3.1. General features of the LSC and identification of different flow modes

We first study the energy of the $n$ th Fourier mode $E_n$ and the total energy of all Fourier modes $E_t$ , which are defined as $E_n = {\delta _n}^2$ and $E_t = {\textstyle \sum _{n=1}^{4}} E_n$ , respectively. Figure 3 shows the ratio of the time-averaged energy of the $n$ th Fourier mode to the total time-averaged Fourier-mode energy, $\langle E_n \rangle /\langle E_t \rangle$ , as a function of $ \textit{Ra}$ in cells with different aspect ratios at the middle height of the sidewall. It can be seen that the first Fourier mode accounts for more than 70 % of the total Fourier-mode energy for all three aspect ratios and over the investigated $ \textit{Ra}$ range. In contrast, the second, third and fourth modes are much smaller. The first mode therefore captures the temperature variation along the sidewall and directly provides a convenient measure of the strength and orientation of the hot-upflow/cold-downflow structure. For this reason, the flow-mode classification and the following analysis are based on the first Fourier mode ( $n=1$ ). In the following, $\delta _i$ and $\theta _i$ denote the strength and orientation of the first Fourier mode at the top ( $i=t$ ), middle ( $i=m$ ) and bottom ( $i=b$ ) heights of the sidewall.

Figure 3.

Ratio of the time-averaged energy of the $n$ th Fourier mode to the time-averaged total Fourier-mode energy, $\langle E_n \rangle /\langle E_t \rangle$ , as a function of $ \textit{Ra}$ in cells with different aspect ratios at the middle height of the sidewall. Here, $\langle E_n \rangle = \langle {\delta _n}^2 \rangle$ is the time-averaged energy of the $n$ th Fourier mode, and $\langle E_t \rangle = {\textstyle \sum _{n=1}^{4}} \langle E_n \rangle$ is the time-averaged total energy of all Fourier modes.

Figure 4 shows some examples of the time series of the flow strength $\delta _i$ ( $i=t,m,b$ ) and orientation $\theta _i$ ( $i=t,m,b$ ) of the LSC measured at the top ( $i=t$ , blue solid lines), middle ( $i=m$ , black dashed lines) and bottom ( $i=b$ , red dash-dotted lines) heights of the sidewall in cells with different $\varGamma$ : ( $a$ , $b$ ) $\varGamma = 1$ , $ \textit{Ra} = 4.01 \times 10^6$ , ( $c$ , $d$ ) $\varGamma = 1/2$ , $ \textit{Ra} = 3.27 \times 10^7$ and ( $e$ , $f$ ) $\varGamma = 1/3$ , $ \textit{Ra} = 1.01 \times 10^8$ . It is seen that the strength $\delta _i$ and orientation $\theta _i$ of the LSC in a $\varGamma =1$ cell at different heights are generally aligned with each other, suggesting that the LSC is in the form of a coherent single-roll structure. In addition, they fluctuate around a preferred direction at $\theta \sim 240^{\circ }$ . A detailed examination of $\theta _i$ shown in the inset of figure 4( $b$ ) suggests that the twisting/torsional oscillation reported in water by Funfschilling & Ahlers (Reference Funfschilling and Ahlers2004) and Brown & Ahlers (Reference Brown and Ahlers2006) is also present in liquid-metal convection. It should be noted that no cessation event, i.e. a complete stop of the LSC at three heights (Brown & Ahlers Reference Brown and Ahlers2006; Xi & Xia Reference Xi and Xia2007; Xie et al. Reference Xie, Wei and Xia2013), was observed during a measurement period of 8.2 h, which corresponds to 10 120 free-fall time units, suggesting that the LSC is very robust in the $\varGamma =1$ cell.

In contrast, in cells with $\varGamma = 1/2$ and $1/3$ , although the strength $\delta _i$ of the LSC at different heights exhibits similar temporal variation, as shown in figures 4( $c$ ) and 4( $e$ ), the flow strength at the mid-height of the cell shows some discrepancy with those at the top and bottom heights. It reaches zero much more frequently in the $\varGamma =1/3$ cell. The orientations $\theta _i$ of the LSC at different heights display obvious irregular fluctuations with time or show occasional large deviations among different heights, as shown in figures 4( $d$ ) and 4( $f$ ). These observations suggest that the flow structure and dynamics have changed qualitatively in cells with $\varGamma = 1/2$ and $1/3$ when compared with the single-roll LSC in the $\varGamma =1$ cell. Notably, these characteristics persist across all $ \textit{Ra}$ studied for a given aspect ratio of the convection cell.

Figure 4.

Examples showing the time series of the flow strength $\delta _i$ and orientation $\theta _i$ of the LSC measured at the top ( $i=t$ , blue solid lines), middle ( $i=m$ , black dashed lines) and bottom ( $i=b$ , red dash-dotted lines) heights of the sidewall in cells with ( $a$ , $b$ ) $\varGamma = 1$ , $ \textit{Ra} = 4.01 \times 10^6$ , ( $c$ , $d$ ) $\varGamma = 1/2$ , $ \textit{Ra} = 3.27 \times 10^7$ and ( $e$ , $f$ ) $\varGamma = 1/3$ , $ \textit{Ra} = 1.01 \times 10^8$ .

To quantitatively determine the flow structure in cells with different $\varGamma$ and study their $ \textit{Ra}$ -dependence, we follow a method proposed by Xi & Xia (Reference Xi and Xia2008b ) and Weiss & Ahlers (Reference Weiss and Ahlers2011), where the angular difference among different heights $| \Delta \theta |_{i,j} = 180^{\circ }-|180^{\circ }-| \theta _i-\theta _j||$ is used as a proxy for the flow mode. The flow mode is classified as being in the DRM when $| \Delta \theta |_{t,b} \gt 120^{\circ }$ and in the SRM when none of $| \Delta \theta |_{i,j}, (i \ne j=t,m,b)$ exceed $60^{\circ }$ . If neither SRM nor DRM is observed, the flow is classified as a transition mode (TRM). The thresholds of $60^\circ$ and $120^\circ$ are adopted following previous studies (Xi & Xia Reference Xi and Xia2008b ; Weiss & Ahlers Reference Weiss and Ahlers2011) to facilitate direct comparison. To assess the robustness of this threshold-based classification, we repeated the analysis using three paired threshold sets: $(50^\circ ,130^\circ )$ , $(60^\circ ,120^\circ )$ and $(70^\circ ,110^\circ )$ , corresponding to deviations of $\pm 16.7\,\%$ and $\pm 8.3\,\%$ from their respective reference values of $60^\circ$ and $120^\circ$ . We find that the relative weights and $1-\textit{Nu}_D/\textit{Nu}_S$ are insensitive to these threshold variations. More importantly, the identified dominant modes and the main $ \textit{Ra}$ -dependent trends remain unchanged. These results demonstrate that our conclusions are robust with respect to the choice of the thresholds.

Figure 5( $a{,}b$ ) shows the time-averaged value of $\langle |\Delta \theta |_{t,b}\rangle$ , and its standard deviation $\sigma _{|\Delta \theta |_{t,b}}$ as functions of $ \textit{Ra}$ for cells with $\varGamma =1$ , $1/2$ and $1/3$ . In addition to the statistics based on the full time series, the data are conditioned on the SRM and DRM, allowing the characteristics of each flow state to be compared directly. In the $\varGamma =1$ cell, both $\langle |\Delta \theta |_{t,b}\rangle$ and $\sigma _{|\Delta \theta |_{t,b}}$ remain small and increase only weakly with $ \textit{Ra}$ , indicating a coherent single-roll LSC with nearly aligned top and bottom orientations. As $\varGamma$ decreases to $1/2$ and $1/3$ , the overall statistics show much larger angular differences and stronger fluctuations, suggesting an apparent loss of vertical coherence of the LSC. These results are qualitatively consistent with those of Schindler et al. (Reference Schindler, Eckert, Zürner, Schumacher and Vogt2022).

To determine whether the apparent loss of coherence originates from a genuine decorrelation within an individual flow mode or from transitions between distinct large-scale flow states, we condition the angular statistics on the SRM and DRM modes. Figure 5( $a$ ) shows that, for the $\varGamma =1/2$ and $1/3$ cells, the SRM- and DRM-conditioned angular differences vary only weakly with $ \textit{Ra}$ . The overall statistics are closer to the SRM statistics in the $\varGamma =1/2$ cell, but closer to the DRM statistics in the $\varGamma =1/3$ cell, reflecting the different relative weights of the two modes. This interpretation is further supported by the relative weight analysis presented later. In addition, figure 5( $b$ ) shows that the standard deviations within the individual SRM and DRM modes are considerably smaller than those obtained from the overall statistics. Thus, the apparent loss of coherence in the overall statistics mainly results from transitions between different flow modes rather than from a progressive loss of coherence within either the SRM or the DRM. These mode-separated statistics further highlight the dominant role of aspect ratio in selecting the preferred large-scale flow structure and its apparent coherence.

Figure 5.

( $a$ ) Mean value $\langle | \Delta \theta |_{t,b} \rangle$ and ( $b$ ) standard deviation $\sigma _{| \Delta \theta |_{t,b}}$ of the angular difference $|\Delta \theta |_{t,b}$ between the top and bottom heights as functions of $ \textit{Ra}$ in cells with $\varGamma =1$ , $1/2$ and $1/3$ for the full time series (ALL) and for the conditioned SRM and DRM modes.

Figure 5. Long description

The image contains two scatter plots labeled as Panel A and Panel B. Panel A shows the mean value of the angular difference between the top and bottom heights as a function of Ra. The x-axis represents Ra on a logarithmic scale, and the y-axis represents the mean angular difference in degrees. The data points are categorized into three groups: ALL (black squares), SRM (red circles), and DRM (blue triangles). Each group is further divided into three subsets based on different conditions: 1, 1/2, and 1/3. Panel B shows the standard deviation of the angular difference between the top and bottom heights as a function of Ra. The x-axis represents Ra on a logarithmic scale, and the y-axis represents the standard deviation in degrees. The data points are categorized similarly to Panel A. Both panels illustrate how the mean value and standard deviation of the angular difference vary with Ra in different modes.

3.2. Statistical properties of flow modes and their dependence on $\varGamma$

3.2.1. Relative weight of different flow modes

To quantitatively characterise the relative weights $w_i (i=s,d,t)$ of the different flow modes, we investigate the fractions of time the flow spends in each mode over the measurement period. The relative weights $w_s$ , $w_d$ and $w_t$ represent the fractions of time when the flow is in the SRM, DRM and TRM, respectively. Figure 6( $a$ – $c$ ) shows $w_s$ (black squares), $w_d$ (red circles) and $w_t$ (blue triangles) as functions of $ \textit{Ra}$ in the cells with ( $a$ ) $\varGamma = 1$ , ( $b$ ) $\varGamma = 1/2$ and ( $c$ ) $\varGamma = 1/3$ . In the $\varGamma =1$ cell, as shown in figure 6( $a$ ), one observes that $w_s$ is $100\,\%$ , while $w_d$ and $w_t$ are $0\,\%$ , suggesting that the SRM is the dominant flow mode and there is almost no DRM or TRM. Due to the dominating role of the SRM, the relative weights $w_i$ of the different flow modes are independent of $ \textit{Ra}$ for $\varGamma = 1$ . In the $\varGamma =1/2$ cell, as shown in figure 6( $b$ ), $w_s$ is approximately 60 %, and it is larger than $w_d$ and $w_t$ , which are close to each other with a value of ${\sim} 20\,\%$ . Meanwhile, $w_s$ increases with increasing $ \textit{Ra}$ ; in contrast, $w_d$ decreases with increasing $ \textit{Ra}$ . These observations suggest that the SRM in the $\varGamma = 1/2$ cell becomes more robust and coherent with increasing $ \textit{Ra}$ , while the DRM becomes weaker. As for the data obtained in the $\varGamma =1/3$ cell shown in figure 6( $c$ ), $w_d$ , $w_t$ and $w_s$ show almost no variation with increasing $ \textit{Ra}$ and are close to 58 %, 27 % and 15 %, respectively, suggesting that the DRM is the dominant flow mode in this configuration.

Although the relative weights of the different flow modes exhibit some variation with $ \textit{Ra}$ , their dependence on $ \textit{Ra}$ remains relatively weak over the parameter range investigated. We therefore calculated the Rayleigh-number-averaged relative weights of the SRM ( $\langle w_s \rangle$ ), the DRM ( $\langle w_d \rangle$ ) and the TRM ( $\langle w_t \rangle$ ) for different $\varGamma$ (considering all $ \textit{Ra}$ ). These results are listed in table 2 and plotted as functions of $\varGamma$ in figure 6( $d$ ). It is seen that as $\varGamma$ increases, $\langle w_s \rangle$ increases, while both $\langle w_d \rangle$ and $\langle w_t \rangle$ decrease, implying that the SRM becomes more prevalent, while the DRM and the TRM gradually diminish with increasing $\varGamma$ , consistent with Wondrak et al. (Reference Wondrak, Sieger, Mitra, Schindler, Stefani, Vogt and Eckert2023). The data in figure 6( $d$ ) suggest that in cells with $\varGamma =1$ and $\varGamma =1/2$ , the SRM is the dominant flow mode. Conversely, in the cell with $\varGamma =1/3$ , the DRM is the dominant flow mode.

Table 2.

Rayleigh-number-averaged relative weights of the SRM ( $\langle w_s \rangle$ ), DRM ( $\langle w_d \rangle$ ) and TRM ( $\langle w_t \rangle$ ) for different $\varGamma$ in the present liquid metal convection.

Table 2. Long description

A table with three columns and four rows, including a header row. The columns are labeled with Greek letters and symbols, and the rows are labeled with numerical values. The table presents the Rayleigh-number-averaged relative weights of the SRM, DRM, and TRM for different values. Row 1: Gamma, {w_s}, {w_d}, {w_t}. Row 2: 1, 100 percent, 0 percent, 0 percent. Row 3: 1/2, 63.6 percent, 17.5 percent, 18.9 percent. Row 4: 1/3, 14.5 percent, 58.1 percent, 27.4 percent.

Figure 6.

Relative weights of the SRM ( $w_s$ , black squares), the DRM ( $w_d$ , red circles) and the TRM ( $w_t$ , blue triangles) as a function of $ \textit{Ra}$ in the cells with ( $a$ ) $\varGamma =1$ , ( $b$ ) $\varGamma =1/2$ and ( $c$ ) $\varGamma =1/3$ . Lines are provided to guide the eye. ( $d$ ) Rayleigh-number-averaged relative weights of the SRM ( $\langle w_s \rangle$ ), the DRM ( $\langle w_d \rangle$ ) and the TRM ( $\langle w_t \rangle$ ) for different $\varGamma$ . The published data using water as the working fluid are obtained from Xi & Xia (Reference Xi and Xia2008b ) for $ \textit{Pr} = 4.9{-} 5.3$ and Weiss & Ahlers (Reference Weiss and Ahlers2011) for $ \textit{Pr} = 4.38$ .

Figure 6. Long description

The image contains four panels, each depicting a line graph. Panel A shows the relative weights of the SRM (black squares), the DRM (red circles), and the TRM (blue triangles) as a function of the Rayleigh number (Ra) for an aspect ratio (Γ) of 1. The x-axis represents the Rayleigh number (Ra) ranging from 10^6 to 10^7, and the y-axis represents the relative weight (w_–i) in percent. The SRM dominates with values close to 100 percent, while the DRM and TRM have significantly lower values. Panel B shows the same data for an aspect ratio of 1/2. The SRM still dominates but with a decreasing trend, while the DRM and TRM show slight variations. Panel C depicts the data for an aspect ratio of 1/3, where the SRM shows a decreasing trend, and the DRM and TRM exhibit more pronounced variations. Panel D shows the Rayleigh-number-averaged relative weights of the SRM, DRM, and TRM for different aspect ratios (Γ). The x-axis represents the aspect ratio (Γ) ranging from 0.3 to 1.0, and the y-axis represents the averaged relative weight (w_–i) in percent. The SRM shows a decreasing trend with increasing aspect ratio, while the DRM and TRM show varying trends. The published data are obtained from Xi & Xia (2008b) for Γ = 1/2 and Weiss & Ahlers (2011) for Γ = 1, using water as the working fluid.

We now compare our data with those reported by Xi & Xia (Reference Xi and Xia2008b ) for $ \textit{Pr} = 4.9{-} 5.3$ and Weiss & Ahlers (Reference Weiss and Ahlers2011) for $ \textit{Pr} = 4.38$ , using water as the working fluid, as shown in figure 6( $d$ ). Despite a difference of nearly two orders of magnitude in $ \textit{Pr}$ , the SRM remains the dominant flow mode in both liquid-metal and water convection for $\varGamma =1$ and $1/2$ , indicating that flow-mode selection at these aspect ratios is governed primarily by geometrical confinement and depends only weakly on $ \textit{Pr}$ . However, in the $\varGamma = 1/3$ cell, the system using water as the working fluid exhibits no clearly dominant mode, whereas the DRM becomes the prevailing mode in the low- $ \textit{Pr}$ liquid-metal system. A possible explanation is that the DRM in the liquid-metal convection forms a stronger flow structure that is relatively resistant to background turbulent perturbations. However, the weaker flow structures in water are more readily disrupted, leading to frequent switching among modes and hence no clear modal dominance. This contrast suggests that the $ \textit{Pr}$ dependence of flow-mode selection may become more evident under stronger confinement, although differences in experimental boundary conditions and apparatus from the present study and those from Xi & Xia (Reference Xi and Xia2008b ) may also contribute.

3.2.2. Time-averaged strength for different flow modes

Figure 7 plots the normalised time-averaged strength $\langle \delta _i \rangle /\Delta T$ at the top ( $i=t$ , blue squares), middle ( $i=m$ , black circles) and bottom ( $i=b$ , red triangles) heights of the sidewall as functions of $ \textit{Ra}$ in cells with different $\varGamma$ . Here, the entire time series for $\delta _i/\Delta T$ for a given $ \textit{Ra}$ were averaged for panels ( $a$ ) and ( $b$ ), and the time series for $\delta _i/\Delta T$ in the conditioned SRM and DRM were averaged for panels ( $c$ ) and ( $d$ ), respectively. A power-law fit to the SRM data in figure 7( $c$ ) yields $\langle \delta _i \rangle /\Delta T = 9.41Ra^{-0.30}$ . The resulting fit is shown as the black solid line in figure 7( $c$ ), and the same line is reproduced in figures 7( $a$ ), 7( $b$ ) and 7( $d$ ) as a common reference for comparison. From figure 7( $a$ ), in the $\varGamma =1$ cell, $\langle \delta _m \rangle /\Delta T$ at the middle height is slightly larger than that at the top and bottom heights, which are approximately equal to each other. It is suggested that the LSC with a tilted elliptical flow structure has the greatest temperature contrast at the middle height (Xi & Xia Reference Xi and Xia2008b ). However, $\langle \delta _m \rangle /\Delta T$ at the middle height is the smallest compared with that at the other heights in the $\varGamma =1/2$ and $1/3$ cells, which is consistent with the relatively large weight of the DRM in these cells. When the flow is in the DRM, the flow mixes well in the region at the middle height of the sidewall, so there are no large temperature differences in this region. In addition, despite differences in magnitude among the different flow modes and aspect ratios, the data generally exhibit an $ \textit{Ra}$ -dependence consistent with the effective exponent $-0.30$ obtained from the SRM data.

Figure 7.

Normalised time-averaged strength $\langle \delta _i \rangle /\Delta T$ at the top ( $i=t$ , blue squares), middle ( $i=m$ , black circles) and bottom ( $i=b$ , red triangles) heights of the sidewall as a function of $ \textit{Ra}$ in cells with different $\varGamma$ : ( $a$ ) data from all flow modes; ( $b$ ) present data together with those of Xi & Xia (Reference Xi and Xia2008b ); ( $c$ ) data in the SRM and ( $d$ ) data in the DRM. The black solid line represents a power-law fit to the SRM data in panel ( $c$ ), $\langle \delta _i \rangle /\Delta T = 9.41Ra^{-0.30}$ , and the same line is shown in panels ( $a$ ), ( $b$ ) and ( $d$ ) for reference. Note panels (a) and (b) share the same legend and panels (c) and (d) share the same legend.

Next, we compare these results with previous experimental results in cells with the same $\varGamma$ reported by Xi & Xia (Reference Xi and Xia2008b ) using water as the working fluid with $ \textit{Pr}=4.9{-}5.3$ , as shown in figure 7( $b$ ). It is found that these results obtained in the $\varGamma =1$ and $\varGamma =1/3$ cells are consistent with those obtained from Xi & Xia (Reference Xi and Xia2008b ), except for the result obtained in the $\varGamma =1/2$ cell, in which the flow strength measured at the three heights is essentially the same in water convection. Notably, the power-law function with an effective exponent $-0.30$ can also describe these data, although the $ \textit{Pr}$ of the two datasets differ by two orders of magnitude, which indicates that $\langle \delta _i \rangle /\Delta T$ has a weak dependence on $ \textit{Pr}$ .

More insight into the normalised time-averaged strength can be obtained by separating the different flow modes, as shown in figures 7( $c$ ) and 7( $d$ ). From figure 7( $c$ ), one observes that all of the data at the three heights in the SRM in cells with different $\varGamma$ agree well with each other and nicely collapse on the power-law curve with an effective exponent $-0.30$ , which indicates that $\langle \delta _i \rangle /\Delta T$ in the SRM in different $\varGamma$ cells has universal characteristics and a weak dependence on $\varGamma$ . However, for $\langle \delta _i \rangle /\Delta T$ in the DRM, as shown in figure 7( $d$ ), in both the $\varGamma =1/2$ and $1/3$ cells, the amplitudes at the top and bottom heights are equal to each other and larger than that at the middle height. The amplitudes at the different heights in the $\varGamma =1/3$ cell are slightly larger than those in the $\varGamma = 1/2$ cell, where there is an overlap in $ \textit{Ra}$ , suggesting that the DRM in the $\varGamma=1/3$ cell may be stronger than that in the $\varGamma=1/2$ cell.

3.3. Different routes of the flow mode transitions in the $\varGamma = 1/3$ cell

Although the present results show that the flow in the $\varGamma = 1/3$ cell undergoes frequent transitions between the SRM and the DRM, the detailed process by which the DRM transforms into the SRM and then returns to the DRM (DRM–SRM–DRM) remains unclear. In other words, the transition routes between different flow modes have not yet been well investigated. Since a statistical characterisation of these transition routes would require a much larger number of long-duration measurements, this subsection focuses on several typical DRM–SRM–DRM transition routes rather than their statistical properties. Therefore, the examples discussed in the following should be regarded as representative phenomenological routes observed in selected events, rather than as statistically established or dominant pathways.

Figure 8.

Route 0: representative DRM-SRM-DRM transition routes involving cessation of the flow. Panels $(a)$ , $(d)$ and $(g)$ show the time series of the normalised flow strength $\delta _i/\langle \delta _i \rangle$ , while panels $(b)$ , $(e)$ and $(h)$ show the corresponding orientation $\theta _i$ at the top ( $i=t$ , blue squares) and bottom ( $i=b$ , red triangles) heights of the sidewall. The green dashed horizontal line marks the threshold $\delta _i/\langle \delta _i \rangle = 0.15$ for flow cessation and the shaded regions denote periods during which the flow is classified as being in the SRM. Panels $(c)$ , $(f)$ and $(i)$ show the corresponding polar plots of $\delta _i/\langle \delta _i \rangle$ and $\theta _i$ at the top ( $i=t$ , blue squares) and bottom ( $i=b$ , red triangles) heights of the sidewall. In the polar plots, the green dashed circle marks the same threshold $\delta _i/\langle \delta _i \rangle = 0.15$ . The colour bars indicate the temporal evolution in units of $t/t_{\textit{ff}}$ . Panels ( $a$ – $c$ ), ( $d$ – $f$ ) and ( $g$ – $i$ ) correspond to the simultaneous cessation of both rolls, the cessation of the top roll and the cessation of the bottom roll, respectively.

Figure 8. Long description

Panel A: A line graph shows the time series of the normalized flow strength for Route 0a. The x-axis represents time in units of t/t_–ff, and the y-axis represents the normalized flow strength. The graph includes two data series: one for the top height (blue squares) and one for the bottom height (red triangles). Panel B: Another line graph shows the corresponding orientation at the top and bottom heights. The x-axis is the same as in Panel A, and the y-axis represents the orientation in units of 2π. Panel C: A polar plot displays the normalized flow strength at the top and bottom heights. The radial axis represents the normalized flow strength, and the angular axis represents the orientation. The color bar indicates the temporal evolution in t/t_–ff. Panel D: A line graph shows the time series of the normalized flow strength for Route 0b. The x-axis and y-axis are the same as in Panel A. Panel E: Another line graph shows the corresponding orientation at the top and bottom heights. The x-axis and y-axis are the same as in Panel B. Panel F: A polar plot displays the normalized flow strength at the top and bottom heights. The radial and angular axes are the same as in Panel C. Panel G: A line graph shows the time series of the normalized flow strength for Route 0c. The x-axis and y-axis are the same as in Panel A. Panel H: Another line graph shows the corresponding orientation at the top and bottom heights. The x-axis and y-axis are the same as in Panel B. Panel I: A polar plot displays the normalized flow strength at the top and bottom heights. The radial and angular axes are the same as in Panel C.

Figure 9.

Route 1: representative DRM-SRM-DRM transition routes in which the double-roll structure returns with an azimuthal rotation direction opposite to that before the transition. Panels $(a)$ , $(d)$ and $(g)$ show the time series of the normalised flow strength $\delta _i/\langle \delta _i \rangle$ , while panels $(b)$ , $(e)$ and $(h)$ show the corresponding orientation $\theta _i$ at the top ( $i=t$ , blue squares) and bottom ( $i=b$ , red triangles) heights of the sidewall. The green dashed horizontal line marks the threshold $\delta _i/\langle \delta _i \rangle = 0.15$ for flow cessations and the shaded regions denote periods during which the flow is classified as being in the SRM. Panels $(c)$ , $(f)$ and $(i)$ show the corresponding polar plots of $\delta _i/\langle \delta _i \rangle$ and $\theta _i$ at the top ( $i=t$ , blue squares) and bottom ( $i=b$ , red triangles) heights of the sidewall. In the polar plots, the green dashed circle marks the same threshold $\delta _i/\langle \delta _i \rangle = 0.15$ . The colour bars indicate the temporal evolution in units of $t/t_{\textit{ff}}$ . Panels ( $a$ – $c$ ), ( $d$ – $f$ ) and ( $g$ – $i$ ) correspond to the simultaneous rotation of both rolls, the rotation of the top roll, and the rotation of the bottom roll, respectively.

Figure 10.

Route 2: representative DRM-SRM-DRM transition routes in which the double-roll structure returns with the same azimuthal rotation direction as before the transition. Panels $(a)$ , $(d)$ and $(g)$ show the time series of the normalised flow strength $\delta _i/\langle \delta _i \rangle$ , while panels $(b)$ , $(e)$ and $(h)$ show the corresponding orientation $\theta _i$ at the top ( $i=t$ , blue squares) and bottom ( $i=b$ , red triangles) heights of the sidewall. The green dashed horizontal line marks the threshold $\delta _i/\langle \delta _i \rangle = 0.15$ for flow cessation and the shaded regions denote periods during which the flow is classified as being in the SRM. Panels $(c)$ , $(f)$ and $(i)$ show the corresponding polar plots of $\delta _i/\langle \delta _i \rangle$ and $\theta _i$ at the top ( $i=t$ , blue squares) and bottom ( $i=b$ , red triangles) heights of the sidewall. In the polar plots, the green dashed circle marks the same threshold $\delta _i/\langle \delta _i \rangle = 0.15$ . The colour bars indicate the temporal evolution in units of $t/t_{\textit{ff}}$ . Panels ( $a$ – $c$ ), ( $d$ – $f$ ) and ( $g$ – $i$ ) correspond to the simultaneous rotation of both rolls, the rotation of the top roll and the rotation of the bottom roll, respectively.

Figure 10. Long description

Panel A: A line graph shows the time series of the normalized flow strength for R2a. The x-axis represents time in units of t/t_–ff, and the y-axis represents the normalized flow strength. The graph includes two data series: one for the top height (blue squares) and one for the bottom height (red triangles). The green dashed horizontal line marks the threshold, and the shaded regions denote periods during which the flow is classified as being in the SRM. Panel B: Another line graph shows the corresponding orientation at the top and bottom heights of the sidewall for R2a. The x-axis represents time in units of t/t_–ff, and the y-axis represents the orientation in units of theta_i (2pi). Panel C: A polar plot shows the orientation and strength of the LSC at the top and bottom heights of the sidewall for R2a. The green dashed circle marks the threshold, and the color bar indicates the temporal evolution in t/t_–ff. Panel D: A line graph shows the time series of the normalized flow strength for R2b. The x-axis represents time in units of t/t_–ff, and the y-axis represents the normalized flow strength. The graph includes two data series: one for the top height (blue squares) and one for the bottom height (red triangles). The green dashed horizontal line marks the threshold, and the shaded regions denote periods during which the flow is classified as being in the SRM. Panel E: Another line graph shows the corresponding orientation at the top and bottom heights of the sidewall for R2b. The x-axis represents time in units of t/t_–ff, and the y-axis represents the orientation in units of theta_i (2pi). Panel F: A polar plot shows the orientation and strength of the LSC at the top and bottom heights of the sidewall for R2b. The green dashed circle marks the threshold, and the color bar indicates the temporal evolution in t/t_–ff. Panel G: A line graph shows the time series of the normalized flow strength for R2c. The x-axis represents time in units of t/t_–ff, and the y-axis represents the normalized flow strength. The graph includes two data series: one for the top height (blue squares) and one for the bottom height (red triangles). The green dashed horizontal line marks the threshold, and the shaded regions denote periods during which the flow is classified as being in the SRM. Panel H: Another line graph shows the corresponding orientation at the top and bottom heights of the sidewall for R2c. The x-axis represents time in units of t/tff, and the y-axis represents the orientation in units of theta_i (2pi). Panel I: A polar plot shows the orientation and strength of the LSC at the top and bottom heights of the sidewall for R2c. The green dashed circle marks the threshold, and the color bar indicates the temporal evolution in t/t_–ff.

Three different DRM–SRM–DRM transition routes are identified: Route 0 (R0, as shown in figure 8) is characterised by the flow mode transitions caused by the cessation of the flow; Route 1 (R1, as shown in figure 9) and Route 2 (R2, as shown in figure 10) are characterised by the flow mode transitions caused by the azimuthal rotation of the flow without cessation. The key difference between R1 and R2 lies in whether the azimuthal rotation direction of the double roll reverses across the transition. For R1, the azimuthal rotation direction of the double roll reverses before and after the flow mode transition. In contrast, for R2, the azimuthal rotation direction of the double roll remains unchanged throughout the flow mode transition process.

3.3.2. Route 1 and route 2

As previously mentioned, the key difference between the R1 and R2 routes is whether the azimuthal rotation direction after the transition is opposite to (R1) or the same as (R2) that of the initial DRM, i.e. the top and bottom rolls rotate in the opposite direction (R1) or the same direction (R2) to the initial DRM. Since the transition processes of R1 (as shown in figure 9) and R2 (as shown in figure 10) are similar, we will primarily focus on R1. Depending on which roll or rolls undergo azimuthal rotation, R1 is further classified into R1a (rotation of both top and bottom rolls, as shown in figure 9 $a$ – $c$ ), R1b (rotation of the top roll, as shown in figure 9 $d$ – $f$ ) and R1c (rotation of the bottom roll, as shown in figure 9 $g$ – $i$ ).

We first analyse the transition process of R1a, as shown in figure 9( $a$ – $c$ ). During the period $0 \lesssim t/t_{\textit{ff}} \lesssim 20$ , the strength of the top and bottom rolls remains close to their own average value, while their flow directions change by approximately $0.5 \pi$ in opposite directions, resulting in a change in the flow mode from the DRM to the SRM. The SRM regenerates in a direction perpendicular to the original DRM flow plane and persists for approximately $40 t_{\textit{ff}}$ . Then, the top and bottom parts of the SRM rotate back by approximately $0.5 \pi$ in the reverse direction, forming the DRM. In R2a, by contrast, the top and bottom parts of the SRM continue rotating in the same direction by approximately $0.5 \pi$ before returning to the DRM, as shown in figure 10( $a$ – $c$ ).

Next, we examine the transition processes of R1b and R1c. Figures 9( $d$ – $f$ ) and 9( $g$ – $i$ ) show R1b and R1c, respectively. The strength of the top (bottom) roll is weaker than that of the bottom (top) roll. Over approximately $20 t_{\textit{ff}}$ , the flow direction of the top (bottom) roll changes by approximately $\pi$ , eventually merging with the bottom (top) roll to form the SRM. After persisting for a period of approximately $30 t_{\textit{ff}}$ , the top (bottom) part of the SRM rotates back by approximately $\pi$ in the reverse direction, re-establishing the DRM. It should be noted that for R2b, as shown in figure 10( $d$ – $f$ ) (R2c, as shown in figure 10 $g$ – $i$ ), the top (bottom) part of the SRM continues rotating in the same direction by approximately $\pi$ before returning to the DRM.

Figure 11.

Probability density functions (p.d.f.s) of the orientation $\theta _i$ at the top ( $i = t$ , blue squares), middle ( $i = m$ , black circles) and bottom ( $i = b$ , red triangles) heights of the sidewall when the flow is in the ( $a$ ) SRM and the ( $b$ ) DRM at $ \textit{Ra}=1.01 \times 10^8$ in the $\varGamma =1/3$ cell.

3.4. Structure and asymmetry of the DRM in the $\varGamma = 1/3$ cell

We now focus on the internal organisation of the DRM in the strongly confined cell with $\varGamma = 1/3$ , where this mode is statistically dominant. Figure 11 shows the p.d.f.s of the orientation $\theta _i$ at the top ( $i = t$ , blue squares), middle ( $i = m$ , black circles) and bottom ( $i = b$ , red triangles) heights of the sidewall when the flow is in the ( $a$ ) SRM and the ( $b$ ) DRM at $ \textit{Ra}=1.01 \times 10^8$ in the $\varGamma =1/3$ cell. In figure 11( $a$ ), the p.d.f.s of the orientation $\theta _i$ measured at the different heights are close to each other and peak at $\theta \sim 0.2\pi$ , indicating a roughly vertically coherent LSC with a well-defined preferred orientation when the flow is in the SRM. This finding is consistent with previous studies (Xi & Xia Reference Xi and Xia2008b ; Weiss & Ahlers Reference Weiss and Ahlers2011). In contrast, when the flow is in the DRM, as shown in figure 11( $b$ ), significant differences appear in the p.d.f.s of the orientation $\theta _i$ at the different heights. At the top and bottom heights, the p.d.f.s of the orientation $\theta _i$ exhibit peaks near $\theta \sim 0.2\pi$ and $\theta \sim 1.3\pi$ , respectively, with a difference of nearly $\pi$ , suggesting the presence of two vertically stacked rolls rotating in opposite directions.

At the middle height, the p.d.f. of the orientation $\theta _m$ in the DRM is relatively uniform across orientations, but retains a weak preference for $\theta \sim 0.2\pi$ , consistent with the preferred orientation at the top height. This suggests that the two vertically stacked rolls are unequal in size, with a larger roll in the upper part of the cell and a smaller one in the lower part. Notably, the preferred orientation of the SRM coincides with that of the upper roll in the DRM, suggesting that the flow structure in the upper part of the cell is more robust during flow mode transitions. This, in turn, implies an asymmetry in roll strength. This asymmetry may originate from the different thermal boundary conditions at the top and bottom plates. The top plate is maintained at a constant temperature, while the bottom plate is subjected to a constant heat flux, which may lead to different boundary-layer dynamics and plume-emission behaviours, thereby breaking the vertical symmetry of the flow.

Figure 12.

( $a$ ) The p.d.f. of the structure factor $S=| \Delta \theta |_{b,m}/( | \Delta \theta |_{b,m}+| \Delta \theta |_{t,m})$ . The solid line is a sixth-order polynomial fit to the data and is provided as a guide to the eye. ( $b$ ) Normalised conditional average of the strength $\langle \delta _i | S \rangle / \langle \delta _i \rangle$ at the top ( $i = t$ , blue squares) and bottom ( $i = b$ , red triangles) heights of the sidewall.

To quantify this asymmetry when the flow is in the DRM, following Xi & Xia (Reference Xi and Xia2008b ), we employ a structure factor $S=\left | \Delta \theta \right |_{b,m}/(\left | \Delta \theta \right |_{b,m}+\left | \Delta \theta \right |_{t,m})$ as a quantitative indicator. Here, DRM1 with $S \sim 0$ denotes a double-roll structure with a smaller upper roll and a larger lower roll, whereas DRM2 with $S \sim 1$ characterises a double-roll structure with a larger upper roll and a smaller lower roll. DRM3 with $S \sim 0.5$ represents a double-roll structure with two equal-sized rolls. First, we examine the size of the rolls. Figure 12( $a$ ) presents the p.d.f. of the structure factor $S$ at $ \textit{Ra}=1.01 \times 10^8$ in the $\varGamma =1/3$ cell. This plot reveals two prominent peaks near 0 and 1, suggesting that DRM1 and DRM2 are the preferred flow structures when the flow is in the DRM in a $\varGamma = 1/3$ cell. This finding is broadly consistent with the results reported by Xi & Xia (Reference Xi and Xia2008b ), where asymmetric double-roll states were also observed. However, the relative preference between the two asymmetric DRM states differs. In the present liquid-metal experiment, DRM2 is more probable than DRM1, whereas the opposite preference was reported in the previous water experiment (Xi & Xia Reference Xi and Xia2008b ). This difference may reflect the combined influence of $ \textit{Pr}$ , experimental boundary conditions, residual cell asymmetries, and possible spontaneous symmetry breaking. The origin of the asymmetric DRM states and the mechanism controlling the relative preference between DRM1 and DRM2 at different $ \textit{Pr}$ therefore remain open questions.

To compare the strength of the upper and lower rolls when the flow is in the DRM1, DRM2 and DRM3, we examine the normalised conditional average of the strength $\langle \delta _i | S \rangle / \langle \delta _i \rangle$ at the top ( $i = t$ , blue squares) and bottom ( $i = b$ , red triangles) heights of the sidewall at $ \textit{Ra}=1.01 \times 10^8$ in the $\varGamma =1/3$ cell, as shown in figure 12( $b$ ). For $S \sim 0$ , the value $\langle \delta _b | S \rangle / \langle \delta _b \rangle$ at the bottom height is larger than $\langle \delta _t | S \rangle / \langle \delta _t \rangle$ at the top height, indicating that the strength of the larger-lower roll is stronger than that of the smaller-upper roll in the DRM1. Conversely, near $S \sim 1$ , $\langle \delta _b | S \rangle / \langle \delta _b \rangle$ at the bottom height is smaller than $\langle \delta _t | S \rangle / \langle \delta _t \rangle$ at the top height, suggesting that the strength of the larger-upper roll is stronger than that of the smaller-lower roll in the DRM2. In addition, near $S \sim 0.5$ , the values $\langle \delta _i | S \rangle / \langle \delta _i \rangle$ at the top and bottom heights are close to each other, implying that the strengths of the upper and lower equal-sized rolls become comparable when the flow is in the DRM3. These results demonstrate a clear coupling between roll size and roll strength, i.e. the larger roll is associated with stronger strength, while the smaller roll is weaker.

This coupling provides a natural interpretation of the preferred orientation observed in figure 11. The preferred SRM orientation coincides with that of the upper roll in the DRM, indicating that the upper part of the circulation is dynamically more robust during mode transitions. Since DRM2, characterised by a larger and stronger upper roll and a smaller and weaker lower roll, is the most probable double-roll state, the lower roll is more susceptible to disturbance and reorganisation in the process of the flow mode transition, whereas the upper roll remains comparatively stable.

3.5. Heat transport associated with different flow modes

Previous studies using water as the working fluid have demonstrated a clear correlation between flow mode and heat transport (Xi & Xia Reference Xi and Xia2008b ; Weiss & Ahlers Reference Weiss and Ahlers2011; Xie et al. Reference Xie, Ding and Xia2018). It has been shown that the maximum difference in heat transport between DRM and SRM is approximately $1.7\,\%$ , occurring at $ \textit{Ra} = 4.76\times 10^9$ , while the differences at larger $ \textit{Ra}$ are smaller (Weiss & Ahlers Reference Weiss and Ahlers2011). Since heat transport in liquid metal is correlated with its internal large-scale flow, we examine the influence of flow mode on heat transport in liquid metal convection by combining the multi-thermal probe technique with simultaneous heat-transport measurements. For this purpose, following Weiss & Ahlers (Reference Weiss and Ahlers2011), we construct binary state functions $F$ from the time series of the identified flow modes. Specifically, $F_{\textit{SRM}}=1$ when the flow is in the SRM and $F_{\textit{SRM}}=0$ otherwise (i.e. for the DRM and TRM). Similarly, $F_{\textit{DRM}}=1$ when the flow is in the DRM and $F_{\textit{DRM}}=0$ otherwise (i.e. for the SRM and TRM).

Figure 13( $a$ ) shows representative time series of $F_{\textit{SRM}}$ (blue line), $-F_{\textit{DRM}}$ (red line) and $ \textit{Nu}$ (black line) measured in a cell with $\varGamma =1/3$ at $ \textit{Ra}=1.01\times 10^8$ . For visual clarity, $F_{\textit{DRM}}$ is plotted with a negative sign, such that the SRM and DRM appear at $+1$ and $-1$ , respectively. This sign reversal is used only for visualisation; the original binary state function $F_{\textit{DRM}}$ is used in the cross-correlation analysis in the following. It can be seen qualitatively that the SRM is associated with a larger $ \textit{Nu}$ , whereas the DRM is associated with a smaller $ \textit{Nu}$ . To quantify this relationship, we calculate the cross-correlation functions between the two state functions and $ \textit{Nu}$ , denoted by $C_{F_{\textit{SRM}}, \textit{Nu}}$ and $C_{F_{\textit{DRM}}, \textit{Nu}}$ , respectively. The normalised cross-correlation function is defined as

(3.1) \begin{align} C_{F,\textit{Nu}}(\tau )=\frac {\left \langle \left [F(t+\tau )-\langle F\rangle \right ]\left [\textit{Nu}(t)-\langle \textit{Nu}\rangle \right ]\right \rangle }{\sigma _F \sigma _{\textit{Nu}}}, \end{align}

where $F$ denotes either $F_{\textit{SRM}}$ or $F_{\textit{DRM}}$ , $\langle \boldsymbol{\cdot }\rangle$ denotes time-averaging, and $\sigma _F$ and $\sigma _{Nu}$ are the corresponding standard deviations.

The results are shown in figure 13( $b$ ), with the SRM represented by a blue line and the DRM by a red line. With the definition of $C_{F,\textit{Nu}}$ in (3.1), an extremum at a negative time lag indicates that changes in the flow mode precede the corresponding heat-transport response. The time lag is normalised by the free-fall time $t_{\textit{ff}}$ . Both cross-correlation functions reach their extrema at approximately $\tau /t_{\textit{ff}} \approx -5$ , corresponding to a positive maximum for $C_{F_{\textit{SRM}}, \textit{Nu}}$ and a negative minimum for $C_{F_{\textit{DRM}}, \textit{Nu}}$ . The thermal diffusion time between the fluid-plate interface and the embedded thermistors is estimated as $\tau _{\textit{Cu}} \sim d^2 / \kappa _{\textit{Cu}}$ , where $\kappa _{\textit{Cu}} = 1.1 \times 10^{-4}\,\mathrm{m^2s^{-1}}$ and $d = 3\,\mathrm{mm}$ , giving $\tau _{\textit{Cu}} \approx 0.08\,\mathrm{s}$ , which is much shorter than the observed delay, i.e. $|\tau | \approx 26.25\,\mathrm{s}$ , so that the measured plate temperature could respond rapidly to changes in the fluid temperature. Thus, the negative-lag peak is unlikely to arise from heat diffusion within the copper plate, but is instead consistent with the interpretation that the heat-transport response follows the flow-mode transition after a finite delay, rather than changing instantaneously. Furthermore, $C_{F_{\textit{SRM}}, \textit{Nu}}$ reaches a positive maximum at $\tau /t_{\textit{ff}} \approx -5$ , whereas $C_{F_{\textit{DRM}}, \textit{Nu}}$ reaches a negative minimum at approximately the same lag. This implies that heat transport is slightly more efficient in the SRM than in the DRM. These observations are consistent with findings reported by Weiss & Ahlers (Reference Weiss and Ahlers2011) in cells with water as the working fluid.

Figure 13.

( $a$ ) Representative time series of the SRM state function $F_{\textit{SRM}}$ (blue line), the sign-reversed DRM state function $-F_{\textit{DRM}}$ (red line) and $ \textit{Nu}$ (black line). For visual clarity, $F_{\textit{DRM}}$ is plotted with a negative sign, such that the SRM and DRM appear at $+1$ and $-1$ , respectively. ( $b$ ) The cross-correlation functions $C_{F_{\textit{SRM}}, \textit{Nu}}$ (blue line) and $C_{F_{\textit{DRM}}, \textit{Nu}}$ (red line). The vertical dashed line marks $\tau /t_{\textit{ff}}=0$ . Measurements are obtained in a cell with $\varGamma = 1/3$ at $ \textit{Ra} = 1.01 \times 10^8$ .

Figure 14.

Normalised difference in $ \textit{Nu}$ between the SRM and DRM in cells with $\varGamma = 1/2$ (red circles) and $\varGamma = 1/3$ (blue triangles). The error bars denote the uncertainty of $1- \textit{Nu}_D/\textit{Nu}_S$ estimated from the $ \textit{Nu}(t)$ time series and the corresponding flow-state labels. The black squares are data from Weiss & Ahlers (Reference Weiss and Ahlers2011) in a cell with $\varGamma = 1/2$ ( $ \textit{Pr} = 4.38$ ). The red, blue and black solid lines represent power-law fits $1-\textit{Nu}_{D}/\textit{Nu}_{S} = 2.87Ra^{-0.22}$ , $1-\textit{Nu}_{D}/\textit{Nu}_{S} = 0.36Ra^{-0.12}$ and $1-\textit{Nu}_{D}/\textit{Nu}_{S} = 4.91Ra^{-0.25}$ in cells with $\varGamma = 1/2$ ( $ \textit{Pr} = 0.029$ ), $\varGamma = 1/3$ ( $ \textit{Pr} = 0.029$ ) and $\varGamma = 1/2$ ( $ \textit{Pr} = 4.38$ ), respectively. The red dashed line represents the extrapolation of the $-0.22$ scaling.

To quantify the difference in heat transport between the SRM and the DRM, figure 14 presents the normalised difference in $ \textit{Nu}$ , $1-\textit{Nu}_{D}/\textit{Nu}_{S}$ as a function of $ \textit{Ra}$ in cells with $\varGamma = 1/2$ (red circles) and $\varGamma = 1/3$ (blue triangles). Here, $ \textit{Nu}_{S}$ and $ \textit{Nu}_{D}$ represent heat transport conditioned on the SRM and DRM, respectively. The error bars represent the standard deviations of the corresponding estimates of $1-\textit{Nu}_D/\textit{Nu}_S$ and provide a quantitative measure of the uncertainty. The uncertainty is relatively larger for the $\varGamma =1/3$ cell, likely because the number of SRM samples is smaller when DRM is the dominant flow mode. The normalised difference in $ \textit{Nu}$ shows a systematic dependence on $ \textit{Ra}$ : it decreases with increasing $ \textit{Ra}$ . This suggests that the heat-transport difference between the two flow modes is larger at smaller $ \textit{Ra}$ . Although the magnitude of this difference is small, the positive values of $1-\textit{Nu}_D/\textit{Nu}_S$ indicate that the SRM is generally associated with a slightly larger $ \textit{Nu}$ than the DRM. Power-law fits to the data yield $1-\textit{Nu}_{D}/\textit{Nu}_{S} = 2.87Ra^{-0.22}$ for $\varGamma = 1/2$ and $1-\textit{Nu}_{D}/\textit{Nu}_{S} = 0.36Ra^{-0.12}$ for $\varGamma = 1/3$ , as indicated by the red and blue solid lines in figure 14, respectively.

We next examine the effect of $ \textit{Pr}$ on the difference in heat transport between the SRM and the DRM. Data (black squares) obtained from Weiss & Ahlers (Reference Weiss and Ahlers2011) in a $\varGamma = 1/2$ cell using water as the working fluid with $ \textit{Pr}=4.38$ are also presented in figure 14. Consistent with the present results, the difference in $ \textit{Nu}$ decreases with increasing $ \textit{Ra}$ and follows a power-law $1-\textit{Nu}_{D}/\textit{Nu}_{S} = 4.91Ra^{-0.25}$ , as shown by the black solid line. This exponent is close to the value $-0.22$ obtained in liquid metal convection with $ \textit{Pr} = 0.029$ . When the $-0.22$ scaling is extrapolated to the $ \textit{Ra}$ range covered by the water measurements (red dashed line), the predicted value of $1-\textit{Nu}_{D}/\textit{Nu}_{S}$ for liquid-metal convection is, on average, approximately $19\,\%$ higher than that for water at the same $ \textit{Ra}$ , with the relative difference increasing weakly with $ \textit{Ra}$ . This suggests that the difference in $ \textit{Nu}$ between different flow modes may depend weakly on $ \textit{Pr}$ , in contrast to its much stronger dependence on $ \textit{Ra}$ .