The acoustics of cooled Helmholtz resonators

Helmholtz resonators (HRs) are commonly used to damp thermoacous- tic oscillations in aero-engine and gas turbine combustors. In practice the HR is often maintained at a cooler temperature than the combustor, but this tends to be neglected when modelling. In this work, we derive an analytical model for both the acoustic damping and the effect on the com- bustor thermoacoustics of HRs whose volume is at a different temperature to the combustor. The energy conservation equation based on stagnation enthalpy, together with the mass and momentum conservation equations, is incorporated into a linear HR sound absorption model. These are coupled with linear wave-based models for the plane acoustic waves in the system to which the HR is attached – here a combustor. The temperature differ- ence is found to generate an entropy wave which advects downstream of the resonator (and may generate further acoustic waves if accelerated), and to change the acoustic damping near the resonant frequency. This model for the HR acoustics is then incorporated into a low-order network model for combustor thermoacoustics. The importance of capturing the effect of the HR temperature difference on the overall thermoacoustic modes is clearly demonstrated