Analytical approximations for heat release rate laws in the time- and frequency-domains

This paper focusses on the relationship between the heat release rate and the acoustic field, which is a crucial element in modelling thermoacoustic instabilities. The aim of the paper is twofold. The first aim is to develop a transformation tool, which makes it easy to switch between the time-domain representation (typically a heat release law involving time-lags) and the frequency-domain representation (typically a flame transfer function) of this relationship. Both representations are characterised by the same set of parameters n1, n2, …, nk. Their number is quite small, and they have a clear physical meaning: they are time-lag dependent coupling coefficients. They are closely linked to the impulse response of the flame in the linear regime in that they are proportional to the discretised (with respect to time) impulse response. In the nonlinear regime, the parameters n1, n2, …, nk become amplitude-dependent. Their interpretation as time-lag dependent coupling coefficients prevails; however, the link with the impulse response is lost. Nonlinear flames are commonly described in the frequency-domain by an amplitude-dependent flame transfer function, the so-called flame describing function. The time-domain equivalent of the flame describing function is sometimes mistaken for a ‘nonlinear impulse response’, but this is not correct. The second aim of this paper is to highlight this misconception and to provide the correct interpretation of the time-domain equivalent of the flame describing function.