We investigate the mechanisms leading to acoustic whistling for a jet passing through a circular hole in a thick plate connecting two domains. Two generic situations are considered. In the first one, the upstream domain is a closed cavity while the downstream domain is open, leading to a class of conditionally unstable modes. In this case, the instability source lies in the recirculation region within the thickness of the plate, but coupling with a conveniently tuned resonator is needed to select the conditional instability range. In the second situation, the two regions, upstream and downstream of the hole, are considered as open, leading to a class of hydrodynamic modes where instability of the recirculation region is sufficient to generate self-oscillations without the need of any resonator. A matched asymptotic model, valid in the low Mach limit, is used to derive a global impedance of the system, combining the impedance of the hole and the modelled impedances of the upstream and downstream domains. It is shown that the knowledge of this global impedance along the real ω-axis provides an instability criterion and a prediction of the eigenvalues of the full system. Validations against the solutions of the eigenvalue problem obtained from the linearized fully compressible formulation confirm the accuracy of the approach. Then, it is subsequently used to characterise the range of existence of instabilities as a function of the Reynolds number, the Mach number, the aspect ratio of the hole and (for the cavity configuration) the dimensionless volume of the cavity.
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