3.1. Effects of $\textit {Re}_{\tau}$ on wall shear stress fluctuations

The PDF of the streamwise wall shear stress $\tau _{x}$ for different Reynolds numbers is shown as solid lines in figure 1(a). The width increment of the tails at different $\textit {Re}_{\tau}$ shows a clear influence of the Reynolds number on the occurrence of both extremely positive and backflow events. These results show good agreement with the findings reported by Lenaers et al. (Reference Lenaers, Li, Brethouwer, Schlatter and Örlü2012) for channel flows at $\textit {Re}_{\tau} = 180$, 590 and 1000 represented by ($\times$) symbols. The vertical dashed lines in figure 1(a) indicate the thresholds used to define the tails of the PDF as defined in § 2. In figure 1(a,b), an exponential decay on the tails is observed, which is an expected behaviour of systems where large deviations exist (Touchette Reference Touchette2009). Although the central region of the PDF, close to the peak, can be modelled using the central limit theorem, the behaviour of the extreme tails can be more precisely analysed using the large deviation theory (Majumdar & Schehr Reference Majumdar and Schehr2017). The work by Diaz-Daniel et al. (Reference Diaz-Daniel, Laizet and Vassilicos2017) proved that a log-normal distribution fits the central region of the probability distribution for the streamwise wall shear stress. However, the log-normal distribution is not suitable to fit the extreme tails. Here, we propose a combined distribution for the streamwise wall shear stress considering a truncated log-normal fit in the range $0 \leq \tau _x^+ \leq 3.1$, and an asymmetric double exponential fit for the extreme tails. The results of the exponential fits are plotted in figure 1(a) using the ($\Box$) symbol. The function used to fit the left tail of the streamwise wall shear stress PDF in terms of $\textit {Re}_{\tau}$ has the form

(3.1)\begin{equation} P(\tau_x^+)_{BF} = \xi_{BF} \exp{(\phi_{BF} \tau_x^+\exp{(\psi_{BF}}\tau_x^+))}, \end{equation}

where the coefficients $\xi _{BF}$, $\phi _{BF}$ and $\psi _{BF}$ are functions of the Reynolds number as follows

(3.2)\begin{gather} \xi_{BF} = -0.0053+0.0011 \ln{(\textit{Re}_{\tau})}, \end{gather}

(3.3)\begin{gather}1/\phi_{BF} = 0.114+0.018\ln{(\textit{Re}_{\tau})}, \end{gather}

(3.4)\begin{gather}\psi_{BF} = -0.153+0.040\ln{(\textit{Re}_{\tau})}. \end{gather}

Similarly, the right tail for extreme events follows the correlation

(3.5)\begin{equation} P(\tau_x^+)_{EP} = \xi_{EP} \exp{(\phi_{EP} \tau_x^+\exp{(\psi_{EP}}\tau_x^+))}, \end{equation}

with coefficients

(3.6)\begin{gather} 1/\xi_{EP} = -0.059+0.012\ln{(\textit{Re}_{\tau})}, \end{gather}

(3.7)\begin{gather}1/\phi_{EP} = -0.034 - 0.040\ln{(\textit{Re}_{\tau})}, \end{gather}

(3.8)\begin{gather}\psi_{EP} = 0.034-0.0064\ln{(\textit{Re}_{\tau})}. \end{gather}

Figure 1. (a) Probability density function of the streamwise wall shear stress $\tau _{x}^+$ for different $\textit{Re}_{\tau}$: (solid line) present results, where (green) $\textit {Re}_{\tau} \approx 170$, (red) $\textit {Re}_{\tau} \approx 500$, (black) $\textit {Re}_{\tau} \approx 1000$, (blue) $\textit {Re}_{\tau} \approx 2000$; ($\times$) channel flow results from (Lenaers et al. Reference Lenaers, Li, Brethouwer, Schlatter and Örlü2012) at $\textit {Re}_{\tau} = 180$, 590 and 1000; ($\Box$) correlation presented in (3.1) and (3.5). The dashed lines are the cutoff conditions used to define the extreme events. The arrow represents an increment in $\textit{Re}_{\tau}$. (b) Probability density function of the azimuthal wall shear stress $\tau _{x}^+$ for the present results.

The PDF of the azimuthal $\tau _{\theta }^+$ fluctuations for different Reynolds numbers is observed in figure 1(b) where a similar behaviour is observed in terms of $\textit{Re}_{\tau}$, with symmetric exponential behaviour at the tails. A symmetric behaviour is also found in the full conditional averages of $\tau _\theta ^+$ for both EP and BF events, as will be discussed later.

Although the contribution of the extreme tails of the $\tau _x^+$ PDF is less than 0.2 % when compared with the total contribution of the most probable events (close to the peak of the distributions), the understanding of the dynamics related to EP and BF events can provide useful insights for drag control purposes since it contributes importantly to the deficit or surplus of the components of the Reynolds stress tensor (Gomit et al. Reference Gomit, De Kat and Ganapathisubramani2018). Consequently, extreme events are related to high magnitudes of turbulent kinetic energy production and viscous dissipation, as analysed and presented in § 5.

The percentage of extreme events at different $\textit {Re}_{\tau}$ in terms of the thresholds used in this work is shown in figure 2(a). The present results for BF events at different Reynolds numbers ranging from $\textit {Re}_{\tau} \approx 170$ up to $2000$ are represented by ($\Diamond$). Our data have been compared with the studies conducted by Lenaers et al. (Reference Lenaers, Li, Brethouwer, Schlatter and Örlü2012) and Hu, Morfey & Sandham (Reference Hu, Morfey and Sandham2006) for channel flows, showing good agreement. It is also interesting to notice that the percentage of BF events increases as a logarithmic correlation of the form

(3.9)\begin{equation} \% {BF} = -0.147 + 0.031 \ln{(\textit{Re}_{\tau})}.\end{equation}

Figure 2. (a) Per cent of backflow events at the wall; ($\Diamond$) present, ($\times$) channel flow by Lenaers et al. (Reference Lenaers, Li, Brethouwer, Schlatter and Örlü2012), ($\triangle$) channel flow by Hu et al. (Reference Hu, Morfey and Sandham2006), (solid) correlation of the form $\% {BF} = -0.147 + 0.031 \ln {(\textit{Re}_{\tau})}$. The red symbols represent EP events being ($\circ$) present, and ($\dotsm$) linear correlation $\%\textit {EP}=7.2\times 10^{-5} \textit {Re}_{\tau}+0.0019$. (b) Root mean square of wall shear stress fluctuations; ($\Diamond$) present, ($+$) pipe flow by El Khoury et al. (Reference El Khoury, Schlatter, Brethouwer and Johansson2014), ($\triangle$) channel flow by Hu et al. (Reference Hu, Morfey and Sandham2006), ($---$) correlation by Schlatter & Örlü (Reference Schlatter and Örlü2010) for $\tau _{x,rms}^+$ in TBL, ($-\cdot -$) correlation by Diaz-Daniel et al. (Reference Diaz-Daniel, Laizet and Vassilicos2017) for $\tau _{\theta ,rms}^+$ in TBL, (solid lines) correlation of the form $\tau _{rms}^+ = a + 0.0355 \ln {(\textit {Re}_{\tau})}$ where $a=0.168$ for the streamwise component and $a=0.025$ for the azimuthal component.

Likewise, the percentage of extreme positive events (%EP) is represented by ($\circ$) in figure 2(a) and seems to follow a linear behaviour for the selected threshold. Therefore, the cutoff $\tau _x^+ > 3.1$ used to characterise the extreme positive events seems to be suitable at $Re_\tau \leq 1000$ since this value is related to the same contributions of EP and BF to the probability distribution of $\tau _x^+$ (see table 2). However, at higher Reynolds numbers, a different value for the threshold might be required to be used to define the EP events.

The value of the normalised root-mean-square of the wall shear stress fluctuations ($\tau _{x,rms}^+$ and $\tau _{\theta ,rms}^+$) has been plotted in figure 2(b). Data from the present study, El Khoury et al. (Reference El Khoury, Schlatter, Brethouwer and Johansson2014) for pipe and Hu et al. (Reference Hu, Morfey and Sandham2006) for channel flows are displayed. The dashed lines represent the correlations proposed by Schlatter & Örlü (Reference Schlatter and Örlü2010) and Diaz-Daniel et al. (Reference Diaz-Daniel, Laizet and Vassilicos2017) for the streamwise and the spanwise wall shear stress fluctuations in TBL, respectively. The $\tau _{rms}^+$ for internal flows (channel and pipe) agrees with the boundary layer correlations at high Reynolds numbers. However, at low $\textit {Re}_{\tau}$, the trend of $\tau ^+_{rms}$ seems to follow a different correlation with respect to the TBL correlations. The differences found between internal and external flows at low $\textit {Re}_{\tau}$ might exist due to the pressure gradient required to drive the flow in internal geometries, as explained by Schlatter & Örlü (Reference Schlatter and Örlü2010). Therefore, for internal flows, the correlation

(3.10)\begin{equation} \tau_{rms}^+ = a + 0.0355 \ln{(\textit{Re}_{\tau})}, \end{equation}

is the best fit for the current data, with $a=0.168$ for the streamwise component and $a=0.025$ for the azimuthal component.

The percentage of backflow events in terms of the wall-normal position is shown in figure 3(a). The results are plotted for the pipe flow case at $\textit {Re}_{\tau} \approx$ 170, 500 and 1000, and exhibit a similar behaviour compared to the channel flow by Lenaers et al. (Reference Lenaers, Li, Brethouwer, Schlatter and Örlü2012). As a complement, figure 3(b) presents the outermost location at which backflow events of the streamwise velocity fields $U_x$ are found. Interestingly, at the lowest Reynolds number studied, it is observed that reverse flow events occur within the viscous sublayer. In contrast, at the higher Reynolds numbers $\textit {Re}_{\tau} \geq 1000$, it is possible to observe backflow events at the buffer region.

Figure 3. (a) Percentage of backflow events as a function of the wall-normal location for $\textit {Re}_{\tau} \approx 170$, 500 and 1000; ($\times$) channel flow by Lenaers et al. (Reference Lenaers, Li, Brethouwer, Schlatter and Örlü2012). The arrow represents increment in $\textit{Re}_{\tau}$. (b) Outermost location of backflow events as a function of $\textit{Re}_{\tau}$; ($\Diamond$) present results, ($\times$) Lenaers et al. (Reference Lenaers, Li, Brethouwer, Schlatter and Örlü2012) and ($\Box$) Jalalabadi & Sung (Reference Jalalabadi and Sung2018), ($---$) limit of the viscous sublayer.

4.1. Conditional average scheme

Volumetric conditional fields have been computed to study the different flow quantities related to extreme events. It is important to note that conditional averages based solely on a single condition (e.g. $\tau _x^+ < 0$), usually lead to symmetric results. Indeed, this problem has already been observed and discussed by Sheng et al. (Reference Sheng, Malkiel and Katz2009). Turbulent structures are seldom symmetric. It is reported that low-velocity structures are characterised by their meandering behaviour (see Hutchins et al. (Reference Hutchins, Monty, Ganapathisubramani, Ng and Marusic2011) or Jiménez (Reference Jiménez2018)). Furthermore, observing symmetrical vortical structures at the near-wall region at high $\textit {Re}_{\tau}$ is unlikely (Robinson Reference Robinson1991; Schoppa & Hussain Reference Schoppa and Hussain2002; Adrian Reference Adrian2007). Hence, it is difficult to capture meaningful kinematics of the fluid motion by conducting conditional averages based only on the EP or BF wall shear stress events using a single threshold. Thus, in this investigation, we have implemented a two-step conditional average considering the following conditions, as illustrated in figure 4:

1. (i) The centroid of the extreme event on the two-dimensional streamwise wall shear stress field is located in terms of the thresholds proposed in § 3. An averaging box is used, with size ${\rm \Delta} x^+=1800$ in the streamwise direction, ${\rm \Delta} R\theta ^+=400$ in the azimuthal direction and $y^+ = 500$ in the wall-normal direction. The centroid of the extreme event is located within the averaging box at the coordinate ${\rm \Delta} x^+ = 0$, ${\rm \Delta} R\theta ^+ = 0$ and $y^+ = 0$ in the streamwise, azimuthal and wall-normal directions respectively.

2. (ii) After finding the centroid of the extreme event, a second condition is used to detect the presence of a left- ($Q_L$) or a right-sided ($Q_R$) vortex in terms of the second invariant of the velocity gradient field at the vicinity of the BF or EP event centroid using a smaller secondary box. As a convention, a left-sided vortex will be defined as a vortex located at the left of the box centreline with respect to the flow direction. The opposite condition defines a right-sided vortex for the purposes of this particular research (see figure 4 for clarity).

Figure 4. Scheme used to compute conditional averages.

A similar approach has been used previously by Lenaers et al. (Reference Lenaers, Li, Brethouwer, Schlatter and Örlü2012), who observed that BF events are related to the existence of an oblique vortex. Likewise, Hutchins et al. (Reference Hutchins, Monty, Ganapathisubramani, Ng and Marusic2011) used a comparable method based on the values of the friction velocity for four points at the vicinity of a negative event in order to characterise the meandering behaviour of low-speed streaks in turbulent boundary layers. This approach has been proven to detect relevant flow features, as will discussed in the following sections.

4.2. Conditional wall shear stresses

Figures 5(a) and 5(d) exhibit the conditional average of $\tau _x^+$ under the effect of a conditionally averaged left-sided vortex (hereinafter $Q_L$) for EP and BF events, respectively. As observed, when the aforementioned scheme is used, the averaged quantities are not symmetrical with respect to the streamwise centreline. Similar results, but under the effect of a right-sided vortex ($Q_R$) are also displayed in figures 5(b) and 5(e). Conversely, when conditional averages do not discriminate the location of the vortices, symmetrical results are obtained, as observed on figures 5(c) and 5(f). More substantial differences when using the proposed scheme are observed when figures 6(d) and 6(e) are compared with figure 6(f), where the footprint of the oblique vortex responsible for the BF events is clearly observed in the two-step conditioned fields.

Figure 5. Contours represent conditional ensemble averages of $\tau _x^+$ conditioned to EP events (a–c) and BF events (d–f). The extreme event centroid is located at ${\rm \Delta} x^+=0$ and ${\rm \Delta} R\theta ^+=0$. (a,d) Under the effects of a left-sided structure (dark isosurface). (b,e) Ensemble averages considering a right-sided structure. (c,f) Full conditional averages (without the consideration of the structure orientation). The dark isosurfaces in all figures are conditional $Q$ vortical structures at a level $Q^+=0.5$.

Figure 6. Conditional averages of $\tau _{\theta }^+$ conditioned to EP events (a–c) and BF events (d–f). Averages have been computed under the same conditions as figure 5.

4.3. Conditional velocity fluctuations

Here, an examination of the three-dimensional velocity structures related to extreme events is presented. Particularly, the focus is on the streamwise and wall-normal turbulent velocity fields. The contour plots are complemented with transverse slices and vector plots which aim to identify the flow dynamics associated with extreme wall shear stress events. It should be mentioned that all the conditional fields presented hereinafter are based on a left-sided ($Q_L$) vortex unless otherwise stated.

As discussed previously, EP events are related to the existence of a strong quasi-streamwise vortex. Figure 7(a) exhibits the existence of a high streamwise momentum region ($u_x^+ \geq 4.5$) located at the buffer region between $-50 \leq {\rm \Delta} x^+ \leq 150$. This small-scale region of high velocity exists due to the transport of streamwise momentum from the overlap and outer layer towards the wall, induced by the aforementioned vortex, agreeing with the buffer structures associated with high skin friction investigated by Sheng et al. (Reference Sheng, Malkiel and Katz2009). In addition, it is noted that the high-momentum region is embedded within a large-scale structure of positive fluctuation whose streamwise extension scales with the characteristic length of the pipe $\delta =R$. This large-scale structure is located at the logarithmic and outer regions, extending up to $y^+\approx 400$ or $y /\delta \approx 0.4$ in the wall-normal direction.

Figure 7. (a) Ensemble average of streamwise velocity fluctuations conditioned by an EP event on the ${\rm \Delta} x^+\hbox{--}y^+$ plane at ${\rm \Delta} R\theta ^+ = 0$. (b) Transverse sections of (a) at different streamwise locations. (c) Ensemble average of wall-normal velocity fluctuations conditioned by an EP event on the ${\rm \Delta} x^+\hbox{--}y^+$ plane at ${\rm \Delta} R\theta ^+ = 0$. (d) Transverse sections of (c).

The transverse slices displayed in figure 7(b) at ${\rm \Delta} x^+=0$ and 100 show that at the left side of the high-momentum region, at the viscous sublayer and the buffer, a low-speed region is created. This structure seems to be a nascent lifted streak with low streamwise momentum. This lifted streak is the result of the transport of momentum from the wall towards the outer layer and is located at the upwash flank of the strong quasi-streamwise vortex. This structure shows coherence, and its topology has similarities to the unstable low-speed streaks related to the near-wall vortex regeneration cycle (Jiménez & Pinelli Reference Jiménez and Pinelli1999; Schoppa & Hussain Reference Schoppa and Hussain2002; Masahito et al. Reference Masahito, Yasufumi, Yuki and Michio2007).

Figures 7(c) and 7(d) show the averaged configuration of the wall-normal velocity ($u_y^+$) structures related to the EP $\tau _x^+$ events under the influence of a $Q_L$ vortex. Figure 7(c) shows that there exists a strong region of downwash motion at the inner region associated with the high streamwise momentum region mentioned above. As a result, a strong Q4 layer is created in the viscous sublayer and buffer region just above the EP event, agreeing with the study conducted by Pan & Kwon (Reference Pan and Kwon2018). Additionally, it is observed that, in the logarithmic and outer layers, there exists a large-scale region of negative wall-normal velocity. This provides further evidence that large-scale outer structures are related to the near-wall intermittency. The mechanism related to the existence of these large scales of motion will later be analysed using vector plots (in § 6).

The transverse sections, displayed in figure 7(d), reveal a pair of opposite wall-normal velocity regions located at $y^+<30$ just above the EP event. These upwash and downwash motions, located at the near-wall region, are produced at both flanks of the quasi-streamwise vortex. It should be noted that the downwash ($u_y^+<0$) side is associated with the high streamwise momentum region whereas the upwash side is related to the lifted low-speed streak (see figure 7b), showing the essential role that streamwise vortices have in the near-wall cycle (Robinson Reference Robinson1991; Jiménez & Pinelli Reference Jiménez and Pinelli1999; Schoppa & Hussain Reference Schoppa and Hussain2002). Surprisingly, the high magnitudes of $\langle u_y^+ \rangle _{EP}$ and its disposition in pairs resemble the studies conducted by Xu et al. (Reference Xu, Zhang, den Toonder and Nieuwstadt1996) and Lenaers et al. (Reference Lenaers, Li, Brethouwer, Schlatter and Örlü2012). Xu et al. (Reference Xu, Zhang, den Toonder and Nieuwstadt1996), defined the extreme wall-normal events as $|u_y / u_{y,rms} | > 5$. Our conditioned values reveal a similar behaviour with negative $u_y/u_{y,rms} \approx -7$ and the positive $u_y/u_{y,rms} \approx 5$. The work by Xu et al. (Reference Xu, Zhang, den Toonder and Nieuwstadt1996) also mentions that extreme wall-normal events are related to a streamwise vortex located on the buffer layer. Similar observations are laid out by Lenaers et al. (Reference Lenaers, Li, Brethouwer, Schlatter and Örlü2012). However, none of the aforementioned authors studied the relationship of extreme wall-normal velocity with extreme positive $\tau _x^+$ events, and the additional coherent outer motions associated with them. As a consequence, our results provide further evidence that streamwise vortices are an essential mechanism to sustain near-wall turbulent energy transfers since they provide large contributions to the total Reynolds shear stress $\langle -u_y u_x \rangle ^+$ (not shown here), turbulent kinetic energy production (PR) and viscous dissipation (DS) as will be further analysed in § 5. Finally, it is important to highlight that, if the location of the vortex were not considered with the conditional averages, symmetric results would be observed in figure 7(d) this being a trio of alternating extreme events of wall-normal velocity.

The existence of near-wall strong sweep events related to high skin friction events is more clearly observed at the zoomed view of the streamwise velocity fluctuation field from figure 8, where the vector plots clearly exhibit the interaction of the streamwise vortex and the high- and low-momentum structures associated with EP wall-friction events.

Figure 8. (a) Extended view of figure 7(b) at ${\rm \Delta} x^+ = 50$, showing the velocity vectors for the $u_\theta ^+$ and $u_y^+$ components. (b) Zoomed view of streamwise turbulent velocity on a ${\rm \Delta} x^+\hbox{--}y^+$ plane sectioned at ${\rm \Delta} R \theta^+ = -15$. Vectors represent the conditional velocity fluctuations for $u_x^+$ and $u_y^+$. The white line contour exhibits the quasi-streamwise vortex.

Similarly, the streamwise velocity fluctuations averaged under negative $\tau _x^+$ events displayed in figure 9(a,b), exhibit a qualitative behaviour that resembles the results for no-slip critical points (points at the wall where the streamwise and the spanwise skin friction are simultaneously zero) obtained by Cardesa et al. (Reference Cardesa, Monty, Soria and Chong2014) and Chin et al. (Reference Chin, Monty, Chong and Marusic2018b). The present results, in addition, show that the reverse flow event occurs below the tail of a large-scale low-speed streak. It is also interesting to observe that the approach used to compute the conditional averages in this study unveils the average meandering of the low- and high-speed structures as exhibited in the transverse sections shown in figure 9(b). These results are consistent with the existence of the strong oblique vortex that seems to be associated with BF events. In figure 9(a,b) it should also be noted that the low-speed streak associated with BF events is located below a large-scale structure of positive streamwise turbulent fluctuation. This shows that backflows are associated with the interaction of two uniform-momentum zones (UMZ) of low and high momentum, respectively, which generate high shear at their interface. The existence of a large structure of positive $u_x^+$ located at the overlap and outer region is in good agreement with the amplitude modulation effect of outer LSM in the near the wall dynamics. These results suggest a different mechanism to the recent study by Bross et al. (Reference Bross, Fuchs and Kähler2019) on APG-TBL who argue that reverse flow events exist below a low-speed superstructure located at the log-law region. These dissimilarities might be due to differences between canonical flows and the large regions with a reverse flow that characterise the velocity profile of an APG-TBL.

Figure 9. (a) Ensemble average of streamwise velocity fluctuations conditioned on BF events on the ${\rm \Delta} x^+\hbox{--}y^+$ plane at ${\rm \Delta} R\theta ^+ = 0$. (b) Transverse sections of (a) at different streamwise locations. (c) Ensemble average of wall-normal velocity fluctuations conditioned under BF events on the ${\rm \Delta} x^+\hbox{--}y^+$ plane at ${\rm \Delta} R\theta ^+ = 0$. (d) Sections of (c) at different streamwise locations.

Figure 9(c,d) displays the behaviour of BF conditioned wall-normal velocity fluctuations. The existence of positive and negative coherent wall-normal velocity fluctuations on the ${\rm \Delta} x^+\hbox{--}y^+$ and ${\rm \Delta} R\theta ^+\hbox{--}y^+$ planes is consistent with the existence of an oblique vortex. However, the magnitudes of $u_y/u_{y,rms}$ are not large enough to assert a relationship between BF events and extreme wall-normal fluctuations. Another interesting observation is that the large-scale low-speed streak in figure 9(a) and the large-scale structures of $\langle \tau _x^+ \rangle _{BF}$ and $\langle \tau _\theta ^+ \rangle _{BF}$, displayed previously in figures 5 and 6, are related to a long region with positive $u_y$ with an approximate streamwise extension of $1000$ wall units as observed in the section displayed in figure 9(c). At the logarithmic and outer layers, in the range $-200 < {\rm \Delta} x^+ < 500$, we note the presence of a region of negative $u_y$, an outer downwash region related to the existence of the large-scale structure of high momentum. This downwash motion can be induced, in part, by the strong oblique vortex. However, the vector plot sections exhibited in § 6 suggest that large-scale rotating motions might also be responsible for the existence of this structure of negative $u_y$.

The transverse velocity sections shown in figure 9(d) at ${\rm \Delta} x^+ = 100$, 200 and 300 reveal that the positive $u_y$ region located at the centre of the buffer region is surrounded by negative wall-normal velocity fluctuations on its sides. This behaviour suggests the presence of large streamwise counter-rotating inclined motions which transport momentum from the inner region towards the overlap and outer layers, and hence, those structures are consistent to the existence of the low-speed streak displayed in figure 9(a,b). Further evidence of these swirl motions is provided in § 4.4. As a result, it is educed that backflow events, on average, require the combination of a complex set of fluid motions to occur at the same time (high-velocity LSM at the outer layer, low-speed streaks, an oblique or a spanwise vortex and large-scale counter-rotating motions). Such a combination of complex kinematics makes reverse flow events extremely rare.

To complement the three-dimensional understanding of the averaged small-scale flow dynamics occurring in BF events, figure 10(a,b), respectively, show an expanded view of a transverse and a longitudinal section of the contours for $\langle u_x^+\rangle _{BF}$ accompanied by the velocity vectors and a line contour of the oblique vortex (white). In figure 10, the centre of the strong vortex and its effects on the high- and low-velocity structures can be clearly observed.

Figure 10. (a) Extended view of figure 9(b) at ${\rm \Delta} x^+ = 0$ showing the vectors for the $u_\theta ^+$ and $u_y^+$ components. (b) Zoomed view of streamwise velocity fluctuations on a ${\rm \Delta} x^+\hbox{--}y^+$ plane sectioned at ${\rm \Delta} R \theta ^+ = 0$ displaying the velocity fluctuation vectors for the $u_x^+$ and $u_y^+$ components. The white line contour exhibits the oblique vortex at $Q^+=0.5$.

4.4. Conditional vorticity

The azimuthal $\omega _\theta$ and wall-normal $\omega _y$ conditional vorticity fields are analysed using a similar approach as in § 4.3 to study the dynamics related to extreme events further.

Extreme positive conditional vorticity fields are shown in figure 11. The azimuthal vorticity component is shown in figure 11(a,b). The conditioned $\langle \omega _\theta \rangle _{EP}$ volumetric field has been normalised by the ensemble average $\langle \omega _\theta \rangle$. This normalisation allows us to observe the regions with surplus or deficit vorticity when compared with the ensemble averaged flow quantities. It is interesting to notice that in the range $-50 \leq {\rm \Delta} x^+ \leq 50$ and below $y^+=2$ the EP conditioned field exhibits a region with $\omega _\theta$ surplus with a similar shape as the conditioned $\langle \tau _x^+ \rangle _{EP}$ (see figure 5a). This region with high magnitudes of $\omega _\theta$ is created due to the high shear produced due to the presence of a small-scale high-speed streak located above the EP event. Above the viscous sublayer ($y^+>5$) and in the range $-50 \leq {\rm \Delta} x^+ \leq 200$, there is a region with an azimuthal vorticity deficit, located over the near-wall high streamwise momentum structure observed in figure 8(a).

Figure 11. (a) Value of $\langle \omega _\theta \rangle _{EP} / \langle \omega _\theta \rangle$ on the ${\rm \Delta} x^+\hbox{--}y^+$ plane at ${\rm \Delta} R\theta ^+ = 0$. The black vertical dashed lines show the streamwise locations of the transverse sections shown in (b). (c,d) Value of $\langle \omega _y^+ \rangle _{EP}$ on the ${\rm \Delta} x^+\hbox{--}{\rm \Delta} R\theta ^+$ plane at $y^+ \approx 5$ and $y^+ \approx 20$ respectively. The $\langle \omega _y^+ \rangle _{EP}$ contours are accompanied by turbulent velocity vector plots. (e) Sections of (d).

In figure 11(a), at the location between $300 < {\rm \Delta} x^+ < 1200$ and $70<y^+<200$ there exits an inclined large-scale region with a slight surplus of spanwise vorticity generated below the large-scale structure of positive streamwise fluctuation discussed in the previous section. In contrast, there exists a region with vorticity deficit over the region with vorticity surplus above $y^+>200$. These large-scale vorticity layers located downstream from the EP event, exist due to the presence of the shear layers that bound large-scale uniform-momentum zones (UMZ).

Another interesting feature is observed in the transverse sections from figure 11(b), particularly at the transverse sections at ${\rm \Delta} x^+ = 0$ and 100, where\query{Q23} a small region of high $\omega _\theta$ appears at the zone where $25<y^+<40$ and $10<{\rm \Delta} R\theta ^+<50$ whose centreline is at the interface located between the lifted streak observed in figure 8(a) and the higher-velocity region above it. This finding highlights the importance of lifted streaks in the generation of new vortical structures as suggested by Jiménez & Pinelli (Reference Jiménez and Pinelli1999), Schoppa & Hussain (Reference Schoppa and Hussain2002) and Adrian (Reference Adrian2007).

The wall-normal vorticity component $\omega _y = -\omega _r$ has also been analysed to understand further the vortical structures related to extreme events. Figure 11(c,d) exhibits wall-parallel sections of $\omega _y$ conditioned for EP events taken at $y^+\approx 5$ and $y^+ \approx 20$ respectively. From the wall-parallel section at $y^+ \approx 5$, it should be noticed that the shape of the wall-normal vorticity field resembles the conditional azimuthal wall shear stress field $\langle \tau _\theta ^+ \rangle _{EP}$, which suggests its correlation with the quasi-streamwise vortex associated with EP events. Moreover, the transverse sections from figure 11(e) show that $\omega _y$ coherent motions are inclined towards the core as ${\rm \Delta} x^+$ increases. Hence, it is educed that $\tau _\theta$ is the footprint of the vortical motions that occur at the buffer and are inclined towards the pipe centreline. These results provide evidence that coherent quasi-streamwise vortices located at the inner region are responsible for significant contributions to the total drag force. Hence, the strong quasi-streamwise vortex would be an essential structure to analyse in terms of drag control. That information is consistent with the recommendations made by Jiménez & Moser (Reference Jiménez and Moser2007) which advocate focussing on the inner region for drag control purposes and Pan & Kwon (Reference Pan and Kwon2018) who states that EP events and its dynamics should be better understood in the same context.

Figures 12(a) and 12(b), similarly, exhibit the $\omega _\theta$ field conditioned for backflow events. The $\langle \omega _\theta \rangle _{BF} / \langle \omega _\theta \rangle$ field shows the existence of a small region with negative $\omega _\theta$ with similar dimensions as the negative $\tau _x^+$ event (see figure 5d). The negative $\omega _\theta$ region extends up to a wall-normal location $y^+ \approx 0.5$. Above the viscous sublayer, there exist two regions with high positive magnitudes of azimuthal vorticity, which are in good agreement with the studies conducted by Cardesa et al. (Reference Cardesa, Monty, Soria and Chong2014) and Cardesa et al. (Reference Cardesa, Monty, Soria and Chong2019). In addition to the studies mentioned above, our conditional averaging method highlights a meandering behaviour of the BF conditioned azimuthal vorticity field observed in the transverse sections taken in figure 12(b). The meandering behaviour of the azimuthal vorticity field is clearly observed in the region of positive $\omega _\theta$ located at $y^+ > 20$, where its spanwise location is shifted approximately 20 wall units from the averaging box centreline at the transverse section at ${\rm \Delta} x^+ = 0$ from figure 12(b). The transverse section at ${\rm \Delta} x^+ = 100$ shows that the positive $\omega _\theta$ region shifts back to $R\theta ^+ = 0$. These coherent structure with high magnitudes of $\omega _\theta$ are the result of the interaction between the lifted low-speed streak and the large-scale high-velocity structure located above it (see figure 9a). Consequently, the coherent $\omega _\theta$ region seems to have a streamwise extension akin to the averaged low-speed streak associated with BF events.

Figure 12. (a) Value of $\langle \omega _\theta \rangle _{BF} / \langle \omega _\theta \rangle$ on the ${\rm \Delta} x^+\hbox{--}y^+$ plane at ${\rm \Delta} R\theta = 0$. The black vertical dashed lines show the locations of the transverse sections of (b). The $\langle \omega _y^+ \rangle _{BF}$ contours in (c,d) are accompanied by conditional velocity fluctuation vector plots that show the existence of large-scale streamwise counter-rotating motions inclined towards the pipe centreline. (e) Transverse sections of (d).

In figure 12(a), it should also be noted that the structure with high $\langle \omega _\theta \rangle _{BF}$ is followed by a large-scale layer of $\langle \omega _\theta \rangle _{BF}\approx 1.5\langle \omega _\theta \rangle$ extending from $400<{\rm \Delta} x^+<1200$. This large-scale layer with high magnitudes of $\langle \omega _\theta \rangle$ shows similarities when compared to the large-scale outer layer of $\langle \omega _\theta \rangle _{EP}$ surplus discussed above. Indeed, this structure seems to be the boundary between two zones of uniform momentum located downstream of the BF event. The similarities observed at the outer region with a surplus of azimuthal vorticity for both extreme events, show that EP and BF skin friction events could be related to the same LSM located at the overlap and outer regions.

Similarly, $\omega _y$ is analysed under BF conditions. To further understand the flow topology of reverse flow events, figure 12(c,d) displays sections of the wall-parallel vorticity contours at the levels $y^+ \approx 5$ and 20 respectively. The wall-parallel section observed in figure 12(c) reveals the existence of two small-scale counter-rotating motions located upstream the BF event with an approximate extension of 90 wall units, and has a similar topology as the wall-normal vorticity field exhibited in figure 3 of Cardesa et al. (Reference Cardesa, Monty, Soria and Chong2019). These counter-rotating motions located close to the wall are not clearly detectable at the buffer, as observed in figure 12(d). Hence, both small-scale swirl motions located within the viscous sublayer ($y^+ \leq 5$) at a streamwise location in the range $-100 \leq {\rm \Delta} x^+ \leq -10$ could be the tails of the strong oblique vortex associated with BF events. Figures 12(c) and 12(d) exhibit two symmetric regions with high magnitudes of $\omega _y$ located downstream of the BF event located at $20 \leq {\rm \Delta} x^+ \leq 400$. These two regions of intense $\omega _y$ are not only the result of large values of $\partial _\theta u_x$, but they also exist due to the presence of two inclined streamwise counter-rotating large-scale coherent motions in addition to the strong oblique vortex, as revealed by the velocity vector plots. The transverse sections from figure 12(e) also show how these swirl motions are inclined towards the outer region. Besides the evidence observed in the vector plots from figure 12(c)–(e), the authors confirmed the existence of these roll modes by examining a set of transverse slices of the ($u_\theta ^+$, $u_y^+$) vector plots (not shown here), where it was observed that these vortical motions have an inclination angle of approximately $11^{\circ }$ with respect to the wall. It is also worth mentioning that these counter-rotating modes are linked to the existence of the large-scale lifted low-speed streak related to BF events and a Q2 event occurring above it (see § 6 for further details). Furthermore, the inclination angle of the high-speed structure (see Marusic & Heuer Reference Marusic and Heuer2007; Cardesa et al. Reference Cardesa, Monty, Soria and Chong2014; Chin et al. Reference Chin, Monty, Chong and Marusic2018b) occurring above the low-speed streak is also consistent with the orientation of these large-scale roll modes. As a result, these findings provide a further understanding of the flow dynamics related to backflow events and its associated large scales of motion.

4.5. Turbulent inertia and velocity-vorticity correlations

Similar to Chin et al. (Reference Chin, Philip, Klewicki, Ooi and Marusic2014b), the equation that describes the streamwise mean momentum balance for fully developed turbulent pipe flows is defined as

(4.1)\begin{equation} \frac{\textrm{d}\langle -u_x u_y\rangle^+}{\textrm{d} y^+} + \frac{\textrm{d}^2 \langle U_x\rangle^+}{\textrm{d} y^{+^2}}+\frac{1}{\textit{Re}_{\tau}} = 0, \end{equation}

where the first term of the left-hand side is known as the mean turbulent inertia (TI), and the two last terms are the mean viscous force and the pressure gradient respectively. The TI term is of particular importance in wall-bounded turbulent flows since this gradient represents the force exerted by the Reynolds shear stress on the mean velocity profile. In other words, it accelerates the flow near the wall, and slows it down at the outer region (Adrian Reference Adrian2007). As a result, the turbulent mean velocity profile is flatter compared to the parabolic laminar mean velocity profile. One way to understand the mechanisms that are responsible for the generation of TI is to decompose it in terms of the streamwise Lamb vector (Klewicki Reference Klewicki1989; Hamman, Klewicki & Kirbi Reference Hamman, Klewicki and Kirbi2008)

(4.2)\begin{equation} \text{TI}=\frac{\textrm{d}\langle -u_x u_y\rangle^+}{\textrm{d} y^+} = \langle -u_y \omega_{\theta} \rangle^+ + \langle u_\theta \omega_y \rangle^+ , \end{equation}

where the terms $\langle u_i \omega _j \rangle$ are known as velocity–vorticity correlations. The correlation $\langle -u_y \omega _{\theta }\rangle ^+$ represents ‘azimuthal vorticity advection in the wall-normal direction’, and $\langle u_\theta \omega _y \rangle ^+$ is known as the ‘vorticity-stretching inertial force’ which is related to the rate of change of the length scale of turbulent eddies within the flow (Chin et al. Reference Chin, Philip, Klewicki, Ooi and Marusic2014b). The combination of (4.1) and (4.2) show how velocity–vorticity correlations influence the mean streamwise velocity profile $\langle U_x \rangle ^+$ (Klewicki Reference Klewicki1989; Chin et al. Reference Chin, Philip, Klewicki, Ooi and Marusic2014b).

The velocity–vorticity correlations from (4.2) have been computed to understand their contribution to the TI. Figure 13 displays the contribution of vorticity advection and vorticity stretching to the TI force conditioned under EP $\tau _x$. These results interestingly reveal that $\langle -u_y \omega _\theta \rangle ^+_{EP}$ makes a high positive contribution to the TI, generating a high positive region of TI of approximately 150 wall units in the streamwise direction, as educed from figure 13(a,c). This observation shows that the quasi-streamwise vortex associated with the EP events works as an important momentum source (acceleration) at the wall. On the other hand, the results presented in figure 13(b) reveal that vorticity stretching provides a larger contribution to TI downstream of the EP event. Additionally, remotely downstream, at $y^+ > 450$, the TI term tends to converge to the unconditional ensemble mean having its peak at $y^+ \approx 8$ (see Chin et al. Reference Chin, Philip, Klewicki, Ooi and Marusic2014b), however, it has a slightly higher value at the peak over a streamwise extension that scales with the pipe radius. The present results provide further evidence of the relationship between EP events and LSM.

Figure 13. Velocity–vorticity correlations conditioned under an EP event. (a) The $\langle -u_y \omega _\theta \rangle ^+_{EP}$ field. (b) The $\langle u_\theta \omega _y \rangle ^+_{EP}$ field. (c) Conditional average for turbulent inertia field computed in terms of the velocity–vorticity correlations.

The contribution of velocity–vorticity correlations to the turbulent inertia conditioned for backflow events is displayed in figure 14. The $\langle -u_y \omega _\theta \rangle ^+_{BF}$ term from figure 14(a) exhibits a complex vorticity-advection field having positive and negative regions. Upstream of the strong oblique vortex centre (i.e. ${\rm \Delta} x^+ < -10$), it can be observed that there is a region with a significant magnitude of negative $\langle -u_y \omega _\theta \rangle ^+_{BF}$ which leads to a region of streamwise deceleration (momentum sink) consistent to the existence of BF events. In contrast, downstream the oblique vortex centre, there is a region with a high positive region of vorticity advection, which acts as a momentum source. Similar to the EP event, the vorticity-stretching term displayed in figure 14(b) provides a larger contribution downstream of the negative $\tau _x$ event when compared with the vorticity advection term, where TI tends to converge to its mean value but with a slightly smaller magnitude. This indicates that BF events have an impact at large streamwise extensions comparable with the normalised characteristic length of the flow $\delta ^+ = \textit {Re}_{\tau}$, and hence evinces the influence of the large-scale vortical motions on the TI term.

Figure 14. Velocity–vorticity correlations conditioned under a BF event. (a) The $\langle -u_y \omega _\theta \rangle ^+_{BF}$ field. (b) The $\langle u_\theta \omega _y \rangle ^+_{BF}$ field. (c) Conditional average for turbulent inertia field computed in terms of the velocity–vorticity correlations.

6.1. Instantaneous visualisations

Based on the flow quantities analysed in the previous sections, next, we aim to provide a three-dimensional illustration of the main coherent structures (high-/low-velocity streaks, strong vortices and counter-rotating motions) associated with both extreme events studied.

Firstly, the instantaneous distribution of extreme events is presented in figure 17 where the colour contours represent the streamwise velocity fluctuations at $y^+ = 15$. It is curious to note that extremely high as well as backflow events are not isolated. Indeed, they are clustered between high- and low-speed streaks. Moreover, it is also noticed that extreme events are unlikely to occur below large-scale structures of negative fluctuation $u_x < 0$. In other words, a highly intermittent behaviour near the wall is generated below large-scale structures of positive streamwise fluctuation. This observation is consistent with the conditional fields discussed in § 4.3 where it was shown that both BF and EP events occur below LSMs of positive streamwise velocity fluctuation. Additionally, it provides further evidence of the amplitude modulation effect of LSMs at the near-wall cycle (Hutchins & Marusic Reference Hutchins and Marusic2007b; Marusic et al. Reference Marusic, Mathis and Hutchins2010).

Figure 17. Instantaneous visualisation of streamwise velocity fluctuations at $y^+ = 15$. The (*) symbol represents the location of instantaneous BF events at the wall and (+) shows the instantaneous location of EP events.

Figure 18 shows a zoomed view of figure 17. Here, both backflow and extreme positive events seem to occur close to regions with high magnitudes of $\partial _\theta u_x$ (i.e. near the interface between high- and low-speed streaks), and their presence is more abundant at or near the tail of low-speed streaks. Figure 18 also illustrates that low-speed streaks meander at BF events, which is consistent to the behaviour observed in the low-speed streak presented in figure 9(b).

Figure 18. Zoomed view of the rectangular region highlighted in figure 17.

Although the amplitude modulation effect of large scales of motion on the near-wall dynamics explains the high intermittency that exists at the near-wall region below a high-velocity LSM or VLSM, it does not explain why both EP and BF extreme events are usually clustered. In figure 19, we present a three-dimensional view of the flow structures existing at the same location as the flow field exhibited in figure 18. The yellow isosurfaces show the $Q$ vortical structures, and the grey isosurfaces show the low-speed streaks at a level $u_x^+ = -0.2$. In the same figure, the red and blue patches show the locations of EP and BF wall-friction events. The positive $u_x^+>0$ structures are not shown here for clarity since they have large extensions that fill most of the field of view. However, it should be noted that most of the void spaces in figure 19 are filled by regions of positive fluctuation.

Figure 19. Instantaneous visualisation of the flow structures that generate clusters of EP and BF events at the same location as figure 18. The grey isosurfaces are low-speed streaks, the yellow isosurfaces are the vortical $Q$ structures, the blue patches show the location of BF events at the wall and the red patches represent EP skin friction events.

The central panel presented in figure 19 reveals that regions, where clusters of extreme events occur are also densely populated by vortical structures within the inner region ($y^+ < 100$). Due to the large magnitudes in the velocity gradients that exist between the high- and low-speed UMZ (Meinhart & Adrian Reference Meinhart and Adrian1995), large-scale layers of high azimuthal vorticity are generated, which were also observed in the conditional $\omega _\theta$ fields exhibited in § 4.4. Those large-scale layers of azimuthal vorticity surplus shown on the conditional $\langle \omega _\theta \rangle _{BF}$ and $\langle \omega _\theta \rangle _{EP}$ might reveal the existence of large populations of spanwise vortical structures and probably hairpins located downstream from the extreme events. Consequently, the present results provide further evidence that packets of spanwise and hairpin-like vortices exist above low-speed streaks (Adrian et al. Reference Adrian, Meinhart and Tomkins2000; Adrian Reference Adrian2007; de Silva, Hutchins & Marusic Reference de Silva, Hutchins and Marusic2016). At the sides of the low-speed streaks, in the buffer region, the most common vortical structures are quasi-streamwise vortices, which in some cases, could be attached to the legs of larger hairpin structures (Robinson Reference Robinson1991; Jiménez & Pinelli Reference Jiménez and Pinelli1999; Adrian et al. Reference Adrian, Meinhart and Tomkins2000). Hence, this high population of vortical structures is the result of the high shear rates existing at the interface of uniform momentum zones of high and low speed, which also lead to the existence of instabilities that sustain near-wall turbulence.

The left lower panel from figure 19 shows a zoomed view of a region clustered with EP and BF events named ‘cluster 1’. Here, three reverse flow events occur below a low-speed streak. The BF event located at ${\rm \Delta} x^+ \approx 1.53\times 10^4$ exists due to the circulation of an inverted hairpin, which resembles the structures revealed by Chin et al. (Reference Chin, Monty, Chong and Marusic2018b) and Wu et al. (Reference Wu, Cruickshank and Ghaemi2020). It is also noted that there exist two EP events approximately 40 wall units downstream the BF event mentioned above, which are generated by two counter-rotating vortices of similar strength. Indeed, one of the vortices seems to be the trailing leg of a hairpin structure. In addition, the distance between the centroids of both EP events in the spanwise direction is approximately 60 wall units. These observations are in good agreement with the conceptual model suggested by Sheng et al. (Reference Sheng, Malkiel and Katz2009) for experimental data of TBL at $\textit {Re}_{\tau} = 1470$, providing further evidence that the buffer structures related to EP and BF events might be $Re$ invariant, self-similar and scale in viscous units.

Downstream, at ${\rm \Delta} x^+ \approx 1.537\times 10^4$ there are observed two other backflow events generated by a single oblique vortex. In the central panel, it is also possible to observe the existence of a hairpin structure related to cluster 1 whose head is located at the log-law region, and at a streamwise location ${\rm \Delta} x^+ \approx 1.55 \times 10^4$. This hairpin structure has a large inclined leg, which transports momentum from the wall towards the outer region at its inner flank. On the other hand, the outer flank of this large leg transports momentum towards the wall and generates the high stress event mentioned above.

The right upper panel from figure 19 displays a zoomed view of the second cluster of extreme wall-friction events. It is observed that a backflow event occurs at ${\rm \Delta} x^+ \approx 1.565 \times 10^4$. This reverse flow event is located below a low-speed streak, and is generated by the circulation of the trailing leg of an oblique vortex located near the wall with a similar topology as the conditional left-sided $Q_L$ vortex analysed in § 3. Downstream the reverse flow event mentioned above, there exist two vortical structures similar to hairpins located at $y^+ \approx 60$. These structures are possibly generated due to the high shear existing between the high and low UMZ and resemble the hairpin packet explained by Adrian et al. (Reference Adrian, Meinhart and Tomkins2000) and Adrian (Reference Adrian2007). In the same panel, it is possible to observe a patch of high skin friction. This patch is located next to two small-scale streamwise vortices that generate a high-momentum region favouring the existence of EP events. The other flank of these streamwise vortices is related to the existence of the lifted low-speed streak located next to them.

The discussion presented above, reveals that extreme events might be clustered since both BF and EP events are located below a large-scale structure of positive velocity fluctuation that interacts with a low-speed streak, generating large-scale shear regions which in turn are densely populated by strong vortical structures. As a result, these regions are highly intermittent, which also increases the probability of occurrence of extreme events.

6.2. Conditional structures

The instantaneous flow fields analysed above showed the complex dynamics underlying the existence of extreme events in a general form. To have a clearer view of the mean structures responsible for extreme events, there have been computed conditionally averaged volumetric flow fields using the same approach explained in § 4. Figure 20 shows isosurfaces of the main coherent structures together with vector plots associated with EP conditional fields related to a left-sided ($Q_L$) vortex. The structure coloured by the wall-normal velocity $\langle u_y^+\rangle _{EP}$ is the energetic quasi-streamwise vortex computed using the second invariant of the velocity gradient $Q$ at a level $Q^+ = 0.5$. At the right-hand side of the vortex (in dark red), the coherent structure of high streamwise velocity fluctuation plotted at a level $u_x ^ + = 2.5$ is a product of streamwise momentum transported from the outer to the near-wall region by the quasi-streamwise vortex. This high streamwise momentum isosurface has a length of approximately 300 wall units at the chosen level and is inclined towards the core in the streamwise direction. The region of EP $\tau _x^+$ appears to be the footprint of the tail of this high-speed structure. In addition, this high-velocity structure is embedded within a large-scale structure of positive fluctuation. The large-scale isosurface herein is plotted at a level $u_x^+ = 1.25$. However, it should be noted that the large-scale structure extends up to $y^+ \approx 400$ or $y/\delta \approx 0.4$ as observed in § 4.3. At the other flank of the strong vortex, (in grey) a lifted streak at a level $u_x^+ \approx -0.1$ can be observed. This small lifted low-speed streak is also observed in figure 8(a). That lifted streak has a similar topology to the structures studied by Schoppa & Hussain (Reference Schoppa and Hussain2002) who demonstrated that these structures are associated with instabilities responsible for the generation of streamwise vortices. Indeed, the present instantaneous fields also show the existence of lifted streaks next to the quasi-streamwise vortices related to EP events. The EP conditional vector plot in the sliced plane to the left of the box exhibits the velocity vectors ($u_x, u_y$) showing that the quasi-streamwise vortex generates a strong Q4 (sweep) event at the buffer and downstream the extremely high $\tau _x$ event. Finally, the transverse vector plot displayed downstream the EP event has been taken at ${\rm \Delta} x^+ \approx 300$ and has been shifted for clarity. This vector plot shows that the large-scale structure of positive $u_x$ related to EP events is also associated with outer counter-rotating motions that transport momentum from the core of the flow towards the overlap region.

Figure 20. Conditioned isosurfaces of different flow parameters under an EP event. The structure on the left-hand side (grey) shows a lifted low-speed streak with $u_x^+ = -0.1$. The structure in the middle (coloured by $u_y^+$) is a left-sided ($Q_L$) quasi-streamwise vortex computed using the $Q$ criterion. The red structure on the right-hand side is a high-speed streak isosurface at $u_x^+ \approx 2.5$. The transparent red isosurface is a structure of positive fluctuation at a level $u_x^+ \approx 1.25$. Vector plots on the two sliced planes are shifted for clarity and show conditional averages at ${\rm \Delta} R\theta ^+ = 0$ and ${\rm \Delta} x^+ =300$.

Likewise, isosurfaces of three-dimensional flow structures related to backflow events are displayed in figure 21. The structure coloured by $\langle u_y^+\rangle _{BF}$ is an oblique left-sided ($Q_L$) vortex at a level $Q^+ = 0.5$. The BF conditional averaged isosurfaces also reveal the topology of a large-scale structure of positive fluctuation and a low-speed streak. The grey isosurface has been computed at a level $u_x^+=-1$ showing that a large-scale region of approximately 1000 wall units of strong negative fluctuation at the sublayer and the buffer is required to engender BF events. The light blue transparent isosurface shows a region close to the boundary of the low-speed streak at a level $u^+=-0.1$ and has a streamwise extension of approximately 1500 wall units or $x/R \approx 1.5$ (note that the complete extension of the low-speed streak is not shown here). This low-speed streak shows a meandering behaviour induced by the oblique vortex at ${\rm \Delta} x^+ \approx 0$. Instantaneous observations of the vector plots taken at several streamwise locations (not shown here) reveal that the low-speed streak is associated with two near-wall large counter-rotating roll modes which are inclined approximately 11$^\circ$, in good agreement with Marusic & Heuer (Reference Marusic and Heuer2007). The transverse vector plot taken at ${\rm \Delta} x^+ \approx 300$ and shifted to the boundary of the box downstream the BF event, reveals an interaction between near-wall and outer counter-rotating roll modes. The presence of this counter-rotating motions is consistent with the conditional structures studied by Hutchins & Marusic (Reference Hutchins and Marusic2007b) and Marusic et al. (Reference Marusic, Mathis and Hutchins2010). Additionally, there exists a large structure containing high values of streamwise velocity fluctuations shown as a transparent red isosurface. This high-speed region is plotted at a level $u_x^+ = 1$. The high-speed structure surrounds the low-speed streak, generating high shear between both structures. This explains the large population of vortical structures at the regions where clusters of extreme events were found. Finally, the velocity vector slice displayed in the ${\rm \Delta} x^+$–$y^+$ plane, is shifted for clarity and is taken at the azimuthal location ${\rm \Delta} R\theta = 0$. The vectors located above ${\rm \Delta} x^+ \approx -20$ exhibit a Q2 (ejection) region followed by a sweep downstream and at the top of the vortex. That behaviour resembles the ‘hairpin vortex signature’ explained by Adrian et al. (Reference Adrian, Meinhart and Tomkins2000) at the outer region of TBLs. Our evidence shows that these events can also occur within the inner layer due to the presence of oblique vortices or inverted hairpin-like structures as revealed by Chin et al. (Reference Chin, Monty, Chong and Marusic2018b) and more recently by Wu et al. (Reference Wu, Cruickshank and Ghaemi2020). Finally, an inclined layer of Q2 events with a streamwise extension that exceeds 1000 wall units is also observed on the same vector plot, generated by the large-scale counter-rotating motions and related to the presence of the low-speed streak.

Figure 21. Conditioned isosurfaces of different flow parameters under a BF event. The grey structure attached to the wall shows a lifted low-speed streak region at a level $u_x^+ = -1.0$. The cyan transparent isosurface that surrounds the grey region is a low-momentum region plotted at $u_x^+ = -0.1$. The structure above the low-speed streak coloured by $u_y^+$ is a left-sided ($Q_L$) oblique vortex computed at $Q^+ = 0.5$. The red isosurface at the top is a large-scale structure of positive streamwise fluctuation computed at $u_x^+ \approx 1.25$. Vector plots on the two sliced planes are shifted for clarity and show conditional averages at ${\rm \Delta} R\theta ^+ = 0$ and ${\rm \Delta} x^+ = 300$.

Due to the complexity of the flow structures associated with extreme events, we finish this study with the conceptual models shown in figures 22 and 23 where most of the coherent motions related to EP and BF events are exhibited. It is important to state that all the flow structures displayed in solid colour are based on the actual post-processed data and results contained in our three-dimensional conditional averages and instantaneous flow visualisations. Note that the structures represented in solid black lines hypothesise the continuation of the coherent structures that are not clearly observed from our data.

Figure 22. Schematic three-dimensional sketch that exhibits in scale the conditional coherent structures (low-/high-speed streaks, vortices, swirl motions) that contribute to the existence of EP events. Coloured structures represent coherent motions observed in the present conditional fields. Structures represented in black lines try to hypothesise the continuation of coherent structures which cannot be clearly observed in the conditional fields.

Figure 23. Schematic three-dimensional sketch that exhibits in scale the conditional coherent structures (low-/high-speed streaks, vortices, swirl motions) that contribute to the existence of BF events. The colour convention is the same as figure 22.

Figure 22 shows the flow structures associated with EP events. The primary structure related to extremely high wall shear stress events is the strong streamwise vortex represented as a dark red isosurface for a left-sided ($Q_L$) vortex and dark blue for a ($Q_R$) vortical structure. The orange isosurface represents a high-speed streak located directly above the extreme $\tau _x$ event, which results from a high rate of momentum transport from the outer to the inner layers produced mainly by the strong streamwise vortex. The green structures represent the low-momentum region generated at the upwash side of the strong streamwise vortex. The light red structures are weaker streamwise vortical structures that extend as part of the strong quasi-streamwise vortex (dark red). Based on our findings, the strong quasi-streamwise vortex is a region with high magnitudes of streamwise vorticity located, on average, at the trailing leg of a larger vortical structure (Jiménez & Pinelli Reference Jiménez and Pinelli1999).

At the right-hand side of the $Q_L$ vortex (dark red isosurface), there exists a counter-rotating motion in light blue. The same phenomenon is observed for a strong $Q_R$ vortex (dark blue isocontour) associated with EP events. Indeed, both counter-rotating swirl motions are not quiescent and contribute to the existence of a large-scale region of positive $u_x$ (see figure 20 for clarity) by transporting momentum from the outer layers towards the inner region. This large-scale structure of positive fluctuation surrounds the small-scale region of high momentum related to the existence of high wall-friction events. At the right-hand side of the light blue vortical structure, there is represented in grey isosurface a low-speed streak which results from the transport of momentum by this structure. The pale yellow isosurface represents a region of $u_x > 0$. Finally, resembling the horseshoe or hairpin models for the outer region motions, we hypothesise that the swirl motions will continue to the outer layers forming hairpin-like structures. Our instantaneous visualisations support this hypothesis (see figure 19), which reveal that hairpin-like structures or hairpin packets occur downstream the clusters of extreme events.

The coherent motions observed under conditional backflow fields are represented in figure 23. In this figure, we illustrate the main coherent structures associated with backflow events generated by both a left-sided ($Q_L$) and a right-sided ($Q_R$) oblique vortex as previously shown in figure 5. Herein, the low-speed streaks are attached to the wall (in grey isosurfaces) and are surrounded by high-momentum regions $u_x>0$ represented in pale yellow isosurface. The strong oblique vortex associated with backflow events is located just above the low-speed streak represented as a light yellow isosurface. It is interesting to notice that between the low-speed streak and the high-momentum regions there exist two inclined counter-rotating roll modes which could be, on average, the tails of a hairpin-like structure. Additionally, these swirl motions are associated with the existence of an inclined Q2 layer that occurs downstream from the BF event as shown previously in figure 21. These more ‘quiescent’ modes are essential structures since they transport streamwise momentum from the inner layer to the overlap region generating the long low-speed streak between them. Furthermore, because of their inclination, they also aid to the existence of backflow being them a source of negative streamwise velocity fluctuations. Both inclined counter-rotating motions leave their footprint at the wall, and can be observed in the alternating behaviour of $\tau _\theta$ exhibited in figure 6(d–f). Our results also indicate that all the near-wall complex dynamics associated with BF events occurs below a LSM of positive fluctuation, which seems to modulate the near-wall cycle, showing consistency with the study conducted by Lenaers et al. (Reference Lenaers, Li, Brethouwer, Schlatter and Örlü2012).

The model presented figure 23 is in good agreement with the conceptual sketch suggested by Sheng et al. (Reference Sheng, Malkiel and Katz2009), who studied in detail the buffer structures responsible for extreme events in TBL using digital holographic microscopy. On the other hand, our model coincides partially to the study conducted by Bross et al. (Reference Bross, Fuchs and Kähler2019) in the sense that BF events occur below low-speed streaks and are related to oblique or spanwise vortical structures. Nevertheless, the outer structures presented by Bross et al. (Reference Bross, Fuchs and Kähler2019) show a different mechanism when compared to the present investigation as they argue that BF events occur below a large-scale structure of low momentum which decelerates the near-wall low-speed streaks. The existence of a low-speed structure above extreme events also disagrees with the amplitude modulation effect of LSM on the near-wall dynamics Hutchins & Marusic (Reference Hutchins and Marusic2007b), as a low-momentum structure would damp out the near-wall turbulent fluctuations, resulting in lower intermittency of the near-wall cycle. The differences mentioned could be conferred to dissimilarities between the APG-TBL and the canonical pipe flow.