This paper is focused on the fundamental mechanism(s) of viscoelastic turbulence that leads to polymer-induced turbulent drag reduction phenomenon. A great challenge in this problem is the computation of viscoelastic turbulent flows, since the understanding of polymer physics is restricted to mechanical models. An effective state-of-the-art numerical method to solve the governing equation for polymers modeled as nonlinear springs, without using any artificial assumptions as usual, was implemented here on a three-dimensional channel flow geometry. The capability of this algorithm to capture the strong polymer-turbulence dynamical interactions is depicted on the results, which are much closer qualitatively to experimental observations. This allowed a more detailed study of the polymer-turbulence interactions, which yields an enhanced picture on a mechanism resulting from the polymer-turbulence energy transfers.

We present a numerical study of the statistical properties of three-dimensional dissipative turbulent flows at scales larger than the forcing scale. Our results indicate that the large scale flow can be described to a large degree by the truncated Euler equations with the predictions of the zero flux solutions given by absolute equilibrium theory, both for helical and non-helical flows. Thus, the functional shape of the large scale spectra can be predicted provided that scales sufficiently larger than the forcing length scale but also sufficiently smaller than the box size are examined. Deviations from the predictions of absolute equilibrium are discussed.Experimental and numerical studies of threedimensional homogeneous hydrodynamic turbulent flows have been so far mostly focused on the finite energy flux solutions of the Navier-Stokes that manifest themselves on scales smaller that the forcing scale for which the Kolmogorov cascade and intermittency take place [1]. This is because the flows of many experiments designed to study statistically stationary turbulent regimes are forced at scales not much smaller than the size of the container. This is also the case of most direct numerical simulations (DNS) for which the flow is often forced in the largest possible modes aiming for the largest scale separation between the forcing scale and the small scales in the dissipative range. A notable exception is of course the limit of two-dimensional flows for which the inverse cascade of energy [2] leads to a negative flux of energy that excites scales larger than the forcing scale.Many flows of geophysical or astrophysical interest far from the two-dimensional limit involve spatial structures at scales larger than the forcing scale. At these scales, no energy flux is expected and the usual Kolmogorov cascade picture does not hold. This is also true for some flows involved in industrial processes, such as large scale turbulent mixing. Dynamical and statistical properties of the zero flux solutions in scales larger than the forcing scale could thus be of interest for many applications in three-dimensional hydrodynamic turbulence.Despite the lack of quantitative studies of the large scales in three-dimensional statistically stationary turbulence, it has been believed since a long time that the scales larger than the forcing scale are in statistical equilibrium (see page 209 of reference [1]). The argument is that the energy driving the flow is transferred from the forcing scale ℓ f to the dissipation scale ℓ η by the Kolmogorov cascade and that no mean energy flux exists toward scales larger than ℓ f . The scales between ℓ f and the container size L, thus do not involve any mean energy flux and could be in statistical equilibrium.With this assumption a k 2 energy spectrum similar to the Rayleigh-Jeans spectrum for blackbody radiation would result with all modes in the range 2π/L < k < 2π/ℓ f being in equipartition. Such a spectrum has been obtained long ago using the Hopf equation for flows without forcing and viscosity [3]...