Vortex Filament Methods for Superfluids

Abstract

Vortex filaments are an idealized form of rotational flow where the vorticity is confined to a small core region, of radius a, around a one dimensional line embedded in the three dimensional flow. Outside of this core region the flow is potential. When the dynamics of the core size are not important these objects may also be referred to as vortex lines.

In classical fluid mechanics, by which I mean solutions of the Navier-Stokes or Euler equations, vortex filaments are a useful tool for understanding the geometry and dynamics of a flow. But after an initial popularity in the early 1980’s [1] [2] [3] [4] [5], the use of vortex filament methods fell out of favor in classical fluid mechanics. Though there has been some slight resurgence in this method recently [6] [7] due to the rapidly increasing computational power available and the development of new computational algorithms, direct numerical simulations and large eddy simulations have become the methods of choice for calculating the motion of fluids. One reason for the decreased use of vortex filament methods is that while they give a clear and intuitive understanding of a flow through the easy visualization of the vortex filaments, this representation was often just a rough cartoon of the true flow. Vortex filaments are only a convenient idealization in a classical flow. The vorticity in a realistic classical flow rarely takes the form of clearly discrete vorticity filaments.