Helical Contour Dynamics

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.