## 2. Throughflow Model and Research Object  
The governing equations of CAM are dependent on the circumferentially averaged Navier-Stokes equations [11; 12; 13]. The equations are solved by a time-marching finite volume method. The JAMESON [14] scheme is adopted to discretize the inviscid fluxes. In order to discretize the viscous flux, the Green-Gauss theorem is applied to solve the gradient of the variable on the edge of a grid cell. As for the time discretization, an explicit Runge-Kutta 4 step method is put into use. In CAM, the Spalart-Allmaras (S-A) turbulence model was applied. Since the Mach number is below 0.3 in the shrouded cavity and above 0.3 in the main passage, preconditioning was applied to avoid the stiffness of compressible equation at low speed. At the same time, some measures to accelerate convergence were adopted, such as implicit residue averaging, local time step. The accuracy verification of the throughflow model by NASA Rotor 67 and a high-loaded transonic axial fan stage (ATS-2) can refer to reference [11]. The throughflow program has been applied to the first stage of a high-loading transonic six-stage compressor with inlet guide vane (IGV). The first stage with IGV has a design mass flow rate of 55.0 kg/s, a total to total pressure ratio of 1.81, and an adiabatic efficiency of 0.81. The flow path with sealing cavity is shown in Figure 2, where the geometric structure of the channel, cavity and labyrinth is simplified to some extent. The detailed geometric parameters of the flow path and blade are given in Tables 1 and 2 respectively. Figure 2. Flow path. In Figure 3, the computational mesh is shown. The channel and the mainstream flow path grid are matched at the junction interface. To prove the reliability of the calculation results, it is necessary to verify the grid independence before the study. Sealing the labyrinth cavity is the key part of this study, so three kinds of cavity grids with different grid numbers were used to verify the grid independence. The topology of the mesh and the boundary conditions were set the same. The results are plotted in Figure 4. The curves of the case with 6975 cavity grid points coincide with that of the case with 15,357 cavity grid points, but they differ from the curves of the case with 4549 cavity grid points in adiabatic efficiency. In conclusion, when the number of grid points exceeds 6975, the results are considered independent of the grid number. In consideration of the calculation's accuracy and the requirement of saving time, the mesh with 6975 cavity grid points was used in this study. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**S0**} & \multicolumn{2}{c}{**R1**} & \multicolumn{2}{c}{**S1**} \\ \cline{3-8}
**Parameters** & **Inlet** & **Leading** & **Trailing** & **Leading** & **Trailing** & **Leading** & **Trailing** & **Outlet** \\  & & **Edge** & **Edge** & **Edge** & **Edge** & **Edge** & **Edge** & **Edge** & \\ \hline hub radius/mm & 198.76 & 173.40 & 172.94 & 177.94 & 203.43 & 206.41 & 216.93 & 217.44 \\ shroud radius/mm & 397.79 & 362.10 & 355.37 & 350.10 & 343.37 & 340.97 & 337.98 & 337.62 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The geometric parameters of flow path. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **S0** & **R1** & **S1** \\ \hline No. of Airfoils & 32 & 28 & 50 \\ Chord/mm & 51.65 & 103.26 & 46.27 \\ Diameter/mm & 724.20 & 700.20 & 681.95 \\ Solidity & 0.86 & 1.74 & 1.35 \\ Aspect ratio & 3.65 & 1.67 & 2.91 \\ Inlet metal angle/(\({}^{\circ}\)) & 0.47 & \(-\)51.22 & 44.76 \\ Outlet metal angle/(\({}^{\circ}\)) & 8.16 & \(-\)40.49 & 9.60 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The main blading parameters of the compressor. Figure 3: The computational mesh of the compressor with seal cavity leakage. The boundary conditions are as follows: for the main flow of the compressor, the total temperature, total pressure and flow direction are given at the inlet, i.e., standard atmospheric conditions, horizontal intake.