## 2 Geometry and Numerical Method
### Numerical Method  
The commercial software ANSYS Fluent was used to conduct the computational fluid dynamic (CFD) simulations of windage loss and flow characteristics in the gap. The steady-state and three-dimensional Reynolds-averaged Navier-Stokes (RANS) equations [12; 17; 30; 31] were solved, using the governing equations as follows [32]:  
Continuity equation:  
\[\frac{\partial}{\partial x_{\rm i}}(\rho u_{\rm i})=0 \tag{3}\]  
Momentum equation:  
\[\frac{\partial}{\partial x_{\rm i}}\left(\rho u_{\rm i}u_{\rm i}\right)+\frac{ \partial}{\partial x_{\rm j}}\left(P+\frac{2}{3}\mu_{\rm e}\frac{\partial u_{ \rm i}}{\partial x_{\rm i}}\right)-\frac{\partial}{\partial x_{\rm i}}\mu_{ \rm e}\left[\left(\frac{\partial u_{\rm i}}{\partial x_{\rm j}}\right)+\left( \frac{\partial u_{\rm j}}{\partial x_{\rm i}}\right)\right]=0 \tag{4}\]  
Energy equation:  
\[\frac{\partial}{\partial x_{\rm i}}(\rho u_{\rm i}H)-\frac{\partial}{\partial x _{\rm i}}\left[\left(\lambda+\frac{C_{\rm p}\mu_{\rm t}}{Pr_{\rm t}}\right) \frac{\partial T}{\partial x_{\rm i}}\right]=0 \tag{5}\]  
where \(u\) is the fluid velocity, \(x\) is the direction of coordinate system, \(P\) is the fluid pressure, \(\mu_{\rm e}\) is the effective viscosity, \(H\) is the total enthalpy, \(\lambda\) is the thermal conductivity, \(C_{\rm p}\) is the  
Figure 3: Schematic diagram of [PERSON]’s experimental apparatus [23]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Description** & **Case I** & **Case II** & **Case III** & **Case IV** & **Unit** \\ \hline Outer radius \(R_{\rm o}\) & 37.5 & 39.5 & 41.5 & 43.5 & mm \\ Gap width \(\delta\) & 2 & 4 & 6 & 8 & mm \\ Radius ratio \(\eta\) & 0.056 & 0.113 & 0.169 & 0.225 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimensions of outer radius \(R_{\rm o}\), gap width \(\delta\), and radius ratio \(\eta\). heat capacity at constant pressure, \(\mu_{\rm t}\) is the turbulent viscosity, \(Pr_{\rm t}\) is the turbulent Prandtl number, \(T\) is the fluid temperature, and i and j are index values. In addition, the above three governing equations needed to be combined with the realizable \(k\)-\(\epsilon\) turbulence model to characterize the turbulence flow in this study. The turbulence kinetic energy \(k\) and dissipation rate \(\epsilon\) can be solved by the following two equations [33]:  
\[\frac{\partial}{\partial x_{\rm j}}\big{(}\rho ku_{\rm j}\big{)}=\frac{ \partial}{\partial x_{\rm j}}\left[\left(\mu+\frac{\mu_{\rm t}}{\sigma_{\rm k} }\right)\frac{\partial k}{\partial x_{\rm j}}\right]+G_{\rm k}+G_{\rm b}-\rho \epsilon-Y_{\rm M} \tag{6}\]  
\[\frac{\partial}{\partial x_{\rm j}}\big{(}\rho\epsilon u_{\rm j}\big{)}=\frac {\partial}{\partial x_{\rm j}}\left[\left(\mu+\frac{\mu_{\rm t}}{\sigma_{\rm \varepsilon}}\right)\frac{\partial\epsilon}{\partial x_{\rm j}}\right]+\rho \epsilon C_{\rm I}S-\rho C_{2}\frac{\epsilon^{2}}{k+\sqrt{\
u\epsilon}}+C_{ \rm 1}\epsilon\frac{\epsilon}{k}C_{\rm 3}G_{\rm b} \tag{7}\]  
where \(\sigma_{\rm k}=1.0\), \(\sigma_{\rm\varepsilon}=1.2\), \(G_{\rm k}\) is the generation item due to the mean velocity gradients, \(G_{\rm b}\) is the generation of buoyancy, \(Y_{\rm M}\) is the ratio of the fluctuating dilatation to the overall dissipation rate, \(S\) is the modulus of the mean rate-of-strain tensor, and \(C_{\rm 2}\), \(C_{\rm 1c}\), and \(C_{\rm 3c}\) are constants. Moreover, \(C_{\rm 1}\) can be obtained based on the following equation:  
\[C_{\rm 1}=\max\left[0.43,\ \frac{\gamma}{\gamma+5}\right],\ \gamma=S\frac{k}{ \epsilon},\ S=\sqrt{2S_{\rm ij}S_{\rm ij}} \tag{8}\]  
In this study, the finite volume approach with the second-order upwind equation was adopted to discretize the RANS equations, and the pressure-based solver was used for numerical integration. The computation stability during simulation was estimated by monitoring the torque. When the torque of the stationary and rotational walls reached a stable value, and the torque difference between the two walls was less than 1% with additional iterations, the numerical results were considered \"steady\".