1. Introduction
Thermal convection occurs widely in nature and industrial processes. Rayleigh–Bénard (RB) convection, a fluid layer confined between two horizontally parallel conducting plates with heating from below and cooling from above, is a paradigmatic model for studying generic thermal convection phenomena. This system is characterized by three dimensionless parameters, namely the Rayleigh number 
$Ra = \alpha g \Delta T\, H^3/(\nu \kappa )$, the Prandtl number 
$Pr = \nu /\kappa$ and the aspect ratio 
$\varGamma$ (the lateral dimension of the fluid layer over its height 
$H$). Here, 
$\Delta T$ is the temperature difference across the fluid layer, 
$g$ is the gravitational acceleration, and 
$\alpha$, 
$\nu$ and 
$\kappa$ are the thermal expansion coefficient, the kinematic viscosity and the thermal diffusivity of the working fluid, respectively. The response of the system is generally quantified by the heat transfer efficiency and flow strength, which are indicated by the Nusselt number 
$Nu$ and the Reynolds number 
$Re$, respectively.
Over the past decades, extensive studies have been devoted to understand how the heat transport and the flow dynamics of turbulent RB convection are determined by the three control parameters mentioned above (Ahlers, Grossmann & Lohse Reference Ahlers, Grossmann and Lohse2009; Lohse & Xia Reference Lohse and Xia2010; Chillà & Schumacher Reference Chillà and Schumacher2012). Recently, there have been increasing interests that go beyond the conventional RB configuration by modifying the boundary condition and geometric structure of the system. Examples include creating different roughness patterns over the conducting plates (Wagner & Shishkina Reference Wagner and Shishkina2015; Toppaladoddi, Succi & Wettlaufer Reference Toppaladoddi, Succi and Wettlaufer2017; Xie & Xia Reference Xie and Xia2017; Rusaouën et al. Reference Rusaouën, Liot, Castaing, Salort and Chillà2018; Zhang et al. Reference Zhang, Sun, Bao and Zhou2018; Foroozani et al. Reference Foroozani, Niemela, Armenio and Sreenivasan2019; Zhu et al. Reference Zhu, Stevens, Shishkina, Verzicco and Lohse2019; Emran & Shishkina Reference Emran and Shishkina2020), tilting the convection cell with a broad range of inclination angle (Guo et al. Reference Guo, Zhou, Cen, Qu, Lu, Sun and Shang2015; Shishkina & Horn Reference Shishkina and Horn2016; Wang et al. Reference Wang, Xia, Wang, Sun, Zhou and Wan2018; Jiang, Sun & Calzavarini Reference Jiang, Sun and Calzavarini2019; Zwirner et al. Reference Zwirner, Khalilov, Kolesnichenko, Mamykin, Mandrykin, Pavlinov, Shestakov, Teimurazov, Frick and Shishkina2020), using mixed or time-modulated thermal boundary conditions (Ripesi et al. Reference Ripesi, Biferale, Sbragaglia and Wirth2014; Wang, Huang & Xia Reference Wang, Huang and Xia2017; Bakhuis et al. Reference Bakhuis, Ostilla-Mónico, van der Poel, Verzicco and Lohse2018; Yang et al. Reference Yang, Chong, Wang, Verzicco, Shishkina and Lohse2020a), adding partitioned structures inside the convection cell (Bao et al. Reference Bao, Chen, Liu, She, Zhang and Zhou2015; Chen et al. Reference Chen, Bao, Yin and She2017; Liu et al. Reference Liu, Jiang, Chong, Zhu, Wan, Verzicco, Stevens, Lohse and Sun2020) and even applying vibration to the system (Wang, Zhou & Sun Reference Wang, Zhou and Sun2020a). Nevertheless, there still remain many issues to be explored for the classical RB system.
One relatively less addressed issue in the study of classical RB convection is the effect of the 
$Pr$ number, especially compared to the large body of studies over a wide range of 
$Ra$ number. The existing studies on this issue were mostly made in convection cells with three-dimensional (3-D) geometry (Globe & Dropkin Reference Globe and Dropkin1959; Verzicco & Camussi Reference Verzicco and Camussi1999; Kerr & Herring Reference Kerr and Herring2000; Ahlers & Xu Reference Ahlers and Xu2001; Lam et al. Reference Lam, Shang, Zhou and Xia2002; Roche et al. Reference Roche, Castaing, Chabaud and Hébral2002; Xia, Lam & Zhou Reference Xia, Lam and Zhou2002; Breuer et al. Reference Breuer, Wessling, Schmalzl and Hansen2004; Silano, Sreenivasan & Verzicco Reference Silano, Sreenivasan and Verzicco2010; Stevens, Lohse & Verzicco Reference Stevens, Lohse and Verzicco2011; Shishkina et al. Reference Shishkina, Emran, Grossmann and Lohse2017; Yang et al. Reference Yang, Zhu, Wang, Liu and Zhou2020b). According to these studies, two different scaling regimes were identified based on the 
$Pr$ dependences of the 
$Nu$ and 
$Re$ numbers: 
$Nu$ first increases strongly with 
$Pr$ for 
$Pr \lesssim 1$, and then, after reaching a maximum at 
$Pr \simeq 3$, it gradually decreases with 
$Pr$ and can be described by a power law as 
$Nu \sim Pr^{-0.03}$ approximately; the 
$Pr$-dependent scaling of 
$Re$ also possesses a transition around 
$Pr \simeq 1$. These findings were in good agreement with the predictions of the Grossmann–Lohse (GL) model (Grossmann & Lohse Reference Grossmann and Lohse2000, Reference Grossmann and Lohse2001, Reference Grossmann and Lohse2002; Stevens et al. Reference Stevens, van der Poel, Grossmann and Lohse2013), which, in essence, are based on two-dimensional (2-D) equations.
However, the numerical simulations in the 2-D system show different behaviours compared to the 3-D case. Schmalzl, Breuer & Hansen (Reference Schmalzl, Breuer and Hansen2004) first found that the 
$Nu$ data in 2-D RB convection at 
$Ra=10^6$ have no obvious 
$Pr$ dependence in the range of 
$10^{-2}\leqslant Pr\leqslant 10^2$, and 
$Re$ differs from the 3-D data when 
$Pr$ is smaller than unity. By extending the 
$Ra$ number up to 
$Ra=10^{10}$, later numerical studies showed that 
$Nu$ is insensitive to 
$Pr$ when 
$Pr$ is greater than 10 (van der Poel et al. Reference van der Poel, Stevens, Sugiyama and Lohse2012; van der Poel, Stevens & Lohse Reference van der Poel, Stevens and Lohse2013; He et al. Reference He, Fang, Gao, Huang and Bao2021). It was further found that the strong counter-gradient local heat flux in the 2-D configuration can lead to an anomalous 
$Nu$–
$Pr$ relation, i.e. 
$Nu$ minimizes at 
$Pr\approx 2\text {--}3$, rather than maximizes as found in the 3-D case (Huang & Zhou Reference Huang and Zhou2013). Moreover, the 
$Re$ number in 2-D RB flow not only has a larger magnitude, but also exhibits a notably different 
$Ra$–
$Pr$ dependence from the 3-D case (van der Poel et al. Reference van der Poel, Stevens and Lohse2013; Zhang, Zhou & Sun Reference Zhang, Zhou and Sun2017).
The reasons for these discrepancies between 2-D and 3-D RB systems were assessed to be twofold (Schmalzl et al. Reference Schmalzl, Breuer and Hansen2004; van der Poel et al. Reference van der Poel, Stevens and Lohse2013; Pandey et al. Reference Pandey, Verma, Chatterjee and Dutta2016). The first one comes from the large-scale circulation (LSC). Although the LSC in the 3-D system has a quasi-2-D geometrical character (Xi & Xia Reference Xi and Xia2008; Sugiyama et al. Reference Sugiyama, Ni, Stevens, Chan, Zhou, Xi, Sun, Grossmann, Xia and Lohse2010), its dynamical features are fully 3-D and cannot be resembled by the 2-D flow. The second one is related to thermal plumes. While the movement and evolution of thermal plumes in 2-D geometry are restrained in the third dimension, these limitations are not as great in 3-D. In fact, the two reasons are correlated with each other, and both could be attributed to the effects of spatial confinement. Recent studies showed that spatial confinement can result in pronounced changes in the morphology and dynamics of thermal plumes, which bring about very different large-scale flow dynamics and unexpected heat transport enhancement (Huang et al. Reference Huang, Kaczorowski, Ni and Xia2013; Chong et al. Reference Chong, Huang, Kaczorowski and Xia2015; Chong & Xia Reference Chong and Xia2016; Huang & Xia Reference Huang and Xia2016). As the LSC in the 3-D RB system is essentially a quasi-2-D structure, and the influences of spatial confinement depend strongly on 
$Pr$ (Wagner & Shishkina Reference Wagner and Shishkina2013; Chong et al. Reference Chong, Wagner, Kaczorowski, Shishkina and Xia2018), it is natural to ask: How does the 
$Pr$ number influence RB convection in quasi-2-D geometry? To the best of our knowledge, this issue has not been investigated experimentally before.
For quasi-2-D RB convection, it is generally considered that the LSC is limited to a quasi-2-D plane without complicated 3-D dynamics, such as azimuthal rotation and torsional motion (Sugiyama et al. Reference Sugiyama, Ni, Stevens, Chan, Zhou, Xi, Sun, Grossmann, Xia and Lohse2010). In the present study, we used rectangular convection cells with the same geometrical features as in previous experiments (Huang et al. Reference Huang, Wang, Xi and Xia2015; Chen et al. Reference Chen, Huang, Xia and Xi2019), i.e. the aspect ratio in the plane parallel to the LSC is unity, whereas that perpendicular to the LSC is 0.3. Four such convection cells with different sizes were used to cover the 
$Ra$ number range of 
$6.0\times 10^8 \leqslant Ra \leqslant 3.0\times 10^{10}$ and the 
$Pr$ number range of 
$11.7 \leqslant Pr \leqslant 650.7$. It is found that the LSC breaks down for higher 
$Pr$, resulting in a regime transition in the 
$Re$ number. For the 
$Pr$ range with a persistent LSC, both the 
$Re$ and 
$Nu$ numbers were investigated systematically and compared to the behaviours in 2-D and 3-D systems. In addition to these commonly measured global properties, we also examined the effect of 
$Pr$ on local temperature fluctuations, which has received little attention so far (Daya & Ecke Reference Daya and Ecke2002; Grossmann & Lohse Reference Grossmann and Lohse2004; Wunsch & Kerstein Reference Wunsch and Kerstein2005; Chong et al. Reference Chong, Wagner, Kaczorowski, Shishkina and Xia2018). These experimental results not only enrich our knowledge of classical RB convection in quasi-2-D geometry, but also provide a useful basis for studying non-traditional thermal convection systems.
The remainder of this paper is organized as follows. The experimental set-up and methods are introduced in § 2. The main results are presented in § 3, which is divided into four parts. In § 3.1, we first show the flow patterns obtained with the shadowgraph technique for different 
$Pr$ numbers. Then we present and discuss the behaviour of the 
$Re$ number in § 3.2, followed by the 
$Nu$ number in § 3.3. The local temperature fluctuations measured in the cell centre and near the sidewall will be presented in § 3.4. Finally, we summarize the findings in § 4.
2. Experimental apparatus and methods
As shown in table 1, there are eight datasets in total obtained in the present study. To vary the 
$Pr$ number from 11.7 to 650.7, seven kinds of silicone oil (KF-96 series, Shin-Etsu Inc.) were used as the working fluids. Their fluid properties were provided by Shin-Etsu and verified by independent measurements from a commercial company (Netzsch Inc.). In contrast to previous experiments where the 
$Ra$ range always changes for different 
$Pr$ (Ahlers & Xu Reference Ahlers and Xu2001; Lam et al. Reference Lam, Shang, Zhou and Xia2002; Roche et al. Reference Roche, Castaing, Chabaud and Hébral2002; Xia et al. Reference Xia, Lam and Zhou2002; Yang et al. Reference Yang, Zhu, Wang, Liu and Zhou2020b), we used four convection cells with the same aspect ratios but different sizes to maintain 
$Ra$ with the same values while varying 
$Pr$ (see table 1 for the geometrical information on the convection cells and the corresponding parameter ranges). Having almost identical 
$Ra$ numbers is one important feature of the present study, as it allows us to compare the results from different 
$Pr$ directly. In addition, we have made measurements at 
$Pr = 20.5$ in two different cells (datasets II and III), so that we can examine the effect of system size if there is any.
Table 1. Geometrical information on the convection cells for different 
$Pr$ numbers and the corresponding 
$Ra$ number ranges. Here, H (cm), L (cm) and W (cm) are the height, length and width of the convection cell, respectively. The fluid properties of the silicone oils used in the present study are also provided, where 
$\alpha$ (
$\textrm {{K}}^{-1}$), 
$\kappa$ (
$\textrm {{m}}^{2}\ \textrm {{s}}^{-1}$) and 
$\nu$ (
$\textrm {{m}}^{2}\ \textrm {{s}}^{-1}$) are the thermal expansion coefficient, the thermal diffusivity and the kinematic viscosity, respectively.
The four convection cells used in the experiments were of identical design (Zhou et al. Reference Zhou, Liu, Li and Zhong2012) and their schematic diagram is shown in figure 1(a). Briefly, they were built independently with their own conducting copper plates and Plexiglas sidewalls. The heat flux 
$Q$ was supplied to the bottom plate by a nichrome wire and extracted at the top plate by a recirculating cooler. The temperatures of the bottom and top plates were monitored by thermistors (Omega, Model 44031) embedded uniformly inside the plates (not shown in the figure), based on which we obtained the temperature difference across the fluid layer 
$\Delta T$, and then the 
$Nu$ number can be calculated as 
$Nu = QH/(\chi \Delta T)$. Here, 
$\chi$ is the thermal conductivity of the working fluid. The temperatures inside the fluid were measured by three small thermistors with a tip size of 
${\sim }200\ \mathrm {\mu }\textrm {m}$ (Measurement Specialties, Model G22K7MCD419). As shown in figure 1(a), they were mounted on stainless-steel tubes of 1 mm diameter. Thermistor 
$T_1$ was used to monitor the temperature in the cell centre, which was maintained at 
$40.0\,^{\circ }\textrm {C}$ for all the measurements.
Figure 1. (a) Schematic diagram of the experimental set-up. Here 
$T_1$, 
$T_2$ and 
$T_3$ indicate three small thermistors used to measure local temperature in the flow. (b) The correlation function between the signals measured by 
$T_2$ and 
$T_3$, based on which we can obtain the characteristic flow velocity. See text for the explanation. Inset: A segment of the temperature time series measured by 
$T_2$ and 
$T_3$ at 
$Ra \simeq 5.9\times 10^9$ and 
$Pr = 49.3$.
In addition to acquiring local temperatures, thermistors 
$T_2$ and 
$T_3$ were further used to detect the motion of thermal plumes near the sidewall of the convection cell. This method (Wang et al. Reference Wang, Huang, Zhou and Xia2016) is based on the idea that, when thermal plumes pass through a region, they will result in appreciable temperature variations in the background flow, and the two temperature sensors with a small separation 
$d$ should experience similar variations with a time delay (see the inset of figure 1b). By calculating the correlation function between the signals of 
$T_2$ and 
$T_3$ (figure 1b), we can identify the mean time delay 
$\tau$ and then obtain the velocity 
$U = d/\tau$. Finally, the corresponding 
$Re$ number based on plume motion can be expressed as 
$Re = 2HU/\nu = 2Hd/\tau \nu$, where 
$2H$ has been used for the typical length scale as in most previous studies (Ahlers et al. Reference Ahlers, Grossmann and Lohse2009). Because the flow velocity strongly depends on the distance away from the sidewall and the flow pattern continuously evolves while varying 
$Ra$ and 
$Pr$ (Niemela & Sreenivasan Reference Niemela and Sreenivasan2003; Xia, Sun & Zhou Reference Xia, Sun and Zhou2003; Belkadi et al. Reference Belkadi, Guislain, Sergent, Podvin, Chillà and Salort2020), both of which will affect the measured value of the 
$Re$ number, we adjusted the positions of 
$T_2$ and 
$T_3$ at different (
$Ra, Pr$) settings to make sure that they were located in the region near the sidewall where most organized plumes passed through (see table 4 in the appendix for their distance away from the sidewall). Therefore, the so-obtained 
$U$ and 
$Re$ can well represent the characteristic velocity and strength of the flow, respectively.
Note that in cases where direct measurement of velocity field is difficult, various other methods have been developed to estimate the 
$Re$ number. For example, Chavanne et al. (Reference Chavanne, Chillà, Chabaud, Castaing and Hébral2001) used the cross-correlation spectrum method, He, He & Tong (Reference He, He and Tong2010) applied the elliptic approximation (He & Zhang Reference He and Zhang2006), and Belkadi et al. (Reference Belkadi, Guislain, Sergent, Podvin, Chillà and Salort2020) even derived an estimate from the shadowgraph images. These different methods would lead to very different values of 
$Re$ and thus it does not help to make a comparison of their magnitudes directly. On the other hand, when considering the scaling exponents of 
$Re(Ra, Pr)$, the differences between the various methods are relatively small (see e.g. Ahlers et al. (Reference Ahlers, Grossmann and Lohse2009) for a review and also the recent study by Musilová et al. (Reference Musilová, Králik, La Mantia, Macek, Urban and Skrbek2017)). Therefore, in the present study, we will not compare the magnitude of our measured 
$Re$ with other published data, but just focus on the scaling exponents of 
$Re(Ra,Pr)$.
Before the measurements at each dataset, we first used the shadowgraph technique (Xi, Lam & Xia Reference Xi, Lam and Xia2004) to visualize the flow and obtain an overall picture of the 
$Pr$ effects on the flow dynamics and plume distribution. This information also helped us to determine the positions of the two thermistors 
$T_2$ and 
$T_3$ mentioned above. Then we began the measurements of 
$Re$, 
$Nu$ and local temperatures, which lasted at least 10 h for each (
$Ra, Pr$) setting. To minimize the heat exchange from system to the surroundings, the convection cell was wrapped by thermal insulating Styrofoam with a thickness of 
${\sim }8\ \textrm {cm}$. Furthermore, a size-matched copper basin was placed under the convection cell with three layers of plywood in between. A minor heat flux was provided to the copper basin to keep its temperature equal to that of the bottom plate, so that the heat loss from the bottom plate can be compensated. Finally, the whole apparatus was placed inside a thermostat to prevent any influence from the environment. Such a configuration has been demonstrated to ensure heat transport measurement with high precision (Ni, Zhou & Xia Reference Ni, Zhou and Xia2011b; Wei, Ni & Xia Reference Wei, Ni and Xia2012) and we have adapted it in recent studies of quasi-2-D RB convection (Huang et al. Reference Huang, Kaczorowski, Ni and Xia2013; Huang & Xia Reference Huang and Xia2016).
In order to satisfy the Oberbeck–Boussinesq (OB) approximation, we paid close attention to the non-OB effect, which is characterized by the ratio of the temperature drops across the bottom and top boundary layers, 
$x = (T_b-T_c)/(T_c-T_t)$ (Wu & Libchaber Reference Wu and Libchaber1991; Zhang, Childress & Libchaber Reference Zhang, Childress and Libchaber1997), where 
$T_b$, 
$T_t$ and 
$T_c$ are the measured temperatures of the bottom plate, the top plate and the cell centre, respectively. It was found that the values of 
$x$ were in the range of 
$0.95 \leqslant x \leqslant 1.02$ without obvious 
$Pr$ dependence. Thus the non-OB effect can be safely neglected in the present study (Ahlers et al. Reference Ahlers, Brown, Fontenele Araujo, Funfschilling, Grossmann and Lohse2006).
3. Results and discussion
3.1. Flow pattern
We first examine how the flow pattern in quasi-2-D geometry responds to the 
$Pr$ number. Figure 2 shows the shadowgraph images of typical flow pattern for different 
$Pr$ at 
$Ra \simeq 3.6\times 10^{9}$. (See also Movie 1 in the supplementary movies available at https://doi.org/10.1017/jfm.2021.21.) The flow pattern at 
$Pr = 49.3$ is similar to that at 
$Pr = 20.5$, so its image is not shown here for the sake of the figure's layout. It is seen that, as 
$Pr$ increases from 11.7 to 145.7, thermal plumes pass through the bulk region much less frequently, and their self-organized large-scale motion is more confined along the periphery of the convection cell. However, as 
$Pr$ further increases to 345.2 and 650.7, the flow patterns totally change. Rather than moving along the sidewalls to form a system-sized LSC, thermal plumes at these two 
$Pr$ preferentially rise up (fall down) in random locations of the convection cell and pass through the bulk region in a more straight way. Some plumes are so long-lived that they extend the entire height of the convection cell.
Figure 2. Shadowgraph images of the flow pattern at 
$Ra \simeq 3.6 \times 10^{9}$ for different 
$Pr$: (a) 
$Pr = 11.7$, (b) 
$Pr = 20.5$, (c) 
$Pr = 81.6$, (d) 
$Pr = 145.7$, (e) 
$Pr = 345.2$ and (f) 
$Pr = 650.7$. See also Movie 1 in the supplementary movies available at https://doi.org/10.1017/jfm.2021.21.
These changes are more evident for larger 
$Ra$ as shown in figure 3. Compared to lower 
$Ra$, a major difference for the larger 
$Ra$ cases is that the number of plumes increases significantly and they are more fragmentary thanks to stronger turbulent mixing. What is interesting here is that, for 
$Pr = 345.2$ at 
$Ra = 1.4\times 10^{10}$ (the highest 
$Ra$ that can be achieved for this 
$Pr$ in the present study), the flow state seems to switch between random motion of thermal plumes and two organized rolls (see Movie 2 in the supplementary movies available at https://doi.org/10.1017/jfm.2021.21). This phenomenon implies that, before building up a system-sized LSC, there could be a metastable state in the form of a two-roll structure during the evolution of the flow pattern. We also increased the 
$Ra$ number for 
$Pr = 650.7$ to as high as 
$5.0\times 10^9$, but the LSC is still absent. Note that the thermal insulating Styrofoam was removed during the flow visualization, so detailed flow patterns might be affected due to heat leakage through sidewalls. Nevertheless, the disappearance of the LSC and the change in plume distribution are well reflected by the behaviours of 
$Re$ and local temperature fluctuations, which will be presented in detail below.
Figure 3. Shadowgraph images of the flow pattern at 
$Ra \simeq 2.1\times 10^{10}$ for (a) 
$Pr = 11.7$, (b) 
$Pr = 20.5$, (c) 
$Pr = 81.6$ and (d) 
$Pr = 145.7$. Shadowgraph images of the flow pattern for (e) 
$Ra =1.4\times 10^{10}$, 
$Pr = 345.2$ and (f) 
$Ra= 5.0\times 10^{9}$, 
$Pr = 650.7$. See also Movie 2 in the supplementary movies available at https://doi.org/10.1017/jfm.2021.21.
3.2. Reynolds number
From the shadowgraph supplementary movies, it is also found that the flow motion becomes slower with increasing 
$Pr$. To quantify this effect, we examine the 
$Re$ number obtained by the method as introduced in § 2. Figure 4(a) shows 
$Re$ as a function of 
$Ra$ for different 
$Pr$. Note that the data for 
$Pr = 20.5$ were measured at two convection cells with different sizes. Their almost identical values in the overlapping 
$Ra$ range confirm that 
$Re$ has no dependence on system size (Huang et al. Reference Huang, Wang, Xi and Xia2015), so we will not distinguish them hereafter. From figure 4(a), it is seen that 
$Re$ keeps decreasing as 
$Pr$ increases, and the decrease becomes much larger when 
$Pr$ goes to 345.2 and 650.7. The dramatic drop in 
$Re$ at larger 
$Pr$ well reflects the breakdown of the LSC. Besides the magnitude, the 
$Ra$-dependent scaling also exhibits a sudden change. For 
$11.7\leqslant Pr\leqslant 145.7$, the scaling exponents (see table 2 in the appendix for detailed values) are close to each other, despite there being a slight change from 0.53 to 0.60. These values are consistent with the experimental finding (
$\sim$0.55) in similar convection cells for 
$Pr \simeq 4\text {--}7$ (Chen et al. Reference Chen, Huang, Xia and Xi2019). However, the dependence becomes noticeably stronger for higher 
$Pr$, with the exponents being 0.73 and 0.74 for 
$Pr = 345.2$ and 650.7, respectively. A similar sudden growth of 
$Re$–
$Ra$ scaling (from 0.5 to 0.68) at high 
$Pr$ has been observed previously in cylindrical convection cells, but there was no explanation for that in those studies (Lam et al. Reference Lam, Shang, Zhou and Xia2002; Silano et al. Reference Silano, Sreenivasan and Verzicco2010). Here, with the information from direct flow visualization, we understand that this is a regime transition that originates from the change in large-scale flow.
Figure 4. Log–log plots of (a) 
$Re$ versus 
$Ra$ for different 
$Pr$ and (b) 
$Re$ versus 
$Pr$ for different 
$Ra$, as indicated by the symbols. For clarity, the data in panel (b) have been shifted by multiplication by different constant factors for different 
$Ra$. The solid lines represent power-law fits to individual datasets. See tables 2 and 3 in the appendix for the fitting results in detail.
Table 2. The power-law fitting results for 
$Re = C_{1}Ra^{\alpha }$, 
$Nu= C_{2}Ra^{\gamma }$, (
$\sigma _c/\Delta T$)
$^{\ast } = C_{3}Ra^{\zeta }$ and (
$\sigma _s/\Delta T$)
$^{\ast } = C_{4}Ra^{\xi }$ at different 
$Pr$.
The existence of two distinct regimes for 
$Re$ is more obvious in figure 4(b). While the 
$Re$ numbers at different 
$Ra$ obey similar power laws in the range of 
$11.7\leqslant Pr\leqslant 145.7$ (see table 3 in the appendix for detailed fitting results), the data at higher 
$Pr$ follow a totally different trend. It could be that the convective flow has entered into the severely confined regime (Chong & Xia Reference Chong and Xia2016), where the flow topology is also characterized by long-lived thermal plumes, and the 
$Re$–
$Ra$ scalings found there (0.69–0.80 for 
$Pr = 4.38$) are compatible with the present steep values. However, the limited data in the high-
$Pr$ regime prevent us from further examining their behaviours quantitatively in the present study. On the other hand, overwhelming studies of turbulent RB convection were conducted in the case with a well-defined LSC (Ahlers et al. Reference Ahlers, Grossmann and Lohse2009). The presence of an LSC is also one central assumption of the widely accepted GL model (Grossmann & Lohse Reference Grossmann and Lohse2000). Therefore, in order to make a meaningful comparison with previous studies and model predictions, we henceforth focus on the flow properties (i.e. 
$Re$, 
$Nu$ and temperature fluctuations) when an LSC is persistent. The 
$Pr$ effects on RB convection without an LSC will be subject to more systematic studies in the future.
Table 3. The power-law fitting results for 
$Re = C_{5}Pr^{\beta }$ and 
$Nu = C_{6}Pr^{\delta }$ at different 
$Ra$.
Before comparing our data with previous results obtained in other geometries, we follow the procedure used by Xia et al. (Reference Xia, Lam and Zhou2002) to obtain a simple 
$Re(Ra, Pr)$ relation in the range of 
$11.7\leqslant Pr\leqslant 145.7$. To be specific, the measured 
$Re$ data are first multiplied by a factor 
$f_{Re}(Pr)$ such that the obtained results 
$Re \, f_{Re}(Pr)$ at different 
$Pr$ collapse onto a single line properly (see figure 5a). Note that the factor 
$f_{Re}(Pr)$ depends on 
$Pr$ only and thus it is a constant for the dataset at each 
$Pr$. After determining the best collapse of 
$Re\, f_{Re}(Pr)$ by a standard least-squares method, we made a power-law fit to all the data points in figure 5(a) and obtained the relation 
$Re\, f_{Re}(Pr) \sim Ra^{0.58\pm 0.01}$. Then we normalize 
$Re$ with 
$Ra^{0.58}$ and average the data over all the 
$Ra$ at each 
$Pr$, with the results shown in figure 5(b). Finally, we obtain a unified description of 
$Re$ in the present geometry and parameter ranges as 
$Re \sim Ra^{0.58\pm 0.01}Pr^{-0.82\pm 0.04}$.
Figure 5. (a) Plot of 
$Re\,f_{Re}(Pr)$ as a function of 
$Ra$. Here, 
$f_{Re}(Pr)$ is a factor used to multiply the measured 
$Re$ data such that the obtained results 
$Re \, f_{Re}(Pr)$ at different 
$Pr$ collapse onto a single line. Note that the factor 
$f_{Re}(Pr)$ depends on 
$Pr$ only and thus it is a constant for the dataset at each 
$Pr$. The solid line is the best power-law fit to all the data points: 
$Re \,f_{Re}(Pr) = 0.006Ra^{0.58\pm 0.01}$. (b) Plot of 
$ReRa^{-0.58}$ as a function of 
$Pr$. The data are mean values of 
$ReRa^{-0.58}$ averaged over all the 
$Ra$ numbers for each 
$Pr$. The solid line represents a power-law fit: 
$ReRa^{-0.58} = 0.074Pr^{-0.82\pm 0.04}$.
The 
$Ra$-dependent scaling is in good agreement with the value of 
${\sim }0.6$ for 2-D RB convection with non-slip walls (Sugiyama et al. Reference Sugiyama, Calzavarini, Grossmann and Lohse2009; van der Poel et al. Reference van der Poel, Stevens and Lohse2013; Zhang et al. Reference Zhang, Zhou and Sun2017; He et al. Reference He, Fang, Gao, Huang and Bao2021), but is noticeably larger than those (varying from 0.4 to 0.5) found in the 3-D case (Ahlers et al. Reference Ahlers, Grossmann and Lohse2009). It is noteworthy that some recent numerical studies of 2-D RB convection with free-slip walls also observed a steep 
$Re$–
$Ra$ scaling close to 0.6 (Wang et al. Reference Wang, Chong, Stevens, Verzicco and Lohse2020b; Wen et al. Reference Wen, Goluskin, LeDuc, Chini and Doering2020).
For the 
$Pr$ dependence, the scaling exponent in 2-D simulation varies from 
$-0.91$ to 
$-0.81$ for 
$10^8\leqslant Ra \leqslant 10^{10}$ (van der Poel et al. Reference van der Poel, Stevens and Lohse2013; He et al. Reference He, Fang, Gao, Huang and Bao2021), which is compatible with the finding here (see also table 3 in the appendix). On the other hand, the 
$Pr$ scaling observed in the 3-D system diverges a lot, depending mainly on the characteristic velocity used and the parameter ranges explored (Lam et al. Reference Lam, Shang, Zhou and Xia2002; Breuer et al. Reference Breuer, Wessling, Schmalzl and Hansen2004; Brown, Funfschilling & Ahlers Reference Brown, Funfschilling and Ahlers2007; Silano et al. Reference Silano, Sreenivasan and Verzicco2010; Yang et al. Reference Yang, Zhu, Wang, Liu and Zhou2020b). If we consider 
$Re$ based on the directed measured LSC velocity (close to the definition of the present study), the 
$Pr$ scaling changes from 
$-0.88$ to 
$-0.95$ in a similar 
$Ra$–
$Pr$ range (Lam et al. Reference Lam, Shang, Zhou and Xia2002). Combining the comparisons above, we conclude that the 
$Re$ number in quasi-2-D RB convection is in line with that in the 2-D system.
To gain a better understanding of the present 
$Re(Ra, Pr)$ relation, it is worth pointing out that, if the characteristic flow velocity is a free-fall type, i.e. 
$V_{f} = \sqrt {\alpha g \Delta T H}$, the corresponding 
$Re$ number scales with 
$Ra$ as 
$Re_{V_f} \sim Ra^{0.5}$. For RB convection in three dimensions, while the scaling of 
$Re(Ra)$ is often found to be similar to the free-fall type, the flow is slowed down considerably by viscous drag and the scaling is in general smaller than 0.5 (Ahlers et al. Reference Ahlers, Grossmann and Lohse2009). Thus, the 
$Ra$ scaling value of 
${\sim }0.6$ observed in (quasi-)2-D RB convection suggests that there could be another mechanism affecting the flow dynamics in these geometries. Interestingly, the 
$Re$–
$Ra$ scaling predicted by the GL model at very large 
$Pr$ (say, 
$Pr \gtrsim O(10^3$)) is around 0.6 (Grossmann & Lohse Reference Grossmann and Lohse2000, Reference Grossmann and Lohse2001, Reference Grossmann and Lohse2002). One important feature of RB convection at very large 
$Pr$ is the vanishing of the toroidal component in the flow, which is also the major difference between the (quasi-)2-D and 3-D systems (Schmalzl et al. Reference Schmalzl, Breuer and Hansen2004; van der Poel et al. Reference van der Poel, Stevens and Lohse2013; Pandey et al. Reference Pandey, Verma, Chatterjee and Dutta2016). Therefore, from the viewpoint of large-scale dynamics, the present relation 
$Re \sim Ra^{0.58}$ could be related to the absence of toroidal motion in the third dimension.
On the other hand, varying 
$Pr$ number not only has an effect on the advective dynamics, but also changes the diffusive property directly. Both processes play a role in the 
$Re$–
$Pr$ dependence. To distinguish between these two effects, we normalize the measured velocity 
$U$ with the free-fall velocity 
$V_{f}$. In this way, the behaviour of 
$U/V_{f}$ mainly reflects the role of the kinematic viscosity. As shown in figure 6(a), 
$U/V_{f}$ is a decreasing function of 
$Pr$ that follows an effective power law 
$Pr^{-0.33}$ approximately. Considering that the 
$Re$ number based on 
$V_{f}$ scales with 
$Pr$ as 
$Re_{V_f} \sim Pr^{-0.5}$, this effective power law 
$Pr^{-0.33}$ represents how the diffusive dynamics contributes the 
$Re$–
$Pr$ dependence in the present 
$Pr$ range, and it is consistent with the measured relation 
$Re \sim Pr^{-0.82}$. In a previous study, it was suggested that the free-fall velocity should be multiplied by 
$1/\sqrt {Pr}$ in order to account for the diffusive dynamics in RB convection at high 
$Pr$ (Silano et al. Reference Silano, Sreenivasan and Verzicco2010). To check this, we normalize the measured velocity 
$U$ accordingly and plot the data in figure 6(b). Indeed, 
$(U/V_{f})Pr^{0.5}$ first increases with 
$Pr$ and then seems to be saturated when 
$Pr \gtrsim 50$, which is in agreement with the numerical results in Silano et al. (Reference Silano, Sreenivasan and Verzicco2010). This saturation provides experimental support that 
$V_{f}/\sqrt {Pr}$ is a more appropriate quantity to represent the characteristic large-scale velocity at high 
$Pr$.
Figure 6. Plots of (a) 
$U/V_{f}$ and (b) (
$U/V_{f})Pr^{0.5}$ as functions of 
$Pr$. Here, 
$U$ is the characteristic velocity as explained in § 2 and 
$V_{f}$ is the free-fall velocity. The dashed line in panel (a) shows the best-fitting exponent if a power-law fit is attempted to all the data points therein.
3.3. Heat transport
In this section, we investigate how 
$Pr$ number affects the heat transfer efficiency in quasi-2-D RB convection. As shown in figure 7(a), the 
$Nu$ for 
$Pr = 20.5$ measured in two different cells fall on top of each other, indicating that the system size has no effect on global heat transport. For the data points at each 
$Pr$, they can be fitted by similar 
$Ra$-dependent power laws, with the scaling exponents varying from 0.27 to 0.30. These exponents are consistent with the results found in quasi-2-D geometry at 
$Pr \simeq 4\text {--}7$ (Zhou et al. Reference Zhou, Liu, Li and Zhong2012; Huang & Xia Reference Huang and Xia2016). When considering the 
$Pr$ dependence (figure 7b), 
$Nu$ decreases slightly as 
$Pr$ increases, and the 
$Pr$ scaling varies only marginally from 
$-0.01$ to 
$-0.03$ in the 
$Ra$ range explored. Note that there are three data points showing obvious deviations from the main power-law trend (see also the open symbols in figure 8a), which could be attributed to the heat leakage through the bottom plate at small 
$\Delta T$ (Wei et al. Reference Wei, Ni and Xia2012; Zhou et al. Reference Zhou, Liu, Li and Zhong2012), so they are not included in the analysis here.
Figure 7. Plots of 
$Nu$ as a function of (a) 
$Ra$ for different 
$Pr$ and (b) 
$Pr$ for different 
$Ra$, as indicated by the symbols. For clarity, the data in panel (a) have been shifted by multiplication by the different constant factors shown for different 
$Pr$. The solid lines represent power-law fits to individual datasets with the fitting results shown in tables 2 and 3.
Figure 8. (a) Plot of 
$Nu\,f_{Nu}(Pr)$ as a function of 
$Ra$. Here, 
$f_{Nu}(Pr)$ is a factor used to multiply the measured 
$Nu$ data such that the obtained results 
$Nu\,f_{Nu}(Pr)$ at different 
$Pr$ collapse onto a single line. Note that the factor 
$f_{Nu}(Pr)$ depends on 
$Pr$ only and thus it is a constant for the dataset at each 
$Pr$. The solid line is the best power-law fit to all the data points except for the open symbols: 
$Nu\,f_{Nu}(Pr)$ = 
$0.16Ra^{0.289\pm 0.002}$. (b) Plot of 
$Nu$ compensated by 
$Ra^{0.289}$ as a function of 
$Pr$. The data are the mean values of 
$Nu Ra^{-0.289}$ averaged over different 
$Ra$ at each 
$Pr$. The solid line is a power-law fit: 
$NuRa^{-0.289} = 0.17Pr^{-0.02\pm 0.01}$.
The similar scaling exponents allow us to extract a simple 
$Nu(Ra, Pr)$ relation as we did for the 
$Re$ number. Briefly, the measured 
$Nu$ data are multiplied by a factor 
$f_{Nu}(Pr)$ at each 
$Pr$, so that the obtained results 
$Nu \, f_{Nu}(Pr)$ at different 
$Pr$ collapse onto a single line and can be well described by 
$Nu\, f_{Nu}(Pr) \sim Ra^{0.289}$ (figure 8a). Then we normalize 
$Nu$ with 
$Ra^{0.289}$ and average the data over different 
$Ra$ for each 
$Pr$. Finally, by making a power-law fit to the so-obtained results (figure 8b), we obtain that 
$Nu \sim Ra^{0.289}Pr^{-0.02}$. This scaling relation is in agreement with the previous finding in the 3-D system (Xia et al. Reference Xia, Lam and Zhou2002). The GL model also predicts that 
$Nu$ is a decreasing function of 
$Pr$ in the present parameter ranges (Grossmann & Lohse Reference Grossmann and Lohse2000; Stevens et al. Reference Stevens, van der Poel, Grossmann and Lohse2013). However, the 2-D numerical data show that 
$Nu$ is independent of 
$Pr$ for 
$Pr \gtrsim 10$ (Schmalzl et al. Reference Schmalzl, Breuer and Hansen2004; van der Poel et al. Reference van der Poel, Stevens and Lohse2013; He et al. Reference He, Fang, Gao, Huang and Bao2021). In this context, we would conclude that the scaling relation of heat transport in quasi-2-D RB convection is compatible with the behaviour in 3-D.
In addition to scaling exponent, it is also interesting to check how the magnitude of 
$Nu$ in the quasi-2-D RB system compares to the prediction of the GL model. In the past two decades, the GL model has achieved great success in describing heat transport behaviour in 3-D RB convection (Grossmann & Lohse Reference Grossmann and Lohse2000; Ahlers et al. Reference Ahlers, Grossmann and Lohse2009; Stevens et al. Reference Stevens, van der Poel, Grossmann and Lohse2013). Recently, it has been further shown that the GL prediction is able to capture the 
$Nu$ data in 2-D simulations, only differing by a constant factor (van der Poel et al. Reference van der Poel, Stevens and Lohse2013; He et al. Reference He, Fang, Gao, Huang and Bao2021). From figure 9(a), it is seen that, once the GL predicting curves are shifted down by certain constant factors, most of the data obtained in the present quasi-2-D system can be described by the GL model as well, but the shift factors (ranging from 0.93 to 0.98) are larger than those values (
${\sim }0.8$) in the 2-D case.
Figure 9. (a) The measured 
$Nu$ compared with the values predicted by the updated GL model (Stevens et al. Reference Stevens, van der Poel, Grossmann and Lohse2013). The dashed lines indicate the updated GL model predictions multiplied by 0.93, 0.93, 0.94, 0.94, 0.95, 0.96, 0.96 and 0.98 from highest to lowest 
$Ra$, respectively. (b) The relative difference in 
$Nu$ between the present study and the GL model prediction, i.e. 
$(Nu - Nu_{GL})/Nu_{GL}$.
To compare our experimental data with the GL model prediction directly, we show their relative difference in figure 9(b). It is seen that the measured 
$Nu$ are systematically smaller than the GL predicted values, except for several data points. However, as these deviations are relatively small, it is unclear whether these differences could suggest that heat transfer in quasi-2-D RB convection is weaker than that in the 3-D situation. More studies and direct evidence are needed. Finally, it is noteworthy that, even after proper shifting, the GL prediction does not capture well most of the data at 
$Pr = 145.7$. A similar situation was found in the 3-D system (Xia et al. Reference Xia, Lam and Zhou2002), which was attributed to the uncertainty in the experimental data at high 
$Pr$ that are used to determine the GL model parameters (Stevens et al. Reference Stevens, van der Poel, Grossmann and Lohse2013). In view of this discrepancy, more experimental measurements at high 
$Pr$ are worthwhile.
3.4. Local temperature fluctuations
As revealed by the shadowgraph images, 
$Pr$ number has a strong impact on the spatial distribution of thermal plumes. This change would consequently affect local temperature fluctuations in the flow. To investigate this effect quantitatively, we measured the temperature fluctuations in two representative regions of the convection cell, i.e. the central region where the turbulent background is dominant and the region near the sidewall with abundant amount of thermal plumes. Since the flow dynamics in these two regions are different, the corresponding flow properties should scale with different functions of 
$Ra$ and 
$Pr$.
Figure 10 shows the probability density functions (p.d.f.s) of temperature fluctuation measured in the central region at two 
$Ra$ numbers. It is seen that all the p.d.f.s exhibit a stretched exponential shape and have no systematic variation with respect to different parameters. However, their root-mean-square (r.m.s.) values are sensitive to the changes in 
$Ra$ and 
$Pr$. One can see in figure 11(a) that the normalized temperature fluctuations 
$\sigma _c/\Delta T$ for all 
$Pr$ decrease with a similar trend as 
$Ra$ increases, but there is no clear 
$Pr$ dependence. By taking a closer look at the seemingly messed data, it is found that the two datasets at 
$Pr = 20.5$ differ from each other unexpectedly. The results measured in the large cell with 
$H = 25.2\ \textrm {cm}$ (red open circles) are consistently larger than those in the small cell with 
$H = 12.6\ \textrm {cm}$ (purple open circles), despite the two convection cells having identical geometrical features (i.e. aspect ratios). Thus, in contrast to global properties, our data indicate that local temperature fluctuation can be affected significantly by the system size.
Figure 10. The p.d.f.s of temperature fluctuation measured in the central region of the convection cell for different 
$Pr$: (a) 
$Ra\simeq 3.6\times 10^{9}$ and (b) 
$Ra\simeq 2.1\times 10^{10}$.
Figure 11. (a) Normalized temperature fluctuation 
$\sigma _{c}/\Delta T$ at the cell centre as a function of 
$Ra$ for different 
$Pr$. (b) Plot of 
$\sigma _{c}/\Delta T$ for 
$Pr = 20.5$ measured at two convection cells. See text for detailed explanation. (c) Plot of 
$(\sigma _{c}/\Delta T)^{\ast }$ versus 
$Ra$ as indicated by the same symbols as used in panel (a). Here, 
$(\sigma _{c}/\Delta T)^{\ast } = (\sigma _{c}/\Delta T)/(H/12.6)^{0.4}$ represents temperature fluctuation with the system size effect compensated. The solid lines are power-law fits to individual datasets with the fitting results listed in table 2. (d) Plot of 
$(\sigma _{c}/\Delta T)^{\ast }$ versus 
$Pr$ for different 
$Ra$. The two dashed lines represent the predictions for interior temperature fluctuation by two different models. See text for explanation.
Similar size-dependent behaviours have been observed in previous experimental studies of other local properties in turbulent RB convection (Ni, Huang & Xia Reference Ni, Huang and Xia2011a; Huang et al. Reference Huang, Wang, Xi and Xia2015), but the physical origin remains elusive. In order to examine the effect of 
$Pr$ number on local temperature fluctuations, we first use an empirical method to find out the size-dependent relation in the present study. To be specific, we shift down the data measured in the large cell by a constant factor, so that the shifted data (solid red circles in figure 11b) can collapse with the results in the small cell in the overlapped 
$Ra$ range. Inspired by the study of Ni et al. (Reference Ni, Huang and Xia2011a), the shift factor used has an 
$H$-dependent power-law form, which is determined by the best collapse of the data and turns out to be 
$(25.2/12.6)^{0.4}$. Following this relation, we rescale all the data by 
$(H/12.6)^{0.4}$, i.e. 
$(\sigma _{c}/\Delta T)^{\ast } = (\sigma _{c}/\Delta T)/(H/12.6)^{0.4}$, and plot them in figure 11(c). Note that this rescaling does not change the 
$Ra$ dependence, but it might have an effect on the outcome of the 
$Pr$-dependent behaviour. After the rescaling, it is seen that the temperature fluctuation in the central region decreases with increasing 
$Pr$. This is consistent with the picture that both the number of thermal plumes and the flow strength are reduced in this region as 
$Pr$ increases. This consistency qualitatively supports the validity of the size-dependent relation acquired here.
For 
$(\sigma _{c}/\Delta T)^{\ast }$ at each 
$Pr$, they can be well fitted by similar power laws with the scaling exponents being around 
$-0.30$ (see table 2 in the appendix for detailed values). These exponents are close to the finding (
$-0.35$) obtained in a similar convection cell for 
$Pr = 4.3$ (Huang & Xia Reference Huang and Xia2016), but are very different from those values (
$-0.10$ to 
$-0.17$) found in the 3-D system (see e.g. Castaing et al. Reference Castaing, Gunaratne, Heslot, Kadanoff, Libchaber, Thomae, Wu, Zaleski and Zanetti1989; Daya & Ecke Reference Daya and Ecke2002; Wei et al. Reference Wei, Ni and Xia2012; Xie et al. Reference Xie, Cheng, Hu and Xia2019) and the predictions (
$-0.11$ to 
$-0.16$) of various existing models (Kraichnan Reference Kraichnan1962; Castaing et al. Reference Castaing, Gunaratne, Heslot, Kadanoff, Libchaber, Thomae, Wu, Zaleski and Zanetti1989; Grossmann & Lohse Reference Grossmann and Lohse2004; Wunsch & Kerstein Reference Wunsch and Kerstein2005). Moreover, the approximate universal 
$Ra$ dependence in the present geometry is in contrast to previous measurements in a cylindrical convection cell (Daya & Ecke Reference Daya and Ecke2002). It was shown there that the 
$Ra$ scaling exponent of the interior temperature fluctuation is a strong function of 
$Pr$. These discrepancies could be manifestations of different large-scale dynamics in different geometries (Daya & Ecke Reference Daya and Ecke2001), which have not been taken into account with existing models.
On the other hand, 
$(\sigma _{c}/\Delta T)^{\ast }$ shown in figure 11(d) precludes a simple scaling with respect to 
$Pr$; instead, they appears to be a cross-over of two different regimes that could be described by two different models. Notwithstanding the data scatter, 
$(\sigma _{c}/\Delta T)^{\ast }$ for 
$Pr \lesssim 50$ at most 
$Ra$ cases follow the trend of a 
$Pr^{-0.24}$ power law, which is the prediction for turbulence-dominant temperature fluctuation by Grossmann & Lohse (Reference Grossmann and Lohse2004). As 
$Pr$ becomes larger, the data seem to agree with the prediction of the mixing zone and sheared boundary layer theories (Castaing et al. Reference Castaing, Gunaratne, Heslot, Kadanoff, Libchaber, Thomae, Wu, Zaleski and Zanetti1989; Shraiman & Siggia Reference Shraiman and Siggia1990; Daya & Ecke Reference Daya and Ecke2002). Although we have no explanation for why 
$(\sigma _{c}/\Delta T)^{\ast }$ at larger 
$Pr$ is in line with the mixing zone and sheared boundary layer theories, the cross-over may be understood with the evolution of the turbulent features in the central region. While the flow in this region at small 
$Pr$ is strongly violent and dominated by turbulent fluctuation as assumed by Grossmann & Lohse (Reference Grossmann and Lohse2004), the bulk fluid becomes relatively quiet as 
$Pr$ increases, and thus the assumption of the GL model is not satisfied in the high-
$Pr$ regime. The agreement between experimental data and model predictions confirms that the empirical relation acquired above is adequate for the first-order approximation.
When it comes to the region near the sidewall where a large amount of thermal plume pass through constantly, the behaviour of the temperature fluctuation becomes a bit more complex. It is seen in figure 12 that their p.d.f.s are not symmetric any more. This asymmetric shape can be understood with the flow pattern as seen in the shadowgraph supplementary movies. Firstly, the left skewed tail is mainly contributed by the cold downwelling plumes dominant in this region, which is similar to the 3-D system (Wei et al. Reference Wei, Ni and Xia2012). For the minor peak in the right tail, it could be explained by the observation that some hot plumes detached from the corner are so strong that they reach the measured region from time to time. Besides the difference in shape, the p.d.f.s for different 
$Pr$ could not collapse as well as that in the central region. To quantify these changes, we plot the normalized temperature fluctuation near the sidewall region 
$\sigma _s/\Delta T$ in figure 13(a). It is found that the temperature fluctuation in this plume-abundant region also depends on the system size, but the data obtained in the large cell have lower values than in the small cell, which is contrary to the dependence in the central region. Following the same procedure as used for 
$\sigma _c/\Delta T$, we find that the size-dependent relation for 
$\sigma _s/\Delta T$ is 
$(H/12.6)^{-0.55}$. The different size-dependent relations for different regions imply that, indeed, a local flow property should be used for the normalization.
Figure 12. The p.d.f.s of temperature fluctuation measured near the sidewall of the convection cell for different 
$Pr$: (a) 
$Ra\simeq 3.6\times 10^{9}$ and (b) 
$Ra\simeq 2.1\times 10^{10}$.
Figure 13. (a) Normalized temperature fluctuation 
$\sigma _{s}/\Delta T$ near the sidewall region as a function of 
$Ra$ for different 
$Pr$. (b) Plot of 
$\sigma _{s}/\Delta T$ versus 
$Ra$ for 
$Pr = 20.5$ measured at two convection cells. The symbols share the same meanings as in figure 11(b). (c) Plot of 
$(\sigma _{s}/\Delta T)^{\ast }$ versus 
$Ra$ for different 
$Pr$ as indicated by the same symbols as used in panel (a). Here, 
$(\sigma _{s}/\Delta T)^{\ast } = (\sigma _{s}/\Delta T)/(H/12.6)^{-0.55}$ represents temperature fluctuation with the system size effect compensated. The solid lines are power-law fits with the fitting results shown in table 2. (d) Plot of 
$(\sigma _{s}/\Delta T)^{\ast }$ versus 
$Pr$ for different 
$Ra$. The dashed line represents the prediction by Grossmann & Lohse (Reference Grossmann and Lohse2004) for temperature fluctuation close to the sidewall.
Figures 13(c) and 13(d) show the temperature fluctuation with the system size effect compensated, i.e. 
$(\sigma _{s}/\Delta T)^{\ast } = (\sigma _{s}/\Delta T)/(H/12.6)^{-0.55}$, as functions of 
$Ra$ and 
$Pr$, respectively. It is obvious that a single scaling is insufficient to describe all the data appropriately. Nevertheless, the data below 
$Ra \simeq 10^{10}$ could be well fitted by similar power laws, with the 
$Ra$-dependent exponents ranging around 
$-0.18$ (see table 2 in the appendix for fitting results). There are very few studies of temperature fluctuation near the sidewall in the literature. The limited results were all obtained in cylindrical convection cells and the 
$Ra$-dependent scalings differ a lot from each other, ranging from 
$-0.15$ to 
$-0.24$ (Shang, Tong & Xia Reference Shang, Tong and Xia2008; Lakkaraju et al. Reference Lakkaraju, Stevens, Verzicco, Grossmann, Prosperetti, Sun and Lohse2012; Wei et al. Reference Wei, Ni and Xia2012). Given the large discrepancy, we cannot conclude that the present 
$Ra$-dependent scaling in quasi-2-D geometry is consistent with the 3-D results, but we cannot discount it either. For the 
$Pr$ dependence, it is seen in figure 13(d) that 
$(\sigma _{s}/\Delta T)^{\ast }$ can also be divided into two regimes at 
$Pr \simeq 50$: while the data for 
$Pr \lesssim 50$ is a weakly dependent function of 
$Pr$, they increase significantly when 
$Pr$ becomes larger. By considering the role of thermal plumes explicitly, Grossmann & Lohse (Reference Grossmann and Lohse2004) predict that the temperature fluctuation close to the sidewall scales with 
$Pr$ as 
$Pr^{0.01}$, which is compatible with the present finding. This consistency suggests that the size-dependent relation 
$(H/12.6)^{-0.55}$ used for 
$\sigma _s/\Delta T$ is also valid, at least for 
$Pr \lesssim 50$.
4. Conclusion and outlook
In summary, we have made a detailed experimental study of 
$Pr$ effects in quasi-2-D RB convection, spanning the 
$Pr$ range of 
$11.7 \leqslant Pr \leqslant 650.7$ and the 
$Ra$ range of 
$6.0\times 10^8\leqslant Ra\leqslant 3.0\times 10^{10}$. To achieve these parameter ranges, seven kinds of silicone oil with different viscosities and four rectangular convection cells with different sizes were used in the experiments. The major findings of the present study are as below.
(i) The flow patterns visualized by the shadowgraph technique show that there always exists a well-defined LSC for 
$11.7\leqslant Pr\leqslant 145.7$. As 
$Pr$ increases in this range, the number of thermal plumes that pass through the central region of the convection cell continuously decreases, and the LSC is more confined along the periphery of the system. With 
$Pr$ further increasing to 345.2 and 650.7, the coherent LSC disappears, which is replaced by slender thermal plumes that move up and down in random locations of the entire cell.
(ii) The breakdown of the LSC results in a regime transition in the 
$Re$ number, manifested by the sharp drop in magnitude and the steep change in 
$Ra$–
$Pr$ dependence. For the 
$Pr$ range with a persistent LSC, the 
$Re$ data can be well represented by a simple power-law relation as 
$Re \sim Ra^{0.58}Pr^{-0.82}$. This result is in line with the behaviour in 2-D RB convection. On the other hand, the measured 
$Nu(Ra, Pr)$ relation is found to be 
$Nu \sim Ra^{0.289}Pr^{-0.02}$ in the 
$11.7\leqslant Pr\leqslant 145.7$ range, which is compatible with the finding in the 3-D system.
(iii) The changes in flow dynamics and plume distribution are also reflected by local temperature fluctuations. It is found that the temperature fluctuation in the cell centre decreases monotonically as 
$Pr$ increases, but that in the region near the sidewall first depends on 
$Pr$ weakly and then increases significantly with 
$Pr$. In contrast to the global properties, the 
$Ra$–
$Pr$ dependence of local temperature fluctuation could not be described by a simple unified relation. Their 
$Ra$-dependent scalings could not be accounted for with existing models, but the 
$Pr$ dependences for 
$Pr \lesssim 50$ are consistent with predictions by Grossmann & Lohse (Reference Grossmann and Lohse2004).
Given the scarce experimental studies of 
$Pr$ effects on RB convection, these results undoubtedly enrich our knowledge of the RB system in quasi-2-D geometry, and the collection of data is thorough enough for model testing/development. Finally, we would like to end this paper by remarking on two preliminary findings that deserve more systematic studies in the future. This first one is the regime transition observed at high 
$Pr$ numbers, which is similar to that in severely confined RB convection. It is interesting to see how this transition occurs and how 
$Pr$ number influences the RB system without an LSC. To answer these questions, it requires physically machining more and much larger apparatus from the experimental side, so this topic is beyond the scope of the present study and would be more suitable for numerical simulations. The second one is the effect of system size on local flow properties. Although we have acquired the size-dependent relations for the present data empirically, and their validity was supported by the consistency between our data and the flow features as well as the model predictions, the physical origin of size-dependent behaviours remains unsolved. As many engineering applications are based on scaled-down experiments, this non-trivial phenomenon should receive more attention in future studies.
Supplementary movies
Supplementary movies are available at https://doi.org/10.1017/jfm.2021.21.
Acknowledgements
We thank J.-C. He for his help in this study. We are also grateful to J. Aurnou and Y.-C. Xie for their helpful discussions.
Funding
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11702128, 11961160719 and 91752201) and the Department of Science and Technology of Guangdong Province (Grant No. 2019B21203001).
Appendix
Tables 2–4 summarize all the experimental parameters (
$Pr, Ra$) and the measured results (
$Re, Nu$), as well as the power-law fitting results for these data as shown in the main text.
Table 4. Experimental parameters (
$Pr, Ra$) and the measured results (
$Re, Nu$) for all the measurements in the present study. The positions of the two thermistors 
$T_2$ and 
$T_3$, i.e. their distance away from the sidewall relative to the cell's length 
$s/L$, are also provided.
