Phase prediction of the response of choked nozzles to entropy and acoustic disturbances

The development and transmission of sound through the exit of an aeroengine combustor is often investigated by modelling the complex geometry as a convergent-divergent nozzle. However, these analytical acoustic predictions are usually limited to the compact case, where the length of the nozzle is insignificant compared to the wavelength of the flow perturbations, or to cases where the variation of the mean velocity through the nozzle may be treated as linear, or piece-wise linear. Considering terms up to first order in frequency for the conservation of mass, momentum and energy, this paper investigates an alternative approach by deriving effective lengths for the passage of the flow perturbations through a supercritical convergent-divergent nozzle. The effects due to the presence of a normal shock wave are also studied using a linearised form of the Rankine-Hugoniot relations. The analyses lead to predictions for the phase and magnitude of the transmitted acoustic waves from finite-length nozzles, and are valid for low non-dimensional frequencies. It has been found that these predictions agree well with the numerical results from inviscid simulations.