## 3 Theoretical Formulation and Numerical Modeling
### Effect of the Locations of Strips
The vorticity contours of the wake flow for different locations of strips shown in Figure 9 may give an explanation. For the smooth cylinder, the wake consists of two symmetrical vortices, and the width of the wake is less than the diameter of the cylinder. For \(\alpha=0^{\circ}\), there is a significant change that the front stagnation is more forward than that of the cylinder without strips. It caused the position of the boundary layer separating from the surface of the cylinder to move backward. Thus, the wake is narrower than that of the smooth cylinder. In the case of \(\alpha=20^{\circ}\), the boundary layer has two energy dissipations at the leading and trailing edges of the strips. When the boundary layer is separated from the separation point, the rotation of the fluid is weaker than that of the cylinder without strips. Obviously, the significant difference appears at \(\alpha=60^{\circ}\) and \(\alpha=90^{\circ}\). The boundary layer is separated from the surface of the cylinder much forward than that of the cylinder without strips, so the wake is wider than the diameter of the cylinder, and the distance between the two vortices increased. The rotation of the fluid and the vortex size are much larger than that of the smooth cylinder. The reason for the two cases is that the position of the strips is close to the separation point of the boundary layer, and there is a large disturbance in the wake. When \(\alpha=120^{\circ}\), the upper and lower boundaries of the wake are parallel to each other, and the size of the vortex is greater than that of the cylinder without strips. For \(\alpha=150^{\circ}\), the strips did not affect the development of the boundary layer due to the installation location, so there is no significant difference compared to the smooth cylinder. Figure 8: The comparison of the lift coefficients between the smooth cylinder and cylinder with strips for Re = 55,000. Figure 9: _Cont._  
Figure 10 gives the comparison of the pressure coefficient between the smooth cylinder and cylinder with strips. Only the pressure of the upper-half cylinder is shown, as a result of the symmetrical distribution. The pressure coefficient is defined as follows:  
\[C_{P}=\frac{2P}{\rho V^{2}} \tag{14}\]  
where \(P\) is the pressure acting on the cylinder, \(V\) is the velocity of the cylinder and \(\rho\) is the density of the water. For the smooth cylinder, the pressure coefficient decreases initially and then increases. Figure 9: The vorticity contours of the wake flow for different locations of strips for Re = 55,000. At the location where the strips are installed, there is a large fluctuation in the pulsating pressure. The explanation for the phenomenon is that the boundary layer will be disturbed at the leading and trailing edges of the strips. More energy exchange takes place during the disturbances, so the pressure acting on the surface is usually greater than that of the smooth cylinder. Among six locations, the two largest increments appear in \(\alpha=60^{\circ}\) and \(\alpha=90^{\circ}\). This indicates that the disturbance of the boundary layer during the movement of the cylinder is relatively large at these two positions. When \(\theta\) varies from \(140^{\circ}\) to \(180^{\circ}\), the pressure coefficient is reduced, except for \(\alpha=0^{\circ}\), indicating that the local response of the cylinder could be suppressed by installing strips selectively. Figure 10: _Cont._  
Figure 11 compares the velocity distribution at different moments on the axis between the smooth cylinder and cylinder with strips installed at different locations. The ordinate is the ratio of the relative velocity of the fluid to the velocity of the cylinder. It can be seen that the velocity distribution on the axis follows the trend of the smooth cylinder, although minimum values are different. The curve below the \(x\) axis corresponds to the returning flow in the wake, and the abscissa of the intersection of the curve with the \(x\) axis corresponds to the length of the wake in the recirculating zone.