Low-order modelling of distributed heat release

In low order modelling of thermoacoustic combustion instabilities the Transfer Matrix of a Flame (FTM) relating the acoustic properties upstream and downstream of the flame is frequently obtained from a Flame Transfer Function (FTF), which relates the acoustic fluctuations of velocity or pressure at the burner with the heat release rate of the flame under the assumption that the flame is acoustically compact. This compactness assumption is only valid for Helmholtz numbers He= κ-L_flame Cl, where κ is the acoustic wave number and L_flame the flame length. This work extends the applicability of low-order models to acoustically non-compact flames and provides an analytical expression for the FTM. The consequences of treating acoustically non-compact heat sources as compact elements on the system stability are shown, with insight into growth/ damping rates through mode shapes, energy balances and local Rayleigh index.