The Design of Shock-Free Compressor Cascades Including Viscous Boundary-Layer Effects

Abstract

A numerical procedure has been developed for the design of shock-free transonic compressor cascades with an allowance for viscous effects, providing that the boundary layer is fully attached over the blade. The method described combines, in an iterative process, a modified inviscid hodograph-based inverse-design algorithm (CIDA), originally developed by the author for the design of shock-free airfoils, and the inverse boundary-layer algorithm (LTBLCEQL) of Miner et al. [22]. In the numerical procedure, the inviscid subsonic and supersonic regions of the flow are decoupled allowing the solution of either an elliptic or hyperbolic-type partial differential equation for the full stream function. For the subcritical portion of the flow, the inviscid calculation is performed in a computational plane which is obtained through a sequence of conformal and numerical transformations of the two-sheeted hodograph plane. For the supercritical portion, a characteristic calculation is carried out in the hodograph plane. The results are then mapped back to the physical plane to determine the inviscid blade configuration. Viscous effects are then incorporated via the boundary-layer displacement surface concept. The boundary-layer algorithm incorporates a two-layer eddy viscosity turbulence model and allows for gradual, rather than instantaneous, transition to turbulence. Two examples of shock-free compressor blades are given to demonstrate the capabilities of the numerical coupling procedure.