Noise generation by high-frequency gusts interacting with an airfoil in transonic flow

The method of matched asymptotic expansions is used to describe the sound g enerated by the interaction between a short-wavelength gust (reduced freque ncy ii, with k >> 1) and an airfoil with small but non-zero thickness, camb er and angle of attack (which are all assumed to be of typical size O(delta ), with delta << 1) in transonic how. The mean-flow Mach number is taken to differ from unity by O(delta(2/3)), so that the steady flow past the airfo il is determined using the transonic small-disturbance equation. The unstea dy analysis is based on a linearization of the Euler equations about the me an flow. High-frequency incident vortical and entropic disturbances are con sidered, and analogous to the subsonic counterpart of this problem, asympto tic regions around the airfoil highlight the mechanisms that produce sound. Notably, the inner region round the leading edge is of size O(k(-1)), and describes the interaction between the mean-flow gradients and the incident gust and the resulting acoustic waves. We consider the preferred limit in w hich k delta(2/3) = O(1), and calculate the first two terms in the phase of the far-field radiation, while for the directivity we determine the first term (delta = 0), together with all higher-order terms which are at most O( delta(2/3)) Smaller - in fact, this involves no fewer than ten terms, due t o the slowly-decaying form of the power series expansion of the steady flow about the leading edge. Particular to transonic how is the locally subsoni c or supersonic region that accounts for the transition between the acousti c field downstream of a source and the possible acoustic field upstream of the source. In the outer region the sound propagation has a geometric-acous tics form and the primary influence of the mean-flow distortion appears in our preferred limit as an O(1) phase term, which is particularly significan t in view of the complicated interference between leading- and trailing-edg e fields. It is argued that weak mean-flow shocks have an influence on the sound generation that is small relative to the effects of the leading-edge singularity.