# Numerical Study on the Yaw Control for Two Wind Turbines under Different Spacings
## 3 Simulation Setting  
In this study, all simulations are investigated with two turbines, which are arranged on the central axis of the wind field as shown in Figure 4. The simulation domain is 2000 m long, 500 m wide and 500 m high. In order to save computing costs, the SnappyHexMesh refinement method of OpenFOAM is carried out on the grid. The uniform hexahedron grids with a side length of 2.5 m are used in the central computing area, and the outer grid sizes are 5 m and 10 m, respectively. The total number of grids is about 13 million. As [PERSON] et al. [28] selected uniform inflow in his research, an idealized uniform inflow is used in our study to focus only on the effects of yaw control and spacing, and the wake fields under the turbulent inflow with the turbine spacing of 7D are investigated to analyze the influences of the turbulent intensity on the yaw control. The uniform inflow speed is 11.4 m/s. Similar to [PERSON] [29] studies, the symmetry boundary conditions are specified on the top, bottom, left and right sides. The hub of the turbine is placed in the middle of the height direction of the calculation domain to minimize any influence of vertical domain boundaries. The turbines are arranged in the midline of the calculation domain, so that the rotor axis is located at the center of the flow field. The upstream turbine is 250 m away from the inlet and the downstream turbine is about 5-8D away from the upstream turbine.  
Figure 4: Schematic diagrams and grids of the calculation domain: (**a**) Top View; (**b**) Side View.  
Figure 3: Time\(-\)averaged wake velocity deficit profiles normalized by the hub-height average freestream speed of 8 m/s: (**a**) horizontal profiles at hub height; (**b**) vertical profiles in the wake centerline [13].  
[PERSON] [13] pointed out that it was needed to use at least 20 grid cells across the rotor diameter. In this paper, we place 50 grid cells across the rotor to ensure an accurate simulation of the tip and root vortices. The time step is given according to the Courant-Friedrichs-Lewy (CFL) criterion, that the Courant number (Co) must be less than 1.  
\[Co=\frac{\mathbf{u}\Delta t}{\Delta x} \tag{7}\]  
where \(\mathbf{u}\) is the local wind speed, which is the vector sum of inflow velocity and blade tip velocity. \(\Delta x\) is the grid size, and \(\Delta t\) is the time step. According to the Courant number criterion, \(\Delta t\) needs to be less than 0.031 s. A series of cases with different mesh resolutions and time are investigated to verify the accuracy of the simulation, as shown in Table 2. We choose the output power of the first turbine and the wake velocity together as the judgment bases for the independence verification. For the five cases, the velocity profiles at hub height at 2D downstream of the downstream turbine are shown in Figure 5. After a comprehensive consideration of computation cost and precision, the mesh length in the dense area is finally chosen as 2.5 m, and the time step is taken as 0.025 s.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Situation** & **Mesh Length (m)** & **Time Step (s)** & **Power (MW)** \\ \hline
1 & 2.0 & 0.025 & 5.302 \\
2 & 2.5 & 0.01 & 5.299 \\
3 & 2.5 & 0.025 & 5.298 \\
4 & 2.5 & 0.05 & 5.185 \\
5 & 4.0 & 0.025 & 5.113 \\ \hline \end{tabular}
\end{table}
Table 2: Grid and time step independent validation.  
Figure 5: Velocity profiles of the hub height plane at 2D downstream of the downstream turbine.