Nonlinear Models of Laminar Premixed Slit Flame Responses Subjected to Two-Way Perturbations

Transversal and longitudinal combustion instabilities typically present in gas turbines and rocket engines. It is thus important to investigate the flame responses of the two directional disturbances. When the disturbance level remains weak, the flame response can be considered linear. The flame transfer function (FTF) is typically used, and the global FTF can be easily obtained by superposition of the FTFs at the two-directional forcing. However, when the disturbances arise, the flame response becomes nonlinear due to the nonlinear responses at the two directions and their couplings. The present work thus presents nonlinear analytical models (flame describing function [FDF]) of a two-dimensional laminar premixed slit flame subjected to simultaneously longitudinal and transverse bulk mean flows and disturbances. These nonlinear models are derived based on the asymptotic expansion of the G-equation model of the dynamic flame front. A velocity perturbation model is proposed by considering the change of disturbance convection speed with forcing frequency. To analyze the nonlinear coupling effect of the two-directional forcing, a partly FDF is introduced by neglecting the coupling term. By comparing the results from the FDF, partly FDF, and the FTF, the nonlinear effects are investigated. As the flame tip movement plays an important role in the FDFs for the slit flame, the global effects of flame tip movement on the FDFs are also quantified.