A Finite Element Method for Three-Dimensional Analysis of Thermo-acoustic Combustion Instability

Abstract

A method for predicting the onset of acoustically driven combustion instabilities in gas turbine combustor is examined. The basic idea is that the governing equations of the acoustic waves can be coupled with a flame heat release model and solved in the frequency domain. The paper shows that a complex eigenvalue problem is obtained that can be solved numerically by implementing the governing equations in a finite element code. This procedure allows one to identify the frequencies at which thermo-acoustic instabilities are expected and the growth rate of the pressure oscillations, at the onset of instability, when the hypothesis of linear behavior of the acoustic waves can be applied. The method can be applied virtually to any three-dimensional geometry, provided the necessary computational resources that are, anyway, much less than those required by computational fluid dynamics methods proposed for analyzing the combustion chamber under instability condition. Furthermore, in comparison with the “lumped” approach that characterizes popular acoustics networks, the proposed method allows one for much more flexibility in defining the geometry of the combustion chamber. The paper shows that different types of heat release laws, for instance, heat release concentrated in a flame sheet, as well as distributed in a larger domain, can be adopted. Moreover, experimentally or numerically determined flame transfer functions, giving the response of heat release to acoustic velocity fluctuations, can be incorporated in the model. To establish proof of concept, the method is validated at the beginning against simple test cases taken from literature. Over the frequency range considered, frequencies and growth rates both of stable and unstable eigenmodes are accurately evaluated. Then the method is applied to a much more complex annular combustor geometry in order to evaluate frequencies and growth rates of the unstable modes and to show how the variation in the parameters of the heat release law can influence the transition to instability.

Intrinsic instability of flame–acoustic coupling

“…In fact, the authors already have verified that a thermo-acoustic FEM model setup in Comsol Multiphysics provides exactly the same results [33]. The model is created similarly as described in [20,34], where the 1D Helmholtz equation is augmented by a velocity dependent source term in order to model the flame. For the configuration of Fig.…”

Combining a Helmholtz solver with the flame describing function to assess combustion instability in a premixed swirled combustor

“…In a second step, this function can be combined to a model of the system acoustics which is often obtained analytically (Dowling and Stow, 2003;Sattelmayer and Polifke, 2003;Poinsot and Veynante, 2005). Also, the FTF can be combined to numerical tools which are used to solve the wave equation, usually expressed in the frequency domain (Roux et al, 2005;Nicoud et al, 2007;Camporeale et al, 2011). These numerical tools allow to account for the geometrical complexity of real systems and rely on Helmholtz solvers to model linear acoustics in low Mach number flows.…”

Prediction and control of combustion instabilities in real engines

“…4 ): the rst category; and the only one until the 2000s, is thermoacoustic codes (called TA here) in which ames are not simulated but replaced by their equivalent FTF or FDF [69][70][71][72] . The mean ow is frozen and solutions are sought for the linearized perturbations.…”