Frequency shift due to azimuthal temperature gradients in an annular combustor

Annular combustors have multiple thermoacoustic modes that are linearly stable or unstable at a wide range of frequencies. Due to azimuthal symmetry, there exist pairs of either degenerate (i.e., two modes with identical frequencies) or weakly degenerate modes (i.e., modes with nearly identical frequency) in the system. This weak but typically nonzero frequency spacing has significant influences on the system’s nonlinear dynamics, such as stability and existence of potential limit cycle solutions. This paper presents a leading order expression for the frequency split and frequency shift in an annular combustor with mean temperature gradients. The analysis is constructed for small values of the relative temperature gradient, $ϵ$. For classical 1-D linear acoustics problems, the frequency shift occurs at $O\left(ϵ\right)$ through integration of the temperature fluctuations in the combustor. However, azimuthal periodicity results in a 0 contribution from this approximation and a detailed linear analysis shows that the linear correction only splits the frequency resulting in a loss of degeneracy. The analysis was extended to second order, as the leading order frequency shift occurs at $O\left({ϵ}^2\right)$. Explicit analytical expressions are presented for this second order correction, as well as example calculations quantifying the effect of different combustor parameters on the shift. These results can then be used by those exploring linear and nonlinear dynamics of annular combustion systems.