# Droplet Impact on Suspended Metallic Meshes: Effects of Wettability, Reynolds and Weber Numbers

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Simulation Framework

#### 2.1. Governing Equations

#### 2.2. Sharpening the Interface

#### 2.3. VOF Smoothing

#### 2.4. Dynamic Contact Angle Treatment

## 3. Validation of the Numerical Simulation Framework

#### 3.1. Low Weber Number Impacts

#### 3.2. High Weber Number Impacts

## 4. Droplet Impact on Metallic Meshes

#### 4.1. Experimental Investigation

#### 4.2. Numerical Investigation

^{®}is used. In Figure 11 the actual metallic mesh geometry from the experiments and the corresponding CAD model, are depicted. For the grid (mesh) generation, the snappyHexMesh (sHM) utility of OpenFOAM is utilised. In Figure 12 the computational domain, the boundary conditions as well as the metallic mesh position after using the snappyHexMesh utility is illustrated. For reducing the total number of cells and hence the computational time and cost, one-fourth of the total 3D domain with symmetry planes is simulated.

#### 4.3. Numerical Simulation Results for Droplet Impact on Metallic Meshes

## 5. Parametric Numerical Simulations

#### 5.1. Influence of Reynolds Number

#### 5.2. Influence of Weber Number

#### 5.3. Influence of Wettability

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Comparison of the numerical results obtained using the Kistler dynamic contact angle (DCA) model with experimental data from the work of Yokoi et al. [14].

**Figure 3.**Qualitative comparison of the results of the 2D ( 5 $\mathsf{\mu}$$\mathrm{m}$) and 3D numerical simulation results with experimental snapshots [15].

**Figure 4.**Comparison of the evolution of the contact diameter (D) of experimental data [15] with the 2D axisymmetric and 3D numerical simulation results over time.

**Figure 5.**Sketch of the experimental setup, consisting of an LED light source and high-speed camera. A top view of the mesh is shown on the bottom of the figure. The velocity of the droplet impact is adjusted by changing the height of the needle from which the droplets are released.

**Figure 6.**(

**a**) Deposition outcome; water & glycerol, $Re=5030$ (

**b**) Partial imbibition outcome; water, $Re=5030$ (

**c**) Partial imbibition outcome; water, $Re=$ 10,778 (

**d**) Partial imbibition outcome; water, $Re=5209$ (

**e**) Penetration outcome; water, $Re=$ 10,419.

**Figure 7.**Liquid penetration on the suspended meshes of water droplets. For case (

**a**) ${D}_{p}=25\mathsf{\mu}\mathrm{m}$ and ${D}_{w}=25\mathsf{\mu}\mathrm{m}$, for case (

**b**) ${D}_{p}=400\mathsf{\mu}\mathrm{m}$, ${D}_{w}=220\mathsf{\mu}\mathrm{m}$.

**Figure 8.**Percentage of liquid penetration as a function of pore size consisting different values of impact velocity, for three different liquids.

**Figure 9.**Percentage of liquid penetration as a function of pore size given different range of impact velocity.

**Figure 10.**Percentage of liquid penetration of water droplet with different initial diameter and impact velocity.

**Figure 11.**Metallic mesh for the considered case (${D}_{p}=400\mathsf{\mu}\mathrm{m}$ and ${D}_{w}=220\mathsf{\mu}\mathrm{m}$. (

**a**) microscopic view of the metallic mesh, (

**b**) zoomed view of the metallic mesh generated in Onshape CAD, (

**c**) $1/{4}^{th}$ of the total solid domain of the metallic mesh used for the simulations as generated in CAD.

**Figure 12.**Computational geometry with pore and wire diameter 400 and 220 $\mathsf{\mu}$$\mathrm{m}$ respectively, mesh and boundary conditions for the 3D domain (one-fourth of the total 3D domain) of all simulation cases I–IV. (

**a**) Case I domain. (

**b**) Case II domain. The details of the mesh of case I are shown in (

**d**,

**e**). The boundary conditions of the numerical cases can be seen in figure (

**b**,

**c**). Finally, in figure (

**f**) release droplet position for cases I and II and (

**g**) release droplet position of cases III and IV can be seen.

**Figure 13.**Side view comparison of experimental snapshots of droplet impacting on metallic mesh (${D}_{0}=$$2.70$ $\mathrm{m}$$\mathrm{s}$, ${U}_{0}=$$1.85$ $\mathrm{m}$$\mathrm{s}$) and numerical simulations I–IV, at different time stages.

**Figure 14.**Dimensionless volume (${v}^{\ast}$) of liquid positioned below the metallic mesh as a function of dimensionless time (${t}^{\ast}$) for the numerical cases I–IV and corresponding experiment. An error bar of 10% of the dimensionless volume value in the experimental results have been added due to the limited estimation of the experimental results volume values.

**Figure 15.**Case II: Droplet details after impact at $t=0.2\mathrm{m}\mathrm{s}$, $t=1.6\mathrm{m}\mathrm{s}$, and $t=3.0\mathrm{m}\mathrm{s}$. (

**a**) 3D representation of the droplet; (

**b**) volume fraction; (

**c**) pressure; (

**d**) velocity magnitude.

**Figure 16.**Comparison of droplet impact output for different Reynolds numbers, for instance ${t}^{\ast}=0.7$, $2.7$, $6.0$, magnitude U is shown in $\mathrm{m}$ ${\mathrm{s}}^{-1}$.

**Figure 17.**Dimensionless liquid volume above the metallic mesh versus the dimensionless time ${t}^{\ast}$ for different Reynolds numbers.

**Figure 18.**Qualitative comparison of droplet impact output for different $We$ number values, at times ${t}^{\ast}=0.7$, ${t}^{\ast}=2.7$, ${t}^{\ast}=6.0$, magnitude U is shown in $\mathrm{m}$ ${\mathrm{s}}^{-1}$.

**Figure 19.**Weber number effects: The numerical case II is compared with virtual cases IV-a and IV-b which has surface tension of approximately one-third and two-thirds surface tension value of case II. The graph depicts the dimensionless volume of liquid (${v}^{\ast}$) above the mesh over the dimensionless time (${t}^{\ast}$).

**Figure 20.**Comparison of droplet impact of different wettability values for time periods ${t}^{\ast}=0.7$, ${t}^{\ast}=2.7$, ${t}^{\ast}=6.0$. The velocity magnitude (U) is also shown in $\mathrm{m}$ ${\mathrm{s}}^{-1}$. The investigated contact angles are: ${\theta}_{a}={60}^{\xb0}$, ${\theta}_{r}={43}^{\xb0}$ (case V-a-hydrophilic surface), ${\theta}_{a}={110}^{\xb0}$, ${\theta}_{r}={93}^{\xb0}$ (case II-hydrophobic surface), ${\theta}_{a}={115}^{\xb0}$, ${\theta}_{r}={98}^{\xb0}$ (case V-b-hydrophobic surface) and ${\theta}_{a}={162}^{\xb0}$ and ${\theta}_{r}={154}^{\xb0}$ (case V-c-superhydrophobic surface).

**Figure 21.**Investigation of wettability effects on the droplet’s behaviour, by altering the advancing ${\theta}_{a}$ and receding ${\theta}_{r}$ contact angles. The graph is showing the dimensionless volume of liquid (${v}^{\ast}$) above the mesh over the dimensionless time ${t}^{\ast}$.

Liquid | $\mathit{\rho}$ (kg m${}^{-3}$) | $\mathit{\mu}$ $\times {10}^{-3}$ (Pa s) | $\mathit{\sigma}$ (N m${}^{-1}$) | $\mathbf{Re}$ Range (-) | $\mathbf{We}$ Range (-) |
---|---|---|---|---|---|

Water | 996.0 | 1.0 | 0.073 | 3501.1–9924.4 | 83.9–499.8 |

Acetone | 793.0 | 0.3 | 0.023 | 7835.9–19,518.3 | 179.3–890.1 |

Water and Glycerol | 1118.6 | 10.0 | 0.067 | 448.2–1042.5 | 120.0–560.3 |

Liquid | Needle 21 (mm) | Needle 26 s (mm) |
---|---|---|

Water | 3.0 | 1.9 |

Acetone | 2.0 | 1.7 |

Water and Glycerol | 2.9 | 1.5 |

Pore Size ($\mathsf{\mu}$m) | Impact Velocity ${\mathit{U}}_{0}$ (m s${}^{-1}$) | % Liquid Penetration Size: 1.5 cm | % Liquid Penetration Size: 2.0 cm | % Liquid Penetration Size: 2.5 cm |
---|---|---|---|---|

25 | 1.86 | 7.0 | 10.0 | 7.0 |

25 | 2.70 | 13.0 | 11.0 | 14.0 |

25 | 3.60 | 46.0 | 42.0 | 43.0 |

200 | 1.86 | 79.0 | 76.0 | 77.0 |

200 | 2.70 | 90.0 | 88.0 | 95.0 |

200 | 3.60 | 100.0 | 98.0 | 99.0 |

400 | 1.86 | 96.0 | 94.0 | 96.0 |

400 | 2.70 | 100.0 | 100.0 | 100.0 |

400 | 3.60 | 100.0 | 100.0 | 100.0 |

Pore Size ($\mathsf{\mu}$m) | Impact Velocity ${\mathit{U}}_{0}$ (m s${}^{-1}$) | % Liquid Penetration Size: 1.5 cm | % Liquid Penetration Size: 2.0 cm | % Liquid Penetration Size: 2.5 cm |
---|---|---|---|---|

25 | 1.85 | 0.0 | 0.0 | 0.0 |

25 | 2.70 | 21.1 | 18.6 | 19.4 |

25 | 3.6 | 51.2 | 56.9 | 54.3 |

200 | 1.85 | 100.0 | 100.0 | 100.0 |

200 | 2.70 | 100.0 | 100.0 | 100.0 |

200 | 3.6 | 100.0 | 100.0 | 100.0 |

400 | 1.85 | 100.0 | 100.0 | 100.0 |

400 | 2.70 | 100.0 | 100.0 | 100.0 |

400 | 3.6 | 100.0 | 100.0 | 100.0 |

Pore Size ($\mathsf{\mu}$m) | Impact Velocity ${\mathit{U}}_{0}$ (m s${}^{-1}$) | % Liquid Penetration Size: 1.5 cm | % Liquid Penetration Size: 2.0 cm | % Liquid Penetration Size: 2.5 cm |
---|---|---|---|---|

25 | 1.85 | 0.0 | 0.0 | 0.0 |

25 | 2.70 | 0.0 | 4.7 | 0.0 |

25 | 3.60 | 0.0 | 10.0 | 0.0 |

200 | 1.85 | 58.4 | 57.6 | 60.0 |

200 | 2.70 | 91.2 | 94.4 | 87.3 |

200 | 3.60 | 95.4 | 97.1 | 97.0 |

400 | 1.85 | 86.4 | 89.9 | 83. 3 |

400 | 2.70 | 100.0 | 100.0 | 100.0 |

400 | 3.60 | 100.0 | 100.0 | 100.0 |

**Table 6.**Initial droplet parameters and characteristics for the considered experiment, at $t=00\mathrm{m}\mathrm{s}$. Initial droplet diameter ${D}_{0}$, impact velocity ${U}_{0}$, $We$, $Re$ numbers and advancing ${\theta}_{a}$ and receding ${\theta}_{r}$ contact angles of the droplet.

${\mathit{D}}_{0}$ (mm) | ${\mathit{U}}_{0}$ (m s${}^{-1}$) | $\mathit{We}$ (-) | $\mathit{Re}$ (-) | ${\mathit{\theta}}_{\mathit{a}}$ (${}^{\circ}$) | ${\mathit{\theta}}_{\mathit{r}}$ (${}^{\circ}$) |
---|---|---|---|---|---|

2.70 | 1.85 | 126 | 4953 | 110 | 93 |

**Table 7.**Computational parameters of the four numerical simulation cases considered. The mesh generation used the snappyHexMesh (sHM) and topoSet utilities of OpenFOAM.

Numerical Case | Droplet Centering at $\mathit{t}=0\phantom{\rule{0.166667em}{0ex}}\mathbf{ms}$ | Computational Domain (mm) | Total no. of Cells (Millions) | Levels of Refinement |
---|---|---|---|---|

I | above two wires | (8.00 8.00 25.00) | 31.9 | 4 |

II | above two wires | (7.54 7.54 20.00) | 4.0 | 3 |

III | above one wire | (8.00 8.00 25.00) | 31.9 | 4 |

IV | above one wire | (7.54 7.54 20.00) | 4.0 | 3 |

**Table 8.**Numerical cases performed for investigating the effects of dynamic viscosity ($\mu $), surface tension ($\sigma $), and wettability (${\theta}_{a}$ and ${\theta}_{r}$). The parameter that is changed in each case is indicated in bold. Case II corresponds to the base case.

Case | $\mathit{\mu}$ $\times {10}^{-3}$ (Pa s) | $\mathit{\sigma}$ (N m${}^{-1}$) | ${\mathit{\theta}}_{\mathit{a}}$ (${}^{\circ}$) | ${\mathit{\theta}}_{\mathit{r}}$ (${}^{\circ}$) | $\mathit{We}$ (-) | $\mathit{Re}$ (-) |
---|---|---|---|---|---|---|

II | 1 | 0.073 | 110 | 93 | 126 | 4985 |

III-a | 2.5 | 0.073 | 110 | 93 | 126 | 1994 |

III-b | 5.0 | 0.073 | 110 | 93 | 126 | 997 |

III-c | 7.5 | 0.073 | 110 | 93 | 126 | 665 |

III-d | 10.0 | 0.073 | 110 | 93 | 126 | 499 |

IV-a | 1.0 | 0.022 | 110 | 93 | 416 | 4985 |

IV-b | 1.0 | 0.048 | 110 | 93 | 194 | 4985 |

V-a | 1.0 | 0.073 | 60 | 43 | 126 | 4985 |

V-b | 1.0 | 0.073 | 115 | 98 | 126 | 4985 |

V-c | 1.0 | 0.073 | 162 | 154 | 126 | 4985 |

**Table 9.**Percentage of liquid that remained above and entrapped within the metallic mesh at different time periods.

Case | $\mathit{t}=$ 0.80 ms | $\mathit{t}=$ 1.6 ms | $\mathit{t}=$ 2.40 ms | $\mathit{t}=$ 4.40 ms | $\mathit{t}=$ 6.60 ms | $\mathit{t}=$ 8.80 ms |
---|---|---|---|---|---|---|

II | 72.0 | 33.0 | 16.1 | 4.3 | 1.8 | 1.7 |

III-a | 72.3 | 38.5 | 26.9 | 21.3 | 22.2 | 21.4 |

III-b | 74.8 | 46.1 | 38.1 | 39.6 | 38.8 | 37.5 |

III-c | 76.3 | 50.7 | 44.9 | 46.5 | 45.4 | 45.6 |

III-d | 77.5 | 54.0 | 49.8 | 57.9 | 58.9 | 58.9 |

IV-a | 68.8 | 27.2 | 9.6 | 2.0 | 2.8 | 2.2 |

IV-b | 69.1 | 29.0 | 11.6 | 3.4 | 1.4 | 1.9 |

V-a | 66.2 | 15.7 | 5.2 | 8.5 | 8.7 | 1.0 |

V-b | 70.4 | 29.6 | 12.6 | 2.7 | 0.7 | 0.1 |

V-c | 73.9 | 47.6 | 42.0 | 43.5 | 43.6 | 43.6 |

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## Share and Cite

**MDPI and ACS Style**

Vontas, K.; Boscariol, C.; Andredaki, M.; Georgoulas, A.; Crua, C.; Walther, J.H.; Marengo, M.
Droplet Impact on Suspended Metallic Meshes: Effects of Wettability, Reynolds and Weber Numbers. *Fluids* **2020**, *5*, 81.
https://doi.org/10.3390/fluids5020081

**AMA Style**

Vontas K, Boscariol C, Andredaki M, Georgoulas A, Crua C, Walther JH, Marengo M.
Droplet Impact on Suspended Metallic Meshes: Effects of Wettability, Reynolds and Weber Numbers. *Fluids*. 2020; 5(2):81.
https://doi.org/10.3390/fluids5020081

**Chicago/Turabian Style**

Vontas, Konstantinos, Cristina Boscariol, Manolia Andredaki, Anastasios Georgoulas, Cyril Crua, Jens Honoré Walther, and Marco Marengo.
2020. "Droplet Impact on Suspended Metallic Meshes: Effects of Wettability, Reynolds and Weber Numbers" *Fluids* 5, no. 2: 81.
https://doi.org/10.3390/fluids5020081