Formation of curvature singularity along vortex line in an axi-symmetric, swirling flow.

Abstract

Conclusions and Discussions

The present singularity occurred along the vortex line and the first derivative of the vortex sheet strength had a cusp-form in the neighborhood of the singular point. This profile of the singularity is unlike the known 2-D singularities, since they appear along curves normal to the vortex lines and the vortex sheet strength, not its first derivative, forms a cusp at the critical time.

The conclusion suggests that two types of singularities are possible in the 3-D cylindrical vortex sheet;

Suppose that a surface ${\bf{C}} (\xi,\eta,t)$ represents the cylindrical vortex sheet, where $\xi$ is the circulation parameter taken along the direction perpendicular to the vortex lines and $\eta$ is a Lagrangian parameter that parameterizes the vortex line.

1. If the surface is assumed to be independent of $\eta$, the motion is equivalent to that of a two-dimensional circular vortex sheet. Then, the 2-D streamwise curvature singularity along the circulation parameter, $\max_{\xi} \left| \frac{\partial^2 \bf{C}}{\partial \xi^2} \right| \rightarrow \infty$, appears in finite time. The singularity exist continuously in a certain vortex line, where the vortex sheet strength has a cusp profile.

2. If the surface is independent of $\xi$, namely it is axi-symmetric, it follows from the present numerical computation that the second derivative along the vortex line, i.e. $\max_\eta \left| \frac{\partial^2 \bf{C}}{\partial \eta^2} \right|$, becomes infinite at some finite time. In this case, the singular points form a circular curve, where the first derivative of the vortex sheet strength develops a cusp-form.

3. For a general initial configuration without the symmetries, more complicated phenomenon are expected. Indeed, Sakajo computed evolution of the 3-D cylindrical vortex sheet for a periodic perturbation and observed that generation of a secondary spanwise roll-up structure followed the streamwise roll-up structure due to Kelvin-Helmholtz instability. The primary streamwise roll-up corresponds to the 2-D double branched spiral which is considered to be a solution of the vortex-sheet motion after the 2-D streamwise singularity formation. He also observed the rapid growth of the second derivative of ${\bf{C}}$ along the vortex line just before the secondary roll-up appeared. Actually, since the numerical computation is regularized by the vortex-blob method, the second derivative remains finite throughout the evolution. However, the rapid growth is apparently related to the present axi-symmetric singularity formation.

Reference