The unsteady lift on a swept blade tip

Abstract

Highly swept blades are now commonly used in modern aeroengines, and in this paper we solve a model problem of relevance to the understanding and prediction of noise generation by the interaction between incident vortical disturbances and such a blade. In order to include the potentially significant effects of the blade-tip region, we consider a (semi-infinite span) quarter-plane aligned at a non-zero sweep angle to supersonic mean flow, with a single harmonic gust incident on the quarterplane from upstream. The solution is completed using a novel application of the Wiener–Hopf technique, in which the usual factorisation is carried out with respect to two independent complex variables separately, and closed-form expressions for the practically important lift per unit span are derived for both the subsonic and the supersonic leading-edge regimes. The dependence of the unsteady response on the sweep angle and the gust wavenumbers is examined, and in particular we demonstrate that the magnitude of the effects of the tip region is significantly reduced by increasing the blade sweep or by considering gusts of higher frequency. It also becomes clear that the magnitude of the unsteady response can be either decreased or increased by sweeping the blade, in a way which proves highly dependent on the particular values of the flow parameters. In addition, we consider the two critical values of the gust trace speed along the leading edge which correspond to the sonic velocities in the two spanwise directions, and for which the chordwise oscillation of the unsteady lift distribution on an infinite-span blade vanishes. In these two cases, the lift per unit span (integrated over the infinite chord) is clearly singular, but we demonstrate that the effect of including the blade tip is to smooth out just one of these singularities, and replace it instead by a relatively large, but finite, value.