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

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. Slowflow 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 slowflow 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. * abelf@ethz.ch † noirayn@ethz.ch arXiv:1911.02830v3 [physics.flu-dyn] 9 Jan 2020

Experiments and low-order modelling of intermittent transitions between clockwise and anticlockwise spinning thermoacoustic modes in annular combustors

“…Indeed, it was shown in previous works that it is the only term having an effect on the dynamics of the first azimuthal mode [8]. In [19], the quaternion formalism of Eq. ( 1) is introduced in the wave equation (2), and methods of spatial and slow time averaging are applied to obtain a first order system of coupled Langevin equations for A, χ, θ and ϕ.…”

“…where c is the speed of sound, α is the acoustic damping, γ is the heat capacity ratio, Ξ is a random forcing representing the turbulent heat release rate fluctuations, anḋ Q is the coherent component of the heat release rate of the flames. The latter is responsible of the thermoacoustic instability phenomenon, and it is modelled with a 3 rd order nonlinearity: (γ − 1)Q = β[1 + c 2 cos(2Θ)]p − κp 3 , where β[1 + c 2 cos(2Θ)]p influences the linear stability of the thermoacoustic modes and κp 3 governs the saturation to limit cycles [3,8,18,19]. To account for spatial asymmetries in the coherent heat release rate fluctuations, only the second term c 2 of the spatial Fourier expansion of the source term distribution was kept.…”

“…where ν = (β − α)/2. The constant Γ is the intensity of the noise coming from the projection of the random fluctuation Ξ on the first azimuthal mode shape [19]. ζ A and ζ χ are uncorrelated gaussian noises of intensities Γ/2ω 2 .…”

Dynamics of azimuthal thermoacoustic modes in imperfectly symmetric annular geometries

“…[24][25][26][27][28], and several reduced-order models [29][30][31][32][33][34][35][36][37][38] were proposed to explain experimental observations with limited success. Notably, a recently proposed ansatz for describing the acoustic field [39] was combined with the wave equation to describe an ideal annular combustor, unifying previous theoretical frameworks [40]. However, the transition from linearly stable to self-oscillating azimuthal thermoacoustic modes has never been investigated.…”

“…In this letter, we fill this gap with experimental data showing that the route to a self-sustained quasi-pure spinning mode passes through stationary states that are characterized by a statistically prevailing standing mode and intermittent reversals of spinning direction. We also build upon the model from [40] to unravel the complex topology of the phase space to demonstrate that symmetrically designed combustors will always display explicit symmetry breaking of the thermoacoustic modes at the bifurcation.…”