*5.2. Dynamic Characteristics Analysis*

By calculating the damping coefficient, mass ratio, and Reynolds number selected in this study, it can be found that the value of Ug/Uv is much less than 1, which means that the square column FIM has a strong interaction between VIV and galloping [37]. It can be considered that galloping continuously affects the amplitude and locking area of VIV, and VIV also changes the form of galloping. In order to better understand FIM and flow structure, this paper introduces four kinds representative simulation examples under various Reynolds numbers (Re = 2054/6161/16,430/20,537). This includes four FIM regions: VIV initial branch, VIV upper branch, VIV-galloping transition, and galloping. These vortex patterns, at different Reynolds numbers, are analyzed. In addition, the effects of the amplitude and frequency responses for square columns are discussed as well.

This paper presents the instantaneous vorticity diagram, velocity cloud diagram, and velocity streamline of the square column at different wind speeds (Reynolds number), including the vortex shedding at the highest and lowest points of the structure. In those figures of the vortex pattern, the red vorticity is positive, which means counterclockwise rotation; the blue vorticity is negative, which means clockwise rotation. There are two vortex shedding modes: 2S mode (in one cycle, two vortices shed from opposite sides

of the cylinder) and 2P mode (in each cycle, two pairs of vortices shed from each side of the cylinder). As the wind speed changes, the vortex shape can change, and the branch transitions affect each other [27]. In this paper, the vortex shedding mode is studied by observing the instantaneous vorticity contours around the square column oscillator, where the vorticity is defined as *ω* = *∂v*/*∂x* − *∂u*/*∂y*.

When the wind speed is 1 m/s (Re = 2054), the harvester is in the initial branch of VIV. As shown in Figure 9, the 2S model of the vortex shedding mode in the wake can be clearly observed; that is, the positive vortex and the negative vortex separate out during the vibration period. This mode is the classical von Kármán Street. In this range, as Re increases, the size of the vortex is larger than when Re is low, which is consistent with the numerical results given by Ding et al. [27].

**Figure 9.** Vortex pattern and Velocity distributions contour (Re = 2054).

As the wind speed increases to 3 m/s (Re = 6161), the number of vortices shed in each oscillation cycle increases. In each oscillation period, 6 vortices shed from the square column and shed in the 2P + 2S mode of two pairs of vortices and one single vortex, as shown in Figure 10. This vortex pattern is known as Quasi-2P, which means two pairs of vortices shedding per cycle.

**Figure 10.** Vortex pattern and velocity distributions contour (Re = 6161).

When the wind speed reaches 8 m/s (Re = 16,430), the vortex pattern in the VIV-galloping transition zone is formed. As shown in Figure 11, ten vortices shed from the square column in an oscillation cycle. According to previous research by Ding [38], a similar near-wake vortex structure was captured as 4P + 2S. There are still many controversies about the identification method of multiple vortices shedding from square columns. At the highest and lowest points of vibration, a single vortex appears first. In the next downward or upward process, a total of two pairs of vortices appear.

**Figure 11.** Vortex pattern and Velocity distributions contour (Re = 16,430).

When the wind speed is 10 m/s (Re = 20,537), the FIM of the square column is in the fully developed galloping state zone. As shown in Figure 12, there is a fundamental difference between galloping vibration and VIV. When entering the galloping state, a total of 14 vortices will shed in the manner of 6P + 2S in each oscillation period, two of which are separate vortices. According to the results of previous studies [39], it can be found that the vortex of the square column is at high Reynolds number. The type is *np* + 2S, where *n* represents the number of *p* and increases with the increase of Reynolds number or flow rate. The 2S vortex appears at the highest point and the lowest point of the displacement and falls off respectively.

**Figure 12.** Vortex pattern and Velocity distributions contour (Re = 16,430).

Within a certain wind speed range (as shown in Figures 13–16), the lift coefficient decreases with the increase of wind speed. When the wind speed is relatively small, the lift coefficient becomes a regular oscillation under low Reynolds number [40]. In the initial stage of VIV, comparing the lift coefficient and the displacement graph, the lift coefficient and the displacement performance are highly consistent. The displacement curve and lift coefficient curve are transformed by FFT to obtain the frequency spectrum. In Figure 15b, it can be seen that both the displacement spectrum and lift spectrum have multiple peak frequencies and that they maintain good consistency and their main frequency is close to the natural frequency. When galloping occurs, the dominant frequency of the displacement spectrum decreases, and the lift spectrum frequency is more obviously controlled by the vortex deflation frequency. In Figure 16b, when the square column enters the state of full galloping, it can be found that the proportion of low frequency in the displacement spectrum increases. The increase in its proportion is also an important signal for the full development of galloping. When galloping occurs, the frequency response of the square column is obviously different; its vibration is hardly controlled by the vortex breakaway frequency and the main frequency is always lower than the natural frequency, which also leads to the occurrence of large vibrations.

**Figure 13.** (**a**) Amplitude and lift coefficient versus time; (**b**) Frequency spectrum (Re = 2054).

**Figure 14.** (**a**) Amplitude and lift coefficient versus time; (**b**) Frequency spectrum (Re = 6160).

**Figure 15.** (**a**) Amplitude and lift coefficient versus time; (**b**) Frequency spectrum (Re = 16,430).

**Figure 16.** (**a**) Amplitude and lift coefficient versus time; (**b**) Frequency spectrum (Re = 20,538).

Through the numerical simulation results, as the wind speed increases (the Reynolds number also increases), although the lift coefficient decreases, the lift applied to the two sides of the bluff body continues to increase. The maximum displacement of the system also increases, which causes an increase in the instantaneous stress of the piezoelectric thin film. Combined with the results of the output power of the energy harvester, it can be found that when the Reynolds number is low, the bluff body's vibration amplitude and frequency are low, so the harvesting efficiency is at a low level. However, when the incoming wind speed increases and begins to enter the galloping state, the coupling effect of VIV and galloping can accelerate the onset of galloping and increase the displacement response of the bluff body at low and medium wind speeds. After entering full galloping, the amplitude of the entire system will increase significantly, and the output power will be increased. In addition, compared with VIV, the vibration of the square column in the galloping process is unstable, and its amplitude will increase with the increase of the incoming flow velocity. This means that the kinetic energy of the fluid captured by the square column FIM harvesting system will continue to increase with the increase of the incoming wind, and there is no limit [41]. Therefore, it can be seen that the square cylinder has obvious advantages in the application of FIM energy conversion, especially under high Reynolds number. The form of wind-induced vibration is closely related to the damping, elastic stiffness, and mass of energy harvester, which is used as the power source of energy collection. The harvesting efficiency of the flexible structure can be further improved only through analysis of its dynamic characteristics and reasonable design.
