## 3 Theoretical Formulation and Numerical Modeling
### Effect of The Shapes of Strips  
In order to investigate the influence of the shapes of strips on the wake flow, the strips with a thickness of 2 mm were selected and installed at \(\alpha=90^{\circ}\). Four different shapes were discussed, namely the arc, triangle, rectangle and trapezoid. The cylinder with a diameter of 50 mm (\(\delta/D=0.04\)) has a starting speed of 1.1 m/s, and the starting is almost instantaneous. The focus of this paper is on the initial stage of transient flow behind the cylinder, so only the results before the dimensionless time, T = 3, are given. The effect of the shapes of strips on the drag coefficient, the lift coefficient, the pressure coefficient, the velocity distribution on the axis and the vorticity contours for the wake flow were analyzed. Figure 12 depicts the drag coefficient of the cylinder with strips of different shapes. It can be seen that the drag coefficient of the cylinder with strips of different shapes are all higher than that of the smooth cylinder. Among four different shapes, the drag coefficient of the arc strips is significantly the smallest. The other three curves have the same trend that increase first and then decrease. A peak appears in the curves at T = 1.8. The maximum values of rectangular strips, trapezoidal strips and triangular strips are 2.79, 2.45 and 2.21, respectively. The results are consistent with the general rules. Energy dissipation occurs when the boundary layer passes through the steps of the strips. The arc-shaped strips have no step, so the drag of the arc-shaped strips is the smallest. Moreover, the triangular strips have one step, so the drag is smaller than that of the rectangular strips and the trapezoidal strips with two steps. Figure 13 shows the lift coefficient of the cylinder with strips of different shapes. The results show that the shapes of the strips have a great influence on the lift coefficient. When the shape of the strips is an arc, the lift coefficient is gradually reduced after T = 2, compared with the smooth cylinder. A new phenomenon has emerged in the lift coefficient curves of the other three shapes. The oscillation amplitude of the lift coefficient is increased, and some peaks and valleys appear as a result of the energy exchange occurring at the steps. The difference is that both the rectangular strips and the trapezoidal strips have a lower lift coefficient than that of the cylinder without strips, while the triangular strips have a higher lift coefficient than that of the smooth cylinder. Figure 12: The comparison of the drag coefficients between the smooth cylinder and cylinder with strips of different shapes for Re = 55,000. Figure 14 shows the vorticity contours of the wake flow for different shapes of strips. After the cylinder is installed with strips of different shapes at \(\alpha=90^{\circ}\), the boundary layer is separated earlier from the surface of the cylinder. When the arc-shaped strips are installed, the boundary layer is attached to the surface of the cylinder again, after passing through the strips and then separating from the surface of the cylinder. While the strips of other three shapes are installed, the boundary layer is separated from the strips directly. The arc-shaped strips have little effect on the rotation of the fluid, but they have a significant influence on the wake. 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 smooth cylinder. A big difference happens in the rectangular strips, the trapezoidal strips and the triangular strips. A pair of primary vortices is formed near the separation point from T = 1. The rotation of the fluid and the width of the wake are larger than that of the smooth cylinder all the time. At T = 3, we can see that many secondary vortices are formed from the separation point and then sequentially absorbed by the pair of primary vortices. Figure 13: The comparison of the lift coefficients between the smooth cylinder and cylinder with strips of different shapes for Re = 55,000.