Wave scattering by an infinite cascade of non-overlapping blades

We consider the scattering of waves by an infinite three-dimensional cascade of finite-length flat blades in subsonic flow at zero angle of attack. This geometry is of specific relevance as it provides a model for the components in turbofan engines. We study the scattering problem analytically, considering both acoustical and vortical incident fields, spanwise wavenumbers and transverse mean flow. Most importantly we extend previous work by lifting the restriction that adjacent blades overlap, a condition that had thus far been crucial for the analytical study of this problem. Our method of solution relies on the solution of three coupled boundary value problems using the Wiener-Hopf technique, corresponding to an uncoupled leading-edge approximation, and a subsequent trailing-edge and leading-edge correction. We provide exact expressions for observables in the system, depending only on the solution of a linear matrix equation. Specifically we find closed-form expressions for the far-field behaviour of the scattered potential upstream and downstream of the cascade, the upstream and downstream sound power, as well as the total unsteady lift on each blade in the cascade. A wide range of results are presented, and we see that the non-overlapping cascade is, as might be expected, typically more transparent to incident disturbances than the previously studied overlapping case.