Acoustic Damper Placement and Tuning for Annular Combustors: An Adjoint-Based Optimization Study

Abstract

Thermoacoustic instabilities pose a major threat to modern gas turbines. The use of acoustic dampers, like Helmholtz resonators, has proven useful for the mitigation of such instabilities. However, assessing the effect of acoustic dampers on thermoacoustic modes in annular combustion chambers remains an intricate task. This results from the implicit nature of the thermoacoustic Helmholtz equation associated with the high number of possible parameter values for the positioning of the dampers and their impedance design. In the present work, the principal challenges of the effective placement and the design of the impedance of acoustic dampers in annular chambers are discussed. This includes the choice of an appropriate objective function for the optimization, the combinatorial challenges when dealing with different possible damper arrangements, and the numerical complexities when using the thermoacoustic Helmholtz equation to approach this issue. As a key aspect, the paper proposes a new adjoint-based approach to tackle these problems. The new algorithm establishes algebraic models that predict the effect of acoustic dampers on the growth rates of the thermoacoustic modes. The theory is exemplified on the basis of a generic annular combustor model with 12 burners.

Adjoint Methods as Design Tools in Thermoacoustics

“…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. Recently, Silva et al [175] applied adjoint methods to calculated the critical flame index with relevance to intrinsic thermoacoustic modes.…”

“…A detailed explanation of the Helmholtz equation as applied to thermoacoustics is given by Nicoud et al [40,197]. The advantage of the Helmholtz-equation framework is that it can tackle three-dimensional geometries, with the con that it holds for very small mean-flow Mach numbers and requires numerical discretization, such as finite elements (e.g., [40,166,172,[199][200][201][202]), finite volumes (e.g., [164]) or finite difference. This model is applied in Sec.…”

“…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 , . .…”

“…To calculate the coefficients of (78)- (79), which are the eigenvalue and eigenvector drifts, respectively, the decompositions (78)-(79) are substituted into the eigenvalue problem (55). Each order of ε defines the equation for the eigenpair drift, which, at k-th order, reads in compact notation [172]…”

“…where N 0 is the unperturbed matrix. This is the usual approach used in hydrodynamic stability [134], which was used in thermoacoustics for the calculation of linearly optimal passive control devices [153,154], base-state sensitivities [158,160,165], uncertainty quantification both in annular [163] and longitudinal combustors [164,167] and optimal acoustic damper placements in annular combustors [172].…”