Transient growth of flow disturbances in triggering a Rijke tube combustion instability

Combustion instabilities in a Rijke tube could be triggered by the transient growth of flow disturbances, which is associated with its non-normality. In this work, a Rijke tube with three different temperature configurations resulting from a laminar premixed flame are considered to investigate its non-normality and the resulting transient growth of flow disturbances in triggering combustion instabilities. For this, a general thermoacoustic model of a Rijke tube is developed. Unsteady heat release from the flame is assumed to be caused by its surface variations, which results from the fluctuations of the oncoming flow velocity. Coupling the flame model with a Galerkin series expansion of the acoustic waves present enables the time evolution of flow disturbances to be calculated, thus providing a platform on which to gain insights on the Rijke tube stability behaviors. Both eigenmodes orthogonality analysis and transient growth analysis of flow disturbances are performed by linearizing the flame model and recasting it into the classical time-lag N-τ formulation. It is shown from both analyses that Rijke tube is a non-normal thermoacoustic system and its non-normality depends strongly on the temperature configurations and the flame position. Furthermore, the most ‘dangerous’ position at which the flame is more susceptible to combustion instabilities are predicted by real-time calculating the maximum transient growth rate of acoustical energy.