Abstract
The present paper investigates experimentally the impact dynamics of crown-type splashing for miscible two- and one-component droplet wall–film interactions over a range of Weber numbers and dimensionless film thicknesses. The splashing outcome is parametrised in terms of a set of quantifiable parameters, such as crown height, top and base diameter, wall inclination, number of fingers, and secondary droplet properties. The results show that the outcome of a splashing event is not affected by the choice of similar or dissimilar fluids, provided the dimensionless film thickness is larger than 0.1. Below this threshold, distinctive features of two-component interactions appear, such as hole formation and crown bottom breakdown. The observation of different crown shapes (e.g. V-shaped, cylindrical, and truncated-cone) confirms that vorticity production induces changes in the crown wall inclination, thus affecting the evolution of the crown height and top diameter. The evolution of the crown base diameter, instead, is mainly dependent on the relative importance of liquid inertia and viscous losses in the wall-film. The maximum number of liquid fingers decreases with increasing wall, film thickness, due to the enhanced attenuation of the effect of surface properties on the fingering process. The formation of secondary droplets is also affected by changes in the crown wall inclination. In particular, for truncated-cone shapes the occurrence of crown rim contraction induces a large scatter in the secondary droplet properties. Consequently, empirical models for the maximum number and mean diameter of the secondary droplets are derived for V-shaped crowns, as observed for the hexadecane-Hyspin interactions.
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Abbreviations
- D :
-
Impinging drop diameter (m)
- \(D_{\text {Top}}\) :
-
Crown top diameter (m)
- \(D^{\star }_{\text {Top}}\) :
-
Non-dimensional top diameter (–)
- \(D_{\text {Base}}\) :
-
Crown base diameter (m)
- \(D^{\star }_{\text {Base}}\) :
-
Non-dimensional base diameter (–)
- \(\overline{D}_{1,0}\) :
-
Arithmetic mean diameter (m)
- \(H_{\text {CR}}\) :
-
Crown height (m)
- \(H^{\star }_{\text {CR}}\) :
-
Non-dimensional crown height (–)
- h :
-
Wall-film height (m)
- K :
-
Splashing threshold, \(We^x\ Oh^y\) (–)
- l :
-
Characteristic length (m)
- \(l_{\text {min}}\) :
-
Minimum finger height (pixel)
- \(N_{i}\) :
-
Number of drops (–)
- \(N_{\text {finger}}\) :
-
Number of liquid fingers (–)
- \(N_{\text {finger,max}}\) :
-
Maximum number of liquid fingers (–)
- \(N_{\text {drops}}\) :
-
Number of secondary droplets (–)
- \(N_{\text {drops,max}}\) :
-
Maximum number of secondary droplets (–)
- \(Oh_l\) :
-
Ohnesorge number, \(\mu\)/\(\sqrt{\rho \sigma l}\) (–)
- \(\overline{Oh}\) :
-
Averaged Ohnesorge number, \(\overline{\mu }\)/\(\sqrt{\overline{\rho } \overline{\sigma } D}\) (–)
- Re :
-
Reynolds number, \(U\ D\)/\(\nu\) (–)
- \(\overline{Re}\) :
-
Averaged Reynolds number, \(U\ D\)/\(\overline{\nu }\) (–)
- t :
-
Time (s)
- U :
-
Terminal velocity primary drop (m/s)
- \(u_{\text {drops}}\) :
-
Velocity of secondary droplets (m/s)
- \(V_{\text {rel}}\) :
-
Relative volume of ejected secondary droplets, \(V_{\text {secdrops,tot}}/V_{\text {drop,0}}\) (–)
- \(V_{\text {secdrops,tot}}\) :
-
Volume of ejected secondary droplets (m\(^3\))
- \(V_{\text {drop,0}}\) :
-
Volume of primary drop (m\(^3\))
- \(We_l\) :
-
Weber number, \(\rho U^2 l\)/\(\sigma\) (–)
- \(We_D\) :
-
Weber number primary drop, \(\rho U^2 D\)/\(\sigma\) (–)
- \(\overline{We}\) :
-
Averaged Weber number \(\overline{\rho } U^2 D\)/\(\overline{\sigma }\) (–)
- \(\alpha\) :
-
Crown inclination (\(^{\circ }\))
- \(\dot{\alpha }\) :
-
Temporal evolution of crown inclination (\(^{\circ }\))
- \(\delta\) :
-
Dimensionless film height, h / D (–)
- \(\mu\) :
-
Dynamic viscosity (Pa s)
- \(\nu\) :
-
Kinematic viscosity (mm\(^2\)/s)
- \(\rho\) :
-
Density (kg/m\(^3\))
- \(\sigma\) :
-
Surface tension (Nm\(^{-1}\))
- \(\tau\) :
-
Dimensionless time, tU/D (–)
- \(\tau _0\) :
-
Time of drop impact (s)
- \(\Omega\) :
-
Vorticity (1/s)
- a :
-
Antisymmetric
- C :
-
Constant parameter
- \({\text {ini}}\) :
-
Initial
- \({\text {max}}\) :
-
Maximum value of a parameter
- n :
-
Exponent
- s :
-
Symmetric
- f :
-
Film
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Acknowledgements
This work has been performed in the framework of the projects LA 2512/2-1 and WE 2549/24-1. The authors kindly acknowledge the financial support of the Deutsche Forschungsgemeinschaft (DFG). In addition, A. T. acknowledges the Alexander von Humboldt foundation for their funding support.
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Appendix
Appendix
Initial experimental parameters | Maximum crown parameters | Secondary droplet parameters | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Droplet | Film | D (mm) | U (m/s) | h (mm) | \(\delta\) (–) | We (–) | \(D_{\text{Top}}/D\) (–) | \(H_\mathrm{CR}/D\) (–) | \(D_{\text {Base}}/D\) (–) | \(N_\mathrm{finger,max}\) | \(N_\mathrm{drops,max}\) | \(V_\mathrm{rel}\) (–) | \(\overline{D}_{1,0}\) (mm) | \(\bar{u}_{\text {drops}}\) (m/s) |
Hexadecane | Hyspin | 2.40 | 4.46 | 0.252 | 0.11 | 1343 | 6.38 | 1.92 | 5.06 | 19 | 131 | 2.03 | 0.533 | 1.05 |
Hexadecane | Hyspin | 2.42 | 4.47 | 0.505 | 0.21 | 1363 | 6.08 | 2.68 | 5.49 | 16 | 47 | 1.7 | 0.766 | 1.08 |
Hexadecane | Hyspin | 2.40 | 4.42 | 0.654 | 0.27 | 1315 | 5.86 | 2.71 | 5.96 | 13 | 36 | 1.63 | 0.833 | 1.17 |
Hexadecane | Hyspin | 2.41 | 4.47 | 0.260 | 0.11 | 1356 | 6.48 | 1.93 | 5.11 | 20 | 210 | 2.77 | 0.522 | 1.21 |
Hexadecane | Hyspin | 2.42 | 4.49 | 0.498 | 0.21 | 1369 | 6.09 | 2.65 | 5.25 | 15 | 54 | 2.25 | 0.811 | 1.08 |
Hexadecane | Hyspin | 2.43 | 4.47 | 0.736 | 0.30 | 1361 | 5.9 | 2.92 | 5.93 | 12 | 50 | 2.49 | 0.925 | 1.03 |
Hexadecane | Hyspin | 2.36 | 4.31 | 0.240 | 0.10 | 1236 | 6.5 | 1.91 | 4.97 | 19 | 84 | 1.85 | 0.604 | 0.92 |
Hexadecane | Hyspin | 2.36 | 4.33 | 0.451 | 0.19 | 1245 | 6.08 | 2.6 | 5.26 | 14 | 50 | 1.72 | 0.793 | 0.80 |
Hexadecane | Hyspin | 2.37 | 4.34 | 0.508 | 0.21 | 1253 | 5.97 | 2.74 | 5.26 | 13 | 35 | 1.59 | 0.829 | 0.54 |
Hexadecane | Hyspin | 2.40 | 4.33 | 0.647 | 0.27 | 1267 | 5.81 | 2.76 | 5.86 | 12 | 21 | 1.48 | 1.019 | 0.53 |
Hexadecane | Hyspin | 2.36 | 4.32 | 0.263 | 0.11 | 1236 | 6.48 | 2.03 | 5.09 | 18 | 115 | 1.84 | 0.548 | 1.07 |
Hexadecane | Hyspin | 2.35 | 4.20 | 0.442 | 0.19 | 1166 | 5.94 | 2.29 | 4.96 | 15 | 39 | 1.76 | 0.77 | 0.65 |
Hexadecane | Hyspin | 2.34 | 4.20 | 0.504 | 0.22 | 1158 | 5.84 | 2.39 | 5.1 | 12 | 30 | 1.75 | 0.875 | 0.48 |
Hexadecane | Hyspin | 2.37 | 4.22 | 0.607 | 0.26 | 1191 | 5.76 | 2.53 | 5.31 | 11 | 17 | 1.13 | 1.007 | 0.57 |
Hexadecane | Hyspin | 2.35 | 4.07 | 0.210 | 0.09 | 1094 | 6.6 | 1.77 | 5.14 | 18 | 138 | 2.09 | 0.517 | 1.11 |
Hexadecane | Hyspin | 2.35 | 4.08 | 0.265 | 0.11 | 1099 | 6.38 | 1.96 | 4.97 | 18 | 69 | 1.86 | 0.69 | 0.89 |
Hexadecane | Hyspin | 2.35 | 4.09 | 0.446 | 0.19 | 1104 | 5.96 | 2.39 | 5.18 | 15 | 34 | 1.59 | 0.842 | 0.57 |
Hyspin | Hexadecane | 2.30 | 4.40 | 0.454 | 0.20 | 1363 | 5.41 | 2.98 | 5.5 | 13 | 73 | 2.71 | 0.72 | 1.48 |
Hyspin | Hexadecane | 2.34 | 4.38 | 0.697 | 0.30 | 1375 | 5.34 | 3.22 | 6.07 | 13 | 43 | 2.09 | 0.813 | 1.61 |
Hyspin | Hexadecane | 2.38 | 4.51 | 0.333 | 0.14 | 1483 | 5.79 | 2.88 | 5.27 | 15 | 133 | 3.07 | 0.514 | 1.21 |
Hyspin | Hexadecane | 2.37 | 4.41 | 0.342 | 0.14 | 1413 | 5.68 | 2.81 | 5.23 | 16 | 173 | 2.78 | 0.467 | 1.33 |
Hyspin | Hexadecane | 2.39 | 4.28 | 0.696 | 0.29 | 1341 | 5.21 | 3.2 | 6.05 | 11 | 37 | 2.06 | 0.898 | 1.25 |
Hyspin | Hexadecane | 2.48 | 4.16 | 0.502 | 0.20 | 1314 | 5.3 | 3.04 | 5.57 | 12 | 101 | 2.06 | 0.488 | 1.28 |
Hyspin | Hexadecane | 2.49 | 4.15 | 0.753 | 0.30 | 1313 | 5.43 | 3.18 | 6.22 | 12 | 55 | 1.87 | 0.668 | 1.02 |
Hyspin | Hexadecane | 2.37 | 4.47 | 0.352 | 0.15 | 1453 | 5.74 | 2.78 | 5.22 | 16 | 160 | 3.1 | 0.498 | 1.52 |
Hyspin | Hexadecane | 2.48 | 4.05 | 0.709 | 0.29 | 1242 | 5.3 | 3.07 | 5.89 | 12 | 84 | 1.53 | 0.831 | 1.34 |
Hyspin | Hexadecane | 2.50 | 4.05 | 0.743 | 0.30 | 1256 | 5.39 | 3.08 | 6.05 | 12 | 34 | 1.83 | 0.515 | 1.23 |
Hyspin | Hexadecane | 2.48 | 4.02 | 0.404 | 0.16 | 1229 | 5.61 | 2.78 | 5.23 | 16 | 67 | 2.26 | 0.772 | 1.05 |
Hyspin | Hexadecane | 2.49 | 3.92 | 0.496 | 0.20 | 1173 | 5.51 | 2.78 | 5.41 | 15 | 64 | 1.87 | 0.688 | 1.44 |
Hyspin | Hexadecane | 2.50 | 3.91 | 0.548 | 0.22 | 1171 | 5.42 | 2.87 | 5.59 | 13 | 46 | 2.1 | 0.803 | 0.94 |
Hyspin | Hexadecane | 2.50 | 3.73 | 0.353 | 0.14 | 1065 | 5.59 | 2.63 | 4.92 | 11 | 19 | 0.88 | 0.885 | 0.54 |
Hyspin | Hexadecane | 2.49 | 3.74 | 0.506 | 0.20 | 1071 | 5.58 | 2.59 | 5.31 | 11 | 41 | 1.82 | 0.853 | 1.34 |
Hyspin | Hexadecane | 2.47 | 3.72 | 0.329 | 0.13 | 1048 | 5.6 | 2.38 | 4.88 | 13 | 109 | 2.07 | 0.517 | 1.08 |
Hexadecane | Hexadecane | 2.48 | 4.46 | 0.256 | 0.10 | 1385 | 4.82 | 2.92 | 5.39 | 13 | 171 | 2.190 | 0.315 | 1.81 |
Hexadecane | Hexadecane | 2.40 | 4.43 | 0.458 | 0.19 | 1325 | 3.93 | 3.39 | 5.50 | 15 | 169 | 1.840 | 0.421 | 1.85 |
Hexadecane | Hexadecane | 2.38 | 4.44 | 0.493 | 0.21 | 1313 | 4.13 | 3.42 | 5.71 | 13 | 122 | 1.850 | 0.499 | 1.68 |
Hexadecane | Hexadecane | 2.38 | 4.41 | 0.693 | 0.29 | 1300 | 3.45 | 3.64 | 6.68 | 12 | 187 | 1.120 | 0.346 | 1.51 |
Hexadecane | Hexadecane | 2.47 | 4.43 | 0.749 | 0.30 | 1363 | 3.24 | 3.34 | 5.76 | 10 | 65 | 0.860 | 0.522 | 1.86 |
Hexadecane | Hexadecane | 2.48 | 4.44 | 0.751 | 0.30 | 1374 | 3.33 | 3.40 | 5.90 | 10 | 155 | 1.360 | 0.349 | 1.81 |
Hexadecane | Hexadecane | 2.45 | 4.35 | 0.251 | 0.10 | 1304 | 5.03 | 2.52 | 5.29 | 15 | 204 | 2.640 | 0.450 | 1.76 |
Hexadecane | Hexadecane | 2.41 | 4.25 | 0.510 | 0.21 | 1223 | 4.24 | 3.29 | 5.61 | 11 | 79 | 2.190 | 0.679 | 1.27 |
Hexadecane | Hexadecane | 2.42 | 4.23 | 0.258 | 0.11 | 1212 | 5.06 | 2.72 | 5.27 | 13 | 141 | 1.890 | 0.482 | 1.55 |
Hexadecane | Hexadecane | 2.36 | 4.23 | 0.725 | 0.31 | 1184 | 3.94 | 3.31 | 6.44 | 9 | 46 | 1.260 | 0.599 | 1.12 |
Hexadecane | Hexadecane | 2.38 | 4.11 | 0.246 | 0.10 | 1131 | 5.27 | 2.62 | 5.39 | 14 | 116 | 2.110 | 0.539 | 1.46 |
Hexadecane | Hexadecane | 2.42 | 4.12 | 0.730 | 0.30 | 1152 | 4.16 | 3.20 | 6.04 | 10 | 63 | 1.420 | 0.482 | 1.44 |
Hexadecane | Hexadecane | 2.41 | 3.98 | 0.246 | 0.10 | 1070 | 5.28 | 2.56 | 5.18 | 12 | 50 | 1.910 | 0.727 | 1.08 |
Hexadecane | Hexadecane | 2.42 | 4.00 | 0.475 | 0.20 | 1085 | 4.40 | 3.05 | 5.43 | 10 | 25 | 1.010 | 0.748 | 0.95 |
Hexadecane | Hexadecane | 2.41 | 3.98 | 0.235 | 0.10 | 1075 | 5.49 | 2.55 | 5.33 | 18 | 91 | 2.010 | 0.553 | 1.16 |
Hexadecane | Hsexadecane | 2.44 | 3.83 | 0.470 | 0.19 | 1006 | 5.10 | 2.58 | 5.22 | 13 | 58 | 2.550 | 0.785 | 1.28 |
Hyspin | Hyspin | 2.40 | 4.40 | 0.193 | 0.08 | 1424 | 6.85 | 1.86 | 5.11 | 19 | 42 | 1.010 | 0.681 | – |
Hyspin | Hyspin | 2.37 | 4.41 | 0.237 | 0.10 | 1407 | 7.01 | 2.13 | 5.02 | 15 | 13 | 0.380 | 0.690 | – |
Hyspin | Hyspin | 2.55 | 4.15 | 0.206 | 0.08 | 1345 | 6.90 | 2.03 | 4.99 | 15 | 27 | 0.730 | 0.787 | – |
Hyspin | Hyspin | 2.48 | 4.15 | 0.292 | 0.12 | 1312 | 6.75 | 2.38 | 4.91 | 16 | 17 | 0.470 | 0.731 | – |
Hyspin | Hyspin | 2.49 | 3.99 | 0.186 | 0.07 | 1212 | 6.76 | 1.86 | 5.08 | 14 | 37 | 0.070 | 0.220 | – |
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Geppert, A., Terzis, A., Lamanna, G. et al. A benchmark study for the crown-type splashing dynamics of one- and two-component droplet wall–film interactions. Exp Fluids 58, 172 (2017). https://doi.org/10.1007/s00348-017-2447-2
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DOI: https://doi.org/10.1007/s00348-017-2447-2