2.1. Validation of the DNS boundary layer

Figure 1 shows the inner-scaling mean velocity profile, turbulent intensities, Reynolds shear stress and the $k$-budget (where $k = \langle u_i u_i \rangle /2$ is turbulent kinetic energy) for ${Re}_\tau \simeq 2020$. The data is compared and validated against the DNS of the TBL from Sillero et al. (Reference Sillero, Jiménez and Moser2013) at ${Re}_\tau \simeq 1998$, showing very good agreement. The one-dimensional spanwise premultiplied velocity spectra are plotted in figure 2 for ${Re}_\tau \simeq 2020$. Overall, the figure is consistent with the consensus in the literature: (i) the inner peak at ${\lambda _z}^{+} \simeq 100$, $y^{+} \simeq 15$ characterises the near-wall structures (Kline et al. Reference Kline, Reynolds, Schraub and Runstadler1967; Smith & Metzler Reference Smith and Metzler1983; Tomkins & Adrian Reference Tomkins and Adrian2003); (ii) the outer peak at ${\lambda _z} \simeq 0.7\delta , y \simeq 0.15\delta$ characterises the very large-scale motions (Kim & Adrian Reference Kim and Adrian1999; Tomkins & Adrian Reference Tomkins and Adrian2005; Guala, Hommema & Adrian Reference Guala, Hommema and Adrian2006; Hutchins & Marusic Reference Hutchins and Marusic2007a; Monty et al. Reference Monty, Hutchins, NG, Marusic and Chong2009); (iii) the two peaks in the premultiplied (either $k_x$ or $k_z$) $E_{uu}$ and $E_{-uv}$ spectra correspond to the small-scale and large-scale structures that contain the streamwise turbulent kinetic energy and Reynolds shear stress, which are not observed in the $E_{vv}$ and $E_{ww}$ spectra (Lee & Moser Reference Lee and Moser2015); (iv) the penetration (through the dashed-line linear ridge $\lambda _z = 8y$) of the large-scale contribution onto the near-wall region in the $E_{uu}$ and $E_{ww}$ but are not observed in $E_{vv}$ and $E_{-uv}$ owing to the impermeability of the wall (Hutchins & Marusic Reference Hutchins and Marusic2007a; Lee & Moser Reference Lee and Moser2015; Hwang Reference Hwang2016).

Figure 1. The inner-scaling of $(a)$ mean velocity $U^{+}$, $(b)$ turbulence intensities ${{\langle ({u_i u_i})^{1/2} \rangle }^{+}}$ and Reynolds shear stress ${\langle uv \rangle }^{+}$, (c) $k$-balance ${\textrm {D} k^{+}}/{\textrm {D} t}=0$. Symbol ($\circ$) for present DNS TBL at ${Re}_\tau \simeq 2020$; (– – –) Sillero, Jiménez & Moser (Reference Sillero, Jiménez and Moser2013) at ${Re}_\tau \simeq 1998$.

Figure 2. The one-dimensional premultiplied spectra at ${Re}_\tau \simeq 2020$: (a) $k_z E_{uu}^{+}$; (b) $k_z E_{vv}^{+}$; (c) $k_z E_{ww}^{+}$; (d) $k_z E_{-uv}^{+}$. Symbol $(\times )$ marks the near-wall streaks at $y^{+} \simeq 15$ with average characteristic length scale $\lambda _z^{+} \simeq 100$ and the large-scale structures at $y \simeq 0.15 \,\delta$ with $\lambda _z \simeq 0.7 \,\delta$.

3.1. Interscale transport equations for boundary layers

An analytical approach for large-scale and small-scale interactions, based on the spectral analysis, was proposed by Kawata & Alfredsson (Reference Kawata and Alfredsson2018) for plane Couette flow. Here, we adopt the approach and extend it to TBLs with zero pressure gradient, as described here. A scale decomposition based on the spanwise Fourier mode and a cutoff length scale $\lambda _z = 2{\rm \pi} /k_z$ is used to separate the velocity fluctuation into large-scale $(u'_i)$ and small-scale components $(u''_i)$:

(3.1)\begin{equation} {u_i}(\boldsymbol{x},t) = u_i'(k_z,\boldsymbol{x},t)+u_i''(k_z,\boldsymbol{x},t).\end{equation}

The large-scale and small-scale components satisfy

(3.2)\begin{equation} \langle u_{i}u_{j} \rangle(y) = \langle u_{i}'u_{j}' \rangle (k_z,y)+ \langle u_{i}''u_{j}'' \rangle(k_z,y).\end{equation}

The governing equations are the incompressible Navier–Stokes equations

(3.3)\begin{equation} {{\partial_t}(U_i + u_i)} + {(U_l + u_l){\partial_{x_l}}(U_i+u_i)} ={-}{\frac{1}{\rho}{\partial_{x_i}}(P+p)}+{\nu{\rm \Delta}(U_i+u_i)}.\end{equation}

Transport equations for large-scale and small-scale velocity fluctuations are obtained by introducing the Reynolds decomposition and scale decomposition (3.1) into (3.3). By multiplying the large-scale components $u'_j$ and taking the average, then interchanging $i$ and $j$ and summing up the resulting equation, the transport equation for large-scale Reynolds stress $\langle u'_i u'_j \rangle$ is obtained as follows:

(3.4)\begin{equation} \frac{\textrm{D} \langle u_i'u_j' \rangle}{\textrm{D} t} = P_{ij}^{'}+D_{ij}^{t,'}+\varPhi_{ij}^{'}+D_{ij}^{\nu,'}-\epsilon_{ij}^{'}-Tr_{ij},\end{equation}

and the small-scale transport equation is obtained in a similar manner:

(3.5)\begin{equation} \frac{\textrm{D} \langle u_i''u_j'' \rangle}{\textrm{D} t} = P^{''}_{ij}+D_{ij}^{t,''}+\varPhi_{ij}^{''}+D_{ij}^{\nu,''}-\epsilon_{ij}^{''}+Tr_{ij}. \end{equation}

The terms in the large-scale (3.4) are

(3.6) \begin{equation} \left. \begin{array}{c@{}} \displaystyle P_{ij}^{'} ={-}\langle u_j' \,u_l' \rangle {\partial_{x_l} U_i} - \langle u_i' \,u_l' \rangle {\partial_{x_l} U_j},\quad \varPhi_{ij}^{'} ={-}\tfrac{1}{\rho}[ \langle {u_j' \partial_{x_i} p} \rangle + \langle {u_i' \partial_{x_j} p} \rangle], \\ D_{ij}^{\nu,'} = {\nu {\rm \Delta} \langle u_i'u_j' \rangle}, \quad \epsilon_{ij}^{'} = 2 \nu \langle {\partial_{x_l} u_j'}{\partial_{x_l} u_i'}\rangle,\\ \displaystyle D_{ij}^{t,'} ={-}{\partial_{x_l} (\langle u_i'u_j'u_l'\rangle +\langle u_i'u_j'u_l''\rangle +\langle u_i''u_j'u_l''\rangle +\langle u_i'u_j''u_l''\rangle )},\\ \displaystyle Tr_{ij} = {\langle u_j'u_l'{\partial_{x_l} u_i''} \rangle + \langle u_i'u_l'{\partial_{x_l} u_j''} \rangle - \langle u_i''u_l''{\partial_{x_l} u_j'} \rangle - \langle u_j''u_l''{\partial_{x_l} u_i'} \rangle}, \end{array} \right\},\end{equation}

where are, on the right-hand side of (3.4), the large-scale parts of production, turbulent transport, pressure transport, viscous transport, dissipation and interscale flux (Kawata & Alfredsson Reference Kawata and Alfredsson2018). All the terms in the small-scale (3.5) are obtained by interchanging the superscripts $'$ and $''$ in (3.6). Note that $Tr_{ij}$ in (3.4) and (3.5) are of opposite signs, representing a local transfer of $\langle u_i u_j \rangle$ between the large-scale $\langle u_i' u_j' \rangle$ and small-scale $\langle u_i'' u_j'' \rangle$ at the cutoff wavelength $\lambda _z$ (or wavenumber $k_z$). The other terms in the (3.4) are the decoupling of large-scale components from their small-scale components. The transport equations for $\langle u_i u_j \rangle$ can be obtained as follows:

(3.7)\begin{equation} \frac{\textrm{D} \langle u_i u_j \rangle}{\textrm{D} t} = \frac{\textrm{D} \langle u_i' u_j' \rangle}{\textrm{D} t} + \frac{\textrm{D} \langle u_i''u_j'' \rangle}{\textrm{D} t}.\end{equation}

The interscale transport equation for the Reynolds stress tensor $E_{ij}(k_z,y)$ can be obtained by differentiating (3.4) with respect to the cutoff wavenumber $k_z$,

(3.8)\begin{equation} \frac{\textrm{D} E_{ij}}{\textrm{D} t}(k_z,y) = pr_{ij}+d_{ij}^{t}+\phi_{ij}+d_{ij}^{\nu}-\varepsilon_{ij}+tr_{ij}, \quad \forall \, k_z,\end{equation}

where the large-scale contribution $\langle u_i' u_j' \rangle$ satisfies

(3.9)\begin{equation} \langle u_i' u_j' \rangle = \int_{0}^{k_z} E_{ij}(\hat{k},y) \, {\textrm{d}} \hat{k}, \quad \forall \,k_z,\end{equation}

where the terms in (3.8) are

(3.10)\begin{equation} \left. \begin{array}{c@{}} \displaystyle pr_{ij} ={-}E_{jl} {\partial_{x_l} U_i} - E_{il} {\partial_{x_l} U_j},\quad \phi_{ij} ={-}\tfrac{1}{\rho}{\partial_{k_z}}[\langle u_j'{\partial_{x_i} p} \rangle + \langle u_i'{\partial_{x_j} p} \rangle], \\ d_{ij}^{\nu} = {\nu {\rm \Delta} E_{ij}}, \quad \varepsilon_{ij} = 2 \nu {\partial_{k_z}}\langle {\partial_{x_l} u_j'} {\partial_{x_l} u_i'} \rangle, \quad d_{ij}^{t} = {\partial_{k_z} D_{ij}^{t,'}}, \quad tr_{ij} ={-}{\partial_{k_z} Tr_{ij}}. \end{array} \right\}\end{equation}

3.2. One-dimensional spatial and scale fluxes

Note that when (3.8) is integrated over the wavenumber $k_z\to \infty$, we have

(3.11)\begin{equation} \int_{0}^{\infty} \frac{\textrm{D} E_{ij}}{\textrm{D} t}(\hat{k},y) \,{\textrm{d}} \hat{k} = \frac{\textrm{D} \langle u_i u_j \rangle}{\textrm{D} t} = P_{ij}+D_{ij}^{t}+\varPhi_{ij}+D_{ij}^{\nu}-\epsilon_{ij}\end{equation}

and

(3.12)\begin{equation} \int_{0}^{\infty} tr_{ij}(\hat{k},y) \,{\textrm{d}} \hat{k} = 0, \quad \forall \,y,\end{equation}

where (3.11) is the Reynolds stress transport equation (3.7), and (3.12) indicates that $Tr_{ij}$ represents the scale flux along the length scale $k_z$, and $tr_{ij}$ represents the interscale transport at each length scale. In addition, the turbulent term $d_{ij}^{t}$ in (3.10) can be written as

(3.13)\begin{equation} d_{ij}^{t} ={-}{\partial_{x_l}}[{\partial_{k_z}} ( \langle {u_i'u_j'u_l'}\rangle + \langle {u_i'u_j'u_l''} \rangle + \langle {u_i''u_j'u_l''} \rangle + \langle {u_i'u_j''u_l''} \rangle )],\end{equation}

and the pressure term $\phi _{ij}$ in (3.8) can be decomposed as

(3.14)\begin{equation} \phi_{ij} = d_{ij}^{p} + \varPi_{ij},\end{equation}

where $d_{ij}^{p}$ and $\varPi _{ij}$ are the pressure transport and pressure strain, respectively,

(3.15a,b)\begin{equation} d_{ij}^{p} ={-}\frac{1}{\rho}{\partial_{k_z}}[{\partial_{x_i} \langle p u_j'\rangle} + {\partial_{x_j} \langle p u_i'\rangle}],\quad \varPi_{ij} = \frac{1}{\rho}{\partial_{k_z}}[\langle p ({\partial_{x_i} u_j'}+{\partial_{x_j} u_i'}) \rangle].\end{equation}

Here we adopt the approximations for boundary layers with the parallel-flow assumption (i.e. $\partial _y \langle \cdot \rangle \gg \partial _x \langle \cdot \rangle$). Then $d^{t}_{ij}$ (3.13) and $d^{p}_{ij}$ in (3.15a,b) can be expressed as

(3.16)$$\begin{gather} d_{ij}^{t}(k_z,{y}) \sim{-}{\partial_y}[{\partial_{k_z}} ( \langle {u_i'u_j'v'}\rangle + \langle {u_i'u_j'v''}\rangle+\langle{u_i''u_j'v''}\rangle+\langle{u_i'u_j''v''} \rangle )] ={-}{\partial_y T_{ij}} |_{{y}}, \end{gather}$$

(3.17)$$\begin{gather}d_{ij}^{p}(k_z,{y}) \sim{-}\frac{1}{\rho}{\partial_y}[{\partial_{k_z}} (\langle p u_j'\rangle \delta^{{\ast}}_{i2} + \langle p u_i'\rangle \delta^{{\ast}}_{j2} ) ] ={-}{\partial_y R_{ij}}|_{{y}}, \end{gather}$$

where $\delta ^{\ast }$ is the Kronecker delta. Here (3.16) and (3.17) indicate that $T_{ij}$ and $R_{ij}$ are the spatial fluxes along the wall-normal direction. Notably, $T_{ij}$ and $R_{ij}$, together with $Tr_{ij}$ (3.10), are the spatial fluxes and scale flux, indicating the directions of the energy flows. The turbulent kinetic energy scale flux $Tr_{k}$ and the wall-normal flux $T_{k}$ are shown in figure 3. The positive flux (black contour lines) indicates that the energy flows along the positive $k_z$-axis direction (where the reversed direction is in the $\lambda _z$-axis) for the scale flux $Tr_k$, or along the positive $y$-axis direction for the wall-normal flux $T_k$. The direction is indicated by the black arrow. The negative flux (white contour lines) indicates energy flows in the opposite direction (indicated by the white arrow). The boundary that separates the positive and negative flux can be interpreted as the driving scale of such energy transport. The turbulent kinetic energy balance can be expressed as

(3.18)\begin{equation} pr_{k}+d_{k}^{\nu}-\varepsilon_{k} = {\partial_y}({T_{k}+R_{k}})+{\partial_{k_z} (Tr_{k})}, \quad \forall \,k_z,\end{equation}

where the pressure strain spectrum $\varPi _{k} = \varPi _{ii}/2 = 0$ transfers energy between the velocity components through the continuity equation. In (3.18), $pr_k$ and $\varepsilon _k$ represent the source and sink of the turbulent kinetic energy, respectively, and $d_k^{t,p} = -{\partial _y (T_k+R_k)}$ represent the spatial transport along the wall-normal direction in the physical space and $tr_k = -{\partial _{k_z} (Tr_k)}$ represents the interscale transport of turbulent kinetic energy in the space of scales. In addition, the term $d_k^{\nu }$ is relatively small and represents the viscous transport at the near-wall region. The Reynolds shear stress balance can be expressed as

(3.19)\begin{equation} pr_{{-}uv}+\varPi_{{-}uv}+d_{{-}uv}^{\nu}- \varepsilon_{{-}uv} = {\partial_y}\,({T_{{-}uv}+R_{{-}uv}})+{\partial_{k_z} (Tr_{{-}uv})}, \quad \forall \,k_z.\end{equation}

Here, as expected, $\varepsilon _{-uv}$ and $d_{-uv}^{\nu }$ are relatively small because viscosity plays little part in Reynolds shear stress dynamics (Mansour, Kim & Moin Reference Mansour, Kim and Moin1988). Here $pr_{-uv}$ represents the source, the pressure strain $\varPi _{-uv}$ represents the sink and suppresses the produced Reynolds shear stress by its strain-rate $({\partial _y u'}+{\partial _x v'})$, and $d_{-uv}^{t,p}$ and $tr_{-uv}$ represent the dominant wall-normal and interscale transport of Reynolds shear stress in physical space and scale, respectively.

Figure 3. Contour maps of premultiplied $(a)$ kinetic energy scale flux, $Tr_{k}$, and $(b)$ wall-normal turbulent flux, $T_k$, for ${Re}_\tau \simeq 2020$, along with isolines showing the positive flux (black) and negative flux (white). The lines are isolines of constant flux with contour levels of (a) $0.005[0.01]0.1$ (black) and $-0.1[0.005]-0.005$ (white) and (b) $1[80]800$ (black) and $-20[2]-0.1$ (white). Black and white arrows indicate the direction of the fluxes.

5.1. Filtering procedure

Filtering of data at suitable spanwise cutoff wavelengths allows us to distinguish the large-scale structures associated with the regions of positive energy density and small-scale structures associated with regions of negative energy where energy is distributed from smaller scales (negative) to larger scales (positive), as shown in figure 4(b). Therefore, spanwise spectral filters were applied to the present dataset in Fourier space and the filtered large-scale and small-scale flow fields in physical space were analysed based on quadrant analysis, with emphasis on the large-scale and small-scale ejection and sweep events. Under one-dimensional spanwise filtering, the information about the length scales of the structures related to the interscale process is retained. In the following sections, the spanwise and streamwise length scales and wall-normal extents of the large-scale and small-scale ejection and sweep events are presented. In the streamwise direction, Taylor's hypothesis is utilised on the temporal dataset $(y,z)$ to evaluate the streamwise length scales of the structures in the filtered fields. In addition, the interscale transport of Reynolds shear stress is tracked in time and the time-resolved information is presented.

The (high-pass and low-pass) filter is defined by the spanwise cutoff wavelength (denoted by $\varLambda _z$ and wavenumber $k_z = 2{\rm \pi} /\varLambda _z$). The cutoff wavelength $\varLambda _z$ is determined based on the observations in figure 4$(b)$ that: (i) the spanwise size of the structures that are associated with interscale transport of positive and negative $tr_{-uv}$ scales with wall distance; (ii) a clear separation of the positive or negative spectral peaks occurs at the considered cutoff wavelengths (which is also the local peak of the interscale flux $Tr_{-uv}$). Hence a filter of varying cutoff wavelengths as a function with wall-normal distance is applied. The cutoff wavelength is defined as

(5.1)\begin{equation} \varLambda_z^{+} \simeq \beta\,(y^{+})^{\alpha},\end{equation}

for the varying cutoff wavelength filter; we will term it as variant cutoff filter. The large-scale (low-pass) and small-scale (high-pass) filtered fields are given by

(5.2)\begin{equation} {u_i}(\boldsymbol{x},t) = u_i'(k_z,\boldsymbol{x},t)+u_i''(k_z,\boldsymbol{x},t),\end{equation}

where the large-scale and small-scale components satisfy

(5.3)$$\begin{gather} \varphi_L(y,\beta,\alpha) = \int_{0}^{k_z} tr_{{-}uv}(\hat{k},y) \,{\textrm{d}} \hat{k} > 0, \quad \forall \, y, \end{gather}$$

(5.4)$$\begin{gather}\varphi_S(y,\beta,\alpha) = \int_{k_z}^{\infty} tr_{{-}uv}(\hat{k},y) \,{\textrm{d}} \hat{k} < 0, \quad \forall \, y, \end{gather}$$

and the mean Reynolds shear stress satisfy

(5.5)\begin{equation} \langle uv \rangle(y) = \langle u'v' \rangle (k_z(y),y)+ \langle u''v'' \rangle(k_z(y),y),\end{equation}

and the Reynolds shear stress with zero mean,

(5.6)\begin{equation} \langle u'v'' \rangle = \langle u''v' \rangle = 0.\end{equation}

To validate the filters, figure 6$(a)$ shows the contributions (5.3) and (5.4) as functions of $\varLambda _z^{+}(\beta ,\alpha )$. The first type of $\varLambda _z^{+}$ with $(\beta ,0)$, i.e. a constant cutoff filter, had been widely used to investigate amplitude modulation between large-scale and small-scale streamwise velocity fluctuations (e.g. Hutchins & Marusic Reference Hutchins and Marusic2007b; Mathis et al. Reference Mathis, Hutchins and Marusic2009; Ganapathisubramani et al. Reference Ganapathisubramani, Hutchins, Monty, Chung and Marusic2012), is deemed less suitable for the present study to investigate interscale transport because conditions (5.3) and (5.4) do not hold (marked with symbols and the arrows indicate increasing $\beta$ in figure 6a). The reduced value of $\varphi$ with increasing $\beta$ is due to the cancellation of energy between large scales and small scales. The second type of $\varLambda _z^{+}$, i.e. the variant cutoff filter, with $(\beta ,\alpha )$ in the combinations ranging from $(10,23,30;\,0.35,0.55,0.65)$ are tested where the conditions (5.3) and (5.4) hold. Note that the choice is not unique but rather appears to be the reasonable range of cutoff wavelengths, which attempts to separate the scales associated with the observations of the interscale process as shown in figure 4$(b)$ at the height considered (marked with a solid line in figure 4b). The rationale here is that if the Reynolds shear stress structures are essentially scaled with wall-distance, a spectral filter with variant cutoff wavenumbers will retain the velocity fluctuations corresponding to large scales and small scales at each height, respectively. This is also true for employing a constant cutoff spectral filter to analyse, for instance, small-scale and large-scale $(u^{2})$ contributions (see for example Lee & Moser Reference Lee and Moser2019).

Figure 6. $(a)$ The large-scale and small-scale contributions (black) $\varphi ^{+}_L$ and (red) $\varphi ^{+}_S$ of the interscale transport $tr_{-uv}$. Lines with symbols are based on the constant cutoff filters: $\varLambda _z^{+}(\beta ,0)= 300, 500$ and $1500$ (indicated by the arrow direction); variant cutoff filter $\varLambda _z^{+}(23,0.55)$ (——); variant cutoff filters for $\varLambda _z^{+}(10,0.55)$, $\varLambda _z^{+}(10,0.65)$, $\varLambda _z^{+}(23,0.35)$ and $\varLambda _z^{+}(30,0.35)$ (– – –). $(b)$ Fractional contribution of the ($\circ$) large-scale and (without symbols) small-scale quadrant events to the total Reynolds shear stress: $Q1$ (——); $Q2$ (– – –); $Q3 \ (\cdot \cdot \cdot \cdot \cdot \cdot )$; $Q4$ (– $\cdot$ – $\cdot$ – $\cdot$); variant cutoff filter $\varLambda _z^{+}(23,0.55)$ (blue); constant cutoff filter $\varLambda _z^{+} = 300$ (black).

5.2. The large-scale and small-scale ejection and sweep events

In this section, quadrant analysis is applied to the large-scale and small-scale filtered fields to identify the structures carrying the Reynolds shear stress. The ejection and sweep events are defined based on the pointwise velocity fluctuations as ${Q2}_{L}(u'<0, v'>0)$ and ${Q4}_{L}(u'>0, v'<0)$ for the large-scale components, and similarly, ${Q2}_{S}(u''<0, v''>0)$ and ${Q4}_{S}(u''>0, v''<0)$ for the small-scale counterparts. Figure 7 shows snapshots of the isocontours of the Reynolds shear stress fields ${Q2}_{L}$ (black), ${Q4}_{L}$ (red), ${Q2}_{S}$ (light blue) and ${Q4}_{S}$ (light red) for the variant cutoff filter $\varLambda _z^{+}(23,0.55)$ (figure 7a,b) and the constant cutoff filter $\varLambda _z^{+}(300,0)$ (figure 7c,d) at the same time instance. A clear distinction between the Reynolds shear stress structures is observed between $\varLambda _z^{+}(23,0.55)$ and $\varLambda _z^{+}(300,0)$. The large scales differ in the near wall, whereas the small scales differ in the logarithmic and outer regions. This is as expected due to the choice of filter bands. For instance, an organised structure (marked with an arrow in figure 7c) isolated by the constant cutoff filter considered to represent a ${Q2}_{L}$ is characterised by a ${Q2}_{S}$ in the variant cutoff filter (marked with an arrow in figure 7b). The filtering process is also reflected in figure 6$(b)$. The mean contributions to the total Reynolds shear stress are plotted for the two different filters ($\varLambda _z^{+}(23,0.55$) and $\varLambda _z^{+}=300$). For the filter $\varLambda _z^{+}(23,0.55)$, at $y^{+}>100$ the negative Reynolds shear stress (ejections and sweeps) in the small-scale components are higher than those in the filter $\varLambda _z^{+}=300$, while the negative Reynolds shear stress in the large-scale components are higher for $\varLambda _z^{+}=300$ when compared with $\varLambda _z^{+}(23,0.55)$.

Figure 7. Instantaneous ($z,y$) sections of large-scale and small-scale $Q2$ and $Q4$ based on (a,b) $\varLambda _z^{+}(23,0.55)$ and (c,d) $\varLambda _z^{+}(300,0)$ with $\text {(black)}\ {Q2}_{{L}}, \text {(red)}\ {Q4}_{{L}}$, $\text {(light blue)}\ {Q2}_{{S}}$ and $\text {(light red)}\ {Q4}_{{S}}$. Contour levels range from $-5[0.5]0$. The box plot in panel $(a)$ illustrates direct measurement of the structure's spanwise width and height based on ${H_0}$ (5.7).

To investigate the physical sizes of the $Q$ events present in the filtered fields, we will undertake two distinct approaches: (i) direct measurement and (ii) two-point correlation. With the first approach, we measure the sizes of the structures in the low- and high-pass filtered fields (such as those in figure 7) of pointwise Reynolds shear stress that satisfy

(5.7)\begin{equation} |u'v'(y,z)| > H|u_{rms}v_{rms}(y)|,\quad |u''v''(y,z)| > H|u_{rms}v_{rms}(y)|,\end{equation}

where $H$ is the Reynolds shear stress threshold. Two cases will be discussed in the following: (i) for ${H=0}$ (hereafter referred to as case ${H_0}$) and (ii) for $H=1.75$ (hereafter referred to as case $H_{1.75}$). The root mean square is defined as $\langle ({u_i u_i})^{1/2} \rangle$, see also figure 1$(b)$. The ratio of standard deviation to its mean is $|\langle uv \rangle (y)|/|u_{rms}v_{rms}(y)| \simeq 0.4$ (Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014), which is true for $y\lesssim \delta$. The criterion (5.7) is similar to those in Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012), Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014) and Fiscaletti et al. (Reference Fiscaletti, de Kat and Ganapathisubramani2018). Measurements of $Q$ events with width or height less than approximately $10$ wall units are excluded to avoid resolution issues. The total numbers of identified quadrant events for both $H_0$ and $H_{1.75}$ are provided in table 1, and the percentages show the fractions of identified events that are used for the analysis. It has also been checked that the contribution of the discarded small-scale events on the statistics did not influence the statistics presented in the following section. Here $\varDelta _z$ and $\varDelta _y$ denote the spanwise width and height of the structures and $y_{max} = y_{min} + \varDelta _y$ is the maximum distance of the structures from the wall (as illustrated in figure 7a) and the results are smoothed using a Gaussian distribution.

Table 1. Total number $(10^{6})$ of identified quadrant events. The percentage below the total number of quadrant events stands for the fraction of the total number of samples contributing to the statistics analysed in each case.

Figure 8 shows the probability distributions of the spanwise width $\varDelta _z$ of the individual structures as the probability density functions (p.d.f.s) and $y$ based on ${H_0}$. The spanwise size distributions between the large-scale ${Q2}_{L}$ and ${Q4}_{L}$ and between the small-scale ${Q2}_{S}$ and ${Q4}_{S}$ are similar. The mean width scales with wall distance (marked with a black dashed line) in the logarithmic region, as may be expected by the filter operation. However, it is clear that the large-scale and small-scale sweeps and ejections follow different scalings, approximately along $\langle {\varDelta _z}\rangle \sim y^{2/3}$ and $y^{1/3}$, respectively. On the other hand, for the constant wavenumber filter ($\varLambda _z^{+} = 300$) the spanwise widths of the large-scale and small-scale events have less dependence on the wall distance $\langle {\varDelta _z}\rangle \sim y^{1/5}$, i.e. the spanwise width distribution does not scale with the wall-normal distance (not shown). Furthermore, figure 9 shows the p.d.f.s of the height $\varDelta _y$ and maximum distance from the wall $y_{max}$ of the individual structures. The mean height of the large-scale ${Q}_{L}$ events scales approximately with the maximum distance away from the wall $\langle {\varDelta }_{y} \rangle \sim {y}_{max}$, while the small-scale ${Q}_{S}$ scale approximately with $(\sim y^{2/3})$. In addition to their spanwise and wall-normal length scales, we measure their Reynolds shear stress intensity, which is defined as

(5.8a)$$\begin{gather} u'v'_{\varDelta_z^{i}} = \frac{\displaystyle \int_{\varDelta_z^{i}} u'v'\, {\textrm{d}}z} {\displaystyle \int_{\varDelta_z^{i}}\, {\textrm{d}}z}, \quad i=1,\ldots,M, \end{gather}$$

(5.8b)$$\begin{gather}u''v''_{\varDelta_z^{j}} = \frac{\displaystyle \int_{\varDelta_z^{j}} u''v''\, {\textrm{d}}z} {\displaystyle \int_{\varDelta_z^{j}}\, {\textrm{d}}z}, \quad j=1, \ldots,N, \end{gather}$$

for M large-scale structures and a similar definition for the small-scale mean intensity. Figure 10 shows the mean $uv$ intensity of $Q2$ and $Q4$ events. The mean intensities of the ${Q4}_{S}$ have generally lower values than ${Q2}_{S}$ at all heights. For the large-scale ${Q2}_{L}$ and ${Q4}_{L}$, the mean intensity of ${Q4}_{L}$ is higher than that of ${Q2}_{L}$, at approximately below $y^{+} \simeq 300$ ($y/\delta \simeq 0.15$) and is opposite above this wall-normal location. This is consistent with the observation that the $Q2$ events have an overall higher contribution than the $Q4$ events, owing to the smaller size but higher intensity and larger number of $Q2$ than $Q4$ (Fiscaletti et al. Reference Fiscaletti, de Kat and Ganapathisubramani2018). The reversed contributions of large-scale ${Q2}_{L}$ and ${Q4}_{L}$ at approximately $y^{+} \simeq 300$ or $y \simeq 0.15 \delta$ may be attributed to their relative changes in mean size within the logarithmic region (Fiscaletti et al. Reference Fiscaletti, de Kat and Ganapathisubramani2018). In addition, the mean intensities of the ${Q}_{S}$ and ${Q}_{L}$ presented here are reminiscent of the total Reynolds shear stress carried by attached $Q2$ and $Q4$ with heights of $100\nu /u_\tau < y_{max} < 0.4h$ (relatively short) and $0.4h < y_{max} < h$ (relatively tall), respectively (see figure 6 in Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012)). This suggests that ${Q}_{S}$ and ${Q}_{L}$ share similar characteristics of Reynolds shear stress contributions with the attached $Q2$ and $Q4$ of intermediate height in channels. In addition, the results may suggest that ${Q}_{L}$ are analogous to the tall-attached $Q2$ and $Q4$, and the ${Q}_{S}$ are similar to the small-scale part of the attached $Q2$ and $Q4$, and they are not likely to be the detached groups because they contribute to the mean Reynolds shear stress (Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012). However, it remains unclear at this point because the ${Q}_{L}$ and ${Q}_{S}$ appear to have distinct differences in the spanwise width and height from the attached $Q2$ and $Q4$ (Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012), as shown in figures 8 and 9.

Figure 8. The p.d.f.s $P(\varDelta _z)$ of the spanwise width of the large-scale and small-scale $Q2$ and $Q4$ events normalised by their local means as a function of the wall-normal distance with $(a)\ {Q2}_{L}$, $(b)\ {Q4}_{L}$, $(c)\ {Q2}_{S}$ and $(d)\ {Q4}_{S}$. (a,b) The line style (——) indicates $\varDelta _z \sim y^{0.65}$, (c,d) (——) indicates $\varDelta _z \sim y^{0.35}$, black dashed lines indicate the mean values and blue dashed lines denote one standard deviation from the mean.

Figure 9. The p.d.f.s of height of the large-scale and small-scale $Q2$ and $Q4$ events normalised by their local means, as a function of their maximum distance from the wall with $(a)\ {Q2}_{L}$, $(b)\ {Q4}_{L}$, $(c)\ {Q2}_{S}$ and $(d)\ {Q4}_{S}$. (a,b) The line style (——) are $\varDelta _y \sim y^{0.95}$, (c,d) (——) are $\varDelta _y \sim y^{0.65}$, black dashed lines are the mean values and blue dashed lines are one standard deviation above the mean.

Figure 10. The p.d.f.s of the mean Reynolds shear stress intensity, as defined in (5.8), of the large-scale and small-scale $Q2$ and $Q4$ events as a function of their wall-normal distances, normalised by the local mean with $(a)\ {Q2}_{L}$, $(b)\ {Q4}_{L}$, $(c)\ {Q2}_{S}$ and $(d)\ {Q4}_{S}$. Solid lines indicate the mean values.

It has been shown that sweeps are taller than ejections in TBLs (Fiscaletti et al. Reference Fiscaletti, de Kat and Ganapathisubramani2018). This finding can be extended to the large-scale and small-scale sweeps and ejections. Figure 11$(a)$ shows the mean height comparison between the large-scale and small-scale events for ${H_0}$ (black lines) at $0.01 < y/\delta < 0.5$. The ratios increased from $1$ to $2.5$ with wall distance until $y/\delta \simeq 0.5$, showing that ${Q}_{L}$ are essentially taller than ${Q}_{S}$, except at $y/\delta \simeq 0.02$ (marked with a vertical line) and that they have approximately the same mean height $\langle {\varDelta }_{y,{L}}\rangle /\langle {\varDelta }_{y,{S}}\rangle \approx 1$. The ratios increased with the distance from the wall, which also implies that ${Q}_{L}$ grow faster than the ${Q}_{S}$. A further finding in figure 11$(a)$ is that ${Q2}_{L}$ and ${Q4}_{L}$ are similar in mean size (black line with symbol), which is true for ${H_0}$. As will be discussed in the next section, the ${Q4}_{L}$ are essentially larger than ${Q2}_{L}$ if a higher threshold of Reynolds shear stress is chosen (e.g. $H=1.75$).

Figure 11. $(a)$ The mean height ratio of $Q2$ and $Q4$ events with (blue) ${H_{1.75}}$ and (black) ${H_0}$ based on direct measurement. Solid lines with symbols in panel $(a)$ are ${{Q4}_{L}}/{{Q2}_{L}}$. A vertical solid line indicates $y/\delta = 0.02$ up to $y/\delta = 0.5$. $(b)$ Self-aspect ratios ${\rm \Delta} x/{\rm \Delta} z$ of sweep and ejection events based on a correlation map with coefficient $C=0.2$ up to $y/\delta =0.25$.

We have investigated the probability distributions of the direct measurements of the large-scale and small-scale Q events, in the wall-normal–spanwise plane, in order to investigate the spanwise and wall-normal physical length scales of structures associated with the positive and negative interscale transport, as shown in figure 4$(b)$. In the second approach, we will adopt the two-dimensional correlations to investigate the spatial coherence of the fluctuating components of the instantaneous Reynolds shear stress. The streamwise–spanwise correlations are reconstructed using Taylor's hypothesis by assuming that the $Q$ events move with the mean convection velocity $U_c$. Using the temporal dataset, the correlation coefficient can be written as

(5.9)\begin{align} C_{\tau\tau}({\rm \Delta} x, {\rm \Delta} z) = \frac{\langle \tau(x,z,t) \tau(x+{\rm \Delta} x,z+{\rm \Delta} z,t)\rangle}{\sigma_\tau \sigma_\tau} \simeq \frac{\langle \tau(x,z,t) \tau(x,z+{\rm \Delta} z,t+{\rm \Delta} t)\rangle}{\sigma_\tau \sigma_\tau}, \end{align}

where ${\rm \Delta} t = {\rm \Delta} x/U_c$ and $\tau = u'v'$ for large-scale Reynolds shear stress, with a similar definition for the small-scale component. There is evidence that ejections move more slowly than the local mean velocity by $1.5u_\tau$, while sweeps move faster than that by the same amount (Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014). We adopt a similar approach here in our analysis. Henceforth, we use the mean convection velocity $U_c = U(y) + 1.5u_\tau$ for the large-scale sweep events and $U_c = U(y) - 1.5u_\tau$ for the large-scale ejection events, while $U_c = U(y)$ is used for the small-scale events, based on the assumption that ${Q2}_{L}$ (${Q4}_{L}$) move slightly more slowly (faster) than the local mean velocity, and the small scales move approximately with the local mean velocity (see for example del Álamo & Jiménez (Reference del Álamo and Jiménez2009), for more details on Taylor's hypothesis).

Figure 12(a,b) shows the streamwise–spanwise correlations for ${Q}_{L}$ and ${Q}_{S}$ at wall-normal locations $y/\delta = 0.05$ and $y/\delta = 0.1$. For the small-scale events (${Q2}_{S}$ and ${Q4}_{S}$), they are similar in size with an aspect ratio of ${\rm \Delta} x/{{\rm \Delta} z}$ ≈2.5 and their ${\rm \Delta} x^{+} \approx 60\eta$ based on a correlation coefficient $C=0.2$ at both $y/\delta = 0.05$ and $y/\delta = 0.1$ (see also figure 11b for their aspect ratios based on $C = 0.2$). The aspect ratio reinforces the idea that that ${Q2}_{S}$ and ${Q4}_{S}$ are not likely to be the detached structures, which are found to be isotropically oriented and the Reynolds shear stress cancels in the mean (del Álamo et al. Reference del Álamo, Jiménez, Zandonade and Moser2006; Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012). On the other hand, ${Q2}_{S}$ and ${Q4}_{S}$ have sizes of the order of the Kolmogorov scale fragments, which are of order $\textit {O}(30\eta )$ and merge with large-scale structures (Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014). For the large-scale events (${Q2}_{L}$ and ${Q4}_{L}$), ${Q2}_{L}$ have similar streamwise length scales to ${Q4}_{L}$ but are narrower using $C \simeq 0.2$ to demarcate the length scales. When using higher correlation to define the length scales of the events, ${Q2}_{L}$ is smaller than ${Q4}_{L}$. Analysing the length scales at different correlation coefficients, the results show that ${Q2}_{L}$ has a longer but narrower low intensity tail ($C \simeq 0.2$) with aspect ratio ${\rm \Delta} x/{{\rm \Delta} z}$ ≈3, while ${Q4}_{L}$ has a wider and longer high-intensity tip ($C \geq 0.4$) and aspect ratio ${\rm \Delta} x/{{\rm \Delta} z}$ ≈1. The large-scale events are not similar to the tall-attached sweeps and ejections as observed in Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014), for example, the aspect ratios are approximately $\varDelta _x/\varDelta _z \approx 1$ for $C\geq 0.4$, which differ from the tall attached structures that follow self-similar aspect ratios of approximately $\varDelta _x/\varDelta _z\approx 2$. A possible reason for the difference is that the threshold for the Reynolds shear stress is set to ${H=0}$, resulting in the large-scale but low Reynolds shear stress intensity structures exhibiting a higher aspect ratio. To investigate this further, we have computed the correlations for $Q2$ and $Q4$ events as shown in figure 12(c,d). The $Q4$ events are generally longer and wider than the $Q2$ events, which is consistent with the findings by Fiscaletti et al. (Reference Fiscaletti, de Kat and Ganapathisubramani2018) for TBLs, suggesting that the threshold value of Reynolds shear stress is not likely to be the primary reason for the observed divergence. In order to facilitate comparison with other studies on intense ejection and sweep events (Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012; Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014; Fiscaletti et al. Reference Fiscaletti, de Kat and Ganapathisubramani2018), we performed similar measurements to the previous case $H_0$, but based on ${H=1.75}$ (see (5.7)).

Figure 12. The $C_{\tau \tau }$ correlation coefficients (a,b) (shaded) ${Q2}_{L}$, (black ——) ${Q4}_{L}$, (light blue ——) ${Q2}_{S}$ and (red – – –) ${Q4}_{S}$. In panels (c,d) (shaded) $Q2$ and (——) $Q4$. The shaded and line contours are levels of $0.2[0.2]0.8$. Here (a,c) $y/\delta = 0.05$ and (b,d) $y/\delta = 0.1$.

By setting the threshold for the Reynolds shear stress (${H_{1.75}}$), both $Q2$ and $Q4$ are distributed over a narrower band of spanwise width $\varDelta _z$, as shown in figure 13. The mean widths of the large-scale events (${Q2}_{L}, {Q4}_{L}$) follow the same ridge of the criterion ${H_0}$ (5.7) as $\varDelta _z\sim y^{2/3}$ but are shifted to the left compared with $H_0$ (to the smaller spanwise width $\varDelta _z$ band), suggesting that they are the self-similar structures with higher intensity as identified in figure 7$(a)$. The result for the small-scale events (${Q2}_{S}, {Q4}_{S}$) is presented in (figure 13c,d). The mean spanwise widths follow the order of the Kolmogorov microscale ($O(10\eta$)). This suggests that the small-scale events with higher intensities have sizes of the order of the Kolmogorov microscale, and differ from the $H_0$ case when the low-intensity small-scale events are included.

Figure 13. The p.d.f.s of spanwise width of the large-scale and small-scale $Q2$ and $Q4$ events normalised by their mean values as a function of the wall-normal distance for ${H=1.75}$. In panels (a,b) (——) indicates $\varDelta _z \sim y^{0.65}$ and in panels (c,d) indicates $\varDelta _z \sim y^{0.24}$. Black dashed lines are the mean values and blue dashed lines are one standard deviation from the mean.

The p.d.f.s of the height $\varDelta _y$ and $y_{max}$ of the $Q$ events are shown in figure 14, and the ratios between their height are shown in figure 11$(a)$. First, similar to $H_0$, the ${Q}_{L}$ grow faster than the ${Q}_{S}$ with increasing wall-normal distance, but the height and mean size ratios imply that an increasing Reynolds shear stress threshold decreases the size of ejections ($Q2$) and increases the size of sweeps ($Q4$). This is true if we consider that $Q4$ events are taller but with lower intensity than $Q2$ (Fiscaletti et al. Reference Fiscaletti, de Kat and Ganapathisubramani2018). The distinct difference in size between the large-scale ejection (${Q2}_{L}$) and sweep events (${Q4}_{L}$) appears more prominent compared with $H_0$, which is up to ${Q4}_{L}/{Q2}_{L} \simeq 1.5$, also supporting this view, as shown in figure 11$(a)$. Second, similar to $H_0$, the ratios begin to drop above $y\simeq 0.5\delta$, which may be due to the interaction with the irrotational fluid at the bottom edge of the intermittent region $(y\simeq 0.5\delta )$ that limits the growth of the large-scale events. The large-scale ejections can extend from the wall across the opposite wall in channels while this is not true for boundary layers due to the existence of the intermittent region (Jiménez et al. Reference Jiménez, Hoyas, Simens and Mizuno2010).

Figure 14. The p.d.f.s of height of the large-scale and small-scale $Q2$ and $Q4$ events normalised by their mean values as a function of their maximum distance from the wall for ${H=1.75}$. In panels (a,b) (——) are $\varDelta _y \sim y^{0.75}$ and in panels (c,d) are $\varDelta _y \sim y^{0.4}$. Black dashed lines are the mean values and blue dashed lines are one standard deviation above the mean.

It is noted that in the logarithmic region, intense sweep events are longer than intense ejection events in TBLs (Fiscaletti et al. Reference Fiscaletti, de Kat and Ganapathisubramani2018). This is also true for the large-scale events in $H_{1.75}$, as shown in figure 15. The ${Q4}_{L}$ are significantly longer and wider than ${Q2}_{L}$, on average are $50\,\%$ longer and $25\,\%$ wider across $0.05\delta \lesssim y \lesssim 0.25\delta$. The aspect ratio of ${Q2}_{L}$ is approximately ${\rm \Delta} x/{{\rm \Delta} z} \approx 1.1$ and the aspect ratio of ${Q4}_{L}$ is approximately ${\rm \Delta} x/{{\rm \Delta} z} \approx 1.5$, at $y/\delta =0.05$ and $y/\delta = 0.1$, as shown in figure 11$(b)$. On the other hand, the small-scale events (${Q2}_{S}$ and ${Q4}_{S}$) are similar in size, with aspect ratio ${\rm \Delta} x/{{\rm \Delta} z}\approx 2$ and their ${\rm \Delta} x^{+} \approx 50\eta$ based on $C=0.2$ at both $y/\delta =0.05$ and $y/\delta = 0.1$, similar to the size and aspect ratio reported at ${H_0}$ (figure 12).

Figure 15. The $C_{\tau \tau }$ correlation coefficients for ${H_{1.75}}$ with (shaded) ${Q2}_{L}$, (black ——) ${Q4}_{L}$, (light blue ——) ${Q2}_{S}$ and (red – – –) ${Q4}_{S}$. The shaded and line contours are levels of $0.2[0.2]0.8$. Here (a) $y/\delta = 0.05$ and (b) $y/\delta = 0.1$.

There are similarities and differences between the ${Q}_{L}$ and ${Q}_{S}$ reported here and the intense sweeps and ejections that are responsible for the total momentum transfer as described by Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014). The small-scale events (${Q2}_{S}$ and ${Q4}_{S}$) have similar orders of size and aspect ratio to the Kolmogorov scale fragments for splitting and merging with the large-scale structures (Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014). The large-scale events (${Q2}_{L}$ and ${Q4}_{L}$), on the other hand, are distinctly different from the tall-attached sweeps and ejections (Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014): (i) the lower aspect ratios based on ${H_{1.75}}$ and (ii) different scaling laws of their sizes with respect to wall distances compared with the tall attached sweeps and ejections. The discrepancies, however, might be related to the fundamental differences between channels and boundary layers at the outer edge of the logarithmic region (Jiménez et al. Reference Jiménez, Hoyas, Simens and Mizuno2010). For instance, we have already shown in figure 4(d,f) that the exchange of Reynolds shear stress occurring between the large-scale structures and the interface is not likely to be observed at the core region of the turbulent channel flows (Kawata & Alfredsson Reference Kawata and Alfredsson2018). To provide further insight into the interactions between large- and small-scale events, the production spectra of the Reynolds shear stress $pr_{-uv} \simeq E_{vv} \partial _{y} U$ (3.10) with respect to the interscale transport is plotted in figure 16. The only energy source in the Reynolds shear stress balance is the production term $pr_{-uv}$ governed by the shear (3.19). The alignment between the negative $tr_{-uv}$ and the positive $pr_{{-uv}}$ at the highlighted region (blue isocontours) in figure 16, confirms that the energy source of the small scales inversely transferring to large scales is $pr_{-uv}$. This provides further evidence that small scales are locally produced by shear stress and the interscale transport can be interpreted as the small-scale (${Q2}_{S}$ and ${Q4}_{S}$) and large-scale (${Q2}_{L}$ and ${Q4}_{L}$) interactions.

Figure 16. The one-dimensional premultiplied production spectra of the Reynolds shear stress $pr_{-uv}$ superimposed by the interscale transport spectra $tr_{-uv}$ (isolines) at ${Re}_\tau \simeq 2020$. The contour lines indicate negative contour-levels (blue) at $-0.3[0.1]-0.1$ and positive contour-levels (red) at $0.1[0.1]0.3$ of $k_zy^{+}tr_{-uv}^{+}$.

5.3. Conditional time evolution

The interscale flux indicates directional information of the energy transfer between scales, negative fluxes indicate the inverse transfer from small scales to large scales, while positive fluxes indicate the opposite (Kawata & Alfredsson Reference Kawata and Alfredsson2018). The interscale transport, which is the derivative of the fluxes, represents the gain or loss of the energy at a local scale (also the loss or gain of energy of the fluxes). To understand the interscale transport process further, we investigate the time scales associated with the fluctuations. The instantaneous interscale transport of Reynolds shear stress $\widetilde {tr}_{-uv}$ can be measured according to the occurrence

(5.10)\begin{equation} \widetilde{tr}_{{-}uv} < 0,\quad \widetilde{tr}_{{-}uv} > 0,\end{equation}

where $\tilde {\cdot }$ denotes the instantaneous quantity and ${tr} = \langle \widetilde {tr} \rangle$ is as defined in (3.10). This measures the time elapsed between a local loss ${\rm \Delta} T_{n}$ and when a local gain appears, and a similar definition for the latter, is denoted as ${\rm \Delta} T_{p}$. This process is illustrated in Appendix B. The above measurements were based on the temporal dataset. To reduce the noise due to the high data acquisition rate, the dataset was first filtered in time with a moving average filter of window length ${\rm \Delta} T^{+} \approx 5$. This value is similar to the reference value of minimum flow time scales in the analysis of experimental boundary layer data, as discussed by Hutchins et al. (Reference Hutchins, Nickels, Marusic and Chong2009). Measurements of ${\rm \Delta} T^{+} < 5$ in the results were further excluded to avoid measurements that are close to the temporal resolution limit of the dataset.

Figure 17 shows snapshots of $\widetilde {Tr}_{-uv}$ (panels (a,c)) and $\widetilde {tr}_{-uv}$ (panels (b,d)), normalised by the absolute value of the local maximum, at near-wall region $y^{+}= 13$ and logarithmic region $y/\delta = 0.15$. It is clear that the instantaneous interscale transports exhibit both positive and negative energy transfer, while the instantaneous interscale fluxes indicate the direction of instantaneous energy flow. Linking the observations of instantaneous interscale energy transfer to the physical processes of merging (inverse) and splitting (direct) of the small-scale and large-scale ejection and sweep events (${Q}_{L}$, ${Q}_{S}$) (see § 5.2), the snapshots suggest that both splitting and merging are prevalent and occur nearly simultaneously. The mean time duration $\langle {\rm \Delta} T_n \rangle$ for $\widetilde {tr}_{-uv} < 0$, and $\langle {\rm \Delta} T_p \rangle$ for $\widetilde {tr}_{-uv} > 0$, at spanwise scales and wall-normal locations are shown in figure 18(a,b). Notice that close to the wall ($y^{+} \approx 13$), the average duration of $\widetilde {tr}_{-uv} > 0$ at larger scales $(\lambda _z^{+} \simeq 300)$ is approximately ${\rm \Delta} T_p^{+} \simeq 20$, which is approximately twice that of the duration of $\widetilde {tr}_{-uv} < 0$ at the smaller scales $\lambda _z^{+} \simeq 50$ (${\rm \Delta} T_n^{+} \simeq 10$). The overall time duration (${\rm \Delta} T^{+} \approx O(10$)) is significantly lower than: (i) the bursting period in the buffer layer of $T^{+} \approx O(100)$ (Flores & Jiménez Reference Flores and Jiménez2010; Jiménez Reference Jiménez2013; Hwang & Bengana Reference Hwang and Bengana2016); (ii) the lifetime of the tall attached structures; (iii) the average time of merging and splitting (Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014). On the other hand, the time scales are comparable with the average residence time of uniform momentum zones states associated with negative large-scale (average-$Q2/Q4$) events in TBLs (Laskari et al. Reference Laskari, de Kat, Hearst and Ganapathisubramani2018) and the autogeneration mechanism cycle based on the hairpin-packets model (Jodai & Elsinga Reference Jodai and Elsinga2016).

Figure 17. Examples of the instantaneous interscale fluxes (a,b) $\widetilde {Tr}_{-uv}$ and transport of Reynolds shear stress (c,d) $\widetilde {tr}_{-uv}$, normalised by their mean maximum at the wall-normal location (a,c) $y^{+}=13$ and (b,d) $y/\delta =0.15$.

Figure 18. The mean time duration $\langle {\rm \Delta} T_n \rangle$ for $(a)$ a local loss $\widetilde {tr}_{-uv} < 0$, and (b) $\langle {\rm \Delta} T_p \rangle$ for a local gain $\widetilde {tr}_{-uv} > 0$. $(c)$ The total time duration ratio $T_s$ (5.11). The dashed lines indicate $\lambda _z \sim y^{0.55}$.

To quantify the balance between the positive and negative interscale transfer, the total time duration ratio is defined as

(5.11)\begin{equation} T_s = \frac{\sum {\rm \Delta} T_{p}-\sum {\rm \Delta} T_{n}}{\sum {\rm \Delta} T_{p}+\sum {\rm \Delta} T_{n}},\end{equation}

where the positive value $(0< T_s<1)$ indicates the prevalence of gain at a local scale during the total time duration and the negative value $(-1< T_s<0)$ suggests the opposite. The result is shown in figure 18$(c)$. In the buffer region ($y^{+} < 100$), the small scales exhibit more frequent $\widetilde {tr}_{-uv} < 0$ than large scales. While for large scales, the opposite trends are observed ($\widetilde {tr}_{-uv} >0$). Further away from the wall ($y^{+} > 100$), both direct and inverse energy transport are almost balanced in a wide range of scales, which is indicated by the dashed lines in figure 18$(c)$. This finding reveals that both processes are prevalent with respect to time above the buffer region. Furthermore, figure 18(a,b) suggests that the interscale transport is of the order of the viscous time scale $T^{+} = O(10)$ and is not proportional with distance from the wall. The time scales identified here differ from the time scales reported for merging and splitting of tall attached branches (Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014). One possible explanation is the different methodologies used to compute the time scales and the definition of the lifetime used by Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014). Their time scales correspond to lifetimes of structures that merge and split with similar sizes; whereas, in our study, we measured the time elapsed between a local gain and loss of energy at a given scale (Fourier mode). To further quantify the net rate in the local energy gain and loss at local scales, we compute the conditional estimates for each time elapsed interval defined as

(5.12)\begin{equation} \widetilde{tr}_n = \frac{\displaystyle \int_{ T_n} \widetilde{tr}_{{-}uv}\,{\textrm{d}}t} {\displaystyle \int_{ T_n} \, {\textrm{d}}t}, \quad \widetilde{tr}_p = \frac{\displaystyle \int_{ T_p} \widetilde{tr}_{{-}uv}\,{\textrm{d}}t} {\displaystyle \int_{ T_p} \, {\textrm{d}}t}.\end{equation}

The averages $tr_n = \langle \widetilde {tr}_n \rangle$ and $tr_p = \langle \widetilde {tr}_p \rangle$ are shown in figure 19(a,b). The local energy gain and loss are quite balanced in their conditional averages. The mean values $tr_n$ and $tr_p$ are also shown in figure 19$(c)$ from the near-wall region to the logarithmic region ($y^{+} = 13$ to $y/\delta \simeq 0.25$). There is asymmetry between $tr_n$ and $tr_p$, with larger wavelengths having higher values of $tr_p$ than the smaller wavelengths, while smaller wavelengths have lower values of $tr_n$ than the higher wavelengths. This trend shifts towards relatively larger scales with increasing distance from the wall (indicated by an arrow). The slightly imbalanced trends between $tr_n$ and $tr_p$ reflect the net gain (loss) of energy at larger (smaller) scales across a broad range of $y$ and scales. This reinforces the idea that interscale transport is a multiscale process reflecting the merging and splitting of Reynolds shear stress structures $({Q}_{L}, {Q}_{S})$ and providing further support for the concept that the merging and splitting are important and non-negligible.

Figure 19. (a,b) The premultiplied conditional averaged loss $tr_n$ and gain $tr_p$. $(c)$ The mean conditional loss (blue) $tr_n$ and gain (red) $tr_p$, at different wall-normal locations for $y^{+} = 13, 50, 100, 200\,\text {and}\,500$ (indicated by the arrow direction). The dashed straight lines in panels (a,b) indicate $\lambda _z \sim y^{0.55}$. In panel $(b)$ dashed and solid isolines are contour levels of $1[1]6$ and $-6[1]-1$ of $tr_p$ and $tr_n$, respectively.