Effects of installation position of fin-shaped rods on wind-induced vibration and energy harvesting of aeroelastic energy converter

In order to further analyze the experimental results and discuss the mechanism of flow-induced vibration of aeroelastic structure, numerical simulations of corresponding experimental cases were carried out in this work. The flow is assumed to be two-dimensional and unsteady, and the fluid is incompressible. The flow field is solved using unsteady Reynolds-averaged Navier–Stokes equations together with the k-ω SST turbulence model [57]. The governing equations are:

$$\begin{equation}\frac{{\partial {{\bar u}_i}}}{{\partial {x_i}}} = 0\end{equation} \tag{ 1 }$$

$$\begin{equation}\frac{{\partial {{\bar u}_i}}}{{\partial t}} + \frac{{\partial \mathop {{u_i}{u_j}}\limits^{\_\_\_\_\_} }}{{\partial {x_j}}} = - \frac{1}{\rho }\frac{{\partial \bar p}}{{\partial {x_i}}} + \frac{\mu }{\rho }{\nabla ^2}{\bar u_i} - \frac{{\partial \mathop {{{u{^{^{\prime}}}}_i}{{u{^{^{\prime}}}}_j}}\limits^{\_\_\_\_\_\_} }}{{\partial {x_j}}}\end{equation} \tag{ 2 }$$

where x_i are the Cartesian coordinates, u_i are the corresponding velocity components, μ is the dynamic viscosity, ρ is the fluid density, t is the time.

The numerical calculation in this paper is based on the finite volume method by Fluent together with user-defined function. The continuity and momentum equations are solved using the pressure-based coupling algorithm. A second-order Gauss integration scheme with a linear interpolation for the face-centered value of the unknown is used for the divergence, gradient, and Laplacian terms in the governing equations. The second-order backward Euler method is adopted for time integration [47, 58]. In this work, the dimensionless time step ΔtU/D is 0.1813, which satisfies the Courant–Friedrichs–Lewy condition (ΔtU/D ⩽ t) [59].

In this work, the vibration system is simplified into a mass-spring-damping model, as shown in figure 2(b). On one hand, the width of the cantilever beam is only 1/10 of the FSR-cylinder length, so the physical model ignores the cantilever beam in the numerical calculation. On the other hand, when the FSR-cylinder reaches the maximum amplitude, the displacement of the FSR-cylinder in the flow direction is very small. Thus, the displacement in the flow direction can be ignored in the simulations. Therefore, the vibration system of this work is simplified as a classic mass-damping-spring system. The motion equation of the FSR-cylinder is

$$\begin{equation}M\ddot y + C\dot y + Ky = {F_{{\text{fluid}},\ y}}\end{equation} \tag{ 3 }$$

where M is the mass of the FSR-cylinder, F_fluid,y is the lift force acting on the cylinder, C($C = 4\pi \xi\, {f_{\text{n}}}M$) is the system damping, K($K = {\left( {2\pi {f_{\text{n}}}} \right)^2}M$) is the elastic coefficient. The system damping C is composed of structural damping and energy harvesting damping. The damping and elastic coefficient can be obtained by free attenuation experiment. The system parameters are listed in table 2. The motion equations of the FSR-cylinder can be discreted by the method of fourth-order Runge–Kutta, the detailed description of numerical solution can be found in our earlier work [58].

Figure 3 shows the computational domain and its size is 50 D× 40 D (length × height). The center of the cylinder is 20 D from the inlet boundary and 20 D from the upper and lower boundaries (Top and Down). The blocking ratio is less than 5%. The inlet boundary condition is considered as uniform velocity and the outlet is a pressure outlet. The top and down boundaries are set as symmetrical boundary conditions. And the non-slip boundary condition is applied for the cylinder.

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The numerical simulation of this study uses the overset grid technology, as shown in figure 4. The overall mesh consists of two parts, including the background mesh and the component mesh (a circular area with a diameter of 3 D) that accompanies the motion of the cylinder. The background grid is generated at first to discretize the entire computational domain, while the component grid is used to deal with the cylinder and generated on the top of the background grid. The gradient grid is used for the background mesh to improve computational efficiency. As shown in figure 4(b), the component mesh is a circular area with a diameter of 3 D, and the local mesh refinement is considered for the near wall of the cylinder. The overset grid can improve the update efficiency of mesh and avoid grid deformation, which can better capture the characteristics of the flow field. The data exchange of the overlapping parts of the grid can be realized by interpolation. In order to improve the efficiency and correctness of the numerical method, three levels of grids with different densities were studied for grid independence verification. The results of force coefficient are shown in table 2, where C_L-RMS is the root-mean-square value of the lift coefficient and C_D-m is the time-averaged value of drag coefficient. It should be noted that the definitions of lift and drag coefficients are C_L= F_fluid,y /(0.5 ρU² DL) and C_D = F_fluid,x /(0.5 ρU² DL), where F_fluid,x is the drag force acting on the FSR-cylinder. The maximum difference between the results obtained by different grids is only 1.4%, which meets requirement of the calculation accuracy. Therefore, the grids with medium densities were selected for numerical calculations in this work. In addition, the numerical simulation results have been verified by experiment, and the comparative analysis of experiment and simulation can be found in sections 4.2 and 4.3.

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The load resistance is an important factor affecting the output power of the aeroelastic energy harvester. It should be noted that the load resistance is the external resistance connected to the output circuit of the wind energy harvester, as shown in figure 2. The optimal resistance is affected by many factors, including aeroelastic structure, wind speed, vibration mode, etc. Even a slight change in the installation angle of the FSR will affect the optimal resistance. In order to obtain the optimum load resistance for this work, the influence of the load resistance on the output power of FSR-cylinder energy harvester under different wind speeds was firstly tested. Figure 5 shows the output power of the FSR-cylinder under different wind speeds and load resistances, where the wind speed range is 0–6.8 m s⁻¹ and the load resistance range is 100 kΩ–1 MΩ. The parameters of the FSR in experiments for load resistance influence study are H= 6 mm, α= 20°, and θ= 40°. It can be observed from figure 5 that the output power increases as the wind speed increases. Due to galloping of FSR-cylinder occurs, the maximum output power reaches 1.4 mW. At the same wind speed, as the load resistance increases, the output power firstly rise and then drops, and the power reaches a maximum value at 200kΩ. The optimal load resistance depends on the capacitance of piezoelectric film and natural frequency of the oscillatory system, which is determined by Gauss law [55, 60]. In this experiment, the capacitance of PZT is 70nF which can be measured by a digital capacitance meter, the natural frequency of the aeroelastic energy harvester is 10.8 Hz which is obtained by the free attenuation method, and the theoretical optimal value of load resistance is 215kΩ. It can be seen that the optimal load resistance obtained from the experiment agrees with the theoretical value. Therefore, the optimum external load resistance of other experiments in this work is considered to be 200 kΩ.

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In order to discuss the influence of the position of the fin-shaped attachment on the wind-induced vibration characteristics of the cylinder, this section analyzes the amplitude response of the FSR-cylinder under different conditions. In the experiment, the height H of FSR is 6 mm, the coverage angle α is 20°. The installation angle θ varies in the range of ±160°, and the interval is 10° in the experiment. Figure 6 shows that the dimensionless amplitudes (A/D) of the cylinder vary with wind speed at different installation angles, where the amplitude A is the average value of the maximum displacement of the cylinder over multiple vibration cycles. Figure 6(a) presents the results for the some representative cases that the interval of θ is 20° and figure 6(b) is the amplitude contour plotted based on all experimental results. Meanwhile, figure 6(a) shows the experimental results of smooth circular cylinder and the numerical results of representative cases for comparison. The numerical (CFD) results of amplitude responses match well with the experimental data, this verifies that the numerical method in this paper is effective. With increasing wind speed, the amplitude of smooth circular cylinder increases first and then decreases. When the wind speed exceeds 3.5 m s⁻¹, the amplitude of smooth cylinder is extremely low, and the vibration is completely suppressed. The oscillation response of the smooth cylinder in the tested wind speed is consistent with the VIV. For the cylinder in VIV, the amplitude is very small when the cylinder starts to vibrate. As the wind speed increases, the vibration frequency of the cylinder approaches the natural frequency, and resonance occurs, that is 'synchronization'. As the wind speed further increases, the vibration amplitude decreases rapidly and enters the 'desynchronization' region. The phenomenon of VIV has been observed in many earlier works [61, 62]. In combination with figures 6(a) and (b), compared to the smooth circular cylinder, the amplitude of the FSR-cylinder increases with increasing wind speed when the installation angle of the fin attachment is 0° < θ< 70° (Region I). For the case of θ= 60°, when the wind speed exceeds the critical velocity, the amplitude of FSR-cylinder increases sharply with the increase of the wind speed, galloping occurs and the maximum amplitude can reach 2.17 D. When the installation angle is located in 70° ⩽ θ < 120° (Region II), it can be seen that the vibration of FSR-cylinder is very weak and the amplitude is almost zero. When the installation angle is 120° < θ ⩽ 160° (Region III), the amplitude response curves of FSR-cylinder is similar to the VIV of smooth cylinders. When the wind speed reaches the critical velocity at which galloping occurs, the amplitude of the cylinder increases sharply. It is worth noting that when the installation angle is 0° or 120°, the obvious vibration can be observed only when the wind speed increases to a large value, and then the amplitude gradually increases. This phenomenon indicates that the large wind speed is required for achieving aerodynamic instability of these two structures.

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Figure 7 shows the vibration frequency ratios (f_v/f_n) of FSR-cylinder with different installation angles vary with wind speed. It should be noted that, because the frequency variation trends of different cases in the same region of installation angle are similar, in order to facilitate the analysis, some representative frequency results are selected and plotted. The results shown in figure 7 include the cases of θ = 40° in Region I, θ = 80° in Region II, θ = 160° in Region III, and the cases of θ = 120° and the smooth cylinder. In addition, the simulation results agree well with the experiments for the frequency response shown in figure 7.

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The dimensionless frequency (f_v/f_n) in figure 7 is the ratio between the main frequency (f_v) of the vibration and the natural frequency (f_n) of the system. When the vibration of smooth circular cylinder occurs, the frequency ratio first increases with the increase of wind speed, and then stabilizes around 1.0. When the wind speed exceeds 3.40 m s⁻¹, the frequency ratio further increases with the increase of wind speed. The variation trend of frequency for smooth cylinder corresponds to the VIV initial branch, synchronous region (also known as locked-in or upper branch), and desynchronized region (also called lower branch). It is worth noting that the wind range where the vibration frequency is locked at the natural frequency is the best region of energy harvesting for smooth cylindrical by wind-induced vibration, but this range is relatively narrow. For FSR-cylinder, when the installation angle is 40°, as the wind speed increases, the vibration frequency increases first and then decreases, but the amount of change is small, and the vibration frequency of the cylinder is slightly lower than the natural frequency. In conjunction with figure 6, it can be seen that galloping occurs for FSR-cylinder at θ = 40°, and the galloping is a low-frequency and high-amplitude vibration response. When θ = 80°, the frequency ratio of FSR-cylinder is maintained near 1.0 at low wind speed. However, when the wind speed exceeds 3.4 m s⁻¹, the vibration frequency drops sharply to nearly zero, the cylinder is in a stable state and has almost no vibration at this time. When θ = 120°, the vibration frequency of FSR-cylinder is similar to that of θ = 40°. The differences are that the vibration frequency of θ = 120° is smaller and the amplitude is much lower than those of θ = 40°. When the installation angle is 160°, the vibration of FSR-cylinder shows a VIV phenomenon in the range of low wind speeds. When the wind speed reaches the critical velocity, the vibration frequency of the cylinder drops sharply and falls to a value near the natural frequency. This is the phenomenon that as the wind speed increases, galloping occurs after the vibration of cylinder is desynchronized.

The experimental results of power harnessing of the aeroelastic energy harvester are shown in figure 8. The output voltages (V_rms) is the root mean square voltage which is usually used to analyze the output performance of the wind energy harvester. It is observed that the output voltage and harvested power (P_h) of the smooth cylinder first increase and then decrease with increasing wind speed. For FSR-cylinder, when the installation angle is in Region I, such as θ =20°, θ =40°, and θ =60° in figure 8, the V_rms and P_h of FSR-cylinders increase continuously with the increase of wind speed, and the maximum voltage and power reach 18.1 V and 1.645 mW at U= 6.8 m s⁻¹. The maximum power of FSR-cylinder is 42 times as high as that of a smooth circular cylinder system. In addition, when the wind speed is the same, the larger the installation angle, the greater the energy output of the FSR-cylinder. Especially when the wind speed is high, the V_rms and P_h of the case at θ =60° is much higher than that of θ =20°, as shown in figure 8. When the FSR installation angle is in zone II, such as θ =80° and θ =100°, the V_rms and P_h of FSR-cylinder are almost zero in the tested wind speed range. The energy conversion characteristics of the FSR-cylinder in this region are poor. When θ =140° and θ =160° (Region III), as the wind speed increases, the V_rms and P_h of FSR-cylinders first increase and then decrease. After reaching the critical wind speed, both the V_rms and P_h increase sharply. The energy response of the FSR-cylinder is similar to the amplitude response (figure 6(a)), because the energy output of the piezoelectric film is closely related to the oscillation amplitude and frequency of FSR-cylinder. From the above analysis, it can be known that when the installation angle is in the Region I, the FSR-cylinder will gallop at higher wind speeds, which can better convert wind energy into vibration energy, and achieve the optimum effect of energy harvesting for the aeroelastic system, especially for the case of θ = 60°.

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The above contents are the experimental results of this work. In order to analyze the mechanism of wind-induced vibration of FSR-cylinder, numerical simulation was performed for the relevant experimental cases. The following contents are the simulation results of this work. The force analysis of structure is helpful for deep understanding the phenomenon of wind-induced vibration. However, it is difficult to test the force of FSR-cylinder experimentally due to it will affect the energy output of the aeroelastic energy harvester. Therefore, this study uses numerical methods to analyze the force of FSR-cylinder. Figure 9 shows the time histories of the force coefficient of FSR-cylinder at different installation angles, where the lift coefficient and drag coefficient are C_L and C_D, respectively. The time history curves of lift and drag coefficients are obtained by numerical calculation.

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As shown in figure 9, when the installation angle is 40°, the amplitude of the FSR-cylinder is high, and the force acting on the cylinder is relatively large. The peak values and frequencies of the lift and drag coefficients increase with the increase of the wind speed. The magnitudes of lift and drag forces are close. When the wind speed is 2.7 m s⁻¹, the lift and drag coefficients change sinusoidally with time. When the wind speeds are 4.1 m s⁻¹ and 5.4 m s⁻¹, the time histories of lift and drag coefficients present multi-cycle characteristic, which contains multiple differences peaks. When the installation angle is 80°, the time histories of lift and drag coefficients show the regular sinusoidal change, but the values of C_L and C_D do not change much with the wind speed. And the drag coefficient is obviously larger than the lift coefficient when θ =80°. According to the theoretical criteria given by Den Hartog [63], the structure is stable in vibration and is not prone to instability, so the amplitude is smaller at θ =80°. When the installation angle is 120°, although the time histories of lift and drag coefficients also show a multi-cycle phenomenon, the peak values of the force coefficients are small. When θ =160°, the time histories of the force coefficients have a multi-cycle phenomenon when the wind speed is 2.7 m s⁻¹. As the wind speed rises to 4.1 m s⁻¹, the curve shows a regular sinusoidal change. And a multi-cycle trend is observed again when the wind speed increases to 6.1 m s⁻¹. It can be seen from figures 6 and 9, when the amplitude of the cylinder is large, the time histories of lift and drag coefficients present a multi-period change phenomenon; when the amplitude is small, the time-history curves of the force coefficients show a regular sinusoidal variation trend.

The near wake of the cylinder can help to analyze the response mechanism of the cylinder in wind-induced vibration. This section selects one period of vorticity contours of the FSR-cylinder with four different installation angles for analysis, as shown in figure 10. Time t corresponds to the equilibrium position of the cylinder, and T is the vibration period of the cylinder. The selected wind speeds are 2.7 m s⁻¹, 4.1 m s⁻¹ and 5.4 m s⁻¹ (except θ =160°). In combination with the previous analysis, the FSR-cylinder is in galloping state at 6.1 m s⁻¹. In order to further analyze the galloping response of the FSR-cylinder when θ =160°, the case with wind speed of 6.1 m s⁻¹ was selected in figure 10(c).

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For the installation angles of 40°, 80° and 120°, the separation points of the shear layers are located at the sharp corners of the FSRs. However, when the installation angle is 160°, the separation point of the shear layer is at the leeward position of the cylinder. It is worth noting that the separation point of the smooth circular cylinder is around 76° [64]. This phenomenon shows that the introduction of the fin-shaped attachment changes the position of the separation point of the shear layer on the cylinder, and when the installation angle is small, the separation point is fixed at the sharp corner of FSR. When the installation angle is large, the separation point moves downstream and approaches the stagnation point of the cylinder. When the installation angle is 40°, the cylinder oscillation has a large amplitude. At the three wind speeds shown in figure 10, the vortex patterns present the 2P mode [65], that is, two vortices shed from both sides of the cylinder in one vibration cycle. With the increase of wind speed, the stronger the entrapment ability of the shear layer, the greater the fluid force acting on the cylinder caused by the vortex shedding, and the vibration of the cylinder is strengthened. When the installation angle is 80°, the vortex pattern shows the 2S mode which indicates two single vortices shed per vibration cycle [65]. This pattern is similar to that of smooth cylinder, and the FSR-cylinder has a poor vibration effect for θ =80°. When the installation angle is 120°, the near-wake structure of the FSR-cylinder is also 2S mode. The difference is that the shear layer is relatively narrow and elongated at θ =120°, and the FSR-cylinder receives less lift and lower amplitude. This is because the separation point is fixed at the sharp corner by the FSR, which further causes the shear layer to be stretched, and the formation and shedding of the vortex moves downstream. For θ =160°, when the wind speeds are 2.7 m s⁻¹ and 4.1 m s⁻¹, the vortex patterns are 2S mode. However, when galloping occurs (the wind speed reaches 6.1 m s⁻¹), the vortex pattern becomes more difficult to capture, which is similar to the 2P mode (four vortex shedding in one cycle), but the vortex pattern is not fixed for this case. The reason is that the separation point is no longer fixed at the sharp corner of the FSR, but is transferred to the surface of the circular cylinder. When the wind speed is small, the cylinder first undergoes VIV, and after the wind speed reaching a critical velocity, galloping occurs. In addition, when the FSR-cylinder is in galloping, the number of vortices shedding during one vibration cycle increases, and the frequency of vortex shedding and the structural vibration frequency is no longer a simple linear relationship.