A simultaneously coupled viscous-inviscid interaction (VII) analysis is used to model the unsteady viscous separated ow through a subsonic compressor. The inner viscous ow around the airfoil and in the wake is modeled using a nite difference discretization of the boundary-layer equations and a one-equation turbulence transport model. The outer inviscid ow is modeled using a variational nite element discretization of the compressible full potential equation. The viscous and inviscid regions are simultaneously coupled using a injection type boundary condition along the airfoil and wake. The resulting nonlinear unsteady equations are linearized about the nonlinear steady ow to obtain a set of linear equations that describe the unsteady small-disturbance behavior of the viscous ow through the cascade. The discretized small-disturbance VII equations are used to form a generalized, quadratic, non-Hermitian eigenvalue problem that describes the eigenmodes (natural modes) and eigenvalues (natural frequencies) of uid motion about the cascade. Using a Lanczos algorithm, the eigeninformation is computed ef ciently for various steady ow in ow angles and unsteady interblade phase angles. The eigenvalues and eigenmodes are then used in conjunction with a classical mode summation technique to construct computationally ef cient reduced-order models of the unsteady ow through the cascade. Using just a few eigenmodes, less than 0.01% of the total number, the unsteady aerodynamic loads acting on vibrating airfoils (the aeroelastic stability problem) can be ef ciently and accurately computed over a relatively wide range of reduced frequencies provided that one or more static corrections are performed. Finally, the eigenvalues and eigenvectors provide physical insight into the unsteady aerodynamic behavior of the cascade. For example, we show the ability of the present eigenanalysis to predict purely uid mechanic instabilities such as rotating stall.
