A predictive model of the turbulent burning velocity for planar and Bunsen flames over a wide range of conditions

The flame stretch factor

characterizes the effect of chemical kinetics and molecular transport on $s_{T}$, and it links the mean laminar flamelet consumption speed $\langle s_{L}\rangle_{A}$ to the unstretched laminar flame speed $s_{L}^{0}$, where $\langle\cdot\rangle_{A}$ denotes the average over the flame surface. The model of $I_{0}$ was proposed by Lu and Yang [25] and is improved in the present study on the modeling of turbulence stretch.

As the first-order approximation, strain rate effects are neglected in many models by simply setting $I_{0}=1$ and $\langle s_{L}\rangle_{A}=s_{L}^{0}$ in Eq. (4) [2, 3]. On the other hand, $I_{0}$ plays an important role on $s_{T}$ in the cases with strong thermal-diffusive effects. For instance, the lean hydrogen/air mixture with $\phi=0.6$ and $p=10\;\mathrm{atm}$ has $I_{0}\approx 3$ in strong turbulence [25]. Comparing with the case at $p=1$ atm, $s_{T}$ of lean hydrogen flames raises apparently with pressure because of the flame stretch effects. Furthermore, the growth of $I_{0}$ with the turbulence intensity suppresses the bending of $s_{T}$ in strong turbulence.

In turbulent flames, $I_{0}$ depends on the distribution of the curvature and strain rate over the local flamelet. The linear model of the laminar flame speed [47] and symmetric distribution of flame curvature [25, 38] suggest that effects of positive and negative flame curvatures tend to cancel out in strong turbulence, and the influence of the strain rate can be approximated with a presumed probability density function. Assuming the Dirac distribution for simplicity, the averaged consumption speed over the flame surface can be approximated as $\langle s_{L}\rangle_{A}=s_{L}$ [47].

The thermal-diffusive effects alter $s_{L}$ with respect to the stretch on flames. Asymptotic analysis [47] showed a simple relation $s_{L}/s_{L}^{0}=1-\mathrm{MaKa}$ for weak or moderate stretch, where Ma is the Markstein number. In order to investigate the stretch effects with detailed chemistry and molecular transport, the response of $s_{L}$ to stretch in one-dimensional stretched flames, such as counterflow and cylindrical flames, can be employed as reference solutions [28, 49].

We model $I_{0}$ using a lookup table $\mathcal{F}$ formed by laminar flame data [25] to capture the effects of detailed fuel chemistry and transport. Laminar counterflow flames with two streams of the cold mixture and the corresponding equilibrium product are simulated to build the table $\mathcal{F}$. For each counterflow flame solution, the consumption speed is calculated to obtain the ratio $s_{L}/s_{L}^{0}$, and the strain rate $a_{t}$ is estimated by the velocity gradient at the location with the maximum fuel consumption rate. Using a series of counterflow simulations from weak stretched flames to extinction, a table of $s_{L}/s_{L}^{0}$ versus $\mathrm{KaLe}$ is obtained, where Le is the effective Lewis number of mixture, and the Karlovitz number of the laminar flame is calculated as

For a certain condition, the stretch factor of the turbulent premixed flame is retrieved from the table as

Here,

is the model proposed by Bradley et al. [50, 51] for the turbulence stretch effect on flames, and $p_{0}=1\;\mathrm{atm}$ is a reference value for normalization, where $\textrm{Re}=u^{\prime}l_{t}/\nu$ denotes the Reynolds number. We find that this model is more generalized than that we used in Eq. (7) in Ref. [25].

By incorporating the effects of detailed chemistry and transport, the $I_{0}$ model in Eq. (6) is able to capture the response of $s_{L}$ to stretch for various reactants. In particular, a preliminary application of the modeled $I_{0}$ on lean hydrogen flames demonstrated that this type of model significantly improves the predication of $s_{T}$ from the simple model of $I_{0}=1$ at a broad range of pressure conditions [25].

To estimate the flame surface ratio $A_{T}/A_{L}$ in Eq. (3), we apply the modeling approach developed by You and Yang [30] based on theoretical analysis on Lagrangian statistics of propagating surfaces [31, 32]. The essence of this modeling framework is summarized below.

In Eq. (3), $A_{T}/A_{L}$ is approximated by the area ratio $A(t^{*})/A_{L}$ of global propagating surfaces at a truncation time $t^{*}$, which signals that the characteristic curvature of initially planar propagating surfaces has reached the statistically stationary state in non-reacting HIT. This state resembles the statistical equilibrium state in combustion between the flame area growth due to turbulent straining and the area reduction due to flame self-propagation.

In the modeling of flame wrinkling in turbulence, the temporal growth of

is approximated by an exponential function, where the constant growth rate

is approximated by a linear model in terms of $I_{0}$ and the dimensionless laminar flame speed $s_{L0}^{0}=s_{L}^{0}/s_{L,\textrm{ref}}$ normalized by a reference value $s_{L,\textrm{ref}}=1$ m/s. The constants $\mathcal{A}=0.317$ and $\mathcal{B}=0.033$ are determined by fitting DNS data for Lagrangian statistics of propagating surfaces in non-reacting HIT.

Accounting for $t^{*}$ at limiting conditions of very weak ($u^{\prime}/s_{L}^{0}\rightarrow 0$) and very strong ($u^{\prime}/s_{L}^{0}\rightarrow\infty$) turbulence, theoretical analysis and data fit of Lagrangian statistics of propagating surfaces yield

where $T_{\infty}^{*}=5.5$ is a universal truncation time describing the stationary state of material surfaces. Substituting Eqs. (9) and (10) into Eq. (8) yields

where turbulence related constants $\mathcal{A}$, $\mathcal{B}$, and $T_{\infty}^{*}$ are universal, and $\mathcal{C}$ is a fuel-dependent coefficient which will be discussed later.

The flame area model in Eq. (11) captures the dependence of $A_{T}$ on $u^{\prime}$ in very weak and strong turbulence [25, 30]. As $u^{\prime}\rightarrow 0$, the Taylor expansion of Eq. (11) predicts a linear growth of $A_{T}$ with $u^{\prime}$. As $u^{\prime}\rightarrow\infty$, the modeled $A_{T}$ reaches an asymptotic state, showing the bending phenomenon of $s_{T}$ with $I_{0}\approx 1$. At $p=1$ atm, the validation with several DNS datasets showed that the model in Eq. (11) captures the variation of $s_{T}$ with $u^{\prime}$, outperforming previous models [30].

It is well recognized that $s_{T}$ has a dependence on length scales for turbulent premixed flames in the thin reaction zone [26, 42, 52], but this dependence is only implicitly characterized by the Reynolds number in Eq. (11). In this work, we improve the area ratio model Eq. (11) by introducing a scaling of the length scales and keeping the merit of Eq. (11) at the two turbulence limits.

In small-scale turbulence, Damköhler argued that turbulence affects scalar mixing only via turbulent transport. In analogy to the scaling relation $s_{L}^{0}\sim\sqrt{D}$ of the laminar burning velocity and the molecular diffusivity $D$ [47], the turbulent burning velocity is assumed to be proportional to the square root of turbulent diffusivity $D_{T}\sim u^{\prime}l_{t}$ as

To have a dependence on $l_{t}/\delta_{L}^{0}$ as $u^{\prime}/s_{L}^{0}\rightarrow 0$ in very weak turbulence, Eq. (10) is modified as

and then the Taylor expansion of Eq. (8) for $u^{\prime}\rightarrow 0$ becomes

where $\mathrm{Re}_{F}=s_{L}^{0}\delta_{L}^{0}/\nu$ is the flame Reynolds number with the kinematic viscosity $\nu$, $\mathrm{Da}=\left(l_{t}/\delta_{L}^{0}\right)\left(s_{L}^{0}/u^{\prime}\right)$ is the Damköhler number. Equation (14) has the same scaling $s_{T}\sim\mathrm{Da}^{1/4}u^{\prime}$ in the Zimont model [14]. As $u^{\prime}/s_{L}^{0}\rightarrow\infty$ in very strong turbulence, Eq. (11) becomes $A_{T}/A_{L}=\exp(\xi T_{\infty}^{*})$ and it is a constant for constant $I_{0}$. To match the scaling in Eq. (12), we propose

Combining the corrections in Eqs. (13) and (15), Eq. (11) is improved by considering the scaling with length scales as

We remark that the Taylor expansion of Eq. (16) in very weak turbulence is not exactly the same as Eq. (14) due to the factor of $\ln(l_{t}/\delta_{L}^{0})/2$ introduced for strong turbulence in Eq. (15), whereas the scaling $s_{T}\sim\mathrm{Da}^{1/4}u^{\prime}$ in Eq. (14) is kept. In addition, Eq. (16) leads to a questionable prediction $A_{T}/A_{L}<1$ at the limit $l_{t}/\delta_{L}^{0}\rightarrow 0$, but this modeling defect only happens as $l_{t}/\delta_{L}^{0}<\exp(-2\xi T_{\infty}^{*})=0.03$. Regarding to the length scale $l_{t}/\delta_{L}^{0}\geq O(1)$ in practical cases in Fig. 1, this shortcoming can be neglected for most applications.

We illustrate the importance of the length scale modeling in $A_{T}/A_{L}$ using three sets of DNS of planar flames [53]. It is noted that these DNS cases based on the progress variable are not included in Table 1 and Fig. 1. We approximate $I_{0}=1$ for Le $=1$ for this DNS, and then Eq. (3) is simplified to $s_{T}/s_{L}^{0}=A_{T}/A_{L}$. The cases are set to have the gas expansion $\rho_{u}/\rho_{b}=6$, where $\rho_{b}$ is the density of the burnt gas. By adjusting the one-step chemistry coefficient, $s_{L}^{0}$ and $\delta_{L}^{0}$ in these cases are varied, as listed in Table 2.

Figure 2 compares the predictions of $s_{T}$ from the present model Eq. (16) with the length scale effects and from the previous model Eq. (11), where $\mathcal{C}=0.83$ is further modeled by Eq. (17) with Le $=1$ and $I_{0}=1$. In groups R and L of this DNS series, the laminar flame parameters are the same, while $l_{t}/\delta_{L}^{0}=2.955$ and 1.545 are different. We find that the model prediction from Eq. (16) (solid lines) agrees well with the DNS results (symbols), showing the growth of $s_{T}/s_{L}^{0}$ with $l_{t}/\delta_{L}^{0}$. By contrast, the model in Eq. (11) (dash-dotted lines) fails to predict different $s_{T}/s_{L}^{0}$ with the length scale effect in these groups. In groups T and L, the laminar flame parameters are different and length scale ratios are close. Additionally, since $s_{L}^{0}$ is small in these groups, the contribution from $\xi$ in Eq. (9) is negligible. Thus, the model predictions from Eq. (16) are close for these two groups.

In Eq. (16), the model coefficient $\mathcal{C}$ characterizes the fuel effect on the growth of $A_{T}$ and $s_{T}$ in weak turbulence with $u^{\prime}/s_{L}^{0}=O(1)$. Since the hydrodynamic and thermal-diffusive instabilities drive the growth of $A_{T}$ in weak turbulence [54], unstable flames with Le $<$ 1 have large $\mathcal{C}$, while stable flames with Le $>1$ have small $\mathcal{C}$. One way to determine $\mathcal{C}$ is from one or a few available DNS or experimental data points of $s_{T}$ in weak turbulence. Alternatively, the constant value $\mathcal{C}=2.0$ is suggested for hydrogen mixtures, and $\mathcal{C}=1.0$ is recommended for other fuels such as methane [30].

Towards a universal predictive model of $s_{T}$, we reduce the degree of freedom on the determination of $\mathcal{C}$ in the present work. First, we decompose $\mathcal{C}=\mathcal{C}_{0}I^{\prime}_{0}$, where $I^{\prime}_{0}=I_{0}(K=1)$ is obtained from stretch factor table $\mathcal{F}$, and $\mathcal{C}_{0}$ only depends on the mixture composition. Figure 3 shows the fit of $\mathcal{C}_{0}$ from the DNS and experimental cases listed in Table 1 against the Lewis number of mixtures, where $\mathcal{C}_{0}$ is calculated using the nonlinear least square fit [55] for each case. The size of each symbol in Fig. 3 is proportional to the number of data points in the corresponding dataset. Note there is an apparent trend that $\mathcal{C}_{0}$ decreases with the increase of Le, though the data points are very scattered.

From the data fit of $\mathcal{C}_{0}$ in Fig. 3, we propose an empirical model

for various mixtures. This model implies stronger hydrodynamic and thermal-diffusive instabilities at smaller $\rho_{b}/\rho_{u}$ and Le, respectively, and predicts large $\mathcal{C}_{0}$ for unstable cases. We remark that the severe scattering points of $\mathcal{C}_{0}$ in Fig. 3 suggest that $\mathcal{C}_{0}$ may depend on multiple parameters rather than a simple function in Eq. (17), so the data-driven methods can be used to fit $\mathcal{C}_{0}$ in the future work.

Substituting Eqs. (6) and (16) with all the model constants into Eq. (3), we have a predictive model of the turbulent burning velocity

In Eq. (18), the model constants related to turbulence are universal, and the fuel-dependent parameters are obtained by the lookup table from a separate laminar flame calculation or the fit of DNS and experimental data in the literature. Therefore, the model Eq. (18) has no free parameters.

From the algebraic model in Eq. (18), $s_{T}$ is obtained from a given set of reactant and flow parameters. Specifically, the required inputs for $s_{T}$ predictions are the reactant species, equivalence ratio $\phi$, unburnt temperature $T_{u}$, pressure $p$, turbulence intensity $u^{\prime}$, and integral length $l_{t}$. Moreover, the free propagation of laminar premixed flames and a series of counterflow flames to extinction are calculated to obtain the laminar flame parameters, including laminar flame speed $s_{L}^{0}$, flame thermal thickness $\delta_{L}^{0}$, stretch factor table $\mathcal{F}$, and the Lewis number Le. Finally, $s_{T}/s_{L}^{0}$ is calculated by substituting $u^{\prime}$ and $l_{t}$ into Eq. (18). The above procedure has been implemented by a modularized code available at https://github.com/YYgroup/STmodel.

As mentioned in Section 3, model parameters $\mathcal{A}$, $\mathcal{B}$, and $T_{\infty}^{*}$ are fitted from DNS data of nonreacting HIT [30], the fuel-dependent coefficient $\mathcal{C}_{0}$ is calculated by an empirical expression fitted from a number of combustion DNS/experimental cases. The uncertainty in the fits causes the uncertainty of the model parameters.

As an illustrative example, Fig. 4 shows the uncertainty in the modeling of $\xi$ in Eq. (9). The uncertainty is obtained via the Bayesian linear regression [56, 58] from Lagrangian statistics of propagating surfaces in nonreacting HIT with constant $s_{L0}$ and $I_{0}=1$. In this uncertainty quantification of model predictions, samples on $\mathcal{A}$ and $\mathcal{B}$ are generated with the Markov chain Monte–Carlo method [56, 58]. In Fig. 4, the solid line represents Eq. (9), and the dark and light shades denote the 68% ($\pm\sigma$) and 95% ($\pm 2\sigma$) confidence intervals on the $\xi$ model, where $\sigma$ denotes the standard deviation of model predictions of $\xi$ . We observe that the fit of turbulence statistics can lead to a range of possible parameter values.

Similar to $\mathcal{A}$ and $\mathcal{B}$, other model parameters in Eq. (18) are associated with uncertainties. In particular, the fit of $\mathcal{C}_{0}$ involves a large uncertainty regarding the severe scatter of data points shown in Fig. 3. In subsequential validations, the model parameters $\mathcal{C}_{0}$ and $T_{\infty}$ are presumed to satisfy the independent normal distribution $\mathcal{N}(\mu,\sigma)$ with the mean $\mu$ and the standard deviation $\sigma$, and the Monte–Carlo method is applied to generate samples for these two parameters. Table 3 lists the uncertainty level $\sigma/\mu$ for the model parameters in Eq. (18), which is specified based on sample statistics from DNS datasets. The model prediction of $s_{T}$ is calculated for each sample, and then the uncertainty is calculated from the statistics of $s_{T}$.

The uncertainty in chemical kinetic models is typically associated to the reaction rate coefficient $k$ [59]. It propagates to the model prediction of $s_{T}$ via $s_{L}^{0}$, $\delta_{L}^{0}$, and $I_{0}$ that characterizes effects of detailed chemical kinetics and molecular transport in Eq. (18). In our implementation, one-dimensional freely propagating flames and counterflow flames are calculated to obtain $s_{L}^{0}$, $\delta_{L}^{0}$, and $I_{0}$. Consequently, the uncertainty of chemical kinetic models firstly propagates to the result of laminar flames.

In the uncertainty quantification of chemical kinetic models [59], the reaction rate coefficient $k_{i}$ of the $i$-th reaction is normalized into a factorial variable [60, 61]

where $k_{i,0}$ is the nominal value of $k_{i}$, and $f_{i}$ is the uncertainty factor of the $i$-th reaction. It is assumed that $x_{i}$ of all reaction coefficients satisfies the independent normal distribution $\mathcal{N}(\mu=0,\sigma=1)$.

We conducted chemical kinetic uncertainty quantification for the cases of hydrogen and methane flames with the FFCM-1 mechanism [62]. The uncertainty factors $f_{i}$ for kinetic rates are taken from the FFCM-1 model [62, 63]. The laminar flame parameters $s_{L}^{0}$, $\delta_{L}^{0}$, and $I_{0}$ for each sample are calculated with the Monte–Carlo sampling [64] of $x_{i}$, and are then used in Eq. (18) to propagate the chemical kinetics uncertainty to the $s_{T}$ prediction. Uncertainties of $I_{0}$ for two cases in Refs. [25, 44] are shown in Fig. 5 for example. We observe that the uncertainty range depends on many factors such as the fuel, equivalence ratio, and pressure. In general, the flame speed exhibits large sensitivity on reaction rate coefficients at high pressures and near extinction conditions, so the uncertainty range increases with pressure and Ka. For the hydrogen cases, the large chemical uncertainties of $I_{0}$, $s_{L}^{0}$, and $\delta_{L}^{0}$ lead to 95% confidence intervals larger than 10 times of the predicted $s_{T}$, so it is not presented in the further model assessment for clarity. By contrast, the uncertainty range for the methane flames is relatively small, and it is similar for other methane cases.

Hydrogen is a promising fuel for clean combustion applications without carbon emission. The lean hydrogen flame at a negative Markstein number is thermal-diffusive unstable, and it has $I_{0}>1$ growing with pressure, so its turbulent burning velocity depends on both $I_{0}$ and $A_{T}/A_{L}$ in Eq. (3). In particular, the growth of $I_{0}$ due to the turbulence stretch on flames can significantly enhance $s_{T}$ at high pressures [25].

Aspden et al. [33, 34, 35, 36] conducted a series of DNS on statistically planar turbulent premixed flames with various fuels at a wide range of turbulence intensities and corresponding Karlovitz numbers. For the lean hydrogen flames, $s_{T}$ at different equivalence ratios and length scales was reported. Figure 6 compares the model predictions (lines) and DNS results (symbols) of $s_{T}$. The DNS cases in Fig. 6a have two equivalence ratios $\phi=0.31$ and $\phi=0.4$ and the same length scale ratio $l_{t}/\delta_{L}^{0}=0.5$ [33] with the GRIMech 2.11 [67], and the cases in Fig. 6b have $\phi=0.4$ with a larger $l_{t}/\delta_{L}^{0}=0.66$ [36] with the Li mechanism [68].

As shown in Fig. 6a, the turbulent burning velocity grows almost linearly with the turbulence intensity for the very lean case with $\phi=0.31$ (blue squares), even at a very large $u^{\prime}/s_{L}^{0}$ up to 100. By contrast, the variation of $s_{T}$ with $u^{\prime}$ shows a typical bending phenomenon for $\phi=0.4$ (red circles), where $s_{T}$ stops growing and then decays with $u^{\prime}$ in strong turbulence. The present model (solid lines) well captures the different trends for the two equivalence ratios, which is contributed by the variation of $I_{0}$ in Eq. (18) with laminar flame parameters. Figure 6b validates the model for a larger $l_{t}/\delta_{L}^{0}=0.66$ [36]. With the length scale effects incorporated in the modeling of $A_{T}$ in Eq. (16), our model predicts the difference of $s_{T}$ due to different turbulence length scales. Furthermore, the uncertainty range of model predictions in Fig. 6 basically covers the deviation of $s_{T}$ in DNS results.

Lu and Yang [25] investigated the pressure effects on the model prediction $s_{T}$ for lean hydrogen premixed flames. In the present work, the model of $s_{T}$ is further improved with the length scale effects in Eq. (16) and a universal model of $\mathcal{C}_{0}$ for different fuels in Eq. (17). Figure 7 compares the model predictions (solid lines) and DNS results (symbols) of $s_{T}$ at four pressures from 1 to 10 atm. The present model keeps the good performance [25] on predicting $s_{T}$ in a broad range of pressure conditions.

Methane is the largest component of natural gas, which is one of the major energy sources. The methane/air mixture has Le close to 1, resulting in $I_{0}\approx 1$. Hence, the turbulent burning velocity is mainly controlled by the flame area ratio in Eq. (3).

In Fig. 8, the model predictions of $s_{T}$ (solid lines) are assessed by the DNS results (symbols) of lean methane premixed flames with $\phi=0.7$, $p=1$ atm, and different length ratios $l_{t}/\delta_{L}^{0}=4$ and $l_{t}/\delta_{L}^{0}=1$ from Aspden et al. [35, 36]. The DNS results indicate that the turbulent burning velocity increases with $l_{t}/s_{L}^{0}$. In general, $s_{T}/s_{L}^{0}$ with $l_{t}/\delta_{L}^{0}=4$ is about two times of that with $l_{t}/\delta_{L}^{0}=1$, consistent with the scaling in Eq. (12). This difference is well captured by the model by considering the length scale effect in Eq. (16).

In a series of experiments with a Bunsen burner, Wang and co-workers [43, 44, 46] measured $s_{T}$ of turbulent premixed flames with various fuels at elevated pressures. The present model predicts a linear growth of $s_{T}$ at small and moderate turbulence intensities in Fig. 9, generally agreeing with the experimental result. For the methane flames, the pressure only has a minor influence on $I_{0}$, so the model predictions of $s_{T}/s_{L}^{0}$ are similar at different pressures. Additionally, Fig. 9b presents modeling uncertainties from chemical kinetics (purple shade), model parameters (grey shade), and both ones (light blue shade). For this case, the uncertainty from model parameters plays a dominant role. Note that the chemistry uncertainty can be more significant for lean H₂/air mixtures and high pressures [62], as indicated in Fig. 5.

Another dataset of $s_{T}$ for methane flames with pressure effects was obtained by Fragner et al. [39] via a series of Bunsen flame experiments at different $p$ and $\phi$. As this dataset has roughly the same $u^{\prime}\approx 0.44\;\mathrm{m/s}$ and $l_{t}\approx 5.5\;\mathrm{mm}$, Fig. 10 only plots model predictions versus experiment measurements instead of the plot $s_{T}/s_{L}^{0}$ versus $u^{\prime}/s_{L}^{0}$. As most points lie on the diagonal of the plot, the model predictions generally agree with the experimental results. Similar to the previous validations, the model assessment for this dataset with $p=0.1\sim 0.4$ MPa and $\phi=0.7\sim 1.0$ confirms that the model in Eq. (18) is able to predict $s_{T}$ of methane flames at a wide range of conditions.

For most cases employed for the model validation, the model estimation of $s_{T}$ from Eq. (18) generally agrees with the DNS/experimental data. Although the predictions are not perfectly accurate for some cases, they correctly capture the general variation trends of $s_{T}$. Furthermore, the uncertainties in data statistics and model parameters are shown by error bars and shades in figures, respectively. We observe that the model prediction basically covers the data points with reasonable uncertainty ranges.

Propane is a widely used alternative fuel for transportation. The propane/air mixture can cover a range of Le by tuning the equivalence ratio, facilitating the investigation of thermal-diffusive effects.

Figure 11 plots $s_{T}/s_{L}^{0}$ of propane/air Bunsen flames at three equivalence ratios in the experiment of Tamadonfar and Gülder [41]. It shows that $s_{T}/s_{L}^{0}$ for the propane/air mixture increases with $\phi$, similar to the lean hydrogen flame in Fig. 6a. This effect of the equivalence ratio is captured by the proposed model. For another experiment of propane/air Bunsen flames in Zhang et al. [44], Fig. 12 compares the model prediction of $s_{T}$ against the experimental result. The experiment data were obtained only at weak turbulence, and the present model predicts the linear growth of $s_{T}/s_{L}^{0}$ with $u^{\prime}/s_{L}^{0}<2$.

In practice, most fuels are mixtures of different components. For specific applications, different fuels are mixed to adjust flame properties such as the laminar flame speed. Among various mixed fuels, the hydrocarbon fuel blended with hydrogen has a wide range of Le, so it is particularly useful for the model validation with the effect of $\mathcal{C}_{0}$. Zhang et al. [46] studied the effect of differential diffusion on turbulent lean premixed flames of mixed fuels. In this experiment series, the extent of hydrogen enrichment for the CH₄/H₂/air mixture varies from 0% to 60%. The laminar flame speed of each mixture is kept the same by adjusting $\phi$ in the experiment, whereas the measured turbulent burning velocities are different. As shown in Fig. 13, the mixture with more hydrogen extent has larger $s_{T}$ with stronger thermal-diffusive instability, and this trend is predicted by our model.

Large hydrocarbon molecules can be found in many practical fuels such as gasoline, diesel, and jet fuels. They are also employed to construct the surrogate models for engineering applications. For the large hydrocarbon fuels, the large Le reduces the stretched laminar burning velocity $s_{L}$, and the response of $s_{L}$ to flame stretch is generally not sensitive to pressure [23].

Figure 14 compares model predictions (solid lines) and DNS results (symbols) of iso-octane premixed flames [23] at 1 and 20 atm. The 143-species skeletal mechanism [69] was used in our calculation of $I_{0}$ for this case. With the modeled $I_{0}$ pre-computed from laminar flames, the model correctly predicts that $s_{T}$ for heavy fuels is smaller than that for light fuels. Furthermore, the model prediction is insensitive to pressure, consistent with the DNS observation. Figure 15 validates the model by another DNS result for n-dodecane flames [36]. The small stretch factor for the large hydrocarbon fuel leads to a lower value of $s_{T}$, and the model prediction agrees well with the DNS result.

In the present study, we assess the model Eq. (18) using 285 DNS and experimental results of $s_{T}$ over a wide range of fuels, equivalence ratios, pressures, turbulence intensities, and turbulence length scales. The validations above have demonstrated that the proposed model gives overall good predictions of $s_{T}$.

Besides the model assessment through the representative cases for each type of fuels, Fig. 16 presents a comprehensive comparison between the model and DNS/experimental results for all the datasets listed in Table 1. Here, Fig. 16a is obtained from the model in Eq. (18) without free parameters. The corresponding averaged modeling error $\bar{\varepsilon}$ over all the cases is $25.3\%$. Here, the modeling error for each case is defined by

where $s_{T,\mathrm{model}}$ is the $s_{T}$ predicted by the model in Eq. (18), and $s_{T,\mathrm{data}}$ is the $s_{T}$ data obtained from DNS or experiment.

Considering the scattering of the DNS and experimental data across different groups and the intrinsic measurement and statistical uncertainties involved in the datasets, the averaged modeling error $25.3\%$ appears to be acceptable, and this performance is the best among existing $s_{T}$ models. Here, the existing $s_{T}$ models are also assessed using the 285 cases in Table 1. We find that the combination of sub models of the turbulence intensity, pressure, Lewis number, and fuel chemistry is important for the accurate prediction of $s_{T}$ over a wide range of conditions. For example, the model [6] considering turbulence, pressure, and Lewis-number effects has relatively small $\bar{\varepsilon}=30\%$, and the models [1, 20, 70, 14] only considering turbulence effects have $\bar{\varepsilon}$ larger than $50\%$.

On the other hand, the empirical model of the fuel-dependent coefficient in Eq. (17) can be improved, regarding the severe scatter of $\mathcal{C}_{0}$ in Fig. 3. If we apply $\mathcal{C}_{0}$ presented in Fig. 3 for the corresponding dataset, i.e., the model of Eq. (18) has one free parameter $\mathcal{C}_{0}$, the model prediction is significantly improved in Fig. 16b, and $\bar{\varepsilon}$ is reduced from $25.3\%$ to $13.8\%$. This large reduction demonstrates that the complexity of the fuel property and the hydrodynamic instability in weak turbulence is a major source of the uncertainties in the modeling of $s_{T}$.

Although the present model has been only validated for turbulent planar and Bunsen flames due to the requirement of the same consumption-based $s_{T}$ definition, it can be applied to flames with other geometries. Our preliminary test shows that the present $s_{T}$ model prediction qualitatively agrees with the experimental data of V-flames of the methane/air mixture [71, 72] and counterflow flames of the blended fuel mixture of CH₄/H₂ and C₃H₈/H₂ [24], though different $s_{T}$ definitions were used in these flame data. For spherical flames, most experiments observed the accelerating outwardly propagating flame front [73, 74, 75] which has not reached a statistically stationary state, and $s_{T}$ was reported as an average over a range of flame radii [9, 74, 75]. Since the present model does not include this transient effect in turbulent flame development, it tends to overpredict $s_{T}$ for spherical flames. Thus, the modeling of $s_{T}$ for the flames with complex geometries remains an open problem.