  We use a direct numerical simulations (DNS) database for turbulent flow in a
square duct up to bulk Reynolds number $\Rey_b=40000$, to quantitatively
analyze the role of secondary motions on the mean flow structure. For that
purpose we derive a generalized form of the identity of Fukagata, Iwamoto and
Kasagi (FIK), which allows to quantify the effect of cross-stream convection on
the mean streamwise velocity, wall shear stress and bulk friction coefficient.
Secondary motions are found to contribute for about $6\%$ of total friction,
and to act as a self-regulating mechanism of turbulence whereby wall shear
stress nonuniformities induced by corners are equalized, and universality of
the wall-normal velocity profiles is established. We also carry out numerical
experiments whereby the secondary motions are artificially suppressed, in which
case their equalizing role is partially taken by the turbulent stresses.

Fluid Dynamics