2.1. Direct numerical simulation

The Navier–Stokes equations for incompressible flow are solved using the fully implicit decoupling method by Kim, Baek & Sung (Reference Kim, Baek and Sung2002). The configuration corresponds to fully developed turbulent channel flow, and details of the DNS and the validation are provided in the earlier study by Lee et al. (Reference Lee, Lee, Choi and Sung2014). In the present study, $x$ , $y$ and $z$ denote the streamwise, wall-normal and spanwise directions, respectively, and the associated components of velocity fluctuations are $u$ , $v$ and $w$ ; capital letters are used to indicate mean quantities. An angle bracket $\langle \cdot \rangle$ denotes ensemble averaging. The superscript $+$ represents quantities normalized by the viscous scales, namely the friction velocity, $u_{{\it\tau}}$ , and viscosity, ${\it\nu}$ . The Reynolds number based on the channel half-height, ${\it\delta}$ , and the laminar centreline velocity, $U_{CL}$ , is $Re_{{\it\delta}}=28\,000$ , and the friction Reynolds number is $Re_{{\it\tau}}=930$ . The domain size in the streamwise and spanwise directions is $(L_{x},L_{z})=(10{\rm\pi}{\it\delta},3{\rm\pi}{\it\delta})$ . A staggered mesh is employed with $4993\times 401\times 2497$ grid points in the $x$ , $y$ and $z$ directions, respectively. The grid sizes in the homogeneous directions are ${\rm\Delta}x^{+}=5.86$ and ${\rm\Delta}z^{+}=3.51$ . The minimum and maximum wall-normal grid spacings are ${\rm\Delta}y_{min}^{+}=0.0287$ and ${\rm\Delta}y_{max}^{+}=7.31$ , respectively. The simulation is performed at a constant mass flow rate, and the time step is ${\rm\Delta}t=0.002{\it\delta}/U_{CL}$ , or ${\rm\Delta}t^{+}=0.0618$ , and the averaging period spans $220{\it\delta}/U_{CL}$ , corresponding to seven sweeps across the flow domain. A total of 2010 snapshots are stored, with each two consecutive fields separated by $0.1{\it\delta}/U_{CL}$ (or ${\rm\Delta}t^{+}=3.09$ ). In the present study, the following wall-normal regions are defined: the near-wall region $y^{+}<30$ ; the log layer $30<y^{+}<0.15{\it\delta}^{+}$ ; and the channel core $y^{+}>0.15{\it\delta}^{+}$ .

2.2. Structure-detection method

To investigate the formation of elongated $u$ structures in space and time, streak detection and conditional sampling techniques are applied to the instantaneous fields. The streak-detection algorithm is based on the work by Nolan & Zaki (Reference Nolan and Zaki2013), where local extrema of $u$ are used to reconstruct the three-dimensional (3D) streamwise-elongated structures. Further detailed descriptions of the methodology can be found in Zaki (Reference Zaki2013). Figure 1 displays an example of the outcome of the detection method. The threshold $u_{th,raw}=0.1U_{b}$ , where $U_{b}$ is the bulk velocity, is used in the present study (Dennis & Nickels Reference Dennis and Nickels2011; Baltzer et al. Reference Baltzer, Adrian and Wu2013; Lee et al. Reference Lee, Lee, Choi and Sung2014). The procedure is outlined below.

Step 1: Gaussian filter. A two-dimensional Gaussian filter is applied in the $y$ – $z$ plane to the instantaneous $u$ field. The standard deviation ( ${\it\sigma}$ ) of the Gaussian filter is determined as a function of the wall-normal distance, because the spanwise length scale of the $u$ motions varies with height from the wall. The standard deviation is set to the half-width of the spanwise two-point correlation of $u$ at a level of 0.5. This choice prevents adjacent negative $u$ from counterbalancing the positive $u$ , and vice versa. The curve-fitted standard deviation function ${\it\sigma}(y/{\it\delta})$ is

Step 2: Long-wavelength-pass filter. A long-wavelength-pass filter is applied to the Gaussian-filtered flow fields in order to isolate structures longer than $1{\it\delta}$ in the streamwise direction. If the streamwise filter is applied alone, it leads to discontinuous isosurfaces of the velocity field, which are not suitable for identifying contiguous $u$ structures along the streamwise direction (Lee et al. Reference Lee, Lee, Choi and Sung2014). Hence, the Gaussian filter is also needed in order to ensure that the $u$ isosurfaces are smooth.

The smoothed isosurfaces in figure 1(b) illustrate the filtered flow field, $\hat{u}$ . Owing to the weakened magnitude of $u$ by the filters, the velocity threshold is corrected, $u_{th}(y)=f(y)u_{th,raw}$ , where $f(y)=\langle u(\boldsymbol{x})\hat{u} (\boldsymbol{x})\rangle /\langle u(\boldsymbol{x})u(\boldsymbol{x})\rangle$ . Then local extrema are identified in the filtered fields along lines in the spanwise direction. Note that $\hat{u}$ is a smooth function due to the Gaussian filter and, as a result, the computed local extrema at the core of each flow structure are contiguous (see figure 14 in Lee et al. (Reference Lee, Lee, Choi and Sung2014)). Neighbouring extrema thus mark the cores of the velocity disturbances. The extrema that are connected in the streamwise direction are linked and labelled as a unique object, which can be regarded as the characteristic projections ( $CP$ ) of the flow structures. In this manner, $CP$ represents the streamwise and wall-normal extents of LSMs:

Based on the above definition, $CP=+1$ marks high-speed structures and $CP=-1$ identifies low-speed ones. Figure 1(b) shows a transparent isosurface of the filtered velocity field, along with the surface $CP=-1$ coloured by wall-normal height. Figure 1(c) shows a top view of the $u$ -perturbation contours at $y/{\it\delta}=0.15$ . The thick solid lines show the intersection of the wall-parallel plane with the $CP$ surface and they mark the streamwise and spanwise positions of the organized motion. The most downstream and upstream ends of $CP$ denote the head and the tail of the streak, respectively. The streamwise length, $L_{CP}$ , of each streak is measured and the spanwise boundary of the streak is defined by identifying lower and upper bounds ( $\hat{u} =\pm u_{th}$ ) on either side of the streak core (see figure 1 c). If there are multiple adjacent streaks (for example, near $x/{\it\delta}=4{-}6$ in figure 3 b), the boundaries of each streak are adjusted to the spanwise location where $\partial \hat{u} /\partial z$ changes sign.

Figure 2.

Histograms of the streamwise length ( $L_{CP}$ ) of low- and high-speed streaks at $y/{\it\delta}=0.15$ . (a) The mean number of the streaks per field ( $\langle N_{CP}\rangle$ ) in the wall-parallel plane. (b) The spanwise width ( $W_{CP}$ ) of low- and high-speed streaks. (c) The mean area occupied by the structures per field ( $\langle {\it\Pi}_{CP}\rangle$ ) versus their streamwise length; $\langle {\it\Pi}_{CP}\rangle$ is normalized by the total area of the wall-parallel plane ( $A=L_{x}\times L_{z}$ ). (d) The mean amplitude ( $A_{u}$ ) of the low- and high-speed streaks of a particular length. In (a,c), the closed and open symbols represent low- and high-speed streaks, respectively. The grey lines mark an exponential curve fit, and the insets show the results in linear scale. In (b,d), the contour lines indicate $\langle N_{CP}\rangle$ . The bin size for $L_{CP}$ is $1{\it\delta}$ .

3.1. Long streaks of streamwise velocity fluctuations

In this section, the length and area of long negative- and positive- $u$ regions are examined using the structure-detection algorithm (§ 2.2). Figure 2 shows the populations of negative- and positive- $u$ structures versus their streamwise length, $L_{CP}$ , at $y/{\it\delta}=0.15$ . The mean number ( $\langle N_{CP}\rangle$ ) of negative- $u$ (closed symbol) and positive- $u$ (open symbol) regions is displayed in figure 2(a). A single occurrence, $\langle N_{CP}\rangle =1$ , means that an average of one structure with the specified length is observed in this wall-parallel plane per unit time. Structures that are recorded more than once per field have a streamwise length less than $8{\it\delta}$ for the negative- $u$ regions and $6{\it\delta}$ for the positive counterparts. The vast majority of detected streaks have a streamwise length less than $1{\it\delta}$ . The maximum streamwise extent of negative- $u$ regions is approximately $20{\it\delta}$ , while that of positive structures is only $12{\it\delta}$ . Although the populations of both negative- and positive- $u$ regions decrease logarithmically with their length, the negative- $u$ regions tend to be longer than their positive counterparts and the latter are relatively biased towards shorter lengths. Beyond 3 ${\it\delta}$ in particular, the disparity between the populations of low and high speed becomes more pronounced. The slope of the populations, $d_{i}$ , is found to be steeper for positive- $u$ regions, where $d_{1}=1.01$ and $d_{2}=0.62$ for positive and negative $u$ , respectively. The results based on the present detection algorithm are in agreement with those by Dennis & Nickels (Reference Dennis and Nickels2011) and by Baltzer et al. (Reference Baltzer, Adrian and Wu2013) despite the difference in the structure identification procedure. Dennis & Nickels (Reference Dennis and Nickels2011) counted both low- and high-speed structures whose streamwise lengths were greater than $2{\it\delta}$ in a turbulent boundary layer. For low-speed structures, 44 % of the structures were below 3 ${\it\delta}$ while 54 % of the high-speed structures were in the same range. The distributions evaluated using the present algorithm show similar trends: 43 % and 60 % for low- and high-speed regions, respectively. The predicted ratio of long low- and high-speed structures ( ${>}2{\it\delta}$ ) are also similar in the two studies, approximately 1.3–1.5. Low-speed regions that are longer than $7{\it\delta}$ are less than 5 % of the total population, consistent with Baltzer et al. (Reference Baltzer, Adrian and Wu2013).

Figure 3.

Area identification at $y/{\it\delta}=0.15$ : (a) nLSM ( $u_{1}$ ) and nVLSM ( $u_{2}$ ); (b) pLSM ( $u_{3}$ ) and pVLSM ( $u_{4}$ ); here n = negative and p = positive. The contour levels are the same as in figure 1(c). The solid and dashed lines represent $CP=\pm 1$ and $\hat{u} =\pm u_{th}$ , respectively.

The population trends of the positive- and negative- $u$ regions classified based on their streamwise length ( $L_{CP}$ ) and spanwise width ( $W_{CP}$ ) are displayed in figure 2(b). The width of each region is computed by measuring lower and upper bounds along $CP=\pm 1$ (see figure 1),

where $z_{1}$ and $z_{2}$ are lower and upper bounds along $CP=\pm 1$ and

The contour lines represent $\langle N_{CP}\rangle$ and only streaks with length greater than $1{\it\delta}$ are shown. As the length of the streaks increases, the streaks maintain their spanwise width around $0.3{\it\delta}$ . While the most probable width of positive- and negative- $u$ streaks remains comparable, $W_{CP}\approx 0.3{\it\delta}$ , the maximum width of negative- $u$ streaks is generally narrower than its positive- $u$ counterpart. The width of short positive- $u$ streaks ( ${<}3{\it\delta}$ ) reaches up to $0.8{\it\delta}$ . In general, the positive- $u$ streaks are shorter and wider than the negative- $u$ ones. To analyse the spatial extent occupied by the positive- and negative- $u$ structures, figure 2(c) shows the mean area $\langle {\it\Pi}_{CP}\rangle$ of the streaks in the $y/{\it\delta}=0.15$ plane. The area occupied by streaks of a particular length ${\it\Pi}_{CP}(L_{CP})$ is defined as

where $A_{i}$ is the area for each streak,

The area ${\it\Pi}_{CP}$ is normalized by the total area of the wall-parallel plane ( $A=L_{x}\times L_{z}$ ). The sum of ${\it\Pi}_{CP}$ indicates the total area fraction occupied by streamwise velocity fluctuations greater than the threshold value. The area fraction of the weak region, where the perturbation field is weaker than the threshold value, is not included here. The total area fractions of the negative and positive $u$ are similar in the log layer, $21\,\%$ and $19\,\%$ , respectively, although the area fractions depend on the streamwise length of the structures. Each short-length streak has a smaller area compared to the longer streaks while ${\it\Pi}_{CP}$ for short-length structures is greater than the longer structures due to the higher number density in the flow field. The area fraction has a peak between $1{\it\delta}$ and $3{\it\delta}$ . In addition, the area fraction of the positive- $u$ regions crosses the negative counterpart around $L_{CP}=3{\it\delta}$ , similar to the trend of their respective populations reported in figure 2(a). Owing to the smaller population of the positive- $u$ regions ( ${>}3{\it\delta}$ ), their area fraction rapidly drops in comparison to the negative- $u$ structures. The slopes of the area fraction are $d_{1}=0.80$ and $d_{2}=0.47$ for positive and negative $u$ , respectively. Moreover, the area fraction of the negative- $u$ regions with lengths longer than $3{\it\delta}$ is approximately $10\,\%$ ( $11\,\%$ for $L_{CP}<3{\it\delta}$ ) and of the positive counterparts is $5\,\%$ ( $14.5\,\%$ for $L_{CP}<3{\it\delta}$ ). Although the population of long negative- $u$ structures is small (logarithmic decrease in their population in figure 2 a), they occupy a similar area fraction as the shorter structures. On the other hand, the area of the positive- $u$ regions is mainly occupied by the relatively short structures ( $L_{CP}<3{\it\delta}$ ) at $y/{\it\delta}=0.15$ . It should be noted that the effect of the threshold $u_{th}$ employed in our structure identification was examined (see Appendix). We found that the general trends reported in this section remain qualitatively unchanged.

To compare the magnitude of the streamwise velocity fluctuations within the organized motions, the strength of the streamwise streaks was evaluated along $L_{CP}$ . The mean amplitude for each streak, $A_{u}$ , is

The distributions of $A_{u}(L_{CP},y^{+}=0.15{\it\delta}^{+})$ are shown in figure 2(d). The contours at $A_{u}>0$ illustrate the distribution of the positive- $u$ streaks and those of the negative- $u$ streaks are shown as $A_{u}<0$ . Both distributions are relatively wider for the shorter length and become narrower as the length of the streaks increases. Although the distributions of the streak amplitudes are similar for both negative and positive $u$ , the long streaks ( ${\geqslant}12{\it\delta}$ ) are only observed in negative $u$ . In addition, these long streaks carry significantly higher disturbance amplitude ( $|A_{u}^{+}|\approx 3$ ) compared with $u_{rms}^{+}=1.87$ at $y/{\it\delta}=0.15$ . Their long streamwise extent indicates that the very long- $u$ regions contribute a significant fraction of the Reynolds stresses.

The VLSMs are motions whose streamwise wavelengths are greater than 3 ${\it\delta}$ (Guala et al. Reference Guala, Hommema and Adrian2006; Balakumar & Adrian Reference Balakumar and Adrian2007; Wu et al. Reference Wu, Baltzer and Adrian2012). Several studies have therefore used the energy contained within this wavelength range to differentiate LSMs and VLSMs (Dennis & Nickels Reference Dennis and Nickels2011; Baltzer et al. Reference Baltzer, Adrian and Wu2013; Lee et al. Reference Lee, Lee, Choi and Sung2014; Lee, Ahn & Sung Reference Lee, Ahn and Sung2015). In particular, Baltzer et al. (Reference Baltzer, Adrian and Wu2013) defined the VLSMs as the negative- $u$ streaks longer than 3 ${\it\delta}$ at $y/{\it\delta}=0.15$ , and their contribution to the Reynolds shear stress at that wall-normal position was computed by Lee & Sung (Reference Lee and Sung2011, Reference Lee and Sung2013). In the present study, streamwise-extended regions of contiguous $CP$ in the $x$ – $z$ plane ( $y/{\it\delta}=0.15$ ) were classified into LSM and VLSM using a $3{\it\delta}$ limit, and these motions were further differentiated according to their sign (n $=$ negative and p $=$ positive). This classification yields the following four categories (Lee et al. Reference Lee, Ahn and Sung2015): nLSM ( $1{\it\delta}\leqslant L_{CP}|_{CP=-1}<3{\it\delta}$ ); nVLSM ( $L_{CP}|_{CP=-1}\geqslant 3{\it\delta}$ ); pLSM ( $1{\it\delta}\leqslant L_{CP}|_{CP=+1}<3{\it\delta}$ ); and pVLSM ( $L_{CP}|_{CP=+1}\geqslant 3{\it\delta}$ ). Figure 3 shows an example of these $u$ structures. For convenience, we introduce an area indicator $I(x,z)$ , which represents the bounded area of each motion based on its sign and length:

The velocity perturbations associated with each motion can thus be conditionally sampled based on the area indicator $I$ . For example, the streamwise velocity fluctuation $u$ corresponding to the specific motion ( $I$ ) is expressed as $u_{I}(x,z;y_{ref})=u(x,y_{ref},z)\mid I(x,z)$ , where $y_{ref}$ is 0.15 ${\it\delta}$ . The conditional sampling was performed using the instantaneous raw streamwise velocity fluctuations $u$ , and the results were differentiated into the four categories given by (3.6).

3.2. Conditional velocity correlations

The spatial characteristics of long- $u$ streaks are explored by computing the conditional two-point correlations. For the streamwise structures ( $u_{I}$ ) identified by the area indicator function $I$ , the conditional correlation coefficient $(R)$ with the fluctuating velocity field ( $u_{j}$ ) is defined as

where the subscript $j$ represents different components of the velocity perturbations ( $u,v,w$ ) and ${\it\sigma}$ is the standard deviation. For brevity, we will refer to $R$ simply as the correlation. The present conditional correlations (3.7) are similar to the expression by Ganapathisubramani (Reference Ganapathisubramani2008), who studied the velocity field in the vicinity of momentum sources and sinks. The evaluation of $R[u_{I},u_{j}]$ reflects the fluctuating velocity field associated with the specific events ( $u_{I}$ ), which can reveal the statistical structure of the flow around the long- $u$ motions.

Figure 4.

Conditional correlations of the streamwise velocity fluctuations, $R[u_{I},u]$ , in 3D perspective view: (a) nLSM ( $u_{1}$ ); (b) nVLSM ( $u_{2}$ ); (c) pLSM ( $u_{3}$ ); and (d) pVLSM ( $u_{4}$ ). The isosurfaces mark $R[u_{I},u]=0.03$ (red) and $-$ 0.015 (blue). The inserted contour lines represent $R[u_{I},u]$ in the $x$ – $y$ plane ( $r_{z}/{\it\delta}=0$ ) and range from 0.02 to 0.08 with an increment of 0.01.

Figure 4 shows the 3D representation of the conditional two-point correlations $R[u_{I},u]$ at $y_{ref}/{\it\delta}=0.15$ with the contour lines in the $x$ – $y$ plane ( $r_{z}/{\it\delta}=0$ ). There is a positive isosurface of the correlations elongated in the streamwise direction and inclined with respect to the wall. The positive values have a ramp-like shape, which is a general feature of the unconditional two-point correlations ( $R[u,u]$ ) in the log layer (Hutchins & Marusic Reference Hutchins and Marusic2007a ; Lee & Sung Reference Lee and Sung2011; Baltzer et al. Reference Baltzer, Adrian and Wu2013). Despite the general similarities to the characteristics of the unconditional correlation ( $R[u,u]$ ), the magnitude and shape of $R[u_{I},u]$ depend on the conditions. In particular, the correlations of nVLSM ( $R[u_{2},u]$ ) and pLSM ( $R[u_{3},u]$ ) have higher values and longer isosurfaces than the others, indicating that those structures are the most strongly correlated with the surrounding perturbation fields. The correlation of nVLSMs extends longer than that of nLSMs in the upstream and downstream directions ( $10{\it\delta}$ and $6{\it\delta}$ for each case based on $R=0.02$ ), and the maximum correlation values for nLSM and nVLSM are 0.12 and 0.17, respectively. The streamwise velocity fluctuations are more correlated with nVLSMs in both the streamwise and wall-normal directions, although the population of nVLSMs is less dominant than that of nLSMs in the log region as shown in figure 2(a). In addition, the correlation of nLSMs is inclined at a larger angle ( ${\sim}8^{\circ }$ ) in the streamwise direction compared to the nVLSMs ( ${\sim}6^{\circ }$ ). The ramp-like character of the correlations indicates that nLSMs and nVLSMs are correlated with the streamwise fluctuations in both the near-wall and core regions along the streamwise direction. The correlations are extended as footprints down to the wall and the footprint of nVLSMs is more evident than that of nLSMs. The conditional correlations of the positive- $u$ conditions (pLSM and pVLSM) are illustrated in figure 4(c,d). The correlations of pLSMs are more dominant than those of pVLSMs; the maximum correlation values are 0.16 and 0.09 and persist over streamwise extents of $7{\it\delta}$ and $5{\it\delta}$ , respectively. The pLSMs are organized motions of the positive- $u$ regions in the log layer, and this result is also consistent with the population trend and the dominant area fraction of pLSMs (14.5 %) relative to pVLSMs (5 %) at $y/{\it\delta}=0.15$ (figure 2 a,c).

Figure 5(a,b) displays the spatial organization of the correlations in the cross-flow plane ( $r_{x}/{\it\delta}=0$ ). Note that the correlations are shown side by side in each panel, with LSMs on the left (red) and VLSMs on the right (blue) since each is symmetric in the spanwise direction. The heights of the positive correlations ( $r_{z}/{\it\delta}=0$ ) are $0.55{\it\delta}$ (nLSM) and $0.8{\it\delta}$ (nVLSM), indicating the dominance of nVLSMs up to the core region of the channel flow. The positive correlations with width ${\rm\Delta}z/{\it\delta}\approx 0.3$ are centred at the reference position and are flanked on both sides by weaker negative values at a distance of the order of $1{\it\delta}$ . The anticorrelated regions of nVLSMs are similar in height compared to the positive correlation. The distinct negative patterns observed in the correlations of nVLSMs show that relatively strong positive- $u$ structures are flanked on both sides of nVLSMs. A comparable spanwise size of the conditional correlations of the nLSMs and nVLSMs ( ${\rm\Delta}z/{\it\delta}\sim 0.3$ ) indicates that the two classes of structures have similar spanwise scales, which is a prerequisite for VLSMs to be formed by the streamwise concatenation of LSMs (Kim & Adrian Reference Kim and Adrian1999).

Figure 5.

Conditional correlations $R[u_{I},u]$ , (a,b) in the $y$ – $z$ plane ( $r_{x}/{\it\delta}=0$ ) and (c,d) in the $x$ – $z$ plane ( $y=y_{ref}$ ): (a) nLSM ( $u_{1}$ ) and nVLSM ( $u_{2}$ ); (b) pLSM ( $u_{3}$ ) and pVLSM ( $u_{4}$ ); (c) nLSM ( $u_{1}$ ) and pLSM ( $u_{3}$ ); and (d) nVLSM ( $u_{2}$ ) and pVLSM ( $u_{4}$ ). The blue and red lines represent the negative- and positive- $u$ conditions. The positive contours (solid) range from 0.02 to 0.08 with an increment of 0.01 and the values of the negative contours (dashed) are reported in (a,b).

The conditional correlations of pLSMs and pVLSMs are illustrated in figure 5(b). The correlation of pLSMs reaches up to the core region similar to nVLSMs. However, the patterns of the positive- $u$ correlations (figure 5 b) differ appreciably in the wall-normal direction from the negative- $u$ correlations (figure 5 a). They retain their width along the wall-normal direction while their negative counterparts gradually decrease close to the wall. In other words, the footprint of the positive- $u$ structure is wider than that of the negative- $u$ one, which suggests that the near-wall region which is affected by the outer large scales is larger for positive- $u$ motions; this point will be discussed in detail in § 3.3.

Wall-parallel cross-sections ( $y=y_{ref}$ ) of the correlations are presented in figure 5(c,d). The correlations of LSMs (figure 5 c) show a bias: the pLSM is biased in the upstream direction whereas the nLSM is biased downstream. However, the correlations of VLSMs are nearly symmetric in the streamwise direction (figure 5 d). Lee & Sung (Reference Lee and Sung2013) also reported a bias in the correlation when conditioned on the sign of $u$ only (but not the size of the structure). The present results thus attribute this bias to the dominant contribution by $u$ regions with length 1–3 ${\it\delta}$ (LSMs). In addition, the correlations that represent the prominent length scales in the streamwise and spanwise directions are pLSMs and nVLSMs (figure 5 c,d). Lee & Sung (Reference Lee and Sung2013) and Sillero et al. (Reference Sillero, Jiménez and Moser2014) used the two-point correlation to estimate the streamwise and spanwise length scales at various wall-normal locations. They found that the negative- $u$ structures are slightly longer and narrower than the positive- $u$ ones. The present correlations reveal that the relatively longer (negative- $u$ ) and wider (positive- $u$ ) length scales observed in those previous studies are due to the contributions of nVLSMs and pLSMs, respectively.

Figure 6.

Conditional cross-correlations, $R[u_{I},w]$ , in 3D perspective view: (a) nLSM ( $u_{1}$ ); (b) nVLSM ( $u_{2}$ ); (c) pLSM ( $u_{3}$ ); and (d) pVLSM ( $u_{4}$ ). The red and blue isosurfaces are $R[u_{I},w]=0.01$ and $-0.01$ , respectively.

Next, the conditional cross-correlations between the specific $u$ events and the spanwise velocity fluctuations ( $w$ ), $R[u_{I},w]$ , are examined. The 3D representations of $R[u_{I},w]$ are illustrated in figure 6. Each of figures 6(a)–6(c) includes four isosurfaces, two of which are attached to the wall. Figure 6(d) for pVLSM only displays two isosurfaces due to the weak correlation level. The sign of the correlation reflects the dominant direction of spanwise velocity fluctuations associated with the $u$ events. For the negative- $u$ events, the positive correlation is associated with negative $w$ and the anticorrelation corresponds to positive $w$ . In other words, the attached and detached pairs represent converging and diverging motions in the spanwise direction, respectively. The interpretation of the sign of the correlation reverses for positive- $u$ events. A clear feature of these correlations is their elevated levels below the reference position, close to the wall. The streamwisely organized motions in the log layer have a strong correlation with the near-wall spanwise velocity fluctuations. The $R[u_{2},w]$ (nVLSM), in particular, shows a very long streamwise extent of attached isosurfaces. For positive $u$ , $R[u_{3},w]$ (pLSM) shows highly correlated regions relative to $R[u_{4},w]$ (pVLSM) and the attached pairs are more elongated upstream than downstream.

Figure 7.

Conditional correlations ( $R[u_{I},w]$ ) in the $y$ – $z$ plane ( $r_{x}/{\it\delta}=0$ ): (a) nLSM ( $u_{1}$ ); (b) nVLSM ( $u_{2}$ ); (c) pLSM ( $u_{3}$ ); and (d) pVLSM ( $u_{4}$ ). The filled contours are $R[u_{I},w]=\pm$ (0.005 : 0.005 : 0.03). The inserted line contours represent $R[u_{I},v]$ varied from $-$ 0.015 to $-$ 0.025 with an interval of 0.005 (dashed) and $R[u_{I},v]=0.006$ (solid).

To investigate the cross-correlation $R[u_{I},w]$ in further detail, a cross-section at $r_{x}/{\it\delta}=0$ is shown in figure 7. As illustrated by the filled contours, the attached pairs have larger correlations close to the wall and their peaks occur at $y^{+}=42$ even though the reference position of the events is located at $y/{\it\delta}=0.15$ . Compared to the negative values of $R[u_{I},v]$ (dashed lines), the spanwise velocity fluctuations penetrate more deeply towards the wall, which reflects that the footprints of outer large-scale structures are more directly related to near-wall spanwise velocity fluctuations – a result that is consistent with the amplitude modulation of $w$ recently reported by Talluru et al. (Reference Talluru, Baidya, Hutchins and Marusic2014). The spanwise locations of the peaks are $r_{z}/{\it\delta}=\pm 0.13$ and $\pm 0.18$ for the negative- and positive- $u$ events, respectively. In other words, the peak distance is wider for the positive- $u$ events due to the dispersive motions in the spanwise direction, indicating that the outer large-scale positive $u$ could affect near-wall $w$ over a larger spanwise extent. Additionally, the contours of $R[u_{I},u]$ for negative $u$ are gradually narrower under the reference position whereas those for positive $u$ are relatively wider in figure 5(a,b). This difference in the shape is due to the opposite direction of spanwise motions in the vicinity of the wall; the congregative motions for negative- $u$ and dispersive for positive- $u$ footprints.

The line contours of $R[u_{I},v]$ which represent the wall-normal fluctuations ( $v$ ) associated with specific $u$ events are displayed in figure 7. Around the reference point, the correlations are negative, which is characteristic of Q2 ejections ( $u<0$ and $v>0$ ) for the outward motions and Q4 sweeps ( $u>0$ and $v<0$ ) for the inward motions (Wallace, Eckelmann & Brodkey Reference Wallace, Eckelmann and Brodkey1972). In particular, nVLSMs and pLSMs are most strongly correlated to the surrounding wall-normal fluctuations, $R[u_{2},v]$ and $R[u_{3},v]$ . The Reynolds shear stress associated with the ejection and sweep events is therefore dominated by nVLSMs and pLSMs in the log layer. By comparing $R[u_{I},v]$ with $R[u_{I},w]$ , a pair of large-scale circulations is observed on both sides of the ejection (negative- $u$ condition) and sweep (positive- $u$ condition). The association between low-speed, large-scale structures and a pair of counter-rotating roll modes was reported in previous studies (e.g. Hutchins & Marusic Reference Hutchins and Marusic2007b ; Chung & McKeon Reference Chung and McKeon2010; Baltzer et al. Reference Baltzer, Adrian and Wu2013; Talluru et al. Reference Talluru, Baidya, Hutchins and Marusic2014). Those works mentioned that large-scale circulations have an important influence on the near-wall fluctuations. These opposite wall-normal motions of ejections ( $v>0$ ) and sweeps ( $v<0$ ) each induce opposite spanwise near-wall movements, which can be described as congregative and dispersive, respectively. This highlights an important difference in inner–outer interactions involving large-scale low- versus high-speed events. In addition, the maximum correlation values for nLSM ( $R[u_{1},w]$ ) and nVLSM ( $R[u_{2},w]$ ) are 0.034 and 0.048, respectively, which suggests that the strong congregative motions reside under the nVLSM. In common with $R[u_{I},u]$ , the pLSM has the greater maximum value (0.05) relative to the pVLSM (0.025). Hence, the nVLSM and pLSM are the dominant structures that have an effect on the near-wall spanwise velocity fluctuations by inducing strong converging and diverging motions in the near-wall region. In the next section, the influence of the footprints of nVLSM and pLSM are examined by conditionally sampling the near-wall fluctuations that are associated with outer large-scale structures.

3.3. Footprint of outer large-scale structures

The large-scale features with streamwise extent over $1{\it\delta}$ are frequently observed near the top of the log layer. Very long motions ( ${>}3{\it\delta}$ ), in particular, are predominantly low-speed events compared to high-speed ones (figure 2). Figure 8(a) shows the streamwise velocity fluctuations in the wall-parallel plane $y/{\it\delta}=0.15$ . There are several elongated structures of negative and positive $u$ with stripiness over the plane and long low-speed structures are flanked by relatively short high-speed regions. The inserted line represents a conditionally sampled low-speed event with length over $21{\it\delta}$ . This structure is visibly similar (typically $O(20{\it\delta})$ in length) to that reported by Monty et al. (Reference Monty, Stewart, Williams and Chong2007) and by Hutchins & Marusic (Reference Hutchins and Marusic2007a ), although the present structure is identified using a different technique. Hutchins & Marusic (Reference Hutchins and Marusic2007a ) applied a Gaussian filter in two planes, located in the log and buffer regions of turbulent channel data, and noted the visual similarity between the velocity patterns for both wall-normal positions. They interpreted their observation as the footprints of very long low-speed regions extending down to very near the wall. In figure 8(b) the streamwise velocity fluctuations are plotted at $y^{+}=14.5$ . To identify the near-wall regions associated with outer large-scale events (footprints), we conditionally sampled the near-wall $CP$ lines (shown in figure 8 c), which are vertically connected to the long $CP$ lines ( $L_{CP}>1{\it\delta}$ ) at $y/{\it\delta}=0.15$ . The procedure of the footprint identification is summarized below.

1. (1) As noted in § 2.2, the contiguous regions of streamwise velocity fluctuations are identified based on the $CP$ in the wall-parallel plane at $y/{\it\delta}=0.15$ . By measuring the streamwise length of the signed $CP$ ( $L_{CP}$ ), the $u$ streaks are conditionally sampled in the flow field.

2. (2) To identify the near-wall extent of these long $u$ motions, the connectivity of the $CP$ s is traced in the wall-normal direction by comparing adjacent wall-parallel planes ( $j$ and $j-1$ planes). Neighbouring $CP$ s connected in the $j-1$ plane are continuously identified down to the wall.

3. (3) Among the near-wall $CP$ s (figure 8 c), only those that are vertically connected to the reference $CP$ are conditionally sampled (black lines in figure 8 d). The areas of the near-wall $CP$ s are determined by measuring their lower and upper bounds similar to figure 1(c). In this manner, we quantify the footprints of outer long $u$ motions, as marked by the grey shading in figure 8(d).

The detected footprint of nVLSMs (grey area in figure 8 d) is remarkably similar to its outer low-speed event. The collective length of the near-wall $CP$ s is over 20 ${\it\delta}$ . This near-wall arrangement affirms the correlation between near-wall and outer structures, and the view that the near-wall ones are modulated by the outer large scales, consistent with Hutchins & Marusic (Reference Hutchins and Marusic2007b ) and Mathis et al. (Reference Mathis, Hutchins and Marusic2009). To examine this relation further, the near-wall regions which are connected to outer large-scale structures are conditionally sampled from each instantaneous field and their characteristics are analysed.

Figure 8.

Instantaneous flow fields in the $x$ – $z$ plane. The contours show the streamwise velocity fluctuations at (a)  $y/{\it\delta}=0.15$ and (b–d)  $y^{+}=14.5$ . In (a), the inserted solid line ( $CP=-1$ ) identifies an nVLSM with 21 ${\it\delta}$ . The lines in (c) mark $CP=-1$ in that wall-parallel plane. In (d), at $y^{+}=14.5$ , the solid lines identify the near-wall $CP$ that are vertically connected to the line that marks the nVLSM in (a).

Figure 9.

Joint p.d.f.s of (a) $u$ – $v$ and (b) $u$ – $w$ at $y^{+}=14.5$ . The thinner upper panels show the p.d.f. of $u$ and the thinner right panels show the p.d.f. of $v$ and $w$ . The velocity fluctuations are computed from the detected near-wall $CP$ s, which are vertically connected to the outer structures. The blue and red lines represent the velocity fluctuations associated with the footprints of nVLSM (blue) and pLSM (red), respectively. Contour levels are distributed exponentially from 0.005 to 0.05.

Figure 9 shows the joint probability density function (joint p.d.f.) of (a) $u$ – $v$ and (b)  $u$ – $w$ at $y^{+}=14.5$ . The velocity fluctuations associated with the footprints are identified using the conditionally sampled $CP$ s in the near-wall region. Here, the joint p.d.f.s of the nVLSMs (blue) and pLSMs (red) are shown because these are the dominant motions that have an influence on the near-wall region (cf. § 3.2). The joint p.d.f. of $u$ – $v$ is presented in figure 9(a) and the adjacent upper (right) panel shows the p.d.f. of $u$ ( $v$ ). Ejections and sweeps are the principal perturbations on the footprints of nVLSM and pLSM, respectively. However, their distributions are not symmetric with respect to the origin; the contour line of the pLSMs reaches $u^{+}=8$ and $v^{+}=-1.8$ , whereas that of the nVLSMs reaches $u^{+}=7$ and $v^{+}=-1.8$ . This difference also appears as a spread in the $u$ and $v$ p.d.f.s of the pLSMs. The inserted lines mark the peak locations of the p.d.f.s; the characteristic $u$ of the footprint of nVLSMs ( $u^{+}=-4.5$ ) is greater than that of the pLSMs ( $u^{+}=3.7$ ). The asymmetry between negative- and positive- $u$ events demonstrates that the influence of large-scale structures depends on their sign, consistent with the work by Agostini & Leschziner (Reference Agostini and Leschziner2014). This trend is prominent in the joint p.d.f. of $u$ – $w$ (figure 9 b). The $w$ of pLSMs are widely distributed, while those of nVLSMs are narrowly distributed. The intense $w$ are apparent on the footprints of the large-scale positive- $u$ event. The wider (narrower) distributions for the pLSMs (nVLSMs) arise due to the spanwise dispersive (congregative) motions shown in figure 7.

In order to quantify the contributions of the various outer events to the Reynolds shear stress, the weighted joint p.d.f. of $u$ – $v$ is shown in figure 10(a). The weighted joint p.d.f. is computed by multiplying the joint p.d.f. by $uv$ . Note that the results for nVLSM and pLSM are plotted side by side in each panel, with nVLSMs at left and pLSMs at right. The maximum values of the weighted joint p.d.f. are denoted by the black filled circles. For the nVLSMs, the maximum values are $u^{+}=-4.75$ and $v^{+}=-0.35$ , which are located in the second quadrant. In contrast, the maximum is $v^{+}=-0.5$ for the pLSMs. These strong downward motions driven by sweeps of outer positive $u$ lead to intense velocity fluctuations even inside the viscous sublayer (not shown here). The difference between the nVLSMs and pLSMs is also apparent in the spanwise velocity fluctuations. Figure 10(b) shows the weighted joint p.d.f. of $u$ – $w$ , where $w$ on the footprints of the pLSMs are clearly greater than those of the nVLSMs. The maximum values of the spanwise component are $\pm 1.15$ and $\pm 0.7$ for pLSMs and nVLSMs, respectively. The outer large-scale positive- $u$ event not only strengthens the near-wall $u$ , but also enhances the near-wall $w$ . The $w$ are appreciably modulated by the pLSMs, even more than $u$ and $v$ ; this is the origin of the highest amplitude modulation coefficient for $w$ in the near-wall region, which was reported by Talluru et al. (Reference Talluru, Baidya, Hutchins and Marusic2014). In addition, the discrepancy in the distributions of outer positive- and negative- $u$ events in figure 10(b) reflects that the downward motions carried by the positive- $u$ structures (sweeps) move the higher-momentum fluid towards the wall where it is spread laterally.

Figure 10.

Weighted joint p.d.f.s of (a) $u$ – $v$ and (b) $u$ – $w$ at $y^{+}=14.5$ . The velocity fluctuations are computed from the detected near-wall $CP$ s, which are vertically connected to the outer structures. The blue and red lines represent the velocity fluctuations associated with the footprints of nVLSM (blue) and pLSM (red), respectively. The black circles mark the maximum value of the velocity fluctuations.

Figure 11.

Conditional structures associated with the footprints of nVLSMs (a,b) and pLSMs (c,d) at $y^{+}=14.5$ . (a,c) Isosurfaces of the streamwise velocity fluctuations $u^{+}|_{FP_{I}}=-1.0$ (blue) and $+1.0$ (red). Black isosurfaces are 7 % of the maximum swirling strength. The conditional structures of near-wall streaks that are not connected to outer large scales ( $u^{+}|_{NWS}$ ) are shifted in the spanwise direction by $0.5{\it\delta}$ and their magnified views are shown in wall units. (b,d) Conditionally averaged fields in the $y$ – $z$ plane ( $r_{x}/{\it\delta}=0$ ). Vectors represent the in-plane velocity components.

To examine the flow field around the footprints of nVLSM ( $u_{I}=2$ ) and pLSM ( $u_{I}=3$ ), the conditionally averaged velocity is defined,

where $CP_{I}$ denotes the streaks that are vertically connected to the outer large scales ( $u_{I}$ ). The reference wall-normal position is $y_{ref}^{+}=14.5$ . Figure 11(a,c) illustrates isosurfaces of conditional streamwise velocity fluctuation $u|_{FP_{I}}$ . Although the reference wall-normal position was chosen at $y_{ref}^{+}=14.5$ , the conditional structures have long streamwise lengths, extending over 4 ${\it\delta}$ (nVLSM) and 2 ${\it\delta}$ (pLSM) in the streamwise direction. On the contrary, the conditional averages based on the negative- and positive- $u$ streaks that are not connected to the outer large-scale events ( $u|_{NWS}$ ) have streamwise lengths of approximately 1000 wall units and are confined within the near-wall region. Figure 11 clearly shows the scale separation in the near-wall region. In addition, the black isosurface represents the swirling strength educed of the conditionally averaged fields (Zhou et al. Reference Zhou, Adrian, Balachandar and Kendall1999). Conditional structures of both nVLSMs and pLSMs have a long pair of roll modes on both sides of large negative- and positive- $u$ structures. These large-scale circulations are quite different from the smaller rolls associated with the velocity contours of $\boldsymbol{u}|_{NWS}$ and which are shown in the zoomed view (insets of figure 11 b,d). The centres of the small rolls are located at $y^{+}=40$ and $r_{z}^{+}=50$ while those of large-scale circulations are located at $y/{\it\delta}=0.2$ and their spanwise width is $0.4{\it\delta}$ . Although the large-scale circulations of negative- and positive- $u$ structures have similar widths, the maximum swirling strength of the pLSM is 1.3 times greater than that of the nVLSM.

3.4. Formation of long negative- $u$ structure

The previous section demonstrated that long negative- $u$ structures ( ${>}3{\it\delta}$ ) are frequently observed in the log layer compared to positive- $u$ ones which prevail in relatively short streamwise extent ( ${<}3{\it\delta}$ ). The analysis also showed that the different flow patterns around $u_{I}$ contribute to their distinct near-wall signatures, i.e. congregative versus dispersive. This section focuses on the long negative- $u$ structures ( ${>}3{\it\delta}$ ) linked with a pair of large-scale circulations around those structures in order to explain their dominance compared to the positive counterparts. The work by Lee et al. (Reference Lee, Lee, Choi and Sung2014) provides an important background. They found that the low-speed regions with lengths 1–3 ${\it\delta}$ dominantly undergo merging, and that the merging process was facilitated by the different convection velocities between the upstream and downstream structures. Using conditional spatiotemporal correlations of negative- $u$ structures, they showed that the convection velocities of the structures depend on their strength, $u$ : weak negative- $u$ structures have a relatively higher convection velocity than strong negative- $u$ ones. To understand the merging process, it is necessary to explain the cause of the variation in the strength of $u$ between the tail and the head of the two structures involved.

Figure 12.

Three-dimensional isosurfaces of negative- $u$ structures ( $u=-0.9u_{th,raw}$ ). The time of each instantaneous field is: (a)  $t_{ref}-3.0{\it\delta}/U_{CL}$ , (b)  $t_{ref}-1.5{\it\delta}/U_{CL}$ , (c)  $t_{ref}$ and (d)  $t_{ref}+1.5{\it\delta}/U_{CL}$ . The isosurfaces are coloured according to their wall-normal distance: $y/{\it\delta}<0.1$ (white) and $y/{\it\delta}\geqslant 0.1$ (blue).

The merging process of two low-speed structures is captured in figure 12. The blue isosurface represents the outer region, $y/{\it\delta}\geqslant 0.1$ . Initially a gap with streamwise length of the order of $0.5{\it\delta}$ separates the upstream and downstream structures (figure 12 a). At $t=t_{ref}$ , the outer structures start to merge by narrowing the gap, and ultimately form a very long negative- $u$ structure. During the merging process, the tail of the downstream structure has a slower convective speed than the head of the upstream one. The slower convection speed of the downstream structure is partly due to the relatively higher magnitude of $u$ in that region relative to around the head of the upstream structure. However, the origin of this difference in $u$ is unclear and, in light of the correlation between the events in the outer and near-wall flow, a simultaneous investigation of the two regions during the merging process is required. We will demonstrate that this merging process is tightly connected to the spanwise congregative motion under the footprint of large-scale low-speed event.