Published online by Cambridge University Press: 04 July 2014
In this theoretical and numerical paper, we derive the adjoint equations for a thermo-acoustic system consisting of an infinite-rate chemistry diffusion flame coupled with duct acoustics. We then calculate the thermo-acoustic system’s linear global modes (i.e. the frequency/growth rate of oscillations, together with their mode shapes), and the global modes’ receptivity to species injection, sensitivity to base-state perturbations and structural sensitivity to advective-velocity perturbations. Some of these could be found by finite difference calculations but the adjoint analysis is computationally much cheaper. We then compare these with the Rayleigh index. The receptivity analysis shows the regions of the flame where open-loop injection of fuel or oxidizer will have the greatest influence on the thermo-acoustic oscillation. We find that the flame is most receptive at its tip. The base-state sensitivity analysis shows the influence of each parameter on the frequency/growth rate. We find that perturbations to the stoichiometric mixture fraction, the fuel slot width and the heat-release parameter have most influence, while perturbations to the Péclet number have the least influence for most of the operating points considered. These sensitivities oscillate, e.g. positive perturbations to the fuel slot width either stabilizes or destabilizes the system, depending on the operating point. This analysis reveals that, as expected from a simple model, the phase delay between velocity and heat-release fluctuations is the key parameter in determining the sensitivities. It also reveals that this thermo-acoustic system is exceedingly sensitive to changes in the base state. The structural-sensitivity analysis shows the influence of perturbations to the advective flame velocity. The regions of highest sensitivity are around the stoichiometric line close to the inlet, showing where velocity models need to be most accurate. This analysis can be extended to more accurate models and is a promising new tool for the analysis and control of thermo-acoustic oscillations.
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