## 2 Numerical Modeling
### Physical Model Details
#### 2.1.3 Periodic Modeling of Cooling Channels  
The cooling channels in both Schemes A1 and B1 are arranged periodically in the direction of the rudder axis, excluding the inlet liquid- and outlet liquid-collecting cavities. This arrangement facilitates analysis and design using representative cells with periodic boundary conditions. Furthermore, considering that the aerodynamic heating of the rudder is more severe in the forward half compared to the backward half, this study solely modeled the forward half of the simplified rudder. Figure 5: The structural layout of actively cooled Scheme B1. The physical model depicted in Figure 6 illustrates a representative cell of the cooling channel in Scheme A1, where the \(x\) direction is the chord direction, and the \(z\) direction is the rudder axis direction (also the gravity direction). It consists of an entrance zone, a heating zone, and an exit zone. Both the entrance and exit zones have lengths of 50 mm and are designed to be adiabatic. The purpose of the entrance zone is to ensure complete fuel flow development before heating occurs, while the inclusion of the exit zone aims to minimize any computational impact caused by the outflow boundary conditions. The heating zone corresponds to the forward half of the simplified rudder and consists of three segments: a horizontal segment with a length of 93.75 mm, an inclined segment with a half-inclination angle of 7.41\({}^{\circ}\) (as shown in Figure 3), and a leading-edge segment characterized by the outer chamfering radius R1 = 2.25 mm, the inner chamfering radius R2 = 1 mm, the channel outer radius R3 = 2.5 mm, as well as the channel inner radius R4 = 0.5 mm. From a cross-sectional perspective, the representative unit of the cooling channel consists of a solid wall and fluid cavity. The wall can be further divided into the skin outer wall, the skin inner wall, and the gap wall. It should be noted that the cross-sectional dimensions undergo continuous variation in the leading-edge segment of the heating zone (not depicted in the figure) while remaining consistent in both the horizontal and inclined segments of the heating zone, as well as in the entrance/exit zones. These non-variable cross-section dimensions include skin inner-/outer-wall thicknesses of 0.5 mm, a gap wall thickness of 1 mm (indicating a channel spacing of 2 mm), and a cavity area measuring 1 mm \(\times\) 2 mm. The physical model of the representative cell of the cooling channel in Scheme B1 is illustrated in Figure 7. It comprises the entrance zone, the core channel zone, the heating zone, and the exit zone. Notably, the core channel zone is located near the mid-plane (\(x\)-\(z\) plane) within the rudder and represents the main difference between Scheme B1 and Scheme A1. It connects to the entrance zone at one end and to the leading-edge segment of the heating zone at the other. Furthermore, the heating zone is still composed of a horizontal segment, inclined segment, and leading-edge segment. However, there are variations in terms of the inner chamfering radius (R2) (0.25 mm) and channel inner radius (R4) (0.25 mm) for the leading-edge segment compared to those in Scheme A1. All other aspects including the outer chamfering radius (R1), channel outer radius (R3), as well as the geometry of the entrance/exit zones remain unchanged from those utilized in Scheme A1. Lastly, the dimensions of those non-variable cross sections are the same as in Scheme A1, except that the wall names are different, changing from \"skin inner/outer wall\" to \"core side wall\". Figure 6: The representative cell of the cooling channel in Scheme A1. The boundary conditions (BCs) of the representative units of the cooling channel in Schemes A1 and B1 are shown in Figure 8. In Scheme A1, mass flow inlet and pressure outlet BCs are defined, while heat flux BCs are applied to the surface of the skin outer wall within the heating zone. The surfaces of the gap wall are designated as periodical, translational BCs.