Relationship of gain and phase in the transfer function of swirling flames

Gain and phase are two key characteristics of flame transfer function (FTF) in evaluating unsteady flame response. Few studies have investigated the correlation between gain and phase of FTF compared to the extensive literature on the gain characteristics. In this study, we measured the gain and phase of acoustic-excited swirling flames with different flow rates, fuels, equivalence ratios, and burner structures on a single nozzle premixed swirl burner. We identified, for the first time, the consistent variation of FTF gain and derivative of phase over a wide range of acoustic frequencies, and revealed an essential linear relationship between the extremal frequencies of gain and phase derivative, demonstrating that the gain and phase of FTF are not independent. We further proposed an analytical decomposition of the periodically oscillated swirling flame in which FTF equals the combined complex vector of all the perturbing mechanisms. The synchronized variation characteristics of the gain and phase derivative for different flames and acoustic excitations can be explained using a two vector model based on vector decomposition. The distinct time lags and angular velocities of the perturbing vectors were determined by the different spatial positions of the flame tip and base. The local extrema of the FTF modulus and phase derivative are caused by the phase interference of the sub-vectors. Multiple complex vectors affecting heat release rate should be considered under high frequency.