3.1. Global features of the flow field

Figure 2 illustrates the instantaneous three-dimensional flow fields at five different $Ra$ ranging from $5.0 \times 10^{8}$ to $5.0 \times 10^{9}$ . The flow field is presented by the streamlines of the instantaneous velocity field, and the colour of the streamlines represents both the direction and magnitude of the vertical velocity, with positive (negative) value corresponding to the upward (downward) velocity. While the flow field depicted in figure 2 represents an instantaneous snapshot, it effectively captures the general features of the flow field at different $Ra$ . From figures 2(a–e), it is evident that in such a very slender geometry, a distinct single-roll LSC with ascending flow on one side and descending flow on the other side, which is typically observed in a unit cylindrical or rectangular cell (Xia, Sun & Zhou Reference Xia, Sun and Zhou2003; Xi, Lam & Xia Reference Xi, Lam and Xia2004), is absent. Instead, the flow in such a very slender cell appears to be in the helical form at first glance, as the ascending and descending flows undergo azimuthal motion as they travel vertically along the sidewall; one can see an example of this event at approximately the middle height of figure 2(a). While a closer look shows that sometimes the ascending and descending flows can also intersect and cross each other, one can see an example of this event at approximately the bottom of figure 2(a). An overall picture of the flow structure is that the flow is divided into different segments in the vertical direction irrespective of whether the neighbouring segments are divided by a $180^\circ$ orientation change of the ascending and descending flows or by an intersect and crossing of the ascending and descending flows. And it is evident that as $Ra$ increases, the number of segments gradually increases.

Figure 2.

Instantaneous three-dimensional flow fields at five different $Ra$ ranging from $5.0 \times 10^{8}$ to $5.0 \times 10^{9}$ . The flow field is illustrated by the streamlines of the instantaneous velocity field, with the colour indicating both the direction and magnitude of the vertical velocity. Positive values correspond to upward velocity, while negative values correspond to downward velocity.

Figure 3.

Log-log plot of $Re$ as a function of $Ra$ . The slope of the best fitting (blue) line is 0.43. Inset: the same data as in the main plot, but $Re$ is compensated by $Ra^{0.43}$ .

Figure 3 shows the measured Reynolds number $Re$ as a function of $Ra$ . Here, $Re = UH/\nu$ , where $U = \langle u^{2}+v^{2}+w^{2} \rangle _{Vol,t}^{1/2}$ , with $\langle \rangle _{Vol,t}$ meaning volume (entire cell) and time (duration of the experiment) average. The scaling of $Re$ versus $Ra$ is found to follow $Re \sim Ra^{0.43}$ . According to the Grossmann–Lohse (GL) theory (Ahlers et al. Reference Ahlers, Grossmann and Lohse2009), the $Ra{-}Pr$ space can be categorised into eleven regimes. If we neglect the effect of aspect ratio $\Gamma$ , then the parameters of our experiments fall within regime $IV_{u}$ . In this regime, GL theory predicts a scaling of $Re \sim Ra$ to be $4/9$ , which aligns closely with our experimental findings.

In figure 4, we present the horizontal map of the averaged magnitude of the horizontal and vertical velocities for five $Ra$ values. Here, the horizontal velocity comprises both $u$ and $v$ , and the vertical velocity is represented by $w$ . The horizontal map is obtained by averaging the velocity over the entire height and time. Due to the thin boundary layer compared to the cell height, averaging over the height primarily reflects the flow state in the bulk region of the cell. The top row of figure 4 depicts the horizontal velocity magnitude. In all cases, this value reaches its maximum at the centre and decreases towards the sidewall, indicating that the most significant horizontal movement occurs in the centre of the domain. The horizontal map of the vertical velocity magnitude is shown in the bottom row of figure 4; the distributions exhibit annulus shapes for relative high $Ra$ , such as $Ra = 1.5\times 10^{9}$ , $2.8\times 10^{9}$ and $5\times 10^{9}$ , with equal values in the azimuthal direction. Schmidt et al. (Reference Schmidt, Calzavarini, Lohse, Toschi and Verzicco2012) reported a similar distribution in their simulation with vertically unbounded slender cells, and demonstrated that the position with the maximum value shifts towards the sidewall as $Ra$ increases. In our experiments, within the $Ra$ range, the position of the maximum value remains almost unchanged, which occurs at approximately $2/3$ radius. When $Ra$ is small, such as for $Ra = 5\times 10^{8}$ and $9.3\times 10^{8}$ , unlike the high $Ra$ cases, the vertical velocity shows a saddle-shaped distribution, with the maximum values located at two preferred orientation regions, as shown in figures 4(a) and 4(b). The formation of this distribution is because when $Ra$ is small, the flow orientation remains relatively steady both in height and over time. There are large and persistent separated ascending and descending flows that have preferred orientations, thus contributing to this distribution. When $Ra$ is increased, the ascending and descending flows change their orientations more frequently, and the preference of the orientation disappears. However, this variation does not occur for the horizontal map of the horizontal velocity. It is because, on the one hand, the vertical velocity is the main component of the flow, which can be seen from the magnitude of velocity at figure 4, and on the other hand, the horizontal velocity is naturally small near the sidewall since the horizontal velocity vanishes due to the impenetrable boundary condition at the sidewall.

Figure 4.

Horizontal maps of the averaged magnitudes of the horizontal and vertical velocities. Top row: horizontal velocity magnitude ( $\langle u^{2}+v^{2}\rangle _{V,t}^{1/2}$ ). Bottom row: vertical velocity magnitude ( $\langle w^{2}\rangle _{V,t}^{1/2}$ ), where $\langle \rangle _{V,t}$ denotes averaging over entire height and time.

3.2. The spatial structure of the flow field

We further discuss the spatial flow structure by examining instantaneous flow fields. We begin with examining the case of small $Ra$ . Figure 5(a) presents the instantaneous flow field for $Ra=5\times 10^{8}$ , and figure 5(b) displays the horizontal-averaged vertical and horizontal velocities at different heights. It can be seen that at most heights, the vertical velocity significantly exceeds the horizontal component, indicating that the predominant direction of the flow is vertical. However, at $0.15H$ , the two curves almost intersect, with the horizontal velocity increasing to a maximum value, and the vertical velocity decreasing to a minimum simultaneously. By looking carefully at the horizontal cross-section of the flow field near $0.15H$ , as shown in figure 5(d), it becomes clear that from $0.1H$ to $0.2H$ , the descending flow (blue region) gradually travels from left to right. Simultaneously, the ascending flow (red region) is squeezed to separate and move towards the sidewall, forming an annulus shape. Eventually, the descending flow moves to the right-hand side, the squeezed ascending flow is re-merged together at the left-hand side, completing an azimuthal positional exchange between the descending and ascending flows. We define this azimuthal positional exchange process due to the intersection and crossing of the ascending and descending flows as a crossing event.

Figure 5.

At $Ra=5.0 \times 10^{8}$ , (a) streamlines of the instantaneous flow field, and (b) horizontal-averaged vertical and horizontal velocity magnitudes as functions of height, where $\langle \rangle _{H}$ represents the average over the horizontal cross-section. (c,d) Horizontal cross-sections of the instantaneous flow field at different heights, corresponding to the twisting and crossing events.

There is another scenario for the azimuthal positional exchange between the descending and ascending flows, as shown in figure 5(c). In this scenario, the ascending and descending flows slowly twist azimuthally, with their shapes and flow strength remaining unchanged. For instance, at $0.5H$ , the ascending flow (red region) is situated on the left-hand side. As it ascends, the ascending flow gradually rotates azimuthually in the anticlockwise direction. By the time it reaches $0.7H$ , it has rotated azimuthually by almost $90^\circ$ . At the same time, the descending flow also rotates azimuthally in the clockwise diretion by about $90^\circ$ . From figure 5(b), it can be seen that both the vertical and horizontal velocities do not vary significantly during the event, which distinguishes them from the crossing event. Thus we refer to this behaviour as a twisting event. It should be noted that the azimuthal rotation directions of the flow at different heights are not always the same; for example, from $0.85H$ to $0.9H$ , the ascending/descending flow rotates in the clockwise direction, while the flow around $0.5H$ is rotating in the anticlockwise direction. This implies that the flow cannot be described by a simple helical structure with uni-rotating direction. Given the two events described above, one interesting question arises regarding the number of segments in figure 5(a). At first glance, one might infer that there are four segments, based on the three intersections of the ascending and descending flows in the streamlines. However, it is important to note that a twisting event (such as those at $0.6H$ and $0.85H$ ) does not lead to the formation of a new segment. Only a crossing event can lead to a new segment. Thus there are only two segment in figure 5(a): one is the flow below $0.15H$ , and the other is the flow above $0.15H$ . Using crossing events to separate different segments of the flow is consistent with the method proposed in Zwirner et al. (Reference Zwirner, Tilgner and Shishkina2020). It should be noted that the two segments divided by the crossing event are very different from two vertically stacked rolls, because the streamlines near the boundary of the two rolls will not intersect; instead, they should be separated and largely parallel to each other. To conclude, the measured three-dimensional flow field reveals that the flow in this very slender cell is neither in a single helical form nor in a multi-roll form.

3.3. Flow mode analysis

To gain further insights into the flow field and also effectively identify the crossing and twisting events, we employ the mode decomposition analyses. By projecting the velocity onto different velocity modes, we can extract quantitative information about the flow field. Previous investigations suggest that typically only a few modes are required to quantitatively describe the essential dynamics of the flow in an RBC cell (Xi et al. Reference Xi, Zhang, Hao and Xia2016; Chen et al. Reference Chen, Huang, Xia and Xi2019).

Given the intricate three-dimensional property of the flow field described above, it is natural to consider using mode decomposition of the three-dimensional velocity field into the three-dimensional modes, as was done in Reiter, Zhang & Shishkina (Reference Reiter, Zhang and Shishkina2022) (see figure 17 in their paper). However, employing three-dimensional modal decomposition requires the flow to be periodic and relatively steady, which contrasts with the results presented in the current study. Thanks to the fact that in our experimental results the vertical velocity in the horizontal cross-sections during the events exhibits simple and characteristic patterns, we are able to conduct the modal analysis for the vertical velocity in the horizontal cross-section at different heights. Here, following the approach outlined by Shishkina (Reference Shishkina2021) and Reiter et al. (Reference Reiter, Zhang and Shishkina2022), we construct two-dimensional flow modes on the horizontal cross-section of the cylinder. This involves utilising Bessel functions in the radial direction and Fourier functions in the azimuthal direction. We then decompose the vertical velocity at each horizontal cross-section of different height into these modes.

Figure 6.

(a) Schematic patterns of the five typical modes with azimuthal wavenumbers $m=0, 1, 2, 3, 4$ . (b) Height- and time-averaged energy $\langle E_{m}\rangle$ contained in the $m=0, 1, 2, 3, 4$ modes as functions of $Ra$ . The cumulative energy of these modes is also depicted, denoted by orange squares.

The general form of the vertical velocity modes is

(3.1) \begin{equation} w_{m}(r,\varphi )=J_{m} ( \widetilde {\alpha _{m}}r/R)\cos (m \varphi ), \end{equation}

where $m=0,1,2,3,4$ is the azimuthal wavenumber, $w_{m}$ represents the $m$ th-order vertical velocity mode, $r$ is the radial position, $\varphi$ is the azimuthal angle, $J_{m}$ represents the $m$ th-order Bessel function of the first kind, $\widetilde {\alpha _{m}}$ denotes the first positive root of the first derivative of the $m$ th-order Bessel function, and $R$ is the radius of the cylindrical cell. The patterns corresponding to these five modes (i.e. $m=0,1,2,3,4$ ) are shown in figure 6(a).

To quantify the contribution of each mode in the flow field, we project the flow field of each horizontal cross-section onto the respective flow modes, and calculate the amplitude $A_{m}$ of each mode, which is defined as

(3.2) \begin{equation} A_{m}=\int _{S}^{}\frac {w\,w_{m}(\varphi _m)}{\| w_m \| }\,{\rm d}S, \end{equation}

where $\| w_m \|$ denotes the magnitude of each mode, calculated as $\| w_m \| = w_m w_m^{{\rm T}}$ . For $m = 1, 2, 3, 4$ modes, we adopt the largest value of right-hand side of (3.2) as $A_{m}$ by altering the azimuthal angle $\varphi$ of these modes during the calculation, and the corresponding azimuthal angle is $\varphi _m$ . Following previous studies (e.g. Xi et al. Reference Xi, Zhang, Hao and Xia2016), we evaluate the energy contained in the $m$ th-order mode $E_m$ as $E_m = A_m^2 / w^2$ for each horizontal cross-section, where the energy contained in each mode is normalised by the kinetic energy $w^2$ of each horizontal cross-section. This normalisation ensures that $E_m$ remains within the range 0–1, also facilitating comparisons across different heights and time instants. In figure 6(b), we present the height- and time-averaged energy $\langle E_{m}\rangle$ contained in the $m=0, 1, 2, 3, 4$ modes as functions of $Ra$ , along with their cumulative energy. The height-averaging involves calculating $\langle E_m\rangle _H$ by averaging $E_m$ across all the horizontal cross-sections within a single instantaneous snapshot, yielding one value for each time instant. By time-averaging the time series of these instantaneous $\langle E_m\rangle _H$ , we obtain the height- and time-averaged energy $\langle E_{m}\rangle$ of each mode. It is seen that these five modes account for over $80\,\%$ of the total energy within the current $Ra$ range. The $m = 1$ (first-order) mode dominates the flow field, while the energy contributions from the $m = 3$ and $m = 4$ modes are negligibly small across the entire range of $Ra$ . As $Ra$ increases, the energy contained in the first-order mode decreases, whereas the energy contained in the $m = 0$ (zeroth-order) mode increases. Meanwhile, the energy contained in $m = 2$ (second-order) mode remains relatively constant and smaller than that of the zeroth-order mode. Although the energy in the second-order mode is comparable to that of the zeroth-order mode, the zeroth-order mode exhibits a stronger correlation with the crossing events discussed earlier. Specifically, during the crossing events, the energy contained in the zeroth-order mode can even surpass that of the first-order mode, as we will demonstrate below. Therefore, to simplify the subsequent analysis, we will focus our discussion on the zeroth- and first-order modes exclusively. We have to emphasise that we conduct the modal analysis for the vertical velocity in the horizontal cross-section at different heights. So actually, these modes represent the vertical flow, not the horizontal flow. Through examining the amplitude/energy content of each mode at different heights, we can see how the vertical flow varies in the vertical direction.

Figure 7.

At $Ra=5.0 \times 10^{8}$ , for the same velocity field as shown in figure 5: (a) streamlines and horizontal cross-sections of the instantaneous flow field; (b) horizontal-averaged vertical and horizontal velocity magnitudes as functions of height, it is replotted here for better comparison; (c) energy contained in the $m=0$ and $m=1$ modes as functions of height; (d) variation of azimuthal angle of the $m=1$ mode with height.

Figure 7(c) shows the energy $E_m$ contained in these two modes at different heights for $Ra=5.0 \times 10^{8}$ . It is evident that at most heights, the energy contained in first-order mode is larger than that of the zeroth mode, suggesting that the dominant flow pattern in the cell is represented by ascending flow on one side, and descending flow on the other side. Upon comparing with the velocity magnitude profiles shown in figure 7(b), it is apparent that in the region where the first mode dominates, the magnitude of vertical velocity increases continuously, and reaches its maximum at approximately the middle height of the cell. Conversely, when the zeroth mode dominates, the magnitude of vertical velocity decreases, while the magnitude of horizontal velocity increases. Furthermore, it is found that, inside a crossing event at approximately $0.15H$ , the energy contained in the zeroth mode significantly increases and surpasses that of the first mode. Such evolution of zeroth and first mode with height can be used to identify the crossing event. Thus practically, we define a crossing event where the energy contained in the zeroth mode exceeds that of the first mode. We further verified that relaxing the criterion does not affect the experimental results (see Appendix A).

Given that the first mode is dominant in most of the height, its azimuthal angle can approximately represent the corresponding orientation of the flow at different heights. Figure 7(d) depicts the variation of the flow orientation with height. It can be observed that when a crossing event occurs, the orientation of flow field suddenly changes by $180^\circ$ within a very short vertical distance. In contrast, a twisting event results in continuous changes in orientation with height over the range of $0.4H$ to the top plate. Our measurements reveal that the rotating direction, the duration of a continuous rotation and the maximum rotation angle during the twisting event are all in a random manner. Furthermore, between $0.18H$ and $0.44H$ , the orientation of the flow remains unchanged, indicating a very stable non-twisting roll-like flow structure.

Figure 8.

At $Ra=5 \times 10^{9}$ : (a) streamlines and horizontal cross-sections of the instantaneous flow field; (b) horizontal-averaged vertical and horizontal velocity magnitudes as functions of height; (c) energy contained in the $m=0$ and $m=1$ modes as functions of height; (d) variation of azimuthal angle of the $m=1$ mode with height.

As $Ra$ increases, the flow field experiences more crossing and twisting events. Figures 8(a) and 8(c) show the instantaneous flow field at $Ra=5\times 10^{9}$ and the energy contained in the two dominant flow modes. Similar to the low $Ra$ case, the first mode dominates at most of the heights, while the energy contained in the zeroth mode surpasses that of the first mode at four heights where the crossing occurs. We note that at higher $Ra$ cases, the surpassing might not be as pronounced as that in relative lower $Ra$ . This is because the crossing event is not only the spatial feature of the flow field, but also a dynamic feature evolving with time. It is very likely that these moments are just before or after a crossing event in time. Or it could well be the case that the energy contained in the zeroth-order mode approaches, but eventually does not exceed that of the first-order mode. In current measurements, the sampling frequency is relatively slow, which may be not sufficient to capture all the instantaneous turbulent features, especially at high $Ra$ with smaller characteristic time scale. However, in general, the crossing event occurs more frequently at high $Ra$ , as shown in figure 8 compared to low $Ra$ . Furthermore, in the higher $Ra$ case, the orientation of the flow changes more rapidly, as can be seen in figure 8(d), which indicates that the twisting event also occurs more frequently. The presence of a large and stable roll-like flow structure with fixed orientation is hardly observed, which is evident from the velocity magnitudes and the energy shown in figures 8(b) and 8(c).

Figure 9.

At $Ra=9.3 \times 10^{8}$ : (a) streamlines and horizontal cross-sections of the instantaneous flow field; (b) horizontal-averaged vertical and horizontal velocity magnitudes as functions of height; (c) energy contained in the $m=0$ and $m=1$ modes as functions of height; (d) variation of azimuthal angle of the $m=1$ mode with height.

In addition to the previous findings, one can also see that in figures 7(c) and 8(c), near the boundaries, the zeroth-order mode dominates the flow field. This dominance of the zeroth-order mode can occur over short segments, as seen in top and bottom boundaries of figures 7(c) and 8(c), or it can extend over longer segments, as shown in figure 9(c). In figure 9, we show a typical instantaneous snapshot illustrating long-range dominance of the zeroth-order mode near both the top and bottom boundaries at $Ra = 9.3 \times 10^{8}$ . Upon closer examination of the flow field near the bottom boundaries, we find that such a long segment extends to nearly $0.2H$ , and the flow field exhibits a donut like structure (see the horizontal cross-sections). This donut like flow structure near the boundaries can be described as follows: the ascending/descending flow emerges and moves upwards/downwards near the sidewall, while the descending/ascending flow moves downwards/upwards along the (radial) centre of the cell until it reaches and impinges upon the plate, thus the annulus is formed. In figure 9(b), we observed that although the zeroth-order mode dominates the flow field similar to that in a crossing event, there are only small changes in the magnitude of both the vertical and horizontal velocities, which distinguishes it significantly from the behaviour inside a crossing event. It should be noted that such donut like flow structure was also reported by Hartmann et al. (Reference Hartmann, Chong, Stevens, Verzicco and Lohse2021) in their $\Gamma = 1/10$ cylindrical cell by DNS. Practically, we propose a threshold to identify the donut like structure, that is, if the prevalence of the zeroth mode over the first mode lasts for an extended length in the vertical direction (say, longer than $0.05H$ ) near the boundaries. It is important to emphasise that the donut like flow structure is found exclusively near the boundaries and is almost absent in the bulk region. Additionally, we found that the transition from the donut like flow structure to the roll-like flow structure occurs when the energy contained in the first-order mode exceeds that in the zeroth-order mode, which aligns with the characteristics of a crossing event. Therefore, for clarity, this flow event can also be referred to as a crossing event.

Figure 10.

Spatio-temporal evolution of the nodes and their corresponding PDFs. Each blue point represents a node. Here, (a) $Ra = 5.0 \times 10^{8}$ , (b) $Ra = 9.3 \times 10^{8}$ , (c) $Ra = 1.5 \times 10^{9}$ , (d) $Ra = 2.8 \times 10^{9}$ , (e) $Ra = 5 \times 10^{9}$ . Additionally, in (a), the five dashed lines correspond to the five time points at which the instantaneous flow field is shown in figure 11.

3.4. Temporal and spatial evolution of segments

Let us now discuss the spatio-temporal behaviour of the flow structure, especially focusing on the behaviour of the segment. Based on the results above, here, we divide the flow into segments, where each segment represents the flow portion between two successive crossing events. These segments can manifest as either roll-like flow structures (twisted or non-twisted) or donut like flow structures. To study the length of the segments and its evolution, we define the node separating two successive segments, which is the height where the zeroth mode reaches a local maximum, or the intersection point of the zeroth and first mode when a donut like structure exists near the boundaries. In our experiment, due to the small velocity magnitude near the top and bottom boundaries, we propose setting a threshold $0.05H$ ; segments shorter than the threshold are not identified.

Figure 10 shows a spatio-temporal evolution of nodes at different $Ra$ over 2000 s. Since we set the threshold $0.05H$ , there are no nodes within the region comprising $5\,\%$ of the total height below the top, and $5\,\%$ above the bottom. It can be seen that when $Ra = 5.0 \times 10^{8}$ and $ 9.3 \times 10^{8}$ , the number of the nodes is relative small, and they are mostly concentrated in the regions close to the top and bottom boundaries. From figure 10(a), it is evident that the segments near the bottom grow in length over time. In figures 11(a–d), we present the four instantaneous snapshots near the bottom boundary at four different time points, as labelled in figure 10(a). Over time, the height of the two nodes gradually increases, as shown by the dashed lines. Specifically, the lower segment grows in length, while the upper segment decreases in length. It appears that a cap prevents the lower segments from growing freely or exceeding this cap, which is located at approximately $0.4H$ . At certain time points (see figure 11 e), when the segments near the boundaries reach the cap at approximately $0.4H$ , they suddenly collapse and the nodes disappear. Unlike the continuous growth of segments from the bottom, the segments near the top boundary stay relatively unchanged in length, showing an asymmetry in the flow structure. Such an asymmetry in slender cells has been reported previously by Hartmann et al. (Reference Hartmann, Chong, Stevens, Verzicco and Lohse2021); they reported an asymmetric temperature distribution with height, through DNS with a $\Gamma =0.1$ cylindrical cell. They conjectured that such an asymmetric behaviour is related to the existence of a lower sublayer. Obviously, further understanding is required to elucidate the reasons for this asymmetry. For $Ra=9.3 \times 10^{8}$ , figure 10(b) exhibits relative symmetric distribution of the nodes. These nodes disappear at approximately $0.2H$ away from both top and bottom plates. A common feature of the first two low $Ra$ cases is that the bulk region has fewer nodes. The middle parts of both figures 10(a) and10(b) are relatively blank, indicating that longer segments can exist in the bulk. When $Ra$ is larger, the bulk region starts to exhibit more nodes, as shown in figure 10(c) for $Ra=1.5 \times 10^{9}$ , while for $Ra=2.8 \times 10^{9}$ and $5 \times 10^{9}$ , the entire height of the cell is filled with nodes, indicating that the crossing event occurs more uniformly at all heights. This suggests that the flow enters a more chaotic state, especially for the bulk region.

Figure 11.

Instantaneous three-dimensional flow field near the bottom boundary at the five different time points, corresponding to the dashed lines labelled a–e in figure 10(a). The flow field is visualised using streamlines of the instantaneous velocity field, with colour coding to represent both the direction and magnitude of the vertical velocity. Positive values indicate upward velocity, whereas negative values indicate downward velocity. The two dashed lines depict the tracks of the two nodes.

Figure 12.

Time evolution of the azimuthal angle of the $m=1$ mode at middle height for five different $Ra$ over 2000 s: (a) $Ra = 5.0 \times 10^{8}$ , (b) $Ra = 9.3 \times 10^{8}$ , (c) $Ra = 1.5 \times 10^{9}$ , (d) $Ra = 2.8 \times 10^{9}$ , (e) $Ra = 5 \times 10^{9}$ .

On the right-hand side of figure 10, we present the probability density functions (PDFs) of the heights where the nodes are located for five different $Ra$ values. For all $Ra$ , the PDF has a larger value near the boundaries, which indicates that the regions near boundaries have more nodes, i.e. more crossing events. The PDF in the bulk region is relatively flat, indicating no significant preference for the occurrence of crossing events at specific heights in the bulk region. As $Ra$ increases, the peak value of the PDF near the boundaries decreases, suggesting that more crossing events occur in the bulk region, indicating a more uniform distribution of crossing events across all heights.

In figure 12, we present the time evolution of the flow orientation (the azimuthal angle of the first-order mode) at middle height for five different $Ra$ values. For $Ra=5.0 \times 10^{8}$ and $9.3 \times 10^{8}$ , the orientation changes in only a very limited range. For $Ra=1.5 \times 10^{9}$ , the flow orientation at middle height starts continuous rotating but with no preferred direction. For $Ra=2.8 \times 10^{9}$ and $ 5 \times 10^{9}$ , the continuous rotating becomes much more rapid, and the rotating range becomes much larger, and this indicates a more chaotic state of the flow.

Figure 13.

The PDFs of (a) segment number, and (b) segment length, for the five different $Ra$ values.

Finally, we discuss the statistics of flow segments. Figure 13 presents the PDFs of segment number and segment length for different $Ra$ values. In figure 13(a), it is evident that as $Ra$ increases, the most probable segment number rises slightly, from approximately 2 to 4, with the peak probability decreasing from nearly 0.5 to approximately 0.3. At $Ra = 5.0 \times 10^8$ , the peak heights for segment numbers 2 and 3 are comparable. For $Ra = 9.3 \times 10^8$ and $Ra = 1.5 \times 10^9$ , the PDFs are quite pronounced, with the peak located at segment number 3. As $Ra$ continues to increase, the PDFs broaden, and the peak segment number gradually shifts higher. For example, at $Ra = 5.0 \times 10^9$ , 5–6 segments within the cell are frequently observed, as indicated by the PDF. Figure 13(b) shows the PDFs of segments length at five different $Ra$ values. For $Ra \leqslant 1.5\times 10^{9}$ , the PDF distribution can be interpreted as having two peaks: one peak corresponds to a longer segment, typically between $0.5H$ and $0.7H$ , while the other peak is associated with a shorter segment at approximately $0.1H$ . When $Ra \gt 2.8 \times 10^{9}$ , only shorter segments dominate, with peaks of segment lengths approximately $0.1H$ . One may wonder why the most probable segment length is only $0.1H$ when the most probable segment number is 4 at $Ra = 5.0 \times 10^9$ . This is reasonable since the segments are not equal sized. For example, there can be four segments, with three of them having length approximately $0.1H$ , and the other segment having length approximately $0.7H$ . Regarding the peak located at approximately $0.1H$ observed in all cases with varying $Ra$ , it implies that segments with equal length and width have a high probability of forming within the cell. According to Zwirner et al. (Reference Zwirner, Tilgner and Shishkina2020), mode transitions between multiple-roll states in a $\Gamma = 0.2$ cell are driven by the elliptical instability. For an elongated segment present in our experiments – such as the long segment above $0.15H$ shown in figure 5 – this elliptical instability presumably triggers its break-up, leading to the emergence of multiple-segment flow modes. Additionally, Zwirner et al. (Reference Zwirner, Tilgner and Shishkina2020) highlighted the role of prominent azimuthal flow during the break-up process, suggesting that such azimuthal motion can twist or disrupt single-roll flows. In our paper, we also illustrate twisting events at various heights through mode decomposition. Due to the elliptical instability, longer segments are likely to break into shorter ones. This instability-driven break-up process is constrained by a limit, where the roll/segment achieves a configuration with equal length and width (Waleffe Reference Waleffe1990). In figure 14, we display the average number of segments as a function of $Ra$ . One can see that the average number of segments increases with $Ra$ . This trend is consistent with the results presented in figure 13 and in figure 2.

Figure 14.

The average number of segments as a function of $Ra$ .