# Design and Development of Bladeless Vibration-Based Piezoelectric Energy-Harvesting Wind Turbine
## 2 Design Methodology  
The design methodology of the proposed design can be explained in the following statements. When a steady fluid flow is applied to an elastic bluff body at high enough Reynolds numbers (\(R_{e}>100\)), the flow breaks from the body surface, resulting in an unsteady broad wake. Two unstable shear layers on each side of the body roll up to form vortices, which characterize the flow pattern. These vortices that were produced lead to oscillation (mechanical energy), which is then converted to electric energy through the components of the proposed energy harvesting device. The undisturbed flow speed \(U\) and the size of the cross-sectional body \(D\) are connected by the well-known Strouhal relationship, \(S_{T}=\frac{f_{T}D}{U}\), and with a frequency of \(f_{T}\), these vortices are shed to the wake periodically. The frequency of vortex shedding increases as the flow velocity increases from zero, and there is a flow velocity at which vortex shedding has a frequency close to the body's natural frequency of oscillations \(f_{N}\), and significant oscillations can be induced in the body for low mass and mechanical properties. When a body oscillates, it creates a complex interaction with the flow field surrounding it, which has two unique features: (i) a range of flow velocities exists where the vortex shedding frequency is synced with the oscillation frequency (lock-in regime), resulting in large oscillations; and (ii), with jumps in oscillation amplitude and the fluid forces acting on the body, the cylinder response may exhibit hysteresis. The proposed design incorporated these two qualities. However, at low wind speed, achieving a frequency where \(f_{T}\) equals \(f_{N}\) remains a challenge that the proposed design in this study should be able to address. The wind flow is incompressible with constant density, and the effect of viscosity is negligible far from the walls of the VIV design in this study. The conservation of mass and momentum equations are the most fundamental equations in every fluid flow domain. However, according to the turbulent nature of wind flow in this study, the momentum equation should be written so that the effect of turbulent stresses is considered. Therefore, the conservation of mass seen in Equation (1) and the conservation of momentum (the Cauchy equation) Equation (2) were utilized in this work, as follows:  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot(\rho\mathbf{u})=0 \tag{1}\]  
\[\frac{\partial(\rho\mathbf{u})}{\partial t}+\
abla\cdot(\rho\mathbf{u})=- \
abla p+\
abla\cdot\tau \tag{2}\]  
where \(\rho\) is the density, \(t\) is time, \(\mathbf{u}\) is the velocity field, \(p\) is the pressure, and \(\tau\) is the turbulent stress tensor and would be calculated based on the selected model in the CFD software. Solving these equations, the Lift coefficient is attainable, which is necessary for power generation calculations as in Equation (3):  
\[C_{L}=\frac{F_{L}}{\frac{1}{2}\rho U^{2}A} \tag{3}\]  
where \(F_{L}\) is lift force; \(A\) is reference area, and \(U\) is freestream velocity. Also, the Strouhal number (\(St\)), which is mandatory for finding vortex shedding frequency, can be calculated using Equation (4). \[St=\frac{fL}{U} \tag{4}\]  
where \(St\) is the Strouhal number, \(f\) is the vortex shedding frequency, and \(L\) is the characteristic length. The harvested power of the VIV device can be determined using Equation (5), considering the defined parameters and their values reported in Table 1:  
\[\overline{P}=\frac{\omega^{2}b^{2}h^{2}e_{31}^{2}\overline{A}^{2}R}{4{\left(1 +bL{\varepsilon_{33}}\omega R/\Delta\right)}^{2}} \tag{5}\]