(Reprinted
from
the
Journal
of
Applied
Mechanics
for September, 191,9)
The
Dynamics
of
Cavitation Bubbles
BY
M.
S.
PLESSET,
1
PASADENA,
CALIF.
Three
regimes
of
liquid
flow
over
a
body
are
defined,
namely:
(a)
noncavitating
flow;
(b)
cavitating
flow
with
a
relatively
small
number
of
cavitation
bubbles
in
the
field
of
!}ow;
and
(L)
cavitating
flow
with
a
single
large
cavity
about
the
body.
The
assumption
is
made
that,
for
the
second
regime
of
flow,
the
pressure
coefficient
in
the
flow
field
is
no
different
from
that
in
the
noncavitating
flow.
On
this
basis,
the
equation
of
motion
for
the
growth
and
collapse
of
a
cavitation
bubble
containing
vapor
is
derived
and
applied
to
experimental
observations
on
such
bubbles.
The
limitations
of
this
equation
of
motion
are
pointed
out,
and
include
the
effect
of
the
finite
rate
of
evaporation
and
condensation,
and
compressibility
of
vapor
and
l
iquid.
A
brief
discussion
of
the
role
of
"nuclei"
in
the
li
quid
in
the
rate
of
formation
of
cavitation
bubbles
is
also
given.
INTRODUCTION
A
DISTINCTIVE
feature
of
the
hydrodynamics
of liquids is
the
possibility of
the
coexistence of a
vapor
or
gas
phase
with
the
liquid
phase.
Such
two-phase
flow is
usually
called
cavitating
flow,
although
it
could
as
well
be
characterized
as
liquid
flow
with
boiling.
Cavitating
flow
has
great
theoretical
interest
in
addition
to
the
hydrodynamics
involved
because of
the
relation
of
this
flow condition
to
the
physical-chemical proper-
ties
of
the
liquid.
The
practical
significance of
cavitation
is of
course clear.
The
drag
of
submerged
bodies
moving
through
a
liquid rises
when
cavitation
appears;
similarly,
the
efficiency of
pumps,
turbines,
and
propellers
drops
with
the
development
of
cavitation;
and
the
damage
which
may
be
produced
by
cavita-
1 •
tion
in
these
devices is well known.
The
particular
flow
problem
discussed
in
this
paper
is
the
flow
of a
liquid
(water)
over
a
submerged
body
which will be con-
sidered
to
be
at
rest.
If
Po
denotes
the
static
pressure,
and
Vo
the
uniform
flow velocity of
the
liquid
at
a
great
distance
from
the
body,
then
the
general
character
of
the
flow
in
so
far
as
cavi-
tation
is concerned is
correlated
with
the
cavitation
parameter
po-p.
K=
--
....................
[1]
(p
Vo
2
)
/2
where
p.
is
the
vapor
pressure
of
the
liquid
and
p
its
density.
Obviously,
one
cannot
expect
a single
constant
to
describe so
complex a
phenomenon
as
cavitating
flow
about
a
submerged
body;
however, a correlation
in
a
qualitative
way
may
be
made
with
the
various
types
of
liquid
flow.
Three
flow regimes for a
given
suitably
shaped
body
will
be
indicated
here.
The
first
(K
sufficiently large) is
noncavitating
flow.
This
state
of flow con-
sists
of a
liquid
phase
only
and,
with
neglect of compressibility
1
Associate
Professor
of
Applied
Mechanics,
California
Institute
of
Technology.
Contributed
by
the
Applied
Mechanics
Division
and
presented
at
the
Annual
Meeting,
New
York,
N.Y.,
November
28-December
3,
1948, of THE
AMERICAN
SociETY oF
MECHANICAL
ENGINEERS.
Discussion
of
this
paper
should
be
addressed
to
the
Secretary,
ASME
29
West
39th
Street,
New
York,
N.Y.,
and
will
be
accepted
until
October
10, 1949,
for
publication
at
a
later
date.
Discussion
received
after
the
closing
date
will
be
returned.
NoTE:
Statements
and
opinions
advanced
in
papers
are
to
be
understood
as
individual
expressions
of
their
authors
and
not
those
of
the
Society.
Paper
No.
48-A-107.
effects, follows
the
same
laws
as
are
familiar in
air
flow.
If
now
K is
made
smaller, a
state
of flow is
attained
in which a
relatively
small
number
of
bubbles
appear
near
the
boundary
of
the
body.
This
state
of flow will be
taken
as
the
second regime of flow.
If
K
is
further
reduced,
the
number
of bubbles increases,
until
eventu-
ally
they
merge
into
one
large
cavity
which completely encloses a
portion
of
the
body.
The
state
of flow
with
a single
cavity
about
the
body
is
the
third
flow regime,
and
may
be called
cavity
flow.
A
further
reduction of K
brings
about
only
an
increase
in
the
size
of
the
cavity.
These
three
flow conditions
are
illustrated
in
Fig.
1.
FIG.
1 VIEWS
SHOWING
THE
THREE REGIMES
OF
FLOW
(In
th
e
top
view,
th
e
cavitation
param
e
ter
K =
0.40;
in
the
center
K
0.28;
and
in
the
bottom
K =
0.18.)
In
the
cavity-flow regime,
the
boundary
of
the
cavity
may
be
taken
with
reasonably good
approximation
to
be
a
surface
of con-
stant
pressure
and
of
constant
flow speed.
The
pressure
and
velocity
in
the
flow field
are
fundamentally
different
from
those
in
noncavitating
flow.
It
may
be
remarked
that,
at
least
for
two-dimensional flows,
the
powerful
mathematical
methods
of
the
free
streamline
theory
may
be
applied
to
the
solution
of
cavity
flow problems (1, 2).
2
The
second regime of flow
has
here
been
characterized
some-
what
arbitrarily
as
the
flow condition in which
there
is
only
a
relatively
small
number
of
bubbles
in
the
flow field.
This
limita-
tion
is
made
in
order
to
get
an
analytic
simplification.
If
there
are
only
a few
small
bubbles,
the
effect of
the
pressure
disturbance
of
one
bubble
upon
another
may
be neglected.
Further,
one
may
suppose
that
the
pressure
field,
except
at
the
bubble, is
deter-
mined
in
the
same
way
as
if
there
were no
bubble
cavitation.
As is well
known
for
noncavitating
flow, if p is
the
static
pres-
sure
at
any
point
in
the
flow field,
and
if
po
and
Vo
are
the
static
pressure
and
flow velocity
in
the
uniform
flow
at
a
distance
from
the
body,
then
with
neglect of viscous effects,
the
pressure
coefficient
•
Numbers
in
parentheses
refer
to
the
Bibliography
at
the
end
of
the
paper.
HYL
,
-%77
-
A
.)
l.AG·J
i
.A;
•r
r
I'
Tf.ChNO
I.CJ
I.S
'f
\
7 0
278
JOURNAL
OF
APPLIED
MECHANICS
SEPTEMBER,
1949
p
-po
cp
=
--
...........
....
.
....
[21
(pVo')/2
is
indep
e
ndent
of p
0
and
V
0
.
The
pre
se
nt
assumption
cons
ist
s
in
the
ca
l
culation
of
the
stat
ic
pre
ss
ure
p in
the
sec
ond
How
regime
with
the
appropr
i
ate
value
s of
Po
and
Vo
from the
pres-
s
ur
e coefficient C P
determin
ed for
noncavitating
fl
ow.
Thi
s
ass
umpti
on
that
the
pressure
coeffieien t is
essentia
lly
the
same
just
before
the
first few
cav
it
ation
bubbl
es
appear
as
it
is
after
of course is
subject
to
experimental
verification,
and
the
neces-
sary
experiments
arc
planned
for
the
high-speed
water
tun
nel in
the
Hydrodynamic
s
Laboratory
of
the
Ca
lifornia
In
st
itut
e of
T
echno
logy.
For
the
present,
t.his
assumption
is considered a
reasonable
one.
It
may
be
remarked
also
that
as
the
numb
er
of
bubbl
es increases
with
decreasing
K,
the
pressure
field
should
go
over
into
that
characteristic
of
the
cavity-flow field;
but
, in
the
transition,
the
pre
ss
ure
di
st
ribution
over
the
body s
hould
s
how
sma
ll-
sca
le
spatial
variations
betw
een
the
limi
ts
of
the
pre
s-
s
ur
e field of
noncavitating
fiow
and
that
of
the
fully developed
cavity
How.
ExPERIMENTAL
0B
SJ>
RVAT!ON
S
OF
CAVITATION
BuBBLE
S
In
the
present
paper
an
equation
of
motion
will
be
developed
for a
cavitation
bubbl
e in a fiow regime of
the
second
type
.
Thi
s
eq
uation
of
motion
will
be
applied
to
an
ana
lysis of
exper
im
ental
observations
made
in
the
high-
speed
water
tunnel.
Si nce a
di
s-
cussion
of
these
expe
rimen
ts
has
been
given
recently
by
Knapp
and
Holland
er
(3),
only
general
feature
s will
be
mentioned
hen•.
Th
e
cavitation
experiments
were
made
with
a 1.5-calihcr ogivc
for
which
the
noncavitating
pre
ss
ure
distribution
had
been
measured,
Fig.
2.
Run
s
were
made
with
tunnel
velocities
l',
1.0
~lp_-Qt,~e
'
\
_,J~oE:.'-
1---
0
1'\..
/
/~
0 .6
0 .
<0
o.•
/
~
.
'>
1
0 0 .2
~~
0
ft-
f+<~
~
AXIAL,
'ttt~;tft~~~t-
.
e......._r-....
L2
'('
2
10
::f:<?;f:-:H"~~-+
+
2 .4 2.e
::~2
0.
'-..
-v-o.2
-o.•
-0.6
-o.e
-•.o
'
'
~
v:"
',
e~3u~cu"'"..,.
.....
.....
t--.
-
--
-
-
PRESSURE
OISTP.IBUTION
ON
1.5
CALIBER
OGIVE
1. 0
o.e
0
...
d)
w
O.
<CI
3
z
0
.2
: ·
...1
0
u:
fi
0.2
a.
...1
0 .4
~
0
0 .
6£
o.e
1
.o
FIG.
2
ExPERIMENTALLY
D
E'l'ER
MINED
PnEssunE
CoEFFJCII·:l':T.
Cv
=
(p
-
po)/2(p
Vo
2
),
Is
SHOWN
AS
A
FuNcTION
0 1' 1
\XJAL
DJ
~
TANCE
ALONG
Moo"L
(The
mode
l profile is s
hown
in
the
dotted
cn
r
vL~s
with
the
associatf•d
~wa
i
f'
for
the
pr
ofile on t.hP
ri~ht.)
from 40 fps to 70 fps,
and
the
static
pressure
p
0
,
was
reduced
until
a few
cavitation
bubbles
appeared.
Photographs
of
these
bub-
bles were
taken
on
a
moving
film
at
a
rat
e of 15,000
per
sec to
20,000
per
sec; a
reproduction
of
an
example
of
these
photo
-
graphs
is
shown
in
Fig.
3.
EQUATION
OF
MOTION
FOR
A
CA
VITA
'!'
!ON
BUBBLB
Frequent
reference
has
been
made
in
the
literature
on
cavita-
tion
to
Rayleigh's
so
lution
for
the
problem
of
the
coll
apse
of a
spherical
cav
it
y
in
a
liquid
(4).
Rayleigh
consid
ered
the
si
tua-
tion
in
which
the
pr
ess
ure
at
a
distance
from
the
bubble
was
constant.
With
this
assumpt
i
on,
the
variation
of
the
bubbl
e
radius
with
time
may
be
simply
and
elegantly
deduced
from
the
Fra.
3
Tars
SERIES
OF
FRAMES
SHows
THE
BuBBLE
DBNOTED
AS
BuBBLE
1
PLESSET-TI-IE
DYNAMICS
OF
CAVITATION
BUBBLES
279
energy
integral
of
the
motion.
In
the
present
problem,
the
bubble
moves
through
a region in which
the
pressure varies
quite
rapidly
so
that
an
extension of
Rayleigh's
theory
is re-
quired.
This
extension
may
be readily carried
out
as follows:
Consider a
spherical
bubble
in
a perfect, incompressible liquid of
infinite
extent,
and
Jet
the
origin of co-ordinates
be
at
the
bubble
center
which is
at
rest.
The
radiu
s of
the
bubble
at
any
time
tis
R,
and
r is
the
radius
to
any
point
in
the
liquid.
Then,
as
is
well known (5),
the
velocity
potential
for
motion
of
the
liquid
with
spherical
symmetry
is
.......
[3]
and
the
Bernoulli
integral
of
the
motion
is
_
9:J!
+!
('V
q,)•
+
p(r)
=
P(t)
..
at
2 P P
......
[4]
where
k
dR/dt,
p(r)
is
the
static
pressure
at
r,
and
P(t)
is
the
static
pressure
at
a
distance
from
the
bubble. Also, from
Equa-
tion
[3]
('V¢)
2
=
R'
R
2
/r
4
...
.
...
....
.....
[5]
0</>
1 .
..
- = - (2 R R
2
+ R
2
R)
....
•
..........
[6]
ot r
Equation
[4]
will
be
applied
at
T = R so
that
the
equation
of
motion
for
the
bubble
radius
is
determined
(5). One notes
that
(oq,
jot)r=n = 2 k• + R R
so
that
Equation
[4]
becomes
p(R)
-
P(t)
3 .
..
2 U
2
+ R R . . . .
....
. .
...
[7]
p
Equation
[7]
is
the
general
equation
of
motion
for a spherical
bubble
in
a
liquid
with
given
external
pressure
P(t),
and
with
the
pressure
at
the
bubble
boundary
p(R)
. One
gets
Rayleigh's
solution
as
a special case
with
P(t)
-
p(R)
= Po
(a
constant)
and
with
the
aid
of
the
relation
3 . . 1 d .
- R
2
+ R R = - .- -
(R
3
R
2
)
2
2R
R
2
dt
Equation
[7]
is
adapted
to
the
present
problem
with
the
as-
sumption
that
p(R) =
p.
- 2u/ R
........
.
.........
[8]
where
Pv
is
the
vapor
pressure
of
the
water
at
the
appropriate
temperature
and
u is
the
surface-tension
constant
for
water.
It
is
thus
supposed
that
one
has
to
deal
with
the
growth
and
collapse of a
"vapor"
bubble.
The
problem
is defined
when
P(t)
is known.
It
will be assumed,
as
just
discussed,
that
P(t)
is
determined
from
the
noncavitating
pressure
distribution
over
the
body.
The
analysis of
the
experimental
data,
and
the
comparison
with
the
theory,
are
carried
out
in
the
following
manner:
The
experimental
data
given include
bubble
photographs,
Fig. 3,
which
determine
the
following:
1
The
position of
the
bubble
relative
to
the
body
profile
as
a
function
of
time.
2
The
radius
R of
the
bubble
as
a function of time.
Further,
the
tunnel
temperature
(and
hence
p.)
are
given
as
well
as
Po
and
Vo;
these
data
are
usually combined
in
the
specifi-
cation of
the
cavitation
parameter
K
and
the
tunnel
tempera-
ture.
From
this
information,
and
the
knowledge of
the
pressure
distribution
over
the
body,
Fig. 2,
the
absolute
pressure
at
the
model surface is
determined.
This
absolute
pressure
as
a
function
of position
on
the
model
is now
transformed
into
the
func-
tion
P(t)
from
the
correlation of
the
bubble
position
on
the
model
with
time. vVhen
P(t)
has
been
determined,
the
integration
of
the
equation
of
motion
(Equations
[7]
and
[8])
may
be carried
out
to
get
the
radius
of
the
bubbleR
as
a
function
of time.
The
equation
of
motion
cannot
be
integrated
analytically,
and
its
integration
was
performed
numerically.
The
solution is
deter-
m:
ncd
when two
constants
are
specified,
and
these
were
taken
to
be
the
observed
value
of
the
maximum
radius
Rm
where
R = 0.
Thus
the
theoretical
solution
has
been
fitted
to
the
experimental
curve
only
at
the
peak
of
the
radius-time
curve.
The
theoretical
curve
was
then
determined
by
integrating
forward
(the
collapse
portion)
and
backward
(the
growth
portion) from
this
one
point.
A comparison of
the
calculations
with
the
measured
values is
shown in Figs. 4, 5,
6,
and
7.
The
agreement
is considered
satis-
factory,
particularly
since
it
must
be emphasized
that
precise
experimental
data
are
difficult
to
obtain.
The
theoretical
radius-time
curve
is
quite
sensitive
to
the
P(t)
function; for
the
experiments
thus
far analyzed,
it
is believed
by
the
experimental
workers
that
the
cavitation
parameter
K
has
not
been
deter-
mined
with
quite
the
necessary accuracy.
Further,
there
are
some difficulties
in
the
determination
of
the
bubble
outlines
with
premswn.
That
this
is
the
case is
not
surprising
since
one
is
requiring considerable
photographic
detail
throughout
a
proc-
ess which lasts for a
time
of
the
order
of a millisecond.
It
must
also be
pointed
out
that
there
are
approximations
involved
in
applying
the
theoretical
equation
to
the
experimental
situation.
These
approximations
will now
be
considered.
THEORETICAL
APPROXIMATIONS
The
PressnTe Field.
It
has
been
supposed
that
the
pressure
field,
P(t),
acting
on
the
bubble
is
determined
from
the
pres-
sure
distribution
over
the
model.
It
is clear
that,
in
the
initial
stages
of
cavitation
of
present
interest,
the
bubbles
will
form
as
close
to
the
model surface
as
possible since
the
pressures
take
their
lowest values
there.
However,
it
also
has
been
assumed
that
the
bubble
is
acted
on
by
a spherically
symmetric
field. Since
the
bub-
ble is of finite
extent
and
since
the
pressure field
has
definite
pressure
gradients
both
along
the
model
and
normal
to
it,
it
is
clear
that
a simplification
has
been
introduced.
These
pres-
sure
gradients
would
be
a source of
asymmetry
in
bubble
shape,
and
there
is some evidence of
this
asymmetry.
It
is believed
that
the
approximation
made
is
not
such
as
to
obscure
the
es-
sential
details
of
the
growth
and
collapse;
space
gradients
in
the
pressure field
are
here
regarded
as
a second-order effect.
It
also
has
been
assumed
that
the
bubble
is
in
a
liquid
of
in-
finite
extent,
and
it
is
evident
that
the
bubble
grows
and
col-
lapses in
the
neighborhood of
the
model surface.
This
asym-
metry
in
the
fluid field
has
an
effect which
may
be
pointed
out
as
follows: As
compared
with
the
experimental
situation,
the
theory
would
exaggerate
the
importance
of
the
liquid
inertia
(this
inertia
leads
to
the
term
in
R'
in
Equation
[7]).
Comparison
of
the
theoretical
curve
with
the
experimental
points
would
seem
.
to
indicate
some
overestimate
of
this
inertia
term
where
R
is
small, i.e.,
near
the
beginning of
the
growth
and
toward
the
end
of
the
collapse.
The
presence of
the
model surface
has
an
additional
effect
on
the
flow field in
its
neighborhood which arises from
the
boundary
layer.
The
thickness of
this
boundary
layer
may
be
estimated
from
the
Blasius formula,
and
for
the
present
flow
conditions
280
JOURNAL OF
APPLIED
MECHANICS
SEPTEMBER,
1949
>6
I
r
BU~BLE
ll
J
-
~
E!KPeRIMENTAL
"''INT$
-.!
...
~
I
.14f--~:
~~
PT/SEC.
Lv:oe
1\
r-
-
1:2
v
I
I-
;J
.
-
.I
\ .
I
0
\.
"'
I-
lj
-
cvt:lve..
....
8
·f
<r~-
-
\
-
~...)/
.,...
.
-
OG
,p
. I
a"l
\
-
I
. -
.
I
.0
4
4
2
0
I
-
-
.
02
.
I
I
2
'--
/
0
I I I I I I I I
'l
·l'
NUM
1
BEP.
;F
FR~~E5l
3r
T
-
--
_..,..,.
\FRArE
""
5x
\0
-~
SEC
-
0
FIG. 4
12
-
B!.!~BLEJ3
J
...
TS
c:.'-'JJE
...,_
C G
I!
l(Pt;;P.IM~NTA,I,_
I
-v-
4
0FT/SEC
K•
.3
3
,'-
.
~
-
~
0
G
8
<?-"'-
0
4
.
-:,.o
v o
i\
-
;:~
,-{>.
c,.U~=-
'{
c-
/
.:><r;
..-"
4
.V'
.,.,
\
"·"'
.
-
Q<r
.
/
2
'
.
I
/
5
-
/
0
I I I I
!--;""
I I I I I
I
--t--r-
lj'
<>p
2
...
r
'T
3i'
NUMBER
0~
FRAMES
\FRAME•6G7
x
10
-~SEC
0
- I
FIG
. 6
l
eads
to
a
thickness
of
the
order
of 6 X
I0
-
3
in. On
Lhe
basis of
th
is
estimate,
the
effect of
the
boundary
layer
will be neglected.
It
should
be
noted
that
the
present
measurements
extend
to
minimum
bubble
sizes
larger
than
this
boundary-layer
thickness
although
some
reduction
in
the
effective
value
of R
should
be
ex-
pected
for
very
small
R.
An
exper
imental
source
of
apparent
asymmetry
in
bubble
shape
might
be
s
upp
osed
to
arise from
an
overestimate
of
the
bubble
dimension
in
the
direction
of
its
motion
which
would
be
produced
by
its
motion
during
the
time
of
li
ght
exposure
(1.5 X
IO
-•
sec).
However,
this
blurring
would give
an
apparent
ex-
tension of
the
imag
e
by
approximately
I0
-
3
in. so
that
this
error
is
not
particularly
significant.
Temperature and Pressure Conditions
in
Bubble.
It
has
been
ass
um
ed
in
the
theoreLical
calculation
s
that
the
vapor
pre
ssure,
p.,
in
the
bubble,
and
hence
the
bubble
temperatur
e,
remain
constant
. Clearly,
heat
must
be
applied
to
the
bubble
to
evapo-
rate
water
and
maintain
the
vapor
pressure
during
growth,
and
heat
of
condensation
must
be
removed
during
collapse.
The
.
temperature
changes
required
may
be
estimated
readily.
Con-
sider
a
bubble
with
maximum
radius
Rm
which
has
a
growth
time
r.
The
total
mass
of
vapor
which
is
evaporated
into
the
bubble
is (4.,./3)Rm
3
p',
where
p'
is
the
vapor
density.
The
total
heat
r
eq
uired is
4
f-
BU
~BLE
J 2
_l
#r.-
-
QE)(P&.P.IM~T""'l..
~
Z-V~~~;~/9EC
~
~
\
10
~
0
Ill
::>
G
~
0
4
a
0 2
0
K•:>O
I
i/
.
-
lfj
-
,_~
.l·~
-
c..uP-"'e.
I-
i
Q-.¢.,..-
\
-
.i/
,ff
I-
·7
Q~
~
\
-
I
.
r-
. I
I
\
-
I
I
I
I
-
I I
.{
I I I I I l
_l
J//
'[
lp
2,0
2,4
2i
T i
NUMBER
OF
FRAMES
-
I
FRAME
•
5.:
10·~
SEC
Fra.
5
.
l<tr-
-
....,---r--....,--,..----,--,..--
-,--,..--"""114
I-
B!.!BBLEI
•
;;e,
t~~
•
exPEAIMeNTAI._
u;r;'"'
I
....
,
L2 - v
=~~:;_
/Ss.c.
.
-t-f..rr_:G~-If--
-"k\
--\--+--l'2
K-
.3o
0
V"
_
r-
t-..'
•
-
lo
otr"f.'t-9-+---+--+--T-
'f+-
---t--
ilo
lli
r-
:z-"'/
_\ -
I
I
2
I 0
e
(
coii
Q.
2
0
2.
3
·oe
o,y~~~--~----;~~~v~--~~--~
--
-;e
s
I-
Q"'<;
-
i:
.cx;
J
--+---
~~+
,
-t--
-tft./
f G
~
~.o<~
l
-
-----
\-~·
~
·~l/~
--
t---~aqt
1
----t----t----+~\
__
,_
__
_,
- 4
-
~
- Q /
\•
.02
\-
--
+--~l-----t
--
,f-t-
---t----+----+
~
--+---
-i2
- , I -
o~uy~I~~~~~/LLLl~luy~l~~~~~~~~~O
/12
II
~0
"'J4
218
2
/
lNUMBER
oF
FRAME.S
- / I
FRAME.-
5
I(
10-~
Sli.C
.
L---~~~--~----L---~---L--~----L---J-2
Fra.
7
where L is
Lhe
l
atent
heat
of
evaporat
ion.
Thu
s for a
bubble
which grows
to
a
maximum
radius
Rm
= 0.10 in. in 20 frames
( r = w-
3
sec),
the
mass
of
vapor
is 1.17 X w-• grams,
and
Q =
6.8 X w-• calories.
This
heat
is
taken
out
of a
water
layer
sur-
rounding
the
bubble.
If
the
thermal
diffusivity of
water
is D
(D
= 1.43 X w-•
sq
em
per
sec),
then
the
order
of
magnitude
of
the
thickness
d of
the
water
l
ayer
from
which
this
heat
is con-
ducted
is
and
forT
=
w-a
sec, d = 1.2 X
w-
a em.
The
volume
of
the
water
l
ayer
from
which
this
h
eat
comes is of
the
order
of
magni-
tude
4.,.Rm
2
d,
and,
in
the
present
examp
le,
the
corresponding
mas
s of
water
is 1.0 X
w-ag.
Finally,
the
temperature
drop
of
this
water
l
ayer
is
(
4-n-
/3)
Rm
3
p'
L
4.,.Rm
2
dpc
Rm
p'L
3d
pC
where
cis
the
specific
heat
of
water.
In
the
present
exam
ple,
6.T
(growth) = 0.7 deg C 1.3
deg
F.
A
typioal
value
of collapse
time
is r = 10
fram
es = 0.5 X
w-a
sec,
and
the
corresponding
PLESSET-THE
DYNAMICS
OF
CAVITATION
BUBBLES
281
temperature
change,
estimated
in
this
same
way,
is
t;.T (collapse)
""
1
deg
C = 1.8
deg
F
It
is
apparent
that
these
temperature
changes
are
insignifi-
cant
so
that
one
may
take
the
bubble
boundary
to
have
a con-
stant
temperature,
essentially
the
same
as
the
water
tempera-
ture,
and
a
constant
value
of p •.
This
conclusion
cannot
be
accepted
unconditionally,
how-
ever, since
evaporation,
or
condensation,
is a process
which
takes
place
at
a finite
rate
and,
if
this
rate
is
not
sufficiently
high
to
keep
up
with
the
rate
of volume
change
of
the
bubble,
the
vapor
in
the
bubble
will
behave
more
like a
permanent
than
a
condensable
gas.
This
effect definitely
limits
the
range
of
valid-
ity
of
the
particular
assumption,
p.
=
const,
toward
the
end
of
the
collapse
phase
where
the
radial
velocity
il
increases
rapidly.
This
trend
toward
rapid
increase in
the
calculated
radial
velocity
is
illustrated
in
Fig. 8.
The
rate
of
evaporation,
60
~UBBILE•I
I
f-
I\
"
... 2
----
~0
\
"
...
<>------
"
·~e-·--··-
\
-
:30
\.
r-
"
~
r-
'
'
~
I
--
--
I'~
-
~
...
0
I I
I I
I \ I
~DIU~t'R.
(;~C.14~0~
)
';';4
"
-I
L?
v
1-
/
/
/
20
/
I~
v
/
/
30
/}
/
1- /
.oiO
I
1-
I
j
l7
~0
f
f-
60
I'
FIG.
8
CALCULATED
RADIAL
VELOCITIES
R,
AnE
SHOWN
AS
A
FuNc-
TION
oF
BuBBLE
RADIUS
R
or
condensation,
can
be
estimated
from
elementary
kinetic
theory
which
says
that
the
mass
of
gas
evaporated
(or
con-
densed)
per
unit
area
per
unit
time
at
an
absolute
temperature
Tis
i =
Pv~2~T·
....
.
.............
[9)
where
p.
again
denotes
the
vapor
pressure
for a
vapor
with
molar
mass
M,
and
B is
the
gas
constant.
If
one
assumes
that
the
vapor
obeys
the
perfect,
gas
law
P
= i_
BT
• M
which
is
reasonably
accurate
in
the
temperature
range
of
inter-
est
(6),
Equation
[9)
may
be
written
j
~
p'~
2
~~
=
p'V
................
[10]
where V = V
BT
/27rM is
the
desired
ve
l
ocity
to
be
associated
with
the
rate
of
the
evaporation
or
condensation
process.
For
the
present
problem,
at
22.2 C = 72
F,
V is
approximately
150
mps
""
500 fps.
Hence
unless
i~
is
appreciably
less
than
this
value, one
may
not
assume
the
constant
value
for
p..
During
the
collapse, when R
approaches
or
exceeds
this
value,
the
col-
lapse
velocity would
tend
to
be decreased because
the
vapor
will begin to show a rising
pressure
as
it
behaves
like a
permanent
gas.
A
further
effect of
interest
is
the
shock
loss
which
will
appear
in
the
vapor
when R
reaches
the
gas acoustic velocity.
The
ef-
fects of compressibility
both
in
the
vapor
and
in
the
liquid
will
not
be
considered here,
although
the
problems posed
by
them
are
of
great
interest.
A so
lution
of
these
problems will
be
decisive
for
the
quantitative
determination
of
the
high pressures arising
toward
the
end
of
the
bubble
collapse,
the
regrowth
or
subse-
quent
oscillations
of
the
bubble,
and
the
sound
energy
radi-
ated.
Air
Content
in
Bubble and Role
of
Nuclei
in
Formation
of
Bubbles.
The
assumption
has
been
made
that
any
air
con-
tained
in
the
bubble
does
not
affect
the
dynamics
of
the
bubble
growth
and
collapse
over
the
range
of
bubble
sizes which
have
been
measured
and
analyzed
here.
This
assumption
might
be
considered
questionable
since
the
water-tunnel
flow
experiments
are
made
with
water
containing
an
appreciable
concentration
of
dissolved air.
Furthermore
in
the
region of
flow
in which
the
bubble
behavior
is
studied,
the
liquid pressure is considerably
below
the
liquid
static
pressure
Po
at
a
distance
from
the
model.
Hence
one
should
expect
that
the
water
is
supersaturated
with
dissolved
air
and
that
diffusion of
air
into
the
bubble
would
take
place.
An
analytic
solution for
such
a diffusion
problem
has
been
carried
out
by
P. S.
Epstein
and
the
author,
the
details
of which
will
be
presented
elsewhere.
For
the
present
discussion
it
is
necessary
only
to
say
that
the
diffusion process is so slow
that
it
does
not
contribute
appreciably
to
any
alteration
in
the
air
content
of
the
bubble.
As will be
pointed
out
later,
the
initial
air
content
of a
bubble
is
very
small
so
that
over
the
range
of
bubble
sizes
which
are
ob-
served
and
to
which
the
present
calculations
have
been
applied,
the
effect of
the
air
may
be
neglected.
It
must
be
emphasized,
however,
that
the
small
mass
of
air
in
the
bubble
plays
a
most
important
role in
the
initial
stages
of
bubble
growth,
and
also
may
enter
in
the
final
stages
of
the
bubble
collapse.
The
initial
stages
of
bubble
growth
in
which
the
air
content
would
be
of
significance, refer
to
bubble
dimensions
which
are
beyond
the
present
range
of
experimental
observation.
Similarly,
the
final
stages
of
bubble
collapse
in
which
the
compressibility of air,
water
vapor,
and
liquid
are
of
importance,
refer
to
bubble
dimensions
which lie
within
the
last
frame
photographed.
3
A few
remarks,
nevertheless,
may
be
made
concerning
the
initial
formation
of
the
bubble.
It
is
the
present
view
that
the
formation
of a
bubble
in
c~vitating
flow,
or
in boiling,
begins
from
a nucleus
within
the
liquid
containing
air,
or
vapor,
or
both.
Such
gas-phase
nuclei
are
ordinarily
sub1nicroscopic in size,
and
become
ev
id
ent
only
upon
growth
of
the
nuclei
through
pressure
reduction
in
the
liquid
(reduction
in
the
function
denoted
pre-
viously
by
P [t]),
or
through
elevation
of
temperature
(increase
in
the
function
denoted
by
p [R]).
The
absence
of
such
nuclei
means
that
the
very
large
forces of
surface
tension
must
be
overcome
to
initiate
cavitation
or
boiling.
It
is well
known
that
degassed
pure
liquids
can
withstand
very
large
tensions,
'Knapp
and Hollander
(3)
assumed
that,
over the present range
of observation,
the
bubble contains essentially only water vapor.
The
present discussion supports this view.