## 4 Mesh Independence and Validation of the CFD Simulation  
### Mesh Independence Study  
The mesh created was quadrilateral structured mesh. For accurate results, k-omega SST and transition SST turbulence models require near-wall meshes with \(y^{+}\sim 1\) and \(y^{+}<1\), respectively [29]. Therefore, the final meshes used for the simulations were generated with \(y^{+}\) average value (i.e., area-weighted average) of less than 1. The \(y^{+}\) maximum value was greater than but close to 1 such as 1.27 to mention a value from one of the geometries. To obtain the mesh mentioned above, four meshes were created with different numbers of cells for the grid independence study. They were referred to as coarse (112,959 cells), medium (151,834 cells), semi-fine (452,760 cells), and fine (902,069 cells) meshes. The parameters used to judge if the CFD results were mesh independent were: 1. The secondary mass flow rate, \(\dot{m}_{s}\); 2. The primary mass flow rate, \(\dot{m}_{p}\); 3. The axis static pressure along the eject; 4. The wall static pressure along the eject; 5. The pressure profiles across the primary nozzle outlet; 6. The pressure profiles across a location at the end of the converging mixing section (i.e., before the ejector throat inlet). The primary inlet pressure, \(P_{0}\), used for the grid independence study was 618 kPa. The comparison of the secondary and primary mass flow rates (parameters no. 1 and 2) for these four meshes can be seen in Table 2. The mass flow rates of all mesh sizes are nearly identical with a maximum change of around 0.61% for the secondary mass flow rate and nearly no change in the primary mass flow rate. For parameters no. 3, 4, 5, and 6, it was found that they did not change significantly with respect to the mesh size increase. As an example, Figure 6 shows the static pressure profile along the axis of the ejector for the four meshes. Based on these comparisons and a reasonable computational time needed, we decided to choose the medium mesh (151,834 cells) for further CFD study. The sample of the chosen mesh (the medium mesh) around the outlet of the primary nozzle as well as the inlet of the secondary flow is presented in Figure 7. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Mesh Name**} & \multirow{2}{*}{**Number**} & \multicolumn{2}{c}{**k-Omega SST**} & \multicolumn{2}{c}{**Transition SST**} & \multicolumn{2}{c}{**k-Omega SST**} & \multicolumn{2}{c}{**Transition SST**} \\ \cline{3-10}  & & \(\dot{m}_{p}\)[kg/s] & \(\dot{m}_{s}\)[kg/s] & \(\dot{m}_{p}\)[kg/s] & \(\dot{m}_{s}\)[kg/s] & \(\Delta\dot{m}_{p}\)[1 \%] & \(\Delta\dot{m}_{s}\)[2 \%] & \(\Delta\dot{m}_{p}\)[1 \%] & \(\Delta\dot{m}_{s}\)[2 \%] \\ \hline Coarse & 112,959 & 0.07104 & 0.01212 & 0.07107 & 0.01156 & 0.00 & 0.00 & 0.00 & 0.00 \\ Medium & 151,834 & 0.07104 & 0.01211 & 0.07108 & 0.01157 & 0.00 & -0.08 & 0.01 & 0.09 \\ Semi-fine & 452,760 & 0.07102 & 0.01212 & 0.07106 & 0.01161 & \(-0.03\) & 0.00 & \(-0.01\) & 0.43 \\ Fine & 902,069 & 0.07102 & 0.01211 & 0.07106 & 0.01163 & \(-0.03\) & \(-0.08\) & \(-0.01\) & 0.61 \\ \hline \hline \end{tabular}  
* \(\Delta\dot{m}_{p}=\left(\dot{m}_{p}-\dot{m}_{p,coarse}\right)\times 100/\dot{m}_{p,coarse}\) [%]; \(\dot{m}_{p}\) and \(\dot{m}_{p,coarse}\) are the primary mass flow rates of a certain mesh and the coarse mesh, respectively.
* \(\Delta\dot{m}_{s}=\left(\dot{m}_{s}-\dot{m}_{s,coarse}\right)\times 100/\dot{m}_{s,coarse}\) [%]; \(\dot{m}_{s}\) and \(\dot{m}_{s,coarse}\) are the secondary mass flow rates of a certain mesh and the coarse mesh, respectively.  
\end{table} 
Table 2: The comparison of mass flow rates change with mesh size increase. Figure 6: Absolute static pressure along the axis (top). The ejector internal contour (bottom) is also presented to see the corresponding location of the static pressure. The turbulence model is the transition SST model.