Characterizing energy growth during combustion instabilities: Singularvalues or eigenvalues?

Abstract

Non-normality in thermoacoustic systems has received attention recently. It has been shown that in a non-normal but classically linearly stable system, there can be significant transient energy growth of small perturbations before their eventual decay. This growth occurs in the absence of non-linear effects. This phenomenon can be explained by the non-normality of the governing linear operator or the non-orthogonality of the eigenvectors of the system. In this paper, we study various aspects of this transient energy growth for a general combustor, with localized heat release approximated by the popular n-τ model. Galerkin technique is used to simplify the governing acoustic equations in a duct in the presence of a localized heat source with appropriate boundary conditions. Singularvalue Decomposition (SVD) is used to compute the transient energy growth. SVD is used as a tool to obtain the maximum possible energy amplification and the optimal initial conditions required for this amplification. The necessary and sufficient conditions for no energy growth are discussed. A parametric analysis is performed to highlight the effect of system parameters on the maximum transient growth rate and to obtain regions of stability of the thermoacoustic system.