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Free-Lagrange simulations of the expansion and jetting collapse of air bubbles in water

Published online by Cambridge University Press:  25 February 2008

C. K. TURANGAN ,
A. R. JAMALUDDIN ,
G. J. BALL  and
T. G. LEIGHTON
C. K. TURANGAN
Affiliation:
School of Engineering Sciences, University of Southampton, Highfield, SO17 1BJ, UK
A. R. JAMALUDDIN
Affiliation:
School of Engineering Sciences, University of Southampton, Highfield, SO17 1BJ, UK
G. J. BALL
Affiliation:
Atomic Weapons Establishment, Aldermaston, Reading, RG7 4PR, UK
T. G. LEIGHTON*
Affiliation:
Institute of Sound & Vibration Research, University of Southampton, Highfield, SO17 1BJ, UK
*
Author to whom correspondence should be addressed: tgl@soton.ac.uk
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Abstract

A free-Lagrange numerical method is implemented to simulate the axisymmetric jetting collapse of air bubbles in water. This is performed for both lithotripter shock-induced collapses of initially stable bubbles, and for free-running cases where the bubble initially contains an overpressure. The code is validated using two test problems (shock-induced bubble collapse using a step shock, and shock–water column interaction) and the results are compared to numerical and experimental results. For the free-running cases, simulations are conducted for a bubble of initial radius 0.3 mm located near a rigid boundary and near an aluminium layer (planar and notched surfaces). The simulations suggest that the boundary and its distance from the bubble influence the flow dynamics, inducing bubble elongation and jetting. They also indicate stress concentration in the aluminium and the likelihood of aluminium deformation associated with bubble collapse events. For the shock-induced collapse, a lithotripter shock, consisting of 56 MPa compressive and −10 MPa tensile waves, interacts with a bubble of initial radius 0.04 mm located in a free field (case 1) and near a rigid boundary (case 2). The interaction of the shock with the bubble causes it to involute and a liquid jet is formed that achieves a velocity exceeding 1.2 km s−1 for case 1 and 2.6 km s−1 for case 2. The impact of the jet on the downstream wall of the bubble generates a blast wave with peak overpressure exceeding 1 GPa and 1.75 GPa for cases 1 and 2, respectively. The results show that the simulation technique retains sharply resolved gas/liquid interfaces regardless of the degree of geometric deformation, and reveal details of the dynamics of bubble collapse. The effects of compressibility are included for both liquid and gas phases, whereas stress distributions can be predicted within elastic–plastic solid surfaces (both planar and notched) in proximity to cavitation events. There is a movie with the online version of the paper.

JFM classification

Compressible Flows: Shock waves Drops and Bubbles: Bubble dynamics Drops and Bubbles: Cavitation
Type
Papers
Information
Journal of Fluid Mechanics , Volume 598 , 10 March 2008 , pp. 1 - 25
Copyright
Copyright © Cambridge University Press 2008

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Footnotes

†

Present address: Institute of High Performance Computing, Singapore Science Park II, Singapore 117528.

‡

Present address: Romax Technology Limited, Nottingham Science & Technology Park, Nottingham, NG7 2PZ, UK.

References

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Turangan et al. supplementary movie

Movie 1. This animation of simulated Schlieren images shows the collapse of a bubble in an infinite body of liquid, induced by the passage over the bubble of the pressure pulse from an extracorporeal shock wave lithotripter (travelling from left to right, and labelled IS in the movie and in figure 1(a), below). Except for the wall of the air bubble (which is initially spherical with radius 0.06 mm) and the horizontal line just below the middle of the image (which indicates the axis of rotational symmetry), the darker shading indicates perturbations of refractive index in the usual manner for Schlieren images. An expansion wave (travelling upwards and to the left; and labelled EX in the movie and in figure 1(a), below) is generated when the lithotripter pulse first impacts the bubble. The upstream bubble wall (on the left) then involutes and forms a liquid jet (labelled LJ), which accelerates to over 1200 m s-1 (figure 1(b), below) before impacting the downstream bubble wall. This impact produces a blast wave (labelled BW in the movie and figure 1(a)) that generates a spatial peak pressure of over 1 GPa. Further details on this animation can be found below the figures, at the base of this page.

Download Turangan et al. supplementary movie(Video)
Video 827.4 KB

Turangan et al. supplementary movie

Movie 1. This animation of simulated Schlieren images shows the collapse of a bubble in an infinite body of liquid, induced by the passage over the bubble of the pressure pulse from an extracorporeal shock wave lithotripter (travelling from left to right, and labelled IS in the movie and in figure 1(a), below). Except for the wall of the air bubble (which is initially spherical with radius 0.06 mm) and the horizontal line just below the middle of the image (which indicates the axis of rotational symmetry), the darker shading indicates perturbations of refractive index in the usual manner for Schlieren images. An expansion wave (travelling upwards and to the left; and labelled EX in the movie and in figure 1(a), below) is generated when the lithotripter pulse first impacts the bubble. The upstream bubble wall (on the left) then involutes and forms a liquid jet (labelled LJ), which accelerates to over 1200 m s-1 (figure 1(b), below) before impacting the downstream bubble wall. This impact produces a blast wave (labelled BW in the movie and figure 1(a)) that generates a spatial peak pressure of over 1 GPa. Further details on this animation can be found below the figures, at the base of this page.

Download Turangan et al. supplementary movie(Video)
Video 420.3 KB
Supplementary material: Image

Turangan et al. supplementary material

Figure 1. (a) The time histories of bubble volume (solid line) and the pressure as measured at a point on the axis of symmetry, 0.18 mm upstream of the initial position of the centre of the bubble (broken line). The labels show the moments when the pressure at that point is dominated by the lithotripter shock (IS), the expansion wave (EX), and the blast wave (BW). The datum of time is the moment when the lithotripter shock first impacts the upstream bubble wall (corresponding to t=0.25 μs in the above animation). (b) The time histories of the speeds of the points on the bubble wall where the upstream (solid line) and downstream (broken line) walls cross the axis of rotational symmetry. Once the jet has formed, the speed of its tip corresponds to the speed of the upstream wall (solid line). The datum of time is as for part (a)

Download Turangan et al. supplementary material(Image)
Image 476.5 KB