3.1. Heat transfer features

We first study the effect of the obstructed centre on the heat transport efficiency of the system. Figure 2(a) shows a log–log plot of Nusselt number $Nu$ as a function of $Ra$ for the four different cylindrical obstructions. Here, $Nu$ is calculated by $Nu = Q/(\lambda \Delta T/H)$, where $Q$ is the heat flux, which is the power provided by the heaters embedded in the bottom plate divided by its surface area, and $\lambda$ is the thermal conductivity of the working fluid. For comparison, we also plot the Nusselt number $Nu_0$ measured in the normal (unobstructed) cell (Chen et al. Reference Chen, Huang, Xia and Xi2019), and it can be described well by a single power law: $Nu_0 \sim Ra^{0.30}$. It can be clearly seen from figure 2(a) that when the cell centre is obstructed, the heat transport efficiency $Nu$ exhibits an overall enhancement, especially in the low-$Ra$ range. To gain a clearer insight of how the cylindrical obstruction affects the heat transport efficiency, we plot the normalized Nusselt number ($Nu/Nu_0$) as a function of $Ra$, as shown in figure 2(b). The value of $Nu_0$ used to compensate $Nu$ in figure 2(b) is determined from a power-law fit of measured $Nu_0$. We can see that in the low-$Ra$ range, $Nu$ is greatly enhanced, the maximal enhancements being roughly $21\,\%$, $16\,\%$, $3\,\%$ and $8\,\%$, respectively, for $D = 0.8L$, $0.5L$, $0.2L$ and $0.1L$. The error bars shown in figure 2 represent the root-mean-square values of $Nu$. As can be seen in figure 2(a), the error is very small which remains inside the symbols. Even though the error bars in figure 2(b) appear to be larger, it is essential to note that the magnitude of the heat transfer enhancement cannot be attributed to the uncertainty. It also should be mentioned that the enhancement of $Nu$ in the low-$Ra$ range exhibits a non-monotonic decrease with a decrease of the diameter of cylinder obstruction and the $D = 0.2L$ case, rather than the $D = 0.1L$ case, gives a minimum enhancement of $Nu$. And with an increase of $Ra$, the enhancement becomes weaker and $Nu$ eventually converges to $Nu_0$. The obstructed cell centre does not always result in an enhancement of $Nu$: one can see that for the $D = 0.2L$ case, in the range of $10^8 \leq Ra \leq 2 \times 10^8$ $Nu$ shows a reduction of up to $3\,\%$ of heat transfer which is not observed in the $D = 0.8L$ and $D = 0.5L$ cases.

Figure 2.

(a) Log–log plot of $Nu$ as a function of $Ra$ measured in the centre-obstructed cell. Here $Nu_0$ measured in the unobstructed normal cell is also plotted for comparison. (b) Semi-log plot of $Nu/Nu_0$ as a function of $Ra$.

It is unforeseen and rather counterintuitive to see such a large enhancement of heat transport for the $D = 0.8L$ case, because in this case about $50\,\%$ of the volume of the cell is obstructed by the cylindrical obstruction and only about $50\,\%$ of working fluid (compared with the unobstructed case) works as heat carrier to transport the heat from bottom plate to top plate. Similar heat transport enhancement with less working fluid was also revealed by Huang et al. (Reference Huang, Kaczorowski, Ni and Xia2013). They found the heat transport efficiency is greatly enhanced by introducing lateral confinement to the cell, i.e. by keeping the length and height of the cell unchanged, but decrease the width of the convection cell and correspondingly change the lateral aspect ratio $\varGamma _{lateral}$. A $15\,\%$ heat transport enhancement is achieved by confining the convection cell from $\varGamma _{lateral} = 0.6$ to $0.1$. It is worth emphasizing that this $21\,\%$ heat transfer enhancement is achieved by a simple obstruction inserted into the convection cell.

3.2. Flow topology

To reveal how this enhancement of heat transfer efficiency is achieved, we look closely into the flow structures in the convection cells with cylindrical obstructions. Figure 3 shows the evolution of the typical flow fields with increasing diameter of the cylindrical obstruction around three typical $Ra$ values, and the flow fields presented here are averaged over a short time period of one LSC turnover time $t_{E}$ to better illustrate the flow topology. Here, $t_{E}$ is obtained from the cross-correlation of temperature signals inside the top and the bottom plates. We first focus on figure 3(a–c), which shows the flow evolution with $Ra$ in the normal cell. When $Ra$ is small ($Ra = 6.36 \times 10^7$), the flow consists of four small rolls at each corner (see figure 3a), which is referred to as a four-roll state (FRS). It should be mentioned that the FRS shown here is not the outcome of superposition of two flow fields with opposite flow directions (Chong et al. Reference Chong, Wagner, Kaczorowski, Shishkina and Xia2018). The average time used to compute figure 3(a) is 80 s. In fact, the FRS is quite stable over the entire measurement duration. Even if we average the flow field over the entire measurement period, the average flow field remains unchanged as the FRS. It is worth noting that, according to the study by Chandra & Verma (Reference Chandra and Verma2013), the direction of the FRS cannot change due to the symmetry feature in the system. When $Ra$ increases to $Ra = 1.87 \times 10^8$, as shown in figure 3(b), the flow takes a form of a tilted-ellipse-shape single roll with two small rolls enclosed inside and two other small corner rolls diagonally sitting in opposite corners, which is named an abnormal single-roll state (ASRS) (Chen et al. Reference Chen, Huang, Xia and Xi2019). When $Ra$ further increases to $Ra = 7.94 \times 10^8$, it can be seen from figure 3(c) that the flow structure is made up of a tilted-ellipse-shape single roll without any substructures in it and two corner rolls diagonally sitting in opposite corners, this kind of flow structure being referred to as a single-roll state (SRS). This transition from FRS to ASRS and then to SRS in the normal cell was previously observed by Chen et al. (Reference Chen, Huang, Xia and Xi2019).

Figure 3.

Evolution of flow topology in cells with obstructed centre with different size of obstruction at/around the three typical $Ra$ values. The flow topology is illustrated by the streamlines of the short-time-averaged velocity field (averaged within one LSC turnover time $t_E$). The diameter of the cylindrical insert is (a–c) $0$ (normal cell), (d–f) $0.1L$, (g–i) $0.2L$, (j–l) $0.5L$ and (m–o) $0.8L$.

We then examine how the flow topology is modified by the cylindrical obstruction in the low-$Ra$ range ($Ra$ is around $1 \times 10^8$). In this low-$Ra$ range the flow is in the FRS in the normal cell as shown in figure 3(a). One can clearly see from figure 3(d) that the flow becomes an ASRS when the $D = 0.1L$ cylindrical obstruction is present. For the $D = 0.2L$ case (figure 3g), the flow also takes the form of an ASRS. When the diameter of the cylinder further increases ($D = 0.5L$, figure 3i), the flow pattern becomes a single-roll form without any substructures, i.e. SRS, and the corner rolls can also be clearly observed. If the diameter of the cylinder becomes even larger ($D = 0.8L$, figure 3m), due to the large volume of the central region being obstructed by the cylinder, the hot (cold) plumes are forced to move up (down) peripherally with two smaller but visible corner vortices located at top-left and bottom-right corners. We conclude that in the low-$Ra$ range, the presence of the cylindrical obstruction promotes the transition of the flow from FRS to ASRS (when the cylindrical obstruction is small), and from FRS to the SRS (when the cylindrical obstruction is large). We then study the modification of the flow topology due to the cylindrical obstruction in the intermediate-$Ra$ range ($Ra$ around $2 \times 10^8$). At this $Ra$, the flow field is in the ASRS in the normal cell. When the $D = 0.1L$ cylindrical obstruction is present (figure 3e), the flow field is promoted to the SRS. When $D$ increases to $0.2L$ (figure 3h), the main vortex becomes stronger and the two corner vortices become smaller. When $D$ further increases to $0.5L$ and $0.8L$ (figure 3k,n), the hot (cold) plumes are forced to move up (down) peripherally with two very small corner vortices. We conclude that in the intermediate-$Ra$ range, the presence of the cylindrical obstruction promotes the transition of the flow from ASRS to the SRS. When it comes to the large-$Ra$ range, the modification of flow topology is less significant, as shown in the third column of figure 3. One can see that the flow stays in the SRS when $D$ increases from $0.1L$ to $0.8L$, but with the increase of $D$, the main vortex becomes stronger, and the corner vortices become weaker.

Now, with the observed change in flow topology due to the presence of the obstruction, we can understand how the heat transport enhancement is achieved. For low $Ra$, i.e. $Ra \leq 1 \times 10^8$, a previous study has shown that in this regime the FRS dominates the flow field (Chen et al. Reference Chen, Huang, Xia and Xi2019). While when the centre of the cell is obstructed, FRS is replaced by ASRS or SRS (depending on the diameter of cylindrical obstruction). It has been shown that the SRS is more efficient in heat transfer than the FRS (Xu et al. Reference Xu, Chen, Wang and Xi2020, Reference Xu, Xu and Xi2023); thus this change of flow topology from FRS to ASRS/SRS leads to a prominent heat transfer enhancement in the low-$Ra$ regime. For high $Ra$, i.e. $Ra \geq 5 \times 10^8$, heat transfer measurements indicate that no significant enhancement is observed and this can be explained by the fact that in this regime the flow topology does not change much (always in SRS) when the cell centre is obstructed. In the normal cell, the LSC mostly moves in the periphery of the cell (Xia, Sun & Zhou Reference Xia, Sun and Zhou2003), and very few plumes travel through the central region directly; thus the obstruction added at the central region has negligible effects on the flow.

To quantitatively study the changes in flow topology due to the addition of the cylindrical obstructions, we decompose the measured PIV velocity field using Fourier mode decomposition. Fourier mode decomposition has been widely used in the RBC community to study the evolution of flow topology (Chandra & Verma Reference Chandra and Verma2013; Wagner & Shishkina Reference Wagner and Shishkina2013; Chong et al. Reference Chong, Wagner, Kaczorowski, Shishkina and Xia2018; Wang et al. Reference Wang, Xia, Wang, Sun, Zhou and Wan2018; Chen et al. Reference Chen, Huang, Xia and Xi2019, Reference Chen, Wang and Xi2020). Here we briefly introduce some key points of Fourier mode decomposition. By projecting the flow field $(u_x, u_z)$ onto the following artificially constructed Fourier basis:

(3.1)$$\begin{gather} u_x^{m,n} = 2\sin (m{\rm \pi} x)\cos (n{\rm \pi} z), \end{gather}$$

(3.2)$$\begin{gather}u_z^{m,n} ={-} 2\cos (m{\rm \pi} x)\sin (n{\rm \pi} z), \end{gather}$$

where $m, n = 1, 2, 3$ are considered, we can obtain the corresponding amplitude ${A^{m,n}}(t) = {\langle {{u_x}(t)u_x^{m,n}} \rangle _{x,z}} + {\langle {{u_z}(t)u_z^{m,n}} \rangle _{x,z}}$ of Fourier mode $(m,n)$. Here $x$ and $z$ are in units of $L$. To apply Fourier mode decomposition in the centre-obstructed cell, we set the velocity vectors inside the obstructed centre to be zero. Here the Fourier mode $(m,n)$ corresponds to the flow field with $m$ rolls in the $x$ direction and $n$ rolls in the $z$ direction (see their schematic patterns in figure 4a–d) . For example, the ($1,1$) mode represents a single-roll flow mode which has one roll in both $x$ and $z$ directions. As the first four flow modes, i.e. ($1,1$), ($2,1$), ($1,2$), ($2,2$), usually contain most of the energy in the flow, in practice we mainly focus on how these four Fourier modes evolve with $Ra$.

Figure 4.

(a–d) Schematic patterns of the first four Fourier modes of the flow in the convection cell. Time-averaged energy contained in the ($1, 1$) mode (e) and in the ($2, 2$) mode (f) as a function of $Ra$ for different diameter $D$ of the obstruction at the cell centre.

In figure 4(e), we plot the energy contained in the ($1,1$) flow mode $\langle E_{1,1} \rangle$ (normalized by the total flow energy $\langle E_{total}\rangle$) as a function of $Ra$ measured in different cases. Here, the energy contained in each of the Fourier modes is defined as $E^{m,n}(t) = {[A^{m,n}(t)]}^2$ (Xi et al. Reference Xi, Zhang, Hao and Xia2016). First of all, we can clearly see that the energy contained in the ($1,1$) mode increases with increasing $Ra$ in both the normal cell and the centre-obstructed cell. Besides, in the centre-obstructed cells, more and more energy is contained in the ($1,1$) mode than that in the normal cell, which implies that the main vortex is more and more energetic, especially for low $Ra$. For example, when the diameter of the cylinder insert is $D = 0.5L$, more than $70\,\%$ of energy is in the ($1,1$) mode at $Ra \simeq 1 \times 10^8$, while in the normal cell, only $20\,\%$ of energy is in the ($1,1$) mode. Here we do not have the data for the $D = 0.8L$ case, due to the fact that in this case the majority of the area of the cell is obstructed by the cylinder. Similarly in figure 4(f), we plot the energy contained in the ($2,2$) mode $\langle E_{2,2} \rangle$ as a function of $Ra$ measured in different cases. It is found that the energy contained in the ($2,2$) mode decreases with increasing $Ra$ in both the normal cell and the centre-obstructed cell. Besides, in the centre-obstructed cells, less energy is contained in the ($2,2$) mode than that in the normal cell, which implies that the corner vortices are less energetic in the centre-obstructed cell, especially for low $Ra$.

From the analysis above we have found a positive correlation between the ($1,1$) mode and $Nu$, i.e. the enhancement of $Nu$ is due to the stronger ($1,1$) mode as this mode is more efficient for heat transfer. Similarly we have found a negative correlation between the ($2,2$) mode and $Nu$, i.e. the enhancement of $Nu$ is also due to the weaker ($2,2$) mode as this mode is less efficient for heat transfer. As a result, in the low-$Ra$ range, we would expect that the enhancement of $Nu$ would decrease with diameter $D$ of the cylindrical obstruction. While the data for $D= 0.2L$ are not consistent with our expectation, the $Nu$ enhancement for $D= 0.2L$ is less than that for $D= 0.1L$. Obviously the change in flow topology (FRS is replaced by ASRS or SRS) alone cannot account for this non-monotonic decrease of maximum heat transport enhancement with decreasing $D$. The net viscous damping imposed by the cylinder surface ($A_{vis}$) very likely plays a role. The net viscous damping can be simply evaluated by the net increased surface area induced by the cylindrical obstruction, which is $(0.3L {\rm \pi}D - 0.5 {\rm \pi}D^2) = 0.5 {\rm \pi}(0.6LD - D^2)$, where $0.3L {\rm \pi}D$ is the curved surface area of the cylinder which is directly exposed to the working fluid and $0.5 {\rm \pi}D^2$ is the area of the two flat end surfaces of the cylinder which are exposed to the working fluid. As $A_{vis} \sim 0.6LD - D^2$, it is a quadratic function. The maximum $A_{vis}$ occurs at $D = 0.3L$, where the viscous damping would be the strongest. As a result, the heat transfer efficiency $Nu$ would be reduced the most. When $D$ becomes larger or smaller than $0.3L$, $A_{vis}$ decreases; thus the reduction of $Nu$ due to the viscous damping would be less. With the combined effects of the monotonic increase of the coherence of the flow and the quadratic dependence of the increased net surface area with the diameter of the cylinder, now we can understand why the $D = 0.2L$ case, instead of the $D = 0.1L$ case, gives the minimum heat transport enhancement. Certainly, the evaluation of net viscous damping here is quite crude and we do not take the change of flow topology into account, but the simple assumption generally explains our experimental results.

3.3. Flow reversals

In this section, we study how the reversal of the LSC is affected by the obstruction at the cell centre. To detect the reversal of the LSC, we use the temperature contrast method (Sugiyama et al. Reference Sugiyama, Ni, Stevens, Chan, Zhou, Xi, Sun, Grossmann, Xia and Lohse2010; Ni et al. Reference Ni, Huang and Xia2015; Huang & Xia Reference Huang and Xia2016; Chen et al. Reference Chen, Huang, Xia and Xi2019), i.e. identify the reversal event through the temperature contrast $\delta _T = T_{{left}} - T_{{right}}$, where $T_{{left}}$ and $T_{{right}}$ are the temperatures measured by the thermistors embedded on the left- and right-hand sides of the bottom plate. We employed commonly used threshold criteria, just as in many previous studies (Huang & Xia Reference Huang and Xia2016; Chen et al. Reference Chen, Huang, Xia and Xi2019, Reference Chen, Wang and Xi2020). The criteria, which are based on the amplitude and duration of the temperature contrast $\delta _T$, are the following. The first criterion is that the two circulation states of the LSC (clockwise direction and counter-clockwise direction) should be clearly distinguished from each other. This implies that the probability distribution function (PDF) of $\delta _T$ should have two peaks (two most probable values) with each peak corresponding to one circulation state, and the distance between the two peaks in the PDF should be larger than the root-mean-squared value of $\delta _T$, so that the two peaks can be clearly distinguished. The second criterion is that when the flow changes from one state to the other, $\delta _T$ should first cross over the peak corresponding to the current state then cross over the peak corresponding to the other state, i.e. the value of $\delta _T$ should be larger (smaller) than the higher (lower) peak value of $\delta _T$ in the PDF. The third criterion is that the flow should stay at the new state for more than one turnover time of the LSC. Figure 5 shows 50-hour-long time traces of temperature contrast $\delta _T$ measured in cells with four different sizes of cylindrical obstructions. From top to bottom, the diameter of cylindrical obstruction increases from $D = 0.1L$ to $D = 0.8L$. One can easily see that with increasing $D$, it is harder for reversal to occur. To compare, it was previously found in normal cells (Chen et al. Reference Chen, Huang, Xia and Xi2019), for $Ra = 1.1 \times 10^8$, $Pr = 5.7$, that more than 100 flow reversals occur in a single day. With the $D = 0.1L$ cylindrical obstruction, there are still 37.5 reversals within a day on average. When the diameter of the cylindrical obstruction increases to $D = 0.2L$, only 14 reversals have been identified in 50 hours. When $D$ becomes $0.5L$ and $0.8L$, no reversal can be observed in the 50-hour-long measurements. In addition to the reduced flow reversal rates, one can also observe the larger magnitude of $\delta _T$ in figure 5(d). This larger magnitude results from the much more coherent flow structure induced by the $0.8L$ obstruction, which is shown in figure 3.

Figure 5.

Time traces of temperature contrast $\delta _T$ for different diameter $D$ of the cylindrical obstruction but almost the same $Ra$: (a) $Ra = 9.74 \times 10^7$, $D = 0.1L$; (b) $Ra = 9.12 \times 10^7$, $D = 0.2L$; (c) $Ra = 9.41 \times 10^7$, $D = 0.5L$; (d) $Ra = 9.51 \times 10^7$, $D = 0.8L$. The dashed lines in (a,b) indicate the two flow states.

Figure 6.

Flow reversal frequency $f$ (normalized by $1/t_E$) as a function of $Ra$ for the $D = 0.1L$ obstruction. Reversal frequency measured in a normal cell (Chen et al. Reference Chen, Huang, Xia and Xi2019) is also plotted for comparison. The two lines are the power-law fits to the normal cell data and the centre-obstructed cell data and the vertical dashed lines indicate the two transitional $Ra$ values, respectively.

It was previously found that the flow reversal frequency decreases in a power-law manner with increasing $Ra$ (Chen et al. Reference Chen, Huang, Xia and Xi2019), while even at the lowest $Ra$ in our experiments, the reversal frequencies of $D = 0.2L$, $0.5L$ and $0.8L$ cases are very low; therefore it is time-consuming to acquire enough reversal statistics for these cases. In our measurements, in cases with higher reversal frequency (with low $Ra$ or small $D$), we typically require at least 100 reversal events for analysis. However, for those cases with lower reversal frequency (high $Ra$ or large $D$), acquiring a large number of reversal events can be extremely challenging. In such a situation, we extend the measurement to at least two days to obtain at least five events. Thus below we only show the results of reversal frequency measured in the $D = 0.1L$ case. Figure 6 shows the normalized reversal frequency $f$ as a function of $Ra$ for $D = 0.1L$, $Pr = 5.7$, where $f$ is defined as the average number of reversal events per second. To compare the reversal frequency $f$ measured from different cells, we use $1/t_E$ to non-dimensionalize the data. For comparison, the data measured in the normal cell are also plotted. The first feature one can see from figure 6 is that the reversal frequency is greatly reduced in cells with an obstructed central region, reduced by $80\,\%$ on average. The second feature one can see is that $f$ in the $D = 0.1L$ cell decreases with $Ra$, i.e. it is harder for reversal to occur with increasing $Ra$, which is consistent with that in the normal cell (Sugiyama et al. Reference Sugiyama, Ni, Stevens, Chan, Zhou, Xi, Sun, Grossmann, Xia and Lohse2010; Chen et al. Reference Chen, Huang, Xia and Xi2019) and also in a corner-less cell (Chen et al. Reference Chen, Wang and Xi2020). Meanwhile, one can see that $f$ in the $D = 0.1L$ cell extends to lower $Ra$ range which agrees with the flow topology change due to the central obstruction. In this low-$Ra$ range in the normal cell the flow is in the FRS, and thus there is no reversal of the flow. In the centre-obstructed cell the FRS is promoted to the SRS, and thus flow reversal is possible. The third feature one can see is that, similar to the normal cell case, the reversal frequency data in the $D = 0.1L$ cell exhibit two distinct regimes separated by a transitional $Ra$ ($Ra_{t}$): a slow-decay regime and a fast-decay regime, with different scaling relationships. The scaling exponents of the two distinct regimes are $ft_E \sim Ra^{-1.73}$ ($\,ft_E \sim Ra^{-1.06}$ in the normal cell) and $ft_E \sim Ra^{-2.87}$ ($\,ft_E \sim Ra^{-3.03}$ in the normal cell) and $Ra_{t}$ is around $1 \times 10^8$ ($Ra_{t} = 2 \times 10^8$ in the normal cell). One can see that the scaling exponents in the corresponding regime do not change too much from the normal cell to the centre-obstructed cell.

To gain quantitative relationships between the flow topology and reversal frequency, we study the evolution of different Fourier modes with $Ra$ for different sizes of the cylindrical obstructions. In figure 7, we show the mean energy $\langle E^{m,n}(t) \rangle$ contained in each of the first four modes as a function of $Ra$ for both the $D = 0.1L$ cell and the normal cell. Previously we know that in the normal cell (as shown in figure 7b) both the ($2,1$) and ($1,2$) modes are negligibly weak over the whole $Ra$ range and the ($1,1$) and ($2,2$) modes dominate the flow field. Meanwhile, the $Ra$ dependencies of the ($1,1$) and ($2,2$) modes show opposite trends and consequently there exists a cross-over $Ra$ which had been found to be identical to the transitional $Ra$ in the reversal frequency (Chen et al. Reference Chen, Huang, Xia and Xi2019). It should be noted that the dominance of the ($2,2$) mode does not lead to a FRS. Previous study (Chen et al. Reference Chen, Huang, Xia and Xi2019) had revealed that although the ($2,2$) mode overtakes the ($1,1$) mode at low $Ra$, the flow structure still takes the form of a single roll (LSC) but with substructures inside, and thus this flow state was named as an ASRS. In the ASRS regime, the ($2,2$) mode dominates the flow field, yet the energy contained in the ($2,2$) mode remains below 50 $\%$. As $Ra$ decreases further, more of the energy shifts to the ($2,2$) mode. For instance, when 60 $\%$ of the energy is contained in the ($2,2$) mode, the flow field is in the FRS, and LSC no longer exists. Because the ($1,1$) and the ($2,2$) modes are mostly contributed by the main vortex and corner vortices respectively, the evolution of the ($1,1$) and the ($2,2$) modes is essentially consistent with the flow topology transition shown in figure 3(a–c), i.e. the transition from FRS to ASRS and finally to SRS. Hence one important conclusion drawn for the normal cell is the one-to-one correspondence between the transition of reversal rate and the transition of flow topology, which implies the transition of reversal frequency originates from the transition of flow topology (Chen et al. Reference Chen, Huang, Xia and Xi2019). In the cell with the obstructed centre, similar trends of the ($1,1$) and ($2,2$) modes can be found, as shown in figure 7(a) for the $D = 0.1L$ case. There is also a cross-over $Ra$ between the ($1,1$) and ($2,2$) modes, and the cross-over $Ra$ is very close to $Ra_{t}$ of the reversal frequency (also plotted in figure 7(a) as a dot-dashed line). Besides, the flow topology evolution exhibited in figure 3 can also be essentially described by the Fourier mode evolution with $Ra$. This implies that the one-to-one correspondence between the transition of reversal frequency and the transition of flow topology found in the normal cell also pertains in the centre-obstructed cell and this provides further evidence of the general mechanism responsible for the transition of the reversal frequency.

Figure 7.

Time-averaged energy contained in each flow mode as a function of $Ra$ measured in (a) centre-obstructed cell ($D = 0.1L$) and (b) normal cell. The dot-dashed lines indicate the transitional $Ra$ for the reversal frequency obtained from figure 6. The normal cell data are from Chen et al. (Reference Chen, Huang, Xia and Xi2019) for comparison.

Previously we have found that the transition of the scaling of the frequency of the flow reversal is controlled by the transition of the stability of the ($1,1$) mode in the unobstructed normal cell (Chen et al. Reference Chen, Huang, Xia and Xi2019). Now in the centre-obstructed cell we have also found a transition of the flow reversal frequency, it is natural to ask whether this transition is also controlled by the same mechanism, i.e. the transition of the stability of the ($1,1$) mode. To answer this question we examine the stability of the Fourier modes in the obstructed cells. The stability $S^{m,n}$ of the $(m,n)$ mode is defined as the ratio of the mean and the root-mean-square value of the amplitude of the $(m,n)$ mode $\langle A^{m,n}\rangle / \sigma _{A^{m,n}}$ (Chen et al. Reference Chen, Huang, Xia and Xi2019). According to the definition, a larger $S^{1,1}$ means a more stable main vortex; similarly a larger $S^{2,2}$ means more stable corner vortices. In figure 8, we plot the normalized $Ra$ dependence of the stability of the ($1,1$) and ($2,2$) modes in the $D = 0.1L$ cell, and for comparison the data measured in the normal cell are also plotted. One can see that the magnitude of $S^{1,1}$ in the $D = 0.1L$ obstruction case is very close to that in the normal cell case. In addition, the scaling of $S^{1,1}$ (with $Ra$) is also divided into two scaling regimes and the exponents of the slow increase and the fast increase are very close to those of the normal cell case. This implies that the transition of the reversal frequency scaling (with $Ra$) in the centre-obstructed cell is indeed also controlled by the transition of the stability of the ($1,1$) mode. Moreover, in the centre-obstructed cell the transitional Rayleigh number $Ra_{tr}$ for the reversal frequency and that for the change of the stability of the ($1,1$) mode are also almost the same.

Figure 8.

(a) The stability of the ($1,1$) mode $S^{1,1}$ and (b) the stability of the ($2,2$) mode $S^{2,2}$ as functions of the normalized $Ra$ for the centre-obstructed cell ($D = 0.1L$) and the normal cell. The solid lines in (a) are the power-law fits. Data from both the $D = 0.1L$ cell and the normal cell are included for the fit and two blue points which are very close to the transition $Ra$ are not included in the fitting. The normal cell data are from Chen et al. (Reference Chen, Huang, Xia and Xi2019) for comparison.

If we check figure 4 again, one may realize that the small enhancement of $\langle E^{1,1}\rangle$ and small reduction of $\langle E^{2,2}\rangle$ in the $D = 0.1L$ cell (compared with that in the normal cell) may not match the very large reduction in reversal frequency shown in figure 6. In figure 8(b), we show $S^{2,2}$ as a function of $Ra$, and one can see that the values of $S^{2,2}$ show a great reduction compared with those in the normal cell, which means that the ($2,2$) mode in the obstructed cell is less stable than that in the normal cell. As the reversal is accomplished by the corner vortices squeezing the main vortex, a stronger main vortex (($1,1$) mode), weaker corner vortices (($2,2$) mode), together with the less stable corner vortex (($2,2$) mode) will result in a great reduction of the reversal frequency. Thus the great reduction of the reversal frequency in the obstructed cell is attributed to the combined effects of the enhanced energy contained in the ($1,1$) mode, reduced energy contained in the ($2,2$) mode and the reduced stability of the ($2,2$) mode.

Moreover, it should be noted that the above discussion only gives an implication that the main-vortex-breaking stage of the reversal process is strongly affected by the cylindrical obstruction; however, whether the obstruction affects the corner vortex connection process is still not known to us. Our conjecture is that the existence of the obstruction at the centre of the cell would obstruct the two corner vortices from connecting. Unfortunately due to the extremely low frequency of the reversal, we are not able to catch one reversal process during our flow visualization; thus we are not able to give an accurate answer to this question and this is definitely an important aspect waiting for further study.