Influence of mean loading on noise generated by the interaction of gusts with a cascade: downstream radiation

Abstract

We consider the effects of blade mean loading on the noise generated by the interaction between convected vorticity and a blade row. The blades are treated as flat plates aligned at a non-zero incidence angle, $\delta$, to the oncoming stream, and we take harmonic components of the incident vorticity field with reduced frequency $k$, and use asymptotic analysis in the realistic limit $k\,{\gg}\, 1$, $\delta \,{\ll}\, 1$ with $k\delta=O(1)$. In a previous paper (Peake & Kerschen, J. Fluid Mech., vol. 347 (1997), pp. 315–346) we have analysed the sound radiated back upstream, but the field in the blade passages and the sound radiated downstream are also of considerable practical interest, and are considered in this paper. The flow is seen to consist of inner regions around each leading edge, in which sound is generated by the local gust–airfoil and gust–flow interactions, and an outer region in which the incident gust and the acoustic radiation interact with the non-uniform mean flow and the other blades. It is shown that the complicated multiple interactions between the blades can be represented by images in potential–streamfunction space, yielding closed-form expressions for the phase distortion experienced by sound waves propagating down the blade passages. The acoustic radiation downstream of the cascade at $O(1)$ distances is dominated by the duct-mode beams that emanate from the passages, while the far downstream field is generated by the diffraction of the duct modes by the trailing edges. The modal amplitudes of the radiation field far downstream tend to be largest when the mode direction is close to the propagation direction of the duct mode which generated it, corresponding to the way (in uniform flow) in which the radiation from a single blade passage tends to be beamed in the duct-mode directions. Although the diffraction coefficient for the scattering from a single trailing edge is singular in these directions, we show how uniformly valid expressions can be derived by combining the trailing-edge fields in an appropriate way, thereby describing the larger amplitude in the beam directions. The steady non-uniform flow downstream has the effect of tilting the directions of the beams by $O(\delta)$ angles away from the duct-mode directions, which are explicitly determined. Throughout this analysis it will be seen that the interaction with the non-uniform mean flow introduces phase corrections of size $O(k\delta)$, which, given the way in which interference effects between the multiple blades dominate unsteady cascade flow, proves to be highly significant.