Radiation properties of the semi-infinite vortex sheet

Abstract

An exact calculation is given of the acoustic radiation from a time dependent flow coupled to an inhomogeneous solid surface. Specifically, the flow consists of a vortex sheet leaving a semi-infinite plate and undergoing a two-dimensional spatial Kelvin-Helmholtz instability. In the absence of the plate, such an instability mode of the vortex sheet generates no sound. In the presence of a rigid plate, it is found that the intensity-directivity law is $I\sim U^{4}$sin$^{2}\frac{1}{2}\theta $, with $\theta $ measured from the downstream direction. If the plate is compliant and fluid loading effects high, the radiation is weaker, with $I\sim U^{5}$sin$^{2}\theta $. These results agree completely with those predicted from general theories of the scattering of the near-field of point quadrupoles by large wedge-shaped surfaces (Ffowcs Williams & Hall 1970; Crighton & Leppington 1970, 1971). Imposition of the 'rectified' Kutta condition of Orszag & Crow (1970) does not modify the sound field. Application of the 'full' Kutta condition, that the sheet leaves the plate at zero gradient, results in an enormous increase in the radiation, with $I\sim U^{2}$cosec$^{2}\frac{1}{2}\theta $.