Isotropy vs anisotropy in small-scale turbulence

The decay of large-scale anisotropies in small-scale turbulent flow is investigated. By introducing two different kinds of estimators we discuss the relation between the presence of a hierarchy for the isotropic and the anisotropic scaling exponents and the persistence of anisotropies. Direct measurements from a channel flow numerical simulation are presented. One of the main assumptions made by A.N. Kolmogorov in his 1941 theory is the restoring of universality and isotropy at small scales in turbulent flows. The idea is that the effects of a large-scale anisotropic forcing and/or boundary conditions are rapidly lost during the process of energy transfer toward the small scales. The overall result is that the isotropy and the universality of turbulent fluctuations should be locally restored at small enough scales and large enough Reynolds numbers. The rate of convergence toward isotropy can be quantitatively predicted within the K41 theory both as a function of the scale, e.g. for the structure functions, and as a function of the Reynolds numbers, e.g. for the single-point moments of the velocity gradients. Experiments [1] and numerical simulations [2, 3, 4] do not confirm those predictions. The skewness of the transversal gradients, S3 = 〈(∂yux) 3〉/〈(∂yux) 〉 is for example found to have a very slow decay with Reλ. The effect is even stronger for the fifth-order skewness S5 = 〈(∂yux) 5〉/〈(∂yux) 〉, observed to remain O(1) for all available Reλ. Similar results were recently reported on a series of hydrodynamical problems. The most striking ones were obtained analytically in passive scalar/vector models advected by isotropic, Gaussian and white-in-time velocity fields (the so-called Kraichnan model [5]) with a large scale anisotropic forcing [6, 7]. Numerical [8, 9] and experimental (see, e.g., [10, 11]) evidences of persistence of anisotropies in real passive scalars have also been reported. On one hand, there are then strong indications in favor of a persistent memory of the large-scale anisotropies even at the smallest scales of a turbulent flow. On the other hand, there are theoretical arguments [12] going in the opposite direction, i.e. that anisotropic fluctuations are sub-dominant with respect to the