A numerical study on the underwater explosion bubble pulsationand the collapse process

Ching-Yu Hsu a, Cho-Chung Liang b, Anh-Tu Nguyen b,n, Tso-Liang Teng c

a Department of Marine Mechanical Engineering, ROC Naval Academy, Taiwan, ROCb Department of Mechanical and Automation Engineering, Da-Yeh University, Changhua, Taiwan, ROCc Hsiuping University of Science and Technology, Taichung, Taiwan, ROC

a r t i c l e i n f o

Article history:Received 29 March 2013Accepted 19 February 2014Available online 7 March 2014

Keywords:Underwater explosion bubbleWater jetToroidal

a b s t r a c t

The dynamic underwater explosion bubble process is a complex phenomenon with many facets toconsider. After detonation, the shockwave quickly strikes nearby structures. The bubble oscillation maythen substantially damage the structures, even if the bubble pulse is not large. This is because othereffects, such as whipping or water jet impact, often accompany the bubble pulse. This study applied theEulerian Technique using ABAQUS software to simulate the process of underwater bubble explosion,pulsation and collapse. This approach allows many materials to be used in an Eulerian element andmanages with large deformation of materials such as flows and gases. The simulated bubble is equivalentto a bubble generated by 55 g of TNT. Although boundary conditions simplify the model, the method isfeasible for simulating bubble dynamics and provides acceptably accurate results on bubble migration inwater, pressure pulse, water jet formation, and flow field velocity surrounding the bubble. Futureresearch can use this method to study bubbles, including their interaction with free surfaces and thesubmerged structures or floating structures.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Two crucial and distinct phenomena are usually involved in anunderwater explosion. The first phenomenon is the quick propa-gation of an underwater shockwave in the water. If the shockwaveencounters the free surface of water and structures, the reflectionand refraction of the shockwave may cause other phenomena,such as local cavitation and bulk cavitation. Such phenomena canreproduce loading pressure on the structures. The second phe-nomenon involves the motion of high temperature and pressuregas bubble, which occurs over a longer period (Cole, 1948).Because the initial pressure of gas bubble is much higher thanthe equilibrium hydrostatic pressure, the bubble expands rapidlyand its internal pressure decreases gradually, but the motionpersists because of the inertia of the outward flowing water. At alater time, the gas pressure falls below the hydrostatic pressure ofthe surrounding water. The bubble expansion continues until thevelocity of the fluid surrounding the bubble vanishes, the bubblebegins to contract at an increasing rate. The inward motion of thefluid remains until the gas is compressed, the bubble contraction isreversed abruptly. The bubble can experience repeated circles of

expansion and contraction. As the bubble approaches minimumvolume, the bubble produces secondary pressure pulses. Further-more, under the buoyancy effect, the bubble tends to migratetoward the water surface during its process. Fig. 1 depicts thissequence of events (Cole, 1948).

Many studies have examined underwater explosion bubblephenomena using experiments and simulations. These studieshave explored the non-compressive radial motion of bubbleswithout gravity, bubble motion with gravity, the effects of com-pressibility and non-spherical form on bubble motion, and theeffects of different boundaries, free surfaces, rigid structures anddeformable structures. According to Cole (1948), Ramsauer (1923)is cited in Cole (1948) conducted one of the earliest systematicbubble motion measurements. Ramsauer identified the upwardbubble motion and the relationship among maximal radius, depthand charge weight. Although his experimental results correspondto the simple theory, his method provides limited information onisolated bubble points. High-speed motion photographs are amore powerful tool for studying bubble migration. Edgerton firstapplied this method by using such photographs to determine thatbubbles rise slightly when they first expand and rapidly sinkbelow their initial position. Recently, numerous researchers haveconducted theoretical and experimental bubble studies.

Spark and laser techniques have recently become more popularfor generating bubbles because of their advantages, which includerelative safety, cleanliness, and compactness. Such techniques

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Ocean Engineering

http://dx.doi.org/10.1016/j.oceaneng.2014.02.0180029-8018 & 2014 Elsevier Ltd. All rights reserved.

n Corresponding author. Tel.: þ886 4 8511223; fax: þ886 4 8511223.E-mail addresses: tuna_3003@yahoo.com,

tuna3003.vn@gmail.com (A.-T. Nguyen).

Ocean Engineering 81 (2014) 29–38

enable high-quality bubble dynamic observation at a lower costthan do experiments involving explosives. Gibson and Blake(1980) and Lew et al. (2007) have used spark-generated bubbles.Bubble images were captured on high-speed cameras at a rate ofover 10,000 frames per second (FPS). Gibson and Blake (1980)focused on the behavior of bubbles adjacent to flexible and rigidboundaries, and Lew et al. (2007) studied the development ofwater jets produced by bubbles near plates with holes and withoutholes. Remarkable laser technology applications include researchby Brujan et al. (2002) and Lee et al. (2011). An AQ-switched Nd-YAG laser system was applied to generate the bubbles for variousstudy purposes. Brujan et al. (2002) investigated the temporalevolution of liquid jet shockwave emission in the bubble collapsephase near a rigid body and examined the splash effect using high-speed photography. They also used the boundary integral methodto determine velocity fields and pressure contours in the liquidsurrounding a bubble. Lee et al. (2011) successfully measuredshockwaves and bubble wall speeds by using an ICCD camerademonstrating that the first shockwave generates strong pressure,followed by a weak rebound of a secondary shockwave.

Although spark and laser techniques have benefits, experi-ments using real charges are crucial because they help prevent theeffect of unpredictable factors on research results. Klaseboer et al.(2005) conducted one of the few charge experiments relevant tothe current study. They experimentally and numerically examinedthe interaction between underwater bubbles and nearby rigid andresilient submerged structures in a test pond. Hexocire chargeweights ranged from 10 to 50 g and were placed at variousdistances from the structures. Bubble migration was observed ina free-field environment and near horizontal and vertical plates.

Many researchers have used the numerical method, which isessential technique for solving bubble problems. The boundaryelement method (BEM) is a numerical computational method forsolving linear partial differential equations formulated as integralequations. The BEM uses only surface elements that are materialinterfaces or assigned boundary conditions. Therefore, this methodis used to simulate bubble dynamics because it requires less timefor data preparation; is easier to change the applied mesh; isuseful for problems that require re-meshing; is highly accurate;requires less computer time and storage by using fewer nodes andelements; and allows users to filter out unwanted information andfocus on a particular internal region, thereby reducing computeruse time.

A bubble oscillation close to the free surface creates a freesurface spike. Wang et al. (1996), Zhang et al. (1998), and Pearson

et al. (2004) have examined this phenomenon. Wang et al. (1996)provided valuable information on the effects of strength para-meters and on the initial pressure at various buoyancy parametersof the bubble rebounding in connected form. Zhang et al. (1998)proposed a directed procedure for computing solid angles on thefree surface. This method was successfully incorporated into a 3Dmodel and applied to the interaction between two bubbles parallelto the free surface. Pearson et al. (2004) used a nonlinear nodedistribution on the free surface to calculate the necessary ellipticintegrals. Their refined method accurately calculated the motion ofone or more bubbles near an infinite surface.

During the bubble collapse phase, bubbles become toroidalduring water jet formation. Various techniques have been used toaddress simulation difficulties, such as converting a single-connected bubble to a double-connected toroidal bubble andencountering overcrowded elements near the water jet tip.Wang et al. (1996) placed a vortex ring at a geometric center ofthe bubble cross-section on impact (a vortex ring can be placedanywhere within the bubble). The vortex ring accounted for thedouble connectivity of the bubble. Zhang et al. (2001) extendedthis research and used a 3D vortex ring to represent the flowcirculation from jet impact and to facilitate flow calculation in thepost-impact toroidal phase. Zhang et al. (2001) used several noveltechniques, including a weighted-average scheme, adaptive meshrefinement, and a smoother scheme based on the least-squaresprinciple to simulate the essential physics of bubble jetting, flowcirculation surrounding the toroidal tube, and the rebounding ofthe toroidal bubble near a rigid wall with the effect of buoyancy.

To ensure that the mesh was stable and smooth during bubbleevolution, Wang et al. (2003) introduced a new regulation meshtechnique, the elastic mesh technique (EMT). The EMT allows asignificantly larger time increment to be used and effectivelyavoids the mesh refinement requirement within a reasonableprecision range. Wang et al. (2005) seeded a vortex ring insidethe bubble form to account for circulation and introduced a modelfor gas bubbles close to a rigid wall during the collapse phase.Investigating the high-pressure impulse acting on the wall duringjet impact indicated that the high-speed water jet may damage asurface.

By using advanced techniques, previous research, for exampleKlaseboer et al. (2005) and Zhang et al. (2008a, 2009), has nearlyclarified the interaction between bubbles and complex floating orsubmerged structures. Studies have also examined the effect offactors such as gravity and buoyancy on the bubble dynamicprocess. Zhang et al. (2008a) coupled a BEM solver, which is usedfor calculating bubbles, with the finite element method (FEM)solver, ABAQUS, which solves the structure part response. Theirresults showed an agreement with experimental results. They alsofound that the whipping effect of a ship when interacting with abubble occurs when the pulsating frequency of the bubblematches the low-order eigenfrequency of the ship.

The application of finite element program for solving UNDEXbubble problem has been addressed in the previous researches(Chisum and Shin, 1997; Abe et al., 2007; Barras et al., 2012), inwhich multiple materials in Eulerian solver was used to simulatethe water, the air, and detonation gas products. The behavior of gasbubbles in the vicinity of simple boundaries was simulated and theeffects of mesh on the accuracy of numerical calculation wereestimated. However, the early works only focused on the 2-Dcalculation and the interaction between UNDEX bubble andcomplex structures are still controversial.

This study used the Eulerian technique in ABAQUS to simulatethe first oscillation cycle of a 3-D UNDEX bubble in a free field. Theavailability of many materials in this technique allowed thesimultaneous simulation of three materials, explosive gas product,air, and water, in the same Eulerian domain. Using the Eulerian

Fig. 1. Pressure waves and bubble phenomena of underwater explosion.

C.-Y. Hsu et al. / Ocean Engineering 81 (2014) 29–3830

element helps prevent excessive distortion of elements duringanalysis. This study investigated the motion of bubbles under theeffect of gravity, as well as major bubble characteristics during thecollapse phase. The numerical results correspond to the experi-mental results. This study provides another approach for studyingunderwater explosion bubbles. The proposed method can be usedto study the migration of underwater explosion bubbles in otherboundary types.

2. Theoretical background

2.1. Spherical bubble dynamics theory

The dynamics of cavitation and bubbles in fluid is a relevantresearch topic because it has military, medical, and engineeringapplications. In consideration of cavitation and bubble dynamics ina large mass of incompressible liquid, gas is isothermally com-pressed. By ignoring the effects of surface tension and liquidviscosity, Rayleigh first derived the bubble boundary momentumequation in 1917 (Plesset and Prosperetti, 1977).

R €Rþ32ð _RÞ2 ¼ pðRÞ�p1

ρlð1Þ

where R is the bubble radius, ρl is the liquid density, p1 is thepressure in the liquid at infinity, and p(R) is the pressure in theliquid at the bubble boundary. The dots denote differentiationwithrespect to time.

Considering the viscosity effects and surface tension, Plessetgeneralized the Rayleigh equation for bubble dynamics and used itto derive the fundamental equation governing the dynamics of aspherical cavity of a bubble radius, R(t)

R €Rþ32ð _RÞ2 ¼ 1

ρpi�p1�2s

R�4μ

R_R��

ð2Þ

where s and m are the surface-tension constant and liquidviscosity coefficient, respectively, pi is pressure in the gas at thebubble wall, and p1 is the pressure at infinity, which may be afunction of time. Eq. (2) is valid if other effects are neglected, suchas interface stability, liquid compressibility, and the effect ofenergy flowing in or out of the bubble. This equation is calledthe Rayleigh–Plesset equation (Plesset and Prosperetti, 1977).

Because a shockwave occurs several milliseconds after detona-tion while the underwater explosion bubble oscillates for a second,the effect of a shockwave on bubble dynamics is negligible. Theheat exchange between the explosion gas product and the waterdomain is not considered because of the short bubble oscillationduration. Therefore, the Rayleigh–Plesset equation is usuallywritten in a simple and dimensionless form, making simulationmore convenient

R0 €R

0þ1:5ð _R

0Þ2 ¼ ε

R00

R0

� �3γ

�1 ð3Þ

where R0 is the dimensionless radius of the bubble, R00 is the initial

dimensionless radius of the bubble, γ is the special heat ratio, and εis the strength parameter (the ratio of initial bubble pressure toambient pressure) (Zhang et al., 2008b).

The gas pressure p inside the bubble consists of constant vaporpressure pc and non-condensable gas pressure that conforms tothe adiabatic law and the gas pressure may be expressed in term ofits volume V as

p¼ pcþpoVo

V

� �γ

ð4Þ

where Vo and po are the initial volume and pressure of the non-condensable gas, respectively.

For the concerning theoretical models of UNDEX bubble, thereader may refer to the original citations

Herring (1941)

a€a 1�2_acl

� �þ32_a2 1� 4_a

3cl

� �¼ ρ�1

l Pg�plþacl_Pg

� �ð5Þ

Keller and Kolodner (1956)

a€a 1� _acl

� �þ32_a2 1� _a

3cl

� �¼ ρ�1

l ðPg�plÞ 1þ _acl

� �þ acl_Pg

� �ð6Þ

Geers and Hunter (2002)

a€a 1þζ� 1�ρgρl

� �_acl

� �þ32_a2 1þ3

2ζ�1

3_acl

� �þ13

ρgρl

� �1þ _a

clþ acl

_ρgρg

!" #

þðζclþ _ζaÞ_a¼ ρ�1l ðPg�pIÞ 1þ _a

cl

� �þ acl_Pg

� �ð7Þ

where a¼ aðtÞ; _a¼ da=dt; €a¼ d2a=dt2 are the liquid/gas interfacelocation, velocity, and acceleration respectively.

The hydrostatic pressure in the water is pI ¼ patmþρlgdlwhere patm is the atmospheric pressure, g is the gravitationalacceleration, dI and is the depth of the explosion, ρl the density ofwater, ρg is the gas density, _ρ is the gas density rate, andζ¼ ρgcg=ρlcl.

The empirical equations commonly used to describe themagnitude and period of underwater explosion dynamics havebeen derived from underwater explosion experiments (Prior andBrown, 2010).

The maximal bubble radius:

R¼ 3:4W

Zþ10:1

� �1=3

ð8Þ

The time of the first pulse:

T ¼ KW1=3

ðZþ10:1Þ5=61� R

5Z

� �ð9Þ

where K is the constant depending on the explosive type, Z is theexplosive depth, and W is the charge weight.

2.2. Numerical background

2.2.1. Finite element methodABAQUS is a suite of powerful engineering simulation programs

based on the FEM that is used to solve problems, ranging fromrelatively simple linear analyses to the most challenging nonlinearsimulations. ABAQUS is applicable to a wide range of fields such asstructural stress or displacement, heat transfer, mass diffusion,thermal management of electrical components (coupled thermal–electrical analysis), acoustics, soil mechanics (coupled pore fluid-stress analysis), and piezoelectric analysis. ABAQUS 6.11 consists ofthree main analysis modules: ABAQUS/Standard, ABAQUS/Explicit,and ABAQUS/CFD. ABAQUS/Standard is a general-purpose analysismodule that can solve a range of linear and nonlinear problemsinvolving the static, dynamic, thermal, and electrical response ofcomponents. ABAQUS/Explicit is a special-purpose analysis mod-ule that uses an explicit dynamic finite element formulation. It issuitable for short, transient, dynamic events, such as impact andblast problems, and is also efficient at solving highly nonlinearproblems involving changing contact conditions, such as creatingsimulations. ABAQUS/CFD is a computational fluid dynamics analysisproduct.

The FEM is based on two classical formulations, Lagragian andEulerian formulations. The Lagrangian solver is particularly wellsuited to describing solid behavior because all nodes are fixedwithin the material, facilitating the tracking of the free surface and

C.-Y. Hsu et al. / Ocean Engineering 81 (2014) 29–38 31

applying boundary conditions. However, the main drawback of thismethod is that, for high-strain gradients, numerical mesh maybecome overly distorted, causing a short duration step and possibleaccuracy loss.

By contrast, the Eulerian solver is well suited to large deformationproblems or flow situations, such as fluid and gas behaviors, becausenodes remain fixed while material flows through the mesh. However,tracking free surfaces using the Eulerian technique is difficult becausetracking the material motion requires a sophisticated technique. Inaddition, using the Eulerian technique often increases memory andcalculation time. The Eulerian method in ABAQUS/Explicit is basedon the volume-of-fluid method and is implemented only for 3Delements. Material is tracked by computing the Eulerian volumefraction of each element when the material flows through the mesh.Eulerian elements may contain many materials simultaneously, andEulerian analysis automatically accounts for contact between differ-ent Eulerian materials.

2.2.2. Equation of state2.2.2.1. Us–UP equation of state. In simulations, simulating theentire process in the most refined and comprehensive manner isoptimal. The numerical code of ABAQUS/Explicit provides a linearUs–UP equation of state (EOS) model that can simulate incom-pressible viscous and inviscid laminar flow governed by theNavier–Stokes equation of motion. This is available in the Mie-Grüneisen EOS (ABAQUS Analysis User0s Manual, 2011), using theprincipal Hugoniot as a reference curve. For water, the shearmodulus or viscosity should be small to avoid an overly stiffresponse because these factors act as a penalty parameter. The EOSis assumed for the pressure as a function of density and theinternal energy per unit mass. The most common form of the Mie-Grüneisen equation (ABAQUS Analysis User0s Manual, 2011) is

p�pH ¼ ΓρðEm�EHÞ ð10Þwhere Em is the internal energy per unit mass, pH and EH are theHugoniot pressure and specific energy, respectively, Γ is theGrüneisen ratio, defined as Γ ¼ Γ0

ρ0ρ (Γ0 is a material constant,

and ρ0 is the reference density).Hugoniot pressure, pH, is a function of density only, and this is a

linear relationship between the shock and particle velocity, withcoefficients from relevant experimental data (ABAQUS AnalysisUser0s Manual, 2011).

Hugoniot energy EH is related to Hugoniot pressure pH using

EH ¼ pHη2ρ0

where η is the nominal volumetric compressive strain

η¼ 1�ρ0ρ

Substituting these factors into (10) produces

p�pH ¼ Γ0ρ0ðEm�pHη2ρ0

Þ

) p¼ pHð1�Γ0η

2ÞþΓ0ρ0Em ð11Þ

A common fit to the Hugoniot data is

pH ¼ ρ0c02η

ð1�sηÞ2ð12Þ

Substituting this into (11) yields a linear Us–Up Hugoniot form:

) p¼ ρ0c02η

ð1�sηÞ2ð1�Γ0η

2ÞþΓ0ρ0Em ð13Þ

where c0 and s define the linear relationship between linear shockvelocity Us and particle velocity Up as follows: Us ¼ c0þsUp (ρ0c02η isequivalent to the elastic bulk modulus with small nominal strains).

2.2.2.2. Ideal gas equation of state. ABAQUS software uses the EOSto simulate air using the following equation:

pþpA ¼ ρRðθ�θzÞ ð14Þwhere pA is ambient pressure, R is the gas constant, θ is the currenttemperature, and θz is absolute zero on the temperature scale.

Specific energy depends on temperature.

Em ¼ Emo þZ θ�θz

θo � θzcvðTÞdT ð15Þ

where Em is the initial specific energy at initial temperature θo, andcv is the specific heat at a constant volume.

Modeling an ideal gas EOS is typically performed adiabatically;the temperature increase is calculated directly at the materialintegration points based on the adiabatic thermal energy increasecaused by work pdv, where v is the specific volume. Therefore,unless fully coupled temperature–displacement analysis is per-formed, ABAQUS/Explicit always assumes that an adiabatic condi-tion exists.

The initial gas state is determined from initial density, ρ andeither initial pressure stress, p0 or initial temperature, θo.

The initial specific energy for an ideal gas EOS affects only theoutput of total specific energy and is calculated using

Emo ¼Z θo �θz

0cvðTÞdT ð16Þ

In the special case of adiabatic analysis with constant specific heat,the specific energy is linearly related to temperature.

Em ¼ cvðθ�θzÞ ð17ÞPressure stress is calculated using

pþpA ¼ ðγ�1ÞρEm ð18Þwhere γ is the specific heat ratio.

3. Numerical study

3.1. Model description

This simulation replicates an experiment by Klaseboer et al.(2005), which was conducted in a pond to study the dynamicprocess of bubbles produced in an underwater explosion. Hexocire(55 g) was placed at a depth of 3.5 m and approximately in themiddle of the pond. The pond sidewalls were designed to diffractexplosion shockwaves and to absorb the shock explosion energy.A high-speed camera above the water surface recorded images ofthe bubble explosion phenomena. Tourmaline piezoelectric sen-sors were used to measure pressure in the water and on thestructures.

Because the dynamics of underwater explosion bubbles arecomplex, this study focused on the dynamic process of bubbles ina free field, especially the bubble collapse phase. Therefore, thisstudy used a simplified model, which assumes that the shockwaveeffect on bubble migration is negligible because of its shortduration; an underwater explosion bubble is initially a sphericalcompressed gas bubble at high pressure; and water is inviscid,irrotational, and incompressible.

Because the simulation model involves the flow of liquid andgases, using a standard Lagragian-based FEM is impractical. Thenumerical analysis was therefore conducted using a Eulerian-basedfinite volume program. Eulerian analysis in ABAQUS manages theexcessive deformation of fluid and allows many materials to beincluded in the same element. The numerical model includes threeregions of material, water, air, and highly compressed detonationgas, in a rectangular parallel-piped domain. Fig. 2 shows a sche-matic of the numerical model. Because the hydrostatic pressure in

C.-Y. Hsu et al. / Ocean Engineering 81 (2014) 29–3832

the water surrounding the bubble plays an essential role in thedynamic bubble process, the height of the water region was thesame as that of the water in the pond (7 m), and the gas bubble wasinitiated at the same position as the charge in the experiment(3.5 m). An air region was created in the model to make thesimulation more realistic; this also facilitated managing the risingwater surface when the bubble expanded. All faces of the Euleriandomain were fixed to prevent materials flowing out, and a non-reflecting Eulerian boundary was used for the side boundaries ofthe Eulerian domain to eliminate the reflection of pressure wavesthat may affect the simulation result accuracy. To obtain accurateresults and conserve computational resources, only the initialbubble region was meshed at the small size of 5 mm. The regionof 700 mm from the bubble center had a mesh size of 20 mm, andthe remaining model region had a large mesh size. Fig. 3 shows halfof the model and an enlarged view of the mesh surrounding theoriginal bubble. The overall model element mesh consisted of 1 173872 EC3D8R elements, eight nodes, and multi-material elements.

The pressure in the air domain was proportional to the atmo-spheric pressure. The hydrostatic pressure in the water began atthe same pressure as the air at the contact surface between the airand water regions. Hydrostatic pressure increased according to theincrease in water depth. To obtain the initial conditions of thesimulated bubble, this study used data calculated by Cole using

TNT (Cole, 1948). If the charge is a sphere, then the charge radius iscalculated using

rch ¼34Wπρ0

� �1=3

ð19Þ

where W is the charge weight, and ρ0 is the charge density.Based on calculations by Cole, the relationship between pres-

sure and specific density is shown in Fig. 4. At higher densities, thepressures are much higher, corresponding to the increased effectof repulsive forces on the internal energy. If the specific density isadequately large, the detonation gas product can be considered anideal gas with a specific heat ratio of 1.25. This study assumes thatthe bubble initiates at twice the charge radius, or 40 mm; there-fore, the initial bubble volume becomes eight times larger, and the

Fig. 2. The arrangement of the model.

Fig. 3. Finite element model; (a) the half of the model and (b) an enlarged view of the mesh surrounding the original bubble.

Fig. 4. Cole calculation for TNT on the logarithmic plot.

C.-Y. Hsu et al. / Ocean Engineering 81 (2014) 29–38 33

bubble density becomes eight times smaller.

ρg ¼ ρ0=8¼ 1630=8¼ 203:75 kg=m3

The specific density of gas proportional to this density value is

vg ¼ 1=ρg ¼ 0:00491m3=kg¼ 4:91 cm3=g:

The initial bubble pressure proportional to this specific densityvalue is Pg¼5.704E7 Pa.

Because the gas bubble simulation time is short, the gas bubbleexpansion is considered an adiabatic process. Therefore, the idealgas EOS in ABAQUS can be used for the air and explosive gasmaterial. The UP–US EOS was used for the water in which thesimulation bubble occurred. Table 1 shows the physical parameterdetails of the numerical model.

3.2. Results and discussion

Analyzing the dynamic process of underwater explosion bub-bles is complex, requiring the consideration of numerous facetsand theories. The proposed method uses commercial finite ele-ment software ABAQUS 6.11 to simulate the dynamic process ofan underwater explosion bubble in three dimensions. The FEMresults were verified by the BEM results and experimental results(Klaseboer et al., 2005).

Although the bubble oscillation simulation was simplified infree-field conditions and the detonation and shockwave propaga-tion phase was not calculated precisely because of the assumption,the simulation provided valuable information on bubble oscilla-tion, especially during the collapse phase by reflecting the influ-ence of buoyancy on bubble motion. Simulated and experimentalbubbles show similar bubble shapes at different intervals (Fig. 5).Fig. 6 shows the bubble radius time history during the first cycle.The bubble radius increased at a high speed at the early stagebecause the internal pressure of compressive gas was high, andthe internal pressure decreased quickly. Then the bubble radiuschanged slightly near the first maximal radius of approximately470 mm at 48 ms. However, the bubble radius decreased quickly atthe final bubble contraction stage, and the first oscillation cycleended after approximately 93 ms. These features are similar to theresults derived by Cole (1948). For TNT charge, K¼2.11, (Prior andBrown, 2010), Eqs. (8) and (9) produced a maximal radius of541.7 mm and a first bubble period oscillation of 88.16 ms. Thesevalues are approximate and are close to the values predicted byusing the FEM. However, it should be remembered that the bubblein the present study was assumed to be initiated at twice thecharge radius with a null-velocity of the bubble wall. These

simplifications obviously influence the accuracy of FEM, thus thegood results obtained may be slightly changed if the initial bubbleis considered in the real conditions. The Earth0s gravitational fieldaffects the migration of underwater explosion bubbles in a freefield, especially during the bubble collapse phase. The numericalmodel accounted for this by setting the gravitational constant forthe entire domain. The static equilibrium of the water domain wasquickly obtained during the analysis by using the GEOSTATIC*Keyword embedded in ABAQUS 6.11. Fig. 7 shows typical timehistories for the FEM upper and lower bubble margin distances, aswell as the results from Klaseboer et al. (2005). Fig. 8 shows acomparison of the bubble mass center to the initial bubbleposition. The time history profiles are similar, despite a slightdifference between the numerical and the experimental results.The upward migration of the simulation and experiment bubbleswere small during the expansion and contraction, but the upwardmigration became noticeable after the first bubble minimum. Thischaracter confirms the results obtained by Cole (1948). After thefirst minimal volume, FEM results show a rise of 0.27 m, and theexperimental and BEM results are 0.28 m and 0.25 m, respectively.

A remarkable feature of the Eulerian technique proposed in thispaper is that the method provides valuable information on thebubble collapse phase. During the bubble collapse phase, differentparts of the bubble contract at various rates. The part below thebubble moves the fastest because of the higher ambient hydro-static pressure. This imbalance results in the surface on the bottomdeforming into an upward water jet. Fluid at the bottom of thebubble rushes into this developing water jet, thus creating animpact on the opposite bubble wall. During this transition frombeing simply connected to doubly connected, the spherical bubbletransforms into a toroidal bubble. The shape of the bubble in thetoroidal form is shown in Fig. 9. Fig. 10 shows images of bubbleswith the water jet formation at different intervals during thecollapse phase. At this stage, intensive flow changes in the bubblearea occur. Fig. 11 shows the velocity field at different timeintervals corresponding to the intervals in Fig. 10. At 93.75 ms,when the bubble becomes doubly connected, the flow velocity inthe middle of the bubble reaches approximately 130 m/s in anupward direction; this velocity field represents the magnitude anddirection of the water jet. To investigate the mechanism damage ofstructures near the collapsing bubble, considering the pressureimpulse emitting during the bubble collapse phase is crucial. Thepressures at three points around the bubble were also determined.P1 is the point at a horizontal distance of 0.7 m from the initialbubble position. P2 is the point 0.3 m above P1, and P3 is 0.5 mbelow P1 (Fig. 12). Fig. 13 shows a bubble pulse comparison of theFEM, BEM, and experimental results. Although the magnitudes ofpressure pulses vary considerably between the numerical and theexperimental results, they reflect a similar trend. The highestpressure pulse occurred at the nearest point, P1, and the smallestpressure pulse occurred at the farthest point, P3. Because thepressure pulse is propagated as a pressure wave in the waterdomain and because of energy lost during transmission, bubblepressure pulse magnitude decreases as the standoff distancesincrease. The higher experiment pressure may be caused by thecharge weight approximation. The initial bubble conditions in thisstudy were based on TNT calculations, but the experiment wasconducted using Hexocire. Although the same type of charge wasused as in the BEM, the initial bubble pressure calculation reportedin this paper was based on an ideal gas line with a specific heat of1.25 (Fig. 4). This approximation also reduced the initial bubblepressure.

Computational modeling has been used to predict manyaspects of an underwater explosion event. To solve problems suchas the dynamics of explosion bubbles in water, the BEM isadvantageous because it requires only information that is on the

Table 1Physical parameters used in the simulation.

Material Parameter Symbol Value Unit

Water Density ρw 1000 kg/m3

Sound speed Cw 1500 m/s

Air Density ρa 1.17 kg/m3

Ratio of specific heat γa 1.4 –

Initial pressure Pa 1.0E5 Pa

Explosive (TNT) Weigh W 0.055 kgDensity ρch 1630 kg/m3

Charge radius rch 0.02 m

Gas bubble Density ρg 203.75 kg/m3

Ratio of specific heat γg 0.04 –

1.25Bubble radius rg 0.04 mInitial pressure Pg 5.704E7 Pa

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bubble wall; whereas the FEM calculates the entire domainsolution and requires abundant computational resources. How-ever, the FEM has advantages and has become standard amongengineers for solving nonlinear problems. When studying under-water explosion bubbles, the dynamic behavior of bubbles at theend of the first expansion and contraction cycle is crucial becauseit is associated with water jet formation. The water jet usuallycauses substantial mechanical damage to nearby structures; there-fore, information on the flow surrounding the bubble during thecollapse phase is essential, thus indicating the advantages of usingthe FEM. Furthermore, recent advances in computer technologyhave considerably reduced the time required to perform suchsimulations, and the main weakness of the FEM is no longer amajor concern. The model simulation in this paper was run on aworkstation with an Intel(R) Xeon(R) E5 CPU, 2 GHz processors, 32Gbytes of RAM, 64-bit Microsoft Windows 7, and 500 Gbytes ofhard drive storage capacity. ABAQUS version 6.11-1 was thesimulation code used, and the total CPU time was 8 h 25 min.

Fig. 5. Bubble shape of a 55 g TNT at different intervals, increasing from left to right: (a) Exp. results at: t¼1, 7, 50, 85, 93, 94, 95 and t¼96 ms; (b) BEM results at: t¼0, 7.5,50.2, 80.1, 89.1, 90.61, 95 and t¼91.58 ms; (c) FEM results at: t¼0, 7.5, 50.25, 80.25, 92.25, 93.75, 94.5 and t¼96 ms.

Fig. 6. Diameter vs. time obtained in first circle of bubble oscillation.

Fig. 7. The position of the upper bubble surface and the lower bubble surface.

Fig. 8. The position of the center of mass as a function of time.

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By using a modern numerical analysis computer program basedon an Eulerian method that allows elements to contain multiplematerials, it is possible to investigate explosion gas bubblephenomena with acceptable accuracy. Thus far, such phenomenahave been understood only qualitatively.

4. Conclusion

The Eulerian analysis technique in the ABAQUS/Explicit 6.11code presented in this paper was used to reproduce the mainbubble behavior in an experiment. A comparison with the experi-ment results shows that the model reliably reproduced essentialbubble characteristics. The period of first circle and maximalradius of the bubble were reasonably similar to the experimental

data. The first bubble pulse, velocity field of flow around thebubble, and water jet formation during the collapse phase werealso investigated.

Despite limitations related to simplifications such as laminar,incompressible viscous, and inviscid flow behaviors, as well as theidealization of explosion gas products, the simplified modelsimulated most of the crucial physics of an underwater explosionbubble. This includes the pulsation of bubble, the generation andpropagation of bubble pressure waves and the water jet producedby a bubble collapse.

This study investigated the potential application of an Euleriantechnique in a finite element analysis code, such as ABAQUS 6.11,to quantitatively model an underwater explosive bubble in afree field. The quantitative characterization is suitable for other

Fig. 9. The 3D surface of bubble with the formation of water jet; (a) top view and (b) side view.

Fig. 10. Simulated bubble motion during the collapse phase at: (a) t¼93 ms, (b) t¼93.75 ms, (c) t¼94.5 ms and (d) t¼95.25 ms.

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explosive charge types, weights, and depths. Future studies shouldexpand the applicability of the results to real conditions, such asthe interaction between underwater explosion bubbles and struc-tures composed of different materials. The influence of the meshsolution, fluid compressibility, and heat exchange should be care-fully considered to obtain results that are more accurate. Suchstudies would be challenging and require deep theoretical content,but would also yield valuable and compelling results.

Acknowledgment

The authors wish to acknowledge the financial support pro-vided by the NSC 100-2221-E-212-001-MY3. We would also liketo thank my friends for their invaluable efforts in the construc-tion, assembly, and the commissioning and operation of thefacility.

As with any paper of this nature, there are references I mayhave neglected to cite or have missed in a literature search. Any

Fig. 11. Velocity field in the bubble area during the collapse phase at: (a) t¼93 ms, (b) t¼93.75 ms, (c) t¼94.5 ms and (d) t¼95.25 ms.

Fig. 12. Schematic location of P1, P2 and P3.

Fig. 13. The comparison of pressure pulse at different points.

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unintentional omission should not be an inference that suchuncited works are unimportant. Discussions with many friendsand colleagues were both stimulating and helpful.

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