3.1 Flow dynamics

It is one’s intuition that the flow dynamics should strongly depend on geometry. To find out how the flow dynamics responds to geometric confinement, we show in figure 1(a–c) the PIV-measured mean velocity fields for ${\it\Gamma}=0.3$ (the unconfined case) and 0.15. The first thing we can see is that there exists a well-defined LSC with two counter-rotating corner rolls in the ${\it\Gamma}=0.3$ cell, as found in previous studies with similar aspect ratios (Xia et al. Reference Xia, Sun and Zhou2003; Sugiyama et al. Reference Sugiyama, Ni, Stevens, Chan, Zhou, Xi, Sun, Grossmann, Xia and Lohse2010). For the ${\it\Gamma}=0.15$ cell, however, one finds a quite different flow pattern in the long time-averaged field: The single-roll structure has disappeared, instead, we have a four-roll structure. One way such a four-roll pattern in the mean field can arise is for the LSC to frequently reverse its flow direction, which had been experimentally demonstrated in an earlier study (Sun, Xia & Tong Reference Sun, Xia and Tong2005b ). Indeed, by taking short time averages of the instantaneous velocity fields, we can obtain LSCs with both clockwise and counter-clockwise flow directions in the ${\it\Gamma}=0.15$ cell (see figure 1 c for the counter-clockwise case), which was also illustrated in a recent numerical study with a similar ${\it\Gamma}$ (Kaczorowski, Chong & Xia Reference Kaczorowski, Chong and Xia2014). The second difference between these two ${\it\Gamma}$ cells is that the magnitude of the time-averaged LSC is smaller for ${\it\Gamma}=0.15$ , as indicated clearly by the scale bar. While the increased drag force from the walls can certainly slow down the flow, a more fluctuating flow can also result in a smaller mean velocity. Larger fluctuations in the flow field for small ${\it\Gamma}$ are evident in the root-mean-square (r.m.s.) velocity field (see figure 1 d–f), which shows that the fluctuations in the bulk of the ${\it\Gamma}=0.15$ cell become stronger and the region of strong fluctuations also becomes larger. It has been found in Xia et al. (Reference Xia, Sun and Zhou2003) that larger velocity fluctuations are associated with plume clusters, therefore the r.m.s. velocity fields measured here indicate that there are more thermal plumes going through the bulk region in the highly confined geometry, rather than rising and falling along the side walls as is the case for larger ${\it\Gamma}$ . Note that the reversal of the LSC could also contribute to a larger velocity fluctuation in the bulk region. However, we found that the r.m.s. velocity field in the ${\it\Gamma}=0.15$ cell for the situation when a LSC can still be identified by short time averaging is quite similar to the one with a four-roll pattern (see figure 1 e,f), suggesting that the major contributors for larger fluctuations in the bulk flow are the thermal plumes.

Figure 1. Coarse-grained vector maps of mean velocity field (a–c) and the corresponding contour maps of the r.m.s. velocity field (d–f) at $Ra=1.1\times 10^{9}$ . (a,d)  ${\it\Gamma}=0.3$ cell by long time averaging; (b,e) ${\it\Gamma}=0.15$ cell with a four-roll pattern by long time averaging; (c,f) ${\it\Gamma}=0.15$ cell with a well-identified LSC by short time averaging. The velocity is coded in both colour scale and vector length in units of $\text{cm}~\text{s}^{-1}$ . The r.m.s. velocity is represented by the colour coding in units of $\text{cm}~\text{s}^{-1}$ .

Figure 2. A schematic sketch of the reversal process changing from (a) state I to (b) state II. (c) A typical time series of the temperature contrast ${\it\delta}$ between the two sensors embedded in the bottom plate for $Ra=1.1\times 10^{9}$ . The red solid circles indicate detections of flow reversal events based on the criteria described in the text. The red dashed lines indicate the two flow states I and II shown in (a) and (b), respectively. (d) The corresponding PDF of the data shown in (c). The blue dashed lines are two independent Gaussian function fits to the data and the red solid line is the superposition of them. It is seen that the data can be excellently fitted with a double-Gaussian function. See text for the explanations of $d$ and ${\it\sigma}$ . Note that similar results were obtained from the top plate.

That flow reversals become more probable in quasi-2-D cells with smaller ${\it\Gamma}$ was also observed in some previous studies (Vasilev & Frick Reference Vasilev and Frick2011; Wagner & Shishkina Reference Wagner and Shishkina2013; Kaczorowski et al. Reference Kaczorowski, Chong and Xia2014), but their statistical properties have not been examined. Here, we used the temperature contrast method (Sugiyama et al. Reference Sugiyama, Ni, Stevens, Chan, Zhou, Xi, Sun, Grossmann, Xia and Lohse2010; Ni et al. Reference Ni, Huang and Xia2015) to detect flow reversal events over long time periods. This method is based on the idea that a region where cold plumes impinge should have a lower temperature than that where hot plumes emit. As illustrated in figure 2, when the LSC circulates in the counter-clockwise direction (state I as sketched in figure 2 a), the temperature in the left side of the bottom plate is lower than that in the right side; when the flow reverses its circulation directing and changes to state II (figure 2 b), the temperature in the left side becomes higher. Therefore, the temperature contrast ${\it\delta}=T_{right}-T_{left}$ will change its sign during a reversal event and thus is a good indicator of reversals (see figure 2 c). Besides the sign change of ${\it\delta}$ , we set three more criteria to count true reversal events. The first one is that the two circulation states can be clearly distinguished from each other. This is quantified with the PDF of ${\it\delta}$ , as shown in figure 2(d). To be specific, the distance $d$ between the two most probable values in the PDF should be larger than the r.m.s. value ${\it\sigma}$ so that the two peaks can be clearly distinguished, where $d$ and ${\it\sigma}$ were obtained by a double-Gaussian function fit as shown in figure 2(d). The second criterion is that when the flow changes from one state to the other, ${\it\delta}$ should first crossover the peak, i.e. the value of ${\it\delta}$ should be larger (smaller) than the peak value in the PDF corresponding to state I (II). The third one is that the flow should stay at the new state for more than one turnover time $t_{LSC}$ of the LSC, which was obtained from the correlation function of the temperature signals inside the plates (Brown, Funfschilling & Ahlers Reference Brown, Funfschilling and Ahlers2007). With these criteria, we are ready to examine the flow reversals quantitatively.

Figure 3. Examples of flow reversals at $Ra=1.1\times 10^{9}$ for (a) ${\it\Gamma}=0.3$ , (b) ${\it\Gamma}=0.2$ and (c) ${\it\Gamma}=0.15$ . The red solid circles indicate detections of flow reversal events based on the criteria described in the text. (d) The normalized flow reversal frequency $f_{rev}/f_{LSC}$ as a function of ${\it\Gamma}$ for different $Ra$ , where $f_{rev}=1/\langle {\it\tau}\rangle$ , $f_{LSC}=1/t_{LSC}$ , and $\langle {\it\tau}\rangle$ is the mean time interval between successive reversals.

We first show in figure 3(a–c) some examples of flow reversals at $Ra=1.1\times 10^{9}$ for ${\it\Gamma}=0.3$ , 0.2 and 0.15, respectively. It is seen clearly that the LSC reverses its circulating direction with a drastically increased frequency as ${\it\Gamma}$ decreases, which is quantified in figure 3(d). However, when ${\it\Gamma}$ is further reduced from 0.15 to 0.1, the reversal frequency does not show a significant increase anymore, so we do not show the examples for ${\it\Gamma}=0.1$ here. It is further observed in figure 3(d) that the increase in the reversal frequency is $Ra$ -dependent: the higher Ra becomes, the larger the increase (by more than 1000-fold for the highest value of Rayleigh number reached in the experiment), suggesting that the impact of confinement on the LSC dynamics is stronger for higher $Ra$ . Note that the measurements in smaller ${\it\Gamma}$ cells did not go to lower Ra, because ${\it\sigma}$ becomes comparable with $d$ in these cases, thus the first criterion is not satisfied anymore. This is a limitation of the measurement technique. We also remark that no flow reversal was observed in the ${\it\Gamma}=0.6$ cell, even with a long time measurement of up to 15 000 $t_{LSC}$ (more than two weeks) at $Ra=2.8\times 10^{8}$ , at which flow reversals are supposed to have a higher probability of occurrence than at higher Ra (Sugiyama et al. Reference Sugiyama, Ni, Stevens, Chan, Zhou, Xi, Sun, Grossmann, Xia and Lohse2010; Ni et al. Reference Ni, Huang and Xia2015).

Figure 4. A semi-log plot of the normalized mean time interval between successive reversals $\langle {\it\tau}\rangle /t_{LSC}$ as a function of $(d/2{\it\sigma})^{2}$ for different ${\it\Gamma}$ . The solid lines are fittings to individual data sets according to the formula $\langle {\it\tau}\rangle /t_{LSC}=C_{p}({\it\Gamma})\exp [(d/2{\it\sigma})^{2}]$ obtained in a model developed by Ni et al. (Reference Ni, Huang and Xia2015) recently. Inset: The fitting coefficient $C_{p}$ versus ${\it\Gamma}$ .

In a recent study using quasi-2-D cells similar to the ones used in the present experiment, Ni et al. (Reference Ni, Huang and Xia2015) have shown that the mean time interval between successive reversals $\langle {\it\tau}\rangle$ follows a stochastic process, i.e. $\langle {\it\tau}\rangle /t_{LSC}=C_{p}({\it\Gamma})\exp [(d/2{\it\sigma})^{2}]$ , where $d$ is taken as the relative strength of the LSC and ${\it\sigma}$ is its fluctuation. Figure 4 shows the data $\langle {\it\tau}\rangle /t_{LSC}$ versus $(d/2{\it\sigma})^{2}$ for different ${\it\Gamma}$ cells in a semi-log plot. It is seen that the data can be nicely fitted with the formula above, with a strong ${\it\Gamma}$ -dependent prefactor $C_{p}$ , as shown in the inset, indicting that the model developed in Ni et al. (Reference Ni, Huang and Xia2015) can also well describe the reversal behaviours in a highly confined geometry. Since the value of $d/{\it\sigma}$ can be used to characterize the stability of the LSC, we further examine their values separately to see how confinement affects the properties of the LSC. It is seen in figure 5 that as ${\it\Gamma}$ decreases, $d$ decreases on one hand, and ${\it\sigma}$ increases on the other, which suggests that LSC becomes weaker and experiences stronger fluctuations in highly confined geometry, thus confirming the PIV-measured results. As $d$ is essentially the temperature difference between the left and right sides of the conducting plates, smaller $d$ also suggests that the temperature is more uniform in the plate for small ${\it\Gamma}$ cells, consistent with the findings in Huang et al. (Reference Huang, Kaczorowski, Ni and Xia2013).

Figure 5. (a) The strength of the LSC $d$ and (b) its fluctuation ${\it\sigma}$ normalized by the global temperature difference ${\rm\Delta}T$ as a function of ${\it\Gamma}$ for different $Ra$ . Note that ${\rm\Delta}T$ for different ${\it\Gamma}$ cells are the same for the same $Ra$ , as they have the same height.

Figure 6. PDFs of the normalized time interval between successive flow reversals at $Ra=1.1\times 10^{9}$ for (a) ${\it\Gamma}=0.2$ and (b) ${\it\Gamma}=0.15$ , respectively. The straight lines indicate exponential fittings.

In figure 6, we show the PDFs of time interval between successive reversals ${\it\tau}/t_{LSC}$ for ${\it\Gamma}=0.2$ and 0.15 cells on a semi-log scale. It is seen that the data for both ${\it\Gamma}$ are in good agreement with the exponential function $p({\it\tau}/t_{LSC})\sim \exp (-{\it\tau}/t_{LSC})$ . (The levelling off of the PDFs at the large values of ${\it\tau}/t_{LSC}$ is caused by the insufficient statistics.) These results indicate that flow reversals in the quasi-2-D geometry obey Poisson statistics as in the 3-D case (Sreenivasan et al. Reference Sreenivasan, Bershadskii and Niemela2002; Brown & Ahlers Reference Brown and Ahlers2007; Xi & Xia Reference Xi and Xia2007), which suggests that it may be possible to develop a geometry-independent model to explain the apparently different LSC dynamics in different geometries.

Figure 7. (a,b) Time series of temperatures measured at the cell centre for (a) $Ra=1.1\times 10^{9}$ and (b) $Ra=8.6\times 10^{7}$ . From top to bottom, ${\it\Gamma}=0.1$ , 0.15, 0.2, 0.3 and 0.6. (c,d) The corresponding PDFs of temperature fluctuations shown in (a,b): (c) $Ra=1.1\times 10^{9}$ and (d) $Ra=8.6\times 10^{7}$ . Note that the data for different ${\it\Gamma}$ have been upshifted for clarity. The green and orange lines represent exponential and Gaussian fits, respectively.

3.2 Local temperature fluctuations

In Huang et al. (Reference Huang, Kaczorowski, Ni and Xia2013), it was simply mentioned that with decreasing ${\it\Gamma}$ the local temperature fluctuation in the cell centre increases significantly and its PDF changes from exponential to Gaussian. Here, we present the details. Figure 7(a,b) shows the time series of temperatures measured at the cell centre for $Ra=1.1\times 10^{9}$ and $Ra=8.6\times 10^{7}$ , respectively. It is seen clearly that the fluctuations in small ${\it\Gamma}$ cells are stronger than those in large ones. Besides the change in fluctuation magnitude, the corresponding PDFs (shown in figure 7 c,d) also change their forms: as ${\it\Gamma}$ is reduced, there is a transition from (stretched) exponential to Gaussian-like. The strong geometric dependency of the temperature fluctuation PDF found in the quasi-2-D system is consistent with a recent numerical simulation using similar geometries (Kaczorowski et al. Reference Kaczorowski, Chong and Xia2014), but quite different from previous studies conducted in 3-D systems, in which a universal form for different geometries was observed (Daya & Ecke Reference Daya and Ecke2001; Song & Tong Reference Song and Tong2010). The functional form of the temperature fluctuation PDF is one important hallmark for different flow states (Heslot, Castaing & Libchaber Reference Heslot, Castaing and Libchaber1987; Castaing et al. Reference Castaing, Gunaratne, Kadanoff, Libchaber and Heslot1989; Massaioli, Benzi & Succi Reference Massaioli, Benzi and Succi1993) and the plume distribution in the convection cell (Xia & Lui Reference Xia and Lui1997), thus these changes confirm that the flow state has been changed and/or there are more plumes passing though the cell centre as a result of confinement.

Figure 8. (a) Normalized temperature fluctuation ${\it\sigma}_{c}/{\rm\Delta}T$ at the cell centre as a function of $Ra$ for different ${\it\Gamma}$ . The solid lines represent power-law fits to the individual data sets. (b) The corresponding flatness versus $Ra$ .

To examine these changes quantitatively, we plot the Ra-dependence of the normalized r.m.s. temperature ${\it\sigma}_{c}/{\rm\Delta}T$ and their flatness values in figure 8. From figure 8(a), one can see that while the magnitude of ${\it\sigma}_{c}/{\rm\Delta}T$ increases as ${\it\Gamma}$ decreases, its $Ra$ -scaling exponent in general decreases (see table 1 for detailed values). Large variations in the Ra dependency of temperature fluctuations due to geometric effects have also been found in earlier studies (Daya & Ecke Reference Daya and Ecke2001; Song & Tong Reference Song and Tong2010). We note that the scalings for the ${\it\Gamma}=0.6$ and 0.3 cells are the same as those found in a recent simulation of turbulent RB convection in a cubic geometry (Kaczorowski et al. Reference Kaczorowski, Chong and Xia2014), but very different from the value $-$ 0.49 reported in Daya & Ecke (Reference Daya and Ecke2001) in a convection cell with square cross-section. The ${\it\Gamma}=0.3$ cell also has similar flatness values with the ${\it\Gamma}=0.6$ cell (see figure 8 b), suggesting that the convective flow has not sensed the confinement effects from the walls for ${\it\Gamma}=0.3$ . However, when ${\it\Gamma}$ is further reduced, there is a sudden drop in the flatness values; and these values gradually decrease to ${\sim}$ 3 (Gaussian case) for smaller ${\it\Gamma}$ and lower $Ra$ , which is consistent with the change in the form of temperature fluctuation PDF shown in figure 7. Special attention is paid to the abnormal $Ra$ dependence of flatness in the ${\it\Gamma}=0.1$ cell, which is also reflected in its very different time series signals and the bimodal shaped PDF at $Ra=8.6\times 10^{7}$ . Based on the recent finding by Chong et al. (Reference Chong, Huang, Kaczorowski and Xia2015), these strange behaviours for ${\it\Gamma}=0.1$ could be due to the fact that the LSC no longer exists in this case.

3.3 Heat transport

We now examine the influences of spatial confinement on the heat transport properties. Figure 9 shows a semi-log plot of the compensated Nu versus Ra for different ${\it\Gamma}$ cells. As the heat transfer in a cubic cell (i.e. ${\it\Gamma}=1$ ) is identical to that in a cylinder (Qiu & Xia Reference Qiu and Xia1998b ), for clarity and easy comparison, we plot in figure 9 the results measured in a cylindrical cell of ${\it\Gamma}=1$ (Wei, Ni & Xia Reference Wei, Ni and Xia2012), which is the most commonly used geometry in the study of turbulent RB convection. From figure 9, we find that: (i) the data for the ${\it\Gamma}=0.6$ cell collapse well with those measured in the ${\it\Gamma}=1$ cell; (ii) Nu first decreases a little when ${\it\Gamma}$ is changed from 0.6 to 0.3 and then increases significantly afterwards, up to 17 % for the parameter range explored in the present study. Note that the Nu data have been reported in Huang et al. (Reference Huang, Kaczorowski, Ni and Xia2013). We plot them here in a different form to understand the confinement effects in turbulent thermal convection from a different point of view.

Figure 9. Compensated $Nu$ as a function of $Ra$ from different ${\it\Gamma}$ cells as indicated by different symbols. The solid lines are power-law fits to individual data sets. The solid hexagons are the results measured in a cylindrical cell of ${\it\Gamma}=1$ (Wei, Ni & Xia Reference Wei, Ni and Xia2012).

Figure 10. Fitting coefficients for the combined power law $Nu=C_{1}Ra^{1/3}+C_{2}Ra^{1/4}$ as a function of ${\it\Gamma}$ : (a) $C_{1}$ and (b) $C_{2}$ . (c) The ratio between $C_{2}Ra^{1/4}$ and $C_{1}Ra^{1/3}$ versus $Ra$ in different ${\it\Gamma}$ cells.

The solid lines in figure 9 represent power-law fits $Nu=ARa^{{\it\alpha}}$ to the measured data with the scaling exponent ${\it\alpha}$ varying from 0.27 to 0.29 (see table 1 for detailed values). These values are in general agreement with those found in previous studies in the same parameter ranges of $Ra$ and $Pr$ (Ahlers, Grossmann & Lohse Reference Ahlers, Grossmann and Lohse2009). We recall that the Grossmann–Lohse (GL) model splits the globally averaged dissipation rate into bulk and boundary layer (BL) contributions and suggests that different linear combinations of power laws for different parameter ranges should be used for the $Nu$ – $Ra$ fitting (Grossmann & Lohse Reference Grossmann and Lohse2000). Based on this idea, we fit to each set of data with a combination of two power laws proposed in the GL model, i.e. $Nu=C_{1}Ra^{1/3}+C_{2}Ra^{1/4}$ , in which $C_{1}Ra^{1/3}$ and $C_{2}Ra^{1/4}$ represent the contributions from the bulk and BLs, respectively. The obtained fitting coefficients $C_{1}$ and $C_{2}$ for different ${\it\Gamma}$ are plotted in figures 10(a) and 10(b), respectively (see table 1 for detailed values). It is seen that the ${\it\Gamma}$ dependencies of $C_{1}$ and $C_{2}$ have an opposite trend: the decrease (increase) in $C_{1}$ is accompanied by an increase (decrease) in $C_{2}$ . Interestingly, the behaviour of $C_{2}$ is similar to that of the global Nu: It first decreases when ${\it\Gamma}$ is changed from 0.6 to 0.3 and then increases with decreasing ${\it\Gamma}$ afterwards (see figure 1 in Huang et al. (Reference Huang, Kaczorowski, Ni and Xia2013)). This is because in the present parameter range, the heat transport, in terms of the energy dissipation rate, is dominated by the contributions from the boundary layers, which is supported by comparing the magnitudes between $C_{1}Ra^{1/3}$ and $C_{2}Ra^{1/4}$ as shown in figure 10(c). The change in the ratio of $C_{2}Ra^{1/4}$ over $C_{1}Ra^{1/3}$ as ${\it\Gamma}$ is decreased suggests that the confinement increases the contribution from the BLs in the global heat transport and weakens that from the bulk. This result is consistent with the finding in Huang et al. (Reference Huang, Kaczorowski, Ni and Xia2013), in which it was found that the confinement suppresses the large-scale turbulent flow in the bulk and enables thermal plumes (commonly viewed as detached thermal BLs) to pass through the entire cell, rather than being confined in its periphery.

Why have such significant changes in Nu not been observed in previous studies in 3-D and true 2-D systems? Note that even for the smallest ${\it\Gamma}$ explored in the present study, the width of the cell (its smallest dimension) is still much larger than the thermal boundary layer thickness, therefore it is unlikely that the geometric effects will directly lead to any change in the BLs and hence Nu. Thus, the reasons must be connected to the flow structures. It has been shown by Huang et al. (Reference Huang, Kaczorowski, Ni and Xia2013) that the confinement restricts the plume’s movement near the BLs into a 1-D motion, resulting in the formation of highly coherent plumes. These highly coherent plumes turn out to have a stronger interaction with the BLs, which is supported by the mean velocity fields as shown in figure 1. It is seen that in the ${\it\Gamma}=0.3$ cell the mean flow near the boundary layers has a large horizontal component, whereas in the ${\it\Gamma}=0.15$ case, the mean flow is largely perpendicular and thus perturbs the BLs directly. Therefore, geometric confinement in the quasi-2-D geometry not only changes the morphology and dynamics of thermal plumes, but also strengthens the interaction between the flow structures in the bulk and the BLs, both of which have not been observed in previous studies in 3-D systems. This explains why these studies found a weak ${\it\Gamma}$ effect on heat transport. In the true 2-D case, there is no lateral confinement of the plumes when the aspect ratio is changed. As the coherent properties of plumes are not modified in this case but the flow structures are changed as a result of the aspect ratio variation, $Nu$ is found to be sensitive to the nature of the flow structures in the 2-D geometry (van der Poel et al. Reference van der Poel, Stevens and Lohse2011, Reference van der Poel, Stevens, Sugiyama and Lohse2012).

In a recent numerical simulation of 2-D turbulent convection (Chandra & Verma Reference Chandra and Verma2013), extremely large fluctuations and even negative values of global Nu were observed during the flow reversal processes, which was attributed to the strong geometric constraints to the flow. Since the confinement also leads to more frequent flow reversals, it would be interesting to examine the instantaneous Nu for different ${\it\Gamma}$ during reversals. As we can see in figure 11(a,b), neither negative values nor very large fluctuations are observed in the time series of instantaneous Nu for all the cases, which is also reflected by their PDFs, shown in figure 11(c,d). Although there is a trend that the fluctuations of instantaneous Nu increase as ${\it\Gamma}$ decreases, the fluctuation values for all the ${\it\Gamma}$ and the differences between them are so small that they are all within 1 % of the mean values. These results further suggest that the quasi-2-D geometry is different from the true 2-D case.

Figure 11. (a,b) Time series of instantaneous Nu for (a) $Ra=1.1\times 10^{9}$ and (b) $Ra=8.6\times 10^{7}$ . From top to bottom, ${\it\Gamma}=0.1$ , 0.15, 0.2, 0.6 and 0.3. (c,d) The corresponding PDFs of the Nu fluctuations shown in (a,b): (c) $Ra=1.1\times 10^{9}$ and (d)  $Ra=8.6\times 10^{7}$ .