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G. L.
CHAHINE
Research Engineer,
Ecole Nationale
Superieure
de Techniques
Avancees,
Paris,
France;
National
Council of Scientific Research,
Beirut, Lebanon
Introduction
The erosion of solid
boundaries
subjected
to cavitation
is
believed to be
primarily
due
to the impingment
of high-speed
microjets
forned during
the collapse of
bubbles"
When
a
col-
lapsing bubble
is close
to the solid boundary
it implodes
asym-
metrically.
A microjet initiated
by
the deformation
of the
bub-
ble wall
penetrates
the
bubble
and
strikes
the solid
boundary
with
a veloeity capable of
damaging
the wall
[1-8].r
Thus
an
eventual
deviation of
this
jet
or
the attenuation
of its
energy
would be of
great
interest.
'
In
some
practieal
applications
and
experiments
reported
by
Smith
and
Mes1er
[9]
this was obtained
by introducing
gases
into
the cavitation
region
or injecting
gas
bubbles in
the vicinity
of
solid boundaries
to be
protected.
In
another
set of
experiments
by Gibson
[10],
the
cavitation
damage
resistance
capability of
flexible
membranes
and elastomeric
coatings
was reported.
In
all these cases, the behavior
of the collapsing
bubble is
similar to
thst
of a
bubble
near
& free
sudace.
When
the effect of
gravity
is negligible
an
oscillating
bubble is repelled
by
a
free
surfacg or
a
flexible
membrane,
and
its
microjet is
oriented
away from
this
surface
Ul,
l5l,
whereas
it is
attracted
by a solid
boundary
and
its microjet is
oriented
toward
it.
Thus, the investigation
of
the interaction
of
a
free
surface
and
e
collapsing bubble,
with low
gravity
effect,
would
be a first
l**o
in
brackete
deaignate
Refereuces
st end
of
peper.
Contributcd
by the Ftuide
Engiaeeriag
Division
of
Tns
AuBareex
Socrerr
or
Mncsearrcer.
Exsnrppns
and
presented
at
ttre Joint Applie.d
Mechanics,
Fluids
Engineering and Bioeogineering
Conference, New Hsven,
Conn.,
June
t5-17,
1977.
Manuscript
reeeived
at ASME lfeadquarters
March 24, Lg7T.
Paper
No. 77-FE
6.
Journal
of
Fluids
Engineering
lnteraction
Between
an
0scillating
Bubhle
and
a
Free
Surface
Interant[on
betueen
a collapsi,ng
bubble and,
o
free
wrfane
i*
innsti4ateil tlw,aretically
and erperirnentollg using hi4h speed
plwtographg. A limit
uah,e,
for
tlw
d;istonce
front
tlw
lree
wrfone to thc cenler
of tlw
bubblp
reported
to
its
rod;itts
is
fwnd,.
Unilnr
this
limit
tlw
free
surfo*e is nat distwbed
befme
thn, enl, of
thc
collapw
in
tlw
first
approri,-
mation.
OnIy
in
tlvis
case,
tfu
methnd
of images ean
be
used ard tlw
free
surface
fu
repla,ced
bU
an
ima,ge-sotnu, symmetrinal
ulith
respect
to tlw
free
su,rfane,
to th,e
si,nk
representirry
th.e bubble. Abwe this
limit,
preci,w ntpo,stLrem,ents of bubble ilzformati,on
and moti,on
o,re
giuen.
Just
attzr tlw
colla,pse
of
thn bubble
befir*, obserualinns
slww
o
si,n4ular
perturbatinn
on tlw
free
wxta&,
with
tlw
formatinn
o{
a thin spilce directeil
twsards
tlrc
ai,r. In all
ceses, bu,ayarcy
has
rw
timc tn toJce
effeet,
end
tltp
bubblp is re-
pelled
from
tlw
free
atrfwe
whiln, tlw
re<nteri,n4
jet,
fmmcd
d,uri,na
collapw,
is
orinr$pd
owuJ
from
il.
step
in
the
understanding
of
the
other similar cas€r,.
Moreover,
the
observation
of the
collapse
of
a
vapor bubble in
the
vicinity
of a
free surfaee
(or
the
interface
of two
nonmiscible liquids)
should be
very
interesting
for the
me&surement
of
the energy o{
collapse.
Actualln
as feported
in a
prior
work
by
the
author
[16],
il the
two
liquids are
colored
differentlS the mieroiet can
be
visualize4 and
the
measurement
of
the
veloeity
and the
cross
section
of this
jet
would
therefore
give
us
a
basis
for calculating
its
energy.
Theoretically,
the
problem of the
growth
of a bubble
nea,r
&
free
surface has
been
largely investigated
in the case of under-
water explosions
[17-20].
The compressibility
of
the
liquid is
ta'ken
into account,
and the
explosion
is assumed to
produce
a
spherical
blast
wave,
which
propagates
leaving
behind
it a vapor-
filled cavity.
This blast
wave reaches
the
free
surface,
deforms
it
and an expansion
w&ve is reflected
back
in
the water,
interacts
with the
bubble
and so
on. .. .
These studies
ane
very interesting,
in regard to the bubble
deformations,
when its
period
of
oscillation
is of
the same
order of
magnitude
as the
time
needed by
the
blast
wave
to
reach
the
freo
surface.
For small
oscillating
bubbles,
like cavitation bubbles
close to the
surface, the
velocity
of the
bubble
-wall
is
largely
subsonic,
except
for
the
end of
collapse.
On
the
other hand, the
ratio
of
the
free surface to
bubble
distance to the
velocity of
sound
is
very
small
compared
to the
period
of
oscillation of
the
cavitation
bubble.
So, the assumption
of incompressibility
is
a
good
approximation.
Although the
problem
of
the
deformation
of the free surfaco
has not been,
in this
case, investigated
theoretically, the
method
of images
(often
used
to
study
potential flows near solid
or
freo
surfaces)
has been
proposed
by several
authors
in order
to
study
the
influence
of the
free surface
on the
behavior
of
the bubble
[11,
15,211.
DECEMBER
1,e77
l7w
When
the surface
is
solid,
the
flow erested
by a sink
at
a dis-
tanee
I
from
the
st:rface
is identieel
to that ereated
b;r two
sinks
placed
at
s
distance
2l apart.
This result
ha^c
been
extended
ef-
fectively
to the
collapse
of a
bubble near
& solid
boundary, the
growing
bubble
being
represented
by a source
and
the
collapsing
by
a
siak.
Its
validity
has been confirmed
experimentally by
Tim
gnd
l{ammitt
[6]
who reobtained
a collapse simi]ar
to
thst
near a solid
boundary
with
two simultaneout spark-generated
bubbles collapsing
together,
each
one behaving
as the image
of
the other.
Using
the
same
method, Kling and Eammitt
lil
analyzed
theoretically
the
attraction of
a
collapsing
bubble
towards
a
solid
wall. Numerically, the same method
has
been
recently
used to simulate tbe
asymmetrical collapse
of e bubble
near a
solid
boundary
[81.
The
bubble
w'ast
approximated
by a
sp&tiel
distribution
of
sourees and the
solid boundar5r was nF
placed
by a
symmetrical
distribution of identical sourc€s.
The extension
of
the
method
of
images to the case of
a
free
surtace
is often
done in naval
hydrodynamics
[221.
In
this
case,
the free surface
is replaced in
a
first approximation
by an
image
sin\
symmetrical
with respect to this free surface, to each
souroe.
This
approximation is vslid when
the
perturbation
of the
free surface
is small eompared to its initial distance to the
source.
We can extend
this approximation to the collapse of
a
bubble
which
does not significantly disturb the initial free sur-
fsce. Ho\rever,
&!l wiII
be
shown in this
paper,
this
approximation
is no longer
valid when the initial distance between
the
bubble
and the
free surface
is
under
a
limit
value
l*. In
this
case
the
deformation
of
the
free
surface
can be several times larger than
the
distancs
l*.
In
this
peper
we determine the limiting distance experimentally
and
theoretically,
and show the
different
types of interaction
between
the free surface and
tho
oscillating
bubble.
Theoretical
Study
In order to
determine
theoretically
the limiting distance of
the
interaction
between a
collapsing bubble
and a
free
surface, and
to
delineate the domain
in which the
theory
of images
can bs
ipplied the
following
assumptions are used:
the
bubble is as.
sumed
to
be sufficiently
far from the free
surface to allow
rep-
resenting
it as an
oscillating source of variable intensity
with
time.
Yiscosity and
surface tension
of the liquid are neglected. As
shown by
computations
made by
Poritsky
[23]
and reported by
Knapp,
Daily, and
llammitt
[24],
viscosity and
surface tension
have appreciable effects
on
the
growth
and
eollapse
of the bubble
for
liquids which
have very much
greater
viscosities
and surface
tensions
than
water. Later Chahine
[16,
25]
using matched
asymptotic
expansions
concluded that both viscosity
and surface
tension
can be
neglected
for
liquids of
the s&me r&nge
as water
during
the
period
of oscillation of the
bubble.
Ifowever,
surface
tension
on
the
free
surface
is no
longer negligible
when
the sur-
face is singularly
disturbed.
The
relatively slow
motion
of
the
wall
of
the
bubble,
just
a
short time after
its creation
and up
until
the last
phase
of its
collapse,
can
justify
the
approximation
of liquid
ineompressibility.
Pritchett
[14]
showed that,
if f0 is the explosion
energy,
and C
the speed
of sound" the
Mach
number based
upon
the interface
velocity
can be
written
In our
ca.se: cavitation bubbles
of maximum
radius 2 cm
and
10i s maximum
period
of oscillation,
or for
our experiments
Eo
11
100
joules.
M<l;
if
l)10rts.
which
is
extremely
small
compared
.to
the
period
of oscillation
of the
bubble.
The
same
conclusion
is
to be
made
for
the time needed
by an
eventual
blast wave to
reach the
free surface: if
the
distance
free
surface
to the
center
of
the
bubble
isl0times its
maximum radius
the time
needed
is
about
10{ s.
Gravity is the
only external
force
taken
into
acount. In
this
case
the
problem
is axially symmetric
andif the
veloeity
potential
6'(r',
z',1')
(see
Fig. L) is introduce4
the equation
to be
solved
is:
,.
z
I
25
-\trt
tyl-_,a-
4;tna
.fl/t
5C
\8rp
-l
Fig.l
-J{9
c
:
radius of
equivalent
sphere
to the
bubble
d
:
do/ilt
&^, dn
:
v&lueg of o, d
correspond-
ing to the
maximum
value
of
(or
d)
c:(rr+r2)/2
db ih, rb rt
:
distances
from character-
istic
points
of
the bubble
to the coordinate
axis
Eo
-
energy of
the
discharge
g
-
gravitational
constant
K
:
4trd^T/a^
Kth
:
theoretical
value of K
710
|
DEcEMBER re77
I
:
electrodo
gap
to
fres sur-
face distance
-
initial
I
:
lirnit
I
:
reference
longth
of the free
surface
perturbation
L
-
honzontal
reference
length
:
y'r+l-+
:
ambient
pressure
in the
veesel
Po
:
VApOr
presSUre
B
:
maximum value
of rr
r',
zt
:
cylindrical
coordinates
r, z
:
dimensiouless
rt, zt
ff
-
time
referance time
dimensionless
time
,/m
tlh
a^/lo
flow
poiential
dimensionless flow
poten-
fo
t,
It
p
Po
f-
t-
Tt-
o!:
B:
d':
0:
tisl
\
:
L/Io
?
:
fiee
surface
perturbation
p
:
water density
r,
-
Rayleigh
collapse
time
r
:
T/T'
E:KF
Transactions
of
the ASME
Vlft(rt,
at, lf)
=
g
(1)
This
potentiul
is zubiected
to the
kinems,tic
and
dynamical
con-
dition
on the
free
surface.
Tr"
+
4,rt$rtt
-
{rrr
:
O; on
zt
:
h *
T'(rt,
,')
(2)
2gq'
+
26t'
*
$''E
+
6,"r
:
0;
on z'
:
h*
rt'(rt,
ft)
(3)
The
initial
aud
boundary
conditions ct infinity
are
written as
?'(rt,
0)
:
g,
Q'(r',
E',0)
:
0
(4)
z:t+
W.
qt(?, r)
*
o(F) (13)
Now
{
< <
I,
means
thst
la/a^
}
)
KrR;
therefore, the
method
s,{
imgg€s
can
be applied
only when
h
>>
It
:
o*Fn
To
give
gn
order
of
magnitude
of
l*, let
us
note
that
intuitively
K is
greater when
the
collapse
is
more
violent,
and that,
for ex-
ample,
in
the
eaite
presented
in
Fig.
4,
K
-
38 and
Ir
-
3.4on
-
2.5R.
As in this
perticular case
/o
-
3.2R,
and the
free
surface
is not
disturbed,
we
s€€ that
this
is
in
*gleement
with the theo-
retical
result.
To evaluate
K in a more
general
ease
let us look
at the
theoretical
empty
cavity
collapse
modd
first worked on
by
Rayleigh
[28]
and
which
leads
to
the
evalu8tion
of the
time
of collapse.
r
:
0.915
Ro(p/PiLn
Twice
.tr,
is
well-known
to
thoso
interested
in underwst€r
ex-
plosions
as the
Willis
[29]
formuls:
?
:
1.83
poqp/PtYn
(14)
frt
and
6.t--0
as
z'+-
-
or
lrrl
+$.o (5)
All
the
quantities
in
equations
(1F(5)
are
made
dimensionless
by
means
of
the
following
relations
t'-T.t
z'-h.z
r':L't
nt:lt'q
.
4ro*&*
ond
6t
:
-=-4.
'Q
VD
+
L;
c.
and
d-
arer
respectively,
the
values
of
a(r)
and
ih/ilt
w*
responding
to
the
maximum
value
of
(ald),
where
o(t) is the radius
of the
spherc
equivalent to
the
oscillatl.g
bubbla
In
this
case
(1), (3)
can be
written as follows:
6-+
-t.d,+Ir.0rr:o
(6)
f-Lftrc
.n
+';.0*;
-
$
rl-o,
*
d,r)
:
o;
on
z-I*c.q(r,t)
(7)
Insortins
equbtion
(3)
in
equation
(2)
give:
where
?
is the
period of oscillation,
E6 the
maximum
radius, and
Po the
ambient
pressure.
These
values
are
obtained
by
the in-
tegration
of
the
equation:
on:?f(*
_,
)
(15)
t-trV-*E.'Qu+2+
p
6,en
*'I
**,+
T
Eo-e,,
The
solution
of
(15)
gives
the
variation
o(t)
of the
radius
of the
bubbla
K
is oagily
obtained
from
(15)
by
cdculating
the
value
of
o(t)
for
which
d/tu(la'al)
:
0,
end
which
is:
at:
ftr.
4.rn
(16)
fi,n:
(2Polp}.n
(17)
Finally
(14),
(16),
snd
(1?)
insert€d
in the
expression
(10)
of
K
gives
us:
K
:
4r
X
\/2X
1.8:t
X
4tn
-
51.6
*
)r-?pn[^6"
*2
}1j
&6,6-
:
A;
on
z
:
I
*
aq
(8)
T
Titb
p-vi+I-|
7-fu7'
Ar
and
lt
:
a^Krn
=.8.7 o-
;
It
sz
2.$
no
(19)
Thus
the
value
of
K
is
constant
in
this
ease
and
a unique
value
for
lr
is
obtained.
For
the
more
realistic
case
of a
gas-filled
loUUtu,
msny
authors
bave
pointed
to the
faci that
tho
pressure
of
a
gas
inside
the
bubble
introduces
e
damprng
effect.
But
in sll
cas€fl-
the
correction
introduced
by
the
gas
presence and
its be-
havior
does
not
exceed
a
few
percent
[141.
on
the
other
hand'
experimental
results
on
underwater
explosions
sre
always
re'
ported
in agreement
with
the
simplest
model
[14'
1U. Meanwhile,
" -or"
recent
trsshnical
note
by
Gibson
[30]
reported
significant
deviation
from
the
Willis
formuia
(14),
for
electric
spark-gen-
erated
bubbles,
when
the
liquid
is
close
to
its
boiling
point.
This
deviation,
as
well
as
the
errors
due
to the
empty
cavity
model
and
measurement
errors,
added
to
the
fact that
the ex-
ample
presented
above
is
not
the
exact
limit case,
can explain
the
di-ffereoce
between
the
theoretical
value
of
K and
the
value
K
=r
38
in
this
example.
However,
we
should
notice
that
the
two
values
obtained
for
l* differ
only
by 7
percen-t'
Experimental
EquiPment
vapor
bubbles
&re
gener&ted
in
water
by
dischorging
for I
.,ru"y
iri*f
timo
(approx.
10-5
s),
a
capacitor
acrossl
a
pair
of
ptaii"um
electrode,
The
maximum
charge
of the
capacitor
is
approximately
20m
volts.
The
shortness
of the
dischsrgo
is ob-
tainea
by
means
of a
flash
bulb
which
acts
as an
interruptor.
The elecirodes
are
mounted
in
a
vessel
whose
horizontal
cross
section
is
sufficiently
large
to
prevent
an
appreciable
variation
of
the
fr.ee
level
surface
due
to the
addition
of the
volume
of
the
(18)
r-T/T*
;
T'-{6ls
(11)
r
and
I
are the
given quantities
for the
physical
problem.
tr,
?,
nnd
c must
be
chosen such thst
the
maximum
number of terms
are
retained
in the equations
to
be solved
by
a
perturbation
method
[261.
In
this
problem we ar€ interested
in the behavior
of the
free
surface
during
the
oscillation of the
bubble. On the
other
hand,
as
aforementioned,
the
only
ca^se
whero
gravity
effects
are
negligible
is considered.
Thus,
the
refercnce
time
?, is the
period
of
the
oscillatrie*
of
the
bubble
and r
is always
< <
I
in the
case
of a cavitation
bubble
where
gravity
has no time to
occur.
The order
of magnitude
of the
different terrrs
in
equations
(6F(8)
depends
only
upon
the
order of magnitude
of
[.
A8
shown
in another
report
[27]
linearization
is
only
possible
for
g
<
<
1,
and
potential
flow
obtained by
placing
a
source at
(0
0) and
a sink
at
(2,
0) satisfies
the
kinematie end dynamical
conditions
on the
freo
surface
z
:
I
*
a
'
nQ,
t).
Iu
this
case
the
free surface
is
not disturbed
in the
first approximation
and
the
order
of maguitude
of
c4
is
(F
").
In
tho
second approxi-
mation
nr|,
t')
can
be written
lr,
:
L/h
i
f-Kff
i
i
d,-hh
(9)
, P:a-flo
(10)
?r(r,
t)
-
2
f'
+*o*
.
au
'-
(*
*
rF"Jn
The
equation
of
tho
free surfsce
ctn
then
be written
Journal
of Fluids
Engineering
(12)
DECEMBER
Le77
I
7tL
bubble.
ThLq
variation
was
limited
to 0.1 mm
in
the worst
,:r-se.
The eiectrode
gep'to-boundary
distance
was
incres,sed
by
rdding
water
or by
moring
the
electr,:des. The
pressure
in the
r-essel
and
its
variation
&re
meesured
by a
pressure
tap *nd a
differential
transducer.
The maximum
value of
the capacitor charge
limits
the
maxi-
mum electrode
gap.
In order to
increase
this limit
and
to com-
pensafe
for
ihe
erosion
of the electrodes by
t'he effect
of
the
sparks,
ihe
vessel
is
heremetically seeled
end connected
to a
vacuum
pump capable
of
a minimum
pressure
of 0.06 atm.
)Ioreover,
lowering
the
ambient
pressure
in
the
vessel
has
two
other
favorable
effects:
waler
degasing
is faciiited before the
beginning
of the experiment,
an4 more
importantly
the
period
of collapse
is
signifieantly
increased
(i.e.,
if
ihe imposed
perssure
were 0.06
atm
instead of
1
atm this
period
would be
muitiplied
by
a
factor
of
five). This slowing
down of the
phenomenon
w&s
directly
responsible
for
obtaining valuable information
with
a
high-speed
c&mera
with a
limiting framing
rate of
20,000 frames
per
seeond.
A HYCAM
Model
K20S4 AW camera capable
of
10,000
frames
per
second
was
used to
photograph
the bubble
and free surface
evolution.
The rate of the c&mer&
can be
doubled
by taking a
haU
frame
for
each exposule.
All
the
figures
presented
here
are
half-frame
photographs.
-With
a
telephoto
lens
having
a
150-cm
focal
length,
and
a
set of extension
tubes, the
minimum size
of
the
field
can
be re-
duced to
19
X
7 mmr.
In
this
ca.se,
the
distance from
the
c&mera
to
the
electrodes
is about 70 cm, and this
distance
is sufficient
to
plaee
flash
bulbs
in
front of
the
tank and
on
the
side of the
csmers.
In
all
cases focus
lighting was
provided
by eight
750
watts
flash bulbs
arranged as mentioned above.
-Once
the
films
were
developd,
an
LW
photo
Optical
Analyser
was
used to
study
the
bubble
and the free surface
evolution
by
projecting
frame by
frame
the
part
of
the
frlm
which
is
of interest.
Experimental
Results
and Comparison With Other
Results
When
viscosity
and surface tension
are
neglected
the
p&ram-
eter
which characterizes
the
interaction between
the free surface
and the
oscillating
bubble
(the
deformation of the
free
surface
for
example,
or any
other
parameter)
depends on
gravitS
radius and
growth
rate
of
the bubble, the
initial
free surface to
bubble
distance
lo,
and
time.
This
leads us to three adimensional
parameters,
a Froude
number
(r-t
in our analysis), the
growth
rate
nnmber
K
-
t&^T/a^, and
F
:
a^/la. In
the
case of
cavitation
bubbles, r
( (
1, during the
period
of
oscilletion
f,
gravity
hqs
no
time to take
efrect, So that:
T
-
f(fr,
K,
p)
can
be written
q
.
T(K,
FT
Let's
note that
B
and
K
are not
independent:
p
-
(a^/h)
varies
with the
growth
rate
like
o-, Bnd
K
depends on
Io because d-,
o-, and
?
sre dependent
upon the eapacitor
cherge
C, the
op-
erating
pressure
P6, and
/o.
In addition, the analysis above
shows
that the two
parameters
K
and
p
always
appear as the
grouping
Kpr: so
that
finall5
if
we are interested
qualitatively
in
whether
a
jet
arises
from
the
free
surface
or
not.
q
-
l(KpE)
Theoretically
in the simplest cese of
an
empty cavity model
K
is
proved
to
be s constaat. Experimentally
observations
of
periods
of
oscillation show a deviation
of only a
few
percent'
from
this
model.
So that in
this cese,
?
is
only a
function of
B
in
the
first approximation.
In order
to
obtain
simultaneously a comparison between the
different
behaviors of
bubbles
grown
virtually at the same rate
7n
I
DEcEMBER ts77
at
difrerrent
depths,
and
to
verify
the
theoretftsl
result
thst
Kfl'
proceed in this
way:
(c)
Maintaining C and
Po
eonstant, the interaction
was
con-
sidered
for a
range of
values
of /o
The value of Kff
limi{,,
which
implies
a
value of It function
of
{!a w&s
deduced and
seen
to
be
close to
l.
(b)
As
the
criterion
of little
disturbatrc€
is not very
precisg
and
as the
real
value of K for
the
experiment
is not
known
in
advance,
it
wa^s
verified
that
for different
values
of
Pc
and for
0
slightly
greater
than
p
limit
:
(Kth)tn
:
(51.$)-tr.;
n/h
<
0.05.
As
we
were
limited by the
maxi*um value
of
the
capacitor,
a
range
of
values
for
Po
between
0.05 and 0.&1
atm were
investigated
wtfhn/h
-
0.4.
In
all
cases the
perturbation
generated
on the
free surface
wes
only a slight
orLe
(n/h
<
0. 05).
Below are
presented
the
results
for
the experiments
(o).
A series
of
photographic-experimental
mns were
made
t,o
examine the
effects
of initial
free surface electrode-gap
distance
I
upon bubble
collapse
behavior
and free
surfece disturbanca
In all cases
here
presented
the
magnitudes
of
the
capacitance,
the
charging
voltage,
and the
damping resistor were
the sarne.
The
pFessure
Po was maintained
at approximatively
0.
1
atm
with
only slight
variations
in
the
experimental conditions,
due
to
fluctuations
in the atmospheric
pressure,
which
introduced
varia-
tions in
Po less than
10
percent.
From the
photographic
sequences,
messurements
were
taken
of
the
positions
of the
difrerent characteristic
points
of the
bubble
and
of
the
free surface.
Five
points
were chosen
for
this
purpose
(see
FiS.2). d is the
highest
point
of the
free
surface. The
vertical
distance from
,4.
to
0
(electrode gap)
is denoted
by L B
and
C
are
horizontally the
farthest
points
from 0. The
horizontal
dis-
tances
from
0
to
B
and C
are
denoted
by dr
and
ila.
ln
all cases
we
have dr
cc
d2
because
of the
axial symmetry
of
the
problem.
.E
and
D are the
extreme
vertical
poink
of
the bubble.
OD
and
OE are
defined
ss
rr
end
&,
respectively.
It
should be noted
that
OE
is characteristic
of
the
microjet
penetrating
the bubble.
In
order to
compare the
different
cases, all variables
&ro nor-
malized by
dividing
distances
by
B,
the maximum value
of 11,
and
the
times &re
normalized by the theoretical Rayleigh
[28]
collapse
time
of a spherical
bubble
with
a radius B,
an
external
pressure
Po, and an
internal
pressure
Po.
As mentioned
above
B
is a
function
of lo
and
(P0
-
P,).
From
I
I
fo
A
>
B
o
7"\
C
r
d
V
d2
-
D
Fig. 2
Dcfinitlon
sketch
for thr charactrrisuc
polntr
of
thr
bubbb
and
the frce surface
Transactions
of
the ASME
Fig.
3.
I
is shown
to
be
a decreasing function
of
lo for
(Po
-
Pu)
approximately
constant.
However,
an asyrnptotic
value
of B
is
attained
f.or
R/t,t <
0. 3.
fhis
vaiue
sppears
also to be a limiting
value
for
the
bubble.free
surface interaction.
As
a matter of
fact,
we can
see
from
Fig. 4. that f.or Rlh
-
0.
31 the free
surface
is
eompletely
undisturbed
by the deformation
of
the bubble
while
this
deformation
is
practically
spherical. In
this
case
K
.-
38
and
Kpr
-
0.32
(
1.
llaen
p
increases, the calculation
of K,
based
on
the equivalent
sphere
radius,
shows
an increase
in its value.
So
Kpr
increases
more
rapidly
than
ff.
I{owever,
when
the
interaction
free sur-
fac+bubble
is
very
weak
all
values
obtained
for K
are close to
the
theoretical
value
35
<
K
(
65. As rKff
is increased
from
bhe
previous value
(0.32)
to the value Kpr
=.
4,
as
in Fig.
5,
where
R/h
:0.6,
the
deformation
of the free
surface
increase
slowly
and attains the
value
I/lo
-
0.23.
For
this r&nge of values
of. KPE,
the
free
surface remains
smooth
and
no
angular deformation
is seen in
the region
above
the
center
of
the bubble. This
bubble remains
approximately
spherical
during 70
percent
of
its life,
then coUapses
nonspherically
and its
upper side moves
away from
the free surface.
However,
this side
reaches
the
opposite
one
before
the formation
of
a
microjet.
.{s
assumed,
the
effeet of
grar-ity
is
negligible. :ad
ihe
;epulsion
due
to the
presence
rlf
rhe
iree surface
is
preri.ominant.
In
rhe
casse
presented
in Fig.
6, R,/1,1
-
0.3.
K(ta
-
22,
and free-sungce
is
more seriously dlsturbed. During rhe
growrh itf
rhe
bubbte.
its
upper side is attracted by
t'he
free surfr,ce,
:lnd
the
briobie
is
elongated towards thi-c surface
itself being
disrurbed
regularly.
Then
the
bubble collapses
with the
formation
of
a reentrant
jet,
while
the
disturb&nce
on
the
surface becomes
more
engular.
The deformation
of the bubble and
its influence
on
t,he tree
sur-
face during the
6rst few milliseconds
was
well-explained
uheo-
retically
and
numerically by
Ball Hauss and
Holt
ltgl:
The
spherical
biast wave
produced
after the erplosion
reaches
the
free
surface and
deforms it.
Then,
the
reflected
expansion
w&ve
deforms, in turn,
the upper side
of ihe bubble. Ilowever,
this
study did
not
go
further, and
a satisfactory explanation
of
the
singular
disturbance
on
the
free surface,
which
appears
a long
time after the spherical blast
has reached this
free
surface,
moFB
precisely
just
after
the
beginning
of the
collapse,
cannot
be
given.
In
Fig.7,
selected
frames of the
last case
where
Rllo
-
1.5,
KF,
=.130
are shown.
The
perturbation
of the free
surface be,
comes
great
even
during
the
bubble
growth.
Then
as more clearly
seen in Fig. 9(c),
when
the bubble attains
its
meximum
radius,
for
a fapse
of
time
of
approximately
2
milliseconds during which
the
displacements
of
the
wall
of
the
bubble
cannot
be detected,
a
thin
jet
comparable to
Taylor's instability
[21]
arises from
the
free
surfacg
while
another
very clearly
visible
microjet
(see
FiS. 8)
penetrates
the
bubble,
pierces
it, and continuee
to
be seen
moving
&way
from the
free surface.
This
jet
seems
to carqr
array
Flg.5
Bubble
No THE
3$ R/lc
:
0.t
Fig.7 Bubblc N0
THE 252
Rllo: 125i
t
-
0.161
5.5' lz.t'
L6.8,2;22'25.a,
2t.2, {t
ms
.o
5Ee
-(\1
E5
o=
.=6
EL
3e2
e€
€5
Eb.
bE'
99
=F
H€
€c
o
raintv
_!,+\_
"'U.,y---
--€-----:2l-
lo
2345
distance
from electrode
gap
to the
inilhl
frae
surface
lo
,
cm.
Flg.3 Eubblr radlur v.r3u3 rlcctrodt
lrp
to thc
Initlal
frct surfacc
distancr
Flg.
I
Bubblc
il. THE
4,
bz Rlh: 0.:n;
t
:
0J&
7.91i,
11.7, 1i!.6 mr
Eubblr
l{r,
THE 20; Rlls
-
0.5;
t
-
0J3r
t.t,15:,
17.1 nr
W
m
ffi
W
ffi
M
:-i
i;pv\|
.:'
.
::;:'i1
Journal of
Fluids
Engineering
DECEMBER te77
|
713
<"#
Flg. t Bubblc
Nc
THE
2r[:
I||tcrojrt
with it a
quantity
of
gases
which later
collapses,
while
two lateral
jets
are
produced
on
the
free surface
on
both
sides
of the first
jet.
Let's again note
that the
jet
formed
on
the free
surfacg
usually
known
in
underwater explosions
phenomena
as
the
"sult&n"
or the
"plume"
[12,
14J,
appeers here
a very long
time
after
the explosion, during
the
eollapse
and
not
the
growth
of
the
bubble
as
generally
reported in underwater
explosions,
In Fig,
I
(o), (b),
c),
the outlines of
a
growing
and collapsing
bubble are
drawn
at
difrerent
selected
times, for
three different
values
of. R/h. In
this
figure
the timing
of
the creation
of the
jet
and the
variation of
the bubble
radius
are more
clearly seen.
We
can conelude
that
the time between
the beginning
of collapse
and
the initiation of the
jet
decreases
with
(Kpr;-r.
When
/o/R
becomes
much less
than ong
venting
of
the interior
of
the bubble
occurs and
our study is no longer
valid-
fn
order to
make
quantitative
comparisons
between
the
dif-
ferent cases
presented
above,
the
variation
of
the normalized
values of l, 11, 12, with nornalized
times are
plotted
in Fig. 10.
As in all cases dr has
approximatively the
same magnitude
as
I
and
for the sake
of
simplicity, its variations
are not
plotted
in
FiS.
10.
In
the
Fig. 11, the normalized
magnitude
of C
:
(rt
+
rl)/Z,which represents
the
position
of
the center
of
the
bubble,
is
plotted.
It
shows the motion of
this center,
and cao be com-
pared
to that
of /..
As shown in
Fig. fQ
the
variations
of normalized
rr(l)
are
practically
independent
of the electrode
gapfree
surface distance
and
of Kpt.
In
the
four
cases
presented
the
four
curves
are
nearly
identical, espeeially
if
wo take into
account
errors
of measure-
ment and
calculation. The
difrerent
cases difrer
only
in the nor-
malized time
beyond which
the
definition
of
the
radius
1,
is no
longer
valid. As
in addition dr
=
d:
has
approximately
the
same
magnitude
8s
11, w€
see that
the behavior of
this
part
is not in-
fluenced by
the
presence
of
the free surface.
This
conclusion is in
agreement
with
observations
done on
underwater
explosions,
even
if venting does oecur,
as in upper
eritical
depth
problems
t201.
On the contrary,
the
behavior
of
the upper
portion
of the
bubble
is
highly
dependent
upon
the
relative
distance
from
the
free
surface.
This dependence
is
unchanged
for values
of Kff
<
l;l*/R
)
3.
In
these cases, as shown in
Fig. lQ
for lo
-
3.2 R,
the curve
-rr(d)
is very
close
to the curve r2(t)
and the bubble is
practically
spherical.
As
the
norrnalized
distance /o is reduced
from this
limit we
observe
(o)
an increase in
the slope of rr(t)
during
the
growth
as well
as
during
the
collapse.
(b)
an
inerease
in
the
maximum
value
of rr(t)
(c)
a decrease in
the
period
of oscillation
of
the bubble.
As a consequence
of
these
obsen'ations
we see
that the speed
of
the
mierojet increases
with the
proximity
of
the bubble
to the
free surface.
The
maximum
value
attained in
the case /o
-
0.88
is 4 m/s.
In
this
casg
the
speed
of
the
jet
on
the
free
surface
is
3.5
m/s.
On the
other hand
the migration
of
the center of
the bubble
from
the
free surface
sppe&F larger
ash/R is
reduced. The
center
714
I
DEoEMBER
ts77
oi
r|t
€l
JI
rl
sl
I
I
Fig.
9(a) Shapc
of
thc bubblc and
thc frec surfacr
at diffcrtrnt thna$
Bubblc
t{c
THE 2O:
R",'.,lls:
0.60r
tlmc scalc:
13.10- c.
Flg. 9(b)
Shape of
thc bubble and
thc frcc surfacc
at diffarent
tmcs.
B-bbt.
No
THE 23: Rrner/to
:
0.!15, timc
scalc: t.lf
:
FlC.
t(c)
Shapa
of
thc bubblc
and
thc frcc surfacc at
dlffcrcnt tlmcs.
BubbfG
Nf
THE Ez Rmt-llo:
l.5r
tlme scal.
:
t.10{ r
is seen to
move slightly toward
the
free surface during
the
growth
phasg
then
during
collopse the
center
is
violently repelled
from
the free surface
and this
repulsion
is
higher
gg
l"/R
decreases.
The
time
interval
between the
two
passages
of
the
center
of the
bubble through
the
initial
point
0 is shown
to
decrease with
Io/R
and.
with the
period
of oscillation
of the bubble.
Although a
very large amount
of experimental
data has
been
acquired
over the
ye&rs
concerning
underwater
explosions,
pre-
eise measurement
of
the bubble
motion
are lees
available. Most
eI
ol
3l
Il
El
-I
Transactions
of
the ASME
|\|
;
n
-,
;
?
g3
CD
B
3
-t
3
.s?
c
E
c
E
I
Ffg.
l0 Yarlation
ol dv
dars
rrp lrd
I
vcrsus
time
for
dlffcrcnt
dcpths
of
thc bubblc
r, t -
\s
k'Jo
I
gtresse4
in
all
cases
we
have
presented
here
this
jet
eppearc
a
long
time
afterwards.
This
difrerence seems
to
be
due
to
the
formation
of
a
stronger
shock-wave after
the
exprosion
than
after the
generation
of
the spark.
craig
also reporled
the
pres-
ence of
e root
in
his
experiments,
which has never
been
observed
in ours
before
the
collapse
of
the bubble.
Another
interesting
report
on
the
behavior of
the
bubble
radius
which
has
already
been
quoted
tlgl
is on
the
wrGwAM
nuclesr
explosion
of
1g5b.
unfortunately
comparisons
are again
not
possible
because
of
the
important
buoyancy efrect
in
that
case,
which
gave
the
bubble
an
upward
motion.
To
our knowledgg
no
precise
measurements
of
the iuward
microjet
motion
in
the
presenc€
of
a
free
surface
are
available.
rn
the case
of
the
presence
of
a
solid
wall,
such
measurements
have
been
presented,
however,
by several
researchers
on
cavit&-
tion cited
in
the
introduction.
The
limit
distance
of
the action
of
the solid
boundary
on
the
bubble reported
is about
three
times
the radius
14,
5,71.
This value
is
quite
close
to
the one
we
have
presented
here.
Conclusion
I
Theoretical
and
experimental
resultg
ghow
that
the
free
surface
and
the bubble
Dre
no
longer
disturbed
by each
other
when
their
distance
apart is larger
than KF
-
l.
'Whea
the
in-
teraction
isvery
weak,Kis
between
85
and
Gb,sothat
Bisbetween
0.25
and
0.3.
we may
conclude
that in
all
cases l*
is about
three
times
the
maximum
radiusof
thesphere
equivarent
tothe bubble.
2
Jets which
enter
cavities
sre
observed
experimentally
and
they
are directed
ewey
from
the free
surface. The
speeds
of
these
jets,
even
if
we
take into
account
the small
difierence
of
pressure
between
the exterior
and
the interior
of the
bubble,
are
smaller
then
those observed
previously
near
a
solid boundary:
the
maximum
speed
of
6 m/s observed
corresponds
to B0 m/s,
when
the
pressure
difrerence
is I
atm, wheress
speeds
reported
ne&r
a boundary
are
greater
then
100
m/s.
3
The
bubbles
moving
slightly
toward
the free
surface
during
their
gnawth
are
repelled
during
collapse
and rebound.
The
repulsioo
increases
when
lo/B
decreases.
In
all
cases
buoyancy
hss
no
time
to take
effect.
4 The
free'surface
is
regularly
disturbed
when lo is not
much
less
than l*.
It
is very
singularly
deforured
when lo
(
(
l*.
A
thin
jet
is
formed
on
this free
surface
and the
time scale of
tbis
deformation
is
much
greater
than
the
period
of collpase.
This
jet
occurs
a short
time
after
the beginning
of
the
collapse.
This
delay
decreases
with ro/t*.
This work
is
to be
continued
through
the
investigation
of the
case
of different
liquid
interfaces
and
a
flexible
boundary,
and
must
be
enlarged
to include
a wider
renge of values
of
the
energy of
discharge.
Acknowledgments
I
wish
to thank Dr.
Hammitt
for his
interest
in this
work.
This
work
w&s
supported
by
the
Direction
des
Recherches
et
Moyens
d'Essais
(Delegation
Ministerielle pour
l,Armement),
Franee.
References
_
I l{aude,
C.
F.,
and
Ellis, A.
T.,
.,On
the Mechanism
of
Cavitation
P_?tr"g"_
bv
Nonhemispheiieal
Car-ities
Collapsing
in Contact
With
a
Solid Boundar5,,"
Journal of Basic
Enginen-
in4, Tn.rxs.
AS\[E,
Series
D, Vol.
bg,
fgOt,
ppl
O+S+SO.
. _! _Pfesset, \!.
S., and
Chapman,
R. B.,
i,Collapse
of an
Ini-
tialll'
Spherical
Vapour
Cavitl-'in
the Neighbourhobd
of
a Solid
Bounda,r.v,"
Journnl
of Flui.it
Meclnnics,-Vol.
4?,
Part
2,
1g7l,
pp.2E3-290.
3
Mitchel,
T. I{., end
Hammitt,
F.
G.,
,,Asvmetric
Cavita-
tion Bubble
Collapse,"
Jocnx.rL or
Flurn
ExarNlrnrsG,
Tn.rNs.
AS\IE,
Series
I, Vol.
gS,
No.
t,
1gZB,
pp.
2g-3?.
4
Shima,
A.,
"The
Behavior
of
a Spherical
Bubble in the
R/to
=0.3
Bllo
= 0,6
tilr
: t.t
t.
=
a,lt I
to'l
t
Rllo
= 0,3
Rllo
= 0,95
d1,d2,11
ltl
of
the
experimental
data pubrished
&re coneerned
with
the
period
of
oscillation
of
the
bubblg
and
the
influsnc€
of
a free
surtace
or
a
solid
wall
on
it.
Thus
the shortening
effect
of
the
presence
of
the
free
surface,
observed
here,
has
been
largely
'verified.
Amo.g
reeent
experimental
studies,
the
one
which
bears
the
mosr
resemblanee
to
the
present
study,
to our
knowledge,
is
that
of
craig
[12].
His
experiments
were
performed
with"sphericar
ex-
plosive
charges
placed
at
different
depths
in
a
l,arge
tank.
He
was
especially
interested
in
the
upper
critical
aeptu problem
where
venting
occurs;
and
he
naa-to
simultaneously
use
six
ca,mer&s,
to
obtain
precise
results.
Although
our
study
is
not
coneerned
with
this problem,
it
is
possible
to
make
some
in-
teresting
comparisons.
First,
he
reported
that:
,,the
gas
bub.
ble
reaches
its
maximum
size,
and
Legins
to
collapse
d"rt
fro-
ttre
bottom,"
which
indicates
that
the
efiect
of
gra'ity
is
pre.
dominant
in
his
experiments.
This
is confirmed
by
the iact
that
the
period
of
oseiuation
in
the
exampres presented
is
about
0.
5
seconds
q-hile
the
time
y*
:
{Is,i,g
<
0.0i .r
.
(10
distanee
from
the
center
of
the
explosion
to
the
initiar
free
surface.
onr1,
in
one
c&se
(Fig.
13
in
[i2])
is
downn'ard
motion
of
the
oscillating
bubble
reported.
This
case
eorresponds
t'
a
period
of oscilIation
(0'1
s)
smaller
than
r*
(0.2is),
where
our
study
is
more'arid..
However,
erren
in
thi.s
cs.se
nhich
corresponds
Lo
a
Rihof
about
0.8,
the
water
jet
on
the
free
surface
L.
reported
t,o
be
erearcd
just
a
short
time
after
the
exprosion.
As
we
ha'e
prertousrl-
Journal
of
Fluids
Engineering
B/ls
=t,5
f,llo
- 0,6
l
.r
Fig.
tl
Yarlation
of
/
and
c vorsus
timc
for
different
yatues
ot
Rlro
DE0EMBER re77
I
7L5
Vicjry!]'-of
a Solid_-Wxll," J_utrna!
9f
Basig
Enginering,
TR.r.-vs.
.\S\[8,
Series
D,
Vol.
90,
]io.
l, ][ar. 1968,
pp.
7;-89:
5
rhima, l-,
"The
Behavior of
:r
:pheiiial
Brrbble
in t.he
Yicinitl'of
e Solid
lValli Report
{,"
The
Reports
oJ
the
[nstitu,te
?t,!iglr^Speed.l[echanrcs,
Tohoku t-niversirv,
Vo[.
!l],
lgTl.
pp.
19;:120.
S
Ti**,
E,
_E.,_1,14
Hemmitt,
-1. 9.,
,,Bubble
Collapse
Ad-
jacenr
to.g_liigiq
ll'all,..a
Flexible lVell
r,nd
a
Second
Birbble,"
1971 lS-1.18 Catvitation
Foru,m,
pp.
18-20.
- !
Kling,
C.
!.rand
H.amTitt,
f.9.*-.,A
photographic
Study
of Spark-f*duced
Cavitatio*
Bubble-Collapse,"
tourriat
of
Basie
Engineer'ing, Tn.lss.
A-{}[E,
Series
D,
Dec.
tl]7?,
pp.
82.:ri.!33.
8 Lenoir \[.,
",iiTrtlllllol
Numeriqrre
du
C6[apse
d'une
Brrlle de Cavitation," -A^\S?'{,
Rapport
de
Recherche,'064,
Feb.
1976.
-
.-!
qgitn,
8.H.,.?n{
}Iesler,
R,
B._.
',A
photographic
Study
of the Effect of
an Air Bubble
on
_rhe
Growth
and
coflapse
of i
Iapor
g:,!plf
ngar.
a
:Surface,"
Iournal
qf-
Bosic
Engineerin4,
Tnexs.
ASIIE, Series D,
Dec. 1972,
pp.
933-942.
10
Gibson,
?.p.,-:The
Kingtic
and
Thermal
Exparnion
of
Ygqgr
Bubb.les,"
9.81F9r^Division
of
)Iechanical
Erigineering,
]'lelbourne Australia, 1970.
1t Herring, C.,
"Theory
of the
Pulsations
of
the
Gas Bubble
Produced.bf_
11 !!{egwater
Explosion,
"
C
olumbia
U niu.
){ D
RC,
rep. C-4-dr
20-010,
1941.
12
Craig,
B.
C.,
"Experimental
Observations
of
Underwater
fJetonations
Near
the Water
Surfacg"
LTniv.
of
Californi*,
Los
Alamos inf. Report, LA-5548-\'IS,
1974.
l3 Pritchett, J. W.,
"fncompressible
Calculations
of
Under-
water Explosiou
Phenomena,"
Proceeding
of the
?nd
Int.
Conf
.
on
Nunterical l[ethods in
Fluid Jlechanics,
lgTQ
Univ.
of Cali-
fornia,
Berkeley,
pp.
422428.
14 Pritchett,
J.
W.,
"An
Evaluation
of Various
Theoretical
l'Iodels
for Underwater Explosion
Bubble.Pulsations,,,
Informa-
tion-Eesearch Associates, Berkeley,
IRATR-2-Zf-Api.
1971.
-ild
Cole,
R.
H.,
Unclerwater Eiploseons,
Dover
?ublications,
1948.
16 Chahine,
G.
-L.,^"Etude
Asyrnptotique
et
Experimentale
des
Oscillations et du Collapse
des Billes
d-e
Cavitation."
These
de Docteur Ing.enie^u.r,
Universite
Paris
VI,
Dec.
lg74.'Rapport
externe E N ST A, 11042, lg7 4.
_-17__Collins,-
R.,
.'jlnlgrye_ Fxplosions
at
the Ocean
Surface,,'
The
Phystcs
of
Fluids,
Vol.
11,
No. 4,
Apr.
1908.
pp.
7OL-Tlg.'
18 Chan, B.C., Holt, NI.,
and Welsh,
R. L.,.,,Explosions
due
to Pressurized S-pheres.at
the_OceanSurface,t'
Thi
physics
of
Fluids, Vol. 11, No. 4, Apr. 1968, pp.714-722.
19 Ballhaus,
W.
F.,
Jr.,
and
Holt,
M.,
"fnteraction
pil-ween
thd
Ocean
Surface and
Underwater
Spherical
Blast
W'&ves,"
The
Physics
of
Fluiils,
Vol. 17, No.
6,
June
1974,
pp.
1068-102'g.
_20 -
Hall,
-R. V.,^'j-Numgri_c_al
Solution
of the
Upper
Critical
P:l$^Problem,"
Office
of Naval
Research, Mai-1924,
AD-
782
282.
2I
Birkhoff,
G,_Zarantonello,
E.
H.,
Jd,s,
Wakes and
Cautties,
Academic
Press,
1957.
22
Wehausen,
J. Y., and Laitone,
E. V.,
"surface
Waves."
Handubuck
der Physi&,
(Encyclopedia
of Physics),
Vol.IX,
1g60.
23
Poritsky,.
H,
"Tbg
Collapse
or
Growth
of a
Spherical
Bubble
or Cavity in a Viscous
Fluid,"
Proceeding
of the First
U.S.
National
Congress in Applied Mechanics
(CSME),
lgb2,
pp.
813-821.
.24 _ $1app,_8,-I.rPr.!y,.1.
W., and
Hammitt,
F.
G., Cauila-
.
tion,
II_ccraw-Hill,
New
York., 1970, pp.
104-1i1.
2it
Chahine,-G._L.,-"Et_ude
Asymptofique
du
Comportement
d'une
Btrlle
de Cavitation
dans un
C!-amp
de
Pression
Variable,"
Journal, de
Llecanique,
Yoi.
15,
No.
2,
1070,
pp.
287-806.
26
l)arrozes,
J. S.,
"The
Method
of
Nlatched
Asvmptotic
Expansions Applied
to
Problem
Involving
two
Singular
Perdurba-
tion P-arsmeters,"
Fluid,
Dynatnic
Tranx., Yol.
6, Part
2, pp.
11 9-1:9.
'27
Clrehine,
G. L.,
"Coilapse
d'une
bulle au
voisinage
d'une
surtrrce
libre,'.'
Rapport
rle
Recierche
EYS?1,
No.
{)ig,
Dee.
197i.
_
28 Pg$igh,
L-ord,*"On
rhe
Pressure
Developed'in a
Liquid
During Collapse
of a Sphericai
Cavity,,,
PhiI. rtog.,
lgl7,
tol.
:14,
pp.
94-98.
?9.
\tillis,
H. F.,
"Underwater
Explosions:
The Time
In-
iervei
Berween S
uecessive
Explosio
ns,"-
B
rilish
R
eport,
WA-42-2
t,
i941.
30 Gibson,
D.
G.,
"The
Prrlsation
Time
of Spark
Induced
lgpor-
B_ubbles,"
..l^oy1ryaL
of
Basic
Engineering,
Ti,.rxs.
ASltE,
llar.
1972,
pp.
248-249.
31 Rejappa,
-,\.
R.,
"On
the Instability
of Fluid
Surfaces
when
Aceelerated
Perpendicular
to
their
Planea,,,
Aclo
ilIectw
anico, Yol.
10, 1970, pp.
193-205.
DtscussloN
F.
G.
Hammittr
I
would
like
to congratulate
Dr. Chahine
for
this excellent
work on bubble dynamics,
particularly
the interaction
between
free
surfaces and
growing
and collapsing bubbles.
I
would
also
like to thank him very
much
for his
acknowledging
my
pre.
vious
and continuing interest in
the
excellent
work
in his
labora-
tory at
ENSTA in Paris.
The
present
work is a most valuable
contribution
to the
grow-
ing literature in
the
field of
individual bubble
dynamics.
I
think
Dr. Chahine's study is
particularly
important
in
comparing
relatively simplified
analytical results with high-speed
photo-
graphic
observations,
and thus showing and verifying
that various
of
possible pertinent parameters
can be safely
neglected
in
certain
cases. Of
particular
interest
are the
results
relating
distance
to a
nearby free
surface, and
its
effect
on bubble
collapse
and
growth
behavior.
Dr. Chahine's
technique
of
reducing
pressure
to
retard
col-
lapse, and thus improving
photographic
opportunities,
is
most
useful,
and
I believe
was earlier
pioneered
by Benjamin
and
Elliss in Cambridge.
Author's
Closure
The author
wishes
to thank
Dr. Hammitt
for his
interesting
'liscussion
and
the
additional reference.
The
pioneering
character
of
the work
of
Benjamin
and Ellis
should
be noted,
and
the
tech-
nique
of reducing pressure
to slow
down
collapse
has
surely
been
used
since
in
several laboratories
working
on
cavitation.
some
of
the authors
have
been directly
or indirectly
cited
in
the
article
particularly
Naude,
Ellis,
and
Gibson.
tProfessor,
Dept.
of Mech. Engr., University of Michigan,
Ana
Arbor,
Mieh.
tBenjarnin,
T. 8.,
and
Ellis, A. T.,
"The
Collapse
of Covitation
Bubbles
and
the Pressures
thereby Produced
against Solid
Boundaria,"
Phil.
Ttar*.
Rogal
Soc.
of
London,4,
Yol.
260,
1966,
pp.22L-224.
7tG
I
DEcEMBER
te77
Transactions of
the ASME
