Single-scale two-dimensional-three-component generalized-Beltrami-flow solutions of incompressible Navier-Stokes equations

Abstract

Generalized-Beltrami-flow (GBF) solutions, which are exact solutions of incompressible Navier-Stokes equations (NSE), are still rare. Most existing GBF solutions are either planar or axisymmetric cases. We derive analytically a series of single-scale two-dimensional-three-component (2D3C) GBF solutions under the framework of helical decomposition. These solutions yield a manifold of fixed points with infinite degrees of freedom in the solution space. The key of the derivation is to arbitrarily put different wave vectors at the same wave length, and to apply a novel parallel relation to any pair of these wave vectors. Although these solutions belong to a general class of 2D3C Euler solutions, to our knowledge there has been no publication focusing on these particular GBF forms. The significance of these GBF solutions is that the novel parallel relation implies new statistical relations on turbulence energy transfer and velocity phases.