3.1. Turbulent burning velocities and conditioned profiles

Figure 1 reports evolution of turbulent burning velocities (i) $U_t^F(t)$ evaluated by integrating the fuel consumption rate $\dot {\omega }_{\textrm {H}_2} (\boldsymbol {x},t)$ over the computational domain, i.e.

(3.1)\begin{equation} U_t^F(t) ={-} \frac{1}{(\rho Y_{\textrm{H}_2})_u \varLambda^2} \int\!\!\int\!\!\int \dot{\omega}_{\textrm{H}_2} (\boldsymbol{x},t)\,\textrm{d} \boldsymbol{x}, \end{equation}

or (ii) $U_t^{T}(t)$ obtained by integrating the heat release rate $\dot {\omega }_{T} (\boldsymbol {x},t)=\sum _{k=1}^{N_{sp}} h_k \dot {\omega }_k (\boldsymbol {x},t)$ over the computational domain, i.e.

(3.2)\begin{equation} U_t^{T}(t) = \frac{1}{\varLambda^2} \int\!\!\int\!\!\int \frac{\dot{\omega}_T (\boldsymbol{x},t)}{c_p (T_b-T_u)}\,\textrm{d} \boldsymbol{x}. \end{equation}

Here, $\rho$ and $Y$ designate density and mass fraction, respectively; $h_k$ and $\dot {\omega }_k$ are the enthalpy and the rate of production, respectively, of species $k$; $c_p$ is the heat capacity of the mixture; $N_{sp}=9$ is the number of species. These results have been obtained from flames (a) A and A1, (b) B and B1, (c) C and C1. Results computed in cases C1/H and C1/H2 will be discussed later.

After an initial stage, each computed curve oscillates around a mean value of $U_t^F/S_L$ or $U_t^{T}/S_L$ – see horizontal straight lines in figure 1. These mean values averaged at $t/\tau _t>t^*$ and the dimensional mean values of $U_t^F$ and $U_t^{T}$ are reported in table 1. The results obtained for $U_t^F$ and $U_t^{T}$ are close to one another, thus, indicating that fuel consumption and heat release are in an equilibrium state on the global level, as expected. However, such an equilibrium state is not observed on the local level, as will be discussed later.

Figure 1 and table 1 show that, in line with numerous experimental and DNS data discussed in the introduction, the present simulations do predict a significant increase in $U_t^F/S_L$ or $U_t^{T}/S_L$ with decreasing $Le$. It is of interest to note that the effect magnitude is increased by $Ka$ under conditions of the present study. Indeed, a ratio of $(U_t^F/S_L)_{Le<1}/(U_t^F/S_L)_{Le=1}$ is equal to 1.5, 1.7 and 2.1 in the A, B and C, pairs of flames, respectively. Thus, the computed bulk turbulent burning velocities do not indicate that molecular transport is dominated by turbulent mixing in the studied flames.

On the contrary, the conditioned profiles of the temperature, equivalence ratio, fuel consumption rate, heat release rate and species mass fractions, e.g. mass fraction of the radical H, extracted from the entire computational domain at $t/\tau _t>t^*$ and plotted in figures 2(a), 3(a), 4(a), 5(a) and 6(a), respectively, suggest the opposite conclusion, i.e. molecular transport is dominated by turbulent mixing at least in case C characterized by the highest Karlovitz number. Indeed, in this case, the conditioned profiles of $\langle T \,|\, c \rangle (c)$, $\langle \phi \,|\, c \rangle (c)$, $\langle \dot {\omega }_{\textrm {H}_2} \,|\, c \rangle (c)$, $\langle \dot {\omega }_T \,|\, c \rangle (c)$ and $\langle Y_{\textrm {H}} \,|\, c \rangle (c)$ (see blue dotted–dashed lines) differ significantly from the counterpart profiles obtained from the unperturbed (see squares) or highly strained (see circles) stationary planar laminar premixed flames characterized by $Le=0.32$. However, the conditioned profiles are sufficiently close to the counterpart profiles obtained from the unperturbed unity Lewis number flame (see pentagons). In particular, the profiles of $\langle \dot {\omega }_{\textrm {H}_2} \,|\, c \rangle (c)$ are very close in the low Lewis number turbulent flame C and in the unity Lewis number unperturbed laminar flame, cf. blue dotted–dashed line and pentagons in figure 4(a). Even in cases B and A, the conditioned profiles are closer to the laminar-flame profiles computed for $Le=1$ when compared with the low Lewis number flame profiles.

Figure 2. Time-averaged conditioned profiles of temperature extracted (a) from the entire computational domain, and at (b) $\langle c_F \rangle =0.1 \pm 0.02$, (c) $\langle c_F \rangle =0.5 \pm 0.02$, (d) $\langle c_F \rangle =0.9 \pm 0.02$. Results computed in cases A, B and C are plotted using black solid, red dashed and blue dotted–dashed lines, respectively. Squares, pentagons and circles show profiles computed for the unperturbed laminar low Lewis number flame, the unperturbed laminar unity Lewis number flame, and critically strained (the strain rate is equal to $11.3/ \tau _f$), planar, stationary laminar low Lewis number flame, respectively.

Figure 3. Time-averaged conditioned profiles of equivalence ratio. Legends are explained in the caption of figure 2.

Figure 4. Time-averaged conditioned profiles of fuel consumption rate. Legends are explained in the caption of figure 2.

Figure 5. Time-averaged conditioned profiles of heat release rate. Legends are explained in the caption of figure 2.

Figure 6. Time-averaged conditioned profiles of hydrogen mass fraction $Y_{\textrm {H}}$. Legends are explained in the caption of figure 2.

It is worth remembering that if $Le$ is increased from 0.32 to 1.0, $S_L$ is also increased – see table 1. Such an effect might be assumed to cause a higher $U_t$ if molecular transport associated with a low Lewis number for lean hydrogen–air flames is dominated by turbulent mixing associated with $Le=1$. However, the last two columns in table 1 show that the dimensional $\bar {U}_t^F$ or $\bar {U}_t^{T}$ is significantly higher in case C when compared with case C1 characterized by a larger $S_L$. Therefore, the high $\bar {U}_t^F$ or $\bar {U}_t^{T}$ in case C should not be attributed to an increase from $S_L(Le=0.32)=0.58$ to $S_L(Le=1.0)=0.78$ m s$^{-1}$.

Thus, the DNS data involve both findings emphasized in the introduction and, consequently, are suitable for exploring their consistency, e.g. consistency of figure 1, which shows a significant influence of differential diffusion on the integrated fuel consumption or heat release rate, and figure 4(a) or 5(a), which shows that the conditioned fuel consumption rate $\langle \dot {\omega }_{\textrm {H}_2} \,|\, c \rangle (c)$ or heat release rate $\langle \dot {\omega }_T \,|\, c \rangle (c)$, respectively, sampled from the entire flame brush in case C approaches $\dot {\omega }_{\textrm {H}_2}(c)$ or $\dot {\omega }_T(c)$, respectively, in the unperturbed unity Lewis number laminar flame.

The apparent inconsistency between the DNS results plotted in figure 1 and figures 2(a), 3(a), 4(a), 5(a) and 6(a) is explained in figures 2–6, where conditioned profiles extracted from (b) leading, (c) median and (d) trailing layers of the mean flame brushes are reported. These three layers are characterized by the transverse-averaged combustion progress variable $\langle c \rangle (y,t)$ approximately ($\pm 0.02$) 0.1, 0.5 and 0.9, respectively. While the median conditioned profiles plotted in figures 2(c), 3(c), 4(c), 5(c) and 6(c) look similar to the conditioned profiles sampled from the entire computational domain, this is not so for the leading or trailing conditioned profiles. More specifically, the conditioned profiles (with the exception of $\langle \dot {\omega }_T \,|\, c \rangle$ in case C) sampled from the trailing layer are closer to the counterpart conditioned profiles obtained from the unperturbed low Lewis number laminar flame, cf. lines and squares in figures 2(d), 3(d), 4(d), 5(d) and 6(d). On the contrary, the conditioned profiles of $\langle \phi \,|\, c \rangle (c)$, $\langle \dot {\omega }_{\textrm {H}_2} \,|\, c \rangle$ and $\langle Y_{\textrm {H}} \,|\, c \rangle$ sampled from the leading layer are closer (for temperature, this trend is only pronounced at $c>0.7$, cf. lines and circles in figure 2b, whereas the profiles of $T(c)$ obtained from the unperturbed unity Lewis number and highly strained low Lewis number laminar flames are close to one another at $c<0.7$; for heat release rate, the discussed trend is weakly (if any) pronounced – see figure 5b) to the counterpart conditioned profiles obtained from the highly strained low Lewis number laminar flame, cf. lines and circles in figures 3(b), 4(b), and 6(b). It is also worth stressing that the peak fuel consumption rate evaluated at $\langle c \rangle =0.1 \pm 0.02$ is significantly higher than the peak rates in the unperturbed laminar flames (both $Le=0.32$ and $Le=1.0$), but is close to the peak rate in the highly strained low Lewis number laminar flame. Thus, under conditions of the present simulations, apparent dominance of turbulent mixing, implied by the results shown in figures 2(a), 3(a), 4(a), 5(a) and 6(a) is an artefact of sampling statistics from the entire computational domain, whereas molecular transport plays an important role in the leading and trailing zones of the mean flame brush and significantly affects bulk turbulent burning velocity.

The results reported in figures 1–6 raise at least three questions.

1. (i) Why are differential diffusion effects more pronounced at the edges of the mean flame brush?

2. (ii) Why is turbulent burning velocity more sensitive to differential diffusion effects localized to the leading edge?

3. (iii) Why do trends observed for the conditioned profile of heat release rate, sampled from the leading edge (see figure 5b) differ from trends observed for the conditioned profiles of fuel consumption rate (see figure 4b)?

These three questions, and one more, are discussed in the next four subsections, with the focus of discussion being placed on a family of C-flames (C and C1, as well as C1/H and C1/H$_2$, which will be discussed later), because they are characterized by higher Karlovitz numbers when compared with four other flames (A1, A1, B and B1).

3.2. Why are differential diffusion effects more pronounced at the leading edge?

The most evident explanation of a significant increase in the equivalence ratio, fuel consumption rate or mass fraction of H at the leading edge of a mean flame brush consists of highlighting effects that stem from the local curvature of reaction zones. Indeed, first, from purely topological reasoning, local flame curvature should be predominantly positive at low $\langle c \rangle$ and predominantly negative at large $\langle c \rangle$. A decrease (increase) in the probability of finding positively (negatively) curved flames with $\langle c \rangle$ was documented in earlier DNS studies (Sabelnikov et al. Reference Sabelnikov, Lipatnikov, Nishiki, Dave, Hernández-Pérez, Song and Im2021, figure 8a). This trend is also observed in figure 7, with the probability of finding highly curved reaction zones being large at low $\bar {c}$.

Figure 7. Probabilities of finding $\delta _L h_m>1$ (black solid line) and $\delta _L h_m<-1$ (red dashed line) versus transverse and time-averaged combustion progress variable $\bar {c}$. The probabilities have been sampled from flame zones characterized by a sufficiently high local fuel consumption rate (larger than 10 % of its peak value in the unperturbed laminar flame); case C.

Second, theories of laminar premixed flames stretched by large-scale flow perturbations (Matalon & Matkowsky Reference Matalon and Matkowsky1982; Pelcé & Clavin Reference Pelcé and Clavin1982; Zel'dovich et al. Reference Zel'dovich, Barenblatt, Librovich and Makhviladze1985; Class et al. Reference Class, Matkowsky and Klimenko2003; Kelley et al. Reference Kelley, Bechtold and Law2012) predict a significant influence of flame curvature on the local burning rate and, in particular, an increase in the burning rate in positively curved flames if $Le<1$. This increase stems from the local (i) enrichment (for lean hydrogen–air flames) and (ii) preheating of the reaction zone due, respectively, to imbalances of (i) molecular fluxes of fuel and oxygen towards the reaction zone and (ii) molecular fluxes of the reactants towards the zone and molecular heat transfer from it.

For instance, the molecular diffusion term in the transport equation for a species mass fraction $Y_k$ can be decomposed as follows (Peters Reference Peters2000):

(3.3)\begin{equation} \underbrace{\boldsymbol{\nabla}\boldsymbol{\cdot} \left(\rho D_k \boldsymbol{\nabla} Y_k \right)}_{T_0} = \underbrace{\boldsymbol{n}_k \boldsymbol{\cdot} \boldsymbol{\nabla} \left(\rho D_k \boldsymbol{n}_k \boldsymbol{\cdot}\boldsymbol{\nabla} Y_k\right)}_{T_1} \underbrace{-\rho D_k |\boldsymbol{\nabla} Y_k| \boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{n}_k}_{T_2}, \end{equation}

where $\boldsymbol {n}_k=-\boldsymbol {\nabla } Y_k/|\boldsymbol {\nabla } Y_k|$ is the unit vector normal to an isosurface of $Y_k(\boldsymbol {x},t)=$const. Term $T_2$ is directly proportional to curvature of the isosurface and is known as curvature term. For major reactants (H$_2$ and O$_2$), this term is positive in a positively curved (i.e. $\boldsymbol {\nabla }\boldsymbol {\cdot } \boldsymbol {n}>0$) reaction zone, because $\boldsymbol {n} \boldsymbol {\cdot } \boldsymbol {n}_k < 0$ for these reactants. Recall that symbol $\boldsymbol {n}$ designates the unit vector normal to an isosurface of $c(\boldsymbol {x},t)=$const, i.e. $\boldsymbol {n} = - \boldsymbol {\nabla } c/|\boldsymbol {\nabla } c|$. Moreover, the curvature term $T_2$ is larger for the fuel, because $D_{\textrm {H}_2}$ is significantly larger than $D_{\textrm {O}_2}$. As a result, the mass fraction of H$_2$ and, hence, the local equivalence ratio is increased in positively curved reaction zones in the case of a lean hydrogen–air mixture. Since $D_{\textrm {H}_2}$ is also significantly larger than the molecular heat diffusivity $\kappa$ of such a mixture, similar arguments could be used to emphasize an increase in the local temperature in positively curved reaction zones. On the contrary, for atomic hydrogen, $T_2>0$ in negatively curved reaction zones, because $\boldsymbol {n} \boldsymbol {\cdot } \boldsymbol {n}_\textrm {H} > 0$ in the largest part of a reaction zone, with the exception of a radical recombination region (Williams Reference Williams2000) characterized by $c$ close to unity.

An important role played by term $T_2$ is shown in figures 8 and 9, which report dependencies of doubly conditioned (on the local combustion progress variable $c$ and curvature $h_m$) terms $T_j$ on $h_m$ and $c$, respectively. In particular, figure 8(a) shows that, for H$_2$, both the total diffusion term $\langle T_0 | c=0.75 | h_m \rangle$ (black solid line) and the curvature term $\langle T_2 | c=0.75 | h_m \rangle$ (blue dotted line) are significantly increased by curvature if $|\delta _L h_m|<2$, whereas dependence of the normal diffusion term $\langle T_1 | c=0.75 | h_m \rangle$ on $h_m$ is less pronounced and non-monotonous, with $T_1$ peaking at a slightly negative curvature. Note that these results are conditioned to $c=0.75 \pm 0.02$, because the highest (over the computational domain and time) fuel consumption rate is found at a close value of the combustion progress variable in flame C. For the radical H, the curvature term $T_2$ also plays an important role, but results in decreasing $T_0$ with $h_m$, as already noted when discussing (3.3).

Figure 8. Time-averaged dependencies of molecular diffusion terms $\langle T_j | c | h_m \rangle$ on the normalized local curvature $\delta _L h_m$, sampled for (a) H$_2$ and (b) H at $c = 0.75 \pm 0.02$ from the entire flame brush in case C. Black solid, red dashed and blue dotted lines show $T_0$, $T_1$ and $T_2$, respectively.

Figure 9. Time-averaged dependencies of molecular diffusion terms on the local combustion progress variable $c$, sampled at $\delta _L h_m = -1 \pm 0.005$ (blue dotted lines), $0 \pm 0.005$ (red dashed lines) and $1 \pm 0.005$ (black solid lines) from the entire flame brush in case C. Here (a)  $\langle T_0 | c | h_m \rangle$ for H$_2$, (b)  $\langle T_0 | c | h_m \rangle$ for H, (c)  $\langle T_1 | c | h_m \rangle$ for H$_2$, (d)  $\langle T_1 | c | h_m \rangle$ for H, (e)  $\langle T_2 | c | h_m \rangle$ for H$_2$ and ( f)  $\langle T_2 | c | h_m \rangle$ for H.

The same trends are observed in figure 9, which shows dependencies of $\langle T_j | c | h_m \rangle$ on $c$, conditioned to $\delta _L h_m = -1 \pm 0.005$ (blue dotted lines), $0 \pm 0.005$ (red dashed lines) and $1 \pm 0.005$ (black solid lines). The curvature term $T_2$ results in increasing the molecular diffusion flux $\boldsymbol {\nabla }\boldsymbol {\cdot }(\rho D_k \boldsymbol {\nabla } Y_k)$ (i) for H$_2$ if the local curvature is positive and (ii) for H if the local curvature is negative.

It is also of interest to note that time-averaged dependencies of $\langle T_j \,|\, c \rangle$ on $c$, sampled from the leading and trailing zones of the mean flame brush, look similar to dependencies of $\langle T_j | c | h_m \rangle$ on $c$, conditioned to $\delta _L h_m = 1 \pm 0.005$ and $-1 \pm 0.005$, respectively, cf. curves plotted in black solid and blue dotted lines, respectively, in figures 9 and 10. This similarity (i) is associated with predominance of the positive and negative curvature at $\langle c \rangle =0.1 \pm 0.02$ and $0.9 \pm 0.02$, respectively (see figure 7) and (ii) supports a hypothesis that differential diffusion effects are more pronounced at the leading and trailing edges of the mean flame brush, because the local reaction zones are highly curved at small and large $\langle c \rangle$, with the curvature signs being opposite at these two values of $\langle c \rangle$.

Figure 10. Time-averaged dependencies of conditioned molecular diffusion terms on the local combustion progress variable, sampled at $\langle c \rangle = 0.1 \pm 0.02$ (black solid lines), $0.5 \pm 0.02$ (red dashed lines) and $0.9 \pm 0.02$ (blue dotted lines) in flame C. Here (a)  $\langle T_0 \,|\, c \rangle$ for H$_2$, (b)  $\langle T_0 \,|\, c \rangle$ for H, (c)  $\langle T_1 \,|\, c \rangle$ for H$_2$, (d)  $\langle T_1 \,|\, c \rangle$ for H, (e)  $\langle T_2 \,|\, c \rangle$ for H$_2$ and ( f)  $\langle T_2 \,|\, c \rangle$ for H.

Due to an imbalance of molecular diffusion fluxes of H$_2$, O$_2$, and heat to and from curved reaction zones, the local temperature, equivalence ratio, fuel consumption rate and mass fraction of H are increased by curvature, as reported in figure 11. However, these quantities are substantially affected not only by the curvature. For instance, dependencies of $\langle T | c=0.75 | h_m \rangle$, $\langle \phi | c=0.75 | h_m \rangle$, $\langle \dot {\omega }_{\textrm {H}_2} | c=0.75 | h_m \rangle$ and $\langle Y_{\textrm {H}} | c=0.75 | h_m \rangle$ on $\delta _L h_m$ sampled at $\langle c \rangle = 0.1 \pm 0.02$ and $0.9 \pm 0.02$ are substantially different, cf. black solid and blue dotted lines. Accordingly, dependencies of $\langle T | c | h_m \rangle (c)$, $\langle \phi | c | h_m \rangle (c)$, $\langle \dot {\omega }_{\textrm {H}_2} | c | h_m \rangle (c)$ and $\langle Y_{\textrm {H}} | c | h_m \rangle (c)$ on $c$, sampled from these two zones, are also substantially different (cf. left and right columns in figure 12). Effects of other flow characteristics on the conditioned quantities $\langle q | c \rangle$ and behaviour of the doubly conditioned heat release rate $\langle \dot {\omega }_{T} | c | h_m \rangle$ will further be addressed in §§ 3.4 and 3.5, respectively.

Figure 11. Time-averaged dependencies of doubly conditioned (a)  temperature $\langle T | c=0.75 | h_m \rangle$, (b) equivalence ratio $\langle \phi | c=0.75 | h_m \rangle$, (c)  fuel consumption rate $\langle \dot {\omega }_{\textrm {H}_2} | c=0.75 | h_m \rangle$ and (d)  hydrogen mass fraction $\langle Y_{\textrm {H}} | c=0.75 | h_m \rangle$ on the normalized local curvature $\delta _L h_m$, sampled from the entire flame brush (orange dotted–dashed lines), its leading zone characterized by $\langle c \rangle (y,t)=0.1 \pm 0.02$ (black solid lines), middle zone ($\langle c \rangle =0.5 \pm 0.02$, red dashed lines) and trailing zone ($\langle c \rangle =0.9 \pm 0.02$, blue dotted lines); case C.

Figure 12. Time-averaged dependencies of doubly conditioned (a,b) temperature $\langle T | c | h_m \rangle$, (c,d) equivalence ratio $\langle \phi | c | h_m \rangle$, (e, f) fuel consumption rate $\langle \dot {\omega }_{\textrm {H}_2} | c | h_m \rangle$ and (g,h) hydrogen mass fraction $\langle Y_{\textrm {H}} | c=| h_m \rangle$ on $c$, obtained at $\delta _L h_m=-0.1 \pm 0.005$ (blue dotted lines), $0 \pm 0.005$ (red dashed lines) and $1 \pm 0.005$ (black solid lines) in case C. Results sampled from the leading and trailing zones of the flame brush are reported in cells (a,c,e,g) and (b,d, f,h), respectively.

3.3. Leading point concept

The DNS results plotted in figures 1–6 indicate that both turbulent burning velocities and the conditioned profiles of $T$, $\phi$, $\dot {\omega }_{\textrm {H}_2}$, $\dot {\omega }_T$ and $Y_{\textrm {H}}$, sampled at $\langle c \rangle =0.1 \pm 0.02$, are significantly affected by $Le$, whereas the conditioned profiles of the same quantities, sampled from the middle of the C-flame brush, are associated with $Le=1$. This observation not only shows that differences in molecular transport coefficients of reactants and/or heat play a more important role at the leading edge of a mean turbulent flame brush, but also implies that the turbulent burning velocities are controlled by processes localized to the leading edge. This implication is in line with (i) the so-called KPP theory of convection–diffusion–reaction waves developed by Kolmogorov, Petrovsky & Piskounov (Reference Kolmogorov, Petrovsky and Piskounov1937) and extended in subsequent studies reviewed elsewhere (Ebert & van Saarlos Reference Ebert and van Saarlos2000; Sabelnikov, Petrova & Lipatnikov Reference Sabelnikov, Petrova and Lipatnikov2016) and (ii) Zel'dovich's idea about the crucial role played by leading points in propagation of premixed turbulent flames. That idea was developed in the former USSR several decades ago, as reviewed elsewhere (Kuznetsov & Sabelnikov Reference Kuznetsov and Sabelnikov1990; Lipatnikov & Chomiak Reference Lipatnikov and Chomiak2005), and was supported in recent theoretical (Sabelnikov & Lipatnikov Reference Sabelnikov and Lipatnikov2013, Reference Sabelnikov and Lipatnikov2015; Kha et al. Reference Kha, Robin, Mura and Champion2016), experimental (Venkateswaran et al. Reference Venkateswaran, Marshall, Shin, Noble, Seitzman and Lieuwen2011, Reference Venkateswaran, Marshall, Seitzman and Lieuwen2013, Reference Venkateswaran, Marshall, Seitzman and Lieuwen2015; Zhang et al. Reference Zhang, Wang, Yu, Jin, Zhang and Huang2018) and DNS (Amato et al. Reference Amato, Day, Cheng, Bell, Dasgupta and Lieuwen2015a,Reference Amato, Day, Cheng, Bell and Lieuwenb; Kim Reference Kim2017; Dave, Mohan & Chaudhuri Reference Dave, Mohan and Chaudhuri2018; Lipatnikov, Chakraborty & Sabelnikov Reference Lipatnikov, Chakraborty and Sabelnikov2018) studies, as well as in earlier single-step chemistry (Karpov, Lipatnikov & Zimont Reference Karpov, Lipatnikov and Zimont1996) or recent complex chemistry (Verma, Monnier & Lipatnikov Reference Verma, Monnier and Lipatnikov2021) Reynolds-averaged Navier–Stokes (RANS) simulations of experiments with lean H$_2$–air (Karpov & Severin Reference Karpov and Severin1980) or H$_2$/CO–air (Venkateswaran et al. Reference Venkateswaran, Marshall, Shin, Noble, Seitzman and Lieuwen2011, Reference Venkateswaran, Marshall, Seitzman and Lieuwen2013) flames, respectively.

The present work further supports the leading point concept by showing its consistency with recent DNS data that indicate mitigation of the influence of $Le$ on the conditioned profiles of local flame characteristics in intense turbulence. Conversely, the leading point concept supports consistency of the two investigated findings by hypothesizing that $U_t$ is controlled by processes localized to the leading edge of the mean flame brush.

Moreover, complex-chemistry DNS of lean hydrogen–air turbulent flames offer the following opportunity for probing the leading point concept. Among nine species included into the state-of-the-art chemical mechanisms of hydrogen combustion and in the mechanism by Kéromnès et al. (Reference Kéromnès2013) used in the present study, only atomic hydrogen H and molecular hydrogen H$_2$ are characterized by molecular diffusivities much larger than the diffusivity of O$_2$ or heat diffusivity of the mixture. At the same time, as discussed earlier, molecular diffusion of H$_2$ results in increasing (when compared with the unperturbed laminar flame) equivalence ratio in positively curved reaction zones, whereas molecular diffusion of H from recombination to reaction zones results in increasing $Y_{\textrm {H}}$ in negatively curved reaction zones. Accordingly, one may assume that an increase in local burning rate should predominantly be observed (i) in negatively curved reaction zones concentrating in the trailing edge of a mean flame brush if Lewis numbers are equal to unity for all species with the exception of H, but (ii) in positively curved reaction zones concentrating in the leading edge of a mean flame brush if Lewis numbers are equal to unity for all species with the exception of H$_2$.

Based on the above reasoning, two extra DNS cases, C1/H and C1/H2, were run by using the mixture-averaged molecular diffusivities of H and H$_2$, respectively, and setting Lewis numbers equal to unity for all other species. The computed results sampled from the entire flame brushes do show that the equivalence ratio, fuel consumption rate, heat release rate and mass fraction of H conditioned to positive curvature are larger than their values conditioned to negative curvature in case C1/H2 (cf. blue solid and dashed lines in figure 13), whereas the opposite trend is observed in case C1/H (cf. red solid and dashed lines).

Figure 13. Time-averaged dependencies of doubly conditioned (a) equivalence ratio $\langle \phi | c | h_m \rangle$, (b) fuel consumption rate $\langle \dot {\omega }_{\textrm {H}_2} | c | h_m \rangle$, (c) heat release rate $\langle \dot {\omega }_T | c | h_m \rangle$, and (d) hydrogen mass fraction $\langle Y_{\textrm {H}} | c | h_m \rangle$ on combustion progress variable. Results conditioned to $\delta _L h_m = -1 \pm 0.005$ and $\delta _L h_m = 1 \pm 0.005$ are shown in dashed and solid lines, respectively. Results obtained from flames C1/H and C1/H2 are plotted in red and blue lines, respectively.

Significant influence of molecular diffusion of atomic hydrogen on local burning structures was recently explored by Rieth et al. (Reference Rieth, Gruber, Williams and Chen2021) by analysing DNS data obtained from (i) preheated lean NH$_3$/H$_2$/N$_2$–air flames propagating in a turbulent shear layer and (ii) statistically planar highly preheated lean hydrogen–air turbulent flames. In particular, by turning off the molecular diffusivity of atomic hydrogen, Rieth et al. (Reference Rieth, Gruber, Williams and Chen2021) have shown that enhanced molecular transport of H into negatively curved low-temperature heat release zones results in significantly increasing local heat release rate. A similar effect is observed in figure 13(c).

The qualitative difference between the present cases C1/H and C1/H2 is also demonstrated in figure 14, where doubly conditioned profiles of $\langle \dot {\omega }_{\textrm {H}_2} | c | h_m \rangle (c)$, $\langle \dot {\omega }_T | c | h_m \rangle (c)$ and $\langle Y_{\textrm {H}} | c | h_m \rangle (c)$ sampled from the leading (red or blue lines) and trailing (black lines) zones are reported. Moreover, figures 14(b), 14(d) and 14( f) show that fuel consumption rate, heat release rate and mass fraction of H, respectively, conditioned to $\langle c \rangle = 0.1 \pm 0.02$, are larger than their values conditioned to $\langle c \rangle = 0.9 \pm 0.02$ in flame C1/H2 (cf. blue and black curves), respectively. In flame C1/H, the opposite trend is documented (cf. red and black curves in figures 14a, 14c or 14e).

Figure 14. Time-averaged dependencies of doubly conditioned (a,b)  fuel consumption rate $\langle \dot {\omega }_{\textrm {H}_2} | c | h_m \rangle$, (c,d) heat release rate $\langle \dot {\omega }_T | c | h_m \rangle$ and (e, f)  hydrogen mass fraction $\langle Y_{\textrm {H}} | c | h_m \rangle$ on combustion progress variable, sampled at $\langle c \rangle = 0.1 \pm 0.02$ (red or blue lines) and $0.9 \pm 0.02$ (black lines). Results conditioned to the negative $\delta _L h_m = -1 \pm 0.005$ or positive $\delta _L h_m = 1 \pm 0.005$ are shown in dashed or solid lines, respectively. Results are for (a,c,e) case C1/H and (b,d, f) case C1/H2.

In addition to figures 13 and 14, which present doubly conditioned $c$-profiles for two intervals of curvature (negative and positive), figure 15 reports simply conditioned profiles of $\langle \dot {\omega }_{\textrm {H}_2} \,|\, c \rangle (c)$, $\langle \dot {\omega }_T \,|\, c \rangle (c)$ and $\langle Y_{\textrm {H}} \,|\, c \rangle (c)$ sampled from the leading (panels (a,c,e)) and trailing (panels (b,d, f)) zones of mean flame brushes in four C-cases. Comparison of results plotted in blue dotted and orange dotted–dashed lines show that preferential diffusion of molecular hydrogen results in increasing (decreasing) fuel consumption rate, heat release rate and mass fraction of H at $\langle c \rangle = 0.1 \pm 0.02$ ($0.9 \pm 0.02$, respectively), with differential diffusion of other species acting in the opposite direction (cf. blue dotted and black solid lines). Comparison of results plotted in red dashed and orange dotted–dashed lines show that preferential diffusion of atomic hydrogen results in increasing (decreasing) $\langle \dot {\omega }_{\textrm {H}_2} \,|\, c \rangle (c)$, $\langle \dot {\omega }_T \,|\, c \rangle (c)$, and $\langle Y_{\textrm {H}} | c | h_m \rangle (c)$ at $c<0.6$ and $\langle c \rangle = 0.9 \pm 0.02$ (various $c$ and $\langle c \rangle = 0.1 \pm 0.02$, respectively).

Figure 15. Time-averaged dependencies of conditioned (a,b) fuel consumption rate $\langle \dot {\omega }_{\textrm {H}_2} | c \rangle$, (c,d) heat release rate $\langle \dot {\omega }_T | c \rangle$ and (e, f) hydrogen mass fraction $\langle Y_{\textrm {H}} | c \rangle$ on combustion progress variable, sampled (a,c,e) at $\langle c \rangle = 0.1 \pm 0.02$ or (b,d, f) at $0.9 \pm 0.02$ Results obtained in cases C, C1, C1/H and C1/H2 are plotted in black solid, orange dotted–dashed, red dashed and blue dotted lines, respectively.

Thus, figures 13–15 imply that differential diffusion effects accelerate burning at the leading edge of the C1/H2 flame brush or at the trailing edge of the C1/H flame brush (if $c<0.6$). Therefore, turbulent burning velocities computed in the two cases offer an opportunity to compare the contributions of the leading and trailing edges to the bulk combustion rate. Figure 16 shows that, among four C-cases, the highest and lowest turbulent burning velocities are obtained in cases C1/H2 (blue dotted lines) and C1/H (red dashed lines), respectively. This result highlights the leading edge. Moreover, comparison of blue dotted and black solid lines indicates that differential diffusion effects for all species with the exception of H$_2$ reduce turbulent burning velocity. In particular, differential diffusion effects for H do so (cf. red dashed and orange dotted–dashed lines). The decrease in the turbulent burning velocity in case C1/H when compared with case C1 is associated with lower fuel consumption and heat release rates at $\langle c \rangle =0.1 \pm 0.02$ (cf. curves plotted in red dashed and orange dotted–dashed lines, respectively, in figures 15a and 15c). While these rates are higher in case C1/H at $\langle c \rangle =0.9 \pm 0.02$ (see figures 15b and 15d), the turbulent burning velocities show the opposite trend, thus, indicating the special role played by the leading edge.

Figure 16. Evolution of the normalized turbulent burning velocities (a) $U_t^F/S_L$ and (b) $U_t^{T}/S_L$ in flames C (black solid lines), C1 (orange dotted–dashed lines), C1/H (red dashed lines) and C1/H2 (blue dotted lines). Horizontal straight lines show mean values.

It is worth remembering, however, that the KPP theory was developed for a specific set of source terms in the convection-diffusion-reaction equation (Kolmogorov et al. Reference Kolmogorov, Petrovsky and Piskounov1937). For another set of source terms, wave propagation can be controlled by processes localized to its trailing edge. A laminar premixed flame described by the classical Zel’dovich–Frank-Kamenetskii theory (Zel'dovich & Frank-Kamenetskii Reference Zel'dovich and Frank-Kamenetskii1938; Zel'dovich et al. Reference Zel'dovich, Barenblatt, Librovich and Makhviladze1985) is an example of such a pushed wave (van Saarloos Reference van Saarloos2003). Transition from pulled flames whose speed is controlled by processes localized to its leading edge to pushed flames was earlier explored by Aldushin, Zel'dovich & Khudyaev (Reference Aldushin, Zel'dovich and Khudyaev1979), Zel'dovich (Reference Zel'dovich1980), Clavin & Liñán (Reference Clavin and Liñán1984), Sabelnikov & Lipatnikov (Reference Sabelnikov and Lipatnikov2015) and Sabelnikov et al. (Reference Sabelnikov, Petrova and Lipatnikov2016). Such a transition can also occur in turbulent flows. For instance, differential diffusion of atomic hydrogen into negatively curved cusps close to the trailing zone of a turbulent flame brush could promote local autoignition, which substantially accelerates flame propagation. However, such an effect has yet to be found in highly preheated mixtures only (Gruber et al. Reference Gruber, Bothien, Ciani, Aditya, Chen and Williams2021; Rieth et al. Reference Rieth, Gruber, Williams and Chen2021). At the room temperature and, in particular, under conditions of the present study, turbulent flames appear to be pulled waves.

3.4. Turbulent combustion and perturbed laminar flames

Within the framework of the leading point concept, an increase in turbulent burning velocity with decreasing Lewis number in lean hydrogen–air turbulent flames is proposed to be modelled by substituting characteristics of the unperturbed laminar flame with characteristics of a highly perturbed laminar flame (Kuznetsov & Sabelnikov Reference Kuznetsov and Sabelnikov1990; Lipatnikov & Chomiak Reference Lipatnikov and Chomiak2005). Accordingly, selection of a laminar flame configuration that is most appropriate for modelling the local structure of the leading points in a turbulent flow (Lipatnikov & Chomiak Reference Lipatnikov and Chomiak1998) is of significant fundamental and applied interest. While this issue is beyond the major scope of the present paper, a few comments follow.

First, certain results discussed earlier highlight local flame curvature, but also show that the list of important local flame perturbation characteristics is not limited to the curvature, e.g. see figure 11, where dependencies of various quantities on curvature are different for different zones of the mean flame brush. Such a list should involve other local flame or flow characteristics, e.g. strain rate. Indeed, since the pioneering study by Klimov (Reference Klimov1963), local strain rate is well known to significantly affect flame structure and burning rate, with such effects expected to be of great importance in highly turbulent flows (Abdel-Gayed, Al-Khishali & Bradley Reference Abdel-Gayed, Al-Khishali and Bradley1984b; Bray & Cant Reference Bray and Cant1991; Bradley et al. Reference Bradley, Gaskell, Gu and Sedaghat2005; Lipatnikov & Chomiak Reference Lipatnikov and Chomiak2005), where the normalized strain rates $\tau _f a_t$ can be large.

Figure 17 shows that joint probability density function (p.d.f.) for normalized curvature $\delta _L h_m$ and normalized strain rate $\tau _f a_t$ varies substantially with $\langle c \rangle$. Accordingly, differences between four curves in figure 11 could be attributed to strain-rate effects, which should be different in the leading, middle and trailing zones of the mean flame brush due to significant differences between the p.d.f.s plotted in figures 17(a), 17(b) and 17(c), respectively. Such an explanation could also be interpreted as an indication that strain rate is one more important characteristic of the local structure of the leading points, in line with the theories of perturbed laminar flames (Matalon & Matkowsky Reference Matalon and Matkowsky1982; Pelcé & Clavin Reference Pelcé and Clavin1982; Zel'dovich et al. Reference Zel'dovich, Barenblatt, Librovich and Makhviladze1985; Class et al. Reference Class, Matkowsky and Klimenko2003; Kelley et al. Reference Kelley, Bechtold and Law2012).

Figure 17. Joint p.d.f.s for normalized curvature $\delta _L h_m$ and normalized strain rate $\tau _f a_t$ sampled at (a) $\langle c \rangle = 0.1 \pm 0.02$, (b) $\langle c \rangle = 0.5 \pm 0.02$ and (c) $\langle c \rangle = 0.9 \pm 0.02$; case C.

However, second, it is worth remembering that the aforementioned theories address large-scale (when compared with the laminar flame thickness) perturbations. Under such conditions, a perturbation is quantified using a single value of curvature, a single value of strain rate, etc., i.e. the perturbation characteristics are considered to be the same in different flame zones. In a highly turbulent flame characterized by $Ka \gg 1$, the Kolmogorov length scale is much smaller than the local flame thickness and, consequently, the local flow characteristics such as strain rate vary significantly within the flame. To demonstrate such variations, points and instants characterized by the smallest values of $|n_x| \ll 1$ and $|n_z| \ll 1$ in the local reaction zones were selected in order for the local profiles of $\delta _L h_m(y)$ or $\tau _f a_t(y)$ to show variations of the curvature and strain rate, respectively, along the normal to the reaction zone.

Such profiles obtained in case B at three instants are plotted in figure 18. While both curvature and strain rate vary with $c(y)$, variations in $\tau _f a_t$ are much more pronounced, with even sign of the local strain rate changing in the reaction zone. Accordingly, it is not clear what value of $\tau _f a_t$ should be used (i) to characterize turbulence-induced perturbations of the local flame structure or (ii) to calculate local combustion characteristics conditioned to strain rate.

Figure 18. Instantaneous axial profiles of (a) normalized curvature $\delta _L h_m(y)$ and (b) normalized strain rate $\tau _f a_t(y)$, obtained at $t/\tau _t=27.51$ (black solid lines), 65.67 (red dashed lines) and 70.20 (blue dotted lines) and presented versus $c(y)$. Vertical straight lines show positions of the peak fuel consumption rate; case B.

Since the locally maximal fuel consumption or heat release rate is controlled by the local mixture composition and temperature, which, in their turn, are affected by molecular transport in upstream flame zones, strain rates evaluated at certain distance upstream of leading points are likely to be more appropriate for characterizing perturbations in the leading points at $Ka \gg 1$. Probably for these reasons, doubly conditioned profiles of $\langle q | a_t | c \rangle$ obtained by evaluating $q$, $a_t$ and $c$ in the same point (not shown) do not reveal any clear trend regarding the influence of the strain rate on $\langle q | a_t | c \rangle$ in case C. This issue definitely requires further research and will be a subject for future study.

Moreover, it is also worth stressing that the influence of intense small-scale turbulence ($Ka \gg 1$) on a premixed flame is a highly unsteady phenomenon, because the lifetime of the smallest eddies is much shorter than the laminar flame time scale under such conditions. Therefore, a simple laminar flame configuration (stationary counter-flow flames, expanding spherical or cylindrical flame, a single cusp, etc.) is unlikely to allow researchers to model all major characteristics (temperature, equivalence ratio, fuel consumption and heat release rates, radical concentrations, etc.) of leading points in a highly turbulent flame.

Nevertheless, third, the fact that the profiles of $\langle \dot {\omega }_{\textrm {H}_2} \,|\, c \rangle (c)$ sampled at $\langle c \rangle = 0.1 \pm 0.02$ from flames A, B and C are close to the profile of $\dot {\omega }_{\textrm {H}_2}(c)$ obtained from a highly strained planar stationary laminar flame (cf. lines with circles in figure 4b), should not be ignored. This result does not mean that the highly strained planar stationary laminar flame is an appropriate model of the local structure of leading points in turbulent flames, e.g. the laminar-flame profiles of $\phi (c)$ and, especially, $\dot {\omega }_T(c)$ differ from $\langle \phi \,|\, c \rangle (c)$ and $\langle \dot {\omega }_T \,|\, c \rangle (c)$, respectively (cf. circles with lines in figures 3b and 5b), respectively. However, the peak fuel consumption rate computed in a highly strained planar stationary laminar flame could still be useful for modelling turbulent burning velocity, as hypothesized by Kuznetsov & Sabelnikov (Reference Kuznetsov and Sabelnikov1990) and confirmed in recent RANS simulations (Verma et al. Reference Verma, Monnier and Lipatnikov2021) of experiments with lean H$_2$/CO–air flames (Venkateswaran et al. Reference Venkateswaran, Marshall, Shin, Noble, Seitzman and Lieuwen2011, Reference Venkateswaran, Marshall, Seitzman and Lieuwen2013).

For instance, a recent analysis (Lee et al. Reference Lee, Dai, Wan and Lipatnikov2021a,Reference Lee, Dai, Wan and Lipatnikovb) of extreme points, i.e. points characterized by the highest instantaneous fuel consumption or heat release rate over the entire computational domain, has shown that the extreme $\dot {\omega }_{\textrm {H}_2}(t)$ fluctuates weakly with time around a mean value, which is almost the same for the three flames A, B and C characterized by significantly different Karlovitz numbers, see table 1. On the contrary, fluctuations of $\dot {\omega }_T(t)$ are more pronounced, with its mean value being substantially increased by $Ka$. These and other results discussed by Lee et al. (Reference Lee, Dai, Wan and Lipatnikov2021a,Reference Lee, Dai, Wan and Lipatnikovb) suggest that there is a maximal possible (for a specific mixture of unburned reactants under specific temperature and pressure) increase in the peak fuel consumption rate, with this extreme increase occurring in various highly perturbed reaction zones, including critically strained ones. This hypothesis requires further research and will be a subject for future study.

3.5. Fuel consumption and heat release rates

As noted at the end of the previous subsection, there are substantial differences in the behaviour of fuel consumption and heat release rates in highly perturbed reactions zones. Similar differences can be observed by comparing figures 4 and 5 or figures 11(c) and 19. These differences are associated with a phenomenon of decorrelation between local fuel consumption and heat release rate at low temperatures, explored by Carlsson, Yu & Bai (Reference Carlsson, Yu and Bai2014), Aspden, Day & Bell (Reference Aspden, Day and Bell2015), Dasgupta et al. (Reference Dasgupta, Sun, Day and Lieuwen2017) and also observed by analysing the present DNS data obtained from flames A, B and C (Lee et al. Reference Lee, Dai, Wan and Lipatnikov2021a). For instance, curves plotted in figures 4 and 5 show that $\langle \dot {\omega }_{\textrm {H}_2} \,|\, c \rangle (c)$ and $\langle \dot {\omega }_T \,|\, c \rangle (c)$ peak at significantly different $c$ (the latter rate peaks at a lower $c$) in the turbulent flames A, B and C, whereas fuel consumption and heat release rates peak at sufficiently close $c$ in the unperturbed laminar flame, see squares.

Figure 19. Time-averaged dependencies of doubly conditioned heat release rate $\langle \dot {\omega }_T | c=0.75 | h_m \rangle$ on the normalized local curvature $\delta _L h_m$, sampled from the entire flame brush (orange dotted–dashed lines), its leading zone characterized by $\langle c \rangle\ (y,t)=0.1 \pm 0.02$ (black solid lines), middle zone ($\langle c \rangle =0.5 \pm 0.02$, red dashed lines) and trailing zone ($\langle c \rangle =0.9 \pm 0.02$, blue dotted lines); case C.

The decorrelation stems from the fact that three reactions that control heat release rate at low temperatures in lean hydrogen–air flames, i.e. H+O$_2$+M$\rightleftharpoons$HO$_2$+M, HO$_2$+H$\rightleftharpoons$2OH, and HO$_2$+OH$\rightleftharpoons$H$_2$O+O$_2$ do not involve H$_2$, with the rates of these reactions being significantly increased in negatively curved reaction zones due to preferential diffusion of atomic hydrogen into them (Aspden et al. Reference Aspden, Day and Bell2015).

Such effects manifest themselves in several figures. First, a decrease in $\langle \dot {\omega }_T | c=0.75 | h_m \rangle$ with increasing curvature is observed in figure 19. Second, results plotted as red lines in figure 13(c) also show that heat release rate is significantly higher in negatively curved reaction zones, with the effect being most pronounced at $c<0.4$. Third, the same trend is observed in figure 14(c), where results obtained from flame C1/H are reported. Fourth, due to preferential diffusion of atomic hydrogen into negatively curved reaction zones, the rate $\langle \dot {\omega }_{\textrm {H}_2} \,|\, c \rangle (c)$ sampled at $\langle c \rangle = 0.9 \pm 0.02$ from flame C1/H is larger (at $c \le 0.6$) than $\langle \dot {\omega }_{\textrm {H}_2} \,|\, c \rangle (c)$ sampled at $\langle c \rangle = 0.9 \pm 0.02$ from flame C1 (cf. red dashed and orange dotted–dashed lines, respectively, in figure 15d). For these rates sampled at $\langle c \rangle = 0.1 \pm 0.02$, the opposite trend is observed in figure 15(c), because the probability of finding negatively curved reaction zones is low at low $\langle c \rangle$. Fifth, figures 20(a) and 20(b) show that $\langle \dot {\omega }_{\textrm {H}_2} | c | h_m \rangle$ is higher for positive curvatures, whereas $\langle \dot {\omega }_T | c | h_m \rangle$ is higher for negative curvatures, respectively.

Figure 20. Time-averaged dependencies of doubly conditioned (a) fuel consumption rate $\langle \dot {\omega }_{\textrm {H}_2} | c | h_m \rangle$ and (b) heat release $\langle \dot {\omega }_T | c | h_m \rangle$ on $c$, obtained at $\delta _L h_m=-0.1 \pm 0.005$ (blue dotted lines), $0 \pm 0.005$ (red dashed lines) and $1 \pm 0.005$ (black solid lines) in case C. Symbols are explained in the caption for figure 2.

The noted substantial differences in the behaviours of fuel consumption and heat release rates in turbulent flames imply that the basic idea of the leading point concept, i.e. the use of characteristics of highly perturbed laminar flames for evaluation of turbulent burning velocity (Kuznetsov & Sabelnikov Reference Kuznetsov and Sabelnikov1990; Lipatnikov Reference Lipatnikov2012), (i) is appropriate for the former rate $\dot {\omega }_{\textrm {H}_2}$, which is increased in positively curved reaction zones concentrating at the leading edge (see black solid line in figure 7), but (ii) does not seem to be appropriate for the latter rate $\dot {\omega }_T$ in lean hydrogen–air turbulent flames. Indeed, the highest heat release rate is reached in negatively curved reaction zones (see figures 19 and 20b), but the probability of finding such zones is low at low $\langle c \rangle$ for purely topological reasons (see red dashed line in figure 7).

Nevertheless, the leading point concept could still be useful for indirectly evaluating the heat-release-based $U_t^{T}$, as $U_t^{T} \approx U_t^F$, see figures 1 and 16, as well as table 1. Therefore, to evaluate $U_t^{T}$, the concept should be applied to $U_t^F$. Similarly, the speed $S_t$ of a fully developed turbulent flame may be evaluated either by (i) integrating $\dot {\omega }_{\textrm {H}_2}$ (or $\dot {\omega }_T$) along the normal to the mean flame brush or (ii) by exploring the speed of the leading edge of the flame brush. Both problems (evaluation of $U_t^{T}$ or evaluation of $S_t$) could hypothetically be solved by selecting one of two alternative methods, but a more elaborated method (i.e. the leading point concept and a constraint of $U_t^{T} \approx U_t^F$) should be adopted to evaluate $U_t^{T}$ due to the lack of models with documented capabilities for directly predicting a significant increase in $U_t^{T}$ with decreasing Lewis number in lean hydrogen–air turbulent flames.