Nonlinear flame describing function and mean shift kinematics of slit flames under combined axial-transverse forcing

This study investigates the nonlinear kinematics of a premixed slit flame using a two-dimensional G -equation level-set framework. Results show that combined forcing induces nonlinear saturation in the FDF, characterized by early gain flattening and premature phase drops, which intensify with the transverse forcing amplitude. Kinematic analysis reveals that this geometric nonlinearity manifests as a reduction in the time-averaged flame height, defined as the mean shift. In the quasi-steady limit, this mean shift is analytically quantified via a multivariate asymptotic expansion, where fourth-order terms successfully capture the saturation mechanism at elevated amplitudes. By introducing a scaling parameter to account for transverse dominance, the frequency-dependent decay of the mean shift in the compact limit collapses onto a single master curve, enabling the derivation of a unified theoretical model that integrates this asymptotic response with a second-order low-pass filter. Furthermore, because the mean shift reduces the physical extent of the flame, it alters the wrinkle propagation time. Correcting the Strouhal number using the measured mean shift collapses the dispersed nonlinear FDF curves onto the linear theory prediction. The analysis is further extended to disturbances convected at a finite speed, for which the linear transfer function is derived analytically and the correction with the measured mean shift continues to collapse the nonlinear FDF. These findings establish that the nonlinear FDF behavior under multidimensional forcing is fundamentally governed by the kinematic mean shift, providing a theoretical baseline for decoupling geometric nonlinearities from other thermo-diffusive or hydrodynamic instabilities in turbulent flames. Novelty and significance statement The novelty of this study lies in identifying the kinematic mean shift, the change of the time-averaged flame height, as the mechanism governing the nonlinear flame describing function under combined axial-transverse forcing. The transverse perturbations dominate the mean shift and thereby the gain flattening. The mean shift is quantified analytically via an asymptotic expansion, and a Strouhal number correction based on the mean shift collapses the nonlinear FDF onto the linear prediction, including disturbances convected at a finite speed. By isolating pure kinematic interactions, this work provides a theoretical baseline for decoupling geometric nonlinearities from thermo-diffusive or hydrodynamic instabilities in reduced-order models.