are presented in Section \[sec:minimal\]. Section \[sec:Reynolds\] describes the effect of the Reynolds number on the drag reduction. Section \[sec:validation-datasets\] presents the drag reduction data available in the literature, providing an evaluative overview on the state of the art in terms of a comparison between the amounts of drag reduction and the estimates based on $S$. Section \[sec:conclusions\] is devoted to a summary.

Numerical procedure {#sec:procedure}
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We have studied the turbulent flow in a channel with moving walls through direct numerical simulations of the incompressible Navier–Stokes equations. The walls move in phase along the spanwise direction with velocity $$W(t;W_m,T) = W_m \sin \left( \frac{2 \pi t}{T} \right).$$ Figure \[fig:figure\_1\] displays a sketch of the computational domain.

Our pseudo-spectral solver is described in @luchini-quadrio-2006: it is based on Fourier expansions in the homogeneous directions and on fourth-order accurate, compact finite-difference schemes for the discretization of the differential operators in the wall-normal direction. Aliasing errors in the computation of the non-linear terms are eliminated by expanding the flow variables into a (at least) $3/2$ larger number of modes for each homogeneous direction before transforming from the Fourier to the physical space. The temporal integration is carried out with a partially-implicit procedure: 