A theoretical study of mean azimuthal flow and asymmetry effects on thermo-acoustic modes in annular combustors

Abstract

To cite this version:Michaël Bauerheim, Michel Cazalens, Thierry Poinsot. A theoretical study of mean azimuthal flow and asymmetry effects on thermo-acoustic modes in annular combustors. Proceedings of the Combustion Institute, Elsevier, 2015, vol. 35 (n°3) AbstractThe objective of this paper is to develop an analytical model to capture two symmetry breaking effects controlling the frequency and nature (spinning, standing or mixed) of azimuthal modes appearing in annular chambers: (1) Using two different burner types distributed along the chamber (2) Considering the mean azimuthal flow due to the swirlers or to effusion cooling. The ATACAMAC (Analytical Tool to Analyze and Control Azimuthal Modes in Annular Chambers) methodology is applied using the linearized acoustic equations with a steady and uniform azimuthal mean flow. It provides an analytical implicit dispersion relation which can be solved numerically. A fully analytical resolution is possible when the annular chamber is weakly coupled to the burners. Results show that symmetry breaking, either by mixing burners types or with a mean azimuthal flow, splits the azimuthal modes into two waves with different frequencies and structures. Breaking symmetry promotes standing modes but adding even a low azimuthal mean flow fosters spinning modes so that the azimuthal mean flow must be taken into account to study azimuthal modes.

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

“…The frequency gap between the two components is 2M ω n , which is small compared to the pure acoustic frequency ω n . This frequency splitting due to the azimuthal flow has been previously observed experimentally in [16] and predicted from a linear stability analysis of a lumped-element description of the thermoacoustics of an annular combustor [19]. It has to be noted that the phenomenon of frequency splitting does not depend on the model adopted for the flame response.…”

“…In previous studies dealing with thermoacoustic instabilities in annular configurations, the time delay between the acoustic field and the heat release rate oscillations has been accounted for in different ways: i) by using the simple n-τ formulation (e.g. [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.…”

“…In the field of thermoacoustics, Bauerheim et al [19] showed that a mean swirl in annular combustors affects the solutions of the linear eigenvalue problem and that it leads to counter-rotating modes with different linear growth rates and frequencies.…”

“…Such predictions can be achieved using reduced-order thermoacoustic network models which include modelled or measured flame transfer functions and acoustic propagation in 3D complex geometries -e.g. [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.…”

Weakly nonlinear analysis of thermoacoustic instabilities in annular combustors

“…Firstly, a non-zero U θ makes one of the two spinning modes rotate faster and the other slower, and makes standing modes slowly rotate with pressure and velocity nodes moving at the speed of the mean azimuthal flow. See for example Wolf et al (2010) for numerical evidence and a discussion, and refer to Bauerheim et al (2014), Bauerheim, Cazalens & Poinsot (2015) for a detailed analysis of this first effect of U θ in a linear framework. Secondly, a non-zero U θ bends the flames in the azimuthal direction, orthogonally to the burner's axis.…”