## 4 Conclusions  
The 2D and 3D numerical simulations of pressure-swirl atomizer internal flow were compared in transient and steady states with experimental data for \(Re\) = 1000, 2000 and 4000. Various numerical setups, including six turbulence models and several geometrical simplifications, were investigated. The air-core was found stable and developed in all the tested regimes. The discharge coefficient C\({}_{D}\) slightly decreased with \(Re\) as a result of lower relative viscous losses, which was confirmed by a slight increase in the measured relative swirl velocity. The results from steady simulations are comparable with the time-averaged transient ones, but their convergence rate is at least about an order of magnitude higher. No steady simulation setup was found stable for the 2D model. From the geometry simplification study, it is evident that the inlet port must be modelled in full to assure reliable results. The full 3D model can give only a little advantage compared to the periodical geometry. The laminar solver was capable to predict the \(C_{D}\), air-core dimensions, and velocity profiles with an error of less than 5% compared with the experimental results in both 2D and 3D simulations for the whole range of \(Re\). The LES model performed similarly to the laminar solver for low \(Re\) and was slightly more accurate for \(Re\) = 4000. The two-equation models, \(k\)-\(\varepsilon\) and \(k\)-\(\omega\), were sensitive to proper solving of the near wall flow and were not accurate for low \(Re\). However, all the models captured well the trends. The worst results were surprisingly obtained for RSM (Reynolds Stress model), which diverged for \(Re\) = 1000 and 2000 or predicted the undeveloped air-core. For an unknown reason, the only parameter predicted with a large error was the spray cone angle (\(SCA\)). The air-core diameter fluctuations were observed and the dominant frequency was found to rise with \(Re\) to keep a constant value of the Strouhal number. However, the transient simulations did not match the frequencies and turbulence models suppressed the amplitude of the surface waves. For the atomizer rapid CFD prototyping, the steady 3D simulation with a laminar solver is the most efficient approach. The transient simulations may be helpful in predicting unstable atomizers for some more extreme geometries, but experimental verification is still required. Figure 13: Frequency spectra of air-core diameter fluctuations at location (**left**): \(d_{ac2}\), (**right**): \(d_{ac11}\). Laminar simulation. **Author Contributions:** Conceptualization, M.M. and J.J.; methodology, M.M. and J.S. ; software, J.S. ; validation, J.S. and O.C. ; investigation, M.M., J.S. and O.C. ; data curation, M.M., J.S. and O.C. ; writing--original draft preparation, M.M. and J.S. ; writing--review and editing, M.M. ; visualization, M.M. ; supervision, J.J.; project administration, J.J.; funding acquisition, J.J. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by the project \"Computer Simulations for Effective Low-Emission Energy Engineering\" funded as project No. CZ.02.1.01/0.0/0.0/16_026/0008392 by Operational Programme Research, Development and Education, Priority axis 1: Strengthening capacity for high-quality research. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** The experimental data that support the findings of this study are openly available here: [[https://lurl.cz/WKLF9](https://lurl.cz/WKLF9)]([https://lurl.cz/WKLF9](https://lurl.cz/WKLF9)) (accessed on 19 June 2022). **Conflicts of Interest:** The authors declare no conflict of interest. **Nomenclature**