only explored a portion of the $(T,W_m)$ space. It would be of interest to verify the existence of a [*maximum*]{} Reynolds number above which the wall-shear stress is unchanged, and to investigate how it varies with $T$ and $W_m$.

The effect of the Reynolds number on the maximum net energy saving can be estimated by considering separately the changes on the power spent $P_{sp}$ to move the walls and on the power saved through the wall motion, i.e. the drag reduction. As shown by (\[eq:power-spent-percent\]), $P_{sp}(\%)$ varies with the Reynolds number only because of a change of $U_b^+$, which can be expressed as follows: $$\label{eq:Ub}
U_b^+ = \frac{Re}{2 Re_\tau} = 7.715 Re_\tau^{0.136},$$ where we have used $Re=15.43 Re_\tau^{1.136}$ given by [@pope-2000] at page 279. ($U_b^+$ at $Re_\tau=200, 400$ obtained by our calculations is less than $1$% different from the one given by (\[eq:Ub\])). It follows that $$\label{eq:Psp}
P_{sp}(\%) = 6.481 (W_m^+)^2 \sqrt{\frac{\pi}{T^+}} Re_\tau^{-0.136}.$$ The power spent for $W_m^+=4.5$ and $T^+=125$, namely for the oscillating conditions at which the maximum net energy saving occurs (see Section \[sec:minimal\]), decreases from $P_{sp}$=10.1% at $Re_\tau=200$ to $P_{sp}$=9.2% at $Re_\tau=400$. By assuming that the drag reduction for these oscillation conditions decreases by the same amount 