# Scalability of Mach Number Effects on Noise Emitted by Side-by-Side Propellers
## 3 Numerical Results  
The numerical investigation herein proposed is aimed at assessing the effect of the propeller blade tip Mach number, \(M_{t}\), on the noise emitted by two adjacent co-rotating and counter-rotating propellers in forward-flight conditions. The examined configuration consists of two side-by-side, five-bladed propellers, with parallel rotational axes and diameter \(d=0.67\) m. The chord length of the untapered blades is equal to 0.025 m. For the aerodynamic analysis, each blade surface is discretized by 840 panels, with 30 panels in the chordwise direction (15 on the upper side and 15 on the lower side of the airfoil) and 28 panels spanwise, whereas 5040 panels are used to discretise the corresponding two-revolution length free wake surface (28 panels in the radial direction and 180 panels along the azimuth direction). The time discretisation consists of 360 time-steps per rotor revolution, corresponding to 1\({}^{\circ}\)-azimuth resolution. Aeroacoustic calculations are performed using the same blade mesh and time step used for the aerodynamic analysis. These discretization parameters are obtained as the best compromise between accuracy and computational efficiency, chosen as the result of a convergence analysis on the rotor thrust performed by gradually increasing their values. The investigation concerns the effect of the propeller blade Mach number on the near-field and far-field noise emitted by the propeller-wing system when the relative distance and phase between the propeller blades change. Two angular velocities are examined, 5000 rpm and 6730 rpm, corresponding to \(M_{t}=0.52\) and \(M_{t}=0.7\), respectively. Note that the first velocity is typical of scaled models for wind-tunnel tests, whereas the second one is consistent with standard operative conditions of real-life DEP configurations. The effects of the distance between the propeller hubs and of the propeller blade phasing are investigated considering three tip-clearances, \(TC_{1}=0.05d\), \(TC_{2}=0.125d\), and \(TC_{3}=d\) and three phase shifts (\(12^{\circ},24^{\circ}\), and \(36^{\circ}\)). All configurations are tested by keeping unchanged both the advance ratio (increasing the freestream velocity from 25 m/s to 33.65 m/s as \(M_{t}\) increases) and the thrust (equal to 189 N for each propeller), in that the proposed investigation aims at studying the noise provided by similarly loaded propellers. A preliminary trim analysis is performed for each configuration to identify the collective angle providing the required thrust. The results of the numerical investigations are shown in terms of Overall Sound Pressure Levels (OASPLs, evaluated on the basis of the first nine BPFs) on a sphere with radius \(R=40d\) for the far-field investigation and \(R=5d\) for the near-field analysis, centered at the midpoint of the two propeller hubs. Specifically, two views of the sphere are shown. Considering a right-handed frame of reference with origin at the center of the sphere, the \(z\)-axis coincident with the advancing direction and pointing forward and the \(y\)-axis directed starboard (see Figure 1), the northern view (NV) consists of the view of the sphere as seen from the positive \(x\)-axis, whereas in the southern view (SV) it is observed from the negative \(x\)-axis. First, the effect of the blade tip Mach number on the far-field noise produced by counter-rotating propellers is evaluated for the defined values of TCs. The corresponding predicted OASPL are compared in Figure 2.  
These results show that, although an OASPL magnitude difference of about 10 dB is observed to be induced by the Mach number change analyzed, the directivities remain similar for all TCs, especially in the region closer to the rotor disk.