Averaging of thermoacoustic azimuthal instabilities

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

“…The only explanation for the repeatable preferred orientation is the presence of an explicit rotational symmetry breaking in the thermoacoustic system. In addition to this explicit rotational symmetry breaking, the thermoacoustic dynamics exhibits a spontaneous reflectional symmetry breaking, which has been investigated by Faure-Beaulieu et al (2021a) and which manifests itself in figure 6(b): When Φ < 0.55, P(χ ) is unimodal, with a maximum close to zero, indicating that the thermoacoustic mode is mixed, with predominant standing wave component, which corroborates the theoretical studies from Faure-Beaulieu & Noiray (2020) and Ghirardo & Gant (2021). Considering figure 7, one can see that the underlying time traces exhibit erratic fluctuations of χ around 0.…”

Dynamics of azimuthal thermoacoustic modes in imperfectly symmetric annular geometries

“…Furthermore, in addition to this explicit symmetry breaking, the system exhibits a spontaneous symmetry breaking in terms of the nature of the thermoacoustic mode. When Φ < 0.55, P (χ) is unimodal, with a maximum close to zero, indicating that the thermoacous-tic mode is predominantly standing, which corroborates two recent theoretical studies [40,45] that predicted the predominance of standing modes when the normalized stochastic forcing amplitude is large compared to the one of the limit cycle. For Φ = 0.55 and Φ = 0.5625, P (χ) is bimodal: the spinning component of the mixed mode intermittently changes direction as illustrated in Fig.…”

Exact potentials in multivariate Langevin equations

“…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].…”