Exact relationship for third-order structure functions in helical flows

An exact law for turbulent flows is written for third-order structure functions taking into account the invariance of helicity, a law akin to the so-called “4/5 law” of Kolmogorov. Here, the flow is assumed to be homogeneous, incompressible and isotropic but not invariant under reflectional symmetry. Our result is consistent with the derivation by O. Chkhetiani [JETP Lett. 10, 808, (1996)] of the von Kármán–Howarth equation in the helical case, leading to a linear scaling relation for the third-order velocity correlation function. The alternative relation of the Kolmogorov type we derive here is written in terms of mixed structure functions involving combinations of differences of all components for both the velocity and vorticity fields. This relationship could prove to be a stringent test for the measuring of vorticity in the laboratory, and provide a supplementary tool for the study of the properties of helical flows.