The compact Green’s function for a semi-infinite elastic plate, with application to trailing edge noise and blade–vortex interaction noise

Abstract

Analytic representations are derived for the acoustic Green’s function governing the production of aerodynamic sound by sources near the edge of a thin, semi-infinite elastic plate. Simple closed-form expressions are obtained for the compact Green’s function, which is applicable when the length scale of the source flow near the edge is much smaller than both the acoustic and bending wavelengths. This requires that the characteristic flow Mach number M and frequency f satisfy M2≪f/fc≪1, where fc is the coincidence frequency, above which the bending wave speed in vacuo exceeds the speed of sound in the fluid. At such frequencies the flexural response of the plate can significantly affect the level of sound generation by sources in the neighborhood of the edge. This is illustrated by applications to sound production by (i) turbulent boundary layer flow over an elastic trailing edge (which includes a comparison with more complete predictions applicable for arbitrary values of f/fc≲1), and (ii) the interaction of a rectilinear vortex with the leading and trailing edges of a large plate. Comparisons with analogous predictions for a rigid edge indicate that surface compliance can result in a significant reduction in the intensity of edge-generated sound.