2.1 Rectangular cell

A schematic drawing of the rectangular convection cell used in the first experiment is shown in figure 1, which has dimensions (in cm) of $25~(L)\times 12.5~(H)\times 7.5~(W)$ . A constant heat flux was uniformly supplied at the bottom plate, which is made of copper. The top boundary was a combination of copper plates (denoted by ‘C’) and adiabatic plates (denoted by ‘A’). Specifically, the ‘C’ plate was connected to a refrigerated circulator to maintain a constant temperature, and the ‘A’ plate was made of Plexiglass with a thermal conductivity approximately three orders of magnitude lower than that of copper. Therefore, the top boundary covered by the ‘A’ plate can be viewed as thermally insulating and the heat flux inputted at the bottom plate is primarily extracted from the ‘C’ plate at the top.

Figure 1.

A schematic drawing of the rectangular cell with a CAC configuration for the top boundary. See text for details.

To investigate the insulating effects of the ‘A’ plates on the convection, five configurations of the top boundary, i.e. CCC, CCA, CAC, CAA and ACA, corresponding to $\unicode[STIX]{x1D709}=0$ , $1/3$ , $1/3$ , $2/3$ and $2/3$ , were studied in the parameter range $2.3\times 10^{9}\leqslant Ra_{F}\leqslant 1.8\times 10^{11}$ with the bulk temperature of the fluid maintained at $40.0\,^{\circ }\text{C}$ , corresponding to $Pr=4.3$ . The lateral aspect ratio in the experiments was fixed at $\unicode[STIX]{x1D6E4}=2$ .

The influence of the insulating lids (i.e. ‘A’ plates) on the flow dynamics of the convection was investigated by carrying out two-dimensional (2D) velocity measurements in the mid-plane between the front and back sidewalls of the convection cell at $Ra_{F}=1.1\times 10^{11}$ , using a particle image velocimetry (PIV) system that has been introduced elsewhere (Xia, Sun & Zhou Reference Xia, Sun and Zhou2003). Briefly, the flow was seeded with particles of $50~\unicode[STIX]{x03BC}\text{m}$ diameter and was illuminated by a laser sheet of thickness ${\sim}1$  mm. Each velocity map contains $117\times 57$ vectors and 7200 velocity maps were acquired with a sampling rate of ${\sim}2$  Hz. The response of heat transfer to the insulating lids relies on precise temperature measurement. The temperatures of all the plates (including the ‘A’ plates) were measured by embedded thermistors with a head of 2.4 mm in diameter (Omega 44 031), and the temperature of the bulk fluid was measured by a small thermistor with a head of diameter 0.38 mm and a time constant of 30 ms in water. All the thermistors were calibrated individually to ensure an accuracy better than $0.01\,^{\circ }\text{C}$ . To eliminate any influence from the environment, the convection cells were wrapped in several layers of polystyrene foam sheets and placed in a thermostat box with a temperature stability better than $0.1\,^{\circ }\text{C}$ .

2.2 Cylindrical cell

Cylindrical convection cells with diameter $D$ and height $H$ both equal to 20 cm were used in the second experiment and their schematic drawings are shown in figure 2. Different from the thermal insulating method for the rectangular cell (using Plexiglass plates), part of the top plate of the cylindrical cell was insulated by coating a material with thermal conductivity $\unicode[STIX]{x1D712}=0.016~\text{W}~(\text{m}~\text{K})^{-1}$ to either the central region or the edge of the top plate, as shown in figure 2(b–e). Similarly, there are totally five configurations of the top plate used in this experiment: ‘ $\unicode[STIX]{x1D709}=0$ homo.’, ‘ $1/4$  centre’, ‘ $1/4$  edge’, ‘ $3/4$  centre’, and ‘ $3/4$  edge’.

Figure 2.

Schematic drawings of (a) the convection cell with homogeneous boundary condition at the top, i.e. $\unicode[STIX]{x1D709}=0$ , and (b–e) convection cells with different insulating patterns at the top plates. Each pattern is named as ‘insulating area $+$ insulating location’. For example, ‘ $1/4$ centre’ means that the insulating material covers the central region with an area equal to $1/4$ of the total area of the top plate.

The values of the control parameters are: $1.4\times 10^{10}\leqslant Ra_{F}\leqslant 1.2\times 10^{12}$ , $\unicode[STIX]{x1D709}=0$ , $1/4$ and $3/4$ , $Pr=4.3$ and $\unicode[STIX]{x1D6E4}=1$ . Only heat transport measurements were made in this second experiment using cylindrical cells.

Figure 3.

(a) The global heat transfer efficiency $Nu$ as a function of the flux Rayleigh number $Ra_{F}$ for different configurations of the rectangular cell. (b) The corresponding data normalised by $Nu$ in the ‘CCC’ case. (c) $Nu$ and (d) normalised $Nu$ for the cylindrical cells. The dashed line with the value $Nu/Nu_{Homo.}=1$ in (d) is plotted for reference.

3.1 Heat transfer

We first examine the influence of insulating lids on the global heat transfer efficiency, defined by the Nusselt number $Nu=F/(\unicode[STIX]{x1D712}\unicode[STIX]{x0394}T/H)$ , where $\unicode[STIX]{x0394}T$ is the temperature difference between the ‘C’ plates and the bottom plate. As shown in figure 3(a,b) for $Nu$ in the rectangular cells, the heat transfer efficiency decreases in the presence of the ‘A’ plate, and more ‘A’ plates result in a larger reduction of $Nu$ (the average reduction of $Nu$ is approximately 16 % when adding one ‘A’ plate and 40 % when adding two ‘A’ plates); however, the location of ‘A’ plates seems to play a negligible role on the heat transport reduction behaviour. (The relative $Nu$ differences between the ‘CCA’ and ‘CAC’ cases are within 3 %, with the exception of the datum at the lowest $Ra_{F}$ , which is approximately 5 %. The $Nu$ difference between the ‘CAA’ and ‘ACA’ cases is also within 3 %.) For the cylindrical cells (see figure 3 c,d), similar phenomenon is observed: i.e. the leading-order change of $Nu$ is insensitive to the spatial arrangement of the insulating lids, but primarily depends on the fractional insulating area. One may find that the reductions of $Nu$ in the cylindrical cell are not as much as those in the rectangular cell even though the fractional insulating areas in the cylindrical cell are slightly larger. Possible reasons for this are that the insulating materials as well as the construction of the insulating lids used for the cylindrical and rectangular cells are different. For example, the ‘A’ plates (Plexiglass) used in the rectangular case are much thicker than the coating of the insulating material in the cylindrical cell.

To find out how the insulating lids change the scaling behaviour of global heat transfer, we show in figure 4 log–log plots of $Nu$ versus the Rayleigh number $Ra$ ( $=\unicode[STIX]{x1D6FC}g\unicode[STIX]{x0394}TH^{3}/(\unicode[STIX]{x1D708}\unicode[STIX]{x1D705})$ ) for rectangular and cylindrical cells respectively. The corresponding power-law fitting parameters (scaling exponent and prefactor) are listed in table 1. It is seen that, in general, the addition of insulating lids at the top not only decreases the magnitude of $Nu$ but also reduces its scaling exponent with $Ra$ . An exception is in cylindrical cells with a $1/4$ insulating area – within experimental uncertainty both the scaling exponent and the magnitude of $Nu$ are essentially the same as the uncovered (homogeneous) case. Clearly, more precise measurements are needed to ascertain what exactly happens in this case.

Figure 4.

The global heat transfer efficiency $Nu$ as a function of the Rayleigh number $Ra$ for different configurations in (a) the rectangular cells and (b) the cylindrical cells. The solid lines correspond to power-law fittings with the fitting parameters listed in table 1.

Table 1.

Fitting parameters of the $Nu{-}Ra$ scaling in figure 4.

The weak dependence of heat transfer efficiency $Nu$ on the insulating pattern contrasts strongly with the simulation work by Cooper et al. (Reference Cooper, Moresi and Lenardic2013), who find that the dependence of heat transfer on the configurations of the lids is strong for $\unicode[STIX]{x1D709}>50\,\%$ . In their study, a significant enhancement of heat transport up to 44 % was found due to the breakup of a super lid ( $\unicode[STIX]{x1D709}=75\,\%$ ) on the top surface into several smaller lids. Another 2D simulation work by Ripesi et al. (Reference Ripesi, Biferale, Sbragaglia and Wirth2014) gives a similar conclusion. They found that for a constant $\unicode[STIX]{x1D709}=0.5$ , there will be a systematical increase of the heat flux with a decrease of the lid size $\unicode[STIX]{x1D706}$ until $\unicode[STIX]{x1D706}$ becomes comparable to the thermal boundary layer thickness. According to their findings, one would expect an enhancement of $Nu$ in the ‘ACA’ configuration compared to that in the ‘CAA’ case, since the size of the ‘A’ plate in the present case is much larger than the thermal boundary layer thickness. However, the $Nu$ data in the two configurations are almost the same in our experiment. The reason for this discrepancy between the experiment and simulation is unclear and requires further study.

Figure 5.

(a) The bulk temperature r.m.s. $\unicode[STIX]{x1D70E}_{T}$ normalised by the global temperature difference $\unicode[STIX]{x0394}T$ as a function of  $Ra_{F}$ . (b) The probability density functions (PDFs) of the normalised bulk temperature fluctuations at $Ra_{F}=1.1\times 10^{11}$ for five different configurations of the rectangular cell. The green solid and red dashed lines indicate the exponential distribution and Gaussian distribution, respectively.

To see how the insulating lids affect the statistical properties of the local temperature field, we examine the temperature fluctuations measured in the bulk region. It is seen in figure 5(a) that the magnitude of temperature fluctuations behaves similarly to the Nusselt number: i.e. at the leading-order level, the root-mean-square (r.m.s.) temperature is insensitive to the insulating pattern and depends mainly on the insulating area. Furthermore, a larger insulating area leads to a larger reduction in the r.m.s. temperature (average reduction of approximately 35 % when inserting one ‘A’ plate and 74 % when inserting two ‘A’ plates). The temperature PDF, on the other hand, is essentially invariant with respect to both the spatial arrangement and area of the lids. This suggests that the presence of insulating lids does not strongly change the plume dynamics in the bulk of the convective flow.

3.2 Flow dynamics

Figure 6.

(a–e) Time-averaged velocity fields for five configurations at $Ra_{F}=1.1\times 10^{11}$ . The scale bar represents $(U^{2}+V^{2})^{1/2}$ in units of $\text{cm}~\text{s}^{-1}$ . (f–j) The corresponding contour maps of the stream function with the scale bar in unit of $\text{cm}^{2}~\text{s}^{-1}$ . The stream function values are set to zero at the cell boundaries. The contour interval is $0.6~\text{cm}^{2}~\text{s}^{-1}$ . Note that the ‘CAC’ and ‘ACA’ configurations correspond to symmetric insulating patterns ( $\unicode[STIX]{x1D6EC}=0$ ) at the top, while the ‘CCA’ and ‘CAA’ configurations correspond to asymmetric insulating patterns ( $\unicode[STIX]{x1D6EC}=1$ ).

Table 2.

Flow-field quantities obtained from the PIV-measured time-averaged velocity field for different configurations in the rectangular cell at $Ra_{F}=1.1\times 10^{11}$ . $\unicode[STIX]{x1D709}$ : fractional insulating area; $\unicode[STIX]{x1D6EC}$ : symmetry parameter with $\unicode[STIX]{x1D6EC}=0$ ( $\unicode[STIX]{x1D6EC}=1$ ) standing for symmetric (asymmetric) insulating pattern; structure: indicating whether the flow field has a single-roll (SR) or a double-roll (DR) structure; $V_{max}$ : the maximum value of the velocity magnitude; KE: mean kinetic energy per unit mass $\mathit{KE}=(U^{2}+V^{2})/2$ ; $\unicode[STIX]{x1D6F9}_{max}$ ( $\unicode[STIX]{x1D6F9}_{min}$ ): the maximum (minimum) stream function with positive (negative) value stands for overturning in the counter-clockwise (clockwise) direction.

To gain a direct picture on how the convective flow has been changed due to the insulating lids, we show in figure 6 the PIV-measured velocity fields of the rectangular cell in five configurations at $Ra_{F}=1.1\times 10^{11}$ , along with the corresponding contour maps of the stream function. (Note that the stream function value is set to zero at the cell boundaries, as it is the stream function difference rather than its absolute value that determines the flow rate.) It is seen that the influences of the insulating lids on the flow dynamics are primarily twofold. The first effect is that with the same insulating fractional area $\unicode[STIX]{x1D709}$ , different arrangements of the insulating lids will result in different flow patterns. Specifically, a symmetric/asymmetric insulating pattern leads to a symmetric/asymmetric flow pattern, i.e. an asymmetric single-roll (SR) flow structure emerges in the asymmetric CCA and CAA configurations and a symmetric double-roll (DR) flow structure emerges in the symmetric CAC and ACA cases. It is further seen from table 2 that the flows subjected to a symmetric insulating pattern ( $\unicode[STIX]{x1D6EC}=0$ ) are generally less energetic than those subjected to an asymmetric insulating pattern ( $\unicode[STIX]{x1D6EC}=1$ ), in terms of maximum velocity $V_{max}$ , mean kinetic energy per unit mass $\mathit{KE}$ and the maximum stream function. The second effect is that for the same value of symmetry parameter $\unicode[STIX]{x1D6EC}$ the flow becomes stronger with increasing $\unicode[STIX]{x1D709}$ . For example, for the cases with asymmetric insulating pattern ( $\unicode[STIX]{x1D6EC}=1$ ), $\mathit{KE}$ , $V_{max}$ and $\unicode[STIX]{x1D6F9}_{max}$ in CCA ( $\unicode[STIX]{x1D709}=1/3$ ) are all larger than in CCC ( $\unicode[STIX]{x1D709}=0$ ); and they further increase in the CAA configuration (i.e. $\unicode[STIX]{x1D709}=2/3$ , see table 2 for detailed values). Similarly, for the symmetric insulating pattern ( $\unicode[STIX]{x1D6EC}=0$ ), these flow-strength-related quantities in the ACA configuration ( $\unicode[STIX]{x1D709}=2/3$ ) are larger than those in the CAC case ( $\unicode[STIX]{x1D709}=1/3$ ). These changes in the flow strength owing to either the area or spatial arrangement of the insulating lids are in sharp contrast to the behaviour of $Nu$ , which decreases as $\unicode[STIX]{x1D709}$ increases, but are almost the same with the same $\unicode[STIX]{x1D709}$ . To understand this, we note that in the presence of the insulating lids, the flow must be stronger so that it can more efficiently remove the same amount of heat flux through a smaller cooling area at the top. Therefore, at the same $Ra_{F}$ , the flow strength is larger for a larger insulating area $\unicode[STIX]{x1D709}$ . As for the situation with the same $\unicode[STIX]{x1D709}$ , notice that in the symmetric flow structure (DR structure) there is more fluid ‘participating’ in the heat transport process (i.e. one more branch in the middle region of the convection cell, see figure 6), and also its circulating path is shorter, so the overall flow can circulate with a smaller velocity to transport the same amount of heat, compared to that in the asymmetric flow structure (SR structure). Another notable feature is that for all the cases the upwelling hot fluids are always beneath the insulating lids, whereas the downwelling cold fluids are always beneath the conducting plates irrespective of how the lids are arranged, which is somewhat expected.

Figure 7.

(a) Horizontal cuts of the vertical velocity $V(x)$ at $Y=0$  cm. (b) Vertical cuts of the horizontal velocity $U(y)$ at $X=0$  cm for CCA and CAA cases, $X=-L/4$ for CAC case, and $X=L/4$ for the ACA case.

In order to further examine how the flow strength is changed in detail, we show in figure 7 the velocity profiles along the horizontal and vertical cuts. Note that the different shapes between the solid and dashed lines in figure 7(a) represent the differences between the asymmetric (SR) and symmetric (DR) flow structures. For the cases with different $\unicode[STIX]{x1D709}$ but the same flow structure (same value of the symmetry parameter $\unicode[STIX]{x1D6EC}$ ), we can see that both the vertical and horizontal velocities increase as $\unicode[STIX]{x1D709}$ increases, and the increases are much more significant in the regions away from the insulating lids. This is likely caused by a larger temperature gradient in both horizontal and vertical directions arising from the insulating patterns. When it comes to the cases with the same $\unicode[STIX]{x1D709}$ but different values of the symmetry parameter $\unicode[STIX]{x1D6EC}$ , things are a little different. It is found that the differences in the magnitude of the horizontal velocity between the asymmetric (solid lines) and symmetric (dashed lines) structures are more pronounced than those in the vertical directions. This indicates that the symmetry breaking of the insulating pattern enhances the horizontal velocity more than it does for the vertical velocity.