## 4 Results
### Performance Measurements and Predictions
#### 4.1.2 Dual Rotor  
To explore the impact of rotor interactions, a pair of co-rotating counter-clockwise rotors and a pair of counter-rotating rotors were simulated using Helios. The 6000 RPM case was chosen, and the rotors were placed at tip-to-tip separation distances of \(t=0.04D\), \(t=0.16D\), and \(t=0.24D\), where t is the tip-to-tip separation distance and D is the rotor diameter. The mean thrust of the entire system and the sum of the torques of the two rotors were normalized by the single rotor values and plotted in Figure 9. As expected, the interactions of the dual rotors reduce the performance of the individual rotors and lead to lower thrust and higher torque than the ideal value of 2. The expected trend is captured in which both performance parameters approach the ideal value of 2 with increasing  
Figure 8: Figure of merit measurements and predictions for the APC8x4.5-MR rotor in hover. Figure 7: A collection of measured and computed thrust coefficient and torque coefficient values for the APC8x4.5-MR rotor in hover. distance. Another interesting result is the sensitivity of the performance parameters to the interactions. The thrust reduced by more than 1% for each rotor in the dual rotor system, compared to the isolated rotor, but the torque is only reduced by less than 0.5%, on average. Although these differences are negligible and may not be noticeable on a real propeller subject to external factors, the computations indicate that the interactions and induced flow are stronger perpendicular to the rotor plane. The counter-rotating configuration demonstrates lower thrust and greater torque overall, due to a higher number of interactions. The trends of the counter-rotating configuration are not as well behaved as those of the co-rotating case, due to the unsteadiness in the thrust signal caused by the interactions. To show this, the thrust signal over the final rotor revolution is plotted in Figure 10 for both configurations. The thrust signal of the single rotor was doubled and plotted in black for comparison. All values were again normalized by the mean thrust over the final rotor revolution of the single rotor. The amplitude of the thrust oscillation clearly increases with the presence of a second rotor. The results show that increasing the rotor proximity increases the amplitude of the thrust oscillation, which will increase the noise. In addition, the unsteady behavior of the counter-rotating configuration is highlighted by a strong impulsive increase in thrust directly after the blade crossing, seen in Figure 10. This impulse is not seen in the torque signal, emphasizing the larger interactional effects perpendicular to the rotor plane. Figure 10: Computed time-varying thrust using Helios of the dual rotor systems operating at 6000 RPM normalized by the mean thrust value of the single rotor. The thrust signal of the single rotor was doubled and plotted in black for comparison. Figure 9: Helios predictions of the performance of the dual rotor systems operating at 6000 RPM in hover normalized by isolated rotor values. The flowfields of the dual rotors at the closest separation distance of \(t=0.04D\) were extracted from the simulation at the last time step to provide insight into the flow physics. A planar slice of vorticity magnitude across the rotors' centerline can be seen in Figure 11. Iso-surfaces of q-criterion = 0.001 colored by the vorticity magnitude are shown in Figure 12. Both the co-rotating configuration and counter-rotating configuration are pictured. The same omega-shaped, horseshoe vortex predicted and measured in Refs. [27; 31; 33; 35] can be seen in the counter-rotating configuration between the two rotors in Figure 12b. This vortex is generated by the interaction between the tip vortices of the two rotors; here, it is shown to be stronger for the counter-rotating case, indicated by the high vorticity band noticeable in Figure 11b, which is not visible in Figure 11a.