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RESEARCH ARTICLE
Experiments on bubble dynamics between a free surface
and a rigid wall
A. M. Zhang •P. Cui •Y. Wang
Received: 8 April 2013 / Revised: 17 August 2013 / Accepted: 14 September 2013 / Published online: 28 September 2013
ÓSpringer-Verlag Berlin Heidelberg 2013
Abstract Experiments were conducted where the under-
water bubble oscillates between two boundaries, a free
surface and a horizontal rigid wall. The motion features of
both the bubble and the free surface were investigated, via
the consideration of two key factors, i.e., the non-dimen-
sional distances from the bubble to the two boundaries. To
support the investigation, experiments were conducted in
the first place where the bubble oscillates near only one of
the two boundaries. Then the other boundary was inserted
at different positions to observe the changes in the motion
features, including the types, maximum speed and height of
the water spike and skirt, the form and speed of the jets,
and bubble shapes. Correspondence is found between the
motion features of the free surface and different stages of
bubble oscillation. Intriguing details such as gas torus
around the jet, double jets, bubble entrapment, and microjet
of the water spike, etc., are observed.
1 Introduction
Cavitation bubble has been studied for more than a century
and still remains at the center of attention in various fields.
The interest in bubble interaction with boundaries mainly
arose from the damage effect to nearly all known surfaces,
such as ship propellers, turbines, and other fluid machinery.
Researches are also motivated by the important role of
bubble in a wide variety of applications from ocean engi-
neering to medical science.
Spherical bubble collapse in infinite free fluid field can
be roughly modeled by the Rayleigh–Plesset equations
(Rayleigh 1917) or by other models involving compress-
ibility, viscosity, heat transfer, etc. (Gilmore 1952; Prosp-
eretti and Lezzi 1986). On the other hand, the collapse of
bubbles near boundaries will be asymmetrical, with the
formation of liquid jet in suitable conditions. The physical
property of the boundary is crucial to bubble collapse,
especially the direction of bubble jet. Various experiments
of bubble collapse near free surface were carried out (Blake
and Gibson 1981; Chahine 1977; Dadvand et al. 2009;
Robinson et al. 2001, etc.) in which the bubble migrates
and produces a jet away from the boundary and becomes
toroidal. Numerical simulations were also carried out on
bubble-free surface interaction using the boundary integral
method (Blake and Gibson 1981; Blake et al. 1987; Pear-
son et al. 2004b; Wang et al. 1996, etc.). Most of the
simulations were focused on the collapse phase before jet
penetration, while Pearson et al. (2004b), Wang et al.
(1996), etc., obtained the subsequent toroidal bubble, and
the simulations stopped before bubble rebound. In the
current work, more detailed observations are made to the
late collapse phase of the bubble, including a second
toroidal bubble, bubble rebounds, as well as the scenario
where two jets were found within a single collapse. The
free surface deformation motivated by the bubble was also
discussed in the above works and by Longuet-Higgins
(1983), etc. Variations in the width, height, and speed of
the free surface liquid jet (referred to as water ‘‘spike’’)
with the standoff distance of the bubble were discussed. In
suitable conditions, the ‘‘lateral jets’’ (lateral deformation
on the spike, referred to as water ‘‘skirt’’) were observed.
The numerical simulations were mostly focused on the
A. M. Zhang (&)P. Cui
College of Shipbuilding Engineering, Harbin Engineering
University, Harbin 150001, China
e-mail: zhangaman@hrbeu.edu.cn
Y. Wang
Naval Academy of Armament, Beijing 100161, China
123
Exp Fluids (2013) 54:1602
DOI 10.1007/s00348-013-1602-7
bubble rather than on the water spike that keeps evolving
for a long time after bubble rebound; fewer simulations
were focused on the water skirt. In this paper, a number of
experiments focused on free surface motions are carried
out over a range of standoff distance, in order to obtain a
collection of typical morphologies of the water spike and
skirt. Besides, a coincidence between the different stages of
bubble oscillation and the features of free surface motion is
found, which leads to a discussion on the formation of the
water skirt.
More extensive and abundant researches were carried
out on bubble collapse in the neighborhood of solid
boundaries (see, e.g., Naude
´and Ellis (1961), Lauterborn
and Bolle (1975), Blake and Gibson (1987), Philipp and
Lauterborn (1998), Shima et al. (1981), etc.). Apart from
the bubble jet directed toward the boundary, efforts were
also focused on the formation and collapse of toroidal
bubble (Brujan et al. 2002; Lindau and Lauterborn 2003;
Vogel et al. 1989), shock wave emission by bubble
rebound (Blake et al. 1998; Lindau and Lauterborn 2003;
Shima et al. 1983), and damage done to the boundary
(Bourne and Field 1995; Philipp and Lauterborn 1998;
Tomita and Shima 1986). In the above studies, a second
toroidal cavity was observed after jet penetration with
larger non-dimensional standoff; with smaller standoff,
the splashing of the jet against the boundary was studied.
Numerical simulations related to solid boundaries are
also prosperous. Many have successfully simulated the
toroidal bubble after jet penetration and the splashing
(Best 1993; Blake et al. 1997,1999; Brujan et al. 2002;
Pearson et al. 2004a; Tong et al. 1999; Wang et al.
2005; Zhang et al. 1993), though it is still hard to
numerically treat small-standoff cases and the bubble
rebound accurately. Interest of this paper is therefore on
the effect of the bubble rebound on the water skirt,
especially from those against the solid wall in the case
of small standoff distances.
In addition to the free surface and the solid boundary,
bubble collapse near elastic boundary was also examined
both experimentally and numerically (Brujan et al. 2001;
Shaw et al. 1999; Turangan et al. 2006, etc.), where the
bubble may produce a jet away from the boundary or split.
Bubble dynamics were well studied in the neighborhood of
a single boundary in the above-mentioned papers. Never-
theless, works are rare involving multiple boundaries,
especially those of different natures. In this paper, experi-
ments will be conducted where a bubble is initiated
between a free surface and a horizontal rigid wall. The
forms of the water spike and skirt, their height and speed,
the speed of the reentrant jets, and other bubble behaviors
are investigated by keeping one of the two standoffs con-
stant and the other varied.
2 Experimental setup
2.1 Bubble generation and recording
The low-voltage electric-spark bubble-generating method is
adopted in the current work. It was initially applied by Tur-
angan et al. (2006) and then modified over time (Dadvand
et al. 2009;Fongetal.2009). This method is suitable for the
current experiment because the period of the bubble generated
is longer (*5 ms in the following experiments) and the radius
is larger (*10 mm) compared to a laser-induced bubble [e.g.,
Lindau and Lauterborn (2003)], which will lower the
requirements in photographing and enables us to capture more
motion details with higher spatial and temporal resolution,
especially tocapture the bubble and the water spike withinthe
same video. The circuit adopted in the current work is based
on that of Zhang et al. (2011), as demonstrated in Fig. 1.The
simple circuit includes a 6,600 lF capacitor and a 200 V DC
power supply.Two copper-wire electrodes from the two poles
of the capacitor are in contact with each other at the ‘‘con-
nection point’’. With the discharge of the capacitor,the copper
wires are ignited into bright plasma and a bubble is generated.
According to repetitive experiments, the maximum average
radius of a bubble is closely related to the duration of the
plasma and falls between 11.50 ±0.60 mm when the dura-
tion is controlled within 2.5–3.0 ms. In very few cases is the
average radius or the plasma duration out of the above-men-
tioned ranges, and such results are discarded to ensure the
repeatability of the maximum radius. Despite the unstable
plasma, the bubble center (in free field) always coincides with
the connection point of the copper wire, which is, therefore,
referred to as the ‘‘initial bubble center’’.
Fig. 1 The experimental setup
Page 2 of 18 Exp Fluids (2013) 54:1602
123
To capture the motion of the bubble and the free surface,
a high-speed camera (Phantom V12.1) is adopted. The
camera works at 24,000 frames per second, with an
exposure time of 10 ls for each frame. The whole exper-
iment section is placed in a 0.5 90.5 90.5 m glass tank
and illuminated by a continuous light source from oppose
to the camera. A piece of polished glass is used as the rigid
bottom wall, which is adjusted to parallelism with the static
water surface. The distance from the tank boundary to the
bubble is more than 20 times of the bubble maximum
radius; hence, its effect could be neglected. Also ignored is
the effect of the electrodes, which have a diameter of
0.25 mm, only about 1.3 % of the bubble diameter, and
unlikely to cause substantial influence to the bubble
behaviors. The experiments are conducted at room tem-
perature (22 °C) and atmospheric pressure; the pressure
discrepancy between bubble top and bottom is negligible
given the small radius, and hence, the buoyancy or gravity
effect will be overcome by inertia. In fact, neither jet nor
upward migration of the bubbles is observed in free fluid
field.
The content of the bubble may include water vapor and
electrolysis products; hence, the non-equilibrium vapor
condensation could exist and cause additional cushioning
of the bubble collapse as argued by Fujikawa and Ak-
amatsu (1980).
2.2 Measurements and accuracy
The capturing time of the last image before the plasma is
taken as time zero. The maximum error in time measuring
is the image interval, 41.66 ls, which is small enough
compared to the first bubble oscillation period (*5 ms).
Before bubble generation, a ruler is placed at the initial
center perpendicular to the lens axle and recorded as length
calibration with which the spatial measurements are
directly carried out over discrete pixels of the captured
images. Hence, the special precision is up to the pixel
length, calculated to be 0.12 and 0.30 mm for the cases
with and without water spike, respectively. In addition,
there are blurs in some images on object edges with a
typical thickness of 1–2 pixels; therefore, the spatial error
could be 0.12–0.24 mm (for bubble) and 0.30–0.60 mm
(for water spike).
An equivalent maximum radius R
eq
is adopted since in
many cases the bubbles develop into non-spherical shapes.
Thirty points are picked around the bubble’s circumference
on the image captured at maximum expansion to calculate
the enclosed area A. The equivalent radius is then defined,
inversing the formula of circle area, as Req ¼ffiffiffiffiffiffiffiffiffi
A=p
p. For
spherical bubble, R
eq
is 0.37 % smaller than actual radius
due to the use of discrete points to represent the circle.
Considering the error in length (0.12–0.24 mm), the final
error in R
eq
is calculated to be 0.17–0.34 mm, i.e.,
1.5–3.0 % for an 11.50-mm bubble. As displayed in the
camera view in Fig. 1, there are two length parameters in
the current work, namely
1. the non-dimensional standoff distance from the free
surface, defined as c
f
=d
f
/R
eq
2. the non-dimensional standoff distance from the rigid
wall, defined as c
b
=d
b
/R
eq.
where d
f
and d
b
are the vertical distances from the initial
center of the bubble to the free surface and the rigid bot-
tom, respectively. The two parameters are found to be the
major factors affecting the motion features of the bubble
and the free surface.
The speed of a jet is measured at its tip. The speed of a
water spike is measured at its highest point or the upper edge
of the first droplet (if present) is pinched off from the spike.
The spike height is measured vertically from the static free
surface to the top. As for the water skirt, its speed and height
are measured at the highest point of the integrated liquid bulk,
ignoring the liquid droplets sprayed around.
The Weber number (We), the Reynolds number (Re), the
Froude number (Fr), and the Ohnesorge number (Oh) are
introduced as follows: We =qDU
2
/r,Re =UD/v,
Oh ¼vffiffiffiffiffiffiffiffiffiffiffi
q=rL
p, and Fr ¼U=ffiffiffiffiffiffi
gD
pto delineate the differ-
ent regimes of the free surface motions observed, where q,
v,r,D, and Uare density, kinetic viscosity and surface
tension coefficient of water (at 22 °C), characteristic
length, and characteristic velocity, respectively. In the
current experiments, Re is between 10
4
and 10
5
for bubble
jets of 10–20 m/s in velocity with Dbeing jet width
(1–5 mm), up to 1.2 910
5
for bubble-wall motion with
Dbeing instantaneous bubble diameter, and of order 10
4
for jet torus migration with Dbeing torus diameter; hence,
the viscous effect of the bubble is less significant against
inertia. For the motion of water spikes and skirts with
maximum speed being 1–10 m/s, Fr is relatively low
(approximately 2.0–21.0) with Dset as the maximum
bubble diameter.
3 Experiment results
The aim of the experiments is to observe the dynamics of
the bubble and the free surface when the bubble oscillates
between two different boundaries; however, the experi-
ments where only one of the boundaries exists are carried
out in the first place, for the following reasons.
1. Some major features are not unique to the two-
boundary cases and will be better illustrated with
single-boundary experiments.
Exp Fluids (2013) 54:1602 Page 3 of 18
123
2. The single-boundary cases are necessary references to
investigate the changes of these features when a
second boundary is inserted.
3. Some of the features in single-boundary cases that are
not emphasized or observed in previous studies are
presented in the current work to facilitate the mor-
phology analysis in two-boundary cases.
4. The temporal and spatial scale and the generation
method can be kept identical between the single-
boundary and the two-boundary cases.
3.1 Single boundary: the free surface
3.1.1 Bubble motion
First of all, the behavior of the spark-generated oscillatory
bubble near the free surface is investigated. The time
evaluation of a typical case, with c
f
=0.72, is demon-
strated in Fig. 2. Sphericity of the bubble is retained (frame
1) until the end of the growth. The upper wall of the bubble
involutes away from the surface as the contraction begins
and then produces a reentrant jet (frame 2) with a maxi-
mum speed of 17.2 m/s.
The liquid jet hits the lower side of the collapsing
bubble which then becomes toroidal. A protrusion is pro-
duced (frame 4 and onward) which is also a toroidal gas
bubble centered around the ongoing jet. To avoid confu-
sion, we refer to the main bubble as the ‘‘toroidal bubble’’
and the protrusion as the ‘‘jet torus’’ since it is drawn by the
jet. The morphology of two parts (frames 1–5) linked was
numerically simulated by Pearson et al. (2004b) (see
Figs. 5,6therein).
The motion of the toroidal bubble is characterized by a
rapid diminishing in volume (frames 5–7) until rebounding
in frame 7. The content of the main bubble is being
squeezed into the jet torus possibly under high pressure of
non-condensable gas and uncondensed vapor inside the
bubble. At the rebound, the jet torus breaks away from the
toroidal bubble (frame 7), migrates downward, and invo-
lutes. The involution can be told by the motion of the
fringes of the torus, as marked in frames 8 and 10.
A collapse generating two tori was reported experimen-
tally in the vicinity of solid boundaries (Brujan et al. 2002;
Zhang et al. 1993). However, to the best knowledge of the
authors, there is a lack of report on such collapse near the
free surface, except for that of Robinson et al. (2001), but
for detailed observation, higher resolution would be pre-
ferred. This can be achieved by, e.g., generating a larger
bubble and lowering the frame rate requirement, as carried
out in this paper. In addition, a schematic diagram is drawn
in Fig. 3a to demonstrate the sectional profiles of the bubble
corresponding to the captured frames. In the 3-D toroidal
bubble simulations by Li et al. (2012), the jet torus is not
present probably due to mesh-related issues.
3.1.2 Instability and shock wave emission
It should be noted that instability of the bubble surface
(Brennen 2002; Menon and Lal 1998) occurs during the
collapse phase. The collapse pauses, and the bubble
undergoes a slight re-expansion with its surface becoming
less smooth between frames 4 and 5, probably a result of
non-equilibrium condensation (i.e., vapor condensation
rate being lower than reducing rate of bubble volume) and
Fig. 2 Bubble collapse near free surface, c
f
=0.72. The frame number is placed at the corner of each frame, with its capturing time (in ms) in
the square bracket. The motion features are labeled in italics; bubble rebounds are captured at minimum volume. Same for the figures hereinafter
Page 4 of 18 Exp Fluids (2013) 54:1602
123
existence of non-condensable gas contents. Brennen (2002)
suggested that the Rayleigh–Taylor instability is likely to
be excited. Very soon the collapse resumes.
It was theoretically predicted that a local high pressure is
induced in the final stage of bubble collapse (Rayleigh 1917).
The inertia of the contracting liquid is overcome by the gas
pressure inside the bubble, which increases dramatically with
the bubble collapse. The bubble then rebounds, and a shock
wave is emitted to the liquid. The shock wave has been
visualized by Shima et al. (1981), Shaw et al. (1996), Tomita
and Shima (1986), Ward and Emmony (1991), Ohl et al.
(1995), and others in later works. For the toroidal bubble here
near the free surface, a ‘‘rebound’’ is also observed (see
frames 7–8). Shock wave is not directly seen due to exposure
limitations, but can be indicated. Debris of copper was cata-
pulted out during the plasma stage, piercing through the
bubble surface (see the spots around the bubblein frames 1–2
in Figs. 2and 4, especially on the bottom-left side). The
debris is merely *0.5 mm in diameter and unlikely to have
substantial influence on bubble motion, but brings minor
portions of gas away from the bubble to form microbubbles
appearing as the dark spots suspending around the main
bubble. Right after the rebound, the microbubbles are sud-
denly excited into pulsations. For instance, the microbubble
boxed out in frame 7 in Fig. 2can be seen to expand, contract,
and expand again in a chronological series of magnified
subgraphs on the bottom-left side of the figure. According to
Ohl and Ikink (2003) and Tomita and Shima (1986), micro-
bubbles were found to produce forced pulsation when
impacted by shock waves. Therefore, it is reasonable to
assume that in the current experiments, a shock wave is
radiated and acts upon the microbubbles, especially consid-
ering that shock wave emission was readily observed by
Brujan et al. (2002), Lindau and Lauterborn (2003), etc., from
similar collapsing toroidal bubble near a rigid boundary.
3.1.3 Thin jet formation and the double-jet scenario
With different c
f
, the bubble motion possesses practically
the same features as described above. Nevertheless, there is
(a) (b)
Fig. 3 Schematic diagram for bubble collapse near free surface, ac
f
=0.72; profiles A–Ecorrespond to frames 1, 2, 4, 6, and 7, respectively, in
Fig. 2.bc
f
=0.51; profiles A–Ecorrespond to frames 3–5 and 8–9, respectively, in Fig. 4
Fig. 4 Bubble collapse with two jets (the double-jet scenario), c
f
=0.51. The pulsation of the microbubbles (e.g., marked out in frame 9) is
shown in a series of magnified subgraphs with capturing time labeled at the corner
Exp Fluids (2013) 54:1602 Page 5 of 18
123
one more special behavior that needs to be pointed out. In
previous experiments and simulations (Blake and Gibson
1981,1987; Chahine 1977; Pearson et al. 2004b), thinner
reentrant jet was observed with smaller standoff distance.
Nevertheless, in the present experiments, there are not only
changes in jet width but also a coexistence of two jets with
proper standoff distances, as demonstrated in Fig. 4in an
overall view with c
f
.
The first jet is formed near the end of growth, originated,
and migrates away from the liquid veneer above the bubble
(frame 1); this narrow jet is referred to as ‘‘the thin jet’’ to
avoid confusion. It penetrates the bubble in early collapse
phase, producing a protrusion, i.e., a jet torus similar to that
in Fig. 2, frame 5. The upper side of the bubble continues
to involute during bubble collapse and turns into a ‘‘con-
ventional’’ reentrant jet in frame 5. This jet follows and
maintains coaxial with the thin jet through the bubble.
Their coexistence is referred to as the ‘‘double-jet’’ sce-
nario. The toroidal bubble formed as the reentrant (second)
jet reaches the bubble bottom, and another jet torus is
formed (frame 7 and onward).
The average width of the thin jet is 0.86 mm, about 7 %
of the maximum bubble radius and 26 % of the reentrant jet
width. The maximum jet velocities are about 9.6 and
15.4 m/s. It is found in further experiments that as c
f
increases, the formation of the thin jet is delayed while the
reentrant jet becomes advanced in time. A schematic dia-
gram is given in Fig. 3b to demonstrate the thin jet for-
mation and the double-jet scenario in Fig. 4.
More details of the thin jet formation are provided in
Fig. 5. In the first frame, a flimsy liquid veneer is left between
the bubble and the free surface. Next, the veneer develops a
conelike shape with liquid accumulating at its top which is
subsequently elongated (frame 2).In the meantime, the ‘‘thin
jet’’ is initiated from the top (see frames 3–6 (magnified)).
The downward migration of the thin jet is coupled with the
rising of the top (frame 4 and onward).
3.1.4 Free surface motion
A liquid jet migrating upward was found when the free
surface is disturbed by rapid bubble expansion. There
might be obvious distortions on the lateral part of such
liquid jets as well when c
f
changes, contributing to various
water surface morphologies. The free surface dynamics
affected by small-scale (neglecting buoyancy) oscillating
bubble have been reported experimentally by Gibson
(1968), Chahine (1977), Blake and Gibson (1981), etc.,
featuring a main water jet from the surface, and also by
Dadvand et al. (2009) where ‘‘lateral jets’’ (deformation on
lateral parts of the main jet) were also noticed. Neverthe-
less, a more comprehensive study on the configuration of
the main or lateral jets is still preferable since in most
works the number of cases carried out was limited in
addition to the difficulty in enclosing both the bubble and
the entire spike in the images.
In this section, we first try to explain the key free surface
motion features with bubble behaviors, present images with
fair details, and then give a collection of the distinctive
morphologies of the free surface with different c
f
. To begin
with, a typical example with c
f
=0.80 is demonstrated in
Fig. 6. For convenience, the main part of the upward sur-
face jet is referred to as the ‘‘water spike’’ and its deformed
lateral parts as the ‘‘water skirt’’, as marked out in Fig. 6
(continued), frame 11. According to repeated observations,
we suggest that the spike is motivated by the expansion of
the bubble, while the skirt is induced mainly by the
‘‘rebound’’ of the bubble, which will be detailed below.
The form of the free surface motion is directly affected by
the standoff distance c
f
.
In Fig. 6, the free surface rises as a dome with bubble
expansion and becomes conical with its lateral part falling;
as the bubble collapses, the top rises pertinaciously prob-
ably due to the high water pressure at the rear of the
reentrant jet, as simulated by Blake and Gibson (1987), Li
et al. (2012), and Robinson et al. (2001); the water dome
then elongates into a spike. Right following the instability
stage (introduced in Sect. 3.1.2) between frames 4 and 5, a
minor ‘‘fold’’ appears around the spike (frame 6).
As the toroidal bubble rebounds, the liquid motion is
inverted from contracting toward the bubble center to
expanding away from it. This expansion of liquid becomes
visible at the free surface, resulting in the up-lifting of a
relatively large liquid bulk underneath the fold (frame 8
and onward). This portion of liquid collides with the liquid
above it and produces a water ‘‘skirt’’ at the position where
Fig. 5 Details of thin jet formation, c
f
=0.50. Frames 3–6 are magnified
Page 6 of 18 Exp Fluids (2013) 54:1602
123
Fig. 6 a Evolution of the water spike and the initial water skirt along with bubble oscillation, c
f
=0.80; b(continued) evolution of the water
spike and the integrated water skirt after bubble collapse, c
f
=0.80
Exp Fluids (2013) 54:1602 Page 7 of 18
123
the fold was (frame 9 and onward). In addition, the ‘‘fold’’
was possibly caused similarly by the slight re-expansion of
the bubble during the instability stage.
Frames 9–17 capture the formation and development of
the skirt; with the bubble diameter (2R
eq
) and maximum
skirt speed being the characteristic length and speed,
respectively, We &4.01 910
2
and Re &2.6 910
4
. The
‘‘skirt’’ remains integrated as it evolves. In this process, the
spike out-speeds the skirt (We &4.50 910
3
,
Re &8.4 910
4
around 6–8 ms with the spike speed being
the characteristic speed). Droplets pinch off from the top
due to surface tension at the end of the inertial upward
migration (frame 13 and onward. With the droplet diameter
being characteristic length, Oh &2.13 910
-3
and
We &2.42 910
2
). With an upward motion, the skirt
(especially the top part) radially converges toward the
spike (frames 14–16).
Correspondence between bubble oscillation and free
surface motion is reflected in Fig. 7. A sharp increase
(peaking *6.9 m/s) in spike speed is found in alignment
with high-speed bubble expansion; the first decay in spike
speed (0.7–2.0 ms) can be accounted for by the decelera-
tion in bubble-wall speed in late expansion phase. The
spike speed passes an inflection point at *2 ms and rises
to a second peak (*3.5 m/s), which is simultaneous with
the bubble jet and hence probably accelerated by the high
pressure between the spike and the rear of the jet as
mentioned before.
Acceleration of the ‘‘fold’’ (to *1.1 m/s) is ensued from
the toroidal bubble rebound; the ‘‘fold’’ then bursts into a
water skirt, while the decay in spike speed is not disturbed
since its top is adequately far from the bubble.
More types of free surface motion are shown in Figs. 8,
9,10,11,12,withc
f
ranging from 0.31 to 1.60. In Fig. 8,
the effect of the rebound on the liquid surface is magnified
due to a smaller standoff distance c
f
=0.54. Consequently,
the water skirt acquires a higher speed of *2.6 m/s and
sprays around into droplets (frames 7–10) rather than stays
integrated, with We =2.24 910
3
, one order of magnitude
larger than the last case, and Re =6.24 910
4
.Oh for
these droplets (generally 0.5–1 mm in diameter) reaches
3.0 910
-3
–5.0 910
-3
. Nevertheless, the lower part of
the skirt top still converges toward the spike. In addition,
the double-jet scenario occurs in this case.
c
f
is reduced to 0.31 in Fig. 9. The bubble bursts into
open air; a screen of liquid is catapulted above the surface
and soon converges radially to form a very unstable water
spike, which is in rapid rise up to 16 m/s (We reaches
8.49 910
4
). No bubble rebound is observed; a possible
reason is that the pressure inside the bubble is presumably
normalized to atmospheric pressure when channeled to
open air. Consequently, there is no water skirt formed later
on.
With farther distance, the water spike is less pro-
nounced (see Fig. 10 where c
f
=1.12), acquiring lower
maximum speed (*1.0 and *0.31 m/s at two peaks) and
height. On the contrary, the effect of the rebounding is
less attenuated, and the skirt obtains a faster speed and
‘‘swallows’’ the spike (frames 4–6, first row); the top of
the spike then turns into a pit (see frames 1–2, second
Fig. 7 Speed variations in the
spike and skirt with time. The
oscillation stages of the bubble
are also marked along the time
axis, and its correspondence
with the speed curves is
analyzed in the text. The spike
speed is measured at the top
point of the spike or at the upper
edge of the first droplet (if
present) pinched off from the
spike. The skirt speed is
measured at the highest point of
the integrated skirt
Page 8 of 18 Exp Fluids (2013) 54:1602
123
row). The skirt merges over the pit, entrapping a bubble in
the liquid. In the meantime, a microliquid jet is catapulted
upward (frames 3–5, second row); then droplets pinch off
from the top of the jet (with Oh &6.10 910
-3
, frames
6–8, second row) with maximum speed of 4.5 m/s. Later,
the tip of the microjet inflates into a small dome (frames
8–9, second row). The high-speed microjet may be
explained as the consequence of the high-pressure stag-
nation resulting from the radial focusing of the bulk flow
during the merge of the skirt over the pit (Deng et al.
2007). The microjet formation is depicted schematically
in Fig. 11.
In this case, lower We is retained as 1.65 910
2
/
2.55 910
2
for the spike (at second speed peak)/skirt (at
maximum speed), while Re is still of order 10
4
. It is indi-
cated that the spike and the skirt migrate independently and
their relative heights are quite different from that in other
cases; this adds to the conjecture that they are generated by
different sources (i.e., by the bubble expansion and the
‘‘rebound’’, respectively).
When c
f
grows even larger to 1.60, neither the spike nor
the skirt is pronounced (Fig. 12). Though still generated
separately if observed very carefully (and the skirt is
dominating in the early stage), they rise as a whole at
*0.54 m/s maximum, in a cone shape where surface ten-
sion plays a more important role (We &96.7).
A summary is given in Fig. 13, where the maximum spike
heights rise exponentially with decreasing c
f
and reach more
than 200 times of R
eq
above the static surface. Five regions are
identified for the five typical free surface morphologies.
3.2 Single boundary: rigid bottom
The bubble behaviors (neglecting buoyancy effect) near
rigid boundaries with different c
b
have been comprehen-
sively studied (see, e.g., Benjamin and Ellis (1966), Blake
Fig. 8 Evolution of the water spike and the spraying skirt, c
f
=0.54
Fig. 9 Evolution of the bursting bubble and the unstable water spike, c
f
=0.31
Exp Fluids (2013) 54:1602 Page 9 of 18
123
and Gibson (1987), Shima et al.(1981), Tomita and Shima
(1986), Lindau and Lauterborn (2003), etc., and the refer-
ences therein). The late collapse phase of the bubble with
suitable c
b
(e.g., *1.0) is usually characterized by the two
toroidal fractions, i.e., the toroidal bubble and the jet torus,
similar to the free surface cases in Sect. 3.1, as captured in
Fig. 4a in Brujan et al. (2002) and Fig. 4a in Tomita and
Shima (1986), etc. Shock wave emissions are recorded
from the rebound of both fractions in Fig. 4a in Brujan
et al. (2002). However, in general, a jet induced by a rigid
wall forms at a later stage during bubble collapse than a jet
induced by a free surface and acquires a higher speed. In
addition, the formation of a jet torus is not necessary. In
some experiments, e.g., those by Lauterborn and Bolle
(1975) and Philipp and Lauterborn (1998), the bubble
contracts to minimum as a sole torus probably due to a
higher vapor condensation rate.
With a smaller c
b
(e.g., \0.6), the liquid between the
bubble and the rigid boundary was completely evacuated;
hence, the jet impacts directly onto the boundary rather
than to form a jet torus. Afterward, a toroidal bubble is
formed in close contact with the rigid boundary, collapses
and rebounds against it, and also radially expands along the
rigid surface and breaks into small segments. Typical cases
were presented in Fig. 2g–i by Philipp and Lauterborn
(1998), Fig. 14 by Lindau and Lauterborn (2003), and
Fig. 4d by Tomita and Shima (1986). In the subsequent
sections, the effect of the above bubble behaviors on the
free surface will be discussed.
3.3 Two boundaries: the insertion of a rigid bottom
The following two-boundary experiments are carried out
on the joint dynamics of the bubble, the rigid bottom, and
the free surface. To investigate its effect, the rigid bottom
wall is inserted at different c
b
(ranging from 0.18 to 4.80),
while c
f
remains constant. The image series of three typical
cases are listed in Figs. 14,15 and 16.
Figure 14 shows the comparative case without a rigid
bottom. The maximum speed and height of the spike are
6.8 m/s and 183 mm (16.6 times of R
eq
), respectively; the
skirt stays integrated with maximum height and speed being
54 mm and 1.2 m/s, respectively (We &4.77 910
2
,
Re &2.88 910
4
); the thin jet as in Sect. 3.1.3 is not
produced.
The case with c
b
being 0.97 is presented in Fig. 15. The
jet torus is blocked by the wall from further migration.
Fig. 10 Evolution of the lower water spike ‘‘swallowed’’ by the skirt, c
f
=1.10; the swallowing, bubble entrapment, and the microjet are shown
in the magnified images in the second row
D
C
B
F
E
A
Fig. 11 Schematic diagram for the microjet; profiles A–Fcorrespond
to frames 1–5 and 9 in Fig. 10, respectively
Page 10 of 18 Exp Fluids (2013) 54:1602
123
Fig. 12 Evolution of the lower water spike and skirt, c
f
=1.60
Bursting
bubble
&
unstable
high spike
Higher
spike
&
spraying
skirt
Higher
spike
&
integrated
skirt
Lower
spike
swallowed
by
skirt
Lower
spike
&
skirt
no buttom wall
Fig. 13 Variations in the non-
dimensional maximum spike
height with c
f
, no rigid bottom.
The maximum spike height is
measured as the vertical
distance from the highest
position of the spike top or the
first droplet pinched off from
the spike to the static water
surface. The non-dimensional
height H is the ratio between the
maximum spike height and the
maximum bubble radius. The
horizontal axis can be divided
into five regions according to
five different combinations of
water spikes and skirts. A
typical snapshot of each is given
in the corresponding region
Fig. 14 Evolution of the water spike and skirt along with bubble oscillation, without bottom wall, c
f
=0.89
Exp Fluids (2013) 54:1602 Page 11 of 18
123
Different from the previous case, a thin jet is produced
(frame 4) because the upper bubble wall is pressed closer to
the free surface by the rigid bottom during expansion,
although the initial bubble center is not changed. This also
caused the water spike to acquire a 31 % higher maximum
speed, 8.9 m/s, and a 164 % higher height, 484 mm, i.e.,
44 times of R
eq
, when compared to the non-bottom case,
with droplets pinching off with Oh between 2.0 910
-3
and 3.7 910
-3
. On the other hand, with We =5.60 910
2
and Re =3.12 910
4
, the form and height (48 mm) of the
water skirt are not much altered, probably because the
rebound of the toroidal bubble (frame 7) is not strength-
ened by the presence of the rigid wall and the effect of the
jet torus rebounding against the bottom is weak and late in
time.
The bottom is lifted to c
f
=0.23 in Fig. 16. As a result,
the bubble expands in direct contact with the bottom wall
into a hemisphere (frames 2–3); its upper wall is pressed
even closer to the free surface. Predictably, a thin jet is
generated (frame 3); following the thin jet, the upper
bubble wall involutes from above (frames 4–5) to produce
a reentrant jet which is induced by the joint effect of both
the free surface and the rigid bottom since the same
occurred with the presence of either boundary.
In frame 6 where the toroidal bubble rebounds against
the wall, dark spots are observed in surrounding fluid
which, as explained in Sect. 3.1.2, are microbubbles split
from the main bubble. The microbubbles are exited into
abrupt pulsation (subgraph B), and the copper remnants are
catapulted outward with their tracks captured as the radial
dark stripes (subgraph A); hence, there is possibly, fol-
lowing the same inference in Sect. 3.1.2, a shock wave
emitted by the rebound.
It is seen from Fig. 17 that with c
b
=0.23 the spike/
skirt acquires a maximum speed of 9.8/2.9 m/s and a height
of 608/91 mm (55/8.2 times of R
eq
). The insertion of the
rigid wall elevated the maximum spike speed by *41 %,
accompanied with a thin jet. Again, the second rise in spike
speed coincides with the jet, possibly due to the high liquid
pressure generated at jet rear. The skirt, initiated as bubble
rebound against the wall, has a 140 % increase in maxi-
mum speed compared to the non-bottom case. This could
be attributed to that more gas is retained in the toroidal
bubble rather than injected into a jet torus, contributing to a
stronger gas compression in the collapse. Thus, a stronger
radial liquid expansion is triggered, producing a more
vigorous water skirt at the free surface which sprays its
fringes into water drops (frame 9 and onward) with Oh
ranging from 2.5 910
-3
to 5.5 910
-3
. For the skirt, We
is elevated to 2.78 910
3
. The increase in spike height, on
the other hand, is more likely to be caused by that the
bubble is pushed closer to the free surface (equivalent to a
decrease in c
f
), since the toroidal bubble is less effective to
the spike top as concluded in Sect. 3.1.4.
Measurements from more cases are presented below. It
is displayed in Fig. 18 that with a fixed c
f
, the maximum
Fig. 15 Evolution of the water spike and skirt along with bubble oscillation with both free surface and rigid bottom; c
f
=0.89, c
b
=0.97
Page 12 of 18 Exp Fluids (2013) 54:1602
123
Fig. 16 Evolution of the water spike and skirt along with bubble oscillation, with both the free surface and the rigid bottom; c
f
=0.89,
c
b
=0.23
Fig. 17 Speed variations in the
spike and skirt with time. The
oscillation stages of the bubble
are marked along the time axis
Exp Fluids (2013) 54:1602 Page 13 of 18
123
height of the water spike rises sharply as c
b
falls below 0.5.
The trend is very similar to that in Fig. 13, and hence, we
suggest that both are results of the increased proximity
between the upper bubble wall and the free surface.
Figure 19 demonstrates the change in bubble jet with
different c
b
. Pushed nearer to the free surface by small c
b
,
the bubble is enabled to produce a thin jet. Generally, for c
b
above 1.2, only reentrant jet is produced; for c
b
under 0.4,
only thin jet is observed since there is no enough space for
the reentrant jet to develop. Between 0.4 and 1.2 there exist
the double-jet scenarios, in which the reentrant jet is
always faster. Both kinds of jets gain in speed as c
b
decreases.
In conclusion, the insertion of a rigid wall in the prox-
imity of an oscillating bubble near free surface will result
in (1) a smaller distance between the upper bubble wall and
the free surface which may cause a thin jet and a higher
water spike and (2) rebound of the toroidal bubble against
the surface of the rigid wall which possibly causes a more
vigorous water skirt.
Displacement curves of the top and the bottom points
of the bubble from its initial center are plotted in Fig. 20;
the top bubble boundary will be partially blocked by a
dark strip on the image caused by meniscus effect when
surpassing the static water surface. To overcome this
problem, a 2-degree spline is fitted to the visible part of
the upper bubble boundary (see Fig. 14, frames 2–3) to
represent the blocked part. Fortunately, this meniscus
problem occurs merely in a few images near bubble
maximum, and the displacement of the top point mea-
sured on the spline fits well to the whole displacement
curve. Back to Fig. 20, with the rigid wall being the only
boundary, the bubble jet is developed at a relatively late
stage and appears in the figure as the sharp downfall of
the bubble’s top point between 3.5 and 4.0 ms. On the
other hand, for the free surface-only case, the reentrant jet
is produced earlier and possesses a lower speed. Further,
when both boundaries are present, the thin jet appears in
the late expansion phase, earlier but slower than the
reentrant jet, as reflected by the lower steepness of the
curve. In addition, the maximum displacement of bubble
top is slightly larger, as the top part is elongated at the
free surface; the bubble bottom is pressed when the rigid
wall is inserted.
3.4 Two boundaries: the insertion of the free surface
The effect of the free surface is investigated with different
heights (c
f
) above the bubble initial center, while the
bubble-wall distance (c
b
) is kept constant. Two groups of
experiments are conducted with c
b
being 0.97 and 0.10,
respectively. For each group, the snapshots of the selected
cases at the same moment or stage are displayed together
for comparison.
In the first group, c
f
ranges from 1.75 to 0.20 and six
cases are chosen for display. Images in the first row of
Fig. 21 are captured at maximum bubble radius. The free
surface becomes more disturbed with smaller c
f
. Thin jets
are observed with c
f
=0.37 and 0.58, which are initiated at
approximately 1.9 and 1.3 ms, respectively, earlier with
lower free surface (see case (e) and (f)).
The second row is captured at 5.00 ±0.08 ms. Bubble
oscillation is more advanced with smaller c
f
. In the first two
cases with large c
f
, the upper bubble walls are flattened but
have not yet turned into reentrant jets, while in case (c) and
(d) the reentrant jets are penetrating the bubble. The bub-
bles in the last two cases are at later collapse stages, where
the jet tori are expanding along the rigid wall. The spike
acquires a greater height as c
f
decreases.
Fig. 18 Variations in non-dimensional maximum spike height with
c
b
when c
f
=0.89. Non-dimensional height H is the ratio between the
maximum spike height and R
eq
Fig. 19 Variations in jet speed with c
b
when c
f
=0.89. For double-
jet scenarios, the coexisting thin jet and reentrant jet of the same
bubble are linked by the dot line
Page 14 of 18 Exp Fluids (2013) 54:1602
123
In the third row, the free surface motion is fully devel-
oped. For (a) and (b), the motions are not pronounced. In
case (c), the fringe of the skirt is smooth and integrated,
while it starts to revolute outward in case (d) and sprays
radially into droplets in case (e); finally, the skirt is dis-
torted in case (f).
Fig. 20 The displacement
curves of the top and the bottom
point of the bubble in different
boundary conditions. The free
surface is inserted at c
f
=0.89;
the rigid bottom is inserted at
c
b
=0.97
Fig. 21 Bubble and free surface motion with different c
f
while c
b
=0.97. First row captured at maximum bubble radius. Second row captured at
5.00 ±0.08 ms. Third row captured at 13.33 ±0.08 ms. c
f
in the six cases are a1.69, b1.35, c0.97, d0.81, e0.58, and f0.37
Exp Fluids (2013) 54:1602 Page 15 of 18
123
For a more comprehensive investigation, a smaller
c
b
(0.10) is assigned to the second group. The c
f
ranges from
2.03 to 1.06 in the cases captured in Fig. 22.
The maximum bubbles are shown in the first row of
Fig. 22, while the second row is captured right before the
water skirts become visible. With falling c
f
, the spike is
initiated with higher speed. Next, the skirts emerge in the
third row. In case (b) and (c) where the spikes are less
pronounced due to larger c
f
(1.65 and 1.50), the skirts
overpass the spikes, similar to the case in Fig. 10. The
shape to which a skirt evolves into is less integrated as c
f
reduces.
There are two factors contributing to the intensified free
surface motion, i.e., the proximity of the upper bubble wall
to the free surface during expansion, which is also attrib-
uted to the pressing from the rigid wall, and the outward
fluid motion at bubble rebound. As has been discussed in
Sect. 3.1.4, the spike height is not likely to be affected by
the jet torus; hence, the elevation of spike heights (shown
later in Fig. 23) is mainly a result of smaller c
f
. As for the
water skirts, vigorous ones are observed in the non-bottom
cases with small c
f
; besides, it is indicated by both the
images and the speed curves that the water skirt formation
is closely linked to the rebound of the toroidal bubble
Fig. 22 Bubble and free
surface motion with different c
f
when c
b
=0.10, captured at
maximum bubble radius (first
row) and right before water skirt
formation (second row)
Page 16 of 18 Exp Fluids (2013) 54:1602
123
(either the one with a jet torus or the one clinging to the
wall); therefore, the intensified skirts should be attributed
to both factors mentioned above.
In the third row of Fig. 22, the concealed spikes in
(b) and (c) re-emerge above the skirts. Again, a microjet is
catapulted upward in case (c), with a speed up to 7.5 m/s;
only this time the gas entrapment is missing.
Figure 23 demonstrates the maximum spike height
variation with c
f
, with the presence of the rigid wall. Three
groups of experiments are conducted where c
b
is kept at
0.10, 0.63, and [10, respectively, and c
f
falls from 2.0 to
0.3. The maximum height of the spike always decreases
exponentially with growing c
f
. Compared to the group
where c
b
[10, the maximum heights at different c
f
are
elevated with c
b
=0.63 and are elevated again when c
b
is
reduced to 0.10.
4 Conclusions
The dynamics of the bubble and the free surface have been
experimentally studied using high-speed photography. The
bubbles are generated by underwater electric discharge and
pulsate in the vicinity of the free surface and/or a hori-
zontal rigid boundary, with varying bubble-boundary dis-
tances (c
b
and c
f
). Intriguing motion features have been
found with both single and double boundaries. The current
results benefit from higher temporal and spatial resolution
allowed by the relatively large scale (*10 mm for radius,
*5 ms for first period) of the spark-generated bubbles.
The bubble splits into two toroidal bubbles due to the
reentrant jet in late collapse phase near the free surface as
well as at certain distances from the rigid boundary.
Besides, with proper c
f
, the bubble was found to produce
two successive jets in the first collapse phase. According to
configurations of the water spike and skirt, the free surface
motions are categorized into five types. The maximum
height of the spike rises exponentially with decreasing c
f
.
There are two peaks in the upward speed of the spike; the
first is induced by the initial growth of the bubble, while
the second is probably induced by the high fluid pressure at
the rear of the bubble jet. Based on the observations of the
transient pulsation of microbubbles suspending around the
main bubble, shock wave emissions were inferred at the
rebound of the toroidal bubble or the jet torus.
For a constant bubble-free surface distance (c
f
), the
decrease in bubble-wall distance (c
b
) leads to elevations of
the speed and maximum height of the water spike and skirt,
increase in reentrant jet speed, and formation of the thin jet.
For a constant bubble-rigid bottom distance (c
b
) and a
descending free surface, the bubble is advanced in oscil-
lation stages, and an exponential rise in maximum spike
height was found.
Most of the motion features observed in the double-
boundary cases are inherited from the single-boundary
cases but change in speed, height, etc. Therefore, additional
considerations are required in bubble applications with
multiple boundaries, especially those of different natures.
Acknowledgments This work was supported by the project of the
Outstanding Youth Fund of China (Grant No. 51222904) and the
National Security Major Fundamental Research Program of China
(Grant No. 613157). The authors are also grateful for the precious
advice and support from Dr. Q. X. Wang from University of
Birmingham.
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