the oscillation the Reynolds number was too low and the ones by [@dhanak-si-1999], whose analysis was based on a simplified model and not on the Navier-Stokes equations. The new correlation parameters are $C_S=0.92$, $S_1$=135.11, $S_2$=-0.85, $\ell_a^+=6.2$ and $W_{th}^+=1.7$, which are essentially unchanged from the analysis in QR for $Re_\tau=200$. These new values have been used for the present data analysis.

The results of the present simulations, designed to identify $T_{op t,D}$, are compared in Fig. \[fig:figure\_4\] with the prediction obtained by (\[eq:S\]) and (\[eq:fit\]). Good agreement occurs except for three data points at low $T$ (open symbols in Fig. \[fig:figure\_4\] and black squares in Fig. \[fig:figure\_3\]). For these points, the predicted values are lower than the actual DNS data. We have not been able to explain this behaviour at very small $T$, which corresponds – being the displacement fixed – to very large $W_m$ ($W_m^+>$40). We have first verified that the space-averaged spanwise velocity profile at various phases still agrees with the laminar solution, from which (\[eq:S\]) is determined. As a further check, the Reynolds number $Re_\delta$ for the spanwise oscillating flow, based on $W_m$ and on the Stokes layer thickness $\delta_s = \sqrt{\nu T/\pi}$ (where $\nu$ is the kinematic 