## 5 Results and Discussion
### Aeroacoustic Results  
This section presents the acoustic predictions for the twin- and three-propeller configurations and those for the isolated propeller, as well as the sensitivity analysis of the emitted noise of the multi-propeller configurations to the rotors tip-clearance. All the numerical results concern the noise emitted by rotor #1 and are evaluated over a sphere of radius equals to 5 m, centred at the hub of the rotor #1 (depicted in grey in Figure 7). In practical applications, the acoustic characterisation of single elements of an array allows scattering and propagation analysis considering multiple equivalent sources. The underlying assumption is that the equivalent sources may be considered equal, except for those of the first and last element of the array. Thus, obtaining results for each propeller of a three-propeller configuration, allows the evaluation of the noise emitted by an array of arbitrary number of propellers. For all cases shown in the next section, the results have been presented from two views: the northern view (where the sphere is observed from above) depicted in Figure 7a, and the southern view (where the sphere is observed from below) depicted in Figure 7b. For the sake of clearness, in both pictures the flow direction \(U_{\infty}\) is shown in black (-\(x\) direction). First of all, the sound directivity of the isolated rotor is shown in Figure 8, which depicts (from left to right) the sound pressure levels (SPL) of the first three Blade Passing Frequencies and the overall sound pressure level (OASPL), evaluated by using the first nine BPFs. Due to axisymmetry of the problem, only the southern hemisphere of the radiation pattern is shown. As expected, the noise emitted by the propeller is dominated by thickness contribution (results separately showing thickness and loading contributions are not shown here for the sake of conciseness) and concentrated about the rotation plane. When a twin-propeller configuration is considered, the axisymmetry of the problem is lost, and hence the acoustic results have to be shown on the whole sphere. Figure 9 shows the noise hemispheres observed from below (southern) and above (northern) for the case of the minimum distance between rotors (\(dist=0\)). In the northern hemisphere, the noise directivity, in terms of 1 st BPF, is very similar to that of the isolated rotor. Whereas in the southern hemisphere (i.e., the hemisphere where the blades of the two propellers approach), it is really different. In the northern hemisphere, the  
Figure 6: Aerodynamic pressure evaluated at \(y=0.75R\) and \(x=0.25c\) on rotor #1: comparison between twin- and three-propeller configurations. 1 st BPF appears to have a banded shape. Regarding the southern hemisphere, the noise directivity presents two-lobe noise pattern with maximum noise level slightly increased. These considerations can be extended to the OASPL, which is dominated by the pattern of the 1 st BPF (in this case, the maximum noise level on the southern hemisphere is about 2 dB higher than that of the northern). On the contrary, the higher harmonics hemispheres significantly differ with respect to those relative to the isolated rotor, with associated maximum noise levels 10 to 20 dB higher. Then, results concerning the sensitivity analysis to the distance are summarised in Figure 10, which shows the comparison in terms of OASPL among the isolated rotor and the four twin-propeller configurations analysed (characterised by \(dist=\{0,0.125d,0.25d,0.5d\}\)). As expected, the northern hemisphere (second row) remains unaffected by the relative distance between rotors, whereas the southern (first row) hemisphere is significantly modified for \(dist<0.25d\). Note that, this threshold might seem to contradict the aerodynamic results shown in Figure 5a, where the effects of rotor proximity are already evident for \(dist=0.5d\). This is because the thickness contribution dominates the emitted noise, hiding the loading term, which is the only contribution affected by aerodynamic interaction effects.