Joukowski’s wake model for tip-splitting rotors.

Abstract

The interaction between a tip vortex and a solid surface is responsible for premature structural component fatigue in wind turbines and undesirable noise in helicopter rotors during low speed and descending flight. One noise reduction strategy uses a modified airfoil to split and spread the vorticity in two tip vortices. The present paper aims to provide the wake structure produced by such a rotor for wind turbine and helicopter regimes. We use a filamentary approach, such that vortices are assumed to roll-up quickly to form thin vortex filaments of finite but small size and compute the induced velocity using a cut-o↵ method. The structure of the wake is analyzed in the near- and far-fields separately. It is found to have a dual nature and to be well-described by a twisted vortex pair locally aligned along with a larger helical structure. The linear stability of the far-wake with respect to long-wave displacements is also analyzed. Two kinds of instability modes are obtained associated with a pairing between successive turns of the large helical structure and a pairing between successive turns of the vortex pair.