## 3 Verification of Accuracy of FODNOPP
### Rotor Geometry Model and Observer Point Coordinates  
Figure 1 depicts the propeller PROP [11] and the counter-rotating propfan models CRPFt and CRPFc [13] along with the pressure distributions on their blade surfaces. Among them, only PROP and CRPFt are utilized in the code verifications. In 2016, based on the compressible lifting surface theory, [PERSON] [13] developed an inverse design problem approach and achieved the design of CRPFc at a cruise point with an altitude \(H=11\) km and a flight Mach number \(M=0.8\). Now, this paper meticulously recalibrates the orientations of the front and rear propfans of CRPFc by resetting the pitch angles of two airfoil sections, at a relative radius of \(r/R=0.75\), to \(37.9^{\circ}\) for the front propfan blades and to \(37.4^{\circ}\) for the rear propfan blades, respectively. Via this calibration, CRPFt is derived, specifically for the noise prediction under the take-off conditions at an altitude \(H=300\) m and a flight Mach number \(M=0.24\). The two pitch angles chosen correspond to those employed in the wind tunnel experiments by Sullivan [14], in which the counter-rotating propfan model F7A7 at an incoming Mach number \(M=0.24\) is tested. The pressure distributions on the blade surfaces of PROP are directly derived from the specific load distribution equation presented in [11]. In contrast, for CRPFt and CRPFc, the time-dependent pressure distributions on the surfaces of each blade are reconstructed from the approximate unsteady solutions of the flow fields simulated by the NLH solver. The blades of the front propfan have one distinct time sequence, while the blades of the rear propfan have another distinct time sequence. We take the time interval between the maximum moment and the minimum moment in the discrete-time sequence to be equal to the change cycle of the pressure of the blade surface. This pressure reconstruction process enables capturing the pressure distributions at each moment within these respective time sequences for each propfan. The pressure distributions on the blades of CRPFt and CRPFc, at \(t=0\) s, are shown in Figure 1b,c, in which the plain blades are the reference blades F1t, R1t, F1c, and R1c. (F, R, B, 1, t, and c denote, respectively, the blade of front propfan of either CRPFt or CRPFc, blade of rear propfan of either CRPFt or CRPFc, blade of PROP, first blade, take-off conditions, and cruise conditions). In the flow field simulations, the coordinate system of the model may differ from that used in the acoustic calculations. In the software NUMECA FINET(tm)/Turbo 14.2, the incoming flow is oriented along the \(+z\) axis, so the flight direction of the rotor is along the \(-z\) axis, as illustrated in Figure 1b,c. In the orthogonal coordinate systems \(Oxyz\), shown in Figure 1a-c, the geometric centers of PROP, the front propfan of CRPFt, and the front propfan of CRPFc are all situated at their respective origin \(O\), and their rotor discs lie within their respective \(Oxy\) plane. Additionally, the pitch change axes of the reference blades B1 of PROP, F1t of CRPFt, and F1c of CRPFc coincide with their respective \(+y\) axis, while the pitch change axes of the reference blades R1t of CRPFt and R1c of CRPFc are parallel to their respective \(+y\) axis. In the noise calculation using FODNOPP, the flight direction of the rotor is set along the \(+z\) axis, and, at time \(t=0\) s, the pitch change axes of the reference blades B1, F1t, and F1c coincide with their respective \(+y\) axis. The parameters for the noise calculations, including the flight speeds, rotational speeds, and noise observer coordinates for PROP, CRPFt, and CRPFc, are detailed in Table 2. The observer coordinates, set at \(O\) (\(0,7.28\) m, \(0\)), align with those employed in [11], ensuring consistency in the comparative analysis.