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

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Our theoretical understanding of 2D and 3D turbulence and their basic distinguishing feature of vortex stretching reveals fundamental differences. This disparity is remarkably unified in atmospheric flows that exhibit 2D mechanisms on large scales, but 3D turbulence manifests itself on small scales. The main goal of the present proposal is to understand the fundamental differences and transitions between 2D and 3D turbulence. For this purpose, helical-symmetric flows, which occur ubiquitous in fluid physics, will be used as a model problem. These flows "live" on a 2D manifold but have three independent velocity vectors and are therefore often called 2 1/2-dimensional flows. Helical flows decompose into those with and without velocity vectors along the helix. If the velocity vector is non-zero, it induces a vortex stretching. The central questions in the present proposal are grouped around the underlying invariants that determine parts of the spectra or structure functions. E.g. in 3D turbulence and on small length scales this is, according to Kolmogorov's theory, the dissipation, whereas in 2D turbulence the spectrum is determined by the integral invariant of the enstrophy. A central discovery in Kelbin et al. (2013) was that helical flows admit infinite classes of conservation laws and thus integral invariants. The central new invariants for helical turbulence are generalized helicity, when vortex stretching is present, and generalized enstrophy, without vortex stretching. The fundamental working hypothesis is that both of these are key determinants of the respective spectra. In addition, the applicant has recently developed a statistical theory of helical turbulence that includes vortex stretching in Deussen et al. (2020). Therein potentials of the correlations were introduced which are to be examined for their significance of the spectrum on the basis of simulation data. In contrast, we expect for helical flows without vortex stretching the infinite dimensional conformal group, which the applicant could prove theoretically for the first time in 2D turbulence in Grebenev et al. (2017). The working hypothesis is the potential link between the infinite dimensional conformal group and the infinite number of conservation laws (Casimirs) with the enstrophy being the first in this row and which has a close analogy to generalized enstrophy for helical flows. In order to support the theoretical questions with highly accurate data, a large set of simulations for helical flows at very high Reynolds numbers shall be performed. The code is based on the discontinuous Galerkin method and has been developed in the context of the DFG project OB 96/41-1 for exactly this purpose. Finally, the transition 2D / 3D by asymptotically small velocities along the helix will be investigated, so that asymptotically small vortex stretching exist. Central questions here are the changes of the above-mentioned integral invariants and their validation by related simulations.