3.1. Large-scale motions

In our previous work (Zhang et al. Reference Zhang, Xia and Chen2022), it was demonstrated that plume currents generated by rising plumes from the unstable side are the most significant large-scale structures in RPPF. For a given small or medium $Ro_\tau$, the number and pattern of plume currents vary continuously with time $t$. To illustrate the patterns and dynamic behaviours of plume currents, figure 4 presents the contours of $v'(t,z)$ in the $t$–$z$ plane with $x=y=0$ at $Ro_\tau =5$. From figure 4(a), it can be observed that in the absence of control, two or more plume currents may exist, and they are constantly splitting, merging and oscillating. The supplementary movie 1 available at https://doi.org/10.1017/jfm.2023.1091 displays the instantaneous distributions of the wall-normal velocity in the $y$–$z$ plane, demonstrating the chaotic behaviours of plumes and plume currents. With the control of two (R5J2B) and three (R5J3B) injection slots (in a spanwise period) on the unstable side, the number of plume currents (in a spanwise period) is fixed at two and three, respectively, as depicted in figure 4(b,c) and supplementary movie 2. In addition, the plume currents oscillate near the injection slots. It is worth noting that the injection/suction velocity is significantly smaller than the wall-normal velocity in plume currents, since the r.m.s. wall-normal velocity on the unstable side is approximately $0.6\,\%$ of the bulk mean velocity. The relatively weak control exerts a considerable influence by enhancing the self-sustaining process of plume currents (Zhang et al. Reference Zhang, Xia and Chen2022), making the large-scale motions much stronger and more stable. Injection provokes the separation of plumes near the injection slots, and the separated plumes converge into plume currents that induce Taylor–Görtler vortices. These vortices sweep more plumes towards the injection slots. However, with four injection slots (R5J4B), the number of plume currents cannot be fixed and can be two, three or four. The splitting and merging of plume currents is not so frequent as the uncontrolled case, and the spanwise locations of plume currents are often near the injection slots or in the middle of two injection slots. A plume current in the middle of two injection slots is likely to be the result of two neighbouring plume currents that have merged. The relatively smaller influence of four injection slots is probably due to the fact that the intrinsic spanwise distance between plume currents in RPPF at $Re_\tau =180$ and $Ro_\tau =5$ is around ${\rm \pi}$ and $2{\rm \pi} /3$. Therefore, four plume currents (with an average distance of ${\rm \pi} /2$) induced by four injection slots tend to merge. Supplementary movie 3 provides insight into the complex dynamics of plumes and plume currents in R5J4B, which is rather different from that of R5J2B.

Figure 4.

Contours of $v'(0,0,z,t)$ in the $Ro_\tau =5$ cases. The origin of $t$ is chosen within a statistically steady state: (a) R5J0; (b) R5J2B; (c) R5J3B; (d) R5J4B; (e) R5J2T; (f) R5J3T; (g) R5J4T.

Figure 5 shows the contours of $v'(t,z)$ at $Ro_\tau =10$. The control of two (R10J2B) and three (R10J3B) injection slots on the unstable side have a similar effect on plume currents as those at $Ro_\tau =5$, as depicted in figure 5(b,c). Nevertheless, four injection slots applied to the unstable side can induce four relatively stable plume currents in the $Ro_\tau =10$ case, which is not achieved in the $Ro_\tau =5$ case. We attribute this to the difference in the intrinsic pattern of plume currents at two rotation numbers. As displayed in figures 4(a) and 5(a), the distances between plume currents at $Ro_\tau =10$ are generally smaller than those at $Ro_\tau =5$, which is explained in our previous work (Zhang et al. Reference Zhang, Xia and Chen2022).

Figure 5.

Contours of $v'(0,0,z,t)$ in the $Ro_\tau =10$ cases. The origin of $t$ is chosen within a statistically steady state: (a) R10J0; (b) R10J2B; (c) R10J3B; (d) R10J4B; (e) R10J2T; (f) R10J3T; (g) R10J4T.

For comparison, we present the plume current behaviours under injection/suction control on the stable side in figures 4(e–g) and 5(e–g). It can be observed that the large-scale motions including splitting, merging and spanwise oscillation remain active and appear similar to those in the uncontrolled case. This indicates that the injection/suction control has a negligible influence on the large-scale motions. The primary reason for this is that the separation of plumes on the unstable side is not significantly influenced by the stable side, which is a shared property of both RPPF and penetrative convection.

Figure 6 depicts $\langle \varPhi _{vv}\rangle _y$, which is the spanwise energy spectra $\varPhi _{vv}(k_z;y)$ averaged along the $y$ direction. The large-scale modes (with $k_z\leq 8$) of the uncontrolled $Ro_\tau =0$ case are found to be much weaker than those of the $Ro_\tau =5$, 10, 30 cases. This can be inferred from figure 6, showing that the peak values in the non-rotating case and the rotating cases are around $0.03$ and within $[0.15,0.25]$, respectively. For the controlled cases, the maximum enhancement of the large-scale modes in the $Ro_\tau =0$ case with injection/suction control is small ($\sim$10 %) compared with that in the $Ro_\tau =5$ and $Ro_\tau =10$ cases ($\sim$50 %) under injection/suction control on the unstable side. We attribute this observation to the sensitivity of large-scale modes to the injection/suction control in the presence of plumes and plume currents in rotating cases.

Figure 6.

Here $\varPhi _{vv}(k_z;y)$ averaged over $y$: (a) $Ro_\tau =0$; (b) $Ro_\tau =5$; (c) $Ro_\tau =10$; (d) $Ro_\tau =30$.

Supplemental to figures 4(d) and 5(d), 6(b–d) show the selectivity of large-scale modes to the injection/suction control. Figure 6(b) shows that at $Ro_\tau =5$, the control with four injection slots has greater enhancement to the $k_z=2$ mode instead of the expected $k_z=4$ mode. Such discrepancy between the control configuration and the plume current distribution disappears in the R10J4B case. It is likely that the dominating mode of the uncontrolled case represents the distribution of plume currents that is most efficient in collecting plumes (totally). In other words, the distances between neighbouring plume currents with the dominating patterns in uncontrolled cases tend to be as small as possible (close to the limit of stability), so that each plume is close enough to a plume current and gets pulled towards it. Therefore, a pattern of plume currents (e.g. four plume currents induced by four injection slots at $Ro_\tau =5$) that is significantly more compact than the dominating pattern in the uncontrolled case (e.g. $k_z=3$ mode at $Ro_\tau =5$) is likely to exceed the maximum ratio between height and distance, which will result in the merging of plume currents.

The above results are still valid in other $Ro_\tau$, $Re_\tau$ and $L_z$, as shown in figure 7, where different numbers of injection/suction slots are applied at the unstable wall. Figure 7(a), showing the results at $Ro_\tau =2$, indicates that the dominating mode is $k_z=2$ in the uncontrolled case, and only $N_J=2$ can achieve significant enhancement at $k_z=N_J$. In comparison, $N_J=3$ and $N_J=4$ mainly enhance the $k_z=2$ and $k_z=1$ modes, respectively, indicating the merging of too compact plume currents. Figure 7(b), showing the results at $Ro_\tau =5$ at a higher $Re_\tau =300$, indicates that the dominating pattern in the uncontrolled case is also $k_z=3$, which is the same as the case at $Re_\tau =180$ (figure 6b). As a result, the wavenumber of the maximally enhanced mode of each $N_J$ is also the same, which can be found by comparing figures 6(b) and 7(b). It is also worth mentioning that, although $v'_{rms}(-1)/U_b$ is smaller at larger $Re_\tau$, the increase of the $k_z=3$ mode by the $N_J=3$ control becomes even more prominent. This indicates that the present control strategy still works under the increase of Reynolds number. Figure 7(c,d) shows the control effect in a larger $L_z=4{\rm \pi}$ at $Re_\tau =180, Ro_\tau =5$, where the dominant distribution of plume currents has $k_z=2.5$. Figure 7(c) shows that doubling the spanwise domain does not significantly change the locations and heights of the peaks in the spectra, if we assign the energy at each non-integer wavenumber to the nearest integer wavenumber smaller than it. This suggests that $L_z=2{\rm \pi}$ is sufficient for capturing energy spectra at $k_z\geq 2$ and that the control with $L_z/D_J\in \mathbb {Z}$ is robust. With $L_z=4{\rm \pi}$, the control set-ups allow $N_J=2.5$ and $N_J=3.5$, which means that there are 5 and 7, respectively, pairs of control slots in the spanwise domain. Figure 7(d) shows that the control with $2 \leq N_J \leq 3.5$ can still fix and enhance the large-scale motions with $k_z=N_J$. However, it should be noted that the enhancement of large-scale motions does not necessarily lead to very strong enhancement of the turbulent transport, which can be seen from the values of $Nu$ listed in table 1.

Figure 7.

Here $\varPhi _{vv}(k_z;y)$ averaged over $y$: (a) $Re_\tau =180$, $Ro_\tau =2$ and $L_z=2{\rm \pi}$; (b) $Re_\tau =300$, $Ro_\tau =5$ and $L_z=2{\rm \pi}$; (c,d) $Re_\tau =180$, $Ro_\tau =5$ and $L_z=4{\rm \pi}$. The results at $Re_\tau =180$, $Ro_\tau =5$ and $L_z=2{\rm \pi}$ are also shown in (c) for comparison. The results with $L_z=4{\rm \pi}$ in (c) are derived by assigning the value at each non-integer wavenumber to the nearest integer wavenumber smaller than it.

Another interesting phenomenon can be inferred from figure 6(c) corresponding to $Ro_\tau =10$. It shows that the control with three and four injection slots on the unstable side can induce a large increase in the energy of $k_z=3$ and $k_z=4$ modes, respectively, while the increase of the $k_z=2$ mode caused by two injection slots is relatively small. This indicates that a control pattern which is too sparse compared with the dominating pattern in the uncontrolled case would be less capable of enhancing large-scale motions. The underlying mechanism should be that a sparse pattern of plume currents may fail to collect some near-wall plumes. Those plumes will be dissipated by viscosity rapidly or form new plume currents. Such a mechanism is further supported by figure 6(d), corresponding to $Ro_\tau =30$ at which the $k_z=7$ mode is dominating in the uncontrolled case. It can be seen that three injection slots on the unstable side are less effective than four injection slots. Moreover, two injection slots on the unstable side mainly promote the $k_z=6$ mode, indicating that there are more plume currents between injection slots.

The above discussion can be distilled into a straightforward guideline for the current injection/suction control strategy aimed at enhancing large-scale motions in RPPF. For a fixed $Re_\tau$ and $Ro_\tau$, utilizing $N_J L_z/2{\rm \pi}$ injection slots on the unstable side with $N_J$ neither significantly smaller nor much larger than the $k_z$ value of the dominant large-scale pattern in the uncontrolled case can substantially amplify plume currents with the corresponding pattern. Injection slots that are too closely spaced will cause plume currents to merge, while excessively sparse injection slots will result in the loss of energy from some plumes or the generation of new plume currents. Consequently, injection/suction control patterns that are either too compact or too sparse, with $N_J L_z/2{\rm \pi}$ injection slots on the unstable side, will not effectively amplify the $k_z=N_J$ mode. Injection/suction control on the stable side has a limited impact on enhancing large-scale motions, as plumes do not originate or detach from this region.

Although spectral analysis is simple to perform, we should notice from figures 4 and 5 that the large-scale plume currents are non-sinusoidal in the spanwise direction, which means that Fourier spectra do not fully characterize the properties and contributions of different plume current patterns. A complementary analysis method should be introduced to specify how frequently each plume current pattern occurs and how much it contributes to turbulent kinetic energy and scalar transport. Also, as observed from figures 4 and 5, the plume currents in uncontrolled cases are continually changing in both number and spanwise locations, and even in controlled cases with pressure side injection/suction (R5J3B and R5J4B), they are not always fully fixed. Consequently, the commonly employed velocity decomposition method by Bech & Andersson (Reference Bech and Andersson1996), which relies on streamwise and time averaging, is unsuitable for RPPF. This is because the secondary field will depend on the start time of averaging if the time interval for averaging is not large enough, or vanish when it is too large. Furthermore, the conditional averaging technique with spanwise shifting introduced by Dai et al. (Reference Dai, Huang and Xu2016) is also not suitable since it assumes that the number of large-scale structures remains fixed. Instead, using the clustering algorithm introduced in our previous work (Zhang et al. Reference Zhang, Xia and Chen2022) and briefly described in § 2.3, we identified the large-scale patterns in RPPF. Figure 8 displays the cluster centroids of R5J0 and R5J2B, which indicates that these two cases differ in the number and shapes of large-scale patterns. Here R5J0 has five large-scale patterns, whereas R5J2B only has four patterns, with the pattern of five plume currents missing. In the R5J2B $C_1^{{\dagger}}$ pattern, the plume current in the middle is much stronger than the other two. Figure 9(a) illustrates the occurrence possibilities $P_N^{{\dagger}}$ of cluster $C_N^{{\dagger}}$ in different cases. Here $N$ is the number of spanwise periods of the corresponding cluster, and follows the top-down order in figure 8 as well as the supplementary document. As expected, three and four pressure side injection slots promote the pattern of three and four plume currents, respectively. Moreover, two injection slots on the pressure side significantly increase the occurrence possibilities $P_2^{{\dagger}}$ of the $C^{{\dagger}} _2$ pattern to almost double that in the uncontrolled case. In contrast, injection on the suction side has a weaker influence on the number and occurrence possibilities $P_N^{{\dagger}}$ of patterns, consistent with figures 4 and 6. Figure 9(a) also perfectly demonstrates the necessity of introducing cluster analysis as a complement of Fourier spectral analysis. By comparing figures 6(b) and 9(a), it can be found that the control with $N_J=2$ strongly suppressed the pattern of four plume currents (characterized by $C_4^{{\dagger}}$), which seems contradictory to the increase of the $k_z=4$ mode. This is because such increase is achieved by the enhancement of the $C_2^{{\dagger}}$ pattern which is highly non-sinusoidal and contributes not only to the $k_z=2$ mode but also to the $k_z=4$ mode.

Figure 8.

Velocity vectors in the $y$–$z$ plane and contours of the wall-normal component of cluster centroids at $Ro_\tau =5$: (a–e) R5J0, clusters $C_1^{{\dagger}} -C_5^{{\dagger}}$; (f–i) R5J2B, clusters $C_1^{{\dagger}} -C_4^{{\dagger}}$.

Figure 9.

(a) Occurrence possibilities $P_N^{{\dagger}}$ of clusters $C_N^{{\dagger}}$ and (b) the $y$–$z$ plane average of the conditional average of $v'v'$ within clusters $C_N^{{\dagger}}$ at $Ro_\tau =5$.

Figure 9(b) presents the $y$–$z$ plane average of the conditional average of $v'v'$ within clusters $C_N^{{\dagger}}$ in $Ro_\tau =5$ cases. The results confirm that the pressure side control, featuring two, three or four injection slots, can significantly enhance the large-scale pattern with $N=N_J$, as anticipated. In contrast, the control on the stable side has a negligible effect on the strength of large-scale motions. Although the pattern corresponding to $C_5^{{\dagger}}$ appears to be enhanced by two injection slots on the stable side, its occurrence probability is too low to significantly influence the overall $\langle v'v'\rangle$.

Despite the non-sinusoidal characteristics of large-scale motions, analysis in the spectral perspective may still facilitate intuitive understanding of the complex dynamics, especially interscale energy transfer (Cho, Hwang & Choi Reference Cho, Hwang and Choi2018; Kawata & Alfredsson Reference Kawata and Alfredsson2018; Lee & Moser Reference Lee and Moser2019; Gatti et al. Reference Gatti, Chiarini, Cimarelli and Quadrio2020). Figure 10 displays contours of spanwise spectral turbulent kinetic energy, the related turbulent production term and the interscale transport term at $Re_\tau =180$ and $Ro_\tau =5$. Here, $e^+(y;k_z)=E^+(y;k_z)-E^+(y;k_z-2{\rm \pi} /L_z)$ and $p^+(y;k_z)=P^+(y;k_z)-P^+(y;k_z-2{\rm \pi} /L_z)$ are the increments of the corresponding cumulative terms (please note that for the two cases considered in figure 10, $L_z=2{\rm \pi}$). As shown in figure 10(a), large-scale energy mainly concentrates in $2\leq k_z\leq 4$, and the $k_z=4$ mode is relatively closer to the pressure side. Figure 10(c) shows that the kinetic energy production mainly occurs at the scales with $5 \leq k_z\leq 12$ locating at wall distances within $10.9\leq y^+\leq 17.4$. The dominating scales of $p^+$ are inconsistent with the spectral distribution of kinetic energy. Figure 10(e) shows an inverse energy cascade at $k_z<12$ near the pressure side, which is in accordance with the self-sustaining process introduced in our previous work (Zhang et al. Reference Zhang, Xia and Chen2022) and is responsible for the inconsistency of dominating scales between $p^+$ and $e^+$. Therefore, the large-scale turbulent kinetic energy in the uncontrolled case, R5J0, is generated at $5 \leq k_z\leq 12$ in the near-wall region and then transferred to $2\leq k_z\leq 4$ away from the wall through the mechanisms of interscale energy transfer and turbulent spatial transport.

Figure 10.

Contours of (a,b) spanwise spectra of turbulent kinetic energy $e^+$; (c,d) spanwise spectra of production term $p^+$; and (e,f) cumulative spanwise spectra of interscale transport term $T^+$ at $Re_\tau =180$ and $Ro_\tau =5$. (a,c,e) for R5J0 and (b,d,f) for R5J2B.

With two injection slots applied to the pressure side, the $e^+$ of the $k_z=2$ mode increases significantly as shown in figure 10(b), which is consistent with the results in figures 6(b) and 8(b,g). In addition, the large-scale energy spectra of R5J2B become more concentrated in $k_z=2n$ modes with $n\in \mathbb {Z}$. This can be attributed to the non-sinusoidal property of the pattern of two plume currents. Consistently, as depicted in figure 10(d), the kinetic energy production mainly occurs at the scales with $k_z=2n$ ($n\in \mathbb {Z}$) near the pressure side. However, figure 10(f) shows that the inverse cascade is weakened in R5J2B, although the $e^+$ of the $k_z=2$ mode is significantly enhanced as shown in figures 10(b) and 6(b). This is not surprising because $T^+$ only represents the net energy transfer. When the large-scale energy increases, the interaction between large-scale modes will be stronger and consequently enhances the forward cascade. This neutralizes part of the inverse cascade and results in the decrease of positive $T^+$. Therefore, the enhancement of the $k_z=2$ mode in R5J2B should be directly attributed to the increase of the production term, which is due to the implementation of the two injection slots.

3.2. Turbulent statistics

Figure 11 depicts the mean velocity profiles $\langle u\rangle$ for all cases. It is observed from figure 11(a) that the mean velocity profiles at $Ro_\tau =0$ become asymmetric in controlled cases, while in the uncontrolled case, it remains symmetric. This is because the injection/suction amplifies the near-wall motions and increases the friction. Figure 11(a–c) illustrate that the pressure side injection/suction can decrease the mean flux, and the decrease is insensitive to the distribution of injection/suction slots. The maximum relative decrease of $U_b$ is $5.93\,\%$, achieved in R5J2B, as listed in table 1. In $Ro_\tau =30$ cases, the influence of the present weak injection/suction is indiscernible, where the relative decrease of $U_b$ is less than $1\,\%$, as shown in figure 11(d) and listed in table 1. For $Ro_\tau >0$ cases, the impact of the suction side injection/suction is small. It is also worth noting that the controlled cases maintain the local linear law of mean velocity profiles, whose slope is approximately $Ro_\tau$. Figure 12 displays the Reynolds shear stress $\langle u'v'\rangle$ for all cases. For the non-rotating cases, as displayed in figure 12(a), the $\langle u'v'\rangle$ profiles of the controlled cases with two, three and four injection slots almost coincide with each other, and they differ from that of the uncontrolled case. However, for the rotating cases, although the mean velocity profiles of the uncontrolled cases and the suction-side controlled cases deviate from those of the pressure-side controlled cases at $Ro_\tau =5$ and $10$, the $\langle u'v'\rangle$ profiles do not show obvious deviation, as displayed in figure 12(b–d). From the mean momentum equation (Xia et al. Reference Xia, Shi and Chen2016)

(3.1)\begin{equation} \langle u'v'\rangle=y+(1-u_{\tau p}^2)+\frac{1}{Re_\tau}\frac{{\rm d}\langle u\rangle}{{\rm d}\kern0.05em y}, \end{equation}

it can be inferred that $u_{\tau p}$ varies very slightly in the controlled and uncontrolled cases at the same $Ro_\tau$, as listed in table 1.

Figure 11.

Mean velocity profiles $\langle u\rangle$ in uncontrolled and controlled cases: (a) $Ro_\tau =0$; (b) $Ro_\tau =5$; (c) $Ro_\tau =10$; (d) $Ro_\tau =30$.

Figure 12.

Reynolds shear stress $\langle u'v'\rangle$ in uncontrolled and controlled cases: (a) $Ro_\tau =0$; (b) $Ro_\tau =5$; (c) $Ro_\tau =10$; (d) $Ro_\tau =30$.

According to the definition in Brethouwer (Reference Brethouwer2018) and equation (2.1), the Nusselt number of $\phi$ can be defined equivalently (assuming an infinite averaging time) as

(3.2)\begin{equation} Nu=\left.-2\frac{\partial\langle\phi\rangle}{\partial y}\right|_{y=\pm1} =1+PrRe_\tau\int_{{-}1}^1{\langle v'\phi'\rangle \,{{\rm d}\kern0.05em y}}. \end{equation}

It is evident that to increase scalar transport, one must increase the convective scalar flux $\langle v'\phi '\rangle$, as the contribution of molecular diffusion remains constant. Therefore, enhancing the strength of wall-normal motions can be a promising approach.

Figure 13 presents the $\langle v'v'\rangle$ profiles for all cases. From the values of $\langle v'v'\rangle$ at $y=\pm 1$, it is evident that the present injection/suction is too weak to be noticed at the walls. However, it has a significant impact on the $\langle v'v'\rangle$ profiles for $Ro_\tau \leq 10$ cases, especially in the locations and heights of the peaks. Such influences can be attributed to the difference in strength and height of plume current between clusters, and the variance of the clusters’ occurrence possibilities. For example, $\langle v'v'\rangle$ of case R5J2B has a peak closest to the suction side, and it is the largest in the upper-half of the channel except for an extremely small region at the suction side (where $\langle v'v'\rangle =0.01$ for suction side control cases). This is because the $C_2^{{\dagger}}$ pattern of plume currents has the largest averaged strength and height of plume currents as shown in figure 8(a–e), and such a pattern is dominating (figure 9a) and further strengthened (figure 8g) in R5J2B. At $Ro_\tau =30$, the control has minor influence on $\langle v'v'\rangle$ profiles. It is observed that the effect of the control in non-rotating cases is weakly dependent on the slot distributions. In contrast, the effect of the control is strongly dependent on the slot distributions on the pressure side for $Ro_\tau =5$ and $Ro_\tau =10$ cases.

Figure 13.

Here $\langle v'v'\rangle$ in uncontrolled and controlled cases: (a) $Ro_\tau =0$; (b) $Ro_\tau =5$; (c) $Ro_\tau =10$; (d) $Ro_\tau =30$.

The profiles of convective scalar flux denoted by $\langle v'\phi '\rangle$ are presented in figure 14. Figure 14(a) demonstrates that the increase of $\langle v'\phi '\rangle$ is not strongly dependent on the distribution of injection/suction slots in non-rotating cases. This behaviour is similar to $\langle u\rangle$, $\langle u'v'\rangle$ and $\langle v'v'\rangle$ in non-rotating cases. However, when $Ro_\tau$ is non-zero and not very large, the distribution of injection/suction slots on the unstable side becomes a crucial factor in turbulent scalar transport. Figure 14(b) reveals that the largest relative enhancement of $\langle \langle v'\phi '\rangle \rangle _y$ is $17.2\,\%$ in R5J2B, which is almost three times the relative decrease of $U_b$ in this case. There are two main reasons for this observation. Firstly, case R5J2B exhibits a significant increase of large-scale wall-normal kinetic energy, similar to case R5J3B, as shown in figure 6(b). Secondly, two plume currents have a larger spanwise distance than three plume currents, and a greater distance between plume currents may lead to higher efficiency of scalar transport, as discussed in our previous work (Zhang et al. Reference Zhang, Xia and Chen2022). This is supported by figure 15(a), where the $y$–$z$ plane average of the conditional average of $v'\phi '$ within clusters $C_N^{{\dagger}}$ at $Ro_\tau =5$ is shown. This can be used to illustrate the scalar transport capability of different large-scale patterns. Notably, two injection slots on the pressure side can significantly increase the scalar transport capability of the pattern with two plume currents ($C_2^{{\dagger}}$ shown in figure 8g). Under the control of three injection slots on the pressure side, the pattern with three plumes becomes the most efficient in scalar transport. However, the impact of achieving this pattern through the use of three injection slots on the pressure side is relatively minor, combined with the occurrence possibility $P_3^{{\dagger}}$ of cluster $C_3^{{\dagger}}$ shown in figure 9(a). More quantitatively, figure 15(b) shows the weighted contribution of $\langle v'\phi '\rangle$ from each cluster $C_N^{{\dagger}}$. It shows that the main contribution to $\langle v'\phi '\rangle$ comes from cluster $C_2^{{\dagger}}$ in R5J2B, while it is from clusters $C_1^{{\dagger}}$ for other cases. At $Ro_\tau =10$, three injection slots on the unstable side slightly outperform two injection slots, as shown in figure 14(c). This is because $\langle v'v'\rangle$ in case R10J3B is generally larger than that in case R10J2B, as demonstrated in figure 13(c), and the higher efficiency of two plume currents in scalar transport cannot compensate for their lower strength. When $Ro_\tau =30$, the increase of $\langle v'v'\rangle$ and $\langle v'\phi '\rangle$ induced by injection/suction on the unstable side is relatively small compared with the corresponding cases at $Ro_\tau =5$ or $Ro_\tau =10$, as shown in figure 14(d). This suggests that turbulent kinetic energy and scalar transport are only sensitive to the control when the scale of the control pattern is close to the intrinsic scale of large-scale motions. Similar to $\langle u\rangle$, $\langle u'v'\rangle$ and $\langle v'v'\rangle$, convective scalar transport is weakly influenced by injection/suction on the stable side. This also indicates that the present control strategy increases the scalar transport not simply by blowing $\phi$ at the boundary into the fluid, but by varying the distribution and strength of plume currents.

Figure 14.

Convective scalar flux $\langle v'\phi '\rangle$ in uncontrolled and controlled cases: (a) $Ro_\tau =0$; (b) $Ro_\tau =5$; (c) $Ro_\tau =10$; (d) $Ro_\tau =30$.

Figure 15.

(a) The $y$–$z$ plane average of the conditional average of $v'\phi '$ within clusters $C_N^{{\dagger}}$ and (b) the weighted contribution of $\langle v'\phi '\rangle$ from each cluster$C_N^{{\dagger}}$ at $Ro_\tau =5$.

For a better demonstration of the local impact of injection/suction, the relative enhancement of convective scalar flux $\langle v'\phi '\rangle$ is presented in figure 16. It is evident from figure 16(b–d) that pressure side injection/suction usually induces a significant local enhancement of convective scalar transport. The corresponding peaks locate at approximately $y=0.7$ for $Ro_\tau =5$ and $10$, and at around $y=0.4$ for $Ro_\tau =30$. This is due to the rise in both the mean strength and height of plume currents through effective injections, evident from the comparison between figures 8(b) and 8(g). With stable plume currents, plumes are more likely to penetrate deeper into the stable region near the suction side. Since the mean scalar gradient is relatively large in the stable region due to the weaker convective flux, plumes that approach the suction side will induce significant local enhancement of scalar transport. From another perspective, the uncontrolled cases exhibit a rapid drop in $\langle v'\phi '\rangle$ near the suction side, and the injections can strengthen the plume currents and consequently delay the decline.

Figure 16.

Relative deviation of $\langle v'\phi '\rangle$ from that of the corresponding uncontrolled cases: (a) $Ro_\tau =0$; (b) $Ro_\tau =5$; (c) $Ro_\tau =10$; (d) $Ro_\tau =30$.

Based on the discussions presented above, we can claim that the present injection/suction control on the unstable wall can achieve an $O(10\,\%)$ enhancement of turbulent scalar transport with control velocity $O(1\,\%) U_b$. When $v'_{rms}(-1)$ is fixed at $0.1u_\tau$, only in rotating cases is scalar transport sensitive to injection/suction velocity distribution. It is recommended that the distance between injection slots be equal to or slightly larger than the dominant spanwise wavelength $\lambda _z=2{\rm \pi} /k_z$ observed in the uncontrolled case, in order to ensure a satisfactory enhancement of plume currents and scalar transport efficiency.

The statement on the injection slot spacing for optimal scalar transport enhancement is still valid at different $Ro_\tau$, $Re_\tau$ or in a larger spanwise domain. This is demonstrated in figure 17, which exhibits the profiles of the convective scalar flux $\langle v'\phi '\rangle$ at three new sets of parameters. As shown in figure 7, the wavenumbers $k_z$ of the dominating spanwise Fourier modes of the three uncontrolled cases, R2J0, R5J0_Re+ and R5J0_Lz+, are 2, 3 and 2.5, respectively. Correspondingly, at $Re_\tau =180$ and $Ro_\tau =2$, two injection slots achieve maximum scalar transport in a spanwise domain of $L_z=2{\rm \pi}$, as shown in figure 17(a). Likewise, two injection slots in $L_z=2{\rm \pi}$ achieve maximum scalar transport at $Re_\tau =300$ and $Ro_\tau =5$, as shown in figure 17(b). Figure 17(c) demonstrates that the largest enhancement of $\langle v'\phi '\rangle$ and $Nu$ at $Re_\tau =180, Ro_\tau =5$ and $L_z=4{\rm \pi}$ is achieved by using four injection slots ($N_J=2$). Figure 17(c) demonstrates that four injection slots ($N_J=2$) provides the largest enhancement of $\langle v'\phi '\rangle$ and $Nu$ at $Re_\tau =180$, $Ro_\tau =5$ and $L_z=4{\rm \pi}$, where the relative increase of turbulent transport is 16.1 %. For the control with five injection slots ($N_J=2.5$), the enhancement of the turbulent scalar transport is also significant, which is 14.5 %. It is worth noting that, although the controls with $N_J=3$ and 3.5 can greatly amplify the wall-normal kinetic energy at $k_z=3$ and 3.5, respectively, as shown in figure 7(d), they can only enhance the turbulent scalar transport by 5.1 % and 2.7 %. These results not only demonstrate the robustness of the control strategy, but also strongly support our criterion on the optimal choice of injection/suction distribution for scalar transport enhancement, i.e. the distance between injection slots should be equal to or slightly larger than the dominant spanwise wavelength in the uncontrolled case.

Figure 17.

Convective scalar flux $\langle v'\phi '\rangle$ in uncontrolled and controlled cases: (a) $Re_\tau =180$, $Ro_\tau =2$, $L_z=2{\rm \pi}$; (b) $Re_\tau =300$, $Ro_\tau =5$, $L_z=2{\rm \pi}$; (c) $Re_\tau =180$, $Ro_\tau =5$, $L_z=4{\rm \pi}$.