Assessing non-normal effects in thermoacoustic systems with mean flow

In this paper, non-normal interactions in a thermoacoustic system are studied, using a low-order expansion of the state variables in terms of eigenmodes. The thermoacoustic eigenmodes are determined as solutions of the Helmholtz equation or the linearized Euler equations, respectively, in the presence of a time-lagged heat source. Subsequently, non-normal effects are evaluated in a post-processing analysis based on the computed eigenmodes. In the case where the eigenmode analysis is based on the linearized Euler equations, effects of a non-zero mean flow velocity can be taken into account. The energy associated with the eigenmodes may then contain contributions of convected entropy and vorticity modes as well as the acoustic field. The notion of transient growth of perturbation energy is thus extended from an expression based on the classical acoustic energy density to a form based on a generalized disturbance energy. The expansion in terms of eigenmodes is computationally efficient, making the approach potentially applicable to complex, 3D configurations including non-trivial boundary conditions and spatio-temporal distributions of heat release fluctuations. In the present paper, the method is applied to a 1D configuration that consists of a duct including a 1D heat source, followed by a choked isentropic nozzle. It is shown that for such a case, it is essential to include the contribution of entropy perturbations in the calculation of the optimal initial perturbation and the maximum transient energy growth. Subsequently, the impact of increasing mean flow Mach number and increasing strength of flame/acoustic interaction on non normal effects is assessed in a parameter study.