The effect of hydrogen enrichment on the conical premixed methane–air flame response and thermoacoustic modes coupling

This paper investigates the thermoacoustic dynamic responses of hydrogen-enriched laminar premixed conical methane–air flames under dual-mode coupling. Utilizing the level set method and the $G$-equation model, one meticulously derives the flame describing function (FDF) in relation to variations in hydrogen enrichment levels (${\eta }_H$). This precisely derived FDF is then integrated into the low-order network model of the Rijke tube to analyze the thermoacoustic unstable modes of the system. Finally, dual-frequencies incoming flow velocity perturbations are reintroduced as inputs to obtain the flame response under thermoacoustic modes coupling. Results show that while keeping the unstretched steady flame aspect ratio constant, an increase in ${\eta }_H$ not only raises the FDF’s cut-off frequency but also enhances the FDF gain within the nonlinear frequency region, leading to more unstable modes, especially high frequency modes within the Rijke tube system. Furthermore, the flame response is further altered under the excitation of dual unstable modes. When both modes are within the linear frequency region of the FDF, the flame response is co-controlled by the two modes, predominantly by the mode with a larger velocity perturbation amplitude, with weaker modes coupling leading to the additional frequency perturbation having a suppressive effect on the flame response at the original frequency. Conversely, when one or two modes are within the nonlinear frequency region of the FDF, the flame response is dominated by the lower-frequency mode, with higher nonlinear modes coupling allowing the additional frequency perturbation to both promote and suppress the response at the original frequency and also couple to produce a significant difference frequency response. Novelty and significance The novelty of this paper lies in the determination of the flame describing function (FDF) with varying hydrogen enrichment levels and the stability of thermoacoustic systems, and more significantly, conducting an in-depth and comprehensive investigation into the nonlinear dynamic responses of hydrogen-enriched flames to different types of dual-mode coupling perturbations. The dual-mode perturbations result in either attenuation or amplification of the flame response, depending on whether the corresponding frequencies lie within the linear or nonlinear frequency regions of the FDF, which innovatively incorporates the influence of the FDF phase and examines the responses at the difference frequencies generated by the coupling. This lays a foundation for further exploration into the mechanisms of modes coupling in hydrogen-enriched and other thermoacoustic systems.