Exact relationship for third-order structure functions in helical flows

An exact law for turbulent flows is written for third-order structure funct ions taking into account the invariance of helicity, a law akin to the so-c alled "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 Karman-Howarth equation in the helical case, l eading to a linear scaling relation for the third-order velocity correlatio n 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. T his relationship could prove to be a stringent test for the measuring of vo rticity in the laboratory, and provide a supplementary tool for the study o f the properties of helical flows.