Application of global stability approaches to whistling jets and wind instruments

We discuss the application of the so-called global approaches, arising from the field of hydrodynamical instabilities, to aeroacoustic resonators. We illustrate the potential of the approach for the case of a jet passing through two successive holes (”hole-tone” configuration) which is relevant to human whistling, birdcalls and tea kettles. First, treating the hydrodynamic system as locally incompressible and linearized around a base flow, we compute the conductance of the double aperture and show that this one can provide positive energy feedback to an external system. Secondly, introducing the coupling with an acoustical resonator through convenient impedances imposed as boundary conditions and solving an eigenvalue problem, we show that the full system is e ectively linearly unstable and able to support self-sustained oscillations. The results compare favorably with recent experiments, and the analysis yields novel insight into the nature of the feedback mechanism responsible for the whistling. The application to the related situation of a corrugated pipe, and to more realistic instruments such as ocarinas and free reeds, will also be discussed.