# A Thickened Flame Model adaptation to weakly stretched flames for non-unity Lewis number mixtures

S. Poncet, Cedric Mehl, K. Truffin, Olivier Colin

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## To cite this version:

S. Poncet, Cedric Mehl, K. Truffin, Olivier Colin. A Thickened Flame Model adaptation to weakly stretched flames for non-unity Lewis number mixtures. Combustion and Flame, 2024, 270, pp.113758. 10.1016/j.combustflame.2024.113758. hal-04874911

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Submitted on 8 Jan 2025

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# A Thickened Flame Model adaptation to weakly stretched flames for non-unity Lewis number mixtures

S. Poncet *, C. Mehl, K. Truffin, O. Colin

IFP Energies Nouvelles, Institut Carnot IFPEN Transports Energie, 1-4 Avenue de Bois-Preau, 92500 Rueil-Malmaison, France

## ARTICLE INFO

**Keywords:**

Stretch

Markstein length

Non-unity Lewis number

Laminar premixed flame

Thickened Flame Model

Hydrogen

## ABSTRACT

For the study and design of industrial scale combustion systems, the Thickened Flame Model (TFM) is a widely used turbulent combustion model. It allows for the direct resolution of the flame front on Large Eddy Simulation (LES) meshes by artificially thickening the flame front by a factor *F*, i.e. $\delta_L = F\delta_L^0$, while the unstretched adiabatic flame speed $S_L^0$ is preserved. However, when considering differential diffusion effects, the modification of flame reactivity induced by strain rate and curvature is enhanced by the flame thickening process. Especially, at low stretch rates, the derivative of the flame speed with stretch, i.e. the Markstein length *L*, is multiplied by *F*. This induces large errors on the stretched flame speed estimation for mixtures with Lewis numbers *Le* far from unity, such as for lean hydrogen/air combustion. The present work proposes a methodology to recover the exact Markstein length of thickened flames, called *Ma-TFM*. This easy-to-implement method relies on a 2-parameters evaluation, which monitors *L* and $S_L^0$. Various Markstein length definitions from literature are considered and estimated using two laminar stretched flame configurations: (i) reactants-to-products counter-flow and (ii) spherical flames. *Ma-TFM* is evaluated for (i) lean $H_2$/air ($Le < 1$) and (ii) stoichiometric $C_8H_{18}$/air ($Le > 1$) mixtures at ambient conditions. *Ma-TFM* accurately recovers the targeted Markstein lengths for both mixtures, enabling precise estimation of either consumption or displacement flame speed at low stretch rates. *Ma-TFM* remains fairly accurate even at large strain rates for the $Le > 1$ flame, while on the contrary for the $Le < 1$ flame, the consumption speed is highly under-predicted. The extinction strain rate with *Ma-TFM* is then assessed, showing a systematic under-prediction of the extinction strain rate for both flames, as with standard TFM. Additional work is thus needed to shed light on $Le < 1$ flame conditions. *Ma-TFM* is finally evaluated on a flame-vortex configuration with the stoichiometric $C_8H_{18}$/air mixture. The evolution of flame geometry and flame front reactivity are both better estimated with *Ma-TFM* than with standard TFM.

**Novelty and Significance Statement**

This paper presents a novel method to correct the over-sensitivity of the Thickened Flame Model to low stretch rates by adjusting the species diffusivities and reaction rates via two multiplicative parameters. While previous studies only considered counter-flow strained flames to establish their model parameters, this study assesses the impact of the canonical flame configuration employed by comparing strained and curved (spherical) flames. The choice of the laminar flame speed considered (consumption or displacement speed) is also assessed for the first time. Most importantly, while previous stretch correction methods were applied exclusively on positive Markstein length mixtures ($Le > 1$ cases), it is shown that the proposed stretch correction can be applied to negative Markstein length mixtures as well (here a lean $H_2$/air mixture), provided the stretch rate remains low. Finally, as a first step towards flame/turbulence interaction modeling, the stretch correction efficiency is quantified for the first time on a canonical flame-vortex configuration.

## 0. Introduction

In Large Eddy Simulations (LES) of reactive flows, the flame thickness is usually small compared to the mesh size ($\Delta x \gg \delta_L^0$). The Thickened Flame Model (TFM), consisting in the artificial thickening of

the flame front by a factor *F*, is a competitive approach which ensures the proper resolution of flame fronts [1] while preserving the correct propagation speed for an unstrained flame. The artificial thickening is achieved by multiplying species and temperature diffusivities by *F* while dividing the reaction rates of these quantities by *F*. This allows

<footnote>* Corresponding author.
E-mail address: s.poncet368@laposte.net (S. Poncet).</footnote>
exact conservation of the laminar flame speed $S_L^0$ and a thickening of the laminar flame thickness such that $\delta_L = F\delta_L^0$. More specifically, if $v$ is the gas kinematic viscosity and $S_{ck}$ the species Schmidt numbers, the diffusivity of species $k$ in TFM becomes $D_k = Fv/S_{ck}$. In the same way, if we denote the Prandtl number by $Pr$, the temperature diffusivity becomes $D_T = Fv/Pr$.

Early works with TFM [2] considered a unity Lewis assumption, for which $Le_k = Sc_k/Pr = 1$. In this case, the flame reactivity is not altered by flame stretching and the flame consumption speed is insensitive to any variation in strain or curvature applied to the flame front. This is also verified for the artificially thickened flame, which therefore exhibits a consistent behavior. Unfortunately, the unity Lewis assumption fails at recovering the correct flame behavior in most practical flame configurations like spark ignition [3] or flame extinction [4], and more generally, for any turbulent or laminar flame subject to an important stretch. This assumption is also very inaccurate for decarbonized fuels such as $H_2$ and $NH_3$ which are currently attracting a lot of interest in the research and industry communities. Although TFM does not alter the species Lewis numbers which remain equal to $Le_k = Sc_k/Pr$ regardless of the thickening factor, Quillatrel [5] showed that thickening alters the laminar flame speed of a stretched flame. More precisely, if we consider a flame subject to a stretch $K$, the thickened flame will react to this stretch as if it were $FK$ [6]. Several developments were recently proposed in the literature to correct these effects, and are briefly reviewed here.

A first category of methods focused on the improvement of TFM in a tabulated chemistry context, where reaction rates are computed on canonical laminar flames prior to the actual 3-D simulation [7–9]. In the work of Proch et al. [7], premixed unstrained laminar flames are used to build the progress variable reaction rate table, along with a strain correction factor applied to the reaction rates. This correction is estimated using a fit of laminar strained premixed flames. This method requires the computation of the local strain which is often imprecise and, as argued in [10], it does not account for the time response of the flame to strain. The actual strain experienced by the flame does not correspond to the instantaneous strain at the flame location. In Han et al. [8], a solution to this problem is proposed by considering strained laminar flames to build the reaction rate table. The selection of the correct strained flame in the table is performed by adding the mass fraction of radical H as a third input parameter (in addition to mixture fraction and progress variable). As this quantity is given by a transport equation, this model is able to take into account the flame-response time to stretch. An alternative option is to add a strain correction to the species diffusivities rather than to the reaction rates [9]. This is achieved by solving an optimization problem to recover the correct flame profile, and thus the flame speed. This approach is more general, but requires a complex optimization process, and is exact only for the steady strained flames which are used during the optimization.

Evolutions of TFM, used along with reduced chemistry, have also been investigated by several authors. Quillatrel [5] proposed a simple stretch correction method based on asymptotic flame theory, for which species diffusivities are computed using modified Schmidt numbers. The model leads to an improvement of the flame behavior in some cases [3,5], but is too approximate to be generalized to any configuration [10]. Recently, Comer et al. [4] proposed a new species diffusion correction in conjunction with a reaction rate correction which was designed to recover the extinction limits of a lean $C_3H_8$/air flame ($Le > 1$) computed with TFM. As emphasized by Detomaso et al. [10], this model does not guarantee that the unstrained laminar flame speed is recovered, which is a serious limitation. Finally in [10], diffusivities and reaction rates are also modified, but this time the constraints are (i) to recover the unstretched laminar flame speed and (ii) to recover the correct consumption speed $S_c(K_s)$ at a prescribed stretch $K_s$. This requirement comes from the fact that according to asymptotic flame theories, the flame consumption speed $S_c$ is linked to the stretch $K$ by

a linear relation that is valid only when $K\tau_f^0 \ll 1$, where $\tau_f^0 = \delta_L^0/S_L^0$ is the characteristic laminar flame time:

$$S_c = S_L^0 - L_c K. \quad (1)$$

The slope of this linear regression is the consumption Markstein length $L_c$. For $K\tau_f^0 \ll 1$, recovering the correct $S_c$ is therefore equivalent to recovering the correct Markstein length $L_c$. But in practical combustion devices, $K\tau_f^0$ might be superior to unity, and consequently, Detomaso et al. [10] propose to recover directly $S_c(K_s)$ rather than $L_c$. Although the argument is sound, the authors recognize that targeting a large value of $K_s$ means that the dependency of $S_c$ with stretch is imprecise at low stretch values $K \ll K_s$. It is still unclear whether an accurate behavior at low or high stretch target should be preferred for practical applications. Moreover, the calculation of the correction relies on a complex optimization process and on the choice of an arbitrary target stretch $K_s$, which might affect the generalization capabilities of the model in industrial cases, where a wide range of thermo-chemical conditions are met.

In this study, we propose an alternative approach to correct the sensitivity of artificially thickened flames to stretch in the context of reduced chemistry. The so-called Ma-TFM model features a correction of diffusivities and reaction rates similar to Detomaso et al. [10] and Comer et al. [4]. However, the correction is here set by imposing the correct Markstein length to the thickened flame, therefore guaranteeing its correct behavior for low to moderate stretching. A distinction between consumption and displacement Markstein lengths is made. While all previous studies considered strained flame, we also discuss here the possibility to use spherical expanding flames to determine the corrections. A simple algorithm, easily applicable to any conditions encountered in practical systems, is then proposed to estimate the Ma-TFM correction. Contrarily to previous works [4,10], a particular attention is here paid to $H_2$ flames, whose behavior in the presence of stretch significantly differs from classical fuels.

In Section 1, we briefly recall the theory of the stretch influence on flame speed and the impact of TFM on stretched flame configurations. In Section 2, we present the Ma-TFM model, based on the modification of species diffusivities, to correct the over-sensitivity of thickened flame fronts to stretch. Ma-TFM is assessed on lean $H_2$/air ($Le < 1$) and stoichiometric $C_8H_{18}$/air ($Le > 1$) mixtures in Section 3. First a spherical flame configuration is considered, assessing the impact of Ma-TFM on both the flame speed and flame structure. Then two counter-flow flame configurations are considered, namely a single reactants-to-products and twin flames, allowing to assess the thickened flame response to high strain rates and its extinction limit. Finally, Section 3 presents a first evaluation of Ma-TFM on a flame-vortex interaction case.

# 1. Theoretical background

## 1.1. Premixed flame speeds and sensitivity to stretch

The balance between thermal energy released by oxidation reactions and the supply of chemical energy from reactive species, sustaining the combustion process in premixed flames, can be altered by flame stretching. The flame stretch is defined as the rate of change of the surface A of an element of the flame front normalized by A and is evaluated as $K = 1/A \times dA/dt$. Under asymptotic theory assumptions, the flame speed scales linearly with the stretch [11]. The linear scaling is verified for both the flame consumption speed $S_c$ or the flame displacement speed $S_d$ in the vicinity of burnt gases [12]. On the one hand, the flame consumption speed represents the speed at which the flame consumes the reactants [12]:

$$S_c = - \frac{1}{\rho_u (Y_{k_u} - Y_{k_b})} \int_{x^-}^{x^+} \dot{\omega}_k \, dn \quad (2)$$

where $\rho_u$, $Y_{k_u}$, $\dot{\omega}_k$ and $n$ are respectively the density, the mass fraction of species $k$, the reaction rate of species $k$ and the distance along the flame
normal. $x^-$ and $x^+$ represent the coordinates at the unburnt and burnt gas side of the reactive front. Indexes u and b refer to the unburnt and burnt gases, respectively. On the other hand, the flame displacement speed $S_d^b$ corresponds to the velocity of an iso-surface of the flame considered in the vicinity of burnt gases. A normalized displacement speed is here defined as $\tilde{S}_d^b = \sigma^{-1}S_d^b$, where $\sigma = \rho_u/\rho_b$ is the expansion ratio. The linear scaling of $S_c$ and $\tilde{S}_d^b = \sigma^{-1}S_d^b$ with the stretch K are expressed as:

$$S_c = S_L^0 - L_c K \quad (3)$$

$$\tilde{S}_d^b = S_L^0 - L_d^b K \quad (4)$$

where $L_c$ and $L_d^b$ are respectively the consumption and displacement Markstein lengths [12]. The difference $L_d^b - L_c$ is of the order of the laminar flame thickness $\delta_L^0$ and is always positive (see Eq. (34) in [12]), i.e. $L_d^b > L_c$.

## 1.2. Stretch sensitivity of artificially thickened flames

Based on asymptotic theory, the consumption Markstein length scales with the flame thickness as $L_c \propto \delta_L^0 (Le_d - 1)$, where $Le_d$ is the Lewis number of the limiting reactant. Hence, for mixtures with large disparities between species mass diffusion and thermal diffusion, i.e $Le_i$ far from unity, the flame speed is significantly modified by stretch. As a consequence, with TFM, $L_c$ is proportional to the thickened flame thickness $F\delta_L^0$, where F is the thickening factor. A modification of the original TFM model is then necessary to account for stretch effects in flame simulations which take preferential diffusion into account. Quillatrel [5] proposed a corrected Lewis number $Le_k^*$ for all species k based on the conservation of $L_c$, estimated from the asymptotic theory:

$$Le_k^* = 1 + \frac{Le_k - 1}{F} \Leftrightarrow Sc_k^* = Pr + \frac{Sc_k - Pr}{F} \quad (5)$$

This model was shown to reduce the over-estimation of the stretch effect with TFM [5]. However, the asymptotic theory being only qualitative for real multi-species chemistry, this approach still leads to important errors. Therefore, we propose here a model based on flame simulations which allows to accurately recover either the displacement or the consumption Markstein length of the flame.

# 2. The Ma-TFM model

## 2.1. Model formulation

Following previous approaches [4,5,10], we propose to modify species diffusivities to correct the flame response to stretch. We define effective diffusion coefficients $D_k^*$, and thus Lewis numbers $Le_k^*$, as follows:

$$D_k^* = \frac{D_k}{\gamma} \quad \text{or} \quad Le_k^* = \frac{\alpha}{D_k^*} = \gamma Le_k \quad (6)$$

where $\gamma$ is a model parameter set so that the flame sensitivity to stretch is correctly retrieved. $\gamma$ is dependent on the local thermo-chemical conditions (pressure, temperature, equivalence ratio, dilution, ...) and the thickening factor F. The modification of the diffusive fluxes however leads to an alteration of the flame speed and thickness. To recover the unstretched laminar flame speed, the pre-exponential factors of all reaction rates are multiplied by a parameter B. It is evaluated as the ratio $B = (S_L^0/S_L^\gamma)^2$, with $S_L^\gamma$ the laminar flame speed obtained by using the effective diffusivity $D_k^*$ without correcting the pre-exponential factor.

## 2.2. Model parameter evaluation

The model parameter $\gamma$ is determined using a method based on canonical stretched flame simulations. We want to find the value of $\gamma$ for each thickening factor F, written $\gamma^F$, which satisfies:

$$L(F, \gamma^F) = L(F = 1, \gamma = 1) \quad (7)$$

where $L(F, \gamma)$ denotes the Markstein length of a flame thickened by a factor F and using a diffusivity scaled by $\gamma$. The Markstein length L can be either based on consumption or displacement flame speeds. We consider the Markstein number Ma defined as:

$$Ma(\gamma) = \frac{L(F, \gamma)}{F\delta_L^0} \quad (8)$$

As argued by Giannakopoulos et al. [12], asymptotic theory leads to a proportionality between Markstein lengths and the flame thickness. In turn, this leads to the relationship $L(F, \gamma) = F L(F = 1, \gamma)$. Although only exact for simplified one-step high activation energy chemistry, the equality remains reasonable for more complex chemistry as will be shown in Section 3. Eq. (7) is then equivalent to:

$$Ma(\gamma = \gamma^F) = \frac{Ma(\gamma = 1)}{F} \quad (9)$$

The methodology adopted to determine $\gamma^F$ is then as follows:

(i) We compute a series of stretched flames with varying values of $\gamma$, considering the correction factor $B(\gamma)$ estimated from planar unstretched flames. This enables to determine the value of $Ma(\gamma)$ for a given range of $\gamma$. These flames may be computed at any fixed thickening F by virtue of the proportionality relationship given in Eq. (8).

(ii) The unknown variable $\gamma^F$ is determined by solving Eq. (9). This can be repeated for several values of F in order to deal with practical simulations where F spatially varies.

Step (i) might be performed by considering either canonical curved or strained flames. In Section 3, both strained and curved laminar flames will be considered to obtain $Ma(\gamma)$, leading to different values for $\gamma^F$. Additionally, $\gamma_F$ might be determined considering either the consumption or the displacement Markstein number, leading to different estimations. A discussion on this aspect will also be given in Section 3.

# 3. Results and discussion

The proposed methodology is assessed on stoichiometric $C_8H_{18}$/air ($Le > 1$) and lean $H_2$/air ($Le < 1$) mixtures. Both mixtures are considered at temperature $T_u = 300$ K and pressure $P = 1$ bar. A 2-step mechanism involving 6 species [13] is used for the stoichiometric isooctane case while a detailed mechanism (21 reactions, 10 species) [14] is used for the lean hydrogen mixture ($\Phi = 0.47$). The characteristics of premixed adiabatic laminar flames are (i) $S_L^0 = 0.37$ m/s and $\delta_L^0 = 380$ $\mu$m for the $C_8H_{18}$ case (ii) $S_L^0 = 0.43$ m/s and $\delta_L^0 = 474$ $\mu$m for the $H_2$ case.

The canonical strained and curved flames considered in this work are first detailed in Section 3.1. The determination of the factor $\gamma_F$ with the Ma-TFM correction is then explained in Section 3.2. Finally, a posteriori simulations using the Ma-TFM approach are performed on the canonical flames themselves in Section 3.3 and on a flame-vortex interaction simulation in Section 3.4.

## 3.1. Canonical stretched flame configurations

### 3.1.1. Single counter-flow flames

Counter flow reactants-to-products flames, illustrated in Fig. 1, are selected to compute strain-based propagation speed and Markstein length.
<table><caption>**Table 1**
Consumption strained based Markstein lengths $L_{\alpha_T}^c$ evaluated from strained single counter-flow flames.</caption><thead><tr><th>Fuel</th><th><i>L</i><sub>aT</sub><sup>c</sup> (F1) [mm]</th><th><i>L</i><sub>aT</sub><sup>c</sup> (F4) [mm]</th><th><i>L</i>(F4) / <i>L</i>(F1)</th><th><i>L</i><sub>aT</sub><sup>c</sup> (F8) [mm]</th><th><i>L</i>(F8) / <i>L</i>(F1)</th><th><i>L</i><sub>aT</sub><sup>c</sup> (F16) [mm]</th><th><i>L</i>(F16) / <i>L</i>(F1)</th></tr></thead><tbody><tr><td>C<sub>8</sub>H<sub>18</sub></td><td>0.070</td><td>0.269</td><td>3.8</td><td>0.533</td><td>7.6</td><td>1.041</td><td>14.9</td></tr><tr><td>H<sub>2</sub></td><td>-0.461</td><td>-1.828</td><td>4.0</td><td>-3.444</td><td>7.5</td><td>-7.235</td><td>15.7</td></tr></tbody></table>

<figure><img src="image_5.png" /><figcaption>**Fig. 1.** Illustration of reactants-to-products counter-flow flame configuration.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Fig. 2.** Illustration of the spherical flame configuration.</figcaption></figure>

Counter-flow flames are solved using the Cantera open-source software [15]. The distance between injection inlets $l$ scales with the thickening factor such that $l = 158F\delta_L^0$ and $l = 127F\delta_L^0$ respectively for the C<sub>8</sub>H<sub>18</sub> and H<sub>2</sub> case. Transport properties are computed using the mixture-averaged model [15]. In this configuration, the total stretch rate $K$ corresponds to the global strain rate $a_T$ estimated as:

$$K = a_T = \frac{|u_{FG}| + |u_{BG}|}{l} \qquad (10)$$

where $u_{FG}$ and $u_{BG}$ are the fresh and burnt gas inlet velocities, respectively. The strain rate is set by varying the mass flux of fresh gases at the inlet from an initial value $\dot{m}_{min}^{FG} = 0.6 \text{ kg m}^{-2} \text{ s}^{-1}$ and using a constant step $\Delta\dot{m}^{FG} = 0.1 \text{ kg m}^{-2} \text{ s}^{-1}$. The burnt gases mass flux is set to $\dot{m}^{BG} = \sqrt{\rho_h/\rho_u}\dot{m}^{FG}$ to maintain the flame front in the computational domain. The flame front resolution is set by the automated refining mesh algorithm of Cantera, based on the gradients of variables. It guarantees a proper resolution of the flame for both the reference and thickened flame cases, with around 100 points in the front.

For single counter-flow flames, the analysis is here focused on the strain-based consumption Markstein length $L_{aT}^c$. In order to determine $L_{aT}^c$, we consider the linear regression of the flame consumption speed $S_c$ determined from Eq. (2) with respect to the strain rate $a_T$ on the range $a_T = [20/F; 150/F]$ s<sup>-1</sup>:

$$S_c = S_L^0 - L_{aT}^c a_T \qquad (11)$$

The values of $L_{aT}^c$ obtained for both C<sub>8</sub>H<sub>18</sub> and H<sub>2</sub> cases with $F \in [1, 4, 8, 16]$, without the Ma-TFM correction, are provided in Table 1. For the thickened strained single flames configuration, we verify that $L_{aT}^c$ is overestimated by a factor $F$ as suggested by asymptotic theory.

### 3.1.2. Spherical flames

The flame response to curvature is investigated using spherical laminar flames propagating in the outwards direction into quiescent premixed fresh gases, as illustrated on Fig. 2.

The spherical flame propagation is here solved under its 1-D formulation, which enables an efficient computation and filters out intrinsic flame instabilities. The computations are performed using the CONVERGE CFD software [16]. The simulation domain length $l$ is scaled with $F$ such that $l = 1921F\delta_L^0$ and $l = 1540F\delta_L^0$ respectively for C<sub>8</sub>H<sub>18</sub> and H<sub>2</sub> mixtures. The large domain size enables to suppress the impact of reflecting pressure waves at the outlet. The cell size is set to have 10 points in the flame thickness: $\Delta_x = F\delta_L^0/10$. Species diffusion coefficients are determined from $D_k = Fv/Sc_k$. The species

Schmidt numbers $Sc_k$ are constant and determined in the burnt gases of a premixed adiabatic laminar flame. Unlike with the mixture-averaged model, this approach eliminates the need to evaluate local binary diffusion coefficients, thus reducing CPU costs by 10 to 20% in industrial applications. The Prandtl number is considered constant for C<sub>8</sub>H<sub>18</sub>/air flame ($Pr = 0.7$). On the contrary, as illustrated in Appendix A of Supplementary data, the Prandtl number varies significantly throughout the flame front for the lean H<sub>2</sub>/air flame case. Therefore, the Prandtl number for H<sub>2</sub> is computed using the expression below:

$$Pr = \frac{\mu_{air} C_{p,mix}}{\lambda_{mix}} \qquad (12)$$

where $\mu_{air}$ and $C_{p,mix}$ are respectively the dynamic viscosity of air and the mixture's heat capacity, and $\lambda_{mix}$ is the conductivity computed using a mixture-averaged formulation [17]. This hybrid transport properties model gives results nearly identical to those of the mixture-averaged model as shown in Appendix A of Supplementary data.

For spherical flames, the stretch $K$ is directly related to the flame radius $R_{BG}$ determined in the burnt gas [18]:

$$K = \frac{2}{R_{BG}} \frac{d R_{BG}}{dt} = \kappa S_d^b \qquad (13)$$

where $\kappa$ is the flame curvature. $R_{BG}$ is here evaluated by computing the total mass of burnt gas $m_{BG}$ in the domain:

$$R_{BG} = \left( \frac{3}{4\pi} \frac{m_{BG}}{\rho_b} \right)^{1/3} \qquad (14)$$

where $m_{BG} = m_{H_2O}/Y_{H_2O}^b$ with $m_{H_2O}$ the total mass of H<sub>2</sub>O in the domain and $Y_{H_2O}^b$ its mass fraction in the equilibrium burnt gas state.

The flame displacement speed in the burnt gas side is $S_d^b = dR_{BG}/dt$. The consumption and displacement Markstein lengths are determined by a linear regression of $S_c$ and $\tilde{S}_d^b$ on the stretch rate interval $\kappa S_d^b = [20/F; 100/F]$ s<sup>-1</sup>. Table 2 provides values of the Markstein lengths, for thickening factors ranging from 1 to 16, without the Ma-TFM correction. Similarly to Table 1, it confirms in particular the impact of the thickening factor $F$ on Markstein lengths, with $L(F) \approx FL(F = 1)$ for both $L_d^b$ and $L_c$. We note that the difference between the displacement and consumption Markstein lengths $L_d^b - L_c$ is 3 times larger for the H<sub>2</sub>/air than the C<sub>8</sub>H<sub>18</sub>/air mixture. The value of $L_d^b - L_c$ is mainly a function of the thermal expansion ratio, the thermal conductivity across the flame and scales with the flame thickness [12].

Tables 1 and 2 show that $L(F)/L(F = 1) = F$ with less than 7% deviation no matter the Markstein length $L$ considered. This illustrates the validity of the assumption $L(F, \gamma) = FL(F = 1, \gamma)$, on which the model is based, even for detailed chemical mechanisms.
<table><caption>**Table 2**

Curvature-based Markstein lengths evaluated from consumption speed $S_c$ (Eq. (3)) and flame displacement speed $S_d^b$ (Eq. (4)).</caption><thead><tr><th>Fuel</th><th>Variable [mm]</th><th>F1</th><th>F4</th><th><i>L</i>(F4)/<i>L</i>(F1)</th><th>F8</th><th><i>L</i>(F8)/<i>L</i>(F1)</th><th>F16</th><th><i>L</i>(F16)/<i>L</i>(F1)</th></tr></thead><tbody><tr><td rowspan="3">C<sub>8</sub>H<sub>18</sub></td><td><i>L</i><sup>b</sup><sub>d</sub>(1, F)</td><td>0.099</td><td>0.389</td><td>3.9</td><td>0.768</td><td>7.8</td><td>1.549</td><td>15.6</td></tr><tr><td><i>L</i><sub>c</sub>(1, F)</td><td>0.070</td><td>0.275</td><td>3.9</td><td>0.540</td><td>7.7</td><td>1.103</td><td>15.8</td></tr><tr><td><i>L</i><sup>b</sup><sub>d</sub> - <i>L</i><sub>c</sub></td><td>0.029</td><td>0.114</td><td>3.9</td><td>0.228</td><td>7.9</td><td>0.446</td><td>15.4</td></tr><tr><td rowspan="3">H<sub>2</sub></td><td><i>L</i><sup>b</sup><sub>d</sub>(1, F)</td><td>-0.361</td><td>-1.422</td><td>3.9</td><td>-2.857</td><td>7.9</td><td>-5.745</td><td>15.9</td></tr><tr><td><i>L</i><sub>c</sub>(1, F)</td><td>-0.457</td><td>-1.805</td><td>3.9</td><td>-3.622</td><td>7.9</td><td>-7.277</td><td>15.9</td></tr><tr><td><i>L</i><sup>b</sup><sub>d</sub> - <i>L</i><sub>c</sub></td><td>0.096</td><td>0.383</td><td>4.0</td><td>0.765</td><td>8.0</td><td>1.532</td><td>16.0</td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>**Fig. 3.** Markstein numbers obtained from spherical and counter-flow flame configurations with respect to the model parameter $\gamma$ considering the H₂/air mixture ($\Phi = 0.47, T_u = 300$ K, $P = 1$ bar).</figcaption></figure>

## 3.2. Determination of $\gamma_F$ from canonical flames

In this section, we illustrate the procedure to obtain the Ma-TFM correction parameter $\gamma_F$ from the strained and curved laminar flames presented above. The procedure detailed in Section 2.2 is followed to that purpose.

Series of (i) reactants-to-products counter-flow and (ii) spherical flames are run using the effective diffusivities $D_k^* = D_k/\gamma$ for several values of $\gamma$. Figs. 3 and 4 show the evolution of Markstein numbers deduced from these flames for H₂ and C₈H₁₈, respectively. For spherical flames, the displacement Markstein number $Ma_d^b$ is larger than the consumption Markstein $Ma_c$ by an approximately constant amount. The difference $L_d^b - L_c$ is preserved by the modification of diffusivities as shown in the plots for both H₂ and C₈H₁₈. The strain Markstein number $Ma_{dT}^c$ is globally close to $Ma_c$, except for lower values of $\gamma$ in the C₈H₁₈ case, where it approaches $Ma_d^b$. We see in particular that $Ma_c$ and $Ma_{dT}^c$ are similar for $\gamma = 1$ for both H₂ and C₈H₁₈, which is in line with asymptotic theories that predict a unique consumption Markstein number, valid for both curvature and strain [19]. Regardless of its definition, the Markstein number monotonically increases with the value of $\gamma$. Depending on the flame speed definition used to compute the Markstein number, several values for $\gamma^F$ can be obtained. This aspect is investigated in the next paragraph.

The methodology presented in Section 2, aimed at preserving the correct Markstein length for the thickened flame, is used to determine $\gamma^F$. When increasing the thickening factor, the estimated $\gamma^F$ tends towards the value for which $Ma(\gamma) = 0$. Note that the interval of $\gamma$ considered is such that $Ma(\gamma) < 0$ for the H₂ case and $Ma(\gamma) > 0$ for the C₈H₁₈ case. This ensures that the effective Lewis number remains below 1 for the lean H₂ mixture and above 1 for the C₈H₁₈ mixture.

Figs. 5 and 6 show the evolution of the diffusion coefficient $\gamma^F$ as a function of the thickening factor $F$ for the H₂/air and C₈H₁₈/air

<figure><img src="image_8.png" /><figcaption>**Fig. 4.** Markstein numbers obtained from spherical and counter-flow flame configurations with respect to the model parameter $\gamma$ considering the C₈H₁₈/air mixture ($\Phi = 1, T_u = 300$ K, $P = 1$ bar).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Fig. 5.** Estimated $\gamma^F$ with respect to the thickening factor $F$ considering the H₂/air mixture ($\Phi = 0.47, T_u = 300$ K, $P = 1$ bar).</figcaption></figure>

mixtures. The values of $\gamma^F$ based on the conservation of $L_d^b$, $L_c$ or $L_{dT}^c$ are provided in Appendix B of Supplementary data. We first observe that $\gamma^F$ is such that $\gamma^F > 1$ (resp. $\gamma^F < 1$) when the mixture Markstein number is negative (resp. positive). In agreement with the results shown in Fig. 3, the values of $\gamma^F$ considering $Ma_c$ or $Ma_{dT}^c$ are relatively close for the H₂/air case, while those obtained considering $Ma_d^b$ are lower than for $Ma_c$ (between -14% for $F = 4$ and -29% for $F = 16$). For the C₈H₁₈/air case, as expected from Fig. 4, $\gamma^F$ values obtained considering $Ma_d^b$ or $Ma_{dT}^c$ are very similar (less than 5% variation), those obtained considering $Ma_c$ being approximately 15% larger. In the next section, we will look at the impact of $\gamma^F$ variation, depending on the Markstein length targeted, on the flame's response to stretch considering canonical flames.
<figure><img src="image_10.png" /><figcaption>**Fig. 6.** Estimated $\gamma^F$ with respect to the thickening factor $F$ considering the C$_8$H$_{18}$/air mixture ($\Phi = 1, T_u = 300$ K, $P = 1$ bar).</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 7.** Normalized flame consumption speed $S_c/S_L^0$ with respect to normalized stretch $K^* = \kappa S_d^b \tau_f^0$ for the H$_2$/air mixture ($\Phi = 0.47, T_u = 300$ K, $P = 1$ bar).</figcaption></figure>

### 3.3. A posteriori evaluation of Ma-TFM on canonical flames

Ma-TFM is then tested on a posteriori on laminar canonical flames. We first focus on the spherical flames, investigating the propagation speed for low stretch rates and illustrating the impact of the diffusivity correction on the flame structure. Subsequently, the evolution of flame speed with respect to stretch is evaluated for two counter-flow flame configurations, namely single reactants-to-products and twin flames. These configurations examine how the flame speed predicted with Ma-TFM responds to high strain rates and affects the extinction strain rate.

#### 3.3.1. Spherical flames propagation speed analysis

The simulation of the spherical flames is first considered. Figs. 7 and 8 show respectively the evolution of $S_c/S_L^0$ and $\tilde{S}_d^b/S_L^0$ with respect to the stretch normalized by the flame time $K^* = \kappa S_d^b \tau_f^0$ for the lean H$_2$/air case. A comparison is proposed between the reference flame ($F = 1$), represented by a continuous black line, and different formulations of TFM at $F = 8$. For the present mixture, the Lewis number is smaller than 1 and thus the consumption Markstein length is negative. As a consequence, $S_c$ is overestimated for the thickened flame at $F = 8$ (+40% at $K^* = 0.1$). A simulation with the TFM correction proposed by Quillatre [5] is also performed. According to Eq. (5), the corrected Lewis numbers rapidly tend towards 1 as $F$

<figure><img src="image_12.png" /><figcaption>**Fig. 8.** Normalized flame displacement speed $\tilde{S}_d^b/S_L^0$ with respect to normalized stretch $K^* = \kappa S_d^b \tau_f^0$ for the H$_2$/air mixture ($\Phi = 0.47, T_u = 300$ K, $P = 1$ bar).</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Fig. 9.** Normalized flame consumption speed $S_c/S_L^0$ with respect to normalized stretch $K^* = \kappa S_d^b \tau_f^0$ for the C$_8$H$_{18}$/air mixture ($\Phi = 1, T_u = 300$ K, $P = 1$ bar).</figcaption></figure>

increases. This is evidenced here in Fig. 7 by the fact that $S_c$ computed with Quillatre's correction is barely sensitive to curvature, leading to an underestimation of $S_c$ by -7% at $K^* = 0.1$. As seen in Fig. 7, $S_c$ is overestimated when using $\gamma^F=8$ estimated from $Ma_d^b$ (+11% at $K^* = 0.1$). As expected, $S_c$ is perfectly recovered using $\gamma^F$ obtained from $Ma_c$ (only +2% on $S_c$ at $K^* = 0.1$). We note that the evolution of $S_c$ and $\tilde{S}_d^b$ obtained with $\gamma^F$ from $Ma_d^c$ and $Ma_c$ overlap on both Figs. 7 and 8. As reported in Appendix B of Supplementary data, for the H$_2$/air mixture and $F = 8$, the diffusion factor $\gamma^F$ estimated to maintain $L_c$ is 3% larger than the one obtained for preserving $L_{dT}^c$. Hence, for the H$_2$/air mixture considered, a 3% variation of $\gamma^F=8$ does not imply a significant change on the flame response to stretch. This suggests that even with a rough estimation of $\gamma^F$, the flame response to stretch remains adequately recovered at this thickening factor value. In the same way, according to Fig. 8, the flame displacement speed is overestimated for the non-corrected thickened flame (by +29% at $K^* = 0.1$) while it is underestimated with Quillatre correction and diffusion correction based on $Ma_c$ (by respectively -17% and -9% at $K^* = 0.1$). As expected, the evolution of $\tilde{S}_d^b$ is accurately recovered by the Ma-TFM correction based on $Ma_d^b$ (+0% on $\tilde{S}_d^b$ at $K^* = 0.1$).

Similarly, Figs. 9 and 10 illustrate the variation of $S_c/S_L^0$ and $\tilde{S}_d^b/S_L^0$ as a function of the normalized stretch $K^*$ for the stoichiometric C$_8$H$_{18}$/air case. In that case, the Lewis number of the mixture is larger than 1 leading to a positive consumption Markstein length. Hence, the
<figure><img src="image_14.png" /><figcaption>**Fig. 10.** Normalized flame displacement speed $\hat{S}_d^b/S_L^0$ with respect to normalized stretch $K^* = \kappa S_d^b \tau_f^0$ for the C$_8$H$_{18}$/air mixture ($\Phi = 1, T_u = 300$ K, P = 1 bar).</figcaption></figure>

standard TFM approach with $F = 8$ underestimates $S_c$ (-16% at $K^* = 0.1$). The diffusion corrections based on $Ma_d^b$ and $Ma_{dT}^c$ overestimate the consumption speed (resp. only +3% and +2% at $K^* = 0.1$). On the contrary, Quillatre [5] approach and the correction based on $Ma_c$ slightly underestimate $S_c$ (resp. -3% and -2% at $K^* = 0.1$). Hence, for the C$_8$H$_{18}$/air case, the consumption speed of the thickened spherical flame is accurately estimated no matter the diffusion correction applied evaluated in the present study. According to the evolution of the flame displacement speed, in Fig. 10, the standard TFM model largely underestimates $\hat{S}_d^b$ (-23% at $K^* = 0.1$). The diffusion correction of Quillatre implies an error of -11% on $\hat{S}_d^b$ while the approaches based on $Ma_c$ and $Ma_{dT}^c$ induce respectively a -8% and -4% error at $K^* = 0.1$. As expected, the best estimation of the flame displacement speed is performed with the diffusion correction considering $Ma_d^b$ with only a -3% error at $K^* = 0.1$.

Ma-TFM thus enables to recover an accurate flame response to low stretch, either preserving the flame consumption speed or the flame displacement speed of an iso-surface in the vicinity of burnt gases. However, as anticipated from the different values of $Ma_c$, $Ma_d^b$ and $Ma_{dT}^c$, preserving all Markstein lengths simultaneously is not possible with the present model.

### 3.3.2. Spherical flame profiles analysis

As species diffusivities are modified by Ma-TFM, it is necessary to understand the extent to which the flame structure is altered. This assessment is performed on spherical laminar flames, focusing on the lean H$_2$/air mixture, as similar observations were made for iso-octane.

**Spherical flame profiles with standard TFM.** Fig. 11 shows the heat release rate (HRR) as a function of temperature across a spherical flame front for thickening factors $F = [1, 8, 16]$ at different stretch rates corresponding to different flame radius. The adiabatic planar flame structure is also added and represented by the solid black line. We see that for a low stretch rate $K_1 = 30$ s$^{-1}$ (which corresponds to a normalized stretch $K_1^* = 0.03$), the structure of the spherical flame with $F = 1$ remains similar to the adiabatic flat flame with $K = 0$ s$^{-1}$. In contrast, the maximum heat release rate $HRR_{max}$ and the temperature of burnt gases $T_b$ are overestimated for the thickened spherical flame fronts in stretch $K_1$. Fig. 11 also shows the HRR of the thickened spherical flame fronts with $F = 8$ at $K_2 = K_1/8 = 3.8$ s$^{-1}$ and with $F = 16$ at $K_3 = K_1/16 = 1.9$ s$^{-1}$, which both collapse with the reference spherical flame at $K_1$. This confirms that the thickened spherical stretched flame fronts behave like a non-thickened flame front at a stretch rate $F$ times larger. This is consistent with the results of Section 3.1 showing a multiplication of the Markstein lengths by $F$.

<figure><img src="image_15.png" /><figcaption>**Fig. 11.** Effect of the standard TFM on the heat release rate with respect to the temperature for the lean H$_2$/air mixture. Stretch rates considered: $K_1 = 30$ s$^{-1}$, $K_2 = K_1/8$ and $K_3 = K_1/16$.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>**Fig. 12.** Effect of the standard TFM on the equivalence ratio profile with respect to the temperature across flame front in the case of the lean H$_2$/air mixture. Stretch rates considered: $K_1 = 30$ s$^{-1}$, $K_2 = K_1/8$ and $K_3 = K_1/16$.</figcaption></figure>

Popp et al. [9] reported similar results for thickened strained flames considering a counter-flow reactants-to-products configuration.

Fig. 12 provides the evolution of the equivalence ratio, estimated from a local atomic balance, as a function of temperature for spherical and planar laminar flame profiles ($K = 0$ s$^{-1}$), considering either differential diffusion (continuous line) or a unity-Lewis number approach (dashed line). For the cases computed with differential diffusion, the equivalence ratio strongly varies across the flame front. For the thickened flame fronts considered at $K_1$, $\Phi$ is overestimated throughout the flame front, and the error increases with the thickening factor. The excessive variation of $\Phi$ across the thickened flame front is a direct consequence of the increased molecular fluxes by a factor $F$ at a given stretch rate. Like for the HRR, if the stretch is divided by $F$, the fuel/air equivalence ratio of the non thickened flame at $K$ is recovered.

**Spherical flame profiles with Ma-TFM.** Fig. 13 shows profiles of temperature, H$_2$, O$_2$ and H$_2$O mass fractions and equivalence ratio with respect to the scaled position and temperature considering various diffusion models, thickening factors and configurations. The spatial position in Figs. 13(a) and 13(c) is normalized by the thermal thickness $\delta_{th}$ of the flame and shifted so that the zero matches the flame's leading edge position $x_{l.e.}: x^*/\delta_{th} = (x - x_{l.e.})/\delta_{th}$.

In Fig. 13(a), we observe that the profile of $Y_{H_2}$ is steepened with Ma-TFM as well as with Quillatre's correction [5]. This is explained by
<figure><img src="image_17.png" /><figcaption>**Fig. 13.** Comparison of the flame structure considering multiple stretch values, diffusion models and thickening factors for the H₂/air mixture case. (a) Spatial H₂ mass fraction and temperature profiles (b) O₂ mass fraction with respect to the temperature (c) Spatial evolution of the equivalence ratio (d) H₂O mass fraction profile as the function of temperature. Stretch rate considered: $K_1 = 30$ s⁻¹.</figcaption></figure>

a reduction of H₂ diffusivity in both models. We also see that the YH₂ profile of stretched thickened flame fronts lies between the unstretched adiabatic profiles obtained with non-unity and unity Lewis numbers. This is consistent with the fact that (i) for all modified diffusion approaches, the H₂ Lewis number is either preserved (standard TFM) or increased towards unity and that (ii) the diffusivity of the limiting reactant drives the combustion process. Note that although the effective Lewis number could theoretically become larger than unity for a large value of γ, for the considered cases the value γF leads to H₂ Lewis numbers smaller than unity. Unlike the H₂ profile, the temperature evolution in Fig. 13(a) coincides for all the flames considered, except for the standard TFM which over-estimates it towards the burnt gases.

The evolution of the mass fractions of O₂ and H₂O are plotted in Figs. 13(b) and 13(d) as a function of temperature. Below T = 1400 K, the profiles of YO₂ and YH₂O across the flame front obtained with the planar adiabatic flame, the reference (F = 1) and the thickened stretched flames with standard TFM are similar. When considering the diffusion correction of Quillatre et al. [5], the species profiles tend towards that of an unstretched unity Lewis flame. This is expected as all species Schmidt numbers tend towards unity as F increases with Quillatre's model. On the contrary, with Ma-TFM, profiles are steepened spatially like for YH₂ shown in Fig. 13(a). It results from the increase of all species Lewis numbers by a factor γF > 1 for this mixture. While the Lewis number of H₂ tends towards unity, it increases above unity for O₂ and H₂O as their original Lewis number is already larger than unity. This explains the behavior observed in Figs. 13(b) and 13(d), where YO₂ and YH₂O depart from the unity Lewis profile. Above T = 1400 K, YO₂ is underestimated and YH₂O overestimated with the standard TFM and Ma-TFM. This change of the equilibrium state composition is also visible in Fig. 13(c), where the standard and

modified TFM models over-predict Φ in the burnt gases. In this figure, we also note that Ma-TFM over-predicts Φ more than standard TFM in the preheat zone, while in the reaction zone located around T = 1100 K, i.e. x*/δᵢth ≈ 0.6, the value of Φ gets closer to that of the reference flame.

These results show that both standard and modified TFM alter the species, temperature and equivalence ratio profiles in the flame front. Ma-TFM can therefore recover correctly the displacement or consumption speed for low stretch rates, but only approximately the flame structure as expected.

### 3.3.3. Counter-flow single-flames propagation speed analysis

As for spherical flames in Section 3.3.1, the different diffusion correction approaches are here applied to the reactants-to-products counter-flow flames described in Section 3.1.1. Results for both H₂ and C₈H₁₈ are analyzed as they lead to different observations. An analysis of the model behavior for high strain rates is also provided here.

Fig. 14 presents the normalized flame consumption speed $S_c/S_L^0$ with respect to the normalized strain rate $a_T^* = a_T T_f^0$ for the C₈H₁₈/air mixture. We first see that $S_c$ is drastically underestimated for the thickened flame front with F = 8 using the standard TFM over the considered strain rates range (−55% at $a_T^* = 0.5$ and −60% at $a_T^* = 1$). All species diffusion corrections deduced from $Ma_c^c$, $Ma_c$ and $Ma_d^b$ lead to a significant reduction of the error made on $S_c$ at all strain rates (the under-estimation is respectively −10%, −18%, −6% at $a_T^* = 0.5$ and −7%, −19%, −2% at $a_T^* = 1$).

In this section, we estimate $L_{aT}^c$ according to Section 3.1.1 from the linear regression of $S_c(a_T)$ on interval $a_T = [20/F; 150/F]$ s⁻¹, which corresponds to $a_T^* = [0.02/F, 0.15/F]$ for the C₈H₁₈/air case. As expected, the Markstein length $L_{aT}^c$ is recovered exactly when applying the correction factor γF=8 deduced from $Ma_{aT}^c$. When targeting $Ma_c$,
<figure><img src="image_18.png" /><figcaption>**Fig. 14.** Normalized flame consumption speed $S_c/S_L^0$ with respect to normalized strain rate $a_T^* = a_T \tau_f^0$ for the C$_8$H$_{18}$/air mixture ($\Phi = 1, T_u = 300$ K, $P = 1$ bar). Counter-flow single-flame configuration.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>**Fig. 15.** Normalized flame consumption speed $S_c/S_L^0$ with respect to normalized strain rate $a_T^* = a_T \tau_f^0$ for the H$_2$/air mixture ($\Phi = 0.47, T_u = 300$ K, $P = 1$ bar). Counter-flow single-flame configuration.</figcaption></figure>

$L_{aT}^c$ is underestimated. Over the whole strain rate range, the correction factor estimated with $Ma_d^b$ brings the best results. This is explained by the fact that $Ma_d^b$ correction overestimates $L_{aT}^c$ and increases the most flame reactivity among diffusion correction approaches.

While Ma-TFM targets the Markstein lengths, in other words the response to stretch at low strain rates, it still performs satisfyingly well at large strain rates in the C$_8$H$_{18}$/air case. We note that the agreement

obtained by $Ma_d^b$-TFM on $S_c/S_L^0$ at $a_T^* = 1$ and on its evolution over the total range of strain rates considered here is similar to the observations of Detomaso et al. on a comparable $Le > 1$ propane/air flame (see Fig. 10 in [10]). This is expected since both models rely on similar species and reaction rate corrections. In both cases, the flame speed is fairly estimated for $a_T^* \in [0, 1]$ due to the recovery of (i) $S_L^0$ at zero stretch, (ii) corrected $S_c$ at targeted stretch and (iii) the monotony of flame response to stretch for the C$_8$H$_{18}$/air case of the present work and the C$_3$H$_8$/air mixture in [10].

The consumption speed for the lean H$_2$/air mixture is shown in Fig. 15. Similarly to the C$_8$H$_{18}$/air flame, the standard thickened flame at $F = 8$ fails to predict $S_c$ at all strain rates while Ma-TFM shows an improved prediction of $S_c$ at low strain rates ($a_T^* < 0.2$), for the three target Markstein numbers. Again, $L_{aT}^c$ (estimated on $a_T^* = [0.02/F, 0.17/F]$ for the H$_2$/air case) is recovered exactly when applying the correction factor $\gamma^{F=8}$ deduced from $Ma_{aT}^c$.

Unlike for the C$_8$H$_{18}$/air flame, Ma-TFM fails at keeping a correct accuracy for $a_T^* > 0.2$. In particular, we see that $S_c$ reaches a maximal value for $a_T^*$ close to 0.2–0.3 and then decreases, while $S_c$ keeps increasing for the reference flame. According to Garcia et al. [20] the consumption flame speed is no longer monotonic with respect to stretch for lean mixtures containing hydrogen. However, the strain rate interval shown in Fig. 15 is too narrow to observe this change in the reference case.

We note that, as the diffusion correction parameter $\gamma$ increases ($\gamma^{F=8} > \gamma^{F=8}_{Ma_c}$), the threshold $a_T^*$ at which $S_c$ exhibits a negative slope

decreases. This earlier decrease leads to lower $S_c$ values at high strain rates (for instance, an error of –12% at $a_T^* = 0.5$ for the $Ma_d^b$ correction compared with –25% for the $Ma_c$ and $Ma_{aT}^c$ corrections). On the contrary, the standard TFM overestimates $S_c$ by +36% at $a_T^* = 0.5$. For this lean hydrogen/air counter-flow flame, the diffusion correction deduced from $Ma_d^b$ is also the most suitable to recover the consumption speed evolution with stretch for the largest strain rate interval.

The S-TF model of Detomaso et al. aims at recovering the correct flame response at a given stretch value (typically $a_T^* \approx 1$) [10]. An application of the S-TF model is here proposed to assess its ability to get the correct lean H$_2$/air flame behavior in comparison with the presently proposed Ma-TFM. The S-TF is also based on an adjustment of thermal and species diffusion along an adaptation of reaction rates. The distinction therefore lies only in the selected optimization objective function. In Fig. 15, for the H$_2$/air mixture, the flame consumption speed is recovered at $a_T^* = 1$ when setting $\gamma = 1.2$ for a thickened flame front with $F = 8$. Hence, using an optimized species diffusion correction and reaction rate adaptation, the thickened flame speed can be corrected at a targeted stretch rate while preserving the un-stretched laminar flame speed for any mixture. However, unlike for the hydrocarbon-fuel mixtures mentioned above, the evolution of $S_c$ with strain rate does not remain monotonic on the interval $a_T^* \in [0, 1]$ for the lean H$_2$/air thickened flames, which leads to much larger errors at intermediate strain rates. For instance, with $\gamma = 1.2$, Fig. 15 shows a peak overestimation of $S_c$ by 25% at $a_T^* = 0.3$. This study highlights that diffusion corrections presented in the present paper or in [10] do indeed ensure the recovery of flame speed at low stretch rates or for a given strain rate value respectively, but neither can claim to predict accurately $S_c$ at any strain rates for lean H$_2$/air mixtures.

### 3.3.4. Counter-flow twin-flames propagation speed and extinction analysis

A schematic representation of the twin-flames configuration is illustrated in Fig. 16. It is classically used to study extinction limits and is comparable to a flame-flame interaction configuration. We recall that Ma-TFM is built to correct the flame speed at low stretch rates, it is thus not expected to predict the extinction strain rate correctly. Therefore our aim here is solely to provide a more complete understanding of the model's capabilities and limitations in predicting the flame behavior across different configurations.
<figure><img src="image_20.png" /><figcaption>Fig. 16. Illustration of the twin counter-flow flame configuration.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Fig. 17. Normalized flame consumption speed $S_c/S_L^0$ with respect to normalized strain rate $a_T^* = a_T \tau_f^0$ for the C$_8$H$_{18}$/air mixture ($\Phi = 1, T_w = 300$ K, $P = 1$ bar). Counter-flow twin-flames configuration.</figcaption></figure>

Simulations of premixed counter-flow twin-flames are performed with Cantera at $F = 1$ (reference case) and $F = 8$. The evolution of the normalized flame consumption speed as a function of the normalized strain rate are plotted in Figs. 17 and 18 for C$_8$H$_{18}$/air and H$_2$/air flames, respectively.

For both fuels, unlike for the counter-flow single-flame configuration, extinction is observed for all flames. It occurs at a strain rate about $F$ times lower for the thickened flames compared to the extinction value of the reference case [4], which confirms standard TFM highly under-predicts extinction strain rates. Compared to standard TFM, Ma-TFM provides a much better evolution of the flame consumption speed with strain for the C$_8$H$_{18}$/air flame, as was observed for the single-flame case in Fig. 14. In accordance with Garcia et al. [20], for lean H$_2$/air counter-flow twin-flames, the consumption speed is seen to increase monotonically in Fig. 18 unlike for single counter-flow flames (Fig. 15). This monotonic trend remains correct for the thickened flames and the reference case in Fig. 18. Unlike what was observed on single flames (Fig. 15), for twin counter-flow thickened H$_2$ flames with Ma-TFM correction, the response to stretch is accurate up to its extinction limit.

For C$_8$H$_{18}$/air, the correction increases the extinction strain rate, which is still largely under-predicted. Conversely for the H$_2$/air flame, extinction occurs at a lower strain rate with Ma-TFM compared to standard TFM. This result could be expected as Ma-TFM presents a much smaller increase of the consumption speed with strain compared to the standard TFM in Fig. 18. The case $\gamma = 1.2$ is also shown, which corresponds to the modified TFM as proposed in [10], recovering the correct consumption speed at a high strain rate close to $a_T^* = 1$ for single counter-flow flames (Fig. 15). In this case, the model strongly

over-predicts the increase of the flame consumption speed with strain rate while the extinction strain rate is still under-predicted by a factor larger than F. These results illustrate the fact that the correction of species diffusivities proposed in this paper, by Detomaso et al. [10] or by Comer et al. [4] allows recovery of only one flame property at a time and for a given stretched flame configuration, whether it be the Markstein length, the flame consumption speed at a specific stretch rate value, or the extinction limit. For Ma-TFM, in the case of positive Markstein length flames (here the C$_8$H$_{18}$/air flame) the agreement in consumption speed is also improved at all strain rates, and the extinction strain rate is closer to the reference value (Figs. 14 and 17). Conversely with negative Markstein length flames, the agreement at intermediate strain rates is improved or worsened depending on the flame topology (resp. twin or single counter-flow flames), and the extinction strain rate is decreased.

This suggests that the functional form of the model, based on a modification of diffusivities and reaction rates by constant factors and used in both S-TF [10] and Ma-TFM, is not adapted to predict the correct lean H$_2$ flame response for large interval of stretch values. A possible solution to this issue could be to adapt the correction coefficient $\gamma$ to the instantaneous stretch rate seen by the flame. Although this could help achieve better results, it would add computational complexity, as the stretch rate is not accurately computed in LES and also varies along the flame front. For this reason, the Ma-TFM formulation can be considered as a first approach, simple to implement and to use.

### 3.4. Model evaluation on flame-vortex interactions

Ma-TFM is finally evaluated on a flame/vortex interaction case, where both strain and curvature are present. The analysis is here made for the C$_8$H$_{18}$ case, since lean H$_2$/air flames are subject to thermo-diffusive flame instabilities due to the low Lewis number. Hence, the methodology cannot be assessed on the lean H$_2$ case because small-scale instability structures are nearly suppressed at $F > 1$ while they will appear at $F = 1$.

The selected configuration is a planar flame propagating towards a vortex, as illustrated in Fig. 19. The 2D computational domain is square with side $l = 200\delta_L^0$. The incoming vortex is defined with radius $r = 10\delta_L^0$ and its azimuthal velocity at $r$ is $u' = \|\vec{V}\| = 5S_L^0$. Hence, the characteristic vortex time is $\tau_r = 2\tau_f^0$, i.e. $K_r^* = K_r \tau_f^0 = 0.5$, with $K_r = 1/\tau_r$ the characteristic vortex stretch. The cell size of the base mesh is such that the vortex radius $r$ is resolved with 12 points, i.e. $\Delta x = 0.8\delta_L^0$. In addition, TFM-AMR is used to ensure a resolution of $n = 10$ points in the flame using Adaptive Mesh Refinement (AMR) [21]. Five simulations are carried out to evaluate the performance of Ma-TFM: (i) a reference simulation with $F = 1$; (ii) a simulation with $F = 4$ and standard TFM; (iii) a simulation with $F = 4$ and $Ma_c$-TFM; (iv) a simulation with $F = 4$ and $Ma_d^b$-TFM; (v) a simulation with $F = 4$ and $Ma_{at}^c$-TFM. The initial vortex position is adjusted so that the center of the vortex is equidistant from the peak reaction location for the real and thickened flame cases.

Fig. 20 shows the snapshots of the progress variable field $c$ at $t = 17\tau_f^0$ of the 5 simulations. The progress variable is defined as $c = 1 - Y_{C_8H_{18}/C_8H_{18}}^u / Y_{C_8H_{18}}^u$, where $Y_{C_8H_{18}}^u$ corresponds to the mass fraction of C$_8$H$_{18}$ in fresh gases. By construction, $c = 0$ in fresh gases and $c = 1$ in burnt gases. A clear difference in the flame front surface is observed between the $F = 1$ and $F = 4$ cases. The concave region towards the burnt gases enters deeper into the fresh gases for the reference flame than for the standard TFM with $F = 4$. This comes from the fact that the local flame speed is underestimated in this region since Markstein lengths are overestimated by a factor $F = 4$ in the standard thickened case. Using the Ma-TFM corrections based on $Ma_c$, $Ma_d^b$ and $Ma_{at}^c$, the surface of the thickened flame is much better retrieved, with only slight differences between the 3 simulations.
<figure><img src="image_22.png" /><figcaption>Fig. 18. Normalized flame consumption speed $S_c/S_L^0$ with respect to normalized strain rate $a_T^* = a_T \tau_f^0$ for the H₂/air mixture ($\Phi = 0.47$, $T_u = 300$ K, $P = 1$ bar). Counter-flow twin-flames configuration.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Fig. 19. Illustration of the flame-vortex interaction configuration.</figcaption></figure>

A more quantitative comparison is performed by considering the space-averaged properties of the flame. We compute the flame length as:

$$L_f = \int_S |\nabla c| dS \qquad (15)$$

where S refers to the 2-D computational domain.

The evolution of $L_f$ with respect to the normalized time $t/\tau_f^0$ is represented in Fig. 21. We see a significant underestimation of the flame length for the thickened case with standard TFM compared to the non-thickened case. For all thickened flame fronts, with or without the corrected diffusion coefficients, the maximal flame length is the same. The underestimation of the maximum flame length value for thickened flames is similar with all approaches and 10% smaller than that of the real flame at $F = 1$. This is due to a loss of flame surface in the highly curved zone folding into itself and visible in Fig. 20. For the standard thickened flame front, the maximum flame length is reached about $1.5\tau_f^0$ later than in the reference case (Fig. 21). In contrast, the corrected TFM case based on displacement Markstein numbers $Ma_d^b$ reaches the maximal peak of the flame length with an advance of $0.5\tau_f^0$. The maximal flame length is obtained simultaneously for the real flame and the thickened front with diffusion correction based on the consumption Markstein number $Ma_c$. Another way to compare these simulations is to compare the flame length in Fig. 21 at the instant $t = 17\tau_f^0$ shown in Fig. 20. For the standard TFM, $L_f$ is under-predicted by 18% while $Ma_d^b$, $Ma_c$ and $Ma_{aT}^c$ corrections respectively underestimate it by only 4%, 8% and 5%. These results show that the use of the Ma-TFM correction drastically improves the flame propagation compared to standard TFM. Although the result

<figure><img src="image_24.png" /><figcaption>Fig. 20. Progress variable c spatial field at $t = 17\tau_f^0$ with (i) $F = 1$, (ii) $F = 4$, (iii) $F = 4$; $\gamma^F = 0.622$, (iv) $F = 4$; $\gamma^F = 0.545$ and (v) $F = 4$; $\gamma^F = 0.561$.</figcaption></figure>
<figure><img src="image_25.png" /><figcaption>Fig. 21. Flame length evolution as a function of the flame time.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>Fig. 22. Normalized averaged consumption speed $\overline{S_c}/S_L^0$ evolution as a function of the flame length $L_f$.</figcaption></figure>

obtained using $Ma_d^b$ is slightly better, no significant differences are observed here between the different Ma-TFM corrections.

For a given flame geometry, here estimated by the flame length $L_f$, an averaged flame consumption speed can be defined as:

$$ \overline{S_c} = \frac{1}{L_f} \times \frac{-1}{\rho_u (Y_{fu} - Y_{fb})} \int_S \dot{\omega}_{C_8H_{18}} dS \quad (16) $$

Fig. 22 shows the evolution of the averaged consumption speed, normalized by the laminar flame speed $S_L^0 = 0.37$ m/s, as a function of the flame length. The flame-vortex interaction leads to an increase in the flame length with a predominantly convex curvature towards the fresh gases. As $Le > 1$, this configuration leads to a decrease of the flame speed (of maximum -3% compared to $S_L^0$) as illustrated by the reference case with $F = 1$ in Fig. 22. The standard thickened flame front with $F = 4$ features an overestimated decrease of the averaged speed (maximal decrease of $\overline{S_c}$ by -8% compared to $S_L^0$), in line with the over-sensitivity of the uncorrected TFM model to stretch. On the contrary, the averaged consumption speed obtained with the correction based on $Ma_d^b$ is overestimated with a maximal drop of $\overline{S_c}$ by only -1% compared to $S_L^0$. This is consistent with conclusions drawn from Fig. 9, namely, that preserving $L_d^b$ leads to an over-prediction of the flame consumption speed. Finally, the evolution of $\overline{S_c}/S_L^0$ with the thickened flame front using the $Ma_c$-TFM correction coincides with the reference case, showing the ability of this correction to recover the consumption speed of a stretched flame.

## Conclusions

The present study proposes a framework to correctly take into account stretch effects on artificially thickened flames. The standard TFM approach induces an over-sensitivity of thickened flame fronts to stretch, which leads to a multiplication of the Markstein length by the thickening factor $F$. To overcome this limitation, we propose a model named Ma-TFM, to recover the flame speed of weakly stretched flames. It is based on a modification of species diffusivities to modify the response of the flame to stretch, along with a correction of the reaction rates to keep the unstretched flame speed unaltered. The present work distinguishes the influence of stretch (i) on flame consumption speed and (ii) on flame displacement speed, characterized respectively by a consumption Markstein length $L_c$ and a displacement Markstein length $L_d^b$ [12]. The model correction is then built either from canonical spherical flames, involving curvature, or canonical single counter-flow flames, involving pure strain.

The newly developed Ma-TFM was evaluated both for a $Le < 1$ mixture (lean H$_2$/air case) and for a $Le > 1$ mixture (stoichiometric C$_8$H$_{18}$/air case). Each of the model formulations, depending on the targeted flame speed, was tested a posteriori on spherical and counter-flow flames. We observed that the Ma-TFM correction globally enables a significant improvement of the propagation speed evolution with low stretch compared to standard TFM. More specifically, each model formulation demonstrates precise recovery of the targeted speed. For example, the model based on consumption speed $S_c$ is able to perfectly predict the evolution of consumption speed $S_c(K)$. However, it is not possible, with the current formulation, to retrieve both consumption and displacement speeds at the same time. Additionally, the behavior of strained flames is satisfyingly recovered at low stretch rates using corrections based on spherical flames, which suggests that Ma-TFM may be appropriately built on spherical flames for practical applications.

The behavior at large strain rates was also investigated. The results show that for iso-octane Ma-TFM is able to retrieve the correct behavior over a wide range of strain values, although it was designed to perform well at low stretch. This paves the way towards the use of Ma-TFM for complex turbulent $Le > 1$ flames. A first promising result was obtained in the context of ignition simulations in a gas turbine fueled with liquid n-heptane, where the use of Ma-TFM was found necessary to avoid nonphysical flame extinctions [22]. With this model, the experimental ignition probability map could be reasonably recovered for the first time in CFD. However, for single counter-flow flames, the behavior at high strain rates is not satisfying for H$_2$, which features a more peculiar evolution. We also performed counter-flow twin flames to determine the quenching strain rate of each formulation. As expected, Ma-TFM, which targets the flame speed at low stretch, does not improve the prediction of standard TFM on this aspect. But this is also the case of the formulation of Detomaso et al. [10], which was expected to perform better at high strain rates. Furthermore, the evolution of the consumption speed for the H$_2$/air case with Ma-TFM is found to be much more in agreement with the reference solution than for the single counter-flow flame case. This result indicates that the response to stretch cannot only be classified according to strain and curvature, but also highly depends on the configuration considered. Finally, Ma-TFM, like standard TFM, leads to an alteration of the flame species and temperature profiles when submitted to stretch, as species diffusivities are modified.

A first evaluation was proposed considering a flame/vortex interaction showing both curvature and strain. Simulations were performed for iso-octane only because the lean H$_2$/air mixture would present laminar flame instabilities at $F = 1$ not reproduced by the thickened flame. Results showed a significant improvement of the flame evolution with Ma-TFM. No clear difference was however seen for Ma-TFM based on consumption or displacement speeds. Future work will be needed to challenge both formulations in more complex cases.
Ma-TFM proposed here therefore leads to a significant improvement of the thickened flame response to stretch, and guarantees in particular the correct behavior for $Le > 1$ flames for a large range of stretch values. For lean hydrogen mixtures ($Le < 1$), the model leads to a good behavior at low to moderate stretch values but is unable to perform well at large stretch rates. Additional work is needed to shed light on this issue and improve the model.

### CRediT authorship contribution statement

**S. Poncet:** Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Conceptualization. **C. Mehl:** Writing – review & editing, Validation, Supervision, Software, Methodology, Investigation, Conceptualization. **K. Truffin:** Writing – review & editing, Validation, Supervision, Methodology, Investigation, Conceptualization. **O. Colin:** Writing – review & editing, Validation, Supervision, Software, Methodology, Investigation, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgments

This work was granted access to the HPC resources of TGCC under allocation no. A0162B10763 from the GENCI (Grand Equipment National de Calcul Intensif) eDARI program. The authors gratefully acknowledge Damien Aubagnac-Karkar (IFPEN) for his technical and scientific assistance on the implementation of the model in the Cantera software.

### Appendix A. Supplementary data

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.combustflame.2024.113758.

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