3. Results and discussion

For the first step, we plot the variation of mean temperature $\overline {T}$ with $z$ for six $Ra$ in figure 2, where $z$ represents the distance from the bottom plate. As the temperature probe moves away from the bottom plate, the mean temperature $\overline {T}$ drops sharply over a short distance. However, $\overline {T}$ does not decrease to a level near the centre temperature; instead, it continues to drop below the centre temperature before gradually rising back to the centre temperature as it approaches the cell centre. By examining the point at which the temperature is at its lowest, we observe that it falls within the range of $z/H=0.1$ and $z/H=0.3$ , shifting closer to the bottom plate as $Ra$ increases. This inverse temperature gradient is similar to those reported in previous studies (Chu & Goldstein Reference Chu and Goldstein1973; Segawa et al. Reference Segawa, Naert and Sano1998; Wang & Xia Reference Wang and Xia2003; Wan et al. Reference Wan, Wei, Verzicco, Lohse, Ahlers and Stevens2019).

Figure 2.

(a–f) The measured temperature profiles along the centreline in the convection cell under six different $Ra$ . Here $z$ is normalised by the height of the cell $H$ and $\overline {T}$ represents the mean temperature at each measured location.

The temperature profile is further non-dimensionalised and compared against theoretical predictions. Figure 3(a) shows the temperature distribution at six different Ra values, where the temperature distribution is represented by $\theta$ , with $\theta =[T_{\textit{bot}}-\overline {T}(z)]/ (T_{\textit{bot}}-T_{\textit{bulk}})$ and $T_{\textit{bulk}}=(T_{\textit{bot}}+T_{\textit{top}})/2$ . The non-dimensional temperature profile $\theta$ also exhibits an inverse temperature gradient. Our experimental data show good agreement with the rising segment of the profile reported by Shishkina et al. (Reference Shishkina, Horn, Emran and Ching2017), while the theoretical prediction does not capture the inverse gradient but instead approaches $T_{\textit{bulk}}$ monotonically. To reveal temperature fluctuations at different positions and better understand the characteristics of the boundary layer, we plot the profiles of root-mean-square (RMS) temperature $\sigma _T$ for six $Ra$ values in figure 3(b). It can be seen that there is a peak value of $\sigma _T/\Delta T$ near the bottom wall. The distance between the hot wall and the position of the peak is generally defined as the thickness of the thermal boundary layer (Belmonte, Tilgner & Libchaber Reference Belmonte, Tilgner and Libchaber1994; Wang & Xia Reference Wang and Xia2003; Zhou & Xia Reference Zhou and Xia2013). It is surprising that the temperature fluctuation along the $z$ direction in figure 3(b) does not continue to decrease gradually after a peak, but rather increases gradually to form a second peak and then decreases to an almost constant value. This feature has not been observed in previous studies and is explored in detail in the following discussions. To verify the reliability of the double-peak feature observed in the $\sigma _T/\Delta T$ profiles, we also conduct experiments using water in the same set-up. No such double-peak behaviour is observed in the water experiments. Figures 4(a) and 4(b) present a representative condition for liquid metal and water under the same temperature difference of 9.98 K, respectively. In water, the temperature sharply decreases away from the bottom plate and then stabilises near the bulk temperature of the cell, with no inverse temperature gradient observed. Meanwhile, $\sigma _T / \Delta T$ exhibits a single peak, with the maximum located near the thermal boundary layer. In the same convection cell, the temperature distribution characteristics of liquid metal and water show significant differences. This suggests that the occurrence of the inverse temperature gradient and double-peak feature may be attributed to the extremely low $\textit{Pr}$ of GaInSn ( $\textit{Pr} = 0.029$ ), which is two orders of magnitude lower than that of water.

Figure 3.

(a) The profiles of the normalised mean temperature $\theta$ under six different $Ra$ . The red solid line represents the result calculated using equation (43) from Shishkina et al. (Reference Shishkina, Horn, Emran and Ching2017), with the coefficients corresponding to the case of $\textit{Pr} = 0.032$ and $Ra = 10^7$ . (b) The profiles of the normalised RMS temperature $\sigma _T/\Delta T$ under the six different $Ra$ . Here $z$ is normalised by the height of the cell $H$ .

Figure 4.

Comparison of normalised temperature $\theta$ and RMS temperature $\sigma _T/\Delta T$ between liquid metal and water under the same temperature difference of 9.98 K. (a) Liquid metal at $Ra = 7.20 \times 10^6$ and (b) water at $Ra = 7.56 \times 10^8$ . The inset shows a logarithmic plot in the range $0 \leqslant z/H \leqslant 0.3$ .

The commonly accepted explanation for the inverse temperature gradient is the formation of a stable cold-fluid cluster near the centreline, caused by cold fluid descending from the top plate and accumulating in the lower half of the cell (Chu & Goldstein Reference Chu and Goldstein1973; Wan et al. Reference Wan, Wei, Verzicco, Lohse, Ahlers and Stevens2019). Building on the flow structure analysis of Sugiyama et al. (Reference Sugiyama, Ni, Stevens, Chan, Zhou, Xi, Sun, Grossmann, Xia and Lohse2010) in a quasi-two-dimensional cell, it was shown that for low- $\textit{Pr}$ fluids ( $\textit{Pr} \lt 0.8$ ), the large-scale circulation tends to remain in a single-roll state without undergoing reversals. In their supplementary materials, temperature and velocity fields for $\textit{Pr}=0.7$ illustrate a persistent band of cold fluid extending through the lower half of the cell. By contrast, for water, where reversals occur more often, the flow structures become more complex during these events, leading to strong mixing between hot and cold fluid and preventing the long-lived accumulation of either. This difference in flow organisation is consistent with our experimental observation that the inverse temperature gradient emerges in liquid metals but not in water.

We further calculated the thermal boundary-layer thickness using two different methods: the slope method yields $\delta _{\textit{th}}$ , while the position of the first peak in the RMS temperature profile gives $\delta _{\sigma }$ . These results, along with comparisons with previous studies (Naert, Segawa & Sano Reference Naert, Segawa and Sano1997; Scheel, Kim & White Reference Scheel, Kim and White2012; Pandey Reference Pandey2021; Schindler et al. Reference Schindler, Kim, Vogt and Eckert2025), are shown in figure 5. Both the thermal boundary-layer thickness $\delta _{\textit{th}}$ , $\delta _{\sigma }$ and the location of the second peak decrease with increasing $Ra$ , with the scaling of $\delta _{\textit{th}}/H \approx (2.5642 \pm 0.0154)Ra^{-0.2776 \,\pm\, 0.0331}$ , $\delta _{\sigma }/H \approx (2.6812 \pm 0.0231)Ra^{-0.2617 \,\pm\, 0.0213}$ and second peak $/H\approx (3.5564 \pm 0.0443)Ra^{-0.2168 \,\pm\, 0.0186}$ . Our measured boundary-layer thickness shows the closest agreement with the results reported by Scheel et al. (Reference Scheel, Kim and White2012). We also performed detailed measurements of the heat transfer efficiency $\textit{Nu}$ . Figure 6 shows our measured $\textit{Nu}$ along with data from previous studies (Takeshita et al. Reference Takeshita, Segawa, Glazier and Sano1996; Cioni, Ciliberto & Sommeria Reference Cioni, Ciliberto and Sommeria1997; Glazier et al. Reference Glazier, Segawa, Naert and Sano1999; Stevens et al. Reference Stevens, van der Poel, Grossmann and Lohse2013; King & Aurnou Reference King and Aurnou2015; Zürner et al. Reference Zürner, Schindler, Vogt, Eckert and Schumacher2019). It can be seen that our results show fairly good agreement with the experimental data from Glazier et al. (Reference Glazier, Segawa, Naert and Sano1999). The scaling relation we obtained is $\textit{Nu} \approx (0.1589 \pm 0.0151)Ra^{0.2622 \,\pm\, 0.0061}$ , which satisfies the theoretical prediction of $\textit{Nu} \sim Ra^{1/4}$ in regime $I_l$ proposed by Grossmann & Lohse (Reference Grossmann and Lohse2000, Reference Grossmann and Lohse2001). The scaling behaviour of $\textit{Nu}$ is found to be similar to that of $1/\delta _{\sigma }$ and $1/\delta _{\textit{th}}$ . Since the thermal boundary layer limits the global heat transport in the cell, the heat transfer efficiency of the system may be expected to exhibit a similar $Ra$ dependence to that of the boundary-layer thickness. Figure 6 also presents the $\textit{Nu}$ values estimated from the boundary-layer thicknesses, expressed as $H/(2\delta _{\sigma })$ and $H/(2\delta _{\textit{th}})$ . These estimates are higher than the directly measured $\textit{Nu}$ values. In early experiments using air, Belmonte et al. (Reference Belmonte, Tilgner and Libchaber1994) found that the scaling and magnitude of $H/(2\delta _{\textit{th}})$ for $Ra$ are similar to those of the global $\textit{Nu}$ . However, Lui & Xia (Reference Lui and Xia1998) reported in measurements using water that the thermal boundary-layer thickness measured at the plate centre is insufficient to quantify the global $\textit{Nu}$ of the cell. Furthermore, in the water measurements by Zhou & Xia (Reference Zhou and Xia2013), significant discrepancies are observed between $H/(2\delta _{\textit{th}})$ , $H/(2\delta _{\sigma })$ and $\textit{Nu}$ . Within the range $2.6 \times 10^{10} \lt Ra \lt 6.4 \times 10^{11}$ , the average values of $H/(2\delta _{\textit{th}})$ , $H/(2\delta _{\sigma })$ and $\textit{Nu}$ are 416.2, 619.9 and 350.2, respectively.

Figure 5.

The measured thermal boundary-layer thickness $\delta _{\sigma }$ , $\delta _{\textit{th}}$ and the location of the second peak. Our data are compared with those of previous work, as indicated in the legend. The solid lines represent fitted results. The red solid line corresponds to $\delta_{(\sigma)} /H \approx (2.6812 \pm 0.0231)Ra^{-0.2617 \,\pm\, 0.0213}$ , the blue solid line to $\delta _{\textit{th}} /H \approx (2.5642 \pm 0.0154)Ra^{-0.2776 \,\pm\, 0.0331}$ and the black solid line to the dimensionless second peak location by $H$ $\approx (3.5564 \pm 0.0443)Ra^{-0.2168 \,\pm\, 0.0186}$ . The inset shows a magnified view of $\delta _{\textit{th}}$ and $\delta _{\sigma }$ .

Figure 6.

Heat transfer in liquid metal convection. The scaling of $\textit{Nu}$ versus $Ra$ is shown. Our data are compared with those of previous work, as indicated in the legend. The dashed line represents fitted results.

For a better understanding of the second peak, we extract the temperature information that changed with time at typical locations as shown in figure 7(a). Figure 7(b) presents the temperature time series at a position within the thermal boundary layer, and the dashed line indicates the mean temperature of the bottom plate, which is $38.18\,^{\circ }\rm C$ . The corresponding probability density function (PDF) is shown in figure 7(c). The red line and vertical red dashed line in the PDF indicate the Gaussian distribution and mean temperature, respectively. The PDF at this position shows that the temperature distribution is approximately Gaussian, which is consistent with the temperature characteristics of water, where the temperature distribution within the boundary layer exhibits a Gaussian distribution (Zhou & Xia Reference Zhou and Xia2013). Figure 7(d) shows the temperature time series near the thermal boundary layer. The corresponding PDF is shown in figure 7(e). Different from the temperature characteristics in the boundary layer, its PDF deviates from a Gaussian distribution with an elevated tail in the range above the mean temperature, which indicates that there is a higher probability of temperatures above the mean temperature. This is believed to be caused by hot plume bursts (Zhou & Xia Reference Zhou and Xia2002; Xi et al. Reference Xi, Zhou, Zhou, Chan and Xia2009). As the temperature probe gradually moves away from the thermal boundary layer, figure 7( f–m) shows that the number of temperature spikes decreases, and they almost disappear near the cell centre. Position (3) in figure 7(a) is between the first and the second peak. The temperature time series and its PDF are shown in figures 7( f) and 7(g). The PDF shows that the temperature recovers a Gaussian distribution. When the temperature probe reaches the second peak position, namely position (4), the temperature PDF in figure 7(i) reveals a peak in the range below the mean temperature, with a probability slightly higher than that of a Gaussian distribution, which indicates a higher probability of detecting fluid with temperatures below the mean at this location. When the probe reaches position (5), figure 7( j) shows that the temperature bursts rarely occur. However, a small number of plumes still pass through the probe, which makes the temperature deviate from the Gaussian distribution. After reaching the centre of the cell, which is location (6), the temperature shows a Gaussian distribution, indicating that the plume is not detected in the cell centre. In water, the temperature at the cell centre also exhibits a Gaussian distribution (Zhou, Sun & Xia Reference Zhou, Sun and Xia2007). In summary, our experimental measurements indicate that there is a certain region where the probability of observing temperatures below the mean is relatively higher.

Figure 7.

The time series of temperature at different heights. (a) Profile of the mean temperature $\overline {T}$ and the degree of temperature fluctuation $\sigma _T/\Delta T$ measured at $Ra=4.54\times 10^6$ . The vertical red dashed lines indicate the six selected positions (1)–(6) from which the data in (b–m) are derived. (b) The temperature $T$ and (c) its PDF for $z=2.22$ mm $(0.49\delta _{\textit{th}})$ . The dashed line in (b) is the measured mean temperature of the bottom plate. (d,e) For $z=4.80$ mm $(1.08\delta _{\textit{th}})$ ; (f,g) for $z=9.45$ mm $(2.13\delta _{\textit{th}})$ ; (h,i) for $z=18.33$ mm $(4.12\delta _{\textit{th}})$ ; ( j,k) for $z=31.58$ mm $(7.11\delta _{\textit{th}})$ ; (l,m) for $z=63.00$ mm (cell centre).

We further analyse the higher-order characteristics of the temperature distribution, specifically the skewness and flatness. The skewness and flatness are defined as $T_S=\langle (T-\langle T\rangle ^3)\rangle /{\sigma _T}^3$ and $T_F=\langle (T-\langle T\rangle ^4)\rangle /{\sigma _T}^4$ . Based on these definitions, a symmetric temperature distribution around its mean corresponds to a skewness value near $0$ , while a temperature distribution resembling a Gaussian profile exhibits a flatness value approaching $3$ .

Figure 8 shows the skewness $T_S$ and flatness $T_F$ at different positions for the selected three $Ra$ conditions. Here, the distance $z$ is non-dimensionalised by the thermal boundary-layer thickness $\delta _{\textit{th}}$ . According to the skewness and flatness, the positions from the bottom plate to the centre of the cell can be divided into five regions. Region I is inside the boundary layer. The skewness and flatness are at relatively small values, with skewness close to $0$ and flatness close to $3$ at the beginning, indicating that the temperature distribution approximates a Gaussian distribution within the boundary layer. However, the closer to the boundary-layer location, the more $T_S$ and $T_F$ deviate from $0$ and $3$ , and the deviation is the largest for a distance beyond the boundary layer. Region II extends from the boundary layer to the vicinity of the intersection where hot fluid rises and cold fluid descends. Here $T_S$ and $T_F$ quickly reach their maximum values, and the temperature distribution’s PDF exhibits higher probability in the range above the mean temperature. Then they start to decrease, entering region III. In this region, the hot and cold fluids mix intensely as they rise and fall, leading to a more symmetric temperature distribution. In region III, $T_S$ and $T_F$ decrease to values near $0$ and $3$ , respectively, indicating a return to a Gaussian-like distribution. As the probe continues to move upward beyond region III and enters region IV, it detects descending cold fluid. Influenced by the descending cold fluid, $T_S$ and $T_F$ rise again, and when they reach their maximum values, they begin to decrease once more. Region V is located near the centre of the cell. At the centre of the cell, $T_S$ and $T_F$ decrease to values close to $0$ and $3$ , indicating that the centre is rarely affected by the ascending hot fluid or descending cold fluid and that the temperature distribution again approximates a Gaussian distribution. This phenomenon is distinctly different from the results observed in water. In water, the positions from the bottom plate to the centre of the cell can be divided into three regions based on $T_S$ and $T_F$ . The first region is within the boundary layer, where $T_S$ and $T_F$ are close to $0$ and $3$ , respectively. The second region is the central core region, where $T_S$ and $T_F$ gradually increase with increasing distance from the bottom plate, reaching their maximum values. The third region is the centre of the cell, where $T_S$ and $T_F$ gradually decrease to $0$ and $3$ , respectively (Zhou et al. Reference Zhou, Sun and Xia2008, Reference Zhou, Li, Lu and Liu2011; Zhou & Xia Reference Zhou and Xia2013).

Figure 8.

Profiles of the skewness $T_S$ (a) and the flatness $T_F$ (b) of temperature. Here the vertical distance $z$ is normalised by the thermal boundary-layer thickness $\delta _{\sigma }$ . Five different coloured regions divide the lower half of the cell into five zones: region I to region V. The six red vertical dashed lines correspond to the same positions (locations (1)–(6)) as those shown in figure 7 for $Ra=4.54\times 10^6$ .

To further determine the five regions of temperature distribution in liquid metal, we also use two vertically arranged T-type thermocouples to analyse the delay time and correlation coefficient at different positions. A schematic diagram of the probe is shown in figure 9(a). By placing two temperature probes vertically, one above the other, it is possible to determine the flow direction. If a fluid parcel moves upward, it will first pass the lower probe and then the upper one, resulting in a negative delay time. Conversely, downward motion will produce a positive delay time (Zhou & Xia Reference Zhou and Xia2001). Moreover, this probe arrangement can also be used to identify flow regions. When both probes are located within the same flow region, the temperature signals exhibit high correlation. If the probes are situated in different regions, the correlation will be reduced. The cross-correlation coefficient $R(z)=C_z(\tau _0)$ is used to characterise the strength of the correlation, where $C_z(\tau _0)$ is the maximum value of the cross-correlation function. The cross-correlation function is defined as $C_z(\tau )=\langle \delta T_1(t)\delta T_2(t+\tau )\rangle /[\langle (\delta T_1)^2\rangle \langle (\delta T_2)^2\rangle ]^{1/2}$ , where $\delta T_1(t)$ and $\delta T_2(t)$ are the temperature fluctuations at the lower and upper probes, respectively. At each position, we perform a correlation analysis between the temperature time series at two locations. The maximum value of the correlation function is taken as the correlation coefficient $R$ for that position, and the time lag corresponding to this maximum value is defined as the delay time $\tau$ .

Figure 9.

Correlation analysis of two thermocouples arranged up and down. (a) Schematic of the probe in the flow field. (b) The distribution of the delay time $\tau$ with position. (c) The distribution of the correlation coefficient $R$ with position. The region divisions and red vertical dashed lines in (b,c) are the same as those in figure 8.

Figure 10.

Plume information obtained from single-thermistor measurements. (a) Method for identifying plumes. (b) Parameters used to characterise plume properties. The distributions of (c) plume number and (d) plume strength, both normalised by their maximum values. The region divisions and red vertical dashed lines in (c,d) are the same as those in figure 9.

Figure 9(b) shows the variation of the delay time $\tau$ with a position over half the height of the cell. At the bottom plate, the delay time is distinctly negative across different $Ra$ values, gradually approaching zero as the distance from the bottom plate increases. However, unlike situations in water, where the delay time stabilises near zero and then becomes negative again, the delay time in liquid metal becomes positive after reaching near zero and continues to increase. Negative delay times are caused by the upward movement of tilted plumes, whereas positive delay times correspond to the downward movement of tilted plumes. After reaching a maximum value, the delay time rapidly decreases to around zero because the probe has reached the centre of the cell, where it is difficult to detect plume movement, and the temperature signals from the two probes have almost no time difference. The delay time in region II and region IV exhibits negative and positive values, respectively, indicating the detection of rising hot plumes and descending cold plumes in these regions. In the intermediate region III, the delay time is close to zero, suggesting that the mixing of hot and cold plumes results in a weak vertical velocity in this region. Figure 9(c) shows the corresponding correlation coefficient. When leaving the temperature boundary layer, one probe is inside the boundary layer and the other is outside, so the correlation coefficient first decreases. When both probes are located in the rising thermal plume region (region II), the correlation coefficient increases continuously and reaches its first peak as the upper probe approaches the descending cold plume region (region IV). As the upper probe enters region IV while the lower probe remains outside, the correlation coefficient gradually decreases. Once both probes are located within the descending cold plume region, the correlation begins to rise again and reaches its maximum near the centre of the cell. The spatial distributions of delay time and correlation provide additional evidence for the division of the five regions.

We further differentiate the identified regions based on the statistical characteristics of the thermal plumes. Figure 10 shows the plume information obtained from single-thermistor measurements. When a plume passes through the temperature probe, it causes a fluctuation in temperature, so we extract plume information based on these temperature fluctuations. Figure 10(a) illustrates the process of plume identification. The red dashed line represents the centre value of the temperature PDF, denoted as $T_{bg}$ . In figure 10(b), the dashed line on the horizontal axis corresponds to $T_{bg}$ . The blue dashed line in figure 10(a) represents $T_{bg}$ , where $\sigma _{bg}$ is the width of the Gaussian distribution, as shown in figure 10(b). Not all temperature fluctuations in figure 10(a) are caused by plumes, so a threshold is set here: temperature fluctuations exceeding $T_{bg}+\sigma _{bg}$ are recorded as plume fluctuations. The position of the blue triangle in figure 10(a) indicates the peak exceeding $T_{bg}+\sigma _{bg}$ , which marks the plume’s head. The subsequent temperature trough indicates the plume’s tail, while the preceding temperature trough marks the plume’s front end. The difference between the maximum and minimum temperatures of a plume represents the plume’s strength. Figures 10(c) and 10(d) respectively present the variation of the dimensionless plume number $P_n$ and plume strength $P_s$ with position, where $P_n$ and $P_s$ are normalised by the maximum value under each condition. It can be observed that both $P_n$ and $P_s$ exhibit a double-peak structure. The number of plumes reaches its first peak at the temperature boundary layer (the interface between region I and region II), but the plume strength reaches its first peak near the outside of the boundary layer (region II). The second peaks of both $P_n$ and $P_s$ are located within region IV, where prominent plume activity can be detected. In region V near the centre of the cell, both $P_n$ and $P_s$ continue to decrease and reach relatively low levels, indicating that plume activity is not significant in the central region of the cell. The analysis of plume characteristics also provides a basis for the division of the five regions.