Vortex merging is a basic fluid phenomenon which has been much studied for two-dimensional flows. Here we extend such a study to a specific class of three-dimensional flows, namely to vortices possessing a helical symmetry. In addition to the standard Reynolds number, this introduces another dimensionless control number, the pitch, which quantifies the periodicity length along the helix axis. Helical vortices with large pitches merge very much as in a two-dimension setting. However, their rotation speed is reduced and the merging period is delayed. These effects, caused by the presence of a self-induction velocity in curved three-dimensional vortices, are understood by computing the streamfunction in the frame of reference rotating with the two vortices, and by inspecting the locations of its hyperbolic points. At intermediate pitch values, only viscous diffusion acts, resulting in a slow viscous type of merging. Finally for small pitches, the system is unstable resulting, at the nonlinear stage, in a different type of merging which breaks the initial central symmetry.

We herein present a direct numerical simulation method aimed at describing the dynamics of helical vortices such as those developing in the wake of propellers and wind turbine or helicopter rotors. By enforcing a helical symmetry, the 3D incompressible Navier-Stokes equations are reduced to a 2D problem which we solve using a generalised vorticity/streamfunction formulation. In this framework, we simulate the viscous dynamics of one or several helical vortices and describe quasi-steady states as well as long-time (or far-wake) dynamics. In particular, several types of merging in the two helical vortex systems are identified.