  We perform direct numerical simulations of rotating turbulent Waleffe flow,
the flow between two parallel plates with a sinusoidal streamwise shear driving
force, to study the formation of large-scale structures and the mechanisms for
momentum transport. We simulate different cyclonic and anti-cyclonic rotations
in the range of dimensionless rotation numbers (inverse Rossby numbers)
$R_\Omega$ $\in$ $[-0.16, 2.21]$, and fix the Reynolds number to $Re=3.16\times
10^3$, large enough such that the shear transport is almost entirely due to
Reynolds stresses and viscous transport is negligible. We find an optimum
rotation in anti-cyclonic regime at $R_\Omega=0.63$, where a given streamwise
momentum transport in the wall-normal direction is achieved with minimum mean
energy of the streamwise flow. We link this optimal transport to the strength
of large scale structures, as was done in plane Couette by Brauckmann \&
Eckhardt (J. Fluid Mech., 815, 2017). Furthermore, we explore the large-scale
structures and their behaviour under spanwise rotation, and find disorganized
large structures at $R_\Omega =0$ but highly organized structures in the
anti-cyclonic regime, similar to the rolls in rotating plane Couette and
turbulent Taylor Couette flow. We compare the large scale structures of plane
Couette flow and Waleffe flow, and observe that the streamwise vorticity is
localized inside the cores of the rolls. We show that the rolls take energy
from the mean flow at long time-scales, and relate these structures to
eigenvalues of the streamfunction.
