Motion of a helical vortex

TL;DR

This paper derives new, more accurate formulas for the velocities of a helical vortex in an ideal fluid and analyzes how its geometry affects fluid transport and vortex motion.

Contribution

It introduces improved expressions for the vortex's velocities using Hardin's solution, verified by numerical methods, enhancing understanding of vortex dynamics and fluid transport.

Findings

Thinner vortices move faster with less fluid transport.

Thicker vortices of small pitch have intermediate velocities and push fluid forward.

Large pitch vortices move slowly and carry significant fluid volume.

Abstract

We study the motion of a single helical vortex in an unbounded, inviscid, incompressible fluid. The vortex is an infinite tube whose centerline is a helix and whose cross section is a circle of small radius (compared to the radius of curvature) where the vorticity is uniform and parallel to the centerline. Ever since Joukowsky (1912) deduced that this vortex translates and rotates steadily without change of form, numerous attempts have been made to compute these self-induced velocities. Here we use Hardin's (1982) solution for the velocity field to find new expressions for the vortex's linear and angular velocities. Our results, verified by numerically computing the Helmholtz integral and the Rosenhead-Moore approximation to the Biot-Savart law, are more accurate than previous results over the whole range of values of the vortex pitch and cross-section. We then use the new formulas to study the advection of passive particles near the vortex; we find that the vortex's motion and capacity to transport fluid depend on its pitch and cross section as follows: a thin vortex of small pitch moves fast and carries a small amount of fluid; a thick vortex of small pitch moves at intermediate velocities and is a moderate carrier itself but it pushes fluid forward along the helix axis; and a vortex of large pitch, whether thin or thick, moves slowly and carries a large amount of fluid.