## 1 Introduction  
The supercritical CO\({}_{2}\) (sCO\({}_{2}\)) Brayton cycle first proposed by [PERSON] and [PERSON] [1,2] has, in recent years, been a research hotspot due to its higher efficiency than power cycles using conventional fluid as working fluid above 600 \({}^{\circ}\)C [3, 4, 5, 6]. Additionally, a higher density of sCO\({}_{2}\) allows for compact power devices, all of which can be incorporated into a turbine-alternator-compressor (TAC) unit for the sCO\({}_{2}\) Brayton cycle of output power below 1 MW [7, 8, 9], reducing cycle costs and enabling modular construction technology [10]. Figure 1 depicts the structure of a sCO\({}_{2}\) (TAC) unit, in which the fluid leakage from the compressor into the impeller back gap and labyrinth seal cavities can also be seen. According to Refs. [11, 12, 13], it is pointed out that the leakage flow in the gap and cavity generates windage losses due to friction when the rotor rotates and a cubic relationship between the windage losses and the rotational speed, indicating that the windage losses increase rapidly with the increase of rotational speed. [PERSON] found that the windage  
loss accounts for up to 37.5% of the total loss in a sCO\({}_{2}\) TAC unit because the rotational speed is undoubtedly higher for smaller power devices [14], reducing the output power and potentially leading to rotor overheating [15]. The research conducted thus far has revealed that the windage loss in the impeller back gap and labyrinth seal cavities, which corresponds to the blue and red regions in Figure 1, respectively, significantly impacts the aerodynamic performances of sCO\({}_{2}\) centrifugal compressors and even TAC units [16; 17]. Thus, it is essential to research windage loss, flow, and the model of \(C_{\text{f}}\) in the regions. Moreover, the impeller back gap is considered a disk-type gap, commonly regard as a rotor-stator disk system [14]. The labyrinth seal is similar to the shaft-type gap, which can be assumed to be a concentric cylinder system [18]. In response to the difference in loss mechanisms in the two gap types, their windage losses need to be studied separately. For disk-type gaps, [PERSON] et al. first identified the four flow regimes in enclosed disk-type gaps under smooth wall conditions, based on Reynolds number \(Re\) and radius ratio \(\delta/R_{\text{o}}\), as highlighted in Figure 2[19]. Models for calculating the skin friction coefficient \(C_{\text{f}}\), a parameter that is influenced by several key variables related to windage loss, were proposed for each of the four regimes. Nowadays, the models have been widely used to predict disk-type windage loss [20]. However, [PERSON] et al. confirmed that the \(C_{\text{f}}\) grows with surface roughness, and they imply that [PERSON]'s model is inapplicable to rough walls [21]. The evaluation of windage losses for disk-type gaps was conducted by [PERSON] et al., and the results show that windage loss is proportional to the square of throughflow and a dependency on throughflow direction [22]. In addition, further research on flow characteristics in disk-type gaps has revealed new insights. A variety of vortex structures, such as primary and secondary vortices and the vortex in turbulent boundary layers, were observed by [PERSON] et al. [23]. At higher \(Re\), vortex cores are denser in the boundary layer and vortex intensity decreases, consistent with the conclusion that the streamline instability becomes dominant in the transition boundary layer [24]. [PERSON] et al. developed a one-dimensional pressure model and a design tool for solving the flow in pump side chambers by investigating leakage and pressure distribution in disk-type gaps of centrifugal pumps with water as working fluid [25]. Figure 1: Structure of a sCO\({}_{2}\) TAC unit. For shaft-type gaps, there has been extensive research on windage loss and flow in enclosed shaft-type gaps and labyrinth seals. [PERSON] et al.