Assessment of the accuracy of representing a helical vortex by straight segments

This paper considers the accuracy of representing a helical vortex, as found in the wakes of helicopters, wind turbines, and propellers, by a sequence of straight segments. The accuracy is assessed by comparison with recent results for the induced velocity of a helix of constant pitch and radius. This comparison is motivated by the small values of the vortex pitch behind wind turbines and hovering rotors; small pitch leads to errors associated with the proximity of subsequent turns of the helix to the control point at which the velocity is required. Three cases are considered. The first, the velocity on the helix axis, has an analytic solution which is used to demonstrate that the general accuracy of the straight segment approximation is second order, as has been found in previous comparisons with the velocity field of a vortex ring. For the second case, where the control point has the same radius as the vortex, the segments aligned with the control point are mainly responsible for the error. The error varies from first to third order as the number of segments per revolution of the helix is increased. Thirdly, the self-induced velocity is determined to within an accuracy comparable with the effects of the vortex structure, of which little is presently known in general. The effects of vortex curvature are not significant and easily dealt with.