On the motion of multiple helical vortices

The analysis of the self-induced velocity of a single helical vortex (Boers ma & Wood 1999) is extended to include equally spaced multiple vortices. Th is arrangement approximates the tip vortices in the far wake of multi-blade d wind turbines, propellers, or rotors in ascending, descending, or hoverin g flight. The problem is reduced to finding, from the Biot-Savart law, the additional velocity of a helix due to an identical helix displaced azimutha lly. The resulting Biot-Savart integral is further reduced to a Mellin-Barn es integral representation which allows the asymptotic expansions to be det ermined for small and for large pitch. The Biot-Savart integral is also eva luated numerically for a total of two, three and four vortices over a range of pitch values. The previous finding that the self-induced velocity at sm all pitch is dominated by a term inversely proportional to the pitch carrie s over to multiple vortices. It is shown that a far wake dominated by helic al tip vortices is consistent with the one-dimensional representation that leads to the Betz limit on the power output of wind turbines. The small-pit ch approximation then allows the determination of the blade's bound vortici ty for optimum power extraction. The present analysis is shown to give reas onable estimates for the vortex circulation in experiments using a single h overing rotor and a four-bladed propeller.