On the self-induced motion of a helical vortex

The velocity field in the immediate vicinity of a curved vortex comprises a circulation around the vortex, a component due to the vortex curvature, an d a 'remainder' due to the more distant parts of the vortex. The first two components are relatively well understood but the remainder is known only f or a few specific vortex geometries, most notably, the vortex ring. In this paper we derive a closed form for the remainder that is valid for all valu es of the pitch of an infinite helical vortex. The remainder is obtained fi rstly from Hardin's (1982) solution for the flow induced by a helical line vortex (of zero thickness). We then use Ricca's (1994) implementation of th e Moore & Saffman (1972) formulation to obtain the remainder for a helical vortex with a finite circular core over which the circulation is distribute d uniformly. It is shown analytically that the two remainders differ by 1/4 for all values of the pitch. This generalizes the results of Kuibin & Okul ov (1998) who obtained the remainders and their difference asymptotically f or small and large pitch. An asymptotic analysis of the new closed-form rem ainders using Mellin transforms provides a complete representation by a res idue series and reveals a minor correction to the asymptotic expression of Kuibin & Okulov (1998) for the remainder at small pitch.