## 3 Results and Discussion
### Mesh Independence Test and Comparison of Steady and Transient Models  
Several different mesh sizes were compared at \(Re\) = 2000 using the laminar solver for both steady and transient simulation and steady RANS simulation using a realizable \(k\)-\(\epsilon\) model with scalable wall function. The key parameter that determines the simulation accuracy is the number of mesh cells across the exit orifice; see typical results in Figure 7. With a low number of cells, the error of estimating the air-core dimension rises, which may harm the overall result of the simulation. The effect of cell number on \(C_{D}\), \(SCA\) and the air-core dimension is shown in Figure 8 for 2D transient and 3D steady and transient simulations. Note that the meshes were evenly sized. Therefore, the overall number of cells rises with a second power for 2D meshes from 6784 to 106,206 cells for 11 and 42 cells within \(r_{o}\) distance, respectively, and with a third power for 3D meshes from 131,948 to 686,300 for 16 and 27 cells in \(r_{o}\), respectively. The wall \(y^{+}\) for the finest mesh in the range from 6 to 40 for \(Re\) = 2000. The 3D mesh with 35 cells uses a similar cell size as the 27 cells  
Figure 5: Swirl velocity profiles from LDA measurement, left for \(Re\) = 2000 and various \(Z\) positions. Right for \(Z\) = 2 and 13 mm for various \(Re\).  
Figure 6: Axial velocity profiles from LDA measurements.  
version but contains 6 prismatic cells in the wall boundary layer to reach the wall \(y^{+}\sim 1\) along the exit orifice. Moreover, three different dimensions of the outer outflow area were also compared, but the effect on the results was negligible. Thus, the middle size area was used for all the cases.  
The steady laminar simulation was found to be more prone to mesh sizing and reached a maximum difference in \(C_{D}\) of 7%, while the RANS simulation showed lower dependency on cell size. The transient laminar simulation overperformed the steady one in proximity of results to the experimental data and showed a small effect of the mesh size. The sensitivity of the steady model can be explained by different VOF schemes; the steady simulations used a Compressive scheme, while the transient one used a more accurate geo-reconstruct scheme [30]. Moreover, the steady laminar simulations overestimated the air-core size, which leads to a decrease in \(C_{D}\). The optimal mesh for both 2D and 3D has at least 20 cells in \(r_{o}\), which is in line with other authors; however, a large variance was found in the literature. [PERSON] et al. [19] suggest a mesh with roughly 13 cells in \(r_{o}\), [PERSON] et al. [19] used approximately 25 cells in \(r_{o}\), [PERSON] et al. [16] similarly used around 25 polyhedral cells in \(r_{o}\) plus 7 cells of prismatic layer, while [PERSON] et al. [12] found a mesh independence for a much finer mesh, with roughly 40-50 cells in \(r_{o}\) and [PERSON] et al. [14] used even finer mesh with 80 cells in \(r_{o}\). However, the overall effect of the mesh size is rather small  
Figure 8: Mesh independence test: the effect of cell number in \(r_{o}\). Laminar model, \(Re=2000\).  
Figure 7: Typical results obtained from 3D periodic mesh with 27 cells in \(r_{o}\), transient, Laminar simulation. \(Re=2000\), instantaneous image. Blue and red represent liquid and air, respectively, and green is an interface.  
and other factors, such as a turbulence model, time statistics or initial conditions, may have a greater effect.  
The main advantage of the steady simulation is its time efficiency. It took between 5-9 h for mesh with 686,300 cells on a 12-cores machine (Intel(r) Xeon(r) Silver 4214 @ 2.2 GHz, Santa Clara County, CA, USA) to reach the quasi-steady state. The same case but transient took about 3-4 weeks to produce the same level of convergence. For practical usage, the steady simulations are much more suitable.