On Vortex Methods

Hairpin removal in vortex interactions

“…A smoothing parameter is needed for conve.rgence [2,5,11,19]. The use of a constant u produces an apparent paradox: it is well known ( [45]; see also the analysis in [16,18]) that the stretching of a vortex tube requires work that is used to spin up the fluid around the reduced cross-section.…”

“…More accurate methods that use higher order cores [5,6] and filaments [2,30] are available, but will be seen not to be appropriate for use together with the hairpin removal algorithm as we shall present it.…”

“…The rule t/>(a) =a~ corresponds to the usual Biot-Savart interaction and the modification for a < u is the smoothing needed for convergence [2,13], see also below.…”

“…picks their cores so that the spectrum is O(k-3 ), as in [11], one obtains a vortex method that contains a turbulence model. However, other, physically less well motivated, cores [2,6] may well provide a better numerical approximation. Nevertheless, this link with turbulence theory will lead to interesting speculations.…”

Vortex method for two dimensional Euler equations in bounded domains with boundary correction

“…The consistency, stability and convergence of vortex methods for the initial value problems of the Euler equations was first obtained by Hald and DelPrete [9], [10], [11]; then the results were improved and different proofs were given by Anderson and Greengard [4], Beale and Majda [5], [6], and Raviart [13]. However there are some essential difficulties in the study of the vortex method for initial-boundary value problems.…”