Non-orthogonality and transient growth analysis of a premixed flame-acoustic interaction in a choked combustor

Transient growth of flow disturbances occurred in a non-normal combustion system could trigger combustion instability. In this work, non-orthogonality and transient growth analyses of a choked combustor with a gutter confined are conducted. A thermoacoustic model is developed first to study the non-normal interaction between acoustic disturbances and a premixed V-shaped flame anchored to the tip of the gutter. Eigenmodes non-orthogonality analysis is then conducted. The thermoacoustic system is shown to be non-normal and characterized by non-orthogonal eigenmodes. The non-orthogonality arises from both the complex boundary condition and the monopole-like flame. However, the contribution from the flame is identified to play a dominant role. Transient growth analysis is then performed to gain insights on its finite-time stability behavior, which cannot be predicted by classical linear theory. To characterize the non-normality, two different energy measures are defined and estimated. One involves with acoustic travelling waves. The other is concerned with not only the travelling waves but the monopole-like flame. Comparison is then made between the two measures. It is found that the maximum transient growth rate of combustion-excited disturbances is about 10² - 10⁴ time greater than that of acoustic disturbances. Furthermore, the 'critical' time taken to reach the maximum transient growth rate is about half of period of the fundamental mode, which is about 90% shorter than that when only acoustic disturbances are considered. Finally, the most 'dangerous' location at which the flame is more susceptible to combustion instability is estimated.