# Impact of Mesh Resolution and Temperature Effects in Jet Ejector CFD Calculations
## 2 Materials and Methods
### Meslt  
A mesh sensitivity study has been conducted. The governing parameter for this study was the entrainment ratio, which is a common approach used, for example, by [PERSON] et al. [PERSON] et al. (2015), and the meshes were compared based on the criteria of a number of cells and the value of \(y+\). CFD use a non-dimensional wall distance to describe how close the first grid point--cell center--is to a wall in terms of viscous effects. It helps determine the resolution of the boundary layer near the wall and whether the mesh is fine enough to capture wall-bounded flow phenomena accurately. To define the \(y+\) parameter the following formula is used  
\[y^{+}=\frac{yu_{T}}{\
u} \tag{1}\]  
where \(y\) is the distance from the wall to the center of the first computational cell, and \(u_{T}\) is the friction velocity, calculated from the wall shear stress divided by density:  
\[u_{T}=\sqrt{\frac{\tau_{w}}{\rho}} \tag{2}\]  
where \(\
u\) is the kinematic viscosity of the fluid ([PERSON], 1958). Since in our calculations we have wall cells with very low \(y+\) values together with others with higher values, we have selected the \"All \(y+\) Wall Treatment\" option available in StarCCM+ that blends wall functions for those cells with high \(y+\) values with a low Reynolds treatment for low \(y+\) values. This strategy gives reasonable results for \(y+\) values in the buffer zone (\(1<y+<30\)). The starting case consisted of 16,274 cells in the 2D, axisymmetric geometry. The prism layer mesh was used with geometric progression, stretch factor distribution mode, and a gap fill of 25%. While the \(y+\) factor is a common criterion for assessing mesh quality, jet ejector studies require a more nuanced approach. In this case, the entrainment ratio--defining the mixing of flows--was the primary indicator, while the \(y+\) factor played a secondary role. This mainly applies to near-wall conditions, but in our case, the key objective was to accurately represent both flow mixing and wall friction effects, which guided the mesh selection. To conduct the mesh sensitivity study, the number of cells in the initial reference case was progressively increased until stability was observed in the studied variable (entrainment ratio, \(\omega\)). Figure 4 presents a comparison of entrainment ratios as a function of the number of cells for each case. The corresponding numerical results are detailed in Table 3, where case no. 1 represents the starting mesh, and subsequent cases show progressively refined versions. The last four cases, featuring the densest meshes, yielded similar entrainment ratio results, indicating convergence. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mesh Version Case Number** & **Entrainment Ratio** & **Number of Cells** & **Max. \(y+\)** \\ \hline
1 & 1.146127 & 16,274 & 500 \\
2 & 1.146135 & 16,274 & 500 \\
3 & 1.166865 & 103,191 & 160 \\
4 & 1.175862 & 214,554 & 240 \\
5 & 1.178889 & 568,805 & 8.5 \\
6 & 1.185003 & 1,625,678 & 45 \\
7 & 1.172905 & 2,910,279 & 55 \\
8 & 1.176702 & 3,475,381 & 20 \\
9 & 1.176702 & 3,608,526 & 20 \\
10 & 1.176702 & 3,785,626 & 20 \\
11 & 1.176702 & 3,985,944 & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of different meshes at one working point. To balance accuracy with reasonable computation time, case no. 8--comprising 3,475,381 cells--was selected as the reference mesh for this study and is referred to as _refined_. The _refined_ mesh features a notably high resolution, with over 3,470,000 cells. This is significantly greater than the cell counts in other studies on supersonic jet ejectors, where 2D axisymmetric models can range from 20,300 cells [29] to 330,000 cells [24], and even than detailed 3D models, such as those by [PERSON] et al. (982,362 cells) [30]. While this resolution increases computational demands, it improves simulation accuracy. Using a less refined mesh in CFD simulations can lead to several issues that compromise the accuracy of results. One major consequence is the inaccuracy in capturing Mach number distributions.