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Generation and breakup of Worthington jets after cavity collapse. Part 1. Jet formation

  • STEPHAN GEKLE (a1) (a2) and J. M. GORDILLO (a3)


At the beginning of the last century Worthington and Cole discovered that the high-speed jets ejected after the impact of an axisymmetric solid on a liquid surface are intimately related to the formation and collapse of an air cavity created in the wake of the impactor. In this paper, we combine detailed boundary-integral simulations with analytical modelling to describe the formation of such Worthington jets after the impact of a circular disk on water. We extend our earlier model in Gekle et al. (Phys. Rev. Lett., vol. 102, 2009a, 034502), valid for describing only the jet base dynamics, to describe the whole jet. We find that the flow structure inside the jet may be divided into three different regions: the axial acceleration region, where the radial momentum of the incoming liquid is converted to axial momentum; the ballistic region, where fluid particles experience no further acceleration and move constantly with the velocity obtained at the end of the acceleration region; and the jet tip region, where the jet eventually breaks into droplets. From our modelling of the ballistic region we conclude that, contrary to the case of other physical situations where high-speed jets are also ejected, the types of Worthington jets studied here cannot be described using the theory of hyperbolic jets of Longuet-Higgins (J. Fluid Mech., vol. 127, 1983, p. 103). Most importantly, we find that the velocity and the shape of the ejected jets can be well predicted at any instant in time with the only knowledge of quantities obtained before pinch-off occurs. This fact allows us to provide closed expressions for the jet velocity and the sizes of the ejected droplets as a function of the velocity and the size of the impactor. We show that our results are also applicable to Worthington jets emerging after the collapse of a bubble growing from an underwater nozzle, although this system creates thicker jets than the disk impact.


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Antkowiak, A., Bremond, N., Dizès, S. L. & Villermaux, E. 2007 Short-term dynamics of a density interface following an impact. J. Fluid Mech. 577, 241250.
Aristoff, J. M. & Bush, J. W. M. 2009 Water entry of small hydrophobic spheres. J. Fluid Mech. 619, 4578.
Ashley, H. & Landahl, M. 1965 Aerodynamics of Wings and Bodies. Addison-Wesley.
Bartolo, D., Josserand, C. & Bonn, D. 2006 Singular jets and bubbles in drop impact. Phys. Rev. Lett. 96, 124501.
Bergmann, R., van der Meer, D., Gekle, S., van der Bos, A. & Lohse, D. 2009 Controlled impact of a disc on a water surface: cavity dynamics. J. Fluid Mech. 633, 381409.
Bergmann, R., van der Meer, D., Stijnman, M., Sandtke, M., Prosperetti, A. & Lohse, D. 2006 Giant bubble pinch-off. Phys. Rev. Lett. 96, 154505.
Birkhoff, G. D., MacDonald, D. P., Pugh, W. M. & Taylor, G. I. 1948 Explosives with lined cavities. J. Appl. Phys. 19, 563582.
Blake, J. R. & Gibson, D. C. 1981 Growth and collapse of a vapour cavity near a free surface. J. Fluid Mech. 111, 123140.
Blake, J. R., Robinson, P. B., Shima, A. & Tomita, Y. 1993 Interaction of two cavitation bubbles with a rigid boundary. J. Fluid Mech. 255, 707721.
Bolanos-Jiménez, R., Sevilla, A., Martínez-Bazán, C. & Gordillo, J. M. 2008 Axisymmetric bubble collapse in a quiescent liquid pool. Part II. Experimental study. Phys. Fluids 20, 112104.
Boulton-Stone, J. M. & Blake, J. R. 1993 Gas bubbles bursting at a free surface. J. Fluid Mech. 254, 437466.
Burton, J., Waldrep, R. & Taborek, P. 2005 Scaling instabilities in bubble pinch-off. Phys. Rev. Lett. 94, 184502.
Burton, J. C. & Taborek, P. 2008 Bifurcation from bubble to droplet in inviscid pinch-off. Phys. Rev. Lett. 101, 214502.
Deng, Q., Anilkumar, A. V. & Wang, T. G. 2007 The role of viscosity and surface tension in bubble entrapment during drop impact onto a deep liquid pool. J. Fluid Mech. 578, 119138.
Do-Quang, M. & Amberg, G. 2009 The splash of a solid sphere impacting on a liquid surface: numerical simulation of the influence of wetting. Phys. Fluids 21, 022102.
Duchemin, L., Popinet, S., Josserand, C. & Zaleski, S. 2002 Jet formation in bubbles bursting at a free surface. Phys. Fluids 14, 30003008.
Duclaux, V., Caillé, F., Duez, C., Ybert, C., Bocquet, L. & Clanet, C. 2007 Dynamics of transient cavities. J. Fluid Mech. 591, 119.
Duez, C., Ybert, C., Clanet, C. & Bocquet, L. 2007 Making a splash with water repellency. Nat. Phys. 3, 180183.
Gekle, S., van der Bos, A., Bergmann, R., van der Meer, D. & Lohse, D. 2008 Non-continuous Froude number scaling for the closure depth of a cylindrical cavity. Phys. Rev. Lett. 100, 084502.
Gekle, S. & Gordillo, J. M. 2010 Compressible air flow through a collapsing liquid cavity. arXiv:1001.5402v1.
Gekle, S., Gordillo, J. M., van der Meer, D. & Lohse, D. 2009 a High-speed jet formation after solid object impact. Phys. Rev. Lett. 102, 034502.
Gekle, S., Peters, I., Gordillo, J. M., van der Meer, D. & Lohse, D. 2010 Supersonic air flow due to solid–liquid impact. Phys. Rev. Lett. 104, 024501.
Gekle, S., Snoeijer, J. H., Lohse, D. & van der Meer, D. 2009 b Approach to universality in axisymmetric bubble pinch-off. Phys. Rev. E 80, 036305.
Georgescu, S.-C., Achard, J.-L. & Canot, É. 2002 Jet drops ejection in bursting gas bubble processes. Eur. J. Mech. B 21, 265280.
Glasheen, J. W. & McMahon, T. A. 1996 Vertical water entry of disks at low Froude numbers. Phys. Fluids 8, 20782083.
Gordillo, J. M. 2008 Axisymmetric bubble collapse in a quiescent liquid pool. Part I. Theory and numerical simulations. Phys. Fluids 20, 112103.
Gordillo, J. M. & Gekle, S. 2010 Generation and breakup of Worthington jets after cavity collapse. Part 2. Tip breakup of stretched jets. J. Fluid Mech. doi:10.1017/S0022112010003538.
Gordillo, J. M., Sevilla, A. & Martínez-Bazán, C. 2007 Bubbling in a co-flow at high Reynolds numbers. Phys. Fluids 19, 077102.
Gordillo, J. M., Sevilla, A., Rodríguez-Rodríguez, J. & Martínez-Bazán, C. 2005 Axisymmetric bubble pinch-off at high Reynolds numbers. Phys. Rev. Lett. 95, 194501.
Grumstrup, T., Keller, J. B. & Belmonte, A. 2007 Cavity ripples observed during the impact of solid objects into liquids. Phys. Rev. Lett. 99, 114502.
Gurevich, M. I. 1966 The Theory of Jets in an Ideal Fluid. Pergamon.
Hogrefe, J. E., Peffley, N. L., Goodridge, C. L., Shi, W. T., Hentschel, H. G. E. & Lathrop, D. P. 1998 Power-law singularities in gravity–capillary waves. Physica D 123, 183205.
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, 123.
Keim, N. C., Møller, P., Zhang, W. W. & Nagel, S. R. 2006 Breakup of air bubbles in water: breakdown of cylindrical symmetry. Phys. Rev. Lett. 97, 144503.
Leppinen, D. & Lister, J. R. 2003 Capillary pinch-off in inviscid fluids. Phys. Fluids 15, 568578.
Liger-Belair, G., Polidori, G. & Jeandet, P. 2008 Recent advances in the science of champagne bubbles. Chem. Soc. Rev. 37, 24902511.
Lohse, D., Bergmann, R., Mikkelsen, R., Zeilstra, C., van der Meer, D., Versluis, M., van der Weele, K., van der Hoef, M. & Kuipers, H. 2004 Impact on soft sand: void collapse and jet formation. Phys. Rev. Lett. 93, 198003.
Longuet-Higgins, M. S. 1983 Bubbles, breaking waves and hyperbolic jets at a free surface. J. Fluid Mech. 127, 103121.
Longuet-Higgins, M. S., Kerman, B. R. & Lunde, K. 1991 The release of air bubbles from an underwater nozzle. J. Fluid Mech. 230, 365390.
Longuet-Higgins, M. S. & Oguz, H. 1995 Critical microjets in collapsing cavities. J. Fluid Mech. 290, 183201.
MacIntyre, F. 1968 Bubbles: a boundary-layer ‘microtome’ for micron-thick samples of a liquid surface. J. Phys. Chem. 72, 589592.
Manasseh, R., Yoshida, S. & Rudman, M. 1998 Bubble formation processes and bubble acoustic signals. In Third International Conference on Multiphase Flow, ICMF'98 Lyon, France, pp. 1–8.
May, A. 1951 The effect of surface conditions of a sphere on its water-entry cavity. J. Appl. Phys. 22, 12191222.
Morton, D., Rudman, M. & Liow, J. L. 2000 An investigation of the flow regimes resulting from splashing drops. Phys. Fluids 12, 747763.
Oguz, H. N. & Prosperetti, A. 1990 Bubble entrainment by the impact of drops on liquid surfaces. J. Fluid Mech. 219, 143179.
Oguz, H. N. & Prosperetti, A. 1993 Dynamics of bubble growth and detachment from a needle. J. Fluid Mech. 257, 111145.
Ohl, C. D. & Ikink, R. 2003 Shock-wave-induced jetting of micron-sized bubbles. Phys. Rev. Lett. 90, 214502.
Pozrikidis, C. 1997 Introduction to Theoretical and Computational Fluid Dynamics. Oxford University Press.
Rein, M. 1993 Phenomena of liquid drop impact on solid and liquid surfaces. Fluid. Dyn. Res. 12, 6193.
Schmidt, L. E., Keim, N. C., Zhang, W. W. & Nagel, S. R. 2009 Memory-encoding vibrations in a disconnecting air bubble. Nature Phys. 5, 343346.
Shin, J. & McMahon, T. A. 1990 The tuning of a splash. Phys. Fluids A 2, 13121317.
Thoroddsen, S. T. 2002 The ejecta sheet generated by the impact of a drop. J. Fluid Mech. 451, 373381.
Thoroddsen, S., Etoh, T. & Takehara, K. 2007 a Experiments on bubble pinch-off. Phys. Fluids 19, 042101.
Thoroddsen, S. T., Etoh, T. G. & Takehara, K. 2007 b Microjetting from wave focussing on oscillating drops. Phys. Fluids 19, 052101.
Thoroddsen, S., Etoh, T. & Takehara, K. 2008 High-speed imaging of drops and bubbles. Annu. Rev. Fluid Mech. 40, 257285.
Thoroddsen, S. T., Etoh, T. G., Takehara, K. & Takano, Y. 2004 Impact jetting by a solid sphere. J. Fluid Mech. 499, 139148.
Thoroddsen, S. T. & Shen, A. Q. 2001 Granular jets. Phys. Fluids 13, 46.
Thoroddsen, S. T., Takehara, K., Etoh, T. G. & Ohl, C. D. 2009 Spray and microjets produced by focusing a laser pulse into a hemispherical drop. Phys. Fluids 21, 112101.
Tjan, K. K. & Phillips, W. R. C. 2007 On impulsively generated inviscid axisymmetric surface jets, waves and drops. J. Fluid Mech. 576, 377403.
Turitsyn, K. S., Lai, L. & Zhang, W. W. 2009 Asymmetric bubble disconnection: persistent vibration evolves into smooth contact. Phys. Rev. Lett. 103, 124501.
Weiss, D. A. & Yarin, A. L. 1999 Single drop impact onto liquid films: neck distortion, jetting, tiny bubble entrainment, and crown formation. J. Fluid Mech. 385, 229254.
Worthington, A. M. & Cole, R. S. 1897 Impact with a liquid surface studied by the aid of instantaneous photography. Phil. Trans. R. Soc. Ser. A 189, 137148.
Worthington, A. M. & Cole, R. S. 1900 Impact with a liquid surface studied by the aid of instantaneous photography. Paper II. Phil. Trans. R. Soc. Ser. A 194, 175199.
Yarin, A. L. 2006 Drop impact dynamics: splashing, spreading, receding, bouncing. Annu. Rev. Fluid Mech. 38, 159192.
Zeff, B. W., Kleber, B., Fineberg, J. & Lathrop, D. P. 2000 Singularity dynamics in curvature collapse and jet eruption on a fluid surface. Nature 403, 401404.
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Generation and breakup of Worthington jets after cavity collapse. Part 1. Jet formation

  • STEPHAN GEKLE (a1) (a2) and J. M. GORDILLO (a3)


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