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.

在零马赫数流的假设下，这项工作引入了一种新颖的方法来表示热声不稳定性低阶模型中衬砌和边界的拓扑复杂性。在典型的工业燃烧设备中，很难对这些元素进行建模是双重的：（1）它们具有复数值瑞利电导率或声阻抗的特征，（2）它们由大型的弯曲面板组成，其几何形状对燃烧室的热声稳定性。为了解决第一点，本方法利用了Laurent等人最近引入的帧模态扩展。（2019）[41]，它是经典刚性壁Galerkin展开的推广，旨在处理非平凡的边界条件。这项工作的核心在于第二个困难：复杂形状的衬砌和边界被建模为二维流形，为此可导出一组特定的曲线控制方程。在这些方程式中包含声阻抗或瑞利电导率，可以在相邻体积的边界处强制执行适当的守恒方程式。然后引入表面模态投影以将声学变量扩展到曲线的亥姆霍兹本征问题的模态解的正交基础上。最终的动力学系统被嵌入到状态空间框架中以构建声学网络。研究了第一个非反应性规范测试案例，该案例由圆柱形几何形状的多孔衬套组成，以评估该方法的收敛性和精度。然后，通过对工业燃气轮机典型的多扇区环形燃烧器的部分反射出口进行建模，来说明该方法处理实际反应情况的能力。这种方法可以在基于模态投影的热声低阶模型中包含任意几何复杂度的衬层和其他边界。