3-D thermoacoustic instability analysis based on Green's function approach

Abstract

A fundamental analysis is made of the thermoacoustic instability in a hard-walled box. We model the flame as an acoustically compact source with a heat release characteristic described by a directional nτ-law. This has the following features: it gives the heat release rate in terms of the acoustic velocity at an earlier time τ; it is linear with coupling coefficient n; the "flame surface" is a small flat patch with variable orientation.We derive an integral equation for the acoustic field by using a Green's function tailored to a 3-D rectangular box with hard-wall boundary conditions. The integral equation is solved by two methods. Firstly, an iteration method, stepping forward in time, is used to give the time history of the acoustic velocity. By analysing this time history, we investigate the interference between two (or more) thermoacoustic modes. In the second method, we apply a Laplace transform to determine the thermoacoustic eigenfrequency and growth rate of thermoacoustic modes. This method is suitable for parameter studies, and we use it to investigate the effect of the flame orientation and flame position on the thermoacoustic instability. We show results for the 2-D case. They reveal that the stability behaviour depends strongly on the flame orientation and on the flame position in the xy-plane. We also show results for the interference between different thermoacoustic modes, especially for cases where there are two acoustic modes with similar frequencies.