Self-induced motion and asymptotic expansion of the velocity field in the vicinity of a helical vortex filament

Abstract

A derivation of an expression for the self-induced velocity of a helical vortex filament with moderate and small pitch is presented for the cases of an infinite space as well as a space bounded by a cylindrical tube. This result supplements the known Kelvin’s formula for helix with large pitch in an infinite space.

The second curvature correction for the straight segment approximation of periodic vortex wakes

“…This again implies that U 2c U 1c . Since most rotor wakes are helices of some form, it is important to note that the equivalent inverse pitch term dominates U for a helical vortex of sufficiently small p (Kuibin and Okulov, 1998). There is a further reason to expect U ≈ U 2c U 1c for many turbine wakes: U 1c , but not U 2c , is associated with the impulse necessary to form a vortex ring.…”

The velocity field induced by a helical vortex tube

“…One of the prevailing methods is the cut-off method and its refinement, for a slender vortex tube, in which the Biot-Savart integral is approximated by a line integral and the logarithmic infinity of the integral on the core itself is disingularized so as not to be in contradiction in the known cases of vortex rings and infinitesimal vibrations of the Rankine vortex. 4,5,7,14 This approximation fails to take account of the influence of finite thickness of the distant parts. This paper presents an efficient machinery to make feasible a systematic handling of the Biot-Savart law for vorticity distributions localized in a tube-like structure.…”

“…The Kapteyn series gets along with a circular cylindrical boundary, coaxial to the axis of the supporting cylinder. 7,9 A flow with helical symmetry is amenable to a detailed analysis of the Euler equations. A theorem on the existence of a slender helical vortex tube was given by Adebiyi.…”

“…An efficient technique for elaborating the principal parts of the Kapteyn series was developed by Kuibin and Okulov. 7 Boersma and Wood 8 manipulated an integral representation of the Kapteyn series and made a thorough analysis of it. The Kapteyn series gets along with a circular cylindrical boundary, coaxial to the axis of the supporting cylinder.…”

The Second Curvature Correction for the Straight Segment Approximation of Periodic Vortex Wakes

“…This again implies that U 2c >> U 1c . Since most rotor wakes are helices of some form, it is important to note that the equivalent inverse pitch term dominates U for a helical vortex of sufficiently small 5 p, Kuibin & Okulov (1998). There is a further reason to expect U ≈ U 2c >> U 1c for many turbine wakes: U 1c , but not U 2c , is associated with the impulse necessary to form a vortex ring.…”