Structures and structure functions in the inertial range of turbulence

The deviations from the Kolmogorov 1941 laws of inertial range of turbulence are investigated using the results from the direct numerical simulations of an unforced flow starting from a high-symmetry initial condition by Kida [J. Phys. Soc. Jpn. 54, 2132 (1985)]. The resolution is ${300}^3$ points ${\text{(1200}}^3$ with symmetries, maximum wavenumber 400 after dealiasing), and the Taylor scale Reynolds number is in the order of 100. The scaling exponents of the $p$ th order longitudinal and lateral structure function (for $p$ between 2 and 16) are computed using different methods with particular focus on a recent method by Benzi and collaborators [Phys. Rev. E 48, R29 (1993); Europhys. Lett. 32, 709 (1995)]. Both longitudinal and lateral scaling exponents deviate considerably from Kolmogorov 1941 (K-41) scaling laws, the lateral deviating much more than the longitudinal. A systematic methodology (strain–enstrophy state) is developed to relate the K-41 deviations to different structures in the field. Enstrophy-dominated structures are found to contribute mainly to the deviations in lateral direction whereas the strain-dominated structures to longitudinal direction, albeit in an imbalanced proportion, the lateral deviations being much stronger. Structures whose enstrophy and strain are comparable in magnitude contribute to deviations in both directions. Results are compared to several intermittency models and experiments. Special focus is given to the recent She–Lévêque model [Phys. Rev. Lett. 72, 336 (1994)] whose predictions gave very good agreement if compared to the longitudinal exponents. The model is rewritten for a family of free parameters, giving predictions as good as the original one. The lateral scaling exponents disagree with both the She–Lévêque model and the experimental results (from longitudinal velocity measurements) suggesting that the dominant contribution to intermittency can only be detected from the lateral structure function measurements.