The velocity field in the immediate vicinity of a curved vortex comprises a circulation around the vortex, a component due to the vortex curvature, and 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 for 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 values of the pitch of an infinite helical vortex. The remainder is obtained firstly 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 the Moore & Saffman (1972) formulation to obtain the remainder for a helical vortex with a finite circular core over which the circulation is distributed 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 & Okulov (1998) who obtained the remainders and their difference asymptotically for small and large pitch. An asymptotic analysis of the new closed-form remainders using Mellin transforms provides a complete representation by a residue series and reveals a minor correction to the asymptotic expression of Kuibin & Okulov (1998) for the remainder at small pitch.

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Euler considered sums of the form ∞ m=1 1 m s m−1 n=1 1 n t .Here natural generalizations of these sums namely [p, q] := [p, q](s, t) = ∞ m=1 χ p (m) m s m−1 n=1 χ q (n) n t ,are investigated, where χ p and χ q are characters, and s and t are positive integers. The cases when p and q are either 1, 2a, 2b or −4 are examined in detail, and closedform expressions are found for t = 1 and general s in terms of the Riemann zeta function and the Catalan zeta function-the Dirichlet series L −4 (s) = 1 −s − 3 −s + 5 −s − 7 −s + · · · . Some results for arbitrary p and q are obtained as well.In Memoriam: Between the submission and acceptance of this report we greatly regret that our esteemed colleague John Boersma passed away. This paper is dedicated to his memory.

The analysis of the self-induced velocity of a single helical vortex (Boersma & Wood 1999) is extended to include equally spaced multiple vortices. This arrangement approximates the tip vortices in the far wake of multi-bladed wind turbines, propellers, or rotors in ascending, descending, or hovering flight. The problem is reduced to finding, from the Biot–Savart law, the additional velocity of a helix due to an identical helix displaced azimuthally. The resulting Biot–Savart integral is further reduced to a Mellin–Barnes integral representation which allows the asymptotic expansions to be determined for small and for large pitch. The Biot–Savart integral is also evaluated 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 small pitch is dominated by a term inversely proportional to the pitch carries over to multiple vortices. It is shown that a far wake dominated by helical tip vortices is consistent with the one-dimensional representation that leads to the Betz limit on the power output of wind turbines. The small-pitch approximation then allows the determination of the blade&s bound vorticity for optimum power extraction. The present analysis is shown to give reasonable estimates for the vortex circulation in experiments using a single hovering rotor and a four-bladed propeller.

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