# Scalability of Mach Number Effects on Noise Emitted by Side-by-Side Propellers
## 3 Numerical Results
Note that, for \(\psi=0^{\circ}\) the observer is placed at \((x=-R,y=0)\), and that positive \(\psi\) corresponds to positive rotation around the \(z\)-axis. For all the configurations examined, the effect of the phase angle between the blades of the two propellers is a shift of the azimuth positions where the signals from the two propellers generate constructive or destructive interference. Furthermore, Figure 6 shows that, for the counter-rotating configurations, \(M_{t}=0.52\) and \(TC=0.125d\), a significant reduction of noise (up to 25 dB) is achieved for \(\gamma=36^{\circ}\) and \(120^{\circ}\leq\psi\leq 240^{\circ}\) (in the same azimuth region, the radiated noise presents a visible reduction also for \(\gamma=24^{\circ}\)). A similar behavior is also observed for \(M_{t}=0.7\), and for all the tip-clearances, although the reductions are much smaller and appearing in a narrower azimuth range. The results for co-rotating propellers presented in Figure 7 show that the phase angle \(\gamma=36^{\circ}\) produces a noise magnitude reduction of about 10 dB for TC = \(d\) and azimuth ranges \(60^{\circ}\leq\psi\leq 120^{\circ}\) and \(240^{\circ}\leq\psi\leq 300^{\circ}\). For all the other cases, the noise increases in some directions and decreases in others. Thus, at least for the configurations herein investigated, the strategy of reducing the emitted noise by a proper choice of the relative phase angle between propellers seems to be more effective for the counter-rotating configuration than for the co-rotating one. This conclusion is confirmed by the examination of the values of the averaged OASPL computed over the sphere that are reported in Table 1. However, further considerations can be drawn from Table 1.  
Figure 5: Co-rotating configuration, OASPL difference vs. TC difference, for \(M_{t}=0.52\) (upper) and \(M_{t}=0.7\) (lower). First, regardless of the propeller direction of rotation and the tip Mach number, for \(\gamma=0^{\circ}\), the highest averaged OASPL is obtained for the maximum TC, whereas the minimum averaged OASPL is obtained for \(\text{TC}=0.125d\). This behavior can be the result of a combination of source-observer distance (inversely proportional to the tip-clearance) and unsteady propeller loading (directly proportional to the tip-clearance). Second, regardless of the tip Mach number and tip clearance, increasing the relative phase angle yields a noise increase for the co-rotating configuration and a noise decrease  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(M_{t}\) & **Tip-Clearance** & \(\gamma=0^{\circ}\) & \(\gamma=12^{\circ}\) & \(\gamma=24^{\circ}\) & \(\gamma=36^{\circ}\) \\ \hline \multirow{6}{*}{Counter-rotating} & & 0.05\(d\) & 68.78 dB & 68.51 dB & 67.90 dB & 66.97 dB \\  & 0.52 & 0.125\(d\) & 68.67 dB & 68.43 dB & 67.80 dB & 66.97 dB \\  & & 1\(d\)  
for the counter-rotating one (thus confirming the overall trend observed in Figures 6 and 7). For the counter-rotating configuration, the maximum relative phase angle seems to be the most effective in reducing the noise at both Mach numbers. Finally, the above investigations are also performed for near-field observers (located on the sphere of radius \(5d\)). First, for the counter-rotating configuration, Figure 8 shows the difference between the OASPL evaluated in the far-field hemisphere (radius \(40d\)) and the near-field one (radius \(5d\)), indicated as \(\Delta\,OASPL\), for the defined TCs and Mach numbers. Note that for this comparison, a suitable scaling of the OASPLs in terms of the ratio between the sphere radii is performed. In this case, the differences between near-field and far-field noise distribution are limited to a few dBs in wide areas of the northern hemisphere, especially in regions close to the rotor disk plane. Greater discrepancies arise near the poles (where the noise emitted is less significant) and in the southern hemisphere. From these results it is inferred that the dipole contribution to the noise is negligible also on the hemisphere of radius \(5d\).