A fast summation method and long time mixing of 2-D vortex sheet
Abstract:
The fast summation method of Draghicescu is applied to computation of 2D vortex sheet motion. We report the numerical experiments which shows the effectiveness as well as difficulties of Draghicescu's fast summation method in 2D vortex-sheet computations. For instance, the fast method is nearly thirty times faster than the direct summation method when we use 65536 vortex blobs. As a test problem, we re-examine Krasny's problem of computing a vortex sheet for a fairly long time.
Conclusion:
We have applied the Draghicescu's fast algorithm with adaptive programming technique to the equation of a vortex sheet with periodic boundary condition. Transformation of the independent as well as dependent variable are used. Fourier filter is applied, too. As a result, we can compute the evolution of the vortex sheet up to time=12.0. In particularly, by taking care of uneven distribution of vorticity, we can compute the velocity field more effectively.
As for the physical properties of the vortex sheet, we can summarize the long time evolution as follows; While the vortex sheet keeps rolling-up at the center of the spiral, the trailing arms repeat following processes.
- stretched by the shear flow, and migrate to next vortex sheet,
- caught by the adjacent spiral,
- split into two arms,
- one arm goes into spiral further next and repeat step 1. The other arm goes back and below the spiral and eventually get back to the first spiral.
We guess that the vortex sheet grows repeatedly to the second, the third spiral and so on. However, since the length of vortex sheet grows exponentially, more vortex blobs are needed to compute further evolution.
Pictures:
Reference:
T. Sakajo and H. Okamoto, An application of Draghicescu's fast method to motion of vortex sheet, Japan Journal of Physical Society (1998), vol 67. No.2.
Back
