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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: [email protected]
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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

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