# Effect of Rotational Speed Variation on the Flow Characteristics in the Rotor-Stator System Cavity
## 3 Results and Discussion
### Flow Field Distribution Characteristics  
Under the condition of a constant inlet flow coefficient, \(Re_{\Phi}\) increases from 2.12 \(\times\) 10\({}^{6}\) to 4.23 \(\times\) 10\({}^{6}\). In Figure 9, the distributions of the \(r\)-\(z\)-plane streamlines in the cavity at different rotating speeds are compared. The legend represents the absolute velocity \(V_{\rm tan}\) of the fluid microelement at this point:  
\[V_{\rm tan}=\left(V_{r}^{2}+V_{z}^{2}\right)^{\frac{1}{2}} \tag{6}\]  
Here, \(V_{r}\) is the radial component of the velocity vector, and \(V_{z}\) is the axial component of the velocity vector. The figure shows that the vortex in the cavity is mainly the swirling vortex formed by the entrainment of the outlet and the interference of the rotor-stator wall, that is, the main vortex in the cavity. The position of the main vortex core hardly changes in the radial direction with an increase in \(Re_{\Phi}\). However, under the action of inertial force, the trend of separating the secondary vortex can be seen in the streamline diagram of Case 3. Until \(Re_{\Phi}\) increases to 4.23 \(\times\) 10\({}^{6}\), the secondary vortex is completely separated. The distribution of \(V_{\rm tan}\) on the streamline shows that the absolute velocity of the main vortex core corresponding to the position in the rotor boundary layer is smaller than that at other positions in the boundary layer. Therefore, \(V_{\rm tan}\) in the rotor boundary layer decreases first and then increases along the radial direction as a whole. When the disc rotating speed increases, \(V_{\rm tan}\) in the rotor boundary layer gradually increases under the action of wall adhesion and fluid viscosity. In addition, the streamlines near the stator wall become sparse with the increase in \(Re_{\Phi}\), which indicates that the increase in the rotor rotating speed causes  
Figure 8: Radial distribution of torque coefficient. Figure 9: Flow structure with increasing \(Re_{\Phi}\). the structure of the main vortex formed by the entrainment of the fluid in the cavity to decrease gradually. Figure 10 shows the distribution of the circumferential velocity along the axis of the rotor-stator system on the monitoring line. The abscissa indicates the position of a certain point on the line, and the ordinate indicates the circumferential velocity component of the fluid at that point. The abscissa and ordinate are subjected to the following dimensionless processing:  
\[z^{*}=\frac{z}{h} \tag{7}\]  
\[V_{u}^{*}=\frac{V_{u}}{\Omega r} \tag{8}\]  
where \(z\) is the distance between the point on the monitoring line and the stator wall, \(h\) is the length of the monitoring line, \(V_{u}\) is the circumferential velocity of the point on the monitoring line, and \(r\) is \(1/5r\), \(2/5r\), \(3/5r\), and \(4/5r\). Observing the \({V_{u}}^{*}\) distribution on different monitoring lines, when \(z^{*}<0.7\), the \({V_{u}}^{*}\) value order of the different cases does not change with the position of the monitoring line. It always decreases as \(Re_{\Phi}\) gradually increases, and \({V_{u}}^{*}\) in this area shows a monotonous increasing trend. At \(z^{*}<0.03\), the stator wall and recirculation zone produce a more obvious dimensionless velocity gradient because the \({V_{u}}^{*}\) of the fluid near the wall becomes zero at the wall under the action of fluid viscosity and no-slip wall. A thin layer of fluid near the wall of the rotor has the same circumferential velocity as the rotating disc, so a large velocity gradient is generated when \(0.95<z^{*}<1\). As Figure 10a,b show, when \(z^{*}>0.7\), \({V_{u}}^{*}\) begins to decrease because the inertial force of the fluid in the rotor boundary layer  
Figure 10: Distribution of circumferential velocity (**a**) monitoring line of L\({}_{\text{A1}}\); (**b**) monitoring line of L\({}_{\text{A2}}\); (**c**) monitoring line of L\({}_{\text{A3}}\); (**d**) monitoring line of L\({}_{\text{A4}}\).