# Investigation of Point-Contact Strategies for CFD Simulations of Pebble-Bed Reactor [PERSON]
## 2 Materials and Methods
### Numerical Solvers  
The heat-transfer and fluid-flow physics were coupled in a segregated manner of energy and flow transport. A detailed sensitivity study of turbulence models has been investigated but is not presented here in detail because discussions regarding RANS [13], URANS [14], hybrid RANS/LES [15], and DNS [16] turbulence modeling comparisons already exist in the literature for body-centered cubic, face-centered cubic, structured, and unstructured. Most of the non-linear numerical methods generally predict similar mean velocity and temperature fields but differ significantly on the prediction of the RMS velocity and temperature fields with the bed. It has been suggested that the reason for poor RMS agreement is related to a significant amount of transitional flow affected by the pebble curvature. However, validation data are lacking that are available to perform a thorough evaluation of turbulence modeling for the pebble-bed reactor application. Singular experimental configurations of heated beds that enable simultaneous measurement of temperature and time-resolved velocity fields are not easily accomplished. Prior isothermal efforts focused on PIV for time-resolved velocimetry [17; 18] but lack temperature gradients. In this study, RANS standard \(k\)-\(\epsilon\), realizable \(k\)-\(\epsilon\), and \(k\)-\(\omega\) SST models were investigated. Relative differences in the predicted mean velocity fields between these three models could be decreased by  
1. Enabling the standard \(k\)-\(\epsilon\) cubic constitutive relationship to improve anisotopic turbulence prediction of secondary flows and streamline curvature [19]. 2. Enabling the \(k\)-\(\omega\) SST cubic constitutive relationship [20; 21], which adds non-linear functional representations of the strain and vorticity tensors. Across these closure models, the volume-averaged mean velocity field and bed pressure drop spanned \(\pm 4\%\) and the volume-averaged mean pebble fuel temperature spanned \(\pm 2\%\). The steady RANS realizable \(k\)-\(\epsilon\) turbulence model [22] with a second-order convection scheme, secondary gradients, and Wolfstein shear-driven two-layer all \(y^{+}\) wall treatment [23] were selected. This decision was based on its wide applicability and the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & **Pebble** & **Pebble** & **Heat Transport** \\
**Property** & **Fuel Region** & **Fuel-Free Region** & **Fluid** \\ \hline Density (kg m\({}^{-3}\)) & 1700 & 1700 & Ideal gas \\ Specific heat (J kg\({}^{-1}\) K\({}^{-1}\)) & 700 & 700 & 5200 \\ Thermal conductivity (W m\({}^{-1}\) K\({}^{-1}\)) & 25 & 25 & 0.155 \\ Dynamic viscosity (Pa s) & N/A & N/A & \(1.98\times 10^{-5}\) \\ Molecular weight (kg kmol\({}^{-1}\)) & N/A & N/A & 4.00 \\ Turbulent Prandtl number [12] & N/A & N/A & 0.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Specified temperature- and fluence-independent material properties. performance observed by [PERSON] et al. [14] with acceptable agreement of mean fields with higher-order methods. It is important to note that, regardless of the span of absolute values of velocity, pressure, and temperature fields, the conclusions regarding the geometric approach and selection of point-contact strategies remain unaffected. The turbulent fluid motion and heat transfer are governed by mass continuity, solid energy conservation, fluid energy conservation, Navier-Stokes, and Poisson equations, provided as  
\[\
abla\cdot\mathbf{U}=0, \tag{6}\]  
\[\frac{d}{dt}\iiint_{V}\rho_{s}c_{p,s}T\,dV+\oint_{A}\rho c_{p,s}T\mathbf{v_{s}} \cdot d\mathbf{a}=-\oint_{A}\mathbf{q^{\prime\prime}}\cdot d\mathbf{a}+\iiint_{V}S_{u}dV \tag{7}\]  
\[\frac{d}{dt}\iiint_{V}\rho_{f}E\,dV+\oint_{A}\rho_{f}H\mathbf{v}\cdot d \mathbf{a}= \tag{8}\] \[ \begin{split}-\oint_{A}\mathbf{q^{\prime\prime}}\cdot d\mathbf{a}+ \oint_{A}\mathbf{T_{f}}\cdot\mathbf{U}d\mathbf{a}\\ +\iiint_{V}\mathbf{f_{b}}\cdot dV+\oint_{A}\sum_{i}h_{i}I_{i}d\mathbf{a} \\ +\iiint_{V}S_{u,f}dV\end{split}