Timescales Associated with the Evolution of Reactive Scalar Gradient in Premixed Turbulent Combustion: A Direct Numerical Simulation Analysis

Abstract

The fractional change in the reaction progress variable gradient depends on the flow normal straining within the flame and also upon the corresponding normal gradients of the reaction rate and its molecular diffusion transport. The statistical behaviours of the normal strain rate and the contributions arising from the normal gradients of the reaction rate and molecular diffusion rate within the flame were analysed by means of a Direct Numerical Simulation (DNS) database of statistically planar turbulent premixed flames ranging from the wrinkled/corrugated flamelets regime to the thin reaction zones regime. The interaction of flame-normal straining with the flame-normal gradient of molecular diffusion rate was found to govern the reactive scalar gradient transport in the preheat zone, where comparable timescales for turbulent straining and molecular diffusion are obtained for small values of Karlovitz numbers. However, the molecular diffusion timescale turns out to be smaller than the turbulent straining timescale for high values of Karlovitz numbers. By contrast, the reaction and hot product zones of the flame remain mostly unaffected by turbulence, and the reactive scalar gradient transport in this zone is determined by the interaction between the flame-normal gradients of molecular diffusion and chemical reaction rates.

1. Introduction

The evolution of the reactive scalar gradient magnitude is related to the inverse of three characteristic times: a turbulence timescale, a molecular diffusion time and chemical time. This paper examines the relative magnitudes of these timescales associated with local normal straining, molecular diffusion and chemical reaction contributions to the magnitude of the reaction progress variable gradient evolution for different values of the Karlovitz number. Although several previous studies [1,2,3,4,5,6,7,8,9] focused on the various aspects of the transport of the reactive scalar gradient, the characteristic times associated with different physical processes associated with its evolution are yet to be scrutinized for different regimes of premixed turbulent combustion. This gap in the existing literature is addressed in this paper by considering the intertwining of small-scale turbulent convection, molecular diffusion and chemical reaction to generate/destroy scalar gradients for different Karlovitz numbers using DNS data of freely propagating statistically planar turbulent premixed flames. In this respect, the main objectives of the present analysis are as follows: (1) to quantify the mean behaviour of the terms of the transport equation of the reactive scalar gradient originating from fluid-dynamic straining, molecular diffusion rate and chemical reaction rate; (2) to identify the timescales associated with the different terms of the transport equation of the reactive scalar gradient using DNS data; and (3) to indicate the leading order contributions at different regions of the flame front for turbulent premixed flames in different combustion regimes.

The physical behaviour and statistical variations of the reactive scalar gradient are of pivotal importance in the fundamental understanding and modelling of premixed turbulent combustion [1]. The reactive scalar gradient is closely related to the Flame Surface Density (FSD) [2] and Scalar Dissipation Rate (SDR) [2], which are widely used for the modelling of turbulent premixed combustion. The SDR characterises the rate of micro-mixing of scalars. Thus, scalar fluctuations and scalar-fluctuation gradients are intimately linked in turbulent mixing. The pocket formation in premixed turbulent combustion was analysed using the transport equation of the magnitude of the reactive scalar gradient by Kollmann and Chen [3]. The statistical behaviours of the terms of the transport equation of the magnitude of the reaction progress variable gradient and their local curvature and tangential strain rate dependences were investigated by Chakraborty and Cant [4] using Direct Numerical Simulation (DNS) data. The results of Chakraborty and Cant [4] were found to be in qualitative agreement with the detailed chemistry DNS analysis by Chakraborty et al. [5]. Sankaran et al. [6] also analysed the transport characteristics of the reactive scalar gradient magnitude using DNS data of turbulent slot jet premixed flames. Dopazo et al. [7] demonstrated the role of flame propagation in the flame normal straining in the reactive scalar transport in premixed turbulent flames. Wang et al. [8] and Sandeep et al. [9] studied the behaviours of strain rates influencing the evolution of the magnitude of the reactive scalar gradient in laboratory-scale configurations for premixed turbulent flames.

The rest of the paper is organised in the following manner. The mathematical background and numerical implementation related to the current analysis are provided in Section 2 and Section 3, respectively. The results are presented and discussed in Section 4 and, finally, the main findings are summarised and conclusions are drawn in Section 5.

2. Mathematical Background

The chemical state within a turbulent premixed flame is characterised by a reaction progress variable $c$, which can be defined based on a suitable major species mass fraction $Y_{\alpha }$ in the following manner: $c=(Y_{\alpha }-Y_{\alpha u})/(Y_{\alpha b}-Y_{\alpha u})$ (here, subscripts ‘u’ and ‘b’ refer to the values in unburned and burned gases, respectively). The reaction progress variable $c$ increases monotonically from 0 to 1.0 from the unburned gas to the fully burned products. The transport equation of $c$ is given by [1,2,3,4,5,6,7,8,9]:

$$\rho \frac{\partial c}{\partial t}+\rho u_k\frac{\partial c}{\partial x_k}=\frac{\partial }{\partial x_k}\left(\rho D\frac{\partial c}{\partial x_k}\right)+{\dot{\omega }}_c$$

(1)

where $\rho$ is the gas density, $u_k$ is the kth component of fluid velocity, $D$ is the diffusivity and ${\dot{\omega }}_c$ is the reaction rate of reaction progress variable. Equation (1) can be written in kinematic mode for a given $c$ isosurface as [7,8,9]:

$$d^{c}c/dt=\left[\partial c/\partial t+V_j\partial c/\partial x_j\right]=0$$

(2)

where $d^c(\dots )/dt$ corresponds to a total derivative in a reference frame, which is attached to the corresponding $c$ isosurface and $V_j=u_j+S_d{N}_j$ is the jth component of the isosurface propagation velocity, with $S_d$ and $N_j=-(\partial c/\partial x_j)/|\nabla c|$ being the displacement speed and jth component of the normal vector, respectively. Using $V_j=u_j+S_d{N}_j$ in Equation (2) yields [7,8,9]:

$$\left[\partial c/\partial t+u_j\partial c/\partial x_j\right]=S_d|\nabla c|$$

(3)

A comparison between Equations (1) and (3) provides the expression for the flame displacement speed $S_d$ for a given $c$ isosurface, as follows:

$$S_d=[{\dot{\omega }}_c+\nabla ·(\rho D\nabla c)]/\rho |\nabla c|$$

(4)

Upon taking the gradient of both sides of Equation (2), it is possible to define a transport equation of $|\nabla c|$ in the following manner [7,8,9]:

$$d^c|\nabla c|/dt=-\left(a_N+N_j\partial S_d/\partial x_j\right)|\nabla c|$$

(5)

where $a_N=N_i{N}_j\partial u_i/\partial x_j=N_i{N}_j{S}_{ij}$ is the fluid-dynamic strain rate normal to the isosurface $c$, with $S_{ij}=0.5(\partial u_i/\partial x_j+\partial u_j/\partial x_i)$ being the ijth component of the strain rate tensor. Equation (5) can be rewritten as follows:

$$\frac{1}{\left|\nabla c\right|}\left(\frac{\partial \left|\nabla c\right|}{\partial t}+u_j\frac{\partial \left|\nabla c\right|}{\partial x_j}\right)=-a_N+T_1+T_2$$

(6)

where $T_1$ and $T_2$ are given by the following:

$$T_1=-\frac{1}{\left|\nabla c\right|}\frac{\partial }{\partial x_N}\left[\frac{1}{\rho }\frac{\partial }{\partial x_j}\left(\rho D\frac{\partial c}{\partial x_j}\right)\right]=-\frac{1}{\left|\nabla c\right|}\frac{\partial }{\partial x_N}\left[\frac{1}{\rho }\frac{\partial }{\partial x_N}\left(\rho D\frac{\partial c}{\partial x_N}\right)+D\frac{\partial c}{\partial x_N}2{\kappa }_m\right]$$

(7)

$$T_2=-\frac{1}{\left|\nabla c\right|}\frac{\partial }{\partial x_N}\left(\frac{{\dot{\omega }}_c}{\rho }\right)$$

(8)

Here, $T_1$ and $T_2$ represent the generation or destruction of $\left|\nabla c\right|$ by molecular diffusion and chemical reaction, respectively, $x_N$ is the local coordinate normal to a given $c$ isosurface pointing toward the fresh reactants and ${\kappa }_m=0.5(\partial N_i/\partial x_i)$ is the local isosurface curvature. The magnitudes of $T_1$ and $T_2$ can be taken to represent the inverse of timescales associated with molecular diffusion and chemical reaction rates (i.e., ${\tau }_{md}=1/|T_1|$ and ${\tau }_{ch}=1/|T_2|$), respectively. Equation (7) also allows us to distinguish between two molecular diffusion times, associated with normal and tangential (related to curvature) diffusions, respectively.

From Equations (6)–(8), one might conjecture the existence of two possible zones within the flame:

(1)

Reaction–diffusion zone: Wherever $⌊a_N⌋\ll ⌊T_2⌋$ within the flow domain, the chemical reaction would dominate over the fluid-dynamic strain rate to create or destruct $\left|\nabla c\right|$. Molecular diffusion and chemical reaction would be of the same order, which leads to

$$\frac{1}{\left|\nabla c\right|}\left(\frac{\partial \left|\nabla c\right|}{\partial t}+u_j\frac{\partial \left|\nabla c\right|}{\partial x_j}\right)=T_1+T_2$$

(9)

The turbulent time would be much greater than the reaction and the molecular diffusion times in this zone.

(2)

Convective–diffusive zone: Wherever, $⌊a_N⌋\gg ⌊T_2⌋$ within the flow domain, the fluid-dynamic strain rate would dominate over the chemical reaction to create or destruct $\left|\nabla c\right|$. Molecular diffusion and fluid-dynamic strain rate would plausibly be of the same order, which leads to

$$\frac{1}{\left|\nabla c\right|}\left(\frac{\partial \left|\nabla c\right|}{\partial t}+u_j\frac{\partial \left|\nabla c\right|}{\partial x_j}\right)=-a_N+T_1$$

(10)

The reaction time would be much greater than the turbulent straining and the molecular diffusion times in this zone. Pure mixing is expected to occur in this zone.

The timescales ${\tau }_{md}=1/|T_1|$ and ${\tau }_{ch}=1/|T_2|$ are local and instantaneous times, which can be averaged. Given that molecular diffusion and chemical reactions are small-scale processes, it is worthwhile to assess whether $a_N$ should be of the order of the smallest turbulent strain micro-scale. Therefore, it needs to be assessed if $a_N$ is of the order of a Kolmogorov strain rate, that is, the inverse of the Kolmogorov time micro-scale, ${\tau }_K$. The local and instantaneous ${\tau }_K$ is defined as ${\tau }_K={\left(\nu /\epsilon \right)}^{1/2}$, where the instantaneous kinetic energy dissipation rate is defined as $\epsilon =2\nu S_{ij}{S}_{ij}-(2/3)\nu {\left(S_{kk}\right)}^2$ for variable density flows. Note that for variable density flows, ${\tau }_K$ is closely related to the volumetric dilatation rate, $\nabla ·\mathit{u}$, and it should also be recalled that $2S_{ij}{S}_{ij}>(2/3){\left(\nabla ·\mathit{u}\right)}^2$. The relative importance of turbulent strain micro-scale and chemical reaction to generate/destroy scalar gradients, measured by the ratio $\left|a_N\right|/\left|T_2\right|$, can then be expressed as $\left|a_N\right|/\left|T_2\right|={\tau }_{ch}/{\tau }_{flow}$, where ${\tau }_{flow}~1/|a_N|$ is the flow timescale. This is a dimensionless variable, similar to a gradient-related local Karlovitz number if ${\tau }_{flow}~{\tau }_K$ but with a rather different definition of the characteristic chemical reaction time as ${\tau }_{ch}=1/|T_2|=1/\left|-(1/\left|\nabla c\right|)d\left({\dot{\omega }}_c/\rho \right)/d{x}_N\right|$. For both ${\tau }_{flow}\gg {\tau }_{ch}$ (within the reaction zone of the flame) and ${\tau }_{flow}\ll {\tau }_{ch}$ (within the preheat zone of the flame), the direct interaction between turbulence and reaction would be very weak. The molecular diffusion time, ${\tau }_{md}$, should plausibly be of the same order as the smaller of ${\tau }_{flow}$ and ${\tau }_{ch}$. Small-scale turbulence interacts directly with molecular diffusion in the preheat zone, whereas the reaction directly interacts with molecular diffusion in the reaction zone.

The magnitudes of these timescales and the aforementioned assumptions will be explored in Section 4 of this paper using DNS data of statistically planar turbulent premixed flames for a range of different values of conventional global Karlovitz numbers.

3. Numerical Implementation

The statistical behaviours of $a_N,T_1$ and $T_2$ are analysed in this paper using DNS data of statistically planar premixed flames. The simulations used for the current analysis are conducted using a 3D compressible code known as SENGA+ [10], where all the standard governing equations of reacting flow [11] are solved in non-dimensional form. In SENGA+, all the spatial differentiations are approximated by a 10th-order central difference scheme for the internal grid points, but the order of accuracy drops gradually to a one-sided 2nd-order scheme at the non-periodic boundaries. The time advancement is achieved using an explicit third-order Runge–Kutta scheme [12]. The simulation domain consists of inlet and outlet boundaries in the direction of mean flame propagation (i.e., $x-$direction), and the transverse boundaries are considered to be periodic. A partially non-reflecting outlet boundary condition is specified in accordance with the Navier–Stokes Characteristic Boundary Conditions (NSCBC) technique [13]. The mean inlet velocity $U_{mean}$ at the inlet boundary is modified during the simulation in order to make it equal to the turbulent burning velocity so that the flame remains stationary inside the simulation domain. The fluctuating velocity field is initialised by a homogeneous isotropic incompressible field generated by a pseudo-spectral method [14] following the Batchelor–Townsend spectrum [15] for desired values of root mean square turbulent velocity $u^{\prime }$ and the integral length scale of turbulence $l$. A steady, unstretched laminar flame solution is used to initialize the scalar field. A modified bandwidth forcing [16] in physical space is used to force the turbulence in the unburned gas to maintain the desired values of root mean square velocity $u^{\prime }$ and the integral length scale of turbulence $l$ in the unburned gas. The domain size, the uniform Cartesian grid for discretization and the unburned gas values of root mean square turbulent velocity fluctuation normalised by the unstrained laminar burning velocity $u^{\prime }/S_L$, integral length scale to the Zel’dovich flame thickness ratio $l/{\delta }_z$, Damköhler number $Da=l{S}_L/u^{\prime }{\delta }_z$, Karlovitz number $Ka={\left(u^{\prime }/S_L\right)}^{3/2}{\left(l/{\delta }_z\right)}^{-1/2}$ and heat release parameter $\sigma =(T_{ad}-T_0)/T_0$ are listed in

Table 1. Simulation parameters for all cases considered in this analysis.

Case	${\mathit{u}}^{\mathbf{\prime }}\mathbf{/}{\mathit{S}}_{\mathit{L}}$	$\mathit{l}\mathbf{/}{\mathit{\delta }}_{\mathit{z}}$	$\mathit{D}\mathit{a}$	$\mathit{K}\mathit{a}$	$\mathit{\sigma }$	Domain Size	Grid Size
Case—A	1.0	5.25	5.25	0.44	4.5	140.7${\delta }_z$ $\times$ (70.35${\delta }_z$)2	800 $\times$ (400)2
Case—B	3.0	5.25	1.75	2.27	4.5	140.7${\delta }_z$ $\times$ (70.35${\delta }_z$)2	800 $\times$ (400)2
Case—C	5.0	5.25	1.05	4.88	4.5	140.7${\delta }_z$ $\times$ (70.35${\delta }_z$)2	800 $\times$ (400)2
Case—D	7.5	5.25	0.7	8.96	4.5	140.7${\delta }_z$ $\times$ (70.35${\delta }_z$)2	800 $\times$ (400)2
Case—E	10.0	5.25	0.525	13.80	4.5	140.7${\delta }_z$ $\times$ (70.35${\delta }_z$)2	800 $\times$ (400)2

. Here, the Zel’dovich flame thickness ${\delta }_z$ is defined as ${\delta }_z={\alpha }_{T0}/S_L$ with ${\alpha }_{T0}$ and $S_L$ being the thermal diffusivity and unstrained laminar burning velocity, respectively. The regime of the combustion for the cases according to the Borghi-Peters diagram [17] are also mentioned in Table 1, which indicates that the cases considered here span from the wrinkled flamelet regime to the high Karlovitz number thin reaction zones regime. At least 10 grid points are kept within ${\delta }_{th}=(T_{ad}-T_0)/\max {\left|\nabla T\right|}_L$ (where $T,T_0$ and $T_{ad}$ are the dimensional temperature, unburned gas temperature and adiabatic flame temperature, respectively) for the grid used in these simulations, whereas 1.0–2.0 grid points are accommodated within the Kolmogorov length scale $\eta$ for the turbulence parameters considered here.

A generic single-step Arrhenius mechanism (i.e., $Reactants\to Products$) representing stoichiometric methane–air flame is considered in this analysis for the sake of computational economy in the interest of a detailed parametric analysis. The same approach was undertaken by several authors [11,18,19,20,21,22,23,24] in the past. All the gases are taken to be perfect gases, and the Lewis number of all the species is considered to be unity. The Prandtl number is taken to be 0.7, the Zel’dovich number (i.e., $\beta =T_{ac}\left(T_{ad}-T_0\right)/T_{ad}^2$, where $T_{ac}$ is the activation temperature) is 6.0 and the ratio of specific heats for constant pressure and constant volume (i.e., $\gamma =c_p/c_v$) is 1.4 for all the simulations. These non-dimensional parameters appear in the non-dimensional governing equations for compressible reacting flows, and interested readers are referred to Ref. [11] for further discussion in this regard. These values are representative of stoichiometric methane–air combustion for reactants preheated to 415 K. Several previous analyses [4,5,24] demonstrated that the statistics of $|\nabla c|$ obtained from simple chemistry DNS of stoichiometric methane–air premixed flames show good qualitative agreements with the findings from detailed chemistry DNS. Moreover, it was demonstrated by Keil et al. [25] that the turbulent premixed flame propagation is captured qualitatively by the single-step chemistry. The quantitative differences in the flame displacement speed statistics are comparable to the uncertainty that arises due to different choices of reaction progress variables for the detailed chemical mechanism. As the current analysis depends on the statistics of $|\nabla c|$ and displacement speed, it can be expected at least the qualitative behaviours are going to be captured accurately in the presence of simple chemistry.

The simulations for the current analysis were continued until the turbulent burning velocity $S_T$ and flame surface area $A_T$ attained statistically stationary states, and the desired values of both turbulent kinetic energy and integral length scale were obtained in the unburned gas. This simulation time is at least equal to at least 10 eddy turnover times (i.e., $t_{sim}>10l/u\prime$), which also is greater than the through-pass time (i.e., $t_{sim}>L_x/U_{mean}$). Each simulation typically needed 80,000 CPU hours. Interested readers are referred to Ref. [26] for further information regarding this database.

4. Results and Discussion

4.1. Flame Morphology

The reaction progress variable $c$ distributions for cases A–E in the $x-y$ midplane are shown in Figure 1a–e, respectively. It can be seen from Figure 1a–e that the flame wrinkling, in general, increases with increasing turbulence intensity $u^{\prime }/S_L$. It can be seen from Figure 1a that case A exhibits large-scale wrinkles due to background turbulent fluid motion, but the isosurfaces of $c$ remain parallel to each other, as the internal flame structure remains unaffected by turbulent fluid motion in the wrinkled/corrugated flamelets regime [17], characterised by $\eta >{\delta }_{th}$. By contrast, the reaction progress variable isosurfaces are not parallel to each other in cases B–E, and local occurrences of flame thickening can be discerned from Figure 1b–e. This behaviour for cases B–E originates as a result of flame thickness being greater than the Kolmogorov length scale $\eta$ (i.e., ${\delta }_{th}>\eta$), which allows for local perturbations of the preheat zone by energetic eddies in these cases. As the separation between ${\delta }_{th}$ and $\eta$ increases with an increasing Karlovitz number $Ka$, the localised flame thickening and the extent of the perturbation of the flame structure strengthen from case B to case E.

4.2. Mean Behaviours of $\left(-a_N\right),T_1$ and $T_2$ Conditioned upon $c$

Before discussing the behaviours of the timescales associated with $T_1$ and $T_2$, it is worthwhile to present the variations of ${\dot{\omega }}_c$ and $\nabla ·(\rho D\nabla c)$ for the unstretched laminar premixed flame for the thermochemistry of the cases considered here. The variations of ${\dot{\omega }}_c$, $\nabla ·\left(\rho D\nabla c\right)$, ${\dot{\omega }}_c+\nabla ·\left(\rho D\nabla c\right)=\rho S_d|\nabla c|$ and ${\rho }_0{S}_L|\nabla c|$ with $c$ for the unstretched laminar premixed flame are shown in Figure 2. It can be seen from Figure 2 that ${\dot{\omega }}_c+\nabla ·\left(\rho D\nabla c\right)=\rho S_d{\approx \rho }_0{S}_L|\nabla c|$ is maintained throughout the flame front. Figure 2 shows the chemical reaction rate ${\dot{\omega }}_c$ assumes significant values for $c>0.4$, and $\nabla ·\left(\rho D\nabla c\right)$ takes positive (negative) values for $c<0.6$ ($c>0.6)$. The reaction rate ${\dot{\omega }}_c$ and molecular diffusion rate $\nabla ·\left(\rho D\nabla c\right)$ assume values with similar order of magnitude but with different signs towards the burned gas side, but ${\dot{\omega }}_c+\nabla ·\left(\rho D\nabla c\right)$ remains positive throughout the flame front with a peak close to $c=0.6$. Although the variations in Figure 2 are shown for the steady unstretched laminar premixed flame, the same qualitative (and mostly quantitative) behaviour is observed for the turbulent flame cases considered here, which are not shown here for the sake of conciseness.

The variations of ${\dot{\omega }}_c$ and $\nabla ·\left(\rho D\nabla c\right)$ with $c$ determine the behaviour of the mean values of $T_1$ and $T_2$ conditional upon $c$. The variations of the mean values of $\left\{-a_N,T_1,T_2\right\}\times {\delta }_z/S_L$ conditional upon $c$ (i.e., $\left\{-〈a_N|c〉,〈T_1|c〉,〈T_2|c〉\right\}\times {\delta }_z/S_L$) for cases A–E are shown in Figure 3a–e, respectively. The standard deviations of $\left\{-a_N,T_1,T_2\right\}\times {\delta }_z/S_L$, conditional upon $c$ for cases A–E, are also shown with vertical bars in Figure 3a–e. Figure 3a–e show that the mean values of $(-a_N)$, conditional upon $c$, assumes negative values throughout the flame front. It can be appreciated from $\nabla ·\left(\rho D\nabla c\right)$ distribution in Figure 2 that $\partial \left[(1/\rho )\partial \left(\rho D\partial c/\partial x_j\right)/\partial x_j\right]/\partial x_N$ is expected to assume a negative (positive) value where $\nabla ·\left(\rho D\nabla c\right)$ assumes positive (negative) values. Therefore, the mean value of $T_1=-(1/|\nabla c|)\partial \left[(1/\rho )\partial \left(\rho D\partial c/\partial x_j\right)/\partial x_j\right]/\partial x_N$ conditioned upon $c$ assumes a small positive value towards the unburned gas and eventually becomes negative and attains a local minimum around $c=0.8$ before taking large positive values for $c>0.9$.

It can be appreciated from Figure 2 that $\partial \left({\dot{\omega }}_c/\rho \right)/\partial x_N$ is expected to take negative (positive) values before (after) the peak value of ${\dot{\omega }}_c$ is obtained. Therefore, the mean value of $T_2=-(1/|\nabla c|)\partial \left({\dot{\omega }}_c/\rho \right)/\partial x_N$ conditioned upon $c$ assumes small positive value for $c<0.9$ with a local maximum value around $c=0.8$ and the mean value of $T_2$ eventually assuming large negative values for $c>0.9$ (see Figure 3a–e).

Figure 3a–e reveal that the reaction–diffusion balance (i.e., the balance between $T_1$ and $T_2$) is obtained in the reaction zone and the burned gas side (i.e., $c>0.5$), where $⟨T_1|c⟩\times {\delta }_z/S_L$ and $⟨T_2|c⟩\times {\delta }_z/S_L$ are of order 10 for $c\ge 0.4-0.5$, whereas the contribution of $⟨-a_N|c⟩$ is much smaller. This is consistent with the expected behaviour of the reaction–diffusion zone (see Equation (9)). On the other hand, for $c<0.5$, $⟨-a_N|c⟩$ and of $⟨T_1|c⟩$ provide non-zero contributions while $⟨T_2|c⟩$ remains negligible compared to the other two terms.

The terms $T_1$ and $T_2$ are associated with small-scale physics, and thus, it is worthwhile to consider whether $⟨T_1|c⟩$ and $⟨T_2|c⟩$ scale with the Kolmogorov timescale. The variations of the mean values of $\left\{-a_N,T_1,T_2\right\}\times {\tau }_K$ conditional upon $c$ (i.e., $\left\{-〈a_N|c〉,〈T_1|c〉,〈T_2|c〉\right\}\times {\tau }_K$) for cases A–E are shown in Figure 4a–e, respectively, where ${\tau }_k$ is estimated as ${\tau }_K={\left[〈\nu |c〉/〈\epsilon |c〉\right]}^{0.5}$.

The qualitative behaviours and leading order balance between $-〈a_N|c〉,〈T_1|c〉$ and $〈T_2|c〉$ are not affected by the normalization procedure, as expected. Therefore, the conclusions drawn from Figure 3a–e remain valid also for Figure 4a–e, but the contributions of $⟨T_1|c⟩\times {\tau }_K$ and $⟨T_2|c⟩\times {\tau }_K$ are approximately of order 1.0 for $c<0.95$, particularly for high Karlovitz number flames. It can further be seen from Figure 3 and Figure 4 that the standard deviations of $-a_N,T_1,T_2$ conditioned upon $c$ increases from case A to case E with increasing turbulence intensity.

4.3. Magnitudes of $a_N,T_1$ and $T_2$ Conditioned upon $c$

In order to estimate the timescales associated with $-a_N,T_1$ and $T_2$, it is useful to consider the variations of the magnitudes of these quantities across the flame front. The variations of $〈\left|a_N\right||c〉\times {\delta }_z/S_L$ and $〈\left|a_N\right||c〉\times {\tau }_K$ with $c$ are shown in Figure 5 and Figure 6, along with the corresponding standard deviations conditional upon $c$ for cases A–E. It can be seen from both Figure 5 and Figure 6 that $〈\left|a_N\right||c〉$ assumes the peak value around $c=0.7$ for cases A–C, irrespective of the choice of normalization. By contrast, $〈\left|a_N\right||c〉\times {\delta }_z/S_L$ decreases from the unburned gas side of the flame front and fluctuates around of value of unity for the major part of the flame front. In all cases, $〈\left|a_N\right||c〉\times {\delta }_z/S_L$ remains of the order of unity, but the magnitude decreases from case A to case E. In cases D and E, $〈\left|a_N\right||c〉\times {\tau }_K$ fluctuates around 0.29 throughout the flame front, but $〈\left|a_N\right||c〉\times {\tau }_K$ remains of the order of unity for cases A–C.

The normal strain rate $a_N$ can be expressed as [27]

$$a_N=(e_{\alpha }{\cos }^2{\theta }_{\alpha }+e_{\beta }{\cos }^2{\theta }_{\beta }+e_{\gamma }{\cos }^2{\theta }_{\gamma })$$

(11)

where $e_{\alpha },e_{\beta }$ and $e_{\gamma }$ are the most extensive, intermediate and compressive eigenvalues of the strain rate tensor and ${\theta }_{\alpha },{\theta }_{\beta }$ and ${\theta }_{\gamma }$ are the angles between $\nabla c$ and the corresponding eigenvectors. It was demonstrated by Chakraborty and Swaminathan [27] that $\nabla c$ shows preferential alignment with the eigenvector associated with $e_{\alpha }$ (i.e., $\left|cos{\theta }_{\alpha }\right|=1.0$) when the strain rate induced by flame normal acceleration ($~\sigma S_L/{\delta }_z$) overwhelms turbulent straining ($~u\prime /l$) [27]. This situation is characterised by $\sigma Da\gg 1$ as the ratio of the strain rate induced by thermal expansion to turbulent straining can be scaled with $\sigma Da$ [27]. By contrast, turbulent straining dominates over the strain rate induced by flame normal acceleration for $\sigma Da\ll 1$, where $\nabla c$ preferentially aligns with the eigenvector associated with $e_{\gamma }$ (i.e., $\left|cos{\theta }_{\gamma }\right|=1.0$) [27], similar to passive scalar mixing [28,29,30]. Moreover, Chakraborty and Swaminathan [27] also demonstrated, based on DNS data, that the extent of preferential alignment of $\nabla c$ with the eigenvector associated with $e_{\alpha }$ (i.e., $\left|cos{\theta }_{\alpha }\right|=1.0$) increases in the region of the flame where the chemical reaction effects are strong. The PDFs of $\left|cos{\theta }_{\alpha }\right|$ and $\left|cos{\theta }_{\gamma }\right|,$ for the cases A–E, are shown elsewhere [26] and, thus, are not repeated here, but their behaviours are in agreement with the aforementioned findings by Chakraborty and Swaminathan [27].

In cases A–C, where $\sigma Da\gg 1$, $\nabla c$ predominantly aligns with the eigenvector associated with $e_{\alpha }$ (i.e., $\left|cos{\theta }_{\alpha }\right|=1.0$), and the strengthening of this alignment leads to an increase of $〈\left|a_N\right||c〉$ from the unburned gas side until it reaches a peak value close to $c=0.7$ before decreasing towards the burned gas side, where the effect of thermal expansion weakens. In cases D and E, where $\sigma Da$ remains of the order of 1.0, $\nabla c$ predominantly aligns with the eigenvector associated with $e_{\gamma }$ (i.e., $\left|cos{\theta }_{\gamma }\right|=1.0$) on the unburned gas and burned gas sides of the flame, but the extent of $\nabla c$ alignment with the eigenvector associated with $e_{\alpha }$ ($e_{\gamma }$) increases (decreases) in the regions of high chemical reactivity.

Therefore, $〈\left|a_N\right||c〉\times {\delta }_z/S_L$ decreases from the unburned gas side of the flame front in cases D and E where the strain rate induced by thermal expansion is of comparable strength of turbulent straining, and thus, $〈\left|a_N\right||c〉\times {\delta }_z/S_L$ fluctuates around of value of unity for the major part of the flame front (see Figure 5). It can further be seen from Figure 6 that $〈\left|a_N\right||c〉\times {\tau }_K$ is of order unity, fluctuating in cases D and E around 0.29, which is consistent with previous findings by Yeung et al. [31], who reported $〈\left|a_N\right||c〉=0.29/{\tau }_K$ for passive scalar mixing.

It is worth noting that Karlovitz number scales as ${Ka~(\delta }_z/S_L)/{\tau }_K$, which suggests $⟨{|a}_N||c⟩\times {\tau }_K~⟨{|a}_N||c⟩\times {\delta }_z/S_L\times K{a}^{-1}$. For Karlovitz numbers of order unity (cases A and B and probably C), the scaling of $⟨{|a}_N||c⟩$ is somewhat ambiguous (scaling with ${\delta }_z/S_L$ and with ${\tau }_K$ are equivalent). Moreover, $⟨{|a}_N||c⟩,$ in cases D and E, can be scaled with respect to both ${\delta }_z/S_L$ and ${\tau }_K$, and thus, a greater range of $Ka$ values is needed for conclusive evidence.

The variations of $⟨T_1|c⟩\times {\delta }_z/S_L$ and $⟨T_1|c⟩\times {\tau }_K$, with $c$ across the flame front for all cases considered here, are shown in Figure 7 and Figure 8, respectively. The corresponding standard deviations conditional upon $c$ are also shown in Figure 7 and Figure 8, respectively. It can be seen from Figure 7 that $⟨{|T}_1||c⟩\times {\delta }_z/S_L\mathrm{remains}\mathrm{of}\mathrm{the}\mathrm{order}\mathrm{of}$ 1.0 for $c<0.4$ but assumes values of the order of 10 in the reaction zone and of the order of 100 towards the burned gas side. Thus, perhaps $⟨{|T}_1||c⟩$ cannot be scaled with the chemical timescale ${\delta }_z/S_L$ (see Figure 7). Figure 8 shows that $⟨{|T}_1||c⟩\times {\tau }_K\mathrm{is}\mathrm{of}\mathrm{the}\mathrm{order}\mathrm{of}1.0$ for a major part of the flame front and of the order of 10 towards the burned gas side. Thus, the molecular diffusion rate contribution $⟨{|T}_1||c⟩$ can perhaps be scaled with ${\tau }_K$.

The variations of $⟨T_2|c⟩\times {\delta }_z/S_L$ and $⟨T_2|c⟩\times {\tau }_K$ with $c$ across the flame front for all cases considered here are shown in Figure 9 and Figure 10, respectively, along with the standard deviations conditioned upon $c$. Figure 9 shows that $⟨{|T}_2||c⟩\times {\delta }_z/S_L\mathrm{remains}\mathrm{of}\mathrm{the}\mathrm{order}\mathrm{of}10$ for a major part of the flame front and of the order of 100 towards the burned gas side. Therefore, $⟨{|T}_2||c⟩$ is unlikely to be scaled with the chemical timescale ${\delta }_z/S_L$ (see Figure 9). It can be seen from Figure 10 that $⟨{|T}_2||c⟩\times {\tau }_K\mathrm{is}\mathrm{of}\mathrm{the}\mathrm{order}\mathrm{of}1.0$ for a major part of the flame front and of the order of 10 towards the burned gas side. Thus, the reaction rate contribution $⟨{|T}_2||c⟩$ can perhaps be scaled with ${\tau }_K$.

4.4. Relative Magnitudes of Timescales ${\tau }_{flow},{\tau }_{md}$ and ${\tau }_{ch}$

In order to compare the timescales associated with ${|a}_N|,{|T}_1|$ and $|T_2|$, the variations of the ratios $R_1=〈\left|T_1+T_2\right|/{|a}_N||c〉$, $R_2=〈\left|T_1\right|/{|a}_N||c〉$ and $R_3=〈\left|T_2\right|/{|a}_N||c〉$ with $c$ for cases A–E are shown in Figure 11a–e, respectively.

It can be seen from Figure 11 that all the ratios assume large values on both ends of the flame front because, locally, ${|T}_1|$ and $|T_2|$ can assume large values on both unburned and burned gas sides of the flame front because of vanishingly small values of $|\nabla c|$ in these regions (see Equations (7) and (8)). Figure 11a–e suggest that $R_2=〈\left|T_1\right|/{|a}_N||c〉$ and $R_3=〈\left|T_2\right|/{|a}_N||c〉$ assume comparable values for $c>0.4$, which is indicative of the fact that the timescales associated with ${|T}_1|$ and $|T_2|$ are found to be comparable. This is indicative of the reaction–diffusion zone, which exists for all cases considered here. Moreover, a comparison between $R_1$, $R_2$ and $R_3$ reveals that ${|T}_1|$ predominantly determines the behaviour of $\left|T_1+T_2\right|$ towards the unburned gas side (i.e., $c<0.4$), where $|T_2|$ assumes negligible values. Moreover, $R_2$ assumes a value of the order of unity in the region given by $0.2<c<0.4$ in cases A–C, which suggests the existence of a convective–diffusive zone in these cases. However, $R_2$ does not assume a value of the order of magnitude of unity in cases D and E, which suggests that the timescale for turbulent straining becomes greater than the molecular diffusion timescale in the preheat zone for flames with high Karlovitz numbers. For 0.4 $<c<0.5$, the interaction of the flame normal gradient of the reaction rate, quantified by $T_2$, with turbulence and molecular diffusion effects are found to be strong, given that ${\tau }_K~{\tau }_{md}~{\tau }_{ch}$. For $c\ge 0.5$, the turbulence weakly interacts with the flame normal gradients of reaction and the molecular diffusion rates in this case. Thus, turbulence does not significantly modify the reaction and hot product zones of the flame for the parameter range considered here.

5. Conclusions

The statistical behaviours of the terms originating from flame normal straining, molecular diffusion rate and chemical reaction rate in the transport equation of $|\nabla c|$ were analysed using a DNS database of statistically planar premixed turbulent flames with a range of different Karlovitz numbers. It was found that the balance between $T_1$ and $T_2$ is obtained in the reaction zone and the burned gas side (i.e., $c>0.5$), but the reaction rate contribution $T_2$ is negligible for $c<0.4$, where $(-a_N)$ and $T_1$ exhibit non-zero contributions. The mean values of molecular diffusion rate and reaction rate contributions conditional upon $c$ normalised by the chemical timescale remain of the order of 10 for a major part of the flame front, whereas these terms, normalised by the Kolmogorov timescale, remain of the order of unity. The magnitude of the normal strain rate scale with both the chemical timescale and the Kolmogorov timescale for the cases considered here. A larger separation between chemical and Kolmogorov timescales (i.e., a greater range of Karlovitz numbers) is needed for conclusive evidence regarding the timescale associated with flame normal straining, but the current findings seem to suggest that the timescale associated with the molecular diffusion rate and reaction rate contributions in the transport equation of $|\nabla c|$ might be scaled with the Kolmogorov timescale. The interaction of flame normal straining with the flame normal gradient of molecular diffusion rate determines the transport of the reactive scalar gradient in the preheat zone of the flame where the timescales for turbulent straining and molecular diffusion are comparable for small values of Karlovitz numbers, but the molecular diffusion timescale turns out to be smaller than the turbulent straining timescale for high values of Karlovitz numbers. For 0.4 $<c<0.5$, the interaction of the flame normal gradient of the reaction rate, with turbulence and molecular diffusion rate effects, are found to be strong given that the timescales for the chemical reaction rate and molecular diffusion rate gradients scale with the Kolmogorov timescale. The turbulent fluid motion weakly interacts with the flame normal gradients of reaction and the molecular diffusion rates for $c\ge 0.5$, and the reaction and hot product zones of the flame remain mostly unaffected by turbulence for the cases considered here.

Although several previous analyses [4,5,24] demonstrated both qualitative and quantitative similarities between $|\nabla c|$ statistics obtained from simple and detailed chemistry DNS results, the current findings need to be confirmed based on DNS data with detailed chemistry and transport with a larger separation of Karlovitz numbers. This forms the platform for further analysis in the future. Moreover, the statistics of the suitably averaged quantities related to $|\nabla c|$ transport and its transients need to be analysed in the future for deeper understanding.