Interscale dynamics and local isotropy in high Reynolds number turbulence within triadic interactions
Authors: Brasseur James G. , Wei Chao-Hsuan 
Published: July 2002

Interactions among scales of motion within fully developed turbulence are given in the Fourier-space view by the sum of collections of triadic interactions among subsets of Fourier modes in three-dimensional Fourier space. To analyze general characteristics of interscale couplings in high Reynolds number turbulence, ‘‘chains’’ of roughly 20 000 interconnected triads are embedded within a model energy spectrum and intermodal energy exchange is examined within groups of triadic interactions based on triad geometry. Calculations of the inviscid momentum equation are carried out over a large number of realizations with random initial phase, and ensemble-averaged intermodal energy transfers are analyzed to extract general dynamical features associated with differences in relative scale and triad obtuseness. The studies demonstrate different dynamical effects within ‘‘local,’’ ‘‘nonlocal,’’ and ‘‘distant’’ triadic interactions in different spectral regions. The forward cascade within the inertial range, is found to be strongly dominated by local-to-nonlocal triadic interactions with scale separations as high as 10–15.
A consequence is that for α decades of inertial subrange, Rλ=C×10(2/3)α, with C≊310–470, implying that Rλ must be roughly 1450–2200 for a decade of inertial range, and must exceed 300–400 for any natural inertial range to exist. ‘‘Backscatter’’ is primarily within triadic interactions in which energy moves both to larger and to smaller scales and highly local triadic interactions appear to be weakly inverse cascading, on average. Distant interactions are characterized by the asymptotic form of the triad equations whereby structural information is transferred from the largest to smallest scales, but not energy. It is found that direct energy transfer between the largest and smallest scales ceases when the ratio between scales is greater than 15–20, suggesting that a dominantly ‘‘distant’’ triadic characteristic requires scale separations greater than roughly 15. Energy transfer within the high wave-number modes in distant interactions, however, is nonenergy cascading, on average; energy tends to move laterally within thin spectral shells, but not from one shell to the next.
Furthermore, as energy moves from large to small scales within progressively more local triadic interactions, the tendency is to distribute energy more and more uniformly at smaller and smaller scales, providing an increasingly strong isotropizing influence at successively smaller scales. By contrast, distant interactions are highly directional in three-dimensional Fourier space with characteristics that tend to transfer, over time, structural information (including anisotropy) directly from large to small scales. The various isotropizing and anisotropizing influences within local-to-nonlocal and distant scale interactions compete at the small scales. Which influences dominate over what periods of time depend on a number of interacting dynamical factors, including location within the spectrum, energy and temporal coherence within large-scale structure, the presence of direct forcing at different scales (as with mean shear), the stationary or nonstationary character of the turbulence, and time. It is argued that searches for deviations from local isotropy should be carried out (1) within systematically increasing scale separations, (2) in the limit of the smallest dynamical scales, and (3) as a function of systematic variations in large-scale anisotropy.