## 2 Materials and Methods
### CFD Analysis  
The CFD procedure carried out to simulate the VAWT was described in a previous paper [23]. For this reason, it will be briefly described herein. The model is based on the RANS equations of conservation of mass and momentum. The turbulence was treated through the SST k-\(\omega\) turbulence model. The software OpenFOAM was employed. An important step in the CFD analysis was the validation process. To this end, data obtained by Sandia laboratories [24] were employed. These experiments are based on the turbine shown in Figure 1a. The main characteristics are listed in Table 1. This turbine has two blades, 1 m diameter, 1 m height, 0 overlap, 0 separation gap, and 0 twist angle. Two end plates were added to reduce the escape of air from the blades. The mesh employed is shown in Figure 2. Two zones are differenced: static and rotating. The static domain is almost the whole region, while the rotating domain is a cylinder that rotates according to the rotational velocity. The turbine is placed inside the rotating domain. Regarding boundary conditions, a velocity inlet was modeled on the upstream face, while the downstream face was modeled as an outlet. The exterior faces were assumed to free sleep and the turbine surface was modeled as no-slip. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Diameter (m) & 1 \\ \hline Height (m) & 1 \\ \hline Number of blades & 2 \\ \hline Overlap (m) & 0 \\ \hline Separation gap (m) & 0 \\ \hline Twist angle (\({}^{\circ}\)) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Characteristics of the turbine employed for the validation procedure. A comparison between the experimental and numerical results is shown in Figure 3. Particularly, Figure 3a shows the power coefficient (C\({}_{\text{P}}\)) and Figure 3b compares the torque coefficient (C\({}_{\text{T}}\)) against the tip speed ratio (TSR). These parameters represent the non-dimensioned form of the rotational velocity, power, and torque, and are given by Equations (1)-(3), respectively. In these equations, V is the free-stream velocity, \(\omega\) the rotational speed, P the power, \(\rho\) the density of air, \(\
u\) the kinematic viscosity of air, S the cross-section area (S = DH), and T the torque. Figure 3 shows an adequate concordance between experimental and numerical results. Several reasons are responsible for the differences between experimental and numerical data. First of all, CFD is an approximate method and unavoidable errors are introduced due to the discretization processes and hypotheses realized to simplify the model. On the other hand, experimental measurements are affected by the tolerance of the apparatus employed. Moreover, another important source of error is the blockage effect. This effect is produced when a sample is placed in the test section of a wind tunnel; the sample alters the current velocity. Several methods can be found in the literature to correct this modified current velocity, but significant differences can be obtained using different methods [25]. \[\text{TSR}=\frac{\text{blade tip tangential velocity}}{\text{wind speed}}=\frac{\omega\text{R}}{\text{V}} \tag{1}\]  
\[\text{C}_{\text{P}}=\frac{\text{power}}{\text{available power}}=\frac{\text{P}}{0.5 \rho\text{SV}^{3}} \tag{2}\]  
\[\text{C}_{\text{T}}=\frac{\text{torque}}{\text{available torque}}=\frac{\text{T}}{0.25 \rho\text{SV}^{2}\text{D}} \tag{3}\]  
Figure 3: Comparison between numerical and experimental results; (**a**) C\({}_{\text{P}}\) against TSR; (**b**) C\({}_{\text{T}}\) against TSR. Figure 2: Computational mesh. (**a**) 3D view; (**b**) middle plane. Once the model was validated, the first improvement proposed in the present work is to modify the semicircular blade profile originally proposed by [PERSON] [26]. The nature-inspired Fibonacci spiral is proposed to this end.