## 5 Conclusions and Discussion  
A time-marching throughflow method for the centrifugal compressor simulation was presented. To improve the robustness of the throughflow calculation, a method of enforcing the characteristic boundary conditions based on [PERSON]'s iteration method was developed and introduced. The flow deflection inside the impeller passage was achieved by the well-validated inviscid blade force model. The viscous effects were simulated by the classic viscous force model coupling with the complete empirical loss and slip model for the centrifugal compressor. This throughflow solver was finally employed to analyze the performance and flow fields of the LSCC and HPCC cases as validations. The results were compared with the experimental data and 3D calculations. The main conclusions are summarized below.  
1. Compared to the classical Riemann-invariant-based method, the developed enforcement method for the boundary conditions at the inlet and outlet of computation models significantly improved the robustness of the time-marching throughflow simulation of centrifugal compressors. For the high-load test case, i.e., the HPCC case, the  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Mass Flow** & **Incidence** & **Blade** & **Skin** & **Clearance** & **Mixing** & **Disc** & **Recirculation** & **Vaneless** \\
**Rate (kg/s)** & **Loss (\%)** & **Loss (\%)** & **Loss (\%)** & **Loss (\%)** & **Loss (\%)** & **Loss (\%)** & **Loss (\%)** & **Diffuser** \\ \hline
3.60 & 2.210 & 8.287 & 22.541 & 2.541 & 4.751 & 7.348 & 33.646 & 18.674 \\ \hline
4.39 & 0.313 & 11.346 & 26.448 & 3.756 & 9.624 & 8.842 & 18.936 & 20.736 \\ \hline
5.07 & 0.090 & 11.622 & 32.973 & 4.595 & 16.216 & 9.009 & 6.757 & 18.739 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Contributions of different losses for the HPCC case at 100% speed.  
Figure 10: Relative Mach number of the HPCC at design mass flow rate. (**a**) Throughflow; (**b**) 3D CFD.  
latter obtained converged throughflow results while the former encountered blowing up during the computation.
2. Concerning the computational accuracy, the developed throughflow method accurately predicted the overall performance of the centrifugal compressor at design conditions, and also obtained correct off-design characteristics that were close to the experimental or the 3D-CFD results. Reasonable results were also obtained with respect to the spanwise distribution of the flow parameters and even the whole meridional flow field. Particularly, the throughflow method precisely predicted the choke mass flow rate automatically and precisely for the high-load centrifugal compressor.
3. The simulation of the LSCC and HPCC cases using the developed throughflow method only cost less than two minutes, which is approximately two orders of magnitude lower than the 3D-CFD method.  
In conclusion, the throughflow method developed in this study provides an accurate and efficient tool for the performance analysis of centrifugal compressors. Although some kinds of empirical models, such as the recirculation and the slip models, are expected to be further improved, the prediction results demonstrated that this throughflow method is of great value to the preliminary design of centrifugal compressors.  
Conceptualization, C.Y., J.Y. and H.W.; methodology, C.Y. and Q.T.; software, C.Y.; validation, C.Y., Q.T. and H.W.; formal analysis, C.Y.; investigation, C.Y.; resources, C.Y. and J.Y.; data curation, C.Y. and Q.T.; writing--original draft preparation, C.Y.; writing--review and editing, C.Y., J.D. and H.Z.; visualization, C.Y. and Q.T.; supervision, J.D. and H.Z.; project administration, J.D.; funding acquisition, J.D. and H.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by National Science and Technology Major Project, grant number 2017-II-0004-0017, and National Natural Science Foundation of China, grant number 51922098.  
Not applicable.  
Data Availability Statement: Data sharing is not applicable to this article.