# Helical Contour Dynamics

*2021, Volume 26, Number 6, pp. 600-617*

Author(s):

**Chu T., Llewellyn Smith S. G.**

The equations of motion for an incompressible flow with helical symmetry (invariance
under combined axial translation and rotation) can be expressed as nonlinear evolution
laws for two scalars: vorticity and along-helix velocity. A metric term related to the pitch of
the helix enters these equations, which reduce to two-dimensional and axisymmetric dynamics
in appropriate limits. We take the vorticity and along-helix velocity component to be piecewise
constant. In addition to this vortex patch, a vortex sheet develops when the along-helix velocity
is nonzero.We obtain a contour dynamics formulation of the full nonlinear equations of motion,
in which the motion of the boundary is computed in a Lagrangian fashion and the velocity field
can be expressed as contour integrals, reducing the dimensionality of the computation. We
investigate the stability properties of a circular vortex patch along the axis of the helix in the
presence of a vortex sheet and along-helix velocity. A linear stability calculation shows that
the system is stable when the initial vortex sheet is zero, but can be stable or unstable in the
presence of a vortex sheet. Using contour dynamics, we examine the nonlinear evolution of the
system, and show that nonlinear effects become important in unstable cases.

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