BUBBLE DYNAMICS NEARA CYLINDRICAL BODY:3-D BOUNDARY ELEMENT SIMULATION

OF BENCHMARKPROBLEMS1

G. L. Chahine and S. PrabhukumarDvNeruow, INC.

7210 Pindell School RoadFulton, Maryland20759.

E-mail : glchahin e@dynafl ow-inc. comhttp : //www. dynafl ow-inc. c om

Received: Feb. 1999 Revised: Feb.2000

Abstract

In many practical applications involvingunderwater explosions near a solid strucfure,competition between the forces due to gravity andthose due to the presence of a nearby structure leadsto highly distorted bubble shapes. In order toaccurately simulate such problems, a three-dimensional model is required. In this paper, wepresent results of a validation study of DyNerlow's3-D Boundary Element Code, 3DvN.lFS@, which hasbeen successful in predicting and reproducing a hostof Navy underwater explosion problems. We presentcomparisons of these simulations with carefullyconducted and well documented experimentsconducted by the Naval Surface Warfare center thathave been chosen as benchmark problems for theONR Modeling and Simulation Program. Theseinclude the Snay and Goertner explosion bubble testsand some spark-generated bubble tests. The resultsindicate the high accuracy of 3DvNlFS@ even underthese highly three-dimensional bubble dynamicsconditions. This is achieved with significantly smallerCPU time and memory requirements than withgeneral-purpose hydrocodes and other non-BoundaryElement methods.

l.Introduction

An accurate three-dimensional numericalprediction of the dynamics of the interaction betweenan underwater explosion bubble and a nearbystrucfure is valuable in many situations.Consequently, an increasing number of specializedsimulation tools are being developed by variousresearchers to address the problem, and some existinggeneral hydrocodes are being modified and adaptedfor this purpose. Most of the published results ofthese codes appear to be qualitatively realistic, i.e.deformed bubble shapes, the formation of re-enteringjets, etc. However, there is a definite need forbenchmark problems for validation so that theaccuracy of the various codes can be quantified.

Often in practical applications, the combinedeffects of the forces due to gravlty and the presenceof a nearby structure lead to highly distorted bubbleshapes that cannot be predicted with a two-dimensional code. Engineering intuition andexperience cannot determine, even qualitatively, theresults of such complex combinations of forces. It istherefore essential to confront any developed codewith known exact solutions in simple geometries, andwith well-conducted and documented experimentalobservations in more complex geometricalconfigurations. A first test of validity is the simplecase of a spherical bubble's growth and collapse,which has a theoretical exact solution. Thissupposedly simple test is generally good enough toweed out inaccurate codes. Another set of validationtests, prescribed by the ONR Modeling andSimulation Program, refers to carefully conductedtests by the Naval Surface Warfare Center, includingthe so-called 3D Snay and Goertner underwaterexplosions near cylindrical targets [ 1,17].

DvNerr,ow has developed a 3D BoundaryElement Code, 3DvNa,FSt (fot 3D dvr,la,mics of FreeSurfaces), which has been successful in predictingand reproducing several practical underwater

'Distribution authorized to U.S. Government agencies and their contractors; administrative/operational use(February 2000). Other requests shall be referred to NSWCIHDIV Code 420, Indian Head, MD 20640-5035

explosion experiments conducted by the Navy. Inthis paper, we validate 3DvNaFSo against theexperimental tests of Snay and Goertner and againstcontrolled experiments we performed at DYNAFLowusing spark generated bubbles. The results indicatethat the code perfonns very well even under hightythree-dimensional conditions. The code is based on aBoundary Element Method (BEM) that requiresdiscretization of the boundaries only, as opposed toother non-BEM based codes that requirediscretization of the entire 3D fluid domain.Therefore, the BEM requires orders of magnitudeless computational resources (both CPU time andmemory) when compared to most other non-BEMbased codes. As a result, 3DvNnFS@ appears to bevery competitive and promising for such problemscompared to non-BEM codes. In addition,3DynnFS@ is being fully coupled to the well-accepted structures code DYNA3D, developed byLawrence Livermore National Laboratory [8].'

2. Numerical Model

Due to the complexity of the problem athand only specialized numerical methods presentlyoffer hope for efficient and accurate solution to theproblem. One numerical method that has proven tobe very efficient in solving the type of free boundaryproblem associated with bubble dynamics is theBoundary 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 axisymmetricproblems of bubble growth and collapse nearboundaries. Chahine et al [4-10] extended thismethod to irrotational three-dimensional bubbledynamics problems, and more recently to moregeneral three-dimensional flows [3,13-14]. The greatadvantage of numerical methods is that once amethod has been validated, it can with guidance fromanalytical, experimental, and order of magnitude orphenomenological studies enable one to minimize thenumber of physical phenomena or parameters toconsider for testing.

2.1. Bubble Flow Equations

Shortly after the detonation of an explosivecharge, and following propagation of a shock waveaway from the explosion center, a bubble of high-pressure gas is formed with large subsonic bubblewall velocities. Due to the large -velocities andReynolds numbers involved (R">10'), it has beendemonstrated analytically and observedexperimentally that viscous effects can be ignored forthe bubble wall motion studies. This, added to the

1L

fact thatfor the problems studied below, the liquid isinitially at rest, allows us to assume that the fluid isinviscid and the flow irrotational. The relatively slowmotion of the bubble wall in comparison with thesound speed in the liquid, just a short time afterdetonation also justifies the approximation of liquidincompressibility. In fact, for a explosion of energyEs in a liquid of density p, the Mach number of theflow, M, can be written at time t ll6],

, r - l l 5

M -L l ?u^ l r 3s . ( r )5cl8np "J

Because of the dependence to t-''t , ,"ryshortly after the explosion M drops significantlybelow 1. This enables one to model the fluiddynamics of the phenomenon assuming the liquid tobe inviscid and incompressible. These assumptionsresult in a potential flow field (velocity potential, @ )satisffing the Laplace equation,

v '@ - 0 . (2)The potential O must in addition satisfy initialconditions and boundary conditions at infinity, at thebubble walls, and at the boundaries of any nearbybodies.

At all moving or fixed surfaces (such as abubble surface or a nearby structure) an identitybetween fluid velocities normal to the boundary andthe normal velocities of the boundary itself is to besatisfied:

VO .n = \ / , . r , (3)where n is the local unit vector normal to the bubblesurface and V. is the local velocity vector of themoving surface.

The bubble is assumed to contain non-condensable gas as well as vapor from thesurrounding liquid. The pressure within the bubble isconsidered to be the sum of the partial pressures ofthe non-condensable gases, Pr, and that of the liquidvapor, P,. Vaporization of the liquid is assumed tooccur at a fast enough rate so that the vapor pressuremay be considered to remain constant throughout thesimulation and equal to the equilibrium vaporpressure at the liquid ambient temperature. Incontrast, since time scales associated with gasdiffusion are much larger, the amount of non-condensable gas inside the bubbles is assumed toremain constant. The gas is assumed to satisfy thepolytropic relation, P'7 k = constant, where ? is the

bubble volume and k the polytropic constant, withlrl for isothermal behavior and k = crlcu, the gasspecific heat ratio, for adiabatic conditions. Othermodels of gas diffrrsion and vaporization at theinterface can be easily incorporated into the code.

The pressure in the liquid at the bubblesurface, P7, is obtained at any time from the followingpressure balance equation:

/ a, \kP,=1.r,,1t) -0o, (4\

where Pgo and 1o are the initial gas pressure and

volume respectively, o is the surface tension, and 0is the local curvature of the bubble. In the numericalprocedure Pgo and 1/o arc known quantities at t=0

deduced from the available observations of theparticular explosive behavior in free field at largedepths, i.e. for spherical bubbles (see Chahine et al

16,71).

2.2. Boundary Integral Method for 3D BubbleDynamics

In order to enable the simulation of bubblebehavior in complex geometries and flowconfigurations including the full non-linear boundaryconditions, we developed and implemented a three-dimensional Boundary Element Method (BEM). TheBEM was chosen because of its computationalefficiency. By considering only the boundaries of thefluid domain, it reduces the dimension of the problemby one (e.g., a 3D problem is reduced to a 2Dproblem). This method is based on Green's equation,

which provides O anywhere in the domain of the

fluid (field points P) if the velocity potential, @ , andits normal derivatives are known on the fluidboundaries (points M):

, r [ d a r d I Il f l - + O - - h r - a n D ( P \ .t L At lMPl et lN{.IPl)

\ / '

where Atr =O is the solid angle under 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 thesurface.

If the field point is selected to be on the boundariesof the fluid domain (bubble and nearby structures),then a closed set of equations can be obtained andused at each time step to solve for values of 0@l0n(or <D) assuming that all values of O (or 6OlDn) areknown at the preceding step.

To solve Equation (5) numerically, it isnecessary to discretize the boundaries into panels,perform the integration over each panel, and thensum up the contributions of all panels. Triangularpanels were used in our study resulting in a total of N

J

nodes. Equation (5) then becomes a set of Nequations of index i of the type:

J i f , o \ { ,) ' l l

- - ' j l - Y ( n m \ - o o e , . ( 6 )o* , lnu d , ) - ? " \ " r * i t * ' - - I ) \ - /

where A, and B,rare elements of matrices which are

the discrete equivalent of the integrals given inEquation (5).

To evaluate the integrals in (5) over anyparticular panel, a linear variation of the potential andits normal derivative over the panel is assumed. Inthis manner, both <D and AAIAI are continuous overthe bubble surface, and are expressed as functions ofthe values at the three nodes which delimit aparticular panel. In order to compute the curvature ofthe bubble surface a three-dimensional local bubblesurface frt, f(x,y,z)=O, is first computed. The unitnormal at a node can then be expressed as:

vfr : * r r f r '

(7)

with the appropriate sign chosen to insure that thenormal is always directed towards the fluid. The localcurvature is then computed using

C - V . n . ( 8 )

To obtain the total fluid velocity at any pointon the surface of the bubble, the tangential velocity,21, must be computed at each node in addition to thenormal velocity. This is also done using a locaL

surface fit to the velocity potenti al, Q, -- h(*, y, z) .

Taking the gradient of this function at the considerednode, and eliminating any normal component ofvelocity appearing in this gradient gives a goodapproximation for the tangential velocity

V , : a x ( V o , x n ) . (e)The basic procedure can then be

summarized as follows. With the problem initializedand the velocity potential known over the surface ofthe bubble, an updated value of O<D/dn can beobtained by performing the integration in (5) andsolving the corresponding matrix equation (6). Thetime derivative of O while moving with the boundary(D@/D|, needed to update the value of O, is thenobtained using the Bernoulli equation, which can bewritten after accounting for the pressure balance atthe interface:

p+ - n - p - p-^( %\* - uo * *tyr, -ps,.t '

D t - r . o - v . t o \ q t ) \ r v ,

2 t , t f '

(10)

( 1 1 )

where n and e, are the unit normal and tangential

vectors. This time stepping procedure is repeatedthroughout the bubble growth and collapse, resultingin a shape history of the bubble. Time stepping isdone using a simple Euler stepping scheme with time

step, A/ chosen in an adaptive fashion that ensuresthat smaller time steps are used when rapid changesin the potential occur, while larger ones are used forless rapid changes:

dwr =(#n+v, ",)at:,

L t - " * ,

This ensures that the potential is advanced ina Lagrangian fashion. New coordinate positions ofthe nodes are then obtained using the displacement:

4

3.1. Validation in Simple Geometries

The various authors quoted earlier havevalidated the use of the Boundary Element Method tostudy axisymmetric bubble dynamics. This hasincluded both comparisons with a semi-analyticalsolution for spherical bubbles -- the Rayleigh-PlessetEquation and experimental validation for therelatively simple cases of spherical bubble pulsationsand axisymmetric bubble collapse near flat solidwalls. We have conducted similar comparisons usingour axisymmetric code 2DvNlFS@, and our 3D code3DyiraFS@. Figures la and lb show, for example,the comparative results between 3DvnaFS@,2DvN,q.FS@ and the Rayleigh-Plesset semi-analyticalsolution.

As illustrated in the figure, comparison withthe Rayleigh-Plesset solution reveals that numericaleffors fot a very coarse discretization of an 18-nodes(32 panels) bubble is about I percent of the achievedmaximum radius, and is about 2 percent for thebubble period. The error on the maximum radiusdrops to 0.1 percent for a 3D discretized bubble of198 nodes (392 panels), and is less than 0.05 percenton the period. Similar precise results are obtainedwith a 2D 64-panels discretization. Comparisonswere also made with studies of axisymmetric bubblecollapse available in the literature 11,2,12,191, andhave shown, for the coarse discretization, differenceswith these studies on the bubble period of the order of1 percent 16,77. Our reference [7] includes detailedgrid and time-step convergence analyses for

I?*ytctgh*fltcaset

/' """ 2DyorFli 6{ prntls dg*.sixl["

3 - - - - l l ]gnaFS 392 panzl r . t96 nodsr

4 -*

?DynoFS tE piE€lr ,19*=Sri0-r

-5 **-

SDyaafS 3? Ftnats, [E nodqr

c,a r-oJn 5.Tr Ozl! UJJ2

ttme tsec)

(r2)

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, thoughefficient description of the full bubble dynamicshistory.

In more recent versions of 3DyNnFS@ ahigher order scheme for time stepping isimplemented. In this scheme a predictor-correctormethod is implemented, and more importantly, adirect application of the boundary element method to

the potential time derivative, AO I 0t , enables evenmore accurate computations.

3. Numerical Results and Discussion

\

Baylefb-Flesset--""

SDynrF$ 6* glnrl+ dP*=?rl$-r---- 3DlrrrFS s$a prnctr, 190 nDd€e

S0yn*FS !S pa*cl* dg.o=5rlo-3

08p.rFS 3? panetc. l8 aadcs

TI

G r ' !

vl

E ' - o

cl rt

n$

i l ,n

Lirne (ser l

Figure 1: Comparison between Rayleigh-Plesset solution, the a:risymmetric BEM code 2DvrqaFS andthe 3D BEM code 3Drttq,FS. a) Over bubble period, b) End of collapse.

4g L-u,00

*ylindertk

Cylinder

f * r 1 1n'lse*

T x $S.Sm$sc

T * "S.*.trrrss#

\lt r l

, :- : f

| i ' . ' . ti ' ! , ja ' i : : L$ * -s : : : i

: ' ]* ::i ijli

A ' r iro' i,. ! i b ' t , t { t* . , . t . ' " . f - " + * [ * : ; ; ' i i | :

f - ? 3ilTtsss

T * $ ?msss

*

1' : " { " } : o

' ' :. . t i : r ' i t . q * t

f ,* pt"s$Tt*sfr

Figure 2: Comparison of bubble shapes from the Snay/Goertner experiment with those computed using 3DvNlFS@.Experiment was conducted in a small vacuum tank with a pressure of 256 feet of water above the free surface. A

0.2 gm lead azide charge was exploded at a depth of 2 feet below the free surface, at a standoff distance of 5.54inches from the surface of a 5.33 inch diameter rigid cylinder that is barely visible on the left side of the photos

(Taken from Reference [7]).

' i ' c ,nt , r rWord.Picture.6

0.6

- 0 .5

o t '26 0,5 !.?5 ,, ,1,,

l .zt Lt t .?t z

Figure 3: Motion of top and bottom points on the bubble axisfor an explosion in an infinite medium and within the 4-foot

diameter NSWC vacuum tank. Simulations using 3DvxaFS@.

hf f ip@ P#f * 72msss

f * 4S.S m$ssITIRST MAXIMUM}

the axisymmetric, 2DvNnFS@, and the three-dimensional3DvNaFS@, versions of our code.

3.2. Validation for 3D geometry cases

Figure 2 compares the results obtained from3DvN.q.FS@ with those obtained from a small-scaleunderwater explosion test conducted at NSWC in acylindrical metallic vacuum tank by Snay et al ll7l.Under reduced ambient pressure, a 0.2 g lead azidecharge was exploded at a depth of 2 feet below thefree surface at a stand-off distance of 5.54 inchesaway from the surface of a rigid cylinder. This stand-off distance was chosen to be approximately equal tothe expected maximum radius of the generatedbubble. The ambient pressure above the free surfaceof the tank was 2.56 feet of water and the cylinderhad a diameter of 5.33 inches. The figure shows that

under the combined effects of gravity and thepresence of the structure, a highly distorted bubbleshape is produced with a re-entering jet directedmostly upward. In this case, a portion of the bubbletended to adhere to the nearby cylinder, while theremainder of the bubble behaved as if gravity werethe primary influencing factor. The figure alsoillustrates the capability of 3DvNlFS@ to reproducethe highly distorted bubble dynamics. Onediscrepancy between the numerical and experimentalresults is the period of bubble growth and collapse.The measured bubble period is about 9 percentlonger than the computed bubble period. Thediscrepancy arises from the fact that while the

T i m e = 7 7 . 4 0 m s

6

simulation was done in an infinite mediumsurrounding the bubble and body, the experiment wasconducted in a 4-foot diameter cylindrical vacuumtank. Additional numerical studies have shown thatthis confinement has a significant lengthening effecton the bubble period (Chahine et al [6]). This isillustrated in Figure 3, which compares axisymmetricbubble behavior (motion of top and bottom points onthe bubble axis) in a gravity field in the cylindricalvacuum tank and in an infinite medium. Thesecomputations were done using the 3D code,3DyNlFS@, even though the configuration wasaxisymmetric.

T i m e = 1 4 2 . 4 3 m s

-5- 1 . 5

Figure 4. Comparison of bubble shapes from the NSWC Hydrotank experiment with those computed using3DvN.q.FS@

Feet

Fruqolc. Crdr.i@got

tl@!: Dnl.Gdrdaanar delodbr

7 t - ,

0 . 0 0 . 5 t . o t . 5

Amax

o . o 0 . 5 t . o t . 5 ? _ oF e e t

(a) Experiment Front View

Tlr3: rnllracondraltar dalNllo.

1 3 0a9.o

r0r .0r0e .0i lg0r25 0r l r . 0r33.0r39.0tat .0t rJ.0t.!.0lat.0

- t - s h

J

-o .

f L '

(b) Simulation Front View

0 . 5 t . 0 t . 5 ? . 0

AmB x

- r.!i= -o.5

(a) Experiment Side Viewrt.

(b) Simulation Side View

Figure 5: Comparison of Uoi:.:ld side views of bubble contours from the PETN chargeexpenment with those from 3DyN,lFS@.

o.5

Figure 4 shows an explosion near acylindrical body using a I .l gm PETN chargeexploded at a depth of 3.94 feet below the freesurface. This test was conducted in the hydroballistictank at NSWC and was therefore much less prone tocontainer boundary effects. Detailed analysis of thetest results is presented by Goertner et al. llll. Thedistance of the bubble from the surface of thecylinder was once again chosen to be approximatelyequal to the bubble radius at its maximum. The figureshows very good comparisons between theexperiment and the simulation at the two timesshown. Figure 5 compares the front and side view

outlines of the bubble at selected times from theexperiment with those from the 3DvNlFS@simulation. Both figures show a good agreement forthe whole collapse phase of the bubble dynamics.

The sensitivity of the 3DvNnFS@calculations to scatter inherent to the empiricallyderived explosion bubble parameters is illustrated inFigure 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 pressureof Po*6 feet of water (33 feet for an explosion in theocean), two free field spherical bubble parameters are

pre-computed. The maximum and minimum bubbleradii R,,,o, and Ro,i,, are obtained from Navy derivedempirical relationships as follows:

I

( w ) tR , n u * = / l , - r , - , I '

\ / - t a m b )

1Rn,'in : a,W3 .

the values of R,nu* are not toodifference in the computedexperimental errors.

8

significant, and theresults are within

3.3 Spark-Cell Tests

Figure 7 compares the bubble shapes at fourdifferent times for a spark cell simulation [7,8] of anexplosion that takes place above a cylindrical bodysitting on the bottom of a tank with the results from2DvrrtnFSt. Since the bottom is close to the bubble,the 3DvnAFS@ simulation also included the effect ofthe bottom. The experiment was performed in aDyNerr-ow spark cell. The spark was triggercd at 4inches below the free surface, the cell ambientpressure was 0.41 psi, and the standoff distance was0.79 times the maximum radius that would have beenattained by a bubble in an infinite medium with nogravity. The figure shows a very good comparisonbetween the experiment and the 3DvNnFS@simulation. In this case, the competing forces (gravityand attraction to the cylinder) are collinear anddirectly oppose each other. The force due tobuoyancy acts upward and those due to the body andbottom act downward. As a result, the bubblebecomes elongated before finally detaching itselffrom the cylinder. The detachment occurs becausethe force due to buoyancy was, in this case, greaterthan the attraction force actine downward.

(r2)

( l 3 )

where J and cL are characteristics of the particular

explosive. It is obvious that these parameters, / and

C[, are known within some experimental error, and it

is reasonable to question the influence of thisaccuracy on the numerical (and experimental) results.In Figures 6a and 6b, we explore, for illustration, the

influence of J. In the particular case of the

experiments of Figure 4, the value used for -/ was14.5 which leads to R^o, = 1.05 feet. Figure 6aillustrates 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 experimentally observedshapes at two different times. Note that theexperiment does not allow observation of thereentering jet; rather, the picture shows the outsideoutline of the bubble. Figure 6b compares thepredicted movement of the top-most and bottom-most bubble points for the 3DvN,q,FS@ simulations

with he three different / values with the experimentsat two different times. It is obvious from thiscomparison that the deviations due to small errors in

-0.8 -0.6 -o.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2v (rt)

N

- J . b

+ - 3 . 8

N - 4 . 0

-4.2

f . Ig PETN @3.94f t , Pa=2.O5f t ( top & bot tom nodes)

-J=14.52' J = 1 4 . 7-J =14.3

. Experiment

- 5 . 0 Lo 2 5 5 0 7 5 1 0 0 1 2 5 1 5 0 1 7 5

Time (ms)

Figure 6: Sensitivity of the 3DynaFS simulation results to scaffer in explosion bubble parameters:(a) Bubble contours from the experiment at two different times compared with those

from the simulation for three different values of-I@quation 12)(b) Corresponding motion of the top and bottom points on the bubble.

4. Conclusions

We have described in this paper a 3-DBoundary Element Method that is capable of

capturing the details of the behavior of an explosionbubble in complex geometries. Examples of codevalidation tests were presented which included testcases selected as benchmark tests by the Office ofNaval Research. Several other aspects of the code ofgreat interest to the community have not beendiscussed here but are the subject of on-goinginvestigations or development at DvNAFLow. Theseinclude accounting for fluid structure interaction, andcontinuation of the bubble dynamics computationsfor multiple cycles of bubble oscillations. The firstissue has been the subject of previous publications

[0,13,14]), where a full coupling between ouraxisymmetric version 2DvNaFS@ and Nike2D, andbetween 3DvnaFS@ and the Lawrence Livernorestructure code DYNA3D. In 3DvNaFS@.continuation of the computations beyond thereentering jet impact on the other side of the bubbleis a problem for the boundary element methods that

9

has been successfully addressed for the axisymmetriccase (Chahine et al 19,201). We are presentlyimplementing the same concepts in 3DvNaFSo.

5. Acknowledgments

This work was supported by the Mechanicsand Energy Conversion, Science and TechnologyDivision of the Office of Naval Research. We wouldlike to particularly acknowledge very usefuldiscussions and suggestions by Gregory Harris,Naval Surface Warfare Center, Indian Head Division,Code 420, and significant help from the DvNnFLow,INC. staff, and most particularly Gary Frederick.

REFERENCES

1. J.R. Blake and D. C. Gibson. Cavitation bubblesnear boundaries, Annual Review of FluidMechanics, Vol. 19, (1987), pp. 99-123

Time = l5.O ms Time = 51 .9 ms Time = 54.9 ms

@Tirne = 57.8 rns

-0.16 -0.06 0.06 0.16 0.?6-0.c r-

-o.?5

0. t5-o.c' "

-0.26-0.c l-r:

-0.a6- 0 .0q

-0.96 0.16-0.orJ)

-0.26 -0.t6 -0.(b 0,1t6 0.15 0.86

Figure 7: comparison orbubbleshane:ff#;iir1?TJJ::ilil,,1i*il;il,JiTitabove a cvlindricar bodv sitting

@@

1 0

2.

3 .

4.

5 .

6.

7.

8 .

9.

J.R. Blake, B.B. Taib, and G. Doherty, Transient

cavities near boundaries. Part I. Rigid Boundary,

Journal of Fluid Mechanics, Vol. 170, (1986),

pp.479-497.O. Boulon and G.L. Chahine, Numerical

simulation of unsteady cavitation on a 3D

hydrofoil, 3rd International Symposium on

Cavitation, Grenoble, France, Proceedings Vol.

2, pp. 249-257, April I 998.C.L. Ctratrine, Bubble Interaction with Vortices,Chapter 18 pp. 783-819, in: Fluid Vortices, SJ.Green, editor, Kluwer Academic Publishers, TheNetherlands, 1995.G.L. Chahine, and T.O. Perdue, Simulation ofthe 3-D behavior of an unsteady large bubblenear a structure, 3rd Int. Colloquium on Dropsand Bubbles, California, 1988.G.L. Chahine, T.O. Perdue and C.B. Tucker,Interaction between an underwater explosionbubble and a solid submerged body, DvNRrLow,INc. Technical Report, 89001-1, 1988.G.L. Chahine and R. Duraswamio Boundaryelement method for calculating 2D and 3Dunderwater explosion bubble behavior in freewater and near structures, NSWCDDITR-93144(Limited Distributi on), 1994.G.L. Chahine, G.S. Frederick, C.J. Lambrecht,G.S. Harris, and H.U. Mair, Spark-generatedbubbles as laboratory scale models ofunderwater explosions and their use forvalidation of simulation tools, Proceedings of the66th Shock and Vibration Symposium, Biloxi,Mississippi, pp. 265-277, 1995.G. L. Chahine, R. Duraswami, K.M. Kalumuck,Boundary Element Method for calculating 2Dand 3D underwater explosion bubble loading onnearby structures including Fluid-StructureInteraction Effects, NSWC Weapons andTechnology Department Report NSWCDD/TR-93146. (Limited Distribution), I 996.G.L. Chahine, and K. Kalumuck, "BEM

Software for Free Surface Flow SimulationIncluding Fluid Structure Interaction Effects,"International Journal of Computer Applicationsfor Technology, (1998), No. 11, Vol. 31415,pp.l77-198.

I l. J. F. Goertner, G.S. Harris, J. M. Koenig, and R.Thrun, Interaction of an underwater explosion

bubble with a nearby cylinder: 1983 NSWChydrotank test series data report (U),

NSWCDD/TR-93/1 62 (Confidential), 199 5.12. L. Guerri, G. Lucca, and A. Prosperetti, A

Numerical method for the dynamics of non-

spherical cavitation bubbles, Proceedings of the2"d International Colloquium on Drops andBubbles, California, JPL Publication 82-7, I 98 1.

13. K.M. Kalumuck, G.L. Chahine, R. Duraisami,"Bubble Dynamics -Structure Interactionsimulation on coupling fluid BEM and structuralFEM codes'', Journal of Fluids and Structures,(1995), Vol . 9, pp. 861-883

14. K.M. Kalumuck, G.L. Chahine, R. Duraisami,Analysis of the response of a deformablestructure to underwater explosion bubble loadingusing a fully coupled fluid-structure interactionprocedure, Proceedings of the 66th Shock andVibration Symposium, Biloxi, Mississippi, pp.277-287,1995.

15. H.U. Mair, Preliminary compilation ofunderwater explosion benchmark, Shock andVibration Journal, Vol. 6, No. 4, (1999), pp.1 6 9 - 1 8 1 .

16. J. W. Pritchett, "An Evaluation of VariousTheoretical Models for Underwater ExplosionsNear the Water Surface," (Jniv. of California,Los Alamos Inf. Report La-5548-Ms,1974.

I7. G. Snay, J.F. Goertner and R.S. Price, Smallscale experiments to determine migration ofexplosion gas globes towards submarines', U S.Naval Ordnance Laboratory, White Oak, MD,NAVORD Report 2280, 1952.

18. R"G. Whirley, and J.O. Hallquist, Dyna3D anonlinear, explicit, three-dimensional finiteelement code for solid and structural mechanics -

User manual, UCRL-M A-107254, 1991.19. S.A. Wilkerson, Boundary integral technique for

bubble dynamics, Ph.D. Thesis, The JohnsHopkins University, Baltimore, MD., 1989.

20. S. Zhang, J. Duncan, and G.L. Chahine, TheFinal Stage of the Collapse of a CavitationBubble Near a Rigid Wall, (1993), Journal ofFluid Mechanics, Vol. 257,pp. 147-181.

10.