## 3 Results and Discussion
### Gap Flow in the RDT  
There is no tip leakage flow in the RDT as no tip clearance exists between the propeller tip and duct. However, the gap formed by the rim surface and duct interior surface in the RDT can cause interesting flow phenomena as well. The gap flow in the RDT resembles the flow of a viscous fluid between two concentric cylinders, with one rotating and the other one static. The major influence of the gap on the flow field is the creation of a significant friction torque, which is mainly related to the width of the gap, the dimensions of the inner cylinder, and the Reynolds number of the gap flow. In the work of [PERSON]  
Figure 10: Velocity fields near the blade tip for: (**a**) DP at J = 0.1; (**b**) DP at J = 0.5; (**c**) RDT at J = 0.1; and (**d**) RDT at J = 0.5.  
and [PERSON] [19], empirical formulations for the torque coefficient for a flow without an axial pressure gradient were developed based on experimental results. As the flow regime changes from laminar to turbulent flow at different working conditions, the functional relation for the torque coefficient has to be adjusted. Therefore, only the formulas applied to the present work are considered and are listed as follows:  
\[\text{Re}_{r}=\frac{\rho\omega R_{i}(R_{o}-R_{i})}{\mu} \tag{11}\]  
\[C_{Mr}=0.065\bigg{(}\frac{R_{o}-R_{i}}{R_{i}}\bigg{)}^{0.3}\text{Re}_{r}^{-0.2} \ (\text{Re}_{r}>10^{4}) \tag{12}\]  
\[M_{r}=0.5\tau\rho\omega^{2}R_{i}^{4}LC_{Mr} \tag{13}\]  
where \(\text{Re}_{r}\) is the Reynolds number for the radial gap, \(\rho\) is the fluid density, \(\omega\) is the angular velocity, \(R_{i}\) is the radial gap inner radius, \(R_{o}\) is the radial gap outer radius, \(R_{o}-R_{i}\) is the gap width, \(\mu\) is the dynamic viscosity, \(C_{Mr}\) is the torque coefficient, \(M_{r}\) is the torque of the radial gap, and \(L\) is the length of the rim. A comparison of the torque coefficient between the simulations in this paper and the empirical method in Equations (9)-(11) is provided in Table 4. It can be seen that the torque calculated by the empirical formulas is smaller than the one of the numerical simulations. This may be attributed to the existence of an axial pressure gradient, as it was also observed in the study of [PERSON] [20] that the rim torque with a pressure gradient was larger than without. It is assumed that the interaction between the radial gap flow and axial gap flow is enhanced by the pressure gradient, which makes the flow regime more complex, resulting in the production of a larger torque. In the study of [PERSON] [21], the effects of the axial pressure gradient on the torque coefficient were examined under different working conditions. It was found that when the characteristic Reynolds number for the axial pressure gradient is above \(5\times 10^{3}\) the torque coefficient increases steadily with the increase in the pressure gradient. In the present study, the Reynolds number is much higher than the critical value, and the corresponding results \begin{table}
\begin{tabular}{c c} \hline \hline
**Methods** & \(\mathbf{K_{tr}\times 10^{2}}\) \\ \hline Numerical & 0.57 \\ Empirical & 0.37 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of the rim torque coefficient between empirical correlations and simulations. Figure 11: Vortical structures along the propeller tip region at (**a**) J = 0.1 and (**b**) J = 0.5.  
are observed, as shown in Figure 12. When the propeller loading is heavy, as mentioned above, the pressure difference is more significant and the rim torque, due to gap flow, is larger. The velocity profile in the gap along the axial and tangential direction is shown in Figure 13. A small velocity gradient with respect to the wall is observed near the rim outer surface, which is caused by the centrifugal force as the rim rotates. In the axial direction, the flow is pressure driven and due to the location of the rim ends the flow runs in the opposite direction to the main flow.