*7.3. Reynolds Shear Stress*

DNS studies by Park and Graham [13] and Xi and Graham [12], using MFU at *Re<sup>τ</sup>* = 85, showed that the Reynolds shear stress drops to a very low value during the low-drag events. There is still no information in the prior literature regarding the RSS characteristics, during the conditional events, from either physical experiments or from DNS in extended domains. For the experiments (discussed in Section 2), two-component (streamwise and wall-normal) velocity measurements have been made for *Re<sup>τ</sup>* = 70 and 85 to study the behaviour of the Reynolds shear stress during the conditional events. To carry out the conditional sampling, each wall-normal location is sampled for 2 h

while simultaneously measuring the wall shear stress using HFA. DNS study is conducted for *Reτ* = 70 and 85 which provides the streamwise and wall-normal velocity information for various wall-normal locations (discussed in Section 3).

To calculate the conditional RSS, the streamwise velocity fluctuations and the wall-normal velocity fluctuations during the conditional events are calculated by subtracting their time-averaged values from the instantaneous conditional values. Figure 13 shows the ensemble averaged wall-normal velocities (*V<sup>L</sup>* ) and ensemble averaged Reynolds shear stress (−*uvL*). All the quantities are normalised by the time-averaged friction velocity (*uτ*). The threshold and time-duration criteria to detect a low-drag events are *τw*/*τ<sup>w</sup>* < 0.9 and Δ*t* + *cr* = 200, respectively. For *y*<sup>+</sup> < 21, experimental data are not available and therefore only DNS results are shown. A fairly good agreement between the experimentally and numerically obtained ensemble-averaged wall-normal velocity and RSS is observed. From continuity, the time-averaged wall-normal velocity must be zero, as can be observed from the DNS data. There is a slight discrepancy in the time-averaged values for the experimental data which is attributed to the error associated with the LDV measurements (discussed in Section 2). The conditionally averaged wall-normal velocity is higher than the time-averaged value during the low-drag events.

**Figure 13.** Ensemble-averaged wall-normal velocities (**a**,**c**,**e**,**g**,**i**) and Reynolds shear stresses (**b**,**d**,**f**,**h**,**j**) obtained using DNS (red solid lines) and experiment (black solid lines) during low-drag events for *Reτ* = 70. Here, *t* <sup>+</sup> = 0 indicates start of low-drag events. The time-averaged values for the corresponding wall-normal locations are shown using red dashed lines (obtained using DNS) and black dashed lines (obtained using experiment). The criteria to detect a low-drag event is Δ*t* + *cr* = 200 and *τw*/*τ<sup>w</sup>* < 0.9.

The ensemble averaged streamwise velocities have already been shown previously in Section 7.1. Based on the conditionally-averaged streamwise and wall-normal velocities, it can be said that the low-drag events form a subset of so-called Q2 events, i.e., *u* < 0 and *v* > 0. Figure 14 shows the ensemble-averaged wall-normal velocity and RSS during the high-drag events for *y*<sup>+</sup> = 21 and 40. The ensemble averaged wall-normal velocity is lower than the time-averaged wall-normal velocity whereas the ensemble averaged RSS is unchanged. Again, based on the conditionally-averaged streamwise and wall-normal velocities, it can be said that the high-drag events form a subset of Q4 events, i.e., *u* > 0 and *v* < 0. This behaviour will be further investigated in the following discussions.

**Figure 14.** Ensemble-averaged wall-normal velocities (**a**,**c**) and Reynolds shear stresses (**b**,**d**) obtained using experiment during high-drag events for *Reτ* = 70. Here, *t* <sup>+</sup> = 0 indicates start of high-drag events. The time-averaged values for the corresponding wall-normal locations are shown using red dashed lines (obtained using experiment). The criteria to detect a high-drag event is Δ*t* + *cr* = 200 and *τw*/*τ<sup>w</sup>* > 1.1.

The unconditional and conditionally-averaged RSS profiles, obtained for these two Reynolds numbers, shown in Figure 15. A good agreement can be observed between the experimental and DNS unconditional profiles. The conditionally-averaged data are normalised using *uτ* 2, are shown in Figure 15a,b, for low- and high-drag events, respectively. For the low-drag case, both experimental and DNS results are shown, and for high-drag case only experimental results are shown. A good agreement is observed between the conditionally averaged profiles obtained using experiments and DNS, with a slight discrepancy observed for the *Re<sup>τ</sup>* = 85 results. As seen in Figure 15a, the conditionally averaged profiles have slightly lower values than the unconditional profiles for *y*<sup>+</sup> 10. For *y*<sup>+</sup> - 10 the conditionally averaged profiles are higher than the unconditional profiles with the effect being more significant for *y*<sup>+</sup> between 20 and 40. For the high-drag case, as seen in Figure 15b, the conditionally-averaged RSS profiles almost collapse onto the unconditional profiles for all the wall-normal locations measured. This result suggests that the Reynolds shear stress is more affected by the low-drag events compared to the high-drag events. A sensitivity check has been executed to study the effect of changing the criteria for conditional events on the conditional RSS profiles for *Reτ* = 70. No significant dependence of the RSS profiles is observed for the different values of criteria studied here.

A quadrant analysis is conducted to calculate the contribution to the Reynolds shear stress from various turbulent events [49]. In quadrant analysis, the Reynolds shear stress is divided into four quadrants based on the signs of the streamwise and wall-normal velocity fluctuations: Q1 (+*u*, +*v*), Q2 (−*u*, +*v*), Q3 (−*u*, −*v*) and Q4 (+*u*, −*v*). The Q2 and Q4 events are generally related to the ejection and sweep events, respectively [49]. Here, the normalisation of both unconditional and conditional velocity fluctuations is based on the time-averaged friction velocity (*uτ*). For unconditional velocity fluctuations, the time-averaged velocities are subtracted from the instantaneous velocities, and for the conditional velocity fluctuations, the time-averaged velocities are subtracted from the instantaneous conditional velocities during the low- or high-drag events. Figure 16a,d shows the jpdfs (joint probability density functions) of the unconditional streamwise and wall-normal velocity fluctuations for *Re<sup>τ</sup>* = 70 obtained using the experiment (at *y*<sup>+</sup> = 24) and DNS (at *y*<sup>+</sup> = 25). The shape of the unconditional jpdfs are roughly elliptical with their major axes tilted in the direction of Q2 and Q4 motions. During the low-drag events the jpdf shifts towards the Q2 quadrant, whereas during the high-drag events the jpdf shifts towards the Q4 quadrant.

**Figure 15.** (**a**) Unconditional and conditionally averaged RSS profiles for *Reτ* = 70 and 85 during low-drag events. (**b**) Unconditional and conditionally averaged RSS profiles for *Reτ* = 70 and 85 during high-drag events.

**Figure 16.** Unconditional (**a**), low-drag (**b**) and high-drag (**c**) jpdfs of streamwise and wall-normal velocity fluctuations for *y*<sup>+</sup> = 24 at *Re<sup>τ</sup>* = 70 using experiments. Unconditional (**d**) and low-drag (**e**) jpdfs of streamwise and wall-normal velocity fluctuations for *y*<sup>+</sup> = 25 at *Re<sup>τ</sup>* = 70 using DNS. Unconditional and conditional velocity fluctuations are normalised using the time-averaged *uτ*.

This observation is consistent with the previous results where it is shown that during the low-drag events the ensemble-averaged streamwise decreases and wall-normal velocities increases for *<sup>y</sup>*<sup>+</sup> <sup>≈</sup> 20–40, whereas the opposite is true for high-drag events.

Figure 17a,b shows the unconditional and conditional (low-drag) profiles of contribution from the various quadrants in the Reynolds shear stress for *Re<sup>τ</sup>* = 70 and *Re<sup>τ</sup>* = 85, respectively. Figure 16b,d shows the joint distribution of streamwise and wall-normal velocity fluctuations during the low-drag events for *Re<sup>τ</sup>* = 70, obtained using experiments (at *y*<sup>+</sup> = 24) and DNS (at *y*<sup>+</sup> = 25), respectively. Figure 16c shows the joint distribution during the high-drag events for *Re<sup>τ</sup>* = 70 at *y*<sup>+</sup> = 24, obtained using experiments. A good qualitative agreement is observed between the experimental and DNS results for the unconditional data. It can be seen that the major contributors to the Reynolds shear stress are the Q2 and Q4 motions, which explains the reason for the tilted shape of the jpdf shown in Figure 16a,d. These two quadrants are considered to be responsible for the turbulence production [50,51]. It is also observed that the Q4 motions or the "sweep" type motions are the most dominant motions for *y*<sup>+</sup> 20 and for the higher wall-normal locations Q2 motions or the "ejection" type motions are the most dominant. For the low-drag case, the Q2 events contribute more than the other quadrants for all the wall-normal locations at both *Re<sup>τ</sup>* = 70 and 85. Another interesting observation is that the Q4 events contribution decreases to a very low value during these low-drag events. This further reinforces the hypothesis that the low-drag events are composed of low-streamwise speed and upwash motions. There is a good qualitative and also fairly good quantitative (for *y*<sup>+</sup> - 30–40) agreement between the experimental and DNS results. The discrepancies between the experimental and DNS data in the conditional data are aligned with their unconditional values, which suggests that these slight variations are the result of noise in the measurement rather than different physical observations.

**Figure 17.** (**a**) Contribution to −*uv* from different quadrants for the unconditional case and during the low-drag events for (**a**) *Re<sup>τ</sup>* = 70 and (**b**) *Re<sup>τ</sup>* = 85. The criteria to detect a low-drag event is Δ*t* + *cr* = 200 and *τw*/*τ<sup>w</sup>* < 0.9. Thin black dashed line represents a constant value of zero.

The observation from the quadrant contributions is consistent with the previous numerical findings by Kushwaha et al. [22] where it is shown that the low-wall shear stress events are associated with counter-rotating streamwise vortex pairs transferring momentum away from the wall. Park et al. [16] showed in MFU simulations that the low-drag event is the precursor to a strong bursting event which is again consistent with the present result. The low-speed fluid moves away from the wall (ejection process) during these low-drag events which ultimately undergo a bursting process. The ejection and bursting processes are well studied in the past in regards to the low-speed streaks moving away from the wall and bursting in the buffer layer region (for more details, see in [52,53]). Adrian et al. [54] provided a hairpin vortex model in an effort to unify the various previous findings related to the coherent structures observed in the turbulent boundary layer. It was stated that the hairpin vortex originates from the wall inducing a region of low speed between two legs of the vortex which then lifts up by ejection process. The present work suggests that the low wall shear stress events are representative of low-speed regions which are generally observed between the legs of the hairpin vortices in wall-bounded turbulent flows [54,55]. Although it should be noted that the present work employs a different criterion to detect these low-drag events (*τw*/*τ<sup>w</sup>* < 0.9 and Δ*t* <sup>+</sup> > 200) and therefore these conditional events form only a subset of the low-speed streaks/events observed in the past [55].

Results for the high-drag events are shown in Figure 18a,b for *Reτ* = 70 and 85, respectively. It can be observed that during the high-drag events, the Q4 events are the significant contributor to the Reynolds shear stress. This is again expected based on the ensemble-averaged data, i.e., high-drag events are composed of high-speed and downwash motions for *<sup>y</sup>*<sup>+</sup> <sup>≥</sup> 20.

**Figure 18.** (**a**) Contribution to −*uv* from different quadrants for the unconditional case and during the high-drag events for (**a**) *Re<sup>τ</sup>* = 70 and (**b**) *Re<sup>τ</sup>* = 85. The criteria to detect a high-drag event is Δ*t* + *cr* = 200 and *τw*/*τ<sup>w</sup>* < 0.9. Thin black dashed line represents a constant value of zero.
