Linear stability and adjoint sensitivity analysis of thermoacoustic networks with premixed flames

We analyse the linear response of laminar conical premixed flames modelled with the linearised front-track kinematic G-equation. We start by considering the case in which the flame speed is fixed, and travelling wave velocity perturbations are advected at a speed different from the mean flow velocity. A previous study of this case contains a small error in the Flame Transfer Function (FTF), which we correct. We then allow the flame speed to depend on curvature. No analytical solutions for the FTF exist for this case so the FTF has to be calculated numerically as its parameters - aspect ratio, convection speed and Markstein length - are varied. Then we consider the stability and sensitivity of thermoacoustic systems containing these flames. Traditionally, the stability of a thermoacoustic system is found by embedding the FTF within an acoustic network model. This can be expensive, however, because the FTF must be re-calculated whenever a flame parameter is varied. Instead, we couple the linearised G-equation directly with an acoustic network model, creating a linear eigenvalue problem without explicit knowledge of the FTF. This provides a simple and quick way to analyse the stability of thermoacoustic networks. It also allows us to use adjoint sensitivity analysis to examine, at little extra cost, how the system's stability is affected by every parameter of the system.