A weakly nonlinear analytical model for the transversely forced flame describing function of a slit flame

In this work, a weakly nonlinear analytical model of the flame describing function (FDF) of a two-dimensional laminar premixed slit flame subjected to both transverse perturbations and two-dimensional mean flows has been proposed. Two oncoming velocity convection models are used to investigate their effects on the flame tip movements and the nonlinear FDFs. The perturbation method is then used to solve the G-equation. As the flame tip movement plays an important role in the transverse perturbed FDF, its analytical solution is also derived. Validations of these weakly nonlinear solutions are then conducted by comparing them to the previous linear flame transfer function for weak perturbations and results obtained numerically solving the G-equation model. Results show that the transverse oncoming velocity convection models have a significant effect on the nonlinear flame tip dynamics and the nonlinear FDFs, which can also be found in the previous linear model. The transverse nonlinear effects are not so large, as the effect of the transverse perturbation is typically smaller than that of the longitudinal perturbation, which still contributes to reveal the physical mechanisms of the transverse combustion instabilities. For strong transverse perturbations, the analytical solutions of the FDFs derived by the nonlinear model are different from those derived by the linear model both using the uniform model, especially at low frequencies, which indicates that the nonlinear effect is more significant at low frequencies. Transverse mean flow velocity and the amplitude of the transverse disturbed flow velocity can enhance the nonlinearities at low frequencies.