Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-12T19:48:07.317Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  15 October 2010

STEPHAN GEKLE  and
J. M. GORDILLO
STEPHAN GEKLE
Affiliation:
Department of Applied Physics and J. M. Burgers Centre for Fluid Dynamics, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands Physik Department, Technische Universität München, 85748 Garching, Germany
J. M. GORDILLO*
Affiliation:
Área de Mecánica de Fluidos, Departamento de Ingenería Aeroespacial y Mecánica de Fluidos, Universidad de Sevilla, Avenida de los Descubrimientos s/n 41092, Sevilla, Spain
*
†Email address for correspondence: jgordill@us.es
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

JFM classification

Drops and Bubbles: Aerosols/atomization Drops and Bubbles: Breakup/coalescence Drops and Bubbles: Drops
Type
Papers
Information
Journal of Fluid Mechanics , Volume 663 , 25 November 2010 , pp. 293 - 330
Copyright
Copyright © Cambridge University Press 2010

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.)

References

REFERENCES

Antkowiak, A., Bremond, N., Dizès, S. L. & Villermaux, E. 2007 Short-term dynamics of a density interface following an impact. J. Fluid Mech. 577, 241–250.CrossRefGoogle Scholar
Aristoff, J. M. & Bush, J. W. M. 2009 Water entry of small hydrophobic spheres. J. Fluid Mech. 619, 45–78.CrossRefGoogle Scholar
Ashley, H. & Landahl, M. 1965 Aerodynamics of Wings and Bodies. Addison-Wesley.Google Scholar
Bartolo, D., Josserand, C. & Bonn, D. 2006 Singular jets and bubbles in drop impact. Phys. Rev. Lett. 96, 124501.Google Scholar
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, 381–409.Google Scholar
Bergmann, R., van der Meer, D., Stijnman, M., Sandtke, M., Prosperetti, A. & Lohse, D. 2006 Giant bubble pinch-off. Phys. Rev. Lett. 96, 154505.Google Scholar
Birkhoff, G. D., MacDonald, D. P., Pugh, W. M. & Taylor, G. I. 1948 Explosives with lined cavities. J. Appl. Phys. 19, 563–582.Google Scholar
Blake, J. R. & Gibson, D. C. 1981 Growth and collapse of a vapour cavity near a free surface. J. Fluid Mech. 111, 123–140.Google Scholar
Blake, J. R., Robinson, P. B., Shima, A. & Tomita, Y. 1993 Interaction of two cavitation bubbles with a rigid boundary. J. Fluid Mech. 255, 707–721.Google Scholar
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.Google Scholar
Boulton-Stone, J. M. & Blake, J. R. 1993 Gas bubbles bursting at a free surface. J. Fluid Mech. 254, 437–466.CrossRefGoogle Scholar
Burton, J., Waldrep, R. & Taborek, P. 2005 Scaling instabilities in bubble pinch-off. Phys. Rev. Lett. 94, 184502.Google Scholar
Burton, J. C. & Taborek, P. 2008 Bifurcation from bubble to droplet in inviscid pinch-off. Phys. Rev. Lett. 101, 214502.CrossRefGoogle ScholarPubMed
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, 119–138.Google Scholar
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.CrossRefGoogle Scholar
Duchemin, L., Popinet, S., Josserand, C. & Zaleski, S. 2002 Jet formation in bubbles bursting at a free surface. Phys. Fluids 14, 3000–3008.Google Scholar
Duclaux, V., Caillé, F., Duez, C., Ybert, C., Bocquet, L. & Clanet, C. 2007 Dynamics of transient cavities. J. Fluid Mech. 591, 1–19.CrossRefGoogle Scholar
Duez, C., Ybert, C., Clanet, C. & Bocquet, L. 2007 Making a splash with water repellency. Nat. Phys. 3, 180–183.Google Scholar
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.Google Scholar
Gekle, S. & Gordillo, J. M. 2010 Compressible air flow through a collapsing liquid cavity. arXiv:1001.5402v1.CrossRefGoogle Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
Georgescu, S.-C., Achard, J.-L. & Canot, É. 2002 Jet drops ejection in bursting gas bubble processes. Eur. J. Mech. B 21, 265–280.Google Scholar
Glasheen, J. W. & McMahon, T. A. 1996 Vertical water entry of disks at low Froude numbers. Phys. Fluids 8, 2078–2083.CrossRefGoogle Scholar
Gordillo, J. M. 2008 Axisymmetric bubble collapse in a quiescent liquid pool. Part I. Theory and numerical simulations. Phys. Fluids 20, 112103.Google Scholar
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.Google Scholar
Gordillo, J. M., Sevilla, A. & Martínez-Bazán, C. 2007 Bubbling in a co-flow at high Reynolds numbers. Phys. Fluids 19, 077102.CrossRefGoogle Scholar
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.Google Scholar
Grumstrup, T., Keller, J. B. & Belmonte, A. 2007 Cavity ripples observed during the impact of solid objects into liquids. Phys. Rev. Lett. 99, 114502.CrossRefGoogle ScholarPubMed
Gurevich, M. I. 1966 The Theory of Jets in an Ideal Fluid. Pergamon.Google Scholar
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, 183–205.CrossRefGoogle 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.CrossRefGoogle Scholar
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.Google Scholar
Leppinen, D. & Lister, J. R. 2003 Capillary pinch-off in inviscid fluids. Phys. Fluids 15, 568–578.Google Scholar
Liger-Belair, G., Polidori, G. & Jeandet, P. 2008 Recent advances in the science of champagne bubbles. Chem. Soc. Rev. 37, 2490–2511.Google Scholar
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.CrossRefGoogle ScholarPubMed
Longuet-Higgins, M. S. 1983 Bubbles, breaking waves and hyperbolic jets at a free surface. J. Fluid Mech. 127, 103–121.Google Scholar
Longuet-Higgins, M. S., Kerman, B. R. & Lunde, K. 1991 The release of air bubbles from an underwater nozzle. J. Fluid Mech. 230, 365–390.Google Scholar
Longuet-Higgins, M. S. & Oguz, H. 1995 Critical microjets in collapsing cavities. J. Fluid Mech. 290, 183–201.Google Scholar
MacIntyre, F. 1968 Bubbles: a boundary-layer ‘microtome’ for micron-thick samples of a liquid surface. J. Phys. Chem. 72, 589–592.CrossRefGoogle Scholar
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.Google Scholar
May, A. 1951 The effect of surface conditions of a sphere on its water-entry cavity. J. Appl. Phys. 22, 1219–1222.Google Scholar
Morton, D., Rudman, M. & Liow, J. L. 2000 An investigation of the flow regimes resulting from splashing drops. Phys. Fluids 12, 747–763.Google Scholar
Oguz, H. N. & Prosperetti, A. 1990 Bubble entrainment by the impact of drops on liquid surfaces. J. Fluid Mech. 219, 143–179.Google Scholar
Oguz, H. N. & Prosperetti, A. 1993 Dynamics of bubble growth and detachment from a needle. J. Fluid Mech. 257, 111–145.CrossRefGoogle Scholar
Ohl, C. D. & Ikink, R. 2003 Shock-wave-induced jetting of micron-sized bubbles. Phys. Rev. Lett. 90, 214502.Google Scholar
Pozrikidis, C. 1997 Introduction to Theoretical and Computational Fluid Dynamics. Oxford University Press.CrossRefGoogle Scholar
Rein, M. 1993 Phenomena of liquid drop impact on solid and liquid surfaces. Fluid. Dyn. Res. 12, 61–93.Google Scholar
Schmidt, L. E., Keim, N. C., Zhang, W. W. & Nagel, S. R. 2009 Memory-encoding vibrations in a disconnecting air bubble. Nature Phys. 5, 343–346.Google Scholar
Shin, J. & McMahon, T. A. 1990 The tuning of a splash. Phys. Fluids A 2, 1312–1317.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., Etoh, T. & Takehara, K. 2007 a Experiments on bubble pinch-off. Phys. Fluids 19, 042101.CrossRefGoogle Scholar
Thoroddsen, S. T., Etoh, T. G. & Takehara, K. 2007 b Microjetting from wave focussing on oscillating drops. Phys. Fluids 19, 052101.Google Scholar
Thoroddsen, S., Etoh, T. & Takehara, K. 2008 High-speed imaging of drops and bubbles. Annu. Rev. Fluid Mech. 40, 257–285.CrossRefGoogle Scholar
Thoroddsen, S. T., Etoh, T. G., Takehara, K. & Takano, Y. 2004 Impact jetting by a solid sphere. J. Fluid Mech. 499, 139–148.CrossRefGoogle Scholar
Thoroddsen, S. T. & Shen, A. Q. 2001 Granular jets. Phys. Fluids 13, 4–6.CrossRefGoogle Scholar
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.Google Scholar
Tjan, K. K. & Phillips, W. R. C. 2007 On impulsively generated inviscid axisymmetric surface jets, waves and drops. J. Fluid Mech. 576, 377–403.Google Scholar
Turitsyn, K. S., Lai, L. & Zhang, W. W. 2009 Asymmetric bubble disconnection: persistent vibration evolves into smooth contact. Phys. Rev. Lett. 103, 124501.Google Scholar
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, 229–254.CrossRefGoogle Scholar
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, 137–148.Google Scholar
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, 175–199.Google Scholar
Yarin, A. L. 2006 Drop impact dynamics: splashing, spreading, receding, bouncing. Annu. Rev. Fluid Mech. 38, 159–192.Google Scholar
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, 401–404.Google Scholar