*7.1. Streamwise Velocity*

The conditional sampling of the velocity data and their ensemble-averaging is conducted in a similar manner as has been conducted earlier by Whalley et al. [24] and Kushwaha et al. [22]. For the low-drag events, the drop in the ensemble averaged velocities is observed to be more significant near the wall, with the effect disappearing near the centreline. For the high-drag events, an analogous behaviour to low-drag events is observed. Figure 9 shows an example of the ensemble averaged streamwise velocities for various wall-normal locations at *Re<sup>τ</sup>* = 180 during the low- and high-drag events. Here, the ensemble-averaged streamwise velocities (*U<sup>L</sup>* , *<sup>U</sup>H*) are normalised by *<sup>u</sup>τ*. Very similar results were observed for other Reynolds numbers and therefore are not shown. This behavior of the ensemble averaged streamwise velocities is similar to those previously obtained by Whalley et al. [24] and Kushwaha et al. [22] for 70 ≤ *Re<sup>τ</sup>* ≤ 100. Therefore, it can be said that the ensemble-averaged streamwise velocity during the low- and high-drag events, which were previously observed for 70 ≤ *Re<sup>τ</sup>* ≤ 100, shows similar characteristics even for the flow in the fully-turbulent regime.

**Figure 9.** Ensemble-averaged streamwise velocity for *Reτ* = 180 during (**a**) low-drag events and (**b**) high-drag events. Here, *t* <sup>+</sup> = 0 indicates the beginning of a low-drag or a high-drag event. The criteria to detect a low-drag event are Δ*t* + *cr* = 200 and *τw*/*τ<sup>w</sup>* < 0.9, and a high-drag event are Δ*t* + *cr* = 200 and *τw*/*τ<sup>w</sup>* > 1.1.

Figure 10 shows the unconditional and conditionally-averaged streamwise velocity profiles for *Reτ* = 70, 85, 120, 180 and 250 obtained using experiments, and *Reτ* = 70 and 85 obtained using DNS. Here, the normalisation of the unconditional velocity and the corresponding wall-normal locations are carried out using the time-averaged friction velocity (*uτ*). The conditionally-averaged streamwise velocities and the corresponding wall-normal locations are normalised by the conditionally-averaged friction velocities (*u<sup>τ</sup> <sup>L</sup>* for low-drag and *u<sup>τ</sup> <sup>H</sup>* for high-drag). Before studying the profiles during the conditional events, we first focus on the unconditional (time-averaged) profiles. Experimental and DNS results are in good agreement for *Reτ* = 70 and 85. The unconditional profile obtained for *Re<sup>τ</sup>* = 180 is also in good agreement with the DNS profile obtained by [26] for *Re<sup>τ</sup>* = 180, and the velocity profiles for *Re<sup>τ</sup>* of 180 and 250 approximately collapses on the log-law profile (*U*<sup>+</sup> = 2.5 ln *<sup>y</sup>*<sup>+</sup> + 5.5) for *<sup>y</sup>*<sup>+</sup> <sup>≥</sup> 30.

The velocity statistics during the conditional events is investigated in such a way that only the upper (for high-drag) or lower plateau (for low-drag) of the instantaneous wall shear stress and velocity are considered for the conditional sampling. This is done to avoid any transient behaviours (start and end of conditional events) affecting the result. Therefore, only wall shear stress and velocity data between 30 < *t* <sup>+</sup> < *t* + *end* − 30 are used for conditional sampling, where *t* + *end* indicates the end of a low-drag or a high-drag event. For *y*<sup>+</sup> 10, the unconditional and conditional profiles for *Re<sup>τ</sup>* = 70 and 85 obtained using DNS almost collapse on each other. For *y*<sup>+</sup> - 10, the conditionally averaged velocity profiles are closer to Virk's MDR asymptote than their time-averaged values (for all the Reynolds numbers studied). Previously, Kushwaha et al. [22] and Whalley et al. [24] showed that at 70 ≤ *Re<sup>τ</sup>* ≤ 100, the low-drag velocity profiles get closer to the Virk's MDR and the lower-branch of the nonlinear TW solutions (as obtained by Park and Graham [13]) for similar wall-normal locations, *y*<sup>+</sup> 35. Therefore, the present result confirms the validity of this phenomenon for Reynolds numbers in the fully-turbulent regime. There is a very good agreement between the experimental and DNS results for the velocity profiles during the low-drag events at *Reτ* = 70 and 85. For higher wall-normal locations the conditional velocity profiles start to deviate from Virk's MDR profile, and for *y*<sup>+</sup> -100, the conditional velocity profiles have a slightly higher slope as compared to the Prandtl-von Kármán log-law, as seen for *Reτ* = 180 and 250. For the high-drag events, the conditional velocity profiles are lower than the unconditional profiles for all the Reynolds numbers.

**Figure 10.** Unconditional and conditionally averaged streamwise velocity profiles for *Reτ* = 70, 85, 120, 180 and 250 during low-drag and high-drag events. All the symbols represent the experimental data. Here, the conditionally averaged streamwise velocity data is normalised using conditionally averaged friction velocity. Yellow dotted line represents the Prandtl-von Kármán log-law: *U*<sup>+</sup> = 2.5 ln *y*<sup>+</sup> + 5.5 and the black dash-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]. Black dashed line represents the time-averaged velocity profile obtained using DNS at *Reτ* = 180 by Kim et al. [26].

To further investigate the slope of the conditional velocity profiles, the so-called indicator function is calculated, which is generally used to study the logarithmic dependence of the mean velocity profile [44]. For the unconditional velocity data, the indicator function is given by: *<sup>ζ</sup>* = *<sup>y</sup>*+*dU*+/*dy*+. For the conditional velocity data, the indicator functions are given by *ζ <sup>L</sup>* = *<sup>y</sup>*<sup>+</sup>*LdU*<sup>+</sup>*<sup>L</sup>* /*dy*+*<sup>L</sup>* and *ζ <sup>H</sup>* = *<sup>y</sup>*<sup>+</sup>*HdU*<sup>+</sup>*H*/*dy*+*<sup>H</sup>* for the low- and high-drag events, respectively. The profiles of indicator function are shown in Figure 11. It can be seen that that for *Re<sup>τ</sup>* = 70 and 85, the *ζ* profiles do not exhibit a logarithmic dependence. For *Re<sup>τ</sup>* = 120, 180 and 250, the *ζ* profiles approximately collapse on the value of 1/*<sup>κ</sup>* <sup>=</sup> 2.5 for *<sup>y</sup>*<sup>+</sup> <sup>≥</sup> 30, thus suggesting a logarithmic dependence. Here, *<sup>κ</sup>* is the von Kármán constant. It is observed from Figure 11a,b that the *ζ <sup>L</sup>* profiles at all Reynolds numbers are closer to the Virk's MDR (1/*<sup>κ</sup>* <sup>=</sup> 11.7) for *<sup>y</sup>*<sup>+</sup> <sup>≤</sup> 30. For *Re<sup>τ</sup>* = 120, 180 and 250, the *<sup>ζ</sup> <sup>L</sup>* profiles remain above the unconditional profiles for *<sup>y</sup>*<sup>+</sup> <sup>≥</sup> 30, thus showing that the slope of the low-drag velocity profiles is slightly higher than the unconditional profiles in the log-law region. Figure 11c,d shows that the *ζ <sup>H</sup>* profiles at *Re<sup>τ</sup>* = 70 and 85, are lower than the *<sup>ζ</sup>* profiles (except close to the centreline), with the effect being more significant for *<sup>y</sup>*<sup>+</sup> <sup>≤</sup> 30. For *Re<sup>τ</sup>* = 120, 180 and 250, the slope of the *<sup>ζ</sup> <sup>H</sup>* profiles is slightly lower than the *ζ* profiles for all wall-normal locations.

**Figure 11.** Unconditional (open circles) and conditionally averaged (closed squares) indicator functions for (**a**) *Reτ* = 70 and 85, and for (**b**) *Reτ* = 120, 180 and 250 during low-drag events. Unconditional (open circles) and conditionally averaged (closed squares) indicator functions for (**c**) *Reτ* = 70 and 85, and for (**d**) *Reτ* = 120, 180 and 250 during high-drag events. The criteria to detect a low-drag event is Δ*t* + *cr* = 200 and *τw*/*τ<sup>w</sup>* < 0.9, and a high-drag event is Δ*t* + *cr* = 200 and *τw*/*τ<sup>w</sup>* > 1.1. Dashed lines represent 2.5 and dotted lines represent 11.7.
