# Scalability of Mach Number Effects on Noise Emitted by Side-by-Side Propellers
## 3 Numerical Results
In particular, for the lower Mach number conditions, the northern view presents a symmetric distribution of the noise radiated by the two propellers with respect to the \(xz\)-plane and three lobes with high OASPL values that collapse into a single high-intensity region located close to the \(yz\)-plane as the tip-clearance increases. A similar behavior is also observed for the higher Mach number conditions although, in this case, two trilobate structures are present, with the one below the \(yz\)-plane much more intense than that above it. Instead, in the southern view, a more wavy noise distribution is observed, generated by constructive or destructive interference between sound fields emitted by the two propellers. The corresponding spatial frequency increases with the tip clearance and the noise waves remain in the same azimuth positions regardless the tip Mach number. Some differences appear in the region upstream the rotor disk, where the radiated noise is quite uniformly distributed for \(M_{t}=0.52\), whereas wave interference patterns are clearly visible for \(M_{t}=0.7\). For a clearer interpretation of the noise distribution, Figure 3 shows the differences between the OASPL given by different TCs, for both blade tip Mach numbers considered. In this figure and in the following ones, \(\Delta OASPL_{ij}\) is introduced to denote the difference between OASPLs evaluated for tip-clearance \(TC_{i}\) and tip-clearance \(TC_{j}\). Figure 1: Sketch of the examined configurations, counter-rotating propellers. As expected, the most significant difference occurs between the configurations with the higher and lower tip clearances. At \(M_{t}=0.52\) the differences are localized near the  
Figure 3: Counter-rotating configuration, OASPL difference vs. TC difference, for \(M_{t}=0.52\) (upper) and \(M_{t}=0.7\) (lower). Figure 2: Counter-rotating propellers, influence of TC on OASPL\({}_{*}\) for \(M_{t}=0.52\) (upper) and \(M_{t}=0.7\) (lower). rotor disk plane, whereas upstream the radiated noise is almost independent of the tip clearance. At \(M_{t}=0.7\) the \(\Delta\) OASPL is not negligible also in the microphones near the polar region upstream the propeller disk. Furthermore, the increase of the spatial frequency for increasing tip clearance observed in Figure 2 generates \(\Delta\) OASPL interference patterns in the southern view that are similar for both tip Mach numbers. Next, the same analyses are performed for the configuration with co-rotating propellers. The corresponding evaluated OASPL distributions are shown in Figure 4. It is worth noting that, in this case, the SV and the NV coincide due to the geometric symmetry of this configuration with respect to the _xy_-plane. Akin to the counter-rotating propeller analysis, in order to get a better insight about the tip-clearance effect on the emitted noise, Figure 5 shows the differences between the OASPL given by different TCs, for both blade tip Mach numbers considered. These results confirm the general trend observed also for the counter-rotating case: greater differences are present between maximum and minimum tip-clearances, the increase of the spatial frequency for increasing tip clearance observed in Figure 2 generates \(\Delta\) OASPL interference patterns are similar for both tip Mach numbers. Figure 4: Co-rotating propellers, influence of TC on OASPL, for \(M_{t}=0.52\) (upper) and \(M_{t}=0.7\) (lower). The second numerical investigation carried out regards the effect of the phase shift, \(\gamma\), on the emitted noise. The analysis is performed for both co-rotating and counter-rotating configurations, for both Mach numbers, for all the propeller hub distances considered. Specifically, Figures 6 and 7 show, respectively for the counter-rotating and co-rotating configurations, and for increasing tip-clearance, the far-field OASPL directivity patterns on a circular array of observers with radius equal to \(40d\) and centered at the midpoint of the dual propeller system, lying on the rotor disk plane.