Vortex sheets are used in fluid dynamics to model thin shear layers in slightly viscous flow, and understanding vortex sheet motion has been a longstanding challenge. Rosenhead used the point vortex method to compute periodic vortex sheet motion in 2D flow and the results seemed to confirm Prandtl’s idea that vortex sheets roll up smoothly into concentrated spirals. However later simulations by Birkhoff, using higher resolution, produced irregular point motion, leading him to write a paper entitled, “Do Vortex Sheets Roll-Up?”. Eventually Moore clarified the problem, showing that a curvature singularity typically forms in the shape of an evolving vortex sheet, before the sheet actually rolls up. In this talk, I’ll review these basic contributions and then discuss more recent regularized simulations past the critical time. The results have some features in common with the self-similar spiral vortex sheets studied by Pullin, but chaos intervenes unexpectedly. I’ll also describe a method for computing vortex sheet motion in 3D flow, using a Cartesian treecode, with an application to vortex rings.
Robert’s research interests include Lagrangian particle methods in fluid dynamics and treecode algorithms for computing long-range particle interactions. Applications include vortex sheets/rings, and more recently also geophysical flow, as well as problems in molecular dynamics (e.g. Poisson-Boltzmann model for implicit solvation, charge transport in solar cells).