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

TL;DR

This paper introduces a novel theoretical framework using quaternion algebra and Bloch sphere visualization to analyze symmetry-breaking dynamics of azimuthal thermoacoustic modes in annular combustors, accounting for stochastic turbulence effects.

Contribution

It develops a new analytical model for slow-flow dynamics of azimuthal modes, capturing symmetry-breaking bifurcations and mode states in idealized combustors.

Findings

Model describes standing, spinning, and mixed modes.

Accounts for stochastic turbulence forcing.

Reveals conditions for symmetry-breaking bifurcations.

Abstract

Depending on the reflectional and rotational symmetries of annular combustors for aeroengines and gas turbines, self-sustained azimuthal thermoacoustic eigenmodes can be standing, spinning or mix of these two types of waves. These thermoacoustic limit cycles are unwanted because the resulting intense acoustic fields induce high-cycle fatigue of the combustor components. This paper presents a new theoretical framework for describing, in an idealized annular combustor, the dynamics of the slow-flow variables, which define the state of an eigenmode, i.e. if the latter is standing, spinning or mixed. The acoustic pressure is expressed as a hypercomplex field and this ansatz is inserted into a one dimensional wave equation that describes the thermoacoustics of a thin annulus. Slow-flow averaging of this wave equation is performed by adapting the classic Krylov-Bogoliubov method to the quaternion field in order to derive a system of coupled first order differential equations for the four slow-flow variables, i.e. the amplitude, the nature angle, the preferential direction and the temporal phase of the azimuthal thermoacoustic mode. The state of the mode can be conveniently depicted by using the first three slow-flow variables as spherical coordinates for a Bloch sphere representation. Stochastic forcing from the turbulence in annular combustors is also accounted for. This new analytical model describes both rotational and reflectional symmetry breaking bifurcations induced by the non-uniform distribution of thermoacoustic sources along the annulus circumference and by the presence of a mean swirl.