  We have performed direct numerical simulation of the turbulent flow of a
polymer solution in a square duct, with the FENE-P model used to simulate the
presence of polymers. First, a simulation at a fixed moderate Reynolds number
is performed and its results compared with those of a Newtonian fluid to
understand the mechanism of drag reduction and how the secondary motion,
typical of the turbulent flow in non-axisymmetric ducts, is affected by polymer
additives. Our study shows that the Prandtl's secondary flow is modified by the
polymers: the circulation of the streamwise main vortices increases and the
location of the maximum vorticity move towards the center of the duct. In-plane
fluctuations are reduced while the streamwise ones are enhanced in the center
of the duct and dumped in the corners due to a substantial modification of the
quasi-streamwise vortices and the associated near-wall low- and high-speed
streaks; these grow in size and depart from the walls, their streamwise
coherency increasing. Finally, we investigated the effect of the parameters
defining the viscoelastic behavior of the flow and found that the Weissenberg
number strongly influences the flow, with the cross-stream vortical structures
growing in size and the in-plane velocity fluctuations reducing for increasing
flow elasticity.
