates throughout the channel.

Figure 9.

Planar flames propagating in a closed channel: (a) velocity profiles at equal time intervals, for a representative value $q=6$, and (b) flame speed $S_{f}$ (dashed curves) and propagation speed $\dot x_f$ (solid curves) as a function of time for various values of $q$.

6.2. Spherical flame

We consider now the propagation of a centrally ignited flame in a closed spherical vessel of radius $L$, as shown in figure 6(b). The flame front is described by $\psi (r,t) = R(t) - r = 0$, such that $0 < r < R$ corresponds to the burned gas and $R < r < L$ to the unburned gas region. The propagation speed is then given by $V_{f} = - \dot R$. The ratio of the flame surface area to the volume of the vessel, ${\mathbb {A}}_{f}/{\mathbb {V}} = 3 R^2/L^3$, increases in time as the flame propagates outward.

To leading order, the gas velocity determined from (5.13) and satisfying $u(0,t) = u(L,t)=0$ is given by

(6.9)\begin{equation} v(r,t) = \left\{\begin{array}{@{}ll@{}} \dfrac{\dot P}{3 \gamma P} \left(\dfrac{L^3}{r^2} - r \right), & R < r < L, \\ -\dfrac{\dot P}{3 \gamma P} r, & 0 < r < R, \end{array}\right. \end{equation}

with the pressure obtained by solving $\dot P = 3 \gamma q M R^2/L^3$ subject to $P(0) =1$, and $M$ given by (4.22). The RH relation $[[ \rho (\dot R - v) ]] = 0$ then yields the following expression for the flame position:

(6.10)\begin{equation} R(t) = \left(1 - \frac{P_{e} - P}{P_{e} -1} P^{{-}1/\gamma} \right)^{1/3}L. \end{equation}

Determining the state of the burned gas requires solving the advection equation

(6.11)\begin{equation} \frac{\partial\mathcal{E}}{\partial t} - \frac{\dot{P}}{3\gamma P} r \frac{\partial\mathcal{E}}{\partial r} = 0 \end{equation}

for the entropy function, subject to $\mathcal {E} = \mathcal {E}_b$ at $r = R$. The general solution is $\mathcal {E} = f(r P^{1/3\gamma })$ where the function $f(\zeta )$ is obtained from the relation

(6.12)\begin{equation} f(\zeta) = 1+ \frac{q}{P^{(\gamma-1)/\gamma}}\quad \text{for}\ \zeta = \left(P^{1/\gamma}-\frac{P_e-P}{P_e-1}\right)^{1/3} L. \end{equation}

Similar to the equivalent function for the planar solution, $f(0) = 1+q$ and $f(\zeta ) \sim 1$ as $\zeta \to \infty$; its relevance is limited to values of $\zeta$ corresponding to $1 \le P \le P_{e}$.

Next, we examine the influences that result from the finite thickness of the flame. Unlike the planar flame, the surface of the spherical flame is stretched at a rate $\mathbb {K} = 2\dot R/R$ apparent from the fundamental expression (5.20). We recall that for a flame under confinement, the stretch rate is not a simple combination of curvature and strain, and is obtained only when combined with the volume dilatation rate. Moreover, the flame experiences normal strain at different rates on the unburned and burned sides of the flame surface, given by

(6.13a,b)\begin{equation} K_S^{u} = \frac{qM}{P} \left(\frac{2}{R} + \frac{R^2}{L^3}\right),\quad K_S^{b} = \frac{qM}{P} \frac{R^2}{L^3}, \end{equation}

respectively. When substituted in (5.22) with the curvature term $(M/\rho )\kappa$ evaluated appropriately, one finds that

(6.14)\begin{equation} \mathbb{K} = \left(\frac{M}{P^{1/\gamma}} + \frac{qM}{P}\right) \frac{2}{R} - \frac{qM}{P} \frac{2 R^2}{L^3} = \frac{2 \dot R}{R}, \end{equation}

as expected. The flame speed is given by (5.5), with the corrections resulting from flame stretch most significant when the flame is relatively small. Equation (5.18) for the pressure simplifies to

(6.15)\begin{equation} \frac{\mathrm{d}P}{\mathrm{d}t} = \frac{3\gamma q R^2}{L^3} \left\{ M - \delta \left( \frac{T_a}{T_{f} } \right)^{2} \frac{\beta(Le_{eff} - 1)}{2q R^2} \frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{P J_2}{M} R^2 \right) \right\} \end{equation}

with the only changes at $O(\delta )$ resulting from non-unity Lewis number effects. As for the planar flame, their contribution is generally small for a wide range of $Le_{eff}$. The RH condition (5.9) combined with (5.19) yields the equation

(6.16)\begin{equation} R^3 = \left(1 - \frac{P_{e}-P}{P_{e} -1} P^{{-}1/\gamma} \right) L^3 - \delta \left(\frac{3 R^2}{q} \frac{T_f J_1}{M} \right) \end{equation}

relating the flame location to the pressure. The gas velocity is affected to $O(\delta )$ by the pressure variations, with an additional correction in the burned gas region resulting from the right-hand side of (5.13). One finds that

(6.17)\begin{equation} v(r,t) ={-} \frac{\dot{P}}{3 \gamma P} r + \delta \left.\frac{\lambda}{P^{2/3\gamma}} f'(r P^{1/3\gamma})\right|^r_{r=0},\quad 0 < r < R, \end{equation}

where $\lambda = \lambda (T)$ is evaluated at the origin where $T(0) = P^{(\gamma - 1)/\gamma } T_a$.

The results presented below correspond to a spherical flame evolving from a small unconfined flame kernel and, unless otherwise specified, $\delta =0.05$. Although the computations account for the $O(\delta )$ corrections, their contributions are generally small except for the flame speed and propagation speed; the solutions of the state variables remain indistinguishable from their leading-order counterparts.

The dependence of the pressure level $P$ on time is shown in figure 10(a) for several values of the heat release parameter $q$. The initial linear rise in $P$ appears much slower than for a planar flame, but the sharp increase occurring after the flame has travelled beyond the middle of the vessel is more rapid, being proportional to the flame surface area $R^2$. When increasing $q$, the transition appears to occur at an earlier time, but the duration of flame propagation is also shorter. Unlike the planar flame, the effect of flame stretch on the flame speed and, consequently, on the mass burning rate $m^*$ is evident in figure 10(b). In the absence of stretch, with $P \sim 1$ during the early stages of propagation the flame speed would remain close to the laminar flame speed with $m^* \sim 1$. The incipient spherical flame, however, is highly stretched such that $S_{f}$, and consequently $m^*$, are significantly lower for practically all values of the unburned Markstein number $\mathcal {M}_1$. This behaviour changes at later times due to the diminished effect of stretch on one hand and the pressure rise on the other.

Figure 10.

The pressure $P$ and burning rate $m^*$ as a function of time, for a spherically expanding flame in a closed vessel at various values of the heat release parameter $q$ and $Le_{eff} = 1$.

Profiles of density and temperature are shown in figure 11 at equal time intervals for the representative value of $q = 6$. The results are similar to those observed for planar flames. The gas on both sides of the flame is compressed continuously as the pressure rises. In the unburned gas the density and temperature remain spatially uniform, but vary in time. The pressure rise translates primarily in increasing the gas density with the temperature increasing only slightly. In the burned gas the density and temperature are no longer spatially uniform; the temporally varying temperature and density of the moving flame front are advected with the flow in the burned gas region. The temperature increases significantly reaching its maximum value at the centre of the vessel, as observed by Flamm & Mache (Reference Flamm and Mache1917).

Figure 11.

Density and temperature profiles of a spherical flame in a closed vessel at equal time intervals, for a representative value $q=6$.

Velocity profiles at equal time intervals are shown in figure 12 for $q=6$, $Le_{eff} = 1$ and two values of $\delta$. The sharp discontinuity represents the flame location. Initially the burned gas is at rest and the propagating flame drives the fresh mixture ahead into motion. As the flame grows outwards, the velocity across the flame increases because of the higher flame temperature and the burned gas is swiftly advected away from the flame at an increasing speed. The fresh mixture that moves against a denser gas slows down, coming to rest at the end of the vessel. Two profiles are shown at each time; the dashed curves correspond to the leading-order solution, absent of flame stretch effects, and the solid curves to the complete solution. For $\delta = 0.05$, the expanding stretched flame is seen to propagate slower, lagging behind the solution predicted in the absence of stretch. When stretch effects are underplayed, as for $\delta = 0.001$, the solutions with and without the $O(\delta )$ corrections appear closer.

Figure 12.

Velocity profiles of a spherical flame propagating in a closed vessel at equal time intervals. The dashed lines correspond to the leading-order solution, absent of stretch effects, and the solid lines to the solution with $O(\delta )$ contributions included; for $q=6$ and $Le_{eff} = 1$. Results are shown for (a) $\delta = 0.05$ and (b) $\delta = 0.001$.

We now consider the effects of flame stretch on the propagation speed and its dependence on the properties of the combustible mixture. It is instructive to examine first the simplifications that result when the flame propagates in a sufficiently large vessel ($L \gg 1$). In this limit, the equation for the pressure takes the form

(6.18)\begin{equation} \frac{\mathrm{d}P}{\mathrm{d}t} \sim \frac{3 \gamma q}{L^3} \left(1 - \delta \frac{\beta(Le_{eff} - 1)}{2} \hat J_2 \mathbb{K} \right) R^2, \end{equation}

with the velocity field given by

(6.19)\begin{equation} v(r,t) \sim q \left(1 - \delta \frac{\beta(Le_{eff} - 1)}{2} \hat J_2 {\mathbb{K}}\right) \left\{\begin{array}{@{}ll@{}} \dfrac{R^2}{r^2} - \dfrac{R^2}{L^3} r, & R < r < L, \\ -\dfrac{ R^2}{L^3} r, & 0 < r < R. \end{array}\right. \end{equation}

When $L \to \infty$, we recover the solution of a centrally ignited freely propagating spherical flame; the pocket of burned gas encompassed by the flame remains at rest, the unburned gas forced forward decays according to the inverse of the square of radial distance, and the flame propagation speed is given by

(6.20)\begin{equation} \dot R \sim (1+q) \left(1 - \delta \mathcal{M}_1^{b} \frac{2 (1+q)}{R} \right). \end{equation}

The propagation speed highlights the combined effects of stretch and differential diffusion, as shown in figure 13(a). In mixtures with $Le_{eff} > {Le}^{cr}_{eff}$, for which $\mathcal {M}_1^{b} > 0$, the spherical flame expands slower as compared with the unstretched flame (dashed line) and accelerates toward the asymptotic value $\dot R = 1+q$, associated with the (burned) laminar flame speed. In mixtures with $Le_{eff} < {Le}^{cr}_{eff}$ the flame expands at a relatively faster rate that is gradually reduced due to stretching and decelerates towards the asymptotic value $\dot R = 1+q$. This stretch-dominated behaviour was observed experimentally in various mixtures (Strehlow Reference Strehlow1984; Kwon, Tseng & Faeth Reference Kwon, Tseng and Faeth1992; Beeckmann et al. Reference Beeckmann, Hesse, Kruse, Berens, Peters, Pitsch and Matalon2017) by varying the equivalence ratio, and confirmed theoretically by Frankel & Sivashinsky (Reference Frankel and Sivashinsky1983), Bechtold & Matalon (Reference Bechtold and Matalon1987) and Addabbo et al. (Reference Addabbo, Bechtold and Matalon2002). It persists when the propagation occurs in a closed vessel as illustrated in figure 13(b) for $\delta = 0.001$, but is limited to the early time before the flame encounters a significantly denser gas and slows down until reaching the end of the vessel. The intersection of the curves for various $Le_{eff}$ with the dashed line corresponding to the leading-order solution occurs when the flame zone contributions vanish, namely at radii within $O(\delta )$ of each other. For $\delta = 0.05$, stretch effects extend to larger radii as shown in figure 14. For $Le_{eff} > {Le}^{cr}_{eff}$, the flame accelerates, reaches a peak at nearly the midway point to the vessel wall before slowing down, while for smaller values of $Le_{eff}$, the flame undergoes monotonic deceleration throughout the bulk of the propagation. Finally, we note that the small increase in speed observed when $R \approx 100$ is due to the limitation of the model that does not account for the interaction of the flame zone with the vessel's walls.

Figure 13.

Propagation speed of spherical flames for various $Le_{eff}$; plots correspond to $q=6$ and $\delta =0.001$; the dashed line indicates the solution (leading-order) in the absence of stretch effects. (a) Freely propagating flame. (b) Propagation in a closed vessel.

Figure 14.

Propagation speed of spherical flames for various $Le_{eff}$; plots correspond to $q=6$ and $\delta =0.05$; the dashed line indicates the solution (leading-order) in the absence of stretch effects.

Since the problem under consideration is inherently unsteady, it is appropriate to comment on the initial state of the mixture and its effect on the flame propagation. As noted earlier, the present theory does not describe the ignition event and assumes that a flame already exists at $t=0$. However, when igniting the mixture, a small pocket of burned gas of radius $R_0$ and temperature $T_a$ is first formed and then propagates outward. Accounting for this condition when integrating (2.21) leads to

(6.21)\begin{equation} P_{e} = 1 + \gamma q - \gamma q (R_0/L)^3, \end{equation}

which modifies the dependence of the flame size on pressure (6.16) to read

(6.22)\begin{equation} R^3 = \left(1 - \frac{P_{e}-P}{\gamma q} P^{{-}1/\gamma} \right) L^3 - \delta \left(\frac{3 R^2}{q} \frac{T_f J_1}{M} - \frac{3 R_0^2}{q} T_a \hat J_1 P^{{-}1/\gamma}\right). \end{equation}

Since the initial kernel of radius $R_0$ is of the order of the flame thickness $\delta$, the correction term is extremely small and can be neglected.

Constant volume combustion chambers are commonly used in the laboratory to determine the laminar flame speed of different mixtures. In one approach, measurements of centrally ignited expanding flames are taken up to a limited distance and the laminar flame speed is determined by extrapolating the data to the zero-stretch limit; namely from the asymptotic value $\dot R = 1 + q$ shown in figure 13(a). Data are collected during the early stages of propagation when pressure rise is minimal so that the burned gas enclosed by the flame remains essentially at rest and, thus, $\dot {R} = S_{f}^{b}$. The laminar flame speed is then given by $S_{f} = (\rho _b/\rho _u) S_{f}^{b}$ with $\rho _b$ and $\rho _u$ taken as constants. However, unless the experimental set-up ensures that the ambient pressure remains constant, the results may be tainted by small pressure changes, effects due to compression and by the inward flow of burned gas that becomes substantial when the flame grows bigger as shown in figure 12. The present theory provides a means for estimating these effects in sufficiently large chambers. The dependence of the flame radius on the pressure level given by (6.10), and the gas velocity in the burned gas given by (6.9), may be used to more accurately determine the distance where measurements can be taken without being significantly affected by fluid motion and compressibility effects. Figure 13(b) illustrates a scenario where the laminar flame speed can be reasonably estimated from measurements taken up to $R \approx 30$, in contrast to the conditions presented in figure 14.

Another approach to determine the laminar flame speed involves the use of smaller spherical vessels with thick walls that can accommodate higher pressures. Transducers measure the pressure history as the flame propagates outward, and flame location, propagation speed and flow velocities are then determined from theoretically derived relationships, which are essentially the leading-order expressions presented here. Our theory provides corrections to these quantities that account for effects resulting from the internal flame structure, as given by (6.16) and (6.17). When included in data extrapolation these contributions may lead to better accuracy.

The same reasoning to the aforementioned comments applies to the determination of the Markstein length that, as the present theory demonstrates, is affected by the rise in pressure and flame temperature. Additionally, the experimental pursuit may be hampered by onset of the hydrodynamic instability that may be enhanced by the attenuated influence of diffusion on the thinner flames that result under confinement.