Characterizing modal exponential growth behaviors of self-excited transverse and longitudinal thermoacoustic instabilities

Self-excited thermoacoustic instabilities as frequently observed in rocket motors, gas turbines, ramjets, and aeroengine afterburners are highly detrimental and undesirable for engine manufacturers. Conventionally, modal analysis of such combustion instability is conducted by examining the eigenfrequencies. In this work, thermoacoustic dynamics coupling studies are performed as an alternative approach to predict and characterize modal growth behaviors in the presence of transverse and longitudinal combustion instabilities. Unsteady heat release is assumed to depend on the temperature rate of change that results from the chemical reaction. Coupling the unsteady heat release model with traveling waves enables the modal growth rate of acoustic disturbances to be predicted, thus providing a platform to gain insights onto stability behaviors of the combustor. Both modal growth and total energy analyses of acoustic disturbances are performed by linearizing the unsteady heat release model and recasting it into the classical time-lag N - tau formulation with respect to the velocity potential function phi. It is shown from both analyses that the amplitude of any acoustic disturbances tends to increase exponentially with time, until the growth rate is limited by some dissipative process zeta. The chemical reaction rate increase with temperature is shown to be unstable with respect to acoustic wave motions. Furthermore, the maximum modal "growth rate " is determined in the absence of acoustic losses, i.e., zeta = 0. The derived maximum growth rate is experimentally confirmed to be greater than those practically measured ones from both Rijke tubes and swirling combustors. A phase drift is also experimentally observed. Finally, the effects of (1) the interaction index N, (2) the time-delay tau, (3) the ratio gamma of the specific heats, and (4) the acoustic losses/damping zeta are examined via cases studies. They are found to vary the critical temperature rate of change of the chemical reaction or the critical frequency omega cri above which the combustion system becomes unstable.