Regularized vortex sheet evolution in three dimensions

A computational method is presented to follow the evolution of regularized three-dimensional (3D) vortex sheets through an otherwise irrotational, inviscid, constant-density fluid. The sheet surface is represented by a triangulated mesh with interpolating functions locally defined inside each triangle. C^1 continuity is maintained between triangles via combinations of cubic Bézier triangular interpolants. The self-induced sheet motion generally results in a highly deformed surface which is adaptively refined as needed to capture regions of increasing curvature and to avoid severe Lagrangian deformation. Automatic mesh refinement is implemented with an advancing front technique. Sheet motion is regularized by adding a length scale cut-off to the BiotSavart kernel. Velocity evaluation takes less time than the standard O(N^2) scaling, due to utilization of multi-pole expansions of the kernel. Zero, singly, and doubly periodic vortex sheets are simulated, modeling vortex rings, vortex/jet combinations and standard shear layers. Comparisons with previous two-dimensional (2D) results are favorable and 3D simulations are presented. The perturbed 3D planar shear layer is simulated and compared with a similar experiment revealing qualitatively similar results and agreement on the mechanism by which streamwise vorticity is created. We find the ratio of spanwise to streamwise vorticity to vary between 7 and 9 during early stages of roll-up. A new technique for estimating the curvature singularity time of true vortex sheets (i.e., non-regularized motion) is presented. The motion and singularity time of a planar, doubly periodic sheet, evolving from a 3D normal mode perturbation, is found to reduce to that of a well known singly periodic (and only two-dimensional) problem, an unexpected extension of Moore's [38] non-linear analysis for 2D vortex sheets