**6. Wall Shear Stress Statistics during Conditional Events**

To study the statistics of the conditional wall shear stress, the instantaneous wall shear stress during the low-drag or high-drag events are ensemble-averaged. Figure 6 shows the instantaneous and ensemble averaged wall shear stress fluctuations during low- and high-drag events for *Re<sup>τ</sup>* = 180. The ensemble averaging is executed in two ways: by shifting all the instantaneous low- and high-drag events such that *t* <sup>+</sup> = 0 indicates the beginning of a conditional event (shown in Figure 6a,c), and by shifting all the instantaneous low- and high-drag events such that *t* <sup>+</sup> = 0 indicates the end of a conditional event (shown in Figure 6b,d). This has been done to study the time evolution of the ensemble-averaged wall shear stress with respect to the start and the end of a conditional event. It can be seen that during the low-drag events, the ensemble averaged wall shear stress drops approximately 35% below the time-averaged value. During the high-drag events, the ensemble averaged wall shear stress rises approximately 45% above the time-averaged value. This figure also highlights that although the time-duration criteria for the conditional events is Δ*t* + *cr* = 200, these events can last up to Δ*t* <sup>+</sup> <sup>≥</sup> 400.

The effect of the time-duration and magnitude threshold criteria on the conditional wall shear stress is investigated for *Re<sup>τ</sup>* = 180. For the time-duration criterion, Δ*t* + *cr* is varied between 150 and 250 while keeping the threshold criteria constant as *τw*/*τ<sup>w</sup>* < 0.9 and *τw*/*τ<sup>w</sup>* > 1.1 for the lowand high-drag events, respectively. Figure 7a–d shows the ensemble-averaged wall shear stress for the low- and high-drag events at *Re<sup>τ</sup>* = 180 for various time-duration criteria. The figure shows the ensemble-averaged wall shear stress for the conditional events for both methods of ensemble averaging, i.e., *t* <sup>+</sup> = 0 indicates either the start or end of a conditional event. The plateau of the ensemble-averaged wall shear stress during the low- and high-drag events is observed to be insensitive to the time-duration criteria when varying Δ*t* <sup>+</sup> from 150 to 250, but the duration of these conditional events itself becomes smaller when making the criteria less stringent. A spike in the ensemble-averaged wall shear stress can be observed near the start and end of the low-drag events and similarly, a dip can be seen near the start and end of the high-drag events. Analogous results corresponding to the ensemble-averaged wall shear stress during the low-drag events were also obtained by Kushwaha et al. [22] in channel flow using DNS for *Re<sup>τ</sup>* = 100. They employed mixed scaling (Δ*t* ∗ = 2 and 3) as the time-duration criteria to detect low-drag events. Similar results were obtained for the other Reynolds numbers studied here and are not shown for brevity.

It can be said that the time-duration criteria, either based on mixed or inner scaling (for the range studied), does not affect the strength of the low- or high-drag events. For the rest of this paper, the time-duration criteria for the both low- and high-drag events is fixed at Δ*t* + *cr* = 200 unless stated otherwise. Next, the effect of changing the threshold criteria on the conditional wall shear stress is investigated while keeping the time-duration criterion constant at Δ*t* + *cr* = 200. The threshold criteria used for low-drag events are *τw*/*τ<sup>w</sup>* < 0.8, *τw*/*τ<sup>w</sup>* < 0.9 and *τw*/*τ<sup>w</sup>* < 1, and for the high-drag events are *τw*/*τ<sup>w</sup>* > 1, *τw*/*τ<sup>w</sup>* > 1.1 and *τw*/*τ<sup>w</sup>* > 1.2. The most stringent limits for the strength in the threshold criteria are chosen based on the availability of a sufficient number of conditional events to obtain well-resolved ensemble-averaged wall shear stress results. As the threshold criterion is made more stringent, for the low-drag events (shown in Figure 7e,f), the lower plateau of the ensemble-averaged wall shear stress decreases. Similarly, for the high-drag events (shown in Figure 7g,h), the upper plateau of the ensemble-averaged wall shear stress increases. Similar results were observed for low-drag events only by Kushwaha et al. [22] at *Reτ* = 100. The results are shown only for *Re<sup>τ</sup>* = 180 as very similar results were obtained for the other Reynolds numbers studied.

**Figure 6.** (**a**,**b**) Instantaneous normalised wall shear stress (thin grey lines) and ensemble-averaged wall shear stress (thick black line) during the low-drag events for *Reτ* = 180 where *t* <sup>+</sup> = 0 indicates (**a**) start of a low-drag event and (**b**) end of a low-drag event. Red line highlights an instantaneous low-drag event with a duration of Δ*t* <sup>+</sup> <sup>≈</sup> 410. Purple line and dashed blue line represent the time-averaged value and the threshold value of *τw*/*τ<sup>w</sup>* < 0.9, respectively. (**c**,**d**) Instantaneous normalised wall shear stress (thin grey lines) and ensemble-averaged wall shear stress (thick black line) during the high-drag events for *Reτ* = 180 where *t* <sup>+</sup> = 0 indicates (**a**) start of a high-drag event and (**b**) end of a high-drag event. Red line highlights an instantaneous low-drag event with a duration of Δ*t* <sup>+</sup> <sup>≈</sup> 400. Purple line and dashed blue line represent the time-averaged value and the threshold value of *τw*/*τ<sup>w</sup>* > 1.1, respectively.

Interestingly, as can be seen from Figure 7e–h, the spike in the ensemble-averaged wall shear stress for the low-drag events and dip in the ensemble-averaged wall shear stress for the high-drag events seems to be less significant with increasingly strict threshold criteria. Kushwaha et al. [22] mentions that they have no physical explanation for the existence of the spike or dip in the ensemble-averaged wall shear stress data. To investigate the reason for the spike or dip in the ensemble-averaged data during the conditional events, two artificially generated time series have been produced where one signal is Gaussian and the other signal has the same first four moments as the wall shear stress moments for *Re<sup>τ</sup>* = 180 obtained in the present experiment. The Gaussian signal has a rms value the same as the wall shear stress for *Re<sup>τ</sup>* = 180. This has been conducted to understand if the reason for the spike or the dip is unique to the wall shear stress signals or is merely a statistical artefact of the conditioning. An equal number of samples (*<sup>N</sup>* = 2 × 108) are generated for both of the artificially generated signals using the inbuilt MATLAB function: "pearsrnd".

**Figure 7.** Ensemble-averaged wall shear stress for various time-duration criteria at *Reτ* = 180 for (**a**) start and (**b**) end of low-drag events, and (**c**) start and (**d**) end of high-drag events. The threshold criteria to detect a low- and high-drag event are *τ<sup>w</sup>* /*τ<sup>w</sup>* < 0.9 and *τ<sup>w</sup>* /*τ<sup>w</sup>* > 1.1, respectively. Ensemble-averaged wall shear stress for various threshold criteria at *Reτ* = 180 for (**e**) start and (**f**) end of low-drag events, and (**g**) start and (**h**) end of high-drag events. The time-duration criteria to detect a low-drag or a high-drag event is kept constant at Δ*t* + *cr* = 200.

A comparison of the ensemble averaged data during the conditional events is made between the two artificially generated signals. The time duration is kept the same as Δ*t* + *cr* = 200 to detect the lowand high-drag events. The threshold criteria are varied to study their effect on the ensemble averaged values. For the low-drag events, the threshold criteria are *τw*/*τ<sup>w</sup>* < 0.925, *τw*/*τ<sup>w</sup>* < 0.95, *τw*/*τ<sup>w</sup>* < 0.975 and *τw*/*τ<sup>w</sup>* < 1, and for the high-drag events, the threshold criteria are *τw*/*τ<sup>w</sup>* > 1, *τw*/*τ<sup>w</sup>* > 1.025, *τw*/*τ<sup>w</sup>* > 1.05 and *τw*/*τ<sup>w</sup>* > 1.075. Figure 8 shows the ensemble averaged wall shear stress during low- and high-drag events obtained from the two artificially generated signals. There is a spike (and dip) in the ensemble-averaged wall shear stress near the start of the low-drag (and high-drag) events for both artificially generated signals. The existence of spikes or dips in the ensemble-averaged data from the artificially-generated signals, even in the limit of a Gaussian signal, suggest that these are artefacts of the conditional sampling and ensemble averaging and are not unique to the wall shear stress signals. It is also seen that the spikes (and dips) in the ensemble-averaged data from the low-drag events (and high-drag events) becomes less significant when making the threshold criteria more stringent. This further reinforces the idea that these spikes and dips in the ensemble averaged data are the consequence of the conditional sampling of any time-series signal. Thus, these spikes or dips cannot be used to identify the onset/footprint of low- or high-drag events. Park et al. [16], using MFU simulations, observed that many of the low-drag events are followed by strong turbulent bursts which were detected based on an increase in the volume-averaged energy dissipation rate. There may exist a relation between these turbulent bursts and spikes in the ensemble-averaged wall shear stress data after low-drag events which needs further investigation.

**Figure 8.** Ensemble-averaged wall shear stress during (**a**) low-drag events and (**b**) high-drag events for the artificially generated wall shear stress signal with same first four moments as one measured for *Re<sup>τ</sup>* = 180. Ensemble-averaged wall shear stress during (**c**) low-drag events and (**d**) high-drag events for a Gaussian signal. The time-duration criteria to detect a low-drag or a high-drag event is kept constant at Δ*t* + *cr* = 200. Inset plots show the same data as the main plot but only near the spike or dip in the ensemble averaged data.
