## 2 Numerical Methods  
### Compressor Geometry  
The centrifugal compressor model used in the numerical simulations is the single-stage NASA CC3 centrifugal compressor [29]. The NASA CC3 is a transonic centrifugal compressor with high-compression ratio (4:1). The meridional view of the compressor stage is shown in Figure 1. The computational domain includes three separated parts: stationary inlet domain including a pipe and impeller eye; rotating impeller, with 15 main blades and 15 splitter blades; stationary vaned diffuser domain, including 24 two-dimensional wedge-type vanes. The leading edges of the splitter blades are located at the 50% and 30% of the pitch and chord of the main blades, respectively. In addition, to provide a uniform split flow between the main blade and splitter passages, the splitters are offset toward the suction side of the main blades. The leading edge of the diffuser vanes is placed at 108% of the impeller exit radius with a 7.8\({}^{\circ}\) divergence angle. [PERSON] and [PERSON] [30] describe the compressor design in detail, consisting of the geometrical parameters of the impeller and diffuser. The impeller rotational speed and the compressor mass flow rate are 21,789 rpm and 4.536 kg/s, respectively, at the design point. Table 1 summarizes the centrifugal compressor operating conditions as well as the dimensions of the impeller and diffuser. Figure 2 shows the numerical domain used in the current study for the computational simulations. As shown in the figure, the inlet pipe has been extended in the axial direction (about two times of the diameter of the impeller inlet) to provide a uniform and developed flow at the inlet of the compressor impeller. In addition, in order to minimize the influences of the imposed boundary condition at the diffuser outlet on the flow-field solution within the diffuser passages, the exit domain has been also extended in the radial direction about one chord of the diffuser vanes. A 120\({}^{\circ}\) sector computational domain (1/3 model) was applied for the numerical simulations instead of full-annulus modelling to reduce the computational cost and time. A partial computational model consists of five impeller passages and eight diffuser passages as illustrated in Figure 2a. Applying a partial numerical domain could reduce the computational time and on the other side by using more mesh points, the resolution of the computational grid was also improved. The full-annulus of the compressor stage including the impeller and diffuser blades are shown in Figure 2b. \begin{table}
\begin{tabular}{c c} \hline \hline
**Operating condition** & \\ \hline Inlet temperature & 288 K \\ Inlet pressure & 101,325 Pa \\ Total pressure ratio & 4:1 \\ Rotational speed at design point & 21,789 rpm \\ Mass flow rate at design point & 4.536 kg/s \\ \hline
**Impeller configuration** & \\ \hline Number of main blades & 15 \\ Number of splitter blades & 15 \\ Radius at impeller inlet (\(R_{1}\)) & 105 mm \\ Inlet blade height & 64 mm \\ Radius at impeller exit (\(R_{2}\)) & 215.5 mm \\ Tip speed & 492 m/s \\ Blade height at exit & 17 mm \\ Backsweep angle & 50\({}^{\circ}\) \\ \hline
**Vaned diffuser configuration** & \\ \hline Number of vanes & 24 \\ Divergence angle & 7.8\({}^{\circ}\) \\ Vane height & 17 mm \\ Inlet diameter & 431 mm \\ Outlet diameter & 714 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: NASA CC3 centrifugal compressor specifications. Figure 1: Meridional view of the NASA CC3 centrifugal compressor.