Helical Turbulence - the Transition between 2D and 3D Turbulence

Based on vortex stretching, our theoretical understanding of 2D and 3D turbulence reveals fundamental differences. To gain further insights, we study helical-symmetric flows. The helical coordinate system (r, ζ, η) is given by r, ζ = az + bΦ and η = −bz + ar²Φ, where a, b = const, a² + b² > 0 and (r, Φ, z) are the common cylindrical coordinates. Helical symmetry implies, all dependent variables are independent of η. Helical flows differ whether they have a velocity u^η along the helix or not. In both cases, helical flows admit infinite classes of conservation laws and thus integral invariants exist. The central new invariants for helical turbulence are generalized helicity, when vortex stretching is present, and generalized enstrophy, without vortex stretching. The findings from 2D and 3D turbulence show that global invariants play a central role in turbulence and this is also expected for helical turbulence. Appropriate largescale simulations are conducted for this purpose. The helically reduced Navier-Stokes equations are discretized using high-order discontinuous Galerkin scheme. This numerical framework is utilized to study energy transport in helically symmetric flows for high Reynolds numbers.