Adjoint-Based Calculation of Parametric Thermoacoustic Maps of an Industrial Combustion Chamber

The aim of the present study is to efficiently calculate parametric thermoacoustic maps of typical combustion chambers. Two configurations are considered: an academic configuration based on a Rijke tube, and an industrial combustion chamber, which is the core of a recently developed micro-turbine for power generation. Such maps can be understood as the collection of loci of thermoacoustic eigen frequencies obtained under systematic variations of some defined parameters, while considering the Helmholtz equation as the thermoacoustic model of interest. In this study we consider variations on two parameters: the gain n and time-delay τ associated with a generic flame response model. We also show the feasibility of the proposed approach when considering more realistic flame responses. A straight-forward way to calculate such a thermoacoustic map is by solving the Helmholtz equation, and, thus, the corresponding non-linear eigenvalue problem, one time per parameter combination. With that approach, the non-linear eigenvalue problem needs to be solved hundreds or thousands of times if an adequate resolution of the thermoacoustic map is sought. Such a strategy may be computationally unaffordable. In order to overcome this difficulty, the present work utilizes an adjoint-based, high-order perturbation method. The actual eigenvalue problem is only solved once at a baseline point. After applying the perturbation equations at that point, a polynomial rational function — the Padé approximant — is obtained to estimate the eigen-frequency drift that results for a small or large perturbation in the flame response. It is demonstrated, for both academic and industrial test cases, that the obtained maps are accurate. Additionally, it is shown that these maps reveal a large variety of thermoacoustic features, such as stability boundaries, intrinsic thermoacoustic modes, and exceptional points. The numerical costs for such calculations are negligible even for the industrial combustion chamber investigated.