3.1. Hierarchy of vortices and low-speed structures

As mentioned in the introduction, when we visualize the positive isosurfaces of the second invariant $Q$ of the velocity gradient tensor, only the smallest-scale vortices are captured (yellow objects in figure 1). This is because the smallest-scale flow structures are relevant to quantities related to the velocity gradient. On the other hand, when we visualize, for example, the negative isosurfaces of the fluctuating streamwise velocity, structures as large as the distance from the wall are captured (blue objects in figure 1). This is because quantities related to the velocity are predominantly determined by the largest-scale structures.

Thus, figure 1 shows a clear scale-separation between the structures associated with velocity and its gradients. This is consistent with the spectral distributions of energy and dissipation (Jiménez Reference Jiménez2012). To extract arbitrary-scale structures, we employ the band-pass filter defined by (2.2). Figure 2 shows the isosurfaces of the second invariant $Q^{(\sigma )}$ (yellow) of the velocity gradient tensor and the streamwise velocity $u^{(\sigma )}$ (blue), which are evaluated from the fields filtered at three different scales: (a) $\sigma ^{+}=960$, (b) $240$ and (c) $60$. Note that each of them contains the largest-scale ($\sigma \approx y$) structures at a height around the upper boundary (panel a) and lower boundary (panel b) of the log layer, and in the buffer layer (panel c). Note also that $\sigma ^{+}=60$ is larger than the diameter (approximately $10\eta$) of the smallest-scale vortices, where $\eta$ is the Kolmogorov scale (e.g. $\sigma ^{+}=60$ at $y=h$ corresponds to $\sigma =11\eta$). In figure 2, by using a percolation analysis, we choose the thresholds ($Q^{(\sigma )}_{per}$ and $u^{(\sigma )}_{per}$) of the isosurfaces in an objective manner (see appendix A for the details), which enables us to identify separated structures. We can see that vortices and low-speed structures identified from the band-pass filtered fields at different scales are hierarchical.

Figure 2. The hierarchy of vortices and low-speed structures at the same instant and location as in figure 1. Isosurfaces of the second invariant $Q^{(\sigma )}$ (yellow) of the velocity gradient tensor and the streamwise velocity $u^{(\sigma )}$ (blue) filtered at (a) $\sigma ^{+}=960$, (b) $240$ and (c) $60$. The thresholds $Q^{(\sigma )}_{per}$ are determined by the percolation analyses (see appendix A), namely, (a) $Q^{(\sigma )}/Q^{(\sigma )}_{rms}=0.25$ and $u^{(\sigma )}/u^{(\sigma )}_{rms}=-1$, (b) $2$ and $-2$, (c) $8$ and $-3$. The grid width on the wall indicates 1000 wall units. The flow is from lower left to upper right.

Looking at the scale comparable to approximately $0.2h$ (figure 2a), we notice that vortices (yellow) are quasi-streamwise but they are also inclined to the wall-normal direction, and that largest-scale streaks (blue) are located beside these quasi-streamwise vortices. This combination of quasi-streamwise vortices and meandering streaks is reminiscent of the coherent structures in the buffer layer that were found by Jeong et al. (Reference Jeong, Hussain, Schoppa and Kim1997) for low-Reynolds-number turbulence. The observed largest streaks correspond to VLSM and the quasi-streamwise vortices to LSM, which were observed by Hwang (Reference Hwang2015) in the overdamped LES, though the streamwise length of the observed streaks is limited by the streamwise length ($2{\rm \pi} h$) of the computational domain and shorter than its real length (approximately $15h$) of the VLSM. Furthermore, it is important that similar wall-attached structures are observed in the buffer and log layers. Evidence is given in figure 3, where we crop such structures in rectangular boxes whose faces are parallel to the computational domain. We see that the combination of vortices and streaks is similar in the three panels in figure 3. In other words, the largest-scale structures at each height ($\sigma \sim y$), i.e. wall-attached structures, are the coherent structures composed of quasi-streamwise vortices and low-speed streaks irrespective of the height $y$. Although we cannot observe such hierarchical structures in the turbulent fields without scale decomposition (figure 1), once we decompose the velocity field, it is easy to find the self-similar hierarchy of the flow structures.

Figure 3. An example of the hierarchy of quasi-streamwise vortices and a low-speed streak for the scales (a) $\sigma ^{+}=960$, (b) $\sigma ^{+}=240$ and (c) $\sigma ^{+}=60$. Subdomains in figure 2 are shown, but the thresholds are different ((a) $Q^{(\sigma )+}=6.0\times 10^{-7}$ and $u^{(\sigma )+}=-0.5$, (b) $1.0\times 10^{-5}$ and $-0.5$, (c) $3.0\times 10^{-4}$ and $-0.7$). Here, we choose these thresholds so that we can visualize attached structures. The lighter yellow indicates the isosurfaces with the positive $\omega _x$, whereas the darker yellow indicates those with the negative $\omega _x$. The grid width on the wall indicates $1000$ wall units. The flow is from lower left to upper right.

Next, to show that the observed structures are dominant, we evaluate averaged distributions of $Q^{(\sigma )}$ and $u^{(\sigma )}$ with a given scale $\sigma$ at a given height $y_r$ around intense vortices. We take averages of $Q^{(\sigma )}$ and $u^{(\sigma )}$ around the points which satisfy the conditions that $Q^{(\sigma )}$ at a fixed height $y_r$ is larger than isosurfaces level $Q^{(\sigma )}_{per}$ in figure 2 and the streamwise vorticity $\omega _x^{(\sigma )}$ is positive. Note that the former condition is determined objectively by the percolation analyses (appendix A), and the latter condition ($\omega _x^{(\sigma )}(y_r)>0$) breaks the spanwise symmetry. We show in figure 4 the isosurfaces of the conditional averages of $Q^{(\sigma )}$ (yellow) and $u^{(\sigma )}$ (blue) for the filter scales (a) $\sigma ^{+}=960$, (b) $240$ and (c) $60$. Since the heights $y_r^{+}$ where the conditions are imposed are as large as the filter scales (i.e. (a) $y_r^{+}=960$, (b) $240$ and (c) $60$), the obtained structures are the largest scale possible at each height. As expected from the previous observation in figure 3, the averaged structures for the largest scale are similar irrespective of the heights. The vortices are in the quasi-streamwise direction and inclined to the wall-normal and slightly to the spanwise directions (the different views of figure 4c are shown in figure 5g–i). Incidentally, the inclination angle (i.e., the angle with respect to the wall of the line connecting the two points with the maximum of $Q^{(\sigma )}$ on the cross-sections at $x=\pm \sigma$) are (a) $40$, (b) $35$ and (c) $27^{\circ }$. The low-speed structures are also consistent with the observation in figure 3; a low-speed streak at each height tends to locate on the left-hand side of the vortices (lighter yellow) at the scale with positive streamwise vorticity. We have confirmed that a low-speed structure is located on the right-hand side of the vortex with negative streamwise vorticity under the condition $\omega _x^{(\sigma )}<0$ (figure is omitted). This observation in the real space not only supports the results of the studies on the wall-attached clusters (e.g. del Álamo et al. Reference del Álamo, Jiménez, Zandonade and Moser2006; Lozano-Durán, Flores & Jiménez Reference Lozano-Durán, Flores and Jiménez2012) and of the spectral analysis (e.g. Baars, Hutchins & Marusic Reference Baars, Hutchins and Marusic2017), but also clarifies the details of the wall-attached structures; for example, it is evident that the largest-scale structures are similar to the well known buffer-layer coherent structures. This implies that they are likely to be maintained by the SSP. Since such processes occur simultaneously at different heights, we refer to them as the hierarchical SSP. This self-similar nature of the coherent structures was also observed in the overdamped LES (Hwang Reference Hwang2015; Hwang & Bengana Reference Hwang and Bengana2016).

Figure 4. Averaged distribution of $Q^{(\sigma )}$ and $u^{(\sigma )}$ around the intense largest-scale vortical structures at each height: (a) $\sigma ^{+}=y_r^{+}=960$, (b) $\sigma ^{+}=y_r^{+}=240$, (c) $\sigma ^{+}=y_r^{+}=60$. The thresholds of the yellow isosurfaces are $Q^{(\sigma )}_{per}/5$, and those of the blue ones are $-u^{(\sigma )}_{rms}/2$. The black arrows indicate the centre $(0,y_r,0)$ of the reference frame in which the conditional average is taken. The grid width on the wall is $\sigma ({=}y_r)$. The flow is from lower left to upper right. The blue arrow in panel (a) indicates the direction of ${\boldsymbol {\omega }}$ with positive $\omega _x$.

Figure 5. Different views of the conditionally averaged objects of the vortex with scale $\sigma ^{+}=60$ at heights (a–c) $y_r^{+}=960$, (d–f) $240$ and (g–i) $60$. The thresholds are set to be $Q_{per}^{(\sigma )}/5=1.6Q_{rms}^{(\sigma )}(y_r)$.

Next, let us investigate small-scale vortices away from the wall. We show in figure 5 the different views of the isosurfaces of $Q^{(\sigma )}$ for a scale $\sigma ^{+}=60$ at different heights: (a–c) $y_r^{+}=960$, (d–f) $240$ and (g–i) $60$. Since the vortex shown in figure 5(g–i), which is at the largest scale in the sense that $\sigma =y_r$, is identical to the one in figure 4(c), we may confirm that the vortex is inclined to both the wall-normal (see figure 5h) and spanwise directions (see figure 5i). In contrast, detached structures ($\sigma <y_r$) are more spherical (see figure 5a–c). However, this observation does not imply that the vortical structures themselves are spherical but implies that they are distributed isotropically. This fact is consistent with the conclusion by Jiménez (Reference Jiménez2013) that smaller-scale vortices away from the wall decouple from the mean shear and their orientation becomes isotropic. In our previous study (Motoori & Goto Reference Motoori and Goto2019a), we also showed that smaller-scale vorticity became less aligned to the mean-flow stretching direction in a turbulent boundary layer.

Before closing this subsection, we investigate the spatial relation between small-scale vortices and large-scale structures. We take an average of the large-scale ($\sigma ^{+}=960$) fields under the condition that small-scale ($\sigma _{cond}^{+}=60$) vortices away from the wall ($y_r^{+}=960$) exist. We set $Q^{(\sigma _{cond})}(y_r) > Q^{(\sigma _{cond})}_{per} = 8Q_{rms}^{(\sigma _{cond})}(y_r)$ as the condition so that we can examine correlation between large-scale structures and intense small-scale vortices. Note that, since this condition cannot break the spanwise symmetry, obtained structures are always symmetric in the spanwise direction. Figure 6 shows the transparent isosurfaces of the conditionally averaged quantities of $\omega _x^{(\sigma )}$, $u^{(\sigma )}$ and $v^{(\sigma )}$. Looking at the yellow (positive $\omega _x^{(\sigma )}$) and green (negative $\omega _x^{(\sigma )}$) objects and the dot $(0,y_r,0)$ indicated by the black arrow where the small-scale vortices exist, we can see that small-scale vortices away from the wall are more likely to exist in the ejection (upflow, red; low-speed streak, blue) induced by these large-scale streamwise vortices (yellow and green). Note again that this does not necessarily mean the existence of counter-rotating vortices. The conclusion drawn from figure 6 that small-scale vortices tend to be in large-scale streaks is consistent with the observation by Tanahashi et al. (Reference Tanahashi, Kang, Miyamoto, Shiokawa and Miyauchi2004). A similar relation between vortex clusters and large-scale streaks was also shown by del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2006) and Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012).

Figure 6. Average large-scale ($\sigma ^{+}=960$) structures under the condition of the existence of intense small-scale ($\sigma _{cond}^{+}=60$) vortices ($Q^{(\sigma _{cond})} > Q^{(\sigma _{cond})}_{per} = 8Q_{rms}^{(\sigma _{cond})}$) at the height $y_r^{+}=960$. Thresholds are $25\omega ^{(\sigma )}_{x, {rms}}$ (yellow), $-35\omega ^{(\sigma )}_{x, {rms}}$ (green), $-0.3u^{(\sigma )}_{rms}$ (blue) and $0.3v^{(\sigma )}_{rms}$ (red). The black arrow indicates the centre $(0,y_r,0)$ of the reference frame in which the conditional average is taken. The grid width on the wall indicates $\sigma ^{+}(=y_r^{+}=960)$. The flow is from lower left to upper right. The blue arrow indicates the direction of ${\boldsymbol {\omega }}$ with positive $\omega _x$.

In this subsection, two main conclusions are obtained from the scale-dependent analyses in the real space. One is that the largest-scale structures at each height (i.e., wall-attached structures) are hierarchically composed of quasi-streamwise vortices and low-speed streaks (figures 3 and 4). This implies that they are simultaneously maintained by the SSP at different heights, that is, the hierarchical SSP. The other is that the orientation of small-scale vortices away from the wall is isotropic (figure 5), and they tend to exist in the large-scale ejection induced by quasi-streamwise vortices (figure 6). However, as will be shown below, the last conclusion does not imply that small-scale vortices are carried from the wall. In the following two subsections, we will develop more quantitative arguments on their sustaining mechanism.

3.2. Sustaining mechanism: vortex stretching

The transport equation for the enstrophy $\omega _i^{2}/2$ reads

(3.1)\begin{equation} \frac{1}{2}\frac{\mathrm{D} \omega_i^{2}}{\mathrm{D} t}=\omega_iS_{ij}\omega_j+\nu\omega_i\nabla^{2}\omega_i, \end{equation}

where $\omega _i$ is the vorticity, $S_{ij}$ is the strain-rate tensor and $\nu$ is the kinematic viscosity. Only when the vortex stretching term is positive, the enstrophy is amplified, because the viscous term $\nu \omega _i\nabla ^{2}\omega _i$ weakens the enstrophy. Therefore, to investigate the generation mechanism of vortices, we focus on the vortex stretching term. Since strain rates at all scales simultaneously contribute to the stretching of the vorticity at a given scale, we decompose $\omega _i$ and $S_{ij}$ into various scales to define

(3.2)\begin{equation} g_f(\sigma_S\to\sigma_\omega;y)= \frac{ \omega_i^{(\sigma_\omega)}S_{ij}^{(\sigma_S)}\omega_j^{(\sigma_\omega)} } { \omega_i^{(\sigma_\omega)2} }. \end{equation}

Here, $\omega _i^{(\sigma _\omega )}$ is the fluctuating vorticity filtered at the scale $\sigma _\omega$, and $S_{ij}^{(\sigma _S)}$ is the fluctuating strain rates filtered at $\sigma _S$. Hence, $g_f(\sigma _S\to \sigma _\omega )$ indicates the contribution of the production rates of the enstrophy at $\sigma _\omega$ from fluctuating strain rates at $\sigma _S$. If $g_f(\sigma _S\to \sigma _\omega )$ is positive (or negative), it implies the stretching (or contraction) of the vorticity at $\sigma _\omega$ by the strain rate at $\sigma _S$. Similar quantities were defined by Goto et al. (Reference Goto, Saito and Kawahara2017) for periodic turbulence and by ourselves (Motoori & Goto Reference Motoori and Goto2019a) for a turbulent boundary layer. We also define the contribution from the mean shear to the stretching of vortices at $\sigma _\omega$ by

(3.3)\begin{equation} g_m(M\to\sigma_\omega;y)= \frac{ \omega_i^{(\sigma_\omega)}\overline{S_{ij}}\omega_j^{(\sigma_\omega)} } { \omega_i^{(\sigma_\omega)2} }, \end{equation}

where $\overline {S_{ij}}$ is the mean rate of strain. Note that only $\overline {S_{12}}$ and $\overline {S_{21}}$ are non-zero in the channel flow. Note also that (3.2) and (3.3) differ only in the source of the stretching (i.e. $S_{ij}^{(\sigma _S)}$ and $\overline {S_{ij}}$), and that, for example, when $g_m(M\to \sigma _\omega;y)$ is positive, the mean shear stretches the vorticity at $\sigma _\omega$. Here, when taking the average in the streamwise and spanwise directions and time at a fixed $y$, we impose two conditions. One is that the stretched vorticity is in a rotational region ($Q^{(\sigma _\omega )} > Q^{(\sigma _\omega )}_{rms}$). The other is the condition whether the stretched vorticity is in the upflow (i.e. large-scale ejection; $v^{(\sigma _{cond})} > v^{(\sigma _{cond})}_{rms}$) or downflow (i.e. large-scale sweep; $v^{(\sigma _{cond})} < -v^{(\sigma _{cond})}_{rms}$) for a large scale $\sigma _{cond}^{+}=960$. We impose the latter condition to examine the origin of the observation (figure 6) that smaller-scale vortices away from the wall are more likely to exist in the largest-scale upflow regions.

We show, in figure 7, $\overline {g_f}$ (black or grey) and $\overline {g_m}$ (blue or light blue) in the upflow case (darker colours) and the downflow case (lighter colours) for (a) $y^{+}=960$, (b) $240$ and (c) $60$. These are normalized by the channel half-width $h$ and the skin-friction velocity $u_\tau$. Since the contributions in the upflow and downflow cases are quantitatively similar, first we describe observations common in the both cases. Looking at the open circles in figure 7(a) for a small-scale ($\sigma _\omega ^{+}=30$) at a location ($y^{+}=960$) near the upper boundary of the log layer, the contributions from fluctuating strain rates at the scales one to eight times larger than $\sigma _\omega ^{+}(=30)$ are larger than $\overline {g_m}$. In particular, the contribution $\overline {g_f}(2\sigma _\omega \to \sigma _\omega )$ from the twice-larger scale is the largest. This is also the case for other small scales $\sigma _\omega ^{+}=60$ (open squares) and $120$ (open triangles). It is also interesting to observe that $\overline {g_f}<0$ for $\sigma _S \leqslant \sigma _\omega /2$. This implies that the vortices are contracted by smaller-scale strain rates on average. On the other hand, for the vortices ($\sigma \sim y$, for example, closed circles in figure 7a) whose size is of the order of the distance from the wall, the contributions from the mean shear are larger than those from the fluctuating strain rates ($\overline {g_f}\lesssim \overline {g_m}$) at any scale.

Figure 7. Averaged contribution $\overline {g_f}$ defined by (3.2) from the strain rate at $\sigma _S$ to the stretching of vortices at $\sigma _\omega ^{+}=30$ ($\circ$), $60$ ($\square$), $120$ ($\triangle$) and $240$ ($\bullet$) at heights (a) $y^{+}=960$, (b) $240$ and (c) $60$. The larger symbols correspond to the self-contribution ($\sigma _S=\sigma _\omega$). Blue and light blue symbols indicate the contribution $\overline {g_m}$ defined by (3.3) from the mean shear. Here, $\overline {g_f}$ and $\overline {g_m}$ are normalized by $h$ and $u_\tau$. The averages are taken under the conditions $v^{(\sigma _{cond})}>v_{rms}^{(\sigma _{cond})}$ (black and blue) and $v^{(\sigma _{cond})}<-v_{rms}^{(\sigma _{cond})}$ (grey and light blue) at $\sigma _{cond}^{+}=960$ for each height $y$.

The results for a location ($y^{+}=240$) around the lower boundary of the log layer (figure 7b) show a similar tendency to the above observations for $y^{+}=960$. Small-scale vortices (open squares) are stretched most significantly by the twice-larger scale, whereas large-scale vortices (open triangles and closed circles) are stretched directly by the mean shear.

Next, let us look at the contributions in the buffer layer ($y^{+}=60$; figure 7c) where the hierarchy of vortices is absent. If we refer to $\sigma \sim y$ as a large scale, there are only large-scale vortices in the buffer layer. Although, among the contributions from the fluctuating strain rates, the twice-larger scale contributes most significantly, the contributions from the mean shear are always more important ($\overline {g_f}<\overline {g_m}$). These observations for $y^{+}=960$, $240$ and $60$ are similar to those in the log and buffer layers in a turbulent boundary layer (Motoori & Goto Reference Motoori and Goto2019a).

We have thus shown that, at any height in the buffer and log layers, the largest-scale ($\sigma \sim y$) vortices are stretched by the mean shear. This result supports the speculation on the basis of the spatial structures (figures 3 and 4) that the largest-scale structures are maintained by the hierarchical SSP. This is because $g_m$ (the stretching by the mean shear) corresponds to the lift-up (tilting) of quasi-streamwise vortices in the SSP (Hamilton et al. Reference Hamilton, Kim and Waleffe1995; Hwang & Bengana Reference Hwang and Bengana2016).

Next, we discuss the difference in the large-scale upflow and downflow regions. We observe in figure 7 that $\overline {g_f}(2\sigma _\omega \to \sigma _\omega )$ shown by the black lines (in the upflow case) for small-scale vortices in the log layer (i.e., $\sigma _\omega ^{+}=30$, $60$ and $120$ in figure 7a) are larger than the values by the grey lines (in the downflow case). This implies that the small-scale enstrophy production rates in upflow regions are larger than those in downflow regions. This is reasonable because, in upflow regions, the source of the vorticity is carried from the wall. Note that, small-scale enstrophy in the log layer is not simply advected by the largest-scale upflow but they are amplified due to the stretching by one to eight times larger-scale strain rates. This point will be further verified in the next subsection.

3.3. Sustaining mechanism: real-space energy transfer

We have evaluated the scale-dependent enstrophy production rates and shown that small-scale vortices away from the wall are stretched predominantly by the twice-larger-scale vortices. Although this generation mechanism of the hierarchy of vortices seems consistent with the notion of the energy cascade, the creation of smaller-scale vortices may not necessarily correspond to the energy transfer to smaller scales. This issue was investigated by Goto (Reference Goto2008, Reference Goto2012) for periodic turbulence, but here we employ a different approach. The transport equation for the mean turbulent kinetic energy $K = \overline {\check {u}_i \check {u}_i}/2$ is given by

(3.4)\begin{align} \frac{\partial K}{\partial t} &= \frac{\partial}{\partial x_j} \left\{ -\bar{u}_j K -\frac{1}{2}\overline{\check{u}_j\check{u}_i^{2}} -\frac{1}{\rho}\overline{\check{u}_j\check{p}} +\nu \left( \frac{\partial K}{\partial x_j} +\frac{\partial}{\partial x_i}\overline{\check{u}_j\check{u}_i} \right)\right\} \nonumber\\ &\quad -\overline{\check{u}_i\check{u}_j}\frac{\partial \bar{u}_i}{\partial x_j} -2\nu\overline{\check{S}_{ij}\check{S}_{ij}}, \end{align}

where $\check {S}_{ij}$ is the fluctuating strain-rate tensor $(\partial \check {u}_j/\partial x_i+\partial \check {u}_i/\partial x_j)/2$. The terms in ${\partial }/{\partial x_j}\{\cdots \}$ in (3.4) denote the mean-flow advection, turbulent advection, velocity-pressure correlation and viscous diffusion, respectively. The term $-2\nu \overline {\check {S}_{ij}\check {S}_{ij}}$ denotes the viscous dissipation. These terms do not contribute to the production of the turbulent energy $K$ on average. On the other hand, $-\overline {\check {u}_i\check {u}_j}{\partial \bar {u}_i}/{\partial x_j}$ is the production of $K$ by the mean flow. To investigate interscale energy transfer, we must decompose the velocity in (3.4) into scales. Recently, Kawata & Alfredsson (Reference Kawata and Alfredsson2018) used a decomposition to analyse interscale transfer of the Reynolds stress in plane Couette flow. They decompose the fluctuating velocity into large-scale $u^{(L)}$ and small-scale $u^{(S)}$ parts by sharp Fourier filtering in the spanwise wavenumber space. Since the Fourier filter is orthogonal, the cross-correlation $\overline {u^{(L)}u^{(S)}}$ vanishes. Then, we can derive the transport equation for the energy of the large- and small-scale parts. For example, the equation for the small-scale energy ($K^{(S)}=\overline {{u}_i^{(S)} {u}_i^{(S)} }/2$) is given by

(3.5)\begin{align} \frac{\partial K^{(S)}}{\partial t} &= \frac{\partial}{\partial x_j} \left\{ -\bar{u}_j K^{(S)} -\frac{1}{\rho}\overline{{u}_j^{(S)} {p}^{(S)}} +\nu \left( \frac{\partial K^{(S)}}{\partial x_j} +\frac{\partial}{\partial x_i}\overline{{u}_j^{(S)} {u}_i^{(S)}} \right) \right\} \nonumber\\ &\quad -\overline{u_i^{(S)} u_j^{(S)}}\frac{\partial \overline{u_i}}{\partial x_j} -2\nu\overline{{S}_{ij}^{(S)} {S}_{ij}^{(S)}} +Ad^{\prime}(L\to S)+Tr^{\prime}(L\to S), \end{align}

where

(3.6)\begin{equation} Ad^{\prime}(L\to S)= -\frac{1}{2}\frac{\partial}{\partial x_j} \left\{ \overline{u_i^{(S)} u_i^{(S)} u_j^{(S)}} + \overline{u_i^{(S)} u_i^{(S)} u_j^{(L)}} + 2\overline{u_i^{(S)} u_i^{(L)} u_j^{(L)}} \right\} \end{equation}

and

(3.7)\begin{equation} Tr^{\prime}(L\to S) = -\overline{u_i^{(S)} u_j^{(S)} \frac{\partial u_i^{(L)}}{\partial x_j}} -\left( -\overline{u_i^{(L)} u_j^{(L)} \frac{\partial u_i^{(S)}}{\partial x_j}} \right) \end{equation}

denote the interscale interaction related to the spatial redistribution of $K^{(S)}$ and the energy transfer from large to small scales, respectively. In particular, we can interpret that $-\overline {u_i^{(S)} u_j^{(S)} \partial u_i^{(L)}/\partial x_j}$ in (3.7) is the energy transfer from the large to small scales, while $-\overline{u_i^{(L)} u_j^{(L)} \partial u_i^{(S)}/\partial x_j}$ is from small to large scales. In the present study, although we cannot describe the equation for $K^{(\sigma )}=\overline {u_i^{(\sigma )}u_i^{(\sigma )}}/2$ in a simple form due to the non-orthogonality of the filter, by making use of the terms composed of two different scales, we quantify real-space energy transfers between these scales as follows. First, we define the production

(3.8)\begin{equation} Pr(M \to \sigma_{to};y) = -\overline{u_i^{(\sigma_{to})} u_j^{(\sigma_{to})}}\frac{\partial \overline{u_i}}{\partial x_j} = -\overline{u^{(\sigma_{to})} v^{(\sigma_{to})}}\frac{\partial \bar{u}}{\partial y} \end{equation}

of the $\sigma _{to}$-scale energy $K^{(\sigma _{to})}$ by the mean flow. This term is obtained by replacing $u_i^{(S)}$ with $u_i^{(\sigma _{to})}$ in (3.5). We also quantify the interscale energy transfer

(3.9)\begin{equation} Tr(\sigma_{fr} \to \sigma_{to};y) = -\overline{u_i^{(\sigma_{to})} u_j^{(\sigma_{to})} \frac{\partial u_i^{(\sigma_{fr})}}{\partial x_j}} -\left( -\overline{u_i^{(\sigma_{fr})} u_j^{(\sigma_{fr})} \frac{\partial u_i^{(\sigma_{ to})}}{\partial x_j}} \right) \end{equation}

from scale $\sigma _{fr}$ to $\sigma _{to}$ and the turbulent advection

(3.10)\begin{align} Ad(\sigma_{fr} \to \sigma_{to};y) &= -\frac{1}{2}\frac{\partial}{\partial x_j} \left\{ \overline{u_i^{(\sigma_{to})} u_i^{(\sigma_{to})} u_j^{(\sigma_{to})}}\right. \nonumber\\ &\quad + \left. \overline{u_i^{(\sigma_{to})} u_i^{(\sigma_{to})} u_j^{(\sigma_{fr})}} +2\overline{u_i^{(\sigma_{to})} u_i^{(\sigma_{fr})} u_j^{(\sigma_{fr})}} \right\} \end{align}

of $K^{(\sigma _{to})}$. These terms are obtained by replacing $L$ and $S$ in (3.5) with $\sigma _{fr}$ and $\sigma _{to}$, respectively, and they represent the contributions from scale $\sigma _{fr}$ to scale $\sigma _{to}$. We note again that, although non-diagonal parts ($\overline {u^{(\sigma _{to})}_i u^{(\sigma )}}$ for $\sigma _{to}\neq \sigma$) do not vanish, the terms defined by (3.8)–(3.10) approximate contributions to the energy around the scale $\sigma _{to}$. We confirm this in figure 8, which shows the ratio of non-diagonal parts $K(\sigma _{to},\sigma )=\overline {u_i^{(\sigma _{to})} u_i^{(\sigma )}}/2$ to the diagonal part $K(\sigma _{to},\sigma _{to})$ (i.e. $K^{(\sigma _{to})}$) as a function of $\sigma$ and $y$. We see that the most correlated scales are distributed in a range around $\sigma \approx \sigma _{to}$ irrespective of the height $y$. Therefore, the terms (3.8)–(3.10) may evaluate the contributions to the energy around $\sigma _{to}$. Note also that, although many nonlinear terms related to energy transfers through triad interactions also contribute to the change of $K^{(\sigma _{ to})}$, we assume in the present study that $Tr$ and $Ad$ composed of $\sigma _{fr}$ and $\sigma _{to}$ play an essential role in the energy transfer. Thus, by evaluating the terms ($Pr$, $Tr$ and $Ad$) as functions of $\sigma _{fr}$, $\sigma _{to}$ and $y$, we will show in the following how the energy with a scale $\sigma _{to}$ at a height $y$ is sustained.

Figure 8. The ratio of the non-diagonal parts of $K(\sigma _{to},\sigma )$ to the diagonal part $K(\sigma _{to},\sigma _{to})$ at (a) $\sigma _{to}^{+}=60$, (b) $240$ and (c) $960$ as a function of $\sigma$ and $y$. We interpolate the values evaluated in the six cases ($\sigma ^{+}=30$, $60$, $120$, $240$, $480$ and $960$) to draw these plots.

3.3.1. Production by the mean shear

The blue line in figure 9 shows the total production rate of turbulent kinematic energy as a function of $y$. Although it is a monotonically decreasing function of $y$, its decomposition into scales makes the mechanism of the energy production much clearer. We show in figure 9 the production $Pr(M\to \sigma _{to})$ by the mean flow for six different scales $\sigma _{to}^{+}=30$, $60$, $120$, $240$, $480$ and $960$ as a function of $y$. The thinner (and darker) lines indicate the contribution to the smaller scales. It is remarkable that, for a given height $y$, $Pr(M\to \sigma _{to}\approx y)$ is always largest (see the open squares on the lines) among different $\sigma _{to}$. This implies that the mean shear always produces most significantly the largest-scale (i.e. $\sigma \approx y$) structures at each height. In other words, since the largest-scale structures compose a hierarchy of streaks located beside quasi-streamwise vortices (figures 3 and 4), the mean flow transfers the energy to the streaks induced by these vortices. This sustaining mechanism of the hierarchical streaks composes a part of the hierarchical SSP.

Figure 9. Production (3.8) of $K^{(\sigma _{to})}$ by the mean flow. From the thinner (and darker) to the thicker (and lighter) lines, $\sigma _{to}^{+}=30$, $60$, $120$, $240$, $480$ and $960$. The open squares indicate $Pr(M\to \sigma _{to})$ at $y^{+}=\sigma _{to}^{+}$. The blue line shows the production from the mean flow without scale decomposition. Here, $h$ and $u_\tau$ are used for the normalization. The grey dashed line indicates $(\kappa y^{+})^{-1}$ and the black dashed one indicates $(13\kappa y^{+})^{-1}$ with $\kappa =0.38$.

Next, let us look at the envelope (black dashed line in figure 9) of the curves of $Pr$ for $\sigma _{to}^{+}=120$, $240$, $480$ and $960$. This line is proportional to $y^{-1}$, which implies the self-similarity of the largest structures in the log layer because each scale of the structures equally contributes to the total production rate:

(3.11)\begin{equation} \int^{2\sigma_{to}}_{\sigma_{to}} Pr \, \textrm{d}y\approx C\int^{2\sigma_{to}}_{\sigma_{to}} y^{-1} \, \textrm{d} y = \textrm{const.} \end{equation}

Here, the coefficient $C=(13\kappa )^{-1}$, with $\kappa$ being the Kármán constant, gives the best fit of the envelope. Incidentally, the production rate of the energy around the scale $\sigma _{to}^{+}=30$ in the buffer layer (see the open square for $y^{+}=30$) is larger than the envelope. This implies that the sustaining mechanisms of the largest scales by the mean shear are qualitatively similar but quantitatively different in the buffer and log layers. The difference will explain the statistics discussed in § 3.3.6.

3.3.2. Interscale energy transfer

Next, we evaluate the interscale energy transfer $Tr(\sigma _{fr}\to \sigma _{to};y)$ defined by (3.9). Figure 10 shows $Tr$ evaluated for (a) $y^{+}=960$, (b) $240$ and (c) $60$. Looking at the results for $y^{+}=240$ in the log layer, the contribution $Tr(2\sigma _{to}\to \sigma _{to})$ from the twice larger-scale strain rate is the most significant for all scales. On the other hand, the contributions from the smaller scale ($\sigma _{fr}<\sigma _{to}$) are negative. This implies that the smaller-scale ($\sigma _{fr}$) flow reduces the energy at the larger scale ($\sigma _{to}$). In other words, the direction of the energy transfer is forward on average. This observation suggests that the creation of small-scale vortices indeed corresponds to energy cascading events. Note that, for large scales ($\sigma _{to}^{+}=120$ ($\triangle$) and $240$ ($\bullet$)), $Pr$ (blue dots) is larger than $Tr$. This is because the mean flow transfers its energy to the largest scale at each height as already observed in figure 9. We emphasize that, for smaller scales, the contribution $Pr$ from the mean flow becomes less important, and $Tr(2\sigma _{to}\to \sigma _{to})$ plays a major role.

Figure 10. Scale-dependent energy transfer $Tr$ defined by (3.9) to $K^{(\sigma _{to})}$ with scales $\sigma _{to}^{+}=30$ ($\circ$), $60$ ($\square$), $120$ ($\triangle$) and $240$ ($\bullet$) at heights (a) $y^{+}=960$, (b) $240$ and (c) $60$. Blue symbols indicate $Pr$ defined by (3.8). Here, $h$ and $u_\tau$ are used for the normalization.

The results for the other heights $y^{+}=960$ (in the upper log layer) and $y^{+}=60$ (in the buffer layer) are similar to those for $y^{+}=240$, if we refer to the scale as large as the height as large. In other words, in the buffer and log layers in turbulent channel flow, the large-scale energy is directly transferred from the mean flow, whereas smaller-scale energy is generated by energy cascading process. Incidentally, we also see that the interscale energy transfers to structures larger than the height (e.g. $\sigma _{to}^{+}=240$ in figure 10c) are small.

It is also interesting to observe that the scale-dependent contributions to the vortex stretching (figure 7) and to the energy transfer (figure 10) are similar to each other. This result gives direct evidence that the vortex-stretching process plays a major role in the energy cascade (Tennekes & Lumley Reference Tennekes and Lumley1972; Goto Reference Goto2008, Reference Goto2012; Lozano-Durán et al. Reference Lozano-Durán, Holzner and Jiménez2016; Goto et al. Reference Goto, Saito and Kawahara2017; Doan et al. Reference Doan, Swaminathan, Davidson and Tanahashi2018; Motoori & Goto Reference Motoori and Goto2019a,Reference Motoori and Gotob; Dong et al. Reference Dong, Huang, Yuan and Lozano-Durán2020; Hirota et al. Reference Hirota, Nishio, Izawa and Fukunishi2020).

Thus, we have shown the interscale energy transfer by evaluating $Tr$ (3.9) at a given height $y$. The small-scale energy in the log layer is transferred predominantly from the twice-larger-scale structures and reduced by the half-scale ones. Here, we plot the wall-normal distribution of $Tr(2\sigma _{to}\to \sigma _{to})$ and $Tr(\sigma _{to}/2\to \sigma _{to})$ in figure 11 by black solid and dashed lines, respectively, for (a) $\sigma _{to}^{+}=240$ and (b) $60$. In both cases, at any height, the former is always positive, whereas the latter is negative. In addition, their positive and negative peaks are located around $y\approx 2\sigma _{to}$ and $\sigma _{to}$ (indicated by the open squares on these lines), respectively. Note that, around the negative peak ($y \approx \sigma _{to}$) in the log layer (figure 11a), the production $Pr(M\to \sigma _{ to}\approx y)$ (blue solid lines) gets large. This is consistent with the conclusion drawn from figure 9 and implies that the mean shear first transfers its energy to structures with the scale $\sigma$ comparable to $y$, and then the energy is transferred to the half-scale structures.

Figure 11. Energy production $Pr$ by the mean flow (blue solid lines), energy dissipation $-2\nu S_{ij}^{(\sigma _{to})} S_{ij}^{(\sigma _{to})}$ (blue dashed lines), interscale energy transfer $Tr(2\sigma _{to}\to \sigma _{to})$ from the twice-larger scale (black solid lines) and $Tr(\sigma _{to}/2\to \sigma _{to})$ from the half scale (black dashed lines), and scale-dependent energy advection $Ad$ from scales $\sigma _{fr}^{+}=240$ (red solid lines), $120$ (red dashed lines), $60$ (red dot-dashed lines) and $30$ (red dotted lines) to $\sigma _{to}$ in the equation for the energy with the scale (a) $\sigma _{to}^{+}=240$ and (b) $60$. Two open squares on the black solid and dashed lines indicate $Tr(2\sigma _{to}\to \sigma _{to})$ at $y^{+}=2\sigma _{to}^{+}$ and $Tr(\sigma _{to}/2\to \sigma _{to})$ at $y^{+}=\sigma _{to}^{+}$, respectively. Here, $h$ and $u_\tau$ are used for the normalization. The inset shows only $Ad$.

3.3.6. Reynolds stress

The results shown above suggest that the energy cascade in the log layer occur within a local layer, and the energy source of the cascade is the largest structures at each height produced by the mean shear. We have shown in figure 9 that the envelope (black dashed line) for the curves of $Pr$ in the log layer is proportional to $y^{-1}$. This scaling is the same as that of the total production without filtering (the grey dashed line in figure 9), which is the product of the mean velocity gradient and the Reynolds stress. In this subsection, we show that the behaviour of the Reynolds stress is also explained by the hierarchy of coherent structures.

For turbulent channel flow, if we assume the log law of the mean velocity profile, the Reynolds stress in the log layer obeys

(3.12)\begin{equation} -\overline{\check{u}\check{v}}^{+} = 1-\frac{y^{+}}{Re_\tau}-\frac{1}{\kappa y^{+}} \end{equation}

(see, for e.g. Rotta Reference Rotta1962). For $1/\kappa \ll y^{+} \ll Re_\tau$, $-\overline {\check {u}\check {v}}^{+}$ is approximately $1$. Looking at the Reynolds-stress profile without filtering (the blue line in figure 12a), we can see that this scaling indeed holds in the log layer. To explain this behaviour, we show in figure 12(a) the scale-decomposed Reynolds stress $-\overline {u^{(\sigma )}v^{(\sigma )}}$ for the scales $\sigma ^{+}=30$, $60$, $120$, $240$, $480$ and $960$. We can see that $-\overline {u^{(\sigma )}v^{(\sigma )}}$ always takes a maximum around $y\approx \sigma$, and, in the log layer, the maxima (indicated by the black dashed line) are constant (${\approx }1/13$) irrespective of $\sigma$. For the other heights ($y\not \approx \sigma$), the Reynolds stress is much smaller. Thus, if we assume that the largest-scale coherent structures are most relevant to the Reynolds stress at each height and they are self-similar in the log layer, we may demonstrate that the total Reynolds stress becomes constant. To show this, we plot in figure 12(b) the summation $\tau ^{<}$ of the contributions from the scales smaller than and equal to $\sigma$ (e.g. for $\sigma ^{+}=120$, $\tau ^{<}= -\overline {u^{(120)}v^{(120)}} -\overline {u^{(60)}v^{(60)}} -\overline {u^{(30)}v^{(30)}}$). Although we do not take into account the cross-correlations (e.g. $-\overline {u^{(30)}v^{(60)}}$) between different scales which contribute to the Reynolds stress, the region with constant $\tau ^{<}$ indeed expands for larger $\sigma$.

Figure 12. (a) Scale-decomposed Reynolds stress $-\overline {u^{(\sigma )}v^{(\sigma )}}$ and (b) the summation $\tau ^{<}(y;\sigma )$ of the contributions from the scales smaller than and equal to $\sigma$. From the thinner (and darker) to thicker (and lighter) lines, $\sigma ^{+}=30$, $60$, $120$, $240$, $480$ and $960$. The blue line shows the Reynolds stress without the scale decomposition. The grey dashed line indicates $1$ and the black dashed one indicates $1/13$.

The scaling of the location $y_p$ of the peak of the Reynolds stress with respect to $Re_\tau$ is also important. Chen, Hussain & She (Reference Chen, Hussain and She2019) showed that the scaling of $y_p$ changed around $Re_\tau = 3000$: $y_p^{+}\propto Re_\tau ^{1/3}$ for $Re_\tau \lesssim 3000$, while $y_p^{+}\propto Re_\tau ^{1/2}$ for $Re_\tau \gtrsim 3000$. We examine this scaling transition by using the database with a single $Re_\tau$($=4179$). For this purpose, we again focus on $\tau ^{<}$ (figure 12b) assuming that the increase of $Re_\tau$ corresponds to the increase of the levels of the hierarchy of the largest-scale ($\sigma \approx y$) flow structures, and that the largest-scale flow structures most contribute to the Reynolds stress (figure 12a). Since the hierarchical structures are absent in low-Reynolds-number turbulence, only the buffer-layer coherent structures exist. Therefore, the peak of the Reynolds stress for low Reynolds numbers is related to the peaks of $\tau ^{<}$ for $\sigma ^{+}=30$ and $60$, for which the buffer-layer structures are dominant. As the Reynolds number increases, since the hierarchy of structures appears, larger-scale motions contribute to the Reynolds stress. For example, the curve of $\tau ^{<}$ for $\sigma ^{+}=960$ (the thickest line in figure 12b) is similar to that of the Reynolds stress (blue line) without filtering. Thus, we can interpret $\tau ^{<}(\sigma )$ as a distribution of the Reynolds stress at the Reynolds number $\sigma$. In fact, the peaks of $\tau ^{<}$ demonstrate a scaling transition when increasing $\sigma$ (figure 13). The peak location $y_p$ of $\tau ^{<}(\sigma )$ is proportional to $\sigma ^{1/3}$ for $\sigma ^{+}\lesssim 240$ (see blue solid line), whereas the scaling changes to $y_p\propto \sigma ^{1/2}$ at $\sigma ^{+}\approx 240$ (see red dashed line). Although the peak $y_p^{+}=50$ at the transition point $\sigma ^{+}\approx 240$ is not in exact agreement with the peak location ($y_p^{+}\approx 80$) shown by Chen et al. (Reference Chen, Hussain and She2019), the scaling is reproduced by the hierarchy in the fully developed turbulence ($Re_\tau =4179>3000$). We therefore conclude that the scaling transition for the Reynolds number occurs due to the quantitative difference in the generation mechanism of the largest-scale structures in the buffer and log layers (figure 9).

Figure 13. The height $y_p$ where $\tau ^{<}$ (shown as in figure 12b) is maximum. Blue solid and red dashed lines are proportional to $\sigma ^{1/3}$ and $\sigma ^{1/2}$, respectively.