# Optimizing Bladeless Wind Turbines: Morphological Analysis and Lock-In Range Variations
## 5 Discussion  
As mentioned in the Section 4, it is evident that the initial prototype shape (variation in convergence/divergence angle) holds significant influence, while surface roughness has minimal impact. Figure 12 clearly illustrates how diverse geometries shift the Lock-In range  
Figure 11: CV of the signals illustrated in Figure 9 subjected to different Reynolds numbers. Figure 12: CV of the signals illustrated in Figure 10 subjected to different Reynolds numbers. towards higher Reynolds numbers as the upper diameter expands due to geometric changes. This relationship aligns with Equation (3), linking wind velocity, Reynolds number, cylinder diameter, and the natural frequency of the prototype. The diameter variation outweighs the impact of mass differences among the prototypes. The mass disparity directly affects the natural frequency, making the mass impact less significant across the different prototype types. Consequently, this shift towards higher Reynolds numbers results in an increased Lock-In range, as evidenced by higher RMS values and peak readings recorded by the piezoelectric sensor. Table 4 showcases Reynolds numbers corresponding to the maximum Lock-In range values. The validity of these findings is shown by the results of the Coefficient of Variation (CV) around a value of 10% in maximal Re Lock-In points, which provide a nuanced interpretation within the context of experimental data analysis. A CV value close to 10% suggests a moderate level of variability relative to the mean of the data set. This indicates that while there is some dispersion in the data points, it is not excessively high, suggesting a relatively stable set of measurements or observations. Based on these findings, Figure 13 delineates the relationship between convergence/ divergence angles of the prototype and their corresponding peak Lock-In range points. A linear regression analysis led to Equation (4):  
\[\text{Re (max Lock-In value)}=0.30\ \alpha+4.06,\ \text{with an}\ \text{R}^{2}\ \text{value of}\ 0.99 \tag{4}\]  
Lock-In values increase for higher Re values that suppose higher wind velocities (Equation (3)), increasing the energy available due to the fundamental principles governing the conversion of wind energy into electrical energy. Equation (5) indicates that the energy \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Prototype** & **Angle** & **VRMS [mV]** & **Re (Max Lock-In)** \\ \hline Convergent \(-0.8^{\circ}\) & \(-0.8^{\circ}\) & 1601 & 3800 \\ Straight \(0^{\circ}\) & \(0.0^{\circ}\) & 1631 & 4050 \\ Divergent \(0.8^{\circ}\) & \(0.8^{\circ}\) & 1708 & 4300 \\ Divergent \(1.6^{\circ}\) & \(1.6^{\circ}\) & 1766 & 4600 \\ Divergent \(3.2^{\circ}\) & \(3.2^{\circ}\) & 1829 & 5000 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Max Re of the Lock-In range. Figure 13: Relationship between angle and max Lock-In value. content of the wind striking a wind turbine (\(P\)) is directly related to the air density (\(\rho\)), the area (\(A\)) swept by the turbine's blades, and, most critically, the cube of the wind velocity (\(V\)). \[P=1/2\;\rho\;A\;V^{3} \tag{5}\]  
Assuming all parameters as constants, and relating Equations (4) and (5), the relationship between the power and the angle could be expressed by Equation (6), where \(P\) is the power, \(\alpha\) is the angle, and \(k\) stands in this case as a constant value that relates both variables:  
\[P=k\;(0.30\;\alpha+4.06)^{3} \tag{6}\]  
Considering surface type, despite its relatively minor influence, observations in Figure 11 depict VIV tending to decrease as Re rises for rough surfaces within each geometry. This phenomenon aligns with the behaviour of the wind turbine, which can be assimilated to a mass-spring system. As previously mentioned, the spring constant relies on mast stiffness (constant across all models). Therefore, the oscillation frequency inversely relates to the square root of the body's mass (Equation (1)); thus, when roughness increases the mass, the natural frequency decreases.