0,$$ the maximum wall velocity and displacement which give $P_{net,max}$, the maximum $P_{net}$ at fixed $T^+$, are found: $$\left\{
\begin{array}{l l}
W_{net,max}^+ \\
D_{net,max}^+ \\
\end{array}
\right\}
=
W_1
\left\{
\begin{array}{l l}
1 \\
T^+/\pi \\
\end{array}
\right\}
\exp \left( - \frac{\ell_a^+}{2} \sqrt{\frac{\pi}{T^+}} \right),$$ where $W_1=(S_1 U_b^+/50)^{1/2}=6.55$ for $Re_\tau=200$. It follows that $$P_{net,max}(T^+)=S_1 \sqrt{\frac{\pi}{T^+}}  \exp \left( - \ell_a^+ \sqrt{\frac{\pi}{T^+}} \right)\left(P_1 - \ell_a^+ \sqrt{\frac{\pi}{T^+}}\right) + S_2,$$ where $P_1 = 2\ln(W_1/W_{th}^+) - 1 = 1.7$ for $Re_\tau=200$. The location of the overall maximum $P_{net}$ is well estimated at $T^+=150$ and $W_m^+=4.2$, while its value is slightly underpredicted: 5.6% instead of 7.3% (QR).

Effect of Reynolds number {#sec:Reynolds}
=========================

Our calculations show that slightly lower amounts of drag reduction are obtained by doubling the value of the Reynolds number from $Re_\tau=200$ to $Re_\tau=400$ for $W_m^+=12$ and $T^+=30$, 125 and 200, as shown in Fig. \[fig:figure\_8\]. The wall-shear stress reduction decreases from 21.7% to 20.3% at $T^+=30$ (6.6% change), from 32.5% to 28.1% at $T^+=125$ (13.4% change), and from 27.2% to 22% at $T^+=200$ (19.2% change). These variations thus increase with $T$ for fixed $W_m$. This result is in broad agreement with the analysis by @choi-xu-sung-2002, although a smaller computational domain and a lower 