This study presents a computational investigation of the interactions of a single shock wave with multiple gas-filled bubbles in a liquid medium. This work illustrates how multiple bubbles may be used in shock-bubble interactions to intensify the process on a local level. A high resolution front-tracking approach is used, which enables explicit tracking of the gas-liquid interface. The collapse of two identical bubbles, one placed behind the other is investigated in detail, demonstrating that peak pressures in a two bubble arrangement can exceed those seen in single bubble collapse. Additionally, a parametric investigation into the effect of bubble separation is presented. It is found that the separation distance has a significant effect on both the shape and velocity of the main transverse jet of the second bubble. Extending this analysis to effects of relative bubble size, we show that if the first bubble is sufficiently small relative to the second, it may become entirely entrained in the second bubble main transverse jet. In contrast, if the first bubble is substantially larger than the second, it may offer it significant protection from the incident shock. This protection is utilised in the study of a triangular array of three bubbles, with the central bubble being significantly smaller than the outer bubbles. It is demonstrated that, through shielding of bubbles until later in the collapse process, pressures over five times higher than the maximum pressure observed in the single bubble case may be achieved. This corresponds to a peak pressure that is approximately 40 times more intense than the incident shock wave. This work has applications in a number of different fields, including cavitation erosion, explosives, targeted drug delivery/intensification, and shock wave lithotripsy.

1.
L.
Rayleigh
, “
On the pressure developed in a liquid during the collapse of a spherical cavity
,”
Philos. Mag.
34
,
94
98
(
1917
).
2.
T. B.
Benjamin
and
A. T.
Ellis
, “
The collapse of cavitation bubbles and the pressures thereby produced against solid boundaries
,”
Philos. Trans. R. Soc., A
260
,
221
240
(
1966
).
3.
W.
Lauterborn
and
H.
Bolle
, “
Experimental investigations of cavitation-bubble collapse in the neighbourhood of a solid boundary
,”
J. Fluid Mech.
72
,
391
399
(
1975
).
4.
M. S.
Plesset
and
R. B.
Chapman
, “
Collapse of an initially spherical vapour cavity in the neighbourhood of a solid boundary
,”
J. Fluid Mech.
47
,
283
290
(
1971
).
5.
F. P.
Bowden
and
O. A.
Gurton
, “
Birth and growth of explosion in liquids and solids initiated by impact and friction
,”
Proc. R. Soc. A
198
,
350
372
(
1949
).
6.
N. K.
Bourne
and
J. E.
Field
, “
Bubble collapse and the initiation of explosion
,”
Proc. R. Soc. A
435
,
423
435
(
1991
).
7.
N. K.
Bourne
and
J. E.
Field
, “
Explosive ignition by the collapse of cavities
,”
Proc. R. Soc. A
455
,
2411
2426
(
1999
).
8.
N.
Bourne
and
J.
Field
, “
Shock-induced collapse and luminescence by cavities
,”
Philos. Trans. R. Soc., A
357
,
295
(
1999
).
9.
A. J.
Coleman
,
J. E.
Saunders
,
L. A.
Crum
, and
M.
Dyson
, “
Acoustic cavitation generated by an extracorporeal shockwave lithotripter
,”
Ultrasound Med. Biol.
13
,
69
76
(
1987
).
10.
S.
Zhu
,
F. H.
Cocks
,
G. M.
Preminger
, and
P.
Zhong
, “
The role of stress waves and cavitation in stone comminution in shock wave lithotripsy
,”
Ultrasound Med. Biol.
28
,
661
671
(
2002
).
11.
K. S.
Suslick
and
D. J.
Flannigan
, “
Inside a collapsing bubble: Sonoluminescence and the conditions during cavitation
,”
Annu. Rev. Phys. Chem.
59
,
659
683
(
2008
).
12.
K. S.
Suslick
,
S. J.
Doktycz
, and
E. B.
Flint
, “
On the origin of sonoluminescence and sonochemistry
,”
Ultrasonics
28
,
280
290
(
1990
).
13.
K. S.
Suslick
and
G. J.
Price
, “
Applications of ultrasound to materials chemistry
,”
Annu. Rev. Mater. Sci.
29
,
295
326
(
1999
).
14.
M.
Versluis
, “
How snapping shrimp snap: Through cavitating bubbles
,”
Science
289
,
2114
2117
(
2000
).
15.
D.
Lohse
,
B.
Schmitz
, and
M.
Versluis
, “
Snapping shrimp make flashing bubbles
,”
Nature
413
,
477
478
(
2001
).
16.
H.
Frenzel
and
H.
Schultes
, “
Lumineszenz im ultraschall-beschickten Wasser
,”
Z. Phys. Chem.
27B
,
421
424
(
1934
).
17.
D. F.
Gaitan
,
L. A.
Crum
,
C. C.
Church
, and
R. A.
Roy
, “
Sonoluminescence and bubble dynamics for a single, stable, cavitation bubble
,”
J. Acoust. Soc. Am.
91
,
3166
3183
(
1992
).
18.
M. P.
Brenner
,
S.
Hilgenfeldt
, and
D.
Lohse
, “
Single bubble sonoluminescence
,”
Rev. Mod. Phys.
74
,
425
484
(
2002
).
19.
D. J.
Flannigan
and
K. S.
Suslick
, “
Inertially confined plasma in an imploding bubble
,”
Nat. Phys.
6
,
598
601
(
2010
).
20.
C. G.
Camara
,
S. J.
Putterman
, and
E.
Kirilov
, “
Sonoluminescence from a single bubble driven at 1 Megahertz
,”
Phys. Rev. Lett.
92
,
1
4
(
2004
).
21.
G. E.
Vazquez
,
C. G.
Camara
,
S. J.
Putterman
, and
K. R.
Weninger
, “
Sonoluminescence: Natures smallest blackbody
,”
Opt. Lett.
26
,
575
(
2001
).
22.
J. P.
Dear
and
J. E.
Field
, “
A study of the collapse of arrays of cavities
,”
J. Fluid Mech.
190
,
409
425
(
1988
).
23.
J. P.
Dear
,
J. E.
Field
, and
A. J.
Walton
, “
Gas compression and jet formation in cavities collapsed by a shock wave
,”
Nature
332
,
505
508
(
1988
).
24.
A. B.
Swantek
and
J. M.
Austin
, “
Collapse of void arrays under stress wave loading
,”
J. Fluid Mech.
649
,
399
427
(
2010
).
25.
P.
Testud-Giovanneschi
,
A. P.
Alloncle
, and
D.
Dufresne
, “
Collective effects of cavitation: Experimental study of bubble-bubble and bubble-shock wave interactions
,”
J. Appl. Phys.
67
,
3560
3564
(
1990
).
26.
Y.
Tomita
and
A.
Shima
, “
High-speed photographic observations of laser-induced cavitation bubbles in water
,”
Acustica
71
,
161
171
(
1990
).
27.
S. W.
Fong
,
D.
Adhikari
,
E.
Klaseboer
, and
B. C.
Khoo
, “
Interactions of multiple spark-generated bubbles with phase differences
,”
Exp. Fluids
46
,
705
724
(
2009
).
28.
E.
Klaseboer
,
S. W.
Fong
,
C. K.
Turangan
,
B. C.
Khoo
,
A. J.
Szeri
,
M. L.
Calvisi
,
G. N.
Sankin
, and
P.
Zhong
, “
Interaction of lithotripter shockwaves with single inertial cavitation bubbles
,”
J. Fluid Mech.
593
,
33
56
(
2007
).
29.
A.
Prosperetti
, “
A new mechanism for sonoluminescence
,”
J. Acoust. Soc. Am.
101
,
2003
2007
(
1997
).
30.
J. P.
Best
, “
The formation of toroidal bubbles upon the collapse of transient cavities
,”
J. Fluid Mech.
251
,
79
107
(
1993
).
31.
Z.
Ding
and
S. M.
Gracewski
, “
The behaviour of a gas cavity impacted by a weak or strong shock wave
,”
J. Fluid Mech.
309
,
183
209
(
1996
).
32.
C. W.
Hirt
,
A. A.
Amsden
, and
J. L.
Cook
, “
An arbitrary Lagrangian–Eulerian computing method for all flow speeds
,”
J. Comput. Phys.
14
,
227
253
(
1974
).
33.
G. J.
Ball
,
B. P.
Howell
,
T. G.
Leighton
, and
M. J.
Schofield
, “
Shock-induced collapse of a cylindrical air cavity in water: A free-Lagrange simulation
,”
Shock Waves
10
,
265
276
(
2000
).
34.
C. K.
Turangan
,
A. R.
Jamaluddin
,
G. J.
Ball
, and
T. G.
Leighton
, “
Free-Lagrange simulations of the expansion and jetting collapse of air bubbles in water
,”
J. Fluid Mech.
598
,
1
25
(
2008
).
35.
A. R.
Jamaluddin
,
G. J.
Ball
,
C. K.
Turangan
, and
T. G.
Leighton
, “
The collapse of single bubbles and approximation of the far-field acoustic emissions for cavitation induced by shock wave lithotripsy
,”
J. Fluid Mech.
677
,
305
341
(
2011
).
36.
T. G.
Leighton
,
C. K.
Turangan
,
A. R.
Jamaluddin
,
G. J.
Ball
, and
P. R.
White
, “
Prediction of far-field acoustic emissions from cavitation clouds during shock wave lithotripsy for development of a clinical device
,”
Proc. R. Soc. A
469
,
1
21
(
2013
).
37.
E.
Johnsen
and
T.
Colonius
, “
Implementation of WENO schemes in compressible multicomponent flow problems
,”
J. Comput. Phys.
219
,
715
732
(
2006
).
38.
E.
Johnsen
and
T.
Colonius
, “
Shock-induced collapse of a gas bubble in shockwave lithotripsy
,”
J. Acoust. Soc. Am.
124
,
2011
2020
(
2008
).
39.
E.
Johnsen
and
T.
Colonius
, “
Numerical simulations of non-spherical bubble collapse
,”
J. Fluid Mech.
629
,
231
262
(
2009
).
40.
C.-H.
Chang
and
M.-S.
Liou
, “
A robust and accurate approach to computing compressible multiphase flow: Stratified flow model and AUSM+-up scheme
,”
J. Comput. Phys.
225
,
840
873
(
2007
).
41.
M.
Sussman
,
P.
Smereka
, and
S.
Osher
, “
A level set approach for computing solutions to incompressible two-phase flow
,”
J. Comput. Phys.
114
,
146
159
(
1994
).
42.
R. P.
Fedkiw
,
T.
Aslam
,
B.
Merriman
, and
S.
Osher
, “
A non-oscillatory eulerian approach to interfaces in multimaterial flows (the ghost fluid method)
,”
J. Comput. Phys.
152
,
457
492
(
1999
).
43.
X. Y.
Hu
and
B. C.
Khoo
, “
An interface interaction method for compressible multifluids
,”
J. Comput. Phys.
198
,
35
64
(
2004
).
44.
R. R.
Nourgaliev
,
S. Y.
Sushchikh
,
T. N.
Dinh
, and
T. G.
Theofanous
, “
Shock wave refraction patterns at interfaces
,”
Int. J. Multiphase Flow
31
,
969
995
(
2005
).
45.
R. R.
Nourgaliev
,
T. N.
Dinh
, and
T. G.
Theofanous
, “
Adaptive characteristics-based matching for compressible multifluid dynamics
,”
J. Comput. Phys.
213
,
500
529
(
2006
).
46.
R. R.
Nourgaliev
and
T. G.
Theofanous
, “
High-fidelity interface tracking in compressible flows: Unlimited anchored adaptive level set
,”
J. Comput. Phys.
224
,
836
866
(
2007
).
47.
J. W.
Grove
and
R.
Menikoff
, “
Anomalous reflection of a shock wave at a fluid interface
,”
J. Fluid Mech.
219
,
313
336
(
1990
).
48.
N. A.
Hawker
and
Y.
Ventikos
, “
Interaction of a strong shockwave with a gas bubble in a liquid medium: A numerical study
,”
J. Fluid Mech.
701
,
59
97
(
2012
).
49.
H.
Grandjean
,
N.
Jacques
, and
S.
Zaleski
, “
Shock propagation in liquids containing bubbly clusters: A continuum approach
,”
J. Fluid Mech.
701
,
304
332
(
2012
).
50.
R. K. S.
Hankin
, “
The Euler equations for multiphase compressible flow in conservation form
,”
J. Comput. Phys.
172
,
808
826
(
2001
).
51.
X. Y.
Hu
,
B. C.
Khoo
,
N. A.
Adams
, and
F. L.
Huang
, “
A conservative interface method for compressible flows
,”
J. Comput. Phys.
219
,
553
578
(
2006
).
52.
S. Y.
Sushchikh
and
R. R.
Nourgaliev
, “
Shock waves and flow patterns in a shock-induced bubble collapse
,” in
43rd AIAA Aerospace Sciences Meeting and Exhibit (AIAA Paper 2005-1291)
(American Institute of Aeronautics and Astronautics, Reno, NV, 2005).
53.
E.
Lauer
,
X. Y.
Hu
,
S.
Hickel
, and
N. A.
Adams
, “
Numerical investigation of collapsing cavity arrays
,”
Phys. Fluids
24
,
1
25
(
2012
).
54.
J.
Glimm
,
C.
Klingenberg
,
O. A.
McBryan
,
B. J.
Plohr
,
D. H.
Sharp
, and
S.
Yaniv
, “
Front tracking and two-dimensional Riemann problems
,”
Adv. Appl. Math.
6
,
259
290
(
1985
).
55.
K. K.
Haller
,
Y.
Ventikos
,
D.
Poulikakos
, and
P.
Monkewitz
, “
Computational study of high-speed liquid droplet impact
,”
J. Appl. Phys.
92
,
2821
(
2002
).
56.
J.
Glimm
,
J. W.
Grove
,
X.
Li
, and
N.
Zhao
, “
Simple front tracking
,” in
Contemporary Mathematics
(
American Mathematical Society
,
1999
), Vol.
238
, pp.
133
149
.
57.
J.
Glimm
,
J. W.
Grove
, and
Y.
Zhang
, “
Interface tracking for axisymmetric flows
,”
SIAM J. Sci. Comput.
24
,
208
236
(
2002
).
58.

Based on a liquid density of 968 kg/m3, jet width of 120 μm, liquid velocity of 2090 ms−1, gas velocity of 1570 ms−1, dynamic viscosity of 1.002 × 10−3 kg/s m and surface tension 72.75 × 10−3 N m−1. The gas density is 14 kg/m3.

59.
X.
Jiao
and
H.
Zha
, “
Consistent computation of first- and second-order differential quantities for surface meshes
,” in
Proceedings of the 2008 ACM Symposium on Solid and Physical Modeling, Stony Brook, NY
(
ACM
,
New York
,
2008
), pp.
159
170
.
60.
B.
van Leer
, “
Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method
,”
J. Comput. Phys.
32
,
101
136
(
1979
).
61.
J.
Glimm
,
J. W.
Grove
,
B.
Lindquist
,
O. A.
Mcbryan
, and
G.
Tryggvason
, “
The bifurcation of tracked scalar waves
,”
SIAM J. Sci. and Stat. Comput.
9
,
61
79
(
1988
).
62.
J.
Du
,
B.
Fix
,
J.
Glimm
,
X.
Jia
,
X.
Li
,
Y.
Li
, and
L.
Wu
, “
A simple package for front tracking
,”
J. Comput. Phys.
213
,
613
628
(
2006
).
63.
P.
Colella
, “
A direct Eulerian MUSCL scheme for gas dynamics
,”
SIAM J. Sci. and Stat. Comput.
6
,
104
117
(
1985
).
64.
G.
Strang
, “
Accurate partial difference methods II. Non-linear problems
,”
Numer. Math.
6
,
37
46
(
1964
).
65.
G.
Strang
, “
On the construction and comparison of difference schemes
,”
SIAM J. Numer. Anal.
5
,
506
517
(
1968
).
66.
R.
Courant
,
K.
Friedrichs
, and
H.
Lewy
, “
On the partial difference equations of mathematical physics
,”
IBM J. Res. Dev.
11
,
215
234
(
1967
).
67.
B.
Tully
,
N.
Hawker
,
M.
Betney
, and
Y.
Ventikos
, “
Simulation of warm dense matter in intense bubble collapse
,” in
Proceedings of Meetings on Acoustics
(
Acoustical Society of America
,
Montreal
,
2013
), Vol.
19
, pp.
1
9
.
68.
R.
Menikoff
and
B. J.
Plohr
, “
The Riemann problem for fluid flows of real materials
,”
Rev. Mod. Phys.
61
,
75
130
(
1989
).
69.
W.
Wagner
and
A.
Pruss
, “
The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use
,”
J. Phys. Chem. Ref. Data
31
,
387
535
(
2002
).
70.
N. K.
Bourne
and
J. E.
Field
, “
Cavity collapse in a liquid with solid particles
,”
J. Fluid Mech.
259
,
149
165
(
1994
).
71.
N. K.
Bourne
and
J. E.
Field
, “
Shock-induced collapse of single cavities in liquids
,”
J. Fluid Mech.
244
,
225
240
(
1992
).
72.
A. R.
Jamaluddin
, “
Free-Lagrange simulations of shock-bubble interaction in extracorporeal shock wave lithotripsy
,” Ph.D. thesis (
University of Southampton
,
2005
).
73.
L. B.
Tran
and
H. S.
Udaykumar
, “
A particle-level set-based sharp interface Cartesian grid method for impact, penetration, and void collapse
,”
J. Comput. Phys.
193
,
469
510
(
2004
).
74.
Y.
Tomita
,
A.
Shima
, and
T.
Ohno
, “
Collapse of multiple gas bubbles by a shock wave and induced impulsive pressure
,”
J. Appl. Phys.
56
,
125
(
1984
).
75.
J. R.
Sukovich
,
A.
Sampathkumar
,
P. A.
Anderson
,
R. G.
Holt
,
Y. A.
Pishchalnikov
, and
D. F.
Gaitan
, “
Temporally and spatially resolved imaging of laser-nucleated bubble cloud sonoluminescence
,”
Phys. Rev. E
85
,
056605
(
2012
).
You do not currently have access to this content.