between $T_{opt,D}$ and $T_{opt,W}$ was left unnoticed until QR pointed out that $T_{opt,D}$ should be smaller than $T_{opt,W}$, which is known to be constant at $T_{opt,W}^+ \approx 125$ (when scaled by the friction velocity of the undisturbed flow, Karniadakis and Choi, 2003). QR’s analysis also showed that $T_{opt,D}$ should be a function of $D_m$ itself. Although their limited data appeared to validate these unconfirmed conjectures, it is an important aim of the present study to verify the existence of $T_{opt,D}$ and to assess its properties.

The analytical expression for $S$ yields additional information. As a further aim, we shall use $S$ to seek the smallest values of the oscillation parameters which guarantee a non-zero drag reduction effect. These oscillating conditions are referred to as “minimal" throughout the paper. The same analysis will also allow us to determine the oscillatory conditions needed to attain a specified amount of drag reduction and of net energy saving.

We shall also address the important issue of the effect of Reynolds number on the drag reduction. This point is still open to discussion, as the conclusions of previous studies do not fully agree. The experimental analysis of a turbulent boundary layer by @ricco-wu-2004 has shown 