## 4 Application of FODNOPP for Transonic Counter-Rotating Propfans at Cruising
### Three Critical Variables in Noise Calculation for Profans with Transonic Helical Tip Speed  
FODNOPP effectively computes the three essential variables: (1) the roots \(\tau\) of the retarded-time equations of acoustic source points, each at a helical speed that is either subsonic or transonic; (2) the intersection curves \(\Gamma\) of the blade surfaces with the collapsing sphere at a specified \(\tau\), within the range \([0,t]\); and (3) the algebraic term \(\partial p_{B}/\partial\sigma_{b}\) in Formu 3. Figure 8 illustrates the solution schematics of the three variables, corresponding to the noise calculation at \(t=5.75T\), where \(T=8.280\) ms, as calculated in Section 4.1. As inferred from the periodic nature of the acoustic pressure time histories, the acoustic pressures at  
Figure 7: Overall noise of counter-rotating propfans CRPFc at cruise altitude \(H=11\) km, flight Mach number \(M=0.8\), at observer point \(O_{1}\) (0, 7.28 m, 0) separately using code a1 and code b for front (**a1,b1**), rear (**a2,b2**), and both profans (**a3,b3**) of CRPFc: (**a1–a3**) Acoustic pressure time histories; (**b1–b3**) Spectra (RPF = 120.77 Hz). \(t=5.75T\) are equivalent to those at the moments \(t/T=0.75\) in Figure 7(a1,a2) and \(t/T=0.25\) in Figure 7(a3). The source nodes, P\({}_{1}\) and P\({}_{2}\), located on the suction surface of blade F1c, are identified with coordinates (0.24, 0.8) and (0.62, 0.8), respectively, in terms of \((x/c,r/R)\). The retarded-time equations of nodes P\({}_{1}\) and P\({}_{2}\) are determined as \(f_{1}(\varphi)=0\) and \(f_{2}(\varphi)=0\), respectively. Results reveal that the two equations have one and three real roots, respectively, as illustrated in Figure 8a. The helical Mach numbers, denoted as \(M_{\text{Hel}}\), for P\({}_{1}\) and P\({}_{2}\) are 1.02. Here, for the airfoil section at a relative radius \(r/R\), the \(x\) denotes the chordwise distance from a node to the leading edge, and \(c\) denotes the chord length. At the retarded time \(\tau=0.7T\) for \(t=5.75T\), in CRPFc, only the blades F1c, F2c, F3c, and F4c of the front propfan, along with the blades R2c and R3c of the rear propfan, intersect with the collapsing sphere surface, as illustrated in Figure 8b. The sphere is centered at \(O_{1v}\) (0, 7.28 m, 0 + \(v_{z}t\) m) = \(O_{1v}\) (0, 7.28 m, 11.23 m) = \(O_{1v}\) (0, 11.70D, 18.05D) and has a radius \(r_{\text{s}}=c_{0}(t-\tau)=12.33\) m = 19.82D. In Figure 8b, on the intersection curve \(\Gamma\) of the suction surface of blade F1c with the sphere surface, the node P is selected as the node most adjacent to the airfoil section at \(r/R=0.8\). Here, 'node' denotes the point where one grid line of the blade surface intersects with the sphere surface or its normal plane. To compute \(\partial p_{B}/\partial\varphi_{b}\) at P, it is essential to first determine the intersection curve \(\Gamma_{\text{P}}\), as shown in Figure 8c, on which \(p_{B}\) and \(\varphi_{b}\) are the variables at each node. Here, \(\Gamma_{\text{P}}\) is the intersection curve of plane A with F1c's suction surface. Plane A is a normal plane of F1c's suction surface at P, defined by the tangent  
Figure 8: Schematic representations of three key variables solved using FODNOPP: (**a**) Roots of transonic retarded-time equations; (**b**) Intersection curves of counter-rotating propfans with collapsing sphere; (**c**) Algebraic term \(\partial p_{B}/\partial\varphi_{b}\) in Formu 3.  
vector \(\stackrel{{\rightarrow}}{{b}}\) and the normal vector \(\stackrel{{\rightarrow}}{{n}}\) of F1c's suction surface at P. \(p_{B}\) denotes the pressure at each node of the curve \(\Gamma_{\text{P}}\) in the blade-fixed coordinate system, while \(\sigma_{b}\) denotes the length of the curve on \(\Gamma_{\text{P}}\), extending from the starting node to each node on \(\Gamma_{\text{P}}\). Tangent vector \(\stackrel{{\rightarrow}}{{b}}\) is equal to \(\lambda\stackrel{{\rightarrow}}{{M}}_{t}\) plus \(\lambda_{1}\stackrel{{\rightarrow}}{{t}}_{1}\).