3.1 Flow reversals in two-dimensional tilted cells

We start by studying the influence of tilt on flow reversals for a typical aspect ratio $\unicode[STIX]{x1D6E4}=1$ . Figure 1 shows snapshots of the temperature and velocity fields during one of the reversals at $Ra=5\times 10^{7}$ , $Pr=2$ and $\unicode[STIX]{x1D6FD}=6^{\circ }$ . Although the symmetry of the system is broken, it is found that the reversal process is similar to the level case (Sugiyama et al. Reference Sugiyama, Ni, Stevens, Chan, Zhou, Xi, Sun, Grossmann, Xia and Lohse2010). The two corner rolls grow simultaneously in kinetic energy and also in size due to the plume detachment from the boundary layers, until they merge as a new LSC with the opposite direction, and a flow reversal happens. It is well known that flow reversals occur in a certain $Ra{-}Pr$ parameter space for $\unicode[STIX]{x1D6E4}=1$ . For a system with only one main roll, the reversal process can be nicely indicated by a global quantity, i.e. the net angular momentum

(3.1) $$\begin{eqnarray}L(t)=\langle -(z-1/2)v(\boldsymbol{x},t)+(y-1/2)w(\boldsymbol{x},t)\rangle _{V},\end{eqnarray}$$

where $\langle \cdot \rangle _{V}$ represents averaging over the total area. Figure 2 shows the time evolution of $L$ for the level case with $\unicode[STIX]{x1D6FD}=0^{\circ }$ and the tilted case with $\unicode[STIX]{x1D6FD}=6^{\circ }$ at $Ra=5\times 10^{7}$ and $Pr=2$ . A sign change of $L$ can reveal a reversal; frequent flow reversals in a level cell can be observed in figure 2(a). From figure 2(b), it is evident that the symmetry breaking induced by tilt greatly suppresses flow reversals for $\unicode[STIX]{x1D6E4}=1$ . One sees that the clockwise main roll with negative $L$ stays for quite a short time before reversing its orientation. This is because, for the clockwise LSC, as shown in figure 1(c), tilt introduces a horizontal buoyancy force along the $y$ direction so that the hot plumes are more likely to move up to the bottom-right corner roll and the cold plumes tend to move down to the top-left corner roll. The two corner rolls have more opportunities to grow and trigger a reversal. Besides, the tilt may weaken the clockwise LSC since along the hot and cold plates the horizontal buoyancy force is in the opposite direction to the flow velocity of LSC. Thus the clockwise LSC is less stable and is easier to reverse its orientation, and it always stays for a much shorter time as shown in figure 2(b). Previous studies have shown that the main roll will be locked in the plane along the tilted direction both in cylindrical cells (Xi, Lam & Xia Reference Xi, Lam and Xia2004; Brown & Ahlers Reference Brown and Ahlers2006; Shishkina & Horn Reference Shishkina and Horn2016) and in rectangular cells (Guo et al. Reference Guo, Zhou, Cen, Qu, Lu, Sun and Shang2015, Reference Guo, Zhou, Qu, Lu, Sun and Shang2017) with $\unicode[STIX]{x1D6E4}=1$ . Thus the suppression of flow reversals by tilt for $\unicode[STIX]{x1D6E4}=1$ is within expectation.

Figure 2.

Time evolution of the angular momentum $L$ for $\unicode[STIX]{x1D6E4}=1,~Ra=5\times 10^{7}$ and $Pr=2$ : (a)  $\unicode[STIX]{x1D6FD}=0^{\circ }$ and (b)  $\unicode[STIX]{x1D6FD}=6^{\circ }$ . The positive and negative signs indicate anticlockwise and clockwise circulations, respectively.

We then map out phase diagrams for $\unicode[STIX]{x1D6E4}=1$ over $\unicode[STIX]{x1D6FD}{-}Pr$ parameter space with $Ra=5\times 10^{7}$ and over $\unicode[STIX]{x1D6FD}{-}Ra$ parameter space with $Pr=2$ in figure 3. We run for at least 5000 large-eddy turnover times $t_{E}$ to detect whether flow reversals exist. Time $t_{E}$ is defined by $t_{E}:=4\unicode[STIX]{x03C0}/\langle |\unicode[STIX]{x1D714}_{c}(t)|\rangle$ , where $\unicode[STIX]{x1D714}_{c}$ denotes vorticity in the centre of the main roll. For some cases (for example, $Ra=10^{8},~Pr=2,~\unicode[STIX]{x1D6E4}=1,~\unicode[STIX]{x1D6FD}=3^{\circ }$ ), we even calculate for more than $10\,000\,t_{E}$ to conclude there are indeed no reversals. As shown in figure 3(a), the critical angle $\unicode[STIX]{x1D6FD}_{c}$ displays a non-monotonic relation with $Pr$ . For larger $Pr$ , the $\unicode[STIX]{x1D6FD}_{c}$ value is smaller; for $Pr=5$ , a minor tilt of $1^{\circ }$ will completely suppress flow reversals due to the fact that the thickness of the thermal BL is much smaller than the thickness of the kinetic BL and thermal coupling between the corner roll and the thermal BL is weakened, as stated previously (Sugiyama et al. Reference Sugiyama, Ni, Stevens, Chan, Zhou, Xi, Sun, Grossmann, Xia and Lohse2010). For small $Pr$ , thermal diffusion becomes dominant and suppresses the growth of the corner roll. Therefore, flow reversals in the tilted cell also occur more easily at intermediate $Pr$ , and the corresponding $\unicode[STIX]{x1D6FD}_{c}$ is also larger. In figure 3(b), $\unicode[STIX]{x1D6FD}_{c}$ also displays a non-monotonic relation with $Ra$ . For intermediate $Ra$ , the corresponding $\unicode[STIX]{x1D6FD}_{c}$ is larger. The reason lies in the non-monotonic dependence of the flow reversal frequency on $Ra$ . It can be seen in figure 3 of Sugiyama et al. (Reference Sugiyama, Ni, Stevens, Chan, Zhou, Xi, Sun, Grossmann, Xia and Lohse2010) that flow reversals are most frequent for intermediate $Ra$ (approximately $7\times 10^{7}$ for $Pr=4.3$ ); any increase or decrease of $Ra$ will reduce the flow reversal frequency. For the $Pr=2$ considered in our work, the reversals are most frequent for $Ra\sim 4\times 10^{7}$ , thus the corresponding $\unicode[STIX]{x1D6FD}_{c}$ is larger, and any increase or decrease of $Ra$ will reduce $\unicode[STIX]{x1D6FD}_{c}$ . The reversals occur less and less frequently with increasing $Ra$ and reversals disappear when $Ra\geqslant 2\times 10^{8}$ for $Pr=2$ (Sugiyama et al. Reference Sugiyama, Ni, Stevens, Chan, Zhou, Xi, Sun, Grossmann, Xia and Lohse2010). Thus the main roll here is more sensitive to tilt at larger $Ra$ , and a minor tilt of the cell will suppress flow reversals at relatively high  $Ra$ .

Figure 3.

(a) Phase diagram in the $\unicode[STIX]{x1D6FD}{-}Pr$ plane for $Ra=5\times 10^{7},~\unicode[STIX]{x1D6E4}=1$ . (b) Phase diagram in the $\unicode[STIX]{x1D6FD}{-}Ra$ plane for $Pr=2,~\unicode[STIX]{x1D6E4}=1$ . Red filled diamonds (♦) correspond to detected flow reversals, and green squares (▪) to a single-roll state without reversals.

Compared with $\unicode[STIX]{x1D6E4}=1$ , there have been fewer studies devoted to horizontally adjacent double-roll reversals in a $\unicode[STIX]{x1D6E4}=2$ cell (Chandra & Verma Reference Chandra and Verma2011; Verma et al. Reference Verma, Ambhire and Pandey2015). Here, we first explore the flow reversals in a $\unicode[STIX]{x1D6E4}=2$ cell at a fixed $Ra=10^{7}$ , but varying $Pr$ and $\unicode[STIX]{x1D6FD}$ , as listed in table 1. The $Ra$ dependence for a fixed $Pr=0.3$ is also studied. Figure 4 shows flow reversal scenarios for a level case with $Ra=10^{7}$ , $Pr=0.71$ and $\unicode[STIX]{x1D6E4}=2$ ; it is seen that the single corner roll growth leads to a flow reversal, which is qualitatively the same as that of Verma et al. (Reference Verma, Ambhire and Pandey2015). Owing to the inherent symmetry of the system, the two bottom corner rolls in figure 4(a) and the two top corner rolls in figure 4(f) all have opportunities to grow and trigger flow reversals. However, when the symmetry of the system is broken by tilt, the two corner rolls will no longer play equivalent roles.

Figure 4.

Six snapshots of the temperature field superimposed with velocity vectors during one of the flow reversals for $Ra=10^{7},~Pr=0.71,~\unicode[STIX]{x1D6FD}=0^{\circ }$ at different times: (a)  $t=12\,283$ , (b)  $t=12\,338$ , (c)  $t=12\,349$ , (d)  $t=12\,354$ , (e)  $t=12\,361$ and (f)  $t=12\,377$ .

Figure 5.

Nine snapshots of the temperature field superimposed with velocity vectors during two of the flow reversals for $Ra=10^{7},~Pr=0.71,~\unicode[STIX]{x1D6FD}=1^{\circ }$ at different times: (a)  $t=1684.5$ , (b)  $t=1701$ , (c)  $t=1704$ , (d)  $t=1710$ , (e)  $t=1711.5$ , (f)  $t=1725$ , (g)  $t=2860.5$ , (h)  $t=2869.5$ and (i)  $t=2890.5$ .

Figure 5 shows snapshots for two successive flow reversals at $Ra=10^{7},~Pr=0.71$ and $\unicode[STIX]{x1D6FD}=1^{\circ }$ . As shown in figure 5(a), the flow field is characterized by two main rolls and two bottom corner rolls. Since tilt will introduce a horizontal buoyancy force along the $y$ direction, the hot plumes detached from the lower BL will bear a top-right buoyancy force and be apt to feed the bottom-right corner roll. This is favourable for the growth of the bottom-right corner roll as illustrated in figure 5(b). The bottom-left corner roll, however, almost remains unchanged. The subsequent reversal process shown in figure 5(c–f) is similar to that in a level case. Similarly, for the next reversal cycle, due to the $y$ direction buoyancy force introduced by tilting the cell, the cold plumes detached from the upper BL are more likely to enter the top-left corner roll. Thus the top-left corner roll growth will lead to a flow reversal, as shown in figure 5(g–i). It should also be noted that the horizontal buoyancy force induced by tilt near the top and bottom walls is always in the opposite direction to the flow velocity of one of the two main rolls. Thus the effect of tilt on the main rolls may always play some roles in flow reversals in a tilted $\unicode[STIX]{x1D6E4}=2$ cell, and the unstable clockwise main roll will always first reverse its direction. Owing to the symmetry breaking, only the bottom-right corner roll in figure 5(a) and the top-left corner roll in figure 5(f) have opportunities to grow and lead to a flow reversal; this is different from a level case where the four corner rolls all have opportunities to grow and trigger a flow reversal. This argument can be quantitatively demonstrated by investigating the long-time evolution of $L$ as shown in figure 6. Note that $L$ is calculated with respect to the centre point $(1,0.5)$ here. The flow is dominated by the double-roll mode with $L\simeq 0$ most of the time due to compensation of two main rolls. When flow reversal occurs, the clockwise/anticlockwise intermediate single-roll mode appears (see figures 4 c and 5 c), and thus large negative/positive $L$ will appear, indicated by sharp spikes in figure 6. The large positive $L$ indicates that it is the growth of the bottom-right corner roll shown in figure 5(a) or the top-left corner roll shown in figure 5(f) that leads to flow reversals, while the large negative $L$ indicates that it is the growth of the bottom-left corner roll shown in figure 4(a) or the top-right corner roll shown in figure 4(f) that leads to a successful flow reversal. It is seen that both positive and negative spikes exist for the level case while only positive spikes exist for the tilted case. Thus only the growth of the bottom-right corner roll or the top-left corner roll will lead to a flow reversal in a tilted cell. From figure 6, we can also see that there are many more flow reversals in tilted cells, demonstrating that tilt will promote flow reversals for $\unicode[STIX]{x1D6E4}=2$ . To further quantify the promoting effect of cell tilt on the flow reversal, the reversal frequencies $f$ defined by reversal numbers per free-fall time unit are calculated, which are plotted in figure 7. Clearly, for $\unicode[STIX]{x1D6E4}=1$ , $f$ does not drop much for small $\unicode[STIX]{x1D6FD}\leqslant 2^{\circ }$ , while for $\unicode[STIX]{x1D6FD}>2^{\circ }$ , $f$ drops rapidly. In contrast, $f$ increases monotonically with $\unicode[STIX]{x1D6FD}$ for $\unicode[STIX]{x1D6E4}=2$ ; it even increases more than an order from $\unicode[STIX]{x1D6FD}=0^{\circ }$ to $1^{\circ }$ . Thus a minor tilt will greatly promote flow reversals for $\unicode[STIX]{x1D6E4}=2$ .

Figure 6.

Time evolution of angular momentum $L$ for $Ra=10^{7},~Pr=0.71$ and $\unicode[STIX]{x1D6E4}=2$ : (a)  $\unicode[STIX]{x1D6FD}=0^{\circ }$ , (b)  $\unicode[STIX]{x1D6FD}=0.5^{\circ }$ and (c)  $\unicode[STIX]{x1D6FD}=1^{\circ }$ .

Figure 7.

Flow reversal frequency $f$ as a function of $\unicode[STIX]{x1D6FD}$ . Red circles (○) for $Ra=5\times 10^{7},~Pr=2,~\unicode[STIX]{x1D6E4}=1$ ; blue diamonds (♢) for $Ra=10^{7},~Pr=0.71,~\unicode[STIX]{x1D6E4}=2$ .

Figure 8.

(a) Phase diagram in the $\unicode[STIX]{x1D6FD}{-}Pr$ plane for $Ra=10^{7},~\unicode[STIX]{x1D6E4}=2$ . (b) Phase diagram in the $\unicode[STIX]{x1D6FD}{-}Ra$ plane for $Pr=0.3,~\unicode[STIX]{x1D6E4}=2$ . Red filled diamonds (♦) correspond to detected flow reversals; blue filled circles (●) denote double-roll state without flow reversals; green squares (▪) represent single-roll state without flow reversals.

Again we map out flow states in $\unicode[STIX]{x1D6FD}{-}Pr$ space for $Ra=10^{7}$ and $\unicode[STIX]{x1D6E4}=2$ in figure 8(a). We run for at least $4000\,t_{E}$ to detect whether flow reversals exist. As the position of the centre of the main roll changes after reversals (this is especially evident for reversals with large $\unicode[STIX]{x1D6FD}$ ), the large-eddy turnover time $t_{E}$ is better defined by $t_{E}:=(W+2H)/u_{LSC}$ , where $u_{LSC}$ is the typical velocity of the LSC measured at ${\sim}H/10$ from the sidewalls and plates (Zhou et al. Reference Zhou, Sugiyama, Stevens, Grossmann, Lohse and Xia2011; Ni et al. Reference Ni, Huang and Xia2015). Three different flow states are identified, which are the double-roll state without flow reversals, the double-roll state with flow reversals, and the single-roll state without flow reversals, respectively. One sees that, for $Pr\leqslant 0.5$ and small $\unicode[STIX]{x1D6FD}$ , the flow stays in the stable double-roll state without flow reversals. When $\unicode[STIX]{x1D6FD}$ is beyond a critical value, the double-roll state is no longer stable and flow reversals occur in the intermediate range of $\unicode[STIX]{x1D6FD}$ . However, when $\unicode[STIX]{x1D6FD}$ increases further, the flow becomes a stable one-roll state and no reversals can be observed any more. For $0.71\leqslant Pr\leqslant 4.38$ , the flow transits directly from reversal state to one-roll state. It is interesting that for relatively small $Pr\leqslant 0.5$ , flow reversals can only occur in tilted cells. It is worth mentioning that the reversals even occur at very small $Pr$ and large $\unicode[STIX]{x1D6FD}$ . For instance, there exist flow reversals for $Pr=0.05$ and $\unicode[STIX]{x1D6FD}=70^{\circ }$ , as seen in figure 8(a). This is in stark contrast to that in a $\unicode[STIX]{x1D6E4}=1$ cell where no reversals exist for $Pr\leqslant 0.71$ (Sugiyama et al. Reference Sugiyama, Ni, Stevens, Chan, Zhou, Xi, Sun, Grossmann, Xia and Lohse2010). Because flow reversals in tilted cells with $\unicode[STIX]{x1D6E4}=2$ tend to occur for small $Pr$ , the $Ra$ dependence of flow reversals is studied for a representative small $Pr=0.3$ . Figure 8(b) maps out flow states over the $\unicode[STIX]{x1D6FD}{-}Ra$ parameter space for fixed $Pr=0.3$ . It is obvious that flow reversals can only be found in tilted cells, which even exist at $Ra=10^{8}$ . Based on the above findings, we show that the symmetry breaking of the system induced by tilt has considerably broadened the original $Ra{-}Pr$ parameter space of flow reversals for $\unicode[STIX]{x1D6E4}=2$ . However, the crucial question left in this section is why reversals only occur in a limited range of $\unicode[STIX]{x1D6FD}$ for $\unicode[STIX]{x1D6E4}=2$ .

3.2 Mode decomposition analysis

To find the answer to the question left in § 3.1, a mode decomposition method is employed. We project the velocity components $v$ and $w$ onto the following Fourier basis:

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{v}^{m,n}=\sqrt{2}\sin (m\unicode[STIX]{x03C0}y/2)\cos (n\unicode[STIX]{x03C0}z), & \displaystyle\end{eqnarray}$$

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle {\hat{w}}^{m,n}=-\sqrt{2}\cos (m\unicode[STIX]{x03C0}y/2)\sin (n\unicode[STIX]{x03C0}z). & \displaystyle\end{eqnarray}$$

The velocity can be recovered by

(3.4) $$\begin{eqnarray}\displaystyle & \displaystyle v(y,z,t)=\mathop{\sum }_{m,n}A_{y}^{m,n}(t)\hat{v}^{m,n}, & \displaystyle\end{eqnarray}$$

(3.5) $$\begin{eqnarray}\displaystyle & \displaystyle w(y,z,t)=\mathop{\sum }_{m,n}A_{z}^{m,n}(t){\hat{w}}^{m,n}. & \displaystyle\end{eqnarray}$$

The projection is done componentwise by a scalar product in the $L_{2}$ -space. The corresponding time series $A_{y}^{m,n}(t)$ and $A_{z}^{m,n}(t)$ can be obtained by

(3.6) $$\begin{eqnarray}\displaystyle & \displaystyle A_{y}^{m,n}(t)=\int _{0}^{y_{max}}\int _{0}^{z_{max}}v(y,z,t)\hat{v}^{m,n}\,\text{d}y\,\text{d}z, & \displaystyle\end{eqnarray}$$

(3.7) $$\begin{eqnarray}\displaystyle & \displaystyle A_{z}^{m,n}(t)=\int _{0}^{y_{max}}\int _{0}^{z_{max}}w(y,z,t){\hat{w}}^{m,n}\,\text{d}y\,\text{d}z. & \displaystyle\end{eqnarray}$$

A value $M^{m,n}(t)$ representing the contribution of different modes can be evaluated by $M^{m,n}(t)=\sqrt{A_{y}^{m,n}(t)^{2}+A_{z}^{m,n}(t)^{2}}$ . The mode $(m,n)$ corresponds to a flow structure with $m$ rolls in the $y$ direction and $n$ rolls in the $z$ direction. Although Fourier basis functions do not satisfy the no-slip boundary conditions, it has been shown that they capture the convective flow profiles well (Chandra & Verma Reference Chandra and Verma2011, Reference Chandra and Verma2013; Petschel et al. Reference Petschel, Wilczek, Breuer, Friedrich and Hansen2011; Verma et al. Reference Verma, Ambhire and Pandey2015). The Fourier mode decomposition method has been employed extensively to study 2D reversals (Chandra & Verma Reference Chandra and Verma2011, Reference Chandra and Verma2013; Petschel et al. Reference Petschel, Wilczek, Breuer, Friedrich and Hansen2011; Verma et al. Reference Verma, Ambhire and Pandey2015) and heat transfer in quasi-2D cells (Wagner & Shishkina Reference Wagner and Shishkina2013; Chong et al. Reference Chong, Wagner, Kaczorowski, Shishkina and Xia2018).

We quantitatively analyse the flow mode competition for a representative case with $Ra=10^{7},~Pr=0.5$ and $\unicode[STIX]{x1D6E4}=2$ where flow reversals only happen in a limited $\unicode[STIX]{x1D6FD}$ range, i.e. $7^{\circ }\leqslant \unicode[STIX]{x1D6FD}\leqslant 13^{\circ }$ . Figure 9 shows the time evolution of the amplitudes of dominated modes $(2,1)$ and $(1,1)$ , and the vertical velocity at the centre of the cell $w_{c}$ for $\unicode[STIX]{x1D6FD}=8^{\circ }$ . A sign change of $w_{c}$ can nicely indicate a reversal. The flow near the times $t=890$ and $2110$ , which are denoted by vertical dash-dotted lines, can be viewed as cessations in a $\unicode[STIX]{x1D6E4}=2$ cell. The growth of corner rolls does not trigger successful reversals near these two moments. In figure 9, it is found that the transient $M^{2,1}(t)$ approaches zero during reversals, while $M^{1,1}(t)$ shows a spike. The vanishing of the $(2,1)$ mode and the appearance of the $(1,1)$ mode correspond to the appearance of the anticlockwise single-roll structures during the reversal. So we conclude that the appearance of single-roll mode $(1,1)$ plays a critical role in triggering a reversal, similar to that in a $\unicode[STIX]{x1D6E4}=1$ cell where the $(2,2)$ mode plays the dominant role.

Figure 9.

For $Ra=10^{7},~Pr=0.5$ and $\unicode[STIX]{x1D6FD}=8^{\circ }$ , the time evolution of (a) flow mode amplitude of $M^{2,1}(t)$ (red line) and $M^{1,1}(t)$ (black line); and (b) vertical velocity component $w_{c}$ at the centre location $(1,0.5)$ .

To understand why the flow reversals occur in a limited range of $\unicode[STIX]{x1D6FD}$ , the bifurcation diagram is further shown in figure 10. One sees that the flow is dominated by the $(2,1)$ mode for $0^{\circ }\leqslant \unicode[STIX]{x1D6FD}\leqslant 6^{\circ }$ and the amplitude of the $(1,1)$ mode approaches 0, which is referred to as regime I. The $M^{1,1}$ starts to grow rapidly beyond $\unicode[STIX]{x1D6FD}=7^{\circ }$ , then flow reversals occur for $7^{\circ }\leqslant \unicode[STIX]{x1D6FD}\leqslant 13^{\circ }$ , and we call this regime II. In this regime, the flow is still dominated by the $(2,1)$ mode, but the $M^{2,1}$ and $M^{1,1}$ are comparable. For $15^{\circ }\leqslant \unicode[STIX]{x1D6FD}\leqslant 90^{\circ }$ , the flow is dominated by the $(1,1)$ mode and the amplitude of the $(2,1)$ mode is close to 0, which is called regime III. It can be seen that the $M^{1,1}$ decreases monotonically with increasing $\unicode[STIX]{x1D6FD}$ in this regime; this is because the flow strength, which is denoted by the Reynolds number $Re$ , decreases monotonically with increasing $\unicode[STIX]{x1D6FD}$ in this regime (Guo et al. Reference Guo, Zhou, Cen, Qu, Lu, Sun and Shang2015; Shishkina & Horn Reference Shishkina and Horn2016). It is clear that the reversals tend to happen in the intermediate $\unicode[STIX]{x1D6FD}$ range where the $(2,1)$ mode and $(1,1)$ mode have comparable strengths. Based on the mode decomposition analysis, we conclude that the flow reversals in tilted cells for $\unicode[STIX]{x1D6E4}=2$ can be viewed as a mode competition process between the $(2,1)$ mode and $(1,1)$ mode.

Figure 10.

Time-averaged flow mode amplitudes $M^{1,1}$ and $M^{2,1}$ as a function of $\unicode[STIX]{x1D6FD}$ for $Ra=10^{7},~Pr=0.5$ and $\unicode[STIX]{x1D6E4}=2$ ; the grey area denotes the angle at which flow reversals appear.

3.3 Heat transfer during flow reversals

Finally, we discuss the heat transfer properties during a flow reversal. It is well recognized that heat transfer is closely connected to the mean flow structures in both two dimensions (van der Poel et al. Reference van der Poel, Stevens and Lohse2011) and three dimensions (Sun et al. Reference Sun, Xi and Xia2005; Xi & Xia Reference Xi and Xia2008b ; Bailon-Cuba, Emran & Schumacher Reference Bailon-Cuba, Emran and Schumacher2010; Weiss & Ahlers Reference Weiss and Ahlers2011; Xia Reference Xia2011; Huang et al. Reference Huang, Kaczorowski, Ni and Xia2013; Chong et al. Reference Chong, Huang, Kaczorowski and Xia2015). As shown above, during a flow reversal, the flow structures change greatly, so a significant change of heat transfer is expected. The Nusselt number ( $Nu$ ), used to measure heat transport, is defined as

(3.8) $$\begin{eqnarray}\displaystyle Nu(t)=\frac{\hat{Q}}{\hat{k}\unicode[STIX]{x0394}\hat{T}/{\hat{H}}}=\frac{(\langle \hat{\unicode[STIX]{x1D70C}}{\hat{c}}_{p}{\hat{w}}\hat{T}\rangle _{A}+\langle \hat{k}\unicode[STIX]{x2202}_{z}\hat{T}\rangle _{A})}{\hat{k}\unicode[STIX]{x0394}\hat{T}/{\hat{H}}}, & & \displaystyle\end{eqnarray}$$

where $\hat{Q}$ is the heat flux across any horizontal plane (3D) or line (2D), and $\langle \cdot \rangle$ denotes averaging over any horizontal plane (3D) or line (2D). Here, $Nu$ averaged over the hot and cold plate are denoted as $Nu_{h}$ and $Nu_{c}$ , respectively.

Figure 11.

Time evolution of (a) vertical velocity at the cell centre $w_{c}$ , (b) Nusselt number at the hot wall $Nu_{h}$ , and (c) Nusselt number at the cold wall $Nu_{c}$ , for $Ra=10^{7},~Pr=0.71$ and $\unicode[STIX]{x1D6FD}=1^{\circ }$ .

Figure 11 shows the time evolution of the vertical velocity component at the cell centre $w_{c}$ , $Nu_{h}$ and $Nu_{c}$ for $Ra=10^{7},~Pr=0.71,~\unicode[STIX]{x1D6E4}=2$ and $\unicode[STIX]{x1D6FD}=1^{\circ }$ . We distinguish the two kinds of reversals by the sign change of $w_{c}$ , referred to as REV1 and REV2. For REV1, $Nu_{h}$ experiences a minor decrease, and then an overshoot; after a relatively long time, it returns to its normal value; $Nu_{c}$ has a drastic decrease during the reversal; after an overshoot, it returns to its normal value quickly. For REV2, the behaviour of $Nu_{h}$ is similar to that of $Nu_{c}$ in REV1 and $Nu_{c}$ is similar to that of $Nu_{h}$ in REV1. These phenomena are similar to that in a level case. The most significant feature for $\unicode[STIX]{x1D6E4}=2$ is that $Nu_{h}$ and $Nu_{c}$ behave differently during a reversal.

Figure 12.

Time evolution of ensemble-averaged $Nu_{c}$ and $Nu_{h}$ during flow reversals for (a,b) $Ra=10^{7}$ , $Pr=0.71$ , $\unicode[STIX]{x1D6FD}=1^{\circ }$ and $\unicode[STIX]{x1D6E4}=2$ and (c,d)  $Ra=5\times 10^{7}$ , $Pr=2$ , $\unicode[STIX]{x1D6FD}=0^{\circ }$ and $\unicode[STIX]{x1D6E4}=1$ . (a) REV1. (b) REV2. (c) The reversal from the clockwise single-roll mode to the anticlockwise single-roll mode. (d) The inverse process of (c).

To better illustrate this point, the time evolutions of ensemble-averaged $Nu_{h}$ and $Nu_{c}$ are shown in figure 12. For $\unicode[STIX]{x1D6E4}=2$ , we first locate the data point where $w_{c}$ becomes zero corresponding to the appearance of a single-roll mode during reversals. Then starting from this data point we go forward and backward for 500 free-fall time units. We averaged over 59/62 samples for REV1/REV2, and the results are plotted in figure 12(a) and (b). Similar procedures were also used before (Castillo-Castellanos et al. Reference Castillo-Castellanos, Sergent and Rossi2016; Xi et al. Reference Xi, Zhang, Hao and Xia2016) and in the study of statistical properties of geomagnetic field reversals (Valet et al. Reference Valet, Fournier, Courtillot and Herrero-Bervera2012). One can see from figure 12(a) that both $Nu_{h}$ and $Nu_{c}$ drop to their minima at $t\approx 0$ corresponding to the appearance of a single-roll mode during REV1. The $Nu_{c}$ has a more drastic decrease than $Nu_{h}$ , due to apparent top-down asymmetric flow structures as shown in figure 5(c). There is also a small overshoot for $Nu_{h}$ before reaching its local minimum. After reaching their local minima, both $Nu_{h}$ and $Nu_{c}$ increase rapidly. The $Nu_{c}$ experiences some fluctuations around its local maximum, and then returns to its normal value. While for $Nu_{h}$ , two local maxima are found, and it takes more time for $Nu_{h}$ to return to its normal value. For REV2, the behaviours of $Nu_{h}$ and $Nu_{c}$ are reversed compared to REV1, as shown in figure 12(b). For comparison, $Nu_{h}$ and $Nu_{c}$ for $\unicode[STIX]{x1D6E4}=1$ during two kinds of flow reversals are also plotted in figures 12(c) and 12(d). The time $t=0$ is the time when the global angular momentum $L$ becomes 0, corresponding to the appearance of the $(2,2)$ mode during the reversal, which is top-down symmetric. It is clear that $Nu_{h}$ and $Nu_{c}$ always have the same trend. It is interesting that the overshoot phenomenon is observed, which is similar to that in a cylindrical cell (Xi et al. Reference Xi, Zhang, Hao and Xia2016). In short, the heat transfer is directly connected with flow structures. The different trends of $Nu_{h}$ and $Nu_{c}$ during a reversal in a $\unicode[STIX]{x1D6E4}=2$ cell are due to the preference for single corner roll growth during a reversal; while for $\unicode[STIX]{x1D6E4}=1$ , two corner rolls grow simultaneously, and the flow structure is symmetrical about the horizontal centreline, thus $Nu_{c}$ and $Nu_{h}$ behave the same.