Scaling Law of Flow and Heat Transfer Characteristics in Turbulent Radiative Rayleigh-Bénard Convection of Optically Thick Media ^†

Abstract

Radiative natural convection is of vital importance in the process of energy storage, power generation, and thermal storage technology. As the attenuation coefficients of many heat transfer media in these fields are high enough to be considered as optically thick media, like nanofluids or molten salts in concentrated solar power or phase change thermal storage, Rosseland approximation is commonly used. In this paper, we delve into the impact of thermal radiation on the Rayleigh-Bénard (RB) convection. Theoretical analysis has been conducted by modifying the Grossmann-Lohse (GL) model. Based on turbulent dissipation theory, the corresponding scaling laws in four main regimes are proposed. Direct numerical simulation (DNS) was also performed, revealing that radiation exerts a notable influence on both flow and heat transfer, particularly on the formation of large-scale circulation. By comparing with DNS results, it is found that due to the presence of radiation, the modified Nu scaling law in small Pr range of the GL model is more suitable for predicting the transport characteristics of optical thick media with large Pr. The maximum deviation between the results of DNS and prediction model is about 10%, suggesting the summarized scaling law can effectively predict the Nu of radiative RB convection.

1. Introduction

The increased uptake of the renewable energy has led to an increased need for updating solar thermal energy utilization and thermal storage technology. For example, by combining concentrated solar power with photovoltaics, both the heat and electricity can be provided at the same system, making the energy maximally utilized [1]. What is more, the combined heat and power system, when paired with a phase change thermal storage system [2,3], can achieve a continuous supply of energy. Obviously, understanding and optimizing these fluid transport characteristics is essential for the development and improvement of these energy conversion and storage systems.

It is worth noting that, in applications of the above fields, the operating temperature is usually high and the temperature difference in the system may also be considerable. This makes the effect of thermal radiation non-negligible, which is a typical radiative convection problem. Different from thermal conduction and convection, the radiative energy can penetrate into the interior of the fluid, and transfer between all the elements, as it propagates as an electromagnetic wave. Besides, the directional and spectral natures of radiation increase the dimensionality of the problem, making the determination of radiative heat flux complex. Therefore, many simplified models have been proposed, and the Rosseland diffusion approximation is the most widely used in the radiative convection of optical thick media, in which the radiative heat flux is supposed to be a function of temperature gradient, i.e., $q^{\mathrm{R}}\sim -{\displaystyle \frac{16\sigma T_{\mathrm{ref}}^3}{3{\beta }_r}}{\displaystyle \frac{\partial T}{\partial x}}$ (linear approximation) or $q^{\mathrm{R}}\sim -{\displaystyle \frac{16\sigma T^3}{3{\beta }_r}}{\displaystyle \frac{\partial T}{\partial x}}$ (nonlinear approximation). For example, by using linear Rosseland approximation, Sheikholeslami et al. [4] studied the nanofluid flow and heat transfer under the effect of electric field in a porous lid driven cavity. They found thermal radiation can enhance the temperature gradient near the positive electrode, as well as the convective mode. Similar conclusions were obtained by many other researchers [5,6,7]. Besides, with the help of nonlinear Rosseland approximation, Shah et al. [8] proposed an artificial neural network model for heat transfer problem in carbon nanotubes-based nanofluid, which shows excellent performances by comparing with numerical simulation. Yousuf Ali et al. [9] investigated the effect of radiation on the heat transfer of nanofluids flowing through a stretching sheet by using both linear and nonlinear Rosseland approximations, and found that the thermal performance is enhanced.

Indeed, the influence of radiation on natural convection has been a subject of extensive research for many years [10,11,12]. Kolsi et al. [10] studied the radiative natural convection for $Ra={10}^5$ and $Pr=13.6$, and proposed that, compared to peripheral spiraling motion, the principal flow structure is more sensitive to radiation effect. What is more, in absence of radiation there exists a zone where the flow structure exhibits quasi-2D characteristics, while the effect of radiation on the 3D behavior of the flow is significant, especially in the core of cavity. Wang et al. [13] discovered that thermal radiation can reduce the spatial spreading of the plumes but has little impact on the kinetic field. However, some other scholars suggested that the thermal radiation shows a stabilizing effect on natural convection. Both the convection and convective heat transfer are dampened by the surface radiation [14,15,16,17]. In fact, it has been observed that radiation emanating from the medium exhibits a similar dampening influence on the convective heat transfer [18,19].

The aforementioned studies hold crucial significance, but most of them have derived qualitative conclusions. In order to quantitatively determine the influence of control parameters on the transport characteristics of natural convection, some prediction models for radiative natural convection have been developed. Gad et al. [20] proposed the scaling law of heat transfer in laminar natural convection with radiation for the range of $5\times {10}^3\le Ra\le {10}^5$, $1\le \Gamma \le 5$, $0.1\le {\epsilon }_e\le 0.85$. By applying k-ε turbulence model, Miroshnichenko et al. [16] investigated the heat transport characteristics for turbulent flow in a cavity, which is heated below and cooled by the environment on the other walls. They also summarized the correlations between Nu and (Ra, ε_e) for ${10}^9\le Ra\le {10}^{10}$ and $0\le {\epsilon }_e\le 0.6$. One of the most prominent works is the POD-based low-order model derived by Soucasse et al. [21,22] in the range of ${10}^6\le Ra\le {10}^8$, $Pr=0.7$. Both the observed and predicted models they derived can capture the dynamics and energy transport features of the natural convection coupled with radiation. More relevant researches can be found in the review provided by Song et al. [23]. Nevertheless, one can easily see that most of the existing scaling laws are applicable within a narrow range of parameters and are insufficient to meet the needs of industrial design.

In fact, in the specific case of Rayleigh-Bénard (RB) convection, several scaling laws have been proposed, including the marginal stability theory [24], Chicago mixing zone model [25,26,27], Shraiman-Siggia model (SS model) [28,29], Grossmann-Lohse theory (GL model) [30,31,32,33], and so on. The GL model is widely recognized due to its wide applicability and high accuracy. Besides, it is used as a basis for many other correction models to describe the transport characteristics of RB convection under additional effects. Chakraborty [34] suggested that, by introducing the concept of magnetic energy dissipation, the arguments and assumptions in GL model can be extended to the case of turbulent magnetohydrodynamic (MHD) RB convection with a vertical magnetic field. Based on this idea, Zürner et al. [35,36] have given a set of implicit scaling equations between (Nu, Re) and (Ra, Pr, Ha) in a limited parameter range ($Pr\le 3\times {10}^{-2}$, $Pm=\mu {\sigma }_b\upsilon \le {10}^{-5}$). Moreover, the scaling behaviors of heat and momentum transport in MHD-RB convection were analyzed by Yan et al. [37] on the basis of the ratio of magnetic energy to kinetic energy and the contribution of viscous and ohmic dissipation. Therefore, analogous methodologies were adopted in this study for modeling the transport characteristics of radiative RB convection for the first time.

In this paper, a comprehensive study of the flow and heat transfer characteristics in radiative Rayleigh-Bénard convection within optically thick media has been conducted by using both theoretical analysis and direct numerical simulation. Based on the GL model, the relevant scaling laws have been derived according to the turbulent dissipation theory and the relative contribution of bulk and boundary layers to dissipation. In order to verify the prediction models, direct numerical simulation for the flow in the range of $Ra=1\times {10}^8~5\times {10}^8$, $Pr=7,30$, $ℝ=1.0~6.0$ (The definition of dimensionless parameters is depicted in Equation (6)) has also been conducted. The results show the proposed scaling law can effectively predict the Nu of radiative Rayleigh-Bénard convection, especially for situations with large optical thicknesses. The conclusions drawn from this paper can furnish a theoretical underpinning for industrial design pertaining to the realms of energy storage, power generation, and thermal storage technology.

Section 2 introduces the model and assumptions used in this theoretical analysis. Section 3 shows the scaling law in different regimes and the boundaries between them. Section 4 displays the direct numerical simulation results used to validate theoretical analysis. Finally, the conclusions of this study are presented in Section 5.

2. The Model

Before the theoretical analysis and numerical simulation, some assumptions need to be introduced first:

• An incompressible and optically thick fluid is considered;
• The Boussinessq approximation, as well as Rosseland diffusion approximation are invoked in this work;
• The flow is in a turbulent state and there exists a large-scale circulation (LSC);
• The kinetic BL is assumed to be of Prandtl-Blasius type.

Therefore, considering a fluid layer differentially heated by two rigid plates, the following governing equations can be obtained, shown as Equations (1)–(3):

$$\nabla \cdot \mathit{u}=0$$

(1)

$${\partial }_{\tau }\mathit{u}+\left(\mathit{u}\cdot \nabla \right)\mathit{u}=-{\displaystyle \frac{1}{\rho }}\nabla p+\nu {\nabla }^2\mathit{u}+g\beta \mathsf{\Delta }{e}_z$$

(2)

$${\partial }_{\tau }{\rm T}+\mathit{u}\cdot \nabla {\rm T}=\kappa {\nabla }^2{\rm T}-{\displaystyle \frac{1}{\rho c_p}}\nabla q^R$$

(3)

The radiative heat flux by employing Rosseland approximation is defined as Equation (4):

$$q^R=-{\displaystyle \frac{4}{3}}{\displaystyle \frac{\sigma }{{\beta }_r}}\nabla T^4$$

(4)

When neglecting the high-order terms from the Taylor series of T⁴ about T_ref (here the linear Rosseland approximation is considered), the radiative heat source in Equation (3) can be obtained as Equation (5):

$$\nabla q^R=-{\displaystyle \frac{16}{3}}{\displaystyle \frac{\sigma T_{\mathrm{ref}}^3}{{\beta }_r}}{\nabla }^{2}T$$

(5)

Here three important criterion numbers, say, the Rayleigh number Ra, Prandtl number Pr and radiation parameter $ℝ$, as well as two response criterion numbers, Nusselt number Nu and Reynold number Re, to characterize the RB convection, are defined as Equation (6):

$$\begin{array}{l}Ra={\displaystyle \frac{g\beta \mathsf{\Delta }{h}^3}{\nu \kappa }},Pr={\displaystyle \frac{\nu }{\kappa }},ℝ=1+{\displaystyle \frac{4}{3}}{\displaystyle \frac{1}{{\tau }_{r}Pl}}\\ Nu={\displaystyle \frac{{〈wT〉}_A-\kappa {\partial }_z{〈T〉}_A}{\kappa \mathsf{\Delta }{h}^{-1}}},Re={\displaystyle \frac{\overline{u}h}{\nu }}\end{array}$$

(6)

In above equations, e_z, $\mathit{u}=\left\{u,v,w\right\}$, t, p, h, T, $\mathsf{\Delta }=T_{\mathrm{h}}-T_{\mathrm{c}}$, g, β, κ, λ, $\overline{u}$, β_r, and σ are the unit vector in the z direction, velocity, time, pressure, distance between two differential heated plates, temperature, temperature difference between the two plates, gravitational acceleration, thermal expansion coefficient, thermal diffusivity, thermal conductivity, the mean large-scale velocity, spectral extinction coefficient, and Stefan-Boltzmann constant ($\sigma =5.67\times {10}^{-8}\mathrm{W}\cdot {\mathrm{m}}^{-2}\cdot {\mathrm{K}}^{-4}$), respectively. The determination of characteristic quantities is illustrated in the Figure 1. Subscript h and c stand for the hot bottom and cold top plates, individually. ${\tau }_r={\beta }_{r}h$ is the optical thickness. $Pl={\displaystyle \frac{\lambda /h}{4\sigma T_{\mathrm{ref}}^3}}$ is the Planck number. T_ref is the reference temperature, which will be discussed later. ${〈\cdot 〉}_A$ denotes the average over any horizonal plane (z-plane).

Generally, the mean temperature, $T_{\mathrm{m}-\mathrm{ref}}=\left(T_{\mathrm{h}}+T_{\mathrm{c}}\right)/2$ is usually considered as the reference temperature. However, due to the dependence of radiative heat transfer on $\left|T_{\mathrm{h}}^4-T_{\mathrm{c}}^4\right|$, another strategy has been proposed [38] as Equation (7):

$$T_{\mathrm{rad}-\mathrm{ref}}={\left({\displaystyle \frac{T_{\mathrm{h}}^4+T_{\mathrm{c}}^4}{2}}\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}$$

(7)

By comparing these two reference temperatures, the relative error between them can be gained, shown as Equation (8):

$$O={\displaystyle \frac{T_{\mathrm{rad}-\mathrm{ref}}-T_{\mathrm{m}-\mathrm{ref}}}{T_{\mathrm{m}-\mathrm{ref}}}}={\left[8\cdot {\displaystyle \frac{1+{\left(T_{\mathrm{h}}/T_{\mathrm{c}}\right)}^4}{{\left(1+T_{\mathrm{h}}/T_{\mathrm{c}}\right)}^4}}\right]}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}-1$$

(8)

Figure 2 depicts the dependence of the temperature difference (Δ and $\mathsf{\Delta }/T_{\mathrm{c}}$) and O upon T_c. It can be easily seen that even if the relative temperature difference $\mathsf{\Delta }/T_{\mathrm{c}}$ reaches 50%, O is only 5.56%. It should be noticed that the basic premise for applying the Boussinesq approximation is that the physical properties of fluid do not change significantly except for the density. As shown in Figure 1, Δ has exceeded 100 K for $T_{\mathrm{c}}>200 \mathrm{K}$ and $\mathsf{\Delta }/T_{\mathrm{c}}=50\%$. Indeed, such a large temperature difference usually leads to significant changes in the physical properties of fluid. Under such conditions, the Boussinesq approximation is no longer valid. Therefore, either of these two reference temperatures can usually be considered for most cases with small temperature differences.

3. Scaling Laws

In correspondence with classical RB convection, one can derive the following rigorous relations for the global averaged kinetic and thermal dissipation rates, ε_u and ε_T, according to the theory of Shraiman and Siggia [28,29], as shown in Equations (9) and (10):

$${\epsilon }_u={〈{\epsilon }_u\left(x,t\right)〉}_V={\displaystyle \frac{{\nu }^3}{h^4}}\left(Nu-1\right)RaP{r}^{-2}$$

(9)

$${\epsilon }_T={〈{\epsilon }_T\left(x,t\right)〉}_V=\kappa {\displaystyle \frac{{\mathsf{\Delta }}^2}{h^2}}Nu$$

(10)

where ${〈\cdot 〉}_V$ denotes the volume average. ${\epsilon }_u\left(x,t\right)=\nu {\left[{\partial }_i{u}_j\left(x,t\right)\right]}^2$ and ${\epsilon }_T\left(x,t\right)=\kappa {\left[{\partial }_i{T}_j\left(x,t\right)\right]}^2$ are the local kinetic and thermal dissipation rates.

The main idea of Grossmann-Lohse theory [30,31,32,33] is that the global averaged energy dissipations can be decomposed into the bulk and boundary layer (BL) contributions, as shown in Equation (11):

$$\epsilon ={\epsilon }_{\mathrm{bulk}}+{\epsilon }_{\mathrm{BL}}$$

(11)

By means of dimensional analysis, the different contributions can be estimated as Equations (12)–(17):

$${\epsilon }_{u-\mathrm{bulk}}\sim {\displaystyle \frac{{\overline{u}}^3}{h}}={\displaystyle \frac{{\nu }^3}{h^4}}R{e}^3$$

(12)

$${\epsilon }_{T-\mathrm{bulk}}\sim {\displaystyle \frac{\overline{u}{\mathsf{\Delta }}^2}{h}}=\kappa {\displaystyle \frac{{\mathsf{\Delta }}^2}{h^2}}PrRe(\mathrm{for}{\displaystyle \frac{{\delta }_u}{{\delta }_T}}<1)$$

(13)

$${\epsilon }_{T-\mathrm{bulk}}\sim \overline{u}{\displaystyle \frac{{\delta }_T}{{\delta }_u}}{\displaystyle \frac{{\mathsf{\Delta }}^2}{h}}=\kappa {\displaystyle \frac{{\mathsf{\Delta }}^2}{h^2}}ℝPrR{e}^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}N{u}^{-1}(\mathrm{for}{\displaystyle \frac{{\delta }_u}{{\delta }_T}}>1)$$

(14)

$${\epsilon }_{u-\mathrm{BL}}\sim \nu {\displaystyle \frac{{\overline{u}}^2}{{\delta }_u^2}}{\displaystyle \frac{{\delta }_u}{h}}\sim {\displaystyle \frac{{\nu }^3}{h^4}}R{e}^{{\displaystyle \raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$$

(15)

$${\epsilon }_{T-\mathrm{BL}}\sim \kappa {\displaystyle \frac{{\mathsf{\Delta }}^2}{{\delta }_T^2}}{\displaystyle \frac{{\delta }_T}{h}}\sim \kappa {\displaystyle \frac{{\mathsf{\Delta }}^2}{h^2}}{ℝ}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}P{r}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}R{e}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}(\mathrm{for}{\displaystyle \frac{{\delta }_u}{{\delta }_T}}<1)$$

(16)

$${\epsilon }_{T-\mathrm{BL}}\sim \kappa {\displaystyle \frac{{\mathsf{\Delta }}^2}{{\delta }_T^2}}{\displaystyle \frac{{\delta }_T}{h}}\sim \kappa {\displaystyle \frac{{\mathsf{\Delta }}^2}{h^2}}{ℝ}^{{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}P{r}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}R{e}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}(\mathrm{for}{\displaystyle \frac{{\delta }_u}{{\delta }_T}}>1)$$

(17)

where ${\delta }_u\sim hR{e}^{-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$ and ${\delta }_T\sim ℝhN{u}^{-1}$ are the thickness of the kinetic and thermal BL, respectively. Their ratio is related to Pr, i.e., $Pr~{\delta }_u/{\delta }_T$. It should be emphasized that the kinetic BL is assumed to be of Prandtl-Blasius type in Grossmann-Lohse theory.

Utilizing the aforementioned relations, four main regimes can be easily identified as shown in

Table 1. The four main regimes and the primary contributors.

	${\mathit{\epsilon }}_{\mathit{u}}$	${\mathit{\epsilon }}_{\mathit{T}}$
Regime I	${\epsilon }_{u-\mathrm{BL}}$	${\epsilon }_{T-\mathrm{BL}}$
Regime II	${\epsilon }_{u-\mathrm{bulk}}$	${\epsilon }_{T-\mathrm{BL}}$
Regime III	${\epsilon }_{u-\mathrm{BL}}$	${\epsilon }_{T-\mathrm{bulk}}$
Regime IV	${\epsilon }_{u-\mathrm{bulk}}$	${\epsilon }_{T-\mathrm{bulk}}$

, contingent upon whether the bulk or BL is the primary contributor to energy dissipation:

Additionally, based on the relative thickness of the kinetic and thermal BL, each main regime can be divided into two sub-regions, i.e., ${\delta }_u/{\delta }_T<1$ or ${\delta }_u/{\delta }_T>1$. The scaling laws are given in each sub-region as following:

• Regime I-a:${\epsilon }_u\sim {\epsilon }_{u-\mathrm{BL}}$, ${\epsilon }_T\sim {\epsilon }_{T-\mathrm{BL}}$, ${\delta }_u/{\delta }_T<1$

Substituting Equations (8) and (9) into Equations (15) and (16), respectively, one can get the following equations as shown in Equations (18) and (19):

$$Re={ℝ}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}R{a}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}P{r}^{-{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}$$

(18)

$$Nu={ℝ}^{{\displaystyle \raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$8$}\right.}}R{a}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}P{r}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.}}$$

(19)

• Regime I-b:${\epsilon }_u\sim {\epsilon }_{u-\mathrm{BL}}$, ${\epsilon }_T\sim {\epsilon }_{T-\mathrm{BL}}$, ${\delta }_u/{\delta }_T>1$

By combining Equations (8) with (15) and (9) with (17), the following relations can be obtained as shown in Equations (20) and (21):

$$Re={ℝ}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}R{a}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}P{r}^{-{\displaystyle \raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}$$

(20)

$$Nu={ℝ}^{{\displaystyle \raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}R{a}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}P{r}^{-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}}$$

(21)

• Regime II-a:${\epsilon }_u\sim {\epsilon }_{u-\mathrm{bulk}}$, ${\epsilon }_T\sim {\epsilon }_{T-\mathrm{BL}}$, ${\delta }_u/{\delta }_T<1$

Using Equations (8), (9), (12) and (16), one can obtain the following equations as shown in Equations (22) and (23):

$$Re={ℝ}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}R{a}^{{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}P{r}^{-{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}$$

(22)

$$Nu={ℝ}^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}R{a}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}P{r}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}$$

(23)

• Regime II-b:${\epsilon }_u\sim {\epsilon }_{u-\mathrm{bulk}}$, ${\epsilon }_T\sim {\epsilon }_{T-\mathrm{BL}}$, ${\delta }_u/{\delta }_T>1$

Using Equation (17) instead of (16) for (9), we can get the following equations as shown in Equations (24) and (25):

$$Re={ℝ}^{{\displaystyle \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$15$}\right.}}R{a}^{{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}P{r}^{-{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$$

(24)

$$Nu={ℝ}^{{\displaystyle \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}R{a}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}$$

(25)

• Regime III-a:${\epsilon }_u\sim {\epsilon }_{u-\mathrm{BL}}$, ${\epsilon }_T\sim {\epsilon }_{T-\mathrm{bulk}}$, ${\delta }_u/{\delta }_T<1$

Combining Equations (8), (9), (15) and (13), one can derive the following relations as shown in Equations (26) and (27):

$$Re=R{a}^{{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}P{r}^{-{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$$

(26)

$$Nu=R{a}^{{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}P{r}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$$

(27)

• Regime III-b:${\epsilon }_u\sim {\epsilon }_{u-\mathrm{BL}}$, ${\epsilon }_T\sim {\epsilon }_{T-\mathrm{bulk}}$, ${\delta }_u/{\delta }_T>1$

Similarly, using Equation (14) instead of (13) for (9), the following relations can be obtained as shown in Equations (28) and (29):

$$Re={ℝ}^{{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}}R{a}^{{\displaystyle \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}}P{r}^{-{\displaystyle \raisebox{1ex}{$6$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}}$$

(28)

$$Nu={ℝ}^{{\displaystyle \raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}}R{a}^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}}P{r}^{-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}}$$

(29)

• Regime IV-a:${\epsilon }_u\sim {\epsilon }_{u-\mathrm{bulk}}$, ${\epsilon }_T\sim {\epsilon }_{T-\mathrm{bulk}}$, ${\delta }_u/{\delta }_T<1$

Substituting Equations (12) and (13) into Equations (8) and (9), respectively, one can get the following relations as shown in Equations (30) and (31):

$$Re=R{a}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}P{r}^{-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$$

(30)

$$Nu=R{a}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}P{r}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$$

(31)

• Regime IV-b:${\epsilon }_u\sim {\epsilon }_{u-\mathrm{bulk}}$, ${\epsilon }_T\sim {\epsilon }_{T-\mathrm{bulk}}$, ${\delta }_u/{\delta }_T>1$

Combining Equations (8), (9), (12) and (14), the following relations can be determined as shown in Equations (32) and (33):

$$Re={ℝ}^{{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$9$}\right.}}R{a}^{{\displaystyle \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$9$}\right.}}P{r}^{-{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$$

(32)

$$Nu={ℝ}^{{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}R{a}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$$

(33)

The boundaries between the two different regimes or sub-regions can be derived by considering ${\epsilon }_{\mathrm{bulk}}~{\epsilon }_{\mathrm{BL}}$ or ${\delta }_u~{\delta }_T$, respectively. Thereby, the following correlations can be obtained as shown in Equations (34)–(43):

$$R{a}_{1a-2a}\sim {ℝ}^{-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}P{r}_{1a-2a}^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$$

(34)

$$R{a}_{1a-3a}\sim {ℝ}^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}P{r}_{1a-3a}^{-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$$

(35)

$$R{a}_{3a-4a}\sim {ℝ}^{0}P{r}_{3a-4a}^1$$

(36)

$$R{a}_{2a-4a}\sim {ℝ}^{2}P{r}_{2a-4a}^{-1}$$

(37)

$$R{a}_{1b-3b}\sim {ℝ}^{{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}P{r}_{1b-3b}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$$

(38)

$$R{a}_{3b-4b}\sim {ℝ}^{-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}P{r}_{3b-4b}^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$$

(39)

$$P{r}_{1a-b}\sim {ℝ}^{1}R{a}^0$$

(40)

$$P{r}_{2a-b}\sim {ℝ}^{1}R{a}^0$$

(41)

$$P{r}_{3a-b}\sim {ℝ}^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}R{a}^{-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$$

(42)

$$P{r}_{4a-b}\sim {ℝ}^{{\displaystyle \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}R{a}^{-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$$

(43)

It should be noticed that the specific correlations for the boundaries require a massive amount of experimental or simulated results, which is currently lacking. As a result, it is recommended to refer to the original model at present. What is more, one can easily see that the effect of aspect ratio, $\Gamma$, is not shown in the above scaling laws. This is due to the fact that no significant dependence of the scaling exponent on the aspect ratio is anticipated, in contrast to the pre-factors [30,39].

4. Direct Numerical Simulations

In order to verify the above scaling laws derived from theorical analysis, direct numerical simulation (DNS) has been conducted at the range of $Pr=\left(7,30\right)$, $Ra=\left(1,3,5\right)\times {10}^8$, $ℝ=1.0~6.0$, which are common ranges of fluid parameters in practical applications [40]. The effect of thermal radiation on flow structure and heat transfer has been analyzed.

It should be noticed that $ℝ=1.0$ stands for the case without considering radiation effects, while $ℝ=6.0$ does not represent the limit case. Actually, according to Equation (7) and the research of Zhang et al. [41,42], the radiation parameter $ℝ$ depends on the reference temperature, characteristic length of cavity, and attenuation coefficient of salt.

4.1. Numerical Model

The convection cell adopted in this paper is a 2D square cavity with length h and aspect ratio $\Gamma =1$, as shown in Figure 3. The temperatures of the top and bottom walls are fixed at T_c and T_h. The sidewalls are taken to be perfectly adiabatic. No-slip velocity boundary conditions are considered for all the walls. By setting the length h, free-fall velocity $u_f=\sqrt{g\beta \mathsf{\Delta }h}$, and temperature difference Δ to the corresponding reference scales, the following dimensionless variables can be obtained, shown as Equation (44).

$$\begin{array}{l}\left\{X,Y,Z\right\}={\displaystyle \frac{\left\{x,y,z\right\}}{h}},\mathit{U}={\displaystyle \frac{\mathit{u}}{u_f}},\tau ={\displaystyle \frac{t}{h/u_f}},P={\displaystyle \frac{p}{\rho u_f^2}}\\ \theta ={\displaystyle \frac{T-T_{\mathrm{ref}}}{\mathsf{\Delta }}},\widetilde{q^R}=-{\displaystyle \frac{16}{3}}{\displaystyle \frac{\sigma T_{\mathrm{ref}}^3}{{\beta }_r\lambda }}\sqrt{{\displaystyle \frac{1}{RaPr}}}\nabla \theta \end{array}$$

(44)

Therefore, the dimensionless governing equations can be conventionally obtained as shown in Equations (45)–(47) (refer to Equations (1)–(3)).

$$\nabla \cdot \mathit{U}=0$$

(45)

$${\partial }_{\tau }\mathit{U}+(\mathit{U}\cdot \nabla )\mathit{U}=-\nabla P+\sqrt{{\displaystyle \frac{Pr}{Ra}}}{\nabla }^2\mathit{U}+\theta e_z$$

(46)

$${\partial }_{\tau }\theta +\mathit{U}\cdot \nabla \theta =\sqrt{{\displaystyle \frac{1}{RaPr}}}{\nabla }^2\theta -\nabla \widetilde{q^R}$$

(47)

where U, P, θ and $\widetilde{q^R}$ are the dimensionless velocity, pressure, temperature, and radiative heat flux, respectively.

The dimensionless boundary conditions are as shown in Equation (48):

$$\begin{array}{l}\begin{array}{cc}X=\pm 0.5& -{\displaystyle \frac{\partial \theta }{\partial X}}=0\end{array}\\ \begin{array}{cc}Y=-0.5& \theta =0.5\end{array}\\ \begin{array}{cc}Y=0.5& \theta =-0.5\end{array}\end{array}$$

(48)

The initial conditions are set to $\tau =0$, $U=0$, $\theta =0$ (for fluid and sidewall), $P=1$. In order to strike the balance between capturing system dynamics accurately and maintaining computational efficiency, the size of the time step is set to be adjustable, and the corresponding maximum Courant number is limited to 0.9. The temporal average is performed over a duration of more than 5000 dimensionless time units after the convective flow has been fully developed, to allow for the averaging out of transient fluctuations.

4.2. Numerical Method and Validation

The governing equations are discretized by adopting the finite volume method. The pressure-velocity coupling is handled by the PISO algorithm. The Crank-Nicolson scheme is used for the transient terms, the second order upwind scheme is applied to treat the convective terms, and the second order central difference scheme is adopted for diffusion terms.

As shown in Figure 4, non-equidistant meshes are applied and the computational meshes are refined close to all the four walls so that the transport characteristics in BLs can be captured. The grid independence test is carried out at $Ra=5\times {10}^8$, $Pr=1$. The average Nusselt numbers for different grids have been displayed in

Table 2. The averaged Nusselt numbers for different grids.

Grid Size	Nu	Error (%)
128 × 128	39.17710	-
256 × 256	39.84992	1.72%
384 × 384	40.03957	0.48%

. It turns out that the solution becomes independent of grid size for 256 × 256, since the deviation is less than 0.5%.

In order to validate the present code, the results for $Ra=1.0\times {10}^8$, $Pr=4.3$ were compared with those in References [43,44], as shown in

Table 3. Comparisons with public results from References.

	Nu	Error (%)
Present	24.9	-
Huang et al. [43]	25.38	1.89%
Yang et al. [44]	258.0	3.49%

. It can be seen that the maximum relative error for Nu is less than 3.5%. Therefore, the correctness of present numerical method code is verified.

4.3. Effect of Radiation on Flow Structure

Figure 5 shows the time-evolution of isotherm contours (background) and streamlines (blue lines) at different $ℝ$ when $Pr=7,30$ and $Ra=\left(1,5\right)\times {10}^8$. It can be seen that the large-scale circulation (LSC) has been formed when the effect of radiation is not taken into account ($ℝ=1.0$) or weak ($ℝ=1.8$). The shape of LSC is approximately elliptical, which follows the diagonal of the cavity as the major axis and is accompanied by two small vortices in the short axis direction. The two small vortices may expand, causing the large-scale circulation to gradually collapse and change its tilt direction. Interestingly, with the increasing $ℝ$, the collapse and redirection frequencies of LSC decrease. However, for the case of $ℝ=6.0$, there are two small vortices arranged vertically. This makes it difficult for heat to exchange between the cold fluid at the top and the hot fluid at the bottom. The flow structure is similar to that of small Ra, indicating that thermal radiation suppresses flow.

When Pr increases to 30, LSC still exists even at $ℝ=6.0$, suggesting that the effect of thermal radiation weakens for larger Pr. Moreover, as Ra increases, the effect of thermal radiation is not sufficient to suppress the formation of LSC, but its stabilizing effect will cause a decrease in the redirection frequency of LSC.

4.4. Comparisons between Scaling Law and DNS Results

In order to quantitatively verify the correctness of Equations (19) and (21) for examples, the comparison of time average Nu between DNS results and the theoretical analysis has been performed, as shown in Figure 6. Consistent with the results of many other literatures, the heat transfer is significantly enhanced with an increase in $ℝ$.

According to the boundary division between the various regimes in the GL model [30,31,32,33], the parameter range in this paper is located in Regime I-b, where the exponent ι of $ℝ$ in the relation $Nu~{ℝ}^{\iota }R{a}^{\alpha }P{r}^{\gamma }$ is 5/6. However, as shown in Figure 6, the relative error between simulation results and the predicted results of the relation $Nu~{ℝ}^{5/8}$ in Regime I-a is relatively small (about 10%). This phenomenon may be attributed to the increase of δ_T, which arises due to the suppression of convection by thermal radiation and the increase in fluid temperature uniformity. Actually, the increase in BL thickness was also observed by Hossain et al. [45].

Figure 7 depicts the exponent ι of $ℝ$ obtained from theoretical analysis and DNS. It can be seen that the deviation in between is about 10%. Besides, the deviation decreases with the increase of Pr and Ra. Similar conclusions for the range of $Pr=1$ (usually gas) and $Ra=\left(3~4\right)\times {10}^8$ can be obtained in our previous work [46]. Both of them indicate that the derived scaling law can effectively predict the Nu of radiative Rayleigh-Bénard convection by applying the Rosseland approximation.

5. Conclusions

In summary, the transport characteristics of radiative Rayleigh-Bénard convection in optically thick media with large Pr have been investigated by both theorical analysis and DNS. Main conclusions can be summarized as follows:

(1)

Based on the turbulent dissipation theory and Rosseland approximation, the scaling laws in the four main regimes have been summarized within the scope of numerical simulation research presented in this paper ($Pr=\left(7,30\right)$, $Ra=\left(1,3,5\right)\times {10}^8$, $ℝ=1.0~6.0$), $Nu={ℝ}^{{\displaystyle \raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$8$}\right.}}R{a}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}P{r}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.}}$.

(2)

It has been found that due to the presence of thermal radiation, the Nu scaling law in the small Pr range of the GL model is more suitable for predicting the transport characteristics of optical thick media with large Pr. The deviation between the results of DNS and prediction model is approximately 10%.

(3)

The determination of reference temperature is discussed, and there is almost no difference between the arithmetic mean temperature and radiative mean temperature, especially within the scope of application of Boussinesq approximation.

(4)

The influence of thermal radiation results in a significant enhancement of heat transfer.

(5)

When $Ra>1\times {10}^8$, the LSC can been formed for weak radiation effect. The effect of radiation can stabilize flow, specifically manifested by a collapse of LSC as $ℝ$ increases. However, the radiation effect weakens for larger Pr.

These conclusions offer a theoretical underpinning for industrial designs related to energy storage, power generation, and thermal storage technology. Future research endeavors can focus on determining the pre-factors of scaling laws to further refine and enhance our understanding of these processes.