Modern heavy-duty gas turbines are commonly designed with can-annular combustors, in which all flames are physically separated. Acoustically, however, the cans communicate via the upstream located compressor plenum, or at the downstream gaps found at the transition to the turbine inlet. Due to the relatively small size of connection areas, the coupling between the cans is weak. A can-annular system can therefore be considered as analogous to a set of weakly coupled, nominally identical oscillators, for which mode clustering is a common dynamical feature. Here, mode clustering refers to multiple modes being close in frequency and growth rate.

  
In the present study, a coupling condition that uses the Rayleigh conductivity and ensures acoustic flux conservation is derived. It enables acoustic communication between adjacent cans, in which one-dimensional acoustic waves propagate. Additionally, because can-annular systems commonly feature a discrete rotational symmetry, the acoustic field can be expressed as a Bloch-periodic wave in the azimuthal direction. We demonstrate how the coupling conditions resulting in a combustion system with *N* cans can be expressed as an effective impedance for a single can. The effective impedance depends on the Bloch wavenumber b, the size of the connection radius *r*, the total number of cans *N*, and the complex frequency *s*. If the effective impedance is applied as a boundary condition to a single can system, the resulting model will behave like an acoustically coupled can-annular system, thus reducing the size of the system to be investigated by factor *N*. On the other hand, in order to determine the acoustic or thermoacoustic modes of the can-annular system, all possible Bloch wavenumbers have to be considered.

  
Using this method, we investigate in frequency domain a generic can-annular combustor model consisting of *N* identical cans that are connected at the downstream end. The unsteady heat sources are assumed to be identical in each can and acoustically compact. Their response is assumed to be analogous to that of a fully premixed laminar conical flame, whose Flame Transfer Function (FTF) has been computed numerically using a G-equation solver.

  
We first investigate the effect of the coupling strength and of the number of cans on the purely acoustic modes of can-annular systems. The weak acoustic coupling leads to the formation of eigenvalue clusters, with eigenfrequencies close to those of an analogous system with a single can. The number of modes in a cluster is determined by the number of cans in the system. The spacing between the modes of a cluster is found to be dependent on the coupling strength and the frequency. When increasing the coupling strength, the eigenfrequencies of the can-annular system in a cluster (all originating from the same eigenvalue in the uncoupled limit) follow the same trajectory, but at different rates. We prove that, in each cluster, the larger is the Bloch wavenumber considered, the higher is the sensitivity of the eigenfrequencies to variations in the coupling.

  
We then include the unsteady flame responses and study the behaviour of the thermoacoustic eigenvalues when varying the coupling strength and the number of cans. Similar results to the acoustic case, viz. the formation of clusters and their motion along specific trajectories, are observed. However, in the thermoacoustic case, the acoustic–flame interaction causes variations in the growth rates of the eigenvalues when increasing the coupling strength that are much larger than those observed in the purely acoustic case. In the weakly coupled limit, it is possible to identify clusters of unstable thermoacoustic modes with closely spaced frequencies but very different modeshapes, which opens up new questions regarding the control and the prediction of the nonlinearly saturated thermoacoustic modeshape in this scenario.  
 


Analysis of Thermoacoustic Modes in Can-Annular Combustors Using Effective Bloch-Type Boundary Conditions

Division / Sponsor:
International Gas Turbine Institute (IGTI)

Paper Proceedings: https://asmedigitalcollection.asme.org/GT/GT2020/volume/84058

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