Quaternion structure of azimuthal instabilities

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

“…In the case of almost harmonic perturbations, the quaternion embedding will be defined by a slow amplitude A, a slow phase drift ϕ and two additional slow variables χ and θ characterizing the spatial dynamics: χ will affect the ellipticity of the oscillations and θ, their direction. Ghirardo and Bothien showed in [35] that this quaternion formalism is perfectly suited to represent azimuthal waves in annular combustion chambers. They emphasize on the fact that this representation overcomes the problem of ill-posedness of the phase in previously adopted projections on standing modes e.g.…”

“…This has to be distinguished from the phenomenon of spinning wave: in the case of standing modes with a slowly drifting preferential direction, the modes do not propagate in the azimuthal direction and the rotation velocity is small compared to the speed of sound. As discussed in [35], the range of θ could be limited to [0, π[ instead of ] − π, π], because θ and θ + π mod 2π represent exactly the same state. However, the range [0, π[ is not chosen since it leads to potentially misleading phase jumps in θ time traces.…”

“…As brightly proposed by Ghirardo and Bothien [35], an appropriate way to represent graphically the state of an azimuthal mode and the trajectories of its slow evolution in time is the Bloch sphere, with the following spherical coordinates {A, nθ, 2χ}. As shown in Fig.…”

“…Accounting for it significantly affects the conclusions drawn in [51], as shown for instance in [52]. In [51], a Galerkin expansion based on the natural acoustic modes of the annular combustor was performed, which differs from [35] and the present work where the quaternion-based acoustic field projection is adopted. The convective phase lag is implemented in the present model by replacing the "non-delayed" thermoacoustic source term (14) by the following delayed model…”

“…Second, the phase space is ill-posed for both approaches, because when the amplitude of one of the modes vanishes, the phase difference between these modes becomes undetermined. These two shortcomings have been raised by Ghirardo and Bothien [35] who proposed an alternative quaternion-based projection of the acoustic field, which results in a well-posed phase space. We show in the present paper that an additional limitation of the aforementioned projections is overcome by using the hypercomplex ansatz from [35].…”