Three-dimensional effects in cascade-gust interaction

Abstract

The interaction of convected vorticity and a blade row is an important area of study in theoretical aeroacoustics, and we adopt the usual approach of modelling the rotating blade row as an infinite linear cascade of staggered flat plates and taking a single harmonic gust component of the incident vorticity. In this paper we extend the two-dimensional “strip theory” used by previous authors to three dimensions by including the blade radial (spanwise) direction. We consider representative spanwise gust distributions, and to simplify the three-dimensional analysis we treat only gusts with a high reduced frequency, for which cascade trailing-edge effects can be neglected. The Wiener-Hopf technique is used to obtain a closed-form expression for the spanwise unsteady lift distribution, and the corresponding factorisation is carried out exactly in terms of infinite products, which are then evaluated numerically using Richardson extrapolation. For comparison, the unsteady lift is also calculated using the strip-theory approximation, and reasonable agreement with our new results is found for cases in which all the propagating duct and radiated modes are well cut-on (i.e. the axial wavenumber k 1 is real), and all the evanescent modes are well cut-off ( k 1 possesses a non-zero imaginary part). However, very significant differences occur when any one of the propagating modes is close to cut-off, or an evanescent mode is near cut-on. We therefore conclude that strip theory does not provide an adequate description of the unsteady cascade flow under such circumstances, and that three-dimensional effects must then be included as described in this paper.