A systems perspective on non-normality in low-order thermoacoustic models: Full norms, semi-norms and transient growth

Abstract

Non-normal transient growth of energy is a feature encountered in many physical systems. Its observation is intimately related to the norm used to describe the system dynamics. For a multi-physics problem such as thermoacoustics, where a heat source is in feedback with acoustic waves and a flow field, the appropriate metric is an ongoing matter of debate. Adopting a systemic perspective, it is argued in the present paper that an energy norm is, in principle, a matter of choice, but one that is critically tied to the dynamics described by the system model. To illustrate our arguments, it is shown that different norms exhibit the non-normal dynamics of thermoacoustic systems differently, but that this difference is fully explicable by the energy flux and source terms related to the formulation of the model. The non-normal dynamics as such is unaffected by the choice of norm, and transient growth merely results from a maximization of the flux and source terms governing the energy balance associated with the specific model formulation. Investigating transient growth for arbitrary energy norms requires the capability to handle semi-norm optimization problems. In the present study, we propose an approach to do so using the singular value decomposition. Non-normal transient growth around a stable fix point is then investigated for a low-order model of a simple thermoacoustic configuration of a premixed flame enclosed in a duct with non-zero mean temperature jump and bulk mean flow. The corresponding optimal mode shapes and pertinent parameters leading to transient growth are identified and discussed. For transient growth resulting from the interaction of the flame with the acoustic field, it is found that heat sources with a fast response lead to more transient growth than slow heat sources, because the system can bear a larger source term before becoming linearly unstable. Furthermore, the amount of transient energy growth does not increase monotonically with the amplitude of the initial perturbation of the flame.