Hostname: page-component-54dcc4c588-64p75 Total loading time: 0 Render date: 2025-10-05T11:12:37.968Z Has data issue: false hasContentIssue false
  • English
  • Français

Early-time jet formation in liquid–liquid impact problems: theory and simulations

Published online by Cambridge University Press:  11 October 2018

R. Cimpeanu Open the ORCID record for R. Cimpeanu [Opens in a new window]  and
Madeleine Rose Moore Open the ORCID record for Madeleine Rose Moore [Opens in a new window]
R. Cimpeanu*
Affiliation:
Mathematical Institute, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
Madeleine Rose Moore
Affiliation:
Mathematical Institute, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
*
†Email address for correspondence: radu.cimpeanu@imperial.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We perform a thorough qualitative and quantitative comparison of theoretical predictions and direct numerical simulations for the two-dimensional, vertical impact of two droplets of the same fluid. In particular, we show that the theoretical predictions for the location and velocity of the jet root are excellent in the early stages of the impact, while the predicted jet velocity and thickness profiles are also in good agreement with the computations before the jet begins to bend. By neglecting the role of the surrounding gas both before and after impact, we are able to use Wagner theory to describe the early-time structure of the impact. We derive the model for general droplet velocities and radii, which encompasses a wide range of impact scenarios from the symmetric impact of identical drops to liquid drops impacting a deep pool. The leading-order solution is sufficient to predict the curve along which the root of the high-speed jet travels. After moving into a frame fixed in this curve, we are able to derive the zero-gravity shallow-water equations governing the leading-order thickness and velocity of the jet. Our numerical simulations are performed in the open-source software Gerris, which allows for the level of local grid refinement necessary for a problem with such a wide variety of length scales. The numerical simulations incorporate more of the physics of the problem, in particular the surrounding gas, the fluid viscosities, gravity and surface tension. We compare the computed and predicted solutions for a range of droplet radii and velocities, finding excellent agreement in the early stage. In light of these successful comparisons, we discuss the tangible benefits of using Wagner theory to confidently track properties such as the jet-root location, jet thickness and jet velocity in future studies of splash jet/ejecta evolution.

JFM classification

Drops and Bubbles: Drops and Bubbles Multiphase and Particle-laden Flows: Gas/liquid flow Interfacial Flows (free surface): Interfacial Flows (free surface)

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 856 , 10 December 2018 , pp. 764 - 796
Check for updates
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

Article last updated 07 March 2023

References

Agbaglah, G., Thoraval, M.-J., Thoroddsen, S. T., Zhang, L. V., Fezzaa, K. & Deegan, R. D. 2015 Drop impact into a deep pool: vortex shedding and jet formation. J. Fluid Mech. 764, R1.Google Scholar
Armand, J.-L. & Cointe, R. 1987 Hydrodynamic impact analysis of a cylinder. Trans. ASME J. Offshore Mech. Arctic Engng 111, 109–114.Google Scholar
Castrejón-Pita, A. A., Castrejón-Pita, J. R. & Hutchings, I. M. 2012 Experimental observation of von Kármán vortices during drop impact. Phys. Rev. E 86 (4), 045301.Google Scholar
Cimpeanu, R. & Papageorgiou, D. T. 2018 Three-dimensional high speed drop impact onto solid surfaces at arbitrary angles. Intl J. Multiphase Flow 107, 192–207.Google Scholar
Coppola, G., Rocco, G. & de Luca, L. 2011 Insights on the impact of a plane drop on a thin liquid film. Phys. Fluids 23 (2), 022105.Google Scholar
Hicks, P. D. & Purvis, R. 2010 Air cushioning and bubble entrapment in three-dimensional droplet impacts. J. Fluid Mech. 649 (1), 135–163.Google Scholar
Hicks, P. D. & Purvis, R. 2013 Liquid–solid impacts with compressible gas cushioning. J. Fluid Mech. 735, 120–149.Google Scholar
Howison, S. D., Ockendon, J. R., Oliver, J. M., Purvis, R. & Smith, F. T. 2005 Droplet impact on a thin fluid layer. J. Fluid Mech. 542, 1–23.Google Scholar
Howison, S. D., Ockendon, J. R. & Wilson, S. K. 1991 Incompressible water-entry problems at small deadrise angles. J. Fluid Mech. 222, 215–230.Google Scholar
Jian, Z., Josserand, C., Popinet, S., Ray, P. & Zaleski, S. 2018 Two mechanisms of droplet splashing on a solid substrate. J. Fluid Mech. 835, 1065–1086.Google Scholar
Josserand, C., Ray, P. & Zaleski, S. 2016 Droplet impact on a thin liquid film: anatomy of the splash. J. Fluid Mech. 802, 775–805.Google Scholar
Josserand, C. & Thoroddsen, S. T. 2016 Drop impact on a solid surface. Annu. Rev. Fluid Mech. 48, 365–391.Google Scholar
Josserand, C. & Zaleski, S. 2003 Droplet splashing on a thin liquid film. Phys. Fluids 15 (6), 1650–1657.Google Scholar
Kolinski, J. M., Mahadevan, L. & Rubinstein, S. M. 2014 Lift-off instability during the impact of a drop on a solid surface. Phys. Rev. Lett. 112, 134501.Google Scholar
Kolinski, J. M., Rubinstein, S. M., Mandre, S., Brenner, M. P., Weitz, D. A. & Mahadevan, L. 2012 Skating on a film of air: drops impacting on a surface. Phys. Rev. Lett. 108, 074503.Google Scholar
Korobkin, A. A. 1985 Initial asymptotic behavior of a solution to the three-dimensional problem concerning the entry of a blunt body into an ideal fluid. Dokl. Akad. Nauk SSSR 283, 838–842.Google Scholar
Korobkin, A. A. 2007 Second-order wagner theory of wave impact. J. Engng Maths 58 (1), 121–139.Google Scholar
Korobkin, A. A. & Scolan, Y.-M. 2006 Three-dimensional theory of water impact. Part 2. Linearized Wagner problem. J. Fluid Mech. 549, 343–374.Google Scholar
Langley, K. R., Li, E. Q. & Thoroddsen, S. T. 2017 Impact of ultra-viscous drops: air-film gliding and extreme wetting. J. Fluid Mech. 813, 647–666.Google Scholar
Li, E. Q, Langley, K. R., Tian, Y. S., Hicks, P. D. & Thoroddsen, S. T. 2017 Double contact during drop impact on a solid under reduced air pressure. Phys. Rev. Lett. 119 (21), 214502.Google Scholar
Li, E. Q. & Thoroddsen, S. T. 2015 Time-resolved imaging of a compressible air disc under a drop impacting on a solid surface. J. Fluid Mech. 780, 636–648.Google Scholar
Li, E. Q., Vakarelski, I. U. & Thoroddsen, S. T. 2015 Probing the nanoscale: the first contact of an impacting drop. J. Fluid Mech. 785, R2.Google Scholar
Liu, T., Tan, P. & Xu, L. 2015 Kelvin–Helmholtz instability in an ultrathin air film causes drop splashing on smooth surfaces. Proc. Natl Acad. Sci. USA 112 (11), 3280–3284.Google Scholar
Mandre, S. & Brenner, M. P. 2012 The mechanism of a splash on a dry solid surface. J. Fluid Mech. 690, 148–172.Google Scholar
Mandre, S., Mani, M. & Brenner, M. P. 2009 Precursors to splashing of liquid droplets on a solid surface. Phys. Rev. Lett. 102, 134502.Google Scholar
Mani, M., Mandre, S. & Brenner, M. P. 2010 Events before droplet splashing on a solid surface. J. Fluid Mech. 647, 163–185.Google Scholar
Martin, G. D., Hoath, S. D. & Hutchings, I. M. 2008 Inkjet printing – the physics of manipulating liquid jets and drops. J. Phys.: Conf. Ser. 105 (1), 012001.Google Scholar
Moore, M. R., Ockendon, H., Ockendon, J. R. & Oliver, J. M. 2014 Capillary and viscous perturbations to Helmholtz flows. J. Fluid Mech. 742, R1.Google Scholar
Moore, M. R., Whiteley, J. P. & Oliver, J. M. 2018 On the deflection of a liquid jet by an air-cushioning layer. J. Fluid Mech. 846, 711–751.Google Scholar
Murphy, D. W., Li, C., d’Albignac, V., Morra, D. & Katz, J. 2015 Splash behaviour and oily marine aerosol production by raindrops impacting oil slicks. J. Fluid Mech. 780, 536–577.Google Scholar
Oliver, J. M.2002 Water entry and related problems. DPhil thesis, University of Oxford.Google Scholar
Oliver, J. M. 2007 Second-order wagner theory for two-dimensional water-entry problems at small deadrise angles. J. Fluid Mech. 572, 59–85.Google Scholar
Philippi, J., Lagrée, P.-Y. & Antkowiak, A. 2016 Drop impact on a solid surface: short-time self-similarity. J. Fluid Mech. 795, 96–135.Google Scholar
Popinet, S. 2003 Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries. J. Comput. Phys. 190, 572–600.Google Scholar
Popinet, S. 2009 An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 228, 5838–5866.Google Scholar
Purvis, R. & Smith, F. T. 2005 Droplet impact on water layers: post-impact analysis and computations. Phil. Trans. R. Soc. Lond. A 363, 1209–1221.Google Scholar
Riboux, G. & Gordillo, J. M. 2014 Experiments of drops impacting a smooth solid surface: a model of the critical impact speed for drop splashing. Phys. Rev. Lett. 113 (2), 024507.Google Scholar
Riboux, G. & Gordillo, J. M. 2015 The diameters and velocities of the droplets ejected after splashing. J. Fluid Mech. 772, 630–648.Google Scholar
Riboux, G. & Gordillo, J. M. 2017 Boundary-layer effects in droplet splashing. Phys. Rev. E 96 (1), 013105.Google Scholar
de Ruiter, J., van den Ende, D. & Mugele, F. 2015 Air cushioning in droplet impact. II. Experimental characterization of the air film evolution. Phys. Fluids 27, 012105.Google Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Scolan, Y.-M. & Korobkin, A. A. 2001 Three-dimensional theory of water impact. Part 1. Inverse Wagner problem. J. Fluid Mech. 440, 293–326.Google Scholar
Semenov, Y. A., Wu, G. X. & Korobkin, A. A. 2015 Impact of liquids with different densities. J. Fluid Mech. 766, 5–27.Google Scholar
Smith, F. T., Li, L. & Wu, G. X. 2003 Air cushioning with a lubrication/inviscid balance. J. Fluid Mech. 482, 291–318.Google Scholar
Thoraval, M.-J.2013 Drop impact splashing and air entrapment. PhD thesis, KAUST.Google Scholar
Thoraval, M.-J., Takehara, K., Etoh, T. G., Popinet, S., Ray, P., Josserand, C., Zaleski, S. & Thoroddsen, S. T. 2012 Von Kármán vortex street within an impacting drop. Phys. Rev. Lett. 108 (26), 264506.Google Scholar
Thoraval, M.-J., Takehara, K., Etoh, T. G. & Thoroddsen, S. T. 2013 Drop impact entrapment of bubble rings. J. Fluid Mech. 724, 234–258.Google Scholar
Thoroddsen, S. T. 2002 The ejecta sheet generated by the impact of a drop. J. Fluid Mech. 451, 373–381.Google Scholar
Thoroddsen, S. T., Etoh, T. G., Takehara, K., Ootsuka, N. & Hatsuki, Y. 2005 The air bubble entrapped under a drop impacting on a solid surface. J. Fluid Mech. 545 (1), 203–212.Google Scholar
Thoroddsen, S. T., Thoraval, M.-J., Takehara, K. & Etoh, T. G. 2011 Droplet splashing by a slingshot mechanism. Phys. Rev. Lett. 106 (3), 034501.Google Scholar
Visser, C. W., Frommhold, P. E., Wildeman, S., Mettin, R., Lohse, D. & Sun, C. 2015 Dynamics of high-speed micro-drop impact: numerical simulations and experiments at frame-to-frame times below 100 ns. Soft Matt. 11, 1708–1722.Google Scholar
Wagner, H. 1932 Über Stoß- und Gleitvorgänge an der Oberfläche von Flüssigkeiten. Z. Angew. Math. Mech. 12, 193–215.Google Scholar
Wildeman, S., Visser, C. W., Sun, C. & Lohse, D. 2016 On the spreading of impacting drops. J. Fluid Mech. 805, 636–655.Google Scholar
Wilson, S. K. 1991 A mathematical model for the initial stages of fluid impact in the presence of a cushioning fluid layer. J. Engng Maths 25 (3), 265–285.Google Scholar
Yarin, A. L. 2006 Drop impact dynamics: splashing, spreading, receding, bouncing. Annu. Rev. Fluid Mech. 38, 159–192.Google Scholar
Zhang, L. V., Toole, J., Fezzaa, K. & Deegan, R. D. 2012a Evolution of the ejecta sheet from the impact of a drop with a deep pool. J. Fluid Mech. 690 (5), 5–15.Google Scholar
Zhang, L. V., Toole, J., Fezzaa, K. & Deegan, R. D. 2012b Splashing from drop impact into a deep pool: multiplicity of jets and the failure of conventional scaling. J. Fluid Mech. 703, 402–413.Google Scholar
Zhao, R. & Faltinsen, O. 1993 Water entry of two-dimensional bodies. J. Fluid Mech. 246 (1), 593–612.Google Scholar