A Coupled Mode Analysis of Unsteady Multistage Flows in Turbomachinery Free

A computational method is presented for predicting the unsteady aerodynamic response of a vibrating blade row which is part of a multistage turbomachine. Most current unsteady aerodynamic theories model a single blade row isolated in an infinitely long duct. This assumption neglects the potentially important influence of neighboring blade rows. The present ‘coupled mode’ analysis is an elegant and computationally efficient method for modelling neighboring blade row effects. Using this approach, the coupling between blade rows is modelled using a subset of the so-called spinning modes, i.e. pressure, vorticity, and entropy waves which propagate between the blade rows. The blade rows themselves are represented by reflection and transmission coefficients. These coefficients describe how spinning modes interact with, and are scattered by, a given blade row. The coefficients can be calculated using any standard isolated blade row model; here we use a linearized full potential flow model together with rapid distortion theory to account for incident vortical gusts. The isolated blade row reflection and transmission coefficients, inter-row coupling relationships, and appropriate boundary conditions are all assembled into a small sparse linear system of equations which describes the unsteady multistage flow. A number of numerical examples are presented to validate the method and to demonstrate the profound influence of neighboring blade rows on the aerodynamic damping of a cascade of vibrating airfoils.