Thermal convection is widespread in nature and daily life, which has drawn great focus among researchers.^[1,2] Rayleigh-Bénard (RB) convection is a classical physical model in the field of natural convection research,^[3] and heat transfer enhancement is one of the hot spots in such field.^[4] In recent years, a rising amount of studies have focused on the non-standard RB convection. Compared with the standard RB convection system, it was found by experiments and numerical simulations that some non-standard RB thermal convection systems can enhance heat transfer efficiency, such as RB convection with rough plates,^[5–7] corilois force,^[8–10] with phase transition,^[11] with polymer addition,^[12,13] partitions.^[14–18] There are both passive and active ways to improve the heat transfer efficiency. Due to the fact that the passive method does not introduce external powers, it has been widely concerned and deeply studied for a long time. However, the heat transfer enhancement is generally less than one time.^[19–21]

As a passive method, the partitioned convection system has achieved an extraordinary heat transfer enhancement. Bao et al.^[14] added vertical partitions at equal distances in the RB convection devices with aspect ratio of 5, and left gaps at both ends of the partitions away from the top and bottom plates. It was found that the heat transfer efficiency increased significantly while the number of partitions increases. Through numerical simulation, Bao et al.^[15] found that when the number of partitions increased to 28, adjusting the height of gaps can further improve the heat transfer efficiency, which was excited up to 3.1 times that of the non-partition device. Lin et al.^[16] further studied the influence of different geometrical parameters on the flow and heat transfer characteristics of the system, and found that the height of gaps is the key geometric parameter in the partitioned convection system. Lin and Bao^[17] focused on the TD number, representing the strength of heat separation in this system, and found that TD reflects the magnitude of buoyancy, which is balanced by the pressure difference between the top and bottom of the heat transfer channels. In addition, there is a scale relation with exponent −1.64 between TD and the height of gaps in a certain range,^[17] and a scale relation with exponent 0.4 between TD and Ra.^[18] Therefore, in this paper, size of TD number will be concerned when the temperature characteristics of systems are studied.

In the partitioned convection system, the height of gaps is usually chosen on the same order of magnitude as the thickness of thermal boundary layer. The change of the thickness of the thermal boundary layer of the RB convection system under different Ra numbers is based on a certain law. In this paper, the height of gaps of the partitioned convection system is related to the thickness of the thermal boundary layer of RB convection system, and the temperature characteristics and heat transfer characteristics of the partitioned convection system are discussed when the ratio of height of gaps to thickness of boundary layer takes some specific values.

RB convection is an enclosed system with fluid heated from the bottom and cooled from the top. The convective cavity has height H, width D, and thickness W. We consider an RB convection system with vertical partitions equally distributed in the cell. As illustrated in Fig. 1(a), the partitions are connected with the front and back sidewalls. Both ends are kept away from the cooling and heating plates in a distance d. Figure 1(b) is a zoom-in view of the convective cell. Here l represents the thickness of the partitions and b the width of the vertical channels.

Fig. 1.

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	Fig. 1. (a) The model diagram of partition convection device. (b) An enlarged portion of the partition convection device.

Previous studies have shown that when the number of partitions is large enough, the flows in the channels will become laminar. In such case, the heat transfer efficiency will be the same for both two- and three-dimensional partitioned RBC systems.^[15] Since the focus of this paper is on the laminar regime, we use two-dimensional direct numerical simulation (DNS) for all cases in this paper, which is good enough to reveal the physics of this system. Under the Boussinesq approximation, the non-dimensional governing equations of two-dimensional RB thermal convection is given by

In the laminar partitioned RB system, the fluid in each channel moves vertically in alternate directions. Due to the continuity, horizontal flow is formed in the gaps, shearing the thermal boundary layer and transporting the hot/cold fluid to the upward/downward channels. Such process makes the upward/downward channels have higher/lower temperature than the average of the cell, as shown in Fig. 2. This separation between hot upward flow and the cold downward one enhances the global heat transfer effectively. Thus, this “heat-separation” phenomenon is essential to this system. In this section we will focus on the influence of gaps’ height d and Ra number over this phenomenon.

Fig. 2.

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	Fig. 2. Temperature field and streamline of partitioned convection system.

We average the temperature in the heat transfer channel in the horizontal direction and then integrate it in the longitudinal direction to obtain the TD number^[15] in each heat transfer channels, which represents how strong the heat is separated. The absolute values of TD number in the channels are averaged to get the total TD number of the system, which is able to reflect the magnitude of temperature drift in the heat transfer channels of the system. The formula for calculating the TD number is expressed as follows:

Figure 3(a) shows the TD number as a function d at Ra = 10⁸ with channel width b = 0.092, 0.116, and 0.145. Here d ranges from 0.01 to 0.025. We observed a scaling law . The dependence of TD on Ra is shown in Fig. 33(b) with d=0.02 and b=0.074. There are two states in the TD–Ra relation. When Ra is small, TD basically maintains between 0.3 and 0.5, increases gradually and then begins to decline. As Ra increases, it enters the second state, in which TD decreases while Ra increases, and there is a scaling relation between TD and Ra ( ). Thus, the TD number varies with d and Ra number, and there are scaling laws with a negative exponent in a certain range of d and Ra.

Fig. 3.

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	Fig. 3. (a) TD number under various geometric parameters of partitions. (b) TD number with the change of Ra, and the limit value of temperature is 0.5.

As mentioned above, the height of gaps is on the same order of magnitude as the thickness of thermal boundary layer. Therefore, the relationship between the height of gaps and the thickness of thermal boundary layer is worth discussing. Zhou et al.^[22] gave the definition of thickness of thermal boundary layer in turbulent thermal convection. Thickness of thermal boundary layer is defined as the height where the linear fitting line of the temperature near the bottom intersects the horizontally averaged temperature of the bulk region. Two-dimensional RBC system with aspect ratio of 2 has been calculated and Ra number ranges from 10⁵ to 10¹⁰. Our calculation results show that thickness of thermal boundary layer decreases as Ra increases under a scaling law .

We next normalize d in Fig. 3 using the thickness of the thermal boundary layer , to reveal the importance of the gap height in the heat separation. We plot the TD number in Fig. 3(a), in which Ra is fixed, together with those in Fig. 3(b), in which d is fixed, as a function of . It is a surprise to observe that these two sets of results collapse well, as shown in Fig. 4. These results indicate that within a certain parameter range, TD number in the heat transfer channel depends only on the ratio between the gap height d and thermal boundary layer thickness .

Fig. 4.

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	Fig. 4. TD number under different heights of gaps d which are indicated by black solid rectangle and different Ra number which are indicated by red solid circles with the change of .

Therefore, it is necessary to further explore the effect of on the temperature and heat transfer characteristics of the partitioned RBC.

4. Effect of on physical properties of partitioned convective system

We then fixed the thickness of partitions l=0.12, numbers of partitions n=8, the width of channels b=0.116, and Ra changes from 10⁷ to 2 ×10⁹. The thickness of thermal boundary layer varies with Ra. By changing the height of gaps d, we focus on those cases with and , and study their temperature distribution and heat transfer.

4.1. Influence of on TD

First, we focus on the change of TD with Ra number under different .

Figure 5 shows TD as a function of Ra with fixed . It can be seen that whether or . In such cases, TD seems to be independent of Ra. TD equals 0.37 when and 0.13 when . As a quantity reflecting the global temperature distribution property of the system, TD is independent of the aspect ratio of the system and the width of the heat transfer channel. We find that for a given thickness of partitions, TD number depends only on the ratio , i.e., , regardless of Ra. This means that the magnitude of determines the strength of heat separation in this system.

Fig. 5.

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	Fig. 5. TD number with the change of Ra number when or 1. The 1d results are indicated by the red solid circles and the 0.5d results are indicated by blue solid triangles.

4.2. Temperature distribution near and within the gaps

The gaps and their adjacent area of the partitioned convection system are very close to the thermal boundary layer. Due to the injection of partitions, the thermal boundary layer is suppressed, which is related to the global heat transfer enhancement. In this section we focus on the temperature fields in the gaps and the vicinity at Ra equals 10⁷, 10⁸, and 10⁹ and or . Figure 6 shows the temperature field in the area of and near the lower gap. Figures 6(a), 6(c), and 6(e) correspond to Ra= 10⁷, 10⁸, and 10⁹ at , and figures 6(b), 6(d), and 6(f) correspond to Ra= 10⁷, 10⁸, and 10⁹ at , respectively. The flows in the gap are all from the cold channels to the hot ones. It is evident in the figure that although the thickness of boundary layer decreases while Ra increases, cases with the same are with similar temperature distribution.

Fig. 6.

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	Fig. 6. Temperature field of the gap’s area below the fourth partition of the system and the bottom area of the channels adjacent to it. Panels (a), (c), and (e) correspond to Ra= 107, 108, and 109 when . Panels (b), (d), and (f) correspond to Ra= 107, 108, and 109 when .

From the temperature distribution shown in Figs. 6(a), 6(c), and 6(e), one can see that when the cold fluid passes through the lower gap from left to right, temperature gradually increases due to heating of the bottom plate, and the direction of the gap flow is horizontal. The finest thermal boundary layer is observed at the entrance of the gaps, where the vertical temperature gradient is the largest. The temperature at the outlet is between 0.3 and 0.4. The temperature of fluid at the bottom of the right channel is also between the two isotherms whose values are equal to 0.3 and 0.4. Since is fixed, the height of the gap is different between Ra numbers. However, their temperatures of the fluid entering the right channels are close, and the heating processes of the fluid in the gaps by the bottom plates are also similar. In Figs. 6(b)–6(d), the flows are similar to the former series, except that the fluid has not been heated completely when they flow through the gaps, which may be caused by larger heights of the gaps. Also, there is still cold fluid in the upper part of the gaps. Thus, the temperature of the fluids exiting the gaps ranges from 0.1 to 0.2, which is lower than that of the cases.

To show the similarity of these cases, we again normalized the heights of the gaps by the thickness of the boundary layer, and the temperature distributions under different Ra in the gaps are discussed in details. Three temperature contours θ=0.1, 0.2, and 0.3 with are extracted from the results shown in Figs. 6(a), 6(c), and 6(e), and presented in Fig. 7(a). The gray rectangular in the figure is the region where the partition is located. Figure 7(b) is the result of , in which three temperature contours of −0.1, 0, and 0.1 are selected.

Fig. 7.

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	Fig. 7. When , three isotherms of 0.1, 0.2, and 0.3 are selected (a) and , three isotherms of −0.1, 0, and 0.1 are selected (b) at Ra=107 (dots), Ra=108 (dash Dot), and Ra=109 (solid). The gray area in the figure is the area where the partition is located.

Figure 7 convinces that the temperature distribution in the gap is similar when is the same, also the fluid is under similar heating process when passing through the gap. As shown in Fig. 7(a), when , the isotherms are divided into two regions. Before entering the gaps, the isotherms basically horizontal. Upon approaching the gap, the heights of isotherms gradually decrease, reaching the lowest position at the entrance. After entering the gap, three isotherms gradually rise and end at the bottom of the partition, and the thickness of the thermal boundary layer also increases. In Fig. 7(b), when , the isotherms are divided into three regions. Before entering the gap, the isotherms of different Ra vary greatly, but they all reach the lowest point at the entrance of the gap. After entering the gap, the isotherms of the same temperature value for different Ra numbers almost collapse, and the height rises gradually. The isotherms of −0.1 end up at the bottom of the partition, and the isotherms of 0.0 and 0.1 can extent out of the gap.

In a word, the thickness of the thermal boundary layer of different Ra numbers varies greatly, but as long as are the same, the distributions of temperature in the gaps are basically the same.

4.3. Effect of on Nu number

As shown in the previous section, in the partitioned convection, we find that for the same , the same TD will be produced. In the last part of this section, we will again focus on the effect on heat transfer efficiency, which is the widespread interest in this field.

Figure 8 shows change of Nu with Ra for the two series of simulations, together with Nu in the non-partitioned convection system as a comparison. As shown in Fig. 8, for and , Nu is much higher than the case without partitions, which is an example of the heat transfer enhancement in the partitioned convection system. Within the given range of Ra, the maximum heat transfer efficiency enhancement happens at Ra= 10⁷, whose Nu is 2.2 times of the non-partitioned one for , and 2.6 times for . With the increase of Ra, the enhancement of heat transfer efficiency decreases gradually. We also observed a scaling relation between Nu and Ra for both and 1, which yields . Even though, the enhancement effect in the partitioned convection is still huge. The smallest heat transfer efficiency at for is nearly twice as large as that without partitions.

Fig. 8.

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	Fig. 8. Semi-log plot of Nu/Ra 0.29 as functions of Ra. 0.5d ( ) and 1d ( ) are indicated by blue solid triangle and red solid circles, respectively. nogb (Nu/Ra 0.29 of system without partitions) are indicated by black solid rectangle. The dash lines represent the best power-law fits of corresponding cases.

A series of studies have been carried out on the physical characteristics of the partitioned convection system. Because the height of the gaps is usually selected within the thickness of thermal boundary layer of RB convection, the characteristics of temperature drift characteristics in the partitioned convection system are studied in detail for a given thickness of partitions l=0.12, and the following conclusions are obtained.

(i) By using the thickness of thermal boundary layer to normalize the height of gaps, TD numbers of different gaps’ heights or different Ra numbers are analyzed, and it is found that the curves of these two changes with coincide basically.

(ii) The ratio of gaps’ heights to thickness of thermal boundary layer is selected as constant value, and it is found that TD numbers have nothing to do with Ra numbers, and when , when .

(iii) Even if the thermal boundary layer will change with different Ra number, the temperature distributions of the gap and its adjacent area are basically the same when the are fixed under different Ra numbers.

(iv) Under the same Ra number, the heat transfer efficiency of systems when is higher than that of the systems when . With a given magnitude of , there is a scale relationship between Nu number and Ra number, .