A categorization of marginally stable thermoacoustic modes based on phasor diagrams

Interplay of Clusters of Acoustic and Intrinsic Thermoacoustic Modes in Can-Annular Combustors

“…It is highlighted that the two modes can switch nature, as previously reported [18,34,46]. However, we will not discuss how the eigenmodes can be classified as ITA and acoustic and when precisely the modes switch nature as this question is out of the scope of this study and already discussed by Yong et al [49]. (10,11).…”

“…(10,11). Such a result can be explained using phasor analysis, which has already been used in thermoacoustics [13,16,17,24,[47][48][49]. For example, if we consider the case of fully reflecting boundaries R i = R o = ±1, the acoustic mode is marginally stable, thus simplifying the phasor analysis with arrows of fixed length.…”

“…The solution of the complex-valued Eqs. (9,13) is the set of parameters (τ * d, XP , n XP ) and the defective eigenvalue s * XP . We highlight here one special configuration where an easy analytical solution is found.…”

“…It is straight forward to demonstrate that the eigenvalue s * XP = iπ and the set of parameters (τ * d, XP = 1/2 , n XP ) satisfy Eqs. (9,13).…”

“…Emmert et al [12] then demonstrated that the ITA feedback loop gives rise to a new set of thermoacoustic modes of different nature also for reflecting boundaries and identified such an ITA mode as the most unstable mode in a longitudinal test-rig. Yong et al [13] showed that for a marginally stable ITA mode, the velocity fluctuations and the gradient of pressure fluctuations change sign across the flame, thus providing a simple identification criterion.…”