## 4 Results and Discussion
### Two-Dimensional Examples
#### 4.1.4 Effect of Reynolds Numbers on Body Shapes and Drag Values  
For 2D objects placed in uniform flow with velocity \(\mathrm{u}_{0}=1\) in X-direction, we investigate the profiles of objects with minimum drag. We employ the objective function \(J_{1}\) and \(J_{2}\), and apply the constraint that the area fraction of an object is a lower bound on the fluid structure in a 2D model. We also apply the constraint that the initial structure of the object is fixed at the origin so that the body location is anchored during the process of optimization. The results of two different objective functions for drag minimization at various Reynolds numbers are presented in Figures 6 and 7. The overall optimized shape of the 2D model is nearly centered in the design domain with a shuttle shape. When the Reynolds number tends to zero, we express it as \(\mathrm{Re}=0\) for convenience. Therefore, \(\mathrm{Re}=0\) is the ideal state, which is an approximate simulation that expresses the ignorance of inertial forces. For the results with \(\mathrm{Re}\) close to 1, they have basically the same shape profile. When the \begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Re = 300** & **Mesh Size** & **Re = 0** & **Re = 1** & **Re = 20** & **Re = 50** & **Re = 100** & **Re = 150** & **Re = 200** & **Re = 250** \\ \hline \(h=1/10\) & 17.5045 & 17.5594 & 26.5496 & 36.7503 & 48.5749 & 57.7612 & 65.4489 & 72.4931 & 78.7289 \\ \(h=1/30\) & 17.4802 & 17.4931 & 26.5447 & 36.8043 & 48.5007 & 57.6024 & 65.3731 & 72.2651 & 78.6652 \\ \(h=1/60\) & 17.4328 & 17.4908 & 26.5356 & 36.7716 & 48.5069 & 57.5899 & 65.3632 & 72.2081 & 78.6638 \\ \hline \end{tabular}
\end{table} 
Table 1: Table of drag values for different mesh densities corresponding to various Reynolds numbers. Figure 5: Convergent histories of the optimization objective and the area constraint, including the snapshots for the evolution of the material density and sensitivity distribution. Re is less than 200, the structure is nearly symmetrical up and down and the two sides of the mid-section are smooth. As the Reynolds number continues to increase, the profile grows laterally, the body shape gradually flattens, and the cone angles at both ends become smaller, while the lower edge changes more than the upper edge and the trailing section are relatively slender. The generated structures become so-called streamlined and have the tendency to go beyond the design domain. The above results are approximate to those obtained by [PERSON] using the phase field method with the different objective function [52]. Comparing the optimization results corresponding to objective functions 1 and 2 with the same area fraction of 0.9 in terms of the shape shown in Figure 6, the latter have a  
Figure 6: 2D optimal body shapes for drag reduction corresponding to two different objective functions at various Reynolds numbers. In the above illustrations from (**a**-**i**), the upper graph corresponds to \(J_{1}\) and the lower one corresponds to \(J_{2}\). The area fraction \(v_{0}\) is set to 0.9 in both cases. Figure 7: Comparison of the drag values for two objective functions at different Reynolds numbers in 2D problem. When the Reynolds number is less than 25, the red line partly overlaps and covers the black line. smaller aspect ratio and the structures are all centered in the design domain; in terms of drag values shown from Figure 7, at Reynolds numbers less than 25, the drag values of both are approximate, and when greater than 50, the objective function with convection term corresponds to a smaller value. It means that as the Reynolds number increases, the effect of convection term on the drag values is also greater and should not be ignored. Therefore, the results of \(J_{1}\) and \(J_{2}\) are not same. The reason for the difference is also caused by adding the convection term. Compared with \(J_{1}\), \(J_{2}\) corresponds to a result that is wider in shape and smaller in drag value.