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

Condensate in quasi two-dimensional turbulence

We investigate the process of formation of large-scale structures in a turbulent flow confined in a thin layer. By means of direct numerical simulations of the Navier-Stokes equations, forced at an intermediate scale, we obtain a split of the energy cascade in which one fraction of the input goes to small scales generating the three-dimensional direct cascade. The remaining energy flows to large scales producing the inverse cascade which eventually causes the formation of a quasi two-dimensional condensed state at the largest horizontal scale. Our results shows that the connection between the two actors of the split energy cascade in thin layers is tighter than what was established before: the small scale three-dimensional turbulence acts as an effective viscosity and dissipates the large-scale energy thus providing a viscosity-independent mechanism for arresting the growth of the condensate. This scenario is supported by quantitative predictions of the saturation energy in the condensate

Split energy cascade in turbulent thin fluid layers

We discuss the phenomenology of the split energy cascade in a three-dimensional thin fluid layer by mean of high resolution numerical simulations of the Navier-Stokes equations. We observe the presence of both an inverse energy cascade at large scales, as predicted for two-dimensional turbu- lence, and of a direct energy cascade at small scales, as in three-dimensional turbulence. The inverse energy cascade is associated with a direct cascade of enstrophy in the intermediate range of scales. Notably, we find that the inverse cascade of energy in this system is not a pure 2D phenomenon, as the coupling with the 3D velocity field is necessary to guarantee the constancy of fluxes

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