Critical Exponents and Universality in Fully Developed Turbulence

Multi-fractal model for hydrodynamic fully-developed turbulence (FDT) has been used to provide a detailed structure for the critical exponent σ describing the scaling form of energy (or enstrophy) dissipation rate ǫ (or τ) that appears to exhibit an interesting universality covering radically different hydrodynamic FDT systems. This result also appears to provide a consistent framework for classification of dissipation field into critical, subcritical and supercritical cases. Some FDT problems that exemplify these cases are discussed. Introduction.-Small-scale structure in three-dimensional (3D) incompressible fully-developed turbulence (FDT), following Kolmogorov's [1] epoch-making work, is believed to possess, in the large Reynolds number (R ⇒ ∞) limit, a certain universality in its scaling properties. This universal scaling behavior depends only on symmetries and conservation laws of the system and is unaffected by the large-scale flow structure. This is reminiscent of the scaling behavior near the critical point where many diverse systems show a striking similarity in their behavior [2,3]. Critical phenomena had a theoretical breakthrough in the renormalization group (RG) [4,5] which was the culmination of the ideas of scaling and universality. The formal application of RG procedure was attempted for the FDT problem [6-9]. The goal is to determine the critical exponents (like those associated with correlation length in critical lattice spin systems) that are intrinsic features of the FDT dynamics and not artifacts of the large-scale stirring mechanisms and hence unify radically different FDT systems near their critical points. Spatial intermittency is a common feature of FDT and implies that turbulence activity at small scales is not distributed uniformly throughout space. This leads to a violation of an assumption in the Kolmogorov [1] theory that the statistical quantities show no dependence in the inertial range L ≫ ℓ ≫ η on the large scale L (where the particular external stirring mechanisms generating FDT become influential) and the Kolmogorov microscale η = (ν 3 /ǫ) 1/4 (where the viscous effects become important). Thus, if one views the Kolmogorov [1] theory as a mean field theory [10], the spatial intermittency aspects will be expected to define at least one additional universal scaling exponent [11]. Spatial intermittency effects can be very conveniently imagined to be related to the fractal aspects of the geometry of FDT [12]. The mean energy dissipation field ǫ may then be assumed, in a first approximation, to be a homogeneous fractal [13], and more generally, a multi-fractal [14-17]. The latter idea has …