*7.2. Similarity between Turbulent Drag Reduction and Low-Drag Events in Newtonian Turbulence*

To quantify the "drag reduction" during the low-drag events a percentage decrease in the wall shear stress, during these low-drag events, is calculated. The comparison with the drag-reduction literature is carried out only for *Re<sup>τ</sup>* = 180 and 250. It is found that the percentage drag reduction is about 36% for *Reτ* = 180 and 250 when calculated using Equation (5).

$$\%DR = \frac{\overline{\tau\_w} - \overline{\tau\_w}^L}{\overline{\tau\_w}} \approx 36\% (\text{Re}\_\tau = 180 \text{ and } 250). \tag{5}$$

This level of drag reduction is similar to some of the other techniques employed previously to reduce drag in channel flows. For example, when using polymer additives at low concentration, the low-drag reduction (LDR) regime is observed [45,46]. A comparison is made with the experimental data obtained by Warholic et al. [45] at *Reh* ≈ 20,000 for the case where a drag reduction of about 33% was observed. Drag reduction due to superhydrophobic surfaces were investigated by Min and Kim [47]. They conducted DNS in a channel flow for *Re<sup>τ</sup>* = 180 (for DR = 0) and by using streamwise slip, they obtained a maximum drag reduction of 29%. Choi et al. [48] implemented DNS in a channel flow at *Re<sup>τ</sup>* = 180 (for DR = 0) to numerically study the effect of blowing and suction on the skin-friction drag. They employed out-of-phase boundary conditions for the spanwise and wall-normal velocities

to simulate the blowing and suction effects on the channel, and obtained a drag reduction of about 26% by applying spanwise control.

In Figure 12, a comparison is shown between the streamwise velocity profiles obtained using these three techniques for turbulent drag reduction and the conditional streamwise velocity profile obtained in the present experiment at *Reτ* = 180 and 250.

**Figure 12.** Conditional streamwise velocity profiles for *Reτ* = 180 and 250 during the low-drag events. Streamwise velocity profiles, where different drag reduction mechanisms are employed previously: Warholic et al. [45] used polymeric additive, Min and Kim [47] used hydrophobic surface in the form of slip-boundary condition for the streamwise direction and Choi et al. [48] applied out-of-phase boundary condition to the spanwise velocity at the surface. Dashed line represents the Prandtl-von Kármán log-law: *U*<sup>+</sup> = 2.5 ln *y*<sup>+</sup> + 5.5 and dotted line represents the lower end of the 95% confidence interval of the Virk's MDR asymptote: *<sup>U</sup>*<sup>+</sup> = 11.4 ln *<sup>y</sup>*<sup>+</sup> <sup>−</sup> 18.5 [43].

A good agreement can be seen between the conditionally averaged profile for *Reτ* = 180 and 250 and the profile obtained by Warholic et al. [45] for *DR* = 33% using polymer additives. The profiles obtained by Min and Kim [47] and Choi et al. [48], and the present experiment are also in relatively good agreement with the obvious difference arising due to the lower levels of drag reduction reported in these cases. One major difference in the result obtained by Min and Kim [47] is that the velocity profile shifts upwards even closer to the wall which is the consequence of the slip boundary condition. Therefore, it suggests that for the fully-turbulent flows (*Re<sup>τ</sup>* = 180 and 250), the conditional streamwise velocity for *y*<sup>+</sup> - 20 during the low-drag events mimics the flow as observed during the LDR phenomenon due to polymer addition or the drag reduction due to spanwise oscillation. For the case of superhydrophobicity, this similarity between the velocity profiles can be observed approximately in the log-law region. Thus, if a method could be found to encourage the turbulent state to enter the low-drag "hibernating" state more often, a significant time-averaged drag reduction would be achievable.
