Turbulence of drag reducing polymer solutions

In wall bounded turbulent flows the addition of few parts per million in weight of long chain polymers in an otherwise Newtonian fluid may easily achieve a very large reduction of the viscous drag even though the polymers are substantially smaller than the typical scales of turbulence. This suggests that their effect should be explained in terms of matching characteristic time scales, rather than lengths. Actually the largest relaxation time of the chains commonly falls within the continuous range of time scales of fluid turbulence thus fixing the length (Lumley) below which the polymers are affected by the turbulent motion. Even in homogeneous isotropic turbulence, the chains are strongly stretched by the small scale turbulent motions to the point that a substantial depletion of the classical Richardson cascade of turbulent kinetic energy occurs (De Angelis et al.). The energy flux intercepted by the polymers is accumulated in the microstructure as elastic energy and dissipated by the relative friction between polymers and solvent. In typical conditions, no evidence of transfer of such energy towards smaller scales has been found, thus indicating that the dissipation in the microstructure is local in wavenumber space. This opens the possibility to model the effect of the polymers on the velocity field as a viscous-like process occurring on the whole range of scales where the polymers are acting. Clearly, near a solid wall, where the phenomena of interest for drag-reduction are occurring, the physics is more complex. Typically the classical mean velocity profile is substantially modified, showing a splitting of the logarithmic layer into two parts with distinguished values of the K´arm´an constants. Closer to the wall, in the so-called elastic layer, the logarithmic slope increases to a limiting value which corresponds to the maximum drag reduction (MDR) asymptote observed for the first time by Virk. Further from the wall, the classical slope is recovered to originate the Newtonian plug. As the amount of drag reduction increases, the region occupied by the elastic layer prevails up to filling the entire region. Typically one observes a reduction of the Reynolds shear stress and a change of other low-order statistical observables. Several phenomenological models (e.g. those of Lumley, De Gennes, L'vov-Procaccia) have been proposed to describe the sequence and the rational of the events taking place in such conditions. However, the related alteration of turbulence still deserves a detailed description for a full comprehension of the subject. Recent large scale simulations have provided a new evidence on the phenomena (e.g., among others, Beris, De Angelis, Choi) and complete and detailed data sets for a proper statistical analysis. A generalized form of scale-energy budget able to discriminate between the different kinds of energy fluxes which occur either in physical and scale space (see Marati et al. for Newtonian wall turbulence) is instrumental for the evaluation of the interaction of polymers with the near-wall environment. In fact, to alter the drag, the polymers must interfere directly with the production of turbulent kinetic energy. As we shall see, this defines a new length scale which characterizes the micro-mechanics of the elastic layer, leading to the notion of a range of scales where the draining of energy by the polymers is prevailing. The existence of such elastic range is shown to be crucial for the establishment of the altered state of turbulence which provides the observed drag-reduction phenomenology.