Helical Contour Dynamics
In an incompressible inviscid flow system, the helical symmetry means it is invariant though combined axial translation and rotation about the same axis. In helical symmetry, the axial vorticity is materially conserved if the velocity components along the helical lines are proportional to 1=(1+e^2r^2), where e is the pitch and r is the distance from the z-axis. Linear instability analysis implies that the circular helical vortex patch centered at the origin is neutrally stable. We present the evolution for a family of helically symmetric vortices through contour dynamics, which is a Lagrangian technique to compute the motion of vortices through the integrals along the contours. The contour perturbed initially with mode 4 is dominated by the mode itself while the first mode becomes the most unstable mode in the latter time for the contour perturbed with mode 9. Adding the vortex sheet on the boundary of the shifted contour accelerate the twisting and rotating process. The sharpening shock of vortex sheet is generated in the evolution and may lead to the discontinuity.