3.1. Volume-averaged dynamics and correlations

The volume-averaged budget equation is obtained by integrating (3.1) from either wall to the centre of the channel:

(3.2)\begin{equation} \frac{\partial}{\partial t} E_V = P_V - \varepsilon_V \pm T|_{y=H}, \end{equation}

where the subscript $V$ denotes averaging over the upper or lower half of the computational domain. Without time averaging, the wall-normal transport terms in (3.1) fluctuate in time. The fluxes $\pm T|_{y=H}$ across the centreline are found to be very small compared with the integrated turbulence production $P_V$ and dissipation $\varepsilon _V$.

Figures 1(a,d,g,j,m) present the time signals of volume-averaged TPR, TKE and TDR for R230, R360, R500, R950 and R2000 (see table 1). While these signals fluctuate heavily around their mean values, TDR seems to have a low frequency content that closely follows TPR and TKE, together with some coexisting high-frequency content, a behaviour that becomes more apparent with increasing Reynolds number. The time signal of TDR in FD TCF more explicitly displays various time scales other than that of TKE, particularly at higher Reynolds numbers, unlike previous observations in homogeneous turbulence (Goto & Vassilicos Reference Goto and Vassilicos2015). This difference might be attributed to the presence of a hierarchy of energy-containing scales or integral length scales in FD TCF.

Figure 1. (a,d,g,j,m) Time evolution of volume-averaged TKE $E_V(t)$ (black lines), volume-averaged TDR $\varepsilon _V(t)$ (blue lines) and volume-averaged TPR $P_V(t)$ (red lines), normalised by their time-averaged mean in the lower half-channelfor five Reynolds numbers. (b,e,h,k,n) Auto-correlations $\rho _{[E_V,E_V]}$ (black lines), $\rho _{[\varepsilon _V,\varepsilon _V]}$ (blue lines) and $\rho _{[P_V,P_V]}$ (red lines). (c,f,i,l,o) Cross-correlations $\rho _{[\varepsilon _V,E_V]}$ (blue lines), $\rho _{[P_V,E_V]}$ (red lines) and $\rho _{[\varepsilon _V,P_V]}$ (green lines).

The time histories of volume-averaged TKE and volume-averaged TDR in (3.2) are subsequently studied using a temporal correlation function. For two random signals that fluctuate only in time, say $X(t)$ and $Y(t)$, the temporal correlation function is defined as

(3.3)\begin{equation} \rho_{[X,Y]}(\tau) = \frac{\overline{X'(t+\tau)\,Y'(t)}}{\overline{\sqrt{X'^2(t)}}\,\overline{\sqrt{Y'^2(t)}}}, \end{equation}

where $X'(t) = X(t) - \bar {X}$ is the fluctuation centred around its mean, and $\bar {X}$ is the time average of $X(t)$. The auto-correlation function is recovered for $X(t) = Y(t)$, in which case a maximum value $\rho _{[X,X]}(\tau )=1$ is reached for time lag $\tau = 0$.

The temporal correlation functions are presented in figures 1(b,c,e,f,h,i,k,l,n,o). Limited by the Reynolds number, R230 and R360 show narrow auto-correlation of volume-averaged TKE. The auto-correlation functions of volume-averaged TKE cross zero at $\tau u_\tau /H \approx \pm 2$ for the higher three Reynolds numbers shown here, suggesting that some aspects, at least, of the dynamics and time scales of large-scale energy-containing structures become approximately independent of Reynolds number when it is high enough (Hwang & Bengana Reference Hwang and Bengana2016). The auto-correlation functions in figures 1(b,e,h,k,n) show sharper correlation peaks for TDR and TPR than for TKE, presumably caused by the high-frequency content in the dissipation and production signals. The zero crossings of the auto-correlation for TDR and TPR widen as the Reynolds number increases, although it remains unclear if this continues with further increasing Reynolds number. Importantly, the peaks of the correlation functions between volume-averaged TKE and volume-averaged TDR are at $\tau u_\tau /H \approx 0.2$, with values of $\rho _{[\varepsilon _V,E_V]}$ between 0.5 and 0.8 (increasing with $\textit {Re}_{\tau }$), indicating the presence of an average time lag that is reminiscent of the observations made in forced periodic/homogeneous turbulence by Goto & Vassilicos (Reference Goto and Vassilicos2015) (see also figure 3(b) in Goto & Vassilicos Reference Goto and Vassilicos2016). A similar time lag is also observed between volume-averaged TPR and volume-averaged TDR, in fact with even higher values $\rho _{[\varepsilon _V,P_V]}$ (consistently approximately 0.8 for all five Reynolds numbers) presumably because TPR and TDR are both rates of change of TKE. This time lag is shorter than the typical time scale of the self-sustaining process at large scale (Hwang & Bengana Reference Hwang and Bengana2016). In addition, the correlation functions $\rho _{[\varepsilon _V,E_V]}$ and $\rho _{[\varepsilon _V,P_V]}$ display increasing skewness towards positive time lags at higher Reynolds numbers, spanning the range $\tau u_\tau /H \in [-1,2]$ for R550 but $\tau u_\tau /H \in [-1,4]$ for R2000 in the case $\rho _{[\varepsilon _V,E_V]}$. The skewed correlation function may be interpreted as a ‘spread’ of time lags (see § 3.3).

While we observe a clear positive time lag both between TKE and TDR and between TPR and TDR in FD TCF, it is important to point out that volume averaging obscures the dynamics associated with inhomogeneity in the interactions between TKE and dissipation. The observed dynamical interplay between TKE and TDR cannot be the same as that found in periodic/homogeneous turbulence (Goto & Vassilicos Reference Goto and Vassilicos2015). We address the impact of vertical inhomogeneity on the interplay between TKE and TDR in the remainder of § 3, and the interplay between TKE and TPR in § 4.

3.2. The 2-time 1-plane correlations

The neglect of inhomogeneity, due to volume averaging, needs to be restored for the detailed picture of the full temporal dynamics between TKE and TDR. We therefore now investigate, as in § 3.1, the correlations between time signals of plane-averaged TKE and plane-averaged TDR at different wall-normal locations. In figure 2, we plot time signals of the plane-averaged TKE and TDR normalised by their plane- and time-averaged mean values at three locations: the near-wall region at the edge of the viscous sublayer ($y^+=10$), the near-equilibrium region where production approximately balances dissipation ($y/H=0.3$), and the core region ($y/H=0.8$).

Figure 2. Time evolution of plane-averaged TKE $E(y,t)$ (black lines) and plane-averaged TDR $\varepsilon (y,t)$ (blue lines), normalised by their plane- and time-averaged mean in the lower half-channel for five Reynolds numbers at three wall-normal locations: (a,d,g,j,m) $y^+ = 10$, (b,e,h,k,n) $y/H = 0.3$, and (c,f,i,l,o) $y/H = 0.8$.

In the near-wall region ($y^+=10$ in figure 2, but similar observations hold for $y^+ \leqslant 100$), TKE and turbulence dissipation fluctuate almost concurrently. This is not surprising as there is little separation between the energy-containing scale and dissipative scale in the near-wall region. As the Reynolds number increases (from top to bottom in figure 2), the low frequency signal appears more pronounced. The near-wall time scale becomes shorter and shorter compared to $H/u_{\tau }$ (used to normalise time in figure 2) as Reynolds number increases. It is interesting to observe that even so, TKE and TDR fluctuate almost concurrently even at the highest Reynolds numbers considered here (see figures 2j,m). This suggests that the near-wall energy-containing structures (potentially part of large-scale structures extending from the near-equilibrium and outer layers) are directly related to dissipation in the near-wall region.

At $y/H=0.3$ (figures 2b,e,h,k,n), the time signals of plane-averaged TKE and plane-averaged TDR do not appear to be well correlated with each other as $\textit {Re}_{\tau }$ increases. The same holds true for any $y/H$ between $0.2$ and $0.4$. Note that this is the region where the large-scale structures are very energetic (Hoyas & Jiménez Reference Hoyas and Jiménez2006).

In the core region at $y/H=0.8$ (though similar observations can be made for $y/H=0.5$–$1$), TPR is known to be very small due to diminishing mean shear and Reynolds shear stress near the channel centre. The dynamics of turbulence energetics is therefore dominated by turbulence transport and dissipation. The plane-averaged TKE and the plane-averaged TDR appear highly correlated instantaneously at the same plane, as shown in figures 2(c,f,i,l,o) for all the Reynolds numbers considered here. As can be seen in figure 3, there is no time lag in these correlations, and there is therefore no cascade directly linking plane-averaged TKE and plane-averaged TDR in the core region.

Figure 3. The 2-time 1-plane correlation function contours $\rho _{[\varepsilon,E]}$ versus distance from the wall in (a,c,e,g,i) outer unit $y/H$ with linear axis, or (b,d,f,h,j) inner unit $y^+$ with log axis, and versus time lag $\tau u_\tau /H$, for (a,b) R230, (c,d) R360, (e,f) R500, (g,h) R950, (i,j) R2000. Vertical dashed lines indicate the position of maximum positive correlation as a function of $y$, showing that it coincides with the vertical $\tau = 0$ line.

Similar to the correlation function defined for volume-averaged variables in time in § 3.1, one can define a temporal correlation function for signals in time that depends on the wall-normal location in FD TCF. Figure 3 plots the contours of 2-time 1-plane correlation functions between plane-averaged TKE and TDR, defined as

(3.4)\begin{equation} \rho_{[\varepsilon,E]}(\tau,y) = \frac{\overline{\varepsilon'(t+\tau,y)\, E'(t,y)}}{\overline{\sqrt{\varepsilon'^2(t,y)}}\,\overline{\sqrt{E'^2(t,y)}}}, \end{equation}

where $\tau$ is a time difference. From the correlation contours in figure 3, high positive correlations between TDR and TKE in the near-wall region and the core region with zero time lag are observed, as expected from figure 2. The positive correlations in the near-wall region are concentrated in a narrow time window for small Reynolds numbers, which widens as Reynolds number increases, reflecting the volume-averaged correlation observation in figures 1(c,f,i,l,o). It may be that as $\textit {Re}_{\tau }$ increases, increasingly energetic structures above the near-wall region influence the near-wall region with fluctuations of increasing length and time scales, causing positive correlations with an increasing range of time lags in the near-wall region. In the near-equilibrium region, the correlation contour shows negative values (anti-correlation) between TDR and TKE with positive time lag, i.e. an increase/decrease in local TKE precedes a decrease/increase in local TDR. Finally, TKE and TDR are strongly correlated throughout the core with both positive and negative time lags $\tau$ in the range $\tau u_\tau /H \in [-1,1]$ nearly independent of Reynolds number.

In both the near-wall and core regions, the location of the maximum correlation in terms of time lag is always clearly at $\tau = 0$. In the near-equilibrium region, it is also near $\tau =0$, though less rigorously at $\tau =0$ because of low correlation values there. These zero time lag results have been reported previously for FD TCF (see figure 5(g,h) of Apostolidis et al. Reference Apostolidis, Laval and Vassilicos2022) using data obtained in a computational domain that is double the size of the present simulations in the streamwise and spanwise directions in terms of multiples of $H$ (Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014a). In the near-wall region, high instantaneous correlations between TKE and TDR (neither time lag nor lead) are expected, since there is no scale separation there. The high instantaneous correlations in the core region (neither time lag nor lead) suggest that high/low values of TKE and TDR occur together perhaps because the core flow flip-flops between quiescent and turbulent states as shown in Kwon et al. (Reference Kwon, Philip, de Silva, Hutchins and Monty2014). However, the zero average time lag of plane-averaged dynamics is seemingly inconsistent with the time lag discovered in volume-averaged dynamics in § 3.1. It is therefore natural to question if plane-averaged TKE at some distance from the wall correlates with plane-averaged TDR at another distance from the wall, with or without time lag.

3.3. The 2-time 2-plane correlations: TKE and TDR

To unveil the time lag observed in figure 1 and to illuminate the dynamic relation between TKE and TDR (as well as TPR and TDR) in space and time, we now investigate 2-time 2-plane correlations. The correlations between TKE at a reference $x$–$z$ plane and TDR at another plane in the channel are defined as

(3.5)\begin{equation} \rho_{[\varepsilon,E^{ref}]}(\tau,y;y^{ref}) = \frac{\overline{\varepsilon'(t+\tau,y)\,E'(t,y^{ref})}}{\overline{\sqrt{\varepsilon'^2(t,y)}}\, \overline{\sqrt{E'^2(t,y^{ref})}}}. \end{equation}

A similar 2-time 2-plane correlation can be defined by replacing TKE with TPR, i.e. $E^{ref}$ with $P^{ref}$, and $E'$ with $P'$. From the 2-time 1-plane correlations between TKE and TDR in figures 3(a,c,e,g,i), we observe a region of low positive correlations between $y/H=0.2$ and $y/H=0.4$, the outer part of the near-equilibrium ($\bar {P} \approx \bar {\varepsilon }$) region. We therefore start by plotting the correlation contours for reference plane $y^{ref}/H = 0.3$, which is where the minimal 2-time 1-plane correlation at zero time lag is found (figure 3). Taking a different reference plane within this low correlation region $y/H=0.2$ to $y/H=0.4$ changes little the observations that follow throughout the paper.

The correlation contours in $(\tau,y)$ space, with TKE at reference plane $y^{ref}/H = 0.3$, are plotted in figure 4 for the five Reynolds numbers considered. It can be observed clearly that the TKE in the near-equilibrium region correlates with the TDR in both the core region and the near-wall region, with a positive time lag and also a wall-distance lag. This observed time lag resolves the discrepancy between volume-averaged correlations and 2-time 1-plane correlations. Along the path of positive correlation from the reference plane towards the channel centre, the peaks of correlations in $(\tau,y)$ space lie very close to a line of slope $\delta y/\delta \tau = (1\pm 0.2)u_{\tau }$ (see figures 4 and 5a). This is a time lag combined with a wall-distance lag, and the ratio of the latter to the former is ${\sim }u_\tau$. Note that any turbulence transport in the wall-normal direction should be through the wall-normal velocity in the turbulence transport term (see $T_t(y,t)$). The mean wall-normal velocity is zero, but the standard deviation of the wall-normal velocity fluctuations $v_{rms}$ is of order $u_{\tau }$ and varies slowly with $y$. The correlations with time lag in figure 4 cannot be caused by the passage of inclined structures observed previously in turbulent boundary layers (e.g. in Marusic & Heuer Reference Marusic and Heuer2007) and FD TCF (e.g. in Cheng, Shyy & Fu Reference Cheng, Shyy and Fu2022). If they were, then TDR in the core region would precede TKE in the near-equilibrium region due to the earlier passage of structures further away from the wall than closer to the wall.

Figure 4. The 2-time 2-plane correlation function contours $\rho _{[\varepsilon,E^{ref}]}$ versus distance from the wall $y^+$ or $y/H$ and time lag $\tau u_\tau /H$ with respect to TKE at the reference plane $y^{ref}/H = 0.3$ (horizontal dash-dotted lines). The slopes of the solid black lines are $\pm u_\tau$, respectively. Plots for (a,b) R230, (c,d) R360, (e,f) R500, (g,h) R950, (i,j) R2000. Vertical dashed lines indicate $\tau = 0$.

Figure 5. For given $y^{ref}$, one defines $\delta y(y,y^{ref}) \equiv |y-y^{ref}|$, and then extracts $\delta \tau$ from the maximum correlation at local $y$ and $y^{ref}$, i.e. $\rho (\delta \tau, y ; y^{ref}) = \sup _\tau \{ \rho (\tau, y ; y^{ref}) \}$, where $\rho$ is defined in (3.5). A best linear fit of $\delta y$ versus $\delta \tau$ gives an average slope $\delta y/\delta \tau$ with its 95 % confidence interval for each $y^{ref}$. We plot the fitted slopes $\delta y/(\delta \tau u_\tau )$ as a function of $y^{ref}$ (a) for the 2-time 2-plane correlations above the reference plane, and (b) for the 2-time 2-plane correlations below the reference plane. In both (a,b), the shadings around the different coloured lines are 95 % confidence intervals for R230 (black line), R360 (red line), R500 (blue line), R950 (yellow line) and R2000 (green line).

The TKE can be transported both upwards and downwards from the reference plane. During wall-normal turbulence transport, high TKE fluctuations are also transferred towards smaller scales, which leads to high TDR fluctuations. This transport through scales is illustrated by the increasing correlation between TKE and TDR further away from the reference plane (figure 4), as well as by the path of the slowest decorrelation of TKE with itself, which is also along the line of slope $\delta y/\delta \tau \approx u_\tau$ (not shown here). This interpretation is supported by figure 6, which shows that the 2-time 2-plane correlation between TPR and TDR behaves differently even though the volume-averaged TPR has similar time lags with volume-averaged TDR and volume-averaged TKE. The difference between figures 4 and 6 can be understood by using the fact that TPR is not transported through space and scales, whereas TKE is. Figure 6 shows that TPR and TDR are maximally correlated locally rather than non-locally along a $\delta y/\delta \tau$ line as in the case of TKE and TDR in figure 4.

Figure 6. The 2-time 2-plane correlation function contours $\rho _{[\varepsilon,P^{ref}]}$ versus distance from the wall $y^+$ or $y/H$ and time lag $\tau u_\tau /H$ with respect to TPR at the reference plane $y^{ref}/H = 0.3$ (horizontal dash-dotted lines). Plots are for (a,b) R230, (c,d) R360, (e,f) R500, (g,h) R950, (i,j) R2000. Vertical dashed lines indicate $\tau = 0$.

Regarding the path of high correlations towards the core region, the velocity $\delta y/\delta \tau$ is close to that of the plane-averaged ejections found by Flores & Jiménez (Reference Flores and Jiménez2010). In the core region, the mean shear is very weak, and the eddies are composed mainly of dissipative ones (see § 3.2). It is therefore conceivable that this path is associated with the energy cascade initiated from the large-scale structure mainly around $y^{ref}/H = 0.3$. This would also imply that the ejections towards the outer region (Flores & Jiménez Reference Flores and Jiménez2010) involve not only momentum transport but also energy transport in space and in scales, which leads to TDR fluctuations through a cascade process. In the path towards the wall, the correlation velocity $\delta y/\delta \tau$ appears to be smaller than $(1\pm 0.2)u_\tau$ and less well-defined (see figure 5b). The contours in figure 4 show that as $\textit {Re}_{\tau }$ increases, the TKE in the near-equilibrium region and the TDR in the near-wall region correlate with a wider and more skewed spread of time lags in $H/u_{\tau }$ units. It is worth emphasising that this TKE to TDR spatio-temporal path towards the wall is fundamentally different from the superposition and modulation effects described in Hutchins & Marusic (Reference Hutchins and Marusic2007b).

As in § 3.2, we observe from the 2-time 2-plane correlation in figure 4 a region of anti-correlation between TDR and TKE around the reference plane at $y^{ref}=0.3H$. The extent of this anti-correlation shows a spread of time lags beyond $\tau u_\tau /H \approx 0.5$, and little wall-distance lag, consistent with the anti-correlation in 2-time 1-plane correlation in figure 3. This anti-correlation signifies a high/low TKE event followed in time by low/high dissipation around a reference plane taken at the outer edge of the near-equilibrium region. A plausible explanation for the local anti-correlation between TKE and TDR with time lag could be that TKE is transported away from the upper near-equilibrium region and therefore does not remain there to cascade and dissipate. A high instance of TKE at the upper near-equilibrium region is therefore followed by no increase in dissipation – in fact, even a decrease in dissipation, given that there is no local cascade at $y\approx y^{ref}$ between $y/H=0.2$ and $y/H=0.4$ to feed it.

As a side note, unlike the peak of 2-time 1-plane correlation located at the channel centre in figure 3, the maximum 2-time 2-plane correlation reaches only a distance from the wall approximately $y \sim 0.75H$, above which TKE at the reference plane decorrelates with TDR in the core region. This observed decorrelation is likely due to the fact that the single core region in FD TCF contains information such as signals of TKE ejected from both the upper channel and the lower channel. It would be interesting to investigate in the future temporal correlations between TKE in the near-equilibrium layer and TDR in the outer layer of wall-bounded turbulence with only one wall, such as that of open channel flow (Yao, Chen & Hussain Reference Yao, Chen and Hussain2022; Pirozzoli Reference Pirozzoli2023) or turbulent boundary layers.

3.4. The 2-time 2-plane correlations: near-wall region and Reynolds number dependence

The observations made in § 3.3 indicate a correlation velocity closely related to the turbulence transport velocities from the near-equilibrium region both towards and away from the wall. The path of correlation away from the wall shows little dependence on the Reynolds number and is always very close to the friction velocity. However, the correlation between the TKE in the near-equilibrium region and the TDR in the near-wall region is heavily dependent on Reynolds number. We therefore turn our focus to the 2-time 2-plane correlations between TKE at a reference plane taken in the near-wall region, and TDR fluctuations elsewhere. The correlation contours with reference plane taken at $y^{ref} = 10 \delta _\nu$ are presented in figure 7. This figure does not change significantly if we take a different reference plane in the range $y^{ref} \in [10,50]\,\delta _\nu$.

Figure 7. The 2-time 2-plane correlation function contours $\rho _{[\varepsilon,E^{ref}]}$ versus distance from the wall $y^+$ or $y/H$ and time lag $\tau u_\tau /H$ with respect to TKE at the reference plane $y^{ref} = 10\delta _\nu$ (horizontal dash-dotted lines). The slopes of the solid black lines are $\pm u_\tau$, respectively. Plots are for (a,b) R230, (c,d) R360, (e,f) R500, (g,h) R950, (i,j) R2000. Vertical dashed lines indicate $\tau = 0$.

The first point made by figure 7 is that TKE and TDR fluctuations are strongly correlated in the near-wall region below $y^+ \approx 100$, without time lag on average, but also, quite symmetrically, with equally positive and negative time lags in the range $-2 \leqslant \tau u_{\tau }/H \leqslant 2$. This strong correlation makes sense given that the length and velocity scales associated with TKE and TDR are not well separated in the near-wall region. The second observation coming out of figure 7 is that, unlike figure 4, no continuous path of positive correlation exists from the near-wall region to the near-equilibrium region, and ultimately to the core region. And the third observation is the significant positive correlation without significant time lag between TKE in the near-wall region and TDR in the core region.

For a start towards an interpretation of this third observation, we plot in figure 8 the ratio between the time-averaged TPR and the time-averaged TDR in inner and outer units obtained from our data. The notion of ‘equilibrium’ is merely an approximation of the actual TKE budget in the overlap region. The outer peak of the production to dissipation ratio increases with Reynolds number, and moves away from the wall in inner units. This is a signature of the very-large-scale motions at moderately high Reynolds number (Kim & Adrian Reference Kim and Adrian1999). As these outer structures become stronger with increasing Reynolds number, their contributions to the fluctuations in the TKE budget may also become more pronounced. These strong outer structures in the near-equilibrium region become increasingly dominant in both outer and near-wall TDR fluctuations. In the near-wall region, TKE and TDR fluctuate almost in the same manner (as seen from the high correlation with no time lag on average in the near-wall region in figure 7). Strong correlations between TKE near the wall and TDR in the core region therefore also indicate strong correlations between TDR near the wall and TDR in the core region with zero or little time lag at high Reynolds number, both presumably controlled by strong energy-containing structures of sizes comparable to the half-channel height. Indeed, the 2-time 2-plane correlations of TDR at $y^+=10$ and TDR at other wall distances $y$ (not shown here for economy of space) are nearly identical to figure 7 except perhaps at the lowest $\textit {Re}_{\tau }=230$, for which they are just very similar. A correlation where the reference TKE is taken in the core region was also computed (not shown here for economy of space), and similar instantaneous correlation values with the near-wall TDR were found, suggesting that the near-wall TDR and the core TDR are both controlled by structures of sizes comparable to the half-channel height.

Figure 8. Time-averaged mean profile of TPR over TDR in (a) inner units and (b) outer units for five Reynolds numbers. Horizontal solid line indicates $\bar {P}/\bar {\varepsilon } = 1$.

3.5. The 2-time 2-plane correlations: ejections and sweeps

As already mentioned in § 3.3, the spatio-temporal path of TKE to TDR from the near-equilibrium region to the outer region observed in that subsection is reminiscent of the spatio-temporal path of ejections/sweeps observed by Flores & Jiménez (Reference Flores and Jiménez2010). Hence we now focus on the regions in the $x$–$z$ plane where ejections and sweeps happen. For any quantity $Z(x,y,z,t)$ in FD TCF, we define a conditional plane average as

(3.6)\begin{equation} Z_{Qs}(y,t) = \frac{1}{A_{Qs}(y,t)} \iint_{Qs} Z(x,y,z,t) \,\mathrm{d}x \,\mathrm{d}z, \end{equation}

where Q$s$, for ${s}=1,2,3,4$, indicates the four quadrants in the $u$–$v$ plane, and $A_{Qs} = A_{Qs}(y,t)$ is the sum of the areas in the $x$–$z$ plane where Q${s}$ events take place (Wallace et al. Reference Wallace, Eckelmann and Brodkey1972; Willmarth & Lu Reference Willmarth and Lu1972). In particular, we investigate 2-time 2-plane correlations of unconditioned TDR with TKE conditionally averaged either over ejections (Q2) or over sweeps (Q4) in each plane. It is well-known that ejections and sweeps are the principal contributors to the Reynolds stress $-\langle u'v'\rangle$ in the near-equilibrium region (Willmarth & Lu Reference Willmarth and Lu1972).

Figure 9 presents the same correlation function as in (3.5), but using unconditioned TDR and TKE conditionally averaged (as in (3.6)) over Q2 ejection events, i.e. locations in the plane where $u'<0$ and $v'>0$. The resulting correlation contours show a trend consistent with the unconditional TKE and TDR in figure 4 (in particular, very similar correlation velocities $\delta y/\delta \tau$ to within 10 %, and correlation levels comparable to the unconditional correlations to within 7 %).

Figure 9. The 2-time 2-plane correlation function contours $\rho _{[\varepsilon,E^{ref}_{{Q2}}]}$ versus distance from the wall $y^+$ or $y/H$ and time lag $\tau u_\tau /H$ with respect to TKE conditioned on ejection events at the reference plane $y^{ref}/H = 0.3$ (horizontal dash-dotted lines). The slopes of the solid black lines are $\pm u_\tau$, respectively. Plots are for (a,b) R230, (c,d) R360, (e,f) R500, (g,h) R950, (i,j) R2000. Vertical dashed lines indicate $\tau = 0$.

When the correlation function is computed using TKE conditionally averaged over Q4 sweep events ($u'>0$ and $v'<0$) as shown in figure 10, noticeable changes in the correlation velocity and in the values of correlation coefficients appear. The spatio-temporal path towards the core region shows a significantly reduced correlation velocity (by up to 33 %) and significantly reduced levels of correlation (by up to 20 %). The perhaps counter-intuitive observation, however, is the difference of the correlation coefficients between the ejection-conditioned and sweep-conditioned correlations in the near-wall region. Sweep events are $v'<0$ events, which in effect transport high streamwise momentum ($u'>0$) towards the wall. They may therefore be expected to be mainly responsible for the dissipation of the near-wall region, hence the TKE and TDR conditionally averaged over sweep events may be expected to be more strongly correlated with higher correlation coefficient near the wall. The present correlation contours are contrary to this expectation. Even though ejection events correspond to $v'>0$ structures that would mainly transport slow ($u'<0$) streamwise momentum away from the wall, they appear to be better correlated with the near-wall dissipation than sweep events. For the higher three Reynolds numbers, the correlation levels between reference ejection-conditioned TKE and near-wall TDR (figures 9f,h,j) vary within 10 % of the unconditional correlation (figures 4f,h,j), whereas the sweep-conditioned correlation levels (figures 10f,h,j) are reduced by 20 %–36 %.

Figure 10. The 2-time 2-plane correlation function contours $\rho _{[\varepsilon,E^{ref}_{{Q4}}]}$ versus distance from the wall $y^+$ or $y/H$ and time lag $\tau u_\tau /H$ with respect to TKE conditioned on sweep events at the reference plane $y^{ref}/H = 0.3$ (horizontal dash-dotted lines). The slopes of the solid black lines are $\pm u_\tau$, respectively. Plots are for (a,b) R230, (c,d) R360, (e,f) R500, (g,h) R950, (i,j) R2000. Vertical dashed lines indicate $\tau = 0$.

In FD TCF, the plane-averaged wall-normal velocity must satisfy $\langle v \rangle (y,t) = 0$, an exact result imposed by the no-penetration boundary condition, homogeneity in the wall-parallel directions and incompressibility. This suggests that locally positive $v'>0$ and negative $v'<0$ must occur in pairs at any wall-parallel plane in the channel. As a result, ejections and sweeps should also happen simultaneously at any plane, and the TKE conditionally averaged over ejections and sweeps should fluctuate more or less concurrently, as appears to be approximately the case (see figure 11). On the other hand, it is worth pointing out that ejections have previously been shown to provide the largest contribution to the overall Reynolds shear stress and TKE (Willmarth & Lu Reference Willmarth and Lu1972). From figure 11, in the near-equilibrium region, the TKE conditionally averaged over ejections is indeed higher than conditionally averaged over sweeps, indicating that local turbulent motions associated with ejections are more energetic than sweeps (Farano et al. Reference Farano, Cherubini, Robinet and De Palma2017), which might be responsible for the faster correlation velocity than that of sweeps towards the core region.

Figure 11. Time evolution of conditionally averaged TKE $E_{Qs}/\bar {E}$ at $y/H = 0.3$, normalised by plane- and time-averaged TKE, in the lower half-channel. Red signals indicate TKE conditionally averaged over ejections ($s=2$), and blue over sweeps ($s=4$). Plots are for (a) R500, (b) R950, (c) R2000.

In the near-wall region, there is little separation between energy containing length scales and dissipative length scales. Turbulence energetics, whether the TKE or dissipation, plane-averaged or conditionally averaged over ejections/sweeps, are expected to fluctuate concurrently near the wall. While we expect the sweeps to be responsible for the near-wall fluctuations, it is actually the ejections in the near-equilibrium regions that correlate with the near-wall dissipation better, contrary to our expectation. We present here a hypothesis to account for this observation. As ejections and sweeps operate in tandem, anything relating to one should also somehow relate to the other. Presumably, sweeps are responsible for the connection and travel to the wall, and ejections for the connection and travel to the core, but correlations conditioned on ejections are stronger than correlations conditioned on sweeps because ejections are more organised/structured than sweeps. In support of this suggestion, we plot in figure 12 the time-averaged mean profile of conditionally plane-averaged Reynolds shear stress over ejections and over sweeps, respectively. One may consider the Reynolds shear stress as an indication of organisation/coherence given that $-\langle u'v' \rangle =0$ if $u'$ and $v'$ are uncorrelated, as expected in the absence of organised coherent flow structure, and therefore higher absolute values of $-\langle u'v' \rangle$ indicate higher order/coherence. In the case of FD TCF, the magnitude of $\overline {-\langle u'v' \rangle _{Q2}}$ (time-averaged $-\langle u'v' \rangle$ conditioned on Q2, i.e. plane-averaged over Q2 regions only) is significantly larger than $\overline {-\langle u'v' \rangle _{Q4}}$ for $y>0.2H$, thereby suggesting that ejections are more organised than sweeps. As a result, the correlations conditioned on ejections in figure 9 are stronger than the correlations conditioned on sweeps in figure 10, both towards the wall and towards the core. Note that significant correlations conditioned on sweeps also appear towards the core region in figure 10, which is as counter-intuitive as the significant correlations conditioned on ejections towards the wall in figure 9.

Figure 12. Time-averaged mean profile of conditionally averaged Reynolds shear stress over ejections (red lines) and sweeps (blue lines) for five Reynolds numbers.

4.1. The 2-time 2-plane correlation: TPR and TKE

We plot in figure 13 the 2-time 2-plane correlation between TKE at reference plane $y^{ref}=0.3H$ and TPR elsewhere. Positive correlations exist between TKE at the reference plane and TPR in the near-wall and TPR in the core region. The correlations in the core region share peak locations similar to those between TKE and TDR (figure 4), but the levels of correlation between TKE and TPR are significantly lower, and the correlation peaks do not follow the $\delta y/\delta \tau \approx u_\tau$ line that characterises the path from TKE to TDR. The core correlation between TKE and TPR must be rooted in the correlation between TKE and Reynolds shear stress (see § 4.2).

Figure 13. The 2-time 2-plane correlation function contours $\rho _{[P,E^{ref}]}$ versus distance from the wall $y^+$ or $y/H$ and time lag $\tau u_\tau /H$ with respect to TKE at the reference plane $y^{ref}/H = 0.3$ (horizontal dash-dotted lines). The slopes of the solid black lines are $\pm u_\tau$, respectively. Plots are for (a,b) R230, (c,d) R360, (e,f) R500, (g,h) R950, (i,j) R2000. Vertical dashed lines indicate $\tau = 0$.

The positive correlations between TKE at the reference plane and TPR in the near-wall region appear similar to the correlations between TKE at the same reference plane and TDR in the near-wall region (see figure 4) in terms of both the levels of correlations and the peak locations of the correlations, i.e. the average time lags. As discussed in §§ 3.2, 3.4 and 3.5, the near-wall dynamics are limited by a small range of scales, and TKE, TDR and TPR fluctuate almost concurrently near the wall. The similarity between correlations $\rho _{[P,E^{ref}]}$ and $\rho _{[\varepsilon,E^{ref}]}$ near the wall is therefore expected. One observation is that the near-wall production depends on the turbulent energy further away from the wall, which becomes more evident at higher Reynolds numbers (increasing levels of correlation in the near-wall region). The peaks of the correlation with time lag instead of time lead indicate that the near-wall TPR lags the reference TKE. Such an observation may be unexpected, since one often views the turbulence production as a source of turbulent energy, and TPR should lead TKE in time, as shown in a volume average sense in § 3.1.

It is worth pointing out that the 2-time 2-plane correlations with time lags in figure 13 are not inconsistent with the time lead of peak $\rho _{[P_V,E_V]}$ observed in the volume-averaged correlation. The observed positive values in figure 13 of 2-time 2-plane correlations with time lag in the near-wall and core regions largely cancel the negative values of 2-time 2-plane correlations with time lag in the near-equilibrium region when integrated in the wall-normal direction, and the overall time-lagged peaks of correlations are attenuated.

4.2. The 2-time 2-plane correlation: Reynolds shear stress and TKE

The time-fluctuating TPR is the product of the time-fluctuating plane average mean streamwise shear and the time-fluctuating plane-averaged Reynolds shear stress $-\langle u'v' \rangle$. (The contribution from the plane-averaged spanwise mean shear and the corresponding Reynolds shear stress component $-\langle u'w' \rangle$ is minimal compared to the streamwise component, i.e. less than 1 % of the overall production.) We therefore analyse the dynamics of the turbulence production in terms of 2-time 2-plane correlations between TKE at the reference plane and these two factors.

The TKE and the Reynolds shear stress are closely related to each other in terms of their time fluctuations. As shown by the correlation plot $\rho _{[-\langle u'v'\rangle,E^{ref}]}$ in figure 14, the TKE and the Reynolds shear stress are maximally correlated around the reference plane with no average time lag. The correlation contours above the reference plane show both wall-distance lag and time lag along a line of slowest decorrelation that is very close to the friction velocity up to approximately $y=0.8H$. In the near-wall region around the viscous sublayer, as Reynolds number increases, positive correlations develop with time lag, the spread of which widens in outer time unit $H/u_\tau$, and the levels of correlations increase.

Figure 14. The 2-time 2-plane correlation function contours $\rho _{[-\langle u'v'\rangle,E^{ref}]}$ versus distance from the wall $y^+$ or $y/H$ and time lag $\tau u_\tau /H$ with respect to TKE at the reference plane $y^{ref}/H = 0.3$ (horizontal dash-dotted lines). The slopes of the solid black lines are $\pm u_\tau$, respectively. Plots are for (a,b) R230, (c,d) R360, (e,f) R500, (g,h) R950, (i,j) R2000. Vertical dashed lines indicate $\tau = 0$.

Comparing the correlation contours around the reference plane between $\rho _{[P,E^{ref}]}$ in figure 13 and $\rho _{[-\langle u'v'\rangle,E^{ref}]}$ in figure 14, TKE is strongly correlated with the local Reynolds shear stress, whereas it is relatively decorrelated with TPR. The difference between the above two correlation contours must be due to the time-fluctuating mean shear. If a time- and plane-averaged mean flow was used, then the plane-averaged mean shear would show no fluctuations in time, and the correlation contours $\rho _{[P,E^{ref}]}$ in figure 13 and $\rho _{[-\langle u'v'\rangle,E^{ref}]}$ in figure 14 would be identical. The dynamics of the plane-averaged mean shear is therefore indispensable for the dynamics of TPR.

4.3. Interplay between the Reynolds shear stress and the mean shear: a mechanistic model

To reveal the dynamics of the plane-averaged mean shear and its relation with the plane-averaged Reynolds shear stress, we plot in figure 15 the correlation between the Reynolds shear stress at the reference plane $y^{ref} = 0.3H$ and the mean shear elsewhere. (As already mentioned, the results for $y^{ref} = 0.3H$ are indicative of any $y^{ref}$ between $0.2$ and $0.4$.) Very similar, if not near-identical, correlation maps (not shown here for brevity) are obtained for the correlation between TKE at the reference plane $y^{ref} = 0.3H$ and the mean shear elsewhere. Around the reference plane, there exists a region of anti-correlation between plane-averaged Reynolds shear stress and mean shear with a spread of time lags. On the other hand, the mean shear in the near-wall region shows a positive correlation with the plane-averaged reference Reynolds shear stress in the near-equilibrium region. High/low TKE or Reynolds shear stress in the near-equilibrium region are followed with some time lag by low/high mean shear in the same region, and by high/low mean shear fluctuations in the near-wall region. The anti-correlation of $\rho _{[{\rm d}U/{{\rm d}y},-\langle u'v'\rangle ^{ref}]}$ in the near-equilibrium region in figure 15 resolves the discrepancy between anti-correlation in the contours of $\rho _{[P,E^{ref}]}$ with time lag and $\rho _{[-\langle u'v'\rangle,E^{ref}]}$.

Figure 15. The 2-time 2-plane correlation function contours $\rho _{[{\rm d}U/{{\rm d}y},-\langle u'v'\rangle ^{ref}]}$ versus distance from the wall $y^+$ or $y/H$ and time lag $\tau u_\tau /H$ with respect to plane averaged Reynolds shear stress at the reference plane $y^{ref}/H = 0.3$ (horizontal dash-dotted lines). The slopes of the solid black lines are $\pm u_\tau$, respectively. Plots are for (a,b) R230, (c,d) R360, (e,f) R500, (g,h) R950, (i,j) R2000. Vertical dashed lines indicate $\tau = 0$.

The above observations may be interpreted by the following mechanistic model. Around the near-equilibrium region, when strong ejections and/or sweeps occur, high streamwise momentum in the outer region is transported towards the wall by sweeps, and low streamwise momentum in the inner region is transported away from the wall by ejections, resulting in a local flattening of the mean velocity profile and reduction of the mean shear $\mathrm {d}U(y,t)/\mathrm {d}y$. Therefore, in the near-equilibrium region where viscosity plays little role, high values of $-\langle u' v' \rangle$, and therefore also TKE, precede low values of $\mathrm {d}U/\mathrm {d}y$, as well as resulting reduced values of TPR around the same $x$–$z$ plane. When high momentum in the outer region is transported to the near-wall region by sweeps, the overall near-wall momentum increases, while the no-slip boundary condition at the wall is unaltered. As a result, the mean shear close to the wall becomes elevated, thereby explaining our observation of a positive correlation with positive time lag between $-\langle u' v' \rangle$ in the near-equilibrium region and $\mathrm {d}U/\mathrm {d}y$ in the near-wall region. Since both the near-wall Reynolds shear stress and the near-wall mean shear are positively correlated with reference TKE in the near-equilibrium region with time lag, the near-wall TPR is also positively correlated with TKE in the near-equilibrium region as in figure 13. This mechanistic model therefore explains how TKE can precede TPR even though volume-averaged TPR obviously precedes, albeit slightly, TKE.