## 3 Methods for Modeling and Simulating Rotational Motion
The initial position (later referred to as position with \(0^{\circ}\)) is shown in Figure 7 and represented as white turbines. The rotated positions are represented with yellow and green colors, as the turbines were rotated by \(15^{\circ}\). We used these \(15^{\circ}\) steps and directions for creating the seven different geometries in which the rotation angles were \(0^{\circ}\), \(15^{\circ}\), \(30^{\circ}\), \(45^{\circ}\), \(60^{\circ}\), \(75^{\circ}\), \(90^{\circ}\), and \(115^{\circ}\). The eighth position for \(120^{\circ}\) could be identical to \(0^{\circ}\)[30, 31, 32]. Before conducting the simulations, two kinds of dependency studies were carried out. The first one focused on determining the appropriate domain size, while the second one was for detecting mesh dependency. For the first series of simulations, various domain sizes were used and compared. In the mesh dependency study, six different meshes were employed, in which we compared the torque in each turbine. All simulations were steady state and utilized the boundary conditions described earlier with the initial (\(0^{\circ}\)) geometry. Figure 6: Turbines, their distances, and their rotational directions. The torque difference between the last two meshes was 3.8% and 1%. The change in the torque for the meshes is shown in Figure 8.  
After the mesh dependency study, the last, viz. sixth mesh, was chosen for running our simulation. During the simulations, various quantities were monitored such as torque, pressure on the surface of the turbines, average and maximum velocities, and the residuals. While torque converged quickly, other quantities required more iterations to converge. Despite using a mesh-independent grid, unexpected changes were observed in the torque; therefore, the simulations were carried out further on with adaptive mesh refinements (AMR) until 2500 iterations. A typical run is shown in Figure 9 below. Figure 8: Torque changes on the turbines during the mesh dependency studies. Figure 7: Direction and steps of the rotation angles. We can see in Figure 9 that the torque in each turbine started with an initial value, which converged rather fast to its final value at around 250-300 th iterations. Following this, the torque remained constant until the 1400 th iteration, at which point the first AMR was applied. The mesh refinement occurred after 100 iterations. Once the torque reached convergence, the simulation was stopped at the 2500 th iteration. The mesh that was used included two different meshing methods. In the \"outer\" region where the freestream is more dominant, a trimmed cell mesher was employed. Near the turbines and in the region of the turbines, a polyhedral mesher was used. The initial mesh is shown in Figure 10.  
As previously mentioned, after the 1400 th iteration, AMR was used. The last refined mesh (which was used for the 2500 th iteration) for the 0\({}^{\circ}\) position is shown in Figure 11.  
Figure 10: Cont. Figure 9: The change in the torque during the simulation with the geometry 30\({}^{\circ}\). Figure 11: _Cont_. Figure 10: Initial mesh for the \(0^{\circ}\) position: (**a**) whole computational domain; (**b**) near the regions of the turbines; (**c**) near the surface of the first turbine. The grid contained prism mesh on the turbines with a mesh size of 0.02 mm. The prism layer mesh contained 15 layers, and the total thickness was 1 mm. The minimum adaptation cell size for the AMR was set to 0.45 mm. If this value was below 5, the AMR coarsened the mesh; if it was between 5 and 40, the mesh was not changed; and if its value was above 40, the AMR refined the mesh. The refinement criterion for the last iteration of the 0\({}^{\circ}\) position is shown in Figure 8.0  
For the position 0\({}^{\circ}\) at the 2500 th iteration, the Y+ value and its distribution are shown in Figure 8.1 The average Y+ value on the first turbine was 0.4236, while the maximum and the minimum were 4.4453 and 0.0212, respectively.