## 2 Geometry and Numerical Method  
### Geometric Model and Boundary Conditions  
The computation domain shown in Figure 2, which is composed of two end walls, a rotational wall and a fixed wall, was adopted as the research object to investigate the skin friction coefficient and flow characteristics in a shaft-type gap with rough walls by performing a three-dimensional numerical technique in this study. The main dimensions of the computation domain are defined by the inner and outer radii of the gap, \(R_{\rm i}\) and \(R_{\rm o}\), the axial length \(L\), and the gap width \(\delta=R_{\rm o}-R_{\rm i}\). Additionally, two dimensionless parameters, Reynolds number \(Re\), which depends on the flow state, and radius ratio \(\eta\), which is related to geometric dimensions, are defined to explore their influences on the skin friction coefficient and flow.  
\(Re\) and \(\eta\) are defined as follows:  
\[Re=\frac{\rho\omega R_{\rm i}(R_{\rm o}-R_{\rm i})}{\mu} \tag{1}\]  
\[\eta=\frac{R_{\rm o}-R_{\rm i}}{R_{\rm i}} \tag{2}\]  
where \(\rho\) and \(\mu\) are the density and dynamic viscosity of fluid, respectively; \(R_{\rm i}\) and \(R_{\rm o}\) are the inner and outer radii of the gap, respectively; and \(\omega\) is the rotational speed.  
For the geometric model, \(R_{\rm i}\) and \(L\) were fixed at 35.5 and 200 mm, respectively, which was consistent with [PERSON]'s experimental apparatus shown in Figure 3[23]. In addition, the gap width \(\delta\) ranged from 2 to 8 mm to investigate the effects of the radius ratio on the skin friction coefficient. \(R_{\rm o}\) and \(\eta\) can be determined by \(R_{\rm i}\) and \(\delta\). The dimensions of \(R_{\rm o}\), \(\delta\), and \(\eta\) from Case I to Case IV are listed in Table 1.  
For each case, the rotational speed of the two end walls and the stationary wall was zero, and that of the rotational wall was changed so that \(Re\) ranged from \(10^{2}\) to \(10^{7}\). It needs to be pointed out that since the fluid velocity is relatively high for sCO\({}_{2}\) power devices and the boundary conditions of the numerical validation have to keep consistent with [PERSON]'s experiment, heat transfer performance was not considered in this paper; thus, all walls were set as adiabatic. Based on the manufacturing accuracy of the turbomachinery, the influence of surface roughness \(Ra\) of 0.8, 3.2, and 6.3 \(\upmu\)m on the skin friction coefficient and  
Figure 2: Schematic diagram of computational domain composed of two end walls, a rotational wall and a fixed wall.  
flow characteristics are discussed and analyzed in detail. Moreover, in order to ensure that the research range of the Reynolds number can apply to more operating conditions of sCO\({}_{2}\) power devices, critical CO\({}_{2}\) (higher density than sCO\({}_{2}\)) with a pressure and temperature of 7.38 MPa and 304.13 K, respectively, was selected as the working fluid from Case I to Case IV. According to the data from NIST REFPROP [28; 29], the density and dynamic viscosity of critical CO\({}_{2}\) are set to be 467.60 kg/m\({}^{3}\) and 33.035 uPa\(\cdot\)s. In addition, the gravity effect was considered to be negligeable due to its little influence on flow while the apparatus was laid horizontally.