## 5 Conclusions  
The windage loss in the impeller back gap and labyrinth seal cavities significantly impacts the aerodynamic performances of supercritical CO\({}_{2}\) (sCO\({}_{2}\)) compressors. To accurately calculate windage loss, essential factors affecting the skin friction coefficients \(C_{\text{f,d}}\) (for the disk-type gap) and \(C_{\text{f,s}}\) (for the shaft-type gap), including the Reynolds number \(Re\), pressure ratio \(\pi\), and radius ratio \(\eta\), are investigated in this paper. The flow characteristics of the gap are analyzed and prediction models are proposed, respectively. The conclusions help to use a more accurate form of predicting windage loss when designing sCO\({}_{2}\) compressors. The main conclusions are as follows:  
1. The slope of skin friction coefficients (\(C_{\text{f,d}}\) and \(C_{\text{f,s}}\)) varies abruptly for the Reynolds number (\(Re\)) from \(10^{3}\) to \(10^{4}\), which is the transition zone from laminar to turbulent flow. Moreover, the disk-type gap exhibits higher windage losses compared to the shaft-type gap, with impeller back gap losses exceeding 90% of total windage losses. Furthermore, the pressure ratio's influence on leakage flow diminishes beyond \(Re\geq 10^{6}\), attributable to emerging flow instability effects;
2. As \(Re\) increases beyond \(10^{6}\), centrifugal dominance causes the vortices to fully occupy the inlet region, reducing mass flow rate (\(m\)). Rotational speed and pressure ratio (\(\pi\)) jointly enhance inlet velocity, though \(Re\) exhibits a stronger correlation. The cavity velocities surpass the inlet values due to the labyrinth seal-induced flow area contraction, showing positive correlation with both \(Re\) and \(\pi\);
3. The radius ratio (\(\eta\)) exhibits negligible influence on skin friction coefficients \(C_{\text{f,d}}\) and \(C_{\text{f,s}}\) compared to \(Re\) and \(\pi\). While \(\eta\) equivalently affects mass flow rate across tested ranges, increasing \(Re\) and \(\eta\) synergistically expands high-vorticity zones. The instability of the flow causes \(C_{\text{f,d}}\) and \(C_{\text{f,s}}\) to increase with \(\eta\). The vorticity near the wall is the principal source of windage loss, which is due to the flow instability of the boundary layers;
4. The models of \(C_{\text{f,d}}\) and \(C_{\text{f,s}}\) for the impeller back gaps and labyrinth seal cavities are proposed and validated across \(Re\) regimes (\(10^{2}\)-\(10^{8}\)). With maximal relative deviations 6.23% and 6.83%, the models demonstrate engineering-grade accuracy for aerodynamic loss prediction in sCO\({}_{2}\) compressors. Incorporating the Reynolds number, pressure ratio, and radius ratio, these correlations enable the pre-optimizationof parameter combinations during preliminary design phases, achieving reduction through parameter selection.  
Conceptualization, B.T. and Q.D.; methodology, J.L. ([PERSON]); software, Z.Z.; validation, Q.D., B.T. and Z.Z.; formal analysis, J.L. ([PERSON]); investigation, Z.Z.; resources, B.T. and J.L. ([PERSON]); data curation, B.T., J.L. ([PERSON]) and Z.Z.; writing--original draft preparation, B.T. and Z.Z.; writing--review and editing, J.L. ([PERSON]) and Q.D.; visualization, Z.Z.; supervision, Q.D., J.L. ([PERSON]) and Z.F.; project administration, J.L. ([PERSON]) and Z.F.; funding acquisition, Q.D., J.L. ([PERSON]) and Z.F. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Joint Funds of the National Natural Science Foundation of China (U20A20303) and the National Key R&D Program of China (2023 YFB4102203).  
Not applicable.  
The raw data supporting the conclusions of this article will be made available by the authors on request.  
Author [PERSON] was employed by the company Dongfang Turbine Co., Ltd., Dongfang Electric Corporation. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.