Spontaneous and explicit symmetry breaking of thermoacoustic eigenmodes in imperfect annular geometries

Abstract

This article deals with the symmetry breaking of azimuthal thermoacoustic modes in annular combustors. Using a nominally symmetric annular combustor, we present experimental evidence of a predicted spontaneous reflectional symmetry breaking, and also an unexpected explicit rotational symmetry breaking in the neighbourhood of the Hopf bifurcation which separates linearly stable azimuthal thermoacoustic modes from self-oscillating modes. We derive and solve a multidimensional Fokker–Planck equation to unravel a unified picture of the phase space topology. We demonstrate that symmetric probability density functions of the thermoacoustic state vector are elusive, because the effect of asymmetries, even imperceptible ones, is magnified close to the bifurcation. This conclusion implies that the thermoacoustic oscillations of azimuthal modes in real combustors will systematically exhibit a statistically dominant orientation of the mode in the vicinity of the Hopf bifurcation.

Exact potentials in multivariate Langevin equations

“…The length of the dashed vertical line and the dashed circle's radius are both equal to 8 ν/15κ. The spatial structure of the analytical Fokker-Planck solution shown in this figure is in excellent agreement with the numerical simulations presented in Fig.11of Ref [20]…”

“…(47)-(49) are equivalent a transformed gradient system with g V E and V F. In Refs. [18,20,29], a self-oscillating mode ψ in an annular cavity is projected onto four variables x = (A, θ, χ, ϕ) T using the basic quaternions (i, j, k) (left inset). By this geometric analogy, different states such as pure spinning and standing waves are represented as different points on the Bloch sphere (right inset).…”

“…In the study of self-oscillating thermoacoustic modes in annular cavities, an alternative projection to the one used in [17] and based on quaternions offers a convenient description of the nature of the modal dynamics, where one of the state variables indicates whether spinning or standing waves govern the dynamics at a given time instant [18][19][20]29]. Indeed, by projecting the acoustic field ψ(Θ, t) depending on the azimuthal angle Θ onto the four alternative state variables x = (A, χ, θ, ϕ) T using the basic quaternions (i, j, k), as proposed first in [29] based on the quaternion Fourier transform for bivariate signals defined in [30], the instantaneous state can be mapped to different points on the Bloch sphere.…”

“…Well known, low-dimensional examples of LPs are the Stuart-Landau oscillator [4][5][6], which represents the normal form of a supercritical Hopf bifurcation [7], and the deterministically and stochastically averaged noisedriven Van der Pol oscillator [3,[8][9][10]. Multivariate systems governed by potentials are also found in the classic Kuramoto model [11,12], swarming oscillators [13], classical many-body time crystals [14], networks of coupled limit cycles [10,15,16] and in models of noise-driven, self-sustained modes of annular cavities [17][18][19][20].…”

“…Figure 1. Illustration of the examples presented in Secs.V E and V F. In Refs [18,20,29],. a self-oscillating mode ψ in an annular cavity is projected onto four variables x = (A, θ, χ, ϕ) T using the basic quaternions (i, j, k) (left inset).…”