2.1 Mean-momentum balance

We consider a turbulent flow bounded by two infinitely long and wide parallel walls. Here, $x$ , $y$ and $z$ denote the streamwise, wall-normal and spanwise direction, and the corresponding velocities are $u$ , $v$ and $w$ , respectively. The two walls are located at $y=-h$ and $y=h$ , and they slide in opposite directions with speeds $u|_{y=-h}=-U_{w}$ and $u|_{y=h}=U_{w}$ . A fundamental feature of the turbulent Couette flow is the mean-momentum balance, obtained by taking average of the streamwise momentum equation in time and two homogeneous directions:

where $U$ is the mean velocity, $\overline{u^{\prime }v^{\prime }}$ the Reynolds shear stress (the overbar denotes the average in time and two homogeneous directions), $\unicode[STIX]{x1D70F}_{0}$ the applied shear stress, and $\unicode[STIX]{x1D70C}$ the density of the fluid. Here, we note that the validity of (2.1) is not limited only to the Couette flow – any parallel flow driven by a shear stress uniform in $y$ should satisfy this equation. Therefore, equation (2.1) must be viewed as the generic mean-momentum equation for flows driven by a locally uniform shear stress.

Equation (2.1) provides some important physical insight into the velocity and length scales relevant to the present investigation. Since (2.1) is valid at every wall-normal location, the local shear flow can be viewed as the outcome of the applied local shear stress $\unicode[STIX]{x1D70F}_{0}$ . This implies that the local velocity scale would be given from the local shear stress, such that $u_{\unicode[STIX]{x1D70F}}=\sqrt{\unicode[STIX]{x1D70F}_{0}/\unicode[STIX]{x1D70C}}$ . We note that if the wall-normal location is chosen to be at the wall (i.e. $y=\pm h$ ), $u_{\unicode[STIX]{x1D70F}}$ becomes identical to the friction velocity in the near-wall region. However, in the present case, $u_{\unicode[STIX]{x1D70F}}$ also becomes the local velocity scale at every wall-normal location, as it originates from the constant shear stress $\unicode[STIX]{x1D70F}_{0}$ in the wall-normal direction. It is also important to note that (2.1) itself does not contain any description of the ‘integral’ length scale associated with the largest admissible eddies, and the scale of the largest eddies is here determined by the size of the computational domain. If the computational domain were sufficiently large horizontally, the relevant length scale of the largest admissible eddies would be given by the height of the channel. However, if the computational domain were artificially narrowed in the spanwise direction down to $O(\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}})$ , as in a typical minimal channel simulation (Jiménez & Moin Reference Jiménez and Moin1991; Hwang Reference Hwang2013), the size of the largest admissible eddies would be restricted to $O(\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}})$ . In this case, the rate of turbulence dissipation would be given by ${\mathcal{E}}\sim u_{\unicode[STIX]{x1D70F}}^{3}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ , and the corresponding Kolmogorov length scale becomes identical to the viscous inner length scale: i.e. $\unicode[STIX]{x1D702}=\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ . Normalising (2.1) with $u_{\unicode[STIX]{x1D70F}}$ and $\unicode[STIX]{x1D702}$ then leads to

where the superscript $^{\ast }$ indicates the resulting dimensionless variables. Here, we note that the use of the superscript $^{\ast }$ , instead of the commonly used $^{+}$ for normalisation by the viscous inner units, is to highlight that the associated velocity and length scales are relevant to all wall-normal locations, including the outer and centreline region, where the use of the viscous inner scales would not usually make sense.

In (2.2), it is evident that the values of $\text{d}U^{\ast }/\text{d}y^{\ast }$ and $\overline{u^{\prime }v^{\prime }}^{\ast }$ from a DNS with a sufficiently large horizontal computational domain would be strongly dependent upon the wall-normal location, as a range of different length scales would come into play at each of the wall-normal locations – this can be easily checked by inspecting the available DNS database (see e.g. Pirozzoli, Bernardini & Orlandi Reference Pirozzoli, Bernardini and Orlandi2014). However, if the size of the eddies were artificially restricted to the given Kolmogorov length scale, the velocity and length scales of the eddies at play would be uniquely given by $u_{\unicode[STIX]{x1D70F}}$ and $\unicode[STIX]{x1D702}$ . Therefore, $\text{d}U^{\ast }/\text{d}y^{\ast }$ and $\overline{u^{\prime }v^{\prime }}^{\ast }$ are expected to be constant along the wall-normal direction, except the near-wall region where the no-slip boundary condition would break the homogeneity of (2.2). One such solution, but trivial, is given by $\text{d}U^{\ast }/\text{d}y^{\ast }=1$ and $\overline{u^{\prime }v^{\prime }}^{\ast }=0$ . However, the presence of non-zero mean shear in the flow system considered here also suggests that there may exist other non-trivial solutions of (2.2) that involve non-zero Reynolds shear stress and the resulting turbulence production. The objective of the present study is essentially to address this point.

2.2 Minimal Couette flow up to $Re_{\unicode[STIX]{x1D70F}}\simeq 800$

To seek the production mechanism at the Kolmogorov microscale, a set of minimal-span direct numerical simulations ( $L_{z}^{\ast }\simeq 100{-}140$ ) are performed. This approach, originally designed to investigate the self-sustaining nature of near-wall turbulence (e.g. Jiménez & Moin Reference Jiménez and Moin1991; Hamilton et al. Reference Hamilton, Kim and Waleffe1995; Hwang Reference Hwang2013), has recently been extended to study low-dimensional dynamics of the structures in the logarithmic and outer regions (Flores & Jiménez Reference Flores and Jiménez2010; Hwang Reference Hwang2015; de Giovanetti, Hwang & Choi Reference de Giovanetti, Hwang and Choi2016; Hwang & Bengana Reference Hwang and Bengana2016) as well as to assess the effect of wall roughness with economical computational cost (Chung et al. Reference Chung, Chan, MacDonald, Hutchins and Ooi2015; MacDonald et al. Reference MacDonald, Chung, Hutchins, Chan, Ooi and Garcia-Mayoral2017). The numerical solver used in this study is diablo, the detailed numerical method of which is well documented in Bewley (Reference Bewley2014). In this solver, the streamwise and spanwise directions are discretised using Fourier series with a 2/3 dealiasing rule, whereas the wall-normal direction is discretised using a second-order central difference. The time integration is performed semi-implicitly based on the fractional-step method (Kim & Moin Reference Kim and Moin1985). All the terms with wall-normal derivatives are implicitly advanced using a second-order Crank–Nicolson method, while the rest of the terms are explicitly integrated using a third-order low-storage Runge–Kutta method. This solver has been verified through our previous studies (e.g. Hwang Reference Hwang2013).

Table 1 summarises the simulation parameters in the present study. The simulations are performed up to $Re=50\,000$ ( $Re\equiv U_{w}h/\unicode[STIX]{x1D708}$ ), and the corresponding friction Reynolds number is found to be $Re_{\unicode[STIX]{x1D70F}}\simeq 800$ ( $Re_{\unicode[STIX]{x1D70F}}\equiv u_{\unicode[STIX]{x1D70F}}h/\unicode[STIX]{x1D708}$ ). We note that this value of friction Reynolds number is much smaller than that expected from a simulation with a sufficiently large computational domain, as will be discussed in detail in § 4.2. Both long and short streamwise computational domains are considered: the former with $L_{x}^{\ast }\simeq 3000$ is used to identify the full statistical features, while the latter with $L_{x}^{\ast }\simeq 400$ is used to inspect the low-dimensional dynamics of the motions in the minimal spanwise domain. As was extensively discussed in Hwang (Reference Hwang2013), numerical simulations with such a minimal spanwise domain contain a non-physical two-dimensional motion resolved by zero spanwise wavenumber (i.e. spanwise-independent cross-flow motion). This motion is essentially driven by the quasi two-dimensional nature of the present computational domain, and has been found to be dependent upon the size of the streamwise domain. This motion is therefore eliminated by adopting the filtering approach in Hwang (Reference Hwang2013). Finally, it should be mentioned that the spanwise domain and the filtering of the two-dimensional motions here do not affect the mean-momentum balance in (2.2), as the equation for zero streamwise and spanwise wavenumbers in our numerical solver (i.e. the spatial mean equation) is not directly modified by them (see also figure 2 c). Therefore, the time average of this equation with the spanwise-minimal domain and the removal of the spanwise-independent cross-flow motion still yields the one identical to (2.2).

3.1 Turbulence statistics

Figure 1. An instantaneous flow field of LC4 simulation. Here, the blue iso-surfaces indicate ${u^{\prime }}^{\ast }=-2$ , while the red ones are ${v^{\prime }}^{\ast }=1.5$ .

We first consider simulations with a long streamwise computational domain (i.e. LC simulations in table 1). Figure 1 is a visualisation of an instantaneous streamwise and wall-normal velocity field from the simulation performed at the highest Reynolds number (i.e. LC4). The flow field is occupied by small-scale eddies, the spanwise size of which just fits in the given spanwise computational domain. The motions associated with the streamwise velocity fluctuation (blue iso-surface in figure 1) tend to be elongated in the streamwise direction, but they are not very long. This feature becomes more evident if their streamwise extent is compared with that of the eddies in the near-wall region: the near-wall streamwise velocity fluctuation appears to be much longer than that in the bulk region of the flow. On the other hand, the streamwise extent of the wall-normal velocity fluctuations in the bulk region is slightly shorter than that for the streamwise velocity, and it does not seem to be greatly changed in the near-wall region (red iso-surface in figure 1).

Figure 2. Turbulence statistics for mean-momentum balance and production: (a)  $U^{\ast }(y^{\ast })$ ; (b)  $-\overline{u^{\prime }v^{\prime }}^{\ast }$ ; (c)  $(\text{d}U^{\ast }/\text{d}y^{\ast })-\overline{u^{\prime }v^{\prime }}^{\ast }$ ; (d)  $-\overline{u^{\prime }v^{\prime }}^{\ast }\,(\text{d}U^{\ast }/\text{d}y^{\ast })$ . Here, $\cdots \cdots$ , LC1; – ⋅ – ⋅ –, LC2; - - - -, LC3; ——, LC4.

Figure 2 shows turbulence statistics involved in the construction of the mean-momentum balance and turbulence production. Here, the wall-normal coordinate in figure 2 is given by $y^{\ast }=y/\unicode[STIX]{x1D702}$ , so that $y^{\ast }=0$ corresponds to the centreline of the channel. This unconventional choice for the wall-normal coordinate is specifically to distinguish the Kolmogorov length scale $\unicode[STIX]{x1D702}$ from the viscous inner length scale $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ . In the present study, we also show the statistics only for the upper half of the channel, as they are symmetric about $y^{\ast }=0$ . The mean velocity profile and the Reynolds shear stress for all the considered Reynolds numbers show impressively clear scaling with $u_{\unicode[STIX]{x1D70F}}$ and $\unicode[STIX]{x1D702}$ at all the wall-normal locations, except in the near-wall region where $y^{\ast }$ is largest (figure 2 a,b). In the bulk region, both the mean shear rate and the Reynolds shear stress are almost constant, being $\text{d}U^{\ast }/\text{d}y^{\ast }\simeq 0.07$ and $-\overline{u^{\prime }v^{\prime }}^{\ast }\simeq 0.93$ . They also exactly satisfy the mean-momentum balance in (2.2) throughout all wall-normal locations (figure 2 c), demonstrating that the spanwise-minimal domain and the filtering of the two-dimensional motions do not affect the mean-momentum balance (2.2). Finally, the constant values of the mean shear rate and the Reynolds shear stress at the Kolmogorov scale also imply a non-zero constant value of turbulence production given by $-\overline{u^{\prime }v^{\prime }}^{\ast }\,(\text{d}U^{\ast }/\text{d}y^{\ast })\simeq 0.065$ (figure 2 d). This evidently indicates that the turbulence production mechanism can indeed be active at the Kolmogorov microscale in the bulk region of the flow.

Figure 3. Turbulent velocity fluctuations: (a) $u_{rms}^{\ast }(y^{\ast })$ ; (b) $v_{rms}^{\ast }(y^{\ast })$ ; (c) $w_{rms}^{\ast }(y^{\ast })$ . Here, $\cdots \cdots$ , LC1; – ⋅ – ⋅ –, LC2; - - - -, LC3; ——, LC4.

The velocity fluctuations are shown in figure 3. As expected, all the turbulent velocity fluctuations exhibit very good scaling with $u_{\unicode[STIX]{x1D70F}}$ and $\unicode[STIX]{x1D702}$ over all wall-normal locations, except the near-wall region. In the bulk region, the velocity fluctuations are found to be $u_{rms}^{\ast }\simeq 1.7$ , $v_{rms}^{\ast }\simeq 0.93$ and $w_{rms}^{\ast }\simeq 1.3$ . The different values of the velocity fluctuations suggest that the eddies populating the bulk are anisotropic, as one might expect given the presence of mean shear. In the near-wall region, the anisotropic nature of the velocity fluctuations becomes very strong: in particular, the magnitude of the near-wall streamwise velocity fluctuation is much larger than that in the bulk region; the other components vary only a little in the near-wall region before the drop in the viscous sub-layer. The near-wall velocity fluctuations are also found to scale very well in the wall-normal coordinate normalised by the viscous inner length scale (i.e. $y^{+}=(y+h)/\unicode[STIX]{x1D6FF}_{v}$ ), consistent with the case of pressure-driven minimal turbulent channel (Hwang Reference Hwang2013). However, this issue does not exactly fall within the scope of the present study, and the related discussion is given in the Appendix.

Figure 4. One-dimensional premultiplied (a,c,e,g) spanwise and (b,d,f,h) streamwise wavenumber spectra of (a,b) streamwise velocity, (c,d) wall-normal velocity, (e,f) spanwise velocity and (g,h) Reynolds stress. Here: shaded, LC1; solid, LC4. In (a,b), the contour levels are 0.125, 0.25 and 0.375 times of each of the maximum, while in (c–h), they are 0.25, 0.5 and 0.75 times of each of the maximum.

One-dimensional spectra also exhibit essentially uniform statistics in the bulk region. Figure 4 shows the spanwise and streamwise wavenumber spectra of streamwise, wall-normal, spanwise velocities and Reynolds shear stress from the lowest and the highest Reynolds numbers (LC1 and LC4). All the spectra in figure 4 exhibit remarkably good scaling with $u_{\unicode[STIX]{x1D70F}}$ and $\unicode[STIX]{x1D702}$ , while being constant at all wall-normal locations except the near-wall region. The peaks in the spanwise wavenumber spectra of all the variables (figure 4 a,c,e,g) appear at the largest spanwise wavelength, indicating that the spanwise size of the motions at all the wall-normal locations in the flow field is restricted by the narrow spanwise computational domain (note that the motions uniform in the spanwise direction are eliminated in the simulations, as discussed in § 2.2). The streamwise wavenumber spectra are also shown to indicate the streamwise length scale of the structures in the bulk region. The streamwise extent of the streamwise velocity structure is characterised by $\unicode[STIX]{x1D706}_{x}^{\ast }\simeq 300$ (figure 4 b), whereas that of the wall-normal and spanwise velocity structures is $\unicode[STIX]{x1D706}_{x}^{\ast }\simeq 200$ (figure 4 d,f). The streamwise length scale of the Reynolds shear stress is rather similar to that of the streamwise velocity spectra, revealing the peak at $\unicode[STIX]{x1D706}_{x}^{\ast }\simeq 300$ (figure 4 h). The spectra in the near-wall region are much more anisotropic than those in the bulk region, and their peaks generally appear at larger streamwise wavelengths. The overall features of the spectra in the near-wall region are almost identical to those observed in the spanwise-minimal unit of pressure-driven channel flow (Hwang Reference Hwang2013), as discussed in the Appendix (see also figure 14).

Figure 5. The wall-normal correlation functions with the reference point at $y_{ref}=0$ : (a) $R_{uu}(r^{\ast };0)$ ; (b) $R_{vv}(r^{\ast };0)$ ; (c) $R_{ww}(r^{\ast };0)$ . Here, $\cdots \cdots$ , LC1; – ⋅ – ⋅ –, LC2; - - - -, LC3; ——, LC4.

Figure 1 clearly shows that the wall-normal size of the eddies in the bulk region is not very large, which follows from the fact that all the turbulence statistics are constant in the wall-normal direction (figures 2–4). Therefore, to characterise the wall-normal size of the eddies, a two-point correlation function in the wall-normal direction is introduced. For example, the correlation function for the streamwise velocity is given by

where $y_{ref}$ is the reference point. Similarly, the correlation functions for the wall-normal and spanwise velocities are computed, and denoted by $R_{vv}(r;y_{ref})$ and $R_{ww}(r;y_{ref})$ , respectively.

The computed correlation functions with $y_{ref}=0$ (i.e. the centre of the channel) are shown in figure 5 at all the Reynolds numbers considered. All the computed correlation functions for each velocity component are almost identical when they are scaled with the Kolmogorov length scale $\unicode[STIX]{x1D702}$ . Their tails are found to reach zero roughly at $r^{\ast }\simeq \pm 50\sim \pm 100$ , indicating that the wall-normal size of the eddies would be $l^{\ast }\simeq 50\sim 100$ . This also implies that the constant turbulence statistics in the bulk region of the flow along the wall-normal direction (figures 2–4) are due to the homogeneous distribution of the statistically and dynamically identical eddies at the Kolmogorov scale at all wall-normal locations except the near-wall. In this respect, it should be mentioned that any solutions of the Navier–Stokes equation with uniform shear flow would be invariant under a wall-normal translation, as long as the solutions are not affected by boundary conditions that would break the invariance (e.g. no-slip condition). It is evident that the eddies in the bulk region of the present numerical simulations belong to this case, and the only expected difference between the eddies at different wall-normal locations should therefore be their mean streamwise advection velocity.

3.2 Self-sustained structures at the Kolmogorov scale

Figure 6. Temporal evolution of flow field in the core region ( $0\leqslant y^{\ast }\leqslant 50$ ) from the SC2 simulation: (a) time trace of $E_{u}^{\ast }$ (solid) and $E_{v}^{\ast }$ (dashed); (b) magnification of (a) for $t^{\ast }\in [1476,1658]$ ; (c–f) the corresponding flow visualisation. In (c–f), the red and blue iso-surfaces indicate ${u^{\prime }}^{\ast }=-1.2$ and ${v^{\prime }}^{\ast }=0.8$ , respectively.

Now, to study low-dimensional dynamics of the motions in the bulk region, we consider a much shorter streamwise computational box size (SC cases in table 1). Particular emphasis of this section is given to find any links between the dynamics of the eddies in the bulk region and the SSP/VWI. Instantaneous flow fields around $y^{\ast }=0$ are first inspected while tracking the turbulent kinetic energy of each velocity component obtained by averaging over a small domain around the centreline: i.e.

where $\unicode[STIX]{x1D6FA}^{\ast }=[0,L_{x}^{\ast }]\times [0,L_{y}^{\ast }]\times [0,L_{z}^{\ast }]$ with $l_{y}^{\ast }=50$ and $V^{\ast }$ is the volume of $\unicode[STIX]{x1D6FA}^{\ast }$ . Here, we note that the size of $\unicode[STIX]{x1D6FA}^{\ast }$ is chosen to be just enough for a single eddy at the Kolmogorov length scale (see also the wall-normal autocorrelation in figure 5).

Figure 7. Temporal auto correlation: (a) $C_{uu}(\unicode[STIX]{x1D70F}^{\ast })$ ; (b) $C_{vv}(\unicode[STIX]{x1D70F}^{\ast })$ ; and (c) $C_{ww}(\unicode[STIX]{x1D70F}^{\ast })$ . Here, $\cdots \cdots$ , SC1; – ⋅ – ⋅ –, SC2; - - - -, SC3; ——, SC4. Note that all the correlation functions are almost identical when normalised by the Kolmogorov scales.

Figure 6 shows the temporal evolution of $E_{u}^{\ast }$ and $E_{v}^{\ast }$ and the corresponding flow visualisation in the region $\unicode[STIX]{x1D6FA}^{\ast }$ . Since $E_{v}^{\ast }$ and $E_{w}^{\ast }$ have been found to show strong correlation (see figure 11 d), $E_{w}^{\ast }$ is not shown here for brevity. Both $E_{u}^{\ast }$ and $E_{v}^{\ast }$ show temporal oscillations with a time scale of $T^{\ast }\simeq 200\sim 300$ (figure 6 a), and occur with some phase difference. Although the flow fields are fairly chaotic in general and the structures are also found to often move up and down across the wall-normal boundary of $\unicode[STIX]{x1D6FA}^{\ast }$ , a careful observation leads us to conclude that the overall behaviour of the flow fields is quite similar to the SSP/VWI observed in Hamilton et al. (Reference Hamilton, Kim and Waleffe1995). The strong wall-normal velocity structures (figure 6 c) are transformed into a highly amplified and elongated streamwise velocity structure, reminiscent of a ‘streak’ (figure 6 d). The amplified streamwise velocity structure meanders along the streamwise direction (figure 6 e), and eventually breaks down with regeneration of the wall-normal velocity structures (figure 6 f). This process occurs in a cyclic manner, detailed later with respect to figure 11.

The eddy turn-over dynamics observed in figure 6 is also found to scale very well with the Kolmogorov microscale. This is shown by computing correlation functions with the variables defined in (3.2):

where $i,j=u,v,w$ and $\langle \cdot \rangle$ indicates average in time. Figure 7 shows the autocorrelations computed for all the Reynolds numbers. They are indeed almost indistinguishable when scaled by $u_{\unicode[STIX]{x1D70F}}^{\ast }$ and $\unicode[STIX]{x1D702}$ . The autocorrelation functions of all the variables reach zero at $t^{\ast }\simeq \pm 100\sim \pm 150$ , suggesting that the time scale of the eddy turn-over dynamics in figure 6 would be $T^{\ast }\simeq 200\sim 300$ . Interestingly, this time scale is almost identical to that of the self-sustaining process in near-wall turbulence (e.g. Jiménez et al. Reference Jiménez, Kawahara, Simens, Nagata and Shiba2005) (see also § 4.1 for further discussion).

The visual inspection of a series of flow fields suggests that the eddies at the Kolmogorov scale in the bulk region of the flow appear to support the SSP/VWI mechanism. It is important to mention that given fluid motions, the dynamics of which is governed by the SSP/VWI mechanism, should be able to sustain themselves in the absence of other motions (e.g. Jiménez & Pinelli Reference Jiménez and Pinelli1999; Hwang & Cossu Reference Hwang and Cossu2010; Hwang Reference Hwang2015). Therefore, to provide more solid evidence on the existence of the SSP/VWI mechanism, we design a numerical experiment which removes all the motions except the ones in the core region. The removal has been performed by applying a damping technique similar to the one described in Jiménez & Simens (Reference Jiménez and Simens2001). At each Runge–Kutta substep for the time integration, the right-hand side of the discretised momentum equation, denoted by $\text{RHS}$ , is multiplied by a damping function $\unicode[STIX]{x1D707}(y)$ : i.e.

where $\widehat{\cdot }$ denotes the Fourier-transformed state, and $k_{x}$ and $k_{z}$ are the streamwise and spanwise wavenumbers, respectively. The damping function $\unicode[STIX]{x1D707}(y)$ is chosen as

where the damping factor $\unicode[STIX]{x1D707}_{0}$ provides removal of the motions at $|y^{\ast }|\geqslant \unicode[STIX]{x1D6FF}^{\ast }$ if $\unicode[STIX]{x1D707}_{0}<1$ . Several different values of $\unicode[STIX]{x1D707}_{0}$ have been tested, and, in the present study, we have chosen $\unicode[STIX]{x1D707}_{0}=0.9$ , which provides a marginal damping just enough to remove the target structures. Finally, it should be mentioned that the damping technique (3.4) is not applied to the mean component for $k_{x}=0$ and $k_{z}=0$ to maintain the mean-momentum balance (2.2) (see also figure 9 c).

Figure 8. Instantaneous flow fields of the FSC2 simulation with different damping height: (a) $\unicode[STIX]{x1D6FF}^{\ast }=333$ ; (b) $\unicode[STIX]{x1D6FF}^{\ast }=126$ ; (c) $\unicode[STIX]{x1D6FF}^{\ast }=70$ . Here, the blue iso-surfaces indicate ${u^{\prime }}^{\ast }=-2$ , while the red ones are ${v^{\prime }}^{\ast }=1.5$ .

The removal of the structures except in the core region is performed by gradually decreasing $\unicode[STIX]{x1D6FF}^{\ast }$ to a value at which turbulence in the core region is only just sustained. Table 2 summarises simulation parameters of this numerical experiment. Here, the aspect ratio of the computational domain is kept to be $L_{x}/L_{z}=3$ . Figure 8 shows visualisations of the remaining turbulent structures for three different $\unicode[STIX]{x1D6FF}^{\ast }$ values. The smallest possible $\unicode[STIX]{x1D6FF}^{\ast }$ that maintains turbulence in the core region is found to be $\unicode[STIX]{x1D6FF}^{\ast }\simeq 70$ . We note that this value is consistent with the wall-normal autocorrelation in figure 5, where the correlation functions reach zero at $y^{\ast }\simeq 50{-}100$ , indicating that the core-region structures, the size of which is $O(100\unicode[STIX]{x1D702})$ , sustain themselves.

Figure 9. Turbulence statistics for mean-momentum balance and production: (a)  $U^{\ast }(y^{\ast })$ ; (b)  $-\overline{u^{\prime }v^{\prime }}^{\ast }$ ; (c)  $(\text{d}U^{\ast }/\text{d}y^{\ast })-\overline{u^{\prime }v^{\prime }}^{\ast }$ ; (d)  $-\overline{u^{\prime }v^{\prime }}^{\ast }(\text{d}U^{\ast }/\text{d}y^{\ast })$ . Here, - - - -, SC2; ——, FSC2.

Turbulence statistics for mean-momentum balance and production from the simulation only with the self-sustaining core structures (FSC2) are compared with those of a minimal channel simulation in figure 9. Despite the harsh nature of the applied damping (3.4), the mean shear rate, Reynolds shear stress and turbulence production in the core region ( $y^{\ast }<30\sim 40$ ) of the FSC2 simulation still show reasonably good agreement with those of the full minimal channel simulation. This suggests that the dynamics of the structures in the core region is not significantly disrupted by damping of the structures at $|y^{\ast }|>70$ . Application of the damping (3.4) is also found to make the mean shear rate $\text{d}U^{\ast }/\text{d}y^{\ast }$ rapidly change for $50\lesssim y^{\ast }\lesssim 70$ (figure 9 a), where the Reynolds shear stress rapidly decays (figure 9 b) to satisfy the mean-momentum balance (2.2) (figure 9 c). In this location, turbulence production also exhibits a large peak (figure 9 d), and, interestingly, its peak value appears to be remarkably similar to what is observed in the near-wall region, where the Reynolds shear stress also decays rapidly due to the no-slip boundary condition (see the dashed line in figure 9 d around $y^{\ast }\simeq 500$ ). A detailed discussion on this observation will be given in § 4.1.

Figure 10. Temporal evolution of flow field in the core region ( $0\leqslant y^{\ast }\leqslant 70$ ) from the FSC2 simulation: (a) time trace of $E_{u}^{\ast }$ (solid) and $E_{v}^{\ast }$ (dashed); (b) magnification of (a) for $t^{\ast }\in [1335,1597]$ ; (c–f) the corresponding flow visualisation. In (c–f), the red and blue iso-surfaces indicate ${u^{\prime }}^{\ast }=-2.5$ and ${v^{\prime }}^{\ast }=0.8$ , respectively.

Temporal evolution of the flow structures in the FSC2 simulation is shown in figure 10. The time trace of $E_{u}^{\ast }$ and $E_{v}^{\ast }$ in the FSC2 simulation also exhibits temporal oscillations, and their time scale appears to be roughly at $T^{\ast }\simeq 200\sim 300$ , not very different from that observed in figure 6 (figure 10 a). The temporal oscillations of $E_{u}^{\ast }$ and $E_{v}^{\ast }$ also show a phase difference (figure 10 b). Visualisation of the instantaneous flow fields along the different phase in a single oscillation period reveals that the motions in the FSC2 simulation also seem to undergo the SSP/VWI, similarly to the observation made in figure 6. The strong wall-normal velocity structures (figure 10 c) highly amplify a streaky structure (figure 10 d). The amplified streak subsequently meanders along the streamwise direction (figure 10 e), resulting in its breakdown with regeneration of the wall-normal velocity structure (figure 10 f).

Figure 11. Comparison of auto- and cross-correlation functions from (a,c,e) FSC2 and (b,d,f) SC2 simulations: (a,b) $C_{uu}$ , $C_{vv}$ , $C_{ww}$ ; (c,d) $C_{uv}$ , $C_{uw}$ , $C_{vw}$ ; (e,f) $C_{0u}$ , $C_{0v}$ , $C_{1v}$ .

To obtain a more quantitative comparison between the dynamics of the FSC2 and SC2 simulations, auto- and cross-correlations of the time-dependent variables such as those defined in (3.2) are examined as in, for example, Hwang & Bengana (Reference Hwang and Bengana2016). Here, we also introduce two additional variables:

for $k_{x}=0$ and $k_{z}=2\unicode[STIX]{x03C0}/L_{z}$ , and

for $k_{x}=2\unicode[STIX]{x03C0}/L_{x}$ and $k_{z}=2\unicode[STIX]{x03C0}/L_{z}$ , where $E_{0}^{\ast }$ represents energy of the elongated motion extending over the entire streamwise domain, while $E_{1}^{\ast }$ measures the strength of wavy meandering motion (i.e. streak instability). The cross-correlation functions are subsequently computed following the definition in (3.3).

Figure 11 compares a number of different auto- and cross-correlations computed from the FSC2 simulation with those from the SC2 simulation. The autocorrelations from both FSC2 (figure 11 a) and SC2 simulations (figure 11 b) show their tails reaching zero at $\unicode[STIX]{x1D70F}^{\ast }\simeq \pm 100\sim \pm 150$ , indicating that the turn-over time scale of the structures in both of the cases would be $T^{\ast }\simeq 200\sim 300$ , consistent with the observations in figures 6 and 10. The cross-correlations between $E_{u}^{\ast }$ , $E_{v}^{\ast }$ and $E_{w}^{\ast }$ of both cases (figure 11 c,d) reveal that the wall-normal and spanwise velocities are well correlated to each other, indicating the presence of streamwise roll-like structures ( $C_{vw}$ in figure 11 c,d). On the other hand, $E_{v}^{\ast }$ and $E_{w}^{\ast }$ exhibit a phase difference from $E_{u}^{\ast }$ : the peaks of $C_{uv}$ and $C_{uw}$ in both cases appear at $\unicode[STIX]{x1D70F}^{\ast }\simeq -60\sim -40$ . The cross-correlations of $E_{u}^{\ast }$ with $E_{0}^{\ast }$ in both cases ( $C_{0u}$ in figure 11 e,f) show a large value around $\unicode[STIX]{x1D70F}^{\ast }\simeq 0$ . This implies that $E_{u}^{\ast }$ represents the elongated motion of the streamwise velocity, indicating the streaky motion in figures 6 and 10. For this reason, the cross-correlation between $E_{0}^{\ast }$ and $E_{v}^{\ast }$ in both cases ( $C_{0v}$ in figure 11 e,f) exhibits a peak at $\unicode[STIX]{x1D70F}^{\ast }\simeq -80\sim -50$ , similarly to $C_{uv}$ . Finally, the cross-correlation between $E_{1}^{\ast }$ and $E_{v}^{\ast }$ in both cases ( $C_{1v}$ in figure 11 e,f) reveals its peak location around $\unicode[STIX]{x1D70F}^{\ast }\simeq -40\sim -20$ . This indicates that a streamwise meandering motion consistently appears before the streamwise roll-like structures are generated.

Overall, the auto- and cross-correlations from the FSC2 and SC2 simulations show qualitatively good agreement with each other, suggesting that the dynamics of the self-sustaining structures in the FSC2 case is not very different from that in the SC2 case. Furthermore, the correlations also suggest that there exists a statistically cyclic process in the core region of both simulations, which occurs in the following order:

with the turn-over time scale, $T^{\ast }\simeq 200\sim 300$ . The fluid motion associated with $E_{u}^{\ast }$ and $E_{0}^{\ast }$ indicates an elongated streamwise velocity structure, very likely to be a ‘streak’, and $E_{1}^{\ast }$ indicates its streak meandering motion is presumably caused by a streak instability. The streak instability process occurs before the generation of streamwise roll-like structures represented by $E_{v}^{\ast }$ and $E_{w}^{\ast }$ . The statistical evidence in figure 11 therefore firmly indicates the existence of the SSP/VWI in the core region of the SC2 simulation. Here, it should be stressed that the existence of the SSP/VWI in the SC2 simulation is not limited only to the core region. The observed dynamics in the core region should be invariant under a wall-normal translation, thus it has to be relevant to other wall-normal locations, except the near-wall region where the translational invariance would be broken due to the no-slip boundary condition. However, also in the near-wall region, it is well understood that the SSP/VWI is the main driving mechanism of turbulence (Jiménez & Moin Reference Jiménez and Moin1991; Jiménez & Pinelli Reference Jiménez and Pinelli1999). In summary, we conclude that the SSP/VWI is the dominant mechanism of turbulence production at all wall-normal locations in the present minimal-span simulations.