Three-dimensional theory for the steady loading on an annular blade row

Abstract

A lifting-surface analysis is presented for the steady, three-dimensional, subsonic flow through an annular blade row. A kernel-function procedure is used to solve the linearized integral equation that relates the unknown blade loading to a specified camber line. The unknown loading is expanded in a finite series of prescribed loading functions, that allows the required integrations to be performed analytically. Numerical results are presented for a range of hub/tip ratios, solidities, and camber line geometries. As expected, two-dimensional strip theory shows good agreement with the present results for high hub/tip ratio, at least near midannulus. However, near the hub and tip, and particularly for low hub/tip ratios, the error introduced by the use of twodimensional strip theory becomes significant. This is traced to the induced angle of attack generated by the trailing vortex wake, that is accounted for only in the three-dimensional theory. Comparisons are also made with the three-dimensional inverse (i.e., design) theory of Okurounmu and McCune. Their analysis yields the blade camber line required to produce a specified load distribution. By inputting their blade geometry to our program, we have been able to predict loadings in good agreement with those originally specified by them for two cases: one a free-vortex design, and another in which significant trailing vorticity is present.