that increasing the Reynolds number based on the momentum thickness and free-stream velocity from $Re_\theta$=500 to 1400 (with $D_m^+ \approx 240$) had no influence on the drag reduction. @choi-graham-1998’s experimental results in a pipe flow modified by circular wall oscillations have also indicated that changing the Reynolds number based on the bulk velocity and the pipe diameter from $Re$=23,300 to 36,300 gives variations of drag reduction which are of the order of the uncertainty range. The DNS of a turbulent channel flow conducted by @choi-xu-sung-2002 has instead revealed that the reduction in wall friction can be halved by increasing the Reynolds number from $Re_\tau=100$ to $Re_\tau=400$. We shall carry out a few DNS to ascertain whether or not increasing $Re_\tau$ affects the drag reduction.

The paper is organized as follows. In Section \[sec:procedure\], the numerical procedure and the discretization parameters are described. The main steps of the analysis by QR are briefly recalled in Section \[sec:laminar-analysis\], where the parameter $S$ and the function $T_{opt,D}$=$T_{opt,D}(D_m)$ are defined. Section \[sec:Topt\] discusses the quality of the analytical prediction of $T_{opt,D}$ on the basis of the DNS results. The minimal wall forcing conditions and the dependence of the drag reduction on the oscillatory parameters 