The need of a specific discussion of vortex schemes in the context of three-dimensional flows stems from the very nature of the vorticity equation that in three dimensions incorporate a stretching term. This term fundamentally affects the dynamics of the flow; it is in particular responsible for vorticity intensification mechanisms that make long-time inviscid calculations very difficult. Vorticity stretching is considered as the mechanism by which energy is being transferred between the large and the small scales in the flow. In order to resolve related phenomena, such as the energy cascade, an adequate treatment of diffusion is thus even more crucial than in two dimensions. However, the recipes for deriving diffusion algorithms are the same in two and three dimensions (they are discussed in Chapter 5), and we focus here on inviscid three-dimensional vortex schemes. Vorticity intensification in general is associated with a rapid stretching of Lagrangian elements, which makes it also crucial to maintain the regularity of the particle mesh; we refer to Chapter 7 for a general discussion of regridding techniques.

We will discuss here two classes of vortex methods that extend to three dimensions the two-dimensional schemes introduced in Chapter 2. In the first one, the vorticity is replaced by a set of points (particles), just as in two dimensions, but these particles carry vectors instead of scalars. The stretching term in the vorticity equation is accounted for by appropriate laws that modify the circulations of the particles. We call these methods vortex particle methods.