Representing the geometrical complexity of liners and boundaries in low-order modeling for thermoacoustic instabilities

Abstract

This work introduces a novel method to represent the topological complexity of liners and boundaries in Low Order Models for thermoacoustic instabilities, under the assumption of zero-Mach number flow. In typical industrial combustion devices, the difficulty to model these elements is twofold: (1) they are characterized by complex-valued Rayleigh conductivities or acoustic impedances, and (2) they consist of large, curved panels whose geometries have first-order effect on the combustor thermoacoustic stability. To deal with the first point, the present approach makes use of a frame modal expansion recently introduced by Laurent et al. (2019) [41], which is a generalization of the classical rigid-wall Galerkin expansion, intended to deal with non-trivial boundary conditions. The core of this work lies in the second difficulty: complex-shaped liners and boundaries are modeled as two-dimensional manifolds, for which a specific set of curvilinear governing equations is derived. The inclusion of acoustic impedance or Rayleigh conductivity into these equations enforces the proper conservation equations at the frontiers of the adjacent volumes. Surface modal projections are then introduced to expand acoustic variables onto an orthogonal basis of modes solutions of a curvilinear Helmholtz eigenproblem. The resulting dynamical system is embedded into a state-space framework to build acoustic networks. A first non-reacting canonical test case, consisting of a multi-perforated liner in a cylindrical geometry is studied to assess the convergence and precision of the method. The ability of the approach to deal with realistic reacting cases is then illustrated by modeling the partially reflecting outlet of a multi-sector annular combustor typical of industrial gas turbines. This methodology enables the inclusion of liners and other boundaries of arbitrary geometrical complexity in modal projection-based thermoacoustic Low Order Models.