# Performance Characteristics of an Orthopter-Type Vertical Axis Wind Turbine in Shear Flows
## 3 Numerical Approach
### Numerical Setup  
The computational domain, the computational meshes and the boundary conditions are shown in Figure 5. As the same as the experimental setup, the origin of the coordinate system is defined at the center of the O-VAWT. The modeled O-VAWT was comprised of three blades, one main shaft and two sets of connecting arms. To reduce the computational cost, other components, such as the chains and sprockets are omitted. The sizes of these components are the same as those used in the experiment. The computational domain consists of three blade domains, one rotor domain and one far-field domain. The blade domain included one of the blades and rotates around each blade axis. The rotor domain included these three blade domains, the main shaft and the connecting arms and rotates around the main shaft. The far-field domain was a stationary domain and its size was \(23.5D\times 17D\times 5D\). Except for uniform flow cases, a splitter plate with the same thickness as the experiment was set at \(y=0\) and \(x=-11.37D\) to \(-1.67D\). In all domains, only unstructured meshes were used. Based on our previous mesh-resolution dependency tests [33], the number of computational cells in each domain were set as shown in Table 2. The total number of computational cells was approximately 10 million. All surfaces of the solid components were covered with boundary-layer meshes. The first grid nodes over the surface of the blades were \(y+<1\) in all run cases. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Domain** & **Mesh on This Research** & **Regular Mesh Used by [PERSON] [33]** \\ \hline Blade 1 & 2,300,530 & 1,610,660 \\ Blade 2 & 2,303,725 & 1,614,903 \\ Blade 3 & 2,304,607 & 1,588,363 \\ Blade 4 & - & 1,614,661 \\ Rotor & 1,756,635 & 2,127,497 \\ Far end & 1,770,583 & 441,196 \\
**TOTAL** & **10,436,080** & **8,997,280** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Numbers of grid sizes. Figure 5: Computational domain, computational mesh and boundary conditions. (**a**) Top view of the computational domain; (**b**) bird’s eye view of the computational domain with computational mesh and boundary conditions; (**c**) modeled rotor; (**d**) blade domains and (**e**) rotor domain. At the inlet boundary, the distributions of the stream-wise wind velocity shown in Table 3 were implemented. These distributions were set so that when the O-VAWT is absent, the distributions of the time-mean values of \(u\) at \(x\approx-0.147D\), which corresponds to 0.1 m downwind of the wind tunnel outlet, matched well with those of the wind tunnel experiment, which were generated by using the three kinds of perforated panels with different shielding ratios \(\Phi\), as shown in Figure 6a,b. Here, \(U_{H}\) and \(U_{L}\) are the maximum and minimum streamwise velocities in the profile of a shear flow measured in the wind tunnel experiment; \(\Gamma\) is the velocity ratio defined as:  
\[\Gamma=\frac{U_{H}-U_{L}}{U_{H}+U_{L}} \tag{17}\]  
It is worth noting that amplification factors of \(\Gamma=0.28\) and \(\Gamma=0.51\) are considered as 1.35 and 1.65, respectively, by computing \(U_{H}\)/\(U_{\infty}\). The amplification factor is defined as the ratio of wind speed in the case where there are buildings to wind speed in the case where these buildings are removed. Figure 6: Horizontal distribution of the time-mean values of streamwise velocity at the mid-height of the wind tunnel outlet at \(x\approx-0.147D\), which corresponds to 0.1 m downwind of the wind tunnel outlet, and \(x=0\), which corresponds to the position of the rotational axis of the O-VAWT, when the O-VAWT is absent. (**a**) ASF-SF cases at \(x\approx-0.147D\), (**b**) RSF-SF cases at \(x\approx-0.147D\), (**c**) ASF-SF cases at \(x=0\) and (**d**) RSF-SF cases \(x=0\).