Bubble Dynamics Near a Cylindrical Body: 3-D

BUBBLE DYNAMICS NEARA CYLINDRICAL BODY:
3-D BOUNDARY ELEMENT SIMULATION
OF BENCHMARKPROBLEMS1
G. L. Chahine and S. Prabhukumar
DvNeruow, INC.
7210Pindell SchoolRoad
Fulton, Maryland20759.
E-mail : [email protected]
http ://www. dynaflow-inc. com
Feb.1999 Revised:Feb.2000
Received:
Abstract
In many practical applications involving
underwater explosions near a solid strucfure,
competition between the forces due to gravity and
those due to the presence of a nearby structure leads
to highly distorted bubble shapes. In order to
accurately simulate such problems, a threedimensional model is required. In this paper, we
present results of a validation study of DyNerlow's
3-D Boundary Element Code, 3DvN.lFS@,which has
been successful in predicting and reproducing a host
of Navy underwater explosion problems. We present
comparisons of these simulations with carefully
conducted and well documented experiments
conducted by the Naval Surface Warfare center that
have been chosen as benchmark problems for the
ONR Modeling and Simulation Program. These
include the Snay and Goertner explosion bubble tests
and some spark-generatedbubble tests. The results
indicate the high accuracy of 3DvNlFS@ even under
these highly three-dimensional bubble dynamics
conditions. This is achieved with significantly smaller
CPU time and memory requirements than with
general-purposehydrocodes and other non-Boundary
Element methods.
l.Introduction
An accurate three-dimensional numerical
prediction of the dynamics of the interaction between
an underwater explosion bubble and a nearby
many
situations.
valuable in
strucfure is
Consequently, an increasing number of specialized
simulation tools are being developed by various
researchersto addressthe problem, and some existing
general hydrocodes are being modified and adapted
for this purpose. Most of the published results of
these codes appear to be qualitatively realistic, i.e.
deformed bubble shapes,the formation of re-entering
jets, etc. However, there is a definite need for
benchmark problems for validation so that the
accuracy of the various codes can be quantified.
Often in practical applications, the combined
effects of the forces due to gravlty and the presence
of a nearby structure lead to highly distorted bubble
shapes that cannot be predicted with a twoand
dimensional code. Engineering intuition
experience cannot determine, even qualitatively, the
results of such complex combinations of forces. It is
therefore essential to confront any developed code
with known exact solutions in simple geometries, and
with well-conducted and documented experimental
geometrical
more
complex
observations in
configurations. A first test of validity is the simple
case of a spherical bubble's growth and collapse,
which has a theoretical exact solution. This
supposedly simple test is generally good enough to
weed out inaccurate codes. Another set of validation
tests, prescribed by the ONR Modeling and
Simulation Program, refers to carefully conducted
tests by the Naval Surface Warfare Center, including
the so-called 3D Snay and Goertner underwater
explosionsnear cylindrical targets[ 1,17].
DvNerr,ow has developed a 3D Boundary
Element Code, 3DvNa,FSt (fot 3D dvr,la,micsof Free
Surfaces), which has been successful in predicting
and reproducing several practical underwater
'Distribution authorized to U.S. Government agenciesand their contractors; administrative/operational use
(February 2000). Other requestsshall be referred to NSWCIHDIV Code 420, Indian Head, MD 20640-5035
1
L
explosion experiments conducted by the Navy. In
this paper, we validate 3DvNaFSo against the
experimental tests of Snay and Goertner and against
controlled experiments we performed at DYNAFLow
using spark generatedbubbles. The results indicate
that the code perfonns very well even under highty
three-dimensionalconditions. The code is basedon a
Boundary Element Method (BEM) that requires
discretization of the boundaries only, as opposed to
based codes that require
other non-BEM
discretization of the entire 3D fluid domain.
Therefore, the BEM requires orders of magnitude
less computational resources (both CPU time and
memory) when compared to most other non-BEM
basedcodes. As a result, 3DvNnFS@appearsto be
very competitive and promising for such problems
In addition,
compared to non-BEM codes.
3DynnFS@ is being fully coupled to the wellaccepted structures code DYNA3D, developed by
Lawrence Livermore National Laboratory [8].'
2. Numerical Model
Due to the complexity of the problem at
hand only specialized numerical methods presently
offer hope for efficient and accurate solution to the
problem. One numerical method that has proven to
be very efficient in solving the type of free boundary
problem associated with bubble dynamics is the
Boundary Element Method. In addition to our work,
Guerri et al [2], Blake et al ll,2f, and Wilkerson
[9] used this method in the solution of axisymmetric
problems of bubble growth and collapse near
boundaries. Chahine et al [4-10] extended this
method to irrotational three-dimensional bubble
dynamics problems, and more recently to more
generalthree-dimensionalflows [3,13-14].The great
advantage of numerical methods is that once a
method has been validated, it can with guidancefrom
analytical, experimental, and order of magnitude or
phenomenologicalstudiesenableone to minimize the
number of physical phenomena or parameters to
considerfor testing.
2.1. Bubble Flow Equations
Shortly after the detonation of an explosive
and
following propagationof a shock wave
charge,
away from the explosion center, a bubble of highpressure gas is formed with large subsonic bubble
wall velocities. Due to the large -velocities and
Reynolds numbers involved (R">10'), it has been
analytically
and
observed
demonstrated
experimentally that viscous effects can be ignored for
the bubble wall motion studies. This, added to the
fact thatfor the problems studied below, the liquid is
initially at rest, allows us to assume that the fluid is
inviscid and the flow irrotational. The relatively slow
motion of the bubble wall in comparison with the
sound speed in the liquid, just a short time after
detonation also justifies the approximation of liquid
incompressibility. In fact, for a explosion of energy
Es in a liquid of density p, the Mach number of the
flow, M, can be written at time t ll6],
,
r
-ll5
l r3s.
M - L l ? u ^ "J
5cl8np
(r)
Because of the dependenceto t-''t , ,"ry
shortly after the explosion M drops significantly
below 1. This enables one to model the fluid
dynamics of the phenomenon assuming the liquid to
be inviscid and incompressible. These assumptions
result in a potential flow field (velocity potential, @ )
satisffing the Laplace equation,
v ' @- 0 .
(2)
The potential O must in addition satisfy initial
conditions and boundary conditions at infinity, at the
bubble walls, and at the boundaries of any nearby
bodies.
At all moving or fixed surfaces(such as a
bubble surface or a nearby structure) an identity
between fluid velocities normal to the boundary and
the normal velocities of the boundary itself is to be
satisfied:
VO .n = \/, .r,
(3)
where n is the local unit vector normal to the bubble
surface and V. is the local velocity vector of the
moving surface.
The bubble is assumed to contain noncondensable gas as well as vapor from the
surroundingliquid. The pressurewithin the bubble is
consideredto be the sum of the partial pressuresof
the non-condensablegases,Pr, and that of the liquid
vapor, P,. Vaporization of the liquid is assumed to
occur at a fast enough rate so that the vapor pressure
may be considered to remain constant throughout the
simulation and equal to the equilibrium vapor
pressure at the liquid ambient temperature. In
contrast, since time scales associated with gas
diffusion are much larger, the amount of noncondensable gas inside the bubbles is assumed to
remain constant. The gas is assumed to satisfy the
polytropic relation, P'7 k = constant, where ? is the
bubble volume and k the polytropic constant, with
lrl for isothermal behavior and k = crlcu, the gas
specific heat ratio, for adiabatic conditions. Other
models of gas diffrrsion and vaporization at the
interfacecan be easily incorporatedinto the code.
J
The pressure in the liquid at the bubble
surface,P7,is obtained at any time from the following
pressurebalanceequation:
/ a, \k
P,=1.r,,1t)-0o,
(4\
nodes. Equation (5) then becomes a set of N
equationsof index i of the type:
,- o- '\j l - Y{ (, n m \ - o o e , .
)o'*l, lln u
* ' - - I ) (\ -6/ )
d, )- ?"\"r*i t
Jif
where A, and B,rare elementsof matriceswhich are
where Pgo and 1o are the initial gas pressure and
volume respectively, o is the surface tension, and 0
is the local curvature of the bubble. In the numerical
procedure Pgo and 1/o arc known quantities at t=0
deduced from the available observations of the
particular explosive behavior in free field at large
depths, i.e. for sphericalbubbles (see Chahine et al
16,71).
2.2. Boundary Integral Method for 3D Bubble
Dynamics
In order to enable the simulation of bubble
behavior in complex geometries and flow
configurations including the full non-linear boundary
conditions, we developed and implemented a threedimensional Boundary Element Method (BEM). The
BEM was chosen because of its computational
efficiency. By considering only the boundariesof the
fluid domain, it reducesthe dimension of the problem
by one (e.g., a 3D problem is reduced to a 2D
problem). This method is based on Green'sequation,
which provides O anywhere in the domain of the
fluid (field points P) if the velocity potential, @ , and
its normal derivatives are known on the fluid
boundaries(points M):
d I I
,r[ da r
lfl+ O - - h r - a n D ( P \ . \ /'
et lN{.IPl)
t L At lMPl
where Atr =O is the solid angleunder which t ,::]
the fluid:
e = 4, if P is a point in the fluid,
a = 2, if P is a point on a smooth surface,and
a < 4, if P is a point at a sharp corner of the
surface.
If the field point is selectedto be on the boundaries
of the fluid domain (bubble and nearby structures),
then a closed set of equations can be obtained and
used at each time step to solve for values of 0@l0n
(or <D)assuming that all values of O (or 6OlDn) are
known at the preceding step.
To solve Equation (5) numerically, it is
necessary to discretize the boundaries into panels,
perform the integration over each panel, and then
sum up the contributions of all panels. Triangular
panels were used in our study resulting in a total of N
the discrete equivalent of the integrals given in
Equation(5).
To evaluate the integrals in (5) over any
particular panel, a linear variation of the potential and
its normal derivative over the panel is assumed.In
this manner, both <Dand AAIAI are continuous over
the bubble surface, and are expressedas functions of
the values at the three nodes which delimit a
particular panel. In order to compute the curvature of
the bubble surface a three-dimensionallocal bubble
surface frt, f(x,y,z)=O, is first computed. The unit
normal at a node can then be expressedas:
r:*
vf
rrfr'
(7)
with the appropriate sign chosen to insure that the
normal is always directed towards the fluid. The local
curvature is then computed using
C - V.n.
(8)
To obtain the total fluid velocity at any point
on the surface of the bubble, the tangential velocity,
21,must be computed at each node in addition to the
normal velocity. This is also done using a locaL
surfacefit to the velocity potenti al, Q, -- h(*, y, z) .
Taking the gradient of this function at the considered
node, and eliminating any normal component of
velocity appearing in this gradient gives a good
approximation for the tangential velocity
V,:ax(Vo,xn).
(e)
The basic procedure can then be
summarizedas follows. With the problem initialized
and the velocity potential known over the surface of
the bubble, an updated value of O<D/dncan be
obtained by performing the integration in (5) and
solving the corresponding matrix equation (6). The
time derivative of O while moving with the boundary
(D@/D|, needed to update the value of O, is then
obtained using the Bernoulli equation, which can be
written after accounting for the pressure balance at
the interface:
-ps,.
- \uo
--r .n
o - p
p+
- v - . tp-^(
o \ q%\*
t)
r v , * *tyr, f '
t'
2t,t
Dt
(10)
4
3.1. Validation in Simple Geometries
This ensuresthat the potential is advancedin
a Lagrangian fashion. New coordinate positions of
the nodesare then obtainedusing the displacement:
=(#n+v, ",)at:,
dwr
(11)
where n and e, are the unit normal and tangential
vectors. This time stepping procedure is repeated
throughout the bubble growth and collapse, resulting
in a shape history of the bubble. Time stepping is
done using a simple Euler stepping schemewith time
step, A/ chosen in an adaptive fashion that ensures
that smaller time steps are used when rapid changes
in the potential occur, while larger ones are used for
less rapid changes:
(r2)
Lt-"*,
where a is a user specified parameter. V*or is the
maximum of all velocities obtained at time t, and l^i,
is the minimum panel length in the discretization.
This adaptive scheme enables accurate, though
efficient description of the full bubble dynamics
history.
In more recent versions of 3DyNnFS@ a
higher order scheme for time stepping is
implemented. In this scheme a predictor-corrector
method is implemented, and more importantly, a
direct application of the boundary element method to
the potential time derivative, AO I 0t , enableseven
more accuratecomputations.
The various authors quoted earlier have
validated the use of the Boundary Element Method to
study axisymmetric bubble dynamics. This has
included both comparisons with a semi-analytical
solution for spherical bubbles -- the Rayleigh-Plesset
and experimental validation for the
Equation
relatively simple casesof spherical bubble pulsations
and axisymmetric bubble collapse near flat solid
walls. We have conducted similar comparisons using
our axisymmetric code 2DvNlFS@, and our 3D code
3DyiraFS@. Figures la and lb show, for example,
the comparative results between 3DvnaFS@,
2DvN,q.FS@
and the Rayleigh-Plesset semi-analytical
solution.
As illustrated in the figure, comparison with
the Rayleigh-Plesset solution reveals that numerical
effors fot a very coarse discretization of an 18-nodes
(32 panels) bubble is about I percent of the achieved
maximum radius, and is about 2 percent for the
bubble period. The error on the maximum radius
drops to 0.1 percent for a 3D discretized bubble of
198 nodes (392 panels),and is less than 0.05 percent
on the period. Similar precise results are obtained
with a 2D 64-panels discretization. Comparisons
were also made with studies of axisymmetric bubble
collapse available in the literature 11,2,12,191,and
have shown, for the coarse discretization, differences
with these studies on the bubble period of the order of
1 percent 16,77. Our reference [7] includes detailed
grid and time-step convergenceanalysesfor
3. Numerical Results and Discussion
I
/'
3
4
cl rt
\
Gr'!
----
SDynrF$ 6* glnrl+
08p.rFS
190 nDd€e
?DynoFS tE piE€lr
,19*=Sri0-r
SDyaafS 3? Ftnats, [E nodqr
T
I
dg.o=5rlo-3
3? panetc. l8 aadcs
c,a roJn
4g
dg*.sixl["
ll]gnaFS 392 panzlr. t96 nodsr
dP*=?rl$-r
3DlrrrFS s$a prnctr,
S0yn*FS !S pa*cl*
-*
**-
2DyorFli 6{ prntls
il,n
Baylefb-Flesset
--""
----
n
$
vl
E'-o
-5
?*ytctgh*fltcaset
"""
5.Tr
Ozl!
UJJ2
ttme tsec)
L-
u,00
Lirne (serl
solution,the a:risymmetricBEM code 2DvrqaFSand
Figure 1: ComparisonbetweenRayleigh-Plesset
the 3D BEM code3Drttq,FS. a) Over bubbleperiod, b) End of collapse.
tk
*ylinder
h f f i p @P #
f*r11
n'lse*
f * 72
msss
T * "S.*.t
rrrss#
T x $S.S
m$sc
Cylinder
\l
:' ]
::i ijli
*
A '
.
ro'
!ib'
*. ,.t .'".f-"+*[
f * 4S.Sm$ss
I T I R S TM A X I M U M }
t
r i
ti,, t {
*
|i'.
i'!,ja
t
$
:;;'i i|:
f-?3
r
'
::
:f
'i
.t
: : : Li
*-s
1
..
'':
':"
{"}:o
.q*t
ti:r'it
f ,* pt"s
T * $ ?
msss
ilTtsss
*
l
,:-
$Tt*sfr
Figure 2: Comparisonof bubble shapesfrom the Snay/Goertnerexperimentwith those computedusing 3DvNlFS@.
Experiment was conductedin a small vacuum tank with a pressureof 256 feet of water above the free surface. A
0.2 gm lead azide chargewas exploded at a depth of 2 feet below the free surface,at a standoff distanceof 5.54
inches from the surfaceof a 5.33 inch diameterrigid cylinder that is barely visible on the left side of the photos
(Taken from Reference[7]).
' i' c,nt,rr
the axisymmetric, 2DvNnFS@, and the threedimensional3DvNaFS@,versions of our code.
Word.Picture.6
3.2. Validation for 3D geometry cases
0.6
Figure 2 comparesthe results obtained from
3DvN.q.FS@with those obtained from a small-scale
underwater explosion test conducted at NSWC in a
cylindrical metallic vacuum tank by Snay et al ll7l.
Under reduced ambient pressure,a 0.2 g lead azide
charge was exploded at a depth of 2 feet below the
free surface at a stand-off distance of 5.54 inches
away from the surface of a rigid cylinder. This standoff distance was chosen to be approximately equal to
the expected maximum radius of the generated
bubble. The ambient pressure above the free surface
of the tank was 2.56 feet of water and the cylinder
had a diameterof 5.33 inches.The figure showsthat
- 0.5
o
t'26
0,5
!.?5
l.zt
Lt
t.?t
z
,,,1,,
Figure 3: Motion of top andbottompointson the bubbleaxis
for an explosionin an infinite mediumandwithin the 4-foot
using3DvxaFS@.
diameterNSWCvacuumtank.Simulations
6
under the combined effects of gravity and the
presenceof the structure, a highly distorted bubble
shape is produced with a re-entering jet directed
mostly upward. In this case, a portion of the bubble
tended to adhere to the nearby cylinder, while the
remainder of the bubble behaved as if gravity were
the primary influencing factor. The figure also
illustrates the capability of 3DvNlFS@ to reproduce
the highly distorted bubble dynamics. One
discrepancy between the numerical and experimental
results is the period of bubble growth and collapse.
The measured bubble period is about 9 percent
longer than the computed bubble period. The
discrepancy arises from the fact that while the
simulation was done in an infinite medium
surroundingthe bubble and body, the experimentwas
conducted in a 4-foot diameter cylindrical vacuum
tank. Additional numerical studies have shown that
this confinement has a significant lengthening effect
on the bubble period (Chahine et al [6]). This is
illustrated in Figure 3, which compares axisymmetric
bubble behavior (motion of top and bottom points on
the bubble axis) in a gravity field in the cylindrical
vacuum tank and in an infinite medium. These
computations were done using the 3D code,
3DyNlFS@, even though the configuration was
axisymmetric.
Time = 77.40 ms
Time = 142.43 ms
-5
- 1.5
Feet
Figure 4. Comparisonof bubble shapesfrom the NSWC Hydrotank experimentwith thosecomputedusing
3DvN.q.FS@
Fruqolc.
i@got
Crdr.
0.0
tl@!:
anar
0.5
t.o
Dnl.Gdrda
delodbr
7t -,
t.5
Amax
o.o
0.5
t.o
Feet
t.5
?_o
-t-sh
fL'
(b) Simulation Front View
(a) Experiment Front View
Tlr3: rnllracondr
altar dalNllo.
130
a9.o
r0r.0
r0e.0
ilg0
r 2 50
rlr.0
r33.0
r39.0
tat.0
t rJ.0
t.!.0
lat.0
J
-o.
0.5
t.0
t.5
?.0
AmBx
- r.!i=
(a) Experiment Side View
-o.5
o.5
rt.
(b) Simulation Side View
Figure5: Comparison
of Uoi:.:ld sideviewsof bubblecontoursfrom thePETNcharge
expenmentwith thosefrom 3DyN,lFS@.
Figure 4 shows an explosion near a
cylindrical body using a I .l gm PETN charge
exploded at a depth of 3.94 feet below the free
surface.This test was conductedin the hydroballistic
tank at NSWC and was therefore much less prone to
container boundary effects. Detailed analysis of the
test results is presentedby Goertner et al. llll. The
distance of the bubble from the surface of the
cylinder was once again chosen to be approximately
equal to the bubble radius at its maximum. The figure
shows very good comparisons between the
experiment and the simulation at the two times
shown. Figure 5 compares the front and side view
outlines of the bubble at selected times from the
experiment with those from the 3DvNlFS@
simulation. Both figures show a good agreementfor
the whole collapsephaseof the bubble dynamics.
the 3DvNnFS@
The sensitivity of
calculations to scatter inherent to the empirically
derived explosion bubble parameters is illustrated in
Figure 6. The code is run using the following input.
In order to simulate an explosion of weight W,
detonated at depth D and at an atmospheric pressure
of Po*6 feet of water (33 feet for an explosion in the
ocean), two free field spherical bubble parametersare
8
the values of R,nu*are not too significant, and the
difference in the computed results are within
experimental errors.
pre-computed. The maximum and minimum bubble
radii R,,,o,and Ro,i,,are obtained from Navy derived
empirical relationshipsas follows:
I
( w ) t
R , n u * = /,l- r , - , I '
- t
\/
(r2)
3.3 Spark-Cell Tests
amb )
1
Rn,'in :
a,W3 .
Figure 7 compares the bubble shapesat four
different times for a spark cell simulation [7,8] of an
explosion that takes place above a cylindrical body
sitting on the bottom of a tank with the results from
2DvrrtnFSt. Since the bottom is close to the bubble,
the 3DvnAFS@simulation also included the effect of
the bottom. The experiment was performed in a
DyNerr-ow spark cell. The spark was triggercd at 4
inches below the free surface, the cell ambient
pressurewas 0.41 psi, and the standoff distancewas
0.79 times the maximum radius that would have been
attained by a bubble in an infinite medium with no
gravity. The figure shows a very good comparison
between the experiment and the 3DvNnFS@
simulation. In this case,the competing forces (gravity
and attraction to the cylinder) are collinear and
The force due to
directly oppose each other.
buoyancy acts upward and those due to the body and
bottom act downward. As a result, the bubble
becomes elongated before finally detaching itself
from the cylinder. The detachment occurs because
the force due to buoyancy was, in this case, greater
than the attraction force actine downward.
(l3)
where J and cL are characteristics of the particular
explosive. It is obvious that theseparameters,/ and
C[, are known within some experimentalerror, and it
is reasonable to question the influence of this
accuracy on the numerical (and experimental) results.
In Figures 6a and 6b, we explore, for illustration, the
In the particular case of the
influence of J.
experiments of Figure 4, the value used for -/ was
14.5 which leads to R^o, = 1.05 feet. Figure 6a
illustrates for the time t=146.6 ms, the influence of
choosing-/ as the nominal value, as well as 0.9 J and
l.l J. Also shown are the experimentallyobserved
shapes at two different times. Note that the
experiment does not allow observation of the
reentering jet; rather, the picture shows the outside
outline of the bubble. Figure 6b compares the
predicted movement of the top-most and bottomsimulations
most bubble points for the 3DvN,q,FS@
with he three different / values with the experiments
at two different times. It is obvious from this
comparison that the deviations due to small errors in
f.Ig
PETN @3.94ft, Pa=2.O5ft
(top
& bottom
nodes)
-J.b
+
-3.8
-J=14.52
'J=14.7
- J= 1 4 . 3
N -4.0
N
. Experiment
-4.2
-0.8 -0.6 -o.4 -0.2 0.0 0.2 0.4 0.6 0.8
v (rt)
1.0 1.2
-5.0 L
o
25
50
75
Time
100
(ms)
125
150
175
Figure 6: Sensitivity of the 3DynaFS simulation resultsto scafferin explosion bubble parameters:
(a) Bubble contoursfrom the experimentat two different times comparedwith those
from the simulation for three different values of-I@quation 12)
(b) Correspondingmotion of the top and bottom points on the bubble.
9
4. Conclusions
We have described in this paper a 3-D
Boundary Element Method that is capable of
capturing the details of the behavior of an explosion
bubble in complex geometries. Examples of code
validation tests were presented which included test
cases selected as benchmark tests by the Office of
Naval Research.Several other aspectsof the code of
great interest to the community have not been
discussed here but are the subject of on-going
investigations or development at DvNAFLow. These
include accounting for fluid structureinteraction,and
continuation of the bubble dynamics computations
for multiple cycles of bubble oscillations. The first
issue has been the subject of previous publications
[0,13,14]), where a full coupling between our
axisymmetric version 2DvNaFS@ and Nike2D, and
between 3DvnaFS@ and the Lawrence Livernore
In 3DvNaFS@.
structure code DYNA3D.
continuation of the computations beyond the
reenteringjet impact on the other side of the bubble
is a problem for the boundary element methods that
for the axisymmetric
addressed
hasbeensuccessfully
case (Chahine et al 19,201). We are presently
the sameconceptsin 3DvNaFSo.
implementing
5. Acknowledgments
This work was supportedby the Mechanics
and Energy Conversion,Scienceand Technology
We would
Divisionof the Office of Naval Research.
like to particularly acknowledge very useful
discussionsand suggestionsby Gregory Harris,
Naval SurfaceWarfareCenter,Indian HeadDivision,
Code420,andsignificanthelpfrom the DvNnFLow,
INC.staff,andmostparticularlyGaryFrederick.
REFERENCES
1.
Tirne = 57.8 rns
Time = 54.9 ms
Time = 51.9 ms
Time = l5.O ms
J.R. Blake and D. C. Gibson. Cavitation bubbles
near boundaries, Annual Review of Fluid
Mechanics,Vol. 19, (1987), pp. 99-123
@
-0.16
-0.06
0.06
0.16
0.?6
-0.c r-o.?5
@
@
-o.c' "
-0.26
0.t5
-0.0q
-0.96
0.16
-0.c l-r:
-0.a6
-0.orJ)
-0.26
-0.t6
-0.(b
0,1t6
0.15
0.86
acvlindricar
bodv
sitting
Figure
7:comparison
orbubbleshane:ff#;iir1?TJJ::ilil,,1i*il;il,JiTitabove
10
2. J.R. Blake, B.B. Taib, and G. Doherty, Transient
cavities near boundaries.Part I. Rigid Boundary,
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