*2.2. CFD Mathematical Model*

As shown in Figure 2, the harvester system can be mathematically simplified as a SDOF system. Taking the middle section of the flexible structure as the simulation research object, it is assumed that the wind field is incompressible and the effect of structural torsion and the inverse piezoelectric effect in the system can be reasonably neglected.

**Figure 2.** 2-dimensinal dynamic model.

Thus, the dynamics of the harvester body over a unit length can be depicted by [22]:

$$\mathbf{M}\ddot{\mathbf{y}} + \mathbf{C}\dot{\mathbf{y}} + \mathbf{K}y = F\_{\mathbf{y}} \tag{1}$$

where *y* denotes the lateral displacement of the square column, *M* is the mass per length, *Fy* is the force acting on the square structure in y direction, *C* is the mechanical damping coefficient, and *K* is the stiffness of the vibration model.

The details of the equation can be referred to in Barreiro-Gill et al. [23].

According to the quasi-steady theory [24], wind force can be expressed as function of wind velocity *U*, the characteristic length of the square section *D* (or the side length in this study) and a dimensionless coefficient, or the lift coefficient *CL* in this study:

$$F\_{\mathcal{Y}} = \frac{1}{2} \rho l I^2 D \mathcal{C}\_L \tag{2}$$

The main parameters involved in this model of harvester are listed in Table 1, where *ζ* represents the damping ratio, which is defined as *<sup>ζ</sup>* <sup>=</sup> *<sup>C</sup>*/2√*MK*. Note that the values of damping ratio and effective stiffness coefficient can be determined through experiment.


**Table 1.** Main physical model parameters of the harvester.

#### *2.3. Piezoelectric Performance Equation*

The energy harvester uses a d31 piezoelectric sheet, i.e., the polarization direction is perpendicular to the applied stress direction, and the working mode of the piezoelectric material is the *d31* mode with the polarized direction parallel to the surface normal. To simplify the analysis, a load resistance (*RL*) was attached to the conductive electrode of the PVDF piezoelectric ceramic film. The piezoelectric constitutive relationship can be directly expressed as [25]:

$$D\_3 = d\_{31} T\_1 + \varepsilon\_{33}^T E\_3 \tag{3}$$

where *d*<sup>31</sup> is the piezoelectric charge constant, *D*<sup>3</sup> is the electric displacement component, *T*<sup>1</sup> is the stress component, *E*<sup>3</sup> is the electric field component, and *ε<sup>T</sup>* <sup>33</sup> is the dielectric constant.
