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

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${}_8$H₁₈/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${}_8$H₁₈/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.