## 5 Results and Discussion
### Aerodynamic Results  
The results pertaining to the aerodynamic analysis of the twin- and three-propeller configurations are presented and discussed in this section. All the analyses are performed by meshing the rotor blades with 840 quadrilateral panels (15 along the chord and 28 along the span), the rotor hub with 2160 elements, and the surface of the wake with 5040 elements (180 in the azimuth direction and 28 in the radial direction) distributed on two spirals. As stated before, in all the cases, the rotors have been trimmed to the thrust of the isolated rotor case (3700 N), by minor changes in the blade pitch. The resulting performance in terms of torque (and thus power) vs. thrust has been verified to remain substantially unchanged.  
First, Figure 4 shows, for the twin-propeller configuration and the case of minimum tip-clearance (_dist_ = 0), the computational grid and the wake shape resulting from the aerodynamic interaction of the two rotors. It is worth noting that, also in the case of the smallest distance between rotors,  
Figure 3: Sketch of the analysed configurations, front view: twin-propellers configuration (#1 and #2); three-propellers configuration (#1, #2 and #3).  
the aerodynamic interaction effects on the wake shape are negligible (i.e., there is no intersection between the wakes of the two rotors as well as between the wake of a rotor and blades of the other one).  
Then, Figure 5a shows the time history over an entire revolution of the perturbation pressure (\(p-p_{\infty}\)) on each blade of rotor #1 (at the quarter-chord of the blade section located at the radial station \(r/R=0.75\)) for \(dist=\{0,0.25d,0.5d,1.25d\}\). As expected, as the distance between the rotors diminishes, a growing (in absolute value) pressure peak appears, having its minimum located approximately at the azimuth position where the blade is closer to the second rotor. The aerodynamic interaction becomes negligible for the tip-clearance \(dist\) of the order of magnitude of the rotor diameter. Figure 5b depicts the spectrum of the pressure signals in terms of Discrete Fourier Transform (DFT) module, showing that, for all the configurations considered, the pressure signals are strictly tonal (due to the aerodynamic formulation hypotheses the analysis is based upon), with frequencies multiple of rotor angular velocity, \(\Omega\). Moreover, the spectrum of the minimum-distance configuration presents a linear behaviour up to \(-20\) dB, with a very weak, secondary peak at \(16\)\(\Omega\), whereas the linear part of the spectrum for larger distances extends up to about \(-40\) dB, followed by an irregular behaviour at the higher frequencies, probably due to numerical errors (given the low values of harmonics amplitude). As expected, when the distance between rotors increases, the pressure signal tends to become uniform during the rotor revolution, as in the case of an isolated propeller (indeed, the spectrum harmonic content tends to vanish).  
Finally, for the minimum-distance configuration, Figure 6a shows the comparison of the time history of the pressure at the quarter-chord of the blade section at \(y=0.75R\) of the rotor #1 for the  
Figure 4: Twin-propeller computational mesh for the case \(dist=0\).  
Figure 5: Aerodynamic pressure evaluated at \(y=0.75R\) and \(x=0.25c\) on rotor #1 for four different distances between rotor #1 and #2 (twin-propeller configuration).  
twin- and three-propeller configurations. As expected, in the three-propeller configuration two peaks with the same intensity appear, at the azimuth positions at which the blade of rotor #1 is closest to those of the rotors #2 and #3. Furthermore, Figure 6b compares the spectrum of the pressure signals. Also in this case the spectrum of the three-propeller configuration is linear, with harmonic content at the only frequencies multiple of 2 \(\Omega\) (no odd harmonics are present). Finally, the even tones of the three-propeller configuration are globally more intense than those of twin-propeller case.