Thermoacoustic stability of quasi-one-dimensional flows - Part I: Analytical and numerical formulation

A numerical method is developed for transient linear analysis of quasi-one-dimensional thermoacoustic systems, with emphasis on stability properties. This approach incorporates the effects of mean flow variation as well as self-excited sources such as the unsteady heat release across a flame. Working in the frequency domain, the perturbation field is represented as a superposition of local wave modes, which enables the linearized equations to be integrated in space. The problem formulation is completed by specifying appropriate boundary conditions. Here, we consider impedance boundary conditions as well as those relevant to choked and shocked flows. For choked flows, the boundary condition follows from the requirement that perturbations remain regular at the sonic point, while the boundary conditions applicable at a normal shock are obtained from the shock jump conditions. The numerical implementation of the proposed formulation is described for the system eigenvalue problem, where the natural modes are sought. The scheme is validated by comparison with analytical and numerical solutions.