Uniform-inlet three-dimensional transonic Beltrami flow through a ducted fan

Abstract

Introduction: This report is a continuation of a line of analytical treatment of three dimensional flows in compressor or ducted fans. In the earlier theories [1,2,9,10], the blade row is modelled as a set of spinning lifting lines and the induced velocities are treated as acoustic perturbations. While these studies have been useful in advancing our knowledge of the three dimensional nature of the flow, it has been difficult until now to correlate the theory with experiments. The reason, of course, is that the strong overall swirl induced by such blade rows is by no means a small perturbation in practical applications. It is realized that the high pressure stage of a compressor usually has a large number of blades (~40 to 100). Therefore although the collective effect of all the bales can be a large disturbance, each blade contributes only a small disturbance. Thus a linearized theory is still possible.

To extend the previous acoustic theory to a rotor with large turning we may still represent the blades as superpositions of source and lifting lines, but we have to calculate the exit flow by linearizing about a non-zero (and large) swirl velocity profile. This is the approach taken by the theory proposed by McCune and Hawthorne [3]. They calculated the velocities induced by the trailing vorticity of a nonuniformly loaded rectilinear cascade for incompressible flow. This work was later generalized to the compressible case by Morton [4]. Cheng [5] treated incompressible flow in an annular geometry. In that analysis the blades are represented as lifting lines of nearly constant circulation, and the exit flow is therefore, to lowest order, of "free vortex" type. Linearizing about the free vortex flow, the velocity induced by the trailing vortex sheets due to nonuniform blade bonding are calculated in [5] to order 61.

It is the purpose of this report to treat the general compressible case, including the results of Ref. [5] as the incompressible limit. The result of this analysis also serves as the Green's function for constructing a lifting surface theory for transonic rotors with practical loading. Before going into the details of the theory, let us examine some simple pictures of its findings. The nonuniformly loaded blades shed off the excess circulation as wakes. The induced velocity of the wakes is found to cause a "downwash" at the blade which has the effect of partially nullifying the nonuniformity in loading. This can easily be understood by a consideration of the wake system of a blade. Fig. 1 shows a blade with the tip region more heavily loaded than the inboard stations. 'We can see that the wake induces a tangential velocity component which lowers the angle of attack at the high work (tip) region and increases that of the low work (hub) region.

Another major development is the mode matching of the upstream and downstream flow. The presence of the strong swirl makes the acoustic mode shapes downstream drastically different from those upstream. For example, we can have upstream hyperbolic modes in a transonic rotor while all the downstream modes are elliptic because the relative velocity is subsonic there. Simple mode-wise matching is no longer possible. In the present work, a method of mixed-mode matching is developed. A result is that a pure tone downstream (upstream) can excite a whole spectrum of tones upstream (downstream). In particular, any source downstream can excite the acoustic radiations upstream of a transonic rotor.