3 Results

Following Reynolds’ strategy (Reynolds Reference Reynolds1894), the instantaneous flow field $u_{i}(x,y,z,t)$ ( $i=1,2$ or $3$ refers to the $x$ , $y$ and $z$ directions, respectively; $u$ , $v$ , $w$ are used interchangeably with $u_{1}$ , $u_{2}$ , $u_{3}$ ) is decomposed into a mean part and a fluctuating part. In the present study,

(3.1) $$\begin{eqnarray}\displaystyle & \displaystyle u_{i}(x,y,z,t)=\langle u_{i}\rangle (y)+u_{i}^{\prime }(x,y,z,t). & \displaystyle\end{eqnarray}$$

Here, $\langle u_{i}\rangle (y)$ is the temporally and spatially averaged velocity (TS mean velocity), while $u_{i}^{\prime }(x,y,z,t)$ are the corresponding fluctuations. In RPCF, $\langle u_{2}\rangle =\langle u_{3}\rangle =0$ , and $\langle u_{1}\rangle (y)=\langle u\rangle (y)$ is the only non-zero TS mean velocity component. Figure 2(a) shows the profiles of $\langle u\rangle (y)$ from all four simulations. It is seen that the lines of $\langle u\rangle$ can be divided into two groups (see the zoomed-in view shown in figure 2(b)). One is the from simulations R2_512 and R2_256, and the other is from R3_512 and R3_256, which illustrate the existence of two states. The lines from simulations with three pairs of roll cells have larger slopes in the region $-0.7h\lesssim y\lesssim 0.7h$ and larger wall friction Reynolds numbers $Re_{\unicode[STIX]{x1D70F}}=u_{\unicode[STIX]{x1D70F}}h/\unicode[STIX]{x1D708}$ . Here $u_{\unicode[STIX]{x1D70F}}=\sqrt{\unicode[STIX]{x1D70F}_{w}/\unicode[STIX]{x1D70C}}$ is the friction velocity with $\unicode[STIX]{x1D70F}_{w}$ and $\unicode[STIX]{x1D70C}$ being the wall shear stress and the fluid density, respectively. The detailed values of $Re_{\unicode[STIX]{x1D70F}}$ and mean shear rate at the centreline ( $\unicode[STIX]{x1D6F9}$ ) from all simulations are listed in table 1. It is evident that the deviations are negligible for $Re_{\unicode[STIX]{x1D70F}}$ and $\unicode[STIX]{x1D6F9}$ within each group, while apparent discrepancies can be detected between groups, around $5.5\,\%$ for $Re_{\unicode[STIX]{x1D70F}}$ and $28.3\,\%$ for $\unicode[STIX]{x1D6F9}$ at 256 resolutions. These non-negligible discrepancies between two groups suggest the existence of the two states.

Figure 3.

Time series of $\unicode[STIX]{x1D713}$ .

In order to explore mean shear rate at the centreline more clearly, the time series of the spatially averaged mean shear rate at the centreline $\unicode[STIX]{x1D713}(t)=\text{d}[u(x,0,z,t)]_{x,z}/\,\text{d}y$ is shown in figure 3. Here, $[\cdot ]_{x,z}$ denotes quantities that are averaged in $x$ and $z$ directions. From the definition, $\unicode[STIX]{x1D6F9}$ can be estimated by

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F9}=\frac{1}{N_{t}}\mathop{\sum }_{i=1}^{N_{t}}\unicode[STIX]{x1D713}(t_{i}). & \displaystyle\end{eqnarray}$$

Here, $N_{t}=1501$ is the number of the sampling field. It is evident from figure 3 that $\unicode[STIX]{x1D713}$ from four simulations oscillates around two different mean values. $\unicode[STIX]{x1D713}$ from R3_512 and R3_256 oscillates around 0.2 while those from R2_512 and R2_256 oscillate around 0.14. This means that the vorticity ratios $S=-2\unicode[STIX]{x1D6FA}/(\text{d}\langle u\rangle /\text{d}y)$ in the centre part of the channel from the three-pair state and the two-pair state are approximately $-1$ and $-1.43$ respectively. The absolute vorticity at the former state is close to zero and the flow is neutrally stable in the centre core region (Gai et al. Reference Gai, Xia, Cai and Chen2016), whereas at the latter state, the system rotation will stabilize the flow in the core region (Kristoffersen & Andersson Reference Kristoffersen and Andersson1993). These might explain the lower turbulent kinetic level ( $\langle u^{\prime }u^{\prime }+v^{\prime }v^{\prime }+w^{\prime }w^{\prime }\rangle /2$ ) in the centre part of the channel. The standard deviation $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D713}}$ of $\unicode[STIX]{x1D713}$ from both states are computed by

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D713}}=\sqrt{\frac{1}{N_{t}}\mathop{\sum }_{i=1}^{N_{t}}(\unicode[STIX]{x1D713}(t_{i})-\unicode[STIX]{x1D6F9})^{2}}, & \displaystyle\end{eqnarray}$$

and are listed in table 1 too. The variances are much smaller (around $1/3$ ) than the differences between two states, which is approximately $5.8\times 10^{-2}$ .

Figure 4.

Reynolds stresses from all four simulations: (a) $\langle u^{\prime }u^{\prime }\rangle$ , (b)  $\langle v^{\prime }v^{\prime }\rangle$ , (c) $\langle w^{\prime }w^{\prime }\rangle$ and (d) $-\langle u^{\prime }v^{\prime }\rangle$ . All quantities are normalized by $4U_{w}^{2}$ .

In fact, the deviations are more obvious in higher-order statistics. Figure 4 shows the normalized diagonal components of Reynolds stresses $\langle u^{\prime }u^{\prime }\rangle$ , $\langle v^{\prime }v^{\prime }\rangle$ and $\langle w^{\prime }w^{\prime }\rangle$ and the normalized Reynolds shear stress $-\langle u^{\prime }v^{\prime }\rangle$ (by $4U_{w}^{2}$ ) from different simulations. Again, apparent deviations can be observed for the Reynolds stresses from the flow states with different pairs of roll cells, which again confirms that these two states are turbulent and statistically different. It should be emphasized here that the lines from the same state but with different grid resolutions match well with each other. This verifies that the present simulation results are valid.

Figure 5.

(a) Instantaneous three-dimensional flow structures shown by iso-surfaces of $Qh^{2}/U_{w}^{2}\geqslant 1.5$ and (b) the instantaneous velocity vectors and contours of $Qh^{2}/U_{w}^{2}$ at the streamwise section $x=0$ from simulation R2_512.

Figure 6.

(a) Instantaneous three-dimensional flow structures shown by iso-surfaces of $Qh^{2}/U_{w}^{2}\geqslant 1.5$ and (b) the instantaneous velocity vectors and contours of $Qh^{2}/U_{w}^{2}$ at the streamwise section $x=0$ from simulation R3_512.

In order to further characterize the flow differences, the flow structures are analysed. Figures 5 and 6 display the instantaneous three-dimensional flow structures represented by iso-surfaces of $Qh^{2}/U_{w}^{2}\geqslant 1.5$ (here $Q=-(\unicode[STIX]{x2202}u_{i}^{\prime }/\unicode[STIX]{x2202}x_{j})(\unicode[STIX]{x2202}u_{j}^{\prime }/\unicode[STIX]{x2202}x_{i})$ ), and the instantaneous velocity vectors $(v,w)$ and the contours of $Qh^{2}/U_{w}^{2}$ at the streamwise section $x=0$ from R2_512 and R3_512, respectively. The iso-surfaces of $Q$ were used to identify the vortex structures by Kasagi, Sumitani & Suzuki (Reference Kasagi, Sumitani, Suzuki and Iida1995), Tsukahara (Reference Tsukahara2011) and Gai et al. (Reference Gai, Xia, Cai and Chen2016). It is seen from figures 5 and 6 that the vortex structures are gathered together into clusters, and most of them are located between two counter-rotating roll cells. Furthermore, two pairs of roll cells (or four roll cells) can be identified from figure 5 while three pairs of roll cells (or six roll cells) can be recognized from figure 6, and these roll cells are straight in the streamwise direction. These flow structures are rather stable and will not disappear during the simulations. This scenario is quite similar to that reported by Huisman et al. (Reference Huisman, van der Veen, Sun and Lohse2014). In their work, they reported different turbulent states in TCF with different number of turbulent Taylor vortices (rolls).

Figure 7.

Contours of $v(0,0,z,t)$ from (a) R2_512 and (b) R3_512.

It is interesting to notice that these large-scale roll cells structures are not fixed in space, but they are moving in the spanwise directions (since the periodic boundary condition is used in the spanwise directions). In the present study, the instantaneous wall-normal velocity at the centre plane is used to indicate the location of the roll cells, since a local peak value, either maximum or minimum, will appear around the boundaries of each roll cell. This is different from those used by Huisman et al. (Reference Huisman, van der Veen, Sun and Lohse2014), where they used the peak locations of azimuthal velocity at the centre between two cylinders as the boundaries of rolls. We must emphasize that the present method can only show the approximate locations of roll cells at different time steps. Figure 7(a,b) shows the contours of $v(0,0,z,t)$ from R2_512 and R3_512 at $(z,t)$ spaces, respectively. Similar results can be obtained for streamwise averaged wall-normal velocity $v^{x}(z,t)=1/L_{x}\int _{0}^{L_{x}}v(x,0,z,t)\,\text{d}x$ (not shown here). It is apparent from figure 7 that four roll cells can be recognized at R2_512 while six roll cells can be identified at R3_512. It can also be observed from figure 7 that the travelling velocities of roll cells during $0<tU_{w}/h\lesssim 300$ are not constant for both cases, but for $300<tU_{w}/h\lesssim 750$ , they vary very slowly and asymptotically go to an almost constant drift velocity. The speed from R2_512 is much larger than that from R3_512; see supplementary movies 1 and 2 (available at https://doi.org/10.1017/jfm.2017.869) of velocity vectors at the slice $x=0$ . We have computed the statistics by using the flow fields at $300<tU_{w}/h\lesssim 750$ , and no clear deviations can be observed between this shorter averaging period and the original $750h/U_{w}$ (not shown here). Nevertheless, the present time averaging period might not be long enough to show the effect of this locomotion, since the travelling speeds in the spanwise direction of the roll cells at both states are of order $0.01U_{w}$ , which are much slower than the speed of the wall. Another important point, which deserves further discussion, noted from figure 7 is that the unidirectional spanwise translation of roll cells breaks the spanwise reflection symmetry. At the present simulation with $tU_{w}/h\lesssim 750$ (1750 for R2_256 cases), the roll cells travel in the same direction. If this unidirectional translation of roll cells was a stable feature of the flow solution, there should be another one that translates in the opposite direction. Alternatively, there could be the third possibility that the spanwise translation oscillates from one direction to the other with an extremely long time scale. In the future, simulations with much longer durations should be performed to investigate the detailed feature of the roll cells’ translation and their influences on the mean statistics. Furthermore, more simulations with different initial flow fields need to be carried out to study the influence of initial conditions on the spanwise motion of the roll cells.

In RPCF, a triple decomposition approach (Moser & Moin Reference Moser and Moin1987; Lee & Kim Reference Lee, Kim and Durst1991; Bech & Andersson Reference Bech and Andersson1996; Gai et al. Reference Gai, Xia, Cai and Chen2016) is usually used to decompose the instantaneous velocity field into three parts in order to see the large-scale structures more clearly. That is,

(3.4) $$\begin{eqnarray}\displaystyle u_{i}(x,y,z,t) & = & \displaystyle \langle u_{i}\rangle (y)+u_{i}^{\prime }(x,y,z,t)\nonumber\\ \displaystyle & = & \displaystyle \bar{u}_{i}(y,z)+u_{i}^{\prime \prime }(x,y,z,t)\nonumber\\ \displaystyle & = & \displaystyle \langle u_{i}\rangle (y)+u_{i}^{s}(y,z)+u_{i}^{\prime \prime }(x,y,z,t).\end{eqnarray}$$

Here, $\bar{u}_{i}(y,z)$ is the averaged velocity with respect to $t$ and $x$ , $u_{i}^{s}(y,z)=\bar{u}_{i}(y,z)-\langle u_{i}\rangle (y)$ corresponds to the secondary flow, usually occurring as roll cells. However, if the traditional triple decomposition is adopted to extract the secondary roll cells, the secondary flows will be weakened apparently due to the locomotion of roll cells in the spanwise direction. In order to correctly extract the secondary flows, the following roll-cell-fixed averaging procedure is adopted. First, the instantaneous flow field $u_{i}(x,y,z,t)$ from each sample is averaged in the $x$ direction to get $u_{i}^{x}(y,z,t)$ . Then, the cross-section field $u_{i}^{x}$ is translated along the $z$ direction with the implementation of the periodic boundary condition to shift the maximum of $v^{x}(0,z,t)$ to $z=0$ . Finally, all the shifted samples are averaged in time to get the mean field $\tilde{u} _{i}(y,z)$ . The secondary flows are then defined as $u_{i}^{s}(y,z)=\tilde{u} _{i}(y,z)-\langle u_{i}\rangle (y)$ . It should be noted that the roll-cell-fixed averaging procedure is the same as the traditional triple decomposition if the roll cells are fixed in the spanwise direction.

Figure 8.

Velocity vectors $(v^{s},w^{s})$ from cases (a) R2_512 and (b) R3_512. The vector lengths are proportional to their magnitudes. The contours show the values of $u^{s}$ .

Figure 8 shows the velocity vectors $(v^{s},w^{s})$ obtained from the roll-cell-fixed averaging process at the $y{-}z$ plane with their corresponding streamwise component $u^{s}$ at R2_512 and R3_512. Consistent with the instantaneous contours shown in figure 5 and figure 6, two and three pairs of roll cells can be identified at these two simulations, respectively. The rolls are more oblate in the state with two pairs of roll cells with width around $\unicode[STIX]{x03C0}h$ , while they are nearly circular with width around $2.09h$ in the state with three pairs of roll cells. Furthermore, the regions with larger values of $u^{s}$ are located between the roll cells. Since the top wall moves with velocity $2U_{w}$ in our studies, the counter-rotating roll cells bring down fluid with higher speed from the top but bring up fluid with lower speed from the bottom, as shown by the sign of $u^{s}$ .