Thermoacoustic stabilization of a longitudinal combustor using adjoint methods

We construct a low-order thermoacoustic network model that contains the most influential physical mechanisms of a thermoacoustic system. We apply it to a laboratory-scale longitudinal combustor that has been found to be thermoacoustically unstable in experiments. We model the flame, which is behind a bluff body, by a geometric level-set method. We obtain the thermoacoustic eigenvalues of this configuration and examine a configuration in which six eigenmodes are unstable. We then derive the adjoint equations of this model and use the corresponding adjoint eigenmodes to obtain the sensitivities of the unstable eigenvalues to modifications of the model geometry. These sensitivities contain contributions from changes to the steady base flow and changes to the fluctuating flow. We find that these two contributions have similar magnitudes, showing that both contributions need to be considered. We then wrap these sensitivities within a gradient-based optimization algorithm and stabilize all six eigenvalues by changing the geometry. The required geometry changes are well approximated by the first step in the optimization process, showing that this sensitivity information is useful even before it is embedded within an optimization algorithm. We examine the acoustic energy balance during the optimization process and identify the physical mechanisms through which the algorithm is stabilizing the combustor. The algorithm works by, for each mode, reducing the work done by the flame while simultaneously increasing the work done by the system on the outlet boundary. We find that only small geometry changes are required in order to stabilize every mode. The network model used in this study deliberately has the same structure as one used in the gas turbine industry in order to ease its implementation in practice.