4.1. On the relation between the spectral Reynolds-stress transport and the self-sustaining process of coherent structures

In § 3.2, we have shown that the streamwise length scales split mainly into two wavelength ranges: the larger-$\lambda _x$ range $\lambda _x^+>400$, in which the energy is produced by $pr^x_{uu}$ and taken by the interscale energy transfer $tr^x_{uu}$, and the smaller-$\lambda _x$ range $\lambda _x^+<400$, where the energy is supplied from larger $\lambda _x$ by $tr^x_{uu}$ and redistributed by the pressure–strain correlation $\pi ^x_{uu}$. The boundary between these two $\lambda _x$ ranges has been found to be nearly constant throughout the channel and to coincide with the streamwise-minimal domain length $L_x^+ \approx 400$. In the streamwise-minimal case, the streamwise wavelengths in the energy-producing (and -donating) $\lambda _x$ range are accounted for only by the streamwise-independent (i.e. $k_x=0$) mode, due to the limited degree of freedom, but otherwise the spectral energy transports observed in the reference case are basically retained, and thus the streamwise-minimal case reproduces the reference-case results fairly well. Given that both the inner and outer structures are retained in the streamwise-minimal case, as shown in figures 1 and 7, these observations indicate that the energy production at very large streamwise wavelengths ($k_x\approx 0$), the interscale energy transfer, and the energy redistribution at small wavelengths ($\lambda _x^+<400$) are the essential energy transport processes for both the inner and outer structures.

In addition to the spectral energy transports described above, it should be further noted that the turbulent energy redistributed from $\langle u^2 \rangle$ to $\langle v^2 \rangle$ results in the production of the Reynolds shear stress $P_{-uv}=\langle v^2 \rangle \, \mathrm {d}U/\mathrm {d} y$, which further leads to the reproduction of the streamwise turbulent energy $\langle u^2 \rangle$ as $P_{uu}=-2\langle uv \rangle \, \mathrm {d}U/\mathrm {d} y$. This closed-loop transport of the Reynolds stresses, consisting of production, interscale transfer and inter-component redistribution, resembles the scenario of the self-sustaining cycle of wall turbulence (Hamilton et al. Reference Hamilton, Kim and Waleffe1995; Waleffe Reference Waleffe1997): the significant $\langle u^2 \rangle$ production on the nearly-$x$-independent modes is very compatible with the generation of the $x$-independent $u$-streaks; the interscale energy transfer by $tr^x_{uu}$ is compatible with the streak instabilities; and the energy redistribution from $\langle u^2 \rangle$ to $\langle v^2 \rangle$ and $\langle w^2 \rangle$ at relatively small $\lambda _x$ is compatible with the generation of vortical structures from the streaks through their breakdown. While the self-sustaining process was originally proposed as the generating mechanism of the near-wall structures, recent studies (e.g. Hwang & Cossu Reference Hwang and Cossu2010, Reference Hwang and Cossu2011; Rawat et al. Reference Rawat, Cossu, Hwang and Rincon2015; Hwang & Bengana Reference Hwang and Bengana2016; Cossu & Hwang Reference Cossu and Hwang2017; de Giovanetti, Sung & Hwang Reference de Giovanetti, Sung and Hwang2017) have provided overwhelming evidence indicating that the large-scale structures in the logarithmic and outer layers are also essentially maintained by similar self-sustaining cycles, rather than by the influence of the smaller-scale structures near the wall. We now conjecture that the spectral energy transports observed in § 3.2 are closely related to each subprocess of the self-sustaining cycles of the inner and outer structures, and we examine this conjecture in detail in the following.

An insight supporting the correspondence between the streak instabilities and the interscale energy transport $tr^x_{uu}$ is obtained by decomposing $tr^x_{uu}$. The interscale transport $tr^x_{ij}$ in (3.6) is obtained as

(4.1)\begin{equation} tr^x_{ij}(y,k_x) = - \frac{\partial Tr^x_{ij} (y,k_x)}{\partial k_x}, \end{equation}

where $Tr^x_{ij}$, which represents the flux of the Reynolds stress in the streamwise wavenumber direction across $k_x$ from the larger- to the smaller-scale side, is defined as

(4.2)\begin{equation} Tr^x_{ij} = -\left \langle u^S_i u^S_k \frac{\partial u^L_j}{\partial x_k} \right \rangle -\left \langle u^S_j u^S_k \frac{\partial u^L_i}{\partial x_k} \right \rangle +\left \langle u^L_i u^L_k \frac{\partial u^S_j}{\partial x_k} \right \rangle +\left \langle u^L_j u^L_k \frac{\partial u^S_i}{\partial x_k} \right \rangle. \end{equation}

Here, $u^L_i$ and $u^S_i$ are respectively the large- and small-scale parts of the fluctuating velocity $u_i$, obtained by applying low- and high-pass spatial filterings based on the streamwise Fourier mode with the cutoff wavenumber $k_x$ (see appendix A for details). Then the interscale flux of the streamwise turbulent energy $Tr^x_{uu}$ is expressed as

(4.3)\begin{align} Tr^x_{uu} &= \underbrace{-2 \left \langle u^S u^S \frac{\partial u^L}{\partial x} \right \rangle}_{Tr^{x,1}_{uu}} - \underbrace{2 \left \langle u^S v^S \frac{\partial u^L}{\partial y} \right \rangle}_{Tr^{x,2}_{uu}} \underbrace{-2 \left \langle u^S w^S \frac{\partial u^L}{\partial z} \right \rangle}_{Tr^{x,3}_{uu}} \nonumber\\ &\quad \underbrace{+2 \left \langle u^L u^L \frac{\partial u^S}{\partial x} \right \rangle}_{Tr^{x,4}_{uu}} + \underbrace{2 \left \langle u^L v^L \frac{\partial u^S}{\partial y} \right \rangle}_{Tr^{x,5}_{uu}} + \underbrace{2 \left \langle u^L w^L \frac{\partial u^S}{\partial z} \right \rangle }_{Tr^{x,6}_{uu}}, \end{align}

and therefore the interscale transport $tr^x_{uu}$ is also decomposed as

(4.4)\begin{equation} tr^x_{uu} = tr^{x,1}_{uu} + tr^{x,2}_{uu} + tr^{x,3}_{uu} + tr^{x,4}_{uu} + tr^{x,5}_{uu} + tr^{x,6}_{uu}, \end{equation}

in accordance with $Tr^x_{uu}$.

Figure 15(a) compares the contributions of two major terms on the right-hand side of (4.4) to the energy transfer on the $x$-independent mode $tr^x_{uu}(y,0)$ in the streamwise-minimal case. Note that only $tr^{x,2}_{uu}(y,0)$ and $tr^{x,3}_{uu}(y,0)$ are presented here because the other terms are all zero, as explained in detail in appendix B. As shown here, the energy is transferred from the $x$-independent mode mostly due to $tr^{x,3}_{uu}$, which represents the effect of the spanwise velocity gradient by the larger structure $\partial u^L/\partial z$, as one can see from (4.3). This corresponds to the spanwise variation of $u$ induced by the $x$-independent structures. In the reference case, on the other hand, the energy is taken from finite wavelengths, unlike in the streamwise-minimal case, and therefore all the terms in the right-hand side of (4.4) can contribute to the interscale energy transfer from the large-$\lambda _x$ range. The contributions of all terms in the reference case are presented in figures 15(b) and 15(c) for a near-wall location $y^+=16$ and the channel centre, respectively. As shown in figure 15(b), the contribution by the third term $tr^{x,3}_{uu}$ is dominant at the near-wall location in the reference case at relatively large wavelengths ($\lambda _x^+>400$). In particular, the energy transfer from very large wavelengths $\lambda _x^+>1000$ is entirely due to $tr^{x,3}_{uu}$, which is consistent with the observations in the streamwise-minimal case in figure 15(a). The other terms have roughly the same magnitude, and all become comparable to $tr^{x,3}_{uu}$ at relatively small wavelengths $\lambda _x^+ \approx 100$. The energy transferred by $tr^{x,3}_{uu}$ from large- to moderate-$\lambda _x$ ranges is further transferred towards the smaller-$\lambda _x$ range by the other terms, which may represent the turbulent energy cascade. The third term $tr^{x,3}_{uu}$ also plays a primary role in the interscale energy transport at the channel centre, as shown in figure 15(c). While the second term $tr^{x,2}_{uu}$ also shows a significant contribution, the contribution by $tr^{x,3}_{uu}$ is dominant at very large wavelengths $\lambda _x/h > 10$. The dominant contributions by $tr^{x,3}_{uu}$ at large $\lambda _x$ are consistent with the observations in earlier studies that the main mechanism of the streak instabilities is inflectional instability due to the spanwise variation of $u$ induced by the streaks (e.g. Swearingen & Blackbelder Reference Swearingen and Blackbelder1987; Hamilton et al. Reference Hamilton, Kim and Waleffe1995; Jiménez & Pinelli Reference Jiménez and Pinelli1999); thus, this supports our claim that $tr^x_{uu}$ in the relatively-large-$\lambda _x$ range is closely related to the streak instabilities in the self-sustaining process in both the near-wall and the central regions of the channel.

Figure 15. Comparison of various terms on the right-hand side of (4.3): (a) in the streamwise-minimal (A3) case, on the $x$-independent mode across the channel; (b,c) in the reference case, (b) at the near-wall location $y^+=16$ ($y/h=0.063$) and (c) at the channel centre $y/h=0.5$ across the whole $\lambda _x$ range investigated.

In order to more directly examine the correspondence between the Reynolds stress budgets and the self-sustaining process, we investigate the instantaneous flow fields obtained in case C, where both $L_x$ and $L_z$ are as small as their minimal sizes; this is similar to the flow configuration employed by Hamilton et al. (Reference Hamilton, Kim and Waleffe1995) to observe the self-sustaining cycle. The instantaneous velocity field is decomposed into the $x$-independent mode $u_i^\prime =\langle u_i \rangle _x$ and the $x$-dependent mode $u_i^{\prime \prime }=u_i-u_i^\prime$ (here $\langle \rangle _x$ represents spatial averaging in the $x$-direction). Now we define the integrated turbulent energy on the $x$-independent mode as

(4.5)\begin{equation} \mathcal{E}^\prime_{uu} = \int^{0.195h}_{0} \langle {u^\prime}^2 \rangle_{xz}\, \mathrm{d} y, \end{equation}

and similarly we define the streamwise and lateral energies on the $x$-dependent modes as

(4.6a,b)\begin{equation} \mathcal{E}^{\prime \prime}_{uu} = \int^{0.195h}_{0} \langle {u^{\prime \prime}}^2 \rangle_{xz}\, \mathrm{d} y, \quad \mathcal{E}^{\prime \prime}_{vw} = \int^{0.195h}_{0} ( \langle {v^{\prime \prime}}^2 \rangle_{xz}+\langle {w^{\prime \prime}}^2 \rangle_{xz} )\, \mathrm{d} y; \end{equation}

their time series are presented in figure 16(a). Here $\langle \rangle _{xz}$ indicates spatial averaging in the $x$- and $z$-directions. The integration range $0 \leq y/h \leq 0.195$ (or $0 \leq y^+ \leq 40$ in the viscous unit) is chosen based on the $\langle u^2 \rangle$ profile obtained in case C, so that most of the turbulent energy related to the coherent structures near the bottom wall is included (see figure 4). The time series of the integrated turbulent energies given in figure 16(a) present a typical periodic behaviour. It appears that they are well correlated with each other, and the fluctuations of the $x$-dependent energies $\mathcal {E}^{\prime \prime }_{uu}$ and $\mathcal {E}^{\prime \prime }_{vw}$ follow that of the $x$-independent streamwise energy $\mathcal {E}^{\prime }_{uu}$ with a slight delay. The time delays are quantified in figure 16(b), which presents the cross-correlation functions between $\mathcal {E}^{\prime }_{uu}$ and $\mathcal {E}^{\prime \prime }_{uu}$, and between $\mathcal {E}^{\prime }_{uu}$ and $\mathcal {E}^{\prime \prime }_{vw}$, defined as

(4.7a,b)\begin{equation} C_{\mathcal{E}^{\prime}_{uu}}^{\mathcal{E}^{\prime \prime}_{uu}} (\tau) = \frac{\langle \mathcal{E}^{\prime}_{uu}(t) \mathcal{E}^{\prime \prime}_{uu} (t + \tau) \rangle}{\sqrt{\langle {\mathcal{E}^{\prime}_{uu}}^2 \rangle} \sqrt{\langle {\mathcal{E}^{\prime \prime}_{uu}}^2 \rangle}}, \quad C_{\mathcal{E}^{\prime}_{uu}}^{\mathcal{E}^{\prime \prime}_{vw}} (\tau) = \frac{\langle \mathcal{E}^{\prime}_{uu}(t) \mathcal{E}^{\prime \prime}_{vw} (t + \tau) \rangle}{\sqrt{\langle {\mathcal{E}^{\prime}_{uu}}^2 \rangle} \sqrt{\langle {\mathcal{E}^{\prime \prime}_{vw}}^2 \rangle}}.\end{equation}

As can be seen here, the correlation peaks of both cross-correlation functions are located at negative $\tau$, around $\tau ^+\approx -50$, and $C_{\mathcal {E}^{\prime }_{uu}}^{\mathcal {E}^{\prime \prime }_{vw}}$ gives larger delay than $C_{\mathcal {E}^{\prime }_{uu}}^{\mathcal {E}^{\prime \prime }_{uu}}$. This indicates that the fluctuation of the $x$-independent streamwise turbulent energy $\mathcal {E}^\prime _{uu}$ is followed by that of the $x$-dependent energy $\mathcal {E}^{\prime \prime }_{uu}$, which is further followed by that of the lateral turbulent energy $\mathcal {E}^{\prime \prime }_{vw}$. This observed sequence of $\mathcal {E}^\prime _{uu}$, $\mathcal {E}^{\prime \prime }_{uu}$ and $\mathcal {E}^{\prime \prime }_{vw}$ indicates that they mainly represent the energies of $x$-independent streaks, wavy streaks and vortical structures of the self-sustaining process, respectively.

Figure 16. Time series and cross-correlation functions of integrated turbulent energies and Reynolds stress budgets obtained in case C: (a) time series of $\mathcal {E}^\prime _{uu}$ (blue), $\mathcal {E}^{\prime \prime }_{uu}$ (red) and $\mathcal {E}^{\prime \prime }_{vw}$ (yellow) scaled by $u_\tau \nu$; (b) cross-correlation functions $C_{\mathcal {E}^\prime _{uu}}^{\mathcal {E}^{\prime \prime }_{uu}}$ (blue) and $C_{\mathcal {E}^\prime _{uu}}^{\mathcal {E}^{\prime \prime }_{vw}}$ (red); (c) time series of $\mathcal {P}^\prime _{uu}$ (blue), $\mathcal {T}_{uu}$ (red) and $\mathcal {R}_{uu}$ (yellow) scaled by $u_\tau ^3$; (d) cross-correlation functions $C_{\mathcal {E}^\prime _{uu}}^{\mathcal {P}^\prime _{uu}}$ (blue), $C_{\mathcal {E}^{\prime \prime }_{uu}}^{\mathcal {T}_{uu}}$ (red) and $C_{\mathcal {E}^{\prime \prime }_{vw}}^{\mathcal {R}_{uu}}$ (yellow). The vertical black dashed lines in panels (a) and (c) indicate the time range of the flow visualisation presented in figure 17.

Similarly, the integrated turbulent energy production on the $x$-independent mode $\mathcal {P}^\prime _{uu}$, the integrated interscale energy flux from the $x$-independent towards the $x$-dependent modes $\mathcal {T}_{uu}$, and the integrated pressure–strain energy redistribution $\mathcal {R}_{uu}$ are defined as

(4.8a)\begin{gather} \mathcal{P}^\prime_{uu} = \int^{0.195h}_0 \langle \widetilde{pr^x_{uu}} | _{k_x=0} {\rm \Delta} k_x \rangle_{xz}\, \mathrm{d} y = \int^{0.195h}_0 - 2 \langle u^\prime v^\prime \rangle_{xz} \frac{\mathrm{d}U}{\mathrm{d} y}\, \mathrm{d} y, \end{gather}

(4.8b)\begin{gather}\mathcal{T}_{uu} = \int^{0.195h}_0 \langle \widetilde{Tr^{x,3}_{uu}} |_{k_x=0} \rangle_{xz}\, \mathrm{d} y = \int^{0.195h}_0 -2 \left \langle u^{\prime \prime} w^{\prime \prime} \frac{\partial u^\prime}{\partial z} \right \rangle_{xz}\, \mathrm{d} y, \end{gather}

(4.8c)\begin{gather}\mathcal{R}_{uu} = \int^{0.195h}_0 \langle \widetilde{\varPi_{uu}} \rangle_{xz} (y)\, \mathrm{d} y =\int^{0.195h}_0 2 \left \langle p^{\prime \prime} \frac{\partial u^{\prime \prime}}{\partial x} \right \rangle_{xz}\, \mathrm{d} y, \end{gather}

where the tilde $\widetilde {\:}$ indicates the instantaneous values of the spectral budget term that are not averaged in the $x$- or $z$-direction or in time. The times series of these integrated budget terms are presented in figure 16(c). These time series also indicate certain periodic behaviours and are significantly correlated with each other, similarly to those of the integrated turbulent energies in panel (a). Figure 16(d) presents cross-correlation functions $C_{\mathcal {P}^{\prime }_{uu}}^{\mathcal {E}^{\prime }_{uu}}$, $C_{\mathcal {T}_{uu}}^{\mathcal {E}^{\prime \prime }_{uu}}$ and $C_{\mathcal {R}_{uu}}^{\mathcal {E}^{\prime \prime }_{vw}}$, defined in the same manner as those in (4.7a,b). As shown here, all three cross-correlation functions have the maximum magnitude of correlation located at negative $\tau$ of the order of $\tau ^+ \approx 10$. This indicates that $\mathcal {E}^\prime _{uu}$, $\mathcal {E}^{\prime \prime }_{uu}$ and $\mathcal {E}^{\prime \prime }_{vw}$ respond immediately to the increase or decrease of the energy production $\mathcal {P}^\prime _{uu}$, the interscale energy transfer $\mathcal {T}_{uu}$ and the pressure–strain energy redistribution $\mathcal {R}_{uu}$, respectively. In particular, the peak magnitudes of these cross-correlation functions are all significantly high (more than 0.5), which also indicates the close causal relationship between these turbulent energies and the Reynolds stress budget terms.

Figure 17 presents flow visualisations of the instantaneous flow field at six different instants during the period indicated by black dashed lines in figure 16(a,c). In each panel, the region near the bottom wall, $0 \leq y^+ \leq 60$, is visualised, and the coloured surfaces represent iso-surfaces of $u^+=-5$ (blue), $Q^+=-0.04$ (red), $\tilde {\varPi }_{uu}^+=-0.75$ (yellow) and $\widetilde {Tr_{uu}^{x,3}}^+ |_{k_x=0}=1.45$ (purple), where $Q$ is the second invariant of the velocity-gradient tensor. The instantaneous flow fields shown here are low-pass-filtered with a cutoff wavelength $k_{x,c}h/2{\rm \pi} =1.88$ ($\lambda _{x,c}^+=118$) to allow clear visualisation of coherent structures, but most of the turbulent energies (more than 90 % of $\langle u^2 \rangle$ and 68 % of $\langle v^2 \rangle + \langle w^2 \rangle$) are still retained in the flow field visualised here. In panels (a–c) (from $t^+=1293$ to $t^+=1370$), the evolution of a low-speed streak is observed, and streamwise-elongated vortices are also found aligned with the streak. In particular, in panel (c) the streak has grown enough to penetrate the computational domain in the $x$-direction. This instant ($t^+=1370$) actually corresponds to the peak of the $x$-independent streamwise turbulent energy $\mathcal {E}^\prime _{uu}$ in figure 16(a), and the $x$-dependent streamwise turbulent energy $\mathcal {E}^{\prime \prime }_{uu}$ is also increasing at this instant. Therefore the streak visualised in figure 17(c) is somewhat wavy. It can also be observed that the regions of significant interscale energy transfer from $x$-independent to $x$-dependent modes (purple) and inter-component energy transfer by pressure–strain correlation (yellow) start to appear around the streak. Panel (d) (at $t^+=1386$) corresponds to the peak of the interscale energy transfer $\mathcal {T}_{uu}$ in figure 16(c), and at this instant the $x$-dependent energy $\mathcal {E}^{\prime \prime }_{uu}$ in figure 16(a) is also saturated. Hence, the low-speed streak visualised in figure 17(d) is strongly wavy and surrounded by regions of significant interscale energy transfer. It is also observed that the pressure–strain energy redistribution is significant, and streamwise-elongated vortices visualised by $Q$ are evolving around the wavy streak. In panel (e) (at $t^+=1400$), the streak is shown to have already broken down. However, the regions of significant pressure–strain energy redistribution are still found, and vortical structures also remain. As time proceeds such vortical structures further evolve; as shown in panel (f), they remain after the low-speed streaks have become hardly visible, eventually giving rise to next streaks.

Figure 17. Visualisations of instantaneous flow fields in the region near the bottom wall ($0 \leq y^+ \leq 60$) observed in case C at (a) $t^+=1253$, (b) $t^+=1293$, (c) $t^+=1370$, (d) $t^+=1386$, (e) $t^+=1400$ and (f) $t^+=1489$. The full range of the computational domain is shown in the $x$- and $z$-directions. The three-dimensional surfaces of different colours in each panel represent iso-surfaces of $u^+=-5$ (blue), $Q^+=-0.04$ (red), $\tilde {\varPi }_{uu}^+=-0.75$ (yellow) and $\widetilde {Tr_{uu}^{x,3}}^+ |_{k_x=0} =1.45$ (purple). The flow fields shown are low-pass-filtered at the cutoff wavenumber $k_{x,c}h/2{\rm \pi} =1.88$ ($\lambda _{x,c}^+=118$) for clear visualisation of coherent structures.

The above observations in case C of the growth/breakdown of a streak and vortices in relation to the turbulent energy budget terms agree fairly well with the scenario of the self-sustaining cycle, and thus support our conjecture about their close relationship. Details of the balance between the spectral budget terms in case C are presented in figure 18(a) for a near-wall location $y^+ = 16$. As shown here, the peak of the energy production, accounting for nearly 70 % of the overall energy production, is located at $k_x=0$. The interscale energy transport $tr^{x,3}_{uu}$ is shown to dominate the energy transport at small wavenumbers near $k_x=0$, transferring nearly half the energy produced at $k_x=0$ mainly to the smallest non-zero wavenumber $k_x =2{\rm \pi} /1.6h$, where the most significant energy redistribution by $\pi ^x_{uu}(y,k_x)$ is found. The viscous dissipation $-\varepsilon ^x_{uu}$ is also presented; it is shown to dissipate the energy broadly throughout the investigated wavenumber range. As the flow fields visualised in figure 17 account for most values of $k_x$ at which $pr^x_{uu}$, $tr^x_{uu}$, and $\pi ^x_{uu}$ are significant in their profiles given in figure 18(a), it is reasonable to infer that the spectral budget balance presented in figure 18(a) mainly represents the energy transport related to the dynamics of the streaks and vortices observed in figure 17, which resembles the self-sustaining cycle.

Figure 18. Spectral budget of streamwise turbulent energy transport: (a) at $y^+=16$ in case C; (b) at $y^+ = 16$ in case A3; (c) at channel centre in case A3. In each panel different line colours represent production $pr^x_{uu}$ (blue, solid), interscale transport $tr^{x,3}_{uu}$ (red, solid), the contribution from terms of the interscale transport other than $tr^{x,3}_{uu}$ (red, dashed), pressure–strain energy redistribution $\pi ^x_{uu}$ (yellow) and viscous dissipation $-\epsilon ^x_{uu}$ (purple). The values are scaled by $u_\tau ^3$. The vertical black dotted lines indicate $k_xh/2{\rm \pi} =1/1.6$, which corresponds to the streamwise domain size in cases A3 and C.

The spectral energy budgets obtained in the streamwise-minimal (A3) case also have similar tendencies as observed in case C. Figures 18(b) and 18(c) present the profiles of the spectral budget terms at a near-wall location $y^+=16$ and at the channel centre in the streamwise-minimal case, respectively. Both budget balances indicate maximum energy production at $k=0$, energy transfer from $k_x=0$ to $k_x=2{\rm \pi} /1.6h$ mostly by $tr^{x,3}_{uu}$, and the most significant energy redistribution by $\pi ^x_{uu}$ at $k_x=2{\rm \pi} /1.6h$, similarly to the budget balance in figure 18(a). From this observation one can infer that these spectral budget terms in the near-wall and central regions of the channel in the streamwise-minimal case represent the energy transports related to the self-sustaining cycle of the near-wall and very-large-scale structures, respectively.

Such a connection between the self-sustaining cycle and the Reynolds stress transports observed through spectral analysis has already been pointed out by Cho et al. (Reference Cho, Hwang and Choi2018). Their discussion was based on the spanwise spectra of the turbulent energy production and the energy redistribution by the pressure–strain correlation. They showed that the peak location in the distributions of these spectra follows the scaling $\lambda _z \sim 5 y$, which is consistent with the attached eddy hypothesis, and thereby conjectured that the energy production and the pressure–strain correlation respectively correspond to the streak generation and the regeneration of the streamwise vortices in the self-sustaining process of each wall-attached eddy. In the present study, we have investigated the spectral transport of the Reynolds stress based on streamwise Fourier mode analysis with reduced-size computational domains, and have deduced a similar interpretation of the major terms of the spectral transport equation of the turbulent energy.

4.2. Inverse interscale energy transport in the regeneration of the streamwise vortices in the self-sustaining process

The closed-loop transport of the Reynolds stresses is schematically summarised in figure 19 with their streamwise length scales, and the subprocesses of the self-sustaining cycle that likely correspond are also included. From figure 19 one can make an interesting observation regarding the reproduction of the streamwise turbulent energy $\langle u^2 \rangle$. As explained in the previous section, the turbulent energy production $pr^x_{uu}=2 E^x_{-uv}\, \mathrm {d}U/\mathrm {d} y$ results from the wall-normal turbulent energy $E^x_{vv}$ through the production of the Reynolds shear stress $pr^x_{-uv}=E^x_{vv}\, \mathrm {d}U/\mathrm {d} y$. What should be noted here is that both productions $pr^x_{-uv}$ and $pr^x_{uu}$ are linear processes that do not involve any scale interactions, which means that there has to exist a certain amount of the wall-normal turbulent energy $E^x_{vv}$ at large $\lambda _x$ in order for the production of $\langle u^2 \rangle$ to take place. However, the energy source for $\langle v^2 \rangle$ is the energy redistribution from $\langle u^2 \rangle$, which, as observed in § 3.2, mainly occurs at relatively small $\lambda _x$ ($\lambda _x^+<400$) and is quite small (in the streamwise-minimal case, zero) in the energy-producing $\lambda _x$ range, as $\partial u/\partial x$ is small at such large $\lambda _x$. Of course there is no direct energy production from the mean flow for $\langle v^2 \rangle$, unlike for $\langle u^2 \rangle$ at large scales. Hence, any typical energy supply to $\langle v^2 \rangle$ is not found at large scales, despite the fact that $\langle v^2 \rangle$ at large scales is indispensable for turbulent energy production. This indicates a certain energy transfer to $\langle v^2 \rangle$ at large scales from smaller scales where the energy redistribution from $\langle u^2 \rangle$ to $\langle v^2 \rangle$ and $\langle w^2 \rangle$ is significant.

Figure 19. Schematics of the closed-loop spectral transport of the Reynolds stresses with streamwise length scales. The subprocesses of the self-sustaining cycle that likely correspond are also shown together. The solid black arrows represent the flow of energy transfer, and the dashed black arrow connecting $\langle v^2 \rangle$ and $\langle uv \rangle$ in $P_{uu}$ means that $\langle v^2 \rangle$ triggers $P_{uu}$ through $P_{-uv}$.

It is, in fact, the inverse interscale energy transfer of $\langle w^2 \rangle$ and the energy redistribution from $\langle w^2 \rangle$ to $\langle v^2 \rangle$ that provides energy to the $\langle v^2 \rangle$ component at large $\lambda _x$. As shown in figure 20(a), the interscale transport $tr^x_{ww}$ in the reference case clearly indicates energy transfer from smaller to larger $\lambda _x$, particularly in the near-wall region, as well as the forward transfer (from larger to smaller $\lambda _x$), and this tendency is also reproduced in the streamwise-minimal case as the transfer from the $\lambda _x^+<400$ range to the $x$-independent mode. The interscale transfer of the wall-normal turbulent energy $tr^x_{vv}$ basically indicates forward interscale transfers, but is one order smaller in magnitude than the reversed energy transfer by $tr^x_{ww}$ (not shown). Such reversed energy cascades in the near-wall region have been repeatedly reported in earlier studies of turbulent channel flow (e.g. Saikrishnan et al. Reference Saikrishnan, De Angelis, Longmire, Marusic, Casciola and Piva2012; Cimarelli et al. Reference Cimarelli, De Angelis and Casciola2013, Reference Cimarelli, De Angelis, Jiménez and Casciola2016; Cho et al. Reference Cho, Hwang and Choi2018; Hamba Reference Hamba2019; Lee & Moser Reference Lee and Moser2019), although their roles in maintaining flow structures are still unclear. Figure 20(b) presents the $\langle w^2 \rangle$ budget of the $x$-independent mode in the streamwise-minimal case. As shown here, the main energy source for $\langle w^2 \rangle$ on this mode is the inverse energy transfer from $k_x>0$ throughout the channel, and the turbulent spatial transport $d^t_{ww}$ carries the energy from the near-wall to the channel-central region. The pressure–strain correlation $\pi ^x_{ww}(y,0)$ represents the energy exchange between $\langle v^2 \rangle$ and $\langle w^2 \rangle$, as $\pi ^x_{uu}=0$ and $\pi ^x_{vv}=-\pi ^x_{ww}$ at $k_x=0$, and the profile of $\pi ^x_{ww}(y,0)$ indicates that energy is transferred from $\langle w^2 \rangle$ to $\langle v^2 \rangle$ in the central region of the channel, while it is transferred in the opposite direction in the near-wall region. Integrating $tr^x_{ww}(y,0)$ and $\pi ^x_{ww}(y,0)$ gives $\int ^{0.5h}_{0} \pi ^x_{ww}(y,0)\, \mathrm {d} y \approx -0.35 \int ^{0.5h}_{0} tr^x_{ww}(y,0)\, \mathrm {d} y$, indicating that the net energy transfer from $\langle w^2 \rangle$ to $\langle v^2 \rangle$ by $\pi ^x_{ww}$ at $k_x=0$ is about 35 % of the total energy gain from smaller scales by $tr^x_{ww}$.

Figure 20. (a) Space–wavelength ($y$–$\lambda _x$) diagrams of the premultiplied interscale transport of the spanwise turbulent energy in the streamwise wavenumber direction $k_x tr^x_{ww}$ obtained in the reference case and the streamwise-minimal (A3) case. The horizontal black dashed lines represent the streamwise domain size in the streamwise-minimal case $\lambda _x/h=1.6$. (b) The spectral budget balance of $\langle w^2 \rangle$ transport at $k_x=0$ in the streamwise-minimal case: the interscale transport $tr^x_{ww}(y,0)$ (blue), the pressure–strain cospectrum $\pi ^x_{ww}(y,0)$ (red), the turbulent spatial transport $d^{t,x}_{ww}(y,0)$ (yellow) and the viscous terms $d^{\nu,x}_{ww}(y,0)+\epsilon ^x_{ww}(y,0)$ (purple). The values are scaled by $u_\tau ^3/h$.

Because of this energy supply to the wall-normal and spanwise turbulent energies at $k_x=0$, the energy spectra $E^x_{vv}(y,0)$ and $E^x_{ww}(y,0)$ show non-zero energy distributions as presented in figure 21(a). Their magnitudes are small compared to the streamwise component $E^x_{uu}(y,0)$ as shown here, since the amount of energy supplied to $E^x_{vv}(y,0)$ and $E^x_{ww}(y,0)$ from smaller scales via $tr^x_{ww}$ is only about 5 % of the total amount of energy produced from the mean flow at $k_x=0$ by $pr^x_{uu}$. The Reynolds shear stress cospectrum $E^x_{-uv}(y,0)$ is also presented in the figure and is shown to be of the same magnitude as $E^x_{vv}(y,0)$ and $E^x_{ww}(y,0)$. The budget balance of $-\langle uv \rangle$ transport at $k_x=0$, presented in figure 21(b), shows that the production term $pr^x_{-uv}(y,0)=E^x_{vv}(y,0)\, \mathrm {d}U/\mathrm {d} y$ is the main energy source throughout the channel. This indicates that the small amount of wall-normal energy $E^x_{vv}(y,0)$ indeed maintains the Reynolds shear stress at $k_x=0$, which subsequently results in the reproduction of $\langle u^2 \rangle$ on the $x$-independent mode via the turbulent energy production $pr^x_{uu}(y,0)=2 E^x_{-uv} (y,0)\, \mathrm {d}U/\mathrm {d} y$. In the reference case the turbulent energy production at large scales is also maintained by similar energy transports, including the inverse interscale energy transfer, as described above (not shown).

Figure 21. (a) Profiles of the Reynolds stress spectra $E^x_{ij}$ and (b) the spectral budget balance of $-\langle uv \rangle$ transport at $k_x=0$ in the streamwise-minimal (A3) case. In panel (b), the lines of different colours represent the production $pr^x_{-uv}(y,0)$ (black), the interscale transport $tr^x_{-uv}(y,0)$ (blue), the pressure-related terms $\pi ^x_{-uv}(y,0)+d^{p,x}_{-uv}(y,0)$ (red), the turbulent spatial transport $d^{t,x}_{-uv}(y,0)$ (yellow) and the viscous terms $d^{\nu,x}_{-uv}(y,0)+\epsilon ^x_{-uv}(y,0)$ (purple), and the values are scaled by $u_\tau ^3/h$.

Thus, it has been shown that the turbulent energy production at large streamwise length scales is maintained by the inverse interscale transfer of spanwise turbulent energy from smaller scales. This also means that, supposing that these Reynolds-stress budget terms are closely related to the self-sustaining process as we describe in figure 19, the regeneration of the streamwise vortices through breakdowns of wavy streaks should include an inverse interscale energy transfer as well as the inter-component energy transfer by the pressure–strain correlation. In figure 19, we highlight this point in part (iii), ‘Regeneration of streamwise vortices by nonlinear mechanism’ (see blue dashed box) by referring to the structures generated from ‘wavy streaks’ through their breakdowns as ‘small-scale vortical structures’, distinguishing them from ‘streamwise vortices’, which directly lead to the generation of ‘streaks’. Small-scale vortical structures are generated at small scales ($\lambda _x^+<400$) through streak breakdowns by the pressure–strain energy redistribution, and streamwise vortices are regenerated at large scales by the inverse interscale energy transfer, which eventually results in the regeneration of streaks by triggering turbulent energy production. Hamba (Reference Hamba2019) focused on the inverse interscale transport of $\langle w^2 \rangle$ similarly to the present study and attempted to extract the related vortical structure by means of conditional averaging. He found a longitudinal streamwise vortex accompanied by a shorter vortex located upstream, which indicates that the interactions between these vortices are responsible for the inverse energy cascade. Although the connection between the structures found in his work and the regeneration process of the streamwise vortices in the self-sustaining cycle is still unclear, the inverse energy transfer of $\langle w^2 \rangle$ towards large $\lambda _x$ observed in the present study may also be caused by similar interactions between the long (at large $\lambda _x$) and short ($\lambda _x^+<400$) vortices.

4.3. On the DNS with streamwise-minimal domain at higher Reynolds numbers

As shown in § 3.2, the reason underlying the good agreement between the reference and streamwise-minimal cases is that the upper boundary of the $\lambda _x$ range where the turbulent energy is supplied from larger $\lambda _x$ via the interscale transport $tr^x_{uu}$ and redistributed by the pressure–strain correlation $\pi ^x_{uu}$ is rather independent of $y$ and is smaller than the streamwise minimal length $L_x=1.6h$ throughout the channel. As our investigations are limited to a low-Reynolds-number range, $Re_\tau \approx 126$, it should be further examined whether or not this is still the case even at higher Reynolds numbers. Regarding this point one may refer to the works of Abe et al. (Reference Abe, Antonia and Toh2018) and Lee & Moser (Reference Lee and Moser2019); Abe et al. (Reference Abe, Antonia and Toh2018) performed DNS of turbulent channel flows at $Re_\tau =1020$, comparing the results obtained with very-large and streamwise-minimal domains, and Lee & Moser (Reference Lee and Moser2019) also carried out DNS of a turbulent channel flow at $Re_\tau = 1000$, providing detailed data on the spectral transport of the Reynolds stresses.

In figure 22(a), we reproduce from Lee & Moser (Reference Lee and Moser2019) the $\lambda _x$–$y$ distributions of the premultiplied interscale transport of the streamwise turbulent energy in the streamwise wavenumber direction, denoted by ${E_{uu,x}^{T^{\|}}}$ in their notation, and of the pressure–strain cospectrum $\pi ^x_{uu}$ in a turbulent channel flow at $Re_\tau =1000$. In these distributions, blue and red represent the energy loss and gain, respectively. In the ${E_{uu,x}^{T^{\|}}}$ distribution (on the left of panel (a)), one can see that there exists a certain $y$ region, up to about $y^+=100$, where the boundary between the energy-donating (blue) and -receiving (red) $\lambda _x$ ranges is hardly dependent on $y$ and coincides with $\lambda _x^+ \approx 400$ (see the vertical black dashed line), similarly to our observation in the turbulent plane Couette flow. Farther away from the wall, however, the peak locations of the ${E_{uu,x}^{T^{\|}}}$ distribution clearly vary depending on the distance from the wall, which appears to be proportional to $y$. Similarly to this, it is shown in the right of panel (a) that the $\pi ^x_{uu}$ distribution also follows the scaling with $y$ away from the wall at about $y^+>100$. This means that the aforementioned observation in the present study, that the distributions of $tr^x_{uu}$ and $\pi ^x_{uu}$ are rather independent of $y$, is merely a low-Reynolds-number effect.

Figure 22. (a) $\lambda _x$–$y$ diagrams of (left) the interscale energy transport and (right) the pressure–strain cospectrum for $\langle u^2 \rangle$, scaled by $u_\tau ^4/\nu$, obtained in a turbulent channel flow at $Re_\tau =1000$, reproduced with permission from figures 26(a) and 13(b) in Lee & Moser (Reference Lee and Moser2019) (the vertical dashed lines at $\lambda _x^+=400$ have been added in the present study). (b) Profiles of streamwise turbulent intensity $\langle u^2 \rangle /u_\tau ^2$ obtained in a turbulent channel flow at $Re_\tau =1020$ with various streamwise domain lengths $L_x$, reproduced with permission from figure 7(a) in Abe et al. (Reference Abe, Antonia and Toh2018).

As shown in the $\pi ^x_{uu}$ distribution in figure 22(a), a significant amount of the pressure–strain cospectrum is distributed in the range $\lambda _x^+>400$ for $y^+>100$ at the Reynolds number $Re_\tau =1000$, which cannot be taken into account in the simulation with the streamwise-minimal domain. This would result in a significant underestimation of the energy redistribution from $\langle u^2 \rangle$ to the other components, and consequently $\langle u^2 \rangle$ would be considerably overestimated. This is indeed observed in figure 22(b), which presents the profiles of $\langle u^2 \rangle$ in turbulent channel flows at a similar Reynolds number $Re_\tau =1020$ computed with different streamwise domain length $L_x$, reproduced from Abe et al. (Reference Abe, Antonia and Toh2018). As shown here, as $L_x$ decreases the peak magnitude of $\langle u^2 \rangle ^+$ increases, consistently with the observation in the present study, and in the case of $L_x=0.4h$, i.e. $L_x^+\approx 400$, the overestimation of the magnitude of $\langle u^2 \rangle$ is significant throughout the channel. This tendency is particularly noticeable in the logarithmic region around $y^+ \approx 100$; a plateau of $\langle u^2 \rangle$ in this $y^+$ range, which would correspond to the outer peak of $\langle u^2 \rangle$ at higher Reynolds numbers, is already observed, whereas in larger-$L_x$ cases it does not emerge. This is likely because in the streamwise-minimal ($L_x=0.4h$) case the significant contribution by $\pi ^x_{uu}$ for $L_x^+>400$ is excluded. Hence, it is expected that $\langle u^2 \rangle$ would be similarly overestimated (and correspondingly $\langle v^2 \rangle$ and $\langle w^2 \rangle$ underestimated), particularly in the relatively far-wall region $y^+>100$, in the DNS of turbulent plane Couette flow at higher Reynolds numbers with the streamwise-minimal domain.

4.4. On the inverse interscale transfer of the Reynolds stresses observed through one-dimensional spanwise Fourier mode analysis

As observed in § 3.3, the distribution of the interscale transport of the Reynolds shear stress $tr^z_{-uv}$ obtained in the spanwise-minimal (B2) case indicates transfer from smaller to larger $\lambda _z$ throughout the channel, similarly to the reference-case results, despite the fact that the very-large-scale structure in the channel-core region does not exist in case B2 due to the small $L_z$. This suggests that the interscale Reynolds-stress transfers observed through one-dimensional spanwise Fourier mode analysis may not represent the interaction between the inner- and outer-layer structures, contrary to the suggestion of Kawata & Alfredsson (Reference Kawata and Alfredsson2018). Furthermore, even in case C (both $L_x$ and $L_z$ minimal), the same tendencies of $tr^z_{ij}$ are retained (not shown). This indicates that the spanwise interscale transport $tr^z_{ij}$ may rather be related to the individual dynamics of each near-wall and very-large-scale structure, such as their self-sustaining processes.

A Fourier-mode interaction that might possibly be related to the inverse interscale transfer of the Reynolds shear stress can be found in the work of Hamilton et al. (Reference Hamilton, Kim and Waleffe1995). In their investigation of the temporal variations of the energies of different Fourier modes during the self-sustaining cycle (in their figure 3), it is shown that when the energy of the $x$-independent streak, $M(0,\beta )$ in their notation ($\beta$ is the primary spanwise wavenumber, $\beta =2 {\rm \pi}/L_z$), decreases through the onset of streak instabilities, the energy of the spanwise-independent mode $M(\alpha,0)$ ($\alpha$ is the primary streamwise wavenumber $\alpha =2{\rm \pi} /L_x$) increases, as well as the energy of the wavy streak $M(\alpha,\beta )$, indicating not only $M(0,\beta )$-to-$M(\alpha,\beta )$ energy transfer but also $M(0,\beta )$-to-$M(\alpha,0)$ transfer via the streak instabilities. If such energy transfers were investigated through one-dimensional spanwise Fourier mode analysis, they would appear as an inverse interscale energy transfer from $\lambda _z=L_z$ to $\lambda _z=\infty$, since only the $M(0,\beta )$-to-$M(\alpha,0)$ transfer is observed. On the other hand, in terms of the streamwise Fourier mode, both the $M(0,\beta )$-to-$M(\alpha,\beta )$ and $M(0,\beta )$-to-$M(\alpha,0)$ energy transfers are observed as a forward energy transfer from $\lambda _x=\infty$ to $\lambda _x=L_x$. This means that the interscale energy transfer associated with the streak instabilities of the self-sustaining cycle may be observed as an inverse interscale energy transfer when observed through one-dimensional spanwise Fourier mode analysis. Lee & Moser (Reference Lee and Moser2019) performed two-dimensional spectral analysis on the turbulent energy transport in channel flows at high Reynolds numbers, and observed that there exist significant energy transfers between different Fourier modes with approximately the same wavenumber magnitudes but different orientations. Such ‘scale transfer in orientation’ is observed as interscale energy transfer in different directions (i.e. either a forward or a reversed cascade) when observed through one-dimensional Fourier mode analysis, depending on whether the analysis is based on the streamwise or spanwise Fourier mode. These observations in earlier studies indicate that the inverse interscale energy transfer observed based on one-dimensional Fourier mode analysis does not always mean that energy is really transported from smaller to larger scales.

Similar tendencies to the interscale energy transfer related to the streak instabilities described above are actually found in the distributions of $tr^x_{uu}$ and $tr^z_{uu}$ presented in figures 11(a) and 13(a), respectively. One can see that the spanwise interscale energy transfer $tr^z_{uu}$ provided by the reference case in figure 13(a) indicates an inverse energy transfer from smaller to larger $\lambda _z$, particularly in the near-wall region, while the streamwise interscale transport $tr^x_{uu}$ given in figure 11(a) shows only forward transfer throughout the channel. These tendencies are retained also in the minimal domains: the $tr^x_{uu}$ obtained in the streamwise-minimal case indicates a forward energy transfer from the $x$-independent mode to the $\lambda _x^+<400$ range (see figure 11a), while the $tr^z_{uu}$ obtained in the spanwise-minimal case presents a reversed energy transfer from $\lambda _z=L_z$ to the $z$-independent mode (figure 13a). These behaviours of the interscale energy transfers $tr^x_{uu}$ and $tr^z_{uu}$ are similar to the above-described energy exchange between the $x$- and $z$-independent Fourier modes in the streak instabilities of the self-sustaining cycle.

Similar tendencies are also found for the interscale transport of the Reynolds shear stress. As can be seen in figure 23(a), the distribution of the interscale transport of the Reynolds shear stress in the streamwise wavenumber direction $tr^x_{-uv}$ obtained in the reference case clearly presents forward transfers from larger- to smaller-$\lambda _x$ ranges throughout the channel, although the spanwise transport $tr^z_{-uv}$ presents inverse interscale transfers (see figure 14a). The $y$–$\lambda _x$ distribution of $tr^x_{-uv}$ provided by the streamwise-minimal case also reproduces fairly well the reference-case results for the relatively-small-wavelength range $\lambda _x^+<400$, and the Reynolds shear stress transferred from the $x$-independent mode, i.e. $tr^x_{-uv}(y,0)$, in the streamwise-minimal case is equivalent to the amount of the Reynolds shear stress removed from the corresponding $\lambda _x$ range in the reference case, $\lambda _x/h>1.6$, as presented in figure 23(b). These behaviours of the interscale transports of the Reynolds shear stress $tr^x_{-uv}$ and $tr^z_{-uv}$ are also qualitatively similar to the above-described energy exchange between the $x$- and $z$-independent modes observed in the self-sustaining cycle by Hamilton et al. (Reference Hamilton, Kim and Waleffe1995).

Figure 23. Distributions of the interscale transport of the Reynolds shear stress in the streamwise wavenumber direction $k_x tr^x_{-uv}$ obtained in the reference case and the streamwise-minimal (A3) case, presented in the same manner as in figure 8(a): $y$–$\lambda _x$ diagrams in premultiplied form $k_x tr^x_{-uv}$; (b) profiles of $tr^x_{-uv}$ integrated for $\lambda _x/h\geqq 1.6$ ($0<k_x/h<2{\rm \pi} /1.6$). Note that, in case A3, $\int ^{2{\rm \pi} /1.6h}_0 pr^x_{-uv}(k_x)\, \mathrm {d} k_x=tr^x_{-uv}(0) {\rm \Delta} k_x$.

It should be emphasised here that the inverse interscale transfer of the spanwise turbulence intensity $\langle w^2 \rangle$, which we suggested in § 4.2 to represent the energy transfers related to the regeneration process of streamwise vortices of the self-sustaining cycle, shows qualitatively different tendencies from the energy transfer by the streak instabilities described above. Figure 24 represents the distributions of the interscale transport of $\langle w^2 \rangle$ in the spanwise wavenumber direction $tr^z_{ww}$ obtained in the reference and streamwise-minimal cases, and one can see here that both results clearly indicate an energy transfer from smaller to larger wavelengths in the near-wall region, similarly to the transfer in the streamwise wavenumber direction $tr^x_{ww}$ presented in figure 20(a). Thus, the spanwise turbulent energy $\langle w^2 \rangle$ is transferred from smaller to larger scales in both the streamwise and spanwise wavenumber directions, unlike the above-described energy exchange between the $x$- and $z$-independent modes in the streak instabilities. It is also worth pointing out that in the $tr^x_{-uv}$ distribution (see figure 23a) one can find a weak reversed interscale transport from smaller to larger $\lambda _x$ in the small-wavelength range $\lambda _x^+<100$ throughout the channel, as well as the significant forward interscale transfers in the larger-$\lambda _x$ range. Hence, the Reynolds shear stress is also transferred from smaller to larger scales in both the streamwise and spanwise wavenumber directions throughout the channel at relatively small scales. While we conjecture that the inverse $\langle w^2 \rangle$ transfer is closely related to the generation process of streamwise vortices in the self-sustaining cycle as discussed in the previous section, it is not clear what the inverse $-\langle uv \rangle$ transfer at small scales represents. This will be focused on in our feature studies.

Figure 24. Space–wavelength ($y$–$\lambda _z$) diagrams of the premultiplied interscale transport of the spanwise turbulent energy in the spanwise wavenumber direction $k_z tr^z_{ww}$ obtained in (a) the reference case and (b) the streamwise-minimal (A3) case. The values are scaled by $u_\tau ^3/h$.