## 3 Results and Discussion
### CFD Analysis of Flow and Thermal Fields in Optimized Geometries  
In this section, to illustrate the performance improvement mechanisms, we compare the local flow and thermal fields of the optimized shapes obtained using the RSM at a wind speed of 1.5 m/s with those of the reference shape. Figure 8 shows the flow fields viewed from the surface near the top and bottom of the fin and from the side. For all three shapes, the boundary layer detaches and re-attaches as the flow passes through the slit in the general area. Beyond the forward stagnation point, the laminar boundary layer develops, leading to separation and the observation of a wake and vortices. However, differences were observed among the three shapes in the size of the forward stagnation point and separation zone, as well as in the mixing characteristics of the flow over the upper and lower surfaces through the slit. First, in the reference shape (Figure 8a), the incoming flow at the front of the first tube row develops a laminar boundary layer at the forward stagnation point, and the main flow passes over the tube at a high velocity of 5 m/s. The stagnation point forms at the center of the tube, after which separation occurs, causing the main flow to split upward and downward as it strikes the slit. The flow passing over the upper surface develops a separation layer, while the flow that has been diverted to the lower surface affects the slit of the next row. A similar flow pattern is observed at the second tube row, but after that, the absence of fins leads to wake formation and the development of vortex layers. The flow passing through the slit fluctuates in the y-direction, creating a velocity difference of more than 1.5 m/s. Figure 7: Comparison of optimized wavy-slit fins for two objective functions with the reference geometry. (**a**) Reference geometry. (**b**) Optimized fin with reduced pumping power. (**c**) Optimized fin with volume goodness factor. In the reduced-pumping-power shape (Figure 8b), the greatest amount of flow enters from the front of the first tube row, and a significant impact can be observed at the stagnation point on the lower surface. In this case, the separation zone develops more significantly than in the reference shape, and the two bifurcated main flows dominate the lower surface, which results in an uneven flow pattern. Strong vortices form after the second tube row, and the flow entering through the slit is less than that of the reference shape, causing the flow velocity on the lower surface to slow down to 0.5-1 m/s. In the Gv shape (Figure 8c), due to the increased slit length (\(L\)) compared to the reference shape, the amount of flow entering over the first row of tubes decreases, and the development of vortices and the laminar boundary layer at the stagnation point are delayed. The main flow passing through the slit has a reduced separation layer and, due to the lower velocity, moves along the lower surface. Additionally, the increased slit height (\(H\)) leads to a greater height in the region between slits, allowing for the development of vortices and the laminar boundary layer. Figure 8: Flow comparison around optimized wavy-slit fins for two objective functions and reference geometry. (**a**) Reference. (**b**) Optimized fin with reduced pumping power. (**c**) Optimized fin with volume goodness factor. relatively uniform flow. After the second row of tubes, wake flow continues, and the flow passing through the slit forms a uniform pattern over both the upper and lower surfaces. The heat transfer coefficients on the fin surfaces have been compared in Figure 9 to verify the heat transfer distribution of the reference shape and optimized shapes at an inlet velocity of 1.5 m/s. In the velocity distribution of all three shapes, recirculation zones form near the tube after flow separation, resulting in areas of low heat transfer near the tube in the heat transfer coefficient distribution. Additionally, since the flow is divided by the slits, the heat transfer is locally higher at the front of each slit.