  We study turbulent flows in pressure-driven ducts with square cross-section
through direct numerical simulation in a wide enough range of Reynolds number
to reach flow conditions which are representative of fully developed
turbulence. Numerical simulations are carried out over extremely long
integration times to get adequate convergence of the flow statistics, and
specifically high-fidelity representation of the secondary motions which arise.
The intensity of the latter is found to be in the order of 1-2% of the bulk
velocity, and unaffected by Reynolds number variations. The smallness of the
mean convection terms in the streamwise vorticity equation points to a simple
characterization of the secondary flows, which in the asymptotic high-Re regime
are found to be approximated with good accuracy by eigenfunctions of the
Laplace operator. Despite their effect of redistributing the wall shear stress
along the duct perimeter, we find that secondary motions do not have large
influence on the mean velocity field, which can be characterized with good
accuracy as that resulting from the concurrent effect of four independent flat
walls, each controlling a quarter of the flow domain. As a consequence, we find
that parametrizations based on the hydraulic diameter concept, and
modifications thereof, are successful in predicting the duct friction
coefficient.
