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The budgets of the Reynolds stress, turbulent kinetic energy and streamwise enstrophy are evaluated through direct numerical simulations for the turbulent channel flow of a viscoelastic polymer solution modeled with the Finitely Extensible Nonlinear Elastic with the Peterlin approximation (FENE-P) constitutive equation. The influence of viscoelasticity on the budgets is examined through a comparison of the Newtonian and the viscoelastic budgets obtained for the same constant pressure drop across the channel. It is observed that as the extensional viscosity of the polymer solution increases there is a consistent decrease in the production of Reynolds stress in all components, as well as in the other terms in the budgets. In particular, the effect of the flow elasticity, which is associated with the reduction in the intensity of the velocity-pressure gradient correlations, potentially leads to a redistribution of the turbulent kinetic energy among the streamwise, the wall-normal and the spanwise directions. In this work, we also show that in the presence of viscoelasticity there is a significant reduction in all components of the production of streamwise enstrophy. This is consistent with a proposed mechanism for polymer-induced drag reduction through the inhibition of vortex stretching by the high extensional viscosity of the polymer solution.

We present a study of the drag reduction induced by rigid fibres in a turbulent channel flow using direct numerical simulation. The extra stresses due to the fibres are calculated with the well-known constitutive equation involving the moments of the orientation vector. Drag reductions of up to 26% are calculated, with the largest drag reductions observed using non-Brownian fibres and semi-dilute concentrations. These findings suggest that elasticity is not necessary to achieve turbulent drag reduction. Flow statistics show trends similar to those observed in simulation of polymeric drag reduction: Reynolds stresses are reduced, velocity fluctuations in the wall-normal and spanwise directions are reduced while streamwise fluctuations are increased, and streamwise vorticity is reduced. We observe strong correlations between the fibre stresses and inter-vortex extensional flow regions. Based on these correlations and instantaneous visualizations of the flow field, we propose a mechanism for turbulent drag reduction by rigid fibre additives.

We describe a method for direct numerical simulation of polymer-induced friction drag reduction in turbulent boundary layers. The effect of the polymer additives that induce spatial variations of skin-friction drag is included in the momentum equation through a continuum constitutive model for the viscoelastic stress, which is based on the evolution of a parameter describing the fluid microstructure. We demonstrate that the turbulence structure and polymer microstructure evolve asynchronously as one moves in the streamwise direction. We observe an initial development length, which is followed by a quasisteady region where variations in drag reduction are weak. High drag reduction behavior can be present at short downstream distances from the inflow plane.