Computational Model Applicability Analysis of RANS Simulation for Helium-Xenon Mixture in a Parallel Channel

Shunqi Wang¹, Yu Ji¹, Jun Sun¹, Yuliang Sun¹
¹Institute of Nuclear and New Energy Technology, Key Laboratory of Advanced Reactor Engineering and Safety of Ministry of Education, Tsinghua University, Beijing 100084, China

Abstract

Small gas-cooled nuclear reactor systems predominantly utilize helium-xenon mixtures (He-Xe) as the circulating working fluid and reactor core coolant. At the recommended composition, the Prandtl number (Pr) of He-Xe can be as low as 0.2. The Reynolds-Averaged Navier-Stokes (RANS) method, which depends on turbulence models and turbulent Prandtl number (Prt) models, is widely employed for thermal-hydraulic analysis. However, existing model development studies have focused primarily on the influence of Prt models on He-Xe heat transfer predictions, while neglecting a thorough analysis of the turbulence model's role in heat transfer calculations. This oversight introduces systematic bias into subsequent Prt model development. Therefore, this study employs the RANS method to investigate He-Xe flow and heat transfer in a parallel channel, using Direct Numerical Simulation (DNS) results as a benchmark. We sequentially evaluate the applicability of turbulence models and Prt models to identify optimal turbulence models for channel flow and elucidate the underlying mechanisms by which turbulence models affect heat transfer predictions. The results demonstrate that k-ω type models are the preferred turbulence models for channel flow simulations. For He-Xe RANS calculations, current Prt models require further improvement. Turbulence model selection proves critical for heat transfer accuracy. We recommend that future Prt model development for He-Xe flow and heat transfer should explicitly specify the companion turbulence model and incorporate correction coefficients that account for deviations between the turbulence model predictions and physical reality.

Keywords: Reynolds-Averaged Numerical Simulation; DNS Standard Data; RANS Models

1 Introduction

Helium-xenon mixtures (He-Xe) serve as the working fluid and core coolant in small gas-cooled nuclear reactor systems [1-3]. Research indicates that the Prandtl number of He-Xe depends solely on the mixing ratio: as the helium fraction decreases, the Prandtl number initially decreases then increases. In practical applications, the helium mole fraction typically approximates 72% [4], yielding a Prandtl number as low as 0.2—a significant departure from conventional working fluids. Since fluids with different Prandtl numbers exhibit distinct heat transfer characteristics [5], investigating He-Xe flow and heat transfer behavior is essential for thermal-hydraulic analysis of small gas-cooled reactor systems.

The Reynolds-Averaged Navier-Stokes (RANS) method finds widespread application in scientific and engineering studies of working fluid flow and heat transfer due to its low computational cost and high efficiency. Previous research [6,7] has demonstrated that the turbulent Prandtl number (Prt) model substantially influences heat transfer prediction accuracy. For fluids with Pr > 0.7, such as water and air, a constant Prt model with Prt = 0.85 provides satisfactory accuracy. However, for low-Pr fluids like He-Xe and liquid metals, constant Prt models fail to predict wall temperature and Nusselt number effectively [6]. Consequently, researchers have developed variable Prt models for low-Pr fluids [6,8-11]. These studies, however, have focused exclusively on the Prt model's influence while neglecting a thorough analysis of the turbulence model's role, introducing systematic bias into subsequent He-Xe Prt model development.

To address this gap, the present study employs DNS data for He-Xe flow and heat transfer in a parallel channel [12] as a benchmark, with a Reynolds number of 13,700. The research methodology proceeds as follows: first, we conduct a turbulence model applicability study to identify optimal turbulence models for channel flow; second, based on these models, we evaluate Prt model applicability; and finally, we clarify the influence mechanisms of both turbulence and Prt models on heat transfer predictions. Based on these findings, we provide recommendations for He-Xe Prt model development.

2.1 Governing Equations

The RANS governing equations are:

$$
\begin{align}
\frac{\partial \bar{u}_i}{\partial x_i} &= 0 \
\frac{\partial \bar{u}_i}{\partial t} + \frac{\partial}{\partial x_j}(\bar{u}_i\bar{u}_j) &= -\frac{1}{\rho}\frac{\partial \bar{p}}{\partial x_i} + \frac{\partial}{\partial x_j}\left[\nu\left(\frac{\partial \bar{u}_i}{\partial x_j} + \frac{\partial \bar{u}_j}{\partial x_i}\right) - \overline{u_i'u_j'}\right] \
\frac{\partial \bar{T}}{\partial t} + \frac{\partial}{\partial x_j}(\bar{u}_j\bar{T}) &= \frac{\partial}{\partial x_j}\left[\alpha\frac{\partial \bar{T}}{\partial x_j} - \overline{u_j'T'}\right]
\end{align}
$$

where the overbar denotes time-averaged quantities, $\nu$ and $\alpha$ represent the kinematic viscosity and thermal diffusivity of the working fluid, respectively, while $\nu_t$ and $\alpha_t$ denote the turbulent kinematic viscosity and turbulent thermal diffusivity—quantities with the same dimensions as $\nu$ and $\alpha$ but related to the turbulent flow field. In the RANS framework, turbulence models and Prt models are essentially developed to solve for $\nu_t$ and $\alpha_t$. The relationship among these three quantities is:

$$
\alpha_t = \frac{\nu_t}{\text{Prt}}
$$

where $\nu_t$ is obtained from the turbulence model, and Prt is specified by the user-defined Prt model, which together determine $\alpha_t$.

2.2 Computational Domain and Boundary Conditions

The computational domain is a parallel channel, as illustrated in [FIGURE:1]. The coordinate directions $x$, $y$, and $z$ correspond to the streamwise, wall-normal, and spanwise directions, respectively. The domain dimensions are $6.4\delta \times 2.0\delta \times 3.2\delta$, where $\delta$ represents the channel half-width, set to 5 mm in this study. The RANS calculations employ constant property assumptions, using the properties of He-Xe (40 g/mol) at 2 MPa and 1000 K. After sufficient development length, the flow and heat transfer become statistically homogeneous in the streamwise and spanwise directions, enabling the application of periodic boundary conditions. The walls employ no-slip velocity boundary conditions and constant heat flux temperature boundary conditions.

2.3 DNS Standard Data

Our research group performed DNS calculations for the domain shown in [FIGURE:1] using the Nek5000 software [12], which employs the spectral element method with seventh-order numerical accuracy. Simulations were conducted at Reynolds numbers of 5,600 and 13,700. For this configuration, Kawamura et al. [5] performed DNS calculations in 1998 and obtained dimensionless velocity and temperature fields for a fluid with Pr = 0.2 at Re = 5,600. By comparing the low-Re Nek5000 results with Kawamura's data, we validated the DNS accuracy [12], as shown in [FIGURE:2]. The present study utilizes the high-Re DNS data to evaluate RANS model applicability.

2.4 RANS Models Evaluated

We conducted RANS calculations using the Fluent software. For channel flow domains, two-equation turbulence models are typically applied. The turbulence models compared in this study include the standard $k$-$\varepsilon$, Realizable $k$-$\varepsilon$, RNG $k$-$\varepsilon$, standard $k$-$\omega$, and SST $k$-$\omega$ models. The Prt models are listed in [TABLE:1]. In the table, $Pe_t$ represents the turbulent Peclet number, defined as:

$$
Pe_t = \frac{\nu_t}{\nu}\text{Pr}
$$

TABLE:1 Prt models compared in this study

Prt Model Formulation Fluent default $\text{Prt} = 0.85$ Jischa et al. [8] $\text{Prt} = 0.8882\left(\frac{\text{Pr}}{\text{Pr}_t}\right)^{0.918} + 2.4/Re$ Reynolds et al. [9] $\text{Prt} = 0.5 + 0.5(1+100Pe_t)^{-0.5}(1+120Pe_t^{-1})^{-0.15}$ Azer et al. [10] $\text{Prt} = 0.45157\left(\frac{y}{R}\right)^{0.46}\left(\frac{\text{Pr}}{\text{Pr}_t}\right)^{0.58} + \frac{1135}{f(y/R,Re)}$ Weigand et al. [11] $\text{Prt} = \text{Prt}\infty + 0.3\left(\frac{\text{Pr}_t}{\text{Pr}\right)\right]$}} - 0.3\right)\left[1 - \exp\left(-\frac{Pe_t}{20 Zhou et al. [6] $\text{Prt} = 0.888 + 1000\left(\frac{\text{Pr}}{\text{Pr}_t}\right)^{0.85}Re^{-0.5}$

2.5 Grid Independence Analysis

This study employs steady-state calculations, requiring grid independence verification. We generated structured grids using ICEM software and performed independence analysis for streamwise, wall-normal, spanwise, and first-layer grid thickness. Using streamwise pressure gradient and Nusselt number as criteria, the final grid parameters are: 22 layers in the streamwise direction, 100 layers in the wall-normal direction, and 11 layers in the spanwise direction, totaling 24,200 cells. The maximum $y^+$ value of the first grid layer is 0.45. A schematic of the grid cross-section in the streamwise-normal plane is shown in [FIGURE:3].

3.1 Turbulence Model Applicability Analysis

The dimensionless velocity ($U^+$) is defined as:

$$
U^+ = \frac{U}{u_\tau}
$$

where $U$ is the fluid velocity and $u_\tau$ is the friction velocity, characterizing near-wall friction, defined as $u_\tau = \sqrt{\tau_w/\rho}$, with $\tau_w$ being the wall shear stress. The $U^+$ profiles are shown in [FIGURE:4], where the abscissa represents the dimensionless wall distance defined as $y^+ = yu_\tau/\nu$, with $y$ being the perpendicular distance from the wall. All turbulence models produce $U^+$ profiles with minor deviations from the DNS data. To more intuitively analyze the differences between RANS and DNS results, we define the relative deviation as:

$$
\text{deviation} = \frac{\phi_{\text{RANS}} - \phi_{\text{DNS}}}{\phi_{\text{DNS}}}
$$

where $\phi$ represents any physical quantity. The results, presented in [FIGURE:5], reveal that for $y^+ < 50$, the three $k$-$\varepsilon$ models show better agreement with DNS data. However, for $y^+ > 50$, the two $k$-$\omega$ models exhibit superior agreement, with relative deviations within 2%.

Under the constant property assumption, the velocity profile depends solely on $\nu_t$. The $\nu_t$ distributions from different turbulence models are shown in [FIGURE:6]. The three $k$-$\varepsilon$ models agree better with DNS data in the region $y^+ < 75$, but show significant trend deviations for $y^+ > 75$. In contrast, the two $k$-$\omega$ models demonstrate better agreement with DNS data throughout the domain, with the standard $k$-$\omega$ model showing the best overall performance.

The shear stress in the flow field, which determines the pressure drop and thus the pumping power required for fluid transport, equals the product of velocity gradient and viscosity. The Reynolds decomposition approach in RANS divides shear stress into two components: molecular shear stress arising from molecular viscosity and turbulent shear stress (Reynolds stress) arising from $\nu_t$. [FIGURE:7]-[FIGURE:9] compare the molecular, turbulent, and total shear stresses with DNS data.

In our RANS calculations, $k$-$\varepsilon$ models employ the "Enhanced Wall Treatment" approach, based on the two-layer model for boundary layer flows. This model assumes molecular shear stress dominates in the region $y^+ < 30$, far exceeding turbulent shear stress, while turbulent shear stress dominates for $y^+ > 30$. As shown in [FIGURE:7], the three $k$-$\varepsilon$ models predict molecular shear stress closer to DNS data in the $y^+ < 30$ region, but their wall shear stress characterization is inferior to the two $k$-$\omega$ models—a critical parameter determining pressure drop.

[FIGURE:8] shows that in the $y^+ > 30$ region, $k$-$\omega$ models predict slightly lower turbulent shear stress than $k$-$\varepsilon$ models, but remain closer to DNS results. However, because $k$-$\varepsilon$ models produce larger turbulent shear stress in the $y^+ < 30$ region, artificially enhancing turbulence effects in the inner boundary layer, their total shear stress predictions are inferior to $k$-$\omega$ models, as demonstrated in [FIGURE:9].

Based on the shear stress data in [FIGURE:7]-[FIGURE:9], we calculated the streamwise pressure gradient and its relative deviation from DNS data, summarized in [TABLE:2]. The $k$-$\omega$ models yield streamwise pressure gradients closest to DNS data, with relative deviations of approximately 2.15%, outperforming other turbulence models.

TABLE:2 Relative deviation of streamwise pressure gradient from DNS data for different turbulence models

Turbulence Model Relative Deviation Standard $k$-$\varepsilon$ 4.92% RNG $k$-$\varepsilon$ 4.47% Realizable $k$-$\varepsilon$ 4.84% Standard $k$-$\omega$ 2.16% SST $k$-$\omega$ 2.15%

Overall, while each turbulence model exhibits strengths and weaknesses in flow field characterization, all deviations remain within engineering-acceptable limits. Based on comprehensive evaluation, $k$-$\omega$ type models are the recommended turbulence models for He-Xe flow and heat transfer calculations in parallel channels.

3.2 Prt Model Applicability Analysis

Building upon the previous section's findings, this section evaluates Prt model applicability using the SST $k$-$\omega$ turbulence model. [FIGURE:10] presents the dimensionless temperature ($T^+$) profiles predicted by different Prt models, where $T^+$ is defined as:

$$
T^+ = \frac{T_w - T}{T_\tau}
$$

with $T_w$ being the wall temperature, $T$ the fluid temperature in the domain, and $T_\tau$ the friction temperature defined as $T_\tau = q_w/(\rho c_p u_\tau)$, where $q_w$ is the wall heat flux density.

All models produce $T^+$ profiles with trends similar to DNS data, but significant numerical differences exist. In the channel flow configuration, the constant Prt = 0.85, Reynolds et al., and Azer et al. models show relatively small deviations from DNS data. This observation contradicts previous studies [6], likely due to differences in geometry and Reynolds number. The Zhou et al. model [6], developed for higher-Re circular pipe flows, shows larger deviations in the present configuration. These results indicate that existing Prt models require further improvement for He-Xe applications.

Under the constant property assumption, temperature behaves as a passive scalar determined by $\alpha_t$. [FIGURE:11] shows the $\alpha_t$ profiles from different Prt models. According to the Enhanced Wall Treatment model, we focus on the $y^+ > 30$ region. Near $y^+ \approx 100$, the constant Prt = 0.85 model provides the best $\alpha_t$ characterization, consistent with the $T^+$ results. However, as the fluid approaches the mainstream region (larger $y^+$), the wall-normal turbulent heat flux exhibits a linear decreasing trend [6], making temperature predictions less sensitive to $\alpha_t$ deviations in this region.

[FIGURE:12] presents the Prt distributions from different models. In the $y^+ \approx 100$ region, the constant Prt = 0.85 model shows the largest relative deviation from DNS data—a trend opposite to that observed in [FIGURE:10] and [FIGURE:11]. This apparent contradiction arises because, according to Equation (4), $\alpha_t$ depends on both $\nu_t$ and Prt. Since the selected turbulence model introduces certain deviations in $\nu_t$ (as shown in [FIGURE:6]), a Prt model that accurately predicts Prt values (such as Weigand et al.'s model) may produce larger $\alpha_t$ deviations, leading to significant temperature field prediction errors.

This analysis demonstrates that turbulence models are equally important for heat transfer calculations. Deviations between turbulence-model-predicted $\nu_t$ and actual values propagate into $\alpha_t$ calculations, affecting heat transfer accuracy. As shown in [FIGURE:6], different turbulence models yield substantially different $\nu_t$ distributions, indicating that the same Prt model combined with different turbulence models will produce different heat transfer results. Previous Prt model development has focused solely on the model's standalone performance without emphasizing turbulence model selection, leading to situations where a model predicts Prt values close to DNS data but yields inferior $\alpha_t$ and temperature field accuracy compared to other models. Therefore, future He-Xe Prt model development must emphasize compatibility with specific turbulence models. The recommended approach involves: (1) identifying optimal turbulence models through applicability studies, and (2) integrating the deviations between turbulence-model-predicted $\nu_t$ and actual values as correction coefficients into the Prt model.

4 Conclusions

Based on DNS standard data for He-Xe, this study evaluates RANS computational model applicability, yielding the following conclusions:

1. Considering both stress field and streamwise pressure gradient accuracy, $k$-$\omega$ type models are the recommended turbulence models for He-Xe channel flow RANS calculations. Regarding temperature field accuracy, existing Prt models require further improvement for effective He-Xe applications.

2. Turbulence and Prt models jointly determine $\alpha_t$, which governs temperature field calculation accuracy. For He-Xe Prt model development, the final model must emphasize compatibility with specific turbulence models, integrating deviations between turbulence-model-predicted $\nu_t$ and actual values as correction coefficients.

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