3 Results

The mean velocity profiles for each Couette flow case are shown in  figure 1. Both the mean velocity and the log-layer diagnostic $\unicode[STIX]{x1D6FD}=y^{+}\langle \unicode[STIX]{x2202}u^{+}/\unicode[STIX]{x2202}y^{+}\rangle$ are shown. As expected, the Reynolds numbers are too low to produce a convincing log layer (constant $\unicode[STIX]{x1D6FD}$ ). Indeed in Poiseuille flow, an unambiguous log layer does not appear until $Re_{\unicode[STIX]{x1D70F}}\approx 5000$ (Lee & Moser Reference Lee and Moser2015). Nonetheless, $\unicode[STIX]{x1D6FD}$ is more sensitive to small variations in the mean profile, and therefore it is a good diagnostic for detecting such differences. The $\unicode[STIX]{x1D6FD}$ profiles are consistent among all the cases up to $y^{+}\approx 50$ . Beyond this, $\unicode[STIX]{x1D6FD}$ develops a plateau as $Re$ increases. However, at $Re_{\unicode[STIX]{x1D70F}}\approx 500$ , $\unicode[STIX]{x1D6FD}$ increases slowly until $y/\unicode[STIX]{x1D6FF}\approx 0.3$ then decreases sharply, as previously observed by Avsarkisov et al. (Reference Avsarkisov, Hoyas, Oberlack and García-Galache2014) and Orlandi, Bernardini & Pirozzoli (Reference Orlandi, Bernardini and Pirozzoli2015). At the low Reynolds numbers, there is no discernible difference in $\unicode[STIX]{x1D6FD}$ between the small and large domain cases, but there is a non-negligible difference away from the wall in the R500 cases. In all cases, $\unicode[STIX]{x1D6FD}$ rolls up at the centre, even in the higher Reynolds number cases in which it drops sharply above $y/\unicode[STIX]{x1D6FF}\approx 0.3$ . This is required since at the centre $\text{d}\unicode[STIX]{x1D6FD}/\text{d}y=\langle \unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y\rangle >0$ because $\langle \unicode[STIX]{x2202}^{2}u/\unicode[STIX]{x2202}y^{2}\rangle =0$ by symmetry.

Figure 1.

Mean velocity profile (a) and log-layer diagnostic function (b); cases are defined in table 1.

Figure 2.

Mean velocity gradient at the centre of channel.

The derivative of the mean velocity at the centre of the domain is always zero in Poiseuille flows by symmetry. But in Couette flow the mean velocity is anti-symmetric, and so the velocity gradient is not necessarily zero at the centre line, and therefore, neither is the production of turbulent kinetic energy. The behaviour of the mean velocity gradient at the centre as $Re$ increases is important to understanding the production of $u^{\prime 2}$ in the outer region, but as noted by Avsarkisov et al. (Reference Avsarkisov, Hoyas, Oberlack and García-Galache2014) this remains controversial (Busse Reference Busse1970; Lund & Bush Reference Lund and Bush1980). Standard defect law scaling with $u_{\unicode[STIX]{x1D70F}}$ and $\unicode[STIX]{x1D6FF}$ , and the common assumption that $C_{f}=2u_{\unicode[STIX]{x1D70F}}^{2}/U_{w}^{2}$ goes to zero as $Re\rightarrow \infty$ suggest that the mean velocity gradient scaled by $U_{w}$ and $\unicode[STIX]{x1D6FF}$ should go to zero at large Reynolds number. Indeed the data shown in  figure 2 indicate that the gradient drops by almost a factor of 3 over the factor of 5 increase in Reynolds number investigated here. However, the Reynolds numbers are too low and the range of Reynolds numbers too small to suggest the high Reynolds number asymptotic behaviour of the centre line derivative. It is also clear that there are modest differences in this derivative between the small and large domain cases, as large as 14 % in the R500 case. Further, there are differences in the centre line gradient between the current simulations and those of Avsarkisov et al. (Reference Avsarkisov, Hoyas, Oberlack and García-Galache2014), for unknown reasons. But this could be due to large uncertainties in the statistics at the centre of channel.

Figure 3.

Profiles of the non-zero Reynolds stress components. Cases are defined in table 1, and the R500S-LS curves are the contribution of large-scale structures to the Reynolds stress.

The non-zero Reynolds stress components appear in figure 3. When plotted in wall units, many features are similar to those in Poiseuille flow, especially near the wall, such as the increase in the peak of ${u^{\prime }}^{2}$ and to a lesser extent ${w^{\prime }}^{2}$ with Reynolds number. Away from the wall, particularly at the centre line, ${u^{\prime }}^{2}$ has a strong dependence on $Re$ , which does not occur in Poiseuille flow (Lee & Moser Reference Lee and Moser2015). This may be related to the fact that the production of ${u^{\prime }}^{2}$ is zero in Poiseuille flow by symmetry, but not Couette flow. Note that there is a mild secondary peak in ${u^{\prime }}^{2}$ at approximately $y^{+}=200$ in the R500 cases. Such an outer peak does not occur in Poiseuille flow for $Re_{\unicode[STIX]{x1D70F}}$ up to 5200 (Lee & Moser Reference Lee and Moser2015), but it is observed in both experiments at very high $Re$ in boundary layers and pipes (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012; Marusic et al. Reference Marusic, Mathis and Hutchins2010). There is no appreciable Reynolds number effect on ${v^{\prime }}^{2}$ or $u^{\prime }v^{\prime }$ in these flows, which is also different from Poiseuille flow. The effect of domain size on the Reynolds stress components is negligible in most cases. The one exception is ${u^{\prime }}^{2}$ , in which there is a modest difference between the large and small domain cases away from the wall. In general, the current Reynolds stress component data agree with that from Avsarkisov et al. (Reference Avsarkisov, Hoyas, Oberlack and García-Galache2014), with only minor discrepancies, which are within the uncertainties described in appendix A.

Figure 4.

Contours of the $k_{x}$ -premultiplied streamwise one-dimensional spectra of $u_{i}^{\prime }u_{j}^{\prime }$ for the long domain Couette flow cases identified in table 1, and the planar Poiseuille flow of Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014) at $Re_{\unicode[STIX]{x1D70F}}=550$ , labelled R550P.

Figure 5.

Contours of the $k_{z}$ -premultiplied spanwise one-dimensional spectra of $u_{i}^{\prime }u_{j}^{\prime }$ for the long domain Couette flow cases identified in table 1, and the planar Poiseuille flow of Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014) at $Re_{\unicode[STIX]{x1D70F}}=550$ , labelled R550P.

While the effects of the computational domain size on the one-point statistics is modest, the presence of large-scale turbulent eddies in the larger domain is expected to have a large effect on the spectra, which are shown as a function of $y$ in figures 4 and 5. The so-called premultiplied spectrum $kE(k)$ is plotted because the wavenumber axis is logarithmic and integrating $kE(k)$ over $d\log (k)$ yields the energy. Spectra of the non-zero components of the Reynolds stress tensor, including the cross-spectra of $uv$ are shown for each of the large domain cases. In addition, for comparison, we include the spectra from the large domain Poiseuille flow at $Re_{\unicode[STIX]{x1D70F}}=550$ conducted by Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014) (labelled R550P) with $L_{x}=60\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}$ and $L_{z}=6\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}$ . A striking feature of these spectra is that for small enough scales (say $k_{x}^{+}>0.002$ and $k_{z}^{+}>0.02$ ) and close enough to the wall (say $y^{+}<60$ ), the spectra of each Reynolds stress component are the same regardless of Reynolds number and including the Poiseuille flow. This is due to the approximate universality of small-scale near-wall turbulence.

Another feature of the Couette flow spectra is the sharp peak in the spanwise ( $k_{z}$ ) spectrum at around $k_{z}\unicode[STIX]{x1D6FF}\approx 1.2$ . In $E_{uu}$ and $E_{ww}$ , this peak spans from the centre line to approximately $y^{+}\approx 5$ , while in $E_{vv}$ and $E_{uv}$ it occurs only further from the wall. This peak does not appear in the Poiseuille flow (R550P). Note that peaks of $E_{uu}$ at low wavenumber in the outer flow are observed in boundary layers and channels at higher $Re$ (say $Re_{\unicode[STIX]{x1D70F}}=5000$ or higher) in experiments and DNS (Hutchins & Marusic Reference Hutchins and Marusic2007; Lee & Moser Reference Lee and Moser2015). In addition, DNS data of channel flow exhibit an outer peak of $E_{uv}$ in flows at higher $Re$ (Lee & Moser Reference Lee and Moser2015). The outer peaks of $E_{uu}$ and $E_{uv}$ in boundary layers and channels do not extend from the outer flow to near the wall, and they are much broader in wavenumber. They seem to be a different feature. So the spectral peak observed here appears to be a consequence of the large-scale motions specific to Couette flow. The spanwise scale of these structures is consistent with previous observations by others (Tsukahara et al. Reference Tsukahara, Kawamura and Shingai2006; Avsarkisov et al. Reference Avsarkisov, Hoyas, Oberlack and García-Galache2014; Pirozzoli et al. Reference Pirozzoli, Bernardini and Orlandi2014).

In the streamwise Couette flow spectra, there is a corresponding peak in $E_{uu}$ and $E_{uv}$ at $k_{x}\unicode[STIX]{x1D6FF}\approx 0.08$ at $Re_{\unicode[STIX]{x1D70F}}=220$ and it is broadly consistent with the experiments of Kitoh et al. (Reference Kitoh, Nakabyashi and Nishimura2005) and Kitoh & Umeki (Reference Kitoh and Umeki2008). In the $Re_{\unicode[STIX]{x1D70F}}=500$ case, the $E_{uv}$ spectrum has a peak near the centre line at $k_{x}\unicode[STIX]{x1D6FF}=0.02$ and there are multiple weaker low wavenumber peaks in $E_{uu}$ , with the lowest also occurring at $k_{x}\unicode[STIX]{x1D6FF}\approx 0.02$ . The energy in the large-scale modes is thus distributed across a range of wavenumbers. Note that $k_{x}\unicode[STIX]{x1D6FF}=0.02$ is the lowest non-zero wavenumber in $x$ , corresponding to a wavelength equal to the domain size. At $Re_{\unicode[STIX]{x1D70F}}=500$ , both $E_{vv}$ and $E_{ww}$ also exhibit low wavenumber peaks, although they are so weak they are difficult to detect in the figure. These weak spectral peaks occur at $k_{x}\unicode[STIX]{x1D6FF}\approx 0.02$ and $0.1$ . The streamwise extent of the structures in Couette flow, as measured by the wavelength of these spectral peaks is quite large, up to $310\unicode[STIX]{x1D6FF}$ at $Re_{\unicode[STIX]{x1D70F}}=500$ . And, the extent appears to increase with Reynolds number, as it is only approximately $78\unicode[STIX]{x1D6FF}$ at $Re_{\unicode[STIX]{x1D70F}}=220$ . Because the low wavenumber peak at $Re_{\unicode[STIX]{x1D70F}}=500$ occurs at the lowest represented wavenumber, the streamwise structure of the these large-scale motions is impacted by the finite domain size, so our assessment of their natural streamwise length-scale cannot be definitive.

Note that in the spanwise spectra, the large-scale spectral peaks we have been discussing have their maximum magnitude at the centre line, except for $E_{ww}$ , where it is very weak at the centre line. Furthermore, these peaks extend much closer to the wall in $E_{uu}$ and $E_{ww}$ , than for $E_{vv}$ . These features are consistent with large-scale motions dominated by large-scale streamwise vortices that extend from one wall to the other, as proposed by Komminaho et al. (Reference Komminaho, Lundbladh and Johansson1996), Papavassiliou & Hanratty (Reference Papavassiliou and Hanratty1997), Tsukahara et al. (Reference Tsukahara, Kawamura and Shingai2006), Avsarkisov et al. (Reference Avsarkisov, Hoyas, Oberlack and García-Galache2014). Similar spectral indicators of large-scale streamwise vortices were observed in transitional Couette flow (Tuckerman & Barkley Reference Tuckerman and Barkley2011).

The two-dimensional energy spectra of the non-zero Reynolds stress components are shown in figure 6. Shown are two $k_{x}$ – $k_{z}$ planes with $y^{+}=30$ and $y/\unicode[STIX]{x1D6FF}=1$ . As expected, the dominant small-scale mass of the spectra shifts to larger $k_{x}\unicode[STIX]{x1D6FF}$ and $k_{z}\unicode[STIX]{x1D6FF}$ (smaller scale) with increase Reynolds number. In addition, the two-dimensional spectra have peaks at $k_{z}\unicode[STIX]{x1D6FF}\approx 1.2$ for both the R220L and R500L cases. While the peak in the R220L case is stronger and localized in streamwise scale at $k_{x}\unicode[STIX]{x1D6FF}\approx 0.08$ , the peak in the R500L case is strongest at $k_{x}\unicode[STIX]{x1D6FF}\approx 0.02$ , with energy distributed along a line at $k_{z}\unicode[STIX]{x1D6FF}\approx 1.2$ , to $k_{x}\unicode[STIX]{x1D6FF}$ as low as 0.6. Note that the spectral peaks in $E_{uu}$ and $E_{ww}$ near the wall are at the same wavenumbers as at the centre line. These observations are consistent with the observations of the one-dimensional spectra.

Figure 6.

Contours of the $k_{x}$ - and $k_{z}$ -premultiplied two-dimensional spectra of $u_{i}^{\prime }u_{j}^{\prime }$ for the R220L and R500L cases identified in table table 1.

Figure 7.

Instantaneous streamwise velocity in a wall-parallel plane in the centre of the domain.

The streamwise velocities of an instantaneous field in the wall-parallel plane at the centre of the domain are shown in figure 7. Note that the scale is compressed by a factor of 5 in the $x$ direction for the long domain cases. In all cases, there are strong streaks in the opposite directions. In the R93S/L cases (figure 7 a,d), the strong streaks are observed but the length of streaky motion is somewhat too short to find patterns.

In the R220L case (figure 7 e), there are low and high speed streaks, which are inclined from the streamwise direction. Further, since the domain is periodic, these tilted streaks have a helical topology, as do, presumably, the large vortical structures underlying them. These tilted streaks are responsible for the presence of the strong low wavenumber peak in the R220L streamwise spectrum (figure 5). In a shorter domain (figure 7 b), a larger tilt angle would be required to produce a helical topology. Apparently, such strongly tilted structures are not preferred, so in the smaller domain, the periodic boundary conditions effectively ‘lock’ the structures into an untilted configuration. Note that rotating the simulation domain may facilitate study of such tilted structures using simulations with limited domain sizes (Barkley & Tuckerman Reference Barkley and Tuckerman2005; Duguet et al. Reference Duguet, Schlatter and Henningson2010).

In the higher Reynolds number R500L case (figure 7 f), there is no dominant inclination to the streaks, or helical topology. Instead, they are predominantly streamwise, with large-scale meanders and distortions that are too large to be seen in the short domain case (figure 7 c). Clearly, it is the lack of a dominant inclination and the presence of the meanders that is responsible for the more complex low wavenumber streamwise spectrum $E_{uu}$ , with multiple weak peaks at the centre line.

Figure 8.

Two-point correlations of streamwise velocities in a wall-parallel plane in the centre of the domain.

To quantitatively examine the nature of the large-scale structures indicated in figure 7, the two-point auto-correlation of the streamwise velocity, $R_{uu}(r_{x},y,r_{z})=\langle u^{\prime }(x,y,z)u^{\prime }(x+r_{x},y,z+r_{z})\rangle /\langle {u^{\prime }}^{2}\rangle$ , for $y=0$ (the centre line) is shown in figure 8 where $r_{x}$ and $r_{z}$ are the separations in the streamwise and spanwise directions, respectively. In general, the strength of the correlation at large separations increases with $Re$ indicating the increased coherence of the underlying structures. At the highest Reynolds number ( $Re_{\unicode[STIX]{x1D70F}}=500$ ), the underlying structure is highly coherent in the spanwise direction as indicated by the fact that the magnitude of the correlation coefficient with $r_{x}=0$ and $r_{z}=5\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}/2$ (half the spanwise domain size) is about 0.5 in both the R500S and R500L cases. In all cases, there is a regular alternation in sign of the correlation in the spanwise direction, with a wavelength of $5\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}/3$ in the R220 and R500 cases and $5\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}/4$ in the R093 case, which are one third or one fourth of the spanwise domain size respectively. This is necessarily consistent with the presence of the strong spectral peaks at $k_{z}\unicode[STIX]{x1D6FF}=1.2$ or 1.6 in the spanwise spectra shown in figure 5. This suggests that the spanwise scale of the underlying large-scale structures is a function of Reynolds number. But, the wavelengths that can be supported in the computational domain are highly constrained, so it is not possible to determine the natural spanwise size of the large-scale structures, or how it depends on Reynolds number. However, given the nature of the underlying structures (streamwise vortices filling the domain in the wall-normal direction, see below), it is not expected that the spanwise wavelength could become many times larger than observed here.

For all three Reynolds numbers, the short domain cases have correlation coefficients that do not change sign over the entire streamwise domain, as shown in figure 8(a–c). However, at the lowest Reynolds number the correlation falls almost to zero at maximum streamwise separation (0.05 at $r_{z}=0$ ). In contrast, for the R500S case, the correlation falls only to 0.4 at $r_{x}=L_{x}/2$ and $r_{z}=0$ . This is consistent with the instantaneous streamwise velocities shown in figure 7(a–c), which have little streamwise coherence for R93S, but significant coherence for R500S. In the long domain cases, the correlation decays in the streamwise direction somewhat faster than in the short domain cases. For example, in R93L, the correlation with $r_{z}=0$ falls to less than 0.05 by $r_{x}=10\unicode[STIX]{x03C0}$ , and, in R500L, the correlation at $r_{x}=10\unicode[STIX]{x03C0}$ is approximately 0.3, rather than 0.4 in R500S. Apparently, the periodic boundary conditions in the short domain help reinforce the structures and thus make them somewhat more streamwise coherent.

In the R220L case, the correlations oscillate in the streamwise direction as they do in the spanwise direction, in this case, with a wavelength of $25\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}$ , which is 15 times the spanwise wavelength. The reason for this streamwise oscillation is the inclination of the low and high speed streaks shown in figure 7(e). A correlation pattern like that in figure 8(e) arises from such inclined structures when inclinations of either positive or negative slope are equally likely. The average slope of the inclination is just the ratio of the spanwise to streamwise wavelengths or $1/15$ , which is approximately $4^{\circ }$ from alignment in the streamwise direction. As with the spanwise wavelength, the finite domain size and periodic boundary conditions in the streamwise direction constrain the possible streamwise wavelengths, and thus the possible inclination slopes. In this case, with a spanwise wavelength of $5\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}/3$ , the possible values of the slope are $5n\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}/(3L_{x})$ . In the large domain this is $n/60$ and in the small domain $n/12$ . The inclination slope arising in the R220L case is thus smaller than the smallest non-zero slope possible in the R220S case. This may be why the inclination is zero in R220S.

The question remains as to why there is an inclination to the structures in the R220L case, but none is apparent in the correlations for R93L or R500L. As discussed above, the coherence of the structures underlying the correlations are strongly dependent on the Reynolds number, and the R93L case appears to be too low in Reynolds number to have sufficient streamwise coherence to exhibit inclined structures in the correlations. For R500L, it may be that the preferred inclination angle is also strongly dependent on $Re$ and at this Reynolds number is too small to be represented in the long spatial domain (i.e. slope less than $1/60$ ). Or, it could be that this tendency to generate inclined structures is an intermediate Reynolds number phenomenon only. In either case, it would seem that very long structures that are significantly inclined from the streamwise direction are unlikely for higher Reynolds numbers than those studied here.

Figure 9.

Averaged velocity fields; R500S. (a) Streamwise velocity at the centerline averaged in the streamwise direction. (b) Velocity averaged in time and the streamwise direction. Shown are the streamwise velocity (colour) and the stream function; —— (positive) – – – (negative).

To investigate the structures underlying the correlations in figure 8, we take advantage of the fact that the R500S case has streamwise coherence that extends through the entire streamwise computational domain, and average the velocity in the streamwise direction in this case. The coherence of this streamwise averaged velocity in time is characterized by the contours of the averaged streamwise velocity as a function of $z$ and time, as shown in figure 9(a). Note that these velocities do appear to be coherent in time, and exhibit no substantial spanwise drift over the time simulated ( $3000\unicode[STIX]{x1D6FF}/U_{w}$ ). This suggests that an average in the streamwise direction and time would yield a good representation of the structures responsible for the observed correlations. The results are shown in figure 9(b), where the streamwise velocity and contours of the streamfunction are plotted. As expected, and as observed previously by others (Papavassiliou & Hanratty Reference Papavassiliou and Hanratty1997; Avsarkisov et al. Reference Avsarkisov, Hoyas, Oberlack and García-Galache2014; Pirozzoli et al. Reference Pirozzoli, Bernardini and Orlandi2014), the structures consist of counter-rotating streamwise vortices that fill the domain between the moving walls. Embedded as they are in the background shear of the Couette flow, they produce large fluctuations in the streamwise velocity. The maximum wall-normal and spanwise velocities associated with these vortices are approximately $0.025U_{w}$ and $0.03U_{w}$ , respectively, while at the centre of the domain, the streamwise velocity varies between $\pm 0.13U_{w}$ . These structures make significant contributions to $\langle {u^{\prime }}^{2}\rangle$ and $\langle u^{\prime }v^{\prime }\rangle$ , and as much as half of the total Reynolds shear stress at the centre of the domain (see R500S-LS profiles in figure 3). The vortices are also clearly responsible for the low wavenumber features of the spectra in figures 4–6 and discussed above. Since the streamwise and temporal averaging has reduced the apparent strength of the structures by averaging out meandering and variability, this estimate of contributions to Reynolds stress are likely understated.

While the large-scale coherent vortices described above are a dominant feature of the turbulent Couette flow, there is no evidence of analogous coherent vortices, perhaps spanning from the wall to the centre line, in turbulent Poiseuille flow (see case R550P in figures 4–6, for example). The reason for this difference is not clear. Some insight might be gained by simulation and analysis of model versions of turbulent Couette and Poiseuille flow that eliminate the complexity of the wall by introducing mean shear through volumetric forcing, as in Waleffe (Reference Waleffe1997) and Chantry, Tuckerman & Barkley (Reference Chantry, Tuckerman and Barkley2016), since it seems unlikely that the near-wall layer plays an important role in the formation, or not, of these coherent vortices.