On the reflection and transmission of circumferential waves through nozzles

Abstract

This paper presents a mathematical model for predicting the propagation of circumferential waves (acoustic, entropy and vorticity waves) through an annular nozzle. Combustion chambers in modern aero-engines are typically annular, and so a model for circumferential waves is essential for understanding and predicting both combustion noise and thermoacoustic instabilities for such geometries. The linearised Euler equations are solved using the Magnus expansion to obtain the reflection and transmission coefficients of the annular nozzle for acoustic, entropy and vorticity perturbations. Predictions which account for flow physics, such as a non-zero mean flow angle and the generation of vorticity noise, are obtained for the first time. Results are compared with two numerical methods, showing that the mathematical model is able to predict the transmission and reflection of waves for both compact and non-compact frequencies. The model is used to prove one particularly interesting and relevant feature of annular geometries: the generation of a vorticity wave by the acceleration of a circumferential entropy wave. It is shown that this phenomenon originates from the baroclinic torque in the vorticity equation.