Transient energy growth analysis of a thermoacoustic system with distributed mean heat input

Transient growth of flow disturbances has great potential to trigger unwanted thermoacoustic instability. So far transient growth analysis has tended to focus on thermoacoustic systems with acoustically compact heat sources, even though many systems are associated with distributed mean heat input, such as a premixed flame. In this work, transient growth analysis of both choked and open-ended thermoacoustic systems in the presence of a mean flow and a spatially distributed mean heat input is conducted. Unsteady heat release is modeled within the classical time-lag ℵ-τ formulation. Both uniform and triangular distributions for the rate of mean heat input are considered. The generation of entropy disturbances with such distributed heat input is studied first. It is shown that the entropy waves generated by the uniform and triangular distributed heat input are increased first and then decreased with increased frequency. This is different from the conventional concentrated heat input, of which the entropy waves produced is frequency-independent. In addition, the entropy eigenmodes are shown to be non-orthogonal. To quantify transient growth of flow disturbances, two energy measures are defined, calculated and compared. One is associated with the conventional acoustical energy. The other is associated with both acoustic and entropy disturbances. It is shown that the maximum transient growth Gacmax of acoustical energy is in the range of 102-103 in the choked system, while 100 ⩽ Gacmax ⩽ 101 in the open-ended system. Furthermore, the longer of the uniform distributed heat input, the larger Gacmax. However, such finding is not observed for the triangular heat input. Further insights are obtained by examining the contribution of eigenmodes in different frequency ranges. It is found that the lower frequency eigenmodes play a dominant role. Finally, the effect of the interaction index ℵ on transient growth is examined. It is found that the maximum transient growth of acoustical energy Gacmax and total energy Gtotmax are decreased with increased ℵ. It is also found that the longer of the uniform distributed heat input, the lower Gtotmax. These findings are consistent with those obtained in our non-orthogonality and entropy generation analyses.