## 2 Research Model and Numerical Methods
### Numerical Method and Mesh  
The Navier-Stokes equation is solved by the FINE/Turbo in NUMECA software, and the Spalart-Allmaras (S-A) equation was selected as the turbulence model; the difference scheme used in the study is a second order central difference scheme, and the fourth order Runge-kutta is used for iterative calculation. In order to improve the computational efficiency, multiple meshing, local time step and implicit residual smoothing were adopted to accelerate the convergence. For the transfer of the physical quantities, the \"rotor freezing method\" was applied to the interface between the rotor and the stator. The compressor inlet condition is set as constant and uniform axial air intake, and the whole solid wall of the compressor is chosen as adiabatic without slip. For the outlet condition, the given MFR is adopted, and then other physical quantities such as pressure ratio and efficiency are derived by the calculations. The governing equations can be expressed as:  
\[\frac{\partial U}{\partial t}+\
abla\overset{\rightarrow}{F_{I}}+\
abla \overset{\rightarrow}{F_{V}}=Q \tag{1}\]  
where \(\overset{\rightarrow}{F_{I}}\) and \(\overset{\rightarrow}{F_{V}}\) are the inviscid and viscous flux vectors, respectively. \(Q\) is the source term. \(\overset{\rightarrow}{F_{I}}\) and \(\overset{\rightarrow}{F_{V}}\) can be further decomposed into Cartesian components as follows:  
\[F_{I}=f_{I1}\overset{\rightarrow}{I}_{x}+f_{I2}\overset{ \rightarrow}{I}_{y}+f_{I3}\overset{\rightarrow}{I}_{z} \tag{2}\] \[F_{V}=f_{V1}\overset{\rightarrow}{I}_{x}+f_{V2}\overset{ \rightarrow}{I}_{y}+f_{V3}\overset{\rightarrow}{I}_{z}\] (3) \[U= \begin{bmatrix}\overset{\rightarrow}{\vec{p}}\\ \vec{p}\tilde{w}_{1}\\ \vec{p}\tilde{w}_{2}\\ \vec{p}\tilde{w}_{3}\\ \vec{p}\tilde{w}_{4}\end{bmatrix}  \tag{4}\]  
Figure 2: Three models of the interstage pipeline. \[F_{ii}= \begin{bmatrix}\dfrac{\rho\widetilde{w}_{i}}{p^{*}\delta_{1i}+\rho\widetilde{ w}_{i}\widetilde{w}_{1}}\\ \dfrac{p^{*}\delta_{2i}+\rho\widetilde{w}_{i}\widetilde{w}_{2}}{p^{*}\delta_{3i} +\rho\widetilde{w}_{i}\widetilde{w}_{3}}\\ \left(\widetilde{p}\widetilde{E}+\overline{p^{*}}\right)\widetilde{w}_{i} \end{bmatrix}  \tag{5}\]  
\[F_{VI}=- \begin{bmatrix}0\\ \widetilde{\tau}_{i1}\\ \widetilde{\tau}_{i2}\\ \widetilde{\tau}_{i3}\\ q_{i}+\widetilde{w}_{i}\widetilde{\tau}_{ij}\end{bmatrix}  \tag{6}\]  
The Spalart-Allmaras turbulence model and the perfect gas equation were applied to enclose the above equations. The perfect gas equation can be written as follow:  
\[p=\rho RT=(\gamma-1)\left[\rho E-\dfrac{(\rho\overline{w})^{2}}{2\rho}\right] \tag{7}\]  
Figure 3 shows the grid of the study model, in which the impeller and diffuser grid is automatically generated, having the O4H topology pattern. A H-type topology surrounded by an O-type topology was applied in the rotor blade tip clearance, with 17 layers in the spanning direction. The boundary between diffuser and volute is generated by perfect matching connection, and the two are integrated into a computing domain. Since the volute as well as the interstage pipe are asymmetric geometry, the butterfly topology pattern is adopted to improve the grid accuracy. A grid independence study was performed by steady CFD methods in which the grid number was gradually increased until no noticeable changes in the compressor efficiency was observed, as shown in Table 2. It can be seen that when the number of grids is increased by roughly the same proportion, the compressor stage pressure ratio as well as the efficiency of the fine grid model only increase by 0.07% and 0.05% compared with that of the medium grid model. In order to capture the details of the flow field more accurately and save the calculation time, the medium grid was chosen for the calculation at last. After the grid independence study, the total number of grids is about 7.4 million; the maximum expansion ratio is less than 3.2; the minimum skew angle of grid is 36\({}^{\circ}\); and maximum aspect ratio is less than 780.