3.1. The onset of the centrifugal effect

The most prominent effect of centrifugal force is to accumulate hot fluid at the cell centre and spread cold fluid around the sidewall, resulting in an increased bulk temperature $T_{bulk}$ and a reduced sidewall temperature $T_{w}$. In this section, we will use $T_{bulk}$ as a proxy to study the onset Froude number $Fr_c$. Figure 3(a) plots the deviation of the time-averaged bulk temperature $\langle T_{bulk}\rangle _t$ from the mean temperature $T_m=(T_h+T_c)/2$ as a function of $1/Ro$, which shows similar behaviours for all $Ra$. It is seen that $\langle T_{bulk}\rangle _t-T_m$ has a plateau at small $1/Ro$, and then it starts to increase at large $1/Ro$. The value of the plateau is almost the same as the non-rotation one at $1/Ro=0$, which, due to the non-Oberbeck–Boussinesq (NOB) effect (Ahlers et al. Reference Ahlers, Brown, Araujo, Funfschilling, Grossmann and Lohse2006), is larger for larger $Ra$. A numerical study on the NOB RRBC showed that, at large $1/Ro$, the value of $\langle T_{bulk}\rangle _t-T_m$ starts to decrease from the plateau as $(1/Ro)^{-0.66}$ (Horn & Shishkina Reference Horn and Shishkina2014), which is in contrast to the increased behaviour observed here. This led us to attribute this increased behaviour to the centrifugal force.

Figure 3. (a) Deviations of the time-averaged bulk temperature $\langle T_{bulk}\rangle _t$ from the mean temperature $T_m$ as a function of $1/Ro$ for different $Ra$. (b) Plot of $\langle T_{bulk}\rangle _t/{\rm \Delta} T$ relative to the corresponding non-rotation values as a function of $Fr$. Inset: an example showing that the onset Froude number $Fr_c$ is defined as the position where a linear fitting meets a base line (grey solid lines). (c) Plot of the so-obtained $Fr_c$ values, which shows a power-law behaviour: $Fr_c=3.79\times 10^{-7}\,Ra^{0.53\pm 0.04}$ (blue solid line). The left-pointing and right-pointing triangles are $Fr_c$ values from Hu et al. (Reference Hu, Huang, Xie and Xia2021) for $Ra=4.4\times 10^9$, which are obtained from global heat transport and local temperature measurements, respectively. (d) The data in panel (b) plotted as a function of $Fr/Fr_c$. The symbols in panels (a), (b) and (d) have the same meaning. (e) The bulk temperature as a function of $Fr$ for four constant $1/Ro$ cases.

To see this point more clearly, the normalized bulk temperature as a function of $Fr$ is displayed in figure 3(b), which is plotted relative to the non-rotation value $\langle T_{bulk}(0)\rangle _t/{\rm \Delta} T$ to exclude the NOB effect. It is seen that, with increasing $Fr$, this temperature anomaly increases gradually. Moreover, the onset Froude number $Fr_c$, at which the enhancement begins, increases with increasing $Ra$. To extract $Fr_c$ for different $Ra$, a linear function is used to describe the enhancement, and $Fr_c$ is defined as the position where this linear function meets a baseline, which is the mean value of the temperature deviations at small $Fr$ (see the inset of figure 3b). The so-obtained $Fr_c$ as a function of $Ra$ is shown in figure 3(c), which has a power-law behaviour, $Fr_c \sim Ra^{0.53\pm 0.04}$. For comparison, the onset values from Hu et al. (Reference Hu, Huang, Xie and Xia2021) for $Ra=4.4\times 10^9$ are also plotted, where $Fr_c=0.04$ (left-pointing triangle) and $0.05$ (right-pointing triangle) were obtained from the global heat transport and local temperature measurements, respectively. It is seen that the results obtained using the different methods and in different experimental configurations are consistent with each other within experimental resolution. If the temperature deviations are replotted against $Fr/Fr_c$, the data are collapsed reasonably well (figure 3d). This self-similarity suggests that the enhancement behaviour is indeed caused by the centrifugal force.

Figure 3(b) shows that, for smaller $Ra$, the temperature enhancement is larger at the same $Fr$ and it also sets in earlier, suggesting that the centrifugal effects are more pronounced for smaller $Ra$. From the $1/Ro$–$Fr$ space in figure 2(b), we notice that the data for the smallest (largest) $Ra$ are almost entirely in the QG (3-D) regime. This implies that flows in the QG regime would be affected more easily by the centrifugal force, i.e. the response of RRBC to the centrifugal force depends on the flow states. To see this more clearly, another set of experiments with fixed $1/Ro$ are performed. The measured bulk temperature is displayed in figure 3(e). It is seen that, for the cases with $1/Ro=6.38$ and $9.02$ (in the 3-D regime), the centrifugal effect is hardly present; while for the $1/Ro=14.95$ case (close to the QG regime), it can be observed easily. The centrifugal effects become more significant for the largest $1/Ro$, which lies completely in the QG regime.

3.2. The effects of centrifugal force on the temperature statistics

Figure 4(a) displays the normalized root-mean-square (r.m.s.) values of bulk temperature $T_{rms}/{\rm \Delta} T$ for different $Ra$. The solid lines are power-law fits, $T_{rms}/{\rm \Delta} T=a(1/Ro)^b$. The fitted values of $a$ and $b$ are listed in table 1. There is no power law for the two smallest $Ra$. It is seen that, for $1.1\times 10^9 \leq Ra \leq 4.5\times 10^9$, the data points at large $1/Ro$ deviate from the power-law behaviour and start to drop. This behaviour was also discussed by Kunnen, Geurts & Clercx (Reference Kunnen, Geurts and Clercx2009), who attributed the decrease of $T_{rms}$ at sufficiently strong rotation to the laminarization of the flow, the effect of which was less profound at larger $Ra$. It seems that the behaviour of the temperature fluctuation is dominated by the competition between the buoyancy and Coriolis forces, rather than by the centrifugal force. For example, for the largest $Ra$, although the background mean flow becomes gradually hotter (figure 3a), $T_{rms}$ still follows the power-law behaviour, showing no indication of the centrifugal effect.

Figure 4. (a) The normalized bulk temperature fluctuations $T_{rms}/{\rm \Delta} T$ for different $Ra$. Solid lines are the corresponding power-law fits. See table 1 for the fitting results. (b) The azimuthally and temporally averaged sidewall temperatures $\langle T^m_{w} \rangle _{\phi,t}$ (normalized by ${\rm \Delta} T$) plotted relative to the corresponding non-rotation values. Here, $T^m_{w}$ is the temperature measured at mid-height. (c) The sidewall temperature gradient $\delta T_{w}^{bt}$ as a function of $1/Ro$. It is normalized by the non-rotation value. (d) The same as panel (c) but plotted as a function of $Fr/Fr_c$. The symbols in all panels share the same meaning.

We next study the centrifugal effects on the sidewall temperatures $T_{w}$. For the two lowest $Ra$ (${\rm \Delta} T\leq 2$ K), the ‘heat content’ or the temperature contrast $T-T_m$ of the flow near the sidewall is too small to produce reliable temperature signals, since they can be easily affected by environmental temperature fluctuations. Therefore, their results of $T_{w}$ are not shown. Figure 4(b) plots the sidewall temperature $\langle T^m_{w} \rangle _{\phi,t}$ relative to the non-rotation value $\langle T^m_{w} \rangle _{\phi,t}(0)$, which is measured at mid-height and normalized by ${\rm \Delta} T$. Here, $\langle \cdot \rangle _{\phi,t}$ means azimuthal and temporal averaging. The centrifugal force will spread the cold flow around the sidewall, resulting in a decreased $T_{w}$ at large $Fr$, as shown in figure 4(b). It can be seen that the decrease begins at a Froude number that is slightly different from $Fr_c$. This difference may be attributed to the fact that the sidewall temperatures are azimuthally averaged and low-pass-filtered by the Plexiglas sidewall. At small $Fr$, hot fluid can also appear near the sidewall region (e.g. Curbelo et al. Reference Curbelo, Lopez, Mancho and Marques2014); thus azimuthal averaging would delay the decreasing of the averaged sidewall temperature. Also, at small $Fr$, the cold flow is weak. The thermistors inside the sidewall actually sense the integrated signal over a finite area of the sidewall. Thus, the in-wall thermistors are not sensitive to the weak cold flow. Only if the flow is cold enough at larger $Fr$ can the thermistors in the sidewall detect the cold flow. Both these effects would give rise to a larger onset Froude number detected by the sidewall temperature probes. But at the leading-order level, it can provide a measure of $Fr_c$ consistent with that from the bulk temperature.

Table 1. The fitting results of $T_{rms}/{\rm \Delta} T=a(1/Ro)^b$ to the data in figure 4(a).

The quantity $\delta T_{w}^{bt}=2\langle T_{w}^b-T_{w}^t\rangle _{\phi,t}$ in figure 4(c) characterizes the vertical temperature gradient near the sidewall, where $T_{w}^b$ and $T_{w}^t$ are the sidewall temperatures measured, respectively, at $H/4$ (bottom) and $3H/4$ (top). It is found that, for $1/Ro \lesssim 10$, $\delta T_{w}^{bt}/\delta T_{w}^{bt}(0)$ increases with increasing $1/Ro$, and this increment is more significant for larger $Ra$. These behaviours are consistent with those found in previous studies (Zhong & Ahlers Reference Zhong and Ahlers2010; Weiss & Ahlers Reference Weiss and Ahlers2011), which can be explained by the effects of the secondary flows inside the Stewartson boundary layers near the sidewall (Kunnen et al. Reference Kunnen, Stevens, Overkamp, Sun, van Heijst and Clercx2011; Favier & Knobloch Reference Favier and Knobloch2020; de Wit et al. Reference de Wit, Guzmán, Madonia, Cheng, Clercx and Kunnen2020; Zhang et al. Reference Zhang, van Gils, Horn, Wedi, Zwirner, Ahlers, Ecke, Weiss, Bodenschatz and Shishkina2020). However, with further increase of $1/Ro$, $\delta T_{w}^{bt}/\delta T_{w}^{bt}(0)$ starts to drop. We attribute this to the centrifugal effects at large rotation speed, which can be seen more clearly when $\delta T_{w}^{bt}/\delta T_{w}^{bt}(0)$ is plotted as a function of $Fr/Fr_c$ in figure 4(d). It is seen that the drop of $\delta T_{w}^{bt}/\delta T_{w}^{bt}(0)$ begins at the onset Froude number $Fr_c$. These results thus imply that, when the centrifugal force sets in, the Stewartson layers are destroyed, and the secondary flows are replaced by cold downwelling flows. The temperature of these cold flows becomes more vertically uniform as $Fr$ increases, resulting in the decreased temperature gradients near the sidewall.

3.3. The $Fr_c$ scaling

We now arrive at a position to explain why the response of the system to the centrifugal force is different in different flow regimes. In RRBC, the vertical motion of a fluid parcel can be regarded as free fall since the buoyancy time scale $\tau _{ff}$ is much less than the thermal/viscous diffusion time scales (Julien et al. Reference Julien, Legg, McWilliams and Werne1996). Similarly, if the centrifugal buoyancy time scale $\tau _{cb}^0=\sqrt {l_0/(\alpha \varOmega ^2 R{\rm \Delta} T)}$ dominates within a horizontal length scale $l_0$, the horizontal motion within $l_0$ can also be regarded as free fall. Then, the onset Froude number $Fr_c \sim l_0/H$ can be obtained immediately from the balance $\tau _{ff}\sim \tau _{cb}^0$ (Hu et al. Reference Hu, Huang, Xie and Xia2021). This local balance means that just before the vertically moving hot (cold) fluid parcel arrives at, and is annihilated by, the cold (hot) plate (i.e. within the lifetime $\tau _{ff}$), the centrifugal force starts to influence its horizontal dynamics.

In RRBC, the competition between Coriolis and buoyancy forces leads to different flow states (e.g. Sprague et al. Reference Sprague, Julien, Knobloch and Werne2006; Nieves et al. Reference Nieves, Rubio and Julien2014; Cheng et al. Reference Cheng, Madonia, Aguirre Guzmán and Kunnen2020; Kunnen Reference Kunnen2021). At small $Ra/Ra_c$ (i.e. QG regime), the motions of fluid parcels are confined in small vertical columns. Chong et al. (Reference Chong, Shi, Ding, Ding, Lu, Zhong and Xia2020) found that the column's horizontal motion is diffusive, and the ‘mean-free distance’ $l_0$ between two adjacent columns roughly equals the column's diameter. Only within this ‘mean-free distance’ does the horizontal motion have free-fall features. As $Ra/Ra_c$ increases, the distance $l_0$ between two columns increases; and when $Ra/Ra_c>6.3$ (Nieves et al. Reference Nieves, Rubio and Julien2014), the columns are replaced by fragmented thermal structures, resulting in larger $l_0$ (refer to the visualization given in Nieves et al. (Reference Nieves, Rubio and Julien2014)). This is the reason why the onset Froude number $Fr_c\sim l_0/H$ increases with increasing $Ra$. When $Ra/Ra_c$ is large enough to reach the rotation-unaffected regime (i.e. 3-D regime), the horizontal dynamics of a thermal plume under the ‘horizontal gravity’ $\varOmega ^2R$ is similar to the vertical dynamics under gravity $g$. In this case, $l_0$ can be as large as the cell's radius $R$ (the upper bound). Thus, as shown in figure 2(c), the scaling $Fr_c \sim Ra^{0.53}$ can be extrapolated to larger $Fr$ until it reaches the value of $R/H$ (the black dashed lines). Previous studies (e.g. Sprague et al. Reference Sprague, Julien, Knobloch and Werne2006) have shown that the change of flow structures with increasing $Ra/Ra_c$ also depends on other control parameters, which suggests that $Fr_c\sim Fr_c(Ra, Ro, Pr, \varGamma )$. To determine the dependence on these parameters, more systematic studies are needed in the future.

Based on the above physical picture, we propose a new division of the $1/Ro$–$Fr$ space in figure 2(c). The whole space can be divided into centrifugally affected (shaded) and unaffected regimes by the black lines, which consist of four parts. The solid line is the experimental result, $Fr_c \sim Ra^{0.53}$. And the dashed lines, as explained above, are the extrapolation of this scaling to larger $Fr$ until it reaches its upper bound $R/H$. As for the dotted line with a question mark at very small $Fr$, the same scaling is assumed to apply there without strong justification. The unaffected regime can be still divided into 3-D and QG subregimes, but more attention is needed for the affected regime. The violet dash-dotted line is derived from $Ra/Ra_c=6.3$ (Nieves et al. Reference Nieves, Rubio and Julien2014). Above this line is the strong-Coriolis–weak-centrifugal (SCWC) subregime, where the flow structure is columns, and their diffusive motion is affected by the centrifugal force (Noto et al. Reference Noto, Tasaka, Yanagisawa and Murai2019; Ding et al. Reference Ding, Chong, Shi, Ding, Lu, Xia and Zhong2021). If the centrifugal force becomes stronger, a transition to the strong-Coriolis–strong-centrifugal (SCSC) subregime occurs. In this subregime, no single column can be identified. They are squeezed by the strong centrifugal force, and then merge together into a large coherent structure, such as the hot one at large $1/Ro$ shown in figure 5 in Horn & Aurnou (Reference Horn and Aurnou2019). Below the dash-dotted line is the weak-Coriolis–weak-centrifugal (WCWC) subregime, in which the columns break down into fragmented thermal structures (i.e. the structures in the plume regime and geostrophic turbulence regime identified in Nieves et al. Reference Nieves, Rubio and Julien2014, for example) and their radial motion is enhanced by the centrifugal force. In the QC subregime, the flow structures are like the thermal plumes in Rayleigh–Bénard convection. In this subregime, except for the vertical free-fall motion under $g$, their horizontal motion is also ‘free fall’ in the entire horizontal plane under the ‘gravity’ $\varOmega ^2R$. The transition line from the QC to the SCSC subregime is the same as that in figure 2(b). We note that there is no convincing argument or solid experimental evidence for the lines with a question mark in figure 2(c). The transitions among these subregimes need further experimental and/or numerical verification.

3.4. The heat transport

In RRBC, there are two popular strategies (Kunnen Reference Kunnen2021) when studying the heat transport. Considering the competition between the Coriolis and buoyancy forces, one can either increase $1/Ro$ at fixed $Ra$ or increase $Ra$ at fixed $Ek$. For the fixed $Ra$ case, one usually studies the reduced Nusselt number $Nu_r=Nu(1/Ro)/Nu(0)$ as a function of $1/Ro$ (e.g. Kunnen et al. Reference Kunnen, Stevens, Overkamp, Sun, van Heijst and Clercx2011; Stevens et al. Reference Stevens, Clercx and Lohse2013; Horn & Shishkina Reference Horn and Shishkina2014; Weiss, Wei & Ahlers Reference Weiss, Wei and Ahlers2016). There are three regimes, which are separated by an onset and an optimal critical $1/Ro$. For the $\varGamma =1$ case, Weiss et al. (Reference Weiss, Wei and Ahlers2016) summarized the $Ra$ and $Pr$ dependences of the optimal state by the following scalings: $1/Ro_{max}=21.4Pr^{1.37}Ra^{-0.18}$ and $Nu_{r,max}-1=44Pr^{0.80}Ra^{-0.35}$. This implies that the influence of different $Ra$ can be eliminated by rescaling the data according to

(3.1a,b)\begin{align} \frac{1}{Ro} \rightarrow \frac{1}{Ro^*} =\frac{1}{Ro}\left[\frac{Ra}{Ra_{ref}}\right]^{0.18} \quad\text{and}\quad Nu_r \rightarrow Nu_r^*=(Nu_r-1)\left[\frac{Ra}{Ra_{ref}}\right]^{0.35}+1. \end{align}

Here, $Ra_{ref}$ is a reference Rayleigh number and we choose it to be $Ra_{ref}=9.5\times 10^9$. Indeed, as shown in figure 5(a), the rescaled data $Nu_r^*$ for different $Ra$ collapse reasonably well. However, the data at large $1/Ro$ should be influenced by the centrifugal force.

Figure 5. (a) The rescaled reduced Nusselt number $Nu_r^*$ as a function of the rescaled inverse Rossby number $1/Ro^*$ for different $Ra$. See the text for detailed information. (b) Plot of $Nu_r$ as a function of $Fr$. The symbols are the same as in panel (a). (c) The Nusselt number $Nu$ as a function of $Ra$ for different $1/Ro$. The grey triangles and red stars are from Liu & Ecke (Reference Liu and Ecke1997) with $3< Pr <7$ in a rectangular convection cell. The orange triangles are from the numerical work in Julien et al. (Reference Julien, Legg, McWilliams and Werne1996) at $Pr=1$, with a horizontally periodic flow domain. The remaining data sets are from the present work, which have power-law behaviours, $Nu=ARa^{\alpha }$. The fitted results are listed in table 2, and are compared with those from Liu & Ecke (Reference Liu and Ecke1997) in panels (d) and (e).

Figure 5(b) plots the reduced Nusselt number $Nu_r$ as a function of $Fr$. The centrifugal effects on heat transport were discussed by Horn & Aurnou (Reference Horn and Aurnou2018) based on the exact relation $Nu=(2Fr\sqrt {PrRa}/\varGamma )\langle T u_r r \rangle _{V,t}+Pr\langle \|\boldsymbol {\nabla }\boldsymbol {u}\|^2 \rangle _{V,t}+1$. The enhanced horizontal motion due to centrifugal force has two effects on $Nu$. The direct effect $Nu_{Fr}=(2Fr\sqrt {PrRa}/\varGamma )\langle T u_r r \rangle _{V,t}$ is to reduce the heat transport efficiency, since the hot (cold) fluid with positive (negative) temperature T always has a negative (positive) radial velocity $u_r$. The indirect effect, i.e. the dissipation term $Nu_{\varepsilon }=Pr\langle \|\boldsymbol {\nabla }\boldsymbol {u}\|^2 \rangle _{V,t}$, depends on flow states. It was found that $Nu_{\varepsilon }$ decreases more if the system is more deeply into the QG regime (Horn & Aurnou Reference Horn and Aurnou2018). Thus, at the same $Fr$, $Nu_r$ decreases more rapidly for smaller $Ra$ (i.e. more deeply into the QG regime), as shown in figure 5(b).

Neither of the above-mentioned strategies reveals a well-defined scaling on heat transport, since the competition between Coriolis and buoyancy forces drives the system into different flow regimes. However, there exists a third strategy, i.e. fixing $Ro$ (Liu & Ecke Reference Liu and Ecke2009). In this strategy, the relative importance of these two forces is constant, which indicates that the flow states are similar when varying $Ra$ (Kunnen Reference Kunnen2021). As shown in figure 5(c), the measured heat transport, together with the data from previous studies (Julien et al. Reference Julien, Legg, McWilliams and Werne1996; Liu & Ecke Reference Liu and Ecke1997, Reference Liu and Ecke2009), all exhibit well-defined scalings, i.e. $Nu=ARa^{\alpha }$. The fitted exponent $\alpha$ and amplitude $A$ are summarized in table 2, and are also plotted in figures 5(d) and 5(e), respectively, together with the results from Liu & Ecke (Reference Liu and Ecke1997). In Liu & Ecke (Reference Liu and Ecke1997), the experiment was conducted with relatively low values of $Ra$ (from $4\times 10^6$ to $6\times 10^8$), and all their data showed a $2/7$ scaling. In the present study, the upper end of the $Ra$ range was extended to $\sim 1\times 10^{10}$. As can be seen from the $1/Ro=9.02$ case in figure 5(d), the data start to deviate from the $2/7$ scaling. We note that the centrifugal force has little effect for the $1/Ro=9.02$ case. When pushing $1/Ro$ to much larger values ($1/Ro=14.95$ and $21.80$), the deviation from the $2/7$ scaling becomes even larger (figure 5d). This may be due to the stronger centrifugal effects for these two cases. Although it is still rather complex for the fixed $Ro$ case, the well-defined power-law scaling can provide a better prediction for the unexplored regions of the parameter space. Thus, fixing $Ro$ may be a better strategy when studying the heat transport property of the RRBC system.