## 1 Introduction
initially validated Bilgen models and discovered that the Bilgen model is not applicable to predict \(\mathrm{C_{f}}\) when \(Re\) falls between \(7\times 10^{3}\) and \(3\times 10^{4}\) or exceeds \(3\times 10^{4}\), and an improved model of \(\mathrm{C_{f}}\) incorporating \(\mathrm{CO_{2}}\) as working fluid was built based on numerical results [26]. [PERSON] et al. analyzed Taylor-Couette flow, which provides insight into the causes of windage loss [27; 28; 29]. [PERSON] et al. reported that the leakage flow in labyrinth seal causes a drop of 1.1% in isentropic efficiency of the s\(\mathrm{CO_{2}}\) centrifugal compressor [30]. It reduces the diffuser inlet radial velocity, increases the flow angle, and reduces the pressure drop, particularly for high flow coefficients. [PERSON] et al. studied the impact of operating conditions on the performance of the labyrinth seal when s\(\mathrm{CO_{2}}\) flows through with the method of experiment investigation [31]. [PERSON] et al. studied a compressor, including a cavity structure, both numerically and experimentally, and found that numerical simulations predicted the overall compressor performance and details of the flow characteristics very effectively [32]. The effect of leakage flow on the aerodynamic performance of centrifugal compressors was investigated using numerical methods by [PERSON] et al. The results show that the direction and location of leakage re-entry affect aerodynamic performance and efficiency [33]. Despite the continuous research on windage loss and \(\mathrm{C_{f}}\) in enclosed disk-type and shaft-type gaps, little attention has been paid to windage loss and \(\mathrm{C_{f}}\) models in open gaps. Leakage flow in open gaps complicates the formation of flow fields and windage losses compared to closed gaps. For example, different vortices will form in an open gap compared to a closed gap. Most of the current studies are focused on windage loss influencing factors [34], and have not modeled windage losses by combining the three key parameters of \(Re\), \(\pi\), and \(\eta\). Furthermore, while the aerodynamic performance of compressors and leakage flow in labyrinth seals have been the focus of attention, less attention has been paid to windage losses in the impeller back gap and labyrinth seal cavities, and few studies have been conducted on the losses with the s\(\mathrm{CO_{2}}\) medium. Therefore, the skin friction coefficients \(\mathrm{C_{f,d}}\) and \(\mathrm{C_{f,s}}\) are studied to predict windage loss in the impeller back gap and labyrinth seal cavities. Above all, the effects of Reynolds number \(Re\) of \(10^{2}\)-\(10^{8}\), pressure ratio \(\pi\) of 4.5-6.0 and radius ratio \(\eta\) of 0.018-0.054 are investigated. The three key parameters are intended to enable the exploration of the relationship between the role of physical properties and rotational speed, leakage flow rate, and geometric features on the skin friction coefficient. Subsequently, the action mechanisms  
Figure 2: Four flow regimes in disk-type gaps based on \(Re\) and \(\delta/R_{0}\). are explored in the form of analyzing flow characteristics. Finally, the models of \(C_{t,}\) are proposed in different \(Re\) ranges, and the models are adapted to different key parameters. The conclusions of this paper help to use a more accurate form of predicting windage loss when designing sCO\({}_{2}\) compressors.