Investigation of nonlinear flame response to dual-frequency disturbances

The two-way interaction between the unsteady flame heat release rate and acoustic waves can lead to combustion instability within combustors. To understand and quantify the flame response to oncoming acoustic waves, previous studies have typically considered the flame dynamic response to pure tone forcing and assumed a dynamically linear or weakly nonlinear response. In this study, the introduction of excitation with two distinct frequencies denoted $St_1$ and $St_2$ is considered, including the effect of excitation amplitude in order to gain more insight into the nature of flame nonlinearities and these associated with combustion instabilities. Corresponding results are obtained by combining a low-order asymptotic analysis (up to third order in normalised excitation amplitude) with numerical methods based on the model framework of the $G$-equation. The influence paths of the disturbance at $St_2$ on the flame dynamic response at $St_1$ are studied in detail. Due to the flame propagating forward normally to itself (named flame kinematic restoration), the perturbation at $St_2$ acts together with that at $St_1$ to induce a third-order nonlinear interaction in the flame kinematics, impressively suppressing the spatial wrinkling of the flame at $St_1$. Additionally, introducing the perturbation at $St_2$ alters the effective flame displacement speed, which is responsible for the calculation of the flame heat release rate and further affects the global response at $St_1$. Taking into account the above two factors, the nonlinear response of the flame at $St_1$ is completely quantified and the corresponding characteristics are clearly interpreted.