3.2.1 Integration over ${\mathcal{V}}_{u}$

In the unburned gases volume ${\mathcal{V}}_{u}$ , we have $\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{u}$ and $Y_{f}^{\ast }=1$ together with $\unicode[STIX]{x1D714}_{f}^{\ast }=0$ everywhere. Therefore, (3.9) writes

(3.10) $$\begin{eqnarray}\displaystyle \text{MV}|_{{\mathcal{V}}_{u}}+\text{MF}|_{{\mathcal{V}}_{u}} & = & \displaystyle \unicode[STIX]{x1D70C}_{u}\frac{\text{d}}{\text{d}t}\iiint _{{\mathcal{V}}_{u}(t)}\,\text{d}^{3}V+\unicode[STIX]{x1D70C}_{u}\unicode[STIX]{x0222F}_{\unicode[STIX]{x2202}{\mathcal{V}}_{u}(t)}\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{n}\,\text{d}^{2}A+\unicode[STIX]{x1D70C}_{u}\unicode[STIX]{x0222F}_{\unicode[STIX]{x2202}{\mathcal{V}}_{u,b}(t)}\underbrace{(\boldsymbol{u}-\boldsymbol{w})\boldsymbol{\cdot }\boldsymbol{n}}_{S_{d}^{u}}\,\text{d}^{2}A\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D70C}_{u}\frac{\text{d}{\mathcal{V}}_{u}}{\text{d}t}+\unicode[STIX]{x1D70C}_{u}\unicode[STIX]{x0222F}_{\unicode[STIX]{x2202}{\mathcal{V}}_{u}(t)}\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{n}\,\text{d}^{2}A+\unicode[STIX]{x1D70C}_{u}\langle S_{d}^{u}\rangle A_{f}=0.\end{eqnarray}$$

The budget in the unburned volume ${\mathcal{V}}_{u}$ yields an expression for the area-weighted displacement speed $\langle S_{d}^{u}\rangle =\langle (\boldsymbol{u}-\boldsymbol{w})\boldsymbol{\cdot }\boldsymbol{n}\rangle$ (note that in this case $\boldsymbol{n}$ is directed towards the burned gases)

(3.11) $$\begin{eqnarray}\displaystyle \langle S_{d}^{u}\rangle =\underbrace{-\frac{1}{A_{f}}\frac{\text{d}{\mathcal{V}}_{u}}{\text{d}t}}_{{\mathcal{U}}_{MV}|_{{\mathcal{V}}_{u}}}\underbrace{-\frac{1}{A_{f}}\unicode[STIX]{x0222F}_{\unicode[STIX]{x2202}{\mathcal{V}}_{u}(t)}\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{n}\,\text{d}^{2}A}_{{\mathcal{U}}_{MF}|_{{\mathcal{V}}_{u}}}. & & \displaystyle\end{eqnarray}$$

The superscript $u$ has been kept for designating $S_{d}^{u}$ as it corresponds to the displacement speed of the leading edge $\unicode[STIX]{x2202}{\mathcal{V}}_{u,b}$ . $A_{f}$ is defined as the area of $\unicode[STIX]{x2202}{\mathcal{V}}_{u,b}$ , i.e. the area of the leading edge. (It was checked from the present measurements and DNSs that translating the leading edge spatial coordinates in the normal direction of a distance equivalent to the flame thickness did not lead to substantial difference in $A_{f}$ . This is indicative that $|\boldsymbol{J}|=1$ is plausible in this case.) The terms ${\mathcal{U}}_{MV}|_{{\mathcal{V}}_{u}}$ and ${\mathcal{U}}_{MF}|_{{\mathcal{V}}_{u}}$ are introduced only for illustrative purposes, and will be useful later when plotting figures 8 and 11.

3.2.2 Integration over ${\mathcal{V}}_{b}$

The MV term integrated over ${\mathcal{V}}_{b}$ cannot be measured as we have no information about $Y_{f}^{\ast }$ and $\unicode[STIX]{x1D70C}$ in the flame. One solution consists in considering an infinitely thin flame so that the time derivative of the integral over ${\mathcal{V}}_{b}$ is negligible compared to that over ${\mathcal{V}}_{u}$ . Consequently, neglecting the mass within the flame leads to

(3.12) $$\begin{eqnarray}\displaystyle \text{MV}|_{{\mathcal{V}}_{b}}=0. & & \displaystyle\end{eqnarray}$$

Another solution consists in expressing the volume integral appearing in $\text{MV}|_{{\mathcal{V}}_{b}}$ in the flame curvilinear basis (with again $|\boldsymbol{J}|=1$ )

(3.13) $$\begin{eqnarray}\displaystyle \iiint _{{\mathcal{V}}_{b}(t)}\unicode[STIX]{x1D70C}Y_{f}^{\ast }\,\text{d}^{3}V=\iint _{A_{f}}\underbrace{\left[\int _{\unicode[STIX]{x1D702}(Y_{f}^{\ast }=0)}^{\unicode[STIX]{x1D702}(Y_{f}^{\ast }=1)}\unicode[STIX]{x1D70C}Y_{f}^{\ast }\,\text{d}\unicode[STIX]{x1D702}\right]}_{\unicode[STIX]{x1D70C}_{u}{\mathcal{F}}}\,\text{d}^{2}A_{f}. & & \displaystyle\end{eqnarray}$$

Then, following Poinsot & Veynante (Reference Poinsot and Veynante2005), one can take the mean value of each variable within the integral, viz.

(3.14) $$\begin{eqnarray}\displaystyle {\mathcal{F}}=\frac{1}{\unicode[STIX]{x1D70C}_{u}}\left(\frac{\unicode[STIX]{x1D70C}_{u}+\unicode[STIX]{x1D70C}_{b}}{2}\right)\left(\frac{Y_{f,u}^{\ast }+Y_{f,b}^{\ast }}{2}\right)\unicode[STIX]{x1D6FF}_{L}=\left(1+\frac{\unicode[STIX]{x1D70C}_{b}}{\unicode[STIX]{x1D70C}_{u}}\right)\frac{\unicode[STIX]{x1D6FF}_{L}}{4}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}_{L}$ is the flame thickness whose appropriate definition may be here $\unicode[STIX]{x1D702}(Y_{f}^{\ast }=0.99)-\unicode[STIX]{x1D702}(Y_{f}^{\ast }=0.01)$ . Otherwise, following Poinsot & Veynante (Reference Poinsot and Veynante2011), we can take the average value of $\unicode[STIX]{x1D70C}Y_{f}^{\ast }$ , viz.

(3.15) $$\begin{eqnarray}\displaystyle {\mathcal{F}}=\frac{1}{\unicode[STIX]{x1D70C}_{u}}\left(\frac{\unicode[STIX]{x1D70C}_{u}Y_{f,u}^{\ast }+\unicode[STIX]{x1D70C}_{b}Y_{f,b}^{\ast }}{2}\right)\unicode[STIX]{x1D6FF}_{L}=\frac{\unicode[STIX]{x1D6FF}_{L}}{2}. & & \displaystyle\end{eqnarray}$$

The mass variation term $\text{MV}|_{{\mathcal{V}}_{b}}$ can thus be rewritten as

(3.16) $$\begin{eqnarray}\displaystyle \text{MV}|_{{\mathcal{V}}_{b}}=\unicode[STIX]{x1D70C}_{u}\frac{\text{d}\langle {\mathcal{F}}\rangle A_{f}}{\text{d}t}. & & \displaystyle\end{eqnarray}$$

Since we do not have models for $\langle {\mathcal{F}}\rangle$ (somehow related to the flame shrinking or thickening) that could emerge from first principles, we will consider $\langle {\mathcal{F}}\rangle ={\mathcal{F}}=\text{const}$ . This hypothesis will be tested using DNS of a FVI in section § 4.2. Then,

(3.17) $$\begin{eqnarray}\displaystyle \text{MV}|_{{\mathcal{V}}_{b}}=\unicode[STIX]{x1D70C}_{u}{\mathcal{F}}\frac{\text{d}A_{f}}{\text{d}t}. & & \displaystyle\end{eqnarray}$$

Here ${\mathcal{F}}=0$ when the contribution of the flame thickness is neglected, ${\mathcal{F}}=(1+\unicode[STIX]{x1D70C}_{b}/\unicode[STIX]{x1D70C}_{u})\unicode[STIX]{x1D6FF}_{L}/4$ (Poinsot & Veynante Reference Poinsot and Veynante2005) or ${\mathcal{F}}=\unicode[STIX]{x1D6FF}_{L}/2$ (Poinsot & Veynante Reference Poinsot and Veynante2011). CHEMKIN simulations indicate that ${\mathcal{F}}=\unicode[STIX]{x1D6FF}_{L}/2$ is more suited than ${\mathcal{F}}=(1+\unicode[STIX]{x1D70C}_{b}/\unicode[STIX]{x1D70C}_{u})\unicode[STIX]{x1D6FF}_{L}/4$ . It is worth further emphasizing that $\unicode[STIX]{x1D70C}_{b}$ is not measured. This parameter needs thus to be estimated a priori from equilibrium calculations which rely on some assumptions such as flame adiabaticity that might not be accurate in general. Some other authors (Bradley et al. Reference Bradley, Gaskell and Gu1996) have proposed a different hypothesis for assessing ${\mathcal{F}}$ but it relies on an estimation of $\unicode[STIX]{x1D70C}_{b}$ and is thus not treated here for the same reason. We will hereafter consider only cases where ${\mathcal{F}}=0$ or ${\mathcal{F}}=\unicode[STIX]{x1D6FF}_{L}/2$ that will be tested by DNS in § 4.2. It is worth stressing that the present method, where use is made of ${\mathcal{F}}=\unicode[STIX]{x1D6FF}_{L}/2=\text{const.}$ is thus not suitable when there is significant flame extinction or large variations of flame thickness. It is restricted to the flamelet regime of combustion. Note however that the integral approach for measuring $\langle S_{d}^{u}\rangle$ does not require any hypothesis regarding the flame structure and can thus be used irrespective of the combustion regime. The integral approach depends on ad hoc coefficients for inferring the mass contained within the flame that can, at best, be estimated. Different estimates may yield different conclusions; this is in contrast to the asymptotic derivation that provides a solution with no undetermined coefficients.

The MF term integrated over ${\mathcal{V}}_{b}$ leads to

(3.18) $$\begin{eqnarray}\displaystyle \text{MF}|_{{\mathcal{V}}_{b}} & = & \displaystyle \unicode[STIX]{x0222F}_{\unicode[STIX]{x2202}{\mathcal{V}}_{u,b}(t)+\unicode[STIX]{x2202}{\mathcal{V}}_{f}(t)}[\unicode[STIX]{x1D70C}Y_{f}^{\ast }(\boldsymbol{u}+\boldsymbol{v}_{f}-\boldsymbol{w})\boldsymbol{\cdot }\boldsymbol{n}]\,\text{d}^{2}A\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D70C}_{u}\unicode[STIX]{x0222F}_{\unicode[STIX]{x2202}{\mathcal{V}}_{u,b}}\underbrace{(\boldsymbol{u}-\boldsymbol{w})\boldsymbol{\cdot }\boldsymbol{n}}_{-S_{d}^{u}}\,\text{d}^{2}A+\unicode[STIX]{x0222F}_{\unicode[STIX]{x2202}{\mathcal{V}}_{f}}[\unicode[STIX]{x1D70C}Y_{f}^{\ast }(\boldsymbol{u}+\boldsymbol{v}_{f}-\boldsymbol{w})\boldsymbol{\cdot }\boldsymbol{n}]\,\text{d}^{2}A\nonumber\\ \displaystyle & = & \displaystyle -\unicode[STIX]{x1D70C}_{u}\langle S_{d}^{u}\rangle A_{f}+\unicode[STIX]{x0222F}_{\unicode[STIX]{x2202}{\mathcal{V}}_{f}}[\unicode[STIX]{x1D70C}Y_{f}^{\ast }(\boldsymbol{u}+\boldsymbol{v}_{f}-\boldsymbol{w})\boldsymbol{\cdot }\boldsymbol{n}]\,\text{d}^{2}A.\end{eqnarray}$$

Here, we neglect the diffusion velocity $\boldsymbol{v}_{f}\boldsymbol{\cdot }\boldsymbol{n}$ at the boundary $\unicode[STIX]{x2202}{\mathcal{V}}_{f}$ assuming that the species gradient is oriented in the flame normal direction. Second, we have $\boldsymbol{w}\boldsymbol{\cdot }\boldsymbol{n}=0$ at $\unicode[STIX]{x2202}{\mathcal{V}}_{f}$ . Finally, we take the average value of $\unicode[STIX]{x1D70C}Y_{f}^{\ast }\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{n}$ between the burned and fresh regions to obtain:

(3.19) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x0222F}_{\unicode[STIX]{x2202}{\mathcal{V}}_{f}}\unicode[STIX]{x1D70C}Y_{f}^{\ast }\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{n}=\frac{\unicode[STIX]{x1D70C}_{u}Y_{f,u}^{\ast }v_{r,u}+\unicode[STIX]{x1D70C}_{b}Y_{f,b}^{\ast }v_{r,b}}{2}\unicode[STIX]{x03C0}\unicode[STIX]{x0394}\unicode[STIX]{x1D6FF}_{L}, & & \displaystyle\end{eqnarray}$$

where $v_{r,u}$ and $v_{r,b}$ is the fluid velocity tangential to the flame at the level of $\unicode[STIX]{x2202}{\mathcal{V}}_{f}$ (whose area is equal to $\unicode[STIX]{x03C0}\unicode[STIX]{x0394}\unicode[STIX]{x1D6FF}_{L}$ , $\unicode[STIX]{x1D6E5}$ is the diameter of the control volume), in the fresh and burned gases, respectively. Recalling that $Y_{f,b}^{\ast }=0$ , we end up with

(3.20) $$\begin{eqnarray}\displaystyle \text{MF}|_{{\mathcal{V}}_{b}}=-\unicode[STIX]{x1D70C}_{u}\langle S_{d}^{u}\rangle A_{f}+\unicode[STIX]{x1D70C}_{u}{\mathcal{F}}\unicode[STIX]{x03C0}\unicode[STIX]{x0394}v_{r,u}. & & \displaystyle\end{eqnarray}$$

With these simplifications, the integration of the mass budget (3.9) over ${\mathcal{V}}_{b}$ leads to

(3.21) $$\begin{eqnarray}\displaystyle \langle S_{c}\rangle =\langle S_{d}^{u}\rangle -{\mathcal{F}}\left(\frac{1}{A_{f}}\frac{\text{d}A_{f}}{\text{d}t}+\frac{1}{A_{f}}\unicode[STIX]{x03C0}\unicode[STIX]{x0394}v_{r,u}\right), & & \displaystyle\end{eqnarray}$$

or equivalently by using (3.11) for $\langle S_{d}^{u}\rangle$

(3.22) $$\begin{eqnarray}\displaystyle \langle S_{c}\rangle =\underbrace{-A_{f}^{-1}\left(\frac{\text{d}{\mathcal{V}}_{u}}{\text{d}t}+{\mathcal{F}}\frac{\text{d}A_{f}}{\text{d}t}\right)}_{{\mathcal{U}}_{MV}}\underbrace{-A_{f}^{-1}\left(\unicode[STIX]{x0222F}_{\unicode[STIX]{x2202}{\mathcal{V}}_{u}}\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{n}\,\text{d}^{2}A+{\mathcal{F}}\unicode[STIX]{x03C0}\unicode[STIX]{x0394}v_{r,u}\right)}_{{\mathcal{U}}_{MF}}. & & \displaystyle\end{eqnarray}$$

The ${\mathcal{U}}_{MV}$ term in (3.22) is sometimes referred to as the mass accumulation process while the ${\mathcal{U}}_{MF}$ term represents the net mass flux flowing through the system. Note that the consumption speed evaluated by considering the thin flame limit ( ${\mathcal{F}}=0$ ) is equal to the displacement speed $\langle S_{d}^{u}\rangle$ . Hereafter, we will note $\langle S_{c}^{0}\rangle =\langle S_{d}^{u}\rangle$ the consumption speed inferred with ${\mathcal{F}}=0$ and $\langle S_{c}^{1}\rangle$ that calculated with ${\mathcal{F}}=\unicode[STIX]{x1D6FF}_{L}/2$ . It is worth stressing that (3.21) highlights an explicit relationship between displacement speed (at the leading edge) and consumption speed and shows that the differences between $\langle S_{c}\rangle$ and $\langle S_{d}^{u}\rangle$ are connected to the mass contained within the flame and to the total stretch rate $1/A_{f}\,\text{d}A_{f}/\text{d}t$ . Another discrepancy between the behaviour of $\langle S_{c}\rangle$ and $\langle S_{d}^{u}\rangle$ relies on the tangential mass flux (through the appearance of $v_{r,u}$ , the tangential velocity) leaving the flame.

5.1.1 The total stretch rate

The time evolution of flame area and stretch rate during FVIs are discussed first. Similarly to Thiesset et al. (Reference Thiesset, Maurice, Halter, Mazellier, Chauveau and Gökalp2017), the stretch rate noted $\langle K\rangle$ is given in terms of relative increase in flame surface area evaluated over a domain of width $\unicode[STIX]{x1D6E5}$ (figure 4 a), viz.

(5.1) $$\begin{eqnarray}\displaystyle \langle K(\unicode[STIX]{x1D6E5},t)\rangle =\frac{1}{A_{f}(\unicode[STIX]{x1D6E5},t)}\frac{\text{d}A_{f}(\unicode[STIX]{x1D6E5},t)}{\text{d}t}. & & \displaystyle\end{eqnarray}$$

$\langle K\rangle$ should be interpreted here as the additional stretch rate due to the interaction with the vortex. Indeed, even when the flame is flat and stationary, stretch is not zero and is simply related to $K_{r}$ introduced in (4.2). The total surface-averaged stretch rate $\langle K_{tot}\rangle$ over a domain of diameter $\unicode[STIX]{x1D6E5}$ is the addition of the stretch rate due to the interaction with the vortex $\langle K\rangle$ and that due to the diverging stagnation point flow. It appears explicitly in (3.21) and reads

(5.2) $$\begin{eqnarray}\displaystyle \langle K_{tot}\rangle =\frac{1}{A_{f}}\frac{\text{d}A_{f}}{\text{d}t}+\frac{1}{A_{f}}\unicode[STIX]{x03C0}\unicode[STIX]{x0394}v_{r,u}. & & \displaystyle\end{eqnarray}$$

Hereafter, because the variations in time of the rightmost term of the right-hand side of (5.2) are much smaller than that of $\langle K\rangle$ , results will be presented as a function of $\langle K\rangle$ and not $\langle K_{tot}\rangle$ . The stretch rate quantity controls efficiency functions which are widely used in LES of turbulent premixed combustion (Colin et al. Reference Colin, Ducros, Veynante and Poinsot2000; Charlette et al. Reference Charlette, Meneveau and Veynante2002; Bougrine et al. Reference Bougrine, Richard, Colin and Veynante2014; Thiesset et al. Reference Thiesset, Maurice, Halter, Mazellier, Chauveau and Gökalp2017). It is worth emphasizing that $\langle K(\unicode[STIX]{x1D6E5},t)\rangle$ is the area-weighted average of the local stretch rate $K=K_{T}+K_{C}$ due to the interaction with the vortex. Indeed, starting from a transport theorem for material surfaces (equation (3) in (Candel & Poinsot Reference Candel and Poinsot1990))

(5.3) $$\begin{eqnarray}\displaystyle \frac{\text{d}}{\text{d}t}\int _{A_{f}}\,\text{d}^{2}A=\int _{A_{f}}(-\boldsymbol{n}\boldsymbol{n}:\unicode[STIX]{x1D735}\boldsymbol{u}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}+2S_{d}\unicode[STIX]{x1D705}_{m})\,\text{d}^{2}A & & \displaystyle\end{eqnarray}$$

or equivalently

(5.4) $$\begin{eqnarray}\displaystyle \frac{\text{d}A_{f}}{\text{d}t}=A_{f}\langle K\rangle , & & \displaystyle\end{eqnarray}$$

which yields (5.1). Thiesset et al. (Reference Thiesset, Maurice, Halter, Mazellier, Chauveau and Gökalp2017) showed that the stretch rate also depends on the width $\unicode[STIX]{x1D6E5}$ of the domain over which $A_{f}$ is measured: obviously for an infinite $\unicode[STIX]{x1D6E5}$ , the portion of the flame interacting with the vortex is negligible and $\langle K\rangle$ should tend to zero. In other words, because the portion of flame interacting with the vortex is constant (i.e. there exists a $\unicode[STIX]{x1D6E5}=\unicode[STIX]{x1D6E5}_{i}$ above which $\text{d}A/\text{d}t$ is independent of $\unicode[STIX]{x1D6E5}$ ), we expect $\langle K\rangle$ to decrease with $\unicode[STIX]{x1D6E5}^{2}$ since $A_{f}(t,\unicode[STIX]{x1D6E5})$ monotonically increases with $\unicode[STIX]{x1D6E5}^{2}$ . The evolution of $\langle K(\unicode[STIX]{x1D6E5},t)\rangle$ with respect to $\unicode[STIX]{x1D6E5}$ can be readily described by the following expression (Thiesset et al. Reference Thiesset, Maurice, Halter, Mazellier, Chauveau and Gökalp2017)

(5.5) $$\begin{eqnarray}\displaystyle \langle K\rangle (\unicode[STIX]{x1D6E5},t)=\langle K_{0}\rangle (t)\left(\frac{\unicode[STIX]{x1D6E5}_{i}-\unicode[STIX]{x1D6E5}_{o}}{\unicode[STIX]{x1D6E5}-\unicode[STIX]{x1D6E5}_{o}}\right)^{2}. & & \displaystyle\end{eqnarray}$$

In (5.5), $\unicode[STIX]{x1D6E5}_{i}$ represents the domain width $\unicode[STIX]{x1D6E5}$ above which $\text{d}A/\text{d}t$ is constant and $\unicode[STIX]{x1D6E5}_{0}$ is interpreted as a virtual origin, i.e. $\langle K\rangle ^{-1}\rightarrow 0$ when $\unicode[STIX]{x1D6E5}\rightarrow \unicode[STIX]{x1D6E5}_{o}$ . Thiesset et al. (Reference Thiesset, Maurice, Halter, Mazellier, Chauveau and Gökalp2017) pointed out that $\unicode[STIX]{x1D6E5}_{i}/R_{v}$ and $\unicode[STIX]{x1D6E5}_{o}/R_{v}$ ( $R_{v}$ is the vortex core-to-core distance) were constant irrespectively of the vortex strength and were equal to $2.5\pm 0.05$ and $-0.5\pm 0.1$ , respectively. In the present study, unless specified, results will be presented for $\unicode[STIX]{x1D6E5}=13~\text{mm}$ which corresponds approximatively to $\unicode[STIX]{x1D6E5}_{i}$ .

Figure 6.

Time evolution of the stretch rate for stoichiometric methane and lean ( $\unicode[STIX]{x1D719}=0.9$ ) propane/air mixtures for three different vortex strengths and $\unicode[STIX]{x1D6E5}=13~\text{mm}$ : (a)  $U_{\unicode[STIX]{x1D703}}/\langle S_{c}^{1}\rangle _{t_{0}}=0.8$ , (b)  $U_{\unicode[STIX]{x1D703}}/\langle S_{c}^{1}\rangle _{t_{0}}=1.2$ , (c)  $U_{\unicode[STIX]{x1D703}}/\langle S_{c}^{1}\rangle _{t_{0}}=2.8$ . The time $t$ has been shifted so that $t=0$ corresponds to the time at which $\langle K\rangle$ is maximum.

We consider FVIs for a stoichiometric methane/air mixture characterized by an effective Lewis number close to unity ( $Le=1.05$ ) and a lean ( $\unicode[STIX]{x1D719}=0.9$ ) propane/air mixture for which the Lewis number is about 1.5. The method of Bechtold & Matalon (Reference Bechtold and Matalon2001) was employed for calculating the effective Lewis numbers. Three different vortex strengths are investigated $U_{\unicode[STIX]{x1D703}}/\langle S_{c}^{1}\rangle _{t_{0}}=0.8$ , 1.2 and 2.8.

Results (figure 6) indicate that for the range of vortex strengths investigated here, the magnitude of the total stretch rate increases with $U_{\unicode[STIX]{x1D703}}/\langle S_{c}^{1}\rangle _{t_{0}}$ . One further notes that there is no significant influence of the Lewis number on the creation of surface area as the time evolution of $\langle K\rangle$ is very similar irrespectively of the type of fuel investigated. The vortex efficiency, i.e. $\max (\langle K\rangle )/(U_{\unicode[STIX]{x1D703}}/R_{v})$ is equal to 0.73, 0.88 and 1.06 for methane/air flames at $U_{\unicode[STIX]{x1D703}}/\langle S_{c}^{1}\rangle _{t_{0}}=0.8$ , 1.2 and 2.8, respectively. For propane/air flames, the stretch efficiency is found to be equal to 0.78, 0.92 and 1.06 for $U_{\unicode[STIX]{x1D703}}/\langle S_{c}^{1}\rangle _{t_{0}}=0.8$ , 1.2 and 2.8, respectively. This observation is in agreement with the DNS data of Bougrine et al. (Reference Bougrine, Richard, Colin and Veynante2014), where the differences in stretch rates between $\text{CH}_{4}$ and $\text{C}_{3}\text{H}_{8}$ were small. This indicates that when considering fuels with $Le\geqslant 1$ , Lewis number corrections to the stretch efficiency functions might not be necessary.

As the flame positions (and thus area and stretch) can be computed from a purely kinematic equation (the level-set equation) (Moureau, Fiorina & Pitsch Reference Moureau, Fiorina and Pitsch2009; Thiesset et al. Reference Thiesset, Maurice, Halter, Mazellier, Chauveau and Gökalp2017), the independence of $\langle K\rangle$ to the Lewis number indicates a priori that the evolution of the flame displacement $S_{d}$ during a FVI remains similar irrespectively of the Lewis number (at least for $Le\geqslant 1$ ).

5.1.2 Flame curvature

In addition to (3.11) which provides a means to experimentally assess the area-weighted flame displacement speed, the simultaneous PIV and tomographic measurements allow local values (i.e. as a function of the position on the interface) of $S_{d}^{u}$ to be evaluated by (Peters Reference Peters2000; Poinsot & Veynante Reference Poinsot and Veynante2011)

(5.6a ) $$\begin{eqnarray}\displaystyle S_{d}^{u}=(\boldsymbol{w}-\boldsymbol{u})\boldsymbol{\cdot }\boldsymbol{n}=w_{n}-u_{n}=\left.\frac{1}{|\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}|}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\right)\right|_{\unicode[STIX]{x1D713}=\unicode[STIX]{x1D713}_{0}}, & & \displaystyle\end{eqnarray}$$

with

(5.6b ) $$\begin{eqnarray}\displaystyle \boldsymbol{n}=-\frac{\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}}{|\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}|}, & & \displaystyle\end{eqnarray}$$

where $w_{n}=\boldsymbol{w}\boldsymbol{\cdot }\boldsymbol{n}$ and $u_{n}=\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{n}$ (by considering (5.6b ), if $\unicode[STIX]{x1D713}$ is increasing from the unburned to the burned gases, $\boldsymbol{n}$ is directed towards the unburned gases). The scalar field $\unicode[STIX]{x1D713}$ appearing in (5.6a ) can be prescribed in different ways as it is uniquely defined only at the interface (Peters Reference Peters2000). It is employed here only for estimating $\boldsymbol{n}=-\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}/|\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}|$ and therefore $w_{n}$ and $u_{n}$ at the interface $\unicode[STIX]{x1D713}=\unicode[STIX]{x1D713}_{0}$ . We compared results when $\unicode[STIX]{x1D713}$ was set as a signed distance (level-set) or a spatially filtered progress variable (using a Gaussian kernel) using the experimental binarized images. It was found that results were very similar irrespectively of the scalar field $\unicode[STIX]{x1D713}(\boldsymbol{x})$ employed.

We can thus provide a cross-check of the adequacy of the measurements of $\langle S_{d}^{u}\rangle$ since it can be estimated independently from either (3.11) or from the area-weighted average of (5.6a ). The comparison of $\langle S_{d}^{u}\rangle$ estimated using either (3.11) or (5.6a ) is given in the appendix B. A more detailed validation of the different terms in either (3.11) or (5.6a ) is also documented in appendix B. The conclusion of these tests is that our methodology is reliable and that we can readily extract information about local values of $S_{d}^{u}$ on the basis of (5.6a ).

Figure 7.

Time evolution of $\langle \unicode[STIX]{x1D705}_{m}\rangle$ (a–c) and $\langle S_{d}^{u}\unicode[STIX]{x1D705}_{m}\rangle$ (d–f) for stoichiometric methane and lean ( $\unicode[STIX]{x1D719}=0.9$ ) propane/air mixtures for three different vortex strengths: (a,d)  $U_{\unicode[STIX]{x1D703}}/\langle S_{c}^{1}\rangle _{t_{0}}=0.8$ , (b,e)  $U_{\unicode[STIX]{x1D703}}/\langle S_{c}^{1}\rangle _{t_{0}}=1.2$ , (c,f)  $U_{\unicode[STIX]{x1D703}}/\langle S_{c}^{1}\rangle _{t_{0}}=2.8$ . The time $t$ has been shifted so that $t=0$ corresponds to the time at which $\langle K\rangle$ is maximum.

Our goal in extracting local values of $S_{d}^{u}$ was to compute the curvature contribution to the total stretch rate, i.e. $S_{d}^{u}\unicode[STIX]{x1D705}_{m}$ , where $\unicode[STIX]{x1D705}_{m}$ is the local mean curvature (see appendix A). Experimental results for $\langle \unicode[STIX]{x1D705}_{m}\rangle$ and $\langle S_{d}^{u}\unicode[STIX]{x1D705}_{m}\rangle$ for either methane/air or propane/air mixtures are shown in figure 7. At steady state, the flames are slightly convex towards the unburned gases resulting in small positive values for the curvature. This particular feature is due to the vortex generator that creates a small velocity deficit in its wake. Note that propane/air flames are less curved than methane/air flames at steady state. Once the flame starts interacting with the vortex, the curvature decreases as the flame becomes concave towards the unburned gases. We observe that the curvature magnitude is roughly the same independently of the type of fuel. It increases in magnitude with increasing $U_{\unicode[STIX]{x1D703}}/\langle S_{c}^{1}\rangle _{t_{0}}$ . Therefore, as found for the total stretch rate, the effective Lewis number does not impact significantly the evolution of curvature during flame–vortex interaction. Note that when increasing the vortex strength $U_{\unicode[STIX]{x1D703}}$ , the contribution of curvature to the total stretch rate progressively decreases by comparison with that of the tangential strain rate $\langle K_{T}\rangle =\langle K\rangle -\langle K_{C}\rangle$ . Indeed, $\langle K\rangle$ has more than doubled when $U_{\unicode[STIX]{x1D703}}/\langle S_{c}^{1}\rangle _{t_{0}}$ is increased from 0.8 to 2.8 (figure 6), whilst variations in $\langle K_{C}\rangle$ are much smaller (figure 7). This indicates that in the configuration of a flame interacting with a vortex, the relative contribution of $K_{C}$ and $K_{T}$ to $K$ can be controlled by adjusting the strength of the vortex. This is thus particularly attracting when considering the item (i) discussed in the Introduction.

5.2.1 Flame displacement speed

The evolution of the area-weighted displacement speed of the methane and propane flames as estimated from (3.11) is presented in figure 8. Again, three vortex strengths are considered $U_{\unicode[STIX]{x1D703}}/\langle S_{c}^{1}\rangle _{t_{0}}=0.8$ , 1.2 and 2.8. Because it is more meaningful to highlight variations in $\langle S_{d}^{u}\rangle$ , we subtract $\langle S_{d}^{u}\rangle _{t_{0}}$ (the value at stationary state) to each side of (3.11) to yield

(5.7) $$\begin{eqnarray}\displaystyle \langle S_{d}^{u}\rangle -\langle S_{d}^{u}\rangle _{t_{0}}=-\frac{1}{A_{f}}\frac{\text{d}{\mathcal{V}}_{u}}{\text{d}t}-\frac{1}{A_{f}}\unicode[STIX]{x0222F}_{\unicode[STIX]{x2202}{\mathcal{V}}_{u}(t)}\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{n}\,\text{d}^{2}A-\langle S_{d}^{u}\rangle _{t_{0}}. & & \displaystyle\end{eqnarray}$$

If there are no variations of $\langle S_{d}^{u}\rangle$ during the interaction with the vortex, then one should expect both left-hand side and right-hand side of (5.7) to be zero. Thus,

(5.8) $$\begin{eqnarray}\displaystyle -{\mathcal{U}}_{MV}|_{{\mathcal{V}}_{u}}={\mathcal{U}}_{MF}|_{{\mathcal{V}}_{u}}-\langle S_{d}^{u}\rangle _{t_{0}}, & & \displaystyle\end{eqnarray}$$

which indicates that, for $\langle S_{d}^{u}\rangle$ to be constant, any change in the mass flux ( ${\mathcal{U}}_{MF}|_{{\mathcal{V}}_{u}}$ ) flowing through the system should translate into an equivalent and simultaneous change in volume ( ${\mathcal{U}}_{MV}|_{{\mathcal{V}}_{u}}$ ) of the system. In the following, in order to emphasize the evolution of the different terms in (3.11), we will systematically plot $-{\mathcal{U}}_{MV}|_{{\mathcal{V}}_{u}}$ and ${\mathcal{U}}_{MF}|_{{\mathcal{V}}_{u}}-\langle S_{d}^{u}\rangle _{t_{0}}$ (all terms are divided also by $\langle S_{d}^{u}\rangle _{t_{0}}$ for them to be non-dimensional) as a function of time. Any difference between these two terms will translate in variations of $\langle S_{d}^{u}\rangle$ . The same treatment will apply later for the consumption speed $\langle S_{c}^{1}\rangle$ .

Figure 8 reveals that qualitatively, $\langle S_{d}^{u}\rangle$ increases with time irrespectively of the vortex strength or type of fuel. The impact of the Lewis number appears rather marginal as there are no substantial differences between methane and propane flames. $\langle S_{d}^{u}\rangle$ starts by increasing concomitantly with the growth of $\langle K\rangle$ and $|\langle \unicode[STIX]{x1D705}_{m}\rangle |$ . After some time, the total stretch rate reaches its maximum value (this is referred to as time $t=0$ ) and starts dropping towards zero (time $t>0$ ). Through (1.5), one should expect $\langle S_{d}^{u}\rangle$ to diminish accordingly. The contrary is observed, i.e. the flame displacement speed continues to augment while the total stretch rate decreases. For time $t>0$ , $|\langle \unicode[STIX]{x1D705}_{m}\rangle |$ increases, the curvature being on average negative (the flame is concave towards the unburned gases). The increase of $\langle S_{d}^{u}\rangle$ during this period follows the increase in $|\langle \unicode[STIX]{x1D705}_{m}\rangle |$ . After interacting with the vortex (not shown), the time needed for $\langle S_{d}^{u}\rangle$ to tend back to its original value is quite long, namely a time $t\approx 10{-}12~\text{ms}$ is needed for $\langle S_{d}^{u}\rangle \approx \langle S_{d}^{u}\rangle _{t_{0}}$ .

At this stage, the conclusion is that the flame displacement speed appears to be more correlated to curvature than the stretch rate and that the latter is not the only scalar influencing the flame displacement speed. This thus precludes using (1.5) for characterizing the evolution of $S_{d}$ with respect to the total stretch rate and/or curvature.

The dependency of the flame displacement speed to the total stretch rate, strain rate or curvature can be further illustrated by plotting the scatter plots of $\langle S_{d}^{u}\rangle$ versus $\langle K\rangle$ , $\langle K_{C}\rangle$ and $\langle K_{T}\rangle$ (figure 9). For the sake of clarity we consider only the methane/air flame at $U_{\unicode[STIX]{x1D703}}/\langle S_{c}^{1}\rangle =2.8$ but similar trends were observed for propane/air flames and different vortex intensities. Different widths $\unicode[STIX]{x1D6E5}$ of the control volume are considered in figure 9 in order to emphasize that the trends in the data are the same irrespective of $\unicode[STIX]{x1D6E5}$ . Quantitatively, it is straightforward that due to averaging, the larger $\unicode[STIX]{x1D6E5}$ , the smaller the variations of $\langle S_{d}^{u}\rangle$ , $\langle K\rangle$ , $\langle K_{T}\rangle$ and $\langle K_{C}\rangle$ . This is what is observed in figure 9 consistently with (5.5).

When $t<0$ (before $\langle K\rangle$ reaches its maximum value), $\langle S_{d}^{u}\rangle$ increases almost linearly with $\langle K\rangle$ (figure 9 a). However, if (1.5) were to apply, for time $t>0$ , $\langle S_{d}^{u}\rangle$ should have been decreasing towards its original value following the decrease of $\langle K\rangle$ . This trend is not observed since $\langle S_{d}^{u}\rangle$ continues to increase while the total stretch rate decreases. This confirms that another phenomenon than the stretch rate is at play.

Figure 9.

Plots of $\langle S_{d}^{u}\rangle$ versus $\langle K\rangle$ (a), $\langle K_{C}\rangle$ (b) and $\langle K_{T}\rangle$ (c) for methane/air mixture at $U_{\unicode[STIX]{x1D703}}/\langle S_{c}^{1}\rangle _{t_{0}}=2.8$ . The arrow indicates the direction of increasing time. The different curves correspond to different width $\unicode[STIX]{x1D6E5}$ of the control volume; the smaller $\unicode[STIX]{x1D6E5}$ the larger the variations of $\langle S_{d}^{u}\rangle$ , $\langle K\rangle$ , $\langle K_{C}\rangle$ and $\langle K_{T}\rangle$ . The dotted lines corresponds to predictions using (5.9) with ${\mathcal{L}}_{\unicode[STIX]{x1D705}}=4.98l_{f}$ and ${\mathcal{L}}_{K}=-1.08l_{f}$ .

When plotted against $\langle K_{C}\rangle$ (figure 9 b), the displacement speed appears to be very nicely correlated with flame curvature. Here $\langle S_{d}^{u}\rangle$ is represented as a function of $\langle K_{C}\rangle$ the propagation/curvature contribution to the stretch rate but a similar (if not better) correlation was found with $\langle \unicode[STIX]{x1D705}_{m}\rangle$ . The correlation $\langle S_{d}^{u}\rangle$ versus $\langle \unicode[STIX]{x1D705}_{m}\rangle$ is almost linear and the slope is found to be marginally dependent on the Lewis number. $\langle S_{d}^{u}\rangle$ is also represented as a function of tangential strain rate $\langle K_{T}\rangle$ in figure 9(c). $\langle S_{d}^{u}\rangle$ follows nicely the increase of $\langle K_{T}\rangle$ in the period $t<0$ . Then, for time $t>0$ , $\langle S_{d}^{u}\rangle$ has a sort of S-shape evolution with respect to $\langle K_{T}\rangle$ meaning that $\langle S_{d}^{u}\rangle$ is not uniquely correlated to $\langle K_{T}\rangle$ neither. As a conclusion, equation (1.5) is not applicable to FVI since the correlation between $\langle S_{d}^{u}\rangle$ and $\langle K\rangle$ is not bijective. Another geometrical scalar than the stretch rate is at play. Our data suggest that curvature could be this leading-order parameter in agreement with (1.7). To confirm this, we now provide comparisons between our data and the predictions of (1.7).

It is worth stressing that the curvature Markstein length ${\mathcal{L}}_{\unicode[STIX]{x1D705}}$ is independent of the type of fuel provided $\unicode[STIX]{x1D6FD}$ and $\unicode[STIX]{x1D70E}$ and the dependence of $\unicode[STIX]{x1D706}$ to $T$ are the same. This is approximatively verified when comparing a stoichiometric methane flame and a $\unicode[STIX]{x1D719}=0.9$ propane mixture ( $\unicode[STIX]{x1D70E}=7.33,\unicode[STIX]{x1D6FD}=8.95$ for methane against $\unicode[STIX]{x1D70E}=7.48$ and $\unicode[STIX]{x1D6FD}=8.23$ for propane at $\unicode[STIX]{x1D719}=0.9$ ). However, the stretch Markstein length highlights an explicit dependence on the effective Lewis number and should thus differ substantially between methane and propane flames. The question is now which of the curvature and stretch Markstein length scale dominates. To answer this question, the typical evolutions of ${\mathcal{L}}_{K}$ and ${\mathcal{L}}_{\unicode[STIX]{x1D705}}$ with respect to the isotherm $T/T_{u}$ are presented in figure 10. The parameters used in figure 10 are $\unicode[STIX]{x1D6FD}=8$ , $\unicode[STIX]{x1D70E}=8$ , $\unicode[STIX]{x1D706}=(T/T_{u})^{0.87}$ (Giannakopoulos et al. Reference Giannakopoulos, Gatzoulis, Frouzakis, Matalon and Tomboulides2015a ,Reference Giannakopoulos, Matalon, Frouzakis and Tomboulides b ) and two effective Lewis numbers are considered $Le=1.05$ and 1.5 corresponding to the methane and propane mixture, respectively. In the experiments, even though it is difficult to infer exactly which isotherm is tracked, the displacement speed is measured close to the unburned gases where the Markstein length highlights strong variations with respect to the isotherm (see figure 10). Independently of the isotherm, there is a wide range of $T/T_{u}$ where ${\mathcal{L}}_{\unicode[STIX]{x1D705}}$ dominates by comparison with the stretch Markstein length ${\mathcal{L}}_{K}$ . From figure 10, $|{\mathcal{L}}_{K}|/{\mathcal{L}}_{\unicode[STIX]{x1D705}}<0.2$ in the range $1<T/T_{u}\lesssim 2.5$ . This will remain qualitatively true when considering other dependences of thermal conductivity to temperature.

Figure 10.

Evolution of the curvature and stretch Markstein length as a function of the reduced temperature as inferred from (1.8) with $\unicode[STIX]{x1D6FD}=8$ , $Le=1.5/1.05$ and $\unicode[STIX]{x1D706}=(T/T_{u})^{0.87}$ .

The fact that $\langle S_{d}^{u}\rangle$ was observed to be strongly correlated with curvature and not substantially affected by the Lewis number can thus be readily explained by the dominance of ${\mathcal{L}}_{\unicode[STIX]{x1D705}}$ which is Lewis number independent by comparison with ${\mathcal{L}}_{K}$ . Our data are thus in total agreement with the results of Bechtold & Matalon (Reference Bechtold and Matalon2001), Giannakopoulos et al. (Reference Giannakopoulos, Gatzoulis, Frouzakis, Matalon and Tomboulides2015a ,Reference Giannakopoulos, Matalon, Frouzakis and Tomboulides b ) and support the distinction between the stretch rate and curvature Marstein lengths.

Figure 9 also displays the displacement speed as inferred by surface averaging (1.7) to yield

(5.9) $$\begin{eqnarray}\displaystyle \langle S_{d}^{u}\rangle =S_{l}^{0}-{\mathcal{L}}_{K}\langle K\rangle -{\mathcal{L}}_{\unicode[STIX]{x1D705}}S_{l}^{0}2\langle \unicode[STIX]{x1D705}_{m}\rangle . & & \displaystyle\end{eqnarray}$$

By fitting to experimental data, ${\mathcal{L}}_{\unicode[STIX]{x1D705}}$ and ${\mathcal{L}}_{K}$ (together with $S_{l}^{0}$ ) can be estimated to $4.98l_{f}$ and $-1.08l_{f}$ , respectively. With these values, the measured evolution of $\langle S_{d}^{u}\rangle$ with respect to $\langle K\rangle$ , $\langle K_{C}\rangle$ and $\langle K_{T}\rangle$ is very nicely reproduced by (5.9) (dashed curves in figure 9). This is the main result of the present study as it provides unprecedented quantitative support for (1.7). For propane flames, similar agreement was observed with ${\mathcal{L}}_{\unicode[STIX]{x1D705}}=5.12l_{f}$ and ${\mathcal{L}}_{K}=-0.25l_{f}$ . Our data are thus consistent with the conclusions of Giannakopoulos et al. (Reference Giannakopoulos, Gatzoulis, Frouzakis, Matalon and Tomboulides2015a ) as ${\mathcal{L}}_{\unicode[STIX]{x1D705}}$ is almost independent of $Le$ while ${\mathcal{L}}_{K}$ is clearly different between methane and propane flames. Furthermore, ${\mathcal{L}}_{K}$ pertaining to methane is smaller than that of propane consistently with figure 10. These results were rather robust and independent of the vortex strengths.

Equation (1.8) with $\unicode[STIX]{x1D6FD}=8$ , $\unicode[STIX]{x1D70E}=8$ and $\unicode[STIX]{x1D706}=(T/T_{u})^{0.87}$ provides larger values of ${\mathcal{L}}_{\unicode[STIX]{x1D705}}$ than that measured (see figure 10). This difference is attributed to the fact that theoretical values depend strongly on the isotherm chosen to track the flame whilst in the experiments, it is not clear which particular isotherm is followed. Second, theoretical values are substantially affected by the choice made for the dependence of $\unicode[STIX]{x1D706}$ with respect to temperature. The assumption used by Giannakopoulos et al. (Reference Giannakopoulos, Gatzoulis, Frouzakis, Matalon and Tomboulides2015a ) regarding notably the one-step chemistry with large activation energy, may be also a source of discrepancy.

A very similar evolution of $S_{d}$ with respect to $K$ was presented by Chen & Im (Reference Chen and Im1998) in a turbulent reacting flow. This suggests that (1.7) might hold also in turbulent flows and that the sensitivity of $S_{d}$ to the stretch rate in such flows might be reinterpreted in the light of the present findings.

5.2.2 Flame consumption speed

Figure 11.

Time evolution of the different terms in (3.22) normalized by $\langle S_{c}^{1}\rangle _{t_{0}}$ . The vortex intensity $U_{\unicode[STIX]{x1D703}}/\langle S_{c}^{1}\rangle _{t_{0}}=0.8$ (a,d), 1.2 (b,e) and 2.8 (c,f). Stoichiometric $\text{CH}_{4}$ /Air mixture (a–c), lean ( $\unicode[STIX]{x1D719}=0.9$ )  $\text{C}_{3}\text{H}_{8}$ /Air mixture (d–f).

Now focusing on the flame consumption speed, the time evolutions of the different terms in (3.22) for the two aforementioned mixtures are portrayed in figure 11. Again, three different vortex intensities are considered, viz. $U_{\unicode[STIX]{x1D703}}/\langle S_{c}^{1}\rangle _{t_{0}}=0.8$ , 1.2 and 2.8. For weak to moderate vortex strengths ( $U_{\unicode[STIX]{x1D703}}/\langle S_{c}^{1}\rangle _{t_{0}}=0.8$ and 1.2), one observes two phases in the evolution of $\langle S_{c}^{1}\rangle$ (figure 11). During the first phase, i.e. before $\langle K\rangle$ reaches its maximum value (corresponding to negative time in figure 11), $\langle S_{c}^{1}\rangle$ remains roughly constant. This remark holds irrespectively of the Lewis number. Then in the second phase, i.e. when $K$ starts decreasing towards zero, the flame curvature $\langle \unicode[STIX]{x1D705}_{m}\rangle$ increases in amplitude – $\langle \unicode[STIX]{x1D705}_{m}\rangle$ being on average negative during this phase – and a systematic increase in $\langle S_{c}^{1}\rangle$ is observed (of the order of 20 % to 25 % depending on $U_{\unicode[STIX]{x1D703}}/\langle S_{c}^{1}\rangle _{t_{0}}$ ). The variations of $\langle S_{c}^{1}\rangle$ during this second phase further appears to be almost independent of the type of fuel. A larger vortex intensity ( $U_{\unicode[STIX]{x1D703}}/\langle S_{c}^{1}\rangle _{t_{0}}=2.8$ ), leads to a slight decrease of $\langle S_{c}^{1}\rangle$ and an increase of $K$ in the first phase. Then in the second phase, $\langle S_{c}^{1}\rangle$ starts to increase as emphasized before for the displacement speed.

Figure 12.

Scatter plot of $\langle S_{c}^{1}\rangle$ versus $\langle K\rangle$ (a), $\langle K_{C}\rangle$ (b) and $\langle K_{T}\rangle$ (c) for methane/air mixture at $U_{\unicode[STIX]{x1D703}}/\langle S_{c}^{1}\rangle _{t_{0}}=2.8$ . The arrows indicate the direction of increasing time. The different curves correspond to different width $\unicode[STIX]{x1D6E5}$ of the control volume; the smaller $\unicode[STIX]{x1D6E5}$ the larger the variations of $\langle S_{c}^{1}\rangle$ , $\langle K\rangle$ , $\langle K_{C}\rangle$ , $\langle K_{T}\rangle$ . The dotted lines corresponds to predictions using (3.22) where $\langle S_{d}^{u}\rangle$ is calculated using (5.9) with ${\mathcal{L}}_{\unicode[STIX]{x1D705}}=4.98l_{f}$ and ${\mathcal{L}}_{K}=-1.08l_{f}$ .

Figure 12 give further insights into the sensitivity of $\langle S_{c}^{1}\rangle$ to the stretch rate (figure 12 a), curvature (figure 12 b) and strain rate (figure 12 c): the variations of the flame consumption speed are much more correlated to flame curvature than the stretch rate. Indeed, analysing figure 12 reveals that the correlation between $\langle S_{c}^{1}\rangle$ and $\langle K\rangle$ is not bijective (as in figure 9 a) while there seems to be a clear dependence of $\langle S_{c}^{1}\rangle$ on $\langle K_{C}\rangle$ . The dependency of $\langle S_{c}^{1}\rangle$ versus $\langle K_{C}\rangle$ was again found to be approximatively the same for both methane and propane mixtures. This remark holds irrespectively of the vortex intensity and the width of the control volume, at least for the cases investigated here.

In asymptotic analysis, the reaction zone is confined to a infinitely thin sheet located in the burned gases side. Therefore, as pointed out by Clavin & Garcia (Reference Clavin and Garcia1983), Bechtold & Matalon (Reference Bechtold and Matalon2001) the consumption speed should be altered by the stretch rate only, since ${\mathcal{L}}_{\unicode[STIX]{x1D705}}$ is zero at the reaction zone. Our data appear to contradict this result as $\langle S_{c}^{1}\rangle$ does not fall into a straight line when plotted against $\langle K\rangle$ (figure 12 a). Using (3.22) and (5.9), we obtain however that

(5.10) $$\begin{eqnarray}\displaystyle \langle S_{c}^{1}\rangle =S_{l}^{0}-{\mathcal{L}}_{\unicode[STIX]{x1D705}}2S_{l}^{0}\langle \unicode[STIX]{x1D705}_{m}\rangle -({\mathcal{L}}_{K}+{\mathcal{F}})\langle K\rangle -{\mathcal{F}}\frac{\unicode[STIX]{x03C0}\unicode[STIX]{x0394}v_{r,u}}{A_{f}}, & & \displaystyle\end{eqnarray}$$

which indicates that $\langle S_{c}^{1}\rangle$ is likely to be influenced by both the stretch rate and curvature. Similar conclusions were reached by Bradley et al. (Reference Bradley, Gaskell and Gu1996) and some DNS data of turbulent premixed flames (Haworth & Poinsot Reference Haworth and Poinsot1992; Rutland & Trouvé Reference Rutland and Trouvé1993; Chen & Im Reference Chen and Im2000; Bell et al. Reference Bell, Cheng, Day and Shepherd2007) also indicate that $S_{c}$ is unequally affected by stretch and curvature. Figure 12 compares experimental data to those inferred from (5.10) keeping the same values as before for ${\mathcal{L}}_{K}$ and ${\mathcal{L}}_{\unicode[STIX]{x1D705}}$ . As $\langle S_{c}^{1}\rangle$ is deducted directly from $\langle S_{d}^{u}\rangle$ , it is not surprising to observe a nice agreement between (5.10) and experimental data. This figure was shown in order to again illustrate the pertinence the distinction between the stretch rate and curvature effects on $S_{c}$ . Our data therefore support the conclusions of Bradley et al. (Reference Bradley, Gaskell and Gu1996) but appear in contradiction with the deduction of Bechtold & Matalon (Reference Bechtold and Matalon2001) concerning the dependence of $S_{c}$ to curvature. Here again, similar agreement between experimental data and (5.10) was observed for the propane cases and different vortex strengths.