## 3 Verification of Accuracy of FODNOPP
### Example Verifications for Program Accuracy
#### 3.2.4 Subsonic Counter-Rotating Profans, Using Code a1 and Code b  
The overall noise generated collectively by both propfans of CRPFit at the observer point \(O_{2}\) (0, 7.28 m, 0), under the take-off conditions at an altitude \(H\) = 300 m and a flight Mach number \(M\) = 0.24, is separately calculated using code a1 and code b. The results of each are depicted in Figure 6. The comparative analysis indicates that the acoustic pressure time histories from the two codes show a similar trend. Moreover, the sound pressure levels of the dominant tones, with the frequencies up to 32 times the rotor shaft frequency, are nearly equal. In this case, while code a1 provides a more accurate and smoother acoustic pressure time history compared to code b, the method of code b involves the intricate \(\Gamma\)-curves at a specified retarded time \(\tau\) within the interval [0, \(t\)], with the possibility of presence or absence. These \(\Gamma\)-curves result from the varied intersection forms between the blade surfaces and the collapsing sphere. At a specified time \(\tau\) within [0, \(t\)], the complexity of the intersection form may necessitate further improvement in the accuracy of the \(\Gamma\)-curves, if \(\Gamma\)-curves are present. Therefore, it can be concluded that the results from code a1 and code b, in calculating the discrete frequency noise of counter-rotating propfans with subsonic helical tip speeds, are in agreement. Figure 5: Overall noise of counter-rotating propfans CRPFit at take-off altitude \(H\) = 300 m, flight Mach number \(M\) = 0.24, at observer point \(O_{1}\) (0, 0.4665 m, 0) using code a2, compared with that using NUMECA Fine\({}^{\text{TM}}\)/Turbo: (**a**) Acoustic pressure time histories; (**b**) Spectra (rotor passing frequency RPF = 163.57 Hz). Figure 6: Overall noise of counter-rotating propfans CRPFit at take-off altitude \(H\) = 300 m, flight Mach number \(M\) = 0.24, at observer point \(O_{2}\) (0, 7.28 m, 0) using code b, compared with that using code a1: (**a**) Acoustic pressure time histories; (**b**) Spectra (RPF = 163.57 Hz). The results from the four aforementioned noise examples, including two subsonic single-rotating cases and two subsonic counter-rotating cases, are subjected to a further analysis. Code a1 and code a2, based on Formu 1-RTE and Formu 1A, respectively, differ mainly in several variables within the integrands of the acoustic pressure integrals, the coefficients of these integrals, and whether or not the first-order partial differentiation with respect to the observer time \(t\) is performed. Other than these differences, the two codes employ the same methods and the same algorithms in calculating the retarded times \(\tau\) and in integrating the control variables for the acoustic pressure calculation of the rear propfan. Therefore, it is presumed that code a1 is also capable of accurately calculating the noise of counter-rotating open rotors at subsonic helical tip speeds. Due to the results calculated using code a1 and code b showing a degree of agreement regarding the noise of CRPFt under take-off conditions, code b is somewhat accurate in computing the noise of counter-rotating open rotors at subsonic helical tip speeds. Code b and code c, based on Formu 1-Sph and Formu 3, respectively, differ mainly regarding several variables within the integrands of the acoustic pressure integrals, the coefficients of these integrals, and whether or not the first-order partial differentiation with respect to the observer time \(t\) is performed. Other than these differences, the two codes employ the same methods and the same algorithms in calculating the intersection \(\Gamma\)-curves between the blade surfaces and the collapsing sphere and in integrating the control variables for the acoustic pressure calculation of the rear propfan. Therefore, it is presumed that code c is also capable of accurately calculating the noise of counter-rotating open rotors at subsonic helical tip speeds.