# Analysis of a Nature-Inspired Shape for a Vertical Axis Wind Turbine
## 2 Materials and Methods  
### Numerical Model  
As mentioned previously, a numerical model was developed to analyze several modifications of a VAWT Savonius type. Numerical models present important advantages over experimental setups. Contrary to the high cost and time necessaries to carry out experimental measurements of different configurations, the CFD model developed constituted a tool to fast and cheaply analyze different geometries performance parameters. Moreover, numerical models provide interesting information regarding the velocity and pressure field that are too complicated to measure experimentally. The software OpenFOAM was employed. The main reason is that this is an open source and thus it allows a total access to the code to understand and manipulate it. The governing equations are the RANS (Reynolds-averaged Navier-Stokes). In particular, the conservation of mass and momentum were implemented. As a turbulence model, the SST k-\(\omega\) was employed since several studies revealed its estimation capabilities in VAWTs [39; 40; 41; 42; 43]. In a previous work [44], several turbulence models were compared, and it was concluded that the SST k-\(\omega\) is the most appropriate for these simulations although the computational cost was higher than other turbulence models, such as the commonly employed k-\(\epsilon\). The PIMPLE algorithm was employed for the pressure-velocity coupling. The temporal treatment was realized through an implicit method, with a constant time step corresponding to 1\({}^{\circ}\) of rotation. In order to reach a quasi-state state, several revolutions must be analyzed. Particularly, the results shown in the present work refer to the 6 th rotation. When modeling wind turbines, it is common to employ the non-dimensional parameters shown in Equations (1)-(3). \[\text{TSR}=\frac{\text{blade tip }\tan\text{gential velocity}}{\text{wind speed}}=\frac{\omega\text{R}}{\text{V}} \tag{1}\]  
\[\text{C}_{\text{P}}=\frac{\text{P}}{\text{P}_{\text{Available}}}=\frac{\text{ P}}{0.5\ \text{gSV}^{3}} \tag{2}\]  
Figure 4: Examples of the Fibonacci spiral in nature. (**a**) Snail shell; (**b**) Fran hurricane, September 1996 [36]; (**c**) Whirlpool Galaxy [37]. \[\mathrm{C_{T}}=\frac{\mathrm{T}}{\mathrm{T_{available}}}=\frac{\mathrm{T}}{0.25\; \mathrm{\rho SV^{2}D}} \tag{3}\]  
In the equations above, V is the undisturbed wind speed, \(\omega\) the rotational speed, P the power, \(\rho\) the density of air, S the cross section area (S = DH), T the torque, \(\mathrm{C_{T}}\) the torque coefficient, and \(\mathrm{C_{P}}\) the power coefficient. TSR is the ratio between the blade tip tangential velocity and the wind speed. The power coefficient represents which fraction of the power in the wind is being extracted by the turbine through the projected area of the rotor at the flow direction. The mass flow rate for an air stream flowing through a surface S is \(\mathrm{\rho SV}\), and therefore the available power comes from its kinetic energy, i.e., \(\mathrm{P_{available}=0.5\;\mathrm{\rho SV^{2}=0.5\;\mathrm{\rho SV^{3}}}}\). The power coefficient represents the ratio of the power to the available power. The available torque for an air stream is \(\mathrm{T_{available}=P_{available}/\omega}\). Taking into account that \(\mathrm{V=\omega R}\), the latter expression yields that \(\mathrm{T_{available}=P_{available}/\omega=0.5\;\mathrm{\rho SV^{3}/(V/R)=0.5\; \mathrm{\rho SV^{2}/(2\;V/D)=0.25\;\mathrm{\rho SV^{2}D}}}}\). The torque coefficient represents the ratio of the torque to the available torque, as shown in Equation (3). Obviously, since the power is P = T\(\cdot\omega\), in non-dimensional form it holds that \(\mathrm{C_{P}=C_{T}.TSR}\), as can be seen in Equation (4). \[\mathrm{C_{P}=\frac{P}{P_{available}}=\frac{P}{0.5\;\mathrm{\rho SV^{3}}}=\frac {T\omega}{0.5\;\mathrm{\rho SV^{3}}}=C_{T}TSR} \tag{4}\]  
The domain and boundary conditions are shown in Figure 5. The dimensions are \(23\;\times\;8\;\times\;8\;\mathrm{m}\).