Effects of Asymmetry on Thermoacoustic Modes in Annular Combustors: A Higher-Order Perturbation Study

Abstract

Gas-turbine combustion chambers typically consist of nominally identical sectors arranged in a rotationally symmetric pattern. However, in practice, the geometry is not perfectly symmetric. This may be due to design decisions, such as placing dampers in an azimuthally nonuniform fashion, or to uncertainties in the design parameters, which break the rotational symmetry of the combustion chamber. The question is whether these deviations from symmetry have impact on the thermoacoustic-stability calculation. The paper addresses this question by proposing a fast adjoint-based perturbation method. This method can be integrated into numerical frameworks that are industrial standard such as lumped-network models, Helmholtz and linearized Euler equations. The thermoacoustic stability of asymmetric combustion chambers is investigated by perturbing rotationally symmetric combustor models. The approach proposed in this paper is applied to a realistic three-dimensional combustion chamber model with an experimentally measured flame transfer function (FTF). The model equations are solved with a Helmholtz solver. Results for modes of zeroth, first, and second azimuthal order are presented and compared to exact solutions of the problem. A focus of the discussion is set on the loss of mode-degeneracy due to symmetry breaking and the capability of the perturbation theory to accurately predict it. In particular, an “inclination rule” that explains the behavior of degenerate eigenvalues at first order is proven.

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

“…Such predictions can be achieved using reduced-order thermoacoustic network models which include modelled or measured flame transfer functions and acoustic propagation in 3D complex geometries -e.g. [18,19,20,21] for linear description of the annular thermoacoustics, with stability analysis and investigation of symmetry breaking, [22] for computationally efficient numerical methods to solve the associated nonlinear eigenvalue problem and [23] to perform sensitivity analysis to perturbations in the heat release rate distribution , [24,25] for frequency domain description of the thermoacoustics in annular combustors with measured and modelled nonlinear flame feedback, [15,26] for state-space description allowing for time and frequency domain analysis. Coming back to the 1D description adopted in this work, previous 1D studies from Ghirardo et al [27,28] use a general formulation with flame describing function for both static and dynamic nonlinearities.…”

Adjoint Methods as Design Tools in Thermoacoustics

“…They extended their analysis to flame describing functions for the calculation of limit cycles [170]. Different symmetry-breaking perturbations to the burners were studied with higher-order perturbation adjoint theory in [171]. In eigenvalue optimization, Mensah and Moeck [172] calculated the optimal placement and tuning of acoustic dampers in an annular combustor; while Aguilar and Juniper [173,174] eliminated thermoacoustic oscillations by shape optimization.…”

“…• there is no flame-to-flame interaction from one sector to another; [202,[217][218][219] as modelled in [171].…”

“…In multi-index notation up to order k, the expansion of the thermoacoustic matrix N around the eigenvalue σ 0 and the set of M parameters cast in a vector p 0 , reads [165,171,172] (70) where (i, µ µ µ) ≡ (i, µ 1 , . .…”

“…9). Mode degeneracy with semi-simple eigenvalues is a problem relevant to annular and can-annular combustors because of their rotationally symmetric geometry (see, among others, [163,165,166,171,221,[270][271][272][273][274][275][276][277][278][279]). The multiplicity of the degeneracy induced by the symmetry is typically g = 2, which physically corresponds to two eigenfunctions that are rotating in opposite azimuthal directions with the same frequency.…”