# Computational Modelling of Rectangular Sub-Boundary Layer Vortex Generators
## 4 Results
Figure 12 shows the normalized peak vorticity \((\omega_{\rm x,max}\cdot\rm H)/U_{\infty}\) development of the three different heights of the VGs, which in fact represents the decay of the vortex along the streamwise direction. As expected, the vortex decay seems to follow an exponential law in all cases. At the closest positions to the VG, the largest value corresponds to the lowest VG height of 4.16 mm, but that trend is turned around at the distance of 50H in which the highest value is that corresponding to the case with a 6.25-mm vane height. The values of the peak vorticity of the different VG cases almost overlap each other, and consequently, no significant variation was found. By observing the location of the vortex center as a function of the downstream axis \(x\), one can determine the path of the primary vortex in both lateral (\(z\)) and vertical (\(y\)) directions. The calculation of the vortex center is made by finding the maximum streamwise vorticity point in the corresponding downstream plane position; see [PERSON] et al. [32] and [PERSON] et al. [21]. Figures 13 and 14 show a comparison between the trajectories described by the primary vortex generated by the different VG heights. Streamwise coordinates and both lateral and vertical coordinates are normalized by the baseline VG height H. As illustrated in Figure 13, the lateral path of the three cases tends to follow the direction in which the device is inclined, and the trajectories roughly collapse in the streamwise plane positions studied. The vortex paths in the vertical direction are represented in Figure 14. They collapse in the near wake, from the formation of the vortex up to approximately the downstream plane 18H. As the distance increases far away in the streamwise direction, the higher vortex trajectory corresponds clearly to the higher VG H\({}_{1}\) = 6.25. In the far wake positions, a clear gap in the vertical paths is visible between the three VG cases, which does not occur in the lateral paths. Figure 10: Comparison of the normalized turbulence kinetic energy (TKE) \(K/(U_{\infty})^{2}\) fields between experimental data and numerical simulations. Streamwise plane station from top to bottom: 5H, 10H, 25H, 50H. Figure 11: Normalized streamwise velocity (\(u_{x}/U_{\rm so}\)) profiles for the 4.16-mm, 5-mm and 6.25-mm VG heights at the 10H, 25H and 50H downstream positions. Figure 7:0 The normalized peak vorticity development for the different VG heights. The WSS is the result of friction within the fluid and between the fluid and the walls and is related to the fluid viscosity. The magnitude of the WSS is dependent on how fast the velocity increases when moving away from the wall. The wall shear stress, \(\tau_{\omega}\), can be determined by Equation (3):  
Figure 7:1  
where \(\mu\) is the dynamic viscosity, \(u_{x}\) is the axial flow velocity and \(y\) is the distance normal to the wall. The units in IS are Pa. Figure 15 represents the comparison of the wall shear stress for the three analyzed VG geometries on a flat plate. Data were extracted from the VG trailing-edge up to the end of the domain and at a spanwise coordinate of z = 0. A dependency between the vane height and the wall shear stress produced in the VG wake is clearly seen in Figure 7:2 The largest value of the WSS is reached by the highest VG H\({}_{1}\) = 6.25 mm. According to [PERSON] [33], transferring momentum towards the near wall region means that the velocity is increased, which leads to an increase in the wall shear stress. In the study of [PERSON] et al. [34] on an airfoil with micro-VGs, it was demonstrated that an increase of the WSS resulted in a lift increasing. Figure 7:3 The development of the vortex center in the vertical direction. Figure 7:4 The development of the vortex center in the horizontal direction.