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

Abstract

The method of matched asymptotic expansions is used to describe the sound generated by the interaction between a short-wavelength gust (reduced frequency k, with k [Gt ] 1) and an airfoil with small but non-zero thickness, camber and angle of attack (which are all assumed to be of typical size O(δ), with δ [Lt ] 1) in transonic flow. The mean-flow Mach number is taken to differ from unity by O(δ2/3), so that the steady flow past the airfoil is determined using the transonic small-disturbance equation. The unsteady analysis is based on a linearization of the Euler equations about the mean flow. High-frequency incident vortical and entropic disturbances are considered, and analogous to the subsonic counterpart of this problem, asymptotic 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 which kδ2/3 = 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 (δ = 0), together with all higher-order terms which are at most O(δ2/3) smaller – in fact, this involves no fewer than ten terms, due to the slowly-decaying form of the power series expansion of the steady flow about the leading edge. Particular to transonic flow is the locally subsonic or supersonic region that accounts for the transition between the acoustic field downstream of a source and the possible acoustic field upstream of the source. In the outer region the sound propagation has a geometric-acoustics form and the primary influence of the mean-flow distortion appears in our preferred limit as an O(1) phase term, which is particularly significant in view of the complicated interference between leading- and trailing-edge 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.