on the linear relation between the drag reduction and $S$ is valid. These quantities are not well correlated when the oscillation uncouples from the near-wall turbulence dynamics, namely when half the period of oscillation is larger than a typical pseudo-Lagrangian time scale representing the survival time of the longest-lived structures ($\approx 60$ time units) [@quadrio-luchini-2003]. When $T$ is large, the near-wall structures have enough time to develop their inner dynamics between successive sweeps of the Stokes layer. In this limit, the flow adapts to a new quasi-steady three-dimensional condition, where time can be treated as a parameter and the drag-reducing effect of the Stokes layer is lost.

Drag reduction scaling and optimum period at fixed $D_m$ {#sec:Topt}
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Fig. \[fig:figure\_3\] shows the drag reduction DNS dataset versus $S$ for $Re_\tau=200$ produced for the present analysis, together with the data by QR for $T^+ \leq 150$. The oscillating conditions are largely different, but the data collapse well on the straight line. The linear regression (\[eq:fit\]) has been recomputed by grouping our DNS data with most of the data available in the literature, which are analysed in Section \[sec:validation-datasets\]. The only discarded datasets were the ones by [@choi-xu-sung-2002] at $Re_\tau=100$ because during 