Abstract:
Shell models of turbulence have been employed as toy models which, in their chaotic states, show statistical properties similar to real fluid turbulence, including Kolmogorov energy spectrum and intermittency. These models are interesting because, at the present stage, it is still quite difficult or almost impossible to study relations between those traditional statistical properties and the structure of the chaos underlying the real fluid turbulence because of huge dimension of the chaotic attractor. In this paper we will give a brief review on the chaotic properties of a shell model (GOY model), with emphasis on its Lyapunov spectrum and unstable periodic orbits, in relation to the Kolmogorov scaling law of the turbulence.

Abstract:
The paper {\it On the multifractal nature of fully developed turbulence and chaotic systems}, by R. Benzi {\it et al.} published in this journal in 1984 (vol {\bf 17}, page 3521) has been a starting point of many investigations on the different faces of selfsimilarity and intermittency in turbulent phenomena. Since then, the multifractal model has become a useful tool for the study of small scale turbulence, in particular for detailed predictions of different Eulerian and Lagrangian statistical properties. In the occasion of the 50-th birthday of our unforgettable friend and colleague Giovanni Paladin (1958-1996), we review here the basic concepts and some applications of the multifractal model for turbulence.

Abstract:
Superfluid turbulence is governed by two dimensionless parameters. One of them is the intrinsic parameter q which characterizes the relative value of the friction force acting on a vortex with respect to the non-dissipative forces. The inverse parameter 1/q plays the same role as the Reynolds number Re=UR/\nu in classical hydrodynamics. It marks the transition between the "laminar" and turbulent regimes of vortex dynamics. The developed turbulence, described by a Kolmogorov cascade, occurs when Re >> 1 in classical hydrodynamics. In superfluids, the developed turbulence occurs at q << 1. Another parameter of superfluid turbulence is the superfluid Reynolds number Re_s=UR/\kappa, which contains the circulation quantum \kappa characterizing quantized vorticity in superfluids. The two parameters q and Re_s control the crossover or transition between two classes of superfluid turbulence: (i) the classical regime, where the Kolmogorov cascade (probably modified by the non-canonical dissipation due to mutual friction) is effective, vortices are locally polarized, and the quantization of vorticity is not important; and (ii) the Vinen quantum turbulence where the properties are determined by the quantization of vorticity. The phase diagram of these dynamical vortex states is suggested.

Abstract:
Phenomenological arguments are used to explore finite-time singularity development in different physical fully-developed turbulence (FDT) situations. The role played by the cascade physics underlying this process is investigated. Such diverse aspects as the effects of spatial intermittency and fluid compressibility in three-dimensional (3D) FDT and the role of the divorticity amplification mechanism in two-dimensional (2D) FDT and quasi-2D quasi-geostrophic FDT and the advection-diffusion mechanism in magnetohydrodynamic turbulence are considered to provide physical insights into this process in variant cascade physics situations. The quasi-geostrophic FDT results connect with the 2D FDT results in the barotropic limit while they connect with 3D FDT results in the baroclinic limit (on doing the necessary interchange of vorticity in the 3D case with divorticity in the quasi-2D case); hence they seem to provide a kind of bridge between 2D FDT and 3D FDT results.

Abstract:
In the inertial range of fully developed turbulence, we model the vertex network dynamics by an iterated unimodular map having the universal behavior. Inertial range anomalous scaling for the pair correlation functions of the velocity and the local energy dissipation is established as a consequence of the chaotic behavior of the unimodular map when the Feigenbaum attractor looses stability. The anomalous exponents determined by the Feigenbaum constant $\eta$ to the Kolmogorov's spectra are larger than those observed in experiments.

Abstract:
The spectrum of turbulence in superfluid liquid is modified by the nonlinear energy dissipation caused by the mutual friction between quantized vortices and the normal component of the liquid. In some region of two Reynolds parameters characterizing the flow of a superfluid, we found the new state of the fully developed turbulence. This state displays both the Kolmogorov-Obukhov 5/3-scaling law $E_k \propto k^{-5/3}$ and the new "3-scaling law" $E_k \propto k^{-3}$, each in a well separated range of $k$.

Abstract:
The problem of intermittency in developed hydrodynamic turbulence is considered. Explicit formulae taking into account effects of finite size of the inertial range are presented for the whole set of intermittency exponents. The formulae fit pretty well experimental data whose apparent discrepancies are attributed to different sizes of the inertial ranges in different experiments. Further predictions are given that can be verified by already existing experimental technique.

Abstract:
Using experimental longitudinal and transverse velocities data for very high Reynolds number turbulence, we study both anisotropy and asymmetry of turbulence. These both seem to be related to small scale turbulent structures, and to intermittency. We may assume that the large scale velocity shear gives an impact into the small scale turbulence, resulting in non-locality, and related anomalous events.

Abstract:
We review the Parisi-Frisch MultiFractal formalism for Navier--Stokes turbulence with particular emphasis on the issue of statistical fluctuations of the dissipative scale. We do it for both Eulerian and Lagrangian Turbulence. We also show new results concerning the application of the formalism to the case of Shell Models for turbulence. The latter case will allow us to discuss the issue of Reynolds number dependence and the role played by vorticity and vortex filaments in real turbulent flows.

Abstract:
The scale dependent intermittency exponents in developed hydrodynamic turbulence are calculated assuming a natural hierarchy of correlations in the turbulence. The major correlations are taken into account explicitly, while the remaining small correlations are considered as perturbations. The results agree very well with the currently available experimental data.