Framework for simulating stationary spherical flames

Understanding and quantifying the effects of flame stretch rate on the laminar flame speed and flame structure plays an important role from interpreting experimentally-measured laminar burning velocities to characterizing the impact of turbulence on premixed flames. Unfortunately, accounting for these effects often requires an unsteady reacting flow solver and may be computationally expensive. In this work, we propose a mathematical framework to perform simulations of stationary spherical flames. The objective is to maintain the flame at a constant radius (and hence a constant stretch rate) by performing a coordinate change. The governing equations in the new flame-attached frame of reference resemble the original equations for freely-propagating spherical flames. The only difference is the presence of additional source terms whose purpose is to drive the numerical solution to a steady state. These source terms involve one free parameter: the flame stretch rate, which may either be computed in real time or imposed by the user. This parameter controls ultimately the steady state flame radius and the steady state flame speed. That is why, at a given stretch rate, the results of the stationary spherical flame simulations match those of a freely-expanding spherical flame. As an illustration, the dependence of the laminar flame speed on the stretch rate is leveraged to extract Markstein lengths for hydrogen/air mixtures at different equivalence ratios, as well as for hydrocarbon/air mixtures (CH4 and C7H16). Numerical predictions are in good agreement with experimental measurements (within experimental uncertainties). Finally, the proposed methodology is implemented in the chemical kinetic software FlameMaster. The use of a dedicated steady-state solver with a non-uniform optimized mesh leads to significant reductions in the computational cost, highlighting that the proposed methodology is ideally suited for other chemical kinetic software such as Chemkin/Premix and Cantera.