Isotropic third-order statistics in turbulence with helicity: the 2/15-law

Abstract

The so-called 2/15-law for two-point third-order velocity statistics in isotropic turbulence with helicity is computed for the first time from a direct numerical simulation of the Navier–Stokes equations in a $512^3$ periodic domain. This law is a statement of helicity conservation in the inertial range, analogous to the benchmark Kolmogorov 4/5-law for energy conservation in high-Reynolds-number turbulence. The appropriately normalized parity-breaking statistics, when measured in an arbitrary direction in the flow, disagree with the theoretical value of 2/15 predicted for isotropic turbulence. They are highly anisotropic and variable and remain so over long times. We employ a recently developed technique to average over many directions and so recover the statistically isotropic component of the flow. The angle-averaged statistics achieve the 2/15 factor to within about 7% instantaneously and about 5% on average over time. The inertial- and viscous-range behaviour of the helicity-dependent statistics and consequently the helicity flux, which appear in the 2/15-law, are shown to be more anisotropic and intermittent than the corresponding energy-dependent reflection-symmetric structure functions, and the energy flux, which appear in the 4/5-law. This suggests that the Kolmogorov assumption of local isotropy at high Reynolds numbers needs to be modified for the helicity-dependent statistics investigated here.