The influence of tangential mean flow on the Rayleigh conductivity of an aperture

An investigation is made of the influence of grazing mean flow on the Rayleigh conductivity of an aperture in a thin rigid plane. The fluid is assumed to be incompressible and inviscid, but the Kutta condition is applied to permit the generation of vorticity at the edge of the aperture by an applied time-varying pressure field. Numerical results are given for a circular aperture in the two cases of (i) one-sided mean flow, when the aperture is spanned by a plane vortex sheet in the undisturbed state, and (ii) two-sided mean flow, when the mean velocity is the same on both sides of the plane, so that the undisturbed motion is irrotational. In both cases there exist frequency ranges within which perturbation energy is either absorbed or generated by the mean flow. The numerical. results are supplemented by an approximate analytical treatment of the same problem for a rectangular aperture of large aspect ratio (with its long edge transverse to the mean flow direction). The aperture flux for one-sided flow is shown to be absolutely unstable, and may in principle be triggered by an arbitrary, small disturbance in the mean stream. For two-sided flow the motion is conditionally unstable, in the sense that perturbations are amplified by the extraction of energy from the mean flow only when the frequency of the applied pressure lies in certain discrete bands.