On sound generation by the interaction between turbulence and a cascade of airfoils with non-uniform mean flow

The sound generated by the interaction between a turbulent rotor wake and a stator is modelled by considering the gust response of a cascade of blades in non-uniform, subsonic mean flow. Previous work by Hanson & Horan (1998) that considers a cascade of flat plates at zero incidence is extended to take into account blade geometry and angle of attack. Our approach is based on the work of Peake & Kerschen (1997), who calculate the forward radiation due to the interaction between a single vortical gust and a cascade of flat plates at non-zero angle of attack. The extensions completed in this present paper are two-fold: first we include the effects of small but non-zero camber and thickness; and second we produce uniformly valid approximations which predict the upstream radiation near modal cut-off. The thin-airfoil singularity in the steady flow at each leading edge is crucial in our model of the sound generation. A new analytical expression for the coefficient of this singularity is derived via a sequence of conformal mappings, and it turns out that in our asymptotic limit this is the only quantity which needs to be calculated from the steady flow in order to predict time-averaged noise levels. Once the response to a single gust has been completed, we use Hanson & Horan (1998)'s approach to determine the response to an incident turbulent spectrum, and find that as well as the noise corresponding to the auto-correlation of the gust velocity component normal to the blade, there is also a contribution from the cross-correlation of the normal and tangential velocities. Predictions are made of the effects of blade geometry on the upstream acoustic power level. The blade geometry can have a very significant effect on the noise generated by interaction with a single gust, with changes of up to 10dB from the flat-plate noise levels. However, once these gust results have been integrated over a full incident turbulence spectrum the effects of the geometry are rather smaller, although still potentially significant, leading to changes of up to about 2dB from the flat-plate results. The implication of all this is that the blade geometry can have a significant effect on the tonal noise components generated by rotor-stator interaction (i.e. by single harmonic gusts), but that the broadband part of the noise spectrum is relatively unaffected.