Vortex and Momentum Theories for Hovering Rotors

Abstract

ILLER1 has called attention to a seeming paradox in the vortex theory for hovering rotors. In the usual application of vortex theory for the limiting case of an infinite number of rotor blades to propellers, the induced velocities in the wake are small compared to the forward velocity and the distributed vortices in the wake may be assumed to move downstream at the propeller forward velocity for lightly loaded propellers. Theodorsen showed,2 however, that in the application of the Goldstein theory for optimum circulation distribution on propellers with a finite number of blades, it is necessary, for more heavily loaded propellers representative of practical designs, to take into account the induced velocities in arriving at vortex geometry in the wake. Theodorsen also gave an approximate method for computing slipstream contraction, but for the cases of propellers in cruise flight which he considered, the slipstream contraction was of the order of 1%; the analysis specifically excluded static thrust conditions. For propeller cruise operating conditions it is well known3'4 that the results of vortex theory for an infinite number of blades are essentially identical to the momentum theory of the actuator disk; this remains true for the case of nonconstant blade circulation if both theories are applied to differential annular strips on the blades. It is also true when induced wake rotation is introduced into both theories. A particular result of practical significance in the theories (neglecting wake rotation) is that the induced axial velocity at the propeller vl is one-half the velocity in the ultimate wake, v2. In the propeller case, with negligible slipstream contraction, this follows from the fact that the vortices are assumed to be uniformly distributed over the circumference of a cylinder of approximately constant diameter that has a semi-infinite length viewed from the plane of the propeller and an infinite length viewed from a transverse plane in the ultimate wake, giving rise to a factor of two in the calculated induced velocities. It should be noted, however, that in the momentum theory of the actuator disk it is assumed that the flow velocity in the wake is constant for constant disk loading. In the vortex theory, this is a conclusion arrived at from analyses of the induced flowfield in the vortex cylinder. (Yet another approach is to represent the acutator disk by a doublet layer; this gives results equivalent to vortex theory.) With large slipstream contractions, such as occur for the static thrust condition of a propeller or for a hovering helicopter rotor, it is no longer valid to make this particular calculation since the wake vortices cannot be considered to lie on cylinders of constant diameter in the vicinity of the propeller plane. Yet, even in this case, vortex theory for an