Geometry of two-point velocity correlation tensor of homogeneous isotropic turbulence with helicity in sol-manifold coordinates

Abstract

In this paper, a geometric model of homogeneous isotropic turbulence with helicity in three dimensions using a family of the quadratic forms of Riemannian metrics generated by the two-point correlation tensor field is investigated. We show that this metric induces a sol-metric on infinite cylinder R×T2 as a sol-manifold. Using the corresponded Hamiltonian equation of the geodesic flow of our sol metric on the infinite cylinder R×T2, we find three functionally independent first integrals and then the geodesic flow is integrable. Moreover, we consider how the terms of this family of quadratic forms influence the length scales of turbulent motion. The conformal invariance of the corresponding manifolds on our Reimanian metric is proved using the symmetry group of the system of von Kármán–Howarth and Chkhetiani–Kurien equations in the two limits of finite and infinite Reynolds numbers. This property corresponds to the conservation of angles between the intersecting curves located on the manifold under the action of these scaling groups.