Nonlinear analysis for the evolution of vortex sheets* |
| |
Authors: | Russel E Caflisch |
| |
Institution: | Courant Institute of Mathematical Sciences, 251 Mercer Street, New York University, New York, NY 10012, USA |
| |
Abstract: | The perturbations in a nearly flat vortex sheet will initially grow due to Kelvin-Helmholtz instability. Asymptotic analysis and numerical computations of the subsequent nonlinear evolution show several interesting features. At some finite time the vortex sheet develops a singularity in its shape; i.e. the curvature becomes infinite at a point. This is immediately followed by roll-up of the sheet into an infinite spiral. This paper presents two mathematical results on nonlinear vortex sheet evolution and singularity formation: First, for sufficiently small analytic perturbations of the flat sheet, existence of smooth solutions of the Birkhoff-Rott equation is proved almost up to the expected time of singularity formation. Second, we present a construction of exact solutions that develop singularities (infinite curvature) in finite time starting from analytic initial data. These results are derived within the framework of analytic function theory. |
| |
Keywords: | |
本文献已被 ScienceDirect 等数据库收录! |
|