Reduced-Order Modeling of Unsteady Viscous Flow in a Compressor Cascade

A simultaneously coupled viscous-inviscid 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 flow to obtain a set of linear equations that describe the unsteady small-disturbance behavior of the viscous flow 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