## 4 Results
With regard to the streamline (Figure 16), it is visible that the transient simulation (sliding mesh) and the frozen rotor solutions (steady state) have similar results near the turbine; however, the time-dependent solution has more vortexes than the steady solution. The other steady solution with the mixing plane approach has a \"smoother\" result, which means that the streamlines have a swirl after turbines due to the averaging. With regard to vorticity (Figure 17), the so-called Q criterion at Q = 800 1/s\({}^{2}\) value was used to model the differences between the methods. Figure 17a shows a fluid structure starting from the first turbine's tip and colliding with the second turbine in the sliding mesh simulation. In this case, it is possible to find another turbulence structure, which starts from the second turbine and flows \"back\". The second picture in Figure 17b shows a less consistent swirling structure starting from the tip of the blades, which can also be found in the sliding mesh simulation. Due to RANS/URANS differences, a vortex appears that starts from the tip of the blade and flows straight \"back\", which is also visible in Figure 17b. The third picture in Figure 17c shows the mixing plane approach, which shows neither a turbulence structure such as the result obtained from the sliding mesh nor the ones obtained from the frozen rotor approach. This indicates that the mixing plane approach averages out the turbulences, as mentioned in the introduction. Figure 16: Streamlines near the turbines: (**a**) transient results at the sixth whole revolution; (**b**) frozen rotor result (position 0\({}^{\circ}\)); (**c**) mixed plane results. In the case of numerical methods, it is advisable to mention the computational costs as well. Two steady-state simulations (mixing plane and frozen rotor) were compared to a transient simulation (sliding mesh). Three simulations were carried out with the same mesh settings for comparing the computational resources. Although the same mesh settings were used for the three methods, different cell numbers were generated due to their distinct characteristics. For the mixing plane method, 4,622,698 cell numbers were generated, while for the frozen rotor, 6,109,872, and for the sliding mesh approach, 6,112,514. The difference between the frozen rotor and the sliding mesh comes from the meshes and their tolerances, while the difference between the mixing plane and the other two approaches comes from the differences in the rotating domain's interfaces. Computational costs were examined based on memory usage per cell and time requirement per cell. With regard to memory usage per cell, it can be generally observed that the sliding mesh solution has some periodicity due to its solving method, which appears in every eighth iteration (Figure 18). The explanation for this periodicity is that the SIMPLE algorithm was used, which has a fixed number (8) of inner iterations. After this inner iteration number, the solver jumped to the next time step and calculated eight iterations within that time step to decrease the residual error. It must be mentioned that a growing trend in memory usage can be noticed as time changes. In the case of the frozen rotor method, the memory usage was constant, while the memory usage decreased in the case of the mixing plane until it reached 300 iterations, then  
Figure 17: Isosurface at Q = 800 \(1/\mathrm{s}^{2}\) for turbulence comparison (colored by vorticity between 30 and 500 \(1/\mathrm{s}\)): (**a**) transient results at the sixth whole revolution; (**b**) frozen rotor result (position 0\({}^{\circ}\)); (**c**) mixed plane result. Figure 18: For the different mesh numbers, we calculated the memory requirements for one element. it became constant. In numbers, the average memory usage for the mixing plane simulation was 1588.247791 bytes/element, for the frozen rotor approach, 1409.438993 bytes/element, while for the sliding mesh solution, 1761.482633 bytes/element.