Resonant acoustic frequencies of a tandem cascade. Part 2. Rotating blade rows

Abstract

In Part 1 (Woodley & Peake 1999) we described a method for predicting the occurrence of resonant states in a system comprising twin cascades in zero relative motion. We now demonstrate how that work can be extended to account for the case of more practical interest, in which the upstream cascade (rotor) is rotating in the transverse direction relative to the downstream cascade (stator). Time periodicity now forces the temporal frequency of any disturbance to be an integer multiple of the rotor passing frequency in the stator frame, and vice versa, and this leads to the requirement to sum over a discrete set of temporal modes, as well as over the spatial modes already described in Part 1. The mechanisms by which temporal and spatial modes are scattered by the blade rows is made clear by the analytical approach adopted here; the scattering of the incident pressure (and, for the stator, vorticity) fields by each row in its own frame is completed using results similar to those presented in Part 1, and the fields in the two frames then matched across the inter-row gap to provide a single matrix equation. Specimen results for the conditioning of this equation are given, and although it seems more difficult to obtain very strong excitation than it was for zero rotation, the significance of Parker resonance of the stator is again apparent.