## 3 Results and Discussion
### Investigation of the Influence of VG Design Parameters  
The design variables and layout of the VG are shown in Figure 3, where the thickness \(t\) is fixed at 0.05 and the length of the upper section \(u\) is fixed at 0.2. A total of 27 CFD analyses were performed, and the variables used in each of these cases are listed in Table 6. In the analyses, the Reynolds number and angle of attack were set to 3.0 \(\times\) 10\({}^{6}\) and 12\({}^{\circ}\), respectively. The results of these case studies are presented in Table 7. Figures 12 and 13 show the vorticity contours for cases 1, 2, 3, 5, 8, 11, and 20 at locations 50% and 100% of the chord length from the leading edge, respectively. In Figure 13, the vorticity contour at the trailing edge can be compared when each parameter is changed based on case 2.  
The response surface methodology (RSM) used a LOWESS (Locally Weighted Scatterplot Smoother) second equation to find combinations that could represent the best VG effect using the three independent design parameters, layout (\(L\)), length (\(l\)), and height (\(h\)), as shown in Figure 3[41]. Four objective variables, \(C_{L}\), \(C_{D}\), and the maximum vorticity at the positions 50% and 100% from the leading edge to the trailing edge, were investigated. The correlation between the \(C_{L}\) values is shown for these cases  
Figure 12: _Cont._  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{**Cl**} & \multirow{2}{*}{**Cd**} & \multicolumn{4}{c}{**Maximum Vorticity at Position in the Direction of the Chord (1/s)**} \\ \cline{3-6}  & & & **25\%** & **50\%** & **75\%** & **100\%** \\ \hline
01 & 2.026 & 0.0285 & 10,363 & 3078 & 3033 & 1625 \\ \hline
02 & 2.010 & 0.0302 & 10,289 & 3370 & 3054 & 1766 \\ \hline
03 & 1.974 & 0.0627 & 13,641 & 3773 & 2535 & 1200 \\ \hline
04 & 2.013 & 0.0293 & 13,554 & 3245 & 2472 & 1154 \\ \hline
05 & 1.990 & 0.0327 & 16,308 & 3712 & 2374 & 975 \\ \hline
06 & 1.978 & 0.0344 & 14,352 & 3912 & 2449 & 1128 \\ \hline
07 & 2.014 & 0.0291 & 13,926 & 3205 & 2335 & 1128 \\ \hline
08 & 1.996 & 0.0319 & 16,066 & 3698 & 2338 & 1101 \\ \hline
09 & 1.978 & 0.0348 & 15,392 & 3675 & 2233 & 1112 \\ \hline
10 & 2.113 & 0.0289 & 12,809 & 3092 & 2497 & 1205 \\ \hline
11 & 2.001 & 0.0315 & 13,846 & 3471 & 2446 & 1061 \\ \hline
12 & 1.991 & 0.0332 & 12,298 & 2876 & 1965 & 969 \\ \hline
13 & 1.993 & 0.0306 & 13,626 & 2890 & 1859 & 830 \\ \hline
14 & 2.004 & 0.0311 & 14,949 & 3463 & 2423 & 1141 \\ \hlineFigure 12: Vorticity contours at the midpoint of the chord: (**a**) Case 1; (**b**) Case 2; (**c**) Case 3; (**d**) Case 5; (**e**) Case 8; (**f**) Case 11; (**g**) Case 20.  
Figure 13: Cont.  
Figure 14: Cont.  
Figure 13: Vorticity contours at the trailing edge: **(a)** Case 1; **(b)** Case 2; **(c)** Case 3; **(d)** Case 5; **(e)** Case 8; **(f)** Case 11; **(g)** Case 20.  
Figure 14: Surface plots of \(\rm C_{L}\) showing the effect of three parameters on the lift coefficient: (**a**) \(l=1.5\); (**b**) \(l=2.0\); (**c**) \(l=2.5\); (**d**) \(L=1.5\); (**e**) \(L=1.75\); (**f**) \(L=2.0\); (**g**) \(h=0.75\); (**h**) \(h=1.0\); (**i**) \(h=1.25\).  
Figure 15: _Cont._