A family of helically symmetric vortex equilibria

Abstract

We present a family of steadily rotating equilibrium states consisting of helically symmetric vortices in an incompressible inviscid irrotational unbounded fluid. These vortices are described by contours bounding regions of uniform axial vorticity. Helical symmetry implies material conservation of axial vorticity (in the absolute frame of reference) when the flow field parallel to vortex lines is proportional to (1 + 2 r 2) −1/2 , where is the pitch and r is the distance from the axis. This material conservation property enables equilibria to be calculated simply by a restriction on the helical stream function. The states are parameterized by their mean radius and centroid position. In the case of a single vortex, parameter space cannot be fully filled by our numerical approach. We conjecture multiply connected contours will characterize equilibria where the algorithm fails. We also consider multiple vortices, evenly azimuthally spaced about the origin. Stability properties are investigated numerically using a helical CASL algorithm.

Topology of helical fluid flow

“…Further, if u ζ = u z = 0, then the flow is two-dimensional. It is common to use a stream function depending on ξ and η spanned in the plane normal to the tangent vector [2,12,16,23,26,39]. References are often given to the works of Landman [22] and Dritschel [14], where a helical, incompressible velocity field is decomposed in the form…”

“…Helical flows (flows with helical symmetry) arise behind propellers and wind turbines [19,26] and helical vortices are important for helicopter rotor performance [18,30,36]. Helical pipe flow has a general engineering interest [11,17,21,24,38], and has been also studied as a model for a blood vessel with non-vanishing curvature and torsion [39,40].…”

Self-similar behaviour of a rotor wake vortex core

“…Furthermore, core radius is fixed in time and vorticity and velocity profiles are not known in the core. Recently, considering uniform vorticity, Lucas & Dritschel (2009) have found the form of the vortex core, for different (infinite) helical vortices, using Euler equations.…”

Merging of two helical vortices

“…These flows however are somewhat simpler: at least locally, they satisfy a helical symmetry, i.e. these vortices are invariant through combined axial translation 5 and rotation. Many works have already been devoted to this specific class of flows.…”