EDGE, CAVITY AND APERTURE TONES AT VERY-LOW MACH NUMBERS

This paper discusses self-sustaining oscillations of high-Reynolds-num ber shear layers and jets incident on edges and corners at infinitesim al Mach number. These oscillations are frequently sources of narrow-ba nd sound, and are usually attributed to the formation of discrete vort ices whose interactions with the edge or corner produce impulsive pres sures that lead to the formation of new vorticity and complete a feedb ack cycle of operation. Linearized analyses of these interactions are presented in which free shear layers are modelled by vortex sheets. De tailed results are given for shear hows over rectangular wall aperture s and shallow cavities, and for the classical jet-edge interaction. Th e operating stages of self-sustained oscillations are identified with poles in the upper half of the complex frequency plane of a certain im pulse response function. It is argued that the real parts of these pol es determine the Strouhal numbers of the operating stages observed exp erimentally for the real, nonlinear system. The response function coin cides with the Rayleigh conductivity of the 'window' spanned by the sh ear flow for wall apertures and jet-edge interactions, and to a freque ncy dependent drag coefficient for shallow wall cavities. When the int eraction occurs in the neighbourhood of an acoustic resonator, exempli fied by the flue organ pipe, the poles are augmented by a sequence of poles whose real parts are close to the resonance frequencies of the r esonator, and the resonator can 'speak' at one of these frequencies (b y extracting energy from the mean flow) provided the corresponding pol e has positive imaginary part. The Strouhal numbers predicted by this theory for a shallow wall cavity agree well with data extrapolated to zero Mach number from measurements in air, and predictions for the jet -edge interaction are in excellent accord with data from various sourc es in the literature. In the latter case, the linear theory also agree s for all operating stages with an empirical, nonlinear model that tak es account of the formation of discrete vortices in the jet.