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 utilizing 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 of M distinct flames, for system acoustic losses by means of an acoustic damping coefficient a and for the turbulent combustion noise, modelled by means of the background noise coefficient s. Keeping the response of the flames arbitrary, we show that, together with a and s, only the sum of their responses and their 2n Fourier component in the azimuthal direction affect the dynamics of the azimuthal instability. A parametric flame model is proposed, depending on a linear driving gain ß and a nonlinear saturation constant k. The model is first mapped from continuous time to discrete time, and then recast as a probabilistic Markovian model. The identification of the parameters {a, ß, k, s} is then carried out on engine timeseries data. The optimal four parameters {a, ß, k, s} 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.

Averaging of thermoacoustic azimuthal instabilities

“…Consistently, experiments show that in the nonlinear regime both the amplitude and the phase of u affect the flame response [58,59,60]. We refer to [61,62] for two recent works modelling this nonlinear effect. In the following we neglect this dependence on the azimuthal velocity and employ a simpler formulation where Q θ depends only on the acoustic pressure p:…”

“…7], by further simplifying equations ( 30) and using a different definition of nondimensional noise intensity. Other works have focused on the identification of the equations proposed in this paper [61] and on the modelling of the flame response and its effects on the dynamics [62]. A recent work [101] considers an azimuthal instability of order n = 1 that is dominantly spinning in one direction.…”

Saturation of flames to multiple inputs at one frequency

“…The structure of (2.19) limits the applicability to flames whose gain does not increase as a function of the forcing amplitude, which is the most typical case. The functional form of the saturation function F is chosen to be (Ghirardo et al 2021a)…”

“…To get a better understanding of the effect of the new terms in (2.26), it is convenient to express the heat release rate of each flame in terms of a Fourier series (Ghirardo et al 2021a)…”

“…with the assumption of ω = ω 0 for simplicity and μ 0,1,2,3 are the individual stochastic components of μ z in (2.26). The black terms are the same as the ones originally derived by Ghirardo & Gant (2021) and Ghirardo et al (2021a), and the coloured terms are the new additions and modifications. In the original equation given by (2.4), the zeroth Fourier component N (0) was only present in the time derivative of the amplitude.…”