Energetically consistent computation of combustor stability with a model consisting of a helmholtz finite element method domain and a low-order network

An efficient approach for the detection of the acoustic damping of gas turbine combustors is the combination of spatially resolved finite element method (FEM) approaches based on the Helmholtz equation with low-order networks for all elements leading to acoustic damping. A fundamental problem of such hybrid approaches is that the flow is considered in the networks, but not in the spatially resolved FEM area. Without special treatment of the coupling plane and the boundary conditions, this leads to serious errors in the calculation of the damping rate. The purpose of the paper is the derivation of the required correction procedures, which allow the energetically consistent formulation of such hybrid models and lead to correct damping rates. The time-averaged equation of acoustic energy flux for nonuniform fluid flows is expressed in terms of reflection coefficients and compared to the equivalent formulation for vanishing mean flows. An existing transformation for boundary conditions to obtain equal energy flux at the interface between network and Helmholtz domain is analyzed in detail. The findings are then used to derive an energetically consistent transformation of transfer matrices to couple two FEM domains via a network model. The relevance of energetically consistent transfer matrices for stability analysis is demonstrated with a generic test case. The central partition is acoustically characterized via a low-order model considering mean flow. The resulting acoustic twoport is transformed to obtain an energetically consistent transfer matrix for a subsequent FEM discretized eigenvalue analysis of the remaining geometry. The eigenvalues of energetically consistent calculations are finally compared to eigenvalues of energetically inconsistent setups.