3.1 The largest attached eddies: large-scale and very-large-scale motions

We first explore the temporal evolution of the largest attached eddies (i.e. the energy-containing motions at ${\it\lambda}_{z}=1.5h$ ) in the minimal unit ( $O950$ and $SO950$ in table 1). As discussed in Hwang (Reference Hwang2015), the largest attached eddy is composed of a VLSM (outer streaky motion) and LSMs (outer streamwise vortical structures) aligned with it. In a large computational domain, the VLSM is typically featured to be a long structure mainly carrying a large amount of the streamwise turbulent kinetic energy, and this is quite similar to the behaviour of the streaks in the near-wall region (Hwang Reference Hwang2015). Therefore, in the outer minimal unit, in which only single VLSM would be resolved mainly by zero streamwise wavenumber due to its long length (i.e. $k_{x}=0$ ; see also figure 6), its temporal evolution would be well tracked by

where $u$ is the streamwise velocity fluctuation, ${\it\Omega}_{h}$ the lower (or upper) half of the computational domain of interest and $V_{h}$ the volume of the half of the computational domain.

On the contrary, the LSM is a relatively short structure given with its streamwise length scale at ${\it\lambda}_{x}\simeq 3{-}4h$ . The structure at this length is identical to that often postulated as a hairpin vortex packet by Adrian and coworkers (Adrian Reference Adrian2007). This structure is relatively isotropic, compared to the VLSM which mostly carries the streamwise velocity fluctuation, in the sense that it contains all the velocity components (Hwang Reference Hwang2015). It is important to mention that this structure was found to share a number of statistical similarities with quasi-streamwise vortex in the near-wall region (Hwang Reference Hwang2015). This suggests that the LSMs would likely to be quasi-streamwise vortical structures in the outer region, although they are much more disorganised due to small-scale turbulence associated with cascade and turbulent dissipation (Jiménez Reference Jiménez2012). It should be mentioned that this view is consistent with del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2006) who called this structure the ‘tall-attached vortex cluster’. Since the VLSM carries very little wall-normal or spanwise turbulent kinetic energy, in the minimal unit, the LSMs would be well characterised by either

or

where $v$ and $w$ are the wall-normal and spanwise velocity fluctuations, respectively. Later, we shall see that these two variables are indeed strongly correlated to each other (figure 5 c,d). Here, it should also be pointed out that the use of the primitive variables in characterising the quasi-streamwise vortical structure instead of enstrophy or streamwise vorticity is intentionally introduced to track their behaviour more precisely. Given the significant amount of small-scale turbulence associated with energy cascade and turbulent dissipation in the logarithmic and the outer regions, the variables such as enstrophy or streamwise vorticity would not faithfully represent the large-scale organised vortical structure. Indeed, Jiménez (Reference Jiménez2012) recently showed that such variables carry dominant energy around turbulent dissipation length scales.

Figure 3.

Time evolution of the flow field of $O950$ : (a) time trace of $E_{u}^{+}$ (solid) and $E_{v}^{+}$ (dashed); (b) magnification of (a) for $tu_{{\it\tau}}/h\in [158,164]$ ; (c–f) the corresponding flow visualisation at $tu_{{\it\tau}}/h=159,160.2,160.9,162$ . In (c–f), the red and blue iso surfaces indicate $u^{+}=-3.2$ and $v^{+}=1.4$ , respectively.

A set of the flow fields of $O950$ , in which the largest attached eddies are contained with other smaller-scale turbulent motions, are first examined. Figure 3 shows the time trace of $E_{u}$ and $E_{v}$ , and the corresponding evolution of the flow field in time. Both $E_{u}$ and $E_{v}$ show large-scale temporal oscillations with a time scale roughly at $Tu_{{\it\tau}}/h\simeq 2{-}5$ (figure 3 a). This is exactly the feature known as ‘bursting’ by Flores & Jiménez (Reference Flores and Jiménez2010). A careful observation reveals that $E_{u}$ and $E_{v}$ oscillate with a certain phase difference: for example, at $tu_{{\it\tau}}/h=159$ , $E_{v}$ is around the local maximum while $E_{u}$ being at fairly low-energy state (figure 3 b), and, at $tu_{{\it\tau}}/h=160.2$ , $E_{v}$ reaches the local minimum whereas $E_{u}$ becomes considerably larger (figure 3 b). Visualisation of the corresponding flow fields in figure 3(c–f) suggests that this is likely due to the interactive dynamics between the VLSM (streak) and the LSMs (streamwise vortical structures). At $tu_{{\it\tau}}/h=159$ (figure 3 c), the flow field exhibits a few fairly strong large-scale $v$ structures, which would be a part composing the LSMs in the minimal unit. On the other hand, $u$ structures in the flow field are fairly weak at this time, and they appear to be collectively gathered around the $v$ structures. At $tu_{{\it\tau}}/h=160.2$ (figure 3 d), the $v$ structures become considerably weakened whereas the $u$ structures are significantly amplified, forming a strong streaky structure extending over the entire streamwise domain. This amplification of the streaky structure appears to be a consequence of the ejection of the streamwise momentum by the strong $v$ structures observed at $tu_{{\it\tau}}/h=159$ (figure 3 c), clearly reminiscent of the ‘lift-up’ effect predicted by previous theoretical studies (del Álamo & Jiménez Reference del Álamo and Jiménez2006; Cossu, Pujals & Depardon Reference Cossu, Pujals and Depardon2009; Pujals et al. Reference Pujals, García-Villalba, Cossu and Depardon2009; Hwang & Cossu Reference Hwang and Cossu2010b ). The amplified streaky structure subsequently meanders along the streamwise direction (figure 3 $e$ ). This process eventually leads to regeneration of new $v$ structures with the breakdown of the streaky motion (figure 3 $f$ ).

Figure 4.

Time evolution of the flow field of $SO950$ : (a) time trace of $E_{u}^{+}$ (solid) and $E_{v}$ (dashed); (b) magnification of (a) for $tu_{{\it\tau}}/h\in [564,572]$ ; (c–f) the corresponding flow visualisation at $tu_{{\it\tau}}/h=565,568.5,569.3,570$ . In (c–f), the red and blue iso surfaces indicate $u^{+}=-4$ and $v^{+}=1.5$ , respectively.

Qualitatively the same behaviour is observed in the over-damped simulation $SO950$ , in which only the energy-containing motions at ${\it\lambda}_{z}=1.5h$ are isolated by replacing all the smaller-scale motions with a crude eddy viscosity given with the increased $C_{s}$ . Figure 4 shows the time trace of $E_{u}$ and $E_{v}$ , and the corresponding evolution of the flow field of $SO950$ . As in $O950$ (figure 3 a), both $E_{u}$ and $E_{v}$ exhibit temporal oscillations with the time scale at $Tu_{{\it\tau}}/h=2{-}5$ (figure 4 a). In this case, a more prominent phase difference is observed between the oscillations of $E_{u}$ and $E_{v}$ (figure 4 a,b), presumably because the smaller-scale surrounding turbulent motions are removed in $SO950$ . However, this could also be partially due to the increased oscillation amplitude of $E_{u}$ in $SO950$ , which has previously been shown to be an artefact caused either by the crude eddy viscosity itself or by the lack of the energy-containing motions in the near-wall and logarithmic regions (Hwang Reference Hwang2015). Not surprisingly, the flow fields of $SO950$ are composed of much smoother structures than those of $O950$ (figure 4 c–f). However, the temporal evolution of the $u$ and $v$ structures of $SO950$ are remarkably similar to that of $O950$ : the $v$ structures significantly amplify the streaky $u$ structure via the lift-up effect (figure 4 c,d); the amplified streaky $u$ motion subsequently meanders along the streamwise direction (figure 4 $e$ ); the $v$ structures are finally regenerated with breakdown of the streaky $u$ motion (figure 4 $f$ ).

To statistically quantify this cyclic dynamical process, auto- and cross-correlations of several variables of interest are computed. It is also useful to introduce two additional variables, such that:

for $k_{x}=0$ and $k_{z}=2{\rm\pi}/L_{z}$ , and

for $k_{x}=2{\rm\pi}/L_{x}$ and $k_{z}=2{\rm\pi}/L_{z}$ . Here, $\hat{\cdot }$ denotes the Fourier-transformed state in the $x$ and $z$ directions, and ${\it\Omega}_{y,h}$ is the lower (or upper) half of the wall-normal domain. We note that, in (3.2), $E_{0}$ is introduced to compute the energy of the streamwise uniform component of the motions at ${\it\lambda}_{z}=1.5h$ , while $E_{1}$ is to compute the energy of the first streamwise Fourier component, which would measure the meandering motion observed in figures 3 and 4. The correlation functions are defined as

where $i,j=u,v,w,0,1$ and $\langle \cdot \rangle$ indicates average in time. In the present study, all the correlation functions are computed by averaging the lower and upper half-channels.

Figure 5.

Time correlation functions of (a,c) $O950$ and (b,d) $SO950$ . In (a,b), ——, $C_{uu}({\it\tau})$ ; – – – –, $C_{vv}({\it\tau})$ ; – - – - –, $C_{ww}({\it\tau})$ . In (c,d), ——, $C_{uv}({\it\tau})$ ; – – – –, $C_{uw}({\it\tau})$ ; – - – - –, $C_{vw}({\it\tau})$ .

Figure 5 shows the correlation functions computed with $E_{u}$ , $E_{v}$ and $E_{w}$ from $O950$ and $SO950$ . In the case of $O950$ , all the auto-correlations decay to zero at $|{\it\tau}u_{{\it\tau}}/h|\simeq 1{-}2$ , although $C_{uu}$ tends to drop a little more slowly than $C_{vv}$ and $C_{ww}$ (figure 5 a). The size of the time interval, in which the auto-correlations remain positive, is $Tu_{{\it\tau}}/h\simeq 2{-}4$ , and this roughly corresponds to the single period of the temporal oscillation of $E_{u}$ and $E_{v}$ (figure 3 a). The cross-correlation $C_{uv}$ of $O950$ also reveals that there is indeed a phase difference between $E_{u}$ and $E_{v}$ (figure 5 c): the peak of $C_{uv}$ is located at ${\it\tau}u_{{\it\tau}}/h\simeq -0.6$ , indicating that $E_{u}$ statistically reaches its local extremum approximately ${\rm\Delta}{\it\tau}u_{{\it\tau}}/h\simeq 0.6$ before $E_{v}$ reaches its local extremum. On the other hand, $E_{v}$ and $E_{w}$ are found to be strongly correlated to each other: $C_{vw}({\it\tau}=0)\simeq 0.7$ and $C_{uw}$ also exhibits a peak at ${\it\tau}u_{{\it\tau}}/h\simeq -0.6$ as $C_{uv}$ does (figure 5 c).

Qualitatively the same behaviour is observed in the correlation functions of $SO950$ in which only the energy-containing motions at ${\it\lambda}_{z}\simeq 1.5h$ are isolated (figure 5 b,d). All the auto-correlations drop to zero near $|{\it\tau}u_{{\it\tau}}/h|\simeq 1.5{-}2$ (figure 5 b), resulting in the time interval of positive auto-correlation to be $Tu_{{\it\tau}}/h\simeq 4$ . As for $O950$ , this time scale reasonably well quantifies a single period of the temporal oscillations of $E_{u}$ and $E_{v}$ (figure 4 a). The phase difference between $E_{u}$ and $E_{v}$ is also seen in $C_{uv}$ of $SO950$ (figure 5 d) which shows the peak at ${\it\tau}u_{{\it\tau}}/h\simeq -1.7$ . Finally, $E_{v}$ and $E_{w}$ are also found to be strongly correlated to each other, as seen in $C_{vw}$ (figure 5 d).

Figure 6.

Time correlation functions of (a,c) $O950$ and (b,d) $SO950$ . In (a,b), ——, $C_{00}({\it\tau})$ ; – – – –, $C_{11}({\it\tau})$ ; – - – - –, $C_{01}({\it\tau})$ . In (c,d), ——, $C_{0u}({\it\tau})$ ; – – – –, $C_{0v}({\it\tau})$ ; – - – - –, $C_{1v}({\it\tau})$ .

The correlation functions with $E_{0}$ and $E_{1}$ are also presented in figure 6. The auto- and cross-correlations obtained only with $E_{0}$ and $E_{1}$ (i.e. $C_{00}$ , $C_{11}$ and $C_{01}$ ) are given in figure 6(a,b). The correlation functions of $O950$ show fairly similar behaviour to those of $SO950$ : $C_{00}$ and $C_{11}$ of $O950$ exhibit similar time scales to those of $SO950$ , and $C_{01}$ of both $O950$ and $SO950$ has a phase difference between $E_{0}$ and $E_{1}$ with ${\rm\Delta}{\it\tau}u_{{\it\tau}}/h=1$ . The phase difference between $E_{0}$ and $E_{1}$ implies that the streamwise uniform motion occurs first and the meandering motion follows, as in the instantaneous fields shown in figures 3 and 4. Here, it is interesting to note that the time scale of $C_{11}$ ( $Tu_{{\it\tau}}/h\simeq 2$ ) appears to be much smaller than that of $C_{00}$ as well as that of $C_{uu}$ , $C_{vv}$ and $C_{ww}$ ( $Tu_{{\it\tau}}/h\simeq 4$ ; see also figure 5 a,b). This suggests that the meandering motion of the streak (figures 5 c and 6 c), which would be characterised by $E_{1}$ , is a rapid process which persists only for roughly half of the bursting period.

Cross-correlations are further computed by also considering $E_{u}$ , $E_{v}$ and $E_{w}$ , as shown in figure 6(c,d). The correlation $C_{0u}$ shows that $E_{0}$ and $E_{u}$ are strongly correlated to each other in both $O950$ and $SO950$ , indicating that the streamwise uniform mode ( $k_{x}=0$ ) at ${\it\lambda}_{z}=1.5h$ well represents the streaky motion (i.e. VLSM) in the minimal unit. This feature can also be confirmed from $C_{0v}$ of $O950$ and $SO950$ , which behaves very similarly to $C_{uv}$ (figure 5 c,d). Finally, the peak of $C_{1v}$ for both $O950$ and $SO950$ is found roughly at ${\it\tau}u_{{\it\tau}}/h\simeq -1$ . This indicates that the meandering motion of the streak statistically appears before the vortical structures (LSMs) are fully amplified, consistent with the instantaneous flow fields given in figures 3 and 4.

Despite many qualitative similarities between the dynamical behaviours of the energy-containing motions in $O950$ and $SO950$ , it should be pointed out that the correlation functions of the two simulations ( $O950$ and $SO950$ ) are not quantitatively the same, although this is not very surprising given the aggressive nature of the present numerical experiment: for example, the time intervals of $C_{vv}>0$ and $C_{ww}>0$ of $O950$ are smaller than those of $SO950$ (figure 5 a,b), and the peak locations of $C_{uv}$ , $C_{uw}$ and $C_{0v}$ of $O950$ are a little different from those of $SO950$ (figures 5 c,d, 6 c,d). Apparently, these differences could  stem from two possible origins: one is that $E_{u}$ , $E_{v}$ and $E_{w}$ of $O950$ contain the effect of the surrounding smaller-scale turbulent motions, and the other is that the dynamical behaviour of the motions in $SO950$ is probably a little distorted by the artificially increased eddy viscosity of $SO950$ . The issue of which of the two more dominantly yields the differences could be clarified by further inspecting $C_{00}$ , $C_{11}$ and $C_{01}$ (figure 6 a,b), as they are, in a way, obtained by applying cutoff filters to both of $O950$ and $SO950$ . It appears that the correlation functions of $O950$ are not exactly the same as those of $SO950$ , suggesting that increasing $C_{s}$ a little distorts the motions at ${\it\lambda}_{z}=1.5h$ . However, the times scales of these correlation functions for $O950$ do not appear to be significantly different from those for $SO950$ . This implies that the correlation functions obtained with $E_{u}$ , $E_{v}$ and $E_{w}$ of $O950$ do not precisely represent the dynamical behaviour of the largest attached eddies at ${\it\lambda}_{z}=1.5h$ due to the contribution by the surrounding smaller-scale turbulent fluctuations.

3.2 The attached eddies in the logarithmic region

Now, we consider the attached eddies in the logarithmic region. As shown in § 2.2, the attached eddies in the logarithmic region generate a self-similar statistical structure with respect to the spanwise length scale ${\it\lambda}_{z,0}~(=L_{z})$ . This self-similar part, which contains the essential physical process of the energy-containing motions in the logarithmic region, appears mainly below $y\simeq 0.3{-}0.4L_{z}$ (figure 2). To trace the dynamical behaviour of this self-similar part more precisely, the definitions of $E_{u}$ , $E_{v}$ , $E_{w}$ , $E_{0}$ and $E_{1}$ given in (3.1) and (3.2) are a little modified, so that the wall-normal domain of the integration becomes only $[0,2/3L_{z}]$ . We note that this new definition is still consistent with (3.1), as the wall-normal domain of the integration becomes $[0,1]$ for $L_{z}=1.5h$ , which is the spanwise length scale of the VLSM and the LSM.

Inspection of the instantaneous flow fields of all the simulations in the logarithmic region (see also table 1) reveals that basically the same dynamical process occurs in the attached eddies, given with streaks and quasi-streamwise vortical structures, at each of the spanwise length scales belonging to the logarithmic region: i.e. amplification of the streaks by the quasi-streamwise vortical structures, subsequent meandering of the streaks along the streamwise direction and breakdown of the streaks with regeneration of the quasi-streamwise vortical structures. To avoid repetition of the same discussion given in § 3.1, here we only report auto- and cross-correlation functions of the simulations concerning the logarithmic region with a focus on the scaling of the computed correlation functions.

Figure 7.

Auto-correlation functions: (a,b)  $C_{uu}({\it\tau})$ ; (c,d)  $C_{vv}({\it\tau})$ ; (e,f)  $C_{11}({\it\tau})$ . In (a,c,e), ——  $L950a$ ; – – – – $L950b$ ; – - – - –  $L1800a$ ; – $\cdot \cdot$ – $\cdot \cdot$ –  $L1800b$ ; $\cdots \cdots$   $L1800c$ , while, in (b,d,f), ——  $LS950a$ ; – – – – $LS950b$ ; – - – - –  $LS1800a$ ; – $\cdot \cdot$ – $\cdot \cdot$ –  $LS1800b$ ; $\cdots \cdots$   $LS1800c$ .

Figure 7 shows auto-correlations obtained from all the simulations concerning the energy-containing motions in the logarithmic region. All the correlation functions from both full (figure 7 a,c,e) and over-damped simulations (figure 7 b,d,f) scale fairly well with the spanwise size of the attached eddies $L_{z}~(={\it\lambda}_{z,0})$ , suggesting that the attached eddies in the minimal unit are ‘dynamically’ self-similar with respect to the spanwise size. The auto-correlation functions $C_{uu}$ and $C_{vv}$ in figure 7(a–d) reach zero at ${\it\tau}u_{{\it\tau}}/L_{z}\simeq \pm 1$ , indicating that the single period of the self-sustaining process would be roughly given by $Tu_{{\it\tau}}/L_{z}\simeq 2$ . As in the case of the outer attached eddies, $C_{11}$ , which characterises the time scale of the meandering motion of the streak, is found to have much shorter time scale $Tu_{{\it\tau}}/L_{z}\simeq 1$ than that of $C_{uu}$ and $C_{vv}$ .

Figure 8.

Cross-correlation functions: (a,b) $C_{uv}({\it\tau})$ ; (c,d) $C_{vw}({\it\tau})$ ; (e,f) $C_{1v}({\it\tau})$ . In (a,c,e), ——  $L950a$ ; – – – – $L950b$ ; – - – - –  $L1800a$ ; – $\cdot \cdot$ – $\cdot \cdot$ –  $L1800b$ ; $\cdots \cdots$   $L1800c$ , while, in (b,d,f), ——  $SL950a$ ; – – – – $SL950b$ ; – - – - –  $SL1800a$ ; – $\cdot \cdot$ – $\cdot \cdot$ –  $SL1800b$ ; $\cdots \cdots$   $SL1800c$ .

Figure 8 shows several cross-correlations obtained from the same simulations. The cross-correlation $C_{uv}$ of both full (figure 8 a) and over-damped simulations (figure 8 b) exhibits a phase difference between $E_{u}$ and $E_{v}$ (or $E_{w}$ ), similar to that of the simulations concerning the outer-scaling motions (figure 5 c,d). A strong correlation between $E_{v}$ and $E_{w}$ is also observed in the motions in the logarithmic region, similarly to those in the outer region (figure 8 c,d). Finally, the cross-correlation $C_{1v}$ also confirms that amplification of $E_{v}$ appears a little after $E_{1}$ is amplified, indicating that the vortical structures in the logarithmic region are regenerated after the meandering motion (figure 8 e,f).