## 1 Introduction  
In order to avoid rotor-stator rubbing of rotating machinery, a certain gap is usually reserved between rotor and stator. The fluid filled in the gap will inevitably form windage loss. When the fluid density is large and the rotational speed is high, the windage loss in the gap will increase sharply, resulting in profound power loss [1]. The research on the windage loss in the rotor-stator gap began in high-speed motors and its related theories have made some progress [2]. An experimental study on windage loss was carried out by [PERSON] and [PERSON] [3]. The skin-friction coefficient of the rotating disk was measured at different rotational speeds. Based on the experiment, four coefficient models for predicting windage loss were proposed. [PERSON] [4] conducted a study on the windage loss in a motor gap, where four flow regions with a Reynolds number and a relative gap as parameters were divided according to the model of [PERSON] et al. [3]. [PERSON] et al. [5] and [PERSON] et al. [6] simplified the models of [PERSON]. [3] by ignoring the influence of the relative gap. The authors found that at high Reynolds numbers, the relative gap has a certain influence on the windage loss in the disk. [PERSON] [7] found that when the flow in the gap is laminar and the gap is much smaller than the boundary layer thickness between two walls, the flow is the same as the Couette flow. The velocity distribution is linear and the skin-friction coefficient equation of disks is proposed. [PERSON] et al. [8] studied the turbulent flow in gaps by numerical method and proposed two methods to calculate the vortex ratio and the radial distribution of static pressure. Turbomachinery is widely used, including steam turbines, gas turbines and compressors [9]. With the deepening of research, scholars have gradually paid attention to the windage loss in turbomachinery. [PERSON] [10] proposed a theoretical model to predict the influence of parameters on disk windage loss, considering parameters such as the impeller rim and surface roughness. [PERSON] et al. [11; 12] studied the influence of outlet structure and established an empirical equation to predict the windage loss of the disk. The results show that the outlet structure has only a small effect on the skin-friction coefficient and the outlet angle cannot be ignored. [PERSON] et al. [13] conducted an experimental study on the internal flow of turbine disk cavities and found that the internal flow can be divided into three vortices distributed along the radius direction. The position and the size of each vortex are closely related to the rotational speed. In recent years, [PERSON] et al. [14] used numerical methods to analyze the influence of an axial gap and surface roughness on the windage loss of the gap behind a centrifugal compressor in 2012. It was found that compared with the change of an axial gap, surface roughness has a greater influence on the windage loss. [PERSON] et al. [15] studied the influence of the curvature of disk surface on the flow characteristics by adding some elliptical convex point structures. The results show that when the convex point is located in the middle of the disk, the flow-blocking effect can be neglected. [PERSON] [16] proposed a solution using fluid-dependent values \(\lambda_{T}\) to explain the core rotation rate in the disk gap. [PERSON] [17] provided data items on the gap of shafts and disks and revealed the phenomena involved in rotating flow. The new power cycle with supercritical carbon dioxide (sCO\({}_{2}\)) as a working medium has been widely investigated over the years [18]. The density of CO\({}_{2}\) is large and the rotational speed of its impeller is higher resulting in larger windage loss in the gap behind the impeller. However, there is little open literature on the windage loss of sCO\({}_{2}\). [PERSON] et al. [19] studied the flow within a scroll compressor working with CO\({}_{2}\) and found that the tangential leakage flow had various characteristics in suction and compression chambers.