Theory of Blade Design for Large Deflections: Part II—Annular Cascades

A method of designing highly loaded blades to give a specified distribution of swirl is presented. The method is based on a newly developed, three-dimensional analysis. In the present application, the flow is assumed to be incompressible and inviscid (the annulus has constant hub and tip radii), and the blades are of negligible thickness. A simple free vortex swirl schedule is assumed. The flow velocity is divided into circumferentially averaged and periodic terms. The Clebsch formulation for the periodic velocities is used, and the singularities are represented by periodic generalized functions so that solutions may be obtained in terms of eigenfunctions. The blade profile is determined iteratively from the blade boundary condition. Results from the computer program show how blade number, aspect, and hub-tip ratios affect the blade shape. The blade profiles for a given swirl schedule depend not only on the aspect ratio but also on the stacking position (i.e., the chordwise location at which this thin blade profile is radial), and so too do the mean axial and radial velocities. These effects occur whether the number of blades is large or small, and we conclude that even in incompressible flow the blade element or strip theory is not generally satisfactory for the design of high-deflection blades. The analysis derives the geometrical conditions for the blade profiles on the walls of the annulus which are needed to satisfy the wall boundary conditions in the idealized flow, but which in any practical example will be modified by the presence of wall boundary layers and blade thickness. In the limit when the number of blades approaches infinity, a bladed actuator duct solution is obtained. The conditions for the blade profile at the walls are absent, but the stacking position and aspect ratio still affect the axial and radial velocity distributions for the same swirl schedule.