# Effect of Rotational Speed Variation on the Flow Characteristics in the Rotor-Stator System Cavity
## 2 Physical Model and Numerical Method  
### Physical Model and Computational Domain  
The same cavity model of the rotor-stator system was established as that in the windage torque experiment of [PERSON] et al. [13; 24]. The basic principle of the experimental method uses the torque balance theory, with the turntable motor as a whole placed on the floating support bearing, the whole only being subject to windage torque between the turntable and the air, and on the side of the torque meter, the measurement obtained is the turntable in the process of rotation by windage torque. This method to achieve the turntable wind resistance torque is measured by the static torque meter. The static torque meter is a Lorenz-Mestechnik model 0150H small-range precision torque meter with the range of \(\pm\)20 Nm and the measurement accuracy of 0.1%. The AD-AM-4018 module was used to collect signals and transmitted them to the IPC for real-time monitoring and data storage. The fluid flows in from the axial direction and out from the radial direction. Figure 1 shows a specific structure. The relevant parameters defined in this study are mainly the geometric dimensions and physical quantities. The dimensionless parameter of the geometric dimensions is the outlet gap ratio, \(G\), which is defined as  
\[G=\frac{s}{r} \tag{1}\]  
where \(s\) is the outlet clearance, \(s=27\) mm, and \(r\) is the radius of the rotating disc (\(r=225\) mm). In addition to the dimensionless parameters, the inlet radius \(r_{\text{in}}=20\) mm and the inlet section length \(l_{\text{in}}=150\) mm. The dimensionless parameters of physical quantities include the rotational Reynolds number \(Re_{\Phi}\), flow coefficient \(C_{\text{w}}\), and torque coefficient \(C_{\text{m}}\), which are defined as  
\[Re_{\Phi}=\frac{\rho\Omega r^{2}}{\mu} \tag{2}\]  
\[C_{\text{w}}=\frac{m}{\mu r} \tag{3}\]  
\[C_{\text{m}}=\frac{T}{\frac{1}{2}\rho\Omega^{2}r^{5}} \tag{4}\]where \(\rho\) is the air density, \(\Omega\) is the rotor angular velocity, \(\mu\) is the dynamic viscosity coefficient, m is the mass flow rate, and \(T\) is the single side torque of the rotating disc. The grid of the computational domain is shown in Figure 2. The first grid height of the rotor and stator was 0.005 mm. [PERSON] [25] proposed to use the grid convergence index (GCI) to calculate discrete errors, and the method requires the results to satisfy monotonic convergence conditions. The detailed principles and methods of GCI calculation can be found in the original literature. The number of grids and related data in the grid independence verification are shown in Table 1. Based on the experimental torque coefficient \(C_{\text{m}}\), the influences of different numbers of grids on the calculation results were compared. As shown in Figure 3, with an increase in the grid number, the torque coefficient \(C_{\text{m}}\) decreases to different degrees at different rotational Reynolds numbers. When the grid number was increased from 5.94 \(\times\) 10\({}^{6}\) to 7.68 \(\times\) 10\({}^{6}\), the variation of \(C_{\text{m}}\) values is less than 1%. Considering the calculation accuracy and efficiency, 5.94 \(\times\) 10\({}^{6}\) grids were selected for calculation. \begin{table}
\begin{tabular}{c c c c c} \hline
**Component** & **Total Elements** & **Min Angle (\({}^{\circ}\))** & **Max Aspect Ratio** & **Quality (3 \(\times\) 3 \(\times\) 3)** & **GCI (\%)** \\ \hline
1 & \(1.78\times 10^{6}\) & 45 & 19 & 0.68 & 9.30 \\ \hline
2 & \(3.14\times 10^{6}\) & 45 & 64 & 0.65 & 3.26 \\ \hline
3 & \(5.94\times 10^{6}\) & 45 & 349 & 0.58 & 1.84 \\ \hline
4 & \(7.68\times 10^{6}\) & 45 & 558 & 0.49 & 1.67 \\ \hline \end{tabular}
\end{table} 
Table 1: Main parameters for different meshes. Figure 1: Physical model. Figure 2: Grid of computational domain. The number of boundary layer grids and the value of y\({}^{+}\) are important for observing the formation and development of vortices in the rotor-stator system. [PERSON] et al.