# 2020 - Protection and identification of thermoacoustic azimuthal modes

Abstract: This paper first characterizes the acoustic field of two annular combustors by means of data from acoustic pressure sensors. In particular the amplitude, orientation, and nature of the acoustic field of azimuthal order n is characterized. The dependence of the pulsation amplitude on the azimuthal location in the chamber is discussed, and a protection scheme making use of just one sensor is proposed. The governing equations are then introduced, and a low-order model of the instabilities is discussed. The model accounts for the nonlinear response ofM distinct flames, for system acoustic losses by means of an acoustic damping coefficient $\alpha$ and for the turbulent combustion noise, modelled by means of the background noise coefficient $\sigma$. Keeping the response of the flames arbitrary and in principle different from flame to flame, we show that, together with $\alpha$ and $\sigma$, only the sum of their responses and their 2n Fourier component in the azimuthal direction affect the dynamics of the azimuthal instability. The existing result that only this 2n Fourier component affects the stability of standing limit-cycle solutions is recovered. It is found that this result applies also to the case of a non-homogeneous flame response in the annulus, and to flame responses that respond to the azimuthal acoustic velocity. Finally, a parametric flame model is proposed, depending on a linear driving gain $\beta$ and a nonlinear saturation constant $\kappa$. The model is first mapped from continuous time to discrete time, and then recast as a probabilistic Markovian model. The identification of the parameters {$\alpha$,$\beta$,$\kappa$,$\sigma$} is then carried out on engine timeseries data. The optimal four parameters {$\alpha$,$\sigma$,$\beta$,$\kappa$} are estimated as the values that maximize the data likelihood. Once the parameters have been estimated, the phase space of the identified low-order problem is discussed on selected invariant manifolds of the dynamical system.