Modern pollutant emission regulation has led to the use of lean premixed combustion in gas turbine combustors, a technology which is prone to develop ther- moacoustic instabilities. This phenomenon is the result of a resonant feedback between combustion, acoustic waves and flow dynamics in confined combustion chambers. In this work, combustion instabilities are studied using a Helmholtz equation with a reactive term that takes into account the coupling between combustion and acoustics. The discretization of the resulting Helmholtz equation on unstructured meshes leads to a large sparse non-symmetric complex nonlinear eigenvalue problem of size N (N is equal to the number of nodes in the mesh). Its solution provides the frequencies and growth rates (complex eigenvalues) and the structure (eigenvectors) of the resonant modes of the combustor. Since dangerous combustion instabilities occur mostly at low frequencies, the nonlinear eigenvalue problem must be solved in order to obtain the smallest magnitude eigenvalues. The nonlinear problem is linearized using a fixed point iteration procedure. This leads to a sequence of linear eigenproblems which must be solved iteratively in order to obtain one nonlinear eigenpair. Therefore, efficient and robust parallel eigensolvers for the solution of linear problems are investigated, and strategies to accelerate the solution of the sequence of linear eigenproblems are also proposed. In modern gas turbines with annular combustors, the most dangerous resonant modes often take the form of azimuthal waves, making their study of first importance. This work focuses on azimuthal modes in annular combustors: their stability and nature (standing, spinning or mixed) are investigated as a function of the symmetry of the configuration. Thanks to the efficiency of the algorithms for the solution of the thermoacoustic eigenproblem, the 3D Helmholtz solver AVSP is used for the study of combustion instabilities of an annular industrial combustor.
