Acoustic Radiation from a Semi-infinite Annular Duct in a Uniform Subsonic Mean Flow

By using a Wiener-Hopf approach, an analytical description is derived of the scattered field of a harmonic sound wave coming out of an open ended annular duct (a semi-infinite cylinder inside of which, coaxially, is a doubly infinite hub), submerged in a subsonic, coaxial, uniform mean flow. The possibility of vortex shedding from the pipe exit edge is included. Explicit expressions are given for the acoustic power inside the pipe, in the acoustic far field and, in the presence of vortex shedding, in the hydrodynamic far field, and of the power absorbed by the vortex sheet. The formulae are evaluated numerically with the aid of asymptotic expansions, and a method in which complex contour deformation is used, which is more convenient than those usually employed for this type of diffraction problem. The equality of power appeared to be an important check on the calculations. A numerical survey is made of the behaviour of the acoustic power loss, due to vortex shedding from the trailing edge, at frequencies near cut-off, as a function of Mach number, mode number of the incident wave, and hub radius. The power loss appears to increase with increasing Mach number, increasing hub radius and with decreasing frequency. Only in the case of the plane wave (where k + 0) does the ratio of radiated and transmitted power become zero; for the other modes (at their cut-off frequencies) this ratio tends to a finite value. Son~ewhat surprising is that, in comparison with the jet, the power loss in a uniform flow is much higher. As a typical example for higher frequencies, the far field radiation pattern of a k = 50, m = 4 wave is considered as a function of the Kutta condition and hub radius.