On the use of a uniformly valid analytical cascade response function for fan broadband noise predictions

The present paper extends an existing analytical model of the aeroacoustic response of a rectilinear cascade of flat-plate blades to three-dimensional incident vortical gusts, to the prediction of the noise generated by a three-dimensional annular blade-row. The extended formulation is meant to be implemented in a fan broadband noise prediction tool. The intended applications include the modern turbofan engines, for which analytical modelling is believed to be a good alternative to more expensive numerical techniques. The prediction noise model resorts to a strip theory approach based on a three-dimensional rectilinear cascade model. The latter is based on the Wiener-Hopf technique, and yields the pressure field in the blade passage and the unsteady blade loading. The analytical pressure solution is derived by making an extensive use of the residue theorem. The obtained unsteady blade loading distribution over the blades is then used as a dipole source distribution in an acoustic analogy applied in the annular rigid duct with uniform mean flow. The new achievements are then tested on three-dimensional annular-benchmark configurations and compared with three-dimensional lifting-surface models and three-dimensional Euler linearized codes available in the literature. The accuracy of the model is shown for high hub-to-tip ratio cases. When used as such in a true rectilinear-cascade configuration, it also reproduces the exact radiated field that can be derived directly. For low hub-to-tip ratio configurations, the model departs from three-dimensional computations, both regarding the blade loading and the acoustic radiation. A correction is proposed to account for the actual annular dispersion relation in the rectilinear-cascade response function. The results suggest that the proposed correction is necessary to get closer to the underlying physics of the annular-space wave equation, but that it is yet not sufficient to fully reproduce three-dimensional results.