Vortex interactions and coherent structures in turbulence

This chapter discusses the vortex interactions and coherent structures in turbulence. The study of vortex dynamics is a challenging subject apart from the applications to transition phenomena and turbulent flow. A useful and popular method of analyzing vortex flows, especially for small viscosity and concentrated vorticity, replaces the actual continuous distribution of vorticity by a finite sum over a number of discrete vortices, defined as a volume of rotational fluid surrounded by irrotational fluid. In the atomic representation, the structure of the vortices is supposed given and their motion is described by the Lagrangian evolution equation. In the molecular representation, the deformation and structure of the vortex are at least as important as the motion of the vortex as a whole. The atomic approach is particularly simple for two-dimensional flows when the vortices are usually either points or circles of small radius. The principal method used so far represents the vorticity as a collection of vortex filaments and the velocity is calculated from either the Biot-Savart law of induction with a cut-off proportional to the filament radius to give a finite value to the self-induced velocity, or by redistributing the vorticity on to mesh points of a grid and using a Poisson solver.