# Analysis of a Nature-Inspired Shape for a Vertical Axis Wind Turbine
## 3 Results and Discussion
### Fibonacci Sequence  
The Fibonacci numbers were first described in 200 BC by Indian mathematicians [46]. In the Middle Ages, they were analyzed by the Italian mathematician [PERSON], who was known as [PERSON] ([PERSON], i.e., the son of [PERSON]). The Fibonacci sequence is a recurrent succession given by Equations (5)-(7), where each term f depends on the two predecessors, \(\mathrm{n-1}\) and \(\mathrm{n-2}\). \[\mathrm{f_{0}=0} \tag{5}\]  
\[\mathrm{f_{1}=1,\;if\;n=1} \tag{6}\]  
\[\mathrm{f_{n}=f_{n-2}+f_{n-1},\;if\;n\;>\;1} \tag{7}\]  
The Fibonacci spiral is based on the Fibonacci sequence. The graphical representation of the Fibonacci sequence consists of establishing squares with the size of the successive terms, situated as shown in Figure 9. The Fibonacci spiral is also shown in this figure,  
Figure 8: \(\mathrm{C_{P}}\) against TSR experimentally and numerically obtained. which is formed by circumference arcs. Following the Fibonacci sequence, the side sizes of these squares are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc. The blade profile proposed in the present work is composed by two consecutive sections of the Fibonacci spiral. Any consecutive terms high enough lead to the same relation, the so-called golden ratio or divine proportion [47], 1.61803398874986. The higher the consecutive terms, the closer their relationship to the golden ratio, as shown in Equation (8). \[\frac{\mathrm{f_{n}}}{\mathrm{f_{n-1}}}=1.618033988749864\ldots\text{ if }n \rightarrow\infty,\text{ if }n\ >\ 1 \tag{8}\]  
According to this, each blade is composed by two curves whose ratios are related by the golden ratio. Depending on the orientation, two possibilities can be applied (sequence low-high-high-low or sequence high-low-high), shown in Figure 10a,b. For comparison purposes, the semicircular Savonius blade profile is also illustrated in these figures though dotted lines. Figure 11a illustrates the average power coefficient and Figure 11b the average thrust coefficient against the tip speed ratio for both Fibonacci blade profiles proposed and the Savonius blade profile. It can be seen that both the power and thrust coefficient increase with the tip speed ratio up to a certain value after which it drops down. Fibonacci I configuration provides a higher power and thrust coefficient than Savonius one. Particularly, a 14.1% improvement on the average power coefficient was obtained and a 13.5% improvement on the average thrust coefficient. Nevertheless, Fibonacci II configuration provides lower power and thrust coefficients. Due to this, the Fibonacci I blade profile will be adopted in the remaining work. The reason of the results obtained in Figure 11 is the principle of operation of a Savonius turbine. The power is mainly produced by drag, and the Fibonacci I configuration increments the drag force of the advancing blade and reduces the drag force of the returning blade, increasing the net driving force. Figure 10: Blade profiles analyzed; (**a**) Fibonacci I (black color dashed line), sequence low-high-high-low, and comparison with Savonius (grey color dotted line); (**b**) Fibonacci II (black color dashed line), sequence high-low-low-high, and comparison with Savonius (grey color dotted line). Figure 9: Fibonacci spiral obtained from the Fibonacci sequence. Other improvements obtained by different authors are summarized in Table 2. It is worth mentioning that C\({}_{\text{P}}\) varies with the Reynolds number. For this reason, it is difficult to compare which shape is the most appropriate using this table since all data should be analyzed under the same Reynolds number. Another interest aspect is related to the turbine fatigue hazard. Many VAWTs fail due to blending. The responsible stresses are cyclic and thus promote fatigue, and a dimensional parameter related to the fatigue damage is C\({}_{\text{F}}\), Equation (9), where F represents the thrust force. Since the torque is T = F\(\cdot\)R, in non-dimensional form it holds that C\({}_{\text{T}}\) = C\({}_{\text{F}}\).