## 4 Results and Discussion
### Flow Characteristic Analysis  
To analyze the influences of \(Re\) and \(\pi\) on \(C_{\mathrm{t}s}\), \(C_{\mathrm{t}d}\), and \(m\) from flow characteristics' point of view, Figure 13 depicts the streamlines and velocity at the inlet regions for Case I, where \(Re=10^{2}\), \(10^{5}\), and \(10^{8}\) and \(\pi=4.5\), 5.0, 5.5, and 6.0. It can be seen that the streamlines exhibit similar flow patterns and indicate that fluid can flow smoothly into the impeller back gap as a whole when \(Re=10^{2}\) and \(10^{5}\). However, there are a small number of vortices in the transition regions between the inlet and impeller back gap, such as Regime I near the stationary walls and Regime II near the rotational walls. The reason for this is that the velocity direction of fluid in transition regions suddenly changes, resulting in a dramatic decrease in the velocity magnitude and generation of vortices. As \(Re\) increases, the centrifugal force becomes more prominent, leading to increased flow instability and the formation of vortices that fill the whole inlet region. As a result, the \(m\) decreases with increasing \(Re\) under \(Re\geq 10^{6}\) conditions, as depicted in Figure 12. Furthermore, it is also worth noting that the size of the vortices in the impeller back gap, which makes disk-type windage loss generated, increases when the \(\pi\) rises from 4.5 to 6.0 under the same \(Re\) conditions. The vortex shape at the inlet of the disk-type gap changes when the \(Re\) increases. It shifts from being attached to the rotor wall (Regime II) to filling the entire inlet region. This is attributed to the fact that, at this point, the influence of leakage flow is weakened and the centrifugal force acts as the main role to enhance the  
Figure 12: Leakage flow versus \(Re\) under. vortex formation. The phenomenon can be attributed to the higher inlet velocity under a larger \(\pi\), which leads to the larger decreasing rate of velocity when the fluid in Regime I and Regime II change velocity direction. The contours reveal that the velocity increases with \(Re\) and \(\pi\) due to a larger centrifugal force that depends on \(Re\) and the pressure difference related to \(\pi\). However, the velocity  
Figure 13: Streamlines and velocity at inlet with different \(Re\) of \(10^{2}\), \(10^{5}\), and \(10^{8}\) and \(\pi\) of 4.5, 5.0, 5.5, and 6.0.  
diversity under different \(Re\) conditions is more pronounced than that of different \(\pi\), suggesting that the centrifugal force has a more significant influence on velocity distribution than the pressure difference because its variation is smaller. Additionally, the velocity gradually decreases from the rotational walls, where fluid velocity equals the linear velocity of the rotor, to the stationary walls, where the fluid velocity is 0 due to the non-slip wall. It is further observed that as \(Re\) and \(\pi\) rise, the high-velocity regions near the rotational walls expand, while the low-velocity regions near the stationary walls shrink. This trend indicates that the high-velocity regions become dominant for a larger \(Re\) and \(\pi\). In addition, the streamlines and velocity contours in the cavities of the seal for Case I, with a different \(Re\) of \(10^{2}\), \(10^{5}\), and \(10^{8}\) and a \(\pi\) of 4.5, 5.0, 5.5, and 6.0 are shown in Figure 14.  
The velocity in the cavities is observed to be higher than that at inlet regions described in Figure 13, which is due to the dramatic decrease in the through-flow area when the fluid passes through the labyrinth seal. Similarly, the velocity increases with \(Re\) and \(\pi\), and the effects of \(Re\) on velocity are still more prominent than \(\pi\), as previously analyzed. Moreover, it is worth noting that there are high-velocity regions near the rotational and stationary walls and low-velocity regions at the center of the cavities. As \(Re\) increases, the high-velocity regions grow, and the low-pressure regions decrease.