Effect of steady and fluctuating heat transfer on acoustic and entropy transfer functions of a nozzle

This paper presents an analytical framework to investigate the effect of both steady (mean) and unsteady (fluctuating) heat release rate on the acoustic response of a quasi-one-dimensional nozzle sustaining a mean flow. Previous models consider either steady or unsteady heat release separately, and established independently that these phenomena can significantly alter the acoustic response of the nozzles. In this work, we develop a new model to account for the effect of both steady and unsteady heat release rate with arbitrary spatial distribution. To this end, we propose a Magnus-expansion-based solution of a linearised form of the Euler equations for a perfect, compressible gas flowing inside such a non-isentropic nozzle. The solution requires an additional constraint that relates the fluctuating unsteady heat release with acoustic oscillations, i.e. a closure model. A simple linear flame transfer function (FTF) with constant gain and phase-lag was considered but the analysis can be extended to consider non-linear flame describing functions. The model predictions of the nozzle's acoustic response is successfully validated against numerical solutions of the linearised Euler equations for different steady and unsteady heat release rate distributions inside the nozzle. It is observed that both steady and unsteady heat release rates significantly affect the unsteady nozzle response.