The interaction between a steady jet flow and a supersonic blade tip

Abstract

The potentially high level of noise generated by modern counter-rotation propellers has attracted considerable interest and concern, and one of the most potent mechanisms involved is the unsteady interaction between the tip vortex shed from the tips of the forward blade row and the rear row. In this paper a model problem is considered, in which the tip vortex is represented by a jet of constant axial velocity, which is convected at right angles to itself by a uniform supersonic mean flow, and which is cut by a rigid airfoil with its chord aligned along the mean flow direction. Ffowcs Williams & Guo have previously considered this problem for an infinite-span airfoil and a circular jet; in this paper we extend their analysis to include the effects of the presence of the second-row blade tip on the interaction, by considering a semi-infinite-span airfoil. As a first attempt, the case of a highly compact jet, represented by a delta-function upwash on the airfoil, is considered, and both the total lift on the airfoil and the radiation are investigated. The presence of the airfoil corner and side edge is seen to cause the lift to decay in time from its infinite-span value towards zero, due to a spanwise motion round the side edge; whilst the radiation is shown to be composed of two Signals, the first received directly from the interaction between the jet and the leading edge, and the second resulting from the diffraction of sound waves emanating from the leading edge by the side edge. The effect of choosing a more diffuse upwash distribution is then considered, in which case it becomes clear that the first signal has a considerably larger amplitude, and shorter duration, than the second, diffracted signal.