3.1. Global heat transport

Figure 2(a) shows the $Nu$–$Ra$ data measured in two sets of asymmetric cells (T1B4 and T4B1). Data taken from the literature for symmetric rough cells (T1B1 and T4B4) (Xie & Xia Reference Xie and Xia2017) and an SS cell (Wei et al. Reference Wei, Chan, Ni, Zhao and Xia2014) are also plotted as references. For the Rayleigh number range explored, the heat transport efficiencies of symmetric and asymmetric rough cells are considerably higher than in the smooth case. For a clearer view of the differences between the symmetric and asymmetric cells, we present the corresponding compensated plots of $NuRa^{-0.5}$ versus $Ra$ in figure 2(b). It is seen that the Nusselt numbers of the two asymmetric sets are just in between the curves for the symmetric rough cells (T1B1 and T4B4), and the four curves cross over at a Rayleigh number of approximately $Ra_c = 1.9\times 10^9$. This implies that at this particular Rayleigh number $Ra_c$, the heat transport ability (or equivalently the thermal impedance) of a $\lambda = 1$ rough plate is the same as that of a $\lambda = 4$ rough plate, regardless of whether they are located at the top or bottom of the cell and whether they are assembled symmetrically or asymmetrically. Whereas for $Ra < Ra_c$, the heat transport in asymmetric cells is dominated by the $\lambda = 1$ rough plate and bottlenecked by the $\lambda = 4$ rough plate, for $Ra > Ra_c$ it is the $\lambda = 4$ plate that prompts the heat transfer, and the overall transport is limited by the $\lambda = 1$ plate. It is also worth noting that the $Nu$–$Ra$ relations measured in the two asymmetric cells (T1B4 and T4B1) show the same behaviour, and they differ only by a factor of approximately 5 % at small Rayleigh numbers. This observation also agrees with the fact that the heat-transfer efficiency in thermal convection is insensitive to the applied temperature boundary conditions (Johnston & Doering Reference Johnston and Doering2009; Stevens et al. Reference Stevens, Lohse and Verzicco2011; Wei et al. Reference Wei, Chan, Ni, Zhao and Xia2014). The solid and dashed lines in figure 2(b) show effective power-law fits with $Nu=BRa^\beta$ to the data before and after the intersection point, respectively. The fitting results are summarized in table 2. As expected, the effective scaling exponents $\beta$ of the rough cells are all larger than the value of 0.30 for the smooth case. With each data set, the scaling exponents for $Ra\le 1.9\times 10^9$ are larger than those for $Ra>1.9\times 10^9$. Xie & Xia (Reference Xie and Xia2017) interpreted such a transition as a process in which the viscous boundary layer thickness gets thinner than the height of the roughness elements. We emphasize that the present data for asymmetric rough cells are consistent with this physical picture. Lastly, the fact that the $Nu$–$Ra$ curves for asymmetric cells are just in between those measured in the symmetric cases suggests that using cells with different roughnesses might be an efficient way to extend the Rayleigh number range with high heat-transfer efficiency (Zhu et al. Reference Zhu, Stevens, Shishkina, Verzicco and Lohse2019; Toppaladoddi et al. Reference Toppaladoddi, Wells, Doering and Wettlaufer2021).

Figure 2.

(a) The $Nu$–$Ra$ relations measured in the T1B4 cell (squares) and the T4B1 cell (diamonds). For comparison, data from the literature for symmetric rough cells (T1B1 and T4B4) and a symmetric smooth cell (SS) are plotted as triangles and circles, respectively. The solid line shows a power-law fit of $Ra^{0.30}$. (b) Compensated plots of $NuRa^{-0.5}$. The legends are the same as those in panel (a). The solid and dashed lines represent power-law fits to the data in two Rayleigh number regimes. The fitting results are listed in table 2.

Table 2.

Fitting parameters of $Nu = BRa^\beta$ and $\sigma _{T_{c}}/\Delta T = CRa^\gamma$ for different cells.

3.2. Temperature at the cell centre

Figure 3(a) shows the difference between the time-averaged centre temperature $T_c$ and the arithmetic mean temperature of the top and bottom plates, $T_m = (T_t+T_b)/2$, as a function of the temperature difference $\Delta T$ applied across the two conducting plates. Here $T_t$ and $T_b$ denote the time-averaged temperatures of the top and bottom plates, respectively. The same quantity measured in a symmetric smooth cell is plotted for reference as well (Ahlers et al. Reference Ahlers, Brown, Araujo, Funfschilling, Grossmann and Lohse2006). It is seen that for $\Delta T\ge 5\,{\rm K}$ (corresponding to $Ra>Ra_c$), the measured centre temperatures $T_c$ show apparent deviations from $T_m$ for all three data sets. For an SS cell, it is known that such deviation results from the NOB effect (Ahlers et al. Reference Ahlers, Brown, Araujo, Funfschilling, Grossmann and Lohse2006). In this regime, we note that the curve of T1B4 is always higher than in the SS case, which suggests that for the T1B4 cell, the heat-transfer capability of the bottom plate ($\lambda = 4$) is greater than that of the top plate ($\lambda = 1$). In other words, the thermal impedance of the bottom plate is lower than that of the top plate in this regime. For the case of T4B1, $T_c-T_m$ is negative when $\Delta T\ge 5\,{\rm K}$, which again suggests that the $\lambda = 4$ plate is more efficient in heat transfer than the $\lambda = 1$ plate. On the other hand, when the applied temperature difference is small ($\Delta T<5\,{\rm K}$), for which the OB approximation is valid, the centre temperatures $T_c$ show no apparent difference with $T_m$ for all three cases. To have a clearer view of this regime, we plot in figure 3(b) the normalized centre temperature deviations $(T_c-T_m)/\Delta T$ as functions of the Rayleigh numbers for the two asymmetric cells. It is seen that for the low-Rayleigh-number regime ($Ra< Ra_c$), the signs of bulk temperature deviations are opposite to those measured in the high-Rayleigh-number regime for both the T1B4 and the T4B1 cases. This indicates that for $Ra< Ra_c$, the $\lambda = 1$ plate, in turn, shows a higher heat-transfer capability, regardless of whether it is assembled on the top or at the bottom.

Figure 3.

(a) The difference between the temperature $T_{c}$ at the cell centre and the arithmetic mean temperature $T_m$ of the conducting plates as a function of the applied temperature difference $\Delta T$ across the top and bottom plates. (b) The corresponding normalized temperature deviations as functions of the Rayleigh number. The legends are the same as those in panel (a).

We can now conclude that the heat-transfer capability of an individual plate is mainly determined by its topography and is not sensitive to the applied temperature boundary condition. For the two types of roughness explored in the present study with $\lambda = 1$ and 4, the former shows a higher heat-transfer efficiency (lower thermal impedance) at small Rayleigh numbers, whereas the latter dominates heat transport at high Rayleigh numbers. The critical Rayleigh number separating these two regimes is approximately $Ra_c=1.9\times 10^9$. At this particular Rayleigh number, the heat transport abilities of the two rough plates are balanced. It is also worth mentioning that the relative strength of heat transport abilities of individual plates deduced from centre temperature behaviours shows qualitative agreement with those measured in symmetric cells (see figure 2). In addition, we remark that when the bulk temperature $T_c$ is higher (lower) than $T_m$, the temperature drop associated with the upper thermal boundary layer must be larger (smaller) than that of the lower thermal boundary layer. In this sense, the up–down symmetry of thermal boundary layers is very vulnerable to the change in boundary geometries.

Figure 4(a) shows the normalized centre temperature fluctuations $\sigma _{T_{c}}/\Delta T$ measured in asymmetric and symmetric rough cells, respectively. The same quantity measured in an SS cell is also plotted for reference (Wei et al. Reference Wei, Chan, Ni, Zhao and Xia2014). It is interesting to note that the centre temperature fluctuations, in general, show similar behaviours to the global heat transports (figure 2). The temperature fluctuations in symmetric/asymmetric rough cells are always larger than those measured in an SS cell. This agrees with the physical picture that roughness elements can trigger more plume-emission events and thus increase the intensity of turbulence. A transition is also discernible around $Ra=1.9\times 10^9$ for each rough data set. In addition, the curves for the two asymmetric rough cells collapse well onto each other, and they are both located in between the two curves representing symmetric rough cells. The solid and dashed lines in figure 4(a) indicate power-law fits of $\sigma _{T_{c}}/\Delta T=CRa^\gamma$ to the data before and after the transitional Rayleigh number, respectively. The fitting parameters are also summarized in table 2.

Figure 4.

(a) The normalized bulk temperature fluctuations $\sigma _{T_{c}}/\Delta T$ as functions of $Ra$. The data for the SS case are adapted from Wei et al. (Reference Wei, Chan, Ni, Zhao and Xia2014). The solid and dashed lines show power-law fits to the data. The fitting parameters are summarized in table 2. (b) The measured centre temperature fluctuations $\sigma _{T_{c}}$ as functions of $Ra_F$. The solid line represents a power-law fit with $Ra_F^{0.68}$. Panels (a,b) share the same legend.

The fact that the centre temperature fluctuations (figure 4) and the global Nusselt numbers (figure 2) show a high degree of similarity indicates that they might be highly correlated. In figure 4(b), we plot the measured centre temperature fluctuation as a function of the flux Rayleigh number $Ra_F=NuRa$ (which is proportional to the total heat flux). Remarkably, all data sets collapse well onto a single master curve, regardless of whether the boundary is symmetric or asymmetric, rough or smooth. A single power-law fit to all the data points yields $\sigma _{T_{c}}\sim Ra_F^{0.68}$. In fact, by assuming $Nu\sim Ra^{\beta }$ and $\sigma _{T_{c}}/\Delta T\sim Ra^{\gamma }$, one can obtain $\Delta T\sim Ra\sim Ra_F^{1/(1+\beta )}$ and $\sigma _{T_{c}}\sim Ra_F^{(1+\gamma )/(1+\beta )}$. Therefore, the universal scaling observed in figure 4(b) implies $(1+\gamma )/(1+\beta ) = 0.68 \approx 2/3$, or equivalently $\gamma \approx (2\beta -1)/3$. The result above reveals that the centre temperature fluctuation is mainly determined by the total heat flux and is robust to the changes in the geometrical boundary conditions.

Next, we plot in figure 5(a–c) the probability density functions (p.d.f.s) of the centre temperature fluctuations measured at three different Rayleigh numbers ($Ra = 5.0\times 10^8$, $2.0\times 10^9$ and $5.0\times 10^9$), which correspond to two heat-transfer regimes and the critical Rayleigh number (figure 2). It is quite remarkable that different combinations of individual rough plates show precisely the same p.d.f. at different Rayleigh numbers, and they all conform to an exponential distribution (see the solid lines in figure 5a–c), suggesting that thermal plumes are vigorous and turbulence is fully developed in the bulk region (Heslot, Castaing & Libchaber Reference Heslot, Castaing and Libchaber1987). More importantly, the p.d.f.s in figure 5(a–c) are almost symmetric. To quantify this point, we plot in figure 5(d) the vertical profiles of the skewness coefficient $S_T$ of the temperature fluctuation ($S_T = \langle (T_c-\langle T_c\rangle )^3\rangle /\sigma _{T_{c}}^{3}$) measured at different Rayleigh numbers. We can see that the value of $S_T$ is around zero in a region of $0.3< z/H<0.7$ for all cases. This is counterintuitive since, for most of the Rayleigh number range explored, the heat-transfer capabilities of the two asymmetric plates are not balanced, as discussed above. For example, when $Ra< Ra_c$, the thermal impedance of the top plate is lower than that of the bottom plate in a T1B4 asymmetric cell, so one might at first expect that the cold plumes emitted from the top plate are more intense than the hot plumes that come from the opposite plate. On the contrary, centre temperature measurements reveal that the symmetry of temperature fluctuation is robust to the symmetry breaking of boundary topography. In other words, for the bulk region of RBC with asymmetric geometrical boundaries, the symmetry of temperature fluctuation is spontaneously restored by adjusting the mean centre temperature $T_c$ to a specific value. Such ‘spontaneous symmetry restoring’ behaviour can also be viewed as an inverse process of spontaneous symmetry breaking, which is often observed in condensed matter physics.

Figure 5.

(a–c) Probability density functions (p.d.f.s) of the centre temperatures measured in different cells at $Ra=5.0\times 10^8$ (a), $Ra=2.0\times 10^9$ (b) and $Ra=5.0\times 10^9$ (c). The legends are the same as those in figure 4. The black lines show exponential distributions. (d) The vertical profiles of the temperature skewness measured in asymmetric rough cells at different Rayleigh numbers.

3.3. Heat transport properties of individual rough plates

With the centre temperature $T_c$, the Rayleigh number and Nusselt number of individual plates can be calculated as follows (Tisserand et al. Reference Tisserand, Creyssels, Gasteuil, Pabiou, Gibert, Castaing and Chillà2011; Wei et al. Reference Wei, Chan, Ni, Zhao and Xia2014):

(3.1)\begin{gather} Ra_{i}=\frac{\alpha g\left(2\Delta T_i\right)H^{3}}{\nu \kappa }, \end{gather}

(3.2)\begin{gather} Nu_{i}=\frac{Q}{\chi\left(2\Delta T_i\right)/H}, \end{gather}

where $\Delta T_i=|T_c-T_i|$, the subscript $i$ represents the top plate ($i=t$) or the bottom plate ($i=b$), and $\chi$ is the thermal conductivity. Since the heat flux $Q$ of the two plates is always balanced, when the centre temperature is higher than the arithmetic mean temperature of the two plates, i.e. $T_c>T_m$, we can readily obtain $\Delta T_t>\Delta T_b$, $Ra_t>Ra_b$ and $Nu_t< Nu_b$, which means that the bottom plate is better in heat transport than the top plate (and vice versa).

Figure 6(a,b) show the $Nu_i$–$Ra_i$ relations of the $\lambda = 1$ plate and the $\lambda = 4$ plate calculated from various data sets. It is evident that for the low-Rayleigh-number regime ($Ra< Ra_c$), the heat-transfer efficiencies of the $\lambda = 1$ plate are higher than those of the $\lambda = 4$ plate, i.e. compensated values of $Nu_iRa_i^{-0.5}$ for the $\lambda =1$ plate are all larger than $3.5\times 10^{-3}$ and those of the $\lambda =4$ plate are all smaller than $3.5\times 10^{-3}$. Moreover, we also note from figure 6(b) that for this regime, the $Nu_i$–$Ra_i$ relations of the $\lambda =4$ plate measured in asymmetric and symmetric cells collapse with each other. But for the $\lambda = 1$ plate, the Nusselt numbers measured in asymmetric RB cells are apparently lower than in the symmetric case of T1B1. On the other hand, for the high-Rayleigh-number regime $Ra>Ra_c$, where the heat-transfer abilities of the $\lambda = 4$ plate dominate over those of the $\lambda = 1$ one, the curves of the $\lambda = 1$ plate in turn collapse (see figure 6a), while for the $\lambda = 4$ rough plate, Nusselt numbers measured in asymmetric cells show deviations from the symmetric case of T4B4. These observations suggest that the heat transport properties of individual rough plates can interact with each other in a ‘one-way’ manner. The Nusselt number of the plate with lower heat transport capability is robust, and it serves as a ‘bottleneck’ for the system's overall heat transfer. Meanwhile, this plate can considerably hinder the heat transport efficiency of the other plate with a higher heat-transfer efficiency. Such a ‘one-way communication’ interpretation is also applicable to a smooth–rough asymmetric turbulent RBC system (Tisserand et al. Reference Tisserand, Creyssels, Gasteuil, Pabiou, Gibert, Castaing and Chillà2011; Wei et al. Reference Wei, Chan, Ni, Zhao and Xia2014), where the smooth plate always acts as the plate with lower heat-transfer ability.

Figure 6.

The $Nu_i$–$Ra_i$ relations of (a) the $\lambda = 1$ rough plate and (b) the $\lambda = 4$ rough plate. The vertical dashed lines in both panels correspond to a transitional Rayleigh number of $Ra_c = 1.9\times 10^9$.

3.4. Bulk temperature inversion

Figure 7 shows the vertical mean temperature profiles $(T(z)-T_c)/\Delta T$ measured at various Rayleigh numbers in the asymmetric rough cells. For the bulk region, all profiles collapse well with each other. A linear fit to all the data points in a range of $0.2< z/H<0.8$ yields $(T(z)-T_c)/\Delta T=0.033z/H-0.016$. The positive slope of the temperature profiles indicates that the flow is stably stratified in a time-averaged sense. A weak temperature inversion has also been reported in the bulk region of SS cells (Chillà et al. Reference Chillà, Ciliberto, Innocenti and Pampaloni1993; Tilgner, Belmonte & Libchaber Reference Tilgner, Belmonte and Libchaber1993; Wang & Xia Reference Wang and Xia2003; Maystrenko, Resagk & Thess Reference Maystrenko, Resagk and Thess2007). However, we note that the non-dimensional temperature gradient obtained in the present asymmetric rough settings ($\partial T/\partial z=0.033\Delta T/H$) is apparently stronger than those observed in SS cells. For example, Tilgner et al. (Reference Tilgner, Belmonte and Libchaber1993) obtained a value of $\partial T/\partial z=0.019\Delta T/H$ at $Ra=1.1\times 10^9$ in a rectangular SS cell.

Figure 7.

The normalized mean temperature profiles measured along the centre line of the asymmetric rough cells at different Rayleigh numbers. The black solid line shows a linear fit of $(T(z)-T_c)/\Delta T=0.033z/H-0.016$.

The bulk temperature inversion can be formed when the diffusion time scale of thermal plumes is much larger than the characteristic time scale of the LSC (Tilgner et al. Reference Tilgner, Belmonte and Libchaber1993). In such a case, when the thermal plumes are emitted from one plate and then moved to the opposite side by the LSC, their enthalpies are not much dissipated and therefore can lead to a temperature inversion. Compared with smooth topography, a rough plate can significantly enhance the plume emission rate and promote the flow's coherency. For the present case, we believe that the thermalization time of the plume is also extended by the presence of both rough plates, resulting in a strong temperature inversion. Lastly, we remark that with the formation of a distinct stratification in a strong turbulent background, the present system might be a promising platform for studying the interplay between internal gravity waves and convection (Barel et al. Reference Barel, Eidelman, Elperin, Fleurov, Kleeorin, Levy, Rogachevskii and Shildkrot2020; He, Cheng & Xia Reference He, Cheng and Xia2022), which is an essential topic in physical oceanography and atmospheric turbulence.

3.5. Reynolds numbers

In this section, we explore the effects of asymmetric topographies on the Reynolds number associated with the LSC. It is known that the LSC undergoes ceaseless oscillations. The oscillation period $\tau _f$ of the LSC is determined by first calculating the autocorrelation function of the temperature time series measured by the thermistors embedded in the conducting plates, and then locating the position of the first peak (see figure 8a). For each case, the oscillation period of the LSC is averaged over all thermistors. With this, we can obtain the Reynolds number by $Re=4H^2/(\tau _f\nu )$, where we have assumed $4H$ to be the typical length of the LSC flow path. It is worth mentioning that the above is only a first-order approximation since the exact length of the LSC flow path should be smaller than $4H$ and in general depends on the Rayleigh number (Niemela & Sreenivasan Reference Niemela and Sreenivasan2003; Sun & Xia Reference Sun and Xia2005).

Figure 8.

(a) An example of the autocorrelation function of the temperature time series. (b) The Reynolds number $Re$ as functions of $Ra$ for different sets of convection cells. (c) Compensated plots of $ReRa^{-0.5}$ versus $Ra$. (d) The Reynolds number as functions of the flux Rayleigh number $Ra_F$. The legends of panels (b–d) are the same.

Figure 8(b) shows the $Re$–$Ra$ relations measured in asymmetric and symmetric rough cells, respectively. The reference SS curve is adopted from Wei et al. (Reference Wei, Chan, Ni, Zhao and Xia2014), which was obtained using the same method as in the present study. It is seen that the presence of rough plates results in a significantly higher $Re$ than in the SS case. Compared with the Nusselt number behaviour (figure 2a), the Reynolds numbers for asymmetric and symmetric cells show a slighter deviation. For a clear view, we compensate $Re$ with $Ra^{0.5}$ in figure 8(c). It is seen that the curve for a symmetric rough cell follows a power law, which is approximately $Re\sim Ra^{0.50}$ for T1B1 (downward triangles) and $Re\sim Ra^{0.57}$ for T4B4 (upward triangles). Similar to global heat transfer, the Reynolds number behaviours of the two asymmetric rough sets (T1B4 and T4B1) can also be divided into two regimes. For small Rayleigh numbers $Ra< Ra_c$, they are close to the T4B4 case. But in the large-Rayleigh-number regime $Ra \ge Ra_c$, their Reynolds numbers both show transitions to a power-law scaling around $Re\sim Ra^{0.5}$. The observation above is also consistent with the heat transport properties of individual plates (figure 6), i.e. when heat transport is dominated by the $\lambda =1$ plate ($Ra< Ra_c$), the bottleneck of the $\lambda =4$ plate lies in its capability of transporting not only heat but also momentum (and vice versa for $Ra\ge Ra_c$).

Lastly, we plot the Reynolds number $Re$ as a function of the flux Rayleigh number $Ra_F$ in figure 8(d). In this plot, all data points again collapse onto a single master curve of $Re=0.34Ra_F^{0.36}$, as the temperature fluctuations do (figure 4b). This also suggests that the typical velocity of the LSC is also, to a large extent, determined by the overall heat flux $Q$ that flows through the convection cell, regardless of whether the geometrical boundary is smooth or rough, symmetric or asymmetric.

3.6. Convective temperature and velocity scales

By dimensional analysis, Deardorff (Reference Deardorff1970) proposed that the convection temperature scale and convective velocity scale, which are expressed as $\theta ^{*}\equiv Q^{2/3}/(\alpha gH)^{1/3}$ and $w_{*}\equiv (\alpha gHQ)^{1/3}$, can be used to describe the fluctuations of temperature and velocity in buoyancy-driven turbulence. More recently, Xie et al. (Reference Xie, Cheng, Hu and Xia2019) reported that, when scaled with the convective temperature scale, the bulk temperature fluctuation in symmetric RBC remains constant ($\sigma _{T_{c}}/\theta ^*=0.85$) over a Rayleigh number range of $10^8\le Ra\le 10^{15}$ and a Prandtl number range of $0.7\le Pr\le 23.34$. In addition, the normalized vertical velocity fluctuation shows a weak positive Rayleigh number dependence.

For RBC with asymmetric topographies, we have shown that the positive and negative temperature fluctuations are symmetric (figure 5). So we can take the standard deviation of the centre temperature $\sigma _T$ as the single typical scale of temperature fluctuation. Figure 9(a) presents the normalized centre temperature fluctuations $\sigma _{Tc}/ \theta ^{*}$ as functions of Rayleigh number measured in symmetric and asymmetric rough cells. It is seen that the curves for different cells approximately collapse onto each other, and the data points all fall around a value of 0.85 as reported by Xie et al. (Reference Xie, Cheng, Hu and Xia2019). A single power-law fit to all the data yields a weak Rayleigh number dependence of $\sigma _{Tc}/\theta ^{*}\sim Ra^{0.03}$. Similarly, we plot in figure 9(b) the ratio between the LSC-based Reynolds number $Re$ and the Reynolds number defined by the convective velocity scale $Re^{*}$. The four rough cells with different symmetries again show the same trend, and a single power-law fit results in $Re/Re^{*}\sim Ra^{0.04}$. The analyses above suggest that, even when the boundary symmetric is broken, the convective scales ($\theta ^{*}$ and $w^{*}$) are reasonable approximations of turbulent fluctuations in the bulk of the convection cell.

Figure 9.

(a) The normalized centre temperature fluctuation $\sigma _{Tc}/\theta ^{*}$ as a function of $Ra$. Here $\theta ^{*}$ is the convective temperature scale. The horizontal dashed line shows $\sigma _{Tc}/\theta ^{*}=0.85$. (b) Plot of $Re/Re^{*}$ as a function of $Ra$, where $Re^{*}$ is defined by the convective velocity scale. The legends are the same as those in panel (a).