## 2 Numerical Methods
### Numerical Sensitivity Analysis  
A computational model was carried out for various numerical grids and various turbulence models and then compared with the measurement [29] to ensure the accuracy of the computational method. For the grid independency test, three different types of computational meshes, coarse (4.2 million grids), moderate (9.7 million grids), and fine (12.15 million grids) were tested with the \(k-\omega\) SST turbulence model using the RANS approach. In order to generate various computational grid domains, the number of grid points was changed in the spanwise, streamwise, and radial directions to ensure \(y^{+}<1\) at the wall surfaces.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Flow Solver** & **ANSYS CFX V19.3** \\ \hline \multirow{2}{*}{Spatial discretization} & \multicolumn{2}{c}{High resolution scheme} \\  & (a hybrid of first order and second order upwind scheme) \\ \hline \multirow{2}{*}{Temporal discretization} & First order backward Euler scheme (steady) \\  & Second order backward Euler scheme (unsteady) \\ \hline \multirow{2}{*}{Turbulence model} & \multirow{2}{*}{\(k-\omega\) SST} \\  & & \\ \hline \multirow{2}{*}{Working fluid} & \multirow{2}{*}{Air (treated as an ideal gas)} \\  & & \\ \hline \multirow{4}{*}{Boundary conditions} & \multirow{2}{*}{Inlet duct/Impeller} & Total pressure (101,325 Pa) \\  & & Total temperature (288 K) \\ \cline{1-1} \cline{2-3}  & Inlet duct/Impeller & Frozen rotor interface (steady) \\ \cline{1-1} \cline{2-3}  & Impeller/Diffuser & Transient rotor-stator interface (unsteady) \\ \cline{1-1} \cline{2-3}  & \multicolumn{2}{c}{Diffuser exit} & Average static pressure \\ \hline \hline \end{tabular}
\end{table}
Table 2: Solver setting specification.  
Table 3 compares the total pressure ratio and isentropic efficiency of the tested compressor at the design point condition, computed by Equations (5) and (6) for different grid resolutions.  
\[TPR=\frac{P_{t,outlet}}{P_{t,inlet}} \tag{5}\]  
\[\eta=\frac{\left(TPR^{\frac{\gamma-1}{\gamma}}-1\right)T_{t,inlet}}{T_{t,outlet }-T_{t,inlet}} \tag{6}\]  
where \(\gamma\) is the specific heat ratio of gas. It should be noted that the mass-flow averaged pressure and temperature are measured at the inlet duct and \(1.7\)\(R_{2}\) planes, respectively. Fine grid case was taken as the reference case and the calculated pressure ratio and efficiency of the fine mesh were compared to the other two meshes. As summarized in Table 3, it is observed that by increasing the number of grid elements, the percentage differences in the total pressure ratio and efficiency are considerably less for the moderate mesh. Therefore, the moderate mesh (9.7 million elements) was selected as the computational grid in the current study for all of the numerical simulations to reduce the numerical cost.  
Moreover, to ensure the verification of the numerical results regarding the turbulence closure, \(k-\varepsilon\) and \(k-\omega\) SST turbulence models were conducted for the selected moderate mesh in the grid independency analysis. Table 4 compares the pressure ratio and efficiency of the compressor for the two various turbulence models. It is observed that the \(k-\varepsilon\) turbulence model overestimates both the pressure ratio and efficiency compared to those of the \(k-\omega\) SST. This could be caused by the \(k-\varepsilon\) turbulence model limitations in predictions of the detached and re-attached regions in the flow separation zones especially in the diffuser passages.