Thermoacoustic instabilities: Should the Rayleigh criterion be extended to include entropy changes?

The Rayleigh criterion (which measures the correlation between pressure and heat release) is the standard tool used to investigate and predict combustion instabilities in both experimental and numerical studies. However, the Rayleigh term is just one of the terms appearing in the acoustic energy equation. The recent development of large eddy simulations for combustion chambers allows complete closure of the budget and analysis of all terms in this equation. This task leads to unexpected difficulties and requires some basic work, as multiple definitions of the energy of fluctuations in a reacting compressible flow can be derived. The objective of this article is to revisit the theoretical derivations of the fluctuation energy equations. Two forms of energy are defined: The first is the classic acoustic energy (AE) introduced by various authors. The second is the fluctuation energy (FE) presented by B.T. Chu [Acta Mech. (1965) 215-234]. Both equations are rederived in a compact manner starting from full nonlinear forms. It is shown that the classic Rayleigh criterion naturally appears as the source term of the AE equation, while the FE form leads to a different criterion stating that temperature and heat release must be in phase for the instability to be fed by the flame/acoustics coupling. The FE form also integrates the fluctuations of three variables (pressure, velocity, entropy), while the AE form uses only pressure and velocity perturbations. It is shown that only the FE form should be used in flames, in contradiction to many current studies performed for combustion instabilities.