## 2 Geometric and Operational Parameters for the Numerical Runs
### Mathematical Modeling  
#### 3.2.1 Turbulence Modeling  
In previous experimental measurements carried out by the authors in a wind tunnel [16], it was confirmed that the flow that is being modeled is fully turbulent throughout the domain under consideration. The Reynolds-averaged Navier-Stokes (RANS) approach was adopted for numerically simulating the flow, and thus the mean flow was calculated by solving Equations (2) and (3):  
\[\frac{\partial U_{i}}{\partial x_{i}}=0 \tag{2}\]  
\[\frac{\partial}{\partial x_{j}}\big{(}\rho U_{i}U_{j}\big{)}=-\frac{\partial P }{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\left[\mu\left(\frac{\partial U _{i}}{\partial x_{j}}+\frac{\partial U_{j}}{\partial x_{i}}\right)-\rho\overline {u^{\prime}_{i}u^{\prime}_{j}}\right]+S_{i} \tag{3}\]  
In order to solve the Reynolds stresses (\(\rho\overline{u^{\prime}_{i}u^{\prime}_{j}}\)) and subsequently close the system of Equations (2) and (3), the standard \(k\)-\(\epsilon\) model was applied [31]. Although the detailed models in [PERSON] et al. [16; 17] used the RSTM as a turbulence model, the standard \(k\)-\(\epsilon\) model could be used in this work as the values of the pressure loss coefficients are those that best fit the results obtained in the detailed simulations (those that model the holes). This may be seen as a limitation, but it was done on purpose. The \(k\)-\(\epsilon\) model was chosen due to its robustness and lower cost from a computational point of view, and, therefore, it is of more interest for the thermal designers. Since the procedure consists of an iterative process that serves as a calibration of the coefficients for each of the 81 cases, it can be tuned in such a way that takes into consideration not only the porosity physics but also the difference between the two turbulence models.  
Within the standard \(k\)-\(\epsilon\) model, the Reynolds stresses are calculated by means of the Boussinesq hypothesis:  
\[\rho\overline{u^{\prime}_{i}u^{\prime}_{j}}=-\mu_{t}\left(\frac{\partial U_{i }}{\partial x_{j}}+\frac{\partial U_{j}}{\partial x_{i}}\right)+\frac{2}{3} \rho k\delta_{ij} \tag{4}\]  
So that the Reynolds stress tensor is aligned with the mean strain-rate tensor (\(S_{ij}\)). The standard \(k\)-\(\epsilon\) model consists of two transport equations for both the turbulent kinetic energy (\(k\)) and its dissipation rate (\(\epsilon\)), which can be expressed in the following compact way:  
\[\frac{\partial}{\partial x_{i}}(\rho U_{i}k)=\frac{\partial}{\partial x_{j}} \Bigg{(}\Gamma_{k}\frac{\partial k}{\partial x_{j}}\Bigg{)}+P_{k}-D_{k} \tag{5}\]  
\[\frac{\partial}{\partial x_{i}}(\rho U_{i}\epsilon)=\frac{\partial}{\partial x _{j}}\Bigg{(}\Gamma_{\epsilon}\frac{\partial\epsilon}{\partial x_{j}}\Bigg{)} +P_{\epsilon}-D_{\epsilon} \tag{6}\]  
The terms used in both equations are defined in Table 3. The turbulent viscosity is calculated as  
\[\mu_{t}=\rho C_{\mu}\frac{k^{2}}{\epsilon} \tag{7}\]  
with a value of 0.09 for \(C_{\mu}\).  
Standard wall functions (SWF), which are based on the work by [PERSON] and [PERSON] [32], were used for the near-wall treatment of the turbulence model used to predict flow characteristics very close to solid walls. This wall treatment does not require a fine meshing close to the wall, which means there is a significant reduction in the amount of elements in the model and, therefore, a reduction in computational time.