A viscous-inviscid model of unsteady small-disturbance flows in cascades
A simultaneously coupled viscous-inviscid (IVI) procedure for computing the unsteady flow associated with the onset of stall flutter in turbomachinery cascades is presented. Using this method, the flow is divided into two parts: a viscous flow near the airfoil and in the wake, and an inviscid flow in the rest of the flow field. The viscous flow is modeled using a finite difference discretization of the boundary layer equations. The inviscid flow is modeled using a variational finite element discretization of the full potential equation. The viscous and inviscid flow regions are then simultaneously coupled using a transpiration type boundary condition along the airfoil and wake. At the onset of stall flutter, the unsteadiness in the flow is small compared to the mean flow. Therefore, in the present analysis, the governing unsteady flow equations are linearized about the mean flow to obtain a set of unsteady small disturbance equations. The solution procedure is carried out in two steps. First, the nonlinear steady flow is computed using Newton iteration. Next, the resulting mean flow solution is used to form the coefficients of the small disturbance unsteady flow equations, and the resulting system of linear equations is solved using LU decomposition. The small-disturbance IVI method is used to compute the unsteady aerodynamic response of a typical compressor blade vibrating in pitch and plunge. The computed results demonstrate the destabilizing influence of the viscosity on the aeroelastic stability of airfoils in cascade.
