Vortex methods were originally conceived as a tool to model the evolution of unsteady, incompressible, high Reynolds number flows of engineering interest. Examples include bluff-body flows and turbulent mixing layers. Vortex methods simulate flows of this type by discretizing only the vorticity-carrying regions and tracking the computational elements in a Lagrangian frame. They provide automatic grid adaptivity and devote little computational effort to regions devoid of vorticity. Moreover the particle treatment of the convective terms is free of numerical dissipation.

Thirty years ago simulations using inviscid vortex methods predicted the linear growth in the mixing layer and were able to predict the Strouhal frequency in a variety of bluff-body flow simulations. In three dimensions, we have seen that inviscid calculations using the method of vortex filaments have provided us with insight into the evolution of jet and wake flows. However, the inviscid approximation of high Reynolds number flows has its limitations. In bluff-body flows viscous effects are responsible for the generation of vorticity at the boundaries, and a consistent approximation of viscous effects, including diffusion, is necessary at least in the neighborhood of the body. In three-dimensional flows, vortex stretching and the resultant transfer of energy to small scales produce complex patterns of vortex lines. The complexity increases with time, and viscous effects provide the only limit in the increase of complexity and the appropriate mechanism for energy dissipation. In this chapter we discuss the simulation of diffusion effects in the context of vortex methods.