No abstract

Thermoacoustic systems can exhibit self-excited instabilities of two nature, namely cavity modes or intrinsic thermoacoustic (ITA) modes. In heavy-duty land-based gas turbines with can-annular combustors, the cross-talk between cans causes the cavity modes of various azimuthal order to create clusters, i.e. ensembles of modes with close frequencies. Similarly, in systems exhibiting rotational symmetry, ITA modes also have the peculiar behavior of forming clusters. In the present study, we investigate how such clusters interplay when they are located in the same frequency range. We first consider a simple Rijke tube configuration and derive a general analytical low-order network model using only dimensionless numbers. We investigate the trajectories of the eigenmodes when changing the downstream length and the flame position. In particular, we show that ITA and acoustic modes can switch nature and their trajectories are strongly influenced by the presence of exceptional points. We then study a generic can-annular combustor. We show that such configuration can be approximated by an equivalent Rijke tube. We demonstrate that, in the absence of mean flow, the eigenvalues of the system necessarily lie on specific trajectories imposed by the upstream conditions.

Thermoacoustic systems can exhibit self-excited instabilities of two nature, namely cavity modes or intrinsic thermoacoustic (ITA) modes. In heavy-duty land-based gas turbines with canannular combustors, the cross-talk between cans causes the cavity modes of various azimuthal order to create clusters, i.e. ensembles of modes with close frequencies. Similarly, in systems exhibiting rotational symmetry, ITA modes also have the peculiar behavior of forming clusters. In the present study, we investigate how such clusters interplay when they are located in the same frequency range. We first consider a simple Rijke tube configuration and derive a general analytical low-order network model using only dimensionless numbers. We investigate the trajectories of the eigenmodes when changing the downstream length and the flame position. In particular, we show that ITA and acoustic modes can switch nature and their trajectories are strongly influenced by the presence of exceptional points. We then study a generic can-annular combustor. We show that such configuration can be approximated by an equivalent Rijke tube. We demonstrate that, in the absence of mean flow, the eigenvalues of the system necessarily lie on specific trajectories imposed by the upstream conditions.

Exceptional points can be found for specific sets of parameters in thermoacoustic systems. At an exceptional point, two eigenvalues and their corresponding eigenfunctions coalesce. Given that the sensitivity of these eigenvalues to parameter changes becomes infinite at the exceptional point, their occurrence may greatly affect the outcome and reliability of numerical stability analysis. We propose a new method to identify exceptional points in thermoacoustic systems. By iteratively updating the system parameters, two initially selected eigenvalues are shifted toward each other, ultimately colliding and generating the exceptional point. Using this algorithm, we were able to identify for the first time a physically meaningful exceptional point with positive growth rate in a thermoacoustic model. Furthermore, our analysis goes beyond previous studies inasmuch as we employ a more realistic flame transfer function to model flame dynamics. Building on these results, we analyze the effect of exceptional points on the reliability of thermoacoustic stability analysis. In the context of uncertainty quantification, we show that surrogate modeling is not reliable in the vicinity of an exceptional point, even when large sets of training samples are provided. The impact of exceptional points on the propagation of input uncertainties is demonstrated via Monte Carlo computations. The increased sensitivity associated with the exceptional point results in large variances for eigenvalue predictions, which needs to be taken into account for reliable stability analysis.

A method is proposed that allows the computation of the continuous adjoint of a thermoacoustic network model based on the discretized direct equations. This hybrid approach exploits the self-adjoint character of the duct element, which allows all jump conditions to be derived from the direct scattering matrix. In this way, the need to derive the adjoint equations for every element of the network model is eliminated. This methodology combines the advantages of the discrete and continuous adjoint, as the accuracy of the continuous adjoint is achieved whilst maintaining the flexibility of the discrete adjoint. It is demonstrated how the obtained adjoint system may be utilized to optimize a thermoacoustic configuration by determining the optimal damper setting for an annular combustor.