This paper presents a finite element methodology to predict the thermoacoustic eigenmodes of combustion chambers using the linearized Navier-Stokes equations (LNSE) in frequency space. The effect of the mean flow on the acoustics is accounted for. Besides scattering and refraction of acoustic waves in shear layers, this set of equation describes two main damping mechanisms. One is related to the generation of entropy waves, so called hot-spots, in flame regions. The other is related to the transformation of acoustic energy into vorticity waves at sharp leading or trailing edges. Both fluctuation types, i.e. entropy and vorticity, are convected by the mean flow, leading to significant damping when the fluid discharges into an open outlet. In combustion chamber environments these waves are accelerated in the downstream high pressure distributor and are partially transformed back into acoustic waves constituting to the feedback loop of thermo-acoustic instabilities. Accurate prediction of the eigenmodes and eigenfrequencies of instability require therefore to take these interaction effects into account.
First, the accuracy of the LNSE approach, to capture the damping generated by the first mechanism of entropy generation and convection, is investigated for a generic premixed flame configuration. Solutions of the LNSE are compared to the analytic solutions as well as eigenvalues determined by an Helmholtz ansatz. Later methodology assumes a quiescent medium and neglects all interactions of acoustics with the mean flow. It is shown that large errors are introduced with increasing Mach-number.
To illustrate errors assuming a quiescent medium for realistic combustion chambers, the LNSE are used to assess the eigenmodes of a two-dimensional aero-engine combustor including strong shear regions, in the next step. The non-isothermal mean flow field is obtained performing an incompressible RANS simulation. It features an expanding jet with inner and outer recirculation zones. The acoustic computations using LNSE reveal a set of unstable and neutral hydrodynamic modes in addition to acoustic modes. Both damping mechanisms are present and contribute to the overall system stability. Again the obtained solution is compared to the solution of an Helmholtz code and differences are discussed.