## 2 CFD Analysis and Vortex Visualization Method
### Mathematical Model and Boundary Conditions  
In this study, 3D, compressible, Reynolds Averaged Navier-Stokes (RANS) equations are adopted for obtaining steady state results of the centrifugal compressor. For this simulation, it is assumed that mass, momentum, and energies are conserved and by these assumptions, the governing equations can be written as follows in a stationary frame [19].  
The continuity equation is described as Equation (1),  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{j}}\big{(}\rho U_{ j}\big{)}=0 \tag{1}\]  
The momentum equations are described as Equations (2) and (3),  
\[\frac{\partial\rho U_{i}}{\partial t}+\frac{\partial}{\partial x_{j}}\big{(} \rho U_{i}U_{j}\big{)}=-\frac{(\partial p)}{\partial x_{i}}+\frac{\partial}{ \partial x_{j}}\big{(}\tau_{ij}-\rho\overline{u_{i}u_{j}}\big{)}+S_{M} \tag{2}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Design Parameter** & **Value (Dimensionless)** \\ \hline Impeller Diameter & \(1\) \\ Impeller Height & \(0.3546\) \\ Number of Main Blade & \(6\) \\ Number of Splitter Blade & \(6\) \\ Tip Clearance & \(0.00319\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Design specification of centrifugal compressor.  
Figure 1: Centrifugal compressor impeller used in the analysis.  
The total energy equation is described as Equation (4),  
\[\frac{\partial(\rho h_{hot})}{\partial t}-\frac{\partial p}{\partial t}+\frac{ \partial}{\partial x_{j}}\big{(}\rho U_{j}h_{hot}\big{)}=\frac{\partial}{ \partial x_{j}}\bigg{(}\lambda\frac{\partial T}{\partial x_{j}}-\rho u_{j}h \bigg{)}+\frac{\partial}{\partial x_{j}}\big{[}U_{i}\big{(}\tau_{ij}-\rho u_{ i}u_{j}\big{)}\big{]}+S_{E} \tag{3}\]  
where \(h_{hot}\) is the total enthalpy, related to the static enthalpy \(h(T,p)\) by,  
\[h_{hot}=h+\frac{1}{2}U_{i}U_{j}+k \tag{4}\]  
The velocities at the wall can be represented as follows;  
\[U_{null}=0\ (\text{at stationary frame}) \tag{5}\]  
\[U_{null}=\omega R\ (\text{at rotating frame, hub}) \tag{6}\]  
\[U_{null}=-\omega R\ (\text{at rotating frame, shroud}) \tag{7}\]  
Because the shroud is fixed while the passage is rotating, wall velocity is defined as Equation (7). The heat transfer condition of the wall is defined as follows;  
\[q_{null}=0 \tag{8}\]  
For flows in the rotating frame, rotating at a constant angular velocity \(\omega\), the source of rotating momentum is defined as follows;  
\[S_{M,rot}=S_{Cor}+S_{cfg} \tag{9}\]  
\[S_{Cor}=-2\rho\omega\times U \tag{10}\]  
\[S_{cfg}=\rho\omega\times(\omega\times r) \tag{11}\]  
The calculation domains of numerical analysis were configured to contain one main blade and one splitter blade each; 1/6 of the total impellers, as illustrated in Figure 2. The red and blue parts were set as stationary domains, and the yellow blade parts were set as rotating domains.  
Grid sensitivity is conducted with normalized pressure difference. As shown in Figure 3, the normalized pressure difference after about 900,000 grids is converged. Therefore, the computational grid system is consisted of 1,000,000 hexa  
Figure 2: Computational grid system with a specific view at the hub and tip side.  
Turbogrid. The inlet and outlet conditions (temperature, pressure, mass flow rate, and revolutions per minute) of the calculation domains were the same as the experimental conditions. Figure 4 shows the experimental equipment of the centrifugal compressor. Analysis was performed using ANSYS CFX using the Shear Stress Transport (SST) turbulence model [20]. The Shear Stress Transfer (SST) turbulence model is adopted to predict precisely flow separation [20; 21; 22; 23]. In various research studies, the SST turbulence model has been confirmed to be able to predict precisely flow separation under adverse pressure gradients [21; 22; 23]. Additionally, the SST turbulence model has high numerical stability than \(k-\omega\) turbulence model [20]. By using this turbulence model, 3D-RANS simulations are conducted with high accuracy flow separation prediction and high numerical stability.