Predictability of the energy cascade in 2D turbulence

The predictability problem in the inverse energy cascade of two-dimensional turbulence is addressed by means of direct numerical simulations. The growth rate as a function of the error level is determined by means of a finite size extension of the Lyapunov exponent. For error within the inertial range, the linear growth of the error energy, predicted by dimensional argument, is verified with great accuracy. Our numerical findings are in close agreement with the result of TFM closure approximation.Comment: 3 pages, 3 figure

Chaos and predictability of homogeneous-isotropic turbulence

We study the chaoticity and the predictability of a turbulent flow on the basis of high-resolution direct numerical simulations at different Reynolds numbers. We find that the Lyapunov exponent of turbulence, which measures the exponential separation of two initially close solution of the Navier-Stokes equations, grows with the Reynolds number of the flow, with an anomalous scaling exponent, larger than the one obtained on dimensional grounds. For large perturbations, the error is transferred to larger, slower scales where it grows algebraically generating an "inverse cascade" of perturbations in the inertial range. In this regime our simulations confirm the classical predictions based on closure models of turbulence. We show how to link chaoticity and predictability of a turbulent flow in terms of a finite size extension of the Lyapunov exponent.Comment: 5 pages, 5 figure

Turbulent channel without boundaries: The periodic Kolmogorov flow

The Kolmogorov flow provides an ideal instance of a virtual channel flow: It has no boundaries, but nevertheless it possesses well defined mean flow in each half-wavelength. We exploit this remarkable feature for the purpose of investigating the interplay between the mean flow and the turbulent drag of the bulk flow. By means of a set of direct numerical simulations at increasing Reynolds number we show the dependence of the bulk turbulent drag on the amplitude of the mean flow. Further, we present a detailed analysis of the scale-by-scale energy balance, which describes how kinetic energy is redistributed among different regions of the flow while being transported toward small dissipative scales. Our results allow us to obtain an accurate prediction for the spatial energy transport at large scales.Comment: 7 pages, 8 figure

Introduction to chaos and diffusion

This contribution is relative to the opening lectures of the ISSAOS 2001 summer school and it has the aim to provide the reader with some concepts and techniques concerning chaotic dynamics and transport processes in fluids. Our intention is twofold: to give a self-consistent introduction to chaos and diffusion, and to offer a guide for the reading of the rest of this volume.Comment: 39 page