An unsteady inviscid flow through a cascade of oscillating airfoils is investigated. An inviscid nonlinear subsonic and transonic model is used to compute the steady flow solution. Then a small amplitude motion of the airfoils about their steady flow configuration is considered. The unsteady flow is linearized about the nonlinear steady response based on the observation that in many practical cases the unsteadiness in the flow has a substantially smaller magnitude than the steady component. Several reduced-order modal models are constructed in the frequency domain using the proper orthogonal decomposition technique. The dependency of the required number of aerodynamic modes in a reduced-order model on the far-field upstream Mach number is investigated. It is shown that the transonic reduced-order models require a larger number of modes than the subsonic models for a similar geometry, range of reduced frequencies and interblade phase angles. The increased number of modes may be due to the increased Mach number per se, or the presence of the strong spatial gradients in the region of the shock. These two possible causes are investigated. Also, the geometry of the cascade is shown to influence strongly the shape of the aerodynamic modes, but only weakly the required dimension of the reduced-order models.
Bound Coherent Structures Propagating on the Free Surface of Deep Water