Asymptotic solutions for unsteady flow in cascades

Asymptotic methods are presented for obtaining approximate solutions to unsteady flows in oscillating cascades. The formulation is in the framework of linearized potential flow, and the problems considered include low-frequency and low-solidity expansions for subsonic and supersonic cascades. For the supersonic cascade, simple formulas are obtained for the unsteady lift and moment, valid to first order in a frequency parameter. It is shown that terms of successive orders can be obtained by solving a sequence of quasi-static problems, with the effective upwash at each step modified by the lower-order solutions. The approach is in the spirit of matched asymptotic expansions, and different expansions based on different limit processes are sought for the subresonant and superresonant regions. For cascades in subsonic axial flow, the acoustic resonance phenomenon leads to a nonuniformity with respect to the interblade phase angle. The location of the singularity can be moved by suitably redefining the limit process, permitting uniformly valid expansions to be obtained separately for the subresonant and superresonant regions. Numerical comparisons with the full unsteady solution indicate that the present approximations are remarkably accurate in the range of reduced frequencies of interest in aeroelastic analyses.