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Helical Contour Dynamics


In an incompressible inviscid flow system, helical symmetry means invariance 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+epsilon^2r^2), where e is the pitch and r is the distance from the z-axis. Linear instability analysis shows that a circular helical vortex patch centered at the origin is neutrally stable. We present the evolution of a family of helically symmetric vortices using contour dynamics, a Lagrangian technique to compute the motion of vortices via contour integrals. For contours perturbed by both lower and high modes, the first mode always becomes the most unstable mode for large time. We can inspect the features induced by the lower perturbed mode. We take mode 4 and mode 9 as examples in this work. Adding a vortex sheet on the boundary of the shifted contour accelerates the twisting and rotating process. The distribution of vortex sheet forms a sharpening shock in the evolution and may lead to the discontinuity.

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