# Investigation of Point-Contact Strategies for CFD Simulations of Pebble-Bed Reactor [PERSON]
## 2 Materials and Methods
### Numerical Solvers
\]  
\[\frac{\partial U_{i}}{\partial t}+U_{j}\frac{\partial U_{i}}{\partial x_{j} }=-\frac{\partial P}{\partial x_{i}}+\
u\frac{\partial^{2}U_{i}}{\partial x_ {j}\partial x_{j}}, \tag{9}\]  
\[\frac{\partial^{2}p}{\partial x_{i}^{2}}=-\frac{\partial U_{i}U_{j}}{\partial x _{i}\partial x_{j}}. \tag{10}\]  
where \(\mathbf{U}\) is the fluid velocity vector, \(\rho_{s}\) is the solid density, \(c_{p,s}\) is the solid specific heat capacity, \(T\) is the solid temperature, \(\mathbf{q^{\prime\prime}_{s}}\) is the solid heat flux vector, \(\mathbf{v_{s}}\) is the solid convective velocity, \(S_{u}\) is the solid volumetric heat source, \(\rho_{f}\) is the fluid density, \(E\) is the fluid total energy, \(H\) is the fluid total enthalpy, \(\mathbf{q^{\prime\prime}_{f}}\) is the fluid heat flux vector, \(\mathbf{T_{f}}\) isthe viscous stress tensor, \(\mathbf{f_{b}}\) is the fluid body force vector, \(h_{i}\) is the fluid enthalpy of component \(i\), \(j_{i}\) is the diffusive flux of component \(i\), \(S_{u,f}\) is the fluid volumetric heat source, \(P\) is the fluid internal pressure, \(t\) is time, \(x_{i}\) is the spatial coordinate vectors of component \(i\), and \(\
u\) is the fluid kinematic viscosity. The steady RANS-based approach to turbulence modeling was chosen for this study due to its relatively low computational cost to porous media when compared to that of LES and DNS. The temporal averaging methodology, or Reynolds decomposition process [24], suggested by [PERSON] was to separate fluid variables into their time-averaged mean and fluctuating components. Applying this process to Equations (6)-(10) yields  
\[\frac{\partial\langle U_{i}\rangle}{\partial x_{i}}=0, \tag{11}\]  
\[\frac{\partial\langle U_{i}\rangle}{\partial t}+\langle U_{j}\rangle\frac{ \partial\langle U_{i}\rangle}{\partial x_{j}}+\frac{\partial\langle u_{i}u_{j }\rangle}{\partial x_{j}}=-\frac{\partial\langle P\rangle}{\partial x_{i}}+v_ {t}\frac{\partial^{2}\langle U_{i}\rangle}{\partial x_{j}^{2}}, \tag{12}\]  
\[\frac{\partial^{2}\langle P\rangle}{\partial x_{i}^{2}}=-\frac{\partial\langle U _{i}\rangle}{\partial x_{j}}\frac{\partial\langle U_{j}\rangle}{\partial x_{ i}}-\frac{\partial^{2}\langle u_{i}u_{j}\rangle}{\partial x_{i}\partial x_{j}}. \tag{13}\]  
After the Reynolds decomposition of the Navier-Stokes equations, a closure problem is required to model the Reynolds stress tensor, \(\langle u_{i}u_{j}\rangle\). In this study, the applied closure method is the realizable \(k-\epsilon\) model containing a new transport equation for the turbulent dissipation rate \(\epsilon\)[22]. The transport equations for the kinetic energy \(k\) and the turbulent dissipation rate \(\epsilon\) are  
\[\frac{\partial}{\partial t}(\rho k)+\
abla\cdot(\rho k\overline{\varpi})=\
abla \cdot\left[\left(\mu+\frac{\mu_{t}}{\sigma_{k}}\right)\
abla k\right]+P_{k}- \rho(\epsilon-\epsilon_{0})+S_{k}, \tag{14}\]  
\[\frac{\partial}{\partial t}(\rho\epsilon)+\
abla\cdot(\rho e\overline{\varpi} )=\
abla\cdot\left[\left(\mu+\frac{\mu_{t}}{\sigma_{k}}\right)\
abla\epsilon \right]+\frac{1}{T_{e}}C_{e1}P_{\epsilon}-C_{e2}f_{2}\rho(\frac{\epsilon}{T_{e }}-\frac{\epsilon_{0}}{T_{0}})+S_{e}, \tag{15}\]  
where \(\overline{\varpi}\) is the mean fluid velocity vector and \(\mu_{t,f}\) is the fluid turbulent kinematic viscosity. \(\sigma_{k}\), \(\sigma_{e}\), \(C_{e1}\), and \(C_{e2}\) are model coefficients, \(P_{k}\) and \(P_{\epsilon}\) are production terms, \(f_{2}\) is a damping function, \(T_{e}\) is the large-eddy time scale, \(T_{0}\) is the ambient source term time scale, \(S_{k}\) and \(S_{e}\) are the user-specified source terms, and \(\epsilon_{0}\) is the ambient turbulence value in the source terms that counteracts turbulence decay. For conjugate heat transfer of conduction and convection, the numerical method within STAR-CCM+ for adjacent fluid/solid cells create a fluid/solid contact interface and conserves the total heat flux.