Numerical study of the hole-tone feedback cycle based on an axisymmetric formulation

A method for simulating the hole-tone feedback cycle (Rayleigh's bird-call), based on an axisymmetric discrete vortex method, is described. Evaluation of the sound generated by self-sustained flow oscillations is based on the Powell-Howe theory of vortex sound combined with a thin-plate boundary element theory, to account for scattering from the end plate. A model for acoustic feedback is developed. Several numerical examples are presented and discussed. These examples give an understanding of the relative strengths of contributions from monopole, dipole and quadrupole source terms. It is found that the contribution from monopole sources is largest, followed by the dipole contribution. The quadrupole contribution is quite small in comparison. The effect of acoustic feedback is carefully studied. The acoustic feedback velocity components are too small to alter the fundamental hole-tone dynamics. Still, their effect can be seen on the sound pressure frequency spectra. It is found that they do not alter the peak at the fundamental hole-tone frequency f ₀, but reinforce the peaks at higher harmonics 2f₀, 3f₀, ⋯ , and certain combination frequency peaks.