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

In Part 1 (Woodley & Peake 1999) we described a method for predicting the o ccurrence of resonant states in a system comprising twin cascades in zero r elative motion. We now demonstrate how that work can be extended to account for the case of more practical interest, in which the upstream cascade (ro tor) is rotating in the transverse direction relative to the downstream cas cade (stator). Time periodicity now forces the temporal frequency of any di sturbance to be an integer multiple of the rotor passing frequency in the s tator 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 ar e scattered by the blade rows is made clear by the analytical approach adop ted here; the scattering of the incident pressure (and, for the stator, vor ticity) fields by each row in its own frame is completed using results simi lar to those presented in Part 1, and the fields in the two frames then mat ched across the inter-row gap to provide a single matrix equation. Specimen results for the conditioning of this equation are given, and although it s eems more difficult to obtain very strong excitation than it was for zero r otation, the significance of Parker resonance of the stator is again appare nt.