Calculations of Three-Dimensional Transonic Compressor Flowfields by a Relaxation Method

Abstract

A finite-difference technique for describing transonic flow through an axial compressor blade row is described. The method is an adaptation, to turbomachinery flows, of relaxation techniques previously developed for external-aerodynamics studies of flows over wings and bodies. In the work presented here, the flow is treated in the nonlinear small-disturbance approximation, which applies to the case of thin, lightly loaded blades. The modifications of the external-flow techniques necessitated by the blade-row geometry and periodicity conditions are described. This problem formulation is cast in finite-difference form, and details of the relaxation method used to solve the difference equations are given. Results of sample calculations are presented which show interesting features of the three-dimensional interactions that occur between subsonic and supersonic flows at neighboring spanwise stations. Also presented in these results are the locations of shock waves, which decrease in strength and eventually vanish in the subsonic regions near the hub. These shock waves are generated by the artificial viscosity implicit in the finite-difference equations. A method for limiting their thickness to three or four grid points, even in a skew coordinate system, is presented. In addition, a shock-fitting method capable of representing the shock by a discontinuity is described, and results of a two-dimensional application of this method are shown. Recommendations are made for a number of improvements, including the incorporation of higher-order effects not present in the nonlinear small-disturbance approximation.