Assessment of Extrapolation Relations of Displacement Speed for Detailed Chemistry Direct Numerical Simulation Database of Statistically Planar Turbulent Premixed Flames

Abstract

A three-dimensional Direct Numerical Simulation (DNS) database of statistically planar $H_{2} -$ air turbulent premixed flames with an equivalence ratio of 0.7 spanning a large range of Karlovitz number has been utilised to assess the performances of the extrapolation relations, which approximate the stretch rate and curvature dependences of density-weighted displacement speed $S_{d}^{*}$. It has been found that the correlation between $S_{d}^{*}$ and curvature remains negative and a significantly non-linear interrelation between $S_{d}^{*}$ and stretch rate has been observed for all cases considered here. Thus, an extrapolation relation, which assumes a linear stretch rate dependence of density-weighted displacement speed has been found to be inadequate. However, an alternative extrapolation relation, which assumes a linear curvature dependence of $S_{d}^{*}$ but allows for a non-linear stretch rate dependence of $S_{d}^{*}$, has been found to be more successful in capturing local behaviour of the density-weighted displacement speed. The extrapolation relations, which express $S_{d}^{*}$ as non-linear functions of either curvature or stretch rate, have been found to capture qualitatively the non-linear curvature and stretch rate dependences of $S_{d}^{*}$ more satisfactorily than the linear extrapolation relations. However, the improvement comes at the cost of additional tuning parameter. The Markstein lengths L_M for all the extrapolation relations show dependence on the choice of reaction progress variable definition and for some extrapolation relations L_M also varies with the value of reaction progress variable. The predictions of an extrapolation relation which involve solving a non-linear equation in terms of stretch rate have been found to be sensitive to the initial guess value, whereas a high order polynomial-based extrapolation relation may lead to overshoots and undershoots. Thus, a recently proposed extrapolation relation based on the analysis of simple chemistry DNS data, which explicitly accounts for the non-linear curvature dependence of the combined reaction and normal diffusion components of $S_{d}^{*}$, has been shown to exhibit promising predictions of $S_{d}^{*}$ for all cases considered here.

1 Introduction

The flame propagation in premixed combustion is well-known to be affected by the flame surface curvature and stretch rate (Klimov 1963; Markstein 1951; Karlovitz et al. 1953; Istratov and Librovich 1969; Matalon and Matkowsky 1982; Wu and Law 1984; Kelley and Law 2009; Chen 2011; Kelley et al. 2011; Wu et al. 2005), and an extensive review of the subject is provided by Lipatnikov and Chomiak (2005). In premixed turbulent combustion, the flame propagation is quantified by the flame displacement speed $S_{d}$, which represents the instantaneous speed at which a given scalar isosurface moves normal to itself with respect to the background fluid motion. The density change gives rise to a change in $S_{d}$ within the flame but in a planar unstrained laminar premixed flame, the density-weighted displacement speed $S_{d}^{*} = \rho S_{d} /\rho_{0}$ (where $\rho$ and $\rho_{0}$ are instantaneous gas density and unburned gas density, respectively) remains identical to the unstrained laminar burning velocity S_L._. However, this equality (i.e. $S_{d}^{*} = S_{L}$) is invalid for stretched and curved premixed flames (Giannakopoulos et al. 2015; Chakraborty and Cant 2011; Sabelnikov et al. 2017) and the statistics of $S_{d}^{*}$ are often necessary in the level-set (Peters 2000) and Flame Surface Density (FSD) (Pope 1988) based modelling methodologies. The importance of displacement speed in the closure of the FSD transport equation were discussed in detail by Hawkes and Cant (2001), and Chakraborty and Cant (2009) demonstrated that the effects of local flame curvature $\kappa_{m}$ and flame stretch rate $K = \left( {a_{T} + 2S_{d} \kappa_{m} } \right)$ (where $a_{T}$ is the tangential strain rate) on the density-weighted displacement speed $S_{d}^{*}$ need to be addressed for accurate closure of the FSD transport equation in the context of Large Eddy Simulations (LES). One of the simplest extrapolation relations expresses $S_{d}^{*}$ as a linear function of $K$ with the help of a length L_M known as the Markstein length (Wu and Law 1984). This relation will henceforth be referred to as the linear stretch (LS) extrapolation. Several (for a review see Lipatnikov and Chomiak (2005)) experimental (Kelley and Law 2009; Karpov et al. 1997) or Direct Numerical Simulation (DNS) (Dave et al. 2020; Chen and Im 1998; Chakraborty et al. 2007) studies demonstrated non-linear $K$ dependence of $S_{d}^{*}$, and accordingly Kelley and Law (2009) proposed a quasi-steady non-linear extrapolation relation, which will henceforth be referred to as the NQ model (see Table 1). The NQ model was derived based on the study of Ronney and Sivashinsky (1989) on weakly stretched premixed flames, and this extrapolation relation remains valid not only for near unity Lewis numbers, but also is more successful in capturing the stretch rate dependence of $S_{d}^{*}$ than the LS model (Kelley and Law 2009; Chen 2011).

Table 1 Summary of extrapolation relations

Similar to the LS expression, a linear relation between flame speed and flame front curvature was originally proposed by Markstein (1951) based on an empirical assumption involving a Markstein diffusivity $D_{M} = S_{L} L_{M}$ and later observed experimentally (Karpov et al. 1997; Lipatnikov et al. 2015). This relation is referred to as the LC extrapolation in this analysis. The LC extrapolation was subsequently used by Frankel and Sivashinsky (1983) to analyse spherically expanding flames with thermal expansion under the premise of large flame radii. It was found that both NQ and LC extrapolations provide similar results for negative Markstein diffusivity in the case of thermo-diffusively unstable flames, but significant differences have been reported for positive Markstein diffusivities. The LC extrapolation was subsequently extended by Kelley et al. (2011) by including second and third order series contributions of curvature κ_m, which will be referred to as the non-linear equation (NE) extrapolation in this paper. The evaluation of the NE model requires a non-zero value of the term in brackets. The curvature range that ensures a positive value is quite large for the cases considered in this work such that the singularity can effectively be avoided. An alternative variant of the non-linear extrapolation was suggested by Wu et al. (2005) where a parameter C associated with a $\kappa_{m}^{2}$ contribution is considered and this non-linear extrapolation with 3 terms will be referred to as the N3P extrapolation in this paper. While the non-linear extrapolation relations offer the potential to better represent the data, this advantage comes at the cost of additional tuning parameters. All the extrapolation relations discussed above and listed in Table 1 were originally proposed for weakly stretched laminar flames. These expressions are often used to extract the unstrained laminar burning velocity from the experimentally obtained flame propagation measurements (Wu and Law 1984; Kelley and Law 2009; Chen 2011; Kelley et al. 2011; Wu et al. 2005). It is common that the stretched laminar flame speed is determined as a function of the unstretched laminar flame speed $S_{L}$ and the stretch rate. Most of the stretch extrapolations shown in Table 1 represent such an explicit “forward” functional relationship, which is a common practice in the context of Reynolds Averaged Navier–Stokes (RANS) / Large Eddy Simulations (LES). This motivates to assess the validity of the extrapolation relations for a range of turbulence intensities $u^{\prime}/S_{L}$ (where $u^{\prime}$ is the root-mean-square (rms) velocity fluctuation), which is yet to be reported in the existing literature. Furthermore, Clavin (1985) indicated that the Markstein length/diffusivity behaviour may change depending on the reaction progress variable c value within the flame and thus it is necessary to assess the sensitivity of L_M on $u^{\prime}/S_{L}$ and also on the choice of the value of reaction progress variable c within the flame front. This becomes particularly challenging in the presence of detailed chemistry and transport because the definition of $c$ is not unique and can be done in a number of different ways, which also suggests that the statistics of $S_{d}^{*}$ (and therefore for L_M) are likely to be affected by the choice of c. While the Markstein length is rigorously defined as a derivative of a laminar flame speed with respect to the flame stretch rate at the limit of vanishing stretch rate, the terminology is used widely and more generally in the context of turbulent premixed combustion (Brequigny et al. 2016; Venkateswaran et al. 2015; Chaudhuri et al. 2013) and the same approach has been adopted in this work. In this respect, the main objectives of this paper are:

• (a) To assess the validity of the extrapolation relations for different values of $u^{\prime}/S_{L}$ across different regimes of premixed turbulent combustion.

• (b) To analyse the sensitivity of the values of L_M to the value of c and its definition for different regimes of premixed combustion.

• (c) To identify the extrapolation relation which provides the best possible approximation of the local curvature and stretch rate dependences of $S_{d}^{*}$.

The authors recently addressed some of these issues in the context of simple chemistry (Herbert et al. 2020). However, the choice of reaction progress variable on the performance of extrapolation relations could not be addressed there. Hence, this analysis extends these results in the context of detailed chemistry and transport. Thus, the assessment of the extrapolation relations is the main focus of this analysis and the authors are not aware of any study in the existing literature where the aforementioned exercise was undertaken, and the novelty of this work lies in this respect.

In order to meet these objectives, a detailed chemistry DNS database of statistically planar ${\text{H}}_{2} -$ air premixed flames of equivalence ratio 0.7 (i.e. $\phi = 0.7$) spanning different regimes of premixed turbulent combustion has been considered. The choice of $\phi = 0.7$ is driven by the fact that ${\text{H}}_{2} -$ air premixed flames are nominally thermo-diffusively neutral with respect to the stretch effects on $S_{d}^{*}$ at this equivalence ratio (Chen and Im 1998; Im and Chen 2002). However, different species have different Lewis number even through the flame might be nominally thermo-diffusively neutral, which necessitates the assessment of the performance of the extrapolation relations for different species with varying Lewis numbers $Le$, including major species with $Le \ll 1$ (e.g. ${\text{ H}}_{2}$). In order to address this aspect, the aforementioned detailed chemistry DNS database has been utilised to assess the performance of the LS, NQ, LC, NE and N3P extrapolations for multiple values of the progress variable c across the flame for different definitions of reaction progress variable.

The rest of the paper is organised as follows. The mathematical background and numerical implementation pertaining to this analysis are discussed in the next two sections. Following that, results will be presented and subsequently discussed before the conclusions are drawn.

2 Mathematical background

The reaction progress variable $c$ is defined (Bray 1980) based on a suitable reactant or product mass fraction ${Y}_{R}$ as:

$$ c = \left( {Y_{R0} - Y_{R} } \right)/\left( {Y_{R0} - Y_{R\infty } } \right) $$

(1)

where $Y_{R}$ is the mass fraction of the reactant which is used for the definition of reaction progress variable. In this analysis, reaction progress variables based on ${\text{ H}}_{2}$, ${\text{O}}_{2}$ and ${\text{H}}_{2} {\text{O}}$ mass fractions are considered. The differential diffusion between heat and mass due to the presence of light species in ${\mathrm{H}}_{2}$−air premixed flames may give rise to local occurrences of ${\text{H}}_{2} {\text{O}}$ mass fractions higher than the maximal values found in laminar flame data. Indeed, such behaviour can be locally observed but does not affect the current analysis where reaction progress variable values of $0.3,0.5,0.8$ are primarily considered because the peak value of reaction progress variable reaction rate takes place within this region in laminar ${\text{H}}_{2}$-air flames with $\phi = 0.7$ (Klein et al. 2018). Figure 1 shows the reaction rate magnitude $\dot{\omega }_{Y}$ normalised by the maximum absolute value of reaction rate together with the reaction progress variable ${c}_{Y}$ for major species $Y = H_{2} ,H_{2} O,O_{2}$ versus non-dimensional temperature $\theta = \left( {T - T_{0} } \right)/\left( {T_{ad} - T_{0} } \right){ }$ (with $T,T_{0}$ and $T_{ad}$ being the instantaneous temperature, unburned gas temperature and adiabatic flame temperature, respectively) based on laminar ${\text{H}}_{2} -$ air flame data with $\phi = 0.7$. Figure 1 indicates that this peak location corresponds approximately to $c_{{H_{2} }} \approx 0.8, c_{{H_{2} O}} \approx 0.6, c_{{O_{2} }} \approx 0.6$. This demonstrates that the choice of reaction progress variable values selected for this analysis covers the relevant range of $c$ values for which the results are presented in Sect. 4 and roughly corresponds to the location of maximum reaction rates.

Fig. 1

Magnitude of the reaction rate $\dot{\omega }_{Y}$ normalised with the maximum absolute value of reaction rate (solid lines) and reaction progress variable $c_{Y}$ for major species $Y = {\text{H}}_{2} ,{\text{H}}_{2} {\text{O}},{\text{O}}_{2}$ (dashed lines) versus non-dimensional temperature $\theta$ based on temperature for laminar flame data

The transport equation of c is given by (Chakraborty and Cant 2011; Sabelnikov et al. 2017; Peters 2000; Pope 1988; Hawkes and Cant 2001; Chakraborty and Cant 2009; Bray 1980):

$$ \rho \left[ {\partial c/\partial t + \vec{u} \cdot \nabla c} \right] = \dot{w} + \nabla \cdot \left( {\rho D_{c} \nabla c} \right) $$

(2)

where $\vec{u}$ is the fluid velocity and $\dot{w} = - \dot{w}_{Y} /\left( {Y_{R0} - Y_{R\infty } } \right)$ with $\dot{w}_{Y}$ being the net reaction rate of the corresponding species. In the context of mixture-averaged transport, as employed in the present work, the reaction progress diffusivity $D_{c}$ of a species is determined by $D_{c} = \left( {1 - Y_{k} } \right)/\mathop \sum \limits_{j \ne k} X_{j} /D_{jk}$ where $X_{j}$ is the mole fraction of species j, $D_{jk}$ is the binary diffusion coefficient, and species k is used to define the reaction progress variable (where $k \in \left\{ {H_{2} ,O_{2} ,H_{2} O} \right\}$ has been used for the analysis). The thermo-physical properties used for the DNS computations are used for the analysis of this paper. The kinematic form of the transport equation of a given $c$ isosurfaces can be written as (Chakraborty and Cant 2011, 2009; Sabelnikov et al. 2017; Peters 2000; Pope 1988; Hawkes and Cant 2001):

$$ \rho \left[ {\partial c/\partial t + \vec{u} \cdot \nabla c} \right] = \rho_{0} S_{d}^{*} \left| {\nabla c} \right| $$

(3)

A comparison of Eqs. 2 and 3 indicates that:

$$ S_{d}^{*} = \left[ {\dot{w} + \nabla \cdot \left( {\rho D_{c} \nabla c} \right)} \right]/(\rho_{0} \left| {\nabla c} \right|) $$

(4)

Equation 4 has been utilised to obtain displacement speed from DNS data, and thus the choice of the mass fraction for the definition of $c$ affects the statistics of $S_{d}^{*}$. Using the definition of $S_{d}^{*}$, one gets the following expression for the FSD based reaction rate closure (Chakraborty and Cant 2011; Boger et al. 1998):

$$ \overline{{\left[ {\dot{w} + \nabla \cdot \left( {\rho D_{c} \nabla c} \right)} \right]}} = \overline{{\left( {\rho_{0} S_{d}^{*} } \right)}}_{s} {\Sigma }_{gen} $$

(5)

where $\overline{Q}$ and $\overline{\left( Q \right)}_{s} = \overline{{Q\left| {\nabla c} \right|}} /{\Sigma }_{gen}$ are the Reynolds averaged/LES filtered and surface-weighted average/filtered value of a general quantity Q, respectively with ${\Sigma }_{gen} = \overline{{\left| {\nabla c} \right|}}$ being the generalised FSD (Chakraborty and Cant 2011; Boger et al. 1998). Equation 5 shows that it is not sufficient to model the generalised FSD ${\Sigma }_{gen}$ but a robust extrapolation relation for $S_{d}^{*}$ should in addition account for the curvature and stretch rate dependences of $\overline{{\left( {\rho_{0} S_{d}^{*} } \right)}}_{s}$ with sufficient accuracy (Hawkes and Cant 2001; Boger et al. 1998). This is important because the assumption is not valid in turbulent flames, as demonstrated in several previous analyses (Chakraborty and Cant 2011, 2009; Sabelnikov et al. 2017). It is worth mentioning that the modelling of $\overline{{\left( {\rho S_{d} } \right)}}_{s} = \rho_{0} \overline{\left({S_{d}^{*}}\right)}_{s}$ is closely linked to the estimation of the stretch factor $I_{0}$, in the expression ${\overline{{\left( {\rho S_{d} } \right)}}}_s = I_{0} \rho_{0} S_{L}$ introduced by Bray (1980) because $I_{0}$ can be expressed as: $I_{0} = {\overline{(S_{d}^{*})}_{s}} / S_{L}$.

The extrapolation relations in Table 1 involve flame curvature κ_m and stretch rate K, which are defined as (Pope 1988):

$$ \kappa_{m} = 0.5\nabla \cdot \vec{N} \,{\text{and}} \,K = \nabla .\vec{u} - \left( {\vec{N} \cdot \nabla \vec{u}} \right) \cdot \vec{N} + 2S_{d} \kappa_{m} $$

(6)

where $\vec{N} = - \nabla c/\left| {\nabla c} \right|$ is the flame normal vector which points towards the reactants and according to this convention a positive curvature is associated with a flame surface element, which is convex towards the reactants.

The performances of the extrapolation relations have been quantified in terms of the Pearson correlation coefficient between $S_{d}^{*}$ predicted by the extrapolation relations and the same quantity extracted from the DNS data. The Pearson correlation coefficient, which measures the linear dependence between two variables A and B, is given by Ahlgren et al. (2003):

$$ r\left( {A,B} \right) = cov\left( {A,B} \right)/\sigma_{A} \sigma_{B} $$

(7)

where $\sigma_{{\text{A}}} {\text{ and }} \sigma_{{\text{B}}}$ are the standard deviations of A and B, respectively and $cov\left( {A,B} \right)$ is the co-variance of random variables A and B.

3 Numerical implementation

A skeletal chemical mechanism (Burke et al. 2012) with 9 species and 19 chemical reactions has been utilised to develop a three-dimensional DNS (Im et al. 2016; Wacks et al. 2016; Papapostolou et al. 2017) database of ${\text{H}}_{2}$-air flames with an equivalence ratio of 0.7. An atmospheric combustion with unburned gas temperature T₀ = 300K is considered, which gives rise to an unstrained laminar burning velocity S_L = 135.6 cm/s and heat release parameter τ = (T_ad−T₀)/T₀ = 5.71. The information related to numerical implementation of this database has been discussed in detail elsewhere (Im et al. 2016; Wacks et al. 2016; Papapostolou et al. 2017), and thus will not be repeated here. Turbulent inflow and outflow boundaries are specified in the direction of mean flame propagation and transverse boundaries are considered to be periodic. An improved version of the Navier Stokes characteristic boundary conditions (NSCBC) technique (Yoo et al. 2005) is used to specify the inflow and outflow boundaries. The turbulent velocity fluctuations at the inflow are specified by scanning a plane through a frozen homogeneous isotropic incompressible velocity fluctuation field generated using a pseudo-spectral method (Rogallo 1981) following the Passot-Pouquet spectrum (Passot and Pouquet 1987). The temporal evolution of flame area has been monitored and the simulations are performed until the flame area no longer varies with time and the flame is considered to be statistically stationary. The inflow values of normalised root-mean-square turbulent velocity fluctuation $u^{\prime}/S_{L}$, turbulent length scale to flame thickness ratio $l_{T} /\delta_{th}$, Damköhler number Da = l_TS_L/u′δ_th, Karlovitz number $Ka = (\rho_{0} S_{L} \delta_{th}/\mu_{0})^{0.5} (u^{\prime}/S_{L})^{1.5} (l_{T} /\delta_{th})^{-0.5}$ (expressed here as the ratio of chemical $\delta_{th} /S_{L}$ to Kolmogorov timescale $(\mu_{0}l/\rho_{0} u^{\prime 3})^{0.5})$ and turbulent Reynolds number $Re_{t} = \rho_{0} u^{\prime} l_{T} / \mu_{0}$ for all cases are listed in Table 2 where $\mu_{0}$ is the unburned gas viscosity, $\delta_{th} = (T_{ad} - T_{0})/{\text{max}}|\nabla T|_{L}$ is the thermal flame thickness and the subscript ‘L’ is used to refer to unstrained laminar flame quantities. Cases A-C represent the corrugated flamelets ($Ka < 1$), thin reaction zones (1 < Ka < 100) and broken reaction zones ($Ka > 100$) regimes (Peters 2000) of premixed combustion, respectively according to the regime diagram by Peters (Peters 2000).

Table 2 List of inflow turbulence parameters

The domain size is $20mm \times 10mm \times 10mm$ $(8mm \times 2mm \times 2mm)$ in cases A and B (case C), which has been discretised by a uniform Cartesian grid of $512 \times 256 \times 256$ ($1280 \times 320 \times 320$). Simulations have been carried out for $1.0t_{e} ,6.8t_{e}$ and $6.7t_{e}$ (i.e. $t_{e} = l_{T} /u^{\prime}$) for cases A-C respectively, and this simulation time remains comparable to several previous analyses (Hun and Huh 2008; Reddy and Abraham 2012; Pera et al. 2013; Dopazo et al. 2015). In this regard, it is noted that the longitudinal integral scale $L_{11}$ is a factor of 2.5 smaller than the most energetic scale $l_{T}$ for cases A-C. Consequently, the values of Karlovitz (Damköhler) number in Table 1 will increase (decrease) by a factor 1.6 (2.5) if $L_{11}$ instead of $l_{T}$ is used for their definitions. Therefore, the run-time in terms of the initial eddy-turn over time $L_{11} /u^{\prime}$ is at least 2.5 times that of $t_{e} = l_{T} /u^{\prime}$ and thus the run-time is $\left\{ {2.5,17,16.75} \right\}L_{11} /u^{\prime}$ for cases A-C, respectively.

4 Results and discussion

The scatters of the normalised density-weighted displacement speed $S_{d}^{*} /S_{L}$ in response to the variations of normalised curvature $\kappa_{m} \times \delta_{th}$ for $c = 0.3, 0.5$ and 0.8 isosurfaces for cases A-C for reaction progress variable definition based on ${\text{H}}_{2}$ mass fraction are exemplarily shown in Fig. 2. The corresponding normalised stretch rate $K \times \delta_{th} /S_{L}$ dependence of the normalised density-weighted displacement speed $S_{d}^{*} /S_{L}$ are shown in Fig. 3. The corresponding variations for the reaction progress variable definitions based on ${\text{O}}_{2}$ and ${\text{H}}_{2} {\text{O}}$ mass fractions are qualitatively similar to those shown in Figs. 2, 3 and thus are not explicitly shown here for the sake of conciseness. The maximum reaction rates for these major product or reactant species ${\text{(H}}_{2} ,{\text{O}}_{2} ,{\text{H}}_{2} {\text{O)}}$ are (for laminar flame data) found at a value of $\theta \approx 0.6$ (see Fig. 1). Figure 1 shows that the peak location corresponds approximately to reaction progress variable values based on ${\text{H}}_{2} ,{\text{O}}_{2}$ and of ${\text{H}}_{2} {\text{O}}$. This demonstrates that the choice of reaction progress variable values selected for this analysis covers the range of $c$ where the peak reaction rate occurs. The mean values of $S_{d}^{*} /S_{L}$ conditioned upon the $\kappa_{m} \delta_{th}$ and $K\delta_{th} /S_{L}$ values according to the predictions of the extrapolation relations in Table 1 are also shown in Figs. 2, 3, respectively. In Figs. 2, 3, for the extrapolation relations the parameters L_M and C have been estimated based on a non-linear regression analysis. The parameter L_M is fitted to the datapoints in Fig. 2 for the extrapolation relations which expresses $S_{d}^{*}$ as a function of κ_m not those in Fig. 3 (i.e. it is fitted as a function of mean curvature rather than stretch). Similarly, L_M is fitted to the datapoints in Fig. 3 for the LS extrapolation and not to those in Fig. 2. Figures 2, 3 reveal that the LC and LS extrapolations do not adequately capture the non-linear stretch rate dependence of $S_{d}^{*} /S_{L}$ for all cases irrespective of the choice of reaction progress variable definition and its value. In contrast, the LC relation shows a reasonable curvature dependence of $S_{d}^{*} /S_{L}$ and this holds especially true for case C where the influence of stretch rate is diminished, possibly because of the relatively high frequency of velocity fluctuations for this case. This can be substantiated from the correlation coefficients between $S_{d}^{*} /S_{L}$ and $\kappa_{m} \times \delta_{th}$, and between $S_{d}^{*} /S_{L}$ and $K \times \delta_{th} / S_{L}$ for $c = 0.3, 0.5$ and 0.8 isosurfaces for cases A-C for reaction progress variable definitions based on ${\text{H}}_{2} ,{\text{O}}_{2}$ and ${\text{H}}_{2} {\text{O}}$ mass fractions in Table 3.

Fig. 2

Scatter of $S_{d}^{*} /S_{L}$ with $\kappa_{m} \delta_{th}$ along with the mean values of $S_{d}^{*} /S_{L}$ conditioned upon $\kappa_{m} \delta_{th}$ according to LS, LC, NQ, NE, N3P and NEW extrapolations on $c = 0.3, 0.5$ and 0.8 isosurfaces for cases A–C (1st–3rd rows) for reaction progress variable definitions based on $H_{2} ,$ mass fraction. Note that N3P and NEW extrapolations fall almost upon each other in Figs. 2 and 3

Fig. 3

Scatter of $S_{d}^{*} /S_{L}$ with $K\delta_{th} /S_{L}$ along with the mean values of $S_{d}^{*} /S_{L}$ conditioned upon $K\delta_{th} /S_{L}$ according to LS, LC, NQ, NE, N3P and NEW extrapolations on $c = 0.3, 0.5$ and 0.8 isosurfaces for cases A–C (1st–3rd rows) for reaction progress variable definitions based on $H_{2}$ mass fraction

Table 3 Correlation coefficients between $S_{d}^{*}$ and $\kappa_{m}$, and between $S_{d}^{*}$ and $K$ for different values of $c$ for different definitions of reaction progress variable

A note of caution is that the correlation coefficient is invariant to constant multipliers and therefore the results of the correlation analysis cannot be directly compared to Figs. 2, 3. The negative correlation between $S_{d}^{*}$ and κ_m with a correlation coefficient different from –1.0 indicates a negative, nonlinear correlation and is consistent with several previous DNS findings (Chen and Im 1998; Chakraborty et al. 2007, 2011a,b; Venkateswaran et al. 2015; Hun and Huh 2008; Echekki and Chen 1996; Peters et al. 1998; Chakraborty and Cant 2004, 2005; Chakraborty 2007). It can be seen from Figs. 2, 3 that NE and N3P extrapolations qualitatively predict the non-linear κ_m dependences of $S_{d}^{*}$ when the optimum values of the model parameters (i.e. normalised Markstein length $L_{M} /\delta_{th}$ and the model parameter C in the N3P extrapolation) are used because of their functional forms (see Table 1) but locally there are discrepancies between the quantitative agreements between DNS data and the predictions of the extrapolation relations. However, the predictions of NE and N3P extrapolations differ from each other for large curvature magnitudes for both positive and negative κ_m values. On the other hand, the non-linear NQ extrapolation captures the qualitative nature of the stretch rate K dependence of $S_{d}^{*}$, while both NE and N3P extrapolations do not adequately predict the interrelation between K and $S_{d}^{*}$. Similarly, the predictions of the curvature κ_m dependences of $S_{d}^{*}$ by the NQ extrapolation are qualitatively different from that obtained from NE and N3P extrapolations. It can be seen from Fig. 3 that the LC extrapolation allows for some degree of non-linearity in K dependences of $S_{d}^{*}$ but it does not capture the qualitative nature of stretch rate dependence of density-weighted displacement speed.

It is useful to consider different components of density-weighted displacement speed (Chen and Im 1998; Chakraborty et al. 2007, 2011a, b; Venkateswaran et al. 2015; Hun and Huh 2008; Echekki and Chen 1996; Peters et al. 1998; Chakraborty and Cant 2004, 2005; Chakraborty 2007) in order to explain the observations made from Figs. 2, 3 as:

$$ S_{r}^{*} = \dot{w}/\rho_{0} \left| {\nabla c} \right|;\,S_{n}^{*} = \vec{N} \cdot \nabla \left( {\rho D_{c} \vec{N} \cdot \nabla {\text{c}}} \right)/\rho_{0} \left| {\nabla c} \right|;\,S_{t}^{*} = - 2\rho D_{c} \kappa_{m} /\rho_{0} $$

(8)

where $S_{r}^{*} ,S_{n}^{*}$ and $S_{t}^{*}$ are the reaction component, normal diffusion component and tangential diffusion components. It can be seen from Eq. 8 that $S_{t}^{*}$ is linearly related to κ_m, and Eq. 8 can be utilised to demonstrate that the stretch rate can be expressed as: $K = a_{T} + 2(\rho_{0}/\rho) S_{d}^{*} \kappa_{m} = a_{T} + 2(\rho_{0}/\rho) (S_{r}^{*} + S_{n}^{*}) \kappa_{m} - 4D_{c}\kappa_{m}^{2}$. It can be seen from Table 3 that $S_{d}^{*}$ is mostly strongly negatively correlated with curvature, which reveals that the curvature stretch is likely to induce a significant non-linear curvature dependence of K. For thermo-diffusively neutral flames $\left( {S_{r}^{*} + S_{n}^{*} } \right)$ remains weakly correlated with curvature κ_m (Chakraborty and Cant 2004, 2005; Chakraborty 2007; Chakraborty et al. 2011a, 2011b) and thus the correlation coefficient between $S_{d}^{*}$ and κ_m remain mostly close to −1.0 (note a correlation coefficient of −1.0 is indicative of a linear relation) except for $c = 0.8$ for ${\text{O}}_{2}$ based reaction progress variable (see Table 3). This suggests that a non-linear stretch rate K dependence of $S_{d}^{*}$ is expected, as shown in Fig. 3. Furthermore, an assumption of the linear extrapolation (i.e. LS extrapolation) leads to the following expression of L_M (Chen and Im 1998; Chakraborty et al. 2007):

$$ L_{M} = \left[ {S_{L} - S_{d}^{*} \left] / \right[a_{T} + 2S_{d} \kappa_{m} } \right] $$

(9)

It is worth noting that Eq. 9 suffers from a singularity for $K = 0$ and it cannot be used under this condition. However, under turbulent conditions, $K \ne 0$ in most locations. Moreover L_M, has no relevance for $K = 0$ because under that condition one gets: $S_{d}^{*} = S_{L}$. When $K \ne 0$, it can be concluded from Eq. 9 that L_M is expected to scale as: $L_{M} \sim (-1/2 \kappa_{m})$ for $|2S_{d} \kappa_{m}| \gg a_{T}$, and thus a constant value of L_M in the context of LS extrapolation may not be sufficient to represent $S_{d}^{*}$ behaviour. Equation 9 also suggests that it is possible to have negative Markstein length for large positive curvature locations for the non-zero values of stretch rate K.

The variations of the optimised normalised Markstein length $L_{M} /\delta_{th}$ and the model parameter C in the N3P extrapolation based on a regression analysis are shown in Fig. 4 for $c = 0.3$, 0.5 and 0.8 isosurfaces for cases A-C for different choices of reaction progress variable. Figure 4 shows that $L_{M} /\delta_{th}$ for all the extrapolation relations remains of the order of unity (i.e. $L_{M} /\delta_{th} \sim O(1)$) except for the LS extrapolation for all values of c in all cases. The $L_{M} /\delta_{th}$ values for the LC extrapolation have been found to be higher for ${\text{H}}_{2}$ mass fraction based reaction progress variable than in the cases with ${\text{O}}_{2}$ and ${\text{H}}_{2} {\text{O}}$. It can further be seen from Fig. 4 that the LC extrapolation yields the highest value of $L_{M} /\delta_{th}$ among all the extrapolation relations listed in Table 1, whereas the lowest value of $L_{M} /\delta_{th}$ is obtained for the LS extrapolation. This behaviour is observed for all values of $c$ irrespective of the choice of reaction progress variable. It can be discerned from Fig. 4 that the $L_{M} /\delta_{th}$ values do not show any consistent monotonic trend with the change of reaction progress variable $c$ within the flame front. The $L_{M} /\delta_{th}$ values increase with increasing $Ka$ (i.e. from case A to case C) for the NQ extrapolation but no monotonic trend has been observed for the Markstein length for other extrapolations.

Fig. 4

Variations of the optimised values of $L_{M} /\delta_{th}$ and $C$ for $c = 0.3$, 0.5 and 0.8 isosurfaces (1st–3rd column) in cases A–C for reaction progress variable definitions based on $H_{2} ,O_{2}$ and $H_{2} O$ mass fractions (1st–3rd row)

It is difficult to identify the exact reasons for differences between different progress variable definitions. However, it is not surprising that there are differences. For simple chemistry, L_M for $S_{d}^{*}$ according to the analytical relation can be given by Clavin and Joulin (1983):

$$ L_{M} /\delta_{th} = \left( {1 + \tau } \right)\ln \left( {1 + \tau } \right)/\tau + 0.5\left( {\beta_{Z} /\tau } \right)\left( {Le - 1} \right)\mathop \smallint \limits_{0}^{\tau } \{ \ln \left( {1 + x} \right)/x\} dx $$

(10)

where $D_{0}$ is the unburned gas diffusivity, $\tau$ is the heat release parameter $Le$ is the Lewis number and $\beta_{Z}$ the Zel’dovich number (i.e. $\beta_{Z} = T_{ac} (T_{ad} - T_{0})/T^{2}_{ad}$ with T_ac is the activation temperature). While some parameters cannot be defined for individual species, it is obvious that some other parameters are species-dependent. As an example, the effective or global Lewis number can be substantially different to the species Lewis number, and also the flame thickness $\delta_{L} = 1/{\text{max}}|\nabla{c}|_{L}$ of individual species can be substantially different to $\delta_{th}$.

It is worth mentioning that normalisation of L_M by $\delta_{L}$ (with $\delta_{th}/ \delta_{L}$ = 1.03, 1.18, 1.29 for the definition of $c$ based on ${\text{H}}_{2} ,{\text{ H}}_{2} {\text{O}}$ and ${\text{O}}_{2}$ mass fraction respectively) brings $L_{M} /\delta_{L}$ a little bit closer to unity but does not significantly alter the behaviour shown in the bar charts and thus is not explicitly shown here. The coefficient C for the N3P extrapolation remains small in magnitude in comparison to that of $L_{M} /\delta_{th}$ for all the cases irrespective of the choice of the definition of reaction progress variable. The value of C remains negative for $c = 0.5$ for ${\text{H}}_{2}$ mass fraction-based reaction progress variable for case A but for all other definitions and values of c, and also in other cases, only positive values of C are obtained. The correlation coefficients between $S_{d}^{*}$ obtained from DNS data and the predicted values according to the extrapolation relations using the optimum values of $L_{M} /\delta_{th}$ and C (as shown in Fig. 4) are shown in Fig. 5 for different values and definitions of c for cases A-C. The LC, NQ and N3P extrapolations exhibit high correlation coefficient values (i.e. consistently close to unity) with comparable magnitudes for all values and definitions of $c$ for cases B and C, whereas the correlation coefficients for case A remain smaller than 0.4 for ${\text{H}}_{2}$ mass fraction based reaction progress variable. However, the correlation coefficients for the LC, NQ and N3P extrapolations remain high and have comparable values for case A for ${\text{O}}_{2}$ and ${\text{H}}_{2} {\text{O}}$ mass fraction based reaction progress variables. The correlation coefficients for the LS extrapolation decrease with increasing Ka (i.e. from case A to case C) and a similar trend is observed for the NE extrapolation towards the unburned gas side of the flame front. The variations of the correlation coefficient for the NE extrapolation in the middle and towards the burned gas side of the flame front are found to be mostly qualitatively similar to that of the LC, NQ and N3P extrapolations but the magnitude of the correlation coefficient for the NE extrapolation remains smaller than the LC, NQ and N3P extrapolations. The correlation coefficient for the LS extrapolation shows the smallest value amongst all the extrapolations considered here for case C irrespective of the value of definition of c. Further, on average, over all cases and definitions, the LC extrapolation exhibits higher values of the correlation coefficient than the LS extrapolation. This originates from a non-linear K dependence of $S_{d}^{*}$ according to $K = a_{T} + 2(\rho_{0} /\rho) S_{d}^{*} (S_{L} - S_{d}^{*}) / L_{M}$ when using the LC extrapolation which is not captured in the case of the LS extrapolation.

Fig. 5

Variations of correlation coefficients between $S_{d}^{*}$ obtained from DNS data and the predicted values according to the extrapolation relations using the optimum values of $L_{M} /\delta_{th}$ and $C$ (as shown in Fig. 3) for $c = 0.3$, 0.5 and 0.8 isosurfaces (1st–3rd column) in cases A–C for reaction progress variable definitions based on $H_{2} ,O_{2}$ and $H_{2} O$ mass fractions (1st–3rd row)

A correlation coefficient with a magnitude smaller than 1 implies a non-linear relation. Hence, it is evident from Fig. 5 that the interrelation between $S_{d}^{*}$ and the stretch rate (or curvature) becomes significantly non-linear for high values of $u^{\prime}/S_{L}$ (e.g. see Figs. 2, 3 for case C). Because of the non-linear functional form, the NQ, NE and N3P extrapolations perform relatively better for high values of $u^{\prime}/S_{L}$ and $Ka$. However, the practical usage of the NQ extrapolation requires solution of a non-linear equation (see Table 1) and this depends on the choice of the initial guess value of the root. In this paper a combined bisection and Newton–Raphson method is used for obtaining $S_{d}^{*}$ according to the NQ extrapolation by solving $(S_{d}^{*} /S_{L})^{2} \ln(S_{d}^{*} /S_{L})^{2} = -L_{M} K/ S_{L}$ once $L_{M}$ is estimated by a non-linear regression method using DNS data. The sensitivity to the initial guess becomes particularly important for large values of $u^{\prime}/S_{L}$. The higher order polynomial in the NE extrapolation is also prone to provide artificial overshoots and undershoots for large values of $u^{\prime}/S_{L}$ and in this respect the N3P extrapolation performs better than the NE extrapolation. It is worth noting that the NQ extrapolation predicts equal magnitudes of positive and negative $S_{d}^{*}$ according to $(S_{d}^{*} /S_{L})^{2} \ln(S_{d}^{*} /S_{L})^{2} = -L_{M} K/ S_{L}$ (see Fig. 3), whereas the distributions of positive and negative $S_{d}^{*}$ are not symmetric according to DNS data. This suggests that it is not straightforward to identify a function $f$ such that $S_{d}^{*} = f\left( K \right)$ as different values for $S_{d}^{*}$ can occur for the same value of $K$. An implicit functional relation $f\left( {S_{d}^{*} ,K} \right)$ might circumvent this problem, provided there is a criterion to distinguish the branch of the solution to yield the correct $S_{d}^{*}$.

Based on the performances of the extrapolation relations listed in Table 1, an alternative expression was recently proposed by these authors based on a simple chemistry DNS database (Herbert et al. 2020) in the following manner (which will henceforth be referred to as the NEW extrapolation in this paper and Figs. 2–3,5):

$$ S_{d}^{*} = \underbrace {{S_{L} \left[ {1 - L^{\prime}_{M} \kappa_{m} + C^{\prime}\kappa_{m}^{2} \delta_{th}^{2} } \right]}}_{{S_{Ls} }} - 2\left( {\rho D_{c} /\rho_{0} } \right)\kappa_{m} $$

(11)

where $S_{L} [1-L_{M}^{\prime} \kappa_{m} + C^{\prime} \kappa_{m}^{2} \delta_{th}^{2}]$ approximates $\left( {S_{r}^{*} + S_{n}^{*} } \right)$ and $-2(\rho D_{c}/\rho_{0})\kappa_{m}$ is the exact expression for $S_{t}^{*}$. The curvature $\kappa_{m}$ dependence of $\left( {S_{r}^{*} + S_{n}^{*} } \right)$ remains non-linear for cases A-C for all choices of reaction progress variable, which has been demonstrated in several previous analyses (Chakraborty et al. 2007, 2011a, b; Echekki and Chen 1996; Peters et al. 1998; Chakraborty and Cant 2004, 2005; Chakraborty 2007) and thus is not shown for the sake of conciseness. Thus, a N3P type non-linear extrapolation has been chosen to represent $\left( {S_{r}^{*} + S_{n}^{*} } \right)$. The variations of the optimum values of $L^{\prime}_{M} /\delta_{th}$ and $C^{\prime}$ based on a non-linear regression analysis are shown in Fig. 6 for different values and definitions of $c$ in cases A-C. Figure 6 shows that $L^{\prime}_{M} /\delta_{th}$ remains small for cases B and C for all choices and values of $c$ except for case A where $L^{\prime}_{M} /\delta_{th}$ assumes a value close to − 0.5 for $c = 0.8$. However, $L^{\prime}_{M} /\delta_{th}$ assumes small values for $c = 0.3$ and 0.5 for case A. The variation of $C^{\prime}$ has been observed to be qualitatively similar to the corresponding variation of $C$ for the N3P extrapolation (see Fig. 4) and the value of $C^{\prime}$ remains close to $C$ (i.e. $C^{\prime}/C\sim 1$) for all cases irrespective of the choices and values of $c$ in cases A-C.

Fig. 6

Variations of the optimised $L^{\prime}_{M} /\delta_{th} , C^{\prime}$ and $C^{\prime}/C$ for $c = 0.3, 0.5$ and 0.8 isosurfaces (1st–3rd column) in cases A–C for reaction progress variable definitions based on $H_{2} ,O_{2}$ and $H_{2} O$ mass fractions (1st–3rd row)

The predictions of the NEW extrapolation (i.e. Equation 11) for the $L^{\prime}_{M} /\delta_{th}$ and $C^{\prime}$ values shown in Fig. 6 are also shown in Figs. 2, 3. The corresponding correlation coefficients are shown in Fig. 5, respectively. Figures 2, 3, 5 indicate that the performance of the NEW relation remains comparable to the N3P model. However, the NEW extrapolation models only the physical contribution of $S_{d}^{*}$ (i.e. $\left( {S_{r}^{*} + S_{n}^{*} } \right)$), which induces the non-linear curvature dependence of the density-weighted displacement speed. Therefore, it can be expected that the NEW extrapolation might offer less modelling uncertainties in comparison to other alternative extrapolation relations.

5 Conclusions

The performances of various extrapolation relations, which approximate the stretch rate and curvature dependences of density-weighted displacement speed, were assessed based on a detailed chemistry DNS database of statistically planar turbulent ${\text{H}}_{2}$-air premixed flames with an equivalence ratio of 0.7 spanning a range of Karlovitz number values. It was found that density-weighted displacement speed $S_{d}^{*}$ for reaction progress variables defined based on ${\text{H}}_{2} ,{\text{O}}_{2}$ and ${\text{H}}_{2} {\text{O}}$ mass fractions individually, is non-linearly related to both curvature $\kappa_{m}$ and stretch rate $K$, and this trend for stretch rate dependence strengthens with increasing Karlovitz number $Ka$. However, the non-linearity of $K$ dependences of $S_{d}^{*}$ is considerably stronger than its curvature $\kappa_{m}$ dependence. It was also found that the extrapolation relation (LS), which expresses $S_{d}^{*}$ as a linear function of stretch rate $K$ does not satisfactorily capture the statistical variation of $S_{d}^{*}$. In contrast, a linear extrapolation relation (LC) in terms of curvature allows for non-linear stretch rate dependence of $S_{d}^{*}$ and thus has been found to be more successful in capturing the statistical behaviours of $S_{d}^{*}$ than the LS extrapolation.

It was found that the non-linear extrapolations (e.g. NQ, NE and N3P) perform better than the linear extrapolation relations (e.g. LC and LS), but the improved performance comes at the cost of additional tuning constants. However, the performance of NQ extrapolation depends on solving a non-linear equation which is sensitive to the choice of initial guess of its root, whereas an extrapolation relation (i.e. NE) which accounts for $\kappa_{m}^{3}$ contribution, is prone to artificial overshoots and undershoots and as a result their performances have been found to be inferior to the N3P extrapolation. Thus, a recently proposed extrapolation relation, which explicitly models curvature dependence of $\left( {S_{r}^{*} + S_{n}^{*} } \right)$, was found to exhibit promising performance for the range of $Ka$ considered here. Moreover, the parameters of extrapolation relations (i.e. $L_{M}$ and $C$) were found to be sensitive to the choice and definition of $c$ for all extrapolation relations considered here.

Although the non-linear models sometimes show better performance than the LC extrapolation, the performance of the LC extrapolation was found to be remarkably robust, especially from the standpoint of correlation coefficients and does not need any parameter other than the Markstein length. The analysis also demonstrated a non-negligible sensitivity of the Markstein length to the definition of reaction progress variable, which potentially poses a problem for interpretation and comparison of experimental and detailed chemistry simulation data. Furthermore, no analytical expressions are available for calculating Markstein length in the context of detailed chemistry data. Comparison of the present data with the earlier simple chemistry analysis reported in Herbert et al. (2020) reveals that the correlation coefficients between density weighted displacement speed and the investigated extrapolation relations are lower for the detailed chemistry case. The foregoing discussion indicates the need for extending classical results of combustion theory in the context of detailed chemistry and transport for the analysis of experimental data, for modelling purposes and for comparison with simulation data where the complexity of chemistry is reduced. The present analysis was conducted for ${\text{H}}_{2}$-air premixed flames, which behave differently in comparison to hydrocarbon-air premixed flames. Thus, further analysis in this regard will be needed to assess the performances of extrapolation relations for turbulent hydrocarbon-air premixed flames, which will form the basis of future investigations.