Mathematical Problems in Engineering
Volume 2007 (2007), Article ID 68953, 22 pages
Research Article

Adaptive Panel Representation for 3D Vortex Ring Motion and Instability

Leon Kaganovskiy

Department of Natural Sciences, New College of Florida, 5700 N. Tamiami Trail, Sarasota 34243, FL, USA

Received 26 September 2006; Accepted 16 October 2006

Academic Editor: Semyon M. Meerkov

Copyright © 2007 Leon Kaganovskiy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A hierarchical panel method for representing vortex sheet surface motion in 3D flow is presented. Unlike previously employed filament methods, each panel is a leaf of the tree, so it can be subdivided locally, which allows an efficient adaptive point insertion. In addition, we developed curvature-based insertion criteria which allow to localize point insertion to the most complicated curved regions of the sheet. The particles representing the sheet are advected by a regularized Biot-Savart integral with Rosenhead-Moore kernel. The particle velocities are evaluated by an adaptive treecode algorithm based on Taylor expansions in Cartesian coordinates due to Lindsay and Krasny (2001). The method allows to consider much later stages of a vortex ring instability, which may shed light on this complicated flow phase directly leading to the turbulent flow.