# Aerodynamic Characteristics of a Square Cylinder with Vertical-Axis Wind Turbines at Corners
## 2 Numerical Simulation Configurations
### Grid Size and Time Step Configuration  
The hybrid grid scheme is used for discretization, as shown in Figure 1. Because of the geometric irregularity of the wind turbines, an unstructured grid is adopted around the turbines in the vicinity of the cylinder. Structural meshes are used for the whole computational domain except the region containing wind turbines. As shown in Figure 2, the diameter of wind turbines is \(5/100\)\(W\), \(d\) is \(7/300\)\(W\), and \(e\) is \(1/240\)\(W\), which leads to a ratio (\(e/d\)) of \(1/6\). The height of the minimum grid (\(8\)) is \(0.004\) mm, the maximum \(y+\) values for all cases is less than \(1\). The rotation of the wind turbine is implemented via the dynamic grid model, and a pair of coupling interfaces is set between the rotating region and the stationary region.  
Figure 1: Views of grids for a square cylinder with wind turbines at two leading corners: (**a**) Grid distribution in x–y plane; (**b**) Grid surrounding square cylinder; (**c**) Grid surrounding upper wind turbine; (**d**) Grid surrounding lower wind turbine.  
In order to consider the influence of grid size on numerical simulations, five grids (i.e., G1, G2, G3, G4, and G5) of the plain square cylinder with corner turbines are tested. The drag, lift coefficients, and Strouhal number of the square cylinder obtained from the five grid schemes are compared in Table 1. The mean drag, lift coefficients, and the Strouhal number are defined as:  
\[C_{d}=\frac{F_{d}}{1/2\rho U^{2}W} \tag{10}\]  
\[C_{l}=\frac{F_{l}}{1/2\rho U^{2}W} \tag{11}\]  
\[S_{t}=\frac{f_{s}W}{U} \tag{12}\]  
where \(F_{d}\) and \(F_{l}\) are the drag and lift forces of the cylinder, respectively; \(f_{s}\) is the vortex shedding frequency; \(W\) is the width of the cross section; and \(U\) is the oncoming flow velocity set at the inlet boundary condition. The specifications of the grids used are given in Table 1 and wind turbine rotation speed is 11.88 rev/s. The Strouhal number and \(RMS\) lift coefficients \(C_{l}^{RMS}\) of G4 and G5 are identical. Considering accuracy and calculation speed, G4 is selected as the final grid scheme.  
In addition, the simulation result of a plain cylinder without corner wind turbines is a baseline case to compare to the cases with corner wind turbines. As shown in Figure 3, the mean pressure coefficients on the four faces have favorable agreement with the results in the previous studies [22; 23]. Table 2 gives the comparisons of force coefficients between previous studies and present numerical simulations. The definition of \(\delta_{\text{cd}}\) is the relative error of \(\overline{C_{d}}\). As shown in Table 2, the drag coefficient \(\overline{C_{d}}\) for the plain cylinder is 1.99, which agrees well with that of 2.05 for a square cylinder reported in [24]. The relative error \(\delta_{\text{cd}}\) is nearly 5% when compared with all the experimental results. The \(RMS\) lift coefficient for the plain cylinder is 1.29, which matches well with the \(RMS\) lift coefficients of 1.37 for a square cylinder reported in [25].  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Grid** & **Cell** & & & & & \\
**Schemes** & **Numbers** & \(y^{\star}\)**(Max)** & \(\delta\)**/W** & \(\overline{C_{d}}\) & \(C_{l}^{\text{RMS}}\) & \(S_{t}\) \\ \hline G1 & \(1.1\times 10^{5}\) & 0.66 & \(4\times 10^{-5}\) & 1.43 & 0.69 & 0.136 \\ G2 & \(1.3\times 10^{5}\) & 0.52 & \(4\times 10^{-5}\) & 1.41 & 0.63 & 0.136 \\ G3 & \(1.7\times 10^{5}\) & 0.55 & \(4\times 10^{-5}\) & 1.42 & 0.61 & 0.136 \\ G4 & \(2.4\times 10^{5}\) & 0.57 & \(4\times 10^{-5}\) & 1.47 & 0.78 & 0.134 \\ G5 & \(3.0\times 10^{5}\) & 0.61 & \(4\times 10^{-5}\) & 1.48 & 0.78 & 0.134 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Grid schemes for grid independence tests.  
Figure 2: Sketch of Savonius wind turbines.