Turbulence and secondary motions in square duct flow

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 (Rey 1000). Numerical simulations are carried out over very long integration times to get adequate convergence of the flow statistics, and specifically to achieve high-fidelity representation of the secondary motions which arise. The intensity of the latter is found to be on the order of 1 %-2 % of the bulk velocity, and approximately unaffected by Reynolds number variation, at least in the range under scrutiny. 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 approximated with good accuracy by eigenfunctions of the Laplace operator, in the core part of the duct. Despite their effect of redistributing the wall shear stress along the duct perimeter, we find that secondary motions do not have a large influence on the bulk flow properties, and the streamwise velocity field can be characterized with good accuracy as resulting from the superposition of four flat walls in isolation. As a consequence, we find that parametrization based on the hydraulic diameter concept, and modifications thereof, are successful in predicting the duct friction coefficient.