2 Numerical method and parameters

The velocity $\boldsymbol{u}$ in the DNS is governed by the incompressible Navier–Stokes equations

(2.1) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}=-\unicode[STIX]{x1D735}p+\frac{1}{Re}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}-Ro(\boldsymbol{e}_{z}\times \boldsymbol{u}),\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0,\end{eqnarray}$$

where $\boldsymbol{e}_{z}$ is the unit vector in the $z$ -direction and $p$ the pressure, including the centrifugal acceleration. The equations are non-dimensionalized by the mean bulk velocity $U_{b}$ and channel half-gap $h$ ; time $t$ is thus given in terms of a convection time $h/U_{b}$ . Streamwise, wall-normal, spanwise coordinates are denoted by $x$ , $y$ , $z$ , respectively, and boundary conditions are periodic in the streamwise and spanwise directions and no-slip at the walls. A sketch of the geometry is shown in Brethouwer (Reference Brethouwer2016). In the present DNS $y=-1$ and 1 correspond to the wall on the unstable and stable channel side, respectively.

Equations (2.1) are solved with a pseudo-spectral code with Fourier expansions in the homogeneous $x$ - and $z$ -direction and Chebyshev polynomials in the $y$ -direction (Chevalier et al. Reference Chevalier, Schlatter, Lundbladh and Henningson2007), and the spatial resolution is similar to in previous channel flow DNS (Lee & Moser Reference Lee and Moser2015). In all runs, the flow rate and thus $Re$ was kept constant by adapting the mean pressure gradient. $Re$ is varied from 3000 up to 31 600 and $Ro$ from 0 (non-rotating) to approximately $Ro_{c}$ , which is between 2.67–2.87 for this range of $Re$ . In most DNS, the streamwise and spanwise domain size are either $12\unicode[STIX]{x03C0}h\times 10.5h$ or $8\unicode[STIX]{x03C0}h\times 3\unicode[STIX]{x03C0}h$ , but in some DNS the domain size was chosen differently to accommodate TS instabilities, as explained in Brethouwer (Reference Brethouwer2016), but this variation has little effect on results presented here. The runs are sufficiently long to reach a statistically stationary state in all DNS. Parameters of the DNS at $Re=3000$ –31 600 are listed in table 1. The friction velocity is calculated as $u_{\unicode[STIX]{x1D70F}}=[u_{\unicode[STIX]{x1D70F}u}^{2}/2+u_{\unicode[STIX]{x1D70F}s}^{2}/2]^{1/2}$ , where $u_{\unicode[STIX]{x1D70F}u}$ and $u_{\unicode[STIX]{x1D70F}s}$ are the friction velocity of unstable and stable channel side, respectively (Grundestam et al. Reference Grundestam, Wallin and Johansson2008). With this definition the mean dimensional pressure gradient $\unicode[STIX]{x2202}P/\unicode[STIX]{x2202}x=\unicode[STIX]{x1D70C}u_{\unicode[STIX]{x1D70F}}^{2}/h$ , where $\unicode[STIX]{x1D70C}$ is the fluid density. $Re_{\unicode[STIX]{x1D70F}}^{u}$ and $Re_{\unicode[STIX]{x1D70F}}^{s}$ are Reynolds numbers based on $u_{\unicode[STIX]{x1D70F}u}$ and $u_{\unicode[STIX]{x1D70F}s}$ , respectively, and $Ro_{\unicode[STIX]{x1D70F}}=2\unicode[STIX]{x1D6FA}h/u_{\unicode[STIX]{x1D70F}}$ . The DNS studied here are basically the same as those reported in Brethouwer (Reference Brethouwer2016).

Table 1. DNS parameters: $N_{x}$ , $N_{y}$ and $N_{z}$ are the number of modes in the streamwise, wall-normal and spanwise direction, and $L_{x}$ and $L_{z}$ , are the streamwise and spanwise computational domain size, respectively.

Figure 1. Visualizations of the instantaneous streamwise velocity in an $x$ – $z$ plane at $y^{+}\approx 5$ on the stable channel side. Dark colours signify high velocities. $Re=31\,600$ and (a)  $Ro=0$ , (b)  $Ro=0.45$ , (c)  $Ro=0.9$ , (d)  $Ro=1.2$ . (e)  $Re=20\,000$ and $Ro=0.45$ .

3 Flow visualizations

Before discussing flow statistics and spectra, visualizations are presented to get an understanding of the main flow characteristics. In all following two-dimensional visualizations the complete domain is shown. Figure 1(a–d) shows plots of the instantaneous streamwise velocity in an $x$ – $z$ plane close to wall on the stable side at $Re=31\,600$ and $Ro\leqslant 1.2$ . Structures with a width of $O(h)$ are vaguely visible at $Ro=0$ (figure 1 a), indicating near-wall turbulence modulation by large-scale outer structures (Mathis, Hutchins & Marusic Reference Mathis, Hutchins and Marusic2009). In rotating channel flow, turbulence becomes progressively weaker on the stable side with increasing rotation speeds. At low rotation rates, the flow is fully turbulent on the stable side, but at $Ro=0.45$ it is no longer fully turbulent and regions with small-scale turbulence as well as regions where turbulence is mostly absent can be seen in figure 1(b). The regions with small-scale turbulence have a weakly visible oblique band-like structure with an angle of approximately $30^{\circ }$ to the flow direction. Similar oblique patterns have been observed in, for example, transitional plane Couette flow (Barkley & Tuckerman Reference Barkley and Tuckerman2007; Duguet, Schlatter & Henningson Reference Duguet, Schlatter and Henningson2010) as well as in other flows with some stabilizing force like stratified open channel flow, magnetohydrodynamic channel flow and rotating Couette flows (Brethouwer, Duguet & Schlatter Reference Brethouwer, Duguet and Schlatter2012) and stratified Ekman layers (Deusebio et al. Reference Deusebio, Brethouwer, Schlatter and Lindborg2014). In rotating channel flow the turbulent–laminar patterns exist only on the stable side since the other side is fully turbulent. The turbulent–laminar patterns are not transitional but persist in time, like in the aforementioned studies.

When $Ro$ is raised to 0.9 the turbulent fraction becomes smaller and the turbulent regions appear as spots and bounded band-like oblique structures (figure 1 c), while at $Ro=1.2$ and higher no small-scale turbulence is present and only weak larger-scale fluctuations are seen on the stable channel side (figure 1 d). In the latter case, a continuous cycle of strong turbulent bursts with a long period of $O(1000h/U_{b})$ on the stable side occurs caused by a linear TS wave instability (Brethouwer Reference Brethouwer2016) and a vague imprint of the TS wave is visible in figure 1. At other times the TS wave is often more prominent.

At a lower $Re=20\,000$ the flow is fully turbulent on the stable channel side at $Ro=0.15$ (shown later), whereas at $Ro=0.45$ one distinct oblique turbulent and laminar banded pattern can be observed (figure 1 e). At the same $Ro$ , but a higher $Re=31\,600$ in a DNS with a larger domain, turbulent–laminar patterns are more numerous but less explicit (figure 1 b). Although the angle of the oblique pattern in figure 1(e) is determined by the periodic boundary conditions and the domain size, it is similar to the angle of the patterns seen in figure 1(b,c) and the oblique patterns in other flow cases (Duguet et al. Reference Duguet, Schlatter and Henningson2010; Brethouwer et al. Reference Brethouwer, Duguet and Schlatter2012). Duguet & Schlatter (Reference Duguet and Schlatter2013) argue that the obliqueness of the patterns is caused by a large-scale flow with a non-zero spanwise velocity. In previous DNS of rotating channel flow discussed in the Introduction, such patterns have not been observed, which is likely owing to the use of fairly limited computational domains.

However, in a certain $Re$ – $Ro$ range Johnston et al. (Reference Johnston, Halleen and Lezius1972) observed laminar flows interspersed with turbulent spots on the stable channel side in their experiments, confirming that such transitional flows are found at lower $Ro$ and $Re$ . In DNS at $Re=7000$ and $Ro=0.3$ , Dai et al. (Reference Dai, Huang and Xu2016) observed quasi-periodic behaviour of the wall shear stress and turbulence intensity on the stable side because the flow continuously alternated between laminar-like and intermittent with large-scale streamwise bands with either turbulent or laminar features. They attributed this quasi-periodic behaviour to the dynamics of the streamwise roll cells in their DNS. Streamwise turbulent–laminar bands, as observed by Dai et al. (Reference Dai, Huang and Xu2016), are not seen in the present DNS, which may be related to the size of the computational domain. On the other hand, strong quasi-periodic variations of the wall shear stress and turbulence intensity are observed in some of the present DNS, but these were caused by a linear instability of a TS-like wave, as explained in Brethouwer (Reference Brethouwer2016). This quasi-periodic behaviour is not observed in all but one DNS without this linear instability. The exception is the DNS at $Re=10\,000$ and $Ro=0.45$ , where large variations of approximately 30 % are seen in the wall shear stress on the stable channel side. Visualizations (not shown here) reveal that these variations are related to quasi-periodically growing and decaying turbulent spots on this side. When the wall shear stress reaches a minimum, the spots almost disappear. Large cyclic variations in the wall shear stress have also been observed in DNS of transitional strongly stratified channel and plane Couette flows (Garcia-Villalba & del Álamo Reference Garcia-Villalba and del Álamo2011; Deusebio, Caulfield & Taylor Reference Deusebio, Caulfield and Taylor2015), but these variations disappeared when the computational domain was enlarged, indicating that this behaviour is strongly affected by the size of domain. DNS of spanwise rotating channel flow by Hsieh & Biringen (Reference Hsieh and Biringen2016) confirm that the intermittency on the stable channel side can be strongly influenced by the computational domain size when the flow is transitional there.

Figure 2. Isocontours of (a)  $\unicode[STIX]{x1D706}_{2}=-9$ at $Re=31\,600$ and $Ro=0.9$ , (b)  $\unicode[STIX]{x1D706}_{2}=-2.25$ at $Re=31\,600$ and $Ro=1.2$ , (c)  $\unicode[STIX]{x1D706}_{2}=-0.45$ at $Re=30\,000$ and $Ro=2.1$ (side views, stable channel side is at the top) and close up at isocontours of (d)  $\unicode[STIX]{x1D706}_{2}=-5.6$ at $Re=31\,600$ and $Ro=0.9$ , (e)  $\unicode[STIX]{x1D706}_{2}=-2.25$ at $Re=31\,600$ and $Ro=1.2$ . The isosurfaces are coloured with streamwise velocity. The velocity increases from blue to red.

Continuing with the present cases, if $Ro\geqslant 0.9$ and $Re=20\,000$ or lower, no turbulent spots or band-like structures are seen like in figure 1(c), illustrating that $Re$ effects can be appreciable, as discussed in more detail later. Only weak larger-scale fluctuations are observed on the stable side at high $Ro$ , as illustrated in figure 1(d). However, in a number of cases the calm periods on the stable side are interrupted by violent bursts of turbulence triggered by a linear instability, as explained before.

Figure 2 shows $\unicode[STIX]{x1D706}_{2}$ isocontours at three different $Ro$ (Jeong & Hussain Reference Jeong and Hussain1995) coloured with the streamwise velocity to identify vortices. The strongly turbulent unstable side of the channel with intense vortices obviously shrinks with $Ro$ . Between the unstable side with strong turbulence, and clearly identifiable vortices and stable side with weak turbulence, a seemingly sharp and flat border exists (figure 2 a–c), as was noted by Johnston et al. (Reference Johnston, Halleen and Lezius1972), although when $Ro=0.9$ some areas with vortices corresponding to the patterns in figure 1(c) can still be seen on the stable side. Rotation promotes the formation of streamwise vortices on the unstable side (Dai et al. Reference Dai, Huang and Xu2016), especially in the region where the absolute mean vorticity is approximately zero (Lamballais et al. Reference Lamballais, Lesieur and Métais1996). Indeed, close ups of the vortices on the border between the unstable and stable channel side at $Ro=0.9$ and 1.2 in figure 2(d,e) reveal elongated streamwise vortices and, remarkably, packages of hairpin vortices which are detached from the wall. The red colour signifies that hairpin vortices are mostly found near the streamwise velocity maximum on the border between the unstable and stable channel side. At higher $Ro$ , vortices including the head of the hairpin vortices become increasingly aligned with the streamwise direction, and hairpin vortices, as discussed by Adrian (Reference Adrian2007), become less explicitly visible (not shown here). Yang & Wu (Reference Yang and Wu2012) argued that the Coriolis force reduces the inclination angle of the vortices and favours their streamwise elongation on the unstable side, whereas on the stable channel side this force impedes this streamwise elongation. Lamballais et al. (Reference Lamballais, Lesieur and Métais1996) observed that vortices become aligned with the flow direction with $Ro$ , and remarked that, especially in the region where the absolute mean vorticity is approximately zero, streamwise vortex stretching is promoted. These findings are broadly consistent with the present visualizations.

Figure 3. Visualizations of the instantaneous wall-normal velocity in an $x$ – $z$ plane (a) at $y=-0.5$ and $Re=20\,000$ , $Ro=0.15$ , (b) at $y=-0.5$ and $Re=31\,600$ , $Ro=0.45$ , (c) at $y=-0.75$ and $Re=31\,600$ , $Ro=0.9$ and in a $y$ – $z$ plane at (d)  $Re=20\,000$ , $Ro=0.15$ , (e)  $Re=31\,600$ , $Ro=0.45$ , (f)  $Re=31\,600$ , $Ro=0.9$ . Dark colours signify positive velocities. (g) Isocontours of $\unicode[STIX]{x1D706}_{2}=-36$ at $Re=20\,000$ and $Ro=0.15$ near the wall on the unstable side. The flow is from bottom left to upper right corner.

Large, steady streamwise roll cells induced by the Coriolis force have been observed on the unstable side in several experimental and numerical studies of rotating channel flow (Dai et al. Reference Dai, Huang and Xu2016). Steady means here that they have a relatively long lifetime, although in the experiments of Johnston et al. (Reference Johnston, Halleen and Lezius1972) the roll cells changed in time, whereas in a DNS by Kristoffersen & Andersson (Reference Kristoffersen and Andersson1993) they were more coherent and stationary at $Ro=0.15$ and spanned the whole channel.

Visualizations of the instantaneous wall-normal velocity field in the present DNS are shown in figure 3. Narrow elongated streaks with positive wall-normal velocity away from the wall in a wall parallel $x$ – $z$ plane on the unstable side (figure 3 a) and regions with alternating positive and negative wall-normal velocity in a cross-stream $y$ – $z$ plane at $Ro=0.15$ and $Re=20\,000$ (figure 3 d) indicate the presence of large streamwise roll cells on the unstable side that extend up to the border between the unstable and the stable side. The structures are long, but not as coherent as in the DNS by Kristoffersen & Andersson (Reference Kristoffersen and Andersson1993) since they appear to break up or split at some places. Possible reasons for the reduced coherency can be the higher $Re$ and the larger computational domain in the present study, which puts fewer constraints on the dynamics of the structures through the periodic boundary conditions.

The clustering of intense vortices in streamwise near-wall streaks seen in figure 3(g) show that the roll cells modulate the near-wall dynamics on the unstable side. Dai et al. (Reference Dai, Huang and Xu2016) observed in their DNS of spanwise rotating channel flow that turbulence and vortices on the unstable side are stronger in the regions where the fluid is pumped away from the wall by the counter-rotating roll cells and the local mean wall shear stress has a minimum, but this augmentation was weaker at a higher $Re$ . The higher vorticity was found to be caused by the Coriolis term and strong vortex stretching. This apparently contrasts the effect of large-scale motions in non-rotating wall flows. Experiments by Talluru et al. (Reference Talluru, Baidya, Hutchins and Marusic2014) suggest the presence of large-scale counter-rotating vortices in turbulent boundary layer flow. They show that turbulence is weaker in the near-wall regions, where the fluid is pumped away from the wall by these vortices and the local wall shear stress has a minimum, and stronger where the large-scale motions induce a high wall shear stress, in agreement with Agostini & Leschziner (Reference Agostini and Leschziner2016). The different effect of large-scale motions in non-rotating versus rotating wall flows is presumably a result of the Coriolis term, which affects both the Reynolds stresses and vorticity. Roll cells in rotating flows are likely also more coherent and steady, and induce stronger wall-normal velocities.

Streaks with positive wall-normal velocity (figure 3 b) and large-scale regions with positive and negative wall-normal velocity (figure 3 e) at $Ro=0.45$ and $Re=31\,600$ indicate roll cells, but they seem to be smaller and less coherent than at $Ro=0.15$ . Streaky structures with a positive wall-normal velocity are also seen in figure 3(c) in the DNS at $Ro=0.9$ , but it is not clear if these upward and downward motions seen in figure 3(f) can be interpreted as signs of roll cells. The spacing and form of the streaks indicate that the roll cells, if they exist, are smaller and less coherent than at lower $Ro$ . Roll cells also became smaller in the DNS of rotating channel flow at $Re=2800$ by Dai et al. (Reference Dai, Huang and Xu2016) when $Ro$ was raised from 0.1 to 0.5. Signs of roll cells are visible in the DNS at $Ro=0.45$ and 0.9, but for lower $Re$ (not shown here). When $Ro=1.2$ or higher, no visible signs of roll cells are found at $Re=20\,000$ and 31 600. However, at $Re=5000$ , visualizations hint at the existence of roll cells, and Grundestam et al. (Reference Grundestam, Wallin and Johansson2008) observed them at $Ro=1.27$ and $Re_{\unicode[STIX]{x1D70F}}=180$ . These observations suggest that, at lower $Re$ , roll cells exist up to higher $Ro$ . A more quantitative study of the structures in rotating channel flow is presented in § 7.

At higher $Ro$ the unstable turbulent side becomes smaller and smaller and the flow tends to fully laminarize if $Ro$ approaches $Ro_{c}$ (Wallin et al. Reference Wallin, Grundestam and Johansson2013). Yet, even if $Ro\rightarrow Ro_{c}$ , streamwise and oblique modes are still unstable as a result of rotation, according to linear stability theory (Brethouwer Reference Brethouwer2016), and some of the largest linearly unstable modes become noticeable in the DNS when turbulence becomes weak on the unstable channel side. Figure 4 shows the resulting typical oblique waves on the unstable side in a DNS at $Re=30\,000$ and $Ro=2.7$ , close to $Ro_{c}=2.87$ for this  $Re$ . The waves are quite weak, i.e. the wall-normal velocity is approximately 2 %–3 % of $U_{b}$ in this case.

Figure 4. Visualization of the instantaneous wall-normal velocity for $Re=30\,000$ and $Ro=2.7$ in a (a)  $y$ – $z$ plane and in an (b)  $x$ – $z$ plane at $y=-0.8$ . Colour scale ranges from $v=-0.021$ –0.021 in (b).

4 Flow statistics

In this section, one-point statistics of the flow are presented and the effects of rotation are discussed. As mentioned before, linearly unstable TS waves cause strong recurring bursts of turbulence on a long time scale in some DNS. The instabilities and bursts are the topic of another study (Brethouwer Reference Brethouwer2016), and therefore not discussed in detail here. However, it should be noted that they significantly affect flow structures and turbulence, especially around the bursting moment and on the stable side where the bursts are most intense. Time series of, for instance, the volume-averaged turbulent kinetic energy $K_{m}$ show distinct sharp peaks as a result of the bursts, while in the calm periods between the bursts the same quantity shows only small to moderate variations, see e.g. figure 3(b) in Brethouwer et al. (Reference Brethouwer, Schlatter, Duguet, Henningson and Johansson2014). I have excluded the burst periods when computing the statistics by excluding the periods with distinct peaks in $K_{m}$ caused by the linear instability. The statistics are thus based on the long relatively calm periods of $O(100h/U_{b})$ between the bursts when $K_{m}$ only shows small to moderate variations and the turbulence does not appear to be strongly influenced by the instability. The reason for excluding these bursts periods from the statistics is that the bursts are not the subject of this study and that the bursts would obscure the direct effect of rotation on the turbulence. Besides, the bursts occur on a very long time scale of $O(1000h/U_{b})$ , which makes it practically impossible to obtain reasonable converged statistics if they are included, and they cannot be predicted by Reynolds stress and two-equation turbulence models. Brethouwer (Reference Brethouwer2016) lists all the cases when bursts happen. At $Re=30\,000$ –31 600, linear instabilities and bursts develop when $Ro\geqslant 1.2$ . In the next parts, $U$ is the mean streamwise velocity and $u$ , $v$ , $w$ are the streamwise, wall-normal and spanwise velocity fluctuation, respectively. An overline implies temporal and spatial averaging in the homogeneous directions.

Figure 5 shows mean streamwise velocity profiles scaled by $U_{b}$ at the highest $Re$ considered for $Ro$ up to 2.7. As in previous studies of turbulent channel flow subject to spanwise rotation (e.g. Kristoffersen & Andersson Reference Kristoffersen and Andersson1993; Grundestam et al. Reference Grundestam, Wallin and Johansson2008; Xia et al. Reference Xia, Shi and Chen2016), the mean velocity profile becomes asymmetric and develops an extended linear region where the slope $\text{d}U/\text{d}y\simeq 2\unicode[STIX]{x1D6FA}$ , i.e. $S\simeq -1$ , implying an absolute mean vorticity close to zero. If $Ro\leqslant 1.5$ the velocity on the unstable side goes down, while on the stable side it goes up, since the turbulence becomes stronger and weaker on the unstable and stable side, respectively, as will be shown later. At higher $Ro$ , the profile becomes more and more parabolic-like, and beyond $Ro=2.4$ the linear slope region disappears and the velocity profile approaches a laminar Poiseuille profile, as in the DNS by Grundestam et al. (Reference Grundestam, Wallin and Johansson2008) and Xia et al. (Reference Xia, Shi and Chen2016).

Figure 5. Mean velocity profiles scaled by $U_{b}$ . (a)  $Re=31\,600$ . ——, $Ro=0$ ; – – –, $Ro=0.45$ ; $\cdots \cdots$ , $Ro=0.9$ ; – ⋅ – ⋅ –, $Ro=1.2$ . (b)  $Re=30\,000$ . ——, $Ro=1.5$ ; – – –, $Ro=2.1$ ; $\cdots \cdots$ , $Ro=2.4$ ; – ⋅ – ⋅ –, $Ro=2.7$ . Slopes with $S=-1$ are shown by straight red lines.

Profiles of the root-mean-square (r.m.s.) of the streamwise, wall-normal and spanwise velocity fluctuations, $u^{+}$ , $v^{+}$ and $w^{+}$ , respectively, and the Reynolds shear stress $\overline{uv}^{+}$ normalized by $u_{\unicode[STIX]{x1D70F}}[u_{\unicode[STIX]{x1D70F}u}^{2}/2+u_{\unicode[STIX]{x1D70F}s}^{2}/2]^{1/2}$ for $Ro$ up to 2.1 are shown in figure 6. The results are for the highest $Re$ considered here, 31 600 and 30 000. Note that for $Ro\geqslant 1.2$ the flow is subject to a TS wave instability, resulting in intense bursts of turbulence at this $Re$ (Brethouwer Reference Brethouwer2016), but these periods with turbulent bursts are excluded as much as possible when computing the statistics, as mentioned before. The statistics are thus based on the long calm periods between the bursts.

Figure 6. R.m.s. profiles of (a) streamwise, (b) wall-normal (c) spanwise velocity fluctuations and (d)  $\overline{uv}$ profiles in wall units. $Re=31\,600$ and (——)  $Ro=0$ , (– – –)  $Ro=0.45$ , ( $\cdots \cdots$ )  $Ro=0.9$ , (– ⋅ – ⋅ –)  $Ro=1.2$ . $Re=30\,000$ and (——, red)  $Ro=1.5$ , (– – –, red)  $Ro=2.1$ .

Figure 6 shows that rotation causes a reduction of the turbulence intensity on the stable side, as expected (Johnston et al. Reference Johnston, Halleen and Lezius1972). Especially for the wall-normal component (figure 6 b) this reduction is apparent, while for the other two velocity components it becomes most notable for $Ro\geqslant 1.2$ . The turbulence on the unstable channel side displays a more complex behaviour. The peak of $u^{+}$ first rises with $Ro$ but then declines if $Ro>0.45$ , whereas $w^{+}$ , and especially $v^{+}$ , grow strongly with $Ro$ , and only start to notably decline when $Ro>1.5$ . At higher $Ro$ , turbulence is progressively suppressed (not shown here) and the flow approaches more and more a laminar Poiseuille flow, as observed for the mean flow. Rotation thus has not only a marked influence on the turbulence intensity but also on its anisotropy, as in rotating homogeneous shear flows (Brethouwer Reference Brethouwer2005). The observed trends are in qualitative agreement with DNS of rotating channel flow at lower $Re$ (Grundestam et al. Reference Grundestam, Wallin and Johansson2008). The maximum of $v^{+}$ is near the wall at $Ro=0$ , but remarkably far from the wall on the unstable side at $Ro=0.45$ , which may be due to the presence of roll cells, while at higher $Ro$ it approaches the wall again.

From the mean momentum balance for rotating channel flow follows

(4.1) $$\begin{eqnarray}-\overline{uv}+\unicode[STIX]{x1D708}\frac{\text{d}U}{\text{d}y}=u_{\unicode[STIX]{x1D70F}u}^{2}-u_{\unicode[STIX]{x1D70F}}^{2}(y+1),\end{eqnarray}$$

where the velocities are dimensional. The sum of the viscous and turbulent shear stresses is thus linear in $y$ but is shifted owing to the difference in the wall shear stresses on the unstable and stable channel sides in the rotating flow cases. These stresses are naturally higher on the unstable side owing to the more intense turbulence. When viscous stresses are negligible the $\overline{uv}^{+}$ profile is also linear in $y$ according to (4.1). From this equation it follows that in the part of the channel where $\unicode[STIX]{x1D708}\,\text{d}U/\text{d}y\simeq 2\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FA}=U_{b}^{2}Ro/Re$ the turbulent shear stress is approximated as

(4.2) $$\begin{eqnarray}\overline{uv}\simeq u_{\unicode[STIX]{x1D70F}}^{2}(y+1)-u_{\unicode[STIX]{x1D70F}u}^{2}+U_{b}^{2}\frac{Ro}{Re}.\end{eqnarray}$$

This shows that the $\overline{uv}^{+}$ profile is linear in $y$ and has a unit slope even if viscous stresses are not negligible (Xia et al. Reference Xia, Shi and Chen2016). Figure 6(d) confirms that on the unstable side the $\overline{uv}^{+}$ profiles have a unit slope. The turbulent momentum transfer shifts progressively towards the unstable side with $Ro$ , and for $Ro\geqslant 1.2$ it is in fact negligible on the stable side where viscous stresses dominate. On the unstable side the magnitude of $\overline{uv}^{+}$ starts to decline when $Ro>0.9$ , and is small for $Ro\geqslant 2.1$ . Viscous shear stresses are significant on the strongly turbulent unstable channel side at higher $Ro$ because of the steep mean velocity gradient. If the total shear stress is estimated as $u_{\unicode[STIX]{x1D70F}}^{2}$ , it follows that on the unstable side where $\text{d}U/\text{d}y\simeq 2\unicode[STIX]{x1D6FA}$ the ratio between viscous and total stresses is approximately $Ro\,Re/Re_{\unicode[STIX]{x1D70F}}^{2}$ . From that it follows that the viscous contribution grows with $Ro$ and is approximately 12 % and 26 % at $Re=31\,600$ and $Ro=1.2$ and 1.5, respectively, and even higher at the same $Ro$ but lower $Re$ . Once $Ro\gtrsim 2.0$ , viscous stresses dominate also on the unstable side.

To investigate in more detail the turbulence near the wall on the unstable channel side, I present in figure 7 r.m.s.-profiles of the streamwise, wall-normal and spanwise velocity fluctuations, $u^{\ast }$ , $v^{\ast }$ and $w^{\ast }$ , respectively, in viscous wall units of the unstable side using a logscale for $y^{\ast }$ . Velocity fluctuations are thus scaled by $u_{\unicode[STIX]{x1D70F}u}$ and $y^{\ast }=(y+1)u_{\unicode[STIX]{x1D70F}u}/\unicode[STIX]{x1D708}$ since $u_{\unicode[STIX]{x1D70F}u}$ appears to be the most relevant quantity very close to the wall. Note that $u_{\unicode[STIX]{x1D70F}u}$ can deviate quite significantly from $u_{\unicode[STIX]{x1D70F}}$ , see table 1. The peak of $u^{\ast }$ on the unstable side declines and moves towards smaller $y^{\ast }$ with $Ro$ , whereas the peaks of $v^{\ast }$ and $w^{\ast }$ grow with $Ro$ until $Ro=1.5$ and then decline. This reduction and growth, respectively, are caused by an energy redistribution from streamwise to wall-normal fluctuations by the Coriolis term and pressure–strain correlations in the Reynolds stress equations, as will be shown later. The peak of $w^{\ast }$ moves towards the wall with $Ro$ , whereas that of $v^{\ast }$ is found far away from the wall at $Ro=0.45$ and comes closer to the wall with increasing $Ro$ . The profile of $w^{\ast }$ has two peaks for $Ro\geqslant 1.2$ which are accompanied by a double peak in the spectra, as shown later.

Figure 7. R.m.s. profiles of (a) streamwise, (b) wall-normal and (c) spanwise velocity fluctuations in wall units of the unstable side. $Re=31\,600$ and (——)  $Ro=0$ , (– – –)  $Ro=0.45$ , ( $\cdots \cdots$ )  $Ro=0.9$ , (– ⋅ – ⋅ –)  $Ro=1.2$ . $Re=30\,000$ and (——, red)  $Ro=1.5$ , (– – –, red)  $Ro=2.1$ .

The skewness of the streamwise velocity $S(u)$ and wall-normal velocity fluctuations $S(v)$ are presented in figure 8. Profiles of $S(u)$ and $S(v)$ at lower $Re$ are presented by Hsieh & Biringen (Reference Hsieh and Biringen2016). $S(u)$ is typically negative away from the wall in non-rotating channel flow owing to ejections of low-speed fluid (Kim, Moin & Moser Reference Kim, Moin and Moser1987). In rotating channel flow, $S(u)$ becomes considerably more negative near the wall at $Ro=0.45$ and 0.9 on the unstable side, except very near the wall. This could be caused by a reduction of sweeping events of high-speed flow towards the wall as a consequence of rotation (Kristoffersen & Andersson Reference Kristoffersen and Andersson1993). On the other hand, the analysis by Dai et al. (Reference Dai, Huang and Xu2016) indicates that under the influence of rotation the streaks become stronger on the unstable side, at least up to moderate $Ro$ , which implies more or intenser ejections. The roll cells, observed before, could also play a role. If $Ro>0.9$ , $S(u)$ becomes less negative near the wall, which can be related to the weaker streaks and related ejections at high $Ro$ , as suggested by Lamballais, Métais & Lesieur (Reference Lamballais, Métais and Lesieur1998). Further away from the wall, $S(u)$ attains small values in the rotating cases compared to the non-rotating case, indicating that sweeping and ejection events are significantly altered by rotation. $S(v)$ has a large positive value near the wall on the unstable side at $Ro=0.45$ compared to $Ro=0$ , which is likely caused by roll cells that induce high-speed wall-normal velocity away from the wall. Similar behaviour of $S(v)$ was observed in the experiments by Nakabayashi & Kitoh (Reference Nakabayashi and Kitoh2005). Large positive values of $S(v)$ , indicating events with large positive wall-normal velocities, are also found in all rotating cases on the unstable side, quite close to the position where the slope of $U$ begins to deviate from $2\unicode[STIX]{x1D6FA}$ . At $Ro=1.2$ and 1.5, $S(u)$ has large values at slightly larger $y$ , near the position where $U$ has its maximum value. The reason for the large values of $S(u)$ and $S(v)$ around these positions is not fully clear, but visualizations suggest that large values of $v$ are found near the streamwise vortices visualized in figure 2. The large $S(u)$ could be related to the hairpin vortices seen in figure 2, since positive values of $u$ are found above the head of such hairpin vortices (Christensen & Adrian Reference Christensen and Adrian2001; Adrian Reference Adrian2007).

Figure 8. Skewness of (a)  $u$ and (b)  $v$ . $Re=31\,600$ and (——)  $Ro=0$ , (– – –)  $Ro=0.45$ , ( $\cdots \cdots$ )  $Ro=0.9$ , (– ⋅ – ⋅ –)  $Ro=1.2$ . $Re=30\,000$ and (——, red)  $Ro=1.5$ .

The flatness of the wall-normal velocity $F(v)$ attains extreme values in non-rotating channel flow very near the wall as a result of intense near-wall vortices (Lenaers et al. Reference Lenaers, Li, Brethouwer, Schlatter and Örlü2012). The present DNS show that, with faster rotation, $F(v)$ on the unstable channel side decays monotonically with $Ro$ , suggesting that intense near-wall vortices are suppressed.

5 Influence of the Reynolds number

In this section, the influence of the Reynolds number on rotating channel flow at a fixed $Ro$ is examined and shown to be significant. In a later section, this influence on the flow structures found on the unstable channel side is investigated.

Figure 9 shows profiles of the mean velocity $U/U_{b}$ and fluctuations $v^{+}$ at $Ro=0.15$ , 0.45 and 0.9 for three to four $Re$ . In all cases a region with a linear mean velocity profile where $S\simeq -1$ can be readily recognized, demonstrating once more that the appearance of a region with a zero absolute mean vorticity is a fundamental feature of rotating channel flow. Its extent and the closeness of $\text{d}U/\text{d}y$ to $2\unicode[STIX]{x1D6FA}$ do not show much variation with $Re$ , and the same applies to the maximum value of $v^{+}$ on the unstable channel side. On the other hand, $Re$ has an obvious influence on the mean velocity profile as well as $v^{+}$ on the stable channel side where, at higher $Re$ , $v^{+}$ is considerably larger.

Figure 9. (a) Mean velocity profiles scaled by $U_{b}$ and (b) r.m.s. of $v$ in wall units at $Ro=0.15$ and ——, $Re=20\,000$ ; – – –, red,  $Re=5000$ ; $\cdots \cdots$ , blue, $Re=3000$ . (c) Mean velocity profiles scaled by $U_{b}$ and (d) r.m.s. of $v$ in wall units at $Ro=0.45$ and ——, $Re=31\,600$ ; – – –, red, $Re=20\,000$ ; $\cdots \cdots$ , blue, $Re=10\,000$ ; – ⋅ – ⋅ –, green, $Re=5000$ . (e) and (f) are the same as (c) and (d), respectively, but for $Ro=0.9$ . The straight black line has a slope $S=-1$ .

This Reynolds number effect is also observed in visualizations of the instantaneous streamwise velocity in a plane parallel and close to the wall on the stable channel side. At $Ro=0.15$ and $Re=20\,000$ the flow is fully turbulent on the stable channel side (figure 10 a), whereas at a lower $Re=3000$ the flow is largely laminar and contains only some turbulent spots (figure 10 b), resulting in much lower turbulence levels on the stable channel side (figure 9 b). At $Ro=0.45$ and $Re=31\,600$ a large fraction of the flow on the stable side is turbulent and turbulent–laminar patterns develop, as shown before in figure 1(b), whereas at the same $Ro$ but lower $Re=5000$ the flow is predominantly laminar and only one turbulent spot can be observed (figure 9 c). At a lower $Re=3000$ the spot is even smaller. Also at $Ro=0.9$ and $Re=31\,600$ , regions with small-scale turbulence can be observed (figure 1 c), but these regions with small-scale turbulence are completely absent at lower $Re$ (not shown here). $Re$ thus has a marked influence on the flow, especially on the stable channel side, with a growing turbulent fraction and stronger turbulence at fixed $Ro$ when $Re$ gets higher. In the present study, oblique turbulent–laminar patterns have only been observed at higher $Re$ . A speculation is that they also exist at low $Re$ , although at a lower $Ro$ when the stabilizing Coriolis force is less strong. But at a lower $Re$ they can have a longer wavelength, as indicated by Brethouwer et al. (Reference Brethouwer, Duguet and Schlatter2012), implying that very large computational domains are required to resolve them.

Figure 10. Visualizations of the instantaneous streamwise velocity in an $x$ – $z$ plane at $y^{+}\approx 5$ on the stable channel side. Dark colours signify high velocities. (a)  $Ro=0.15$ and $Re=20\,000$ , (b)  $Ro=0.15$ and $Re=3000$ and (c)  $Ro=0.45$ and $Re=5000$ .

Profiles of the r.m.s. of the velocity fluctuations in terms of wall units of the unstable side, i.e. velocity fluctuations scaled by $u_{\unicode[STIX]{x1D70F}u}$ and $y^{\ast }=yu_{\unicode[STIX]{x1D70F}u}/\unicode[STIX]{x1D708}$ , for different $Ro$ and $Re$ are not presented here. However, the peak of the streamwise and spanwise fluctuations in general increases with $Re$ at fixed $Ro$ , whereas the maximum of the wall-normal fluctuations is quite independent of $Re$ but moves towards larger $y^{\ast }$ with $Re$ .

The volume-averaged turbulent kinetic energy $K_{m}$ scaled by $U_{b}^{2}$ , mean wall shear stresses $\unicode[STIX]{x1D70F}_{w}^{u}$ and $\unicode[STIX]{x1D70F}_{w}^{s}$ on the unstable and stable channel sides, respectively, and skin friction coefficient $C_{f}=\unicode[STIX]{x1D70F}_{w}/(\unicode[STIX]{x1D70C}U_{b}^{2}/2)$ with $\unicode[STIX]{x1D70F}_{w}=(\unicode[STIX]{x1D70F}_{w}^{u}+\unicode[STIX]{x1D70F}_{w}^{s})/2$ are shown in figure 11 at different $Ro$ and $Re$ . Figure 11(a) shows that overall the turbulence intensity decays with $Ro$ at fixed $Re$ , and becomes very weak at high $Ro$ , with similar trends for all $Re$ . The data of Xia et al. (Reference Xia, Shi and Chen2016) are also included in the figure, and show first a 15 % growth in $K_{m}/U_{b}^{2}$ until $Ro=0.44$ , and then a quite similar decay as in the other cases; but note that in their DNS series $Re_{\unicode[STIX]{x1D70F}}=180$ and constant, and therefore $Re$ varies. The difference in $\unicode[STIX]{x1D70F}_{w}$ on the stable and unstable side first grows substantially with $Ro$ until its maximum found at approximately $Ro=0.9$ , with the highest $\unicode[STIX]{x1D70F}_{w}$ naturally occurring on the unstable side, but then diminishes and disappears at high $Ro$ when the flow becomes laminar (figure 11 b). Again, the trends are similar at different $Re$ , but the maximum difference appears at lower $Ro$ when $Re$ is lower, and is smaller at the highest $Re$ , possibly because the flow relaminarizes less fast on the stable side at higher $Re$ , as discussed before. The skin friction $C_{f}$ decays monotonically with $Ro$ , like $K_{m}$ , and nearly equals the value for laminar Poiseuille flow for $Ro\geqslant 2.4$ at all $Re$ (figure 11 c).

Figure 11. (a) $K_{m}/(U_{b}^{2})$ , (b)  $\unicode[STIX]{x1D70F}_{w}^{u}/\unicode[STIX]{x1D70F}_{w}$ (upper curves) and $\unicode[STIX]{x1D70F}_{w}^{s}/\unicode[STIX]{x1D70F}_{w}$ , (c)  $C_{f}$ . $\circ$ — $\circ$ , $Re=31\,600$ ; ▫—▫, $Re=30\,000$ ; $\ast$ – – – $\ast$ (red), $Re=20\,000$ ; $\unicode[STIX]{x0394}\cdots \unicode[STIX]{x0394}$ (blue), $Re=10\,000$ ; $\unicode[STIX]{x1D6FB}$ – ⋅ – ⋅ – $\unicode[STIX]{x1D6FB}$ (green), $Re=5000$ ; —— (green), $Re_{\unicode[STIX]{x1D70F}}=180$ (Xia et al. Reference Xia, Shi and Chen2016). The horizontal line in (c) represents $C_{f}$ for a laminar Poiseuille flow.

6 Balances

In this section, the balance terms in the transport equations of the turbulent kinetic energy and Reynolds stresses are studied. Grundestam et al. (Reference Grundestam, Wallin and Johansson2008) have also presented budgets for rotating channel flow, although only for one high $Ro=1.5$ when the turbulence is very weak, while Xia et al. (Reference Xia, Shi and Chen2016) have only presented production terms. The present study contributes with a study of several budget terms at a significantly higher $Re$ covering a wide $Ro$ range.

From (4.2) it follows that the production of turbulent kinetic energy in the part of the channel, where $\text{d}U/\text{d}y\simeq 2\unicode[STIX]{x1D6FA}$ is approximately (see also Xia et al. Reference Xia, Shi and Chen2016)

(6.1) $$\begin{eqnarray}P_{K}=-\overline{uv}\frac{\text{d}U}{\text{d}y}\simeq -2\overline{uv}\unicode[STIX]{x1D6FA}=-\overline{uv}\frac{U_{b}}{h}Ro\simeq \left[u_{\unicode[STIX]{x1D70F}u}^{2}-u_{\unicode[STIX]{x1D70F}}^{2}(y+1)-U_{b}^{2}\frac{Ro}{Re}\right]\frac{U_{b}}{h}Ro.\end{eqnarray}$$

Consequently,

(6.2) $$\begin{eqnarray}\frac{hP_{K}}{u_{\unicode[STIX]{x1D70F}}^{2}U_{b}Ro}\simeq -(y+1)+\left(\frac{u_{\unicode[STIX]{x1D70F}u}}{u_{\unicode[STIX]{x1D70F}}}\right)^{2}-\left(\frac{U_{b}}{u_{\unicode[STIX]{x1D70F}}}\right)^{2}\frac{Ro}{Re}.\end{eqnarray}$$

Thus, the profile of $P_{K}$ scaled with $u_{\unicode[STIX]{x1D70F}}^{2}U_{b}Ro/h=(u_{\unicode[STIX]{x1D70F}}^{4}/\unicode[STIX]{x1D708})(Ro_{\unicode[STIX]{x1D70F}}/Re_{\unicode[STIX]{x1D70F}})$ , where $u_{\unicode[STIX]{x1D70F}}^{4}/\unicode[STIX]{x1D708}$ is the usual viscous wall unit scaling, is expected to be approximately linear in $y$ with a slope $-1$ in the part of the channel where $\text{d}U/\text{d}y\simeq 2\unicode[STIX]{x1D6FA}$ .

Figure 12(a) shows $P_{K}$ scaled by $u_{\unicode[STIX]{x1D70F}}^{2}U_{b}Ro/h$ for $Ro$ up to 2.1. The DNS results for $Ro=0$ are included and scaled by $0.45u_{\unicode[STIX]{x1D70F}}^{2}U_{b}/h$ , i.e. the same scaling as used for the $Ro=0.45$ case. The profiles of $P_{K}$ in the rotating cases show, as expected, a significant linear part with a $-1$ slope. In this part, the scaled $P_{K}$ follows closely the prediction given by the right-hand side of (6.2). Similar linear profiles of $P_{K}$ were observed in DNS of rotating channel flow at $Re_{\unicode[STIX]{x1D70F}}=180$ by Xia et al. (Reference Xia, Shi and Chen2016). Near the maximum of $U$ , slight negative values of $P_{K}$ are observed at some $Ro$ , as in a DNS by Grundestam et al. (Reference Grundestam, Wallin and Johansson2008), which implies that some energy is transferred from turbulence to the mean flow in this region. The maximum value of $P_{K}$ near the wall on the unstable side scales instead with viscous wall units since the peak of $P_{K}$ lies between 0.244 and 0.248 for $Ro\leqslant 1.5$ when scaled by $u_{\unicode[STIX]{x1D70F}u}^{4}/\unicode[STIX]{x1D708}$ (figure 12 b).

Figure 12. (a) $P_{K}$ scaled by $u_{\unicode[STIX]{x1D70F}}^{2}U_{b}Ro/h$ and (b)  $P_{K}$ scaled by $u_{\unicode[STIX]{x1D70F}u}^{4}/\unicode[STIX]{x1D708}$ . (c)  $\unicode[STIX]{x1D700}$ , (d)  $P_{uv}$ , (e)  $P_{uu}^{tot}=P_{uu}+C_{uu}$ and (f)  $P_{uv}^{tot}=P_{uv}+C_{uv}$ , scaled by $u_{\unicode[STIX]{x1D70F}}^{2}U_{b}Ro/h$ . $Re=31\,600$ and (——) $Ro=0$ , (– – –) $Ro=0.45$ , ( $\cdots \cdots$ )  $Ro=0.9$ , (– ⋅ – ⋅ –) $Ro=1.2$ . $Re=30\,000$ and (——, red) $Ro=1.5$ , (– – –, red) $Ro=2.1$ .

In shear flows, the dissipation rate of turbulent kinetic energy, $\unicode[STIX]{x1D700}$ , is often approximately equal to the production of turbulent kinetic energy, $P_{K}$ , which suggests that $\unicode[STIX]{x1D700}$ , like $P_{K}$ , scales with $u_{\unicode[STIX]{x1D70F}}^{2}U_{b}Ro/h$ . Figure 12(c) affirms that, in rotating channel flow, profiles of $\unicode[STIX]{x1D700}$ , scaled by $u_{\unicode[STIX]{x1D70F}}^{2}U_{b}Ro/h$ , display a linear part. The DNS results for $Ro=0$ are again included and scaled by $0.45u_{\unicode[STIX]{x1D70F}}^{2}U_{b}/h$ . The slopes at $Ro=0.45$ and $Ro=2.1$ deviate from $-1$ , but in the other three rotating cases they follow closely the prediction given by the right-hand side of (6.2). For $Ro\geqslant 0.9$ , $P_{K}$ and $\unicode[STIX]{x1D700}$ are in fact strikingly similar in the outer layer on the unstable channel side, much more so than at $Ro=0$ . The close balance between $P_{K}$ and $\unicode[STIX]{x1D700}$ implies that the sum of turbulent, pressure and viscous diffusion in the equation for turbulent kinetic energy is small compared to $P_{K}$ and $\unicode[STIX]{x1D700}$ . In terms of wall units, $P_{K}$ and $\unicode[STIX]{x1D700}$ are large on the unstable side away from the wall in the rotating cases compared to the $Ro=0$ case, whereas on the stable side both are very small if $Ro\geqslant 1.2$ .

The observed scaling of $P_{K}$ and $\unicode[STIX]{x1D700}$ with $u_{\unicode[STIX]{x1D70F}}^{2}U_{b}Ro/h$ suggests that this scaling is meaningful as well for the budget terms in the balance equation for the Reynolds stresses. This equation reads

(6.3) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\overline{u_{i}u_{j}}}{\unicode[STIX]{x2202}t}+U_{k}\frac{\unicode[STIX]{x2202}\overline{u_{i}u_{j}}}{\unicode[STIX]{x2202}x_{k}}=P_{ij}+\unicode[STIX]{x1D700}_{ij}+C_{ij}+\unicode[STIX]{x1D6F1}_{ij}+D_{ij},\end{eqnarray}$$

where the terms on the right-hand side are the production, dissipation, Coriolis, pressure–strain and diffusion term, respectively (Grundestam et al. Reference Grundestam, Wallin and Johansson2008). The Coriolis terms in the equation for $\overline{uu}$ , $\overline{vv}$ , $\overline{ww}$ and $\overline{uv}$ stresses are $C_{uu}=4\overline{uv}\unicode[STIX]{x1D6FA}$ , $C_{vv}=-4\overline{uv}\unicode[STIX]{x1D6FA}$ , $C_{ww}=0$ and $C_{uv}=2(\overline{vv}-\overline{uu})\unicode[STIX]{x1D6FA}$ , respectively. These terms do not perform work but transfer energy between the Reynolds stress components (Kawata & Alfredsson Reference Kawata and Alfredsson2016). Note that $P_{uu}=2P_{K}$ .

Figure 13. (a) $\unicode[STIX]{x1D6F1}_{uu}$ , (b)  $\unicode[STIX]{x1D6F1}_{vv}$ , (c)  $\unicode[STIX]{x1D6F1}_{ww}$ and (d)  $\unicode[STIX]{x1D6F1}_{uv}$ scaled by $u_{\unicode[STIX]{x1D70F}}^{2}U_{b}Ro/h$ . Lines used as in figure 12.

Figure 12(d) shows the production of $\overline{uv}$ Reynolds stresses, $P_{uv}=-\overline{vv}\,\text{d}U/\text{d}y$ , scaled by $u_{\unicode[STIX]{x1D70F}}^{2}U_{b}Ro/h$ . Also in this figure and figure 13, data for $Ro=0$ are included using the scaling $0.45u_{\unicode[STIX]{x1D70F}}^{2}U_{b}/h$ to make a comparison with data of rotating channel flow possible. The profiles of $P_{uv}$ show less spreading than when using the ordinary wall unit scaling, and also reveal an approximately linear slope on the unstable side if $Ro\geqslant 0.9$ . On the stable side, $P_{uv}$ is significant at $Ro=0.45$ but negligible for $Ro\geqslant 1.2$ .

The Coriolis terms in the balance equations of the $\overline{uu}$ and $\overline{vv}$ Reynolds stresses are $C_{uu}=2\overline{uv}/u_{\unicode[STIX]{x1D70F}}^{2}$ and $C_{vv}=-2\overline{uv}/u_{\unicode[STIX]{x1D70F}}^{2}=-C_{uu}$ , respectively, when scaled by $u_{\unicode[STIX]{x1D70F}}^{2}U_{b}Ro/h$ . Profiles of $\overline{uv}/u_{\unicode[STIX]{x1D70F}}^{2}$ were already presented in figure 6(d). From these profiles it follows that on the unstable side the Coriolis term transfers energy from $\overline{uu}$ to the $\overline{vv}$ component in rotating channel flow. On the stable side it is the other way around for $Ro=0.45$ and 0.9, whereas it is small in the other cases.

If $\text{d}U/\text{d}y\simeq 2\unicode[STIX]{x1D6FA}$ , $P_{uu}\simeq -4\overline{uv}\unicode[STIX]{x1D6FA}=-C_{uu}$ , which means that the Coriolis term closely balances the production term and, consequently, the dissipation term approximately balances the pressure–strain term in the equation for the $\overline{uu}$ component since the transport terms are small. This is confirmed by the DNS but not shown here. In fact, the sum $P_{ij}+C_{ij}\equiv P_{ij}^{tot}$ may be considered as a total production term. When $\text{d}U/\text{d}y\simeq 2\unicode[STIX]{x1D6FA}$ , $P_{uu}^{tot}\simeq 0$ , $P_{vv}^{tot}\simeq P_{uu}$ and $P_{ww}^{tot}\simeq 0$ , since $P_{uu}\simeq -C_{uu}=C_{vv}$ as explained above, and $P_{vv}=P_{ww}=C_{ww}=0$ . Figure 12(e) shows $P_{uu}^{tot}$ scaled by $u_{\unicode[STIX]{x1D70F}}^{2}U_{b}Ro/h$ and confirms that in a large part of a rotating channel flow it is nearly zero. Thus, in a non-rotating flow, production feeds energy in the $\overline{uu}$ component and then energy is redistributed to the other components. By contrast, in rotating channel flow on the unstable side, $P_{ij}^{tot}$ feeds energy mainly in the $\overline{vv}$ component and pressure–strain correlations redistribute this energy to the $\overline{uu}$ and $\overline{ww}$ components. This may explain the strong wall-normal velocity fluctuations observed in rotating channel flow on the unstable side. Very near the wall on the unstable side, energy is still fed into the $\overline{uu}$ component. The same applies to the stable side for $Ro=0.45$ and 0.9, but if $Ro\geqslant 1.2$ the sum is nearly zero. Note that $P_{uu}^{tot}$ is slightly negative in rotating channel flow in a region around the maximum of  $U$ .

When scaled by $u_{\unicode[STIX]{x1D70F}}^{2}U_{b}Ro/h$ , $C_{uv}=(\overline{vv}-\overline{uu})/u_{\unicode[STIX]{x1D70F}}^{2}$ . The total production $P_{uv}^{tot}$ scaled by $u_{\unicode[STIX]{x1D70F}}^{2}U_{b}Ro/h$ , shown in figure 12(f), is balanced by $\unicode[STIX]{x1D6F1}_{uv}$ since the dissipation and diffusion terms are relatively small in a large part of the channel, in particular on the unstable side. On the unstable side, $P_{uv}^{tot}$ scaled is negative, but it becomes less negative for $Ro\geqslant 0.9$ . On the stable side, it is positive at $Ro=0.45$ and negative at larger $Ro$ , meaning that it destroys $\overline{uv}$ correlations, leading to the low turbulent shear stresses on the stable side observed before.

Pressure–strain correlations $\unicode[STIX]{x1D6F1}_{uu}$ , $\unicode[STIX]{x1D6F1}_{vv}$ , $\unicode[STIX]{x1D6F1}_{ww}$ and $\unicode[STIX]{x1D6F1}_{uv}$ scaled by $u_{\unicode[STIX]{x1D70F}}^{2}U_{b}Ro/h$ are presented in figure 13. When $Ro=0$ , energy is transferred from $\overline{uu}$ to $\overline{vv}$ and $\overline{ww}$ , whereas in the rotating cases energy is transferred from $\overline{vv}$ to $\overline{uu}$ and $\overline{ww}$ by pressure–strain correlations on the unstable side, except very close to the wall, where $\unicode[STIX]{x1D6F1}_{uu}$ is still negative. Especially at high $Ro$ , $\unicode[STIX]{x1D6F1}_{ww}$ is large, which explains the strong spanwise fluctuations in rapidly rotating channel flows. On the stable side there is a significant energy transfer from $\overline{uu}$ to the other two components by pressure–strain correlations for $Ro\leqslant 0.9$ . At larger $Ro$ , $\unicode[STIX]{x1D6F1}_{uu}$ and $\unicode[STIX]{x1D6F1}_{vv}$ are small but not negligible on the stable side, while $\unicode[STIX]{x1D6F1}_{uv}$ is positive (figure 13 d) and approximately balanced by the Coriolis term $C_{uv}$ , as discussed before. The slope of the pressure–strain profiles on the unstable side is also approximately linear for $Ro\geqslant 0.9$ and scales with $u_{\unicode[STIX]{x1D70F}}^{2}U_{b}Ro/h$ .

Turbulent velocity fluctuations are not insignificant on the stable side (figure 6), although $P_{K}$ is very small there for $Ro\geqslant 1.2$ . Grundestam et al. (Reference Grundestam, Wallin and Johansson2008) have suggested that the turbulence on the stable side is forced by the turbulence on the unstable side through the pressure diffusion term of the $\overline{vv}$ component which then redistributes the energy through pressure–strain correlations. The pressure diffusion of $\overline{vv}$ indeed has a small positive value (not shown here) and figure 13(a,b) show that, albeit small, $\unicode[STIX]{x1D6F1}_{vv}$ is negative and $\unicode[STIX]{x1D6F1}_{uu}$ positive at $Ro=1.2$ and 1.5 on the stable side. The latter is approximately balanced by a negative $P_{uu}^{tot}$ .

7 Spectra and two-point correlations

In order to study the effect of rotation on turbulence structures in a quantitative way, Kristoffersen & Andersson (Reference Kristoffersen and Andersson1993), Alvelius (Reference Alvelius1999) and Grundestam et al. (Reference Grundestam, Wallin and Johansson2008) computed two-point correlations at $Re_{\unicode[STIX]{x1D70F}}=180$ –194. Kristoffersen and Andersson observed that the spanwise near-wall streak spacing in wall units narrows with $Ro$ , whereas Alvelius observed the opposite trend. Generally, relatively short streamwise two-point correlations of streamwise fluctuations on the unstable side and relatively long correlations on the other laminar-like side were observed when the channel was rotating. At low $Ro$ , Kristoffersen and Andersson and Alvelius were also able to notice an impact of Taylor–Görtler vortices on correlations on the unstable side.

Here, I present spanwise two-point correlations of the wall-normal velocity fluctuations as well as spectra, since they better reveal the impact of rotation on the different flow scales, especially the large scales which are known to become important at higher $Re$ in wall flows (Smits, McKeon & Marusic Reference Smits, McKeon and Marusic2011). Spectra of the streamwise velocity fluctuations are presented, since these most clearly reveal the presence of large-scale motions in non-rotating wall flows, as well as spectra of the spanwise and wall-normal velocity fluctuations, since these may uncover the presence of roll cells. The presented cospectra give information on the scales that contribute to the momentum transfer and production of turbulent kinetic energy. The focus is on the strongly turbulent unstable side of the channel. Spectra and two-point correlations are less instructive for the stable side when relaminarization occurs. First, I consider the trends with $Ro$ and next with  $Re$ .

7.1 Spectra at $Re=20\,000$

Spanwise premultiplied energy spectra, $k_{z}\unicode[STIX]{x1D6F7}_{uu}$ , $k_{z}\unicode[STIX]{x1D6F7}_{vv}$ and $k_{z}\unicode[STIX]{x1D6F7}_{ww}$ of the streamwise, wall-normal and spanwise velocity fluctuations, respectively, and cospectra, $k_{z}\unicode[STIX]{x1D6F7}_{uv}$ , at $Re=20\,000$ are shown in figure 14. Streamwise premultiplied energy spectra, $k_{x}\unicode[STIX]{x1D6F7}_{uu}$ , $k_{x}\unicode[STIX]{x1D6F7}_{vv}$ and $k_{x}\unicode[STIX]{x1D6F7}_{ww}$ of the streamwise, wall-normal and spanwise velocity fluctuations, respectively, and cospectra, $k_{x}\unicode[STIX]{x1D6F7}_{uv}$ , at $Re=20\,000$ are shown in figure 15. Spectra are shown for $Ro=0$ , 0.15, 0.45 and 0.9, and are presented as functions of the wall distance $y^{\ast }=(y+1)u_{\unicode[STIX]{x1D70F}u}/\unicode[STIX]{x1D708}$ and the spanwise and streamwise wavelength $\unicode[STIX]{x1D706}_{z}^{\ast }$ and $\unicode[STIX]{x1D706}_{x}^{\ast }$ , respectively, both scaled in term of the viscous length scale of the unstable side, $l^{\ast }=\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}u}$ . The red dashed line in each plot indicates scales with a wavelength $h$ . Spectra are scaled with their maximum values, which are given in terms of $u_{\unicode[STIX]{x1D70F}u}$ in table 2.

Table 2. Maximum values of the premultiplied one-dimensional energy spectra. Superscript $\star$ means the maximum value of the quantity. The spectra are scaled with the friction velocity of the unstable side, $u_{\unicode[STIX]{x1D70F}u}$ .

Figure 14. Maps of premultiplied one-dimensional energy spectra as function of spanwise wavelength $\unicode[STIX]{x1D706}_{z}^{\ast }$ and distance from the wall $y^{\ast }$ at $Re=20\,000$ for $Ro=0$  (a,b),  $Ro=0.15$  (c,d),  $Ro=0.45$  (e,f) and $Ro=0.9$  (g,h). (a,c,e,g) Shows $k_{z}\unicode[STIX]{x1D6F7}_{uu}$ (black lines) and cospectra $k_{z}\unicode[STIX]{x1D6F7}_{uv}$ (blue dashed lines and colours). (b,d,f,h) Shows $k_{z}\unicode[STIX]{x1D6F7}_{vv}$ (blue dashed lines and colours) and $k_{z}\unicode[STIX]{x1D6F7}_{ww}$ (black lines). Contour levels from innermost to outermost are 0.94, 0.8, 0.6, 0.4, 0.28, 0.2, 0.1 and 0.05. Spectra are scaled with their maximum value. The straight black dashed line follows the relation $y=0.1\unicode[STIX]{x1D706}_{z}$ .

Roll cells produce significant wall-normal and spanwise motions in rotating channel flow, as shown, for example, by Dai et al. (Reference Dai, Huang and Xu2016), and that can be expected to lead to energetic large-scale modes in the spectra of the spanwise, and especially the wall-normal velocity, away from the wall. I therefore interpret distinctly energetic large-scale modes in the spanwise $k_{z}\unicode[STIX]{x1D6F7}_{vv}$ and $k_{z}\unicode[STIX]{x1D6F7}_{ww}$ spectra as well as in the streamwise $k_{x}\unicode[STIX]{x1D6F7}_{vv}$ spectra as signs of roll cells. If there are $n$ pairs of counter-rotating roll cells in the computational domain, which has a spanwise size of $L_{z}=3\unicode[STIX]{x03C0}h$ in these DNS, the spanwise spectrum $k_{z}\unicode[STIX]{x1D6F7}_{vv}$ can be expected to have a peak at $\unicode[STIX]{x1D706}_{z}=3\unicode[STIX]{x03C0}h/n$ .

In non-rotating channel flow the spanwise spectra reveal, besides the streaks at $\unicode[STIX]{x1D706}_{z}^{\ast }\approx 100$ indicated by the near-wall peak in $k_{z}\unicode[STIX]{x1D6F7}_{uu}$ , the signature of large wide structures of wavelength $\unicode[STIX]{x1D706}_{z}\approx h$ in $k_{z}\unicode[STIX]{x1D6F7}_{uu}$ and $k_{z}\unicode[STIX]{x1D6F7}_{uv}$ further away from the wall (figure 14 a), like in the DNS by Lee & Moser (Reference Lee and Moser2015). In the streamwise spectra $k_{x}\unicode[STIX]{x1D6F7}_{uu}$ and $k_{x}\unicode[STIX]{x1D6F7}_{uv}$ , energetic peaks are seen at $\unicode[STIX]{x1D706}_{x}^{\ast }\approx 1000$ near the wall, related to the near-wall cycle (Monty et al. Reference Monty, Hutchins, Ng, Marusic and Chong2009). In the outer layer, weakly energetic large-scale structures with wavelengths $\unicode[STIX]{x1D706}_{x}\approx \unicode[STIX]{x03C0}h$ to $4\unicode[STIX]{x03C0}h$ are observed in $k_{x}\unicode[STIX]{x1D6F7}_{uu}$ and $k_{x}\unicode[STIX]{x1D6F7}_{uv}$ (figure 15 a), which become more energetic at higher $Re$ (Lee & Moser Reference Lee and Moser2015).

The near-wall peak in the spanwise spectrum $k_{z}\unicode[STIX]{x1D6F7}_{uu}$ and streamwise spectrum $k_{x}\unicode[STIX]{x1D6F7}_{uu}$ changes little in the rotating cases if $Ro\leqslant 0.45$ , so that a change of the streak spacing induced by rotation, as suggested by Kristoffersen & Andersson (Reference Kristoffersen and Andersson1993) and Alvelius (Reference Alvelius1999), cannot be confirmed. However, at $Ro=0.15$ , $k_{z}\unicode[STIX]{x1D6F7}_{vv}$ has three peaks caused by near-wall structures of wavelength $\unicode[STIX]{x1D706}_{z}^{\ast }\approx 150$ around $y^{\ast }=50$ and large structures with wavelengths $\unicode[STIX]{x1D706}_{z}\approx \unicode[STIX]{x03C0}h/2$ and $\unicode[STIX]{x1D706}_{z}\approx \unicode[STIX]{x03C0}h$ (the latter indicated by the green dashed line) in the outer layer of the unstable side far away from the wall at $y\approx -0.3$ (figure 14 d). A strong large-scale peak is also observed in $k_{z}\unicode[STIX]{x1D6F7}_{ww}$ at wavelength $\unicode[STIX]{x1D706}_{z}\approx \unicode[STIX]{x03C0}h$ in the outer layer. The energetic large-scale modes in the outer layer at $\unicode[STIX]{x1D706}_{z}\approx \unicode[STIX]{x03C0}h$ are almost certainly a consequence of three pairs of counter-rotating roll cells of spanwise size $\unicode[STIX]{x03C0}h/2$ seen previously in figure 3(a). The other peak in $k_{z}\unicode[STIX]{x1D6F7}_{vv}$ at $\unicode[STIX]{x1D706}_{z}=\unicode[STIX]{x03C0}h/2$ shows that there also smaller large-scale structures, possibly roll cells that are half the size of the largest ones, indicating that there might be roll cells of different sizes. Scales with the maximum possible streamwise wavelength, i.e. $\unicode[STIX]{x1D706}_{x}=L_{x}$ , are obviously energetic according to the streamwise spectra in figure 15(c,d), implying that at least some of the roll cells span the whole domain in the streamwise direction.

Figure 15. Maps of premultiplied one-dimensional energy spectra as function of streamwise wavelength $\unicode[STIX]{x1D706}_{x}^{\ast }$ and distance from the wall $y^{\ast }$ at $Re=20\,000$ for $Ro=0$ (a,b), $Ro=0.15$ (c,d), $Ro=0.45$ (e,f) and $Ro=0.9$ (g,h). (a,c,e,g) Shows $k_{x}\unicode[STIX]{x1D6F7}_{uu}$ (black lines) and cospectra $k_{x}\unicode[STIX]{x1D6F7}_{uv}$ (blue dashed lines and colours). (b,d,f,h) Shows $k_{x}\unicode[STIX]{x1D6F7}_{vv}$ (blue dashed lines and colours) and $k_{x}\unicode[STIX]{x1D6F7}_{ww}$ (black lines). Contour levels from innermost to outermost are 0.94, 0.8, 0.6, 0.4, 0.28, 0.2, 0.1 and 0.05. Spectra are scaled with their maximum value.

At a higher $Ro=0.45$ , the peaks in the spanwise spectra $k_{z}\unicode[STIX]{x1D6F7}_{vv}$ , $k_{z}\unicode[STIX]{x1D6F7}_{ww}$ have, compared to the spectra at $Ro=0$ , clearly shifted towards larger scales (figure 14 f), and those of the streamwise spectra $k_{x}\unicode[STIX]{x1D6F7}_{vv}$ , $k_{x}\unicode[STIX]{x1D6F7}_{ww}$ to a streamwise wavelength $\unicode[STIX]{x1D706}_{x}\approx 8\unicode[STIX]{x03C0}h/3$ (figure 15 f) in the outer layer. The energetic large-scale modes observed in the spanwise spectrum $k_{z}\unicode[STIX]{x1D6F7}_{vv}$ at $\unicode[STIX]{x1D706}_{z}\approx 3\unicode[STIX]{x03C0}h/8$ and $y\approx -0.5$ are not present at $Ro=0$ ; therefore, I interpret them as the signatures of roll cells. Two-point correlations, presented later, provide further evidence that they exist. The roll cells are smaller than at $Ro=0.15$ and do not appear to span the whole streamwise domain according to the streamwise spectra, since the peak in the streamwise $k_{x}\unicode[STIX]{x1D6F7}_{vv}$ is at $\unicode[STIX]{x1D706}_{x}<L_{x}$ . These observations are consistent with the results shown by the visualizations in figure 3 earlier. Dai et al. (Reference Dai, Huang and Xu2016) also noted that the roll cell diminished in size with $Ro$ in their DNS at a much lower $Re=2800$ . In the present DNS, energetic wide and long structures in terms of outer units are observed as well in the spanwise and streamwise spectra $\unicode[STIX]{x1D6F7}_{uu}$ , and especially the cospectra $\unicode[STIX]{x1D6F7}_{uv}$ at $Ro=0.45$ , and even more so at $Ro=0.15$ in the outer layer (figures 14 c,e and 15 c,e), showing that the roll cells induce large-scale momentum transfer.

Previous studies have found strong support for the hypothesis that the logarithmic region of wall-bounded turbulent flows is populated by attached eddies (Perry & Chong Reference Perry and Chong1982) whose size grows with $y$ , see e.g. Smits et al. (Reference Smits, McKeon and Marusic2011) and Hwang (Reference Hwang2015). Some evidence of attached eddies can indeed be found in the present spanwise spectrum $k_{z}\unicode[STIX]{x1D6F7}_{uu}$ and cospectrum $k_{z}\unicode[STIX]{x1D6F7}_{uv}$ of the non-rotating channel flow (figure 14 a), since they show that size of the structures grows with the distance to the wall in the logarithmic region. The cospectrum follows roughly the linear relation $y=0.2\unicode[STIX]{x1D706}_{z}-1$ shown by the straight black dashed line in the figure, indicating that the size of eddies is approximately proportional to the distance to the wall, in agreement with the attached eddy hypothesis (Hwang Reference Hwang2015). The spanwise spectra $k_{z}\unicode[STIX]{x1D6F7}_{uu}$ and cospectra $k_{z}\unicode[STIX]{x1D6F7}_{uv}$ at $Ro=0.15$ and 0.45 (figure 14 c,e) show similar characteristics, with growing scales as $y$ increases. The cospectrum also follows in these cases roughly the linear relation $y=0.2\unicode[STIX]{x1D706}_{z}-1$ . This provides support for the idea that rotating wall-bounded flows are also populated by attached eddies, at least up to moderate rotation rates.

At higher rotation rates, large-scale structures become progressively less energetic and smaller. At $Ro=0.9$ no energetic large-scale structures are seen in the spanwise spectra $k_{z}\unicode[STIX]{x1D6F7}_{uu}$ and $k_{z}\unicode[STIX]{x1D6F7}_{uv}$ , while $k_{z}\unicode[STIX]{x1D6F7}_{vv}$ has a peak at spanwise wavelength $\unicode[STIX]{x1D706}_{z}\approx \unicode[STIX]{x03C0}h/6$ in the outer layer at $y\approx -0.8$ (figure 14 g,h). The streamwise spectrum of the streamwise velocity $k_{x}\unicode[STIX]{x1D6F7}_{uu}$ has still a near-wall peak and that of the wall-normal velocity $k_{x}\unicode[STIX]{x1D6F7}_{vv}$ has an outer peak at streamwise wavelength $\unicode[STIX]{x1D706}_{x}\approx 2\unicode[STIX]{x03C0}h/5$ , while the spectrum of the spanwise velocity $k_{x}\unicode[STIX]{x1D6F7}_{ww}$ has two energetic peaks: an outer peak at streamwise wavelength $\unicode[STIX]{x1D706}_{x}\approx \unicode[STIX]{x03C0}h/5$ and a near-wall peak at $\unicode[STIX]{x1D706}_{x}\approx 4\unicode[STIX]{x03C0}h/5$ (figure 15 g,h). Energetic large-scale roll cells are thus not present according to the spectra – correspondingly, no large structures like roll cells are observed in visualizations, and there are fewer energetic large-scale motions that contribute to momentum transfer than in a non-rotating channel flow.

Similar observations (not shown for brevity) are made at yet higher $Ro$ , i.e. no energetic wide or long structures are present according to the spectra and cospectra in terms of outer units. The peak in the spanwise spectrum $k_{z}\unicode[STIX]{x1D6F7}_{vv}$ shifts towards shorter wavelengths and the double peak in the streamwise spectrum $k_{x}\unicode[STIX]{x1D6F7}_{ww}$ persists. The double peak is also observed in the r.m.s. profiles (figure 6 c). The spanwise spectra $k_{z}\unicode[STIX]{x1D6F7}_{uu}$ and $k_{z}\unicode[STIX]{x1D6F7}_{uv}$ still have a spectral peak at $\unicode[STIX]{x1D706}_{z}^{\ast }\approx 100$ , but the peak moves closer to the wall in terms of viscous wall units. The streamwise spectra $k_{x}\unicode[STIX]{x1D6F7}_{uu}$ and $k_{x}\unicode[STIX]{x1D6F7}_{uv}$ have a spectral peak at $\unicode[STIX]{x1D706}_{x}\approx h$ when $Ro\geqslant 0.9$ , but in terms of the viscous wall units the structures become shorter. This means that the near-wall streaky structures move closer to the wall and become shorter, in agreement with the results of Lamballais et al. (Reference Lamballais, Métais and Lesieur1998), but they do not clearly confirm their finding that these structures become weaker at high  $Ro$ .

According to the spanwise spectrum $k_{z}\unicode[STIX]{x1D6F7}_{uu}$ and cospectrum $k_{z}\unicode[STIX]{x1D6F7}_{uv}$ at $Ro=0.9$ (figure 14 g), the structures do not become much larger when the distance to the wall increases, and at higher $Ro$ the scales appear to grow even less with $y$ . The absence of structures whose size grows with the distance to the walls when $Ro\gtrsim 0.9$ implies that attached eddies are much less prominent in very rapidly rotating wall-bounded flows. This suggests that the interaction between the inner and outer layer is weak at high $Ro$ . Pressure and turbulent diffusion of turbulent kinetic energy are accordingly very small on the unstable side in that case.

Spectra at $Re=20\,000$ and $Ro=0.15$ at the stable side of the channel (not shown here) show that the roll cells observed on the unstable side do not deeply penetrate the stable channel side, although the study by Dai et al. (Reference Dai, Huang and Xu2016) shows that they occasionally can become larger and may even affect the flow up to the wall at the stable channel side. The spectra also show energetic near-wall modes at $\unicode[STIX]{x1D706}_{z}^{\ast }\approx 100$ and $\unicode[STIX]{x1D706}_{x}^{\ast }\approx 800$ , as on the unstable side, implying that the near-wall cycle is maintained. On the other hand, although the flow is fully turbulent on the stable side at this low $Ro$ , the spectra on the stable channel side reveal no or much less energetic large-scale modes than at $Ro=0$ , showing that cyclonic rotation weakens large-scale structures in wall-bounded flows. This agrees with the observation that in plane Couette flow even very weak cyclonic spanwise rotation eliminates large-scale structures (Komminaho, Lundbladh & Johansson Reference Komminaho, Lundbladh and Johansson1996).

7.2 Reynolds number dependence

In several other DNS studies of spanwise rotating channel flow, large streamwise roll cells or Taylor–Görtler vortices have been found and examined, as mentioned before. Kristoffersen & Andersson (Reference Kristoffersen and Andersson1993), Lamballais et al. (Reference Lamballais, Métais and Lesieur1998), Dai et al. (Reference Dai, Huang and Xu2016) and Hsieh & Biringen (Reference Hsieh and Biringen2016) have observed them in DNS at $Re_{\unicode[STIX]{x1D70F}}\leqslant 200$ and $Ro\leqslant 0.5$ . Hsieh and Biringen have observed them also at $Re_{\unicode[STIX]{x1D70F}}=406$ and $Ro=0.2$ . It is important that the roll cells are properly resolved since flow statistics deviate significantly if they are suppressed (Hsieh & Biringen Reference Hsieh and Biringen2016). Helical spectra computed from DNS at $Re_{\unicode[STIX]{x1D70F}}=180$ by Yang & Wu (Reference Yang and Wu2012) suggest large roll cells if $0.03\leqslant Ro\leqslant 0.57$ , with the strongest signal at $Ro=0.15$ . The size of the roll cells appears to diminish with $Ro$ for $Ro\geqslant 0.15$ , and there is no sign that they exist at $Ro=0.94$ and higher. On the other hand, visualizations and spanwise two-point correlations computed by Grundestam et al. (Reference Grundestam, Wallin and Johansson2008) at $Re_{\unicode[STIX]{x1D70F}}=180$ indicate that roll cells are present at $Ro=1.27$ , but the statistics show that the roll cells are non-stationary and do not have a long correlation length in the streamwise direction, and their size appears to be smaller than at lower  $Ro$ .

Thus, there is strong evidence that large roll cells exist in spanwise rotating channel flow at low Reynolds numbers for $Ro\lesssim 0.6$ . At higher $Ro$ , roll cells are less certain and more difficult to detect owing to their unsteadiness, but there are indications of their occurrence (Grundestam et al. Reference Grundestam, Wallin and Johansson2008). Spectra presented before reveal large streamwise roll cells in a DNS at $Re=20\,000$ and $Ro=0.15$ , and clearly hint at their occurrence at $Ro=0.45$ , but at $Ro=0.9$ and higher there is no strong evidence of their presence. This suggests that, at lower Reynolds numbers, roll cells exist in a wider $Ro$ range. Note, that in previous DNS of rotating channel flow the computational domain size was $4\unicode[STIX]{x03C0}\times 2\unicode[STIX]{x03C0}$ in the streamwise and spanwise directions, respectively, or less, which is smaller than in the present DNS. This may influence the size and enhance the coherency of the roll cells, especially in combination with periodic boundary conditions. Indeed, in the DNS by Kristoffersen & Andersson (Reference Kristoffersen and Andersson1993) at $Ro=0.15$ the roll cells were very steady, whereas in the experiments by Johnston et al. (Reference Johnston, Halleen and Lezius1972) the boundary conditions were different and the roll cells were much more unsteady.

To answer more unequivocally whether at low $Re$ roll cells exist in a wider $Ro$ range than at higher $Re$ , I have computed premultiplied one-dimensional spanwise and streamwise spectra of the wall-normal and spanwise velocity fluctuations at $Re=5000$ for $Ro=0$ , 0.15, 0.45 and 0.9, see figure 16. The wavelength and the distance to the wall is, as in the previous spectra, scaled by the viscous length scale of the unstable side, $l^{\ast }=\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}u}$ , and the spectra are scaled with their maximum value, see table 2. The red dashed line marks again scales with a wavelength $h$ . Spanwise two-point correlations of the wall-normal velocity fluctuations $R_{vv}=\overline{v(\boldsymbol{x}+z)v(\boldsymbol{x})}/\overline{vv}$ at $Re=3000$ , 5000 and 20 000 and various $Ro$ have also been calculated. If streamwise vortices are present, $R_{vv}$ is expected to display negative correlations with a minimum at a separation distance that is approximately equal to the mean vortex diameter. Accordingly, in non-rotating channel flow, $R_{vv}$ has a minimum at $z^{\ast }\approx 25$ near the wall as a result of near-wall streamwise vortices (Kim et al. Reference Kim, Moin and Moser1987). The two-point correlations are shown at the unstable channel side at the wall-normal position, where $R_{vv}$ approximately has the largest negative values and thus the signs of the roll cells are most clear. In all cases, the mean velocity profile is approximately linear at this position. The domain size is the same in all DNS used to compute the spectra and two-point correlations.

Figure 16. Maps of premultiplied one-dimensional spanwise and streamwise energy spectra at $Re=5000$ for $Ro=0$ (a,b), $Ro=0.15$ (c,d), $Ro=0.45$ (e,f) and $Ro=0.9$ (g,h). (a,c,e,g) Shows spanwise spectra $k_{z}\unicode[STIX]{x1D6F7}_{vv}$ (blue dashed lines and colours) and $k_{z}\unicode[STIX]{x1D6F7}_{ww}$ (black lines). (b,d,f,h) Shows streamwise spectra $k_{x}\unicode[STIX]{x1D6F7}_{vv}$ (blue dashed lines and colours) and $k_{x}\unicode[STIX]{x1D6F7}_{ww}$ (black lines). Contour levels from innermost to outermost are 0.94, 0.8, 0.6, 0.4, 0.28, 0.2, 0.1 and 0.05. Spectra are scaled with their maximum value.

Figure 17. Spanwise two-point correlations of the wall-normal velocity fluctuations at (a) $Ro=0.15$ and $Re=20\,000$ and $y=-0.41$ , $Re=5000$ and $y=0$ , and $Re=3000$ and $y=0$ ; (b)  $Ro=0.45$ and $Re=20\,000$ and $y=-0.54$ , $Re=5000$ and $y=-0.55$ , and $Re=3000$ and $y=-0.5$ ; (c,d) $Ro=0.9$ and $Re=20\,000$ and $y=-0.83$ , $Re=5000$ and $y=-0.75$ , and $Re=3000$ and $y=-0.75$ ; (e,f) $Ro=1.2$ and $Re=20\,000$ and $y=-0.92$ , $Re=5000$ and $y=-0.83$ , and $Re=3000$ and $y=-0.75$ . In (a,b,c,e) $z$ is scaled by $h$ and in (d,f) $z^{\ast }=z/l^{\ast }$ . (——, black) $Re=20\,000$ , (——, red) $Re=5000$ and (——, green) $Re=3000$ .

At $Ro=0.15$ and $Re=5000$ , the spanwise spectrum $k_{z}\unicode[STIX]{x1D6F7}_{ww}$ has a peak at $\unicode[STIX]{x1D706}_{z}=\unicode[STIX]{x03C0}h$ (indicated by the green dashed line) while the peak of $k_{z}\unicode[STIX]{x1D6F7}_{vv}$ extends from approximately $\unicode[STIX]{x1D706}_{z}=\unicode[STIX]{x03C0}h/2$ to $\unicode[STIX]{x03C0}h$ (figure 16 c) and the streamwise spectra reveal structures that span the whole streamwise domain (figure 16 d). Spectra at $Re=3000$ are not shown for brevity, but at $Ro=0.15$ they show a peak at spanwise wavelengths $\unicode[STIX]{x1D706}_{z}=\unicode[STIX]{x03C0}h$ and a lesser one at $3\unicode[STIX]{x03C0}h/5$ . This indicates, similarly as at $Re=20\,000$ , roll cells with a size of $\unicode[STIX]{x03C0}h/2$ and smaller ones. By comparing figure 16(e,f) with figure 16(a,b), it can be concluded that, at $Ro=0.45$ , wider and longer structures exist on the unstable channel side than at $Ro=0$ , implying that large roll cells are present. The spectral peaks in the spanwise $k_{z}\unicode[STIX]{x1D6F7}_{vv}$ and streamwise spectrum $k_{x}\unicode[STIX]{x1D6F7}_{vv}$ are at $\unicode[STIX]{x1D706}_{z}\approx 3\unicode[STIX]{x03C0}h/8$ and $\unicode[STIX]{x1D706}_{x}\approx 2\unicode[STIX]{x03C0}h$ , respectively. At similar wavelengths, peaks are found in the spectra at $Re=20\,000$ , as shown before, and $Re=3000$ .

The spectra suggest that the roll cells at $Ro=0.15$ as well as at $Ro=0.45$ have a similar size for $Re=3000$ –20 000. This is confirmed by the two-point correlations. At $Ro=0.15$ , $R_{vv}$ is clearly negative for $0.5\lesssim z\lesssim 2$ , and its minimum, which is a measure of the mean vortex diameter, is at approximately the same $z$ at all three Reynolds numbers (figure 17 a). The spectra suggest roll cells with a diameter $\unicode[STIX]{x03C0}h/2$ as well as smaller ones, and that appears to be consistent with the two-point correlations which suggest roll cells with a mean diameter of approximately $h$ . Also $R_{vv}$ at $Ro=0.45$ has a minimum at approximately the same $z$ at all three $Re$ (figure 17 b). The spectra indicate that the roll cells have a mean diameter of approximately $3\unicode[STIX]{x03C0}h/16$ , in agreement with the two-point correlations.

These results indicate that for $Re=3000$ –20 000 the size of the roll cells is smaller at $Ro=0.45$ than at $Ro=0.15$ , but approximately independent of $Re$ . The size $\unicode[STIX]{x03C0}h/2$ of the largest roll cells at $Ro=0.15$ indicated by the spanwise spectra is consistent with the size observed in the DNS by Kristoffersen & Andersson (Reference Kristoffersen and Andersson1993) and Yang & Wu (Reference Yang and Wu2012) at the same $Ro$ . Yang and Wu also observed smaller roll cells and found that the size of the roll cells becomes smaller with $Ro$ for $Ro\geqslant 0.15$ , again consistent with the present results. Also the DNS by Dai et al. (Reference Dai, Huang and Xu2016) at $Re=2800$ show that the roll cells become smaller when $Ro$ increases from 0.1 to 0.5. On the other hand, the size $\unicode[STIX]{x03C0}h/2$ of the roll cells in the DNS by Hsieh & Biringen (Reference Hsieh and Biringen2016) at $Ro=0.5$ and $Re_{\unicode[STIX]{x1D70F}}\approx 200$ is considerably larger than in the present DNS at $Ro=0.45$ . This difference might be related to the more restricted computational domain used by Hsieh and Biringen. Besides, they only show visualizations of the time-averaged velocity field, and these naturally emphasize the largest, most steady structures.

At a higher $Ro=0.9$ , the peak in the spanwise spectrum $k_{z}\unicode[STIX]{x1D6F7}_{vv}$ shifts towards a smaller $\unicode[STIX]{x1D706}_{z}\approx 3\unicode[STIX]{x03C0}h/13$ at $Re=5000$ (figure 16 g) and $\unicode[STIX]{x1D706}_{z}\approx 3\unicode[STIX]{x03C0}h/11$ at $Re=3000$ , which is at a larger wavelength than at $Re=20\,000$ (figure 14 h). The large structures are also considerably longer in terms of outer units at lower $Re$ , see figure 16(h) versus figure 15(h). The minimum in $R_{vv}$ shifts towards smaller $z$ at higher $Re$ (figure 17 c), confirming that at $Ro=0.9$ the large scales become smaller at higher $Re$ . The two-point correlations (figure 17 e) and spectra (not shown here) show that at $Ro=1.2$ the differences are even more pronounced. The size of the large flow structures in terms of outer units shrinks monotonically with $Re$ , and this trend continues at higher $Ro$ . On the other hand, if the separation distance $z$ is scaled by the viscous length scale $l^{\ast }$ , the two-point correlations differ considerably at $Ro=0.9$ (figure 17 d) and lower $Ro$ , but much less at $Ro=1.2$ (figure 17 f) and higher $Ro$ . The two-point correlations at $Re=3000$ and $Re=5000$ show in fact an almost perfect collapse. Thus, the size of the largest scales continues to decrease with $Ro$ , but for $Ro\geqslant 0.9$ it becomes dependent on $Re$ as well. Whether these largest structures can be considered to be typical roll cells is not clear since the roll cells and typical near-wall streamwise vortices become less distinguishable at high $Ro$ and low $Re$ . The two-point correlations suggest that their size starts to display a viscous wall unit scaling and therefore may not have the same physical origin as at lower  $Ro$ .