Weakly nonlinear analysis of thermoacoustic instabilities in annular combustors

Abstract

Rotationally symmetric annular combustors are of practical importance because they generically resemble combustion chambers in gas turbines, in which thermoacoustically driven oscillations are a major concern. We focus on azimuthal thermoacoustic oscillations and model the fluctuating heat release rate as being dependent only on the local pressure in the combustion chamber. We study the dynamics of the annular combustor with a finite number of compact flames equispaced around the annulus, and characterize the flames' response with a describing function. We discuss the existence, amplitude and the stability of standing and spinning waves, as a function of: (i) the number of the burners; (ii) the acoustic damping in the chamber; (iii) the flame response. We present the implications for industrial applications and the future direction of investigations. We then present as an example the first theoretical study of thermoacoustic triggering in annular combustors, which shows that rotationally symmetric annular chambers that are thermoacoustically unstable do not experience only stable spinning solutions, but can also experience stable standing solutions. We finally test the theory on one experiment with good agreement.

Symmetry breaking of azimuthal waves: Slow-flow dynamics on the Bloch sphere

“…In the case of subcritical Hopf bifurcations, the topology of the bifurcation diagram is changed (see Figs. 9 and 11 in [39]) and can only be described by using a more complex flame describing functions [28]. Additionally, expression (14) cannot be used to represent dynamic nonlinearities involving an amplitude-dependent delay of the flame response to acoustic perturbation, unlike the general flame describing function considered in [28].…”

“…9 and 11 in [39]) and can only be described by using a more complex flame describing functions [28]. Additionally, expression (14) cannot be used to represent dynamic nonlinearities involving an amplitude-dependent delay of the flame response to acoustic perturbation, unlike the general flame describing function considered in [28]. From now on, the subscript (·) a will be omitted and p approximates the acoustic pressure of azimuthal eigenmodes at a given azimuthal position.…”

“…[48,19]) and its state-space equivalent [49] for modelling the linear thermoacoustic problem in time and frequency domain, ii) by using measured flame describing functions (e.g. [50,28]) for modelling the nonlinear thermoacoustic problem in frequency domain, and iii) by modelling the flame response to acoustic perturbations with a distribution of time delays [14], or by using state-space representations of experimentally measured FTFs [15,26] for simulating the nonlinear thermoacoustic problem in time domain. Recently, Ghirardo, Juniper and Bothien published a very interesting detailed study [51] on the effect of flame phase on thermoacoustic limit cycles in symmetric annular combustors without mean swirl and without stochastic forcing.…”

“…[18,19,20,21] for linear description of the annular thermoacoustics, with stability analysis and investigation of symmetry breaking, [22] for computationally efficient numerical methods to solve the associated nonlinear eigenvalue problem and [23] to perform sensitivity analysis to perturbations in the heat release rate distribution , [24,25] for frequency domain description of the thermoacoustics in annular combustors with measured and modelled nonlinear flame feedback, [15,26] for state-space description allowing for time and frequency domain analysis. Coming back to the 1D description adopted in this work, previous 1D studies from Ghirardo et al [27,28] use a general formulation with flame describing function for both static and dynamic nonlinearities. They show that static nonlinearities associated with non-monotonic describing function gains can explain the occurrence of stable standing modes in perfectly symmetric annular combustors.…”

Data Assimilation in a Nonlinear Time-Delayed Dynamical System with Lagrangian Optimization

“…where u is the acoustic velocity; p is the acoustic pressure; t is the time; x is the axial coordinate of the duct; δ D (x − x f ) is the Dirac delta distribution at the heat source location, x f ; ζ is the damping factor, which models the acoustic energy radiation from the boundaries and thermo-viscous losses; andQ is the heat release rate (or, simply, heat release). The heat release,Q, is modelled by a nonlinear time delayed law [10] Q ≡ βPoly(u f (t − τ )),…”