2. Background and research goals

Premixed turbulent combustion is commonly modelled (Bray Reference Bray1995, Reference Bray1996; Peters Reference Peters2000; Bilger et al. Reference Bilger, Pope, Bray and Driscoll2005; Poinsot & Veynante Reference Poinsot and Veynante2005; Lipatnikov Reference Lipatnikov2012) invoking the following transport equations:

(2.1)\begin{equation} \rho \frac{\partial \psi_k}{\partial t} + \rho \boldsymbol{u}\boldsymbol {\cdot }\boldsymbol{\nabla} \psi_k + \boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{J}_{\psi,k} = \dot{\omega}_{\psi,k} \end{equation}

for a set $\boldsymbol {\psi }=\{\psi _1, \ldots, \psi _N \}$ of scalar characteristics of mixture state in a premixed flame, e.g. mass fractions $Y_l$ of various species, temperature $T$, etc. Here, $t$ designates time; $\rho$ is the density; $\boldsymbol {u}$ is the flow velocity vector; and $\boldsymbol {J}_{\psi,k}$ and $\dot {\omega }_{\psi,k}$ designate the molecular flux of the scalar $\psi _k$ and the rate of its creation, respectively. When developing models of the influence of turbulence on premixed burning, the set $\boldsymbol {\psi }$ is often reduced to a single combustion progress variable $c$, which is equal to zero and unity in unburned reactants and burned products, respectively, and can be defined by properly normalizing temperature or concentration of a major reactant or product (Bray Reference Bray1995, Reference Bray1996; Peters Reference Peters2000; Bilger et al. Reference Bilger, Pope, Bray and Driscoll2005; Poinsot & Veynante Reference Poinsot and Veynante2005; Lipatnikov Reference Lipatnikov2012). For brevity, equations summarized in the present section address this simplest case, but the equations can easily be extended to a more general case, e.g. if the set $\boldsymbol {\psi }$ involves normalized concentrations of fuel and oxidant, as well as normalized temperature.

In the chosen simplest case, a local displacement speed $S_d$ is defined as follows (Gibson Reference Gibson1968; Pope Reference Pope1988; Gran, Echekki & Chen Reference Gran, Echekki and Chen1996):

(2.2)\begin{equation} S_d \equiv \frac{- \boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{J}_c + \dot{\omega}_c}{\rho |\boldsymbol{\nabla} c|} \end{equation}

or

(2.3)\begin{equation} S_d = \underbrace{\frac{\dot{\omega}_c}{\rho |\boldsymbol{\nabla} c|}}_{T_1} + \underbrace{\frac{\boldsymbol{n} \boldsymbol{\cdot}\boldsymbol{\nabla} (\rho D_c \boldsymbol{n} \boldsymbol{\cdot} \boldsymbol{\nabla} c)}{\rho |\boldsymbol{\nabla} c|}}_{T_2} \underbrace{- D_c \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{n}}_{T_3} \end{equation}

if Fickian law is adopted to model the flux $\boldsymbol {J}_c = - \rho D_c \boldsymbol {\nabla } c$. Here, $\boldsymbol {n} = - \boldsymbol {\nabla } c/|\boldsymbol {\nabla } c|$ is the unit vector normal to the iso-surface $c(\boldsymbol {x},t)={\rm const.}$ and pointing to fresh mixture; $D_c$ is molecular diffusivity of $c$; and terms $T_1$, $T_2$ and $T_3$ are associated with reaction, normal diffusion and tangential diffusion, respectively.

The displacement speed is widely used in the premixed turbulent combustion literature for a number of reasons. First, $S_d$ can easily be sampled from DNS data using either (2.2) or

(2.4)\begin{equation} S_d = \frac{1}{|\boldsymbol{\nabla} c|} \left( \frac{\partial c}{\partial t} + \boldsymbol{u} \boldsymbol{\cdot}\boldsymbol{\nabla} c \right). \end{equation}

Second, in an unperturbed (stationary, planar and one-dimensional) laminar premixed flame, the continuity equation, (2.1) with $\psi =c$ and (2.3) directly yield

(2.5)\begin{equation} \rho_u S_L = \rho u = \rho S_d = \rho_u S_d^* = \int_{-\infty}^{\infty} \dot{\omega}_c \,\mathrm{d}\kern0.06em x , \end{equation}

i.e. the density-weighted value

(2.6)\begin{equation} S_d^* \equiv \frac{\rho S_d}{\rho_u} \end{equation}

of displacement speed is equal to the unperturbed laminar flame speed $S_L$ and to burning (or consumption) velocity $\int _{-\infty }^{\infty } \dot {\omega }_c\, \mathrm {d}\kern 0.06em x/\rho _u$ everywhere within such a flame. Here, subscript ‘$u$’ refers to unburned reactants.

Third, substitution of (2.2) into (2.1) results in the following kinematic equation:

(2.7)\begin{equation} \frac{\partial c}{\partial t} + \left( \boldsymbol{u} + S_d \boldsymbol{n} \right) \boldsymbol{\cdot} \boldsymbol{\nabla} c = 0. \end{equation}

Therefore, displacement speed of the iso-scalar surface $c(\boldsymbol {x},t)=\xi$ is equal to the speed of this surface with respect to the local flow. Accordingly, difference in $S_L$ and $S_d^*$ conveys information on the influence of turbulence on the local flame speed.

Fourth, multiplication of (2.2) with $\rho |\boldsymbol {\nabla } c|$ and spatial integration of the obtained equation yield

(2.8)\begin{equation} \int\!\!\int\!\!\int\,\rho S_d |\boldsymbol{\nabla} c| \,\mathrm{d} \boldsymbol{x} =- \int\!\!\int\!\!\int\,\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{J}_c \,\mathrm{d}\boldsymbol{x} + \int\!\!\int\!\!\int\,\dot{\omega}_c \,\mathrm{d}\boldsymbol{x} \end{equation}

or

(2.9)\begin{equation} \int\!\!\int\!\!\int\,\rho S_d |\boldsymbol{\nabla} c| \,\mathrm{d}\boldsymbol{x} = \int\!\!\int\!\!\int\,\dot{\omega}_c \,\mathrm{d}\boldsymbol{x} \end{equation}

if molecular fluxes through opposite boundaries of the considered spatial domain are equal to one another, as occurs in a typical DNS study. Thus, there is a direct link between displacement speed and bulk burning rate $\int \!\!\int \!\!\int \,\dot {\omega }_c \,\mathrm {d}\boldsymbol {x}$. Furthermore, in the simplest case of a statistically planar and one-dimensional flame normal to the $x$ axis, which is considered in the rest of the present paper unless otherwise stated, (2.9) reads

(2.10)\begin{equation} \int_{-\infty}^{\infty} \overline{\rho S_d |\boldsymbol{\nabla} c|} \,\mathrm{d}\kern0.06em x = \int_{-\infty}^{\infty} \overline{\dot{\omega}_c} \,\mathrm{d}\kern0.06em x \equiv \rho_u U_T, \end{equation}

where an overline designates a transverse average. Thus, the integrated product $\overline {\rho S_d |\boldsymbol {\nabla } c|}$ directly characterizes turbulent burning velocity $U_T$, similarly to the integrated mean rate $\overline {\dot {\omega }_c}$.

Fifth, by introducing a flame-conditioned displacement speed

(2.11)\begin{equation} \langle \rho S_d \rangle_f \equiv \frac{\overline{\rho S_d |\boldsymbol{\nabla} c|}}{\overline{|\boldsymbol{\nabla} c|}}, \end{equation}

(2.10) can be rewritten as follows:

(2.12)\begin{equation} \int_{-\infty}^{\infty} \langle \rho S_d \rangle_f \overline{|\boldsymbol{\nabla} c|} \,\mathrm{d}\kern0.06em x = \rho_u \int_{-\infty}^{\infty} \langle S_d^* \rangle_f \overline{|\boldsymbol{\nabla} c|} \,\mathrm{d}\kern0.06em x = \rho_u U_T, \end{equation}

thus further emphasizing the link between turbulent burning velocity and displacement speed.

In the simplest case of $\langle S_d^* \rangle _f=S_L$, i.e. if a local flame in a turbulent flow retains the structure of the unperturbed laminar flame or, in other words, the influence of turbulence on the local flame structure is negligible, (2.12) reads

(2.13)\begin{equation} U_T = S_L \int_{-\infty}^{\infty} \overline{|\boldsymbol{\nabla} c|} \,\mathrm{d}\kern0.06em x \equiv S_L A_{\sigma}. \end{equation}

The integral $A_{\sigma }$ is associated with a relative increase $A_f$ in a turbulent flame surface area when compared with a planar laminar flame, because the quantity $\overline {|\boldsymbol {\nabla } c|}$ in this integral is known as the generalized flame surface density. Indeed, integrating the mean value $\overline {\varSigma _{\xi }}$ of flame surface density $\varSigma _{\xi } \equiv \delta (c-\xi ) |\boldsymbol {\nabla } c|$ (Pope Reference Pope1988) over $\xi$, one can arrive at (Vervisch et al. Reference Vervisch, Bidaux, Bray and Kollmann1995; Veynante & Vervisch Reference Veynante and Vervisch2002)

(2.14)\begin{equation} \int_0^1 \overline{\varSigma_{\xi}} \,\mathrm{d} \xi = \int_0^1 \overline{\delta(c-\xi) |\boldsymbol{\nabla} c|} \,\mathrm{d} \xi = \int_0^1 \langle |\boldsymbol{\nabla} c| | c=\xi \rangle P(\xi) \,\mathrm{d} \xi = \overline{|\boldsymbol{\nabla} c|}. \end{equation}

Here, $\delta (c-\xi )$ is the Dirac delta function. Accordingly, the integral $A_{\sigma }$ on the right-hand side of (2.13) characterizes an increase in a generalized flame surface area and the influence of turbulence on a premixed flame is solely reduced to an increase in this area, in line with the first Damköhler hypothesis (Damköhler Reference Damköhler1940).

If the influence of turbulence on the local burning rate and flame structure is substantial so that $\langle S_d^* \rangle _f \ne S_L$, (2.12) can be rewritten as follows:

(2.15)\begin{equation} U_T = S_L G_0 \int_{-\infty}^{\infty} \overline{|\boldsymbol{\nabla} c|} \,\mathrm{d}\kern0.06em x = S_L G_0 A_{\sigma}, \end{equation}

where

(2.16)\begin{equation} G_0 \equiv \frac{ \displaystyle\int\nolimits_{-\infty}^{\infty} \langle S_d^* \rangle_f \overline{|\boldsymbol{\nabla} c|} \,\mathrm{d}\kern0.06em x}{S_L \displaystyle\int\nolimits_{-\infty}^{\infty} \overline{|\boldsymbol{\nabla} c|} \,\mathrm{d}\kern0.06em x} \end{equation}

is a bulk stretch factor.

Sixth, displacement speed is often used not only to explore turbulent burning velocity $U_T$ adopting (2.15) and (2.16), but also to respond to the major challenge of premixed turbulent combustion (Bray Reference Bray1996), i.e. to arrive at a closure relation for the mean rate $\overline {\dot {\omega }_c}$ of product creation. While (2.2) reads

(2.17)\begin{equation} \overline{\dot{\omega}_c} = \rho_u \overline{S_d^* |\boldsymbol{\nabla} c|} + \boldsymbol{\nabla} \boldsymbol{\cdot} \overline{\boldsymbol{J}_c} = \rho_u \langle S_d^* \rangle_f \overline{|\boldsymbol{\nabla} c|} + \boldsymbol{\nabla}\boldsymbol{\cdot}\overline{\boldsymbol{J}_c}, \end{equation}

the following approximate closure relation:

(2.18)\begin{equation} \overline{\dot{\omega}_c} = \rho_u \overline{u_c} \overline{\varSigma_{\xi}} = \rho_u S_L I_0 \overline{\varSigma_{\xi}} \end{equation}

is widely accepted in the premixed turbulent combustion literature (Bray Reference Bray1990, Reference Bray1996; Peters Reference Peters2000; Veynante & Vervisch Reference Veynante and Vervisch2002; Bilger et al. Reference Bilger, Pope, Bray and Driscoll2005; Poinsot & Veynante Reference Poinsot and Veynante2005; Lipatnikov Reference Lipatnikov2012). Here, a stretch factor $I_0$ is introduced (Cant & Bray Reference Cant and Bray1988; Bray Reference Bray1990; Bray & Cant Reference Bray and Cant1991) to characterize a ratio of the mean local consumption velocity $\overline {u_c}$ to the laminar flame speed $S_L$ and the local consumption velocity $u_c$ (Bray Reference Bray1995, Reference Bray1996; Peters Reference Peters2000; Veynante & Vervisch Reference Veynante and Vervisch2002; Bilger et al. Reference Bilger, Pope, Bray and Driscoll2005; Poinsot & Veynante Reference Poinsot and Veynante2005; Lipatnikov Reference Lipatnikov2012) is a rate of product creation per unit flame surface area, divided by $\rho _u$, or the rate $\dot {\omega }_c/\rho _u$ integrated along the local normal to an infinitesimal flame element in the vicinity of it. This closure relation is based on the first Damköhler hypothesis and reduces the influence of turbulence on a premixed flame to (i) an increase in the flame surface area, characterized locally with $\overline {\varSigma _{\xi }}$, and (ii) a change in the local flame structure, characterized with the stretch factor $I_0$. Therefore, (2.18), in fact, extends the first Damköhler hypothesis by substituting $S_L$ with $\overline {u_c} = S_L I_0$. To adopt such an approach in applied simulations, a model of $I_0$ is required and research into displacement speed aims often at addressing this task by assuming that $S_L I_0$ may be substituted with an appropriately averaged $S_d^*$ in (2.18).

Thus, for the above reasons, statistical behaviour of $S_d$ has been the focus of a number of DNS studies of premixed turbulent combustion. Some of them addressed correlations between $S_d$ or $S_d^*$ and (i) the local strain rate (Chakraborty & Cant Reference Chakraborty and Cant2004; Hawkes & Chen Reference Hawkes and Chen2006; Kim & Pitsch Reference Kim and Pitsch2007; Chaudhuri Reference Chaudhuri2015; Cecere et al. Reference Cecere, Giacomazzi, Arcidiacono and Picchia2016), (ii) the local flame curvature (Echekki & Chen Reference Echekki and Chen1996; Chakraborty & Cant Reference Chakraborty and Cant2004; Sankaran et al. Reference Sankaran, Hawkes, Yoo and Chen2015; Wang, Hawkes & Chen Reference Wang, Hawkes and Chen2017a; Wang et al. Reference Wang, Hawkes, Chen, Zhou, Li and Aldén2017b; Luca et al. Reference Luca, Attili, Schiavo, Creta and Bisetti2019), (iii) alignment characteristics of scalar gradient with principal strain rates (Chakraborty & Swaminathan Reference Chakraborty and Swaminathan2007) or (iv) flame-surface topology (Dopazo, Martín & Hierro Reference Dopazo, Martín and Hierro2007; Cifuentes et al. Reference Cifuentes, Dopazo, Martin, Domingo and Vervisch2014). In other DNS studies, probability density function of $S_d^*$ was investigated (Chakraborty, Klein & Cant Reference Chakraborty, Klein and Cant2007; Nivarti & Cant Reference Nivarti and Cant2017; Song et al. Reference Song, Hernández Pérez, Tingas and Im2021), behaviour of displacement speed conditioned to the local flame curvature was explored (Dave & Chaudhuri Reference Dave and Chaudhuri2020), various terms in an evolution equation for a bulk displacement speed were analysed (Yu et al. Reference Yu, Nilsson, Fureby and Lipatnikov2021), etc. Often, the far-reaching goal of such studies consists of arriving at simple model equations for evaluation of a mean value of $S_d^*$, which could substitute $\overline {u_c}$ in (2.18). Accordingly, the problem of predicting turbulent burning rate is split into two separate tasks: (i) modelling an increase in flame surface area and (ii) modelling the influence of turbulence on local flame speed. Equations sought to solve the latter problem are typically based on the theory of weakly perturbed laminar flames (Matalon & Matkowsky Reference Matalon and Matkowsky1982; Pelcé & Clavin Reference Pelcé and Clavin1982; Class, Matkowsky & Klimenko Reference Class, Matkowsky and Klimenko2003; Kelley, Bechtold & Law Reference Kelley, Bechtold and Law2012), with substantial progress in this research direction being recently made by Dave & Chaudhuri (Reference Dave and Chaudhuri2020).

However, it is worth remembering that the local values of $S_d(\boldsymbol {x},t)$ can be widely scattered and irrelevant to the local burning rate. For instance, in the case of inert turbulent mixing, $S_d$ evaluated using (2.2) does not vanish, whereas $\dot {\omega }_c(\boldsymbol {x},t) \equiv 0$. On the contrary, on the surface of a stationary flame ball (Zel'dovich et al. Reference Zel'dovich, Barenblatt, Librovich and Makhviladze1985), $S_d$ vanishes in spite of a high rate $\dot {\omega }_c$. Furthermore, $S_d$ can be even negative, as predicted by Klimov (Reference Klimov1963) and documented by Gran et al. (Reference Gran, Echekki and Chen1996) and in a number of subsequent DNS studies (Chakraborty et al. Reference Chakraborty, Klein and Cant2007; Nivarti & Cant Reference Nivarti and Cant2017; Luca et al. Reference Luca, Attili, Schiavo, Creta and Bisetti2019; Yu & Lipatnikov Reference Yu and Lipatnikov2019; Song et al. Reference Song, Hernández Pérez, Tingas and Im2021; Berger, Attili & Pitsch Reference Berger, Attili and Pitsch2022). In addition, $S_d$ can tend to infinity in the vicinity of so-called ‘zero gradient points’ (Gibson Reference Gibson1968), e.g. somewhere in preheat zones of two colliding laminar flames. In such a case, large local values of $S_d$ do not indicate high local burning rates $\dot {\omega }_c$. In a general case, a local displacement speed does not characterize the local burning rate $\dot {\omega }_c$, because $S_d$ may be controlled by molecular mixing; see terms $T_2$ and $T_3$ in (2.3).

These fundamental issues are often circumvented by assuming that the local effects emphasized above are suppressed after averaging. Such an assumption appears to be sufficiently plausible if local variations in $S_d^*$ and $u_c$ within reaction zones strained and wrinkled by turbulent eddies are weakly pronounced, e.g. see recent DNS studies by Dave & Chaudhuri (Reference Dave and Chaudhuri2020) or Song et al. (Reference Song, Hernández Pérez, Tingas and Im2021). However, such local variations can be significant if the Lewis number is low. In the latter case, local mixture composition, temperature and, hence, burning rate within reaction zones are significantly affected by the local imbalance of molecular flux of chemical energy to the zones and molecular heat flux from the zones. As reviewed elsewhere (Kuznetsov & Sabelnikov Reference Kuznetsov and Sabelnikov1990; Lipatnikov & Chomiak Reference Lipatnikov and Chomiak2005; Lipatnikov Reference Lipatnikov2012), such differential diffusion effects manifest themselves, e.g. in a large increase in turbulent burning velocity in very lean hydrogen–air mixtures characterized by a low Lewis number. Accordingly, there is a need for exploring behaviour of displacement speed in premixed turbulent flames characterized by a low $Le$ and the present work aims at bridging this knowledge gap.

In this regard, it is worth noting that mean values of displacement speed or flame surface density can be evaluated adopting different methods even in the simplest case of a statistically stationary, one-dimensional and planar mean flame brush addressed here. In particular, if conventionally (time and/or transverse) averaged $\overline {|\boldsymbol {\nabla } c|}$ is considered to measure flame surface density, two different mean values of $S_d^*$, i.e. $\overline {S_d^*}$ or $\langle S_d^* \rangle _f$, can be obtained either by taking the same conventional average or applying (2.11), respectively. For instance, conventionally averaged displacement speed $S_d^*$ was used by Chakraborty & Cant (Reference Chakraborty and Cant2005b), Dave & Chaudhuri (Reference Dave and Chaudhuri2020) and Lu & Yang (Reference Lu and Yang2020), the flame-conditioned $\langle S_d^* \rangle _f$ was adopted by Wang et al. (Reference Wang, Hawkes, Chen, Zhou, Li and Aldén2017b) and difference in $S_L$ and $U_T/\int \overline {|\boldsymbol {\nabla } c|} \,\mathrm {d}\kern 0.06em x$ was used by Awad et al. (Reference Awad, Abo-Amsha, Ahmed, Klein and Chakraborty2022) to assess the validity of the first Damköhler hypothesis. The present work aims specifically at investigating (i) relations between $\overline {\dot {\omega }_c}/\rho _u$ and either $\overline {S_d^*} \,\overline {|\boldsymbol {\nabla } c|}$ or $\langle S_d^* \rangle _f \overline {|\boldsymbol {\nabla } c|}$ and (ii) differences between $\overline {S_d^*}$ and $\langle S_d^* \rangle _f$.

Furthermore, since displacement speed and flame surface density are often used jointly to model burning rate, the sensitivity of a mean displacement speed to methods adopted to average this speed is closely linked with the sensitivity of a mean flame surface density to methods adopted to average this density. From this perspective, the use of the generalized flame surface density $\overline {|\boldsymbol {\nabla } c|}$ may be prioritized by stressing that (2.12) directly shows that the turbulent velocity $U_T$ is controlled by $\langle S_d^* \rangle _f \overline {|\boldsymbol {\nabla } c|}$, whereas (2.18) is just a model. However, contribution of reaction zones to $\overline {S_d^*}$, $\langle S_d^* \rangle _f$ or $\overline {|\boldsymbol {\nabla } c|}$ can be eroded. Indeed, while the local fuel consumption or heat release rate vanishes in zones characterized by a low $c(\boldsymbol {x},t) \ll 1$, such zones could contribute to $\overline {S_d^*}$, $\langle S_d^* \rangle _f$ and $\overline {|\boldsymbol {\nabla } c|}$, because the local $S_d^*(\boldsymbol {x},t)$ and $|\boldsymbol {\nabla } c|(\boldsymbol {x},t)$ are controlled by mixing in such zones and do not vanish there. In particular, due to broadening of local flame preheat zones in sufficiently intense turbulence, which was well documented in recent experimental and numerical studies reviewed elsewhere (Driscoll Reference Driscoll2008; Sabelnikov, Yu & Lipatnikov Reference Sabelnikov, Yu and Lipatnikov2019; Driscoll et al. Reference Driscoll, Chen, Skiba, Carter, Hawkes and Wang2020), volumes characterized by finite $| \boldsymbol {\nabla } c |$ but negligible rate $\dot {\omega }_c$ could substantially contribute to $\overline {| \boldsymbol {\nabla } c |}$ and its integral $A_{\sigma }$, but not to $\overline {\dot {\omega }_c}$ and $U_T$, respectively. Consequently, the use of $\overline {| \boldsymbol {\nabla } c |}$, e.g. for modelling $\overline {\dot {\omega }_c}$ or assessing the first Damköhler hypothesis, does not seem to be fully justified from the fundamental perspective.

Therefore, as far as processes controlling local burning rate are concerned, quantities conditioned to a reaction zone $c_1< c(\boldsymbol {x},t)< c_2$ or to a surface $c(\boldsymbol {x},t)=\xi$ within the zone appear to be of the most interest from the general physics perspective. The latter quantities involve $\overline {\varSigma _{\xi }}=\overline { \delta (c-\xi ) |\nabla c|}$, $\langle S_d^* | \xi \rangle = \overline { \delta (c-\xi ) S_d^*}/\overline { \delta (c-\xi )}$ and $\langle S_d^*\rangle _{\xi }=\overline {S_d^* \delta (c-\xi ) |\boldsymbol {\nabla } c|}/\overline {\varSigma _{\xi }}$. The quantities conditioned to a reaction zone may be evaluated as follows:

(2.19)$$\begin{gather} \langle |\boldsymbol{\nabla} c| \rangle_r(x,t) = \frac{\displaystyle\int\!\!\int\,|\boldsymbol{\nabla} c| \left[ H(c-c_1)-H(c-c_2) \right] \mathrm{d} y \,\mathrm{d} z} {(c_2-c_1)A_0}, \end{gather}$$

(2.20)$$\begin{gather}\langle S_d^* | c_1 \leq c \leq c_2 \rangle(x,t) = \frac{\displaystyle\int\!\!\int\,S_d^* \left[ H(c-c_1)-H(c-c_2) \right] \mathrm{d} y\, \mathrm{d} z} {\displaystyle\int\!\!\int\left[ H(c-c_1)-H(c-c_2) \right] \mathrm{d} y\, \mathrm{d}z}, \end{gather}$$

(2.21)$$\begin{gather}\langle S_d^* |\boldsymbol{\nabla} c| \rangle_r(x) = \frac{\displaystyle\int\!\!\int\,S_d^* |\boldsymbol{\nabla} c| \left[ H(c-c_1)-H(c-c_2) \right] \mathrm{d} y \,\mathrm{d} z} {(c_2-c_1) A_0}, \end{gather}$$

where $H(c)$ is the Heaviside function, $A_0$ is cross-sectional area and subscript $r$ refers to reaction zone. Thus, in addition to the goals stated earlier, the present work aims specifically at (i) investigating the sensitivity of these reaction-zone-conditioned quantities to selection of the zone boundaries $c_1$ and $c_2$, including the case of $c_1=0$ and $c_2=1$, corresponding to $\overline {|\boldsymbol {\nabla } c|}$, $\overline {S_d^*}$ and $\langle S_d^* \rangle _f$, (ii) comparing $\overline {\dot {\omega }_c}/\rho _u$ with either $\langle S_d^* | c_1 \leq c \leq c_2 \rangle \langle |\boldsymbol {\nabla } c| \rangle _r$ or $\langle S_d^* |\boldsymbol {\nabla } c| \rangle _r$ and (iii) studying differences between $\langle S_d^* | c_1 \leq c \leq c_2 \rangle \langle |\boldsymbol {\nabla } c| \rangle _r$ and $\langle S_d^* |\boldsymbol {\nabla } c| \rangle _r$ due to correlation between the local displacement speed and flame surface density.

The above discussion and equations show a close link between a displacement speed and a flame surface density within the framework of models aiming at predicting turbulent burning rate. These local flame characteristics are also linked for the following reason. Evolution of mean flame surface area $\overline {\varSigma _{\xi }} \equiv \overline {\delta (c-\xi ) |\boldsymbol {\nabla } c|}$ is well known (Pope Reference Pope1988; Vervisch et al. Reference Vervisch, Bidaux, Bray and Kollmann1995; Veynante & Vervisch Reference Veynante and Vervisch2002) to be described by the following transport equation:

(2.22)\begin{equation} \frac{\partial \overline{\varSigma_{\xi}}}{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot} ( \langle \boldsymbol{u} + S_d \boldsymbol{n} \rangle_{\xi} \overline{\varSigma_{\xi}} ) = \langle a_t \rangle_{\xi} \overline{\varSigma_{\xi}} + \langle S_d \boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{n} \rangle_{\xi} \overline{\varSigma_{\xi}}, \end{equation}

where $a_t = \boldsymbol {\nabla }\boldsymbol{\cdot }\boldsymbol {u} - \boldsymbol {n} \boldsymbol {\cdot } \boldsymbol {\nabla }\boldsymbol {u} \boldsymbol {\cdot }\boldsymbol {n}$ is the strain rate, $\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {n}$ characterizes the local curvature of the considered flame surface $c(\boldsymbol {x},t)=\xi$ and $\langle Q \rangle _{\xi }$ designates value of the quantity $Q$ conditioned to this surface, i.e.

(2.23)\begin{equation} \langle Q \rangle_{\xi} \equiv \frac{\displaystyle\int\!\!\int\,Q \varSigma_{\xi}\, \mathrm{d}y \,\mathrm{d} z} {\displaystyle\int\!\!\int\,\varSigma_{\xi}\, \mathrm{d} y\,\mathrm{d} z} \end{equation}

in the discussed statistically planar one-dimensional case. Thus, $S_d$ directly affects evolution of flame surface density. Note that, by averaging a transport equation for $|\boldsymbol {\nabla } c|(\boldsymbol {x},t)$, the following counterpart of (2.22) can be obtained:

(2.24)\begin{equation} \frac{\partial \overline{|\boldsymbol{\nabla} c|}}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot} ( \langle \boldsymbol{u} + S_d \boldsymbol{n} \rangle_f \overline{|\boldsymbol{\nabla} c|} ) = \langle a_t \rangle_f \overline{|\boldsymbol{\nabla} c|} + \langle S_d \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{n} \rangle_f \overline{|\boldsymbol{\nabla} c|}, \end{equation}

where $\langle Q \rangle _f$ designates value of the quantity $Q$ conditioned to the flame and defined by an equation similar to (2.11). Both (2.22) and (2.24) were explored in DNS studies (Trouvé & Poinsot Reference Trouvé and Poinsot1994; Chakraborty & Cant Reference Chakraborty and Cant2005a; Wang et al. Reference Wang, Hawkes, Chen, Zhou, Li and Aldén2017b; Luca et al. Reference Luca, Attili, Schiavo, Creta and Bisetti2019; Berger et al. Reference Berger, Attili and Pitsch2022; Suillaud et al. Reference Suillaud, Truffin, Colin and Denis Veynante2022), but eventual differences between results yielded by the two equations have not yet been thoroughly addressed in the literature, to the best of the present authors’ knowledge. Moreover, these equations were rarely applied (Berger et al. Reference Berger, Attili and Pitsch2022; Suillaud et al. Reference Suillaud, Truffin, Colin and Denis Veynante2022) to complex-chemistry flames characterized by $Le \neq 1$. Accordingly, the present work aims also at bridging these knowledge gaps. Relevant specific goals are to analyse DNS data obtained from highly turbulent lean hydrogen–air flames characterized by low $Le$ (i) to compare the strain-rate terms $\langle a_t \rangle _{\xi } \overline {\varSigma _{\xi }}$ and $\langle a_t \rangle _f \overline {|\boldsymbol {\nabla } c|}$ or the curvature terms $\langle S_d \boldsymbol {\nabla }\boldsymbol {\cdot } \boldsymbol {n} \rangle _{\xi } \overline {\varSigma _{\xi }}$ and $\langle S_d \boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {n} \rangle _f \overline {|\boldsymbol {\nabla } c|}$ and (ii) to explore the behaviour of terms $\langle a_t \rangle _{\xi } \overline {\varSigma _{\xi }}$ and $\langle S_d \nabla{\,\cdot\,}\boldsymbol {n} \rangle _{\xi } \overline {\varSigma _{\xi }}$ in (2.22) or terms $\langle a_t \rangle _f \overline {|\boldsymbol {\nabla } c|}$ and $\langle S_d \nabla{\,\cdot\,}\boldsymbol {n} \rangle _f \overline {|\boldsymbol {\nabla } c|}$ in (2.24).

Besides comparison of differently averaged displacement speeds or flame surface densities, as well as strain-rate or curvature terms in (2.22) and (2.24), the present work aims at exploring the influence of differential diffusion on these mean quantities and terms, with the focus of analysis being placed on the leading zone of a premixed flame brush. The point is that there is a growing body of theoretical (Sabelnikov & Lipatnikov Reference Sabelnikov and Lipatnikov2013, Reference Sabelnikov and Lipatnikov2015; Kha et al. Reference Kha, Robin, Mura and Champion2016; Sabelnikov, Petrova & Lipatnikov Reference Sabelnikov, Petrova and Lipatnikov2016; Somappa, Acharya & Lieuwen Reference Somappa, Acharya and Lieuwen2022), experimental (Venkateswaran et al. Reference Venkateswaran, Marshall, Shin, Noble, Seitzman and Lieuwen2011, Reference Venkateswaran, Marshall, Seitzman and Lieuwen2013; Zhang et al. Reference Zhang, Wang, Yu, Jin, Zhang and Huang2018) and DNS (Amato et al. Reference Amato, Day, Cheng, Bell and Lieuwen2015a,Reference Amato, Day, Cheng, Bell, Dasgupta and Lieuwenb; Kim Reference Kim2017; Dave, Mohan & Chaudhuri Reference Dave, Mohan and Chaudhuri2018; Lipatnikov, Chakraborty & Sabelnikov Reference Lipatnikov, Chakraborty and Sabelnikov2018; Lee et al. Reference Lee, Dai, Wan and Lipatnikov2021, Reference Lee, Dai, Wan and Lipatnikov2022d) evidence of a crucial role played by the leading edge of a premixed flame in its propagation. This idea goes back to the so-called KPP theory of convection–diffusion–reaction waves (Kolmogorov, Petrovsky & Piskounov Reference Kolmogorov, Petrovsky and Piskounov1937) and the leading-point concept of premixed turbulent combustion, put forward by Zel'dovich and further developed by the Russian school, as reviewed elsewhere (Kuznetsov & Sabelnikov Reference Kuznetsov and Sabelnikov1990; Lipatnikov & Chomiak Reference Lipatnikov and Chomiak2005; Lipatnikov Reference Lipatnikov2012). Within the framework of the concept, to predict a significant increase in turbulent burning velocity with decreasing $Le$, characteristics of unperturbed laminar flames should be substituted with counterpart characteristics of critically perturbed laminar flames (Kuznetsov & Sabelnikov Reference Kuznetsov and Sabelnikov1990; Lipatnikov & Chomiak Reference Lipatnikov and Chomiak2005), because the latter local flames are hypothesized to pull mean turbulent flame brush. Such a simple method was successfully applied to explain and parametrize challenging experimental data (Karpov, Lipatnikov & Zimont Reference Karpov, Lipatnikov and Zimont1996b; Lipatnikov & Chomiak Reference Lipatnikov and Chomiak2005; 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 to quantitatively predict measured results using Reynolds-averaged Navier–Stokes computations (Karpov, Lipatnikov & Zimont Reference Karpov, Lipatnikov and Zimont1996a; Verma, Monnier & Lipatnikov Reference Verma, Monnier and Lipatnikov2021). The method was also supported in a DNS study (Lee et al. Reference Lee, Abdelsamie, Dai, Wan and Lipatnikov2022e). However, until recently, the leading-point concept was not linked with approaches that highlight flame surface density. A study by Berger et al. (Reference Berger, Attili and Pitsch2022) made a step to bridging this knowledge gap by showing that the curvature term in (2.22) is significantly increased at low $\bar {c}$ if $Le$ is decreased. The present work aims at exploring the issue further.

3. Direct numerical simulation attributes

Since simulations whose data are analysed in the present paper were already discussed earlier (Lee et al. Reference Lee, Dai, Wan and Lipatnikov2021, Reference Lee, Dai, Wan and Lipatnikov2022a,Reference Lee, Dai, Wan and Lipatnikovb,Reference Lee, Dai, Wan and Lipatnikovc,Reference Lee, Dai, Wan and Lipatnikovd,Reference Lee, Abdelsamie, Dai, Wan and Lipatnikove), only a brief summary of the DNS attributes is given below.

Three-dimensional DNS of statistically planar and one-dimensional lean H$_2$–air flames propagating in forced turbulence in a box under room conditions were run using the solver DINO (Abdelsamie et al. Reference Abdelsamie, Fru, Oster, Dietzsch, Janiga and Thévenin2016) and a detailed chemical mechanism (9 species and 22 reversible reactions) by Kéromnès et al. (Reference Kéromnès2013). The solver deals with the low-Mach-number formulation of the Navier–Stokes, energy and species transport equations adopting a semi-implicit third-order Runge–Kutta method for time integration and sixth-order finite-difference central stencil for spatial integration. Mixture-averaged molecular transport and chemical reaction rates are modelled using open-source library Cantera-2.3 (Goodwin et al. Reference Goodwin, Malaya, Moffat and Speth2009).

The transport equations were discretized using a uniform Cartesian grid of $\beta N \times N \times N$ cells in a rectangular computational domain of $\beta \varLambda \times \varLambda \times \varLambda$. The adopted values of $\beta$ are reported in table 1. Inflow and outflow boundary conditions were set along the streamwise $x$ direction, with other boundary conditions being periodic. The inlet root-mean-square velocity was set equal to $0.05\ {\rm m}\ {\rm s}^{-1}$, but turbulence was generated applying the linear velocity forcing method (Lundgren Reference Lundgren2003; Rosales & Meneveau Reference Rosales and Meneveau2005; Carroll & Blanquart Reference Carroll and Blanquart2014) between $x=0.5 \varLambda$ and $x=8 \varLambda$. Before the start of combustion computations, constant-density turbulence was simulated for at least 50 integral time scales $\tau _t=L/u'$. Here, $u'$ designates root-mean-square turbulent velocity and an integral length scale $L$ yielded by the linear velocity forcing method is known to be about $0.19 \varLambda$ (Lundgren Reference Lundgren2003; Rosales & Meneveau Reference Rosales and Meneveau2005; Carroll & Blanquart Reference Carroll and Blanquart2014). Characteristics of the generated turbulence are discussed in detail elsewhere (Lee et al. Reference Lee, Dai, Wan and Lipatnikov2022a).

Table 1. Characteristics of DNS cases.

To initialize combustion DNS, the steady planar laminar flame solution yielded by Cantera-2.3 (Goodwin et al. Reference Goodwin, Malaya, Moffat and Speth2009) was embedded into the computational domain. Subsequently, a flame propagated along the $x$ axis from right to left. To restrict the flame motion and to always keep the flame within the forced-flow subdomain, e.g. at a distance from the inlet boundary larger than $0.5 \varLambda$, the mean inlet velocity was manually changed when necessary. The combustion simulations were run for at least $28 \tau _t$.

The simulation conditions are reported in table 1, where $S_L$, $\delta _L^T=(T_b-T_u )/\max \{|{\rm d}T/{{\rm d} x}|\}$, $\delta _L^F=Y_{H_2,u}/\max \{|{\rm d}Y_{H_2}/{{\rm d} x}|\}$ and $\tau _f=\delta _L^T/S_L$ are the laminar flame speed, two thicknesses and time scale, respectively, computed using open-access code Cantera (Goodwin et al. Reference Goodwin, Malaya, Moffat and Speth2009); subscripts ‘$u$’ and ‘$b$’ designate unburnt and burnt mixture, respectively; $\phi$ is the equivalence ratio; $Re_t=u'L/\nu _u$, $Ka=(u'/S_L)^{3/2} (\delta _L^T/L)^{1/2}$ and $Da=\tau _t/\tau _f$ are turbulent Reynolds, Karlovitz and Damköhler numbers, respectively; $\nu _u$ is the kinematic viscosity of unburnt mixture; $\Delta x$ is the grid size; and $\eta =L \,Re_t^{-3/4}$ is the Kolmogorov length scale. Zel'dovich number $Ze$ is equal to 10.7 (or 18.3) in richer (leaner) low-Lewis-number flames, as discussed in detail elsewhere (Lee et al. Reference Lee, Abdelsamie, Dai, Wan and Lipatnikov2022e). Cases A, A1, C and C1 were addressed in earlier papers (Lee et al. Reference Lee, Dai, Wan and Lipatnikov2021, Reference Lee, Dai, Wan and Lipatnikov2022a,Reference Lee, Dai, Wan and Lipatnikovb). All other cases were studied by Lee et al. (Reference Lee, Dai, Wan and Lipatnikov2022c,Reference Lee, Dai, Wan and Lipatnikovd,Reference Lee, Abdelsamie, Dai, Wan and Lipatnikove).

Cases A and C deal with the same equivalence ratio, but $u'$ is larger in the latter case. To decrease $\phi$ by retaining the same $u'/S_L$, case E has been designed. When compared with a richer flame C, flame E propagates in less intense turbulence (a lower $u'$), because $S_L$ is lower at $\phi =0.35$. Finally, case F has been designed to retain the same $u'/S_L$, $L/\delta _L^T$, $Da$ or $Ka$ when varying $\phi$ (cf. cases C and F). When compared with flame E characterized by the same $\phi$, flame F propagates in turbulence with a larger length scale $L$. In four cases A, C, E and F, the highest Karlovitz number and the lowest Damköhler number are reached in flame E.

Since the four flames A, C, E and F are characterized by a low Lewis number $Le=0.32$, they are associated with strong differential diffusion effects. To suppress such effects, four counterpart cases A1, C1, E1 and F1 have been designed by setting molecular diffusivities of all species equal to heat diffusivity of the mixture. Thus, $Le=1$ in flames A1, C1, E1 and F1. Since an increase in $Le$ results in increasing the unperturbed laminar flame speed (Zel'dovich et al. Reference Zel'dovich, Barenblatt, Librovich and Makhviladze1985), Damköhler or Karlovitz numbers are different in each pair of the considered cases.

The adopted numerical meshes ensure more than 20 grid points across the thickness $\delta _L^T$ in low-$Le$ cases A, C, E and F or at least 10 grid points in cases A1, C1, E1 and F1. In all cases, the Kolmogorov length scale is greater than half the grid size, thus indicating acceptable resolution of the turbulent flow (Yeung & Pope Reference Yeung and Pope1989).

Time-dependent mean quantities $\langle q \rangle (x,t)$ were sampled by averaging the field $q(\boldsymbol {x},t)$ over transverse plane $x={\rm const}$. Stationary mean quantities $\bar {q}(x)$ were obtained by averaging $\langle q \rangle (x,t)$ over time at $t/\tau _t>t^*$, with the normalized transition time $t^*$ being varied from 3 to 15 in different cases (see table 1). The $x$ dependencies $\bar {q}(x)$ were transformed to dependencies of $\bar {q}$ on a mean combustion progress variable defined using either the fuel mass fraction $Y_{H_2}$, i.e. $c_F=(Y_{H_2,u}-Y_{H_2})/Y_{H_2,u}$, or the temperature $T$, i.e. $c_T=(T-T_u)/(T_b-T_u)$. The obtained dependencies $\bar {q}(\bar {c}_F)$ or $\bar {q}(\bar {c}_T)$ were similar to dependencies of $\bar {q}$ on $\bar {c}_F$ or $\bar {c}_T$, respectively, computed (i) by transforming $\langle q \rangle (x,t)$ to $\langle q \rangle (\langle c_F \rangle,t)$ or $\langle q \rangle (\langle c_T \rangle,t)$, respectively, and (ii) by averaging $\langle q \rangle (\langle c_F \rangle,t)$ or $\langle q \rangle (\langle c_T \rangle,t)$, respectively, over time.

In the following, numerical results obtained by analysing $c_F(\boldsymbol {x},t)$ or $c_T(\boldsymbol {x},t)$ field are referred to as results obtained within $c_F$ or $c_T$ framework, respectively, with the same symbol $c$ subsuming both $c_F$ and $c_T$ in relevant equations. Accordingly, the same symbol $S_d$ will subsume displacement speeds of surfaces $c_F(\boldsymbol {x},t)=\xi$ and $c_T(\boldsymbol {x},t)=\theta$, and similarly for other quantities such as $a_t$, $\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {n}$, etc. The two displacement speeds are evaluated as follows:

(3.1)\begin{equation} S_{d,F} = \underbrace{\frac{\dot{\omega}_F}{\rho |\boldsymbol{\nabla} c_F|}}_{T_1} + \underbrace{\frac{\boldsymbol{n}_F \boldsymbol{\cdot}\boldsymbol{\nabla} (\rho D_F \boldsymbol{n}_F \boldsymbol{\cdot} \boldsymbol{\nabla} c_F)}{\rho |\boldsymbol{\nabla} c_F|}}_{T_2} \underbrace{- D_F \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{n}_F}_{T_3} \end{equation}

and

(3.2)\begin{align} S_{d,T} &= \frac{1}{\rho |\boldsymbol{\nabla} c_T|} \left[ \frac{\dot{W}_T}{c_p (T_b-T_u)} + \boldsymbol{\nabla}\boldsymbol{\cdot} \left(\frac{\lambda}{c_p} \boldsymbol{\nabla} c_T \right) + \frac{\rho}{c_p} \boldsymbol{\nabla} c_T \boldsymbol{\cdot} \sum_{k=1}^{N_s} D_k c_p \boldsymbol{\nabla} Y_k \right] \nonumber\\ &= \underbrace{\frac{\dot{W}_T}{\rho c_p (T_b-T_u) |\boldsymbol{\nabla} c_T|}}_{T_1} \nonumber\\ &\quad + \underbrace{\frac{1}{\rho |\boldsymbol{\nabla} c_T|} \left[ \boldsymbol{n}_T \boldsymbol{\cdot}\boldsymbol{\nabla} \left(\frac{\lambda}{c_p} \boldsymbol{n}_T \boldsymbol{\cdot} \boldsymbol{\nabla} c_T \right) + \frac{\rho}{c_p} \boldsymbol{\nabla} c_T \boldsymbol{\cdot} \sum_{k=1}^{N_s} D_k c_p \boldsymbol{\nabla} Y_k \right]}_{T_2} \underbrace{- \frac{\lambda}{c_p} \boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{n}_T}_{T_3}. \end{align}

Here, $\boldsymbol {n}_F=-\boldsymbol {\nabla } c_F/|\boldsymbol {\nabla } c_F|$ and $\boldsymbol {n}_T=-\boldsymbol {\nabla } c_T/|\boldsymbol {\nabla } c_T|$; $\lambda$ and $c_p$ are heat conductivity and capacity of the mixture; $D_k$ designates mixture-averaged molecular diffusivity of species $k$; $N_s=9$ is the number of species; and $\dot {\omega }_F$ and $\dot {W}_T$ are fuel consumption and heat release rates, respectively.

When processing DNS data, (2.23) was substituted with

(3.3)\begin{equation} \langle Q \rangle_{\xi} = \frac{\displaystyle\int\!\!\int\,Q |\boldsymbol{\nabla} c| \left[ H(c-c_1)-H(c-c_2) \right] \mathrm{d} y \,\mathrm{d} z} {\displaystyle\int\!\!\int\,|\boldsymbol{\nabla} c| \left[ H(c-c_1)-H(c-c_2) \right] \mathrm{d} y\, \mathrm{d} z} \end{equation}

and mean flame surface density was evaluated as follows:

(3.4)\begin{equation} \bar{\varSigma} = \frac{\displaystyle\int\!\!\int\!\!\int\,|\boldsymbol{\nabla} c| \left[ H(c-c_1)-H(c-c_2) \right] \mathrm{d} y \,\mathrm{d} z \,\mathrm{d} t}{A_0 (c_2-c_1) \Delta t}. \end{equation}

Here, $\Delta t$ designates the duration of time interval over which the DNS data were averaged. To test this method, bulk surface areas were evaluated for various iso-surfaces $c(\boldsymbol {x},t)=\xi$ using the following equation:

(3.5)\begin{align} &\langle |\boldsymbol{\nabla} c| | \xi - \Delta \xi < c(\boldsymbol{x},t) < \xi + \Delta \xi \rangle (\xi)\nonumber\\ &\quad = \frac{\displaystyle\int\!\!\int\!\!\int\!\!\int\,|\boldsymbol{\nabla} c| \left[ H(c-\xi + \Delta \xi)- H(c-\xi - \Delta \xi) \right] \mathrm{d}\boldsymbol{x} \, \mathrm{d} t} {A_0 (c_2-c_1) \Delta t}, \end{align}

with $\Delta \xi =0.025$. Comparison of obtained results with areas of iso-scalar surfaces calculated using interpolation tools available in Matlab supports the tested method, as shown for two low-$Le$ flames characterized by the highest (for each equivalence ratio) $Ka$ in figure 1.

Figure 1. Surface areas calculated using (3.5) with $\Delta \xi =0.025$ (lines) and interpolation tools available in Matlab (symbols). Results obtained from flame C are plotted in black solid line and circles. Results obtained from flame E are plotted in red dashed line and squares.

When applying (2.19)–(2.21) or (3.3) to the analysis of DNS data, the values of $0 \leq c_1 < c_2 \leq 1$ were evaluated using the following constraint:

(3.6)\begin{equation} \dot{\omega}_{c,L}(c=c_1) = \dot{\omega}_{c,L}(c=c_2) = b \max{ \{\dot{\omega}_{c,L}(c)\} }. \end{equation}

Here, $\dot {\omega }_{c,L}$ designates either fuel consumption or heat release rate in a steady, one-dimensional and planar laminar flame and $c$ refers to either $c_F$ or $c_T$, respectively. Profiles of $\dot {\omega }_{c,L}(c_F)$ obtained from the laminar flames A, C ($\phi =0.5$) and E, F ($\phi =0.35$) are shown in figure 2. These profiles are normalized using $\max { \{\dot {\omega }_{c,L}(c)\} }$ in the low-$Le$ flames. If $b=0$, then $c_1=0$ and $c_2=1$, i.e. time-averaged conditioned quantities $\overline {\langle q | c_1 \leq c \leq c_2 \rangle }$ are equivalent to transverse-averaged quantities $\bar {q}$ in this case. Other values of $b$, $c_1$ and $c_2$ used are reported in table 2.

Figure 2. Dependencies of fuel consumption and heat release rates on fuel-based progress variable, obtained from laminar flames characterized by (a) $\phi =0.5$ and (b) $\phi =0.35$. Black solid and blue dot-dashed lines show fuel consumption rate in low-$Le$ and equidiffusive flames, respectively. Red dashed and yellow dotted lines show heat release rate in low-$Le$ and equidiffusive flames, respectively.

Table 2. Characteristics of laminar flames.

5. Concluding remarks

The presented analysis of complex-chemistry DNS data obtained earlier (Lee et al. Reference Lee, Dai, Wan and Lipatnikov2022a,Reference Lee, Dai, Wan and Lipatnikovd) from four pairs of low-Lewis-number and equidiffusive flames characterized by Karlovitz numbers up to 53 and associated with lean hydrogen–air mixtures with equivalence ratios of 0.5 and 0.35 has shown the following trends, which are listed in the order of importance and based on results obtained within $c_F$ framework.

First, significantly higher ratios of turbulent and laminar burning velocities documented earlier in the low-$Le$ flames (Lee et al. Reference Lee, Dai, Wan and Lipatnikov2022a,Reference Lee, Dai, Wan and Lipatnikovd) result mainly from a local increase in fuel consumption or heat release rate due to preferential diffusion of ${\rm H}_2$ into reaction zones perturbed by turbulent eddies. Such local effects increase not only the peak local values of these rates, but also the widths of zones where the rates are high when compared with the counterpart rate in the equidiffusive flame. (Such an increase in the zone width does not mean an increase in the reaction zone thickness in the low-$Le$ flame, because this thickness is determined by comparing the local rate with its peak value within the same flame, rather than with the peak rate in the equidiffusive flame. The thickness of reaction zones localized to the leading edge of the low-$Le$ flame brush is decreased under the present DNS conditions, as the local gradients of combustion progress variables are steepened under the influence of the local strain rate and flame curvature (Lee et al. Reference Lee, Dai, Wan and Lipatnikov2022b, figure 17).) The latter phenomenon plays a more important role in the richer mixture, with both phenomena being most pronounced at $\overline {c_F}<0.2$. This fact further indicates a crucial role played by the leading zone of a premixed turbulent flame brush in its propagation and is consistent with earlier results (Lee et al. Reference Lee, Dai, Wan and Lipatnikov2021, Reference Lee, Dai, Wan and Lipatnikov2022d), which highlighted this crucial role by analysing the same DNS data, but adopting different diagnostic techniques.

Second, curvature terms in flame-surface-density transport equation (2.22) are significantly larger in the vicinity of the leading edges of low-Lewis-number flames when compared with their equidiffusive counterparts. Nevertheless, the influence of an increase in this term with decreasing $Le$ on mean flame surface density is partially counterbalanced by sensitivity of other terms in the transport equation to $Le$.

Third, as a result, a substantial increase in bulk flame surface area with decreasing $Le$ is solely observed in two leaner flames E and F, with the effect magnitude being significantly lower (especially in flame E characterized by the highest $Ka$) when compared with an increase in $U_T/S_L$ with decreasing $Le$.

Fourth, comparison of mean flame surface densities evaluated at (i) the same distance from the leading edges of low-Lewis-number and equidiffusive flames or (ii) the same mean combustion progress variable $\bar {c}$ may show opposite effects of $Le$ on flame surface area in the vicinity of the leading edges, because $\bar {c}$ increases with the distance faster in the former flames due to a significant increase in the mean rate $\overline {\dot {\omega }_c}(\bar {c})$ of product creation, caused by preferential diffusion of H$_2$.

Fifth, differently averaged displacement speeds (i) are higher in low-Lewis-number flames, especially in leaner flames E and F, when compared with their equidiffusive counterparts and (ii) vary substantially with $\bar {c}$. In equidiffusive flames, such variations are even more pronounced, with the average displacement speeds being negative at the leading edge of a mean flame brush in highly turbulent cases C1 and E1. Therefore, a widely used assumption that a time- and transverse-averaged displacement speed may be substituted with $S_L$ is not warranted even in equidiffusive flames if turbulence is sufficiently intense. Nevertheless, certain bulk displacement speeds sampled from the entire computational domain at different instants are close to $S_L$ in all studied equidiffusive flames, thus implying that the aforementioned simple model could be used to evaluate burning velocity even in intense turbulence if $Le=1$.

Sixth, even if all required quantities are sampled from the same DNS data, the mean rate $\overline {\dot {\omega }_c}(\bar {c})$ of product creation differs substantially from a product of an average flame surface density and an average displacement speed, with the exception of a trivial case of the use of $\overline {S_d |\boldsymbol {\nabla } c|}$, which is almost equivalent to the use of $\overline {\dot {\omega }_c}$. Such differences are more pronounced in low-Lewis-number flames and do not vanish after integration along a normal to the mean flame brush. In equidiffusive flames, (i) the differences are still substantial (with the exception of case A1 characterized by a low $Ka=1.6$), especially at low $\bar {c}$, where certain average displacement speeds are negative in highly turbulent cases C1 and E1, and (ii) the use of $S_d$ and $|\boldsymbol {\nabla } c|$ conditioned to a reaction zone yields worse results when modelling the mean rate $\overline {\dot {\omega }_c}(\bar {c})$. On the contrary, evolution of turbulent burning velocity is well predicted using flame surface density and displacement speeds conditioned to a reaction zone in equidiffusive flames.

Seventh, dependencies of time-averaged conditioned strain rates $\overline {\langle a_t \rangle _{\xi }}$ and $\overline {\langle a_t \rangle _f}$ on $\bar {c}$ are weakly sensitive to $Le$, but differ significantly from one another. Sensitivity of $\overline {\langle a_t \rangle _{\xi }}$ to the choice of boundaries of reaction zone to which the strain rate is conditioned is also non-negligible. The curvature term $\overline {\langle S_d \nabla{\,\cdot\,} \boldsymbol {n} \rangle _f}(\bar {c})$ on the right-hand side of (2.24) is also weakly sensitive to $Le$, whereas the curvature term $\overline {\langle S_d \boldsymbol {\nabla }\boldsymbol {\cdot } \boldsymbol {n} \rangle _{\xi }}(\bar {c})$ on the right-hand side of (2.22) is significantly increased with decreasing $\bar {c}$ in the vicinity of the leading edges of low-Lewis-number flames, as highlighted earlier. Sensitivity of $\overline {\langle S_d \boldsymbol {\nabla }\boldsymbol {\cdot } \boldsymbol {n} \rangle _{\xi }}(\bar {c})$ to the choice of boundaries of reaction zone to which the product $S_d \boldsymbol {\nabla }\boldsymbol {\cdot } \boldsymbol {n}$ is conditioned is weak in equidiffusive flames, but is substantial in low-$Le$ flames, especially at their leading edges. Differences between $\overline {\langle S_d \boldsymbol {\nabla }\boldsymbol {\cdot } \boldsymbol {n} \rangle _f}(\bar {c})$ and $\overline {\langle S_d \boldsymbol {\nabla }\boldsymbol {\cdot } \boldsymbol {n} \rangle _{\xi }}(\bar {c})$ are small in the largest parts of equidiffusive flames A1, E1 and F1, but are substantial in flame C1. In all studied low-$Le$ flames, such differences are significant, thus implying that a transport equation for the generalized flame surface density $\overline {\langle |\boldsymbol {\nabla } c| \rangle _f}$ should not be used to explore behaviour of the mean flame surface density $\overline {\langle |\boldsymbol {\nabla } c| \rangle _{\xi }}$.

Eighth, the influence of differential diffusion on burning rate, flame surface density and displacement speed is more pronounced in leaner flames and is not mitigated by increasing $Ka$.

Ninth, differently averaged displacement speeds are sensitive to (i) averaging method, i.e. either $\langle S_d | c_1 \leq c \leq c_2\rangle$, see (2.20), or $\langle S_d |\boldsymbol {\nabla } c| \rangle _r/\langle |\boldsymbol {\nabla } c| \rangle _r$, see (2.21), and (ii) boundaries $c_1$ and $c_2$ of the reaction zone to which the speed is conditioned. The latter sensitivity is less pronounced for $\langle S_d |\boldsymbol {\nabla } c| \rangle _r/\langle |\boldsymbol {\nabla } c| \rangle _r$ or in equidiffusive flames.

Tenth, while the mean flame surface density $\overline {\varSigma }_{\xi }$, see (3.4), is weakly sensitive to variations in boundaries $c_1$ and $c_2$ of the reaction zone to which $|\boldsymbol {\nabla } c|$ is conditioned, dependencies of $\overline {\varSigma }_{\xi }(\bar {c})$ and $\overline {|\boldsymbol {\nabla } c|}(\bar {c})$ are substantially different. Such differences are more pronounced at higher $Ka$ or in low-Lewis-number flames.

Based on the trends summarized above, the following recommendations for modelling turbulent burning velocity and mean fuel consumption rate could be suggested. Models aiming at predicting $U_T$ in equidiffusive mixtures could (i) place the focus of consideration on an increase in flame surface area by turbulence and (ii) substitute the mean local consumption velocity or displacement speed with $S_L$, with this simplest approach performing well even in sufficiently intense turbulence (case C1). If a model aims at predicting $\overline {\dot {\omega }_c}$, the same approach performs worse, while the use of the reaction zone surface area $\overline {\langle | \boldsymbol {\nabla } c| \rangle _r}$ can yield acceptable results in moderately intense turbulence (cases A1, E1 and F1). Substitution of $S_L$ with any mean displacement speed explored in the present work (with the exception of the trivial case of the use of $\overline {S_d^* | \boldsymbol {\nabla } c|}$) worsens results, especially at the leading edge of a mean flame brush, where mean displacement speeds can be negative in sufficiently intense turbulence.

To predict the influence of differential diffusion on $U_T$ or $\overline {\dot {\omega }_c}$ in mixtures characterized by a low $Le$, the focus of consideration should be placed on the influence of $Le$ on the mean local consumption velocity, whereas the influence of $Le$ on flame surface area seems to be of secondary importance in sufficiently intense turbulence. From this perspective, the leading-point concept, which highlights a strong increase in the mean local consumption velocity near the leading edges of the discussed flames (Kuznetsov & Sabelnikov Reference Kuznetsov and Sabelnikov1990; Lipatnikov & Chomiak Reference Lipatnikov and Chomiak2005; Verma et al. Reference Verma, Monnier and Lipatnikov2021), appears to be the best tool available today, at least for predicting $U_T$. Other approaches that could be invoked to address the influence of $Le$ on the mean local consumption velocity were not addressed in the present work, but could be explored in future studies.

If a model deals with a displacement speed, a significant increase in $S_d$ with decreasing $Le$ should definitely be taken into account, but $\overline {\dot {\omega }_c}$ was poorly predicted even when mean displacement speeds were directly sampled from the present DNS data. For turbulent burning velocity, predictions were substantially better, thus implying that $S_d$-based models could be suitable for predicting $U_T$, rather than $\overline {\dot {\omega }_c}$.

To model the influence of differential diffusion on flame surface area, the transport equation (2.22) for a mean flame surface area $\overline {\varSigma _{\xi }}$ appears to be superior to the transport equation (2.24) for the generalized flame surface area $\overline {| \boldsymbol {\nabla } c|}$. However, sensitivity of unclosed strain rate and curvature terms in the former equation to the choice of reaction zone boundaries should thoroughly be investigated in future studies.

Within $c_T$ framework, the majority of the trends emphasized above are less (if any) pronounced. This difference between results sampled from the same DNS data within $c_F$ and $c_T$ frameworks is associated with the fact that, in lean hydrogen–air flames, heat release rate is significant not only at sufficiently large $c_T$, but also in low-temperature flame zones, where this rate is proportional to the concentration of atomic hydrogen. Since (i) molecular diffusivities of ${\rm H}_2$ and H are large, but (ii) ${\rm H}_2$ and H diffuse in opposite directions, i.e. from reactant and product sides, respectively, an increase in one flux, e.g. due to local flame curvature, is partially counterbalanced (or overwhelmed) by an increase in another flux. For instance, molecular diffusion of H is known to significantly increase the local heat release rate in negatively curved low-temperature reaction zones (Carlsson et al. Reference Carlsson, Yu and Bai2014; Aspden et al. Reference Aspden, Day and Bell2015; Lee et al. Reference Lee, Dai, Wan and Lipatnikov2021; Rieth et al. Reference Rieth, Gruber, Williams and Chen2022; Lee et al. Reference Lee, Dai, Wan and Lipatnikov2022b), whereas molecular diffusion of ${\rm H}_2$ is known to decrease fuel consumption and heat release rate in negatively curved high-temperature reaction zones. Thus, contrary to fuel consumption rate, which is mainly (weakly) affected by preferential diffusion of molecular (atomic) hydrogen, heat release rate is affected by preferential diffusion of both ${\rm H}_2$ and H, with the two effects partially counterbalancing one another. Accordingly, some trends documented within $c_F$ framework and caused by preferential diffusion of molecular hydrogen are less pronounced within $c_T$ framework.

For modelling the strong influence of differential diffusion on burning rate in low-$Le$ turbulent flames, the former framework appears to be superior to the latter one, because, under substantially different conditions, the highest local fuel consumption rate within a turbulent flame brush is close to the fuel consumption rate in the critically strained (close to quenching) laminar premixed flame (Lee et al. Reference Lee, Abdelsamie, Dai, Wan and Lipatnikov2022e, figure 3), whereas a similar trend does not hold for heat release rate (Lee et al. Reference Lee, Dai, Wan and Lipatnikov2022a, figure 6b). Accordingly, a fuel consumption rate pre-computed for the critically strained laminar premixed flame can directly be used as an input parameter for numerical simulations of turbulent combustion of mixtures characterized by a low $Le$ (Verma et al. Reference Verma, Monnier and Lipatnikov2021).