3.1. ‘Optimal’ spectral energy flux boundary

Most of the previous studies used a small-scale isotropic filter ($\lambda ^+_{x|y|z}\sim O(10^2)$) for scale separation (Piomelli et al. Reference Piomelli, Yu and Adrian1996; Natrajan & Christensen Reference Natrajan and Christensen2006; Dong et al. Reference Dong, Huang, Yuan and Lozano-Durán2020). This is not suitable for the purpose of investigating the energy transfer behaviour associated with larger-scale structures. Therefore, we revisit the inertial energy transfer term in the evolution equation of TKE in shear flows, whose formulation in spectral space is

(3.1)\begin{align} & \frac{{\partial \varPhi \left( {{k_x},{k_z},y} \right)}}{{\partial t}} = - \underbrace{\left\langle {{{\hat{u}}^*}\hat{v} + \hat{u}{{\hat{v}}^*}} \right\rangle \frac{{\textrm{{d}}\langle U\rangle }}{{2\textrm{{d}}y}}}_{\hat{P}} \underbrace{-\frac{1}{2}\left\langle {\hat{u}_i^*\widehat{\frac{{\partial {u_i}{u_j}}}{{\partial {x_j}}}} + {{\hat{u}}_i}{{\widehat{\frac{{\partial {u_i}{u_j}}}{{\partial {x_j}}}}}^*}} \right\rangle }_{\hat{T}} \nonumber\\ &\quad - \underbrace{\left\langle {\frac{\textrm{{d}}}{{\textrm{{d}}y}}\left( {\frac{{\hat{p}{{\hat{v}}^*} + {{\hat{p}}^*}\hat{v}}}{2\rho }} \right)} \right\rangle }_{{{\hat{T}}_p}}+ \underbrace{\nu \left\langle {\frac{{{\textrm{{d}}^2}\varPhi }}{{\textrm{{d}}{y^2}}}} \right\rangle }_{{{\hat{T}}_{\nu} }} - \underbrace{\frac{\nu}{2} \left\langle {{{\widehat{\frac{{\partial {u_i}}}{{\partial {x_j}}}}}^*}\widehat{\frac{{\partial {u_i}}}{{\partial {x_j}}}} +\widehat{\frac{{\partial {u_i}}}{{\partial {x_j}}}}{{\widehat{\frac{{\partial {u_i}}}{{\partial {x_j}}}}}^*}} \right\rangle }_{\hat{\varepsilon}}. \end{align}

This formulation can be derived by applying a 2-D Fourier transform to the evolution equation of the in-plane two-point velocity correlation function for shear flows (Monin & Yaglom Reference Monin and Yaglom1975). In (3.1), $\varPhi$ is the spectral TKE, i.e. the density of TKE at a set of wavenumbers ($k_x$, $k_z$) and a particular flow layer of $y$, $\hat {(\cdot )}$ indicates a variable in 2-D spectral space with the dependent variables $k_x$, $k_z$ and $y$ being omitted, $(\cdot )^*$ is the complex conjugate and $\langle \cdot \rangle$ denotes an ensemble average. On the right-hand side of (3.1), $\hat {P}$ is the turbulence production term, $\hat {T}$ is the inertial transfer term caused by turbulent convection, $\hat {T}_p$ is the fluctuation-pressure interaction term, and $\hat {T}_{\nu }$ or $\hat {\varepsilon }$ are the viscous diffusion or dissipation term, respectively.

Note that $\hat {T}$ is the only term that exclusively includes the effect of turbulent fluctuation. It is actually a summation of nine components, i.e.

(3.2)\begin{equation} \hat{T}\left( {{k_x},{k_z}} \right)= \sum_{i,j = 1}^3 {{{\hat{T}}_{ij}}\left( {{k_x},{k_z}} \right)}. \end{equation}

For the present wall-parallel dataset, only four components with $i$, $j=1$ and 3 are kept. This enables us to reformulate ${{\hat {T}}_{ij}}$ in a sense of triadic interaction between three wavenumbers $k$, $k'$ and $k-k'$ (Monin & Yaglom Reference Monin and Yaglom1975), i.e.

(3.3)\begin{equation} {{\hat{T}}_{ij}}({k_x},{k_z}) =\mathrm{Re}\left\langle \sum_{{k_x'},{k_z'}} {\mathrm{i}{k_j}{\hat{u}_i^*( {{k_x},{{k}_z}} )\hat{u}_j( {{k_x} - {{k}_x'},{k_z} - {{k}_z'}} ){{\hat{u}}_i}( {{k_x'},{k_z'}} )} } \right\rangle, \quad i,j \in \{1,3\}. \end{equation}

Such a formulation implies that the exchange of scale energy in a wall-parallel plane by turbulent motion is embodied as the energy flux due to triadic-wave interaction, similar to the scenario in homogeneous and isotropic turbulence (Cardesa, Vela-Martín & Jiménez Reference Cardesa, Vela-Martín and Jiménez2017). Note that this form cannot be obtained if the components associated with the vertical gradient (with $j=2$) in (3.2) are included.

Here, we define the in-plane energy transfer term as

(3.4)\begin{equation} {\hat{T}_{XZ}({k_x},{k_z})} = \sum_{i,j} {{{\hat{T}}_{ij}}({k_x},{k_z})}, \quad i,j \in \{1,3\}, \end{equation}

which measures the rate of change of the spectral TKE (without $v$ component) owing to $u$ and $w$ component turbulent fluctuations in the $x$–$z$ plane. The spectra of ${\hat {T}_{XZ}({k_x},{k_z})}$ present similar patterns among all the studied wall-parallel planes ($y/\delta =0.03\text {--}0.24$). One example at $y/\delta =0.12$ in case $E3$ (with $Re_{\tau }=3500$) is given in figure 3(a), which shows that ${\hat {T}_{XZ}({k_x},{k_z})}$ is characterized as one positive large-scale patch and one negative small-scale patch divided by a zero-crossing line $B$ (bold dashed line in figure 3a). These two patches represent the energy donor modes and energy recipient modes (Lee & Moser Reference Lee and Moser2019), respectively, and are consistent with the formal concept that on average large-scale eddies lose energy, while small-scale fluctuations gain energy.

Figure 3. $(a)$ In-plane energy transfer spectrum ${\hat {T}_{XZ}}(\lambda _x/\delta , \lambda _z/\delta )$ at $y/\delta \approx 0.12$ in case $E3$. Bold dashed line shows the boundary between the energy donor and energy recipient modes at $y/\delta \approx 0.12$, while light dashed lines indicate the corresponding boundaries at other flow layers, i.e. $y/\delta \approx 0.03$, 0.06 and 0.24. The solid line $C$ is an arbitrary path of the energy transfer. $(b)$ Wall-normal variation of the absolute value of the total ${\hat {T}_{XZ}}$ or $\hat {T}$ integrated over the energy donor or energy recipient modes, which are separated by the zero-crossing boundary $B$ shown in panel $(a)$. The upper half-plane is for $\iint |\hat {T}^{+|-}_{XZ}|\,\textrm {d}k_x\,\textrm {d}k_z$ in case $E3$, and the lower half-plane for $\iint |\hat {T}^{+|-}|\,\textrm {d}k_x\,\textrm {d}k_z$ in case $D1$.

The integration of ${\hat {T}_{XZ}({k_x},{k_z})}$ in the whole scale space is close to zero for all the investigated flow layers. This can be evidenced by figure 3(b), in which the magnitudes of $\iint _{\hat {T}_{XZ}\ge 0}|\hat {T}_{XZ}|\,\textrm {d}k_x\,\textrm {d}k_z$ are almost identical to that of $\iint _{\hat {T}_{XZ}<0}|\hat {T}_{XZ}|\,\textrm {d}k_x\,\textrm {d}k_z$. Such an observation seems to be in accordance with the quasi-local assumption of the energy cascade. However, figure 3(b) also shows that when both the $v$ component and the vertical gradient are taken into account in the full resolved $\hat {T}$, slight imbalance between $\iint _{\hat {T}\ge 0}|\hat {T}|\,\textrm {d}k_x\,\textrm {d}k_z$ and $\iint _{\hat {T}<0}|\hat {T}|\,\textrm {d}k_x\,\textrm {d}k_z$ appears in the log layer and below of case $D1$ (with $Re_{\tau }=1750$). This highlights the contribution of the wall-normal turbulent motion on the inertial energy transfer (Cimarelli et al. Reference Cimarelli, De Angelis, Jiménez and Casciola2016; Lee & Moser Reference Lee and Moser2019). Nevertheless, the overall level, as can be estimated by the difference between $\iint _{\hat {T}\ge 0}|\hat {T}|\,\textrm {d}k_x\,\textrm {d}k_z$ and $\iint _{\hat {T}<0}|\hat {T}|\,\textrm {d}k_x\,\textrm {d}k_z$, is relatively small if compared with that of the in-plane energy transfer. This provides a justification for neglecting the wall-normal energy transfer as a leading-order approximation in the present study.

By superimposing the zero-crossing boundary $B$ onto the $\varPhi _{uu}$ spectrum (in figure 2a), it is clearly seen that the energy-bearing large scales (with spectral peak at $\lambda _x\sim 2\delta$ and $\lambda _z \sim 0.5\delta$) reside in the positive $\hat {T}_{XZ}$ region. This is consistent with previous observations that the spectral peak of the turbulent production term locates inside the energy donor modes (Mizuno Reference Mizuno2016; Cho et al. Reference Cho, Hwang and Choi2018; Lee & Moser Reference Lee and Moser2019). Denoting the minimum streamwise and spanwise length scale of $B$ as $\lambda ^B_{x0}$ and $\lambda ^B_{z0}$, respectively. Figure 3(a) shows that $\lambda ^B_{z0}$ in case $E3$ increases from $0.04\delta$ to $0.12\delta$ as the flow layer elevates from $y/\delta \approx 0.03\text {--}0.24$; in contrast, $\lambda ^B_{x0}$ only varies from 0.15$\delta$ to 0.2$\delta$. Figure 15 in appendix A further shows that for all the studied cases, $\lambda ^B_{z0}/\delta$ is quasi-linearly proportional to $y/\delta$ across a wide range of $y$, while the variation of $\lambda ^B_{x0}/\delta$ in the log layer is mild.

The zero-crossing boundary $B$ in the $\hat {T}_{XZ}$ spectrum is regarded as the ‘optimal’ energy flux boundary, across which the spectral energy flux is maximum. As illustrated in figure 3(a), this can be visualized by drawing an arbitrary path $C$ in the 2-D wavenumber space from the origin $(0,0)$ to one set of $(\kappa _x,\kappa _z)$. The net scale energy transferred from larger scales to $(\kappa _x, \kappa _z)$ along $C$ is defined as the following curvilinear integral:

(3.5)\begin{equation} \hat{\varPi} ({\kappa_x},{\kappa_z}) = \int_{\boldsymbol{r}=(0,0)}^{\boldsymbol{r}=(\kappa_x,\kappa_z)}{ - {{\hat{T}}_{XZ}}\left( {{k_x},{k_z}} \right)}\,\textrm{{d}}\boldsymbol{r}. \end{equation}

It is easy to see that $\hat \varPi$ reaches a maximum when $(\kappa _x, \kappa _z)$ falls on $B$. In the following analysis, this ‘optimal’ energy flux boundary is used as the cutting-off threshold of a 2-D sharp-edge filter to separate large-scale modes from smaller ones at each flow layer. The physical significance of such an anisotropic filter is that the yielded net energy exchange is the most intense in an ensemble-averaged sense, as the ‘large’ and ‘small’ scales are actually the energy donor modes or energy recipient modes in the energy transfer spectrum.

3.2. Statistics of energy transfer events

In order to explore the relationship between energy transfer and LSMs, the instantaneous energy transfer event should be properly defined at first. The evolution equation of the ‘resolved’ instantaneous TKE in physical space is

(3.6)\begin{equation} \frac{{\partial \tilde{Q}\left( {x,z,y} \right)}}{{\partial t}} = \underbrace{-\tilde{u}\tilde{v}\frac{{\textrm{{d}}\langle U\rangle }}{{\textrm{{d}}y}}}_P \underbrace{-{{\tilde{u}}_i}\frac{{\partial \widetilde{{u_i}{u_j}}}}{{\partial {x_j}}}}_T \underbrace{-\frac{\textrm{{d}}}{{\textrm{{d}}y}}\left( {\frac{{\tilde{p}\tilde{v}}}{\rho }} \right)}_{{T_p}} + \underbrace{\nu \frac{{{\textrm{{d}}^2}\tilde{Q}}}{{\textrm{{d}}{y^2}}}}_{{T_{\nu} }} \underbrace{-\nu \frac{{\partial {{\tilde{u}}_i}}}{{\partial {x_j}}}\frac{{\partial {{\tilde{u}}_i}}}{{\partial {x_j}}}}_{\varepsilon} , \end{equation}

in which $\tilde {(\cdot )}$ denotes those filtered variables including only ‘resolved’ scales and $\tilde {Q}=\tilde {u}_i \tilde {u}_i/2$ (with $i=1$, 3) is the ‘resolved’ TKE. The terms on the right-hand side of (3.6) correspond to those in (3.1) in spectral space.

According to Piomelli et al. (Reference Piomelli, Cabot, Moin and Lee1991, Reference Piomelli, Yu and Adrian1996) and Natrajan & Christensen (Reference Natrajan and Christensen2006), the inertial transfer term $T$ can be decomposed as follows:

(3.7)\begin{align} T &=- {\tilde{u}_i}\frac{{\partial\widetilde{{u_i}{u_j}}}}{{\partial {x_j}}} = -\left({\tilde{u}_i}\frac{{\partial {{\tilde{u}}_i}{{\tilde{u}}_j}}}{{\partial {x_j}}}+{\tilde{u}_i}\frac{{\partial (\widetilde{{u_i}{u_j}} - {{\tilde{u}}_i}{{\tilde{u}}_j})}}{{\partial {x_j}}}\right) \nonumber\\ &=-\left(\frac{{\partial {{\tilde{Q}}}{{\tilde{u}}_j}}}{{\partial {x_j}}} +\frac{{\partial {{\tilde{u}}_i}\left( {\widetilde{{u_i}{u_j}} - {{\tilde{u}}_i}{{\tilde{u}}_j}} \right)}}{{\partial {x_j}}} -\left( {\widetilde{{u_i}{u_j}} - {{\tilde{u}}_i}{{\tilde{u}}_j}} \right)\frac{{\partial {{\tilde{u}}_i}}}{{\partial {x_j}}}\right). \end{align}

In (3.7), the first two terms on the second line represent the large-scale and subgrid-scale diffusion terms, respectively, while the third term is essentially the same as the subgrid dissipation term, i.e. $\varepsilon _{{SGS}}$, that needs to be modelled in large eddy simulations. Recalling that $T$ is the instantaneous counterpart of $\hat {T}$ being integrated in the large-scale side of the ‘optimal’ energy flux boundary $B$; therefore, it accounts for the effect of triadic interaction among all the scales (as shown in (3.3)). However, previous works (Piomelli et al. Reference Piomelli, Yu and Adrian1996; Liao & Ouellette Reference Liao and Ouellette2013) have shown that only those scale interaction forms represented by $\varepsilon _{{SGS}}$ are responsible for the net energy exchange between ‘resolved’ and ‘unresolved’ scales, which is the reason that $\varepsilon _{{SGS}}$ is named as subgrid energy flux. In contrast, the large-scale and subgrid-scale diffusion terms play a role of redistributing large-scale energy in physical space through turbulent convection.

Following Piomelli et al. (Reference Piomelli, Yu and Adrian1996), we define instantaneous in-plane spatial energy flux as $\varPi _{{d}}$, instantaneous interscale energy flux as $\varPi _{{SGS}}$ and their sum as the total energy flux $\varPi$, i.e.

(3.8)\begin{gather} \varPi_{{d}} =\sum_{i,j}\left(\frac{{\partial {{\tilde{Q}}}{{\tilde{u}}_j}}}{{\partial {x_j}}}+\frac{{\partial {{\tilde{u}}_i}\left( {\widetilde{{u_i}{u_j}} - {{\tilde{u}}_i}{{\tilde{u}}_j}} \right)}}{{\partial {x_j}}}\right),\quad i,j \in \left\{ {1,3} \right\}, \end{gather}

(3.9)\begin{gather} {\varPi_{{SGS}}} =\sum_{i,j} - \left( {\widetilde{{u_i}{u_j}} - {{\tilde{u}}_i}{{\tilde{u}}_j}} \right)\frac{{\partial {{\tilde{u}}_i}}}{{\partial {x_j}}},\quad i,j \in \left\{ {1,3} \right\}, \end{gather}

(3.10)\begin{gather} \varPi= \varPi_{{d}} + {\varPi_{{SGS}}}. \end{gather}

Note that the summation convention is not applied in (3.8)–(3.9), and the indexes $(i,j)$ only account for wall-parallel dimensions. Tests on the DNS dataset given in appendix A show that for the present scale cutting-off boundary, the in-plane components (with $i,j\in \{ {1,3} \}$) dominate over the vertical ones (with $i$ or $j=2$) in the full-resolved $\varPi _{{SGS}}$; however, the contribution of the latter becomes non-negligible when a small-scale isotropic filter is used for scale separation (see figure 16). This justifies the experimental attempt of using only in-plane components to characterize interscale energy flux that are associated with energy-bearing scales. On the other hand, the contributions of both in-plane and vertical components to $\varPi _{{d}}$ are insensitive to the scale filter (see figure 17).

The p.d.f.s of $\varPi$, $\varPi _{{SGS}}$ and $\varPi _{{d}}$ are analysed first. All the studied flow layers have similar p.d.f. distributions; therefore, only the results at $y/\delta \approx 0.12$ in case $E3$ and case $D1$ are illustrated in figure 4. Satisfying collapses are observed between case $E3$ and case $D1$ when $\varPi$, $\varPi _{{SGS}}$ and $\varPi _{{d}}$ are normalized by their own root mean squared (known as RMS) value. The p.d.f. of $\varPi _{{d}}$ (shown in figure 4c) is symmetrical about the origin, indicating a zero net in-plane energy flux, i.e. $\langle \varPi _{{d}}\rangle =0$. It demonstrates that $\varPi _{{d}}$ does not alter the overall level of energy but only redistributes energy in space, consistent with the homogeneous condition in wall-parallel dimension (when only the in-plane turbulent motion is concerned). In contrast, asymmetry between forward scatter events (with positive $\varPi _{{SGS}}$) and backward scatter ones (with negative $\varPi _{{SGS}}$) is seen in the p.d.f. of $\varPi _{{SGS}}$ (in figure 4b), which is inherited by the p.d.f. of $\varPi$ (in figure 4a). The ensemble-averaged $\langle \varPi _{{SGS}}\rangle$ and $\langle \varPi \rangle$ are both positive, consistent with the formal energy cascade and the fact that ‘resolved’ scales are actually those energy donor modes in the in-plane energy transfer spectrum $\hat {T}_{XZ}(k_x, k_z)$.

Figure 4. Premultiplied p.d.f. of (a) $\varPi$, (b) $\varPi _{{SGS}}$ and (c) $\varPi _{{d}}$ at $y/\delta \approx 0.12$ for both case $E3$ (discrete bars) and case $D1$ (solid and dashed curves). Positive and negative events are plotted on the same side to highlight the symmetry condition. Owing to the premultiplication, the height of each bar (or curve segment) measures the relative contribution of the corresponding event to the net energy flux.

4.1. Visualization of $\varPi _{{SGS}}$ event

Figure 5 plots one snapshot of $\tilde {u}(x, y)$, $\varPi _{{SGS}}(x, y)$ and $\varPi _{{d}}(x, y)$ at different flow layers in case $E3$ for a qualitative visualization. Hereinafter, only the results of case $E3$ are to be presented. appendix D deals with the $Re$ effect. It will be shown that for the present studied flow regions, the characteristics of interscale energy transfer and in-plane energy transportation are less dependent on $Re$ (with $Re_{\tau }=1200 \text {--}3500$).

Figure 5. Instantaneous field of $\tilde {u}$ (filled contours), $\varPi _{{SGS}}$ (isolines in a,c,e,g) and $\varPi _{{d}}$ (isolines in b,d,f,h) at (a,b) $y/\delta \approx 0.03$, (c,d) $y/\delta \approx 0.06$, (e,f) $y/\delta \approx 0.12$ and (g,h) $y/\delta \approx 0.24$ in case $E3$. In panel (a,c,e,g), solid (or dashed) isolines indicate forward (or backward) scatter events with $\varPi _{{SGS}}= {\pm}1.5\sigma _{\varPi _{{SGS}}}$, while clusters of forward and backward scatter events are marked by dashed circles, arrows highlight forward scatter events at spanwise interfaces of high- and low-momentum regions, and solid circles highlight those at streamwise interfaces. In panel (b,d,f,h), solid (dashed) isolines indicate positive (or negative) energy transfer events with $\varPi _{{d}}={\pm}1.5 \sigma _{\varPi _{{d}}}$, solid circles indicate pairs of ${\pm }\varPi _d$ events inside low-momentum LSMs.

As shown in figure 5, streaky-like large-scale $\tilde {u}$ structures resembling LSMs are evident from the buffer layer to the lower part of the wake region. They are highly sinusoidal in shape, and their spanwise width presents a significant growth with the increase of $y$. Note that the size of these typical $\tilde {u}$ structures is fairly large compared with the ‘optimal’ energy flux boundary; therefore, their geometries will not be significantly biased by the low-pass sharp-edge filtering.

As for $\varPi _{{SGS}}$, its distribution is more trivial and chaotic in the near-wall region (in figure 5a), but tends to be more compact and coherent in higher layers (in figure 5e,g). Two types of $\varPi _{{SGS}}$ events are seen. Firstly, fragmented regions of $\varPi _{{SGS}}$ with alternating sign aggregate into a pattern of staggered cluster (indicated by dashed circles in figure 5c,e,g). A mutual cancellation of $\pm \varPi _{{SGS}}$ in the cluster yield only weak (if not zero) net interscale energy flux. Such a pattern is similar to the spinwheel-like pattern of alternating lobes of $\pm \varPi _{{SGS}}$ in 2-D turbulence, which were found to be related to strong rotational motions by Liao & Ouellette (Reference Liao and Ouellette2013). Secondly, isolated $\varPi _{{SGS}}$ events scatter in the flow field, among which the population density of positive events are distinctly higher than those of negative ones. It is this kind of event that contribute to the asymmetry of the p.d.f. of $\varPi _{{SGS}}$.

A careful inspection shows that these isolated $\varPi _{{SGS}}$ events are highly correlated with intense local velocity gradients along either the spanwise direction or streamwise direction, i.e. $\partial \tilde {u}/\partial z$ or $\partial \tilde {u}/\partial x$. The former case (arrow-indicated regions in figure 5a,c,e,g) usually appears on the spanwise interface between $\pm \tilde {u}$ structures, while the latter case (regions inside solid circles in figure 5e,g) prefers to appear in-between streamwise-aligned $\tilde {u}$ structures with alternating sign. Note that the latter case has been seen in the $x$–$y$ plane by Natrajan & Christensen (Reference Natrajan and Christensen2006), who attributed it to the opposing Q2 and Q4 events induced by hairpin vortices. Nevertheless, the former one, which seems to have higher occurrence probability in the log layer and above, has not been reported before.

4.2. Cumulative spectrum of $\varPi _{{SGS}}$ events

For a quantitative characterization of the above observation, cumulative energy flux $\sum _{\chi } \varPi _{{SGS}}$ under particular flow conditions, i.e. $\chi =\tilde {u}$, $\partial \tilde {u}/\partial z$ and $\partial \tilde {u}/\partial x$, are shown in figure 6 for case $E3$ to evaluate their individual contributions to the net interscale energy flux $\langle \varPi _{{{SGS}}}\rangle$. Additionally, $\pm \varPi _{{SGS}}$ events are differentiated in the add-up process to indicate their relative contributions. The yielded values are denoted as $\sum _{\chi }\varPi ^+_{{SGS}}$ and $\sum _{\chi }\varPi ^-_{{SGS}}$, respectively, and are also shown in figure 6. Since $\partial {\tilde {w}}/\partial x_j$ are generally smaller than $\partial {\tilde {u}}/\partial x_j$, they are removed from the candidate velocity gradient conditions to be examined. Meanwhile, the condition of $\tilde {u}$ is reserved to quantify the correlation between $\varPi _{{SGS}}$ and LSMs.

Figure 6. The 1-D cumulative spectrum of interscale energy flux $\sum _{\chi }\varPi _{{{SGS}}}$ with respect to flow conditions $\chi$ in case $E3$. $(a)$ $\chi =\tilde {u}$, $(b)$ $\chi =\partial \tilde {u}/{\partial z}$ and $(c)$ $\chi ={\partial \tilde {u}}/{\partial x}$. Solid line, $y/\delta \approx 0.03$; dashed line, $y/\delta \approx 0.06$; dot-dash line, $y/\delta \approx 0.12$; and dotted line, $y/\delta \approx 0.24$. The group of profiles marked by $\varPi ^+_{{SGS}}$, $\varPi ^-_{{SGS}}$ or $\varPi _{{SGS}}$ represent the cumulative contribution from forward scatter events, backward scatter events or all of the events, respectively. Velocity gradients conditions are normalized by their root mean squared value, the same in the following figures.

Figure 6 leads to several interesting observations. Firstly, the profiles of $\sum _{\tilde {u}} \varPi _{{SGS}}(\tilde {u})$ (in figure 6a) centre around $\tilde {u}=0$ with a slight left-offset. Such an asymmetry is seen to be inherited from that of $\sum _{\tilde {u}} \varPi ^+_{{SGS}}(\tilde {u})$ whose peaks locate at $\tilde {u}/\sigma _{\tilde {u}}\approx -0.3$. In contrast, $\sum _{\tilde {u}} \varPi ^-_{{SGS}}(\tilde {u})$ keeps a satisfying symmetry. The high magnitude of $\sum _{\tilde {u}} \varPi ^{\pm }_{{SGS}}$ around $\tilde {u}=0$ suggests a weak correlation between the core region of LSMs and $\varPi _{{SGS}}$ events. Secondly, $\sum _{{\partial \tilde {u}}/{\partial z}} \varPi ^+_{{SGS}}({\partial \tilde {u}}/{\partial z})$ presents a bimodal profile (in figure 6b), except for the highest flow layer (at $y/\delta \approx 0.24$) where a plateau, instead of two isolated peaks, appears in the range of $({\partial \tilde {u}}/{\partial z})/\sigma ({\partial \tilde {u}}/{\partial z}) \in [-1 , +1]$. This bimodal pattern, together with the skewness of $\sum _{\tilde {u}} \varPi ^+_{{SGS}}(\tilde {u})$ towards the $+\tilde {u}$ side, provides a statistical support for the preferential alignment of forward scatter events along the flanks of low-momentum LSMs, as has been visualized in figure 5(a,c,e,g) (as arrow-indicated regions). Finally, the profiles of $\sum _{{\partial \tilde {u}}/{\partial x}}\varPi _{{SGS}}({\partial \tilde {u}}/{\partial x})$ (in figure 6c) present an asymmetric bimodal pattern with two peaks locating at $({\partial \tilde {u}}/{\partial x})/\sigma ({\partial \tilde {u}}/{\partial x})\approx 0.5$ and $-1.5$, respectively. This pattern is jointly contributed by quasi-symmetric single-peak profiles of $\sum _{{\partial \tilde {u}}/{\partial x}}\varPi ^-_{{SGS}}({\partial \tilde {u}}/{\partial x})$ and negative-skewed profiles of $\sum _{{\partial \tilde {u}}/{\partial x}}\varPi ^+_{{SGS}}({\partial \tilde {u}}/{\partial x})$, the latter of which is consistent with the visualization that forward scatter events can be found in the middle of streamwise-aligned positive and negative LSMs (bold solid circles in figure 5e,g) where ${\partial \tilde {u}}/{\partial x}<0$.

The dependency of $\varPi _{{SGS}}$ on both ${\partial \tilde {u}}/{\partial x}$ and ${\partial \tilde {u}}/{\partial z}$ is further considered by constructing a bivariate cumulative spectrum, i.e. $\sum _{\chi }\varPi _{{SGS}}(\chi )$ with $\chi =[{{\partial \tilde {u}}/{\partial x}}, {{\partial \tilde {u}}/{\partial z}}]$. Hereinafter, $\chi$ is omitted in the 2-D summation operator $\sum _{\chi }$ for simplicity. Here, $\tilde {u}$ is not regarded as one of the joint conditions due to its weak correlation with $\varPi ^{\pm }_{{SGS}}$, as has been shown in the univariate cumulative profiles in figure 6(a). Figure 7 shows $\sum \varPi ^{\pm }_{{SGS}}$ and $\sum \varPi _{{SGS}}$ in the phase plane of (${{\partial \tilde {u}}/{\partial x}}, {{\partial \tilde {u}}/{\partial z}})$ at $y/\delta \approx 0.12$ in case $E3$, while figure 19 in appendix D will summarize the cumulative spectra of $\sum \varPi _{{SGS}}({{\partial \tilde {u}}/{\partial x}}, {{\partial \tilde {u}}/{\partial z}})$ in all the studied cases to evidence both a $Re$-independency and a flow-layer insensitivity.

Figure 7. The 2-D cumulative spectrum of $(a)$ positive interscale energy flux events $\sum \varPi ^+_{{SGS}}$, $(b)$ negative events $\sum \varPi ^-_{{SGS}}$ and (c) all of the events $\sum \varPi _{{SGS}}$ in the phase space of $({\partial \tilde {u}}/{\partial x},{\partial \tilde {u}}/{\partial z})$ at $y/\delta \approx 0.12$ in case $E3$. Each spectrum is normalized by its peak value. The subpartitions marked in panel $(c)$ are defined in table 2.

The 2-D contours of both $\sum \varPi ^+_{{SGS}}$ and $\sum \varPi ^-_{{SGS}}$ (in figure 7a,b) are symmetric in the ${{\partial \tilde {u}}/{\partial z}}$ dimension but asymmetric in the ${{\partial \tilde {u}}/{\partial x}}$ dimension. Two peaks in $\sum \varPi ^+_{{SGS}}$ both locate at the ${{+\partial \tilde {u}}/{\partial x}}$ side and straddle on each side of the ${{\partial \tilde {u}}/{\partial z}}$ axis. In contrast, $\sum \varPi ^-_{{SGS}}$ peaks at ${{-\partial \tilde {u}}/{\partial x}}$ side with ${{\partial \tilde {u}}/{\partial z}}=0$. All these observations are consistent with the findings in the univariate cumulative profiles (in figure 6), which can be obtained by integrating the 2-D contours along one dimension. More interestingly, the contour of $\sum \varPi _{{SGS}}({{\partial \tilde {u}}/{\partial x}}, {{\partial \tilde {u}}/{\partial z}})$ (in figure 7c) presents a ‘pacman’ pattern; namely, positive $\sum \varPi _{{SGS}}$ dominates the periphery of the phase plane and encompasses the central negative $\sum \varPi _{{SGS}}$ region, while an extra ‘bean’ of negative $\sum \varPi _{{SGS}}$ appears around $({{\partial \tilde {u}}/{\partial x}})/\sigma ({{\partial \tilde {u}}/{\partial x}})=2$, $({{\partial \tilde {u}}/{\partial z}})/\sigma ({{\partial \tilde {u}}/{\partial z}})=0$. Note that such a complicated pattern cannot be easily inferred from the univariate cumulative profiles.

To measure the contribution of different velocity gradient conditions to the net interscale energy flux, as shown in figure 7(c), the phase plane of $({{\partial \tilde {u}}/{\partial x}}, {{\partial \tilde {u}}/{\partial z}})$ is divided into two groups of subpartitions, i.e. $F1\text {--}F3$ for net forward flux and $B1\text {--}B2$ for net backward flux. The $F1$ (or $B2$) is dominated by forward (or backward) scatter events owing to strong negative (or positive) ${\partial \tilde {u}}/{\partial x}$; $F2$ and $F3$ are the subpartitions where large magnitudes of ${\partial \tilde {u}}/{\partial z}$ and ${\partial \tilde {u}}/{\partial x}$ jointly contribute to the net forward flux; $B1$ locates in the centre of the phase plane, it indicates the dominance of backward scatter events that are independent of in-plane velocity gradient conditions. Table 2 lists the contribution of each subpartition to the total net flux $\langle \varPi _{{SGS}}\rangle$ at $y/\delta \approx 0.12$ in case $E3$, the area ratio occupied by each subpartition in the wall-parallel plane is also included. The corresponding results in all the studied cases are summarized in figure 20 in appendix D. Compared with $F1$ and $F3$ being characterized as strong ${\partial \tilde {u}}/{\partial x}$ condition, $F2$ has the highest contribution to $\langle \varPi _{{SGS}}\rangle$ but a small area fraction. This observation highlights the correlation of intense forward scatter events with strong spanwise shear ${\partial \tilde {u}}/{\partial z}$. On the other hand, among two subpartitions with negative $\sum \varPi _{{SGS}}$, $B1$ has a higher contribution to $\langle \varPi _{{SGS}}\rangle$, but is still one order smaller than those of $F1\text {--}F3$.

Table 2. Subpartitions in the phase plane of (${{\partial \tilde {u}}/{\partial x}}, {{\partial \tilde {u}}/{\partial z}})$ at $y/\delta \approx 0.12$ in case $E3$. The percentage of the contribution to $\langle \varPi _{{SGS}}\rangle$ and the spatial area of each subpartition are also listed.

4.3. Statistical flow structures related to $\varPi _{{SGS}}$

Conditional-averaged flow structures associated with interscale energy flux under different velocity gradient conditions can be approximated by linear stochastic estimation (LSE) (Adrian & Moin Reference Adrian and Moin1988), i.e.

(4.1)\begin{gather} {R^{\gamma}_{{\varPi_{{SGS}}},\boldsymbol{u}}}\left( {{r_x},{r_z}} \right) = \frac{{\left\langle {\boldsymbol{u}\left( {x + {r_x},z + {r_z}} \right) {\varPi_{SGS}^{\gamma}} } \right\rangle }}{\sigma _{\varPi_{{{SGS}}}^{\gamma}}^2}, \end{gather}

(4.2)\begin{gather} \langle \boldsymbol{u}^{\gamma}_{{SGS}} \rangle = R_{{\varPi_{{{SGS}}}},\boldsymbol{u}}^{\gamma}\varPi_{{{SGS}}}^{\gamma}. \end{gather}

In (4.1), $R^{\gamma }_{{\varPi _{{SGS}}},\boldsymbol {u}}$ is the cross-correlation coefficient between $\boldsymbol {u}$ and ${\varPi _{SGS}^{\gamma }}$, $\boldsymbol {u}=[u,w]$ are in-plane velocity vectors without scale filtering, and ${\varPi _{SGS}^{\gamma }}$ denote those events whose velocity gradients belong to one of the subpartitions of the ${{\partial \tilde {u}}/{\partial x}}$–${{\partial \tilde {u}}/{\partial z}}$ phase plane (as shown in figure 7c and table 2), i.e. $\gamma \in \{F1\text {--} F3, B1\text {--}B2\}$. In (4.2), $\langle \boldsymbol {u}^{\gamma }_{{SGS}} \rangle$ is the averaged velocity field subject to the considered condition ${\varPi _{SGS}^{\gamma }}$. It differs from $R^{\gamma }_{{\varPi _{{SGS}}},\boldsymbol {u}}$ by a factor of ${\varPi _{SGS}^{\gamma }}$. Without loss of generality, the magnitude of $\varPi _{{{SGS}}}^{\gamma }$ in (4.2) is set to be ${\pm }1$ for forward or backward scatter events. Moreover, ${\pm }\varPi _{{SGS}}$ events are not differentiated in LSE since the net energy flux is mainly concerned here.

Figure 8 shows the statistical flow structures $\langle \boldsymbol {u}^{\gamma }_{{SGS}} \rangle$ belonging to each of five subpartitions ($F1\text {--}F3$ and $B1\text {--}B2$) at $y/\delta \approx 0.12$ in case $E3$. The averaged flow structures in $F1$ and $B2$ (figure 8a,d) form a pair of mirror images that are anti-symmetric to each other. Both structures are characterized as a pair of streamwise-aligned LSMs with opposite signs of $u$ component fluctuating velocity. The probing position locates at the interface between the upstream and downstream LSMs where $|{\partial \tilde {u}}/{\partial x}|$ is the local maximum. Note that the pattern of $\langle \boldsymbol {u}^{F1}_{{SGS}} \rangle$ is frequently observed in instantaneous flow fields (regions in solid cycles in figure 5g). It is consistent with existing observation in the $x$–$y$ plane, i.e. active scale energy transfer events prefer to reside on the interface of Q2 and Q4 structures (Piomelli et al. Reference Piomelli, Yu and Adrian1996; Carper & Porté-Agel Reference Carper and Porté-Agel2004; Natrajan & Christensen Reference Natrajan and Christensen2006; Dong et al. Reference Dong, Huang, Yuan and Lozano-Durán2020), when regarding the observed pattern of $\langle \boldsymbol {u}^{F1}_{{SGS}} \rangle$ as intersections of large-scale Q2–Q4 structures at a wall-parallel plane.

Figure 8. Statistical flow structures $\langle \boldsymbol {u}^{\gamma }_{{SGS}} \rangle$ obtained by LSE with flow conditions in different subpartitions in the phase plane of $({{\partial \tilde {u}}/{\partial x}}, {{\partial \tilde {u}}/{\partial z}})$ (see table 2) at $y/\delta \approx 0.12$ in case $E3$: $(a)$ $F1$; $(b)$ $B1$; $(c)$ $F2$; $(d)$ $B2$; $(e)$ $F3$. Vectors are normalized to be unit length for a better visualization. Here, $\varPi _{{{SGS}}}^{\gamma }$ in (4.2) is set to be 1 in panel (a,c,e) and $-1$ in panel (b,d). The contour levels are from $-1$ to 1 with a gap of 0.2.

What has not been observed in previous studies is the statistical structures associated with $\varPi _{{SGS}}$ events in $F2$ and $F3$. As shown in figure 8(c,e), the local velocity gradient contributing to forward interscale energy flux is seen to be strongly related to meandering-like distortion of one low-momentum LSM, on the flank of which another high-momentum LSM also presents a synchronized sinusoidal wavy pattern. High magnitude of ${{\partial \tilde {u}}/{\partial z}}$ is generated at the shearing interface between low- and high-momentum LSMs, while the ${{\partial \tilde {u}}/{\partial x}}$ component is attributed to these structures’ wavy geometries, which are comparably mild in the case of $F3$ due to the imposed probing condition of weak ${{\partial \tilde {u}}/{\partial x}}$ there. The instantaneous counterparts of these statistical structures are frequently seen in figure 5(a,c,e,g) (as arrow-indicated regions). Note that only those events with positive sign of ${\partial \tilde {u}}/{\partial z}$ are included in the calculation of $R^{\gamma }_{{\varPi _{{SGS}}},\boldsymbol {u}}$ and $\langle \boldsymbol {u}^{\gamma }_{{SGS}} \rangle$. The LSE in the negative ${{\partial \tilde {u}}/{\partial z}}$ side of $F2$ and $F3$ yields statistical structures that mirror figure 8(c,e) approximately $r_z=0$. It is stressed that such an operation breaks the spanwise symmetry, which will smear out the wavy pattern of individual LSMs to form a straight strip pattern in the unconditional cross-correlation map, as has been widely seen in previous research (Tomkins & Adrian Reference Tomkins and Adrian2003; Hutchins & Marusic Reference Hutchins and Marusic2007). To our knowledge, the meandering feature of LSMs is an important factor to create forward scatter, since the subpartitions of $F2$ and $F3$ jointly contribute approximately $80\,\%$ of the net $\langle \varPi _{{SGS}}\rangle$ (as shown in table 2).

Finally, the statistical structure in $B1$ is characterized as a high-momentum LSM flanked by two low-momentum ones. The flow pattern around the probing point is rather complicated. Recalling that in $B1$, the in-plane velocity gradient is insignificant, while $\sum \varPi ^+_{{SGS}}$ is comparable to $\sum \varPi ^-_{{SGS}}$ (shown in figure 7a,b). It can be inferred that the complicated pattern of $\langle \boldsymbol {u}^{B1}_{{SGS}} \rangle$ is related to the staggered $\varPi _{{SGS}}$ clusters shown in figure 5(a,c,e,g) (in bold dashed circles), in which the mutual cancellation between forward and backward scatter events results in less than 1 % net contribution to $\langle \varPi _{{SGS}}\rangle$.

5.1. Visualization and cumulative spectrum of $\varPi _{d}$

Instantaneous snapshots shown in figure 5(b,d,f,h) visualizes a spatial coherence between LSMs and $\varPi _{{d}}$ events. Namely, pairs of ${\pm }\varPi _{{d}}$ events tend to reside inside LSMs where the sinusoidal wavy pattern is prominent (see elliptical regions in figure 5b,d,f,h for instance). Such a distribution implies the role of velocity gradient associated with LSMs on $\varPi _{{d}}$ events.

To testify to this visualization, figure 9(a–c) shows the profiles of cumulative $\varPi _{{d}}$ as a function of the condition of $\tilde {u}$, $\partial \tilde {u}/\partial z$ and $\partial \tilde {u}/\partial x$, respectively. In the profiles of $\sum _{\tilde {u}} \varPi ^{\pm }_{{d}}(\tilde {u})$, two peaks appear on both sides of $\tilde {u}$ axis, in distinct contrast to the observation in figure 6 that $\sum _{\tilde {u}} \varPi ^{\pm }_{{SGS}}(\tilde {u})$ peaks around $\tilde {u}=0$. It suggests that $\pm \varPi _{{d}}$ events prefer to align with the centre of LSMs, which can be further inferred by the symmetric profiles of $\sum _{{\partial \tilde {u}}/{\partial z}} \varPi ^{\pm }_{{d}}({\partial \tilde {u}}/{\partial z)}$ peaking at ${\partial \tilde {u}}/{\partial z}=0$ (in figure 9b). On the other hand, the shape of $\sum _{{\partial \tilde {u}}/{\partial z}} \varPi _{{d}}({\partial \tilde {u}}/{\partial z})$ and $\sum _{\tilde {u}} \varPi _{{d}}(\tilde {u})$ jointly suggests that in general, ‘resolved’ TKE carried by LSMs are transported from the central region (with intense $|\tilde {u}|$ but weak ${\partial \tilde {u}}/{\partial z}$) to the periphery part (with mild $|\tilde {u}|$ but strong ${\partial \tilde {u}}/{\partial z}$). Lastly, figure 9(c) shows that the sign of ${\partial \tilde {u}}/{\partial x}$ plays a decisive role in determining the direction of the net flux, i.e. positive ${\partial \tilde {u}}/{\partial x}$ corresponds to positive $\sum _{{\partial \tilde {u}}/{\partial z}} \varPi _{{d}}$ and vice versa. This aspect will be addressed in detail in § 5.2.

Figure 9. The 1-D cumulative spectrum of in-plane energy flux $\sum _{\chi }\varPi _{{{d}}}$ with respect to flow conditions of $\chi$ in case $E3$. $(a)$ $\chi =\tilde {u}$, $(b)$ $\chi =\partial \tilde {u}/{\partial z}$ and $(c)$ $\chi ={\partial \tilde {u}}/{\partial x}$. Solid line, $y/\delta \approx 0.03$; dashed line, $y/\delta \approx 0.06$; dot-dashed line, $y/\delta \approx 0.12$; and dotted line, $y/\delta \approx 0.24$. The group of profiles marked by $\varPi ^+_{{d}}$, $\varPi ^-_{{d}}$ or $\varPi _{{d}}$ represent the cumulative contribution from positive, negative and all of the events, respectively.

The 2-D cumulative spectra of $\sum \varPi ^{\pm }_{{d}}$ and $\sum \varPi _{{d}}$ in the phase plane of $(\tilde {u}, \, {\partial \tilde {u}}/{\partial x})$ are shown in figure 10(a–c), respectively. Note that ${\partial \tilde {u}}/{\partial z}$ is excluded from the probing condition due to its insignificant correlation with $\varPi _{{d}}$ (shown in figure 9b). Both $\sum \varPi ^+_{{d}}$ and $\sum \varPi ^-_{{d}}$ present a ‘cashew nut’ shape in the $\tilde {u}, \, {\partial \tilde {u}}$-${\partial x}$ plane but bend towards opposite side of the ${\partial \tilde {u}}/{\partial x}$ axis. They jointly lead to a fragmented pattern of the net flux spectrum $\sum \varPi _{{d}}$ with two losing-energy subpartitions ($L1 \text {--} L2$) and three gaining-energy ones ($G1 \text {--} G3$) (as illustrated in figure 10c). Except for $G3$ that has minor $\tilde {u}$, the rest of the four subpartitions occupy four corners of the phase plane, indicating the mutual effect of ${\partial \tilde {u}}/{\partial x}$ and ${\tilde {u}}$ on the net in-plane energy flux. Table 3 further shows that $L1$ and $G2$ have more intense contributions to $\langle \varPi _{{d}} \rangle$ than their anti-symmetric counterparts of $L2$ and $G1$; meanwhile, the contribution from $G3$ is non-negligible.

Figure 10. The 2-D cumulative spectrum of $(a)$ positive in-plane energy flux events $\sum \varPi ^+_{{d}}$, $(b)$ negative events $\sum \varPi ^-_{{d}}$ and $(c)$ all of the events $\sum \varPi _{{d}}$ in the phase space of $(\tilde {u},{\partial \tilde {u}}/{\partial x})$ at $y/\delta \approx 0.12$ in case $E3$. Each spectrum is normalized by its peak value. The subpartitions marked in panel $(c)$ are defined in table 3.

Table 3. Subpartitions in the phase space of $({\partial \tilde {u}}/{\partial x}, {\tilde {u}})$ at $y/\delta \approx 0.12$ in case $E3$. The percentage of the contribution to $\langle \varPi _{{d}}\rangle$ and the spatial area of each subpartition are also listed.

5.2. Statistical structures related to $\varPi _{{d}}$

The LSE is again invoked to estimate conditional-averaged flow structures ${\langle \boldsymbol {u}^{\gamma }_{{d}}\rangle }$ that are related to ${\varPi ^{\gamma }_{{d}}}$ events with $\gamma$ belonging to five subpartitions ($L1\text {--}L2$, $G1\text {--} G3$) defined in figure 10(c) and table 3. The $\langle {\boldsymbol {u}^{\gamma }_{{d}}\rangle }$ are calculated as

(5.1)\begin{gather} {R^{\gamma}_{{\varPi_{{d}}},\boldsymbol{u}}}\left( {{r_x},{r_z}} \right) = \frac{{\left\langle {\boldsymbol{u}\left( {x + {r_x},z + {r_z}} \right) {\varPi_{d}^{\gamma}} } \right\rangle }}{\sigma _{\varPi_{{{d}}}^{\gamma}}^2}, \end{gather}

(5.2)\begin{gather} \langle {\boldsymbol{u}^{\gamma}_{{d}}\rangle} = R_{{\varPi_{{{d}}}},\boldsymbol{u}}^{\gamma}\varPi_{{{d}}}^{\gamma}. \end{gather}

Similar to § 4.3, $\pm \varPi _{d}$ events are not differentiated in the LSE. To break the spanwise symmetry, only ${\varPi _{{d}}}$ events with ${+\partial \tilde {u}}/{\partial z}$ (or ${-\partial \tilde {u}}/{\partial z}$) are taken into account in ${\langle \boldsymbol {u}^{{L1|L2}}_{{d}}\rangle }$ (or ${\langle \boldsymbol {u}^{{G1|G2|G3}}_{{d}}\rangle }$). The reason for such a consideration will be explained later.

Figure 11 shows $\langle {\boldsymbol {u}^{\gamma }_{{d}}\rangle }$ in all the five subpartitions. The magnitude of $\varPi _{{d}}^{\gamma }$ in (5.2) is set to be 1 for $L1\text {--}L2$ and $-1$ for $G1\text {--}G3$. For the subpartition of $G3$, the yielded ${\langle \boldsymbol {u}^{G3}_{{d}}\rangle }$ (in figure 11e) are characterized as a pair of low- and high-momentum LSMs that are aligned spanwise with each other. The probing point (at $r_x=0$, $r_z=0$) locates at their interface. In contrast, the rest of the four subpartitions ($L1\text {--}L2$ and $G1\text {--}G2$) have statistical structures resembling one single LSM with either high- or low-momentum (in figure 11a–d). The probing point now falls inside them.

Figure 11. Statistical flow structures $\langle \boldsymbol {u}^{\gamma }_{{d}} \rangle$ obtained by LSE with flow conditions in different subpartitions in the phase plane of $({\partial \tilde {u}}/{\partial x}, {\tilde {u}})$ (see table 2) at $y/\delta \approx 0.12$ in case $E3$: $(a)$ $L1$; $(b)$ $G1$; $(c)$ $L2$; $(d)$ $G2$; $(e)$ $G3$. Vectors are normalized to be unit length for a better visualization. Here, $\varPi _{{{d}}}^{\gamma }$ in (5.2) is set to be 1 in panel (a,c) and $-1$ in panel (b,d,e). The contour levels are from $-1$ to 1 with a gap of 0.2.

As has been mentioned in § 5.1, an interesting observation is that the direction of the in-plane energy flux is regulated not by the sign of $\tilde {u}$ at the probing position but by the sign of local ${\partial \tilde {u}}/{\partial x}$. For example, the probed ${\varPi _{{d}}}$ event in $L1$ (or $L2$) locates at the upstream of the centre of a high-momentum LSM (or the downstream of the centre of a low-momentum LSM) to form a positive local ${\partial \tilde {u}}/{\partial x}$. This velocity gradient condition guarantees that no matter what the sign of LSM, $\tilde {u}^2$ at the probing point will always experience a decrease as this LSM convects along time (note that a negative LSM convects upstream if compared with the ambient fluid). The same explanation applies for the phase shift of the probing point from the centre of the identified LSMs in the energy-gaining subpartitions of $G1$ and $G2$. As for $\langle \boldsymbol {u}^{G3}_{{d}} \rangle$ in $G3$, the flow pattern is a bit more complicated, i.e. the local $\tilde {u}$ at the probing point is small while the local shear is intense. The particularity of the in-plane energy transportation in this subpartition will be addressed later.

For all five subpartitions, the slanted pattern of the statistical flow structures around the probing point is attributed to the imposed positive (or negative) ${\partial \tilde {u}}/{\partial z}$ condition. Interestingly, $\langle \boldsymbol {u}^{L1}_{{d}} \rangle$ and $\langle \boldsymbol {u}^{G1}_{{d}} \rangle$ (in figure 11a,b) are anti-symmetric with each other to form a conjugated pair, so do $\langle \boldsymbol {u}^{L2}_{{d}} \rangle$ and $\langle \boldsymbol {u}^{G2}_{{d}} \rangle$ (in figure 11c,d). On considering the different signs of both ${\partial \tilde {u}}/{\partial x}$ and ${\partial \tilde {u}}/{\partial z}$ in the imposed probing conditions, such pairs of $\langle \boldsymbol {u}^{\gamma }_{{d}} \rangle$ structures actually characterize one pair of ${\pm }\varPi _{d}$ events locating inside the wavy part of one instantaneous LSM, as has been visualized in figure 5(b,d,f,h) (in bold elliptical regions). To give statistical evidence, the conditional averaged $\langle \varPi ^{\gamma }_{{d}} \rangle$ structures are estimated by LSE as

(5.3)\begin{gather} {R^{\gamma}_{{\varPi_{{d}}},{\varPi_{{d}}}}}\left( {{r_x},{r_z}} \right) = \frac{{\left\langle {{\varPi_{{d}}}\left( {x + {r_x},z + {r_z}} \right) {\varPi_{d}^{\gamma}} } \right\rangle }}{\sigma _{\varPi_{{{d}}}^{\gamma}}^2} , \end{gather}

(5.4)\begin{gather} {\langle\varPi_{{d}}^{\gamma}\rangle} = R_{{\varPi_{{{d}}}},{\varPi_{{d}}}}^{\gamma}\varPi_{{{d}}}^{\gamma}. \end{gather}

Like (5.1) and (5.2), $\gamma \in \{{L1\text {--}L2, G1\text {--}G3}\}$. An additional condition of either ${\partial \tilde {u}}/{\partial z}>0$ for $L1\text {--}L2$ or ${\partial \tilde {u}}/{\partial z}<0$ for $G1\text {--}G3$ is also implicitly imposed. For generalization, $\varPi ^{\gamma }_{d}$ at the probing point is set as ${\pm }1$ for $L1\text {--}L2$ and $G1\text {--}G3$.

Figure 12 shows the yielded $\langle \varPi _{{d}}^{\gamma }\rangle$ map. For all subpartitions except for $G3$, an additional patch of $\langle \varPi _{{d}}^{\gamma }\rangle$ with opposite sign is always presented on one flank of the central auto-correlation patch. Both patches are slanted towards the same $z$ direction. Their asymmetric strength and size is attributed to the gradual loss of the coherency with the increase of the distance from the probing point. Note that $\langle \varPi _{{d}}^{L1}\rangle$ resembles $\langle \varPi _{{d}}^{G1}\rangle$ with only a flip of the sign (in figure 12a,b), so do $\langle \varPi _{{d}}^{L2}\rangle$ and $\langle \varPi _{{d}}^{G2}\rangle$ (in figure 12c,d). Such a resemblance reminds us that they are actually a pair of mirror images of one physical structure, whose slanted pattern is attributed to the wavy pattern of LSMs that forms the necessary velocity gradient condition contributing to $\varPi _{{d}}$ events.

Figure 12. Statistical structures of $\langle \varPi _{{d}}^{\gamma }\rangle$ corresponding to different condition subpartitions in the space of $({\partial \tilde {u}}/{\partial x}, {\tilde {u}})$ (see table 2): $(a)$ $L1$; $(b)$ $G1$; $(c)$ $L2$; $(d)$ $G2$; $(e)$ $G3$. The event $\varPi _{{{d}}}^{\gamma }$ is 1 for panel (a,c) and $-1$ for panel (b,d,e). The contour levels are $-1$ to 1 with a gap of 0.2.

It is interesting to see that $\langle \varPi _{{d}}^{G3}\rangle$ in figure 12(e) is symmetric with respect to $r_z=0$ even under the probing condition of ${\partial \tilde {u}}/{\partial z}<0$. This is in distinct contrast to the wavy pattern of the $\langle \boldsymbol {u}^{G3}_{{d}} \rangle$ structures in figure 11(e), and suggests a decoupling of $\varPi _{d}$ events from the spanwise velocity condition at the boundary of LSMs. Combing with $\langle {\boldsymbol {u}}_{{d}}^{G3}\rangle$, the pattern of three side-by-side $\langle \varPi _{{d}}^{G3}\rangle$ patches suggests that the energy is transported from both high- and low-momentum LSMs to their outboard peripheries. It can be inferred that vortical motions, instead of the effect of streamwise convection, contribute to the spatial energy transportation around the boundary of LSMs. Due to the lack of information about the wall-normal dimension, the details of this particular energy transfer scenario are still not clear.