Method to Improve the Accuracy of Straight-Segment Representation of Helical Vortices

Abstract

The errors arising from the computational representation of a helical vortex by straight segments, particularly at the small values of pitch that are typical of the wakes of hovering rotors and wind turbines, are considered. For a helix of constant pitch and radius, as in the far wake, the main error in calculating the velocity at a control point of thesameradius arises from the pairs of “ aligned segments,” whose junction is in line with the control point. The aligned segments arethose closest to the control point; for each M segments per revolution of the helix, thereis one pair of aligned segments. They are particularly important at small pitch because of their proximity to the control point. An approximate analytic expression is derived for the velocity due to the section of the helix represented by the aligned segments. Its use is shown to improve signie cantly the accuracy of the induced velocity for the moderate values of M thatare used in many calculations. Fora vortex of changing radius, typical of thenear wake, the aligned segments are also crucial. Replacing them by an extension of the analytic approximation also leads to a signie cant improvement in the accuracy of the induced velocity at moderate values of M. Calculations of the expanding wake of a three-bladed wind turbine are used to compare the improvement in accuracy at moderate M to that obtained by increasing M without replacement. Replacement caused a much smaller increase in execution time per iteration. Nomenclature a = vortex core radius a0;a1;a2 = constants in Eq. (14) b0;b1;b2 = constants in Eq. (14) F1; F2; F3 = functions dee ned in the Appendix k = index of junction of aligned segments L