  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.
