Axisymmetrically stable case:
It evolves like the two-dimensional circular vortex sheet. Three axial spirals appear and roll up due to the Kelvin-Helmholtz instability.
| Time = 0.0 | Time = 0.4 | Time = 0.8 | Time = 1.2 |
| Time = 1.6 | Time = 2.0 | Time = 2.4 | Time = 2.8 |
Axisymmetrically unstable case:
The evolution becomes quite complex and we discovered the secondary roll-up structure. Thanks to fast and accurate computations, we succeed to describe the detailed long-time complex evolution of the three-dimensional vortex sheet including the generation process of the secondary roll-up. I used 4096*128 points to discretize the vortex sheet initially. The computation was carried out by Hewlett-Packard's SPP-1600 (8 processors).
| Stage | Picture | Description |
| The first stage (Time=0.0-1.4) |
Time = 0.0 Time = 0.4 Time = 0.8 Time = 1.2 |
Three axial spiral structures appear owing to Kelvin-Helmholtz instablitiy. The vortex line4s in the vortex sheet concentrate in the spirals, and the high vorticity regions are formed in the center of the spirals. |
| The second stage (Time=1.6-2.2) |
Time = 1.6 Time = 1.8 Time = 2.0 Time = 2.2 |
The vortex sheet distorts in the azimuthal direction, while the three axial roll-ups continue to rolling up. In the stage, due to growth of the axisymmetric disturbances, the high-vorticity regions are distorted azimuthally. Consequently, the high-vorticity regions turn completely to the azimuthal direction in the middle. |
| The third stage (Time=2.4-) |
Time = 2.4 Time = 2.6 |
A new secondary roll-up structure appears in the middle of the vortex sheet. It is because the vortex sheet is rolled up in the azimuthal dirction by the velocity filed induced by the middle parts of the high-vorticity regions. |
I also investigate the maximum of the first derivative and the second derivative of the vortex sheet along the vortex lines, and discuss their behavior when the regularization paramter of the vortex method tends to zero. See the paper for the details.
