## 1 Introduction  
Aerodynamic noise, primarily discrete noise, associated with counter-rotating propfans, limits their application in civil aviation. Numerical prediction of this noise facilitates the development of noise reduction strategies during the early design phase of these propfans, thereby minimizing extensive noise testing later and reducing development costs. Moreover, aircraft equipped with counter-rotating propfan engines can comply with air-worthiness certification noise standards and mitigate noise effects on aircrews, airport personnel, and surrounding communities. Thus, a method to predict the discrete noise of counter-rotating propfans under varying flight conditions, such as take-off, climb, and cruise, is urgently needed in China. Aerodynamic discrete noise from open rotors can currently be computed using computer codes from abroad or commercial software. In 1992, NASA Langley Research Center (LaRC) developed the Advanced Subsonic and Supersonic Propeller Induced Noise prediction program (ASSPIN) [1, 2], using Formu 1A [3] and Formu 3 [1], time-domain solutions of the Ffowcs Williams-Hawkings (FW-H) equation, as derived by Farassat. Previously, based on the two formulations along with Formu 1-RTE [4] and Formu 1-Sph [4], also from Farassat, LaRC had developed WOPWOP [3], as well as two modules of ANOPP [5], and DFP-ATP [6]. Formu 1-RTE and Formu 1A are optimally designed for predicting noise from open rotors at subsonic helical tip speeds, whereas Formu 1-Sph and Formu 3 are applicable for noise prediction across subsonic, transonic, and supersonic helical tip speeds. In 2003, the German Aerospace Centre (DLR) developed APSIM [7], a program for subsonic open rotor aerodynamic noise prediction, based on the FW-H equation and the Kirchhoff formula, using an integral method and featuring a permeable surface function. For analyzing noise from open rotors with helical tip speeds near or beyond the speed of sound, numerical simulation software such as NUMCA FINETM/ Acoustics 8.1 [8], the acoustic module of the software ANSYS Fluent 2021 R1 [9], and Actran 2020 [10] are applicable. However, these tools utilize a permeable surface integral method based on acoustic analogy, capturing noise from the turbulence within the permeable surface and near the rotor. Currently, no published literature about China's domestic program development for predicting the discrete noise of counter-rotating propfans, especially those with helical tip speeds across the speed of sound, is reported. This study introduces FODNOPP, a self-developed program system for predicting the discrete noise of both single- and counter-rotating open rotors with subsonic/transonic/ supersonic helical tip speeds. Its accuracy has been verified through four diverse case studies, along with a specific application for evaluating the cruise noise of the counter-rotating propfans with transonic helical tip speeds. FODNOPP is composed of four codes: code a1, code a2, code b, and code c, each successively utilizing formulations Formu 1-RTE, Formu 1A, Formu 1-Sph, and Formu 3, derived by Farassat. Both code a1 and code a2 employ two methodologies from the literature [5], one for calculating the retarded-time equation roots for subsonic acoustic source points on the blade surfaces of the single rotor and the other for transonic source points, which have been further improved in this study to better accommodate counter-rotating rotors. Observer points are synchronized with the airframe movement. The input data, namely the pressure distributions on the blade surfaces, are calculated using a formulation from [11] or calculated using the nonlinear harmonic (NLH) solver of the CFD software NUMECA FINETM/ Turbo 14.2 [12]. FODNOPP calculates the discrete noise of open rotors by integrating across the solid wall surfaces of the blades, not considering the impact of the rotor flow field on noise propagation.