Finite Chord Research Articles

The Flow of a Perfect Fluid Through an Axial Turbomachine with Prescribed Blade Loading

The theory of the three-dimensional flow through an axial turbomachine, associated with variation of circulation along the blade length, is described as an extension of the classical theory of finite wings and is simplified to a problem in axially symmetric rotational fluid motion by considering an infinite number of blades in each row. The problem is linearized by considering the vorticity generated by the blades to be transported by the mean velocity. The linearized problem leads to well-known partial differential equations and is solved for the radial, tangential, and axial velocity components induced by a single row of stationary or rotating blades with finite chord and prescribed loading. The particular case for which the blade chord approaches zero and the tangential velocity changes discontinuously is associated with the theory of the Prandtl lifting line for finite wings. Because the problem has been linearized, the solution for a multistage turbomachine follows by superposition of appropriate single rows. Analytical expressions and graphical values for the velocity components are given for a single stationary or rotating blade row of given loading with a hub/ t ip ratio of 0.6 and a blade aspect ratio of 2. The corresponding discontinuous approximation is compared with the more nearly exact solution and is shown to constitute a useful approximation to the solution for a finite blade chord when the discontinuity is located appropriately. An exponential approximation for the velocity components, deduced from the analysis, allows rapid estimation of the rate at which the equilibrium velocity profiles develop ahead of, and behind, a blade row and, using the superposition principle, provides a simple means of approximating the velocity distribution in a multistage turbomachine and of discussing mutual interference of blade rows.