Polymer induced drag reduction in exact coherent structures of plane Poiseuille flow

<jats:p>Nonlinear traveling waves that are precursors to laminar-turbulent transition and capture the main structures of the turbulent buffer layer have recently been found to exist in all the canonical parallel flow geometries. The present work examines the effect of polymer additives on these “exact coherent states” (ECS) in the plane Poiseuille geometry, using the FENE-P constitutive model for polymer solutions. In experiments with a given fluid, Reynolds and Weissenberg numbers are linearly related (i.e., Wi∕Re=const). In this situation, we study the effects of viscoelasticity on velocity field and polymer stress field along some experimental paths, which represent different flow behaviors as Re (and Wi) increases. The changes to the velocity field for the viscoelastic nonlinear traveling waves qualitatively capture many of those experimentally observed in fully turbulent flows of polymer solutions at low to moderate levels of drag reduction: drag is reduced, streamwise velocity fluctuations increase, and wall-normal and spanwise velocity fluctuations decrease. The mechanism underlying these observations is the suppression of streamwise vortices by the polymer forces exerted on the fluid. Specifically, at sufficiently high wall shear rates, viscoelasticity completely suppresses these streamwise vortices in the near-wall region, as is found in experiments in the maximum drag reduction regime. The mean shear stress balance for the nonlinear traveling waves shows that Reynolds shear stress decreases and polymer stress increases monotonically with the increase of viscoelasticity, as is found in full turbulence. The study of the influence of the viscoelasticity on the turbulent kinetic energy and Reynolds stress budgets shows that as Re (and Wi) increases, there is a consistent decrease in the production, diffusion, and dissipation of turbulent kinetic energy. The decrease in the velocity pressure gradient term leads to a redistribution of the turbulent kinetic energy among the streamwise, wall-normal and spanwise directions. The influence of the rheological parameters on the viscoelastic ECS is analyzed. It is found that the degree of drag reduction is determined primarily by the extensional viscosity and Weissenberg number. The optimum wavelength conditions under which the viscoelastic ECS first come into existence are also investigated. The wavelengths in streamwise and spanwise directions and the wall-normal extent of the ECS all increase monotonically with the increase of viscoelasticity, as is found in experiments.</jats:p>