## 3 Methods for Modeling and Simulating Rotational Motion
Among the three methods, the sliding mesh approach provides the most realistic results, but it requires the most computational resources too. Force, torque, and other quantities of the performance can be monitored during the time steps, and their values can be plotted against time or angular position. Simplified explanatory diagrams about these modeling approaches are shown in the next figures (Figures 2-4). Figure 3: Frozen rotor approach. Angular velocity from a stationary region and the \(\omega\times r\) component, with its values on the surface (from the stationary viewpoint) [26]. Figure 2: Flow quantities using the mixing plane approach by mesh “layer” to the next “layer” [25]. Commercial CFD software was used for the simulations (Simcenter STAR-CCM+ version V2022.1 from Siemens Digital Industries Software). A rectangle computational domain was employed with dimensions of 10D \(\times\) 15D \(\times\) 20D, where D denotes the diameter of the turbine (D = 200 mm). This computational domain has been tested in [28], and this size proved to be appropriate since neither was the computational cost high nor had the surrounding walls a significant effect on the numerical solution. As for the rotors, the first one was positioned 7.5D from the \"front\" face and in the middle of the other two directions (10D from the sides). The domain with a cutout for the turbine is shown in the following Figure 5.  
In the domain, the positions of the rotors were fixed. The distance between the rotors was 0.5D (100 mm) in both axial and radial directions. The turbines rotated in opposite directions, which is shown in the following Figure 6 with the rotors' distances [29]. Figure 4: Mesh in a transient simulation using the sliding mesh approach: (a) the initial mesh at t = 0 s; (b) the rotated mesh at t > 0 s [27]. Figure 5: Computational domain and its sizes. For the simulations, we used a segregated flow solver (SIMPLE), which solved the flow field and its quantities iteratively. The isotermatic state and constant density were assumed for the fluid domain; therefore, only the continuity and the Navier-Stokes equations were solved. For modeling and simulating the turbulence, the SST \(k\)-\(\omega\) turbulence model was used, which is described by the following equations:  
\[ \begin{array}{c}\frac{\partial(\rho\cdot k)}{\partial t}+div(\rho\cdot k\cdot \boldsymbol{U})=div\Big{(}\Big{(}\mu+\frac{\mu_{t}}{\sigma_{x}}\Big{)}\cdot grad \ k\ \Big{)}+P_{k}-\beta^{*}\cdot\rho\cdot k\cdot\omega\frac{\partial(\rho\cdot \omega)}{\partial t}\\ \frac{\partial(\rho\cdot\omega)}{\partial t}+div(\rho\cdot\omega\cdot \boldsymbol{U})=div\Big{(}\Big{(}\mu+\frac{\mu_{t}}{\sigma_{x}\cdot 1}\Big{)}\cdot grad\ \omega\ \Big{)}+\gamma_{2}\Big{(}2\cdot\rho\cdot s_{ij}\cdot S_{ij}-\frac{2}{3}\cdot \rho\cdot\omega,\frac{\partial U_{i}}{\partial x_{j}}\Big{)}-\\ \beta_{2}\cdot\rho\cdot\omega^{2}+2\cdot\frac{\rho}{\sigma_{x}\cdot 2 }\cdot\frac{\partial k}{\partial x_{k}}\frac{\partial\omega}{\partial x_{k}} \end{array}  \tag{6}\]  
The inlet flow had a velocity of 3.79 m/s. On the sides of the domain, a symmetry boundary condition was applied, while a pressure outlet was employed on the outlet face. The ambient pressure was 1 atm (101,325 Pa). The fluid was \"Air\" from the material database of the Simeenter STAR-CCM+. The angular velocity of the domains was \(-1447.675\) RPM and \(+1447.75\) RPM; therefore, the tip speed ratio was 4 for each turbine. To model the rotational motion, the previously presented approaches (frozen rotor, mixed plane, and sliding mesh) were used. For the mixed plane approach in which the averaging was carried out within the rotating domain, only one geometry position was available since the rotated geometries provided the same results due to the averaging. For the sliding mesh approach in which the simulation was transient, one geometry was used as well since, in this case, the mesh was rotating. For the frozen rotor approach, seven different geometries were created, which were shifted by \(15^{\circ}\) between each step.