Turbulence Driven by a Deterministic Chaotic Dynamics

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 corrections determined by the Feigenbaum constant η to the Kolmogorov’s spectra are larger than those observed in experiments. Turbulence Modeling and Turbulence Control in fluid dynamics is a paradigmatic problem and successful techniques in this domain may serve as models to deal with other complex extended systems. It is generally accepted now that a real understanding of turbulence requires calculations and models that are globally sensitive to all length scales. To elucidate this conviction, let us briefly explain the present situation in the quantum field theory of fully developed turbulence of incompressible fluid which has been developed for the last two decades (see [1] for a review). In order to be specific we consider the correlator K = 〈v(k)v(−k)〉 of the velocity field v in the momentum representation (k ≡ |k| and 〈. . .〉 denote the time averaging). It is found from the Dyson equation K = νk +m − Σ(k), (1) Solid State Electronics Department, Research Institute of Physics, St.-Peterburg State University (Russia) and Centre de Physique Theorique, CNRS; e-mail: [email protected] Centre de Physique Theorique, CNRS, Luminy Case-907, 13288 Marseille (France)