A state-space formulation of a discontinuous Galerkin method for thermoacoustic stability analysis

A hybrid approach for thermoacoustic stability analysis is formulated in a state-space framework. The approach distinguishes between regions of the computational domain with and without important interactions between acoustics and mean flow or unsteady heat release, respectively. The former regions are modeled by a discontinuous Galerkin finite element method (DG-FEM) for the linearized Navier-Stokes equations in conservative form. The latter are represented by reduced-order models of acoustic wave propagation or dissipation, and provide complex-valued, frequency dependent impedance boundary conditions for the DG-FEM domain. The flow-flame coupling is modeled by a flame transfer function that governs a volumetric source term for the fluctuating heat release rate. The respective (sub-)models are formulated and interconnected in a state-space framework, which facilitates the monolithic formulation of hybrid thermoacoustic models. Moreover, the state-space interconnect framework makes it possible to formulate thermoacoustic stability analysis as a linear eigenvalue problem – even if flame transfer function or acoustic boundary conditions depend in a non-trivial manner on frequency. The approach is first verified against analytical solutions for a duct with mean flow across a thin heat source, similar to a Rijke tube. Then the thermoacoustic eigenmodes of a premixed, swirl-stabilized combustor are computed in order to validate the method against experimental data for a configuration of applied interest. For this second validation case, a detailed comparison against predictions of a low-order network-model is also presented.