REDUCED-ORDER MODELING OF UNSTEADY VISCOUS-FLOW IN A COMPRESSOR CASCADE

A simultaneously coupled viscous-viscid interaction (VII) analysis is used to model the unsteady viscous separated flow through a subsonic c ompressor. The inner viscous flow around the airfoil and in the wake i s modeled using a finite difference discretization of the boundary-lay er equations and a one-equation turbulence transport model. The outer inviscid flow is modeled using a variational finite element discretiza tion of the compressible full potential equation. The viscous and invi scid regions are simultaneously coupled using a injection type boundar y condition along the airfoil and wake. The resulting nonlinear unstea dy equations are linearized about the nonlinear steady Bow to obtain a set of linear equations that describe the unsteady small-disturbance behavior of the viscous Bow through the cascade. The discretized small -disturbance VII equations are used to form a generalized, quadratic, non-Hermitian eigenvalue problem that describes the eigenmodes (natura l modes) and eigenvalues (natural frequencies) of fluid motion about t he cascade. Using a Lanczos algorithm, the eigeninformation is compute d efficiently for various steady flow inflow angles and unsteady inter blade phase angles. The eigenvalues and eigenmodes are then used in co njunction with a classical made summation technique to construct compu tationally efficient reduced-order models of the unsteady Bow through the cascade. Using just a few eigenmodes, less than 0.01% of the total number, the unsteady aerodynamic loads acting on vibrating ah-foils ( the aeroelastic stability problem) can be efficiently and accurately c omputed over a relatively wide range of reduced frequencies provided t hat one or more static corrections are performed. Finally, the eigenva lues and eigenvectors provide physical insight into the unsteady aerod ynamic behavior of the cascade. For example, we show the ability of th e present eigenanalysis to predict purely fluid mechanic instabilities such as rotating stall.