## 3 Results and Discussion
### Wind Direction  
Compared with HAWTs, one advantage of a single VAWT operating alone is that there is no need for a wind direction device. However, when a VAWT array operates together, research shows that changes in wind direction have a significant impact on the efficiency of the entire array. In actual scenarios, the wind is not always in a single direction, and prevailing winds often exist. For example, in China's southeastern coastal areas, the prevailing winds are southeast in summer and northwest in winter. Therefore, utilizing wind direction rationally is also crucial [29]. This study investigated four wind directions, \(\alpha=-45^{\circ}\), \(\alpha=0^{\circ}\), \(\alpha=45^{\circ}\), and \(\alpha=90^{\circ}\), where the other wind directions were symmetrical and thus omitted. The spacing between the wind turbines was 2D. To facilitate the full development of the Nautilus wind turbine wake flow and, at the same time, consider the performance of the computer, the outflow field of the cross-section under different wind directions was reasonably arranged, as shown in Figure 5. It is worth noting that, for the \(0^{\circ}\) wind direction, which corresponds to the horizontal wind direction, the wind direction is parallel to the connecting line between the wind turbine array, and the wind turbines are horizontally arranged, which makes the analysis easier. In the exploration of wind turbine aerodynamic performance, the power coefficient, which represents the wind energy conversion efficiency, is of paramount importance. The power coefficient can be expressed as  
\[C_{P}=\frac{T\omega}{\frac{1}{2}\rho HDv^{3}} \tag{3}\]  
In the equation, \(T\) represents torque, \(\omega\) represents angular velocity, \(H\) represents the height of the wind turbine, and \(D\) represents the diameter of the wind turbine. To further investigate the aerodynamic performance of the turbine array, this study proposed a concept of average power coefficient, which is expressed as follows:  
\[C_{P}(average)=\frac{C_{P1}+C_{P2}}{2} \tag{4}\]  
\(C_{P1}\) and \(C_{P2}\), respectively, represent the power coefficients of wind turbines 1 and 2. The average efficiency of the Nautilus wind turbines under different wind directions is shown in Figure 6. From Figure 6a, it can be seen that the average efficiency curve of the wind turbine array under different wind directions is consistent and reaches its maximum value at TSR = 0.6 compared to a single wind turbine. It can be observed from Figure 5a,c that the \(-45^{\circ}\) and \(45^{\circ}\) wind directions are symmetrical about the \(x\)-axis, so the wind energy conversion efficiency curves of these two wind turbine arrays are consistent, with a maximum average power coefficient of about 23%. Compared with other wind directions, the efficiency of the wind turbine array under the \(0^{\circ}\) wind direction is significantly lower, with a maximum average efficiency of only 17%. The maximum conversion efficiency of the \(90^{\circ}\) wind turbine array is higher than that of a single wind turbine. The \(90^{\circ}\) wind direction is perpendicular to the centerline of the wind turbine; that is, the wind turbines are arranged in a transverse configuration, which is consistent with previous research results [30], and the maximum conversion efficiency reaches 27.5%. The optimal efficiency polar coordinate comparison curves of the wind turbine arrays under different wind directions are shown in Figure 6b, where it can be seen that the conversion efficiency is highest for wind directions at the top and bottom, and the efficiency decreases as the wind direction becomes more horizontal. To further explain the above results, the wake velocity fields are quoted. The wake velocity cloud maps of wind turbine arrays under different wind directions are shown in Figure 7, where the turbine numbers are indicated in a and b.