Abstract
This paper presents a method for computing the thermoacoustic modes in combustors. In the case of a nonisothermal reacting medium, the wave equation for the pressure fluctuations contains a forcing term related to the unsteady heat release. Depending on the phase relationship between the acoustics and the flame certain linear modes may become unstable, leading to thermoacoustic instabilities. The relevant Helmholtz equation is derived and two approaches for solving the corresponding nonlinear eigenvalue problem are proposed. The first one is based on an asymptotic expansion of the solution, the baseline being the acoustic modes and frequencies for a steady (or passive) flame and appropriate boundary conditions. This method allows a quick assessment of any acoustic mode stability but is valid only for cases where the coupling between the flame and the acoustic waves is small in amplitude. The second approach is based on an iterative algorithm where a quadratic eigenvalue problem is solved at each subiteration. it is more central processing unit demanding but remains valid even in cases where the response of the flame to acoustic perturbations is large. Frequency-dependent boundary impedances are accounted for in both cases. A parallel implementation of the Arnoldi iterative method is used to solve the large eigenvalue problem that arises from the space discretization of the Helmholtz equation. Several academic and industrial test cases are considered to illustrate the potential of the method.