A unified framework for acoustic instabilities based on the tailored Green’s function

Abstract

This paper presents a unified framework to analyse the role of thermoacoustic and aeroacoustic sources in a one-dimensional acoustic resonator. We introduce two forms of the tailored Green’s function: one that satisfies the boundary conditions of the acoustic pressure and one that satisfies those of the acoustic velocity. We consider acoustic sources, which can enter into a feedback loop with the acoustic field in the resonator and hence give rise to an instability. We assume that they are compact, and we model them as a monopole, dipole or a combination of the two. The reaction of a source to the acoustic field can be described in the time-domain (e.g. by a relationship between the volume outflow of a monopole to the local acoustic velocity at earlier times) or in the frequency domain (e.g. by a transfer matrix). Governing equations for the complete acoustic system are derived from the 1-D version of the acoustic analogy equation, using a Green’s function approach. They turn out to be two coupled Volterra equations for the acoustic pressure and velocity. Their solution is straightforward and numerically inexpensive. We demonstrate our approach with three diverse examples: a nonlinear Rijke tube, a whistling orifice, and a flame anchored on an orifice plate with bias flow. Our framework can be applied to various configurations, in particular to combustion test rigs, which consist of a duct-shaped resonator housing acoustically active elements, such as a flame and an orifice with bias flow.