# Numerical Simulations of Swirling Water Jet Atomization: A Mesh Convergence Study

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## Abstract

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## 1. Introduction

## 2. Computational Details

#### 2.1. Governing Equations

#### 2.2. Computational Domain

#### 2.3. Velocity Fluctuations at the Inlet

#### 2.4. Mesh Refinement

## 3. Results and Discussion

#### 3.1. Averaged Fields

#### 3.2. Instability on the Film Surface

#### 3.3. Droplets Statistics

## 4. Conclusions

- The use of a coarse mesh leads to a significantly erroneous flow structure. The liquid jet does not form a cone, but flows out in the form of a filled swirling jet. An increase in the mesh adaptive refinement level leads to a qualitatively correct flow structure and a formation of a recirculation zone near the nozzle. A further improvement of the mesh quality does not change the mean distribution of flow fields.
- The distribution of the number of drops by size does not seem to converge with an increase in mesh resolution. An increase in the quality of the mesh leads to a shift of the distribution maximum to smaller diameters. When, due to the physics of the process, the value of the computational cell becomes much smaller than the expected mean characteristic size of the droplets, an unphysical peak is observed in the distribution. The droplet diameter corresponding to this peak is close to the value of the minimum calculated cell. The formation of such parasitic drops is associated with a change in a small-scale topology of the jet and a numerical error. At the breakup point, the thickness of the liquid film tends to zero and this raises the numerical errors in the film shape approximation.
- Improving mesh quality leads to the convergence of area and volume distributions, depending on their diameter. Despite the superior number of parasitic droplets, their influence on these parameters is insignificant.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Prescribed longitudinal (

**a**) and tangential (

**b**) mean velocity profiles at the inlet boundary.

**Figure 2.**Jet side view at time 1.5 ms. (

**a**)—Level 7, (

**b**)—Level 8, (

**c**)—Level 9, (

**d**)—Level 10. Isosurfaces of a liquid volume fraction of 0.5 are shown.

**Figure 3.**Time and azimuthal-direction averaged liquid volume fraction. (

**a**)—Level 7, (

**b**)—Level 8, (

**c**)—Level 9. Max value (red): 1, min value (blue): 0.

**Figure 4.**Time and azimuthal-direction averaged longitudinal velocity. (

**a**)—Level 7, (

**b**)—Level 8, (

**c**)—Level 9, (

**d**)— velocity profiles for Levels 8 and 9. Max velocity (red): 10 m/s, min velocity (blue): −5 m/s.

**Figure 5.**Time and azimuthal-direction averaged radial velocity. (

**a**)—Level 7, (

**b**)—Level 8, (

**c**)—Level 9. Max velocity (red): 3 m/s, min velocity (blue): −1 m/s.

**Figure 6.**Time and azimuthal-direction averaged azimuthal velocity. (

**a**)—Level 7, (

**b**)—Level 8, (

**c**)—Level 9. Max velocity (red): 5 m/s, min velocity (blue): −1 m/s.

**Figure 7.**Jet surface at different maximum refinement levels of the computational mesh superimposed with instantaneous computational mesh cross-section. (

**a**)—Level 8, (

**b**)—Level 9, (

**c**)—Level 10, (

**d**)—Level 11.

**Figure 8.**Histograms of the droplets characteristics binned with the droplet diameter. (

**a**)—quantity distribution, (

**b**)—interface area distribution, (

**c**)—volume distribution.

Nozzle Diameter, $\mathsf{\mu}$m | ${\mathit{Re}}_{\mathit{l}}$ | ${\mathit{We}}_{\mathit{l}}$ | Min Linear Cell Size, $\mathsf{\mu}$m | Kolmogorov Scale, $\mathsf{\mu}$m | Hinze Scale ($\mathit{\sigma}{\mathit{We}}_{\mathit{cr}}/{\mathit{\rho}}_{\mathit{l}}{\mathit{U}}^{2}$), $\mathsf{\mu}$m | Sauter Mean Diameter, $\mathsf{\mu}$m | |
---|---|---|---|---|---|---|---|

Present paper | 800 | 6100 | 650 | 6 | 1.2 | 12 | 300 |

Pairetti et al. [7] | 100 | 5800 | 11,600 | 0.37 | 0.13 | 0.08 | <8 |

Constante-Amores et al. [31] | 4000 | 1000–10,000 | 10–1000 | 26 | 5.7–26 | 40–4000 | 300–4000 |

Jiao et al. [16] | 100 | 4300–5800 | 6000–8000 | 5 | 0.81 | 0.1–0.2 | 15 |

Torregrosa et al. [17] | 90 | 5037 | 27,000 | 2.34 | 0.5 | 0.03 | <5 |

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**MDPI and ACS Style**

Vozhakov, I.S.; Hrebtov, M.Y.; Yavorsky, N.I.; Mullyadzhanov, R.I.
Numerical Simulations of Swirling Water Jet Atomization: A Mesh Convergence Study. *Water* **2023**, *15*, 2552.
https://doi.org/10.3390/w15142552

**AMA Style**

Vozhakov IS, Hrebtov MY, Yavorsky NI, Mullyadzhanov RI.
Numerical Simulations of Swirling Water Jet Atomization: A Mesh Convergence Study. *Water*. 2023; 15(14):2552.
https://doi.org/10.3390/w15142552

**Chicago/Turabian Style**

Vozhakov, Ivan S., Mikhail Yu. Hrebtov, Nikolay I. Yavorsky, and Rustam I. Mullyadzhanov.
2023. "Numerical Simulations of Swirling Water Jet Atomization: A Mesh Convergence Study" *Water* 15, no. 14: 2552.
https://doi.org/10.3390/w15142552