# Development and Assessment of an Isotropic Four-Equation Model for Heat Transfer of Low Prandtl Number Fluids

^{1}Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, China^{2}School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing, China^{3}School of Nuclear Science and Technology, Lanzhou University, Lanzhou, China

In the simple gradient diffusion hypothesis, the turbulent Prandtl number (*Pr* fluids are obtained using the present isotropic four-equation model.

## 1 Introduction

Liquid metals are widely considered coolants with good thermal-hydraulic characteristics for many energy systems, such as fast reactors and subcritical reactors (Gu and su, 2021; Wang et al., 2021). However, liquid metals’ low molecular Prandtl number

It is difficult, dangerous, and costly to experiment with liquid metals (Schroer et al., 2012; Ejenstam and Szakálos, 2015), while the computational fluid dynamics (CFD) method is more commonly applied to investigate the thermal-hydraulic properties of these liquids. However, in CFD, the computing cost required by the direct numerical simulation (DNS) method and the large eddy simulations (LES) method is too high to rely on this technique for a quick and economic calculation of heat transfer in complex geometries (Kawamura et al., 1999), while the Reynolds Averaged Navier–Stokes (RANS) method can be promoted. The velocity boundary layer and the temperature layer of traditional fluid (*Pr* ≈ 1) are generally considered to be similar in the RANS framework. In this way, one can obtain a constant turbulent Prandtl number *Pr* fluids (Reynolds, 1975).

In the SGDH framework, Cheng (Cheng and Tak, 2006) derived a

Another popular model in the SGDH work, called the four-equation *k-ε-k*_{θ}*-ε*_{θ} turbulent heat transfer model, is introduced into turbulent and thermal time-scales for the simulation of the explicit first-order turbulent heat diffusivity. The literature (Nagano and Kim, 1988; Abe et al., 1995; Nagano and Shimada. 1996) has contributed extensively to near-wall model closure and thermal turbulence effect for a four-equation model. In recent years, Manservisi (Manservisi and Menghini, 2014a) improved the four-equation *k-ε-k*_{θ}*-ε*_{θ} model proposed by Abe and Nagano. The model predicted the heat transfer process of the plane, circular tube, triangular rod bundles, and quadrilateral rod bundles for *k-ω-k*_{θ}*-ω*_{θ} model with the specific dissipation rates *ω* = *ε*/(*C*_{μ}*k*) and *ω*_{θ} = *ε*_{θ}*/*(*C*_{μ}*k*_{θ}) was developed based on Manservisi’s *k-ε-k*_{θ}*-ε*_{θ} model (Cerroni et al., 2015). Then by the following work of the literatures (Vià et al., 2016; Chierici et al., 2019; Vià and Manservivi, 2019), logarithmic specific dissipation rates of *ω* and *ω*_{θ}, that is, *Ω =* ln(*ω*) and *Ω*_{θ} *=* ln (*ω*_{θ}), have been used to simplify the near-wall boundary conditions and have been introduced into the logarithmic four-equation *k-Ω-k*_{θ}*-Ω*_{θ} model. Other state variables as proposed by Youssef (Youssef, 2006) are the velocity *τ*_{u} *= k/ε* and temperature time-scale *τ*_{θ}*= k*_{θ}*/ε*_{θ} to improve its numerical stability. The proposed *k*_{θ}*-τ*_{θ} model does not suffer from numerical stiffness problems since natural boundary conditions for the variables *k*_{θ} and *τ*_{θ} are used (*k*_{θ} *= τ*_{θ} = 0, at walls). In addition, the static variables *ε* and *ε*_{θ}, can also provide Dirichlet wall boundary conditions and simpler transport modes than the *k*_{θ}*-τ*_{θ} modes (Nagano and Shimada, 1996; Nagano et al., 1997).

However, the application codes of an isotropic four-equation model in complex geometries are still lacking for liquid metals, and its reliability needs to be further verified and evaluated. So, in the present work, the four-equation model in an isotropic dissipation rate

## 2 Mathematical Model

### 2.1 Isotropic Four-Equation Turbulence Model

The incompressible RANS equations with no gravity and constant physical properties for the calculation of velocity, pressure, and temperature fields are considered as follows:

where

The turbulent viscosity

Due to low *Pr* fluids’ physical properties, there are great differences between the momentum and heat. A constant *Pr*_{t} ≈ 0.85–0.9 cannot give acceptable results for liquid metals. It is possible to consider

where

**TABLE 1**. Values for the

Both the turbulent Reynolds number

where

### 2.2 Turbulent Viscosity and Turbulent Thermal Diffusivity

Some dynamic and thermal time-scales should be applied to calculate the turbulent viscosity

where

and

where

and

The above model has been proved to be effective in producing correct turbulence behavior of liquid metal, which draws into the mixing time-scale

### 2.3 Wall Boundary Conditions for Turbulence Models

Appropriate boundary conditions should be applied to solve the four-equation turbulent models without the available wall functions. By using series expansions as shown in Table 2, the near-wall behaviors of dynamical and thermal turbulence variables

According to the definitions of Eqs 7, 8, when

### 2.4 Modified Navier–Stokes Equations for Periodic Boundary Conditions

For a fully developed turbulent field, the pressure gradient term in Eq. 2 is divided into the mean of the pressure gradient

For a fully developed temperature field, temperature *T* is written as follows:

where

## 3 Numerical Results and Analysis

### 3.1 Numerical Solver of the Isotropic Four-Equation Model

For the modified RANS system and the isotropic four-equation

where Q stands for

### 3.2 Numerical Verification of the Plane

The full development process of different *Pr* fluids in a constant heat flow heated plane was studied using DNS (Kawamura et al., 1998; Kawamura et al., 1999; Tiselj and Cizelj, 2012). This study compares the DNS data with *Pr* = 0.01 ∼ 0.05. The plane geometry and mesh parameters are consistent with those in the literature (Su et al., 2021; Su and Gu, 2021). Here, the three-dimensional parallel plane with a flow length of 6.4 h, a spanwise width of 3.2 h, and a plane height of 2 h were selected as the calculation domain, where *h* is the half-height of the plane, 30.25 mm. The wall heat flux is set as *q*_{w} = 360 kW/m^{2}. Finally, 80, 50, and 100 nodes are divided into three directions, respectively: Flow direction, span direction, and height. Due to the symmetry of the plane, half of the plane is taken as the calculation domain, and the structured grid is used. The grid height of the first boundary layer is set to 3 × 10^{−6} m to meet the requirements of the low Reynolds number turbulence model,

##### 3.2.1 Velocity Field Verification

The dimensionless velocity *u*^{+} = *y*^{+} in the linear region (*u*^{+} = *y*^{+}) while it meets *u*^{+} = 2.5ln *y*^{+}+5 in the logarithmic rate region (*y*^{+} > 30).

The distribution of dimensionless Reynolds stress *y*^{+} < 7), this study predicted *y*^{+} > 30, the total shear stress coincides with the Reynolds stress curve.

##### 3.2.2 Temperature Field Verification

1) *Pr* = 0.01

The dimensionless temperature *Pr* = 0.01 fluid along *θ*^{+} = *Pry*^{+} in the linear region (1 < *y*^{+} < 40).

**FIGURE 3**. Non-dimensional temperature and heat flux of the plane for *Pr* = 0.01. **(A)** temperature; **(B)** heat flux of case A; **(C)** heat flux of case B; **(D)** heat flux of case C.

The distribution of dimensionless Reynolds heat flux *Pr* fluid, although the predicted Reynolds heat flux deviates from the DNS results at 1 < *y*^{+} < 30, it does not affect the linear distribution of the temperature field in this region. With the increase in wall distance, although the effect of turbulent heat diffusion is gradually increasing, there is still a difference between the total heat flow and Reynolds heat flux, mainly caused by the strong molecular heat conduction of low *Pr* fluid.

2) *Pr* = 0.025

The distribution of *θ*^{+}/*Pr* of the *θ*^{+}/*Pr* increases with the increase in the Reynolds number. The distribution of *Re*, the

3) *Pr*=0.05

**FIGURE 4**. Non-dimensional temperature and heat flux of the plane for *Pr* = 0.025. **(A)** temperature; **(B)** heat flux.

As shown in Figure 5A, the dimensionless temperature *θ*^{+} of case A of *Pr* = 0.05 fluid is in good agreement with the DNS results. At the same Reynolds number, with the decrease in *Pr*, the molecular thermal conductivity increases and the average *θ*^{+} decreases. As shown in Figure 5B, the dimensionless Reynolds heat flux of case A of *Pr* = 0.05 fluid is in good agreement with the DNS results.

**FIGURE 5**. Non-dimensional temperature and heat flux of the plane for different *Pr* fluids. **(A)** temperature; **(B)** heat flux.

### 3.3 Numerical Analysis of the Square Bundle

To further analyze the applicability of the current isotropic four-equation model in complex geometries, the flow and heat of liquid metal with a Prandtl value of 0.01 in the quadrilateral infinite rod bundle were studied. Figure 6 shows the schematic diagram of the computational domain and local hexahedral mesh of the quadrilateral infinite rod bundle. Lines ab, bo, oc, and arc ca are the local distributions to be analyzed next. The flow parameters are reported in Table 4. The surfaces of rods in contact with the fluids are heated by uniform wall fluxes *D* and *X = P/D* ratio are the same as in Zhukov’s experiment (Zhukov et al., 2002).

**FIGURE 6**. Schematic diagram of the quadrilateral infinite rod bundle. **(A)** flow area; **(B)** local mesh.

##### 3.3.1 Heat Transfer Evaluation and Analysis

In Table 5, a few important heat transfer correlations of Nusselt number *Nu*(*Pe*) data from the heat transfer experiment of 22%Na–78%K in a quadrilateral rod bundle. The correlations obtained by Subbotin (Subbotin et al., 1965) and BREST (Adamov 2001) were derived from triangular bundles, while Mikityuk, from a wide database, conducted experimental results of liquid metals (Mikityuk 2009). It is worth noting that there are no more available experimental data. The correlations should be used with care and be paid more attention to for analysis of the reliability. The

where

First, the simulations for *X* = 1.25, 1.34, and 1.46 with a corresponding hydraulic diameter

**FIGURE 7**. Schematic diagram of Nusselt number for the square bundle. **(A)** *X* = 1.25; **(B)** *X* = 1.34; **(C)** *X* = 1.46; **(D)** *X* = 1.22–1.5.

In Figures 7A–C, the prediction *Nu* of the isotropic *Nu* seems to be more conservative than these experimental correlations. Poor experimental relevance makes this problem’s numerical verification difficult to go on.

As shown in Figure 7D, the numerical slope in the *Nu*(*Pe*) line obtained using the

This present work focuses on the comparison of the effects of two turbulent thermal diffusion models (the

##### 3.3.2 Turbulent Thermal Fields

1) Coolant and wall temperature

Figure 8 shows the dimensionless temperature *X* = 1.25. The overall dimensionless temperature in the channel decreases with the increase in

**FIGURE 8**. Schematic diagram of dimensionless temperature on a slice for square lattice *X* = 1.25. **(A)** *Pe* = 250; **(B)** *Pe* = 2000; **(C)** *Pe* = 4000.

Figure 6 has annotated the line ab, line bo, line oc, and arc ca, where points b and c represent the center of the gap and the channel, respectively. The dimensionless temperature field with *Pe* = 2000 is selected for analysis. In Figure 9A, with the decrease in *P/D*, the gap width decreases, and the overall dimensionless temperature distribution *P/D*. Due to the development of velocity and sufficient cooling of the fluid near the center of the channel, the dimensionless temperature distribution over line oc decreases when the value of *P/D* decreases in Figure 9C.

2) Fluctuation and isotropic dissipation

**FIGURE 9**. Schematic diagram of dimensionless temperature over lines for the square lattice at *Pe* = 2000 with different *P/D* ratios. **(A)** line ab; **(B)** line bo; **(C)** line oc; **(D)** arc ca.

The distribution and shape of

3) Turbulence and molecular thermal diffusivity

**FIGURE 10**. Schematic diagram of average temperature fluctuaion (left) and its isotropic dissipation (right) on a slice for square lattice *X* = 1.25. **(A)** *Pe* = 1000; **(B)** *Pe* = 2000; **(C)** *Pe* = 4000.

**FIGURE 11**. Schematic diagram of dimensionless temperature fluctuation over lines for the square lattice at Pe = 2000 with different P/D ratios. **(A)** line ab; **(B)** line bo; **(C)** line oc.

Figure 12 shows the proportional relationship of turbulent to molecular thermal diffusivity, that is, dimensionless thermal diffusivity *X* = 1.25 at *Pe* = 2000. Reynolds heat flux

**FIGURE 12**. Schematic diagram of dimensionless thermal diffusivity over lines for the square lattice at *Pe* = 2000 with different *P/D* ratios. **(A)** line ab; **(B)** line bo; **(C)** line oc.

Thus, the specific value of turbulent to molecular thermal diffusivity represents the ratio of Reynolds to molecular heat flux. Reynolds heat flux is mainly caused by thermal diffusion caused by turbulent flow, while molecular heat conduction generates molecular heat flux. In Figure 12, in the area closest to the wall point a or c, *P/D* increases. But it decreases over the line bo and line oc when the valve of *P/D* increases because at the same *P/D*, the larger the flow area, the smaller the overall velocity, and the smaller the turbulent heat diffusion over those areas. In Figure 13, the dimensionless thermal diffusivity on a slice for *P/D*, when the

4) Turbulent Prandtl number

**FIGURE 13**. Schematic diagram of dimensionless thermal diffusivity on a slice for square lattice *X* = 1.25. **(A)** *Pe* = 250; **(B)** *Pe* = 1000; **(C)** *Pe* = 2000; **(D)** *Pe* = 4000.

In Figure 14, the *Pr*_{t} with the change of *P/D* is shown over the lines ab, bo, and oc for the square bundle with *X* = 1.25 at *Pe* =2000. The *Pr*_{t} is not constant and linear. It is larger than the general value of 0.85 for liquid metal. The *Pr*_{t} distribution on the line bo is to be smoother than that of lines ab and co. The value of *Pr*_{t} decreases over line ab when the value of *P/D* increases, while it increases over lines bo and oc. This phenomenon echoes the changing trend of dimensionless thermal diffusivity. In Figure 15, the turbulent Prandtl number on a slice for *Pe* = 1,000, 2000, and 4,000 for the square bundle is shown. When *Pr*_{t} near the wall and the center decreases gradually.

**FIGURE 14**. Schematic diagram of turbulent Prandtl number over lines for the square lattice at *Pe* = 2000 with different *P/D* ratios. **(A)** line ab; **(B)** line bo; **(C)** line oc.

**FIGURE 15**. Schematic diagram of turbulent Prandtl number on a slice for square lattice *X* = 1.25. **(A)** *Pe* = 1000; **(B)** *Pe* = 2000; **(C)** *Pe* = 4000.

The average turbulent Prandtl number

Table 6 summarizes the calculation results of average turbulent Prandtl number *P/D* and *Pe* using the present *P/D*, the average turbulent Prandtl number decreases with the increase in *Pe*, while at the same *Pe*, it increases when the value of *P/D* increases.

5) Performance analysis of the present four-equation model

When the same turbulence *Pr*_{t} = 0.85 model. To analyze the calculation speed of the ^{4} MT with an Intel(R) Core(TM) i7-9700 CPU at 3.00 GHz and 16 GB system memory at 2,667 MHz. At present, the model solver is written based on OpenFOAM V6 version, compiled and integrated in Ubuntu 18.04 system environment. As shown in Table 7, the calculation time and memory required by the

## 4 Conclusion

The present work studied an isotropic four-equation model for low Pr number that uses simple Dirichlet wall boundaries. First, the turbulent heat transfer process of *Pr* = 0.01 ∼ 0.05 fluid in the uniformly heated plane is numerically studied on the open-source program OpenFOAM. Then the flow and heat of the isotropic four-equation of the quadrilateral infinite rod bundle region are evaluated and analyzed with low Prandtl number *Pr* = 0.01. In the SGDH framework, the *Pr*_{t} = 0.85 model and the isotropic four-equation model are compared with available experimental correlations in the range of Pe = 250–4,000 and P/D = 1.25–1.46. The numerical results show that.

1) The full development velocity, temperature, Reynolds stress, and Reynolds heat flow of *Pr* = 0.01 ∼ 0.05 fluid in-plane predicted using the isotropic four-equation model are in good agreement with the DNS results.

2) The Nu of the isotropic four-equation model lies between the experimental relationship, more conservative and similar to the Zhukov and BREST correlations for square rod bundles, while the *Pr*_{t} = 0.85 model gives too high a Nusselt number prediction to predict the integra heat properly. The slope of Nu predicted by the present isotropic four-equation model is similar to Manservisi’s model.

3) More detailed heat exchange phenomena and local temperature distribution are obtained using the four-equation model. At the same *P/D*, the average turbulent Prandtl number decreases with the increase in Pe, while at the same Pe, it increases when the value of *P/D* increases.

The isotropic four-equation model can provide more references for calculating the thermal-hydraulic phenomena of liquid metals. But its wider applicability needs further verification.

## Data Availability Statement

The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

## Author Contributions

XS: concept, research, writing, editing, code, and data processing. XL: modification, concept, research, and code. XW: research, modification, and code. YL: concept, editing, and research. QC: editing and research. QS: editing and research. XS: editing and research. GL: funding, project management, concept, and research.

## Funding

This work was supported by the Research on key technology and safety verification of primary circuit, Grant No. 2020YFB1902104, and the Experimental study on thermal-hydraulics of fuel rod bundle, Grant No. Y828020XZ0.

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

## References

Abe, K., Kondoh, T., and Nagano, Y. (1995). A New Turbulence Model for Predicting Fluid Flow and Heat Transfer in Separating and Reattaching Flows-II. Thermal Field Calculations. *Int. J. Heat Mass Transfer* 38 (8), 1467–1481. doi:10.1016/0017-9310(94)00252-q

Adamov, E. O. (2001). *Naturally Safe Lead-Cooled Fast Reactor for LargeScale Nuclear Power*. Moscow: Dollezhal RDIPE.

Cerroni, D., Da Vià, R., Manservisi, S., Menghini, F., Pozzetti, G., and Scardovelli, R. (2015). Numerical Validation of Aκ-ω-Κθ-Ωθheat Transfer Turbulence Model for Heavy Liquid Metals. *J. Phys. Conf. Ser.* 655 (1), 012046. doi:10.1088/1742-6596/655/1/012046

Cheng, X., and Tak, N.-i. (2006). Investigation on Turbulent Heat Transfer to lead-bismuth Eutectic Flows in Circular Tubes for Nuclear Applications. *Nucl. Eng. Des.* 236 (4), 385–393. doi:10.1016/j.nucengdes.2005.09.006

Chierici, A., Chirco, L., Da Vià, R., and Manservisi, S. (2019). Numerical Simulation of a Turbulent Lead Bismuth Eutectic Flow inside a 19 Pin Nuclear Reactor Bundle with a Four Logarithmic Parameter Turbulence Model. *J. Phys. Conf. Ser.* 1224, 012030. doi:10.1088/1742-6596/1224/1/012030

Choi, S. K., and Kim, S. O. (2007). Treatment of Turbulent Heat Fluxes with the Elliptic-Blending Second-Moment Closure for Turbulent Natural Convection Flows. *Int. J. Heat Mass Transfer* 51 (9-10), 2377–2388. doi:10.1016/j.ijheatmasstransfer.2007.08.012

Da Vià, R., Giovacchini, V., and Manservisi, S. (2020). A Logarithmic Turbulent Heat Transfer Model in Applications with Liquid Metals for Pr = 0.01-0.025. *Appl. Sci.* 10 (12), 4337. doi:10.3390/app10124337

Da Vià, R., and Manservisi, S. (2019). Numerical Simulation of Forced and Mixed Convection Turbulent Liquid Sodium Flow over a Vertical Backward Facing Step with a Four Parameter Turbulence Model. *Int. J. Heat Mass Transfer* 135, 591–603. doi:10.1016/j.ijheatmasstransfer.2019.01.129

Da Vià, R., Manservisi, S., and Menghini, F. (2016). A K-Ω-Kθ-Ωθ Four Parameter Logarithmic Turbulence Model for Liquid Metals. *Int. J. Heat Mass Transfer* 101 (10), 1030–1041. doi:10.1016/j.ijheatmasstransfer.2016.05.084

Deng, B., Wu, W., and Xi, S. (2001). A Near-wall Two-Equation Heat Transfer Model for wall Turbulent Flows. *Int. J. Heat Mass Transfer* 44 (4), 691–698. doi:10.1016/s0017-9310(00)00131-9

Duponcheel, M., Bricteux, L., Manconi, M., Winckelmans, G., and Bartosiewicz, Y. (2014). Assessment of RANS and Improved Near-wall Modeling for Forced Convection at Low Prandtl Numbers Based on LES up to Reτ=2000. *Int. J. Heat Mass Transfer* 75, 470–482. doi:10.1016/j.ijheatmasstransfer.2014.03.080

Ejenstam, J., and Szakálos, P. (2015). Long Term Corrosion Resistance of Alumina Forming Austenitic Stainless Steels in Liquid lead. *J. Nucl. Mater.* 461, 164–170. doi:10.1016/j.jnucmat.2015.03.011

Ge, Z., Liu, J., Zhao, P., Nie, X., and Ye, M. (2017). Investigation on the Applicability of Turbulent-Prandtl-Number Models in Bare Rod Bundles for Heavy Liquid Metals. *Nucl. Eng. Des.* 314, 198–206. doi:10.1016/j.nucengdes.2017.01.032

Groetzbach, G. (2013). Challenges in Low-Prandtl Number Heat Transfer Simulation and Modelling. *Nucl. Eng. Des.* 264, 41–55. doi:10.1016/j.nucengdes.2012.09.039

Gu, L., and Su, X. (2021). Latest Research Progress for LBE Coolant Reactor of China Initiative Accelerator Driven System Project. *Front. Energ.* 15, 810–831. doi:10.1007/s11708-021-0760-1

Kawamura, H., Abe, H., and Matsuo, Y. (1999). DNS of Turbulent Heat Transfer in Channel Flow with Respect to Reynolds and Prandtl Number Effects. *Int. J. Heat Fluid Flow* 20 (3), 196–207. doi:10.1016/s0142-727x(99)00014-4

Kawamura, H., Ohsaka, K., Abe, H., and Yamamoto, K. (1998). DNS of Turbulent Heat Transfer in Channel Flow with Low to Medium-High Prandtl Number Fluid. *Int. J. Heat Fluid Flow* 19 (5), 482–491. doi:10.1016/s0142-727x(98)10026-7

Kays, W. M. (1994). Turbulent Prandtl Number-Where Are We? *Asme Trans. J. Heat Transfer* 116 (2), 284–295. doi:10.1115/1.2911398

Lai, Y. G., and So, R. M. C. (1990). Near-wall Modeling of Turbulent Heat Fluxes. *Int. J. Heat Mass Transfer* 33 (7), 1429–1440. doi:10.1016/0017-9310(90)90040-2

Manservisi, S., and Menghini, F. (2014a). A CFD Four Parameter Heat Transfer Turbulence Model for Engineering Applications in Heavy Liquid Metals. *Int. J. Heat Mass Transfer* 69 (2), 312–326. doi:10.1016/j.ijheatmasstransfer.2013.10.017

Manservisi, S., and Menghini, F. (2015). CFD Simulations in Heavy Liquid Metal Flows for Square Lattice Bare Rod Bundle Geometries with a Four Parameter Heat Transfer Turbulence Model. *Nucl. Eng. Des.* 295 (12), 251–260. doi:10.1016/j.nucengdes.2015.10.006

Manservisi, S., and Menghini, F. (2014b). Triangular Rod Bundle Simulations of a CFD κ-ϵ-κ-ϵ Heat Transfer Turbulence Model for Heavy Liquid Metals. *Nucl. Eng. Des.* 273, 251–270. doi:10.1016/j.nucengdes.2014.03.022

Mikityuk, K. (2009). Heat Transfer to Liquid Metal: Review of Data and Correlations for Tube Bundles. *Nucl. Eng. Des.* 239 (4), 680–687. doi:10.1016/j.nucengdes.2008.12.014

Moukalled, F., Mangani, L., and Darwish, M. (2016). *The Finite Volume Method in Computational Fluid Dynamics*. Berlin: Springer International Publishing.

Nagano, Y., Hattori, H., and Abe, K. (1997). Modeling the Turbulent Heat and Momentum Transfer in Flows under Different thermal Conditions. *Fluid Dyn. Res.* 20 (1-6), 127–142. doi:10.1016/s0169-5983(96)00049-4

Nagano, Y., and Kim, C. (1988). A Two-Equation Model for Heat Transport in Wall Turbulent Shear Flows. *J. Heat Transfer* 110 (3), 583–589. doi:10.1115/1.3250532

Nagano, Y., and Shimada, M. (1996). Development of a Two‐equation Heat Transfer Model Based on Direct Simulations of Turbulent Flows with Different Prandtl Numbers. *Phys. Fluids* 8 (12), 3379–3402. doi:10.1063/1.869124

Reynolds, A. J. (1975). The Prediction of Turbulent Prandtl and Schmidt Numbers. *Int. J. Heat Mass Transfer* 18 (9), 1055–1069. doi:10.1016/0017-9310(75)90223-9

Schroer, C., Wedemeyer, O., Skrypnik, A., Novotny, J., and Konys, J. (2012). Corrosion Kinetics of Steel T91 in Flowing Oxygen-Containing lead–bismuth Eutectic at 450 °C. *J. Nucl. Mater.* 431 (1-3), 105–112. doi:10.1016/j.jnucmat.2011.11.014

Shams, A., De Santis, A., and Roelofs, F. (2019). An Overview of the AHFM-NRG Formulations for the Accurate Prediction of Turbulent Flow and Heat Transfer in Low-Prandtl Number Flows. *Nucl. Eng. Des.* 355, 110342. doi:10.1016/j.nucengdes.2019.110342

Shikazono, N., and Kasagi, N. (1996). Second-moment Closure for Turbulent Scalar Transport at Various Prandtl Numbers. *Int. J. Heat Mass Transfer* 39 (14), 2977–2987. doi:10.1016/0017-9310(95)00339-8

Su, X. K., and Gu, L. (2021). *Numerical Study on Turbulent Heat Transfer of LBE in an Annulus Based on A Four-Equation Model*. Chinese Beijing: Atomic Energy Science and technology. Avaialbe at: http://kns.cnki.net/kcms/detail/11.2044.TL.20210720.181. doi:10.13832/j.jnpe.2021.S1.0026

Su, X. K., Gu, L., Peng, T. J., Jiatai, L., Xianwen, L., and Guan, W. (2021). Research on a Four-Equation Model Based on OpenFOAM. *Nucl. Power Eng.* 42 (S1), 26–32. (in chinese).

Subbotin, V. I., Ushakov, P. A., and Kirillov, P. L. (1965). “Heat Transfer in Elements of Reactors with a Liquid Metal Coolant,” in Proceedings of the 3rd International Conference on Peaceful Use of Nuclear Energy, Geneva, 192–200.

Tiselj, I., and Cizelj, L. (2012). DNS of Turbulent Channel Flow with Conjugate Heat Transfer at Prandtl Number 0.01. *Nucl. Eng. Des.* 253 (1), 153–160. doi:10.1016/j.nucengdes.2012.08.008

Wang, G., Gu, L., and Yun, D. (2021). Preliminary Multi-Physics Performance Analysis and Design Evaluation of UO2 Fuel for LBE-Cooled Subcritical Reactor of China Initiative Accelerator Driven System. *Front. Energ. Res.* doi:10.3389/FENRG.202110.3389/fenrg.2021.732801

Weller, H. G., Tabor, G., Jasak, H., and Fureby, C. (1998). A Tensorial Approach to Computational Continuum Mechanics Using Object-Oriented Techniques. *Comput. Phys.* 12 (6), 620–631. doi:10.1063/1.168744

You, B. H., Jeong, Y. H., and Addad, Y. (2019). Assessment of Advanced RANS Models Ability to Predict a Turbulent Swept Liquid Metal Flow over a Wire in a Channel. *Nucl. Eng. Des.* 353 (Nov), 1102061–11020614. doi:10.1016/j.nucengdes.2019.110206

Youssef, M. S. (2006). A Two-Equation Heat Transfer Model for wall Turbulent Shear Flows. *J. Eng. Sci.* 34 (6), 1877–1903. doi:10.21608/jesaun.2006.111184

Keywords: low Pr fluid, four-equation, OpenFOAM, heat transfer, liquid metal

Citation: Su X, Li X, Wang X, Liu Y, Chen Q, Shi Q, Sheng X and Gu L (2022) Development and Assessment of an Isotropic Four-Equation Model for Heat Transfer of Low Prandtl Number Fluids. *Front. Energy Res.* 10:816560. doi: 10.3389/fenrg.2022.816560

Received: 16 November 2021; Accepted: 14 January 2022;

Published: 08 February 2022.

Edited by:

Wei Ding, Helmholtz Association of German Research Centres (HZ), GermanyReviewed by:

Yacine Addad, Khalifa University, United Arab EmiratesLuteng Zhang, Chongqing University, China

Copyright © 2022 Su, Li, Wang, Liu, Chen, Shi, Sheng and Gu. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Long Gu, gulong@impcas.ac.cn