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 compressor. The inner viscous flow around the airfoil and in the wake is modeled using a finite difference discretization of the boundary-layer equations and a one-equation turbulence transport model. The outer inviscid flow is modeled using a variational finite 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 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 (natural modes) and eigenvalues (natural frequencies) of fluid motion about the cascade. Using a Lanczos algorithm, the eigeninformation is computed efficiently for various steady flow inflow angles and unsteady interblade phase angles. The eigenvalues and eigenmodes are then used in conjunction with a classical made summation technique to construct computationally 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 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 fluid mechanic instabilities such as rotating stall.