Research ArticleNumerical Investigation of Water Entry Problem ofPounders with Different Geometric Shapes and Drop Heights forDynamic Compaction of the Seabed

Mohammad Hossein Taghizadeh Valdi 1 Mohammad Reza Atrechian 2

Ata Jafary Shalkoohy 3 and Elham Chavoshi 1

1Department of Civil Engineering Isfahan (Khorasgan) Branch Islamic Azad University Isfahan Iran2Department of Civil Engineering Zanjan Branch Islamic Azad University Zanjan Iran3Department of Civil Engineering Bandar Anzali Branch Islamic Azad University Bandar Anzali Iran

Correspondence should be addressed to Mohammad Reza Atrechian matrechianiauzacir

Received 19 August 2017 Revised 17 October 2017 Accepted 3 January 2018 Published 6 February 2018

Academic Editor Stefano Lo Russo

Copyright copy 2018 Mohammad Hossein Taghizadeh Valdi et al This is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

The water entry problem of three-dimensional pounders with different geometric shapes of cube cylinder sphere pyramid andcone was numerically simulated by the commercial software Abaqus and the effects of pounder shape and drop height from the freesurface of water on deepwater displacement and velocity as well as pinch-off time and depth were investigated An explicit dynamicanalysis method was employed to model fluid-structure interactions using a Coupled Eulerian-Lagrangian (CEL) formulationThesimulation results are verified by showing the computed shape of the air cavity displacement of sphere pinch-off time and depthwhich all agreed with the experimental results The results reveal that the drag force of water has the highest and lowest effect oncubical and conical pounders respectively Increasing the pounder drop height up to the critical height leads to increased poundervelocity while impacting the model bed and more than the critical drop height has a reverse effect on pounder impact velocityPinch-off time is a very weak function of pounder impact velocity but pinch-off depth increases linearly with increased impactvelocity

1 Introduction

Dynamic compaction is regarded as an appropriate methodfor soil improvement after its introduction in 1970 by LouisMenardThismethod is based on the drop of heavy poundersof 8 to 25 tons from a height of 10 to 20m on loose soilsurfaces although lighter or heavier weights and other dropheights are occasionally used [1] The pounders impact onthe soil surface and the resulting vibration on the surface andunderlying layers leads to the compaction of the soil grainsAs a result the relative density of soil and its stiffness and pen-etration resistance increase This leads to increased soil shearstrength and bearing capacity and decreased liquefactionThe pounders used in this method have different shapes andweights due to the operations implementation on the coastand in the sea Onshore dynamic compaction operations

mostly make use of cubical or cylindrical pounders but inoffshore dynamic compaction operations a large amountof pounder velocity depreciates due to its impact on thewaterrsquos surface and deepwater movement meaning that thepounder does not have the energy necessary for soil vibrationand compaction of the surface and its underlying layersTherefore the geometric shape of the pounder must beselected so that the least velocity depreciation occurs duringdeepwater movement to improve a proper radius and depthof soil after seabed impact and soil layer vibration

11 Experimental Studies The water entry problem of solidprojectiles has been an interesting topic in fluid mechan-ics for more than 80 years The study of the underwa-ter motion of projectiles has wide applications in militaryfields especially for naval industries since World War II

HindawiGeofluidsVolume 2018 Article ID 5980386 18 pageshttpsdoiorg10115520185980386

2 Geofluids

Furthermore accurate prediction of water impact forces hasparticular importance in the design of marine structuresand projectiles employed in dynamic compaction of seabedIn the last three decades a lot of researchers addressedthe water entry problem using various methods Watanabe[2 3] investigated the impact of cones upon water Conesof different weights were dropped into water from differentheights while being equipped with a piezoelectric sensorconnected to a vibrometer to register force changes resultingfrom water impact Such experimental results were veryhelpful in approving analytical results and relations May andWoodhull [4ndash6] published three important papers to find thedrag coefficient and the scaling relationship in water entry ofsphere projectiles by considering the Reynolds and FroudenumbersThey used an experimental method to study the aircavity formed during projectile impact on water using high-speed photography Mayrsquos work indicated that Froude scalingis a good first approximation to use when describing cavitybehaviours such as deep closure New et al [7] investigatedthe impact and hydrodynamic characteristics of water entryassociated with prismatic bodies with different fore sectionshapes and entry angles Transient loading on the fore sectionof the prismatic bodywasmeasuredwith a triaxial accelerom-eter Splashes cavity formation and closure procedures wereobservedwith high-speed photography and video recordingsAnghileri and Spizzica [8] studied the normal impact ofa rigid sphere with a water surface using finite elementmethods and certain experiments to gain variations in sphereacceleration during impact in order to verify the appliednumericalmethodTheir results were in good agreementwiththe experimental data Engle and Lewis [9] compared theresults of the numericalmethod for predicting hydrodynamicimpacts with experimental results of symmetrical wedgeswith different initial impact velocities Their studies cateredfor the accuracy and validity of the various methods Aristoffet al [10] performed an experimental and theoretical studyby impacting spheres with different densities on a watersurface A theoretical model was developed that yieldedsimple expressions for pinch-off time and depth as wellas the volume of air entrained by the sphere Their studydemonstrated that the dynamics of the air cavity formed inthe wake of a sphere falling through the water surface can bealtered considerably by the density of the sphere

12 Analytical Solution Von Karman [11] developed analyti-cal solutions for the water entry problem of two-dimensionalwedge bodies He theoretically presented formulas usingasymptotic theory Wagner [12] presented a more preciseanalytical model which took the free surface elevation ofthe water into consideration He developed an asymptoticsolution for water entry by two-dimensional projectiles withsmall local deadrise angles Vorus [13] proposed a novelanalytical model for the water entry problem of bodies In hismodel nonlinearity in the boundary conditions was consid-ered but the geometrical nonlinearity of the impact problemwas neglected Miloh [14] obtained analytical solutions basedon several mathematical and physical assumptions for thesmall-time slamming coefficient and wetting factor of a rigidspherical shape in a vertical and oblique water entry Aristoff

and Bush [15] presented the results of a combined experi-mental and theoretical investigation of the normal impact ofhydrophobic spheres on a water surface by determining theshape of the air cavity behind the sphere pinch-off time anddepth Ghadimi et al [16] employed an analyticalmethod anda numerical solution to study the water entry problem of acircular section They demonstrated a procedure for the stepby step derivation of the analytical formulas and presenteda comparison between the linear and nonlinear solutionsTassin et al [17] investigated the water impact problem ofbodies by CFD code OpenFOAM They studied the waterentry and exit of a two-dimensional bodywith a time-varyingshape by using a modified Logvinovich model during theentry stage and the Von Karman model during the exit stageYao et al [18] developed a theoretical model to describe theevolution of the cavity shape including the time evolution ofthe cavity on fixed locations and its location evolution at fixedtimes

13 Numerical Simulations Park et al [19] presented anumerical method for calculating the impact and bucklingforces of bodies after a quick water entry By neglectingfluid viscosity potential flow was assumed to be nonviscousThis assumption greatly reduced computational time whilesolution accuracy was preserved by considering nonlinearfree surface conditions They solved this issue using a sourcepanel method in which an object is divided into panelsor elements and each panel can be fully floating partiallyfloating or fully out of the water The results were comparedwith laboratory data and valuable results were obtainedBattistin and Iafrati [20] conducted a numerical study onnormal entry into water of a two-dimensional symmetricalor asymmetrical object with a desirable form This studyconsidered potential flow of an incompressible fluid byignoring the effects of gravity and adhesion In this worksimulation was done using the boundary element methodand the boundary conditions of the free surface were fullynonlinear The focus of this investigation was on evaluat-ing pressure distribution and overall hydrodynamic loadsapplied to the colliding body They verified the results forcylinder sphere and cone forms Korobkin and Ohkusu [21]performed an interesting study on the hydroelastic couplingof finite element models using Wagnerrsquos theory for waterimpactsThe aim of the study was to measure direct couplingpossibilities using finite element methods for the structureandWagnerrsquos theory for hydrodynamic loads of impact of anelastic object with the waterrsquos surface Kleefsman et al [22]numerically investigated the dambreak problem and waterentry problemsThe numerical method was based onNavier-Stokes equations that describe the flow of an incompressibleviscous fluid The equations were discretized onto a fixedCartesian grid using the finite volume method Kim et al[23] analyzed the water entry problem of symmetrical bodiesusing the particle hydrodynamic method Yang and Qiu [24]investigated the two-dimensional water entry problems ofsymmetric and asymmetric wedges with various deadriseangles using the Constrained Interpolation Profile (CIP)method The effect of air compressibility for small deadriseangles was also discussed in their work Fairlie-Clarke and

Geofluids 3

Tveitnes [25] conducted a study on impact of wedge-shapedparts with a water surface at a constant velocity In their studyComputational Fluid Dynamic (CFD) analysis was used todetermine the momentum of added mass momentum offlow and gravityrsquos effect during the entry of wedge-shapedobjects at a constant velocity and a dead angle of 5 to 45degrees into water A numerical solution based on finitevolume method was implemented using Fluent software tosolve equations of mass conservation and momentum toobtain flow fields Hydrodynamic forces were not signifi-cantly changed by the effect of gravity on the flow fieldsYang and Qiu [26] computed slamming forces on two-and three-dimensional bodies entering calm water with aconstant velocity The nonlinear water entry problem wasgoverned by Navier-Stokes equations and solved using a CIP-based finite difference method on a fixed Cartesian gridWu [27] developed a high order boundary element methodfor the complex velocity potential problem He focused onapplication of the water entry problem for a wedge withvarying speeds His method ensured not only the continuityof the potential at the nodes of each element but also thevelocity allowing it to be applied to a variety of velocitypotential problems The continuity of the velocity achievedherein was particularly important for this kind of nonlinearfree surface flow problem because when the time steppingmethod is used the free surface is updated using the velocityobtained at each node and the accuracy of the velocityis therefore crucial Ahmadzadeh et al [28] developed athree-dimensional numerical model to simulate a transientfree-falling sphere impacting a free water surface using theCoupled Eulerian-Lagrangian formulations included in thecommercial software Abaqus with consideration for impor-tant parameters such as viscosity compressibility inertiaand variable downward velocity Their study was successfulin determining the shape of the air cavity displacementof sphere versus time pinch-off time and depth comparedto the experimental results found by Aristoff et al [10]Nguyen et al [29] studied the water impact of variousthree-dimensional geometries namely a hemisphere twocones and a free-falling wedge with an implicit algorithmbased on a dual-time pseudo-compressibility method Flowfields of incompressible viscous fluids were solved usingunsteady RANS equations A second-order VOF interfacetracking algorithm was developed in a generalized curvi-linear coordinate system to track the interface between thetwo phases in the computational domain Sensitivity analysiswith respect to the spatial grid resolution and the time stepwas performed to assess the accuracy of the results Freesurface deformation pressure coefficients impact velocitiesand vertical acceleration during impact were compared withavailable experimental data and asymptotic theories showinggood agreement at a reasonable computational cost S Kimand N Kim [30] performed integrated dynamic modelingof supercavitating vehicles and established 6-DOF equationsby defining hydrodynamic forces and moments These sim-ulations also demonstrated the instability of supercavitatingvehicles caused by the asymmetric immersion depth ofcoupled horizontal or vertical fins Nguyen et al [31] used amovingChimera gridmethod to predict the real-timemotion

of water entry bodies by combining the 6-DOF rigid bodymotion model and the numerical calculation of the multi-phase flow field by which the coupled effect of supercavityand moving body was obtained Mirzaei et al [32] held theidea that existing planning force models weaken nonlinearinteractions among the solid liquid and gaseous phases andare often too simple and therefore incorrectThey establishedthe 6-DOF motion model for supercavitating vehicles byredefining the spatial planning force and developed a hybridcontrol scheme based on online planning force identifica-tion which increased the stabilization of supercavitatingvehicles

In this study the water entry problem of three-dimensional pounders with different geometric shapesincluding cube cylinder sphere pyramid and cone wasinvestigated numerically using commercial finite elementcode Abaqus 614-2 and an explicit dynamic analysis methodwas employed to model fluid-structure interactions using aCoupled Eulerian-Lagrangian (CEL) formulation [33] Thisinvestigation focused on the problem of rigid pounderstouching the free surface of a stationary viscous fluid at time119905 = 0 with a variable velocity and finding for 119905 gt 0the shape of cavity pinch-off time and depth of the rigidpounders It should be noted that the fluid was assumedto be viscous and compressible Numerical simulations ofsolid-fluid interactions were performed using the CoupledEulerian-Lagrangian (CEL) formulation and the remeshingtechniques available in Abaqus 614-2 software [34] Thepounder was considered as a rigid solid and its mesh wascreated in Lagrangian form The water was regarded as acompressible fluid and its mesh was created in Eulerianform After validating the numerical results with recentexperimental data the effect of the pounder shape on itsdeepwater displacement and velocity was investigated Thenthe effect of increasing the pounder drop height from thewaterrsquos surface on displacement submersion velocity pinch-off time and depth was investigated The critical drop heightof the pounder which is indicative of the maximum effectof the pounders drop height from the waterrsquos surface ondeepwater movement was finally determined

2 Coupled Eulerian-Lagrangian (CEL) Method

Eulerian materials and Lagrangian elements can interactusing the Eulerian-Lagrangian contact Simulations thatinclude this type of contact are called Coupled Eulerian-Lagrangian (CEL) analyses [28] The CEL method in Abaqusprovides engineers and scientists with the ability to simulatea class of problems where the interaction between structuresand fluids is important This capability does not rely on thecoupling of multiple software products but instead solvesthe fluid-structure interaction (FSI) simultaneously withinAbaqusThis powerful easy to use feature of AbaqusExplicitgeneral contact enables fully coupled multiphysics simula-tions such as fluid-structure interactions The attractivenessof this method is that the fluid is modeled in an Eulerianframe while the structure can be modeled in a Lagrangianframe as is typical for solid mechanics applications Specif-ically in CEL the governing equation of the solid uses

4 Geofluids

the conservation of momentum while the governing equa-tion for the fluid uses a general Navier-Stokes form [36]The Eulerian mesh is typically a simple rectangular gridof elements extending well beyond the Eulerian materialboundaries giving the material space in which to move anddeform If any Eulerian material moves outside the Eulerianmesh it is lost from the simulation The Eulerian materialboundary must therefore be computed during each timeincrement and it generally does not correspond to an elementboundary The interaction between the structure and thewater is handled remarkably well by the solver using thegeneral contact algorithm In traditional Lagrangian analysisnodes are fixed within the material and elements deform asthe material deforms Lagrangian elements are always 100full of a single material so the material boundary coincideswith the element boundary By contrast in Eulerian analysisnodes are fixed in space andmaterial flows through elementsthat do not deform Eulerian elements may not always be100 full ofmaterial andmanymay be partially or completelyvoid

The Eulerian implementation in AbaqusExplicit is basedon the Volume Of Fluid (VOF) method In numericalanalyses using this CEL method the Eulerian material istracked as it flows through the mesh by computing itsEulerian Volume Fraction (EVF) Each Eulerian element isdesignated a percentage which represents the portion ofthat element filled with a material If an Eulerian element iscompletely filled with a material its EVF is 1 and if there isno material in the element its EVF is 0 Eulerian elementsmay simultaneously contain more than one material If thesum of all material volume fractions in an element is less thanone the remainder of the element is automatically filled witha void material that has neither mass nor strength Contactbetween Eulerian materials and Lagrangian materials isenforced using a general contact that is based on a penaltycontact methodThe Lagrangian elements can move throughthe Eulerian mesh without resistance until they encounteran Eulerian element filled with material (EVF = 0) A CELmethod that attempts to capture the advantages of boththe Lagrangian and the Eulerian method is implemented inAbaqus [33]

21 Time Integration Scheme TheCELmethod implementedin AbaqusExplicit uses an explicit time integration schemeThe central difference rule is employed for the solution ofa nonlinear system of differential equations The unknownsolution for the next time step can be found directly fromthe solution of the previous time step so no iteration isneeded Another advantage of choosing an explicit timeintegration scheme for the present problem is robustnessregarding difficult contact conditions [33]

22 Formulation of Eulerian-Lagrangian Contact By usingthe CEL method the contact between the Eulerian domainand Lagrangian domain is discretized using general contactwhich is based on the penalty contactmethodThismethod isless stringent compared to the kinematic contactmethodThepenalty contact method is used in Abaqus to handle contactin CEL analysis [33]

23 Governing Equations in Lagrangian Method The threefundamental conservation equations are the conservation ofmass the conservation of momentum and the conservationof energy The Lagrangian formulation trivially enforces theconservation of mass by defining the density as the ratio ofthe element mass to the current volume The conservation ofenergy in the absence of heat conduction is enforced locallyat selected points within each element or control volumeDifferent finite elements and finite difference methods aretherefore defined in terms of how they handle the conserva-tion of momentum

Finite element methods use the weak form of the equa-tion which is also known as the principle of virtual workor the principle of virtual power within the solid mechanicscommunity In this method the principle of virtual work ispresented as

int119881[120588120575119909 + 120590 120575120576] 119889119881 = int

119881120588119887120575119909 119889119881 + int

119878120591

120591120575119909 119889119878 (1)

where 120588 is density 120590 is the stress 120576 is strain 120591 is shearstress 119881 is the volume of the domain and 119878 is the surfacewhich is divided into 119878120591 and 119878119906 which is where traction anddisplacement boundary conditions are imposed respectivelyEquation (1) can be rewritten as

int119881120588120575119909 119889119881 = 119865ext minus 119865int (2)

where 119865int represents internal force and 119865ext representsexternal force which is expressed as

119865int = int119881120590 120575120576 119889119881

119865ext = int119881120588119887120575119909 119889119881 + int

119878120591

120591120575119909 119889119878(3)

In the finite element method the coordinates velocityacceleration and virtual displacement are interpolated fromtheir values at the nodes using shape functions119873119860 where 119860is the node number They are expressed as

119909 = sum119860

119873119860119909119860 = sum119860

119873119860119860 = sum119860

119873119860119860120575119909 = sum

119860

119873119860120575119909119860

(4)

Substituting the relations from (2) into the inertial termin the weak form gives

int119881120588120575119909 119889119881 = int

119881120588sum119860

119873119860119860sum119861

119873119861120575119909119861119889119881= sum119860

sum119861

[int119881120588119873119860119873119861119889119881] 119860120575119909119861

(5)

Geofluids 5

The external force vector is also derived by a directsubstitution of the appropriate relations from (6)

119865ext = int119881120588119887120575119909 119889119881 + int

119878120591

120591120575119909 119889119878 (6)

119865ext = int119881120588119887sum119861

119873119861120575119909119861119889119881 + int119878120591

120591sum119861

119873119861120575119909119861119889119878 (7)

119865ext = sum119861

int119881120588119887119873119861119889119881 + int

119878120591

120591119873119861119889119878 120575119909119861 (8)

The internal force vector requires the evaluation of thevirtual strain 120575 Again using the relations from (9)

120575120576119894119895 = 12 ( 120597119909119894120597119909119895 +120597119909119895120597119909119894 ) (9)

120575120576119894119895 = sum119861

12 (120597119873119861120597119909119895 120575119909119861119894 +120597119873119861120597119909119894 120575119909119861119895) (10)

These relations may be put in matrix form which definesthe 119861matrix 119861 as

120576120575 = 119861120575119909 (11)

and the internal force may be written as

119865int = int119881120590 120575120576 119889119881

119865int = sum119861

int119881119861119861120590119889119881 120575119909119861

(12)

where 119861119861 is the section of the 119861matrix associated with node119861 To simplify the mass matrix is defined as

119872119888119860119861 = [int119881120588119873119860119873119861119889119881] (13)

where 119872119888 is called the consistent mass matrix because itsderivation is consistent with the finite element method Allthe terms in the weak form have now been written in terms ofunknown coordinates and virtual displacements Collectingthe terms gives

sum119861

119861minus 1119872119861 (int119881 120588119887119873119861119889119881 + int

119878120591

120591119873119861119889119878 minus int119881119861119861120590119889119881)

sdot 120575119909119861 = 0(14)

Since the virtual displacements are arbitrary the 119861 valueis in accordance with

119861 = 1119872119861 (int119881 120588119887119873119861119889119881 + int119878120591

120591119873119861119889119878 minus int119881119861119861120590119889119881) (15)

24 Governing Equations in Eulerian Method For thedescription of fluid flows usually Eulerian formulation isemployed because we are usually interested in the propertiesof the flow at certain locations in the flow domain InEulerian formulation an independent equation is used toexpress conservation of mass Inertia forces also includedisplacement sentences that when present in numericalsolution lead to a matrix with nonsymmetric coefficients thatdepends on calculated velocities Eulerian formulation allowsfor the use of simple strain and stress rates and velocitycalculations instead of displacement values (in the ultrasmalldisplacement analysis) [36 37] Linear viscous isotropic fluidsare known as Newtonian fluids which are by far the mostimportant fluids for practical applications Newtonian fluidsare characterized by the followingmaterial law for theCauchystress tensor 119879

119879119894119895 = 120583( 120597V119894120597119909119895 +120597V119895120597119909119894 minus

23 120597V119896120597119909119896 120575119894119895) minus 119901120575119894119895 (16)

where the velocity vector is V119894 with respect to Cartesiancoordinate 119909119894 the pressure is 119901 the dynamic viscosity is 120583and the Kronecker symbol is 120575119894119895 With this the conservationlaws for mass momentum and energy take the form of

120597119901120597119905 +120597 (119901V119894)120597119909119894 = 0 (17)

120597 (119901V119894)120597119905 + 120597 (119901V119894V119895)120597119909119895= 120597120597119909119895 [120583(

120597V119894120597119909119895 +120597V119895120597119909119894 minus

23 120597V119896120597119909119896 120575119894119895)] minus120597119901120597119909119895 + 120588119891119894

(18)

120597 (119901119890)120597119905 + 120597 (119901V119894119890)120597119909119894= 120583[ 120597V119894120597119909119895 (

120597V119894120597119909119895 +120597V119895120597119909119894) minus 23 ( 120597V119894120597119909119894)

2] minus 119901 120597V119894120597119909119894+ 120597120597119909119894 (119896

120597119879120597119909119894) + 120588119902

(19)

where 120588 is the fluid density 119890 is the specific internal energyand119891119894 and 119902 are external forces and heat sources respectivelyIn energy balance (19) for the heat flux vector ℎ119894 Fourierrsquoslaw with thermal conductivity 120581119891 is used that is a constantspecific heat capacity is assumed and the work done bypressure and friction forces is neglected

ℎ119894 = minus120581119891 120597119879120597119909119894 (20)

System (17)ndash(19) has to be completed by two equations ofstate of the form

119901 = 119901 (120588 119879) 119890 = 119890 (120588 119879) (21)

6 Geofluids

which define the thermodynamic properties of the fluid thatis the thermal and caloric equation of state respectivelyWhen defining the viscosity for the material it is treated as aNewtonian fluid meaning that the viscosity depends only ontemperature This is the default setting in Abaqus Since thetemperature is constant throughout the analysis the viscosityalso is constant

25 Explicit Time-Dependent Solution There are implicitand explicit algorithms for dynamic analysis In the implicitmethod the Newmark method is used the matrices need tobe reversed andmore implementation time is requiredHow-ever this method enjoys high accuracy and its convergenceto linear problems is guaranteed In the explicit method thematrices do not need be reversed and the analysis speed ismuch higher Convergence in the explicit method is achievedif the time step is less than the smallest element divided bythe speed of sound in the material [37]

The solution was advanced in time using the centraldifference method which is also called the Verlet or leap frogmethod It is based on the central difference approximationfor evaluating the derivative of a function 119891(119905) at 119905

119889119891 (119905)119889119905 = 119891 (119905 + (12) Δ119905) minus 119891 (119905 minus (12) Δ119905)Δ119905 (22)

where Δ119905 is the time stepUsing (15) acceleration in time 119905 can be written as

119905 = 119865119905119872119861 (23)

Now using (22) for velocity-acceleration velocity at time119905 + (12)Δ119905 is obtained as

119905+(12)Δ119905 = 119905minus(12)Δ119905 + 119905Δ119905 (24)

Then using (22) for velocity-displacement displacementat time 119905 + Δ119905 is obtained as

119909119905+Δ119905 = 119909119905 + 119905+Δ119905Δ119905 (25)

The central difference method has its step size limited bystability considerations like all explicit integration methodsand the stable time step size was limited to

Δ119905 le 119897119888 (26)

where 119897 is the length of the shortest element and 119888 is the speedof sound

3 Finite Element Modeling

31 Equation of Energy and Hugoniot Curve The equationfor conservation of energy equates the increase in internalenergy per unit mass 119864119898 to the rate at which work is beingdone by stresses and the rate at which heat is being added Inthe absence of heat conduction the energy equation can bewritten as

120588120597119864119898120597119905 = (119901 minus 119901119887]) 1120588 120597120588120597119905 + 119904 119890 + 120588 (27)

PH

PHI1

PHI0

11 101

Figure 1 Schematic representation of a Hugoniot curve [35]

where 119901 is the pressure stress defined as positive in compres-sion 119901119887] is the pressure stress due to the bulk viscosity 119904 isthe deviator stress tensor 119890 is the deviator part of strain rateand is the heat rate per unit mass According to Abaqusdocumentation flow modeling of compressible fluid can beachieved using the linear 119880119904-119880119901 form of the Mie-Gruneisenequation of state The equation of state assumes pressure as afunction of the current density 120588 and the internal energy perunit mass 119864119898 according to

119901 = 119891 (120588 119864119898) (28)

which defines all the equilibrium states that can exist in amaterial The internal energy can be eliminated from (27)to obtain a pressure (119901) versus volume (119881) relationship orequivalently a 119901 versus 1120588 relationship that is unique tothe material described by the equation of state model Thisequation is uniquely dependent on the material defined bythe equation of state This unique relationship is called theHugoniot curve and it is the locus of 119901-119881 states achievablebehind a shockTheHugoniot pressure119901119867 is only a functionof density and can be defined in general from fittingexperimental data An equation of state is said to be linearin energy when it can be written as

119901 = 119891 + 119892119864119898 (29)

where119891(120588) and119892(120588) are only functions of density and dependon the particular equation of state model Figure 1 illustratesa schematic expression of a Hugoniot curve [35]

32 Mie-Gruneisen Equation of State As mentioned previ-ously an equation of state is used to express the behaviourof Eulerian materials For a Mie-Gruneisen equation of statefor linear energy the most common form is written as [38]

119901 minus 119901119867 = Γ120588 (119864119898 minus 119864119867) (30)

where 119901119867 and 119864119867 are the Hugoniot pressure and specificenergy (per unit mass) and are only functions of density andΓ is the Gruneisen ratio defined as

Γ = Γ0 1205880120588 (31)

Geofluids 7

where Γ0 is a material constant and 1205880 is the reference densityHugoniot energy 119864119867 is related to the Hugoniot pressure 119901119867by

119864119867 = 11990111986712057821205880 (32)

where 120578 is the nominal volumetric compressive strain writtenas

120578 = 1 minus 1205880120588 (33)

Elimination of Γ and119864119867 from the above equations yields

119901 = 119901119867 (1 minus Γ01205782 ) + Γ01205880119864119898 (34)

The equation of state and the energy equation representcoupled equations for pressure and internal energy Abaqussolves these equations simultaneously at each material pointusing an explicit method

33 Linear 119880119904-119880119901 Hugoniot Form Normally the 119880119904-119880119901 for-mulation of equation of state (EOS) is used to simulate shocksin solid materials In this study it is used to define fluidmaterials A common fit to the Hugoniot data is given by

119901119867 = 120588011986220120578(1 minus 119904120578)2 (35)

where 1198880 and 119904 define the linear relationship between thelinear shock velocity 119880119904 and the particle velocity 119880119901 using

119880119904 = 1198880 + 119904119880119901 (36)

The Mie-Gruneisen EOS is normally used for processeswith high pressures and high sound velocities The relation-ship between density and pressure is governed by an EOSmodel A linear 119880119904-119880119901 Hugoniot form of the Mie-Gruneisenequation of state is used to evaluate pressure in the waterin the simulation Using this implementation the water ismodeled as a compressible viscous fluid With the aboveassumptions the linear 119880119904-119880119901 Hugoniot form is written as

119901 = 120588011986220120578(1 minus 119904120578)2 (1 minusΓ01205782 ) + Γ01205880119864119898 (37)

where 120588011988820 is equivalent to the elastic bulk modulus at smallnominal strains

When modeling water Abaqus prescribes 119904 = 0 andΓ0 = 0 Filling this simplifies the relation between pressure119901 and density 120588 to an inversely proportional equation and asimplification of the EOS This simplification suffices for themodeling of water [39]

1199011003816100381610038161003816119904=0andΓ=0 = [ 120588011986220120578(1 minus 119904120578)2 (1 minusΓ01205782 ) + Γ01205880119864119898]

119904=0andΓ=0

= 120588011988820120578(38)

Recalling that 120578 = 1 minus 1205880120588 we get

119901 = 120588011986220 (1 minus 1205880120588 ) (39)

When defining a viscosity for thematerial it is treated as aNewtonian fluid meaning that viscosity only depends on thetemperature This is the default setting in Abaqus Since thetemperature is constant throughout the analysis the viscosityis also constant

There is a limiting compression given by the denominatorof this form of the equation of state The denominator of (34)should not be equal to zero therefore an extreme value isdefined for 120588 and 120578 as

120578lim = 1119904 120588lim = 1199041205880119904 minus 1

(40)

At this limit there is a tensile minimum thereafternegative sound speeds are calculated for the material

The linear Hugoniot form 119880119904-119880119901 equation of state canbe employed to model laminar viscous flows governed byNavier-Stokes equations The volumetric response is gov-erned by the equations of state where the bulk modulus actsas a penalty parameter for the constraint [38] The Eulerianpart (water) is simulated using the linear 119880119904-119880119901 Hugoniotform of the Mie-Gruneisen equation of state

The solving method in Eulerian equations is based onthe division of the governing equations In the Lagrangianmethod the nodes are considered to be fixed relative to thematerial at any time interval and elements are deformed withthe material while in the Eulerian method element defor-mation is calculated at any time interval but deformation isapplied as a remeshingThematerial is also displaced betweeneach element and its neighboring elements at any time inter-val Eulerian-Lagrangian contact allows the Eulerian mate-rials to be combined with traditional nonlinear Lagrangiananalyses In the Eulerian-Lagrangian contact the Lagrangianstructure occupies void regions inside the Eulerian meshThe contact algorithm automatically computes and tracks theinterface between the Lagrangian structure and the Eulerianmaterials A great benefit of this method is that there isno need to generate a conforming mesh for the Euleriandomain In fact a simple regular grid of Eulerian elementsoften yields the best accuracy If the Lagrangian body isinitially positioned inside the Eulerian mesh it must beensured that the underlying Eulerian elements contain voidsafter material initialization During analysis the Lagrangianbody pushes material out of the Eulerian elements that itpasses through and they become filled with void SimilarlyEulerian material flowing towards the Lagrangian body isprevented from entering the underlying Eulerian elementsThis formulation ensures that twomaterials never occupy thesame physical space

In this study CEL analysis allowed a direct coupling tosolve this problem The fact that CEL implementation usesthe full Navier-Stokes equations implies that it was able to

8 Geofluids

handle flow appropriately It is important to note that CELis not structured in the same way as a CFD code and wasnot intended to fill this role While it does include viscouseffects with extension to laminar flows it does not include anyturbulence effects but in the context of an impact which has ashort duration and is highly transient (with turbulence effectsbeing of less importance to the structural loading) CEL iswell suited [28]

4 Implementation ofCoupled Eulerian-Lagrangian (CEL)Method in Abaqus

In order to simulate the water entry problem of the pounderand implement the Coupled Eulerian-Lagrangian (CEL)method in Abaqus a geometric model for the pounder andwater was first created A partitionwas created to definewaterand void regions at the water surface Since the fluid (water)undergoes a large deformation an Eulerian grid is suitableThe rigid body (pounder) has small deformations relative tothe fluid making an Lagrangian grid suitable Therefore thewater wasmodeled as Eulerian and the pounder wasmodeledas solid geometry with a rigid body constraint applied to itThen the material properties for the pounder and water weredefined Properties include density (120588) Youngrsquos modulus (119864)and Poissonrsquos ratio (]) for the pounder and density (120588119908) anddynamic viscosity (120578) for the water In the Coupled Eulerian-Lagrangianmethod when the pounder is considered as rigidYoungrsquos modulus and Poisson ratios values should be definedfor the Abaqus but they have no effect on the results Inthe Coupled Eulerian-Lagrangian method 119880119904-119880119901 equationof state (EOS) was selected to define the water properties inAbaqus The 119880119904-119880119901 equation of state is described in detail inAbaqus 614 documentation and it is used to simulateNavier-Stokes flow when turbulence flow is negligible Becausethe major force imposed on the rigid body in water entryproblems is pressure force this equation of state can beused as an appropriate approximation Parameters such as1198620(speed of sound in water) 119904 (constant coefficient in shockvelocity equation) and Γ0 (material constant in Gruneisenequation of state)were defined for the119880119904-119880119901 equation of stateIn the next step the pounder and water were entered intothe assembly section and the pounder was transferred to thewaters center tangent to its surface In order to simulate a free-falling rigid body into water the pounder was first consideredin a position tangent to the water surface then the secondaryvelocity of pounder at the moment of impact was assignedIn the next step explicit dynamic analysis was selected andthe time period for pounder movement between enteringthe water and reaching the model bed was considered as 05seconds Nlgeom was also activated to control the nonlineareffects of large displacements Then by defining a referencepoint and assigning it to the pounder required outputs suchas displacement and velocity values for the vertical direc-tion were selected Based on its defined algorithm Abaqusconsiders the maximum hardness between the Lagrangianelements (pounder) and Eulerian material (water) The Eule-rian solution method uses the meshes defined for poundermovement between water meshes and Abaqus detects theboundary between the Lagrangian and Eulerian materials

Wall

Wall

Wall Water

Sphere

Initial void region

Freesurface

Figure 2 The computational domain boundary conditions andgrid distribution for the water entry problem of sphere

Considering themaximumhardness between the Lagrangianand Eulerian boundaries the contact between these twomaterials is established using the penalty method Concen-trated mass was assigned to the pounder and its moment ofinertia was defined for 119883 119884 and 119885 In the next step thepounder was defined as a rigid body that can move freelyin all directions Gravitational acceleration (Gravity) wasdefined as minus98ms2 in the 119884 direction and a nonreflectingEulerian boundary was assigned to the sides and bottomsurfaces of the computational water domain This boundarycondition is specific to the Eulerian method and preventswave propagation due to rigid body impact on the waterrsquossurface from returning to the computational domain afterreaching the boundariesWith these boundary conditions thewater is regarded as infinite Assuming that the pounder isdropped from a height 119867 above a free water surface withan initial velocity (1198810) of 0ms its secondary velocity (119881) atthe moment of impact was calculated according to the freefall equation of bodies This velocity was vertically applied tothe pounder Eulerian and void regions were specified usingmaterial assignment and the mesh size was defined for thewater and pounder An Eulerian mesh was selected as thewater mesh and an explicit mesh was selected as the poundermesh using structured techniques Finally the problem wassolved and the results were analyzed

41 Model Geometry Before applying the model to pounderwater entry a three-dimensional numerical model was devel-oped to simulate a transient free-falling wedge impactingthe free surface of water in order to check and calibratethe adopted computational model for simpler problemsThen the accuracy of the numerical model was examinedby comparing the numerical results with the experimen-tal data of Aristoff et al [10] for the water entry of asphere Figure 2 illustrates a schematic for the computationaldomain boundary conditions and grid distribution for

Geofluids 9

Table 1 Equation of state parameters for water

Density (120588119908)(kgm3)

Dynamicviscosity (120578)(kgsm2)

Speed of sound(1198880) (msec) Γ0 1199041000 0001 1450 0 0

Water

Aperture release

Z

Diffuser

Lights

Sphere

Camera

Figure 3 Schematic of the experimental apparatus of Aristoff et al[10]

the sphere water entry problem The dimensions for thecomputational domain are considered to be large enoughso that with further increases no considerable changes areobserved in the numerical resultsThe sphere wasmodeled assolid geometry and a rigid body constraint was subsequentlyapplied to it The fluid domain was modeled as a cubewith dimensions of 30 times 50 times 60 cm3 Mesh refinement wasperformed after assigning it to the computational domaintherefore the Eulerianmesh grid size was gradually decreaseduntil no considerable changeswere observed in the numericalresults The Eulerian domain (water) was modeled usingquarter symmetry and contains 746172 elements The spherewas alsomodeled as a Lagrangian solid using 15000 elementsThe Eulerian domain was divided into two parts includingthe upper and lower regions These regions were defined asvoid and stationary water respectively A solid sphere witha diameter of 254 cm and an initial velocity of 217ms inthe normal direction was placed on the free water surfaceThe sphere can move freely in all directions with six degreesof freedom The physical properties of water used in thenumerical simulations are given in Table 1Γ0 and 119904 are material constants in the Gruneisen equationof state and the constant coefficient in shock velocity equa-tion respectively

5 Validation

A schematic of the experimental apparatus of Aristoff et al[10] is presented in Figure 3 A sphere was held by a cameralike aperture at height 119867 above a water tank The tank haddimensions of 30 times 50 times 60 cm3 and was illuminated by a

bank of twenty 32W fluorescent bulbs A diffuser was usedto provide uniform lighting and care was taken to keep thewater surface free of dust Tank dimensions were selectedso that the walls had negligible effects on the results Thesphere was released from rest and fell towards the waterreaching it with an approximate speed119881 asymp radic2119892ℎThe impactsequence was recorded at 2000 frames per second using ahigh-speed camera The resolution was set to 524 times 1280pixels (px) with a field of view of 1128 times 2755 cm2 yieldinga 4646 pxcm magnification The trajectory of the sphereand its impact speed were determined with subpixel accuracythrough a cross correlation andGaussian peak-fittingmethodyielding position estimates accurate to 0025 px (00005 cm)and impact speeds accurate to plusmn4

Figure 4 illustrates a comparison of cavity shapes betweennumerical simulation and the experimental photographstaken by Aristoff et al [10] for a density ratio of 786 and ansimilar period of time The impact velocity of the sphere was217ms and 119905 = 0 refers to when the spheres center passesthrough the free surface This figure shows the cavity shapesfor both experimental and numerical simulation for similartime periods except that cavity shape for the pinch-off timeof the numerical simulation was added to the pictures

An symmetric air cavity formed behind the sphereafter entering the water Air cavity formation consists ofseveral steps including cavity creation cavity expansioncavity contraction behind the sphere and cavity collapseAs the sphere moves deeper through the water it exerts aforce on surrounding fluid in radial directions and transfersits momentum to the fluid However fluid expansion isconfronted with fluid hydrostatic pressure resistance and asa result the radial flow direction is reversed leading to cavitycontraction and collapse Cavity collapse occurs when thecavity wall moves inward until the moment of pinch-off (119905= 708ms) and the cavity is divided into two separated parts[34] While the lower cavity sticks to the sphere and movesalong with it the upper cavity continues its contraction andmoves towards the waterrsquos surface As shown in Figure 4 thenumerical results and experimental observations are in goodagreement

Abaqus numerical simulation methods efficiently modelwater splashes and jet flow Pictures of these phenomenaare depicted in Figure 4 It was concluded from this figurethat pinch-off in the numerical simulation happens with adelay time compared to the experimental method Increasednumerical errors during solution time are one of the causesfor differences between experimental and numerical results

In Figure 5 the numerical results of this study arecompared with the theoretical and experimental data ofAristoff et al as a displacement-time graph of a sphere indifferent water depths [10] As can be seen the numericalresults are in good agreement with those of the analytics andexperiments However the presently computed sphere centerdepth is lower than that found experimentally It seems thatdue to lack of turbulent modeling ability in Abaqus Eulerianformulation and evidently increasing turbulence in flow bytime there is just less difference between experiment andnumerical results in the early time of simulation

10 Geofluids

Experimental results of Aristoff et al (2010)

Pinch-off

Pinch-offNumerical simulation results

t = 59 ＧＭ 129 ＧＭ 199ＧＭ 269ＧＭ 339ＧＭ 409 ＧＭ 479ＧＭ 549ＧＭ 619 ＧＭ 689 ＧＭ 759ＧＭ

t = 59ＧＭ 129 ＧＭ 199ＧＭ 269ＧＭ 339ＧＭ 409 ＧＭ 479ＧＭ 549ＧＭ 619 ＧＭ 689 ＧＭ 708ＧＭ 759ＧＭ

Figure 4 Comparison of numerical results with experimental photographs of Aristoff et al [10] for the sphere water entry problem

002 004 006 0080Time (s)

minus018

minus016

minus014

minus012

minus01

minus008

minus006

minus004

minus002

0

Disp

lace

men

t (m

)

Present studyExperiment (Aristoff et al 2010)Theory (Aristoff et al 2010)

Figure 5 Comparison of displacement-time graph of sphere inwater depth with the theoretical and experimental data of Aristoffet al [10]

6 Materials and Methods

61 Eulerian Model Definition Figure 6 illustrates a sche-matic of the computational domain boundary conditionsand grid distribution for the water entry problem of aspherical pounder The dimensions of the computationaldomain are considered to be large enough so that withfurther increases no considerable changes are observed inthe numerical results The Eulerian domain was dividedinto the upper and lower regions These regions are definedas void and stationary water respectively In water entryproblems for rigid bodies fluid domain dimensions must

Wall

Wall

Wall Water

Spherical pounder

Initial void region

Free surface

Figure 6 The computational domain boundary conditions andgrid distribution for water entry problem of spherical pounder

be at least eight times the dimensions of the rigid bodyHence in this study the fluid domain was modeled as acube with dimensions of 2 times 2 times 1m3 The pounders wereplaced exactly tangent to the water surface and their velocityat the moment of water surface impact was determinedaccording to the free fall equation of bodies The pounderscan move freely in all directions with six degrees of freedom119880119904-119880119901 equation of state was used to define water materialin the Coupled Eulerian-Lagrangian method This equationof state is useful for Navier-Stokes flow simulations whenturbulence flow is negligible In cases where the impact ofa rigid body on a water surface is investigated the majorforce imposed on water body is the pressure force Thereforewith appropriate approximations this equation of state canbe used to investigate the impact of rigid bodies on watersurfaces

62 EulerianMesh Size As shown in Figure 7 with decreasesin the Eulerian domain mesh size to less than 15 cm thedisplacement-time graph of the spherical pounder showsnegligible changes Therefore to improve analysis time the

Geofluids 11

Table 2 Moment of inertia of pounders with different geometric shapes

Geometric shape of the pounder Cube Cylinder Sphere Pyramid ConeMoment of inertia (cm4) 685 5137 411 3082 411

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

01 02 03 040Time (s)

2 ＝Ｇ

175 ＝Ｇ

15 ＝Ｇ

Figure 7 Comparison of the effect of the Eulerian mesh size ondisplacement-time graph of spherical pounder

Eulerian domain was modeled using quarter symmetry witha mesh size of 15 cm and contains 1538943 elements

63 Lagrangian Model Definition Figure 8 illustratespounder dimensions for different geometric shapes Thesepounders are identical in terms of material mass densityYoungrsquos modulus Poissonrsquos ratio volume mass and dropheight from the water surface The only difference is theirgeometric shapeThese parameters were regarded as identicalin all pounders to investigate the effect of pounder shapeon deepwater displacement and velocity air cavity shapepinch-off time and depth and the effect of water drag forceon velocity depreciation when impacting the model bed

Pounder dimensions were selected so that they had avolume of 523 cm3 and equal mass due to their identicalmaterial and mass density Table 2 illustrates the moment ofinertia for pounders with different geometric shapes

Table 3 illustrates thematerial properties of the poundersThese parameters are regarded as identical for all pounderswith the mentioned geometric shapes

64 Lagrangian Mesh Size Due to the negligible effect ofLagrangian mesh size on the results and to improve analysistime the Lagrangian mesh size for all pounders was set to05 cm

Pyramid Cone

Cube Cylinder Sphere

10 ＝Ｇ10 ＝Ｇ10 ＝Ｇ 10 ＝Ｇ

10 ＝Ｇ10 ＝Ｇ10 ＝Ｇ

10 ＝Ｇ5236 ＝Ｇ

20 ＝Ｇ15708 ＝Ｇ

666 ＝Ｇ

Figure 8 The dimensions of pounders with different geometricshapes

Table 3 Material properties of pounders

Materialtype Mass (kg) Density (120588)

(kgm3)

Youngrsquosmodulus (119864)(kgm2)

Poissonrsquosratio (120592)

Steel 411 7850 21 times 1010 03

65 Boundary Condition Wall boundary conditions wereapplied to the sides and bottom of the Eulerian domain tosimulate a water tank Pounder displacement was not con-strained in any direction and the pounders could freely movein all directions The sides and bottom boundaries of thethree-dimensional Eulerian domain were set as nonreflectingboundaries so that the waves released due to pounder impacton water surface did not return to the computational domainafter reaching these boundaries and being absorbed by themThis boundary condition is written as [37]

119889119901 minus 119901119888119889119906 = 0119889119909119889119905 = minus119888 (41)

where 120588 is density 119888 is the speed of sound in the fluid 119901 ispressure 119906 is velocity perpendicular to the wave and 119909 is thedirection perpendicular to the boundary

66 Drop Height and Impact Velocity of Pounders Afterdropping the pounders from a height (ℎ) above the watersurface the velocity of the pounders at themoment of impactwas determined using

1198812 minus 11988120 = 2119892ℎ (42)

where1198810 is the pounder initial velocity before being droppedfrom height ℎ above the waterrsquos surface 119881 is pounder

12 Geofluids

01 02 03 04 050Time (s)

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

CubeCylinderSphere

PyramidCone

Figure 9 Displacement-time graph of pounders with differentgeometric shapes

secondary velocity when impacting the water surface and 119892is the gravitational acceleration of the Earth

From (42) which expresses a time independent equationfor the free fall of objects the initial velocity and the terminalvelocity of the object drop height and the gravitationalacceleration of the Earth are interdependent in the absenceof time In this numerical simulation the pounders weredropped from a height of 1m above thewaterrsquos surfacewith aninitial velocity (1198810) equal to 0ms Therefore the secondaryvelocity of the pounders when impacting the water surfacewas obtained using

119881 = radic2119892ℎ (43)

Instead of simulating a full dropping event from theinitial position the pounders were placed exactly tangent towaterrsquos surface and their secondary velocity at the momentof impact was determined according to (43) A gravitationalacceleration (gravity) of minus98ms2 was applied in the normaldirection of the pounders Therefore a secondary velocityof 443ms in the normal direction of the waterrsquos surfacecorresponds to the speed that would be obtained for thepounders

7 Results and Discussion

71 Effect of the Geometric Shape of the Pounder on Deep-water Displacement Figure 9 illustrates the displacement-time graph of pounders with different geometric shapes Thegraph continues for the cubical pounder from the momentof water surface impact at time zero up to the depth of 1mat 415ms The displacement-time graph of the cylindrical

pounder indicates that after entering the water it reachesthe model bed in a shorter amount of time than the cubicalpounder The graph for this pounder continues from themoment of water surface impact at time zero up to the depthof 1m at 355ms The linear displacement-time graph forthe spherical and pyramidal pounders indicates the fasterdeepwater movement of these pounders compared to thecubical and cylindrical pounders The deepwater movementtime of the spherical and pyramidal pounders after enteringthe water to themodel bed is 285ms and 265ms respectivelyThe linear displacement-time graph of the conical pounderindicates that it has a faster deepwater movement comparedto other pounders as it takes 235ms to reach the model bedafter entering the water

72 Mechanism of Air Cavity Formation and Pinch-Off dueto Deepwater Movement of Pounders with Different Geomet-ric Shapes Figure 10 illustrates the movement trajectory ofpounders with different geometric shapes from the momentof water impact towhen the pounders reached themodel bedAfter entering the water a symmetric air cavity is formedbehind the pounders Air cavity formation involves severalsteps including air cavity creation and expansion behind thepounder air cavity contraction and air cavity collapse [34]As the pounder moves downward in water depth it imposesa force onto surrounding fluid in its radial directions andtransfers its momentum to the fluid This extension is facedwith the hydrostatic pressure resistance of fluidThe directionof the radial flow is reversed and eventually leads to aircavity contraction and collapse In the other words the aircavity contraction is accelerated until pinch-off occurs andthe air cavity is divided into two distinct parts The upperpart of the air cavity continues its contraction and movestowards the waterrsquos surface whereas the lower part of the aircavity clings to the pounder and moves along with it [34]After impacting the water surface the cubical and cylindricalpounders due to their flat contact surfaces lose some of theirvelocityThis velocity depreciation slows down and continuesuntil pinch-offThe spherical pounder with its curved contactsurface has less velocity depreciation compared to the cubicaland cylindrical pounders However some pounder velocityis always depreciated due to impacting the water surfaceThe velocity depreciation process for the spherical pounderduring deepwater movement continues until pinch-off Afterimpacting the waterrsquos surface the pyramidal pounder with itspointy contact surface has a slight velocity depreciation butcontinues to move deeper into the water with approximatelythe same secondary velocity until pinch-off By imposing thedrag force of water onto the pounders surface its velocityis gradually decreased until it reaches the model bed Afterimpacting the water surface the conical pounder with itspointy contact surface continues to move deeper into thewater with an increasing velocity The increasing velocity ofthe conical pounder during deepwater movement continueseven after pinch-off When the pounder moves towards themodel bed the waters drag force is imposed onto poundersurface causing velocity depreciation The pinch-off time forthe pounders with the above-mentioned shapes is between225 and 250ms

Geofluids 13

t = 0 25ＧＭ 50ＧＭ 100ＧＭ 200ＧＭ 225ＧＭ 250ＧＭ 275ＧＭ 300ＧＭ 325ＧＭ 350ＧＭ150ＧＭCu

beCy

linde

rSp

here

Pyra

mid

Con

e

Figure 10 Air cavity formation due to deepwater movement of pounders with different geometric shapes and pinch-off time and depth

73 Effect of the Pounder Geometric Shape onDeepwater Veloc-ity Figure 11 illustrates the velocity-time graph for pounderswith different geometric shapes The cubical and cylindricalpounders lose some of their secondary velocity at the earlymoments of water entry Hence the velocity-time graph ofthese pounders illustrates great velocity depreciation up to15ms The velocity reduction of the cubical and cylindricalpounders gradually continues until pinch-off The velocityof the spherical pounder decreases up to 45ms after waterentry but its velocity depreciation is less than the cubicaland cylindrical pounders This velocity reduction continuesat a slower rate until pinch-off In the initial moments ofwater entry the pyramidal pounder has a slight reduction insecondary velocity up to 55ms Then velocity depreciationcontinues at a faster rate due to the drag force imposed onto

pounders surface and the velocity-time graph of the pyrami-dal pounder shows a gradual velocity reduction until pinch-off After impacting the water surface the conical poundercontinues its deepwater movement with an increasing veloc-ity rate Due to its very low slope the velocity-time graphindicates an increase in velocity for the conical pounder butdue to the waters drag force its velocity begins to decrease at145ms The velocity-time graph indicates a gradual decreasein velocity for conical pounder and velocity depreciationcontinues until pinch-off After pinch-off pounder velocitiesgradually decrease until they reach themodel bed Eventuallywith a velocity of 388ms and 157ms at the moment ofmodel bed impact the conical and cubical pounders have thehighest and lowest velocity respectively Water drag force isimposed in the opposite direction of poundermovement and

14 Geofluids

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Velo

city

(ms

)

01 02 03 040Time (s)

CubeCylinderSphere

PyramidCone

Figure 11 Velocity-time graph of pounders with different geometricshapes

it has the most and least influence on the cubical and conicalpounders respectively

8 Critical Drop Height of the Pounder

In the dynamic compaction of the seabed determining thecritical drop height of the pounder from the freewater surfaceis an effective parameter for the pounder to reach maximumdeepwater velocity After dropping the pounder from height119867 above the free water surface with an initial velocity (1198810)of 0ms its velocity gradually increases until it reaches thesecondary velocity (119881) at the moment of water impact Afterentering the water pounder velocity is decreased due to waterdrag force According to the free fall equation for objectsincreasing the drop height from the waterrsquos surface leads toincreased pounder velocity at the moment of water surfaceimpact but it does not necessarily lead to an increase inpounder velocity during seabed impact because the criticaldrop height of the pounder is a determining factor ofits maximum deepwater velocity According to Figure 12 aconical pounder with a mass of 411 kg was dropped fromdifferent heights (119867) and its velocity when impacting thewaters was measured according to the time independentequation for the free fall of objects Table 4 illustrates thevelocity of the conical pounder with different drop heightswhen impacting the waterrsquos surface

81 Effect of the Drop Height of the Pounder from WaterSurface on Deepwater Displacement Figure 13 illustrates thedisplacement-time graph of the conical pounder with dif-ferent drop heights from the waterrsquos surface As can beseen increasing the drop height of the pounder leads to

Water levelℎ

Figure 12 Schematic of drop height of the conical pounder fromwater surface

Table 4 Velocity of the conical pounder with different drop heightsat the moment of impacting the water surface

Drop height of the pounderfrom water surface (m)

Pounder velocity at themoment of impacting water

surface (ms)025 22105 313075 3831 443125 49515 542175 5852 626225 66425 7

displacement-time graphs that are closer to each other Infact an increase in the drop height of the pounder from thewaterrsquos surface leads to an increase in its secondary velocitywhen impacting the waterrsquos surface Hence the pounderenters the water with a higher velocity and reaches the modelbed in less time

82 Effect of the Drop Height of the Pounder from WaterSurface on Its Deepwater Velocity Increasing the drop heightof the pounder from the waterrsquos surface leads to an increasein its velocity before impacting the water but the poundervelocity gradually decreases after entering the water due tothe its drag force Figure 14 illustrates the velocity-time graphfor the conical pounder with different drop heights from thewaterrsquos surface When the pounder is dropped from a lowheight due to its low velocity it moves deepwater due to itsweight Moreover because of the low velocity of the pounderwhen entering the water pinch-off occurs in less time and atlower depths and its velocity increases after pinch-off until itreaches terminal velocity Increases in pounder drop heightsincrease water entry velocity Hence it will have greatervelocity depreciation in the first moments of water entry

Figure 15 illustrates the terminal velocity-time graph forthe conical pounderwith different drop heightswhen impact-ing the model bed As can be seen increasing the drop height

Geofluids 15

01 02 03 040Time (s)

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

H = 1Ｇ

H = 075Ｇ

H = 05 Ｇ

H = 025 Ｇ

H = 175Ｇ

H = 15 Ｇ

H = 125 Ｇ

H = 25 Ｇ

H = 225 Ｇ

H = 2Ｇ

Figure 13 Displacement-time graph of the conical pounder withdifferent drop heights

of the pounder from the waterrsquos surface leads to an increasein its terminal velocity when impacting the model bedTheseconditions are dominant as long as the pounder reaches acritical drop height from the water surface but increasingthe pounder drop height higher than the critical drop heightleads to decreased terminal velocity when impacting themodel bed In this numerical simulation after dropping thepounder from a height of 2m above the waterrsquos surface itssecondary velocity was 626ms when impacting the watersurface After entering the water the secondary velocity ofthe pounder decreased until it impacts the model bed witha terminal velocity of 454ms Therefore a drop height of2m from above the waterrsquos surface has the highest terminalvelocity when the pounder impacts themodel bed Increasingthe drop height bymore than 2m leads to decreased pounderterminal velocity when impacting the model bed The heightwhich has the greatest effect on pounder terminal velocitywhen impacting the model bed is called the critical dropheight Increasing the drop height of the pounder to greaterthan its critical drop height leads to decreased terminalvelocity when impacting the model bed

Figure 16 illustrates the secondary and terminal velocityof the conical pounder with different drop heights As canbe observed increasing the drop height of the pounder fromwaterrsquos surface leads to increased differences between thesecondary and terminal velocities of the pounder Increasingthe drop height to more than the critical drop height leadsto increased pounder secondary velocity when impactingthe waterrsquos surface and decreased terminal velocity whenimpacting the model bed Hence the difference betweenthe secondary and terminal velocities of the pounder is

01 02 03 040Time (s)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)H = 175Ｇ

H = 125 Ｇ

H = 075Ｇ

H = 025 Ｇ H = 225 Ｇ

H = 1Ｇ

H = 05 Ｇ

H = 25 Ｇ

H = 2Ｇ

H = 15 Ｇ

Figure 14 Velocity-time graph of the conical pounder with differentdrop heights

minus430

minus447 minus454minus415

minus426 minus380

minus367minus355

minus332minus310

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Term

inal

velo

city

(ms

)

01 02 03 040Time (s)

Figure 15 Terminal velocity of conical pounder with different dropheights versus time

not reasonable and a significant amount of the pounderssecondary velocity is depreciated when entering the waterThe great difference between the secondary and terminalvelocities of the pounder reveals the necessity of determiningthe critical drop height of the pounder before dynamic seabedcompaction operations Increasing the pounder drop heightto be greater than the critical drop height not only hasa reverse result on pounder velocity when impacting theseabed but it also requires special equipment such as lengthy

16 Geofluids

Secondary velocityTerminal velocity

025 05 075 1 125 15 175 2 225 250Drop height (m)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)

Figure 16 Secondary and terminal velocities of conical pounderversus drop heights

1 2 3 4 5 6 7 80Impact velocity (ms)

0

005

01

015

02

025

03

Pinc

h-off

tim

e (s)

Figure 17 Pinch-off time versus impact velocity of conical pounder

cranes and huge barges leading to increased implementationcosts

Figure 17 illustrates pinch-off time based on the sec-ondary velocity of the pounder at the moment of watersurface impact As can be seen pinch-off time is a weakfunction of pounder impact velocity and increasing impactvelocity leads to a slight reduction in pinch-off time Thisbehaviour was also observed in the analytical and numericalresults found by Lee et al [40]

Figure 18 illustrates pinch-off depth based on the sec-ondary velocity of pounder at the moment of water surfaceimpact As can be seen the pinch-off depth is a linear

1 2 3 4 5 6 7 80Impact velocity (ms)

minus12

minus1

minus08

minus06

minus04

minus02

0

Pinc

h-off

dep

th (m

)

Figure 18 Pinch-off depth versus impact velocity of conicalpounder

function of pounder impact velocity These results are in linewith the analytical equations of Lee et al [40]

9 Conclusions

In this study the water entry problem of three-dimensionalpounders with different geometric shapes of cube cylindersphere pyramid and cone was numerically simulated bycommercial finite element code Abaqus 614-2 An explicitdynamic analysis method was employed to model the fluid-structure interaction using a Coupled Eulerian-Lagrangian(CEL) method Before applying the model to water entryof the pounders a three-dimensional numerical model wasdeveloped to simulate a transient free-falling wedge impact-ing the free surface of the water in order to calibrate theadopted computationalmodelThe accuracy of the numericalmodel was examined by comparing numerical results withthe experimental data of Aristoff et al The good agreementbetween numerical simulation results with the theoreticaland experimental data of Aristoff et al demonstrates thecapability of the employed numerical methodology to sim-ulate the water entry problem for three-dimensional geome-tries with acceptable accuracy After water entry by a pounderdropped fromaheight of 1m a symmetric air cavity is formedbehind the pounder As the pounder moves downward inwater depth it imposes a force onto surrounding fluid inthe radial directions and transfers its momentum to thefluid This extension is faced with the hydrostatic pressureresistance of the fluid causing the direction of the radialflow to be reversed leading to air cavity contraction andcollapse The results show that the water drag force had thehighest and lowest effect on the cubical and conical poundersrespectively and increasing the drop height to greater thanthe critical drop height leads to increased pounder velocitywhen impacting the waterrsquos surface and decreased velocitywhen impacting the model bed Pinch-off time is a very weak

Geofluids 17

function of pounder velocity when impacting the waterrsquossurface but the pinch-off depth increases linearly along withincreased pounder impact velocities It is recommended thatthe critical drop height for the pounder is determined beforedynamic seabed compaction operations and the adjustmentof crane heights to this height In addition it is also recom-mended that perforated pounders be used in order to reducevelocity depreciation when impacting the waterrsquos surface andto reduce the effect of water drag force

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors warmly appreciate the cooperation and guidanceof Dr B Hamidi and Dr A Mehrabadi in this study

References

[1] B Hamidi J-M Debats H Nikraz and S Varaksin ldquoOffshoreground improvement recordsrdquo Australian Geomechanics Jour-nal vol 48 no 4 pp 111ndash122 2013

[2] SWatanabe ldquoResistance of impact onwater surface part I-cone Institute of Physical andChemical ResearchrdquoTokyo vol 12 pp251ndash267 1930

[3] S Watanabe ldquoResistance of impact on water surface part II-cone (continued)rdquo Institute of Physical and Chemical Researchvol 14 pp 153ndash168 1930

[4] A May and J C Woodhull ldquoDrag coefficients of steel spheresentering water verticallyrdquo Journal of Applied Physics vol 19 no12 pp 1109ndash1121 1948

[5] A May ldquoEffect of surface condition of a sphere on its water-entry cavityrdquo Journal of Applied Physics vol 22 no 10 pp 1219ndash1222 1951

[6] AMay ldquoVertical entry ofmissiles into waterrdquo Journal of AppliedPhysics vol 23 no 12 pp 1362ndash1372 1952

[7] A P New T S Lee and H T Low ldquoImpact loading and waterentrance characteristics of prismatic bodiesrdquo in Proceedings ofthe third international offshore and polar engineering conferencepp 282ndash287 1993

[8] M Anghileri and A Spizzica ldquoExperimental validation of finiteelement models for water impactsrdquo in Proceedings of the secondinternational crash usersrsquo seminar Cranfield Impact Centre LtdEngland UK 1995 second international crash users seminar

[9] A Engle andR Lewis ldquoA comparison of hydrodynamic impactsprediction methods with two dimensional drop test datardquoMarine Structures vol 16 no 2 pp 175ndash182 2003

[10] J M Aristoff T T Truscott A H Techet and J W M BushldquoThe water entry of decelerating spheresrdquo Physics of Fluids vol22 no 3 Article ID 009002PHF pp 1ndash8 2010

[11] T Von Karman The impact of seaplane floats during landingNational Advisory Committee for Aeronautics NACA TN no321 1929

[12] HWagner ldquoPhenomena associatedwith impacts and sliding onliquid surfacesrdquo Journal of Applied Mathematics and Mechanicsvol 12 pp 193ndash215 1932

[13] W S Vorus ldquoA flat cylinder theory for vessel impact and steadyplaning resistancerdquo Journal of Ship Research vol 40 no 2 pp89ndash106 1996

[14] T Miloh ldquoOn the initial-stage slamming of a rigid sphere in avertical water entryrdquo Applied Ocean Research vol 13 no 1 pp13ndash43 1991

[15] J M Aristoff and JW Bush ldquoWater entry of small hydrophobicspheresrdquo Journal of Fluid Mechanics vol 619 pp 45ndash78 2009

[16] P Ghadimi A Dashtimanesh and S R Djeddi ldquoStudy of waterentry of circular cylinder by using analytical and numericalsolutionsrdquo Journal of the Brazilian Society ofMechanical Sciencesamp Engineering vol 34 no 3 pp 225ndash232 2012

[17] A TassinD J PiroAAKorobkinK JMaki andM J CookerldquoTwo-dimensional water entry and exit of a body whose shapevaries in timerdquo Journal of Fluids and Structures vol 40 pp 317ndash336 2013

[18] E Yao H Wang L Pan X Wang and R Woding ldquoVerticalwater-entry of bullet-shaped projectilesrdquo Journal of AppliedMathematics and Physics vol 02 no 06 pp 323ndash334 2014

[19] M-S Park Y-R Jung and W-G Park ldquoNumerical study ofimpact force and ricochet behavior of high speed water-entrybodiesrdquo Computers amp Fluids vol 32 no 7 pp 939ndash951 2003

[20] D Battistin and A Iafrati ldquoHydrodynamic loads during waterenergy of two-dimensional and axisymmetric bodiesrdquo Journalof Fluids and Structures vol 17 no 5 pp 643ndash664 2003

[21] A A Korobkin and M Ohkusu ldquoImpact of two circular platesone of which is floating on a thin layer of liquidrdquo Journal ofEngineering Mathematics vol 50 no 4 pp 343ndash358 2004

[22] K M Kleefsman G Fekken A E Veldman B Iwanowski andB Buchner ldquoA volume-of-fluid based simulation method forwave impact problemsrdquo Journal of Computational Physics vol206 no 1 pp 363ndash393 2005

[23] Y Kim Y Kim Y Liu and D K P Yue ldquoOn the water-entryimpact problem of asymmetric bodiesrdquo in Proceedings of the9th International Conference on Numerical Ship Hydrodynamics(NSH rsquo07) August 2007

[24] Q Yang and W Qiu ldquoNumerical solution of 2D slammingproblem with a CIPmethodrdquo in Proceedings of the InternationalConference on Violent Flows Research Institute for AppliedMechanics Kyushu University Japan 2007

[25] A C Fairlie-Clarke and T Tveitnes ldquoMomentum and gravityeffects during the constant velocitywater entry ofwedge-shapedsectionsrdquo Ocean Engineering vol 35 no 7 pp 706ndash716 2007

[26] Q Yang andW Qiu ldquoNumerical simulation of water impact for2D and 3D bodiesrdquo Ocean Engineering vol 43 pp 82ndash89 2012

[27] G Wu ldquoNumerical simulation for water entry of a wedge atvarying speed by a high order boundary element methodrdquoJournal of Marine Science and Application vol 11 no 2 pp 143ndash149 2012

[28] M Ahmadzadeh B Saranjam A Hoseini Fard and A RBinesh ldquoNumerical simulation of sphere water entry prob-lem using Eulerian-Lagrangian methodrdquo Applied MathematicalModelling vol 38 no 5-6 pp 1673ndash1684 2014

[29] V-T Nguyen D-T Vu W-G Park and Y-R Jung ldquoNumericalanalysis of water impact forces using a dual-time pseudo-compressibility method and volume-of-fluid interface trackingalgorithmrdquo Computers amp Fluids vol 103 pp 18ndash33 2014

[30] S Kim and N Kim ldquoIntegrated dynamics modeling forsupercavitating vehicle systemsrdquo International Journal of NavalArchitecture and Ocean Engineering vol 7 no 2 pp 346ndash3632015

18 Geofluids

[31] V-T Nguyen D-T Vu W-G Park and C-M Jung ldquoNavier-Stokes solver for water entry bodies with moving Chimera gridmethod in 6DOF motionsrdquo Computers amp Fluids vol 140 pp19ndash38 2016

[32] M Mirzaei M Eghtesad and M Alishahi ldquoPlaning forceidentification in high-speed underwater vehiclesrdquo Journal ofVibration and Control vol 22 no 20 pp 4176ndash4191 2016

[33] G Qiu S Henke and J Grabe ldquoApplication of a CoupledEulerian-Lagrangian approach on geomechanical problemsinvolving large deformationsrdquoComputersampGeosciences vol 38no 1 pp 30ndash39 2011

[34] M R Erfanian M Anbarsooz N Rahimi M Zare andM Moghiman ldquoNumerical and experimental investigationof a three dimensional spherical-nose projectile water entryproblemrdquo Ocean Engineering vol 104 pp 397ndash404 2015

[35] W M Isbell F H Shipman and A H Jones ldquoHugoniotequation of state measurements for eleven material to fiveMegabars Materials and Structural Laboratory ManufacturingDevelopments General Motors Technical centerrdquo Tech RepMichigan 1968

[36] M A Jandron R C Hurd J L Belden A F Bower W Fennelland T T Truscott ldquoModeling of hyperelastic water-skippingspheres using AbaqusExplicitrdquo in Proceedings of the SIMULIACommunity Conference 2014

[37] H Forouzani B Saranjam R Kamali and A Abdollahi-far ldquoElasto-plastic time dependent impact analysis of highspeed projectile on water surfacerdquo Journal of Solid and FluidMechanics vol 3 pp 281ndash298 2016

[38] M R Erfanian andMMoghiman ldquoNumerical and experimen-tal investigation of a projectile water entry problem and studyof velocity effect on time and depth of pinch-offrdquo Journal ofModares Mechanical Engineering vol 15 pp 53ndash60 2015

[39] A Sillem Feasibility study of a tire hydroplaning simulation in afinite element code using a coupled Eulerian-Lagrangian method[MSc thesis] 2008

[40] M Lee R G Longoria and D E Wilson ldquoCavity dynamics inhigh-speed water entryrdquo Physics of Fluids vol 9 no 3 pp 540ndash550 1997

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Geofluids 3

Tveitnes [25] conducted a study on impact of wedge-shapedparts with a water surface at a constant velocity In their studyComputational Fluid Dynamic (CFD) analysis was used todetermine the momentum of added mass momentum offlow and gravityrsquos effect during the entry of wedge-shapedobjects at a constant velocity and a dead angle of 5 to 45degrees into water A numerical solution based on finitevolume method was implemented using Fluent software tosolve equations of mass conservation and momentum toobtain flow fields Hydrodynamic forces were not signifi-cantly changed by the effect of gravity on the flow fieldsYang and Qiu [26] computed slamming forces on two-and three-dimensional bodies entering calm water with aconstant velocity The nonlinear water entry problem wasgoverned by Navier-Stokes equations and solved using a CIP-based finite difference method on a fixed Cartesian gridWu [27] developed a high order boundary element methodfor the complex velocity potential problem He focused onapplication of the water entry problem for a wedge withvarying speeds His method ensured not only the continuityof the potential at the nodes of each element but also thevelocity allowing it to be applied to a variety of velocitypotential problems The continuity of the velocity achievedherein was particularly important for this kind of nonlinearfree surface flow problem because when the time steppingmethod is used the free surface is updated using the velocityobtained at each node and the accuracy of the velocityis therefore crucial Ahmadzadeh et al [28] developed athree-dimensional numerical model to simulate a transientfree-falling sphere impacting a free water surface using theCoupled Eulerian-Lagrangian formulations included in thecommercial software Abaqus with consideration for impor-tant parameters such as viscosity compressibility inertiaand variable downward velocity Their study was successfulin determining the shape of the air cavity displacementof sphere versus time pinch-off time and depth comparedto the experimental results found by Aristoff et al [10]Nguyen et al [29] studied the water impact of variousthree-dimensional geometries namely a hemisphere twocones and a free-falling wedge with an implicit algorithmbased on a dual-time pseudo-compressibility method Flowfields of incompressible viscous fluids were solved usingunsteady RANS equations A second-order VOF interfacetracking algorithm was developed in a generalized curvi-linear coordinate system to track the interface between thetwo phases in the computational domain Sensitivity analysiswith respect to the spatial grid resolution and the time stepwas performed to assess the accuracy of the results Freesurface deformation pressure coefficients impact velocitiesand vertical acceleration during impact were compared withavailable experimental data and asymptotic theories showinggood agreement at a reasonable computational cost S Kimand N Kim [30] performed integrated dynamic modelingof supercavitating vehicles and established 6-DOF equationsby defining hydrodynamic forces and moments These sim-ulations also demonstrated the instability of supercavitatingvehicles caused by the asymmetric immersion depth ofcoupled horizontal or vertical fins Nguyen et al [31] used amovingChimera gridmethod to predict the real-timemotion

of water entry bodies by combining the 6-DOF rigid bodymotion model and the numerical calculation of the multi-phase flow field by which the coupled effect of supercavityand moving body was obtained Mirzaei et al [32] held theidea that existing planning force models weaken nonlinearinteractions among the solid liquid and gaseous phases andare often too simple and therefore incorrectThey establishedthe 6-DOF motion model for supercavitating vehicles byredefining the spatial planning force and developed a hybridcontrol scheme based on online planning force identifica-tion which increased the stabilization of supercavitatingvehicles

In this study the water entry problem of three-dimensional pounders with different geometric shapesincluding cube cylinder sphere pyramid and cone wasinvestigated numerically using commercial finite elementcode Abaqus 614-2 and an explicit dynamic analysis methodwas employed to model fluid-structure interactions using aCoupled Eulerian-Lagrangian (CEL) formulation [33] Thisinvestigation focused on the problem of rigid pounderstouching the free surface of a stationary viscous fluid at time119905 = 0 with a variable velocity and finding for 119905 gt 0the shape of cavity pinch-off time and depth of the rigidpounders It should be noted that the fluid was assumedto be viscous and compressible Numerical simulations ofsolid-fluid interactions were performed using the CoupledEulerian-Lagrangian (CEL) formulation and the remeshingtechniques available in Abaqus 614-2 software [34] Thepounder was considered as a rigid solid and its mesh wascreated in Lagrangian form The water was regarded as acompressible fluid and its mesh was created in Eulerianform After validating the numerical results with recentexperimental data the effect of the pounder shape on itsdeepwater displacement and velocity was investigated Thenthe effect of increasing the pounder drop height from thewaterrsquos surface on displacement submersion velocity pinch-off time and depth was investigated The critical drop heightof the pounder which is indicative of the maximum effectof the pounders drop height from the waterrsquos surface ondeepwater movement was finally determined

2 Coupled Eulerian-Lagrangian (CEL) Method

Eulerian materials and Lagrangian elements can interactusing the Eulerian-Lagrangian contact Simulations thatinclude this type of contact are called Coupled Eulerian-Lagrangian (CEL) analyses [28] The CEL method in Abaqusprovides engineers and scientists with the ability to simulatea class of problems where the interaction between structuresand fluids is important This capability does not rely on thecoupling of multiple software products but instead solvesthe fluid-structure interaction (FSI) simultaneously withinAbaqusThis powerful easy to use feature of AbaqusExplicitgeneral contact enables fully coupled multiphysics simula-tions such as fluid-structure interactions The attractivenessof this method is that the fluid is modeled in an Eulerianframe while the structure can be modeled in a Lagrangianframe as is typical for solid mechanics applications Specif-ically in CEL the governing equation of the solid uses

4 Geofluids

the conservation of momentum while the governing equa-tion for the fluid uses a general Navier-Stokes form [36]The Eulerian mesh is typically a simple rectangular gridof elements extending well beyond the Eulerian materialboundaries giving the material space in which to move anddeform If any Eulerian material moves outside the Eulerianmesh it is lost from the simulation The Eulerian materialboundary must therefore be computed during each timeincrement and it generally does not correspond to an elementboundary The interaction between the structure and thewater is handled remarkably well by the solver using thegeneral contact algorithm In traditional Lagrangian analysisnodes are fixed within the material and elements deform asthe material deforms Lagrangian elements are always 100full of a single material so the material boundary coincideswith the element boundary By contrast in Eulerian analysisnodes are fixed in space andmaterial flows through elementsthat do not deform Eulerian elements may not always be100 full ofmaterial andmanymay be partially or completelyvoid

The Eulerian implementation in AbaqusExplicit is basedon the Volume Of Fluid (VOF) method In numericalanalyses using this CEL method the Eulerian material istracked as it flows through the mesh by computing itsEulerian Volume Fraction (EVF) Each Eulerian element isdesignated a percentage which represents the portion ofthat element filled with a material If an Eulerian element iscompletely filled with a material its EVF is 1 and if there isno material in the element its EVF is 0 Eulerian elementsmay simultaneously contain more than one material If thesum of all material volume fractions in an element is less thanone the remainder of the element is automatically filled witha void material that has neither mass nor strength Contactbetween Eulerian materials and Lagrangian materials isenforced using a general contact that is based on a penaltycontact methodThe Lagrangian elements can move throughthe Eulerian mesh without resistance until they encounteran Eulerian element filled with material (EVF = 0) A CELmethod that attempts to capture the advantages of boththe Lagrangian and the Eulerian method is implemented inAbaqus [33]

21 Time Integration Scheme TheCELmethod implementedin AbaqusExplicit uses an explicit time integration schemeThe central difference rule is employed for the solution ofa nonlinear system of differential equations The unknownsolution for the next time step can be found directly fromthe solution of the previous time step so no iteration isneeded Another advantage of choosing an explicit timeintegration scheme for the present problem is robustnessregarding difficult contact conditions [33]

22 Formulation of Eulerian-Lagrangian Contact By usingthe CEL method the contact between the Eulerian domainand Lagrangian domain is discretized using general contactwhich is based on the penalty contactmethodThismethod isless stringent compared to the kinematic contactmethodThepenalty contact method is used in Abaqus to handle contactin CEL analysis [33]

23 Governing Equations in Lagrangian Method The threefundamental conservation equations are the conservation ofmass the conservation of momentum and the conservationof energy The Lagrangian formulation trivially enforces theconservation of mass by defining the density as the ratio ofthe element mass to the current volume The conservation ofenergy in the absence of heat conduction is enforced locallyat selected points within each element or control volumeDifferent finite elements and finite difference methods aretherefore defined in terms of how they handle the conserva-tion of momentum

Finite element methods use the weak form of the equa-tion which is also known as the principle of virtual workor the principle of virtual power within the solid mechanicscommunity In this method the principle of virtual work ispresented as

int119881[120588120575119909 + 120590 120575120576] 119889119881 = int

119881120588119887120575119909 119889119881 + int

119878120591

120591120575119909 119889119878 (1)

where 120588 is density 120590 is the stress 120576 is strain 120591 is shearstress 119881 is the volume of the domain and 119878 is the surfacewhich is divided into 119878120591 and 119878119906 which is where traction anddisplacement boundary conditions are imposed respectivelyEquation (1) can be rewritten as

int119881120588120575119909 119889119881 = 119865ext minus 119865int (2)

where 119865int represents internal force and 119865ext representsexternal force which is expressed as

119865int = int119881120590 120575120576 119889119881

119865ext = int119881120588119887120575119909 119889119881 + int

119878120591

120591120575119909 119889119878(3)

In the finite element method the coordinates velocityacceleration and virtual displacement are interpolated fromtheir values at the nodes using shape functions119873119860 where 119860is the node number They are expressed as

119909 = sum119860

119873119860119909119860 = sum119860

119873119860119860 = sum119860

119873119860119860120575119909 = sum

119860

119873119860120575119909119860

(4)

Substituting the relations from (2) into the inertial termin the weak form gives

int119881120588120575119909 119889119881 = int

119881120588sum119860

119873119860119860sum119861

119873119861120575119909119861119889119881= sum119860

sum119861

[int119881120588119873119860119873119861119889119881] 119860120575119909119861

(5)

Geofluids 5

The external force vector is also derived by a directsubstitution of the appropriate relations from (6)

119865ext = int119881120588119887120575119909 119889119881 + int

119878120591

120591120575119909 119889119878 (6)

119865ext = int119881120588119887sum119861

119873119861120575119909119861119889119881 + int119878120591

120591sum119861

119873119861120575119909119861119889119878 (7)

119865ext = sum119861

int119881120588119887119873119861119889119881 + int

119878120591

120591119873119861119889119878 120575119909119861 (8)

The internal force vector requires the evaluation of thevirtual strain 120575 Again using the relations from (9)

120575120576119894119895 = 12 ( 120597119909119894120597119909119895 +120597119909119895120597119909119894 ) (9)

120575120576119894119895 = sum119861

12 (120597119873119861120597119909119895 120575119909119861119894 +120597119873119861120597119909119894 120575119909119861119895) (10)

These relations may be put in matrix form which definesthe 119861matrix 119861 as

120576120575 = 119861120575119909 (11)

and the internal force may be written as

119865int = int119881120590 120575120576 119889119881

119865int = sum119861

int119881119861119861120590119889119881 120575119909119861

(12)

where 119861119861 is the section of the 119861matrix associated with node119861 To simplify the mass matrix is defined as

119872119888119860119861 = [int119881120588119873119860119873119861119889119881] (13)

where 119872119888 is called the consistent mass matrix because itsderivation is consistent with the finite element method Allthe terms in the weak form have now been written in terms ofunknown coordinates and virtual displacements Collectingthe terms gives

sum119861

119861minus 1119872119861 (int119881 120588119887119873119861119889119881 + int

119878120591

120591119873119861119889119878 minus int119881119861119861120590119889119881)

sdot 120575119909119861 = 0(14)

Since the virtual displacements are arbitrary the 119861 valueis in accordance with

119861 = 1119872119861 (int119881 120588119887119873119861119889119881 + int119878120591

120591119873119861119889119878 minus int119881119861119861120590119889119881) (15)

24 Governing Equations in Eulerian Method For thedescription of fluid flows usually Eulerian formulation isemployed because we are usually interested in the propertiesof the flow at certain locations in the flow domain InEulerian formulation an independent equation is used toexpress conservation of mass Inertia forces also includedisplacement sentences that when present in numericalsolution lead to a matrix with nonsymmetric coefficients thatdepends on calculated velocities Eulerian formulation allowsfor the use of simple strain and stress rates and velocitycalculations instead of displacement values (in the ultrasmalldisplacement analysis) [36 37] Linear viscous isotropic fluidsare known as Newtonian fluids which are by far the mostimportant fluids for practical applications Newtonian fluidsare characterized by the followingmaterial law for theCauchystress tensor 119879

119879119894119895 = 120583( 120597V119894120597119909119895 +120597V119895120597119909119894 minus

23 120597V119896120597119909119896 120575119894119895) minus 119901120575119894119895 (16)

where the velocity vector is V119894 with respect to Cartesiancoordinate 119909119894 the pressure is 119901 the dynamic viscosity is 120583and the Kronecker symbol is 120575119894119895 With this the conservationlaws for mass momentum and energy take the form of

120597119901120597119905 +120597 (119901V119894)120597119909119894 = 0 (17)

120597 (119901V119894)120597119905 + 120597 (119901V119894V119895)120597119909119895= 120597120597119909119895 [120583(

120597V119894120597119909119895 +120597V119895120597119909119894 minus

23 120597V119896120597119909119896 120575119894119895)] minus120597119901120597119909119895 + 120588119891119894

(18)

120597 (119901119890)120597119905 + 120597 (119901V119894119890)120597119909119894= 120583[ 120597V119894120597119909119895 (

120597V119894120597119909119895 +120597V119895120597119909119894) minus 23 ( 120597V119894120597119909119894)

2] minus 119901 120597V119894120597119909119894+ 120597120597119909119894 (119896

120597119879120597119909119894) + 120588119902

(19)

where 120588 is the fluid density 119890 is the specific internal energyand119891119894 and 119902 are external forces and heat sources respectivelyIn energy balance (19) for the heat flux vector ℎ119894 Fourierrsquoslaw with thermal conductivity 120581119891 is used that is a constantspecific heat capacity is assumed and the work done bypressure and friction forces is neglected

ℎ119894 = minus120581119891 120597119879120597119909119894 (20)

System (17)ndash(19) has to be completed by two equations ofstate of the form

119901 = 119901 (120588 119879) 119890 = 119890 (120588 119879) (21)

6 Geofluids

which define the thermodynamic properties of the fluid thatis the thermal and caloric equation of state respectivelyWhen defining the viscosity for the material it is treated as aNewtonian fluid meaning that the viscosity depends only ontemperature This is the default setting in Abaqus Since thetemperature is constant throughout the analysis the viscosityalso is constant

25 Explicit Time-Dependent Solution There are implicitand explicit algorithms for dynamic analysis In the implicitmethod the Newmark method is used the matrices need tobe reversed andmore implementation time is requiredHow-ever this method enjoys high accuracy and its convergenceto linear problems is guaranteed In the explicit method thematrices do not need be reversed and the analysis speed ismuch higher Convergence in the explicit method is achievedif the time step is less than the smallest element divided bythe speed of sound in the material [37]

The solution was advanced in time using the centraldifference method which is also called the Verlet or leap frogmethod It is based on the central difference approximationfor evaluating the derivative of a function 119891(119905) at 119905

119889119891 (119905)119889119905 = 119891 (119905 + (12) Δ119905) minus 119891 (119905 minus (12) Δ119905)Δ119905 (22)

where Δ119905 is the time stepUsing (15) acceleration in time 119905 can be written as

119905 = 119865119905119872119861 (23)

Now using (22) for velocity-acceleration velocity at time119905 + (12)Δ119905 is obtained as

119905+(12)Δ119905 = 119905minus(12)Δ119905 + 119905Δ119905 (24)

Then using (22) for velocity-displacement displacementat time 119905 + Δ119905 is obtained as

119909119905+Δ119905 = 119909119905 + 119905+Δ119905Δ119905 (25)

The central difference method has its step size limited bystability considerations like all explicit integration methodsand the stable time step size was limited to

Δ119905 le 119897119888 (26)

where 119897 is the length of the shortest element and 119888 is the speedof sound

3 Finite Element Modeling

31 Equation of Energy and Hugoniot Curve The equationfor conservation of energy equates the increase in internalenergy per unit mass 119864119898 to the rate at which work is beingdone by stresses and the rate at which heat is being added Inthe absence of heat conduction the energy equation can bewritten as

120588120597119864119898120597119905 = (119901 minus 119901119887]) 1120588 120597120588120597119905 + 119904 119890 + 120588 (27)

PH

PHI1

PHI0

11 101

Figure 1 Schematic representation of a Hugoniot curve [35]

where 119901 is the pressure stress defined as positive in compres-sion 119901119887] is the pressure stress due to the bulk viscosity 119904 isthe deviator stress tensor 119890 is the deviator part of strain rateand is the heat rate per unit mass According to Abaqusdocumentation flow modeling of compressible fluid can beachieved using the linear 119880119904-119880119901 form of the Mie-Gruneisenequation of state The equation of state assumes pressure as afunction of the current density 120588 and the internal energy perunit mass 119864119898 according to

119901 = 119891 (120588 119864119898) (28)

which defines all the equilibrium states that can exist in amaterial The internal energy can be eliminated from (27)to obtain a pressure (119901) versus volume (119881) relationship orequivalently a 119901 versus 1120588 relationship that is unique tothe material described by the equation of state model Thisequation is uniquely dependent on the material defined bythe equation of state This unique relationship is called theHugoniot curve and it is the locus of 119901-119881 states achievablebehind a shockTheHugoniot pressure119901119867 is only a functionof density and can be defined in general from fittingexperimental data An equation of state is said to be linearin energy when it can be written as

119901 = 119891 + 119892119864119898 (29)

where119891(120588) and119892(120588) are only functions of density and dependon the particular equation of state model Figure 1 illustratesa schematic expression of a Hugoniot curve [35]

32 Mie-Gruneisen Equation of State As mentioned previ-ously an equation of state is used to express the behaviourof Eulerian materials For a Mie-Gruneisen equation of statefor linear energy the most common form is written as [38]

119901 minus 119901119867 = Γ120588 (119864119898 minus 119864119867) (30)

where 119901119867 and 119864119867 are the Hugoniot pressure and specificenergy (per unit mass) and are only functions of density andΓ is the Gruneisen ratio defined as

Γ = Γ0 1205880120588 (31)

Geofluids 7

where Γ0 is a material constant and 1205880 is the reference densityHugoniot energy 119864119867 is related to the Hugoniot pressure 119901119867by

119864119867 = 11990111986712057821205880 (32)

where 120578 is the nominal volumetric compressive strain writtenas

120578 = 1 minus 1205880120588 (33)

Elimination of Γ and119864119867 from the above equations yields

119901 = 119901119867 (1 minus Γ01205782 ) + Γ01205880119864119898 (34)

The equation of state and the energy equation representcoupled equations for pressure and internal energy Abaqussolves these equations simultaneously at each material pointusing an explicit method

33 Linear 119880119904-119880119901 Hugoniot Form Normally the 119880119904-119880119901 for-mulation of equation of state (EOS) is used to simulate shocksin solid materials In this study it is used to define fluidmaterials A common fit to the Hugoniot data is given by

119901119867 = 120588011986220120578(1 minus 119904120578)2 (35)

where 1198880 and 119904 define the linear relationship between thelinear shock velocity 119880119904 and the particle velocity 119880119901 using

119880119904 = 1198880 + 119904119880119901 (36)

The Mie-Gruneisen EOS is normally used for processeswith high pressures and high sound velocities The relation-ship between density and pressure is governed by an EOSmodel A linear 119880119904-119880119901 Hugoniot form of the Mie-Gruneisenequation of state is used to evaluate pressure in the waterin the simulation Using this implementation the water ismodeled as a compressible viscous fluid With the aboveassumptions the linear 119880119904-119880119901 Hugoniot form is written as

119901 = 120588011986220120578(1 minus 119904120578)2 (1 minusΓ01205782 ) + Γ01205880119864119898 (37)

where 120588011988820 is equivalent to the elastic bulk modulus at smallnominal strains

When modeling water Abaqus prescribes 119904 = 0 andΓ0 = 0 Filling this simplifies the relation between pressure119901 and density 120588 to an inversely proportional equation and asimplification of the EOS This simplification suffices for themodeling of water [39]

1199011003816100381610038161003816119904=0andΓ=0 = [ 120588011986220120578(1 minus 119904120578)2 (1 minusΓ01205782 ) + Γ01205880119864119898]

119904=0andΓ=0

= 120588011988820120578(38)

Recalling that 120578 = 1 minus 1205880120588 we get

119901 = 120588011986220 (1 minus 1205880120588 ) (39)

When defining a viscosity for thematerial it is treated as aNewtonian fluid meaning that viscosity only depends on thetemperature This is the default setting in Abaqus Since thetemperature is constant throughout the analysis the viscosityis also constant

There is a limiting compression given by the denominatorof this form of the equation of state The denominator of (34)should not be equal to zero therefore an extreme value isdefined for 120588 and 120578 as

120578lim = 1119904 120588lim = 1199041205880119904 minus 1

(40)

At this limit there is a tensile minimum thereafternegative sound speeds are calculated for the material

The linear Hugoniot form 119880119904-119880119901 equation of state canbe employed to model laminar viscous flows governed byNavier-Stokes equations The volumetric response is gov-erned by the equations of state where the bulk modulus actsas a penalty parameter for the constraint [38] The Eulerianpart (water) is simulated using the linear 119880119904-119880119901 Hugoniotform of the Mie-Gruneisen equation of state

The solving method in Eulerian equations is based onthe division of the governing equations In the Lagrangianmethod the nodes are considered to be fixed relative to thematerial at any time interval and elements are deformed withthe material while in the Eulerian method element defor-mation is calculated at any time interval but deformation isapplied as a remeshingThematerial is also displaced betweeneach element and its neighboring elements at any time inter-val Eulerian-Lagrangian contact allows the Eulerian mate-rials to be combined with traditional nonlinear Lagrangiananalyses In the Eulerian-Lagrangian contact the Lagrangianstructure occupies void regions inside the Eulerian meshThe contact algorithm automatically computes and tracks theinterface between the Lagrangian structure and the Eulerianmaterials A great benefit of this method is that there isno need to generate a conforming mesh for the Euleriandomain In fact a simple regular grid of Eulerian elementsoften yields the best accuracy If the Lagrangian body isinitially positioned inside the Eulerian mesh it must beensured that the underlying Eulerian elements contain voidsafter material initialization During analysis the Lagrangianbody pushes material out of the Eulerian elements that itpasses through and they become filled with void SimilarlyEulerian material flowing towards the Lagrangian body isprevented from entering the underlying Eulerian elementsThis formulation ensures that twomaterials never occupy thesame physical space

In this study CEL analysis allowed a direct coupling tosolve this problem The fact that CEL implementation usesthe full Navier-Stokes equations implies that it was able to

8 Geofluids

handle flow appropriately It is important to note that CELis not structured in the same way as a CFD code and wasnot intended to fill this role While it does include viscouseffects with extension to laminar flows it does not include anyturbulence effects but in the context of an impact which has ashort duration and is highly transient (with turbulence effectsbeing of less importance to the structural loading) CEL iswell suited [28]

4 Implementation ofCoupled Eulerian-Lagrangian (CEL)Method in Abaqus

In order to simulate the water entry problem of the pounderand implement the Coupled Eulerian-Lagrangian (CEL)method in Abaqus a geometric model for the pounder andwater was first created A partitionwas created to definewaterand void regions at the water surface Since the fluid (water)undergoes a large deformation an Eulerian grid is suitableThe rigid body (pounder) has small deformations relative tothe fluid making an Lagrangian grid suitable Therefore thewater wasmodeled as Eulerian and the pounder wasmodeledas solid geometry with a rigid body constraint applied to itThen the material properties for the pounder and water weredefined Properties include density (120588) Youngrsquos modulus (119864)and Poissonrsquos ratio (]) for the pounder and density (120588119908) anddynamic viscosity (120578) for the water In the Coupled Eulerian-Lagrangianmethod when the pounder is considered as rigidYoungrsquos modulus and Poisson ratios values should be definedfor the Abaqus but they have no effect on the results Inthe Coupled Eulerian-Lagrangian method 119880119904-119880119901 equationof state (EOS) was selected to define the water properties inAbaqus The 119880119904-119880119901 equation of state is described in detail inAbaqus 614 documentation and it is used to simulateNavier-Stokes flow when turbulence flow is negligible Becausethe major force imposed on the rigid body in water entryproblems is pressure force this equation of state can beused as an appropriate approximation Parameters such as1198620(speed of sound in water) 119904 (constant coefficient in shockvelocity equation) and Γ0 (material constant in Gruneisenequation of state)were defined for the119880119904-119880119901 equation of stateIn the next step the pounder and water were entered intothe assembly section and the pounder was transferred to thewaters center tangent to its surface In order to simulate a free-falling rigid body into water the pounder was first consideredin a position tangent to the water surface then the secondaryvelocity of pounder at the moment of impact was assignedIn the next step explicit dynamic analysis was selected andthe time period for pounder movement between enteringthe water and reaching the model bed was considered as 05seconds Nlgeom was also activated to control the nonlineareffects of large displacements Then by defining a referencepoint and assigning it to the pounder required outputs suchas displacement and velocity values for the vertical direc-tion were selected Based on its defined algorithm Abaqusconsiders the maximum hardness between the Lagrangianelements (pounder) and Eulerian material (water) The Eule-rian solution method uses the meshes defined for poundermovement between water meshes and Abaqus detects theboundary between the Lagrangian and Eulerian materials

Wall

Wall

Wall Water

Sphere

Initial void region

Freesurface

Figure 2 The computational domain boundary conditions andgrid distribution for the water entry problem of sphere

Considering themaximumhardness between the Lagrangianand Eulerian boundaries the contact between these twomaterials is established using the penalty method Concen-trated mass was assigned to the pounder and its moment ofinertia was defined for 119883 119884 and 119885 In the next step thepounder was defined as a rigid body that can move freelyin all directions Gravitational acceleration (Gravity) wasdefined as minus98ms2 in the 119884 direction and a nonreflectingEulerian boundary was assigned to the sides and bottomsurfaces of the computational water domain This boundarycondition is specific to the Eulerian method and preventswave propagation due to rigid body impact on the waterrsquossurface from returning to the computational domain afterreaching the boundariesWith these boundary conditions thewater is regarded as infinite Assuming that the pounder isdropped from a height 119867 above a free water surface withan initial velocity (1198810) of 0ms its secondary velocity (119881) atthe moment of impact was calculated according to the freefall equation of bodies This velocity was vertically applied tothe pounder Eulerian and void regions were specified usingmaterial assignment and the mesh size was defined for thewater and pounder An Eulerian mesh was selected as thewater mesh and an explicit mesh was selected as the poundermesh using structured techniques Finally the problem wassolved and the results were analyzed

41 Model Geometry Before applying the model to pounderwater entry a three-dimensional numerical model was devel-oped to simulate a transient free-falling wedge impactingthe free surface of water in order to check and calibratethe adopted computational model for simpler problemsThen the accuracy of the numerical model was examinedby comparing the numerical results with the experimen-tal data of Aristoff et al [10] for the water entry of asphere Figure 2 illustrates a schematic for the computationaldomain boundary conditions and grid distribution for

Geofluids 9

Table 1 Equation of state parameters for water

Density (120588119908)(kgm3)

Dynamicviscosity (120578)(kgsm2)

Speed of sound(1198880) (msec) Γ0 1199041000 0001 1450 0 0

Water

Aperture release

Z

Diffuser

Lights

Sphere

Camera

Figure 3 Schematic of the experimental apparatus of Aristoff et al[10]

the sphere water entry problem The dimensions for thecomputational domain are considered to be large enoughso that with further increases no considerable changes areobserved in the numerical resultsThe sphere wasmodeled assolid geometry and a rigid body constraint was subsequentlyapplied to it The fluid domain was modeled as a cubewith dimensions of 30 times 50 times 60 cm3 Mesh refinement wasperformed after assigning it to the computational domaintherefore the Eulerianmesh grid size was gradually decreaseduntil no considerable changeswere observed in the numericalresults The Eulerian domain (water) was modeled usingquarter symmetry and contains 746172 elements The spherewas alsomodeled as a Lagrangian solid using 15000 elementsThe Eulerian domain was divided into two parts includingthe upper and lower regions These regions were defined asvoid and stationary water respectively A solid sphere witha diameter of 254 cm and an initial velocity of 217ms inthe normal direction was placed on the free water surfaceThe sphere can move freely in all directions with six degreesof freedom The physical properties of water used in thenumerical simulations are given in Table 1Γ0 and 119904 are material constants in the Gruneisen equationof state and the constant coefficient in shock velocity equa-tion respectively

5 Validation

A schematic of the experimental apparatus of Aristoff et al[10] is presented in Figure 3 A sphere was held by a cameralike aperture at height 119867 above a water tank The tank haddimensions of 30 times 50 times 60 cm3 and was illuminated by a

bank of twenty 32W fluorescent bulbs A diffuser was usedto provide uniform lighting and care was taken to keep thewater surface free of dust Tank dimensions were selectedso that the walls had negligible effects on the results Thesphere was released from rest and fell towards the waterreaching it with an approximate speed119881 asymp radic2119892ℎThe impactsequence was recorded at 2000 frames per second using ahigh-speed camera The resolution was set to 524 times 1280pixels (px) with a field of view of 1128 times 2755 cm2 yieldinga 4646 pxcm magnification The trajectory of the sphereand its impact speed were determined with subpixel accuracythrough a cross correlation andGaussian peak-fittingmethodyielding position estimates accurate to 0025 px (00005 cm)and impact speeds accurate to plusmn4

Figure 4 illustrates a comparison of cavity shapes betweennumerical simulation and the experimental photographstaken by Aristoff et al [10] for a density ratio of 786 and ansimilar period of time The impact velocity of the sphere was217ms and 119905 = 0 refers to when the spheres center passesthrough the free surface This figure shows the cavity shapesfor both experimental and numerical simulation for similartime periods except that cavity shape for the pinch-off timeof the numerical simulation was added to the pictures

An symmetric air cavity formed behind the sphereafter entering the water Air cavity formation consists ofseveral steps including cavity creation cavity expansioncavity contraction behind the sphere and cavity collapseAs the sphere moves deeper through the water it exerts aforce on surrounding fluid in radial directions and transfersits momentum to the fluid However fluid expansion isconfronted with fluid hydrostatic pressure resistance and asa result the radial flow direction is reversed leading to cavitycontraction and collapse Cavity collapse occurs when thecavity wall moves inward until the moment of pinch-off (119905= 708ms) and the cavity is divided into two separated parts[34] While the lower cavity sticks to the sphere and movesalong with it the upper cavity continues its contraction andmoves towards the waterrsquos surface As shown in Figure 4 thenumerical results and experimental observations are in goodagreement

Abaqus numerical simulation methods efficiently modelwater splashes and jet flow Pictures of these phenomenaare depicted in Figure 4 It was concluded from this figurethat pinch-off in the numerical simulation happens with adelay time compared to the experimental method Increasednumerical errors during solution time are one of the causesfor differences between experimental and numerical results

In Figure 5 the numerical results of this study arecompared with the theoretical and experimental data ofAristoff et al as a displacement-time graph of a sphere indifferent water depths [10] As can be seen the numericalresults are in good agreement with those of the analytics andexperiments However the presently computed sphere centerdepth is lower than that found experimentally It seems thatdue to lack of turbulent modeling ability in Abaqus Eulerianformulation and evidently increasing turbulence in flow bytime there is just less difference between experiment andnumerical results in the early time of simulation

10 Geofluids

Experimental results of Aristoff et al (2010)

Pinch-off

Pinch-offNumerical simulation results

t = 59 ＧＭ 129 ＧＭ 199ＧＭ 269ＧＭ 339ＧＭ 409 ＧＭ 479ＧＭ 549ＧＭ 619 ＧＭ 689 ＧＭ 759ＧＭ

t = 59ＧＭ 129 ＧＭ 199ＧＭ 269ＧＭ 339ＧＭ 409 ＧＭ 479ＧＭ 549ＧＭ 619 ＧＭ 689 ＧＭ 708ＧＭ 759ＧＭ

Figure 4 Comparison of numerical results with experimental photographs of Aristoff et al [10] for the sphere water entry problem

002 004 006 0080Time (s)

minus018

minus016

minus014

minus012

minus01

minus008

minus006

minus004

minus002

0

Disp

lace

men

t (m

)

Present studyExperiment (Aristoff et al 2010)Theory (Aristoff et al 2010)

Figure 5 Comparison of displacement-time graph of sphere inwater depth with the theoretical and experimental data of Aristoffet al [10]

6 Materials and Methods

61 Eulerian Model Definition Figure 6 illustrates a sche-matic of the computational domain boundary conditionsand grid distribution for the water entry problem of aspherical pounder The dimensions of the computationaldomain are considered to be large enough so that withfurther increases no considerable changes are observed inthe numerical results The Eulerian domain was dividedinto the upper and lower regions These regions are definedas void and stationary water respectively In water entryproblems for rigid bodies fluid domain dimensions must

Wall

Wall

Wall Water

Spherical pounder

Initial void region

Free surface

Figure 6 The computational domain boundary conditions andgrid distribution for water entry problem of spherical pounder

be at least eight times the dimensions of the rigid bodyHence in this study the fluid domain was modeled as acube with dimensions of 2 times 2 times 1m3 The pounders wereplaced exactly tangent to the water surface and their velocityat the moment of water surface impact was determinedaccording to the free fall equation of bodies The pounderscan move freely in all directions with six degrees of freedom119880119904-119880119901 equation of state was used to define water materialin the Coupled Eulerian-Lagrangian method This equationof state is useful for Navier-Stokes flow simulations whenturbulence flow is negligible In cases where the impact ofa rigid body on a water surface is investigated the majorforce imposed on water body is the pressure force Thereforewith appropriate approximations this equation of state canbe used to investigate the impact of rigid bodies on watersurfaces

62 EulerianMesh Size As shown in Figure 7 with decreasesin the Eulerian domain mesh size to less than 15 cm thedisplacement-time graph of the spherical pounder showsnegligible changes Therefore to improve analysis time the

Geofluids 11

Table 2 Moment of inertia of pounders with different geometric shapes

Geometric shape of the pounder Cube Cylinder Sphere Pyramid ConeMoment of inertia (cm4) 685 5137 411 3082 411

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

01 02 03 040Time (s)

2 ＝Ｇ

175 ＝Ｇ

15 ＝Ｇ

Figure 7 Comparison of the effect of the Eulerian mesh size ondisplacement-time graph of spherical pounder

Eulerian domain was modeled using quarter symmetry witha mesh size of 15 cm and contains 1538943 elements

63 Lagrangian Model Definition Figure 8 illustratespounder dimensions for different geometric shapes Thesepounders are identical in terms of material mass densityYoungrsquos modulus Poissonrsquos ratio volume mass and dropheight from the water surface The only difference is theirgeometric shapeThese parameters were regarded as identicalin all pounders to investigate the effect of pounder shapeon deepwater displacement and velocity air cavity shapepinch-off time and depth and the effect of water drag forceon velocity depreciation when impacting the model bed

Pounder dimensions were selected so that they had avolume of 523 cm3 and equal mass due to their identicalmaterial and mass density Table 2 illustrates the moment ofinertia for pounders with different geometric shapes

Table 3 illustrates thematerial properties of the poundersThese parameters are regarded as identical for all pounderswith the mentioned geometric shapes

64 Lagrangian Mesh Size Due to the negligible effect ofLagrangian mesh size on the results and to improve analysistime the Lagrangian mesh size for all pounders was set to05 cm

Pyramid Cone

Cube Cylinder Sphere

10 ＝Ｇ10 ＝Ｇ10 ＝Ｇ 10 ＝Ｇ

10 ＝Ｇ10 ＝Ｇ10 ＝Ｇ

10 ＝Ｇ5236 ＝Ｇ

20 ＝Ｇ15708 ＝Ｇ

666 ＝Ｇ

Figure 8 The dimensions of pounders with different geometricshapes

Table 3 Material properties of pounders

Materialtype Mass (kg) Density (120588)

(kgm3)

Youngrsquosmodulus (119864)(kgm2)

Poissonrsquosratio (120592)

Steel 411 7850 21 times 1010 03

65 Boundary Condition Wall boundary conditions wereapplied to the sides and bottom of the Eulerian domain tosimulate a water tank Pounder displacement was not con-strained in any direction and the pounders could freely movein all directions The sides and bottom boundaries of thethree-dimensional Eulerian domain were set as nonreflectingboundaries so that the waves released due to pounder impacton water surface did not return to the computational domainafter reaching these boundaries and being absorbed by themThis boundary condition is written as [37]

119889119901 minus 119901119888119889119906 = 0119889119909119889119905 = minus119888 (41)

where 120588 is density 119888 is the speed of sound in the fluid 119901 ispressure 119906 is velocity perpendicular to the wave and 119909 is thedirection perpendicular to the boundary

66 Drop Height and Impact Velocity of Pounders Afterdropping the pounders from a height (ℎ) above the watersurface the velocity of the pounders at themoment of impactwas determined using

1198812 minus 11988120 = 2119892ℎ (42)

where1198810 is the pounder initial velocity before being droppedfrom height ℎ above the waterrsquos surface 119881 is pounder

12 Geofluids

01 02 03 04 050Time (s)

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

CubeCylinderSphere

PyramidCone

Figure 9 Displacement-time graph of pounders with differentgeometric shapes

secondary velocity when impacting the water surface and 119892is the gravitational acceleration of the Earth

From (42) which expresses a time independent equationfor the free fall of objects the initial velocity and the terminalvelocity of the object drop height and the gravitationalacceleration of the Earth are interdependent in the absenceof time In this numerical simulation the pounders weredropped from a height of 1m above thewaterrsquos surfacewith aninitial velocity (1198810) equal to 0ms Therefore the secondaryvelocity of the pounders when impacting the water surfacewas obtained using

119881 = radic2119892ℎ (43)

Instead of simulating a full dropping event from theinitial position the pounders were placed exactly tangent towaterrsquos surface and their secondary velocity at the momentof impact was determined according to (43) A gravitationalacceleration (gravity) of minus98ms2 was applied in the normaldirection of the pounders Therefore a secondary velocityof 443ms in the normal direction of the waterrsquos surfacecorresponds to the speed that would be obtained for thepounders

7 Results and Discussion

71 Effect of the Geometric Shape of the Pounder on Deep-water Displacement Figure 9 illustrates the displacement-time graph of pounders with different geometric shapes Thegraph continues for the cubical pounder from the momentof water surface impact at time zero up to the depth of 1mat 415ms The displacement-time graph of the cylindrical

pounder indicates that after entering the water it reachesthe model bed in a shorter amount of time than the cubicalpounder The graph for this pounder continues from themoment of water surface impact at time zero up to the depthof 1m at 355ms The linear displacement-time graph forthe spherical and pyramidal pounders indicates the fasterdeepwater movement of these pounders compared to thecubical and cylindrical pounders The deepwater movementtime of the spherical and pyramidal pounders after enteringthe water to themodel bed is 285ms and 265ms respectivelyThe linear displacement-time graph of the conical pounderindicates that it has a faster deepwater movement comparedto other pounders as it takes 235ms to reach the model bedafter entering the water

72 Mechanism of Air Cavity Formation and Pinch-Off dueto Deepwater Movement of Pounders with Different Geomet-ric Shapes Figure 10 illustrates the movement trajectory ofpounders with different geometric shapes from the momentof water impact towhen the pounders reached themodel bedAfter entering the water a symmetric air cavity is formedbehind the pounders Air cavity formation involves severalsteps including air cavity creation and expansion behind thepounder air cavity contraction and air cavity collapse [34]As the pounder moves downward in water depth it imposesa force onto surrounding fluid in its radial directions andtransfers its momentum to the fluid This extension is facedwith the hydrostatic pressure resistance of fluidThe directionof the radial flow is reversed and eventually leads to aircavity contraction and collapse In the other words the aircavity contraction is accelerated until pinch-off occurs andthe air cavity is divided into two distinct parts The upperpart of the air cavity continues its contraction and movestowards the waterrsquos surface whereas the lower part of the aircavity clings to the pounder and moves along with it [34]After impacting the water surface the cubical and cylindricalpounders due to their flat contact surfaces lose some of theirvelocityThis velocity depreciation slows down and continuesuntil pinch-offThe spherical pounder with its curved contactsurface has less velocity depreciation compared to the cubicaland cylindrical pounders However some pounder velocityis always depreciated due to impacting the water surfaceThe velocity depreciation process for the spherical pounderduring deepwater movement continues until pinch-off Afterimpacting the waterrsquos surface the pyramidal pounder with itspointy contact surface has a slight velocity depreciation butcontinues to move deeper into the water with approximatelythe same secondary velocity until pinch-off By imposing thedrag force of water onto the pounders surface its velocityis gradually decreased until it reaches the model bed Afterimpacting the water surface the conical pounder with itspointy contact surface continues to move deeper into thewater with an increasing velocity The increasing velocity ofthe conical pounder during deepwater movement continueseven after pinch-off When the pounder moves towards themodel bed the waters drag force is imposed onto poundersurface causing velocity depreciation The pinch-off time forthe pounders with the above-mentioned shapes is between225 and 250ms

Geofluids 13

t = 0 25ＧＭ 50ＧＭ 100ＧＭ 200ＧＭ 225ＧＭ 250ＧＭ 275ＧＭ 300ＧＭ 325ＧＭ 350ＧＭ150ＧＭCu

beCy

linde

rSp

here

Pyra

mid

Con

e

Figure 10 Air cavity formation due to deepwater movement of pounders with different geometric shapes and pinch-off time and depth

73 Effect of the Pounder Geometric Shape onDeepwater Veloc-ity Figure 11 illustrates the velocity-time graph for pounderswith different geometric shapes The cubical and cylindricalpounders lose some of their secondary velocity at the earlymoments of water entry Hence the velocity-time graph ofthese pounders illustrates great velocity depreciation up to15ms The velocity reduction of the cubical and cylindricalpounders gradually continues until pinch-off The velocityof the spherical pounder decreases up to 45ms after waterentry but its velocity depreciation is less than the cubicaland cylindrical pounders This velocity reduction continuesat a slower rate until pinch-off In the initial moments ofwater entry the pyramidal pounder has a slight reduction insecondary velocity up to 55ms Then velocity depreciationcontinues at a faster rate due to the drag force imposed onto

pounders surface and the velocity-time graph of the pyrami-dal pounder shows a gradual velocity reduction until pinch-off After impacting the water surface the conical poundercontinues its deepwater movement with an increasing veloc-ity rate Due to its very low slope the velocity-time graphindicates an increase in velocity for the conical pounder butdue to the waters drag force its velocity begins to decrease at145ms The velocity-time graph indicates a gradual decreasein velocity for conical pounder and velocity depreciationcontinues until pinch-off After pinch-off pounder velocitiesgradually decrease until they reach themodel bed Eventuallywith a velocity of 388ms and 157ms at the moment ofmodel bed impact the conical and cubical pounders have thehighest and lowest velocity respectively Water drag force isimposed in the opposite direction of poundermovement and

14 Geofluids

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Velo

city

(ms

)

01 02 03 040Time (s)

CubeCylinderSphere

PyramidCone

Figure 11 Velocity-time graph of pounders with different geometricshapes

it has the most and least influence on the cubical and conicalpounders respectively

8 Critical Drop Height of the Pounder

In the dynamic compaction of the seabed determining thecritical drop height of the pounder from the freewater surfaceis an effective parameter for the pounder to reach maximumdeepwater velocity After dropping the pounder from height119867 above the free water surface with an initial velocity (1198810)of 0ms its velocity gradually increases until it reaches thesecondary velocity (119881) at the moment of water impact Afterentering the water pounder velocity is decreased due to waterdrag force According to the free fall equation for objectsincreasing the drop height from the waterrsquos surface leads toincreased pounder velocity at the moment of water surfaceimpact but it does not necessarily lead to an increase inpounder velocity during seabed impact because the criticaldrop height of the pounder is a determining factor ofits maximum deepwater velocity According to Figure 12 aconical pounder with a mass of 411 kg was dropped fromdifferent heights (119867) and its velocity when impacting thewaters was measured according to the time independentequation for the free fall of objects Table 4 illustrates thevelocity of the conical pounder with different drop heightswhen impacting the waterrsquos surface

81 Effect of the Drop Height of the Pounder from WaterSurface on Deepwater Displacement Figure 13 illustrates thedisplacement-time graph of the conical pounder with dif-ferent drop heights from the waterrsquos surface As can beseen increasing the drop height of the pounder leads to

Water levelℎ

Figure 12 Schematic of drop height of the conical pounder fromwater surface

Table 4 Velocity of the conical pounder with different drop heightsat the moment of impacting the water surface

Drop height of the pounderfrom water surface (m)

Pounder velocity at themoment of impacting water

surface (ms)025 22105 313075 3831 443125 49515 542175 5852 626225 66425 7

displacement-time graphs that are closer to each other Infact an increase in the drop height of the pounder from thewaterrsquos surface leads to an increase in its secondary velocitywhen impacting the waterrsquos surface Hence the pounderenters the water with a higher velocity and reaches the modelbed in less time

82 Effect of the Drop Height of the Pounder from WaterSurface on Its Deepwater Velocity Increasing the drop heightof the pounder from the waterrsquos surface leads to an increasein its velocity before impacting the water but the poundervelocity gradually decreases after entering the water due tothe its drag force Figure 14 illustrates the velocity-time graphfor the conical pounder with different drop heights from thewaterrsquos surface When the pounder is dropped from a lowheight due to its low velocity it moves deepwater due to itsweight Moreover because of the low velocity of the pounderwhen entering the water pinch-off occurs in less time and atlower depths and its velocity increases after pinch-off until itreaches terminal velocity Increases in pounder drop heightsincrease water entry velocity Hence it will have greatervelocity depreciation in the first moments of water entry

Figure 15 illustrates the terminal velocity-time graph forthe conical pounderwith different drop heightswhen impact-ing the model bed As can be seen increasing the drop height

Geofluids 15

01 02 03 040Time (s)

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

H = 1Ｇ

H = 075Ｇ

H = 05 Ｇ

H = 025 Ｇ

H = 175Ｇ

H = 15 Ｇ

H = 125 Ｇ

H = 25 Ｇ

H = 225 Ｇ

H = 2Ｇ

Figure 13 Displacement-time graph of the conical pounder withdifferent drop heights

of the pounder from the waterrsquos surface leads to an increasein its terminal velocity when impacting the model bedTheseconditions are dominant as long as the pounder reaches acritical drop height from the water surface but increasingthe pounder drop height higher than the critical drop heightleads to decreased terminal velocity when impacting themodel bed In this numerical simulation after dropping thepounder from a height of 2m above the waterrsquos surface itssecondary velocity was 626ms when impacting the watersurface After entering the water the secondary velocity ofthe pounder decreased until it impacts the model bed witha terminal velocity of 454ms Therefore a drop height of2m from above the waterrsquos surface has the highest terminalvelocity when the pounder impacts themodel bed Increasingthe drop height bymore than 2m leads to decreased pounderterminal velocity when impacting the model bed The heightwhich has the greatest effect on pounder terminal velocitywhen impacting the model bed is called the critical dropheight Increasing the drop height of the pounder to greaterthan its critical drop height leads to decreased terminalvelocity when impacting the model bed

Figure 16 illustrates the secondary and terminal velocityof the conical pounder with different drop heights As canbe observed increasing the drop height of the pounder fromwaterrsquos surface leads to increased differences between thesecondary and terminal velocities of the pounder Increasingthe drop height to more than the critical drop height leadsto increased pounder secondary velocity when impactingthe waterrsquos surface and decreased terminal velocity whenimpacting the model bed Hence the difference betweenthe secondary and terminal velocities of the pounder is

01 02 03 040Time (s)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)H = 175Ｇ

H = 125 Ｇ

H = 075Ｇ

H = 025 Ｇ H = 225 Ｇ

H = 1Ｇ

H = 05 Ｇ

H = 25 Ｇ

H = 2Ｇ

H = 15 Ｇ

Figure 14 Velocity-time graph of the conical pounder with differentdrop heights

minus430

minus447 minus454minus415

minus426 minus380

minus367minus355

minus332minus310

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Term

inal

velo

city

(ms

)

01 02 03 040Time (s)

Figure 15 Terminal velocity of conical pounder with different dropheights versus time

not reasonable and a significant amount of the pounderssecondary velocity is depreciated when entering the waterThe great difference between the secondary and terminalvelocities of the pounder reveals the necessity of determiningthe critical drop height of the pounder before dynamic seabedcompaction operations Increasing the pounder drop heightto be greater than the critical drop height not only hasa reverse result on pounder velocity when impacting theseabed but it also requires special equipment such as lengthy

16 Geofluids

Secondary velocityTerminal velocity

025 05 075 1 125 15 175 2 225 250Drop height (m)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)

Figure 16 Secondary and terminal velocities of conical pounderversus drop heights

1 2 3 4 5 6 7 80Impact velocity (ms)

0

005

01

015

02

025

03

Pinc

h-off

tim

e (s)

Figure 17 Pinch-off time versus impact velocity of conical pounder

cranes and huge barges leading to increased implementationcosts

Figure 17 illustrates pinch-off time based on the sec-ondary velocity of the pounder at the moment of watersurface impact As can be seen pinch-off time is a weakfunction of pounder impact velocity and increasing impactvelocity leads to a slight reduction in pinch-off time Thisbehaviour was also observed in the analytical and numericalresults found by Lee et al [40]

Figure 18 illustrates pinch-off depth based on the sec-ondary velocity of pounder at the moment of water surfaceimpact As can be seen the pinch-off depth is a linear

1 2 3 4 5 6 7 80Impact velocity (ms)

minus12

minus1

minus08

minus06

minus04

minus02

0

Pinc

h-off

dep

th (m

)

Figure 18 Pinch-off depth versus impact velocity of conicalpounder

function of pounder impact velocity These results are in linewith the analytical equations of Lee et al [40]

9 Conclusions

In this study the water entry problem of three-dimensionalpounders with different geometric shapes of cube cylindersphere pyramid and cone was numerically simulated bycommercial finite element code Abaqus 614-2 An explicitdynamic analysis method was employed to model the fluid-structure interaction using a Coupled Eulerian-Lagrangian(CEL) method Before applying the model to water entryof the pounders a three-dimensional numerical model wasdeveloped to simulate a transient free-falling wedge impact-ing the free surface of the water in order to calibrate theadopted computationalmodelThe accuracy of the numericalmodel was examined by comparing numerical results withthe experimental data of Aristoff et al The good agreementbetween numerical simulation results with the theoreticaland experimental data of Aristoff et al demonstrates thecapability of the employed numerical methodology to sim-ulate the water entry problem for three-dimensional geome-tries with acceptable accuracy After water entry by a pounderdropped fromaheight of 1m a symmetric air cavity is formedbehind the pounder As the pounder moves downward inwater depth it imposes a force onto surrounding fluid inthe radial directions and transfers its momentum to thefluid This extension is faced with the hydrostatic pressureresistance of the fluid causing the direction of the radialflow to be reversed leading to air cavity contraction andcollapse The results show that the water drag force had thehighest and lowest effect on the cubical and conical poundersrespectively and increasing the drop height to greater thanthe critical drop height leads to increased pounder velocitywhen impacting the waterrsquos surface and decreased velocitywhen impacting the model bed Pinch-off time is a very weak

Geofluids 17

function of pounder velocity when impacting the waterrsquossurface but the pinch-off depth increases linearly along withincreased pounder impact velocities It is recommended thatthe critical drop height for the pounder is determined beforedynamic seabed compaction operations and the adjustmentof crane heights to this height In addition it is also recom-mended that perforated pounders be used in order to reducevelocity depreciation when impacting the waterrsquos surface andto reduce the effect of water drag force

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors warmly appreciate the cooperation and guidanceof Dr B Hamidi and Dr A Mehrabadi in this study

References

[1] B Hamidi J-M Debats H Nikraz and S Varaksin ldquoOffshoreground improvement recordsrdquo Australian Geomechanics Jour-nal vol 48 no 4 pp 111ndash122 2013

[2] SWatanabe ldquoResistance of impact onwater surface part I-cone Institute of Physical andChemical ResearchrdquoTokyo vol 12 pp251ndash267 1930

[3] S Watanabe ldquoResistance of impact on water surface part II-cone (continued)rdquo Institute of Physical and Chemical Researchvol 14 pp 153ndash168 1930

[4] A May and J C Woodhull ldquoDrag coefficients of steel spheresentering water verticallyrdquo Journal of Applied Physics vol 19 no12 pp 1109ndash1121 1948

[5] A May ldquoEffect of surface condition of a sphere on its water-entry cavityrdquo Journal of Applied Physics vol 22 no 10 pp 1219ndash1222 1951

[6] AMay ldquoVertical entry ofmissiles into waterrdquo Journal of AppliedPhysics vol 23 no 12 pp 1362ndash1372 1952

[7] A P New T S Lee and H T Low ldquoImpact loading and waterentrance characteristics of prismatic bodiesrdquo in Proceedings ofthe third international offshore and polar engineering conferencepp 282ndash287 1993

[8] M Anghileri and A Spizzica ldquoExperimental validation of finiteelement models for water impactsrdquo in Proceedings of the secondinternational crash usersrsquo seminar Cranfield Impact Centre LtdEngland UK 1995 second international crash users seminar

[9] A Engle andR Lewis ldquoA comparison of hydrodynamic impactsprediction methods with two dimensional drop test datardquoMarine Structures vol 16 no 2 pp 175ndash182 2003

[10] J M Aristoff T T Truscott A H Techet and J W M BushldquoThe water entry of decelerating spheresrdquo Physics of Fluids vol22 no 3 Article ID 009002PHF pp 1ndash8 2010

[11] T Von Karman The impact of seaplane floats during landingNational Advisory Committee for Aeronautics NACA TN no321 1929

[12] HWagner ldquoPhenomena associatedwith impacts and sliding onliquid surfacesrdquo Journal of Applied Mathematics and Mechanicsvol 12 pp 193ndash215 1932

[13] W S Vorus ldquoA flat cylinder theory for vessel impact and steadyplaning resistancerdquo Journal of Ship Research vol 40 no 2 pp89ndash106 1996

[14] T Miloh ldquoOn the initial-stage slamming of a rigid sphere in avertical water entryrdquo Applied Ocean Research vol 13 no 1 pp13ndash43 1991

[15] J M Aristoff and JW Bush ldquoWater entry of small hydrophobicspheresrdquo Journal of Fluid Mechanics vol 619 pp 45ndash78 2009

[16] P Ghadimi A Dashtimanesh and S R Djeddi ldquoStudy of waterentry of circular cylinder by using analytical and numericalsolutionsrdquo Journal of the Brazilian Society ofMechanical Sciencesamp Engineering vol 34 no 3 pp 225ndash232 2012

[17] A TassinD J PiroAAKorobkinK JMaki andM J CookerldquoTwo-dimensional water entry and exit of a body whose shapevaries in timerdquo Journal of Fluids and Structures vol 40 pp 317ndash336 2013

[18] E Yao H Wang L Pan X Wang and R Woding ldquoVerticalwater-entry of bullet-shaped projectilesrdquo Journal of AppliedMathematics and Physics vol 02 no 06 pp 323ndash334 2014

[19] M-S Park Y-R Jung and W-G Park ldquoNumerical study ofimpact force and ricochet behavior of high speed water-entrybodiesrdquo Computers amp Fluids vol 32 no 7 pp 939ndash951 2003

[20] D Battistin and A Iafrati ldquoHydrodynamic loads during waterenergy of two-dimensional and axisymmetric bodiesrdquo Journalof Fluids and Structures vol 17 no 5 pp 643ndash664 2003

[21] A A Korobkin and M Ohkusu ldquoImpact of two circular platesone of which is floating on a thin layer of liquidrdquo Journal ofEngineering Mathematics vol 50 no 4 pp 343ndash358 2004

[22] K M Kleefsman G Fekken A E Veldman B Iwanowski andB Buchner ldquoA volume-of-fluid based simulation method forwave impact problemsrdquo Journal of Computational Physics vol206 no 1 pp 363ndash393 2005

[23] Y Kim Y Kim Y Liu and D K P Yue ldquoOn the water-entryimpact problem of asymmetric bodiesrdquo in Proceedings of the9th International Conference on Numerical Ship Hydrodynamics(NSH rsquo07) August 2007

[24] Q Yang and W Qiu ldquoNumerical solution of 2D slammingproblem with a CIPmethodrdquo in Proceedings of the InternationalConference on Violent Flows Research Institute for AppliedMechanics Kyushu University Japan 2007

[25] A C Fairlie-Clarke and T Tveitnes ldquoMomentum and gravityeffects during the constant velocitywater entry ofwedge-shapedsectionsrdquo Ocean Engineering vol 35 no 7 pp 706ndash716 2007

[26] Q Yang andW Qiu ldquoNumerical simulation of water impact for2D and 3D bodiesrdquo Ocean Engineering vol 43 pp 82ndash89 2012

[27] G Wu ldquoNumerical simulation for water entry of a wedge atvarying speed by a high order boundary element methodrdquoJournal of Marine Science and Application vol 11 no 2 pp 143ndash149 2012

[28] M Ahmadzadeh B Saranjam A Hoseini Fard and A RBinesh ldquoNumerical simulation of sphere water entry prob-lem using Eulerian-Lagrangian methodrdquo Applied MathematicalModelling vol 38 no 5-6 pp 1673ndash1684 2014

[29] V-T Nguyen D-T Vu W-G Park and Y-R Jung ldquoNumericalanalysis of water impact forces using a dual-time pseudo-compressibility method and volume-of-fluid interface trackingalgorithmrdquo Computers amp Fluids vol 103 pp 18ndash33 2014

[30] S Kim and N Kim ldquoIntegrated dynamics modeling forsupercavitating vehicle systemsrdquo International Journal of NavalArchitecture and Ocean Engineering vol 7 no 2 pp 346ndash3632015

18 Geofluids

[31] V-T Nguyen D-T Vu W-G Park and C-M Jung ldquoNavier-Stokes solver for water entry bodies with moving Chimera gridmethod in 6DOF motionsrdquo Computers amp Fluids vol 140 pp19ndash38 2016

[32] M Mirzaei M Eghtesad and M Alishahi ldquoPlaning forceidentification in high-speed underwater vehiclesrdquo Journal ofVibration and Control vol 22 no 20 pp 4176ndash4191 2016

[33] G Qiu S Henke and J Grabe ldquoApplication of a CoupledEulerian-Lagrangian approach on geomechanical problemsinvolving large deformationsrdquoComputersampGeosciences vol 38no 1 pp 30ndash39 2011

[34] M R Erfanian M Anbarsooz N Rahimi M Zare andM Moghiman ldquoNumerical and experimental investigationof a three dimensional spherical-nose projectile water entryproblemrdquo Ocean Engineering vol 104 pp 397ndash404 2015

[35] W M Isbell F H Shipman and A H Jones ldquoHugoniotequation of state measurements for eleven material to fiveMegabars Materials and Structural Laboratory ManufacturingDevelopments General Motors Technical centerrdquo Tech RepMichigan 1968

[36] M A Jandron R C Hurd J L Belden A F Bower W Fennelland T T Truscott ldquoModeling of hyperelastic water-skippingspheres using AbaqusExplicitrdquo in Proceedings of the SIMULIACommunity Conference 2014

[37] H Forouzani B Saranjam R Kamali and A Abdollahi-far ldquoElasto-plastic time dependent impact analysis of highspeed projectile on water surfacerdquo Journal of Solid and FluidMechanics vol 3 pp 281ndash298 2016

[38] M R Erfanian andMMoghiman ldquoNumerical and experimen-tal investigation of a projectile water entry problem and studyof velocity effect on time and depth of pinch-offrdquo Journal ofModares Mechanical Engineering vol 15 pp 53ndash60 2015

[39] A Sillem Feasibility study of a tire hydroplaning simulation in afinite element code using a coupled Eulerian-Lagrangian method[MSc thesis] 2008

[40] M Lee R G Longoria and D E Wilson ldquoCavity dynamics inhigh-speed water entryrdquo Physics of Fluids vol 9 no 3 pp 540ndash550 1997

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Submit your manuscripts atwwwhindawicom

4 Geofluids

the conservation of momentum while the governing equa-tion for the fluid uses a general Navier-Stokes form [36]The Eulerian mesh is typically a simple rectangular gridof elements extending well beyond the Eulerian materialboundaries giving the material space in which to move anddeform If any Eulerian material moves outside the Eulerianmesh it is lost from the simulation The Eulerian materialboundary must therefore be computed during each timeincrement and it generally does not correspond to an elementboundary The interaction between the structure and thewater is handled remarkably well by the solver using thegeneral contact algorithm In traditional Lagrangian analysisnodes are fixed within the material and elements deform asthe material deforms Lagrangian elements are always 100full of a single material so the material boundary coincideswith the element boundary By contrast in Eulerian analysisnodes are fixed in space andmaterial flows through elementsthat do not deform Eulerian elements may not always be100 full ofmaterial andmanymay be partially or completelyvoid

The Eulerian implementation in AbaqusExplicit is basedon the Volume Of Fluid (VOF) method In numericalanalyses using this CEL method the Eulerian material istracked as it flows through the mesh by computing itsEulerian Volume Fraction (EVF) Each Eulerian element isdesignated a percentage which represents the portion ofthat element filled with a material If an Eulerian element iscompletely filled with a material its EVF is 1 and if there isno material in the element its EVF is 0 Eulerian elementsmay simultaneously contain more than one material If thesum of all material volume fractions in an element is less thanone the remainder of the element is automatically filled witha void material that has neither mass nor strength Contactbetween Eulerian materials and Lagrangian materials isenforced using a general contact that is based on a penaltycontact methodThe Lagrangian elements can move throughthe Eulerian mesh without resistance until they encounteran Eulerian element filled with material (EVF = 0) A CELmethod that attempts to capture the advantages of boththe Lagrangian and the Eulerian method is implemented inAbaqus [33]

21 Time Integration Scheme TheCELmethod implementedin AbaqusExplicit uses an explicit time integration schemeThe central difference rule is employed for the solution ofa nonlinear system of differential equations The unknownsolution for the next time step can be found directly fromthe solution of the previous time step so no iteration isneeded Another advantage of choosing an explicit timeintegration scheme for the present problem is robustnessregarding difficult contact conditions [33]

22 Formulation of Eulerian-Lagrangian Contact By usingthe CEL method the contact between the Eulerian domainand Lagrangian domain is discretized using general contactwhich is based on the penalty contactmethodThismethod isless stringent compared to the kinematic contactmethodThepenalty contact method is used in Abaqus to handle contactin CEL analysis [33]

23 Governing Equations in Lagrangian Method The threefundamental conservation equations are the conservation ofmass the conservation of momentum and the conservationof energy The Lagrangian formulation trivially enforces theconservation of mass by defining the density as the ratio ofthe element mass to the current volume The conservation ofenergy in the absence of heat conduction is enforced locallyat selected points within each element or control volumeDifferent finite elements and finite difference methods aretherefore defined in terms of how they handle the conserva-tion of momentum

Finite element methods use the weak form of the equa-tion which is also known as the principle of virtual workor the principle of virtual power within the solid mechanicscommunity In this method the principle of virtual work ispresented as

int119881[120588120575119909 + 120590 120575120576] 119889119881 = int

119881120588119887120575119909 119889119881 + int

119878120591

120591120575119909 119889119878 (1)

where 120588 is density 120590 is the stress 120576 is strain 120591 is shearstress 119881 is the volume of the domain and 119878 is the surfacewhich is divided into 119878120591 and 119878119906 which is where traction anddisplacement boundary conditions are imposed respectivelyEquation (1) can be rewritten as

int119881120588120575119909 119889119881 = 119865ext minus 119865int (2)

where 119865int represents internal force and 119865ext representsexternal force which is expressed as

119865int = int119881120590 120575120576 119889119881

119865ext = int119881120588119887120575119909 119889119881 + int

119878120591

120591120575119909 119889119878(3)

In the finite element method the coordinates velocityacceleration and virtual displacement are interpolated fromtheir values at the nodes using shape functions119873119860 where 119860is the node number They are expressed as

119909 = sum119860

119873119860119909119860 = sum119860

119873119860119860 = sum119860

119873119860119860120575119909 = sum

119860

119873119860120575119909119860

(4)

Substituting the relations from (2) into the inertial termin the weak form gives

int119881120588120575119909 119889119881 = int

119881120588sum119860

119873119860119860sum119861

119873119861120575119909119861119889119881= sum119860

sum119861

[int119881120588119873119860119873119861119889119881] 119860120575119909119861

(5)

Geofluids 5

The external force vector is also derived by a directsubstitution of the appropriate relations from (6)

119865ext = int119881120588119887120575119909 119889119881 + int

119878120591

120591120575119909 119889119878 (6)

119865ext = int119881120588119887sum119861

119873119861120575119909119861119889119881 + int119878120591

120591sum119861

119873119861120575119909119861119889119878 (7)

119865ext = sum119861

int119881120588119887119873119861119889119881 + int

119878120591

120591119873119861119889119878 120575119909119861 (8)

The internal force vector requires the evaluation of thevirtual strain 120575 Again using the relations from (9)

120575120576119894119895 = 12 ( 120597119909119894120597119909119895 +120597119909119895120597119909119894 ) (9)

120575120576119894119895 = sum119861

12 (120597119873119861120597119909119895 120575119909119861119894 +120597119873119861120597119909119894 120575119909119861119895) (10)

These relations may be put in matrix form which definesthe 119861matrix 119861 as

120576120575 = 119861120575119909 (11)

and the internal force may be written as

119865int = int119881120590 120575120576 119889119881

119865int = sum119861

int119881119861119861120590119889119881 120575119909119861

(12)

where 119861119861 is the section of the 119861matrix associated with node119861 To simplify the mass matrix is defined as

119872119888119860119861 = [int119881120588119873119860119873119861119889119881] (13)

where 119872119888 is called the consistent mass matrix because itsderivation is consistent with the finite element method Allthe terms in the weak form have now been written in terms ofunknown coordinates and virtual displacements Collectingthe terms gives

sum119861

119861minus 1119872119861 (int119881 120588119887119873119861119889119881 + int

119878120591

120591119873119861119889119878 minus int119881119861119861120590119889119881)

sdot 120575119909119861 = 0(14)

Since the virtual displacements are arbitrary the 119861 valueis in accordance with

119861 = 1119872119861 (int119881 120588119887119873119861119889119881 + int119878120591

120591119873119861119889119878 minus int119881119861119861120590119889119881) (15)

24 Governing Equations in Eulerian Method For thedescription of fluid flows usually Eulerian formulation isemployed because we are usually interested in the propertiesof the flow at certain locations in the flow domain InEulerian formulation an independent equation is used toexpress conservation of mass Inertia forces also includedisplacement sentences that when present in numericalsolution lead to a matrix with nonsymmetric coefficients thatdepends on calculated velocities Eulerian formulation allowsfor the use of simple strain and stress rates and velocitycalculations instead of displacement values (in the ultrasmalldisplacement analysis) [36 37] Linear viscous isotropic fluidsare known as Newtonian fluids which are by far the mostimportant fluids for practical applications Newtonian fluidsare characterized by the followingmaterial law for theCauchystress tensor 119879

119879119894119895 = 120583( 120597V119894120597119909119895 +120597V119895120597119909119894 minus

23 120597V119896120597119909119896 120575119894119895) minus 119901120575119894119895 (16)

where the velocity vector is V119894 with respect to Cartesiancoordinate 119909119894 the pressure is 119901 the dynamic viscosity is 120583and the Kronecker symbol is 120575119894119895 With this the conservationlaws for mass momentum and energy take the form of

120597119901120597119905 +120597 (119901V119894)120597119909119894 = 0 (17)

120597 (119901V119894)120597119905 + 120597 (119901V119894V119895)120597119909119895= 120597120597119909119895 [120583(

120597V119894120597119909119895 +120597V119895120597119909119894 minus

23 120597V119896120597119909119896 120575119894119895)] minus120597119901120597119909119895 + 120588119891119894

(18)

120597 (119901119890)120597119905 + 120597 (119901V119894119890)120597119909119894= 120583[ 120597V119894120597119909119895 (

120597V119894120597119909119895 +120597V119895120597119909119894) minus 23 ( 120597V119894120597119909119894)

2] minus 119901 120597V119894120597119909119894+ 120597120597119909119894 (119896

120597119879120597119909119894) + 120588119902

(19)

where 120588 is the fluid density 119890 is the specific internal energyand119891119894 and 119902 are external forces and heat sources respectivelyIn energy balance (19) for the heat flux vector ℎ119894 Fourierrsquoslaw with thermal conductivity 120581119891 is used that is a constantspecific heat capacity is assumed and the work done bypressure and friction forces is neglected

ℎ119894 = minus120581119891 120597119879120597119909119894 (20)

System (17)ndash(19) has to be completed by two equations ofstate of the form

119901 = 119901 (120588 119879) 119890 = 119890 (120588 119879) (21)

6 Geofluids

which define the thermodynamic properties of the fluid thatis the thermal and caloric equation of state respectivelyWhen defining the viscosity for the material it is treated as aNewtonian fluid meaning that the viscosity depends only ontemperature This is the default setting in Abaqus Since thetemperature is constant throughout the analysis the viscosityalso is constant

25 Explicit Time-Dependent Solution There are implicitand explicit algorithms for dynamic analysis In the implicitmethod the Newmark method is used the matrices need tobe reversed andmore implementation time is requiredHow-ever this method enjoys high accuracy and its convergenceto linear problems is guaranteed In the explicit method thematrices do not need be reversed and the analysis speed ismuch higher Convergence in the explicit method is achievedif the time step is less than the smallest element divided bythe speed of sound in the material [37]

The solution was advanced in time using the centraldifference method which is also called the Verlet or leap frogmethod It is based on the central difference approximationfor evaluating the derivative of a function 119891(119905) at 119905

119889119891 (119905)119889119905 = 119891 (119905 + (12) Δ119905) minus 119891 (119905 minus (12) Δ119905)Δ119905 (22)

where Δ119905 is the time stepUsing (15) acceleration in time 119905 can be written as

119905 = 119865119905119872119861 (23)

Now using (22) for velocity-acceleration velocity at time119905 + (12)Δ119905 is obtained as

119905+(12)Δ119905 = 119905minus(12)Δ119905 + 119905Δ119905 (24)

Then using (22) for velocity-displacement displacementat time 119905 + Δ119905 is obtained as

119909119905+Δ119905 = 119909119905 + 119905+Δ119905Δ119905 (25)

The central difference method has its step size limited bystability considerations like all explicit integration methodsand the stable time step size was limited to

Δ119905 le 119897119888 (26)

where 119897 is the length of the shortest element and 119888 is the speedof sound

3 Finite Element Modeling

31 Equation of Energy and Hugoniot Curve The equationfor conservation of energy equates the increase in internalenergy per unit mass 119864119898 to the rate at which work is beingdone by stresses and the rate at which heat is being added Inthe absence of heat conduction the energy equation can bewritten as

120588120597119864119898120597119905 = (119901 minus 119901119887]) 1120588 120597120588120597119905 + 119904 119890 + 120588 (27)

PH

PHI1

PHI0

11 101

Figure 1 Schematic representation of a Hugoniot curve [35]

where 119901 is the pressure stress defined as positive in compres-sion 119901119887] is the pressure stress due to the bulk viscosity 119904 isthe deviator stress tensor 119890 is the deviator part of strain rateand is the heat rate per unit mass According to Abaqusdocumentation flow modeling of compressible fluid can beachieved using the linear 119880119904-119880119901 form of the Mie-Gruneisenequation of state The equation of state assumes pressure as afunction of the current density 120588 and the internal energy perunit mass 119864119898 according to

119901 = 119891 (120588 119864119898) (28)

which defines all the equilibrium states that can exist in amaterial The internal energy can be eliminated from (27)to obtain a pressure (119901) versus volume (119881) relationship orequivalently a 119901 versus 1120588 relationship that is unique tothe material described by the equation of state model Thisequation is uniquely dependent on the material defined bythe equation of state This unique relationship is called theHugoniot curve and it is the locus of 119901-119881 states achievablebehind a shockTheHugoniot pressure119901119867 is only a functionof density and can be defined in general from fittingexperimental data An equation of state is said to be linearin energy when it can be written as

119901 = 119891 + 119892119864119898 (29)

where119891(120588) and119892(120588) are only functions of density and dependon the particular equation of state model Figure 1 illustratesa schematic expression of a Hugoniot curve [35]

32 Mie-Gruneisen Equation of State As mentioned previ-ously an equation of state is used to express the behaviourof Eulerian materials For a Mie-Gruneisen equation of statefor linear energy the most common form is written as [38]

119901 minus 119901119867 = Γ120588 (119864119898 minus 119864119867) (30)

where 119901119867 and 119864119867 are the Hugoniot pressure and specificenergy (per unit mass) and are only functions of density andΓ is the Gruneisen ratio defined as

Γ = Γ0 1205880120588 (31)

Geofluids 7

where Γ0 is a material constant and 1205880 is the reference densityHugoniot energy 119864119867 is related to the Hugoniot pressure 119901119867by

119864119867 = 11990111986712057821205880 (32)

where 120578 is the nominal volumetric compressive strain writtenas

120578 = 1 minus 1205880120588 (33)

Elimination of Γ and119864119867 from the above equations yields

119901 = 119901119867 (1 minus Γ01205782 ) + Γ01205880119864119898 (34)

The equation of state and the energy equation representcoupled equations for pressure and internal energy Abaqussolves these equations simultaneously at each material pointusing an explicit method

33 Linear 119880119904-119880119901 Hugoniot Form Normally the 119880119904-119880119901 for-mulation of equation of state (EOS) is used to simulate shocksin solid materials In this study it is used to define fluidmaterials A common fit to the Hugoniot data is given by

119901119867 = 120588011986220120578(1 minus 119904120578)2 (35)

where 1198880 and 119904 define the linear relationship between thelinear shock velocity 119880119904 and the particle velocity 119880119901 using

119880119904 = 1198880 + 119904119880119901 (36)

The Mie-Gruneisen EOS is normally used for processeswith high pressures and high sound velocities The relation-ship between density and pressure is governed by an EOSmodel A linear 119880119904-119880119901 Hugoniot form of the Mie-Gruneisenequation of state is used to evaluate pressure in the waterin the simulation Using this implementation the water ismodeled as a compressible viscous fluid With the aboveassumptions the linear 119880119904-119880119901 Hugoniot form is written as

119901 = 120588011986220120578(1 minus 119904120578)2 (1 minusΓ01205782 ) + Γ01205880119864119898 (37)

where 120588011988820 is equivalent to the elastic bulk modulus at smallnominal strains

When modeling water Abaqus prescribes 119904 = 0 andΓ0 = 0 Filling this simplifies the relation between pressure119901 and density 120588 to an inversely proportional equation and asimplification of the EOS This simplification suffices for themodeling of water [39]

1199011003816100381610038161003816119904=0andΓ=0 = [ 120588011986220120578(1 minus 119904120578)2 (1 minusΓ01205782 ) + Γ01205880119864119898]

119904=0andΓ=0

= 120588011988820120578(38)

Recalling that 120578 = 1 minus 1205880120588 we get

119901 = 120588011986220 (1 minus 1205880120588 ) (39)

When defining a viscosity for thematerial it is treated as aNewtonian fluid meaning that viscosity only depends on thetemperature This is the default setting in Abaqus Since thetemperature is constant throughout the analysis the viscosityis also constant

There is a limiting compression given by the denominatorof this form of the equation of state The denominator of (34)should not be equal to zero therefore an extreme value isdefined for 120588 and 120578 as

120578lim = 1119904 120588lim = 1199041205880119904 minus 1

(40)

At this limit there is a tensile minimum thereafternegative sound speeds are calculated for the material

The linear Hugoniot form 119880119904-119880119901 equation of state canbe employed to model laminar viscous flows governed byNavier-Stokes equations The volumetric response is gov-erned by the equations of state where the bulk modulus actsas a penalty parameter for the constraint [38] The Eulerianpart (water) is simulated using the linear 119880119904-119880119901 Hugoniotform of the Mie-Gruneisen equation of state

The solving method in Eulerian equations is based onthe division of the governing equations In the Lagrangianmethod the nodes are considered to be fixed relative to thematerial at any time interval and elements are deformed withthe material while in the Eulerian method element defor-mation is calculated at any time interval but deformation isapplied as a remeshingThematerial is also displaced betweeneach element and its neighboring elements at any time inter-val Eulerian-Lagrangian contact allows the Eulerian mate-rials to be combined with traditional nonlinear Lagrangiananalyses In the Eulerian-Lagrangian contact the Lagrangianstructure occupies void regions inside the Eulerian meshThe contact algorithm automatically computes and tracks theinterface between the Lagrangian structure and the Eulerianmaterials A great benefit of this method is that there isno need to generate a conforming mesh for the Euleriandomain In fact a simple regular grid of Eulerian elementsoften yields the best accuracy If the Lagrangian body isinitially positioned inside the Eulerian mesh it must beensured that the underlying Eulerian elements contain voidsafter material initialization During analysis the Lagrangianbody pushes material out of the Eulerian elements that itpasses through and they become filled with void SimilarlyEulerian material flowing towards the Lagrangian body isprevented from entering the underlying Eulerian elementsThis formulation ensures that twomaterials never occupy thesame physical space

In this study CEL analysis allowed a direct coupling tosolve this problem The fact that CEL implementation usesthe full Navier-Stokes equations implies that it was able to

8 Geofluids

handle flow appropriately It is important to note that CELis not structured in the same way as a CFD code and wasnot intended to fill this role While it does include viscouseffects with extension to laminar flows it does not include anyturbulence effects but in the context of an impact which has ashort duration and is highly transient (with turbulence effectsbeing of less importance to the structural loading) CEL iswell suited [28]

4 Implementation ofCoupled Eulerian-Lagrangian (CEL)Method in Abaqus

In order to simulate the water entry problem of the pounderand implement the Coupled Eulerian-Lagrangian (CEL)method in Abaqus a geometric model for the pounder andwater was first created A partitionwas created to definewaterand void regions at the water surface Since the fluid (water)undergoes a large deformation an Eulerian grid is suitableThe rigid body (pounder) has small deformations relative tothe fluid making an Lagrangian grid suitable Therefore thewater wasmodeled as Eulerian and the pounder wasmodeledas solid geometry with a rigid body constraint applied to itThen the material properties for the pounder and water weredefined Properties include density (120588) Youngrsquos modulus (119864)and Poissonrsquos ratio (]) for the pounder and density (120588119908) anddynamic viscosity (120578) for the water In the Coupled Eulerian-Lagrangianmethod when the pounder is considered as rigidYoungrsquos modulus and Poisson ratios values should be definedfor the Abaqus but they have no effect on the results Inthe Coupled Eulerian-Lagrangian method 119880119904-119880119901 equationof state (EOS) was selected to define the water properties inAbaqus The 119880119904-119880119901 equation of state is described in detail inAbaqus 614 documentation and it is used to simulateNavier-Stokes flow when turbulence flow is negligible Becausethe major force imposed on the rigid body in water entryproblems is pressure force this equation of state can beused as an appropriate approximation Parameters such as1198620(speed of sound in water) 119904 (constant coefficient in shockvelocity equation) and Γ0 (material constant in Gruneisenequation of state)were defined for the119880119904-119880119901 equation of stateIn the next step the pounder and water were entered intothe assembly section and the pounder was transferred to thewaters center tangent to its surface In order to simulate a free-falling rigid body into water the pounder was first consideredin a position tangent to the water surface then the secondaryvelocity of pounder at the moment of impact was assignedIn the next step explicit dynamic analysis was selected andthe time period for pounder movement between enteringthe water and reaching the model bed was considered as 05seconds Nlgeom was also activated to control the nonlineareffects of large displacements Then by defining a referencepoint and assigning it to the pounder required outputs suchas displacement and velocity values for the vertical direc-tion were selected Based on its defined algorithm Abaqusconsiders the maximum hardness between the Lagrangianelements (pounder) and Eulerian material (water) The Eule-rian solution method uses the meshes defined for poundermovement between water meshes and Abaqus detects theboundary between the Lagrangian and Eulerian materials

Wall

Wall

Wall Water

Sphere

Initial void region

Freesurface

Figure 2 The computational domain boundary conditions andgrid distribution for the water entry problem of sphere

Considering themaximumhardness between the Lagrangianand Eulerian boundaries the contact between these twomaterials is established using the penalty method Concen-trated mass was assigned to the pounder and its moment ofinertia was defined for 119883 119884 and 119885 In the next step thepounder was defined as a rigid body that can move freelyin all directions Gravitational acceleration (Gravity) wasdefined as minus98ms2 in the 119884 direction and a nonreflectingEulerian boundary was assigned to the sides and bottomsurfaces of the computational water domain This boundarycondition is specific to the Eulerian method and preventswave propagation due to rigid body impact on the waterrsquossurface from returning to the computational domain afterreaching the boundariesWith these boundary conditions thewater is regarded as infinite Assuming that the pounder isdropped from a height 119867 above a free water surface withan initial velocity (1198810) of 0ms its secondary velocity (119881) atthe moment of impact was calculated according to the freefall equation of bodies This velocity was vertically applied tothe pounder Eulerian and void regions were specified usingmaterial assignment and the mesh size was defined for thewater and pounder An Eulerian mesh was selected as thewater mesh and an explicit mesh was selected as the poundermesh using structured techniques Finally the problem wassolved and the results were analyzed

41 Model Geometry Before applying the model to pounderwater entry a three-dimensional numerical model was devel-oped to simulate a transient free-falling wedge impactingthe free surface of water in order to check and calibratethe adopted computational model for simpler problemsThen the accuracy of the numerical model was examinedby comparing the numerical results with the experimen-tal data of Aristoff et al [10] for the water entry of asphere Figure 2 illustrates a schematic for the computationaldomain boundary conditions and grid distribution for

Geofluids 9

Table 1 Equation of state parameters for water

Density (120588119908)(kgm3)

Dynamicviscosity (120578)(kgsm2)

Speed of sound(1198880) (msec) Γ0 1199041000 0001 1450 0 0

Water

Aperture release

Z

Diffuser

Lights

Sphere

Camera

Figure 3 Schematic of the experimental apparatus of Aristoff et al[10]

the sphere water entry problem The dimensions for thecomputational domain are considered to be large enoughso that with further increases no considerable changes areobserved in the numerical resultsThe sphere wasmodeled assolid geometry and a rigid body constraint was subsequentlyapplied to it The fluid domain was modeled as a cubewith dimensions of 30 times 50 times 60 cm3 Mesh refinement wasperformed after assigning it to the computational domaintherefore the Eulerianmesh grid size was gradually decreaseduntil no considerable changeswere observed in the numericalresults The Eulerian domain (water) was modeled usingquarter symmetry and contains 746172 elements The spherewas alsomodeled as a Lagrangian solid using 15000 elementsThe Eulerian domain was divided into two parts includingthe upper and lower regions These regions were defined asvoid and stationary water respectively A solid sphere witha diameter of 254 cm and an initial velocity of 217ms inthe normal direction was placed on the free water surfaceThe sphere can move freely in all directions with six degreesof freedom The physical properties of water used in thenumerical simulations are given in Table 1Γ0 and 119904 are material constants in the Gruneisen equationof state and the constant coefficient in shock velocity equa-tion respectively

5 Validation

A schematic of the experimental apparatus of Aristoff et al[10] is presented in Figure 3 A sphere was held by a cameralike aperture at height 119867 above a water tank The tank haddimensions of 30 times 50 times 60 cm3 and was illuminated by a

bank of twenty 32W fluorescent bulbs A diffuser was usedto provide uniform lighting and care was taken to keep thewater surface free of dust Tank dimensions were selectedso that the walls had negligible effects on the results Thesphere was released from rest and fell towards the waterreaching it with an approximate speed119881 asymp radic2119892ℎThe impactsequence was recorded at 2000 frames per second using ahigh-speed camera The resolution was set to 524 times 1280pixels (px) with a field of view of 1128 times 2755 cm2 yieldinga 4646 pxcm magnification The trajectory of the sphereand its impact speed were determined with subpixel accuracythrough a cross correlation andGaussian peak-fittingmethodyielding position estimates accurate to 0025 px (00005 cm)and impact speeds accurate to plusmn4

Figure 4 illustrates a comparison of cavity shapes betweennumerical simulation and the experimental photographstaken by Aristoff et al [10] for a density ratio of 786 and ansimilar period of time The impact velocity of the sphere was217ms and 119905 = 0 refers to when the spheres center passesthrough the free surface This figure shows the cavity shapesfor both experimental and numerical simulation for similartime periods except that cavity shape for the pinch-off timeof the numerical simulation was added to the pictures

An symmetric air cavity formed behind the sphereafter entering the water Air cavity formation consists ofseveral steps including cavity creation cavity expansioncavity contraction behind the sphere and cavity collapseAs the sphere moves deeper through the water it exerts aforce on surrounding fluid in radial directions and transfersits momentum to the fluid However fluid expansion isconfronted with fluid hydrostatic pressure resistance and asa result the radial flow direction is reversed leading to cavitycontraction and collapse Cavity collapse occurs when thecavity wall moves inward until the moment of pinch-off (119905= 708ms) and the cavity is divided into two separated parts[34] While the lower cavity sticks to the sphere and movesalong with it the upper cavity continues its contraction andmoves towards the waterrsquos surface As shown in Figure 4 thenumerical results and experimental observations are in goodagreement

Abaqus numerical simulation methods efficiently modelwater splashes and jet flow Pictures of these phenomenaare depicted in Figure 4 It was concluded from this figurethat pinch-off in the numerical simulation happens with adelay time compared to the experimental method Increasednumerical errors during solution time are one of the causesfor differences between experimental and numerical results

In Figure 5 the numerical results of this study arecompared with the theoretical and experimental data ofAristoff et al as a displacement-time graph of a sphere indifferent water depths [10] As can be seen the numericalresults are in good agreement with those of the analytics andexperiments However the presently computed sphere centerdepth is lower than that found experimentally It seems thatdue to lack of turbulent modeling ability in Abaqus Eulerianformulation and evidently increasing turbulence in flow bytime there is just less difference between experiment andnumerical results in the early time of simulation

10 Geofluids

Experimental results of Aristoff et al (2010)

Pinch-off

Pinch-offNumerical simulation results

t = 59 ＧＭ 129 ＧＭ 199ＧＭ 269ＧＭ 339ＧＭ 409 ＧＭ 479ＧＭ 549ＧＭ 619 ＧＭ 689 ＧＭ 759ＧＭ

t = 59ＧＭ 129 ＧＭ 199ＧＭ 269ＧＭ 339ＧＭ 409 ＧＭ 479ＧＭ 549ＧＭ 619 ＧＭ 689 ＧＭ 708ＧＭ 759ＧＭ

Figure 4 Comparison of numerical results with experimental photographs of Aristoff et al [10] for the sphere water entry problem

002 004 006 0080Time (s)

minus018

minus016

minus014

minus012

minus01

minus008

minus006

minus004

minus002

0

Disp

lace

men

t (m

)

Present studyExperiment (Aristoff et al 2010)Theory (Aristoff et al 2010)

Figure 5 Comparison of displacement-time graph of sphere inwater depth with the theoretical and experimental data of Aristoffet al [10]

6 Materials and Methods

61 Eulerian Model Definition Figure 6 illustrates a sche-matic of the computational domain boundary conditionsand grid distribution for the water entry problem of aspherical pounder The dimensions of the computationaldomain are considered to be large enough so that withfurther increases no considerable changes are observed inthe numerical results The Eulerian domain was dividedinto the upper and lower regions These regions are definedas void and stationary water respectively In water entryproblems for rigid bodies fluid domain dimensions must

Wall

Wall

Wall Water

Spherical pounder

Initial void region

Free surface

Figure 6 The computational domain boundary conditions andgrid distribution for water entry problem of spherical pounder

be at least eight times the dimensions of the rigid bodyHence in this study the fluid domain was modeled as acube with dimensions of 2 times 2 times 1m3 The pounders wereplaced exactly tangent to the water surface and their velocityat the moment of water surface impact was determinedaccording to the free fall equation of bodies The pounderscan move freely in all directions with six degrees of freedom119880119904-119880119901 equation of state was used to define water materialin the Coupled Eulerian-Lagrangian method This equationof state is useful for Navier-Stokes flow simulations whenturbulence flow is negligible In cases where the impact ofa rigid body on a water surface is investigated the majorforce imposed on water body is the pressure force Thereforewith appropriate approximations this equation of state canbe used to investigate the impact of rigid bodies on watersurfaces

62 EulerianMesh Size As shown in Figure 7 with decreasesin the Eulerian domain mesh size to less than 15 cm thedisplacement-time graph of the spherical pounder showsnegligible changes Therefore to improve analysis time the

Geofluids 11

Table 2 Moment of inertia of pounders with different geometric shapes

Geometric shape of the pounder Cube Cylinder Sphere Pyramid ConeMoment of inertia (cm4) 685 5137 411 3082 411

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

01 02 03 040Time (s)

2 ＝Ｇ

175 ＝Ｇ

15 ＝Ｇ

Figure 7 Comparison of the effect of the Eulerian mesh size ondisplacement-time graph of spherical pounder

Eulerian domain was modeled using quarter symmetry witha mesh size of 15 cm and contains 1538943 elements

63 Lagrangian Model Definition Figure 8 illustratespounder dimensions for different geometric shapes Thesepounders are identical in terms of material mass densityYoungrsquos modulus Poissonrsquos ratio volume mass and dropheight from the water surface The only difference is theirgeometric shapeThese parameters were regarded as identicalin all pounders to investigate the effect of pounder shapeon deepwater displacement and velocity air cavity shapepinch-off time and depth and the effect of water drag forceon velocity depreciation when impacting the model bed

Pounder dimensions were selected so that they had avolume of 523 cm3 and equal mass due to their identicalmaterial and mass density Table 2 illustrates the moment ofinertia for pounders with different geometric shapes

Table 3 illustrates thematerial properties of the poundersThese parameters are regarded as identical for all pounderswith the mentioned geometric shapes

64 Lagrangian Mesh Size Due to the negligible effect ofLagrangian mesh size on the results and to improve analysistime the Lagrangian mesh size for all pounders was set to05 cm

Pyramid Cone

Cube Cylinder Sphere

10 ＝Ｇ10 ＝Ｇ10 ＝Ｇ 10 ＝Ｇ

10 ＝Ｇ10 ＝Ｇ10 ＝Ｇ

10 ＝Ｇ5236 ＝Ｇ

20 ＝Ｇ15708 ＝Ｇ

666 ＝Ｇ

Figure 8 The dimensions of pounders with different geometricshapes

Table 3 Material properties of pounders

Materialtype Mass (kg) Density (120588)

(kgm3)

Youngrsquosmodulus (119864)(kgm2)

Poissonrsquosratio (120592)

Steel 411 7850 21 times 1010 03

65 Boundary Condition Wall boundary conditions wereapplied to the sides and bottom of the Eulerian domain tosimulate a water tank Pounder displacement was not con-strained in any direction and the pounders could freely movein all directions The sides and bottom boundaries of thethree-dimensional Eulerian domain were set as nonreflectingboundaries so that the waves released due to pounder impacton water surface did not return to the computational domainafter reaching these boundaries and being absorbed by themThis boundary condition is written as [37]

119889119901 minus 119901119888119889119906 = 0119889119909119889119905 = minus119888 (41)

where 120588 is density 119888 is the speed of sound in the fluid 119901 ispressure 119906 is velocity perpendicular to the wave and 119909 is thedirection perpendicular to the boundary

66 Drop Height and Impact Velocity of Pounders Afterdropping the pounders from a height (ℎ) above the watersurface the velocity of the pounders at themoment of impactwas determined using

1198812 minus 11988120 = 2119892ℎ (42)

where1198810 is the pounder initial velocity before being droppedfrom height ℎ above the waterrsquos surface 119881 is pounder

12 Geofluids

01 02 03 04 050Time (s)

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

CubeCylinderSphere

PyramidCone

Figure 9 Displacement-time graph of pounders with differentgeometric shapes

secondary velocity when impacting the water surface and 119892is the gravitational acceleration of the Earth

From (42) which expresses a time independent equationfor the free fall of objects the initial velocity and the terminalvelocity of the object drop height and the gravitationalacceleration of the Earth are interdependent in the absenceof time In this numerical simulation the pounders weredropped from a height of 1m above thewaterrsquos surfacewith aninitial velocity (1198810) equal to 0ms Therefore the secondaryvelocity of the pounders when impacting the water surfacewas obtained using

119881 = radic2119892ℎ (43)

Instead of simulating a full dropping event from theinitial position the pounders were placed exactly tangent towaterrsquos surface and their secondary velocity at the momentof impact was determined according to (43) A gravitationalacceleration (gravity) of minus98ms2 was applied in the normaldirection of the pounders Therefore a secondary velocityof 443ms in the normal direction of the waterrsquos surfacecorresponds to the speed that would be obtained for thepounders

7 Results and Discussion

71 Effect of the Geometric Shape of the Pounder on Deep-water Displacement Figure 9 illustrates the displacement-time graph of pounders with different geometric shapes Thegraph continues for the cubical pounder from the momentof water surface impact at time zero up to the depth of 1mat 415ms The displacement-time graph of the cylindrical

pounder indicates that after entering the water it reachesthe model bed in a shorter amount of time than the cubicalpounder The graph for this pounder continues from themoment of water surface impact at time zero up to the depthof 1m at 355ms The linear displacement-time graph forthe spherical and pyramidal pounders indicates the fasterdeepwater movement of these pounders compared to thecubical and cylindrical pounders The deepwater movementtime of the spherical and pyramidal pounders after enteringthe water to themodel bed is 285ms and 265ms respectivelyThe linear displacement-time graph of the conical pounderindicates that it has a faster deepwater movement comparedto other pounders as it takes 235ms to reach the model bedafter entering the water

72 Mechanism of Air Cavity Formation and Pinch-Off dueto Deepwater Movement of Pounders with Different Geomet-ric Shapes Figure 10 illustrates the movement trajectory ofpounders with different geometric shapes from the momentof water impact towhen the pounders reached themodel bedAfter entering the water a symmetric air cavity is formedbehind the pounders Air cavity formation involves severalsteps including air cavity creation and expansion behind thepounder air cavity contraction and air cavity collapse [34]As the pounder moves downward in water depth it imposesa force onto surrounding fluid in its radial directions andtransfers its momentum to the fluid This extension is facedwith the hydrostatic pressure resistance of fluidThe directionof the radial flow is reversed and eventually leads to aircavity contraction and collapse In the other words the aircavity contraction is accelerated until pinch-off occurs andthe air cavity is divided into two distinct parts The upperpart of the air cavity continues its contraction and movestowards the waterrsquos surface whereas the lower part of the aircavity clings to the pounder and moves along with it [34]After impacting the water surface the cubical and cylindricalpounders due to their flat contact surfaces lose some of theirvelocityThis velocity depreciation slows down and continuesuntil pinch-offThe spherical pounder with its curved contactsurface has less velocity depreciation compared to the cubicaland cylindrical pounders However some pounder velocityis always depreciated due to impacting the water surfaceThe velocity depreciation process for the spherical pounderduring deepwater movement continues until pinch-off Afterimpacting the waterrsquos surface the pyramidal pounder with itspointy contact surface has a slight velocity depreciation butcontinues to move deeper into the water with approximatelythe same secondary velocity until pinch-off By imposing thedrag force of water onto the pounders surface its velocityis gradually decreased until it reaches the model bed Afterimpacting the water surface the conical pounder with itspointy contact surface continues to move deeper into thewater with an increasing velocity The increasing velocity ofthe conical pounder during deepwater movement continueseven after pinch-off When the pounder moves towards themodel bed the waters drag force is imposed onto poundersurface causing velocity depreciation The pinch-off time forthe pounders with the above-mentioned shapes is between225 and 250ms

Geofluids 13

t = 0 25ＧＭ 50ＧＭ 100ＧＭ 200ＧＭ 225ＧＭ 250ＧＭ 275ＧＭ 300ＧＭ 325ＧＭ 350ＧＭ150ＧＭCu

beCy

linde

rSp

here

Pyra

mid

Con

e

Figure 10 Air cavity formation due to deepwater movement of pounders with different geometric shapes and pinch-off time and depth

73 Effect of the Pounder Geometric Shape onDeepwater Veloc-ity Figure 11 illustrates the velocity-time graph for pounderswith different geometric shapes The cubical and cylindricalpounders lose some of their secondary velocity at the earlymoments of water entry Hence the velocity-time graph ofthese pounders illustrates great velocity depreciation up to15ms The velocity reduction of the cubical and cylindricalpounders gradually continues until pinch-off The velocityof the spherical pounder decreases up to 45ms after waterentry but its velocity depreciation is less than the cubicaland cylindrical pounders This velocity reduction continuesat a slower rate until pinch-off In the initial moments ofwater entry the pyramidal pounder has a slight reduction insecondary velocity up to 55ms Then velocity depreciationcontinues at a faster rate due to the drag force imposed onto

pounders surface and the velocity-time graph of the pyrami-dal pounder shows a gradual velocity reduction until pinch-off After impacting the water surface the conical poundercontinues its deepwater movement with an increasing veloc-ity rate Due to its very low slope the velocity-time graphindicates an increase in velocity for the conical pounder butdue to the waters drag force its velocity begins to decrease at145ms The velocity-time graph indicates a gradual decreasein velocity for conical pounder and velocity depreciationcontinues until pinch-off After pinch-off pounder velocitiesgradually decrease until they reach themodel bed Eventuallywith a velocity of 388ms and 157ms at the moment ofmodel bed impact the conical and cubical pounders have thehighest and lowest velocity respectively Water drag force isimposed in the opposite direction of poundermovement and

14 Geofluids

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Velo

city

(ms

)

01 02 03 040Time (s)

CubeCylinderSphere

PyramidCone

Figure 11 Velocity-time graph of pounders with different geometricshapes

it has the most and least influence on the cubical and conicalpounders respectively

8 Critical Drop Height of the Pounder

In the dynamic compaction of the seabed determining thecritical drop height of the pounder from the freewater surfaceis an effective parameter for the pounder to reach maximumdeepwater velocity After dropping the pounder from height119867 above the free water surface with an initial velocity (1198810)of 0ms its velocity gradually increases until it reaches thesecondary velocity (119881) at the moment of water impact Afterentering the water pounder velocity is decreased due to waterdrag force According to the free fall equation for objectsincreasing the drop height from the waterrsquos surface leads toincreased pounder velocity at the moment of water surfaceimpact but it does not necessarily lead to an increase inpounder velocity during seabed impact because the criticaldrop height of the pounder is a determining factor ofits maximum deepwater velocity According to Figure 12 aconical pounder with a mass of 411 kg was dropped fromdifferent heights (119867) and its velocity when impacting thewaters was measured according to the time independentequation for the free fall of objects Table 4 illustrates thevelocity of the conical pounder with different drop heightswhen impacting the waterrsquos surface

81 Effect of the Drop Height of the Pounder from WaterSurface on Deepwater Displacement Figure 13 illustrates thedisplacement-time graph of the conical pounder with dif-ferent drop heights from the waterrsquos surface As can beseen increasing the drop height of the pounder leads to

Water levelℎ

Figure 12 Schematic of drop height of the conical pounder fromwater surface

Table 4 Velocity of the conical pounder with different drop heightsat the moment of impacting the water surface

Drop height of the pounderfrom water surface (m)

Pounder velocity at themoment of impacting water

surface (ms)025 22105 313075 3831 443125 49515 542175 5852 626225 66425 7

displacement-time graphs that are closer to each other Infact an increase in the drop height of the pounder from thewaterrsquos surface leads to an increase in its secondary velocitywhen impacting the waterrsquos surface Hence the pounderenters the water with a higher velocity and reaches the modelbed in less time

82 Effect of the Drop Height of the Pounder from WaterSurface on Its Deepwater Velocity Increasing the drop heightof the pounder from the waterrsquos surface leads to an increasein its velocity before impacting the water but the poundervelocity gradually decreases after entering the water due tothe its drag force Figure 14 illustrates the velocity-time graphfor the conical pounder with different drop heights from thewaterrsquos surface When the pounder is dropped from a lowheight due to its low velocity it moves deepwater due to itsweight Moreover because of the low velocity of the pounderwhen entering the water pinch-off occurs in less time and atlower depths and its velocity increases after pinch-off until itreaches terminal velocity Increases in pounder drop heightsincrease water entry velocity Hence it will have greatervelocity depreciation in the first moments of water entry

Figure 15 illustrates the terminal velocity-time graph forthe conical pounderwith different drop heightswhen impact-ing the model bed As can be seen increasing the drop height

Geofluids 15

01 02 03 040Time (s)

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

H = 1Ｇ

H = 075Ｇ

H = 05 Ｇ

H = 025 Ｇ

H = 175Ｇ

H = 15 Ｇ

H = 125 Ｇ

H = 25 Ｇ

H = 225 Ｇ

H = 2Ｇ

Figure 13 Displacement-time graph of the conical pounder withdifferent drop heights

of the pounder from the waterrsquos surface leads to an increasein its terminal velocity when impacting the model bedTheseconditions are dominant as long as the pounder reaches acritical drop height from the water surface but increasingthe pounder drop height higher than the critical drop heightleads to decreased terminal velocity when impacting themodel bed In this numerical simulation after dropping thepounder from a height of 2m above the waterrsquos surface itssecondary velocity was 626ms when impacting the watersurface After entering the water the secondary velocity ofthe pounder decreased until it impacts the model bed witha terminal velocity of 454ms Therefore a drop height of2m from above the waterrsquos surface has the highest terminalvelocity when the pounder impacts themodel bed Increasingthe drop height bymore than 2m leads to decreased pounderterminal velocity when impacting the model bed The heightwhich has the greatest effect on pounder terminal velocitywhen impacting the model bed is called the critical dropheight Increasing the drop height of the pounder to greaterthan its critical drop height leads to decreased terminalvelocity when impacting the model bed

Figure 16 illustrates the secondary and terminal velocityof the conical pounder with different drop heights As canbe observed increasing the drop height of the pounder fromwaterrsquos surface leads to increased differences between thesecondary and terminal velocities of the pounder Increasingthe drop height to more than the critical drop height leadsto increased pounder secondary velocity when impactingthe waterrsquos surface and decreased terminal velocity whenimpacting the model bed Hence the difference betweenthe secondary and terminal velocities of the pounder is

01 02 03 040Time (s)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)H = 175Ｇ

H = 125 Ｇ

H = 075Ｇ

H = 025 Ｇ H = 225 Ｇ

H = 1Ｇ

H = 05 Ｇ

H = 25 Ｇ

H = 2Ｇ

H = 15 Ｇ

Figure 14 Velocity-time graph of the conical pounder with differentdrop heights

minus430

minus447 minus454minus415

minus426 minus380

minus367minus355

minus332minus310

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Term

inal

velo

city

(ms

)

01 02 03 040Time (s)

Figure 15 Terminal velocity of conical pounder with different dropheights versus time

not reasonable and a significant amount of the pounderssecondary velocity is depreciated when entering the waterThe great difference between the secondary and terminalvelocities of the pounder reveals the necessity of determiningthe critical drop height of the pounder before dynamic seabedcompaction operations Increasing the pounder drop heightto be greater than the critical drop height not only hasa reverse result on pounder velocity when impacting theseabed but it also requires special equipment such as lengthy

16 Geofluids

Secondary velocityTerminal velocity

025 05 075 1 125 15 175 2 225 250Drop height (m)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)

Figure 16 Secondary and terminal velocities of conical pounderversus drop heights

1 2 3 4 5 6 7 80Impact velocity (ms)

0

005

01

015

02

025

03

Pinc

h-off

tim

e (s)

Figure 17 Pinch-off time versus impact velocity of conical pounder

cranes and huge barges leading to increased implementationcosts

Figure 17 illustrates pinch-off time based on the sec-ondary velocity of the pounder at the moment of watersurface impact As can be seen pinch-off time is a weakfunction of pounder impact velocity and increasing impactvelocity leads to a slight reduction in pinch-off time Thisbehaviour was also observed in the analytical and numericalresults found by Lee et al [40]

Figure 18 illustrates pinch-off depth based on the sec-ondary velocity of pounder at the moment of water surfaceimpact As can be seen the pinch-off depth is a linear

1 2 3 4 5 6 7 80Impact velocity (ms)

minus12

minus1

minus08

minus06

minus04

minus02

0

Pinc

h-off

dep

th (m

)

Figure 18 Pinch-off depth versus impact velocity of conicalpounder

function of pounder impact velocity These results are in linewith the analytical equations of Lee et al [40]

9 Conclusions

In this study the water entry problem of three-dimensionalpounders with different geometric shapes of cube cylindersphere pyramid and cone was numerically simulated bycommercial finite element code Abaqus 614-2 An explicitdynamic analysis method was employed to model the fluid-structure interaction using a Coupled Eulerian-Lagrangian(CEL) method Before applying the model to water entryof the pounders a three-dimensional numerical model wasdeveloped to simulate a transient free-falling wedge impact-ing the free surface of the water in order to calibrate theadopted computationalmodelThe accuracy of the numericalmodel was examined by comparing numerical results withthe experimental data of Aristoff et al The good agreementbetween numerical simulation results with the theoreticaland experimental data of Aristoff et al demonstrates thecapability of the employed numerical methodology to sim-ulate the water entry problem for three-dimensional geome-tries with acceptable accuracy After water entry by a pounderdropped fromaheight of 1m a symmetric air cavity is formedbehind the pounder As the pounder moves downward inwater depth it imposes a force onto surrounding fluid inthe radial directions and transfers its momentum to thefluid This extension is faced with the hydrostatic pressureresistance of the fluid causing the direction of the radialflow to be reversed leading to air cavity contraction andcollapse The results show that the water drag force had thehighest and lowest effect on the cubical and conical poundersrespectively and increasing the drop height to greater thanthe critical drop height leads to increased pounder velocitywhen impacting the waterrsquos surface and decreased velocitywhen impacting the model bed Pinch-off time is a very weak

Geofluids 17

function of pounder velocity when impacting the waterrsquossurface but the pinch-off depth increases linearly along withincreased pounder impact velocities It is recommended thatthe critical drop height for the pounder is determined beforedynamic seabed compaction operations and the adjustmentof crane heights to this height In addition it is also recom-mended that perforated pounders be used in order to reducevelocity depreciation when impacting the waterrsquos surface andto reduce the effect of water drag force

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors warmly appreciate the cooperation and guidanceof Dr B Hamidi and Dr A Mehrabadi in this study

References

[1] B Hamidi J-M Debats H Nikraz and S Varaksin ldquoOffshoreground improvement recordsrdquo Australian Geomechanics Jour-nal vol 48 no 4 pp 111ndash122 2013

[2] SWatanabe ldquoResistance of impact onwater surface part I-cone Institute of Physical andChemical ResearchrdquoTokyo vol 12 pp251ndash267 1930

[3] S Watanabe ldquoResistance of impact on water surface part II-cone (continued)rdquo Institute of Physical and Chemical Researchvol 14 pp 153ndash168 1930

[4] A May and J C Woodhull ldquoDrag coefficients of steel spheresentering water verticallyrdquo Journal of Applied Physics vol 19 no12 pp 1109ndash1121 1948

[5] A May ldquoEffect of surface condition of a sphere on its water-entry cavityrdquo Journal of Applied Physics vol 22 no 10 pp 1219ndash1222 1951

[6] AMay ldquoVertical entry ofmissiles into waterrdquo Journal of AppliedPhysics vol 23 no 12 pp 1362ndash1372 1952

[7] A P New T S Lee and H T Low ldquoImpact loading and waterentrance characteristics of prismatic bodiesrdquo in Proceedings ofthe third international offshore and polar engineering conferencepp 282ndash287 1993

[8] M Anghileri and A Spizzica ldquoExperimental validation of finiteelement models for water impactsrdquo in Proceedings of the secondinternational crash usersrsquo seminar Cranfield Impact Centre LtdEngland UK 1995 second international crash users seminar

[9] A Engle andR Lewis ldquoA comparison of hydrodynamic impactsprediction methods with two dimensional drop test datardquoMarine Structures vol 16 no 2 pp 175ndash182 2003

[10] J M Aristoff T T Truscott A H Techet and J W M BushldquoThe water entry of decelerating spheresrdquo Physics of Fluids vol22 no 3 Article ID 009002PHF pp 1ndash8 2010

[11] T Von Karman The impact of seaplane floats during landingNational Advisory Committee for Aeronautics NACA TN no321 1929

[12] HWagner ldquoPhenomena associatedwith impacts and sliding onliquid surfacesrdquo Journal of Applied Mathematics and Mechanicsvol 12 pp 193ndash215 1932

[13] W S Vorus ldquoA flat cylinder theory for vessel impact and steadyplaning resistancerdquo Journal of Ship Research vol 40 no 2 pp89ndash106 1996

[14] T Miloh ldquoOn the initial-stage slamming of a rigid sphere in avertical water entryrdquo Applied Ocean Research vol 13 no 1 pp13ndash43 1991

[15] J M Aristoff and JW Bush ldquoWater entry of small hydrophobicspheresrdquo Journal of Fluid Mechanics vol 619 pp 45ndash78 2009

[16] P Ghadimi A Dashtimanesh and S R Djeddi ldquoStudy of waterentry of circular cylinder by using analytical and numericalsolutionsrdquo Journal of the Brazilian Society ofMechanical Sciencesamp Engineering vol 34 no 3 pp 225ndash232 2012

[17] A TassinD J PiroAAKorobkinK JMaki andM J CookerldquoTwo-dimensional water entry and exit of a body whose shapevaries in timerdquo Journal of Fluids and Structures vol 40 pp 317ndash336 2013

[18] E Yao H Wang L Pan X Wang and R Woding ldquoVerticalwater-entry of bullet-shaped projectilesrdquo Journal of AppliedMathematics and Physics vol 02 no 06 pp 323ndash334 2014

[19] M-S Park Y-R Jung and W-G Park ldquoNumerical study ofimpact force and ricochet behavior of high speed water-entrybodiesrdquo Computers amp Fluids vol 32 no 7 pp 939ndash951 2003

[20] D Battistin and A Iafrati ldquoHydrodynamic loads during waterenergy of two-dimensional and axisymmetric bodiesrdquo Journalof Fluids and Structures vol 17 no 5 pp 643ndash664 2003

[21] A A Korobkin and M Ohkusu ldquoImpact of two circular platesone of which is floating on a thin layer of liquidrdquo Journal ofEngineering Mathematics vol 50 no 4 pp 343ndash358 2004

[22] K M Kleefsman G Fekken A E Veldman B Iwanowski andB Buchner ldquoA volume-of-fluid based simulation method forwave impact problemsrdquo Journal of Computational Physics vol206 no 1 pp 363ndash393 2005

[23] Y Kim Y Kim Y Liu and D K P Yue ldquoOn the water-entryimpact problem of asymmetric bodiesrdquo in Proceedings of the9th International Conference on Numerical Ship Hydrodynamics(NSH rsquo07) August 2007

[24] Q Yang and W Qiu ldquoNumerical solution of 2D slammingproblem with a CIPmethodrdquo in Proceedings of the InternationalConference on Violent Flows Research Institute for AppliedMechanics Kyushu University Japan 2007

[25] A C Fairlie-Clarke and T Tveitnes ldquoMomentum and gravityeffects during the constant velocitywater entry ofwedge-shapedsectionsrdquo Ocean Engineering vol 35 no 7 pp 706ndash716 2007

[26] Q Yang andW Qiu ldquoNumerical simulation of water impact for2D and 3D bodiesrdquo Ocean Engineering vol 43 pp 82ndash89 2012

[27] G Wu ldquoNumerical simulation for water entry of a wedge atvarying speed by a high order boundary element methodrdquoJournal of Marine Science and Application vol 11 no 2 pp 143ndash149 2012

[28] M Ahmadzadeh B Saranjam A Hoseini Fard and A RBinesh ldquoNumerical simulation of sphere water entry prob-lem using Eulerian-Lagrangian methodrdquo Applied MathematicalModelling vol 38 no 5-6 pp 1673ndash1684 2014

[29] V-T Nguyen D-T Vu W-G Park and Y-R Jung ldquoNumericalanalysis of water impact forces using a dual-time pseudo-compressibility method and volume-of-fluid interface trackingalgorithmrdquo Computers amp Fluids vol 103 pp 18ndash33 2014

[30] S Kim and N Kim ldquoIntegrated dynamics modeling forsupercavitating vehicle systemsrdquo International Journal of NavalArchitecture and Ocean Engineering vol 7 no 2 pp 346ndash3632015

18 Geofluids

[31] V-T Nguyen D-T Vu W-G Park and C-M Jung ldquoNavier-Stokes solver for water entry bodies with moving Chimera gridmethod in 6DOF motionsrdquo Computers amp Fluids vol 140 pp19ndash38 2016

[32] M Mirzaei M Eghtesad and M Alishahi ldquoPlaning forceidentification in high-speed underwater vehiclesrdquo Journal ofVibration and Control vol 22 no 20 pp 4176ndash4191 2016

[33] G Qiu S Henke and J Grabe ldquoApplication of a CoupledEulerian-Lagrangian approach on geomechanical problemsinvolving large deformationsrdquoComputersampGeosciences vol 38no 1 pp 30ndash39 2011

[34] M R Erfanian M Anbarsooz N Rahimi M Zare andM Moghiman ldquoNumerical and experimental investigationof a three dimensional spherical-nose projectile water entryproblemrdquo Ocean Engineering vol 104 pp 397ndash404 2015

[35] W M Isbell F H Shipman and A H Jones ldquoHugoniotequation of state measurements for eleven material to fiveMegabars Materials and Structural Laboratory ManufacturingDevelopments General Motors Technical centerrdquo Tech RepMichigan 1968

[36] M A Jandron R C Hurd J L Belden A F Bower W Fennelland T T Truscott ldquoModeling of hyperelastic water-skippingspheres using AbaqusExplicitrdquo in Proceedings of the SIMULIACommunity Conference 2014

[37] H Forouzani B Saranjam R Kamali and A Abdollahi-far ldquoElasto-plastic time dependent impact analysis of highspeed projectile on water surfacerdquo Journal of Solid and FluidMechanics vol 3 pp 281ndash298 2016

[38] M R Erfanian andMMoghiman ldquoNumerical and experimen-tal investigation of a projectile water entry problem and studyof velocity effect on time and depth of pinch-offrdquo Journal ofModares Mechanical Engineering vol 15 pp 53ndash60 2015

[39] A Sillem Feasibility study of a tire hydroplaning simulation in afinite element code using a coupled Eulerian-Lagrangian method[MSc thesis] 2008

[40] M Lee R G Longoria and D E Wilson ldquoCavity dynamics inhigh-speed water entryrdquo Physics of Fluids vol 9 no 3 pp 540ndash550 1997

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Submit your manuscripts atwwwhindawicom

Geofluids 5

The external force vector is also derived by a directsubstitution of the appropriate relations from (6)

119865ext = int119881120588119887120575119909 119889119881 + int

119878120591

120591120575119909 119889119878 (6)

119865ext = int119881120588119887sum119861

119873119861120575119909119861119889119881 + int119878120591

120591sum119861

119873119861120575119909119861119889119878 (7)

119865ext = sum119861

int119881120588119887119873119861119889119881 + int

119878120591

120591119873119861119889119878 120575119909119861 (8)

The internal force vector requires the evaluation of thevirtual strain 120575 Again using the relations from (9)

120575120576119894119895 = 12 ( 120597119909119894120597119909119895 +120597119909119895120597119909119894 ) (9)

120575120576119894119895 = sum119861

12 (120597119873119861120597119909119895 120575119909119861119894 +120597119873119861120597119909119894 120575119909119861119895) (10)

These relations may be put in matrix form which definesthe 119861matrix 119861 as

120576120575 = 119861120575119909 (11)

and the internal force may be written as

119865int = int119881120590 120575120576 119889119881

119865int = sum119861

int119881119861119861120590119889119881 120575119909119861

(12)

where 119861119861 is the section of the 119861matrix associated with node119861 To simplify the mass matrix is defined as

119872119888119860119861 = [int119881120588119873119860119873119861119889119881] (13)

where 119872119888 is called the consistent mass matrix because itsderivation is consistent with the finite element method Allthe terms in the weak form have now been written in terms ofunknown coordinates and virtual displacements Collectingthe terms gives

sum119861

119861minus 1119872119861 (int119881 120588119887119873119861119889119881 + int

119878120591

120591119873119861119889119878 minus int119881119861119861120590119889119881)

sdot 120575119909119861 = 0(14)

Since the virtual displacements are arbitrary the 119861 valueis in accordance with

119861 = 1119872119861 (int119881 120588119887119873119861119889119881 + int119878120591

120591119873119861119889119878 minus int119881119861119861120590119889119881) (15)

24 Governing Equations in Eulerian Method For thedescription of fluid flows usually Eulerian formulation isemployed because we are usually interested in the propertiesof the flow at certain locations in the flow domain InEulerian formulation an independent equation is used toexpress conservation of mass Inertia forces also includedisplacement sentences that when present in numericalsolution lead to a matrix with nonsymmetric coefficients thatdepends on calculated velocities Eulerian formulation allowsfor the use of simple strain and stress rates and velocitycalculations instead of displacement values (in the ultrasmalldisplacement analysis) [36 37] Linear viscous isotropic fluidsare known as Newtonian fluids which are by far the mostimportant fluids for practical applications Newtonian fluidsare characterized by the followingmaterial law for theCauchystress tensor 119879

119879119894119895 = 120583( 120597V119894120597119909119895 +120597V119895120597119909119894 minus

23 120597V119896120597119909119896 120575119894119895) minus 119901120575119894119895 (16)

where the velocity vector is V119894 with respect to Cartesiancoordinate 119909119894 the pressure is 119901 the dynamic viscosity is 120583and the Kronecker symbol is 120575119894119895 With this the conservationlaws for mass momentum and energy take the form of

120597119901120597119905 +120597 (119901V119894)120597119909119894 = 0 (17)

120597 (119901V119894)120597119905 + 120597 (119901V119894V119895)120597119909119895= 120597120597119909119895 [120583(

120597V119894120597119909119895 +120597V119895120597119909119894 minus

23 120597V119896120597119909119896 120575119894119895)] minus120597119901120597119909119895 + 120588119891119894

(18)

120597 (119901119890)120597119905 + 120597 (119901V119894119890)120597119909119894= 120583[ 120597V119894120597119909119895 (

120597V119894120597119909119895 +120597V119895120597119909119894) minus 23 ( 120597V119894120597119909119894)

2] minus 119901 120597V119894120597119909119894+ 120597120597119909119894 (119896

120597119879120597119909119894) + 120588119902

(19)

where 120588 is the fluid density 119890 is the specific internal energyand119891119894 and 119902 are external forces and heat sources respectivelyIn energy balance (19) for the heat flux vector ℎ119894 Fourierrsquoslaw with thermal conductivity 120581119891 is used that is a constantspecific heat capacity is assumed and the work done bypressure and friction forces is neglected

ℎ119894 = minus120581119891 120597119879120597119909119894 (20)

System (17)ndash(19) has to be completed by two equations ofstate of the form

119901 = 119901 (120588 119879) 119890 = 119890 (120588 119879) (21)

6 Geofluids

which define the thermodynamic properties of the fluid thatis the thermal and caloric equation of state respectivelyWhen defining the viscosity for the material it is treated as aNewtonian fluid meaning that the viscosity depends only ontemperature This is the default setting in Abaqus Since thetemperature is constant throughout the analysis the viscosityalso is constant

25 Explicit Time-Dependent Solution There are implicitand explicit algorithms for dynamic analysis In the implicitmethod the Newmark method is used the matrices need tobe reversed andmore implementation time is requiredHow-ever this method enjoys high accuracy and its convergenceto linear problems is guaranteed In the explicit method thematrices do not need be reversed and the analysis speed ismuch higher Convergence in the explicit method is achievedif the time step is less than the smallest element divided bythe speed of sound in the material [37]

The solution was advanced in time using the centraldifference method which is also called the Verlet or leap frogmethod It is based on the central difference approximationfor evaluating the derivative of a function 119891(119905) at 119905

119889119891 (119905)119889119905 = 119891 (119905 + (12) Δ119905) minus 119891 (119905 minus (12) Δ119905)Δ119905 (22)

where Δ119905 is the time stepUsing (15) acceleration in time 119905 can be written as

119905 = 119865119905119872119861 (23)

Now using (22) for velocity-acceleration velocity at time119905 + (12)Δ119905 is obtained as

119905+(12)Δ119905 = 119905minus(12)Δ119905 + 119905Δ119905 (24)

Then using (22) for velocity-displacement displacementat time 119905 + Δ119905 is obtained as

119909119905+Δ119905 = 119909119905 + 119905+Δ119905Δ119905 (25)

The central difference method has its step size limited bystability considerations like all explicit integration methodsand the stable time step size was limited to

Δ119905 le 119897119888 (26)

where 119897 is the length of the shortest element and 119888 is the speedof sound

3 Finite Element Modeling

31 Equation of Energy and Hugoniot Curve The equationfor conservation of energy equates the increase in internalenergy per unit mass 119864119898 to the rate at which work is beingdone by stresses and the rate at which heat is being added Inthe absence of heat conduction the energy equation can bewritten as

120588120597119864119898120597119905 = (119901 minus 119901119887]) 1120588 120597120588120597119905 + 119904 119890 + 120588 (27)

PH

PHI1

PHI0

11 101

Figure 1 Schematic representation of a Hugoniot curve [35]

where 119901 is the pressure stress defined as positive in compres-sion 119901119887] is the pressure stress due to the bulk viscosity 119904 isthe deviator stress tensor 119890 is the deviator part of strain rateand is the heat rate per unit mass According to Abaqusdocumentation flow modeling of compressible fluid can beachieved using the linear 119880119904-119880119901 form of the Mie-Gruneisenequation of state The equation of state assumes pressure as afunction of the current density 120588 and the internal energy perunit mass 119864119898 according to

119901 = 119891 (120588 119864119898) (28)

which defines all the equilibrium states that can exist in amaterial The internal energy can be eliminated from (27)to obtain a pressure (119901) versus volume (119881) relationship orequivalently a 119901 versus 1120588 relationship that is unique tothe material described by the equation of state model Thisequation is uniquely dependent on the material defined bythe equation of state This unique relationship is called theHugoniot curve and it is the locus of 119901-119881 states achievablebehind a shockTheHugoniot pressure119901119867 is only a functionof density and can be defined in general from fittingexperimental data An equation of state is said to be linearin energy when it can be written as

119901 = 119891 + 119892119864119898 (29)

where119891(120588) and119892(120588) are only functions of density and dependon the particular equation of state model Figure 1 illustratesa schematic expression of a Hugoniot curve [35]

32 Mie-Gruneisen Equation of State As mentioned previ-ously an equation of state is used to express the behaviourof Eulerian materials For a Mie-Gruneisen equation of statefor linear energy the most common form is written as [38]

119901 minus 119901119867 = Γ120588 (119864119898 minus 119864119867) (30)

where 119901119867 and 119864119867 are the Hugoniot pressure and specificenergy (per unit mass) and are only functions of density andΓ is the Gruneisen ratio defined as

Γ = Γ0 1205880120588 (31)

Geofluids 7

where Γ0 is a material constant and 1205880 is the reference densityHugoniot energy 119864119867 is related to the Hugoniot pressure 119901119867by

119864119867 = 11990111986712057821205880 (32)

where 120578 is the nominal volumetric compressive strain writtenas

120578 = 1 minus 1205880120588 (33)

Elimination of Γ and119864119867 from the above equations yields

119901 = 119901119867 (1 minus Γ01205782 ) + Γ01205880119864119898 (34)

The equation of state and the energy equation representcoupled equations for pressure and internal energy Abaqussolves these equations simultaneously at each material pointusing an explicit method

33 Linear 119880119904-119880119901 Hugoniot Form Normally the 119880119904-119880119901 for-mulation of equation of state (EOS) is used to simulate shocksin solid materials In this study it is used to define fluidmaterials A common fit to the Hugoniot data is given by

119901119867 = 120588011986220120578(1 minus 119904120578)2 (35)

where 1198880 and 119904 define the linear relationship between thelinear shock velocity 119880119904 and the particle velocity 119880119901 using

119880119904 = 1198880 + 119904119880119901 (36)

The Mie-Gruneisen EOS is normally used for processeswith high pressures and high sound velocities The relation-ship between density and pressure is governed by an EOSmodel A linear 119880119904-119880119901 Hugoniot form of the Mie-Gruneisenequation of state is used to evaluate pressure in the waterin the simulation Using this implementation the water ismodeled as a compressible viscous fluid With the aboveassumptions the linear 119880119904-119880119901 Hugoniot form is written as

119901 = 120588011986220120578(1 minus 119904120578)2 (1 minusΓ01205782 ) + Γ01205880119864119898 (37)

where 120588011988820 is equivalent to the elastic bulk modulus at smallnominal strains

When modeling water Abaqus prescribes 119904 = 0 andΓ0 = 0 Filling this simplifies the relation between pressure119901 and density 120588 to an inversely proportional equation and asimplification of the EOS This simplification suffices for themodeling of water [39]

1199011003816100381610038161003816119904=0andΓ=0 = [ 120588011986220120578(1 minus 119904120578)2 (1 minusΓ01205782 ) + Γ01205880119864119898]

119904=0andΓ=0

= 120588011988820120578(38)

Recalling that 120578 = 1 minus 1205880120588 we get

119901 = 120588011986220 (1 minus 1205880120588 ) (39)

When defining a viscosity for thematerial it is treated as aNewtonian fluid meaning that viscosity only depends on thetemperature This is the default setting in Abaqus Since thetemperature is constant throughout the analysis the viscosityis also constant

There is a limiting compression given by the denominatorof this form of the equation of state The denominator of (34)should not be equal to zero therefore an extreme value isdefined for 120588 and 120578 as

120578lim = 1119904 120588lim = 1199041205880119904 minus 1

(40)

At this limit there is a tensile minimum thereafternegative sound speeds are calculated for the material

The linear Hugoniot form 119880119904-119880119901 equation of state canbe employed to model laminar viscous flows governed byNavier-Stokes equations The volumetric response is gov-erned by the equations of state where the bulk modulus actsas a penalty parameter for the constraint [38] The Eulerianpart (water) is simulated using the linear 119880119904-119880119901 Hugoniotform of the Mie-Gruneisen equation of state

The solving method in Eulerian equations is based onthe division of the governing equations In the Lagrangianmethod the nodes are considered to be fixed relative to thematerial at any time interval and elements are deformed withthe material while in the Eulerian method element defor-mation is calculated at any time interval but deformation isapplied as a remeshingThematerial is also displaced betweeneach element and its neighboring elements at any time inter-val Eulerian-Lagrangian contact allows the Eulerian mate-rials to be combined with traditional nonlinear Lagrangiananalyses In the Eulerian-Lagrangian contact the Lagrangianstructure occupies void regions inside the Eulerian meshThe contact algorithm automatically computes and tracks theinterface between the Lagrangian structure and the Eulerianmaterials A great benefit of this method is that there isno need to generate a conforming mesh for the Euleriandomain In fact a simple regular grid of Eulerian elementsoften yields the best accuracy If the Lagrangian body isinitially positioned inside the Eulerian mesh it must beensured that the underlying Eulerian elements contain voidsafter material initialization During analysis the Lagrangianbody pushes material out of the Eulerian elements that itpasses through and they become filled with void SimilarlyEulerian material flowing towards the Lagrangian body isprevented from entering the underlying Eulerian elementsThis formulation ensures that twomaterials never occupy thesame physical space

In this study CEL analysis allowed a direct coupling tosolve this problem The fact that CEL implementation usesthe full Navier-Stokes equations implies that it was able to

8 Geofluids

handle flow appropriately It is important to note that CELis not structured in the same way as a CFD code and wasnot intended to fill this role While it does include viscouseffects with extension to laminar flows it does not include anyturbulence effects but in the context of an impact which has ashort duration and is highly transient (with turbulence effectsbeing of less importance to the structural loading) CEL iswell suited [28]

4 Implementation ofCoupled Eulerian-Lagrangian (CEL)Method in Abaqus

In order to simulate the water entry problem of the pounderand implement the Coupled Eulerian-Lagrangian (CEL)method in Abaqus a geometric model for the pounder andwater was first created A partitionwas created to definewaterand void regions at the water surface Since the fluid (water)undergoes a large deformation an Eulerian grid is suitableThe rigid body (pounder) has small deformations relative tothe fluid making an Lagrangian grid suitable Therefore thewater wasmodeled as Eulerian and the pounder wasmodeledas solid geometry with a rigid body constraint applied to itThen the material properties for the pounder and water weredefined Properties include density (120588) Youngrsquos modulus (119864)and Poissonrsquos ratio (]) for the pounder and density (120588119908) anddynamic viscosity (120578) for the water In the Coupled Eulerian-Lagrangianmethod when the pounder is considered as rigidYoungrsquos modulus and Poisson ratios values should be definedfor the Abaqus but they have no effect on the results Inthe Coupled Eulerian-Lagrangian method 119880119904-119880119901 equationof state (EOS) was selected to define the water properties inAbaqus The 119880119904-119880119901 equation of state is described in detail inAbaqus 614 documentation and it is used to simulateNavier-Stokes flow when turbulence flow is negligible Becausethe major force imposed on the rigid body in water entryproblems is pressure force this equation of state can beused as an appropriate approximation Parameters such as1198620(speed of sound in water) 119904 (constant coefficient in shockvelocity equation) and Γ0 (material constant in Gruneisenequation of state)were defined for the119880119904-119880119901 equation of stateIn the next step the pounder and water were entered intothe assembly section and the pounder was transferred to thewaters center tangent to its surface In order to simulate a free-falling rigid body into water the pounder was first consideredin a position tangent to the water surface then the secondaryvelocity of pounder at the moment of impact was assignedIn the next step explicit dynamic analysis was selected andthe time period for pounder movement between enteringthe water and reaching the model bed was considered as 05seconds Nlgeom was also activated to control the nonlineareffects of large displacements Then by defining a referencepoint and assigning it to the pounder required outputs suchas displacement and velocity values for the vertical direc-tion were selected Based on its defined algorithm Abaqusconsiders the maximum hardness between the Lagrangianelements (pounder) and Eulerian material (water) The Eule-rian solution method uses the meshes defined for poundermovement between water meshes and Abaqus detects theboundary between the Lagrangian and Eulerian materials

Wall

Wall

Wall Water

Sphere

Initial void region

Freesurface

Figure 2 The computational domain boundary conditions andgrid distribution for the water entry problem of sphere

Considering themaximumhardness between the Lagrangianand Eulerian boundaries the contact between these twomaterials is established using the penalty method Concen-trated mass was assigned to the pounder and its moment ofinertia was defined for 119883 119884 and 119885 In the next step thepounder was defined as a rigid body that can move freelyin all directions Gravitational acceleration (Gravity) wasdefined as minus98ms2 in the 119884 direction and a nonreflectingEulerian boundary was assigned to the sides and bottomsurfaces of the computational water domain This boundarycondition is specific to the Eulerian method and preventswave propagation due to rigid body impact on the waterrsquossurface from returning to the computational domain afterreaching the boundariesWith these boundary conditions thewater is regarded as infinite Assuming that the pounder isdropped from a height 119867 above a free water surface withan initial velocity (1198810) of 0ms its secondary velocity (119881) atthe moment of impact was calculated according to the freefall equation of bodies This velocity was vertically applied tothe pounder Eulerian and void regions were specified usingmaterial assignment and the mesh size was defined for thewater and pounder An Eulerian mesh was selected as thewater mesh and an explicit mesh was selected as the poundermesh using structured techniques Finally the problem wassolved and the results were analyzed

41 Model Geometry Before applying the model to pounderwater entry a three-dimensional numerical model was devel-oped to simulate a transient free-falling wedge impactingthe free surface of water in order to check and calibratethe adopted computational model for simpler problemsThen the accuracy of the numerical model was examinedby comparing the numerical results with the experimen-tal data of Aristoff et al [10] for the water entry of asphere Figure 2 illustrates a schematic for the computationaldomain boundary conditions and grid distribution for

Geofluids 9

Table 1 Equation of state parameters for water

Density (120588119908)(kgm3)

Dynamicviscosity (120578)(kgsm2)

Speed of sound(1198880) (msec) Γ0 1199041000 0001 1450 0 0

Water

Aperture release

Z

Diffuser

Lights

Sphere

Camera

Figure 3 Schematic of the experimental apparatus of Aristoff et al[10]

the sphere water entry problem The dimensions for thecomputational domain are considered to be large enoughso that with further increases no considerable changes areobserved in the numerical resultsThe sphere wasmodeled assolid geometry and a rigid body constraint was subsequentlyapplied to it The fluid domain was modeled as a cubewith dimensions of 30 times 50 times 60 cm3 Mesh refinement wasperformed after assigning it to the computational domaintherefore the Eulerianmesh grid size was gradually decreaseduntil no considerable changeswere observed in the numericalresults The Eulerian domain (water) was modeled usingquarter symmetry and contains 746172 elements The spherewas alsomodeled as a Lagrangian solid using 15000 elementsThe Eulerian domain was divided into two parts includingthe upper and lower regions These regions were defined asvoid and stationary water respectively A solid sphere witha diameter of 254 cm and an initial velocity of 217ms inthe normal direction was placed on the free water surfaceThe sphere can move freely in all directions with six degreesof freedom The physical properties of water used in thenumerical simulations are given in Table 1Γ0 and 119904 are material constants in the Gruneisen equationof state and the constant coefficient in shock velocity equa-tion respectively

5 Validation

A schematic of the experimental apparatus of Aristoff et al[10] is presented in Figure 3 A sphere was held by a cameralike aperture at height 119867 above a water tank The tank haddimensions of 30 times 50 times 60 cm3 and was illuminated by a

bank of twenty 32W fluorescent bulbs A diffuser was usedto provide uniform lighting and care was taken to keep thewater surface free of dust Tank dimensions were selectedso that the walls had negligible effects on the results Thesphere was released from rest and fell towards the waterreaching it with an approximate speed119881 asymp radic2119892ℎThe impactsequence was recorded at 2000 frames per second using ahigh-speed camera The resolution was set to 524 times 1280pixels (px) with a field of view of 1128 times 2755 cm2 yieldinga 4646 pxcm magnification The trajectory of the sphereand its impact speed were determined with subpixel accuracythrough a cross correlation andGaussian peak-fittingmethodyielding position estimates accurate to 0025 px (00005 cm)and impact speeds accurate to plusmn4

Figure 4 illustrates a comparison of cavity shapes betweennumerical simulation and the experimental photographstaken by Aristoff et al [10] for a density ratio of 786 and ansimilar period of time The impact velocity of the sphere was217ms and 119905 = 0 refers to when the spheres center passesthrough the free surface This figure shows the cavity shapesfor both experimental and numerical simulation for similartime periods except that cavity shape for the pinch-off timeof the numerical simulation was added to the pictures

An symmetric air cavity formed behind the sphereafter entering the water Air cavity formation consists ofseveral steps including cavity creation cavity expansioncavity contraction behind the sphere and cavity collapseAs the sphere moves deeper through the water it exerts aforce on surrounding fluid in radial directions and transfersits momentum to the fluid However fluid expansion isconfronted with fluid hydrostatic pressure resistance and asa result the radial flow direction is reversed leading to cavitycontraction and collapse Cavity collapse occurs when thecavity wall moves inward until the moment of pinch-off (119905= 708ms) and the cavity is divided into two separated parts[34] While the lower cavity sticks to the sphere and movesalong with it the upper cavity continues its contraction andmoves towards the waterrsquos surface As shown in Figure 4 thenumerical results and experimental observations are in goodagreement

Abaqus numerical simulation methods efficiently modelwater splashes and jet flow Pictures of these phenomenaare depicted in Figure 4 It was concluded from this figurethat pinch-off in the numerical simulation happens with adelay time compared to the experimental method Increasednumerical errors during solution time are one of the causesfor differences between experimental and numerical results

In Figure 5 the numerical results of this study arecompared with the theoretical and experimental data ofAristoff et al as a displacement-time graph of a sphere indifferent water depths [10] As can be seen the numericalresults are in good agreement with those of the analytics andexperiments However the presently computed sphere centerdepth is lower than that found experimentally It seems thatdue to lack of turbulent modeling ability in Abaqus Eulerianformulation and evidently increasing turbulence in flow bytime there is just less difference between experiment andnumerical results in the early time of simulation

10 Geofluids

Experimental results of Aristoff et al (2010)

Pinch-off

Pinch-offNumerical simulation results

t = 59 ＧＭ 129 ＧＭ 199ＧＭ 269ＧＭ 339ＧＭ 409 ＧＭ 479ＧＭ 549ＧＭ 619 ＧＭ 689 ＧＭ 759ＧＭ

t = 59ＧＭ 129 ＧＭ 199ＧＭ 269ＧＭ 339ＧＭ 409 ＧＭ 479ＧＭ 549ＧＭ 619 ＧＭ 689 ＧＭ 708ＧＭ 759ＧＭ

Figure 4 Comparison of numerical results with experimental photographs of Aristoff et al [10] for the sphere water entry problem

002 004 006 0080Time (s)

minus018

minus016

minus014

minus012

minus01

minus008

minus006

minus004

minus002

0

Disp

lace

men

t (m

)

Present studyExperiment (Aristoff et al 2010)Theory (Aristoff et al 2010)

Figure 5 Comparison of displacement-time graph of sphere inwater depth with the theoretical and experimental data of Aristoffet al [10]

6 Materials and Methods

61 Eulerian Model Definition Figure 6 illustrates a sche-matic of the computational domain boundary conditionsand grid distribution for the water entry problem of aspherical pounder The dimensions of the computationaldomain are considered to be large enough so that withfurther increases no considerable changes are observed inthe numerical results The Eulerian domain was dividedinto the upper and lower regions These regions are definedas void and stationary water respectively In water entryproblems for rigid bodies fluid domain dimensions must

Wall

Wall

Wall Water

Spherical pounder

Initial void region

Free surface

Figure 6 The computational domain boundary conditions andgrid distribution for water entry problem of spherical pounder

be at least eight times the dimensions of the rigid bodyHence in this study the fluid domain was modeled as acube with dimensions of 2 times 2 times 1m3 The pounders wereplaced exactly tangent to the water surface and their velocityat the moment of water surface impact was determinedaccording to the free fall equation of bodies The pounderscan move freely in all directions with six degrees of freedom119880119904-119880119901 equation of state was used to define water materialin the Coupled Eulerian-Lagrangian method This equationof state is useful for Navier-Stokes flow simulations whenturbulence flow is negligible In cases where the impact ofa rigid body on a water surface is investigated the majorforce imposed on water body is the pressure force Thereforewith appropriate approximations this equation of state canbe used to investigate the impact of rigid bodies on watersurfaces

62 EulerianMesh Size As shown in Figure 7 with decreasesin the Eulerian domain mesh size to less than 15 cm thedisplacement-time graph of the spherical pounder showsnegligible changes Therefore to improve analysis time the

Geofluids 11

Table 2 Moment of inertia of pounders with different geometric shapes

Geometric shape of the pounder Cube Cylinder Sphere Pyramid ConeMoment of inertia (cm4) 685 5137 411 3082 411

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

01 02 03 040Time (s)

2 ＝Ｇ

175 ＝Ｇ

15 ＝Ｇ

Figure 7 Comparison of the effect of the Eulerian mesh size ondisplacement-time graph of spherical pounder

Eulerian domain was modeled using quarter symmetry witha mesh size of 15 cm and contains 1538943 elements

63 Lagrangian Model Definition Figure 8 illustratespounder dimensions for different geometric shapes Thesepounders are identical in terms of material mass densityYoungrsquos modulus Poissonrsquos ratio volume mass and dropheight from the water surface The only difference is theirgeometric shapeThese parameters were regarded as identicalin all pounders to investigate the effect of pounder shapeon deepwater displacement and velocity air cavity shapepinch-off time and depth and the effect of water drag forceon velocity depreciation when impacting the model bed

Pounder dimensions were selected so that they had avolume of 523 cm3 and equal mass due to their identicalmaterial and mass density Table 2 illustrates the moment ofinertia for pounders with different geometric shapes

Table 3 illustrates thematerial properties of the poundersThese parameters are regarded as identical for all pounderswith the mentioned geometric shapes

64 Lagrangian Mesh Size Due to the negligible effect ofLagrangian mesh size on the results and to improve analysistime the Lagrangian mesh size for all pounders was set to05 cm

Pyramid Cone

Cube Cylinder Sphere

10 ＝Ｇ10 ＝Ｇ10 ＝Ｇ 10 ＝Ｇ

10 ＝Ｇ10 ＝Ｇ10 ＝Ｇ

10 ＝Ｇ5236 ＝Ｇ

20 ＝Ｇ15708 ＝Ｇ

666 ＝Ｇ

Figure 8 The dimensions of pounders with different geometricshapes

Table 3 Material properties of pounders

Materialtype Mass (kg) Density (120588)

(kgm3)

Youngrsquosmodulus (119864)(kgm2)

Poissonrsquosratio (120592)

Steel 411 7850 21 times 1010 03

65 Boundary Condition Wall boundary conditions wereapplied to the sides and bottom of the Eulerian domain tosimulate a water tank Pounder displacement was not con-strained in any direction and the pounders could freely movein all directions The sides and bottom boundaries of thethree-dimensional Eulerian domain were set as nonreflectingboundaries so that the waves released due to pounder impacton water surface did not return to the computational domainafter reaching these boundaries and being absorbed by themThis boundary condition is written as [37]

119889119901 minus 119901119888119889119906 = 0119889119909119889119905 = minus119888 (41)

where 120588 is density 119888 is the speed of sound in the fluid 119901 ispressure 119906 is velocity perpendicular to the wave and 119909 is thedirection perpendicular to the boundary

66 Drop Height and Impact Velocity of Pounders Afterdropping the pounders from a height (ℎ) above the watersurface the velocity of the pounders at themoment of impactwas determined using

1198812 minus 11988120 = 2119892ℎ (42)

where1198810 is the pounder initial velocity before being droppedfrom height ℎ above the waterrsquos surface 119881 is pounder

12 Geofluids

01 02 03 04 050Time (s)

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

CubeCylinderSphere

PyramidCone

Figure 9 Displacement-time graph of pounders with differentgeometric shapes

secondary velocity when impacting the water surface and 119892is the gravitational acceleration of the Earth

From (42) which expresses a time independent equationfor the free fall of objects the initial velocity and the terminalvelocity of the object drop height and the gravitationalacceleration of the Earth are interdependent in the absenceof time In this numerical simulation the pounders weredropped from a height of 1m above thewaterrsquos surfacewith aninitial velocity (1198810) equal to 0ms Therefore the secondaryvelocity of the pounders when impacting the water surfacewas obtained using

119881 = radic2119892ℎ (43)

Instead of simulating a full dropping event from theinitial position the pounders were placed exactly tangent towaterrsquos surface and their secondary velocity at the momentof impact was determined according to (43) A gravitationalacceleration (gravity) of minus98ms2 was applied in the normaldirection of the pounders Therefore a secondary velocityof 443ms in the normal direction of the waterrsquos surfacecorresponds to the speed that would be obtained for thepounders

7 Results and Discussion

71 Effect of the Geometric Shape of the Pounder on Deep-water Displacement Figure 9 illustrates the displacement-time graph of pounders with different geometric shapes Thegraph continues for the cubical pounder from the momentof water surface impact at time zero up to the depth of 1mat 415ms The displacement-time graph of the cylindrical

pounder indicates that after entering the water it reachesthe model bed in a shorter amount of time than the cubicalpounder The graph for this pounder continues from themoment of water surface impact at time zero up to the depthof 1m at 355ms The linear displacement-time graph forthe spherical and pyramidal pounders indicates the fasterdeepwater movement of these pounders compared to thecubical and cylindrical pounders The deepwater movementtime of the spherical and pyramidal pounders after enteringthe water to themodel bed is 285ms and 265ms respectivelyThe linear displacement-time graph of the conical pounderindicates that it has a faster deepwater movement comparedto other pounders as it takes 235ms to reach the model bedafter entering the water

72 Mechanism of Air Cavity Formation and Pinch-Off dueto Deepwater Movement of Pounders with Different Geomet-ric Shapes Figure 10 illustrates the movement trajectory ofpounders with different geometric shapes from the momentof water impact towhen the pounders reached themodel bedAfter entering the water a symmetric air cavity is formedbehind the pounders Air cavity formation involves severalsteps including air cavity creation and expansion behind thepounder air cavity contraction and air cavity collapse [34]As the pounder moves downward in water depth it imposesa force onto surrounding fluid in its radial directions andtransfers its momentum to the fluid This extension is facedwith the hydrostatic pressure resistance of fluidThe directionof the radial flow is reversed and eventually leads to aircavity contraction and collapse In the other words the aircavity contraction is accelerated until pinch-off occurs andthe air cavity is divided into two distinct parts The upperpart of the air cavity continues its contraction and movestowards the waterrsquos surface whereas the lower part of the aircavity clings to the pounder and moves along with it [34]After impacting the water surface the cubical and cylindricalpounders due to their flat contact surfaces lose some of theirvelocityThis velocity depreciation slows down and continuesuntil pinch-offThe spherical pounder with its curved contactsurface has less velocity depreciation compared to the cubicaland cylindrical pounders However some pounder velocityis always depreciated due to impacting the water surfaceThe velocity depreciation process for the spherical pounderduring deepwater movement continues until pinch-off Afterimpacting the waterrsquos surface the pyramidal pounder with itspointy contact surface has a slight velocity depreciation butcontinues to move deeper into the water with approximatelythe same secondary velocity until pinch-off By imposing thedrag force of water onto the pounders surface its velocityis gradually decreased until it reaches the model bed Afterimpacting the water surface the conical pounder with itspointy contact surface continues to move deeper into thewater with an increasing velocity The increasing velocity ofthe conical pounder during deepwater movement continueseven after pinch-off When the pounder moves towards themodel bed the waters drag force is imposed onto poundersurface causing velocity depreciation The pinch-off time forthe pounders with the above-mentioned shapes is between225 and 250ms

Geofluids 13

t = 0 25ＧＭ 50ＧＭ 100ＧＭ 200ＧＭ 225ＧＭ 250ＧＭ 275ＧＭ 300ＧＭ 325ＧＭ 350ＧＭ150ＧＭCu

beCy

linde

rSp

here

Pyra

mid

Con

e

Figure 10 Air cavity formation due to deepwater movement of pounders with different geometric shapes and pinch-off time and depth

73 Effect of the Pounder Geometric Shape onDeepwater Veloc-ity Figure 11 illustrates the velocity-time graph for pounderswith different geometric shapes The cubical and cylindricalpounders lose some of their secondary velocity at the earlymoments of water entry Hence the velocity-time graph ofthese pounders illustrates great velocity depreciation up to15ms The velocity reduction of the cubical and cylindricalpounders gradually continues until pinch-off The velocityof the spherical pounder decreases up to 45ms after waterentry but its velocity depreciation is less than the cubicaland cylindrical pounders This velocity reduction continuesat a slower rate until pinch-off In the initial moments ofwater entry the pyramidal pounder has a slight reduction insecondary velocity up to 55ms Then velocity depreciationcontinues at a faster rate due to the drag force imposed onto

pounders surface and the velocity-time graph of the pyrami-dal pounder shows a gradual velocity reduction until pinch-off After impacting the water surface the conical poundercontinues its deepwater movement with an increasing veloc-ity rate Due to its very low slope the velocity-time graphindicates an increase in velocity for the conical pounder butdue to the waters drag force its velocity begins to decrease at145ms The velocity-time graph indicates a gradual decreasein velocity for conical pounder and velocity depreciationcontinues until pinch-off After pinch-off pounder velocitiesgradually decrease until they reach themodel bed Eventuallywith a velocity of 388ms and 157ms at the moment ofmodel bed impact the conical and cubical pounders have thehighest and lowest velocity respectively Water drag force isimposed in the opposite direction of poundermovement and

14 Geofluids

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Velo

city

(ms

)

01 02 03 040Time (s)

CubeCylinderSphere

PyramidCone

Figure 11 Velocity-time graph of pounders with different geometricshapes

it has the most and least influence on the cubical and conicalpounders respectively

8 Critical Drop Height of the Pounder

In the dynamic compaction of the seabed determining thecritical drop height of the pounder from the freewater surfaceis an effective parameter for the pounder to reach maximumdeepwater velocity After dropping the pounder from height119867 above the free water surface with an initial velocity (1198810)of 0ms its velocity gradually increases until it reaches thesecondary velocity (119881) at the moment of water impact Afterentering the water pounder velocity is decreased due to waterdrag force According to the free fall equation for objectsincreasing the drop height from the waterrsquos surface leads toincreased pounder velocity at the moment of water surfaceimpact but it does not necessarily lead to an increase inpounder velocity during seabed impact because the criticaldrop height of the pounder is a determining factor ofits maximum deepwater velocity According to Figure 12 aconical pounder with a mass of 411 kg was dropped fromdifferent heights (119867) and its velocity when impacting thewaters was measured according to the time independentequation for the free fall of objects Table 4 illustrates thevelocity of the conical pounder with different drop heightswhen impacting the waterrsquos surface

81 Effect of the Drop Height of the Pounder from WaterSurface on Deepwater Displacement Figure 13 illustrates thedisplacement-time graph of the conical pounder with dif-ferent drop heights from the waterrsquos surface As can beseen increasing the drop height of the pounder leads to

Water levelℎ

Figure 12 Schematic of drop height of the conical pounder fromwater surface

Table 4 Velocity of the conical pounder with different drop heightsat the moment of impacting the water surface

Drop height of the pounderfrom water surface (m)

Pounder velocity at themoment of impacting water

surface (ms)025 22105 313075 3831 443125 49515 542175 5852 626225 66425 7

displacement-time graphs that are closer to each other Infact an increase in the drop height of the pounder from thewaterrsquos surface leads to an increase in its secondary velocitywhen impacting the waterrsquos surface Hence the pounderenters the water with a higher velocity and reaches the modelbed in less time

82 Effect of the Drop Height of the Pounder from WaterSurface on Its Deepwater Velocity Increasing the drop heightof the pounder from the waterrsquos surface leads to an increasein its velocity before impacting the water but the poundervelocity gradually decreases after entering the water due tothe its drag force Figure 14 illustrates the velocity-time graphfor the conical pounder with different drop heights from thewaterrsquos surface When the pounder is dropped from a lowheight due to its low velocity it moves deepwater due to itsweight Moreover because of the low velocity of the pounderwhen entering the water pinch-off occurs in less time and atlower depths and its velocity increases after pinch-off until itreaches terminal velocity Increases in pounder drop heightsincrease water entry velocity Hence it will have greatervelocity depreciation in the first moments of water entry

Figure 15 illustrates the terminal velocity-time graph forthe conical pounderwith different drop heightswhen impact-ing the model bed As can be seen increasing the drop height

Geofluids 15

01 02 03 040Time (s)

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

H = 1Ｇ

H = 075Ｇ

H = 05 Ｇ

H = 025 Ｇ

H = 175Ｇ

H = 15 Ｇ

H = 125 Ｇ

H = 25 Ｇ

H = 225 Ｇ

H = 2Ｇ

Figure 13 Displacement-time graph of the conical pounder withdifferent drop heights

of the pounder from the waterrsquos surface leads to an increasein its terminal velocity when impacting the model bedTheseconditions are dominant as long as the pounder reaches acritical drop height from the water surface but increasingthe pounder drop height higher than the critical drop heightleads to decreased terminal velocity when impacting themodel bed In this numerical simulation after dropping thepounder from a height of 2m above the waterrsquos surface itssecondary velocity was 626ms when impacting the watersurface After entering the water the secondary velocity ofthe pounder decreased until it impacts the model bed witha terminal velocity of 454ms Therefore a drop height of2m from above the waterrsquos surface has the highest terminalvelocity when the pounder impacts themodel bed Increasingthe drop height bymore than 2m leads to decreased pounderterminal velocity when impacting the model bed The heightwhich has the greatest effect on pounder terminal velocitywhen impacting the model bed is called the critical dropheight Increasing the drop height of the pounder to greaterthan its critical drop height leads to decreased terminalvelocity when impacting the model bed

Figure 16 illustrates the secondary and terminal velocityof the conical pounder with different drop heights As canbe observed increasing the drop height of the pounder fromwaterrsquos surface leads to increased differences between thesecondary and terminal velocities of the pounder Increasingthe drop height to more than the critical drop height leadsto increased pounder secondary velocity when impactingthe waterrsquos surface and decreased terminal velocity whenimpacting the model bed Hence the difference betweenthe secondary and terminal velocities of the pounder is

01 02 03 040Time (s)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)H = 175Ｇ

H = 125 Ｇ

H = 075Ｇ

H = 025 Ｇ H = 225 Ｇ

H = 1Ｇ

H = 05 Ｇ

H = 25 Ｇ

H = 2Ｇ

H = 15 Ｇ

Figure 14 Velocity-time graph of the conical pounder with differentdrop heights

minus430

minus447 minus454minus415

minus426 minus380

minus367minus355

minus332minus310

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Term

inal

velo

city

(ms

)

01 02 03 040Time (s)

Figure 15 Terminal velocity of conical pounder with different dropheights versus time

not reasonable and a significant amount of the pounderssecondary velocity is depreciated when entering the waterThe great difference between the secondary and terminalvelocities of the pounder reveals the necessity of determiningthe critical drop height of the pounder before dynamic seabedcompaction operations Increasing the pounder drop heightto be greater than the critical drop height not only hasa reverse result on pounder velocity when impacting theseabed but it also requires special equipment such as lengthy

16 Geofluids

Secondary velocityTerminal velocity

025 05 075 1 125 15 175 2 225 250Drop height (m)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)

Figure 16 Secondary and terminal velocities of conical pounderversus drop heights

1 2 3 4 5 6 7 80Impact velocity (ms)

0

005

01

015

02

025

03

Pinc

h-off

tim

e (s)

Figure 17 Pinch-off time versus impact velocity of conical pounder

cranes and huge barges leading to increased implementationcosts

Figure 17 illustrates pinch-off time based on the sec-ondary velocity of the pounder at the moment of watersurface impact As can be seen pinch-off time is a weakfunction of pounder impact velocity and increasing impactvelocity leads to a slight reduction in pinch-off time Thisbehaviour was also observed in the analytical and numericalresults found by Lee et al [40]

Figure 18 illustrates pinch-off depth based on the sec-ondary velocity of pounder at the moment of water surfaceimpact As can be seen the pinch-off depth is a linear

1 2 3 4 5 6 7 80Impact velocity (ms)

minus12

minus1

minus08

minus06

minus04

minus02

0

Pinc

h-off

dep

th (m

)

Figure 18 Pinch-off depth versus impact velocity of conicalpounder

function of pounder impact velocity These results are in linewith the analytical equations of Lee et al [40]

9 Conclusions

In this study the water entry problem of three-dimensionalpounders with different geometric shapes of cube cylindersphere pyramid and cone was numerically simulated bycommercial finite element code Abaqus 614-2 An explicitdynamic analysis method was employed to model the fluid-structure interaction using a Coupled Eulerian-Lagrangian(CEL) method Before applying the model to water entryof the pounders a three-dimensional numerical model wasdeveloped to simulate a transient free-falling wedge impact-ing the free surface of the water in order to calibrate theadopted computationalmodelThe accuracy of the numericalmodel was examined by comparing numerical results withthe experimental data of Aristoff et al The good agreementbetween numerical simulation results with the theoreticaland experimental data of Aristoff et al demonstrates thecapability of the employed numerical methodology to sim-ulate the water entry problem for three-dimensional geome-tries with acceptable accuracy After water entry by a pounderdropped fromaheight of 1m a symmetric air cavity is formedbehind the pounder As the pounder moves downward inwater depth it imposes a force onto surrounding fluid inthe radial directions and transfers its momentum to thefluid This extension is faced with the hydrostatic pressureresistance of the fluid causing the direction of the radialflow to be reversed leading to air cavity contraction andcollapse The results show that the water drag force had thehighest and lowest effect on the cubical and conical poundersrespectively and increasing the drop height to greater thanthe critical drop height leads to increased pounder velocitywhen impacting the waterrsquos surface and decreased velocitywhen impacting the model bed Pinch-off time is a very weak

Geofluids 17

function of pounder velocity when impacting the waterrsquossurface but the pinch-off depth increases linearly along withincreased pounder impact velocities It is recommended thatthe critical drop height for the pounder is determined beforedynamic seabed compaction operations and the adjustmentof crane heights to this height In addition it is also recom-mended that perforated pounders be used in order to reducevelocity depreciation when impacting the waterrsquos surface andto reduce the effect of water drag force

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors warmly appreciate the cooperation and guidanceof Dr B Hamidi and Dr A Mehrabadi in this study

References

[1] B Hamidi J-M Debats H Nikraz and S Varaksin ldquoOffshoreground improvement recordsrdquo Australian Geomechanics Jour-nal vol 48 no 4 pp 111ndash122 2013

[2] SWatanabe ldquoResistance of impact onwater surface part I-cone Institute of Physical andChemical ResearchrdquoTokyo vol 12 pp251ndash267 1930

[3] S Watanabe ldquoResistance of impact on water surface part II-cone (continued)rdquo Institute of Physical and Chemical Researchvol 14 pp 153ndash168 1930

[4] A May and J C Woodhull ldquoDrag coefficients of steel spheresentering water verticallyrdquo Journal of Applied Physics vol 19 no12 pp 1109ndash1121 1948

[5] A May ldquoEffect of surface condition of a sphere on its water-entry cavityrdquo Journal of Applied Physics vol 22 no 10 pp 1219ndash1222 1951

[6] AMay ldquoVertical entry ofmissiles into waterrdquo Journal of AppliedPhysics vol 23 no 12 pp 1362ndash1372 1952

[7] A P New T S Lee and H T Low ldquoImpact loading and waterentrance characteristics of prismatic bodiesrdquo in Proceedings ofthe third international offshore and polar engineering conferencepp 282ndash287 1993

[8] M Anghileri and A Spizzica ldquoExperimental validation of finiteelement models for water impactsrdquo in Proceedings of the secondinternational crash usersrsquo seminar Cranfield Impact Centre LtdEngland UK 1995 second international crash users seminar

[9] A Engle andR Lewis ldquoA comparison of hydrodynamic impactsprediction methods with two dimensional drop test datardquoMarine Structures vol 16 no 2 pp 175ndash182 2003

[10] J M Aristoff T T Truscott A H Techet and J W M BushldquoThe water entry of decelerating spheresrdquo Physics of Fluids vol22 no 3 Article ID 009002PHF pp 1ndash8 2010

[11] T Von Karman The impact of seaplane floats during landingNational Advisory Committee for Aeronautics NACA TN no321 1929

[12] HWagner ldquoPhenomena associatedwith impacts and sliding onliquid surfacesrdquo Journal of Applied Mathematics and Mechanicsvol 12 pp 193ndash215 1932

[13] W S Vorus ldquoA flat cylinder theory for vessel impact and steadyplaning resistancerdquo Journal of Ship Research vol 40 no 2 pp89ndash106 1996

[14] T Miloh ldquoOn the initial-stage slamming of a rigid sphere in avertical water entryrdquo Applied Ocean Research vol 13 no 1 pp13ndash43 1991

[15] J M Aristoff and JW Bush ldquoWater entry of small hydrophobicspheresrdquo Journal of Fluid Mechanics vol 619 pp 45ndash78 2009

[16] P Ghadimi A Dashtimanesh and S R Djeddi ldquoStudy of waterentry of circular cylinder by using analytical and numericalsolutionsrdquo Journal of the Brazilian Society ofMechanical Sciencesamp Engineering vol 34 no 3 pp 225ndash232 2012

[17] A TassinD J PiroAAKorobkinK JMaki andM J CookerldquoTwo-dimensional water entry and exit of a body whose shapevaries in timerdquo Journal of Fluids and Structures vol 40 pp 317ndash336 2013

[18] E Yao H Wang L Pan X Wang and R Woding ldquoVerticalwater-entry of bullet-shaped projectilesrdquo Journal of AppliedMathematics and Physics vol 02 no 06 pp 323ndash334 2014

[19] M-S Park Y-R Jung and W-G Park ldquoNumerical study ofimpact force and ricochet behavior of high speed water-entrybodiesrdquo Computers amp Fluids vol 32 no 7 pp 939ndash951 2003

[20] D Battistin and A Iafrati ldquoHydrodynamic loads during waterenergy of two-dimensional and axisymmetric bodiesrdquo Journalof Fluids and Structures vol 17 no 5 pp 643ndash664 2003

[21] A A Korobkin and M Ohkusu ldquoImpact of two circular platesone of which is floating on a thin layer of liquidrdquo Journal ofEngineering Mathematics vol 50 no 4 pp 343ndash358 2004

[22] K M Kleefsman G Fekken A E Veldman B Iwanowski andB Buchner ldquoA volume-of-fluid based simulation method forwave impact problemsrdquo Journal of Computational Physics vol206 no 1 pp 363ndash393 2005

[23] Y Kim Y Kim Y Liu and D K P Yue ldquoOn the water-entryimpact problem of asymmetric bodiesrdquo in Proceedings of the9th International Conference on Numerical Ship Hydrodynamics(NSH rsquo07) August 2007

[24] Q Yang and W Qiu ldquoNumerical solution of 2D slammingproblem with a CIPmethodrdquo in Proceedings of the InternationalConference on Violent Flows Research Institute for AppliedMechanics Kyushu University Japan 2007

[25] A C Fairlie-Clarke and T Tveitnes ldquoMomentum and gravityeffects during the constant velocitywater entry ofwedge-shapedsectionsrdquo Ocean Engineering vol 35 no 7 pp 706ndash716 2007

[26] Q Yang andW Qiu ldquoNumerical simulation of water impact for2D and 3D bodiesrdquo Ocean Engineering vol 43 pp 82ndash89 2012

[27] G Wu ldquoNumerical simulation for water entry of a wedge atvarying speed by a high order boundary element methodrdquoJournal of Marine Science and Application vol 11 no 2 pp 143ndash149 2012

[28] M Ahmadzadeh B Saranjam A Hoseini Fard and A RBinesh ldquoNumerical simulation of sphere water entry prob-lem using Eulerian-Lagrangian methodrdquo Applied MathematicalModelling vol 38 no 5-6 pp 1673ndash1684 2014

[29] V-T Nguyen D-T Vu W-G Park and Y-R Jung ldquoNumericalanalysis of water impact forces using a dual-time pseudo-compressibility method and volume-of-fluid interface trackingalgorithmrdquo Computers amp Fluids vol 103 pp 18ndash33 2014

[30] S Kim and N Kim ldquoIntegrated dynamics modeling forsupercavitating vehicle systemsrdquo International Journal of NavalArchitecture and Ocean Engineering vol 7 no 2 pp 346ndash3632015

18 Geofluids

[31] V-T Nguyen D-T Vu W-G Park and C-M Jung ldquoNavier-Stokes solver for water entry bodies with moving Chimera gridmethod in 6DOF motionsrdquo Computers amp Fluids vol 140 pp19ndash38 2016

[32] M Mirzaei M Eghtesad and M Alishahi ldquoPlaning forceidentification in high-speed underwater vehiclesrdquo Journal ofVibration and Control vol 22 no 20 pp 4176ndash4191 2016

[33] G Qiu S Henke and J Grabe ldquoApplication of a CoupledEulerian-Lagrangian approach on geomechanical problemsinvolving large deformationsrdquoComputersampGeosciences vol 38no 1 pp 30ndash39 2011

[34] M R Erfanian M Anbarsooz N Rahimi M Zare andM Moghiman ldquoNumerical and experimental investigationof a three dimensional spherical-nose projectile water entryproblemrdquo Ocean Engineering vol 104 pp 397ndash404 2015

[35] W M Isbell F H Shipman and A H Jones ldquoHugoniotequation of state measurements for eleven material to fiveMegabars Materials and Structural Laboratory ManufacturingDevelopments General Motors Technical centerrdquo Tech RepMichigan 1968

[36] M A Jandron R C Hurd J L Belden A F Bower W Fennelland T T Truscott ldquoModeling of hyperelastic water-skippingspheres using AbaqusExplicitrdquo in Proceedings of the SIMULIACommunity Conference 2014

[37] H Forouzani B Saranjam R Kamali and A Abdollahi-far ldquoElasto-plastic time dependent impact analysis of highspeed projectile on water surfacerdquo Journal of Solid and FluidMechanics vol 3 pp 281ndash298 2016

[38] M R Erfanian andMMoghiman ldquoNumerical and experimen-tal investigation of a projectile water entry problem and studyof velocity effect on time and depth of pinch-offrdquo Journal ofModares Mechanical Engineering vol 15 pp 53ndash60 2015

[39] A Sillem Feasibility study of a tire hydroplaning simulation in afinite element code using a coupled Eulerian-Lagrangian method[MSc thesis] 2008

[40] M Lee R G Longoria and D E Wilson ldquoCavity dynamics inhigh-speed water entryrdquo Physics of Fluids vol 9 no 3 pp 540ndash550 1997

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6 Geofluids

which define the thermodynamic properties of the fluid thatis the thermal and caloric equation of state respectivelyWhen defining the viscosity for the material it is treated as aNewtonian fluid meaning that the viscosity depends only ontemperature This is the default setting in Abaqus Since thetemperature is constant throughout the analysis the viscosityalso is constant

25 Explicit Time-Dependent Solution There are implicitand explicit algorithms for dynamic analysis In the implicitmethod the Newmark method is used the matrices need tobe reversed andmore implementation time is requiredHow-ever this method enjoys high accuracy and its convergenceto linear problems is guaranteed In the explicit method thematrices do not need be reversed and the analysis speed ismuch higher Convergence in the explicit method is achievedif the time step is less than the smallest element divided bythe speed of sound in the material [37]

The solution was advanced in time using the centraldifference method which is also called the Verlet or leap frogmethod It is based on the central difference approximationfor evaluating the derivative of a function 119891(119905) at 119905

119889119891 (119905)119889119905 = 119891 (119905 + (12) Δ119905) minus 119891 (119905 minus (12) Δ119905)Δ119905 (22)

where Δ119905 is the time stepUsing (15) acceleration in time 119905 can be written as

119905 = 119865119905119872119861 (23)

Now using (22) for velocity-acceleration velocity at time119905 + (12)Δ119905 is obtained as

119905+(12)Δ119905 = 119905minus(12)Δ119905 + 119905Δ119905 (24)

Then using (22) for velocity-displacement displacementat time 119905 + Δ119905 is obtained as

119909119905+Δ119905 = 119909119905 + 119905+Δ119905Δ119905 (25)

The central difference method has its step size limited bystability considerations like all explicit integration methodsand the stable time step size was limited to

Δ119905 le 119897119888 (26)

where 119897 is the length of the shortest element and 119888 is the speedof sound

3 Finite Element Modeling

31 Equation of Energy and Hugoniot Curve The equationfor conservation of energy equates the increase in internalenergy per unit mass 119864119898 to the rate at which work is beingdone by stresses and the rate at which heat is being added Inthe absence of heat conduction the energy equation can bewritten as

120588120597119864119898120597119905 = (119901 minus 119901119887]) 1120588 120597120588120597119905 + 119904 119890 + 120588 (27)

PH

PHI1

PHI0

11 101

Figure 1 Schematic representation of a Hugoniot curve [35]

where 119901 is the pressure stress defined as positive in compres-sion 119901119887] is the pressure stress due to the bulk viscosity 119904 isthe deviator stress tensor 119890 is the deviator part of strain rateand is the heat rate per unit mass According to Abaqusdocumentation flow modeling of compressible fluid can beachieved using the linear 119880119904-119880119901 form of the Mie-Gruneisenequation of state The equation of state assumes pressure as afunction of the current density 120588 and the internal energy perunit mass 119864119898 according to

119901 = 119891 (120588 119864119898) (28)

which defines all the equilibrium states that can exist in amaterial The internal energy can be eliminated from (27)to obtain a pressure (119901) versus volume (119881) relationship orequivalently a 119901 versus 1120588 relationship that is unique tothe material described by the equation of state model Thisequation is uniquely dependent on the material defined bythe equation of state This unique relationship is called theHugoniot curve and it is the locus of 119901-119881 states achievablebehind a shockTheHugoniot pressure119901119867 is only a functionof density and can be defined in general from fittingexperimental data An equation of state is said to be linearin energy when it can be written as

119901 = 119891 + 119892119864119898 (29)

where119891(120588) and119892(120588) are only functions of density and dependon the particular equation of state model Figure 1 illustratesa schematic expression of a Hugoniot curve [35]

32 Mie-Gruneisen Equation of State As mentioned previ-ously an equation of state is used to express the behaviourof Eulerian materials For a Mie-Gruneisen equation of statefor linear energy the most common form is written as [38]

119901 minus 119901119867 = Γ120588 (119864119898 minus 119864119867) (30)

where 119901119867 and 119864119867 are the Hugoniot pressure and specificenergy (per unit mass) and are only functions of density andΓ is the Gruneisen ratio defined as

Γ = Γ0 1205880120588 (31)

Geofluids 7

where Γ0 is a material constant and 1205880 is the reference densityHugoniot energy 119864119867 is related to the Hugoniot pressure 119901119867by

119864119867 = 11990111986712057821205880 (32)

where 120578 is the nominal volumetric compressive strain writtenas

120578 = 1 minus 1205880120588 (33)

Elimination of Γ and119864119867 from the above equations yields

119901 = 119901119867 (1 minus Γ01205782 ) + Γ01205880119864119898 (34)

The equation of state and the energy equation representcoupled equations for pressure and internal energy Abaqussolves these equations simultaneously at each material pointusing an explicit method

33 Linear 119880119904-119880119901 Hugoniot Form Normally the 119880119904-119880119901 for-mulation of equation of state (EOS) is used to simulate shocksin solid materials In this study it is used to define fluidmaterials A common fit to the Hugoniot data is given by

119901119867 = 120588011986220120578(1 minus 119904120578)2 (35)

where 1198880 and 119904 define the linear relationship between thelinear shock velocity 119880119904 and the particle velocity 119880119901 using

119880119904 = 1198880 + 119904119880119901 (36)

The Mie-Gruneisen EOS is normally used for processeswith high pressures and high sound velocities The relation-ship between density and pressure is governed by an EOSmodel A linear 119880119904-119880119901 Hugoniot form of the Mie-Gruneisenequation of state is used to evaluate pressure in the waterin the simulation Using this implementation the water ismodeled as a compressible viscous fluid With the aboveassumptions the linear 119880119904-119880119901 Hugoniot form is written as

119901 = 120588011986220120578(1 minus 119904120578)2 (1 minusΓ01205782 ) + Γ01205880119864119898 (37)

where 120588011988820 is equivalent to the elastic bulk modulus at smallnominal strains

When modeling water Abaqus prescribes 119904 = 0 andΓ0 = 0 Filling this simplifies the relation between pressure119901 and density 120588 to an inversely proportional equation and asimplification of the EOS This simplification suffices for themodeling of water [39]

1199011003816100381610038161003816119904=0andΓ=0 = [ 120588011986220120578(1 minus 119904120578)2 (1 minusΓ01205782 ) + Γ01205880119864119898]

119904=0andΓ=0

= 120588011988820120578(38)

Recalling that 120578 = 1 minus 1205880120588 we get

119901 = 120588011986220 (1 minus 1205880120588 ) (39)

When defining a viscosity for thematerial it is treated as aNewtonian fluid meaning that viscosity only depends on thetemperature This is the default setting in Abaqus Since thetemperature is constant throughout the analysis the viscosityis also constant

There is a limiting compression given by the denominatorof this form of the equation of state The denominator of (34)should not be equal to zero therefore an extreme value isdefined for 120588 and 120578 as

120578lim = 1119904 120588lim = 1199041205880119904 minus 1

(40)

At this limit there is a tensile minimum thereafternegative sound speeds are calculated for the material

The linear Hugoniot form 119880119904-119880119901 equation of state canbe employed to model laminar viscous flows governed byNavier-Stokes equations The volumetric response is gov-erned by the equations of state where the bulk modulus actsas a penalty parameter for the constraint [38] The Eulerianpart (water) is simulated using the linear 119880119904-119880119901 Hugoniotform of the Mie-Gruneisen equation of state

The solving method in Eulerian equations is based onthe division of the governing equations In the Lagrangianmethod the nodes are considered to be fixed relative to thematerial at any time interval and elements are deformed withthe material while in the Eulerian method element defor-mation is calculated at any time interval but deformation isapplied as a remeshingThematerial is also displaced betweeneach element and its neighboring elements at any time inter-val Eulerian-Lagrangian contact allows the Eulerian mate-rials to be combined with traditional nonlinear Lagrangiananalyses In the Eulerian-Lagrangian contact the Lagrangianstructure occupies void regions inside the Eulerian meshThe contact algorithm automatically computes and tracks theinterface between the Lagrangian structure and the Eulerianmaterials A great benefit of this method is that there isno need to generate a conforming mesh for the Euleriandomain In fact a simple regular grid of Eulerian elementsoften yields the best accuracy If the Lagrangian body isinitially positioned inside the Eulerian mesh it must beensured that the underlying Eulerian elements contain voidsafter material initialization During analysis the Lagrangianbody pushes material out of the Eulerian elements that itpasses through and they become filled with void SimilarlyEulerian material flowing towards the Lagrangian body isprevented from entering the underlying Eulerian elementsThis formulation ensures that twomaterials never occupy thesame physical space

In this study CEL analysis allowed a direct coupling tosolve this problem The fact that CEL implementation usesthe full Navier-Stokes equations implies that it was able to

8 Geofluids

handle flow appropriately It is important to note that CELis not structured in the same way as a CFD code and wasnot intended to fill this role While it does include viscouseffects with extension to laminar flows it does not include anyturbulence effects but in the context of an impact which has ashort duration and is highly transient (with turbulence effectsbeing of less importance to the structural loading) CEL iswell suited [28]

4 Implementation ofCoupled Eulerian-Lagrangian (CEL)Method in Abaqus

In order to simulate the water entry problem of the pounderand implement the Coupled Eulerian-Lagrangian (CEL)method in Abaqus a geometric model for the pounder andwater was first created A partitionwas created to definewaterand void regions at the water surface Since the fluid (water)undergoes a large deformation an Eulerian grid is suitableThe rigid body (pounder) has small deformations relative tothe fluid making an Lagrangian grid suitable Therefore thewater wasmodeled as Eulerian and the pounder wasmodeledas solid geometry with a rigid body constraint applied to itThen the material properties for the pounder and water weredefined Properties include density (120588) Youngrsquos modulus (119864)and Poissonrsquos ratio (]) for the pounder and density (120588119908) anddynamic viscosity (120578) for the water In the Coupled Eulerian-Lagrangianmethod when the pounder is considered as rigidYoungrsquos modulus and Poisson ratios values should be definedfor the Abaqus but they have no effect on the results Inthe Coupled Eulerian-Lagrangian method 119880119904-119880119901 equationof state (EOS) was selected to define the water properties inAbaqus The 119880119904-119880119901 equation of state is described in detail inAbaqus 614 documentation and it is used to simulateNavier-Stokes flow when turbulence flow is negligible Becausethe major force imposed on the rigid body in water entryproblems is pressure force this equation of state can beused as an appropriate approximation Parameters such as1198620(speed of sound in water) 119904 (constant coefficient in shockvelocity equation) and Γ0 (material constant in Gruneisenequation of state)were defined for the119880119904-119880119901 equation of stateIn the next step the pounder and water were entered intothe assembly section and the pounder was transferred to thewaters center tangent to its surface In order to simulate a free-falling rigid body into water the pounder was first consideredin a position tangent to the water surface then the secondaryvelocity of pounder at the moment of impact was assignedIn the next step explicit dynamic analysis was selected andthe time period for pounder movement between enteringthe water and reaching the model bed was considered as 05seconds Nlgeom was also activated to control the nonlineareffects of large displacements Then by defining a referencepoint and assigning it to the pounder required outputs suchas displacement and velocity values for the vertical direc-tion were selected Based on its defined algorithm Abaqusconsiders the maximum hardness between the Lagrangianelements (pounder) and Eulerian material (water) The Eule-rian solution method uses the meshes defined for poundermovement between water meshes and Abaqus detects theboundary between the Lagrangian and Eulerian materials

Wall

Wall

Wall Water

Sphere

Initial void region

Freesurface

Figure 2 The computational domain boundary conditions andgrid distribution for the water entry problem of sphere

Considering themaximumhardness between the Lagrangianand Eulerian boundaries the contact between these twomaterials is established using the penalty method Concen-trated mass was assigned to the pounder and its moment ofinertia was defined for 119883 119884 and 119885 In the next step thepounder was defined as a rigid body that can move freelyin all directions Gravitational acceleration (Gravity) wasdefined as minus98ms2 in the 119884 direction and a nonreflectingEulerian boundary was assigned to the sides and bottomsurfaces of the computational water domain This boundarycondition is specific to the Eulerian method and preventswave propagation due to rigid body impact on the waterrsquossurface from returning to the computational domain afterreaching the boundariesWith these boundary conditions thewater is regarded as infinite Assuming that the pounder isdropped from a height 119867 above a free water surface withan initial velocity (1198810) of 0ms its secondary velocity (119881) atthe moment of impact was calculated according to the freefall equation of bodies This velocity was vertically applied tothe pounder Eulerian and void regions were specified usingmaterial assignment and the mesh size was defined for thewater and pounder An Eulerian mesh was selected as thewater mesh and an explicit mesh was selected as the poundermesh using structured techniques Finally the problem wassolved and the results were analyzed

41 Model Geometry Before applying the model to pounderwater entry a three-dimensional numerical model was devel-oped to simulate a transient free-falling wedge impactingthe free surface of water in order to check and calibratethe adopted computational model for simpler problemsThen the accuracy of the numerical model was examinedby comparing the numerical results with the experimen-tal data of Aristoff et al [10] for the water entry of asphere Figure 2 illustrates a schematic for the computationaldomain boundary conditions and grid distribution for

Geofluids 9

Table 1 Equation of state parameters for water

Density (120588119908)(kgm3)

Dynamicviscosity (120578)(kgsm2)

Speed of sound(1198880) (msec) Γ0 1199041000 0001 1450 0 0

Water

Aperture release

Z

Diffuser

Lights

Sphere

Camera

Figure 3 Schematic of the experimental apparatus of Aristoff et al[10]

the sphere water entry problem The dimensions for thecomputational domain are considered to be large enoughso that with further increases no considerable changes areobserved in the numerical resultsThe sphere wasmodeled assolid geometry and a rigid body constraint was subsequentlyapplied to it The fluid domain was modeled as a cubewith dimensions of 30 times 50 times 60 cm3 Mesh refinement wasperformed after assigning it to the computational domaintherefore the Eulerianmesh grid size was gradually decreaseduntil no considerable changeswere observed in the numericalresults The Eulerian domain (water) was modeled usingquarter symmetry and contains 746172 elements The spherewas alsomodeled as a Lagrangian solid using 15000 elementsThe Eulerian domain was divided into two parts includingthe upper and lower regions These regions were defined asvoid and stationary water respectively A solid sphere witha diameter of 254 cm and an initial velocity of 217ms inthe normal direction was placed on the free water surfaceThe sphere can move freely in all directions with six degreesof freedom The physical properties of water used in thenumerical simulations are given in Table 1Γ0 and 119904 are material constants in the Gruneisen equationof state and the constant coefficient in shock velocity equa-tion respectively

5 Validation

A schematic of the experimental apparatus of Aristoff et al[10] is presented in Figure 3 A sphere was held by a cameralike aperture at height 119867 above a water tank The tank haddimensions of 30 times 50 times 60 cm3 and was illuminated by a

bank of twenty 32W fluorescent bulbs A diffuser was usedto provide uniform lighting and care was taken to keep thewater surface free of dust Tank dimensions were selectedso that the walls had negligible effects on the results Thesphere was released from rest and fell towards the waterreaching it with an approximate speed119881 asymp radic2119892ℎThe impactsequence was recorded at 2000 frames per second using ahigh-speed camera The resolution was set to 524 times 1280pixels (px) with a field of view of 1128 times 2755 cm2 yieldinga 4646 pxcm magnification The trajectory of the sphereand its impact speed were determined with subpixel accuracythrough a cross correlation andGaussian peak-fittingmethodyielding position estimates accurate to 0025 px (00005 cm)and impact speeds accurate to plusmn4

Figure 4 illustrates a comparison of cavity shapes betweennumerical simulation and the experimental photographstaken by Aristoff et al [10] for a density ratio of 786 and ansimilar period of time The impact velocity of the sphere was217ms and 119905 = 0 refers to when the spheres center passesthrough the free surface This figure shows the cavity shapesfor both experimental and numerical simulation for similartime periods except that cavity shape for the pinch-off timeof the numerical simulation was added to the pictures

An symmetric air cavity formed behind the sphereafter entering the water Air cavity formation consists ofseveral steps including cavity creation cavity expansioncavity contraction behind the sphere and cavity collapseAs the sphere moves deeper through the water it exerts aforce on surrounding fluid in radial directions and transfersits momentum to the fluid However fluid expansion isconfronted with fluid hydrostatic pressure resistance and asa result the radial flow direction is reversed leading to cavitycontraction and collapse Cavity collapse occurs when thecavity wall moves inward until the moment of pinch-off (119905= 708ms) and the cavity is divided into two separated parts[34] While the lower cavity sticks to the sphere and movesalong with it the upper cavity continues its contraction andmoves towards the waterrsquos surface As shown in Figure 4 thenumerical results and experimental observations are in goodagreement

Abaqus numerical simulation methods efficiently modelwater splashes and jet flow Pictures of these phenomenaare depicted in Figure 4 It was concluded from this figurethat pinch-off in the numerical simulation happens with adelay time compared to the experimental method Increasednumerical errors during solution time are one of the causesfor differences between experimental and numerical results

In Figure 5 the numerical results of this study arecompared with the theoretical and experimental data ofAristoff et al as a displacement-time graph of a sphere indifferent water depths [10] As can be seen the numericalresults are in good agreement with those of the analytics andexperiments However the presently computed sphere centerdepth is lower than that found experimentally It seems thatdue to lack of turbulent modeling ability in Abaqus Eulerianformulation and evidently increasing turbulence in flow bytime there is just less difference between experiment andnumerical results in the early time of simulation

10 Geofluids

Experimental results of Aristoff et al (2010)

Pinch-off

Pinch-offNumerical simulation results

t = 59 ＧＭ 129 ＧＭ 199ＧＭ 269ＧＭ 339ＧＭ 409 ＧＭ 479ＧＭ 549ＧＭ 619 ＧＭ 689 ＧＭ 759ＧＭ

t = 59ＧＭ 129 ＧＭ 199ＧＭ 269ＧＭ 339ＧＭ 409 ＧＭ 479ＧＭ 549ＧＭ 619 ＧＭ 689 ＧＭ 708ＧＭ 759ＧＭ

Figure 4 Comparison of numerical results with experimental photographs of Aristoff et al [10] for the sphere water entry problem

002 004 006 0080Time (s)

minus018

minus016

minus014

minus012

minus01

minus008

minus006

minus004

minus002

0

Disp

lace

men

t (m

)

Present studyExperiment (Aristoff et al 2010)Theory (Aristoff et al 2010)

Figure 5 Comparison of displacement-time graph of sphere inwater depth with the theoretical and experimental data of Aristoffet al [10]

6 Materials and Methods

61 Eulerian Model Definition Figure 6 illustrates a sche-matic of the computational domain boundary conditionsand grid distribution for the water entry problem of aspherical pounder The dimensions of the computationaldomain are considered to be large enough so that withfurther increases no considerable changes are observed inthe numerical results The Eulerian domain was dividedinto the upper and lower regions These regions are definedas void and stationary water respectively In water entryproblems for rigid bodies fluid domain dimensions must

Wall

Wall

Wall Water

Spherical pounder

Initial void region

Free surface

Figure 6 The computational domain boundary conditions andgrid distribution for water entry problem of spherical pounder

be at least eight times the dimensions of the rigid bodyHence in this study the fluid domain was modeled as acube with dimensions of 2 times 2 times 1m3 The pounders wereplaced exactly tangent to the water surface and their velocityat the moment of water surface impact was determinedaccording to the free fall equation of bodies The pounderscan move freely in all directions with six degrees of freedom119880119904-119880119901 equation of state was used to define water materialin the Coupled Eulerian-Lagrangian method This equationof state is useful for Navier-Stokes flow simulations whenturbulence flow is negligible In cases where the impact ofa rigid body on a water surface is investigated the majorforce imposed on water body is the pressure force Thereforewith appropriate approximations this equation of state canbe used to investigate the impact of rigid bodies on watersurfaces

62 EulerianMesh Size As shown in Figure 7 with decreasesin the Eulerian domain mesh size to less than 15 cm thedisplacement-time graph of the spherical pounder showsnegligible changes Therefore to improve analysis time the

Geofluids 11

Table 2 Moment of inertia of pounders with different geometric shapes

Geometric shape of the pounder Cube Cylinder Sphere Pyramid ConeMoment of inertia (cm4) 685 5137 411 3082 411

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

01 02 03 040Time (s)

2 ＝Ｇ

175 ＝Ｇ

15 ＝Ｇ

Figure 7 Comparison of the effect of the Eulerian mesh size ondisplacement-time graph of spherical pounder

Eulerian domain was modeled using quarter symmetry witha mesh size of 15 cm and contains 1538943 elements

63 Lagrangian Model Definition Figure 8 illustratespounder dimensions for different geometric shapes Thesepounders are identical in terms of material mass densityYoungrsquos modulus Poissonrsquos ratio volume mass and dropheight from the water surface The only difference is theirgeometric shapeThese parameters were regarded as identicalin all pounders to investigate the effect of pounder shapeon deepwater displacement and velocity air cavity shapepinch-off time and depth and the effect of water drag forceon velocity depreciation when impacting the model bed

Pounder dimensions were selected so that they had avolume of 523 cm3 and equal mass due to their identicalmaterial and mass density Table 2 illustrates the moment ofinertia for pounders with different geometric shapes

Table 3 illustrates thematerial properties of the poundersThese parameters are regarded as identical for all pounderswith the mentioned geometric shapes

64 Lagrangian Mesh Size Due to the negligible effect ofLagrangian mesh size on the results and to improve analysistime the Lagrangian mesh size for all pounders was set to05 cm

Pyramid Cone

Cube Cylinder Sphere

10 ＝Ｇ10 ＝Ｇ10 ＝Ｇ 10 ＝Ｇ

10 ＝Ｇ10 ＝Ｇ10 ＝Ｇ

10 ＝Ｇ5236 ＝Ｇ

20 ＝Ｇ15708 ＝Ｇ

666 ＝Ｇ

Figure 8 The dimensions of pounders with different geometricshapes

Table 3 Material properties of pounders

Materialtype Mass (kg) Density (120588)

(kgm3)

Youngrsquosmodulus (119864)(kgm2)

Poissonrsquosratio (120592)

Steel 411 7850 21 times 1010 03

65 Boundary Condition Wall boundary conditions wereapplied to the sides and bottom of the Eulerian domain tosimulate a water tank Pounder displacement was not con-strained in any direction and the pounders could freely movein all directions The sides and bottom boundaries of thethree-dimensional Eulerian domain were set as nonreflectingboundaries so that the waves released due to pounder impacton water surface did not return to the computational domainafter reaching these boundaries and being absorbed by themThis boundary condition is written as [37]

119889119901 minus 119901119888119889119906 = 0119889119909119889119905 = minus119888 (41)

where 120588 is density 119888 is the speed of sound in the fluid 119901 ispressure 119906 is velocity perpendicular to the wave and 119909 is thedirection perpendicular to the boundary

66 Drop Height and Impact Velocity of Pounders Afterdropping the pounders from a height (ℎ) above the watersurface the velocity of the pounders at themoment of impactwas determined using

1198812 minus 11988120 = 2119892ℎ (42)

where1198810 is the pounder initial velocity before being droppedfrom height ℎ above the waterrsquos surface 119881 is pounder

12 Geofluids

01 02 03 04 050Time (s)

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

CubeCylinderSphere

PyramidCone

Figure 9 Displacement-time graph of pounders with differentgeometric shapes

secondary velocity when impacting the water surface and 119892is the gravitational acceleration of the Earth

From (42) which expresses a time independent equationfor the free fall of objects the initial velocity and the terminalvelocity of the object drop height and the gravitationalacceleration of the Earth are interdependent in the absenceof time In this numerical simulation the pounders weredropped from a height of 1m above thewaterrsquos surfacewith aninitial velocity (1198810) equal to 0ms Therefore the secondaryvelocity of the pounders when impacting the water surfacewas obtained using

119881 = radic2119892ℎ (43)

Instead of simulating a full dropping event from theinitial position the pounders were placed exactly tangent towaterrsquos surface and their secondary velocity at the momentof impact was determined according to (43) A gravitationalacceleration (gravity) of minus98ms2 was applied in the normaldirection of the pounders Therefore a secondary velocityof 443ms in the normal direction of the waterrsquos surfacecorresponds to the speed that would be obtained for thepounders

7 Results and Discussion

71 Effect of the Geometric Shape of the Pounder on Deep-water Displacement Figure 9 illustrates the displacement-time graph of pounders with different geometric shapes Thegraph continues for the cubical pounder from the momentof water surface impact at time zero up to the depth of 1mat 415ms The displacement-time graph of the cylindrical

pounder indicates that after entering the water it reachesthe model bed in a shorter amount of time than the cubicalpounder The graph for this pounder continues from themoment of water surface impact at time zero up to the depthof 1m at 355ms The linear displacement-time graph forthe spherical and pyramidal pounders indicates the fasterdeepwater movement of these pounders compared to thecubical and cylindrical pounders The deepwater movementtime of the spherical and pyramidal pounders after enteringthe water to themodel bed is 285ms and 265ms respectivelyThe linear displacement-time graph of the conical pounderindicates that it has a faster deepwater movement comparedto other pounders as it takes 235ms to reach the model bedafter entering the water

72 Mechanism of Air Cavity Formation and Pinch-Off dueto Deepwater Movement of Pounders with Different Geomet-ric Shapes Figure 10 illustrates the movement trajectory ofpounders with different geometric shapes from the momentof water impact towhen the pounders reached themodel bedAfter entering the water a symmetric air cavity is formedbehind the pounders Air cavity formation involves severalsteps including air cavity creation and expansion behind thepounder air cavity contraction and air cavity collapse [34]As the pounder moves downward in water depth it imposesa force onto surrounding fluid in its radial directions andtransfers its momentum to the fluid This extension is facedwith the hydrostatic pressure resistance of fluidThe directionof the radial flow is reversed and eventually leads to aircavity contraction and collapse In the other words the aircavity contraction is accelerated until pinch-off occurs andthe air cavity is divided into two distinct parts The upperpart of the air cavity continues its contraction and movestowards the waterrsquos surface whereas the lower part of the aircavity clings to the pounder and moves along with it [34]After impacting the water surface the cubical and cylindricalpounders due to their flat contact surfaces lose some of theirvelocityThis velocity depreciation slows down and continuesuntil pinch-offThe spherical pounder with its curved contactsurface has less velocity depreciation compared to the cubicaland cylindrical pounders However some pounder velocityis always depreciated due to impacting the water surfaceThe velocity depreciation process for the spherical pounderduring deepwater movement continues until pinch-off Afterimpacting the waterrsquos surface the pyramidal pounder with itspointy contact surface has a slight velocity depreciation butcontinues to move deeper into the water with approximatelythe same secondary velocity until pinch-off By imposing thedrag force of water onto the pounders surface its velocityis gradually decreased until it reaches the model bed Afterimpacting the water surface the conical pounder with itspointy contact surface continues to move deeper into thewater with an increasing velocity The increasing velocity ofthe conical pounder during deepwater movement continueseven after pinch-off When the pounder moves towards themodel bed the waters drag force is imposed onto poundersurface causing velocity depreciation The pinch-off time forthe pounders with the above-mentioned shapes is between225 and 250ms

Geofluids 13

t = 0 25ＧＭ 50ＧＭ 100ＧＭ 200ＧＭ 225ＧＭ 250ＧＭ 275ＧＭ 300ＧＭ 325ＧＭ 350ＧＭ150ＧＭCu

beCy

linde

rSp

here

Pyra

mid

Con

e

Figure 10 Air cavity formation due to deepwater movement of pounders with different geometric shapes and pinch-off time and depth

73 Effect of the Pounder Geometric Shape onDeepwater Veloc-ity Figure 11 illustrates the velocity-time graph for pounderswith different geometric shapes The cubical and cylindricalpounders lose some of their secondary velocity at the earlymoments of water entry Hence the velocity-time graph ofthese pounders illustrates great velocity depreciation up to15ms The velocity reduction of the cubical and cylindricalpounders gradually continues until pinch-off The velocityof the spherical pounder decreases up to 45ms after waterentry but its velocity depreciation is less than the cubicaland cylindrical pounders This velocity reduction continuesat a slower rate until pinch-off In the initial moments ofwater entry the pyramidal pounder has a slight reduction insecondary velocity up to 55ms Then velocity depreciationcontinues at a faster rate due to the drag force imposed onto

pounders surface and the velocity-time graph of the pyrami-dal pounder shows a gradual velocity reduction until pinch-off After impacting the water surface the conical poundercontinues its deepwater movement with an increasing veloc-ity rate Due to its very low slope the velocity-time graphindicates an increase in velocity for the conical pounder butdue to the waters drag force its velocity begins to decrease at145ms The velocity-time graph indicates a gradual decreasein velocity for conical pounder and velocity depreciationcontinues until pinch-off After pinch-off pounder velocitiesgradually decrease until they reach themodel bed Eventuallywith a velocity of 388ms and 157ms at the moment ofmodel bed impact the conical and cubical pounders have thehighest and lowest velocity respectively Water drag force isimposed in the opposite direction of poundermovement and

14 Geofluids

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Velo

city

(ms

)

01 02 03 040Time (s)

CubeCylinderSphere

PyramidCone

Figure 11 Velocity-time graph of pounders with different geometricshapes

it has the most and least influence on the cubical and conicalpounders respectively

8 Critical Drop Height of the Pounder

In the dynamic compaction of the seabed determining thecritical drop height of the pounder from the freewater surfaceis an effective parameter for the pounder to reach maximumdeepwater velocity After dropping the pounder from height119867 above the free water surface with an initial velocity (1198810)of 0ms its velocity gradually increases until it reaches thesecondary velocity (119881) at the moment of water impact Afterentering the water pounder velocity is decreased due to waterdrag force According to the free fall equation for objectsincreasing the drop height from the waterrsquos surface leads toincreased pounder velocity at the moment of water surfaceimpact but it does not necessarily lead to an increase inpounder velocity during seabed impact because the criticaldrop height of the pounder is a determining factor ofits maximum deepwater velocity According to Figure 12 aconical pounder with a mass of 411 kg was dropped fromdifferent heights (119867) and its velocity when impacting thewaters was measured according to the time independentequation for the free fall of objects Table 4 illustrates thevelocity of the conical pounder with different drop heightswhen impacting the waterrsquos surface

81 Effect of the Drop Height of the Pounder from WaterSurface on Deepwater Displacement Figure 13 illustrates thedisplacement-time graph of the conical pounder with dif-ferent drop heights from the waterrsquos surface As can beseen increasing the drop height of the pounder leads to

Water levelℎ

Figure 12 Schematic of drop height of the conical pounder fromwater surface

Table 4 Velocity of the conical pounder with different drop heightsat the moment of impacting the water surface

Drop height of the pounderfrom water surface (m)

Pounder velocity at themoment of impacting water

surface (ms)025 22105 313075 3831 443125 49515 542175 5852 626225 66425 7

displacement-time graphs that are closer to each other Infact an increase in the drop height of the pounder from thewaterrsquos surface leads to an increase in its secondary velocitywhen impacting the waterrsquos surface Hence the pounderenters the water with a higher velocity and reaches the modelbed in less time

82 Effect of the Drop Height of the Pounder from WaterSurface on Its Deepwater Velocity Increasing the drop heightof the pounder from the waterrsquos surface leads to an increasein its velocity before impacting the water but the poundervelocity gradually decreases after entering the water due tothe its drag force Figure 14 illustrates the velocity-time graphfor the conical pounder with different drop heights from thewaterrsquos surface When the pounder is dropped from a lowheight due to its low velocity it moves deepwater due to itsweight Moreover because of the low velocity of the pounderwhen entering the water pinch-off occurs in less time and atlower depths and its velocity increases after pinch-off until itreaches terminal velocity Increases in pounder drop heightsincrease water entry velocity Hence it will have greatervelocity depreciation in the first moments of water entry

Figure 15 illustrates the terminal velocity-time graph forthe conical pounderwith different drop heightswhen impact-ing the model bed As can be seen increasing the drop height

Geofluids 15

01 02 03 040Time (s)

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

H = 1Ｇ

H = 075Ｇ

H = 05 Ｇ

H = 025 Ｇ

H = 175Ｇ

H = 15 Ｇ

H = 125 Ｇ

H = 25 Ｇ

H = 225 Ｇ

H = 2Ｇ

Figure 13 Displacement-time graph of the conical pounder withdifferent drop heights

of the pounder from the waterrsquos surface leads to an increasein its terminal velocity when impacting the model bedTheseconditions are dominant as long as the pounder reaches acritical drop height from the water surface but increasingthe pounder drop height higher than the critical drop heightleads to decreased terminal velocity when impacting themodel bed In this numerical simulation after dropping thepounder from a height of 2m above the waterrsquos surface itssecondary velocity was 626ms when impacting the watersurface After entering the water the secondary velocity ofthe pounder decreased until it impacts the model bed witha terminal velocity of 454ms Therefore a drop height of2m from above the waterrsquos surface has the highest terminalvelocity when the pounder impacts themodel bed Increasingthe drop height bymore than 2m leads to decreased pounderterminal velocity when impacting the model bed The heightwhich has the greatest effect on pounder terminal velocitywhen impacting the model bed is called the critical dropheight Increasing the drop height of the pounder to greaterthan its critical drop height leads to decreased terminalvelocity when impacting the model bed

Figure 16 illustrates the secondary and terminal velocityof the conical pounder with different drop heights As canbe observed increasing the drop height of the pounder fromwaterrsquos surface leads to increased differences between thesecondary and terminal velocities of the pounder Increasingthe drop height to more than the critical drop height leadsto increased pounder secondary velocity when impactingthe waterrsquos surface and decreased terminal velocity whenimpacting the model bed Hence the difference betweenthe secondary and terminal velocities of the pounder is

01 02 03 040Time (s)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)H = 175Ｇ

H = 125 Ｇ

H = 075Ｇ

H = 025 Ｇ H = 225 Ｇ

H = 1Ｇ

H = 05 Ｇ

H = 25 Ｇ

H = 2Ｇ

H = 15 Ｇ

Figure 14 Velocity-time graph of the conical pounder with differentdrop heights

minus430

minus447 minus454minus415

minus426 minus380

minus367minus355

minus332minus310

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Term

inal

velo

city

(ms

)

01 02 03 040Time (s)

Figure 15 Terminal velocity of conical pounder with different dropheights versus time

not reasonable and a significant amount of the pounderssecondary velocity is depreciated when entering the waterThe great difference between the secondary and terminalvelocities of the pounder reveals the necessity of determiningthe critical drop height of the pounder before dynamic seabedcompaction operations Increasing the pounder drop heightto be greater than the critical drop height not only hasa reverse result on pounder velocity when impacting theseabed but it also requires special equipment such as lengthy

16 Geofluids

Secondary velocityTerminal velocity

025 05 075 1 125 15 175 2 225 250Drop height (m)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)

Figure 16 Secondary and terminal velocities of conical pounderversus drop heights

1 2 3 4 5 6 7 80Impact velocity (ms)

0

005

01

015

02

025

03

Pinc

h-off

tim

e (s)

Figure 17 Pinch-off time versus impact velocity of conical pounder

cranes and huge barges leading to increased implementationcosts

Figure 17 illustrates pinch-off time based on the sec-ondary velocity of the pounder at the moment of watersurface impact As can be seen pinch-off time is a weakfunction of pounder impact velocity and increasing impactvelocity leads to a slight reduction in pinch-off time Thisbehaviour was also observed in the analytical and numericalresults found by Lee et al [40]

Figure 18 illustrates pinch-off depth based on the sec-ondary velocity of pounder at the moment of water surfaceimpact As can be seen the pinch-off depth is a linear

1 2 3 4 5 6 7 80Impact velocity (ms)

minus12

minus1

minus08

minus06

minus04

minus02

0

Pinc

h-off

dep

th (m

)

Figure 18 Pinch-off depth versus impact velocity of conicalpounder

function of pounder impact velocity These results are in linewith the analytical equations of Lee et al [40]

9 Conclusions

In this study the water entry problem of three-dimensionalpounders with different geometric shapes of cube cylindersphere pyramid and cone was numerically simulated bycommercial finite element code Abaqus 614-2 An explicitdynamic analysis method was employed to model the fluid-structure interaction using a Coupled Eulerian-Lagrangian(CEL) method Before applying the model to water entryof the pounders a three-dimensional numerical model wasdeveloped to simulate a transient free-falling wedge impact-ing the free surface of the water in order to calibrate theadopted computationalmodelThe accuracy of the numericalmodel was examined by comparing numerical results withthe experimental data of Aristoff et al The good agreementbetween numerical simulation results with the theoreticaland experimental data of Aristoff et al demonstrates thecapability of the employed numerical methodology to sim-ulate the water entry problem for three-dimensional geome-tries with acceptable accuracy After water entry by a pounderdropped fromaheight of 1m a symmetric air cavity is formedbehind the pounder As the pounder moves downward inwater depth it imposes a force onto surrounding fluid inthe radial directions and transfers its momentum to thefluid This extension is faced with the hydrostatic pressureresistance of the fluid causing the direction of the radialflow to be reversed leading to air cavity contraction andcollapse The results show that the water drag force had thehighest and lowest effect on the cubical and conical poundersrespectively and increasing the drop height to greater thanthe critical drop height leads to increased pounder velocitywhen impacting the waterrsquos surface and decreased velocitywhen impacting the model bed Pinch-off time is a very weak

Geofluids 17

function of pounder velocity when impacting the waterrsquossurface but the pinch-off depth increases linearly along withincreased pounder impact velocities It is recommended thatthe critical drop height for the pounder is determined beforedynamic seabed compaction operations and the adjustmentof crane heights to this height In addition it is also recom-mended that perforated pounders be used in order to reducevelocity depreciation when impacting the waterrsquos surface andto reduce the effect of water drag force

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors warmly appreciate the cooperation and guidanceof Dr B Hamidi and Dr A Mehrabadi in this study

References

[1] B Hamidi J-M Debats H Nikraz and S Varaksin ldquoOffshoreground improvement recordsrdquo Australian Geomechanics Jour-nal vol 48 no 4 pp 111ndash122 2013

[2] SWatanabe ldquoResistance of impact onwater surface part I-cone Institute of Physical andChemical ResearchrdquoTokyo vol 12 pp251ndash267 1930

[3] S Watanabe ldquoResistance of impact on water surface part II-cone (continued)rdquo Institute of Physical and Chemical Researchvol 14 pp 153ndash168 1930

[4] A May and J C Woodhull ldquoDrag coefficients of steel spheresentering water verticallyrdquo Journal of Applied Physics vol 19 no12 pp 1109ndash1121 1948

[5] A May ldquoEffect of surface condition of a sphere on its water-entry cavityrdquo Journal of Applied Physics vol 22 no 10 pp 1219ndash1222 1951

[6] AMay ldquoVertical entry ofmissiles into waterrdquo Journal of AppliedPhysics vol 23 no 12 pp 1362ndash1372 1952

[7] A P New T S Lee and H T Low ldquoImpact loading and waterentrance characteristics of prismatic bodiesrdquo in Proceedings ofthe third international offshore and polar engineering conferencepp 282ndash287 1993

[8] M Anghileri and A Spizzica ldquoExperimental validation of finiteelement models for water impactsrdquo in Proceedings of the secondinternational crash usersrsquo seminar Cranfield Impact Centre LtdEngland UK 1995 second international crash users seminar

[9] A Engle andR Lewis ldquoA comparison of hydrodynamic impactsprediction methods with two dimensional drop test datardquoMarine Structures vol 16 no 2 pp 175ndash182 2003

[10] J M Aristoff T T Truscott A H Techet and J W M BushldquoThe water entry of decelerating spheresrdquo Physics of Fluids vol22 no 3 Article ID 009002PHF pp 1ndash8 2010

[11] T Von Karman The impact of seaplane floats during landingNational Advisory Committee for Aeronautics NACA TN no321 1929

[12] HWagner ldquoPhenomena associatedwith impacts and sliding onliquid surfacesrdquo Journal of Applied Mathematics and Mechanicsvol 12 pp 193ndash215 1932

[13] W S Vorus ldquoA flat cylinder theory for vessel impact and steadyplaning resistancerdquo Journal of Ship Research vol 40 no 2 pp89ndash106 1996

[14] T Miloh ldquoOn the initial-stage slamming of a rigid sphere in avertical water entryrdquo Applied Ocean Research vol 13 no 1 pp13ndash43 1991

[15] J M Aristoff and JW Bush ldquoWater entry of small hydrophobicspheresrdquo Journal of Fluid Mechanics vol 619 pp 45ndash78 2009

[16] P Ghadimi A Dashtimanesh and S R Djeddi ldquoStudy of waterentry of circular cylinder by using analytical and numericalsolutionsrdquo Journal of the Brazilian Society ofMechanical Sciencesamp Engineering vol 34 no 3 pp 225ndash232 2012

[17] A TassinD J PiroAAKorobkinK JMaki andM J CookerldquoTwo-dimensional water entry and exit of a body whose shapevaries in timerdquo Journal of Fluids and Structures vol 40 pp 317ndash336 2013

[18] E Yao H Wang L Pan X Wang and R Woding ldquoVerticalwater-entry of bullet-shaped projectilesrdquo Journal of AppliedMathematics and Physics vol 02 no 06 pp 323ndash334 2014

[19] M-S Park Y-R Jung and W-G Park ldquoNumerical study ofimpact force and ricochet behavior of high speed water-entrybodiesrdquo Computers amp Fluids vol 32 no 7 pp 939ndash951 2003

[20] D Battistin and A Iafrati ldquoHydrodynamic loads during waterenergy of two-dimensional and axisymmetric bodiesrdquo Journalof Fluids and Structures vol 17 no 5 pp 643ndash664 2003

[21] A A Korobkin and M Ohkusu ldquoImpact of two circular platesone of which is floating on a thin layer of liquidrdquo Journal ofEngineering Mathematics vol 50 no 4 pp 343ndash358 2004

[22] K M Kleefsman G Fekken A E Veldman B Iwanowski andB Buchner ldquoA volume-of-fluid based simulation method forwave impact problemsrdquo Journal of Computational Physics vol206 no 1 pp 363ndash393 2005

[23] Y Kim Y Kim Y Liu and D K P Yue ldquoOn the water-entryimpact problem of asymmetric bodiesrdquo in Proceedings of the9th International Conference on Numerical Ship Hydrodynamics(NSH rsquo07) August 2007

[24] Q Yang and W Qiu ldquoNumerical solution of 2D slammingproblem with a CIPmethodrdquo in Proceedings of the InternationalConference on Violent Flows Research Institute for AppliedMechanics Kyushu University Japan 2007

[25] A C Fairlie-Clarke and T Tveitnes ldquoMomentum and gravityeffects during the constant velocitywater entry ofwedge-shapedsectionsrdquo Ocean Engineering vol 35 no 7 pp 706ndash716 2007

[26] Q Yang andW Qiu ldquoNumerical simulation of water impact for2D and 3D bodiesrdquo Ocean Engineering vol 43 pp 82ndash89 2012

[27] G Wu ldquoNumerical simulation for water entry of a wedge atvarying speed by a high order boundary element methodrdquoJournal of Marine Science and Application vol 11 no 2 pp 143ndash149 2012

[28] M Ahmadzadeh B Saranjam A Hoseini Fard and A RBinesh ldquoNumerical simulation of sphere water entry prob-lem using Eulerian-Lagrangian methodrdquo Applied MathematicalModelling vol 38 no 5-6 pp 1673ndash1684 2014

[29] V-T Nguyen D-T Vu W-G Park and Y-R Jung ldquoNumericalanalysis of water impact forces using a dual-time pseudo-compressibility method and volume-of-fluid interface trackingalgorithmrdquo Computers amp Fluids vol 103 pp 18ndash33 2014

[30] S Kim and N Kim ldquoIntegrated dynamics modeling forsupercavitating vehicle systemsrdquo International Journal of NavalArchitecture and Ocean Engineering vol 7 no 2 pp 346ndash3632015

18 Geofluids

[31] V-T Nguyen D-T Vu W-G Park and C-M Jung ldquoNavier-Stokes solver for water entry bodies with moving Chimera gridmethod in 6DOF motionsrdquo Computers amp Fluids vol 140 pp19ndash38 2016

[32] M Mirzaei M Eghtesad and M Alishahi ldquoPlaning forceidentification in high-speed underwater vehiclesrdquo Journal ofVibration and Control vol 22 no 20 pp 4176ndash4191 2016

[33] G Qiu S Henke and J Grabe ldquoApplication of a CoupledEulerian-Lagrangian approach on geomechanical problemsinvolving large deformationsrdquoComputersampGeosciences vol 38no 1 pp 30ndash39 2011

[34] M R Erfanian M Anbarsooz N Rahimi M Zare andM Moghiman ldquoNumerical and experimental investigationof a three dimensional spherical-nose projectile water entryproblemrdquo Ocean Engineering vol 104 pp 397ndash404 2015

[35] W M Isbell F H Shipman and A H Jones ldquoHugoniotequation of state measurements for eleven material to fiveMegabars Materials and Structural Laboratory ManufacturingDevelopments General Motors Technical centerrdquo Tech RepMichigan 1968

[36] M A Jandron R C Hurd J L Belden A F Bower W Fennelland T T Truscott ldquoModeling of hyperelastic water-skippingspheres using AbaqusExplicitrdquo in Proceedings of the SIMULIACommunity Conference 2014

[37] H Forouzani B Saranjam R Kamali and A Abdollahi-far ldquoElasto-plastic time dependent impact analysis of highspeed projectile on water surfacerdquo Journal of Solid and FluidMechanics vol 3 pp 281ndash298 2016

[38] M R Erfanian andMMoghiman ldquoNumerical and experimen-tal investigation of a projectile water entry problem and studyof velocity effect on time and depth of pinch-offrdquo Journal ofModares Mechanical Engineering vol 15 pp 53ndash60 2015

[39] A Sillem Feasibility study of a tire hydroplaning simulation in afinite element code using a coupled Eulerian-Lagrangian method[MSc thesis] 2008

[40] M Lee R G Longoria and D E Wilson ldquoCavity dynamics inhigh-speed water entryrdquo Physics of Fluids vol 9 no 3 pp 540ndash550 1997

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Submit your manuscripts atwwwhindawicom

Geofluids 7

where Γ0 is a material constant and 1205880 is the reference densityHugoniot energy 119864119867 is related to the Hugoniot pressure 119901119867by

119864119867 = 11990111986712057821205880 (32)

where 120578 is the nominal volumetric compressive strain writtenas

120578 = 1 minus 1205880120588 (33)

Elimination of Γ and119864119867 from the above equations yields

119901 = 119901119867 (1 minus Γ01205782 ) + Γ01205880119864119898 (34)

The equation of state and the energy equation representcoupled equations for pressure and internal energy Abaqussolves these equations simultaneously at each material pointusing an explicit method

33 Linear 119880119904-119880119901 Hugoniot Form Normally the 119880119904-119880119901 for-mulation of equation of state (EOS) is used to simulate shocksin solid materials In this study it is used to define fluidmaterials A common fit to the Hugoniot data is given by

119901119867 = 120588011986220120578(1 minus 119904120578)2 (35)

where 1198880 and 119904 define the linear relationship between thelinear shock velocity 119880119904 and the particle velocity 119880119901 using

119880119904 = 1198880 + 119904119880119901 (36)

The Mie-Gruneisen EOS is normally used for processeswith high pressures and high sound velocities The relation-ship between density and pressure is governed by an EOSmodel A linear 119880119904-119880119901 Hugoniot form of the Mie-Gruneisenequation of state is used to evaluate pressure in the waterin the simulation Using this implementation the water ismodeled as a compressible viscous fluid With the aboveassumptions the linear 119880119904-119880119901 Hugoniot form is written as

119901 = 120588011986220120578(1 minus 119904120578)2 (1 minusΓ01205782 ) + Γ01205880119864119898 (37)

where 120588011988820 is equivalent to the elastic bulk modulus at smallnominal strains

When modeling water Abaqus prescribes 119904 = 0 andΓ0 = 0 Filling this simplifies the relation between pressure119901 and density 120588 to an inversely proportional equation and asimplification of the EOS This simplification suffices for themodeling of water [39]

1199011003816100381610038161003816119904=0andΓ=0 = [ 120588011986220120578(1 minus 119904120578)2 (1 minusΓ01205782 ) + Γ01205880119864119898]

119904=0andΓ=0

= 120588011988820120578(38)

Recalling that 120578 = 1 minus 1205880120588 we get

119901 = 120588011986220 (1 minus 1205880120588 ) (39)

When defining a viscosity for thematerial it is treated as aNewtonian fluid meaning that viscosity only depends on thetemperature This is the default setting in Abaqus Since thetemperature is constant throughout the analysis the viscosityis also constant

There is a limiting compression given by the denominatorof this form of the equation of state The denominator of (34)should not be equal to zero therefore an extreme value isdefined for 120588 and 120578 as

120578lim = 1119904 120588lim = 1199041205880119904 minus 1

(40)

At this limit there is a tensile minimum thereafternegative sound speeds are calculated for the material

The linear Hugoniot form 119880119904-119880119901 equation of state canbe employed to model laminar viscous flows governed byNavier-Stokes equations The volumetric response is gov-erned by the equations of state where the bulk modulus actsas a penalty parameter for the constraint [38] The Eulerianpart (water) is simulated using the linear 119880119904-119880119901 Hugoniotform of the Mie-Gruneisen equation of state

The solving method in Eulerian equations is based onthe division of the governing equations In the Lagrangianmethod the nodes are considered to be fixed relative to thematerial at any time interval and elements are deformed withthe material while in the Eulerian method element defor-mation is calculated at any time interval but deformation isapplied as a remeshingThematerial is also displaced betweeneach element and its neighboring elements at any time inter-val Eulerian-Lagrangian contact allows the Eulerian mate-rials to be combined with traditional nonlinear Lagrangiananalyses In the Eulerian-Lagrangian contact the Lagrangianstructure occupies void regions inside the Eulerian meshThe contact algorithm automatically computes and tracks theinterface between the Lagrangian structure and the Eulerianmaterials A great benefit of this method is that there isno need to generate a conforming mesh for the Euleriandomain In fact a simple regular grid of Eulerian elementsoften yields the best accuracy If the Lagrangian body isinitially positioned inside the Eulerian mesh it must beensured that the underlying Eulerian elements contain voidsafter material initialization During analysis the Lagrangianbody pushes material out of the Eulerian elements that itpasses through and they become filled with void SimilarlyEulerian material flowing towards the Lagrangian body isprevented from entering the underlying Eulerian elementsThis formulation ensures that twomaterials never occupy thesame physical space

In this study CEL analysis allowed a direct coupling tosolve this problem The fact that CEL implementation usesthe full Navier-Stokes equations implies that it was able to

8 Geofluids

handle flow appropriately It is important to note that CELis not structured in the same way as a CFD code and wasnot intended to fill this role While it does include viscouseffects with extension to laminar flows it does not include anyturbulence effects but in the context of an impact which has ashort duration and is highly transient (with turbulence effectsbeing of less importance to the structural loading) CEL iswell suited [28]

4 Implementation ofCoupled Eulerian-Lagrangian (CEL)Method in Abaqus

In order to simulate the water entry problem of the pounderand implement the Coupled Eulerian-Lagrangian (CEL)method in Abaqus a geometric model for the pounder andwater was first created A partitionwas created to definewaterand void regions at the water surface Since the fluid (water)undergoes a large deformation an Eulerian grid is suitableThe rigid body (pounder) has small deformations relative tothe fluid making an Lagrangian grid suitable Therefore thewater wasmodeled as Eulerian and the pounder wasmodeledas solid geometry with a rigid body constraint applied to itThen the material properties for the pounder and water weredefined Properties include density (120588) Youngrsquos modulus (119864)and Poissonrsquos ratio (]) for the pounder and density (120588119908) anddynamic viscosity (120578) for the water In the Coupled Eulerian-Lagrangianmethod when the pounder is considered as rigidYoungrsquos modulus and Poisson ratios values should be definedfor the Abaqus but they have no effect on the results Inthe Coupled Eulerian-Lagrangian method 119880119904-119880119901 equationof state (EOS) was selected to define the water properties inAbaqus The 119880119904-119880119901 equation of state is described in detail inAbaqus 614 documentation and it is used to simulateNavier-Stokes flow when turbulence flow is negligible Becausethe major force imposed on the rigid body in water entryproblems is pressure force this equation of state can beused as an appropriate approximation Parameters such as1198620(speed of sound in water) 119904 (constant coefficient in shockvelocity equation) and Γ0 (material constant in Gruneisenequation of state)were defined for the119880119904-119880119901 equation of stateIn the next step the pounder and water were entered intothe assembly section and the pounder was transferred to thewaters center tangent to its surface In order to simulate a free-falling rigid body into water the pounder was first consideredin a position tangent to the water surface then the secondaryvelocity of pounder at the moment of impact was assignedIn the next step explicit dynamic analysis was selected andthe time period for pounder movement between enteringthe water and reaching the model bed was considered as 05seconds Nlgeom was also activated to control the nonlineareffects of large displacements Then by defining a referencepoint and assigning it to the pounder required outputs suchas displacement and velocity values for the vertical direc-tion were selected Based on its defined algorithm Abaqusconsiders the maximum hardness between the Lagrangianelements (pounder) and Eulerian material (water) The Eule-rian solution method uses the meshes defined for poundermovement between water meshes and Abaqus detects theboundary between the Lagrangian and Eulerian materials

Wall

Wall

Wall Water

Sphere

Initial void region

Freesurface

Figure 2 The computational domain boundary conditions andgrid distribution for the water entry problem of sphere

Considering themaximumhardness between the Lagrangianand Eulerian boundaries the contact between these twomaterials is established using the penalty method Concen-trated mass was assigned to the pounder and its moment ofinertia was defined for 119883 119884 and 119885 In the next step thepounder was defined as a rigid body that can move freelyin all directions Gravitational acceleration (Gravity) wasdefined as minus98ms2 in the 119884 direction and a nonreflectingEulerian boundary was assigned to the sides and bottomsurfaces of the computational water domain This boundarycondition is specific to the Eulerian method and preventswave propagation due to rigid body impact on the waterrsquossurface from returning to the computational domain afterreaching the boundariesWith these boundary conditions thewater is regarded as infinite Assuming that the pounder isdropped from a height 119867 above a free water surface withan initial velocity (1198810) of 0ms its secondary velocity (119881) atthe moment of impact was calculated according to the freefall equation of bodies This velocity was vertically applied tothe pounder Eulerian and void regions were specified usingmaterial assignment and the mesh size was defined for thewater and pounder An Eulerian mesh was selected as thewater mesh and an explicit mesh was selected as the poundermesh using structured techniques Finally the problem wassolved and the results were analyzed

41 Model Geometry Before applying the model to pounderwater entry a three-dimensional numerical model was devel-oped to simulate a transient free-falling wedge impactingthe free surface of water in order to check and calibratethe adopted computational model for simpler problemsThen the accuracy of the numerical model was examinedby comparing the numerical results with the experimen-tal data of Aristoff et al [10] for the water entry of asphere Figure 2 illustrates a schematic for the computationaldomain boundary conditions and grid distribution for

Geofluids 9

Table 1 Equation of state parameters for water

Density (120588119908)(kgm3)

Dynamicviscosity (120578)(kgsm2)

Speed of sound(1198880) (msec) Γ0 1199041000 0001 1450 0 0

Water

Aperture release

Z

Diffuser

Lights

Sphere

Camera

Figure 3 Schematic of the experimental apparatus of Aristoff et al[10]

the sphere water entry problem The dimensions for thecomputational domain are considered to be large enoughso that with further increases no considerable changes areobserved in the numerical resultsThe sphere wasmodeled assolid geometry and a rigid body constraint was subsequentlyapplied to it The fluid domain was modeled as a cubewith dimensions of 30 times 50 times 60 cm3 Mesh refinement wasperformed after assigning it to the computational domaintherefore the Eulerianmesh grid size was gradually decreaseduntil no considerable changeswere observed in the numericalresults The Eulerian domain (water) was modeled usingquarter symmetry and contains 746172 elements The spherewas alsomodeled as a Lagrangian solid using 15000 elementsThe Eulerian domain was divided into two parts includingthe upper and lower regions These regions were defined asvoid and stationary water respectively A solid sphere witha diameter of 254 cm and an initial velocity of 217ms inthe normal direction was placed on the free water surfaceThe sphere can move freely in all directions with six degreesof freedom The physical properties of water used in thenumerical simulations are given in Table 1Γ0 and 119904 are material constants in the Gruneisen equationof state and the constant coefficient in shock velocity equa-tion respectively

5 Validation

A schematic of the experimental apparatus of Aristoff et al[10] is presented in Figure 3 A sphere was held by a cameralike aperture at height 119867 above a water tank The tank haddimensions of 30 times 50 times 60 cm3 and was illuminated by a

bank of twenty 32W fluorescent bulbs A diffuser was usedto provide uniform lighting and care was taken to keep thewater surface free of dust Tank dimensions were selectedso that the walls had negligible effects on the results Thesphere was released from rest and fell towards the waterreaching it with an approximate speed119881 asymp radic2119892ℎThe impactsequence was recorded at 2000 frames per second using ahigh-speed camera The resolution was set to 524 times 1280pixels (px) with a field of view of 1128 times 2755 cm2 yieldinga 4646 pxcm magnification The trajectory of the sphereand its impact speed were determined with subpixel accuracythrough a cross correlation andGaussian peak-fittingmethodyielding position estimates accurate to 0025 px (00005 cm)and impact speeds accurate to plusmn4

Figure 4 illustrates a comparison of cavity shapes betweennumerical simulation and the experimental photographstaken by Aristoff et al [10] for a density ratio of 786 and ansimilar period of time The impact velocity of the sphere was217ms and 119905 = 0 refers to when the spheres center passesthrough the free surface This figure shows the cavity shapesfor both experimental and numerical simulation for similartime periods except that cavity shape for the pinch-off timeof the numerical simulation was added to the pictures

An symmetric air cavity formed behind the sphereafter entering the water Air cavity formation consists ofseveral steps including cavity creation cavity expansioncavity contraction behind the sphere and cavity collapseAs the sphere moves deeper through the water it exerts aforce on surrounding fluid in radial directions and transfersits momentum to the fluid However fluid expansion isconfronted with fluid hydrostatic pressure resistance and asa result the radial flow direction is reversed leading to cavitycontraction and collapse Cavity collapse occurs when thecavity wall moves inward until the moment of pinch-off (119905= 708ms) and the cavity is divided into two separated parts[34] While the lower cavity sticks to the sphere and movesalong with it the upper cavity continues its contraction andmoves towards the waterrsquos surface As shown in Figure 4 thenumerical results and experimental observations are in goodagreement

Abaqus numerical simulation methods efficiently modelwater splashes and jet flow Pictures of these phenomenaare depicted in Figure 4 It was concluded from this figurethat pinch-off in the numerical simulation happens with adelay time compared to the experimental method Increasednumerical errors during solution time are one of the causesfor differences between experimental and numerical results

In Figure 5 the numerical results of this study arecompared with the theoretical and experimental data ofAristoff et al as a displacement-time graph of a sphere indifferent water depths [10] As can be seen the numericalresults are in good agreement with those of the analytics andexperiments However the presently computed sphere centerdepth is lower than that found experimentally It seems thatdue to lack of turbulent modeling ability in Abaqus Eulerianformulation and evidently increasing turbulence in flow bytime there is just less difference between experiment andnumerical results in the early time of simulation

10 Geofluids

Experimental results of Aristoff et al (2010)

Pinch-off

Pinch-offNumerical simulation results

t = 59 ＧＭ 129 ＧＭ 199ＧＭ 269ＧＭ 339ＧＭ 409 ＧＭ 479ＧＭ 549ＧＭ 619 ＧＭ 689 ＧＭ 759ＧＭ

t = 59ＧＭ 129 ＧＭ 199ＧＭ 269ＧＭ 339ＧＭ 409 ＧＭ 479ＧＭ 549ＧＭ 619 ＧＭ 689 ＧＭ 708ＧＭ 759ＧＭ

Figure 4 Comparison of numerical results with experimental photographs of Aristoff et al [10] for the sphere water entry problem

002 004 006 0080Time (s)

minus018

minus016

minus014

minus012

minus01

minus008

minus006

minus004

minus002

0

Disp

lace

men

t (m

)

Present studyExperiment (Aristoff et al 2010)Theory (Aristoff et al 2010)

Figure 5 Comparison of displacement-time graph of sphere inwater depth with the theoretical and experimental data of Aristoffet al [10]

6 Materials and Methods

61 Eulerian Model Definition Figure 6 illustrates a sche-matic of the computational domain boundary conditionsand grid distribution for the water entry problem of aspherical pounder The dimensions of the computationaldomain are considered to be large enough so that withfurther increases no considerable changes are observed inthe numerical results The Eulerian domain was dividedinto the upper and lower regions These regions are definedas void and stationary water respectively In water entryproblems for rigid bodies fluid domain dimensions must

Wall

Wall

Wall Water

Spherical pounder

Initial void region

Free surface

Figure 6 The computational domain boundary conditions andgrid distribution for water entry problem of spherical pounder

be at least eight times the dimensions of the rigid bodyHence in this study the fluid domain was modeled as acube with dimensions of 2 times 2 times 1m3 The pounders wereplaced exactly tangent to the water surface and their velocityat the moment of water surface impact was determinedaccording to the free fall equation of bodies The pounderscan move freely in all directions with six degrees of freedom119880119904-119880119901 equation of state was used to define water materialin the Coupled Eulerian-Lagrangian method This equationof state is useful for Navier-Stokes flow simulations whenturbulence flow is negligible In cases where the impact ofa rigid body on a water surface is investigated the majorforce imposed on water body is the pressure force Thereforewith appropriate approximations this equation of state canbe used to investigate the impact of rigid bodies on watersurfaces

62 EulerianMesh Size As shown in Figure 7 with decreasesin the Eulerian domain mesh size to less than 15 cm thedisplacement-time graph of the spherical pounder showsnegligible changes Therefore to improve analysis time the

Geofluids 11

Table 2 Moment of inertia of pounders with different geometric shapes

Geometric shape of the pounder Cube Cylinder Sphere Pyramid ConeMoment of inertia (cm4) 685 5137 411 3082 411

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

01 02 03 040Time (s)

2 ＝Ｇ

175 ＝Ｇ

15 ＝Ｇ

Figure 7 Comparison of the effect of the Eulerian mesh size ondisplacement-time graph of spherical pounder

Eulerian domain was modeled using quarter symmetry witha mesh size of 15 cm and contains 1538943 elements

63 Lagrangian Model Definition Figure 8 illustratespounder dimensions for different geometric shapes Thesepounders are identical in terms of material mass densityYoungrsquos modulus Poissonrsquos ratio volume mass and dropheight from the water surface The only difference is theirgeometric shapeThese parameters were regarded as identicalin all pounders to investigate the effect of pounder shapeon deepwater displacement and velocity air cavity shapepinch-off time and depth and the effect of water drag forceon velocity depreciation when impacting the model bed

Pounder dimensions were selected so that they had avolume of 523 cm3 and equal mass due to their identicalmaterial and mass density Table 2 illustrates the moment ofinertia for pounders with different geometric shapes

Table 3 illustrates thematerial properties of the poundersThese parameters are regarded as identical for all pounderswith the mentioned geometric shapes

64 Lagrangian Mesh Size Due to the negligible effect ofLagrangian mesh size on the results and to improve analysistime the Lagrangian mesh size for all pounders was set to05 cm

Pyramid Cone

Cube Cylinder Sphere

10 ＝Ｇ10 ＝Ｇ10 ＝Ｇ 10 ＝Ｇ

10 ＝Ｇ10 ＝Ｇ10 ＝Ｇ

10 ＝Ｇ5236 ＝Ｇ

20 ＝Ｇ15708 ＝Ｇ

666 ＝Ｇ

Figure 8 The dimensions of pounders with different geometricshapes

Table 3 Material properties of pounders

Materialtype Mass (kg) Density (120588)

(kgm3)

Youngrsquosmodulus (119864)(kgm2)

Poissonrsquosratio (120592)

Steel 411 7850 21 times 1010 03

65 Boundary Condition Wall boundary conditions wereapplied to the sides and bottom of the Eulerian domain tosimulate a water tank Pounder displacement was not con-strained in any direction and the pounders could freely movein all directions The sides and bottom boundaries of thethree-dimensional Eulerian domain were set as nonreflectingboundaries so that the waves released due to pounder impacton water surface did not return to the computational domainafter reaching these boundaries and being absorbed by themThis boundary condition is written as [37]

119889119901 minus 119901119888119889119906 = 0119889119909119889119905 = minus119888 (41)

where 120588 is density 119888 is the speed of sound in the fluid 119901 ispressure 119906 is velocity perpendicular to the wave and 119909 is thedirection perpendicular to the boundary

66 Drop Height and Impact Velocity of Pounders Afterdropping the pounders from a height (ℎ) above the watersurface the velocity of the pounders at themoment of impactwas determined using

1198812 minus 11988120 = 2119892ℎ (42)

where1198810 is the pounder initial velocity before being droppedfrom height ℎ above the waterrsquos surface 119881 is pounder

12 Geofluids

01 02 03 04 050Time (s)

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

CubeCylinderSphere

PyramidCone

Figure 9 Displacement-time graph of pounders with differentgeometric shapes

secondary velocity when impacting the water surface and 119892is the gravitational acceleration of the Earth

From (42) which expresses a time independent equationfor the free fall of objects the initial velocity and the terminalvelocity of the object drop height and the gravitationalacceleration of the Earth are interdependent in the absenceof time In this numerical simulation the pounders weredropped from a height of 1m above thewaterrsquos surfacewith aninitial velocity (1198810) equal to 0ms Therefore the secondaryvelocity of the pounders when impacting the water surfacewas obtained using

119881 = radic2119892ℎ (43)

Instead of simulating a full dropping event from theinitial position the pounders were placed exactly tangent towaterrsquos surface and their secondary velocity at the momentof impact was determined according to (43) A gravitationalacceleration (gravity) of minus98ms2 was applied in the normaldirection of the pounders Therefore a secondary velocityof 443ms in the normal direction of the waterrsquos surfacecorresponds to the speed that would be obtained for thepounders

7 Results and Discussion

71 Effect of the Geometric Shape of the Pounder on Deep-water Displacement Figure 9 illustrates the displacement-time graph of pounders with different geometric shapes Thegraph continues for the cubical pounder from the momentof water surface impact at time zero up to the depth of 1mat 415ms The displacement-time graph of the cylindrical

pounder indicates that after entering the water it reachesthe model bed in a shorter amount of time than the cubicalpounder The graph for this pounder continues from themoment of water surface impact at time zero up to the depthof 1m at 355ms The linear displacement-time graph forthe spherical and pyramidal pounders indicates the fasterdeepwater movement of these pounders compared to thecubical and cylindrical pounders The deepwater movementtime of the spherical and pyramidal pounders after enteringthe water to themodel bed is 285ms and 265ms respectivelyThe linear displacement-time graph of the conical pounderindicates that it has a faster deepwater movement comparedto other pounders as it takes 235ms to reach the model bedafter entering the water

72 Mechanism of Air Cavity Formation and Pinch-Off dueto Deepwater Movement of Pounders with Different Geomet-ric Shapes Figure 10 illustrates the movement trajectory ofpounders with different geometric shapes from the momentof water impact towhen the pounders reached themodel bedAfter entering the water a symmetric air cavity is formedbehind the pounders Air cavity formation involves severalsteps including air cavity creation and expansion behind thepounder air cavity contraction and air cavity collapse [34]As the pounder moves downward in water depth it imposesa force onto surrounding fluid in its radial directions andtransfers its momentum to the fluid This extension is facedwith the hydrostatic pressure resistance of fluidThe directionof the radial flow is reversed and eventually leads to aircavity contraction and collapse In the other words the aircavity contraction is accelerated until pinch-off occurs andthe air cavity is divided into two distinct parts The upperpart of the air cavity continues its contraction and movestowards the waterrsquos surface whereas the lower part of the aircavity clings to the pounder and moves along with it [34]After impacting the water surface the cubical and cylindricalpounders due to their flat contact surfaces lose some of theirvelocityThis velocity depreciation slows down and continuesuntil pinch-offThe spherical pounder with its curved contactsurface has less velocity depreciation compared to the cubicaland cylindrical pounders However some pounder velocityis always depreciated due to impacting the water surfaceThe velocity depreciation process for the spherical pounderduring deepwater movement continues until pinch-off Afterimpacting the waterrsquos surface the pyramidal pounder with itspointy contact surface has a slight velocity depreciation butcontinues to move deeper into the water with approximatelythe same secondary velocity until pinch-off By imposing thedrag force of water onto the pounders surface its velocityis gradually decreased until it reaches the model bed Afterimpacting the water surface the conical pounder with itspointy contact surface continues to move deeper into thewater with an increasing velocity The increasing velocity ofthe conical pounder during deepwater movement continueseven after pinch-off When the pounder moves towards themodel bed the waters drag force is imposed onto poundersurface causing velocity depreciation The pinch-off time forthe pounders with the above-mentioned shapes is between225 and 250ms

Geofluids 13

t = 0 25ＧＭ 50ＧＭ 100ＧＭ 200ＧＭ 225ＧＭ 250ＧＭ 275ＧＭ 300ＧＭ 325ＧＭ 350ＧＭ150ＧＭCu

beCy

linde

rSp

here

Pyra

mid

Con

e

Figure 10 Air cavity formation due to deepwater movement of pounders with different geometric shapes and pinch-off time and depth

73 Effect of the Pounder Geometric Shape onDeepwater Veloc-ity Figure 11 illustrates the velocity-time graph for pounderswith different geometric shapes The cubical and cylindricalpounders lose some of their secondary velocity at the earlymoments of water entry Hence the velocity-time graph ofthese pounders illustrates great velocity depreciation up to15ms The velocity reduction of the cubical and cylindricalpounders gradually continues until pinch-off The velocityof the spherical pounder decreases up to 45ms after waterentry but its velocity depreciation is less than the cubicaland cylindrical pounders This velocity reduction continuesat a slower rate until pinch-off In the initial moments ofwater entry the pyramidal pounder has a slight reduction insecondary velocity up to 55ms Then velocity depreciationcontinues at a faster rate due to the drag force imposed onto

pounders surface and the velocity-time graph of the pyrami-dal pounder shows a gradual velocity reduction until pinch-off After impacting the water surface the conical poundercontinues its deepwater movement with an increasing veloc-ity rate Due to its very low slope the velocity-time graphindicates an increase in velocity for the conical pounder butdue to the waters drag force its velocity begins to decrease at145ms The velocity-time graph indicates a gradual decreasein velocity for conical pounder and velocity depreciationcontinues until pinch-off After pinch-off pounder velocitiesgradually decrease until they reach themodel bed Eventuallywith a velocity of 388ms and 157ms at the moment ofmodel bed impact the conical and cubical pounders have thehighest and lowest velocity respectively Water drag force isimposed in the opposite direction of poundermovement and

14 Geofluids

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Velo

city

(ms

)

01 02 03 040Time (s)

CubeCylinderSphere

PyramidCone

Figure 11 Velocity-time graph of pounders with different geometricshapes

it has the most and least influence on the cubical and conicalpounders respectively

8 Critical Drop Height of the Pounder

In the dynamic compaction of the seabed determining thecritical drop height of the pounder from the freewater surfaceis an effective parameter for the pounder to reach maximumdeepwater velocity After dropping the pounder from height119867 above the free water surface with an initial velocity (1198810)of 0ms its velocity gradually increases until it reaches thesecondary velocity (119881) at the moment of water impact Afterentering the water pounder velocity is decreased due to waterdrag force According to the free fall equation for objectsincreasing the drop height from the waterrsquos surface leads toincreased pounder velocity at the moment of water surfaceimpact but it does not necessarily lead to an increase inpounder velocity during seabed impact because the criticaldrop height of the pounder is a determining factor ofits maximum deepwater velocity According to Figure 12 aconical pounder with a mass of 411 kg was dropped fromdifferent heights (119867) and its velocity when impacting thewaters was measured according to the time independentequation for the free fall of objects Table 4 illustrates thevelocity of the conical pounder with different drop heightswhen impacting the waterrsquos surface

81 Effect of the Drop Height of the Pounder from WaterSurface on Deepwater Displacement Figure 13 illustrates thedisplacement-time graph of the conical pounder with dif-ferent drop heights from the waterrsquos surface As can beseen increasing the drop height of the pounder leads to

Water levelℎ

Figure 12 Schematic of drop height of the conical pounder fromwater surface

Table 4 Velocity of the conical pounder with different drop heightsat the moment of impacting the water surface

Drop height of the pounderfrom water surface (m)

Pounder velocity at themoment of impacting water

surface (ms)025 22105 313075 3831 443125 49515 542175 5852 626225 66425 7

displacement-time graphs that are closer to each other Infact an increase in the drop height of the pounder from thewaterrsquos surface leads to an increase in its secondary velocitywhen impacting the waterrsquos surface Hence the pounderenters the water with a higher velocity and reaches the modelbed in less time

82 Effect of the Drop Height of the Pounder from WaterSurface on Its Deepwater Velocity Increasing the drop heightof the pounder from the waterrsquos surface leads to an increasein its velocity before impacting the water but the poundervelocity gradually decreases after entering the water due tothe its drag force Figure 14 illustrates the velocity-time graphfor the conical pounder with different drop heights from thewaterrsquos surface When the pounder is dropped from a lowheight due to its low velocity it moves deepwater due to itsweight Moreover because of the low velocity of the pounderwhen entering the water pinch-off occurs in less time and atlower depths and its velocity increases after pinch-off until itreaches terminal velocity Increases in pounder drop heightsincrease water entry velocity Hence it will have greatervelocity depreciation in the first moments of water entry

Figure 15 illustrates the terminal velocity-time graph forthe conical pounderwith different drop heightswhen impact-ing the model bed As can be seen increasing the drop height

Geofluids 15

01 02 03 040Time (s)

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

H = 1Ｇ

H = 075Ｇ

H = 05 Ｇ

H = 025 Ｇ

H = 175Ｇ

H = 15 Ｇ

H = 125 Ｇ

H = 25 Ｇ

H = 225 Ｇ

H = 2Ｇ

Figure 13 Displacement-time graph of the conical pounder withdifferent drop heights

of the pounder from the waterrsquos surface leads to an increasein its terminal velocity when impacting the model bedTheseconditions are dominant as long as the pounder reaches acritical drop height from the water surface but increasingthe pounder drop height higher than the critical drop heightleads to decreased terminal velocity when impacting themodel bed In this numerical simulation after dropping thepounder from a height of 2m above the waterrsquos surface itssecondary velocity was 626ms when impacting the watersurface After entering the water the secondary velocity ofthe pounder decreased until it impacts the model bed witha terminal velocity of 454ms Therefore a drop height of2m from above the waterrsquos surface has the highest terminalvelocity when the pounder impacts themodel bed Increasingthe drop height bymore than 2m leads to decreased pounderterminal velocity when impacting the model bed The heightwhich has the greatest effect on pounder terminal velocitywhen impacting the model bed is called the critical dropheight Increasing the drop height of the pounder to greaterthan its critical drop height leads to decreased terminalvelocity when impacting the model bed

Figure 16 illustrates the secondary and terminal velocityof the conical pounder with different drop heights As canbe observed increasing the drop height of the pounder fromwaterrsquos surface leads to increased differences between thesecondary and terminal velocities of the pounder Increasingthe drop height to more than the critical drop height leadsto increased pounder secondary velocity when impactingthe waterrsquos surface and decreased terminal velocity whenimpacting the model bed Hence the difference betweenthe secondary and terminal velocities of the pounder is

01 02 03 040Time (s)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)H = 175Ｇ

H = 125 Ｇ

H = 075Ｇ

H = 025 Ｇ H = 225 Ｇ

H = 1Ｇ

H = 05 Ｇ

H = 25 Ｇ

H = 2Ｇ

H = 15 Ｇ

Figure 14 Velocity-time graph of the conical pounder with differentdrop heights

minus430

minus447 minus454minus415

minus426 minus380

minus367minus355

minus332minus310

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Term

inal

velo

city

(ms

)

01 02 03 040Time (s)

Figure 15 Terminal velocity of conical pounder with different dropheights versus time

not reasonable and a significant amount of the pounderssecondary velocity is depreciated when entering the waterThe great difference between the secondary and terminalvelocities of the pounder reveals the necessity of determiningthe critical drop height of the pounder before dynamic seabedcompaction operations Increasing the pounder drop heightto be greater than the critical drop height not only hasa reverse result on pounder velocity when impacting theseabed but it also requires special equipment such as lengthy

16 Geofluids

Secondary velocityTerminal velocity

025 05 075 1 125 15 175 2 225 250Drop height (m)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)

Figure 16 Secondary and terminal velocities of conical pounderversus drop heights

1 2 3 4 5 6 7 80Impact velocity (ms)

0

005

01

015

02

025

03

Pinc

h-off

tim

e (s)

Figure 17 Pinch-off time versus impact velocity of conical pounder

cranes and huge barges leading to increased implementationcosts

Figure 17 illustrates pinch-off time based on the sec-ondary velocity of the pounder at the moment of watersurface impact As can be seen pinch-off time is a weakfunction of pounder impact velocity and increasing impactvelocity leads to a slight reduction in pinch-off time Thisbehaviour was also observed in the analytical and numericalresults found by Lee et al [40]

Figure 18 illustrates pinch-off depth based on the sec-ondary velocity of pounder at the moment of water surfaceimpact As can be seen the pinch-off depth is a linear

1 2 3 4 5 6 7 80Impact velocity (ms)

minus12

minus1

minus08

minus06

minus04

minus02

0

Pinc

h-off

dep

th (m

)

Figure 18 Pinch-off depth versus impact velocity of conicalpounder

function of pounder impact velocity These results are in linewith the analytical equations of Lee et al [40]

9 Conclusions

In this study the water entry problem of three-dimensionalpounders with different geometric shapes of cube cylindersphere pyramid and cone was numerically simulated bycommercial finite element code Abaqus 614-2 An explicitdynamic analysis method was employed to model the fluid-structure interaction using a Coupled Eulerian-Lagrangian(CEL) method Before applying the model to water entryof the pounders a three-dimensional numerical model wasdeveloped to simulate a transient free-falling wedge impact-ing the free surface of the water in order to calibrate theadopted computationalmodelThe accuracy of the numericalmodel was examined by comparing numerical results withthe experimental data of Aristoff et al The good agreementbetween numerical simulation results with the theoreticaland experimental data of Aristoff et al demonstrates thecapability of the employed numerical methodology to sim-ulate the water entry problem for three-dimensional geome-tries with acceptable accuracy After water entry by a pounderdropped fromaheight of 1m a symmetric air cavity is formedbehind the pounder As the pounder moves downward inwater depth it imposes a force onto surrounding fluid inthe radial directions and transfers its momentum to thefluid This extension is faced with the hydrostatic pressureresistance of the fluid causing the direction of the radialflow to be reversed leading to air cavity contraction andcollapse The results show that the water drag force had thehighest and lowest effect on the cubical and conical poundersrespectively and increasing the drop height to greater thanthe critical drop height leads to increased pounder velocitywhen impacting the waterrsquos surface and decreased velocitywhen impacting the model bed Pinch-off time is a very weak

Geofluids 17

function of pounder velocity when impacting the waterrsquossurface but the pinch-off depth increases linearly along withincreased pounder impact velocities It is recommended thatthe critical drop height for the pounder is determined beforedynamic seabed compaction operations and the adjustmentof crane heights to this height In addition it is also recom-mended that perforated pounders be used in order to reducevelocity depreciation when impacting the waterrsquos surface andto reduce the effect of water drag force

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors warmly appreciate the cooperation and guidanceof Dr B Hamidi and Dr A Mehrabadi in this study

References

[1] B Hamidi J-M Debats H Nikraz and S Varaksin ldquoOffshoreground improvement recordsrdquo Australian Geomechanics Jour-nal vol 48 no 4 pp 111ndash122 2013

[2] SWatanabe ldquoResistance of impact onwater surface part I-cone Institute of Physical andChemical ResearchrdquoTokyo vol 12 pp251ndash267 1930

[3] S Watanabe ldquoResistance of impact on water surface part II-cone (continued)rdquo Institute of Physical and Chemical Researchvol 14 pp 153ndash168 1930

[4] A May and J C Woodhull ldquoDrag coefficients of steel spheresentering water verticallyrdquo Journal of Applied Physics vol 19 no12 pp 1109ndash1121 1948

[5] A May ldquoEffect of surface condition of a sphere on its water-entry cavityrdquo Journal of Applied Physics vol 22 no 10 pp 1219ndash1222 1951

[6] AMay ldquoVertical entry ofmissiles into waterrdquo Journal of AppliedPhysics vol 23 no 12 pp 1362ndash1372 1952

[7] A P New T S Lee and H T Low ldquoImpact loading and waterentrance characteristics of prismatic bodiesrdquo in Proceedings ofthe third international offshore and polar engineering conferencepp 282ndash287 1993

[8] M Anghileri and A Spizzica ldquoExperimental validation of finiteelement models for water impactsrdquo in Proceedings of the secondinternational crash usersrsquo seminar Cranfield Impact Centre LtdEngland UK 1995 second international crash users seminar

[9] A Engle andR Lewis ldquoA comparison of hydrodynamic impactsprediction methods with two dimensional drop test datardquoMarine Structures vol 16 no 2 pp 175ndash182 2003

[10] J M Aristoff T T Truscott A H Techet and J W M BushldquoThe water entry of decelerating spheresrdquo Physics of Fluids vol22 no 3 Article ID 009002PHF pp 1ndash8 2010

[11] T Von Karman The impact of seaplane floats during landingNational Advisory Committee for Aeronautics NACA TN no321 1929

[12] HWagner ldquoPhenomena associatedwith impacts and sliding onliquid surfacesrdquo Journal of Applied Mathematics and Mechanicsvol 12 pp 193ndash215 1932

[13] W S Vorus ldquoA flat cylinder theory for vessel impact and steadyplaning resistancerdquo Journal of Ship Research vol 40 no 2 pp89ndash106 1996

[14] T Miloh ldquoOn the initial-stage slamming of a rigid sphere in avertical water entryrdquo Applied Ocean Research vol 13 no 1 pp13ndash43 1991

[15] J M Aristoff and JW Bush ldquoWater entry of small hydrophobicspheresrdquo Journal of Fluid Mechanics vol 619 pp 45ndash78 2009

[16] P Ghadimi A Dashtimanesh and S R Djeddi ldquoStudy of waterentry of circular cylinder by using analytical and numericalsolutionsrdquo Journal of the Brazilian Society ofMechanical Sciencesamp Engineering vol 34 no 3 pp 225ndash232 2012

[17] A TassinD J PiroAAKorobkinK JMaki andM J CookerldquoTwo-dimensional water entry and exit of a body whose shapevaries in timerdquo Journal of Fluids and Structures vol 40 pp 317ndash336 2013

[18] E Yao H Wang L Pan X Wang and R Woding ldquoVerticalwater-entry of bullet-shaped projectilesrdquo Journal of AppliedMathematics and Physics vol 02 no 06 pp 323ndash334 2014

[19] M-S Park Y-R Jung and W-G Park ldquoNumerical study ofimpact force and ricochet behavior of high speed water-entrybodiesrdquo Computers amp Fluids vol 32 no 7 pp 939ndash951 2003

[20] D Battistin and A Iafrati ldquoHydrodynamic loads during waterenergy of two-dimensional and axisymmetric bodiesrdquo Journalof Fluids and Structures vol 17 no 5 pp 643ndash664 2003

[21] A A Korobkin and M Ohkusu ldquoImpact of two circular platesone of which is floating on a thin layer of liquidrdquo Journal ofEngineering Mathematics vol 50 no 4 pp 343ndash358 2004

[22] K M Kleefsman G Fekken A E Veldman B Iwanowski andB Buchner ldquoA volume-of-fluid based simulation method forwave impact problemsrdquo Journal of Computational Physics vol206 no 1 pp 363ndash393 2005

[23] Y Kim Y Kim Y Liu and D K P Yue ldquoOn the water-entryimpact problem of asymmetric bodiesrdquo in Proceedings of the9th International Conference on Numerical Ship Hydrodynamics(NSH rsquo07) August 2007

[24] Q Yang and W Qiu ldquoNumerical solution of 2D slammingproblem with a CIPmethodrdquo in Proceedings of the InternationalConference on Violent Flows Research Institute for AppliedMechanics Kyushu University Japan 2007

[25] A C Fairlie-Clarke and T Tveitnes ldquoMomentum and gravityeffects during the constant velocitywater entry ofwedge-shapedsectionsrdquo Ocean Engineering vol 35 no 7 pp 706ndash716 2007

[26] Q Yang andW Qiu ldquoNumerical simulation of water impact for2D and 3D bodiesrdquo Ocean Engineering vol 43 pp 82ndash89 2012

[27] G Wu ldquoNumerical simulation for water entry of a wedge atvarying speed by a high order boundary element methodrdquoJournal of Marine Science and Application vol 11 no 2 pp 143ndash149 2012

[28] M Ahmadzadeh B Saranjam A Hoseini Fard and A RBinesh ldquoNumerical simulation of sphere water entry prob-lem using Eulerian-Lagrangian methodrdquo Applied MathematicalModelling vol 38 no 5-6 pp 1673ndash1684 2014

[29] V-T Nguyen D-T Vu W-G Park and Y-R Jung ldquoNumericalanalysis of water impact forces using a dual-time pseudo-compressibility method and volume-of-fluid interface trackingalgorithmrdquo Computers amp Fluids vol 103 pp 18ndash33 2014

[30] S Kim and N Kim ldquoIntegrated dynamics modeling forsupercavitating vehicle systemsrdquo International Journal of NavalArchitecture and Ocean Engineering vol 7 no 2 pp 346ndash3632015

18 Geofluids

[31] V-T Nguyen D-T Vu W-G Park and C-M Jung ldquoNavier-Stokes solver for water entry bodies with moving Chimera gridmethod in 6DOF motionsrdquo Computers amp Fluids vol 140 pp19ndash38 2016

[32] M Mirzaei M Eghtesad and M Alishahi ldquoPlaning forceidentification in high-speed underwater vehiclesrdquo Journal ofVibration and Control vol 22 no 20 pp 4176ndash4191 2016

[33] G Qiu S Henke and J Grabe ldquoApplication of a CoupledEulerian-Lagrangian approach on geomechanical problemsinvolving large deformationsrdquoComputersampGeosciences vol 38no 1 pp 30ndash39 2011

[34] M R Erfanian M Anbarsooz N Rahimi M Zare andM Moghiman ldquoNumerical and experimental investigationof a three dimensional spherical-nose projectile water entryproblemrdquo Ocean Engineering vol 104 pp 397ndash404 2015

[35] W M Isbell F H Shipman and A H Jones ldquoHugoniotequation of state measurements for eleven material to fiveMegabars Materials and Structural Laboratory ManufacturingDevelopments General Motors Technical centerrdquo Tech RepMichigan 1968

[36] M A Jandron R C Hurd J L Belden A F Bower W Fennelland T T Truscott ldquoModeling of hyperelastic water-skippingspheres using AbaqusExplicitrdquo in Proceedings of the SIMULIACommunity Conference 2014

[37] H Forouzani B Saranjam R Kamali and A Abdollahi-far ldquoElasto-plastic time dependent impact analysis of highspeed projectile on water surfacerdquo Journal of Solid and FluidMechanics vol 3 pp 281ndash298 2016

[38] M R Erfanian andMMoghiman ldquoNumerical and experimen-tal investigation of a projectile water entry problem and studyof velocity effect on time and depth of pinch-offrdquo Journal ofModares Mechanical Engineering vol 15 pp 53ndash60 2015

[39] A Sillem Feasibility study of a tire hydroplaning simulation in afinite element code using a coupled Eulerian-Lagrangian method[MSc thesis] 2008

[40] M Lee R G Longoria and D E Wilson ldquoCavity dynamics inhigh-speed water entryrdquo Physics of Fluids vol 9 no 3 pp 540ndash550 1997

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Submit your manuscripts atwwwhindawicom

8 Geofluids

handle flow appropriately It is important to note that CELis not structured in the same way as a CFD code and wasnot intended to fill this role While it does include viscouseffects with extension to laminar flows it does not include anyturbulence effects but in the context of an impact which has ashort duration and is highly transient (with turbulence effectsbeing of less importance to the structural loading) CEL iswell suited [28]

4 Implementation ofCoupled Eulerian-Lagrangian (CEL)Method in Abaqus

In order to simulate the water entry problem of the pounderand implement the Coupled Eulerian-Lagrangian (CEL)method in Abaqus a geometric model for the pounder andwater was first created A partitionwas created to definewaterand void regions at the water surface Since the fluid (water)undergoes a large deformation an Eulerian grid is suitableThe rigid body (pounder) has small deformations relative tothe fluid making an Lagrangian grid suitable Therefore thewater wasmodeled as Eulerian and the pounder wasmodeledas solid geometry with a rigid body constraint applied to itThen the material properties for the pounder and water weredefined Properties include density (120588) Youngrsquos modulus (119864)and Poissonrsquos ratio (]) for the pounder and density (120588119908) anddynamic viscosity (120578) for the water In the Coupled Eulerian-Lagrangianmethod when the pounder is considered as rigidYoungrsquos modulus and Poisson ratios values should be definedfor the Abaqus but they have no effect on the results Inthe Coupled Eulerian-Lagrangian method 119880119904-119880119901 equationof state (EOS) was selected to define the water properties inAbaqus The 119880119904-119880119901 equation of state is described in detail inAbaqus 614 documentation and it is used to simulateNavier-Stokes flow when turbulence flow is negligible Becausethe major force imposed on the rigid body in water entryproblems is pressure force this equation of state can beused as an appropriate approximation Parameters such as1198620(speed of sound in water) 119904 (constant coefficient in shockvelocity equation) and Γ0 (material constant in Gruneisenequation of state)were defined for the119880119904-119880119901 equation of stateIn the next step the pounder and water were entered intothe assembly section and the pounder was transferred to thewaters center tangent to its surface In order to simulate a free-falling rigid body into water the pounder was first consideredin a position tangent to the water surface then the secondaryvelocity of pounder at the moment of impact was assignedIn the next step explicit dynamic analysis was selected andthe time period for pounder movement between enteringthe water and reaching the model bed was considered as 05seconds Nlgeom was also activated to control the nonlineareffects of large displacements Then by defining a referencepoint and assigning it to the pounder required outputs suchas displacement and velocity values for the vertical direc-tion were selected Based on its defined algorithm Abaqusconsiders the maximum hardness between the Lagrangianelements (pounder) and Eulerian material (water) The Eule-rian solution method uses the meshes defined for poundermovement between water meshes and Abaqus detects theboundary between the Lagrangian and Eulerian materials

Wall

Wall

Wall Water

Sphere

Initial void region

Freesurface

Figure 2 The computational domain boundary conditions andgrid distribution for the water entry problem of sphere

Considering themaximumhardness between the Lagrangianand Eulerian boundaries the contact between these twomaterials is established using the penalty method Concen-trated mass was assigned to the pounder and its moment ofinertia was defined for 119883 119884 and 119885 In the next step thepounder was defined as a rigid body that can move freelyin all directions Gravitational acceleration (Gravity) wasdefined as minus98ms2 in the 119884 direction and a nonreflectingEulerian boundary was assigned to the sides and bottomsurfaces of the computational water domain This boundarycondition is specific to the Eulerian method and preventswave propagation due to rigid body impact on the waterrsquossurface from returning to the computational domain afterreaching the boundariesWith these boundary conditions thewater is regarded as infinite Assuming that the pounder isdropped from a height 119867 above a free water surface withan initial velocity (1198810) of 0ms its secondary velocity (119881) atthe moment of impact was calculated according to the freefall equation of bodies This velocity was vertically applied tothe pounder Eulerian and void regions were specified usingmaterial assignment and the mesh size was defined for thewater and pounder An Eulerian mesh was selected as thewater mesh and an explicit mesh was selected as the poundermesh using structured techniques Finally the problem wassolved and the results were analyzed

41 Model Geometry Before applying the model to pounderwater entry a three-dimensional numerical model was devel-oped to simulate a transient free-falling wedge impactingthe free surface of water in order to check and calibratethe adopted computational model for simpler problemsThen the accuracy of the numerical model was examinedby comparing the numerical results with the experimen-tal data of Aristoff et al [10] for the water entry of asphere Figure 2 illustrates a schematic for the computationaldomain boundary conditions and grid distribution for

Geofluids 9

Table 1 Equation of state parameters for water

Density (120588119908)(kgm3)

Dynamicviscosity (120578)(kgsm2)

Speed of sound(1198880) (msec) Γ0 1199041000 0001 1450 0 0

Water

Aperture release

Z

Diffuser

Lights

Sphere

Camera

Figure 3 Schematic of the experimental apparatus of Aristoff et al[10]

the sphere water entry problem The dimensions for thecomputational domain are considered to be large enoughso that with further increases no considerable changes areobserved in the numerical resultsThe sphere wasmodeled assolid geometry and a rigid body constraint was subsequentlyapplied to it The fluid domain was modeled as a cubewith dimensions of 30 times 50 times 60 cm3 Mesh refinement wasperformed after assigning it to the computational domaintherefore the Eulerianmesh grid size was gradually decreaseduntil no considerable changeswere observed in the numericalresults The Eulerian domain (water) was modeled usingquarter symmetry and contains 746172 elements The spherewas alsomodeled as a Lagrangian solid using 15000 elementsThe Eulerian domain was divided into two parts includingthe upper and lower regions These regions were defined asvoid and stationary water respectively A solid sphere witha diameter of 254 cm and an initial velocity of 217ms inthe normal direction was placed on the free water surfaceThe sphere can move freely in all directions with six degreesof freedom The physical properties of water used in thenumerical simulations are given in Table 1Γ0 and 119904 are material constants in the Gruneisen equationof state and the constant coefficient in shock velocity equa-tion respectively

5 Validation

A schematic of the experimental apparatus of Aristoff et al[10] is presented in Figure 3 A sphere was held by a cameralike aperture at height 119867 above a water tank The tank haddimensions of 30 times 50 times 60 cm3 and was illuminated by a

bank of twenty 32W fluorescent bulbs A diffuser was usedto provide uniform lighting and care was taken to keep thewater surface free of dust Tank dimensions were selectedso that the walls had negligible effects on the results Thesphere was released from rest and fell towards the waterreaching it with an approximate speed119881 asymp radic2119892ℎThe impactsequence was recorded at 2000 frames per second using ahigh-speed camera The resolution was set to 524 times 1280pixels (px) with a field of view of 1128 times 2755 cm2 yieldinga 4646 pxcm magnification The trajectory of the sphereand its impact speed were determined with subpixel accuracythrough a cross correlation andGaussian peak-fittingmethodyielding position estimates accurate to 0025 px (00005 cm)and impact speeds accurate to plusmn4

Figure 4 illustrates a comparison of cavity shapes betweennumerical simulation and the experimental photographstaken by Aristoff et al [10] for a density ratio of 786 and ansimilar period of time The impact velocity of the sphere was217ms and 119905 = 0 refers to when the spheres center passesthrough the free surface This figure shows the cavity shapesfor both experimental and numerical simulation for similartime periods except that cavity shape for the pinch-off timeof the numerical simulation was added to the pictures

An symmetric air cavity formed behind the sphereafter entering the water Air cavity formation consists ofseveral steps including cavity creation cavity expansioncavity contraction behind the sphere and cavity collapseAs the sphere moves deeper through the water it exerts aforce on surrounding fluid in radial directions and transfersits momentum to the fluid However fluid expansion isconfronted with fluid hydrostatic pressure resistance and asa result the radial flow direction is reversed leading to cavitycontraction and collapse Cavity collapse occurs when thecavity wall moves inward until the moment of pinch-off (119905= 708ms) and the cavity is divided into two separated parts[34] While the lower cavity sticks to the sphere and movesalong with it the upper cavity continues its contraction andmoves towards the waterrsquos surface As shown in Figure 4 thenumerical results and experimental observations are in goodagreement

Abaqus numerical simulation methods efficiently modelwater splashes and jet flow Pictures of these phenomenaare depicted in Figure 4 It was concluded from this figurethat pinch-off in the numerical simulation happens with adelay time compared to the experimental method Increasednumerical errors during solution time are one of the causesfor differences between experimental and numerical results

In Figure 5 the numerical results of this study arecompared with the theoretical and experimental data ofAristoff et al as a displacement-time graph of a sphere indifferent water depths [10] As can be seen the numericalresults are in good agreement with those of the analytics andexperiments However the presently computed sphere centerdepth is lower than that found experimentally It seems thatdue to lack of turbulent modeling ability in Abaqus Eulerianformulation and evidently increasing turbulence in flow bytime there is just less difference between experiment andnumerical results in the early time of simulation

10 Geofluids

Experimental results of Aristoff et al (2010)

Pinch-off

Pinch-offNumerical simulation results

t = 59 ＧＭ 129 ＧＭ 199ＧＭ 269ＧＭ 339ＧＭ 409 ＧＭ 479ＧＭ 549ＧＭ 619 ＧＭ 689 ＧＭ 759ＧＭ

t = 59ＧＭ 129 ＧＭ 199ＧＭ 269ＧＭ 339ＧＭ 409 ＧＭ 479ＧＭ 549ＧＭ 619 ＧＭ 689 ＧＭ 708ＧＭ 759ＧＭ

Figure 4 Comparison of numerical results with experimental photographs of Aristoff et al [10] for the sphere water entry problem

002 004 006 0080Time (s)

minus018

minus016

minus014

minus012

minus01

minus008

minus006

minus004

minus002

0

Disp

lace

men

t (m

)

Present studyExperiment (Aristoff et al 2010)Theory (Aristoff et al 2010)

Figure 5 Comparison of displacement-time graph of sphere inwater depth with the theoretical and experimental data of Aristoffet al [10]

6 Materials and Methods

61 Eulerian Model Definition Figure 6 illustrates a sche-matic of the computational domain boundary conditionsand grid distribution for the water entry problem of aspherical pounder The dimensions of the computationaldomain are considered to be large enough so that withfurther increases no considerable changes are observed inthe numerical results The Eulerian domain was dividedinto the upper and lower regions These regions are definedas void and stationary water respectively In water entryproblems for rigid bodies fluid domain dimensions must

Wall

Wall

Wall Water

Spherical pounder

Initial void region

Free surface

Figure 6 The computational domain boundary conditions andgrid distribution for water entry problem of spherical pounder

be at least eight times the dimensions of the rigid bodyHence in this study the fluid domain was modeled as acube with dimensions of 2 times 2 times 1m3 The pounders wereplaced exactly tangent to the water surface and their velocityat the moment of water surface impact was determinedaccording to the free fall equation of bodies The pounderscan move freely in all directions with six degrees of freedom119880119904-119880119901 equation of state was used to define water materialin the Coupled Eulerian-Lagrangian method This equationof state is useful for Navier-Stokes flow simulations whenturbulence flow is negligible In cases where the impact ofa rigid body on a water surface is investigated the majorforce imposed on water body is the pressure force Thereforewith appropriate approximations this equation of state canbe used to investigate the impact of rigid bodies on watersurfaces

62 EulerianMesh Size As shown in Figure 7 with decreasesin the Eulerian domain mesh size to less than 15 cm thedisplacement-time graph of the spherical pounder showsnegligible changes Therefore to improve analysis time the

Geofluids 11

Table 2 Moment of inertia of pounders with different geometric shapes

Geometric shape of the pounder Cube Cylinder Sphere Pyramid ConeMoment of inertia (cm4) 685 5137 411 3082 411

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

01 02 03 040Time (s)

2 ＝Ｇ

175 ＝Ｇ

15 ＝Ｇ

Figure 7 Comparison of the effect of the Eulerian mesh size ondisplacement-time graph of spherical pounder

Eulerian domain was modeled using quarter symmetry witha mesh size of 15 cm and contains 1538943 elements

63 Lagrangian Model Definition Figure 8 illustratespounder dimensions for different geometric shapes Thesepounders are identical in terms of material mass densityYoungrsquos modulus Poissonrsquos ratio volume mass and dropheight from the water surface The only difference is theirgeometric shapeThese parameters were regarded as identicalin all pounders to investigate the effect of pounder shapeon deepwater displacement and velocity air cavity shapepinch-off time and depth and the effect of water drag forceon velocity depreciation when impacting the model bed

Pounder dimensions were selected so that they had avolume of 523 cm3 and equal mass due to their identicalmaterial and mass density Table 2 illustrates the moment ofinertia for pounders with different geometric shapes

Table 3 illustrates thematerial properties of the poundersThese parameters are regarded as identical for all pounderswith the mentioned geometric shapes

64 Lagrangian Mesh Size Due to the negligible effect ofLagrangian mesh size on the results and to improve analysistime the Lagrangian mesh size for all pounders was set to05 cm

Pyramid Cone

Cube Cylinder Sphere

10 ＝Ｇ10 ＝Ｇ10 ＝Ｇ 10 ＝Ｇ

10 ＝Ｇ10 ＝Ｇ10 ＝Ｇ

10 ＝Ｇ5236 ＝Ｇ

20 ＝Ｇ15708 ＝Ｇ

666 ＝Ｇ

Figure 8 The dimensions of pounders with different geometricshapes

Table 3 Material properties of pounders

Materialtype Mass (kg) Density (120588)

(kgm3)

Youngrsquosmodulus (119864)(kgm2)

Poissonrsquosratio (120592)

Steel 411 7850 21 times 1010 03

65 Boundary Condition Wall boundary conditions wereapplied to the sides and bottom of the Eulerian domain tosimulate a water tank Pounder displacement was not con-strained in any direction and the pounders could freely movein all directions The sides and bottom boundaries of thethree-dimensional Eulerian domain were set as nonreflectingboundaries so that the waves released due to pounder impacton water surface did not return to the computational domainafter reaching these boundaries and being absorbed by themThis boundary condition is written as [37]

119889119901 minus 119901119888119889119906 = 0119889119909119889119905 = minus119888 (41)

where 120588 is density 119888 is the speed of sound in the fluid 119901 ispressure 119906 is velocity perpendicular to the wave and 119909 is thedirection perpendicular to the boundary

66 Drop Height and Impact Velocity of Pounders Afterdropping the pounders from a height (ℎ) above the watersurface the velocity of the pounders at themoment of impactwas determined using

1198812 minus 11988120 = 2119892ℎ (42)

where1198810 is the pounder initial velocity before being droppedfrom height ℎ above the waterrsquos surface 119881 is pounder

12 Geofluids

01 02 03 04 050Time (s)

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

CubeCylinderSphere

PyramidCone

Figure 9 Displacement-time graph of pounders with differentgeometric shapes

secondary velocity when impacting the water surface and 119892is the gravitational acceleration of the Earth

From (42) which expresses a time independent equationfor the free fall of objects the initial velocity and the terminalvelocity of the object drop height and the gravitationalacceleration of the Earth are interdependent in the absenceof time In this numerical simulation the pounders weredropped from a height of 1m above thewaterrsquos surfacewith aninitial velocity (1198810) equal to 0ms Therefore the secondaryvelocity of the pounders when impacting the water surfacewas obtained using

119881 = radic2119892ℎ (43)

Instead of simulating a full dropping event from theinitial position the pounders were placed exactly tangent towaterrsquos surface and their secondary velocity at the momentof impact was determined according to (43) A gravitationalacceleration (gravity) of minus98ms2 was applied in the normaldirection of the pounders Therefore a secondary velocityof 443ms in the normal direction of the waterrsquos surfacecorresponds to the speed that would be obtained for thepounders

7 Results and Discussion

71 Effect of the Geometric Shape of the Pounder on Deep-water Displacement Figure 9 illustrates the displacement-time graph of pounders with different geometric shapes Thegraph continues for the cubical pounder from the momentof water surface impact at time zero up to the depth of 1mat 415ms The displacement-time graph of the cylindrical

pounder indicates that after entering the water it reachesthe model bed in a shorter amount of time than the cubicalpounder The graph for this pounder continues from themoment of water surface impact at time zero up to the depthof 1m at 355ms The linear displacement-time graph forthe spherical and pyramidal pounders indicates the fasterdeepwater movement of these pounders compared to thecubical and cylindrical pounders The deepwater movementtime of the spherical and pyramidal pounders after enteringthe water to themodel bed is 285ms and 265ms respectivelyThe linear displacement-time graph of the conical pounderindicates that it has a faster deepwater movement comparedto other pounders as it takes 235ms to reach the model bedafter entering the water

72 Mechanism of Air Cavity Formation and Pinch-Off dueto Deepwater Movement of Pounders with Different Geomet-ric Shapes Figure 10 illustrates the movement trajectory ofpounders with different geometric shapes from the momentof water impact towhen the pounders reached themodel bedAfter entering the water a symmetric air cavity is formedbehind the pounders Air cavity formation involves severalsteps including air cavity creation and expansion behind thepounder air cavity contraction and air cavity collapse [34]As the pounder moves downward in water depth it imposesa force onto surrounding fluid in its radial directions andtransfers its momentum to the fluid This extension is facedwith the hydrostatic pressure resistance of fluidThe directionof the radial flow is reversed and eventually leads to aircavity contraction and collapse In the other words the aircavity contraction is accelerated until pinch-off occurs andthe air cavity is divided into two distinct parts The upperpart of the air cavity continues its contraction and movestowards the waterrsquos surface whereas the lower part of the aircavity clings to the pounder and moves along with it [34]After impacting the water surface the cubical and cylindricalpounders due to their flat contact surfaces lose some of theirvelocityThis velocity depreciation slows down and continuesuntil pinch-offThe spherical pounder with its curved contactsurface has less velocity depreciation compared to the cubicaland cylindrical pounders However some pounder velocityis always depreciated due to impacting the water surfaceThe velocity depreciation process for the spherical pounderduring deepwater movement continues until pinch-off Afterimpacting the waterrsquos surface the pyramidal pounder with itspointy contact surface has a slight velocity depreciation butcontinues to move deeper into the water with approximatelythe same secondary velocity until pinch-off By imposing thedrag force of water onto the pounders surface its velocityis gradually decreased until it reaches the model bed Afterimpacting the water surface the conical pounder with itspointy contact surface continues to move deeper into thewater with an increasing velocity The increasing velocity ofthe conical pounder during deepwater movement continueseven after pinch-off When the pounder moves towards themodel bed the waters drag force is imposed onto poundersurface causing velocity depreciation The pinch-off time forthe pounders with the above-mentioned shapes is between225 and 250ms

Geofluids 13

t = 0 25ＧＭ 50ＧＭ 100ＧＭ 200ＧＭ 225ＧＭ 250ＧＭ 275ＧＭ 300ＧＭ 325ＧＭ 350ＧＭ150ＧＭCu

beCy

linde

rSp

here

Pyra

mid

Con

e

Figure 10 Air cavity formation due to deepwater movement of pounders with different geometric shapes and pinch-off time and depth

73 Effect of the Pounder Geometric Shape onDeepwater Veloc-ity Figure 11 illustrates the velocity-time graph for pounderswith different geometric shapes The cubical and cylindricalpounders lose some of their secondary velocity at the earlymoments of water entry Hence the velocity-time graph ofthese pounders illustrates great velocity depreciation up to15ms The velocity reduction of the cubical and cylindricalpounders gradually continues until pinch-off The velocityof the spherical pounder decreases up to 45ms after waterentry but its velocity depreciation is less than the cubicaland cylindrical pounders This velocity reduction continuesat a slower rate until pinch-off In the initial moments ofwater entry the pyramidal pounder has a slight reduction insecondary velocity up to 55ms Then velocity depreciationcontinues at a faster rate due to the drag force imposed onto

pounders surface and the velocity-time graph of the pyrami-dal pounder shows a gradual velocity reduction until pinch-off After impacting the water surface the conical poundercontinues its deepwater movement with an increasing veloc-ity rate Due to its very low slope the velocity-time graphindicates an increase in velocity for the conical pounder butdue to the waters drag force its velocity begins to decrease at145ms The velocity-time graph indicates a gradual decreasein velocity for conical pounder and velocity depreciationcontinues until pinch-off After pinch-off pounder velocitiesgradually decrease until they reach themodel bed Eventuallywith a velocity of 388ms and 157ms at the moment ofmodel bed impact the conical and cubical pounders have thehighest and lowest velocity respectively Water drag force isimposed in the opposite direction of poundermovement and

14 Geofluids

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Velo

city

(ms

)

01 02 03 040Time (s)

CubeCylinderSphere

PyramidCone

Figure 11 Velocity-time graph of pounders with different geometricshapes

it has the most and least influence on the cubical and conicalpounders respectively

8 Critical Drop Height of the Pounder

In the dynamic compaction of the seabed determining thecritical drop height of the pounder from the freewater surfaceis an effective parameter for the pounder to reach maximumdeepwater velocity After dropping the pounder from height119867 above the free water surface with an initial velocity (1198810)of 0ms its velocity gradually increases until it reaches thesecondary velocity (119881) at the moment of water impact Afterentering the water pounder velocity is decreased due to waterdrag force According to the free fall equation for objectsincreasing the drop height from the waterrsquos surface leads toincreased pounder velocity at the moment of water surfaceimpact but it does not necessarily lead to an increase inpounder velocity during seabed impact because the criticaldrop height of the pounder is a determining factor ofits maximum deepwater velocity According to Figure 12 aconical pounder with a mass of 411 kg was dropped fromdifferent heights (119867) and its velocity when impacting thewaters was measured according to the time independentequation for the free fall of objects Table 4 illustrates thevelocity of the conical pounder with different drop heightswhen impacting the waterrsquos surface

81 Effect of the Drop Height of the Pounder from WaterSurface on Deepwater Displacement Figure 13 illustrates thedisplacement-time graph of the conical pounder with dif-ferent drop heights from the waterrsquos surface As can beseen increasing the drop height of the pounder leads to

Water levelℎ

Figure 12 Schematic of drop height of the conical pounder fromwater surface

Table 4 Velocity of the conical pounder with different drop heightsat the moment of impacting the water surface

Drop height of the pounderfrom water surface (m)

Pounder velocity at themoment of impacting water

surface (ms)025 22105 313075 3831 443125 49515 542175 5852 626225 66425 7

displacement-time graphs that are closer to each other Infact an increase in the drop height of the pounder from thewaterrsquos surface leads to an increase in its secondary velocitywhen impacting the waterrsquos surface Hence the pounderenters the water with a higher velocity and reaches the modelbed in less time

82 Effect of the Drop Height of the Pounder from WaterSurface on Its Deepwater Velocity Increasing the drop heightof the pounder from the waterrsquos surface leads to an increasein its velocity before impacting the water but the poundervelocity gradually decreases after entering the water due tothe its drag force Figure 14 illustrates the velocity-time graphfor the conical pounder with different drop heights from thewaterrsquos surface When the pounder is dropped from a lowheight due to its low velocity it moves deepwater due to itsweight Moreover because of the low velocity of the pounderwhen entering the water pinch-off occurs in less time and atlower depths and its velocity increases after pinch-off until itreaches terminal velocity Increases in pounder drop heightsincrease water entry velocity Hence it will have greatervelocity depreciation in the first moments of water entry

Figure 15 illustrates the terminal velocity-time graph forthe conical pounderwith different drop heightswhen impact-ing the model bed As can be seen increasing the drop height

Geofluids 15

01 02 03 040Time (s)

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

H = 1Ｇ

H = 075Ｇ

H = 05 Ｇ

H = 025 Ｇ

H = 175Ｇ

H = 15 Ｇ

H = 125 Ｇ

H = 25 Ｇ

H = 225 Ｇ

H = 2Ｇ

Figure 13 Displacement-time graph of the conical pounder withdifferent drop heights

of the pounder from the waterrsquos surface leads to an increasein its terminal velocity when impacting the model bedTheseconditions are dominant as long as the pounder reaches acritical drop height from the water surface but increasingthe pounder drop height higher than the critical drop heightleads to decreased terminal velocity when impacting themodel bed In this numerical simulation after dropping thepounder from a height of 2m above the waterrsquos surface itssecondary velocity was 626ms when impacting the watersurface After entering the water the secondary velocity ofthe pounder decreased until it impacts the model bed witha terminal velocity of 454ms Therefore a drop height of2m from above the waterrsquos surface has the highest terminalvelocity when the pounder impacts themodel bed Increasingthe drop height bymore than 2m leads to decreased pounderterminal velocity when impacting the model bed The heightwhich has the greatest effect on pounder terminal velocitywhen impacting the model bed is called the critical dropheight Increasing the drop height of the pounder to greaterthan its critical drop height leads to decreased terminalvelocity when impacting the model bed

Figure 16 illustrates the secondary and terminal velocityof the conical pounder with different drop heights As canbe observed increasing the drop height of the pounder fromwaterrsquos surface leads to increased differences between thesecondary and terminal velocities of the pounder Increasingthe drop height to more than the critical drop height leadsto increased pounder secondary velocity when impactingthe waterrsquos surface and decreased terminal velocity whenimpacting the model bed Hence the difference betweenthe secondary and terminal velocities of the pounder is

01 02 03 040Time (s)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)H = 175Ｇ

H = 125 Ｇ

H = 075Ｇ

H = 025 Ｇ H = 225 Ｇ

H = 1Ｇ

H = 05 Ｇ

H = 25 Ｇ

H = 2Ｇ

H = 15 Ｇ

Figure 14 Velocity-time graph of the conical pounder with differentdrop heights

minus430

minus447 minus454minus415

minus426 minus380

minus367minus355

minus332minus310

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Term

inal

velo

city

(ms

)

01 02 03 040Time (s)

Figure 15 Terminal velocity of conical pounder with different dropheights versus time

not reasonable and a significant amount of the pounderssecondary velocity is depreciated when entering the waterThe great difference between the secondary and terminalvelocities of the pounder reveals the necessity of determiningthe critical drop height of the pounder before dynamic seabedcompaction operations Increasing the pounder drop heightto be greater than the critical drop height not only hasa reverse result on pounder velocity when impacting theseabed but it also requires special equipment such as lengthy

16 Geofluids

Secondary velocityTerminal velocity

025 05 075 1 125 15 175 2 225 250Drop height (m)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)

Figure 16 Secondary and terminal velocities of conical pounderversus drop heights

1 2 3 4 5 6 7 80Impact velocity (ms)

0

005

01

015

02

025

03

Pinc

h-off

tim

e (s)

Figure 17 Pinch-off time versus impact velocity of conical pounder

cranes and huge barges leading to increased implementationcosts

Figure 17 illustrates pinch-off time based on the sec-ondary velocity of the pounder at the moment of watersurface impact As can be seen pinch-off time is a weakfunction of pounder impact velocity and increasing impactvelocity leads to a slight reduction in pinch-off time Thisbehaviour was also observed in the analytical and numericalresults found by Lee et al [40]

Figure 18 illustrates pinch-off depth based on the sec-ondary velocity of pounder at the moment of water surfaceimpact As can be seen the pinch-off depth is a linear

1 2 3 4 5 6 7 80Impact velocity (ms)

minus12

minus1

minus08

minus06

minus04

minus02

0

Pinc

h-off

dep

th (m

)

Figure 18 Pinch-off depth versus impact velocity of conicalpounder

function of pounder impact velocity These results are in linewith the analytical equations of Lee et al [40]

9 Conclusions

In this study the water entry problem of three-dimensionalpounders with different geometric shapes of cube cylindersphere pyramid and cone was numerically simulated bycommercial finite element code Abaqus 614-2 An explicitdynamic analysis method was employed to model the fluid-structure interaction using a Coupled Eulerian-Lagrangian(CEL) method Before applying the model to water entryof the pounders a three-dimensional numerical model wasdeveloped to simulate a transient free-falling wedge impact-ing the free surface of the water in order to calibrate theadopted computationalmodelThe accuracy of the numericalmodel was examined by comparing numerical results withthe experimental data of Aristoff et al The good agreementbetween numerical simulation results with the theoreticaland experimental data of Aristoff et al demonstrates thecapability of the employed numerical methodology to sim-ulate the water entry problem for three-dimensional geome-tries with acceptable accuracy After water entry by a pounderdropped fromaheight of 1m a symmetric air cavity is formedbehind the pounder As the pounder moves downward inwater depth it imposes a force onto surrounding fluid inthe radial directions and transfers its momentum to thefluid This extension is faced with the hydrostatic pressureresistance of the fluid causing the direction of the radialflow to be reversed leading to air cavity contraction andcollapse The results show that the water drag force had thehighest and lowest effect on the cubical and conical poundersrespectively and increasing the drop height to greater thanthe critical drop height leads to increased pounder velocitywhen impacting the waterrsquos surface and decreased velocitywhen impacting the model bed Pinch-off time is a very weak

Geofluids 17

function of pounder velocity when impacting the waterrsquossurface but the pinch-off depth increases linearly along withincreased pounder impact velocities It is recommended thatthe critical drop height for the pounder is determined beforedynamic seabed compaction operations and the adjustmentof crane heights to this height In addition it is also recom-mended that perforated pounders be used in order to reducevelocity depreciation when impacting the waterrsquos surface andto reduce the effect of water drag force

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors warmly appreciate the cooperation and guidanceof Dr B Hamidi and Dr A Mehrabadi in this study

References

[1] B Hamidi J-M Debats H Nikraz and S Varaksin ldquoOffshoreground improvement recordsrdquo Australian Geomechanics Jour-nal vol 48 no 4 pp 111ndash122 2013

[2] SWatanabe ldquoResistance of impact onwater surface part I-cone Institute of Physical andChemical ResearchrdquoTokyo vol 12 pp251ndash267 1930

[3] S Watanabe ldquoResistance of impact on water surface part II-cone (continued)rdquo Institute of Physical and Chemical Researchvol 14 pp 153ndash168 1930

[4] A May and J C Woodhull ldquoDrag coefficients of steel spheresentering water verticallyrdquo Journal of Applied Physics vol 19 no12 pp 1109ndash1121 1948

[5] A May ldquoEffect of surface condition of a sphere on its water-entry cavityrdquo Journal of Applied Physics vol 22 no 10 pp 1219ndash1222 1951

[6] AMay ldquoVertical entry ofmissiles into waterrdquo Journal of AppliedPhysics vol 23 no 12 pp 1362ndash1372 1952

[7] A P New T S Lee and H T Low ldquoImpact loading and waterentrance characteristics of prismatic bodiesrdquo in Proceedings ofthe third international offshore and polar engineering conferencepp 282ndash287 1993

[8] M Anghileri and A Spizzica ldquoExperimental validation of finiteelement models for water impactsrdquo in Proceedings of the secondinternational crash usersrsquo seminar Cranfield Impact Centre LtdEngland UK 1995 second international crash users seminar

[9] A Engle andR Lewis ldquoA comparison of hydrodynamic impactsprediction methods with two dimensional drop test datardquoMarine Structures vol 16 no 2 pp 175ndash182 2003

[10] J M Aristoff T T Truscott A H Techet and J W M BushldquoThe water entry of decelerating spheresrdquo Physics of Fluids vol22 no 3 Article ID 009002PHF pp 1ndash8 2010

[11] T Von Karman The impact of seaplane floats during landingNational Advisory Committee for Aeronautics NACA TN no321 1929

[12] HWagner ldquoPhenomena associatedwith impacts and sliding onliquid surfacesrdquo Journal of Applied Mathematics and Mechanicsvol 12 pp 193ndash215 1932

[13] W S Vorus ldquoA flat cylinder theory for vessel impact and steadyplaning resistancerdquo Journal of Ship Research vol 40 no 2 pp89ndash106 1996

[14] T Miloh ldquoOn the initial-stage slamming of a rigid sphere in avertical water entryrdquo Applied Ocean Research vol 13 no 1 pp13ndash43 1991

[15] J M Aristoff and JW Bush ldquoWater entry of small hydrophobicspheresrdquo Journal of Fluid Mechanics vol 619 pp 45ndash78 2009

[16] P Ghadimi A Dashtimanesh and S R Djeddi ldquoStudy of waterentry of circular cylinder by using analytical and numericalsolutionsrdquo Journal of the Brazilian Society ofMechanical Sciencesamp Engineering vol 34 no 3 pp 225ndash232 2012

[17] A TassinD J PiroAAKorobkinK JMaki andM J CookerldquoTwo-dimensional water entry and exit of a body whose shapevaries in timerdquo Journal of Fluids and Structures vol 40 pp 317ndash336 2013

[18] E Yao H Wang L Pan X Wang and R Woding ldquoVerticalwater-entry of bullet-shaped projectilesrdquo Journal of AppliedMathematics and Physics vol 02 no 06 pp 323ndash334 2014

[19] M-S Park Y-R Jung and W-G Park ldquoNumerical study ofimpact force and ricochet behavior of high speed water-entrybodiesrdquo Computers amp Fluids vol 32 no 7 pp 939ndash951 2003

[20] D Battistin and A Iafrati ldquoHydrodynamic loads during waterenergy of two-dimensional and axisymmetric bodiesrdquo Journalof Fluids and Structures vol 17 no 5 pp 643ndash664 2003

[21] A A Korobkin and M Ohkusu ldquoImpact of two circular platesone of which is floating on a thin layer of liquidrdquo Journal ofEngineering Mathematics vol 50 no 4 pp 343ndash358 2004

[22] K M Kleefsman G Fekken A E Veldman B Iwanowski andB Buchner ldquoA volume-of-fluid based simulation method forwave impact problemsrdquo Journal of Computational Physics vol206 no 1 pp 363ndash393 2005

[23] Y Kim Y Kim Y Liu and D K P Yue ldquoOn the water-entryimpact problem of asymmetric bodiesrdquo in Proceedings of the9th International Conference on Numerical Ship Hydrodynamics(NSH rsquo07) August 2007

[24] Q Yang and W Qiu ldquoNumerical solution of 2D slammingproblem with a CIPmethodrdquo in Proceedings of the InternationalConference on Violent Flows Research Institute for AppliedMechanics Kyushu University Japan 2007

[25] A C Fairlie-Clarke and T Tveitnes ldquoMomentum and gravityeffects during the constant velocitywater entry ofwedge-shapedsectionsrdquo Ocean Engineering vol 35 no 7 pp 706ndash716 2007

[26] Q Yang andW Qiu ldquoNumerical simulation of water impact for2D and 3D bodiesrdquo Ocean Engineering vol 43 pp 82ndash89 2012

[27] G Wu ldquoNumerical simulation for water entry of a wedge atvarying speed by a high order boundary element methodrdquoJournal of Marine Science and Application vol 11 no 2 pp 143ndash149 2012

[28] M Ahmadzadeh B Saranjam A Hoseini Fard and A RBinesh ldquoNumerical simulation of sphere water entry prob-lem using Eulerian-Lagrangian methodrdquo Applied MathematicalModelling vol 38 no 5-6 pp 1673ndash1684 2014

[29] V-T Nguyen D-T Vu W-G Park and Y-R Jung ldquoNumericalanalysis of water impact forces using a dual-time pseudo-compressibility method and volume-of-fluid interface trackingalgorithmrdquo Computers amp Fluids vol 103 pp 18ndash33 2014

[30] S Kim and N Kim ldquoIntegrated dynamics modeling forsupercavitating vehicle systemsrdquo International Journal of NavalArchitecture and Ocean Engineering vol 7 no 2 pp 346ndash3632015

18 Geofluids

[31] V-T Nguyen D-T Vu W-G Park and C-M Jung ldquoNavier-Stokes solver for water entry bodies with moving Chimera gridmethod in 6DOF motionsrdquo Computers amp Fluids vol 140 pp19ndash38 2016

[32] M Mirzaei M Eghtesad and M Alishahi ldquoPlaning forceidentification in high-speed underwater vehiclesrdquo Journal ofVibration and Control vol 22 no 20 pp 4176ndash4191 2016

[33] G Qiu S Henke and J Grabe ldquoApplication of a CoupledEulerian-Lagrangian approach on geomechanical problemsinvolving large deformationsrdquoComputersampGeosciences vol 38no 1 pp 30ndash39 2011

[34] M R Erfanian M Anbarsooz N Rahimi M Zare andM Moghiman ldquoNumerical and experimental investigationof a three dimensional spherical-nose projectile water entryproblemrdquo Ocean Engineering vol 104 pp 397ndash404 2015

[35] W M Isbell F H Shipman and A H Jones ldquoHugoniotequation of state measurements for eleven material to fiveMegabars Materials and Structural Laboratory ManufacturingDevelopments General Motors Technical centerrdquo Tech RepMichigan 1968

[36] M A Jandron R C Hurd J L Belden A F Bower W Fennelland T T Truscott ldquoModeling of hyperelastic water-skippingspheres using AbaqusExplicitrdquo in Proceedings of the SIMULIACommunity Conference 2014

[37] H Forouzani B Saranjam R Kamali and A Abdollahi-far ldquoElasto-plastic time dependent impact analysis of highspeed projectile on water surfacerdquo Journal of Solid and FluidMechanics vol 3 pp 281ndash298 2016

[38] M R Erfanian andMMoghiman ldquoNumerical and experimen-tal investigation of a projectile water entry problem and studyof velocity effect on time and depth of pinch-offrdquo Journal ofModares Mechanical Engineering vol 15 pp 53ndash60 2015

[39] A Sillem Feasibility study of a tire hydroplaning simulation in afinite element code using a coupled Eulerian-Lagrangian method[MSc thesis] 2008

[40] M Lee R G Longoria and D E Wilson ldquoCavity dynamics inhigh-speed water entryrdquo Physics of Fluids vol 9 no 3 pp 540ndash550 1997

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Submit your manuscripts atwwwhindawicom

Geofluids 9

Table 1 Equation of state parameters for water

Density (120588119908)(kgm3)

Dynamicviscosity (120578)(kgsm2)

Speed of sound(1198880) (msec) Γ0 1199041000 0001 1450 0 0

Water

Aperture release

Z

Diffuser

Lights

Sphere

Camera

Figure 3 Schematic of the experimental apparatus of Aristoff et al[10]

the sphere water entry problem The dimensions for thecomputational domain are considered to be large enoughso that with further increases no considerable changes areobserved in the numerical resultsThe sphere wasmodeled assolid geometry and a rigid body constraint was subsequentlyapplied to it The fluid domain was modeled as a cubewith dimensions of 30 times 50 times 60 cm3 Mesh refinement wasperformed after assigning it to the computational domaintherefore the Eulerianmesh grid size was gradually decreaseduntil no considerable changeswere observed in the numericalresults The Eulerian domain (water) was modeled usingquarter symmetry and contains 746172 elements The spherewas alsomodeled as a Lagrangian solid using 15000 elementsThe Eulerian domain was divided into two parts includingthe upper and lower regions These regions were defined asvoid and stationary water respectively A solid sphere witha diameter of 254 cm and an initial velocity of 217ms inthe normal direction was placed on the free water surfaceThe sphere can move freely in all directions with six degreesof freedom The physical properties of water used in thenumerical simulations are given in Table 1Γ0 and 119904 are material constants in the Gruneisen equationof state and the constant coefficient in shock velocity equa-tion respectively

5 Validation

A schematic of the experimental apparatus of Aristoff et al[10] is presented in Figure 3 A sphere was held by a cameralike aperture at height 119867 above a water tank The tank haddimensions of 30 times 50 times 60 cm3 and was illuminated by a

bank of twenty 32W fluorescent bulbs A diffuser was usedto provide uniform lighting and care was taken to keep thewater surface free of dust Tank dimensions were selectedso that the walls had negligible effects on the results Thesphere was released from rest and fell towards the waterreaching it with an approximate speed119881 asymp radic2119892ℎThe impactsequence was recorded at 2000 frames per second using ahigh-speed camera The resolution was set to 524 times 1280pixels (px) with a field of view of 1128 times 2755 cm2 yieldinga 4646 pxcm magnification The trajectory of the sphereand its impact speed were determined with subpixel accuracythrough a cross correlation andGaussian peak-fittingmethodyielding position estimates accurate to 0025 px (00005 cm)and impact speeds accurate to plusmn4

Figure 4 illustrates a comparison of cavity shapes betweennumerical simulation and the experimental photographstaken by Aristoff et al [10] for a density ratio of 786 and ansimilar period of time The impact velocity of the sphere was217ms and 119905 = 0 refers to when the spheres center passesthrough the free surface This figure shows the cavity shapesfor both experimental and numerical simulation for similartime periods except that cavity shape for the pinch-off timeof the numerical simulation was added to the pictures

An symmetric air cavity formed behind the sphereafter entering the water Air cavity formation consists ofseveral steps including cavity creation cavity expansioncavity contraction behind the sphere and cavity collapseAs the sphere moves deeper through the water it exerts aforce on surrounding fluid in radial directions and transfersits momentum to the fluid However fluid expansion isconfronted with fluid hydrostatic pressure resistance and asa result the radial flow direction is reversed leading to cavitycontraction and collapse Cavity collapse occurs when thecavity wall moves inward until the moment of pinch-off (119905= 708ms) and the cavity is divided into two separated parts[34] While the lower cavity sticks to the sphere and movesalong with it the upper cavity continues its contraction andmoves towards the waterrsquos surface As shown in Figure 4 thenumerical results and experimental observations are in goodagreement

Abaqus numerical simulation methods efficiently modelwater splashes and jet flow Pictures of these phenomenaare depicted in Figure 4 It was concluded from this figurethat pinch-off in the numerical simulation happens with adelay time compared to the experimental method Increasednumerical errors during solution time are one of the causesfor differences between experimental and numerical results

In Figure 5 the numerical results of this study arecompared with the theoretical and experimental data ofAristoff et al as a displacement-time graph of a sphere indifferent water depths [10] As can be seen the numericalresults are in good agreement with those of the analytics andexperiments However the presently computed sphere centerdepth is lower than that found experimentally It seems thatdue to lack of turbulent modeling ability in Abaqus Eulerianformulation and evidently increasing turbulence in flow bytime there is just less difference between experiment andnumerical results in the early time of simulation

10 Geofluids

Experimental results of Aristoff et al (2010)

Pinch-off

Pinch-offNumerical simulation results

t = 59 ＧＭ 129 ＧＭ 199ＧＭ 269ＧＭ 339ＧＭ 409 ＧＭ 479ＧＭ 549ＧＭ 619 ＧＭ 689 ＧＭ 759ＧＭ

t = 59ＧＭ 129 ＧＭ 199ＧＭ 269ＧＭ 339ＧＭ 409 ＧＭ 479ＧＭ 549ＧＭ 619 ＧＭ 689 ＧＭ 708ＧＭ 759ＧＭ

Figure 4 Comparison of numerical results with experimental photographs of Aristoff et al [10] for the sphere water entry problem

002 004 006 0080Time (s)

minus018

minus016

minus014

minus012

minus01

minus008

minus006

minus004

minus002

0

Disp

lace

men

t (m

)

Present studyExperiment (Aristoff et al 2010)Theory (Aristoff et al 2010)

Figure 5 Comparison of displacement-time graph of sphere inwater depth with the theoretical and experimental data of Aristoffet al [10]

6 Materials and Methods

61 Eulerian Model Definition Figure 6 illustrates a sche-matic of the computational domain boundary conditionsand grid distribution for the water entry problem of aspherical pounder The dimensions of the computationaldomain are considered to be large enough so that withfurther increases no considerable changes are observed inthe numerical results The Eulerian domain was dividedinto the upper and lower regions These regions are definedas void and stationary water respectively In water entryproblems for rigid bodies fluid domain dimensions must

Wall

Wall

Wall Water

Spherical pounder

Initial void region

Free surface

Figure 6 The computational domain boundary conditions andgrid distribution for water entry problem of spherical pounder

be at least eight times the dimensions of the rigid bodyHence in this study the fluid domain was modeled as acube with dimensions of 2 times 2 times 1m3 The pounders wereplaced exactly tangent to the water surface and their velocityat the moment of water surface impact was determinedaccording to the free fall equation of bodies The pounderscan move freely in all directions with six degrees of freedom119880119904-119880119901 equation of state was used to define water materialin the Coupled Eulerian-Lagrangian method This equationof state is useful for Navier-Stokes flow simulations whenturbulence flow is negligible In cases where the impact ofa rigid body on a water surface is investigated the majorforce imposed on water body is the pressure force Thereforewith appropriate approximations this equation of state canbe used to investigate the impact of rigid bodies on watersurfaces

62 EulerianMesh Size As shown in Figure 7 with decreasesin the Eulerian domain mesh size to less than 15 cm thedisplacement-time graph of the spherical pounder showsnegligible changes Therefore to improve analysis time the

Geofluids 11

Table 2 Moment of inertia of pounders with different geometric shapes

Geometric shape of the pounder Cube Cylinder Sphere Pyramid ConeMoment of inertia (cm4) 685 5137 411 3082 411

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

01 02 03 040Time (s)

2 ＝Ｇ

175 ＝Ｇ

15 ＝Ｇ

Figure 7 Comparison of the effect of the Eulerian mesh size ondisplacement-time graph of spherical pounder

Eulerian domain was modeled using quarter symmetry witha mesh size of 15 cm and contains 1538943 elements

63 Lagrangian Model Definition Figure 8 illustratespounder dimensions for different geometric shapes Thesepounders are identical in terms of material mass densityYoungrsquos modulus Poissonrsquos ratio volume mass and dropheight from the water surface The only difference is theirgeometric shapeThese parameters were regarded as identicalin all pounders to investigate the effect of pounder shapeon deepwater displacement and velocity air cavity shapepinch-off time and depth and the effect of water drag forceon velocity depreciation when impacting the model bed

Pounder dimensions were selected so that they had avolume of 523 cm3 and equal mass due to their identicalmaterial and mass density Table 2 illustrates the moment ofinertia for pounders with different geometric shapes

Table 3 illustrates thematerial properties of the poundersThese parameters are regarded as identical for all pounderswith the mentioned geometric shapes

64 Lagrangian Mesh Size Due to the negligible effect ofLagrangian mesh size on the results and to improve analysistime the Lagrangian mesh size for all pounders was set to05 cm

Pyramid Cone

Cube Cylinder Sphere

10 ＝Ｇ10 ＝Ｇ10 ＝Ｇ 10 ＝Ｇ

10 ＝Ｇ10 ＝Ｇ10 ＝Ｇ

10 ＝Ｇ5236 ＝Ｇ

20 ＝Ｇ15708 ＝Ｇ

666 ＝Ｇ

Figure 8 The dimensions of pounders with different geometricshapes

Table 3 Material properties of pounders

Materialtype Mass (kg) Density (120588)

(kgm3)

Youngrsquosmodulus (119864)(kgm2)

Poissonrsquosratio (120592)

Steel 411 7850 21 times 1010 03

65 Boundary Condition Wall boundary conditions wereapplied to the sides and bottom of the Eulerian domain tosimulate a water tank Pounder displacement was not con-strained in any direction and the pounders could freely movein all directions The sides and bottom boundaries of thethree-dimensional Eulerian domain were set as nonreflectingboundaries so that the waves released due to pounder impacton water surface did not return to the computational domainafter reaching these boundaries and being absorbed by themThis boundary condition is written as [37]

119889119901 minus 119901119888119889119906 = 0119889119909119889119905 = minus119888 (41)

where 120588 is density 119888 is the speed of sound in the fluid 119901 ispressure 119906 is velocity perpendicular to the wave and 119909 is thedirection perpendicular to the boundary

66 Drop Height and Impact Velocity of Pounders Afterdropping the pounders from a height (ℎ) above the watersurface the velocity of the pounders at themoment of impactwas determined using

1198812 minus 11988120 = 2119892ℎ (42)

where1198810 is the pounder initial velocity before being droppedfrom height ℎ above the waterrsquos surface 119881 is pounder

12 Geofluids

01 02 03 04 050Time (s)

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

CubeCylinderSphere

PyramidCone

Figure 9 Displacement-time graph of pounders with differentgeometric shapes

secondary velocity when impacting the water surface and 119892is the gravitational acceleration of the Earth

From (42) which expresses a time independent equationfor the free fall of objects the initial velocity and the terminalvelocity of the object drop height and the gravitationalacceleration of the Earth are interdependent in the absenceof time In this numerical simulation the pounders weredropped from a height of 1m above thewaterrsquos surfacewith aninitial velocity (1198810) equal to 0ms Therefore the secondaryvelocity of the pounders when impacting the water surfacewas obtained using

119881 = radic2119892ℎ (43)

Instead of simulating a full dropping event from theinitial position the pounders were placed exactly tangent towaterrsquos surface and their secondary velocity at the momentof impact was determined according to (43) A gravitationalacceleration (gravity) of minus98ms2 was applied in the normaldirection of the pounders Therefore a secondary velocityof 443ms in the normal direction of the waterrsquos surfacecorresponds to the speed that would be obtained for thepounders

7 Results and Discussion

71 Effect of the Geometric Shape of the Pounder on Deep-water Displacement Figure 9 illustrates the displacement-time graph of pounders with different geometric shapes Thegraph continues for the cubical pounder from the momentof water surface impact at time zero up to the depth of 1mat 415ms The displacement-time graph of the cylindrical

pounder indicates that after entering the water it reachesthe model bed in a shorter amount of time than the cubicalpounder The graph for this pounder continues from themoment of water surface impact at time zero up to the depthof 1m at 355ms The linear displacement-time graph forthe spherical and pyramidal pounders indicates the fasterdeepwater movement of these pounders compared to thecubical and cylindrical pounders The deepwater movementtime of the spherical and pyramidal pounders after enteringthe water to themodel bed is 285ms and 265ms respectivelyThe linear displacement-time graph of the conical pounderindicates that it has a faster deepwater movement comparedto other pounders as it takes 235ms to reach the model bedafter entering the water

72 Mechanism of Air Cavity Formation and Pinch-Off dueto Deepwater Movement of Pounders with Different Geomet-ric Shapes Figure 10 illustrates the movement trajectory ofpounders with different geometric shapes from the momentof water impact towhen the pounders reached themodel bedAfter entering the water a symmetric air cavity is formedbehind the pounders Air cavity formation involves severalsteps including air cavity creation and expansion behind thepounder air cavity contraction and air cavity collapse [34]As the pounder moves downward in water depth it imposesa force onto surrounding fluid in its radial directions andtransfers its momentum to the fluid This extension is facedwith the hydrostatic pressure resistance of fluidThe directionof the radial flow is reversed and eventually leads to aircavity contraction and collapse In the other words the aircavity contraction is accelerated until pinch-off occurs andthe air cavity is divided into two distinct parts The upperpart of the air cavity continues its contraction and movestowards the waterrsquos surface whereas the lower part of the aircavity clings to the pounder and moves along with it [34]After impacting the water surface the cubical and cylindricalpounders due to their flat contact surfaces lose some of theirvelocityThis velocity depreciation slows down and continuesuntil pinch-offThe spherical pounder with its curved contactsurface has less velocity depreciation compared to the cubicaland cylindrical pounders However some pounder velocityis always depreciated due to impacting the water surfaceThe velocity depreciation process for the spherical pounderduring deepwater movement continues until pinch-off Afterimpacting the waterrsquos surface the pyramidal pounder with itspointy contact surface has a slight velocity depreciation butcontinues to move deeper into the water with approximatelythe same secondary velocity until pinch-off By imposing thedrag force of water onto the pounders surface its velocityis gradually decreased until it reaches the model bed Afterimpacting the water surface the conical pounder with itspointy contact surface continues to move deeper into thewater with an increasing velocity The increasing velocity ofthe conical pounder during deepwater movement continueseven after pinch-off When the pounder moves towards themodel bed the waters drag force is imposed onto poundersurface causing velocity depreciation The pinch-off time forthe pounders with the above-mentioned shapes is between225 and 250ms

Geofluids 13

t = 0 25ＧＭ 50ＧＭ 100ＧＭ 200ＧＭ 225ＧＭ 250ＧＭ 275ＧＭ 300ＧＭ 325ＧＭ 350ＧＭ150ＧＭCu

beCy

linde

rSp

here

Pyra

mid

Con

e

Figure 10 Air cavity formation due to deepwater movement of pounders with different geometric shapes and pinch-off time and depth

73 Effect of the Pounder Geometric Shape onDeepwater Veloc-ity Figure 11 illustrates the velocity-time graph for pounderswith different geometric shapes The cubical and cylindricalpounders lose some of their secondary velocity at the earlymoments of water entry Hence the velocity-time graph ofthese pounders illustrates great velocity depreciation up to15ms The velocity reduction of the cubical and cylindricalpounders gradually continues until pinch-off The velocityof the spherical pounder decreases up to 45ms after waterentry but its velocity depreciation is less than the cubicaland cylindrical pounders This velocity reduction continuesat a slower rate until pinch-off In the initial moments ofwater entry the pyramidal pounder has a slight reduction insecondary velocity up to 55ms Then velocity depreciationcontinues at a faster rate due to the drag force imposed onto

pounders surface and the velocity-time graph of the pyrami-dal pounder shows a gradual velocity reduction until pinch-off After impacting the water surface the conical poundercontinues its deepwater movement with an increasing veloc-ity rate Due to its very low slope the velocity-time graphindicates an increase in velocity for the conical pounder butdue to the waters drag force its velocity begins to decrease at145ms The velocity-time graph indicates a gradual decreasein velocity for conical pounder and velocity depreciationcontinues until pinch-off After pinch-off pounder velocitiesgradually decrease until they reach themodel bed Eventuallywith a velocity of 388ms and 157ms at the moment ofmodel bed impact the conical and cubical pounders have thehighest and lowest velocity respectively Water drag force isimposed in the opposite direction of poundermovement and

14 Geofluids

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Velo

city

(ms

)

01 02 03 040Time (s)

CubeCylinderSphere

PyramidCone

Figure 11 Velocity-time graph of pounders with different geometricshapes

it has the most and least influence on the cubical and conicalpounders respectively

8 Critical Drop Height of the Pounder

In the dynamic compaction of the seabed determining thecritical drop height of the pounder from the freewater surfaceis an effective parameter for the pounder to reach maximumdeepwater velocity After dropping the pounder from height119867 above the free water surface with an initial velocity (1198810)of 0ms its velocity gradually increases until it reaches thesecondary velocity (119881) at the moment of water impact Afterentering the water pounder velocity is decreased due to waterdrag force According to the free fall equation for objectsincreasing the drop height from the waterrsquos surface leads toincreased pounder velocity at the moment of water surfaceimpact but it does not necessarily lead to an increase inpounder velocity during seabed impact because the criticaldrop height of the pounder is a determining factor ofits maximum deepwater velocity According to Figure 12 aconical pounder with a mass of 411 kg was dropped fromdifferent heights (119867) and its velocity when impacting thewaters was measured according to the time independentequation for the free fall of objects Table 4 illustrates thevelocity of the conical pounder with different drop heightswhen impacting the waterrsquos surface

81 Effect of the Drop Height of the Pounder from WaterSurface on Deepwater Displacement Figure 13 illustrates thedisplacement-time graph of the conical pounder with dif-ferent drop heights from the waterrsquos surface As can beseen increasing the drop height of the pounder leads to

Water levelℎ

Figure 12 Schematic of drop height of the conical pounder fromwater surface

Table 4 Velocity of the conical pounder with different drop heightsat the moment of impacting the water surface

Drop height of the pounderfrom water surface (m)

Pounder velocity at themoment of impacting water

surface (ms)025 22105 313075 3831 443125 49515 542175 5852 626225 66425 7

displacement-time graphs that are closer to each other Infact an increase in the drop height of the pounder from thewaterrsquos surface leads to an increase in its secondary velocitywhen impacting the waterrsquos surface Hence the pounderenters the water with a higher velocity and reaches the modelbed in less time

82 Effect of the Drop Height of the Pounder from WaterSurface on Its Deepwater Velocity Increasing the drop heightof the pounder from the waterrsquos surface leads to an increasein its velocity before impacting the water but the poundervelocity gradually decreases after entering the water due tothe its drag force Figure 14 illustrates the velocity-time graphfor the conical pounder with different drop heights from thewaterrsquos surface When the pounder is dropped from a lowheight due to its low velocity it moves deepwater due to itsweight Moreover because of the low velocity of the pounderwhen entering the water pinch-off occurs in less time and atlower depths and its velocity increases after pinch-off until itreaches terminal velocity Increases in pounder drop heightsincrease water entry velocity Hence it will have greatervelocity depreciation in the first moments of water entry

Figure 15 illustrates the terminal velocity-time graph forthe conical pounderwith different drop heightswhen impact-ing the model bed As can be seen increasing the drop height

Geofluids 15

01 02 03 040Time (s)

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

H = 1Ｇ

H = 075Ｇ

H = 05 Ｇ

H = 025 Ｇ

H = 175Ｇ

H = 15 Ｇ

H = 125 Ｇ

H = 25 Ｇ

H = 225 Ｇ

H = 2Ｇ

Figure 13 Displacement-time graph of the conical pounder withdifferent drop heights

of the pounder from the waterrsquos surface leads to an increasein its terminal velocity when impacting the model bedTheseconditions are dominant as long as the pounder reaches acritical drop height from the water surface but increasingthe pounder drop height higher than the critical drop heightleads to decreased terminal velocity when impacting themodel bed In this numerical simulation after dropping thepounder from a height of 2m above the waterrsquos surface itssecondary velocity was 626ms when impacting the watersurface After entering the water the secondary velocity ofthe pounder decreased until it impacts the model bed witha terminal velocity of 454ms Therefore a drop height of2m from above the waterrsquos surface has the highest terminalvelocity when the pounder impacts themodel bed Increasingthe drop height bymore than 2m leads to decreased pounderterminal velocity when impacting the model bed The heightwhich has the greatest effect on pounder terminal velocitywhen impacting the model bed is called the critical dropheight Increasing the drop height of the pounder to greaterthan its critical drop height leads to decreased terminalvelocity when impacting the model bed

Figure 16 illustrates the secondary and terminal velocityof the conical pounder with different drop heights As canbe observed increasing the drop height of the pounder fromwaterrsquos surface leads to increased differences between thesecondary and terminal velocities of the pounder Increasingthe drop height to more than the critical drop height leadsto increased pounder secondary velocity when impactingthe waterrsquos surface and decreased terminal velocity whenimpacting the model bed Hence the difference betweenthe secondary and terminal velocities of the pounder is

01 02 03 040Time (s)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)H = 175Ｇ

H = 125 Ｇ

H = 075Ｇ

H = 025 Ｇ H = 225 Ｇ

H = 1Ｇ

H = 05 Ｇ

H = 25 Ｇ

H = 2Ｇ

H = 15 Ｇ

Figure 14 Velocity-time graph of the conical pounder with differentdrop heights

minus430

minus447 minus454minus415

minus426 minus380

minus367minus355

minus332minus310

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Term

inal

velo

city

(ms

)

01 02 03 040Time (s)

Figure 15 Terminal velocity of conical pounder with different dropheights versus time

not reasonable and a significant amount of the pounderssecondary velocity is depreciated when entering the waterThe great difference between the secondary and terminalvelocities of the pounder reveals the necessity of determiningthe critical drop height of the pounder before dynamic seabedcompaction operations Increasing the pounder drop heightto be greater than the critical drop height not only hasa reverse result on pounder velocity when impacting theseabed but it also requires special equipment such as lengthy

16 Geofluids

Secondary velocityTerminal velocity

025 05 075 1 125 15 175 2 225 250Drop height (m)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)

Figure 16 Secondary and terminal velocities of conical pounderversus drop heights

1 2 3 4 5 6 7 80Impact velocity (ms)

0

005

01

015

02

025

03

Pinc

h-off

tim

e (s)

Figure 17 Pinch-off time versus impact velocity of conical pounder

cranes and huge barges leading to increased implementationcosts

Figure 17 illustrates pinch-off time based on the sec-ondary velocity of the pounder at the moment of watersurface impact As can be seen pinch-off time is a weakfunction of pounder impact velocity and increasing impactvelocity leads to a slight reduction in pinch-off time Thisbehaviour was also observed in the analytical and numericalresults found by Lee et al [40]

Figure 18 illustrates pinch-off depth based on the sec-ondary velocity of pounder at the moment of water surfaceimpact As can be seen the pinch-off depth is a linear

1 2 3 4 5 6 7 80Impact velocity (ms)

minus12

minus1

minus08

minus06

minus04

minus02

0

Pinc

h-off

dep

th (m

)

Figure 18 Pinch-off depth versus impact velocity of conicalpounder

function of pounder impact velocity These results are in linewith the analytical equations of Lee et al [40]

9 Conclusions

In this study the water entry problem of three-dimensionalpounders with different geometric shapes of cube cylindersphere pyramid and cone was numerically simulated bycommercial finite element code Abaqus 614-2 An explicitdynamic analysis method was employed to model the fluid-structure interaction using a Coupled Eulerian-Lagrangian(CEL) method Before applying the model to water entryof the pounders a three-dimensional numerical model wasdeveloped to simulate a transient free-falling wedge impact-ing the free surface of the water in order to calibrate theadopted computationalmodelThe accuracy of the numericalmodel was examined by comparing numerical results withthe experimental data of Aristoff et al The good agreementbetween numerical simulation results with the theoreticaland experimental data of Aristoff et al demonstrates thecapability of the employed numerical methodology to sim-ulate the water entry problem for three-dimensional geome-tries with acceptable accuracy After water entry by a pounderdropped fromaheight of 1m a symmetric air cavity is formedbehind the pounder As the pounder moves downward inwater depth it imposes a force onto surrounding fluid inthe radial directions and transfers its momentum to thefluid This extension is faced with the hydrostatic pressureresistance of the fluid causing the direction of the radialflow to be reversed leading to air cavity contraction andcollapse The results show that the water drag force had thehighest and lowest effect on the cubical and conical poundersrespectively and increasing the drop height to greater thanthe critical drop height leads to increased pounder velocitywhen impacting the waterrsquos surface and decreased velocitywhen impacting the model bed Pinch-off time is a very weak

Geofluids 17

function of pounder velocity when impacting the waterrsquossurface but the pinch-off depth increases linearly along withincreased pounder impact velocities It is recommended thatthe critical drop height for the pounder is determined beforedynamic seabed compaction operations and the adjustmentof crane heights to this height In addition it is also recom-mended that perforated pounders be used in order to reducevelocity depreciation when impacting the waterrsquos surface andto reduce the effect of water drag force

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors warmly appreciate the cooperation and guidanceof Dr B Hamidi and Dr A Mehrabadi in this study

References

[1] B Hamidi J-M Debats H Nikraz and S Varaksin ldquoOffshoreground improvement recordsrdquo Australian Geomechanics Jour-nal vol 48 no 4 pp 111ndash122 2013

[2] SWatanabe ldquoResistance of impact onwater surface part I-cone Institute of Physical andChemical ResearchrdquoTokyo vol 12 pp251ndash267 1930

[3] S Watanabe ldquoResistance of impact on water surface part II-cone (continued)rdquo Institute of Physical and Chemical Researchvol 14 pp 153ndash168 1930

[4] A May and J C Woodhull ldquoDrag coefficients of steel spheresentering water verticallyrdquo Journal of Applied Physics vol 19 no12 pp 1109ndash1121 1948

[5] A May ldquoEffect of surface condition of a sphere on its water-entry cavityrdquo Journal of Applied Physics vol 22 no 10 pp 1219ndash1222 1951

[6] AMay ldquoVertical entry ofmissiles into waterrdquo Journal of AppliedPhysics vol 23 no 12 pp 1362ndash1372 1952

[7] A P New T S Lee and H T Low ldquoImpact loading and waterentrance characteristics of prismatic bodiesrdquo in Proceedings ofthe third international offshore and polar engineering conferencepp 282ndash287 1993

[8] M Anghileri and A Spizzica ldquoExperimental validation of finiteelement models for water impactsrdquo in Proceedings of the secondinternational crash usersrsquo seminar Cranfield Impact Centre LtdEngland UK 1995 second international crash users seminar

[9] A Engle andR Lewis ldquoA comparison of hydrodynamic impactsprediction methods with two dimensional drop test datardquoMarine Structures vol 16 no 2 pp 175ndash182 2003

[10] J M Aristoff T T Truscott A H Techet and J W M BushldquoThe water entry of decelerating spheresrdquo Physics of Fluids vol22 no 3 Article ID 009002PHF pp 1ndash8 2010

[11] T Von Karman The impact of seaplane floats during landingNational Advisory Committee for Aeronautics NACA TN no321 1929

[12] HWagner ldquoPhenomena associatedwith impacts and sliding onliquid surfacesrdquo Journal of Applied Mathematics and Mechanicsvol 12 pp 193ndash215 1932

[13] W S Vorus ldquoA flat cylinder theory for vessel impact and steadyplaning resistancerdquo Journal of Ship Research vol 40 no 2 pp89ndash106 1996

[14] T Miloh ldquoOn the initial-stage slamming of a rigid sphere in avertical water entryrdquo Applied Ocean Research vol 13 no 1 pp13ndash43 1991

[15] J M Aristoff and JW Bush ldquoWater entry of small hydrophobicspheresrdquo Journal of Fluid Mechanics vol 619 pp 45ndash78 2009

[16] P Ghadimi A Dashtimanesh and S R Djeddi ldquoStudy of waterentry of circular cylinder by using analytical and numericalsolutionsrdquo Journal of the Brazilian Society ofMechanical Sciencesamp Engineering vol 34 no 3 pp 225ndash232 2012

[17] A TassinD J PiroAAKorobkinK JMaki andM J CookerldquoTwo-dimensional water entry and exit of a body whose shapevaries in timerdquo Journal of Fluids and Structures vol 40 pp 317ndash336 2013

[18] E Yao H Wang L Pan X Wang and R Woding ldquoVerticalwater-entry of bullet-shaped projectilesrdquo Journal of AppliedMathematics and Physics vol 02 no 06 pp 323ndash334 2014

[19] M-S Park Y-R Jung and W-G Park ldquoNumerical study ofimpact force and ricochet behavior of high speed water-entrybodiesrdquo Computers amp Fluids vol 32 no 7 pp 939ndash951 2003

[20] D Battistin and A Iafrati ldquoHydrodynamic loads during waterenergy of two-dimensional and axisymmetric bodiesrdquo Journalof Fluids and Structures vol 17 no 5 pp 643ndash664 2003

[21] A A Korobkin and M Ohkusu ldquoImpact of two circular platesone of which is floating on a thin layer of liquidrdquo Journal ofEngineering Mathematics vol 50 no 4 pp 343ndash358 2004

[22] K M Kleefsman G Fekken A E Veldman B Iwanowski andB Buchner ldquoA volume-of-fluid based simulation method forwave impact problemsrdquo Journal of Computational Physics vol206 no 1 pp 363ndash393 2005

[23] Y Kim Y Kim Y Liu and D K P Yue ldquoOn the water-entryimpact problem of asymmetric bodiesrdquo in Proceedings of the9th International Conference on Numerical Ship Hydrodynamics(NSH rsquo07) August 2007

[24] Q Yang and W Qiu ldquoNumerical solution of 2D slammingproblem with a CIPmethodrdquo in Proceedings of the InternationalConference on Violent Flows Research Institute for AppliedMechanics Kyushu University Japan 2007

[25] A C Fairlie-Clarke and T Tveitnes ldquoMomentum and gravityeffects during the constant velocitywater entry ofwedge-shapedsectionsrdquo Ocean Engineering vol 35 no 7 pp 706ndash716 2007

[26] Q Yang andW Qiu ldquoNumerical simulation of water impact for2D and 3D bodiesrdquo Ocean Engineering vol 43 pp 82ndash89 2012

[27] G Wu ldquoNumerical simulation for water entry of a wedge atvarying speed by a high order boundary element methodrdquoJournal of Marine Science and Application vol 11 no 2 pp 143ndash149 2012

[28] M Ahmadzadeh B Saranjam A Hoseini Fard and A RBinesh ldquoNumerical simulation of sphere water entry prob-lem using Eulerian-Lagrangian methodrdquo Applied MathematicalModelling vol 38 no 5-6 pp 1673ndash1684 2014

[29] V-T Nguyen D-T Vu W-G Park and Y-R Jung ldquoNumericalanalysis of water impact forces using a dual-time pseudo-compressibility method and volume-of-fluid interface trackingalgorithmrdquo Computers amp Fluids vol 103 pp 18ndash33 2014

[30] S Kim and N Kim ldquoIntegrated dynamics modeling forsupercavitating vehicle systemsrdquo International Journal of NavalArchitecture and Ocean Engineering vol 7 no 2 pp 346ndash3632015

18 Geofluids

[31] V-T Nguyen D-T Vu W-G Park and C-M Jung ldquoNavier-Stokes solver for water entry bodies with moving Chimera gridmethod in 6DOF motionsrdquo Computers amp Fluids vol 140 pp19ndash38 2016

[32] M Mirzaei M Eghtesad and M Alishahi ldquoPlaning forceidentification in high-speed underwater vehiclesrdquo Journal ofVibration and Control vol 22 no 20 pp 4176ndash4191 2016

[33] G Qiu S Henke and J Grabe ldquoApplication of a CoupledEulerian-Lagrangian approach on geomechanical problemsinvolving large deformationsrdquoComputersampGeosciences vol 38no 1 pp 30ndash39 2011

[34] M R Erfanian M Anbarsooz N Rahimi M Zare andM Moghiman ldquoNumerical and experimental investigationof a three dimensional spherical-nose projectile water entryproblemrdquo Ocean Engineering vol 104 pp 397ndash404 2015

[35] W M Isbell F H Shipman and A H Jones ldquoHugoniotequation of state measurements for eleven material to fiveMegabars Materials and Structural Laboratory ManufacturingDevelopments General Motors Technical centerrdquo Tech RepMichigan 1968

[36] M A Jandron R C Hurd J L Belden A F Bower W Fennelland T T Truscott ldquoModeling of hyperelastic water-skippingspheres using AbaqusExplicitrdquo in Proceedings of the SIMULIACommunity Conference 2014

[37] H Forouzani B Saranjam R Kamali and A Abdollahi-far ldquoElasto-plastic time dependent impact analysis of highspeed projectile on water surfacerdquo Journal of Solid and FluidMechanics vol 3 pp 281ndash298 2016

[38] M R Erfanian andMMoghiman ldquoNumerical and experimen-tal investigation of a projectile water entry problem and studyof velocity effect on time and depth of pinch-offrdquo Journal ofModares Mechanical Engineering vol 15 pp 53ndash60 2015

[39] A Sillem Feasibility study of a tire hydroplaning simulation in afinite element code using a coupled Eulerian-Lagrangian method[MSc thesis] 2008

[40] M Lee R G Longoria and D E Wilson ldquoCavity dynamics inhigh-speed water entryrdquo Physics of Fluids vol 9 no 3 pp 540ndash550 1997

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Submit your manuscripts atwwwhindawicom

10 Geofluids

Experimental results of Aristoff et al (2010)

Pinch-off

Pinch-offNumerical simulation results

t = 59 ＧＭ 129 ＧＭ 199ＧＭ 269ＧＭ 339ＧＭ 409 ＧＭ 479ＧＭ 549ＧＭ 619 ＧＭ 689 ＧＭ 759ＧＭ

t = 59ＧＭ 129 ＧＭ 199ＧＭ 269ＧＭ 339ＧＭ 409 ＧＭ 479ＧＭ 549ＧＭ 619 ＧＭ 689 ＧＭ 708ＧＭ 759ＧＭ

Figure 4 Comparison of numerical results with experimental photographs of Aristoff et al [10] for the sphere water entry problem

002 004 006 0080Time (s)

minus018

minus016

minus014

minus012

minus01

minus008

minus006

minus004

minus002

0

Disp

lace

men

t (m

)

Present studyExperiment (Aristoff et al 2010)Theory (Aristoff et al 2010)

Figure 5 Comparison of displacement-time graph of sphere inwater depth with the theoretical and experimental data of Aristoffet al [10]

6 Materials and Methods

61 Eulerian Model Definition Figure 6 illustrates a sche-matic of the computational domain boundary conditionsand grid distribution for the water entry problem of aspherical pounder The dimensions of the computationaldomain are considered to be large enough so that withfurther increases no considerable changes are observed inthe numerical results The Eulerian domain was dividedinto the upper and lower regions These regions are definedas void and stationary water respectively In water entryproblems for rigid bodies fluid domain dimensions must

Wall

Wall

Wall Water

Spherical pounder

Initial void region

Free surface

Figure 6 The computational domain boundary conditions andgrid distribution for water entry problem of spherical pounder

be at least eight times the dimensions of the rigid bodyHence in this study the fluid domain was modeled as acube with dimensions of 2 times 2 times 1m3 The pounders wereplaced exactly tangent to the water surface and their velocityat the moment of water surface impact was determinedaccording to the free fall equation of bodies The pounderscan move freely in all directions with six degrees of freedom119880119904-119880119901 equation of state was used to define water materialin the Coupled Eulerian-Lagrangian method This equationof state is useful for Navier-Stokes flow simulations whenturbulence flow is negligible In cases where the impact ofa rigid body on a water surface is investigated the majorforce imposed on water body is the pressure force Thereforewith appropriate approximations this equation of state canbe used to investigate the impact of rigid bodies on watersurfaces

62 EulerianMesh Size As shown in Figure 7 with decreasesin the Eulerian domain mesh size to less than 15 cm thedisplacement-time graph of the spherical pounder showsnegligible changes Therefore to improve analysis time the

Geofluids 11

Table 2 Moment of inertia of pounders with different geometric shapes

Geometric shape of the pounder Cube Cylinder Sphere Pyramid ConeMoment of inertia (cm4) 685 5137 411 3082 411

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

01 02 03 040Time (s)

2 ＝Ｇ

175 ＝Ｇ

15 ＝Ｇ

Figure 7 Comparison of the effect of the Eulerian mesh size ondisplacement-time graph of spherical pounder

Eulerian domain was modeled using quarter symmetry witha mesh size of 15 cm and contains 1538943 elements

63 Lagrangian Model Definition Figure 8 illustratespounder dimensions for different geometric shapes Thesepounders are identical in terms of material mass densityYoungrsquos modulus Poissonrsquos ratio volume mass and dropheight from the water surface The only difference is theirgeometric shapeThese parameters were regarded as identicalin all pounders to investigate the effect of pounder shapeon deepwater displacement and velocity air cavity shapepinch-off time and depth and the effect of water drag forceon velocity depreciation when impacting the model bed

Pounder dimensions were selected so that they had avolume of 523 cm3 and equal mass due to their identicalmaterial and mass density Table 2 illustrates the moment ofinertia for pounders with different geometric shapes

Table 3 illustrates thematerial properties of the poundersThese parameters are regarded as identical for all pounderswith the mentioned geometric shapes

64 Lagrangian Mesh Size Due to the negligible effect ofLagrangian mesh size on the results and to improve analysistime the Lagrangian mesh size for all pounders was set to05 cm

Pyramid Cone

Cube Cylinder Sphere

10 ＝Ｇ10 ＝Ｇ10 ＝Ｇ 10 ＝Ｇ

10 ＝Ｇ10 ＝Ｇ10 ＝Ｇ

10 ＝Ｇ5236 ＝Ｇ

20 ＝Ｇ15708 ＝Ｇ

666 ＝Ｇ

Figure 8 The dimensions of pounders with different geometricshapes

Table 3 Material properties of pounders

Materialtype Mass (kg) Density (120588)

(kgm3)

Youngrsquosmodulus (119864)(kgm2)

Poissonrsquosratio (120592)

Steel 411 7850 21 times 1010 03

65 Boundary Condition Wall boundary conditions wereapplied to the sides and bottom of the Eulerian domain tosimulate a water tank Pounder displacement was not con-strained in any direction and the pounders could freely movein all directions The sides and bottom boundaries of thethree-dimensional Eulerian domain were set as nonreflectingboundaries so that the waves released due to pounder impacton water surface did not return to the computational domainafter reaching these boundaries and being absorbed by themThis boundary condition is written as [37]

119889119901 minus 119901119888119889119906 = 0119889119909119889119905 = minus119888 (41)

where 120588 is density 119888 is the speed of sound in the fluid 119901 ispressure 119906 is velocity perpendicular to the wave and 119909 is thedirection perpendicular to the boundary

66 Drop Height and Impact Velocity of Pounders Afterdropping the pounders from a height (ℎ) above the watersurface the velocity of the pounders at themoment of impactwas determined using

1198812 minus 11988120 = 2119892ℎ (42)

where1198810 is the pounder initial velocity before being droppedfrom height ℎ above the waterrsquos surface 119881 is pounder

12 Geofluids

01 02 03 04 050Time (s)

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

CubeCylinderSphere

PyramidCone

Figure 9 Displacement-time graph of pounders with differentgeometric shapes

secondary velocity when impacting the water surface and 119892is the gravitational acceleration of the Earth

From (42) which expresses a time independent equationfor the free fall of objects the initial velocity and the terminalvelocity of the object drop height and the gravitationalacceleration of the Earth are interdependent in the absenceof time In this numerical simulation the pounders weredropped from a height of 1m above thewaterrsquos surfacewith aninitial velocity (1198810) equal to 0ms Therefore the secondaryvelocity of the pounders when impacting the water surfacewas obtained using

119881 = radic2119892ℎ (43)

Instead of simulating a full dropping event from theinitial position the pounders were placed exactly tangent towaterrsquos surface and their secondary velocity at the momentof impact was determined according to (43) A gravitationalacceleration (gravity) of minus98ms2 was applied in the normaldirection of the pounders Therefore a secondary velocityof 443ms in the normal direction of the waterrsquos surfacecorresponds to the speed that would be obtained for thepounders

7 Results and Discussion

71 Effect of the Geometric Shape of the Pounder on Deep-water Displacement Figure 9 illustrates the displacement-time graph of pounders with different geometric shapes Thegraph continues for the cubical pounder from the momentof water surface impact at time zero up to the depth of 1mat 415ms The displacement-time graph of the cylindrical

pounder indicates that after entering the water it reachesthe model bed in a shorter amount of time than the cubicalpounder The graph for this pounder continues from themoment of water surface impact at time zero up to the depthof 1m at 355ms The linear displacement-time graph forthe spherical and pyramidal pounders indicates the fasterdeepwater movement of these pounders compared to thecubical and cylindrical pounders The deepwater movementtime of the spherical and pyramidal pounders after enteringthe water to themodel bed is 285ms and 265ms respectivelyThe linear displacement-time graph of the conical pounderindicates that it has a faster deepwater movement comparedto other pounders as it takes 235ms to reach the model bedafter entering the water

72 Mechanism of Air Cavity Formation and Pinch-Off dueto Deepwater Movement of Pounders with Different Geomet-ric Shapes Figure 10 illustrates the movement trajectory ofpounders with different geometric shapes from the momentof water impact towhen the pounders reached themodel bedAfter entering the water a symmetric air cavity is formedbehind the pounders Air cavity formation involves severalsteps including air cavity creation and expansion behind thepounder air cavity contraction and air cavity collapse [34]As the pounder moves downward in water depth it imposesa force onto surrounding fluid in its radial directions andtransfers its momentum to the fluid This extension is facedwith the hydrostatic pressure resistance of fluidThe directionof the radial flow is reversed and eventually leads to aircavity contraction and collapse In the other words the aircavity contraction is accelerated until pinch-off occurs andthe air cavity is divided into two distinct parts The upperpart of the air cavity continues its contraction and movestowards the waterrsquos surface whereas the lower part of the aircavity clings to the pounder and moves along with it [34]After impacting the water surface the cubical and cylindricalpounders due to their flat contact surfaces lose some of theirvelocityThis velocity depreciation slows down and continuesuntil pinch-offThe spherical pounder with its curved contactsurface has less velocity depreciation compared to the cubicaland cylindrical pounders However some pounder velocityis always depreciated due to impacting the water surfaceThe velocity depreciation process for the spherical pounderduring deepwater movement continues until pinch-off Afterimpacting the waterrsquos surface the pyramidal pounder with itspointy contact surface has a slight velocity depreciation butcontinues to move deeper into the water with approximatelythe same secondary velocity until pinch-off By imposing thedrag force of water onto the pounders surface its velocityis gradually decreased until it reaches the model bed Afterimpacting the water surface the conical pounder with itspointy contact surface continues to move deeper into thewater with an increasing velocity The increasing velocity ofthe conical pounder during deepwater movement continueseven after pinch-off When the pounder moves towards themodel bed the waters drag force is imposed onto poundersurface causing velocity depreciation The pinch-off time forthe pounders with the above-mentioned shapes is between225 and 250ms

Geofluids 13

t = 0 25ＧＭ 50ＧＭ 100ＧＭ 200ＧＭ 225ＧＭ 250ＧＭ 275ＧＭ 300ＧＭ 325ＧＭ 350ＧＭ150ＧＭCu

beCy

linde

rSp

here

Pyra

mid

Con

e

Figure 10 Air cavity formation due to deepwater movement of pounders with different geometric shapes and pinch-off time and depth

73 Effect of the Pounder Geometric Shape onDeepwater Veloc-ity Figure 11 illustrates the velocity-time graph for pounderswith different geometric shapes The cubical and cylindricalpounders lose some of their secondary velocity at the earlymoments of water entry Hence the velocity-time graph ofthese pounders illustrates great velocity depreciation up to15ms The velocity reduction of the cubical and cylindricalpounders gradually continues until pinch-off The velocityof the spherical pounder decreases up to 45ms after waterentry but its velocity depreciation is less than the cubicaland cylindrical pounders This velocity reduction continuesat a slower rate until pinch-off In the initial moments ofwater entry the pyramidal pounder has a slight reduction insecondary velocity up to 55ms Then velocity depreciationcontinues at a faster rate due to the drag force imposed onto

pounders surface and the velocity-time graph of the pyrami-dal pounder shows a gradual velocity reduction until pinch-off After impacting the water surface the conical poundercontinues its deepwater movement with an increasing veloc-ity rate Due to its very low slope the velocity-time graphindicates an increase in velocity for the conical pounder butdue to the waters drag force its velocity begins to decrease at145ms The velocity-time graph indicates a gradual decreasein velocity for conical pounder and velocity depreciationcontinues until pinch-off After pinch-off pounder velocitiesgradually decrease until they reach themodel bed Eventuallywith a velocity of 388ms and 157ms at the moment ofmodel bed impact the conical and cubical pounders have thehighest and lowest velocity respectively Water drag force isimposed in the opposite direction of poundermovement and

14 Geofluids

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Velo

city

(ms

)

01 02 03 040Time (s)

CubeCylinderSphere

PyramidCone

Figure 11 Velocity-time graph of pounders with different geometricshapes

it has the most and least influence on the cubical and conicalpounders respectively

8 Critical Drop Height of the Pounder

In the dynamic compaction of the seabed determining thecritical drop height of the pounder from the freewater surfaceis an effective parameter for the pounder to reach maximumdeepwater velocity After dropping the pounder from height119867 above the free water surface with an initial velocity (1198810)of 0ms its velocity gradually increases until it reaches thesecondary velocity (119881) at the moment of water impact Afterentering the water pounder velocity is decreased due to waterdrag force According to the free fall equation for objectsincreasing the drop height from the waterrsquos surface leads toincreased pounder velocity at the moment of water surfaceimpact but it does not necessarily lead to an increase inpounder velocity during seabed impact because the criticaldrop height of the pounder is a determining factor ofits maximum deepwater velocity According to Figure 12 aconical pounder with a mass of 411 kg was dropped fromdifferent heights (119867) and its velocity when impacting thewaters was measured according to the time independentequation for the free fall of objects Table 4 illustrates thevelocity of the conical pounder with different drop heightswhen impacting the waterrsquos surface

81 Effect of the Drop Height of the Pounder from WaterSurface on Deepwater Displacement Figure 13 illustrates thedisplacement-time graph of the conical pounder with dif-ferent drop heights from the waterrsquos surface As can beseen increasing the drop height of the pounder leads to

Water levelℎ

Figure 12 Schematic of drop height of the conical pounder fromwater surface

Table 4 Velocity of the conical pounder with different drop heightsat the moment of impacting the water surface

Drop height of the pounderfrom water surface (m)

Pounder velocity at themoment of impacting water

surface (ms)025 22105 313075 3831 443125 49515 542175 5852 626225 66425 7

displacement-time graphs that are closer to each other Infact an increase in the drop height of the pounder from thewaterrsquos surface leads to an increase in its secondary velocitywhen impacting the waterrsquos surface Hence the pounderenters the water with a higher velocity and reaches the modelbed in less time

82 Effect of the Drop Height of the Pounder from WaterSurface on Its Deepwater Velocity Increasing the drop heightof the pounder from the waterrsquos surface leads to an increasein its velocity before impacting the water but the poundervelocity gradually decreases after entering the water due tothe its drag force Figure 14 illustrates the velocity-time graphfor the conical pounder with different drop heights from thewaterrsquos surface When the pounder is dropped from a lowheight due to its low velocity it moves deepwater due to itsweight Moreover because of the low velocity of the pounderwhen entering the water pinch-off occurs in less time and atlower depths and its velocity increases after pinch-off until itreaches terminal velocity Increases in pounder drop heightsincrease water entry velocity Hence it will have greatervelocity depreciation in the first moments of water entry

Figure 15 illustrates the terminal velocity-time graph forthe conical pounderwith different drop heightswhen impact-ing the model bed As can be seen increasing the drop height

Geofluids 15

01 02 03 040Time (s)

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

H = 1Ｇ

H = 075Ｇ

H = 05 Ｇ

H = 025 Ｇ

H = 175Ｇ

H = 15 Ｇ

H = 125 Ｇ

H = 25 Ｇ

H = 225 Ｇ

H = 2Ｇ

Figure 13 Displacement-time graph of the conical pounder withdifferent drop heights

of the pounder from the waterrsquos surface leads to an increasein its terminal velocity when impacting the model bedTheseconditions are dominant as long as the pounder reaches acritical drop height from the water surface but increasingthe pounder drop height higher than the critical drop heightleads to decreased terminal velocity when impacting themodel bed In this numerical simulation after dropping thepounder from a height of 2m above the waterrsquos surface itssecondary velocity was 626ms when impacting the watersurface After entering the water the secondary velocity ofthe pounder decreased until it impacts the model bed witha terminal velocity of 454ms Therefore a drop height of2m from above the waterrsquos surface has the highest terminalvelocity when the pounder impacts themodel bed Increasingthe drop height bymore than 2m leads to decreased pounderterminal velocity when impacting the model bed The heightwhich has the greatest effect on pounder terminal velocitywhen impacting the model bed is called the critical dropheight Increasing the drop height of the pounder to greaterthan its critical drop height leads to decreased terminalvelocity when impacting the model bed

Figure 16 illustrates the secondary and terminal velocityof the conical pounder with different drop heights As canbe observed increasing the drop height of the pounder fromwaterrsquos surface leads to increased differences between thesecondary and terminal velocities of the pounder Increasingthe drop height to more than the critical drop height leadsto increased pounder secondary velocity when impactingthe waterrsquos surface and decreased terminal velocity whenimpacting the model bed Hence the difference betweenthe secondary and terminal velocities of the pounder is

01 02 03 040Time (s)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)H = 175Ｇ

H = 125 Ｇ

H = 075Ｇ

H = 025 Ｇ H = 225 Ｇ

H = 1Ｇ

H = 05 Ｇ

H = 25 Ｇ

H = 2Ｇ

H = 15 Ｇ

Figure 14 Velocity-time graph of the conical pounder with differentdrop heights

minus430

minus447 minus454minus415

minus426 minus380

minus367minus355

minus332minus310

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Term

inal

velo

city

(ms

)

01 02 03 040Time (s)

Figure 15 Terminal velocity of conical pounder with different dropheights versus time

not reasonable and a significant amount of the pounderssecondary velocity is depreciated when entering the waterThe great difference between the secondary and terminalvelocities of the pounder reveals the necessity of determiningthe critical drop height of the pounder before dynamic seabedcompaction operations Increasing the pounder drop heightto be greater than the critical drop height not only hasa reverse result on pounder velocity when impacting theseabed but it also requires special equipment such as lengthy

16 Geofluids

Secondary velocityTerminal velocity

025 05 075 1 125 15 175 2 225 250Drop height (m)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)

Figure 16 Secondary and terminal velocities of conical pounderversus drop heights

1 2 3 4 5 6 7 80Impact velocity (ms)

0

005

01

015

02

025

03

Pinc

h-off

tim

e (s)

Figure 17 Pinch-off time versus impact velocity of conical pounder

cranes and huge barges leading to increased implementationcosts

Figure 17 illustrates pinch-off time based on the sec-ondary velocity of the pounder at the moment of watersurface impact As can be seen pinch-off time is a weakfunction of pounder impact velocity and increasing impactvelocity leads to a slight reduction in pinch-off time Thisbehaviour was also observed in the analytical and numericalresults found by Lee et al [40]

Figure 18 illustrates pinch-off depth based on the sec-ondary velocity of pounder at the moment of water surfaceimpact As can be seen the pinch-off depth is a linear

1 2 3 4 5 6 7 80Impact velocity (ms)

minus12

minus1

minus08

minus06

minus04

minus02

0

Pinc

h-off

dep

th (m

)

Figure 18 Pinch-off depth versus impact velocity of conicalpounder

function of pounder impact velocity These results are in linewith the analytical equations of Lee et al [40]

9 Conclusions

In this study the water entry problem of three-dimensionalpounders with different geometric shapes of cube cylindersphere pyramid and cone was numerically simulated bycommercial finite element code Abaqus 614-2 An explicitdynamic analysis method was employed to model the fluid-structure interaction using a Coupled Eulerian-Lagrangian(CEL) method Before applying the model to water entryof the pounders a three-dimensional numerical model wasdeveloped to simulate a transient free-falling wedge impact-ing the free surface of the water in order to calibrate theadopted computationalmodelThe accuracy of the numericalmodel was examined by comparing numerical results withthe experimental data of Aristoff et al The good agreementbetween numerical simulation results with the theoreticaland experimental data of Aristoff et al demonstrates thecapability of the employed numerical methodology to sim-ulate the water entry problem for three-dimensional geome-tries with acceptable accuracy After water entry by a pounderdropped fromaheight of 1m a symmetric air cavity is formedbehind the pounder As the pounder moves downward inwater depth it imposes a force onto surrounding fluid inthe radial directions and transfers its momentum to thefluid This extension is faced with the hydrostatic pressureresistance of the fluid causing the direction of the radialflow to be reversed leading to air cavity contraction andcollapse The results show that the water drag force had thehighest and lowest effect on the cubical and conical poundersrespectively and increasing the drop height to greater thanthe critical drop height leads to increased pounder velocitywhen impacting the waterrsquos surface and decreased velocitywhen impacting the model bed Pinch-off time is a very weak

Geofluids 17

function of pounder velocity when impacting the waterrsquossurface but the pinch-off depth increases linearly along withincreased pounder impact velocities It is recommended thatthe critical drop height for the pounder is determined beforedynamic seabed compaction operations and the adjustmentof crane heights to this height In addition it is also recom-mended that perforated pounders be used in order to reducevelocity depreciation when impacting the waterrsquos surface andto reduce the effect of water drag force

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors warmly appreciate the cooperation and guidanceof Dr B Hamidi and Dr A Mehrabadi in this study

References

[1] B Hamidi J-M Debats H Nikraz and S Varaksin ldquoOffshoreground improvement recordsrdquo Australian Geomechanics Jour-nal vol 48 no 4 pp 111ndash122 2013

[2] SWatanabe ldquoResistance of impact onwater surface part I-cone Institute of Physical andChemical ResearchrdquoTokyo vol 12 pp251ndash267 1930

[3] S Watanabe ldquoResistance of impact on water surface part II-cone (continued)rdquo Institute of Physical and Chemical Researchvol 14 pp 153ndash168 1930

[4] A May and J C Woodhull ldquoDrag coefficients of steel spheresentering water verticallyrdquo Journal of Applied Physics vol 19 no12 pp 1109ndash1121 1948

[5] A May ldquoEffect of surface condition of a sphere on its water-entry cavityrdquo Journal of Applied Physics vol 22 no 10 pp 1219ndash1222 1951

[6] AMay ldquoVertical entry ofmissiles into waterrdquo Journal of AppliedPhysics vol 23 no 12 pp 1362ndash1372 1952

[7] A P New T S Lee and H T Low ldquoImpact loading and waterentrance characteristics of prismatic bodiesrdquo in Proceedings ofthe third international offshore and polar engineering conferencepp 282ndash287 1993

[8] M Anghileri and A Spizzica ldquoExperimental validation of finiteelement models for water impactsrdquo in Proceedings of the secondinternational crash usersrsquo seminar Cranfield Impact Centre LtdEngland UK 1995 second international crash users seminar

[9] A Engle andR Lewis ldquoA comparison of hydrodynamic impactsprediction methods with two dimensional drop test datardquoMarine Structures vol 16 no 2 pp 175ndash182 2003

[10] J M Aristoff T T Truscott A H Techet and J W M BushldquoThe water entry of decelerating spheresrdquo Physics of Fluids vol22 no 3 Article ID 009002PHF pp 1ndash8 2010

[11] T Von Karman The impact of seaplane floats during landingNational Advisory Committee for Aeronautics NACA TN no321 1929

[12] HWagner ldquoPhenomena associatedwith impacts and sliding onliquid surfacesrdquo Journal of Applied Mathematics and Mechanicsvol 12 pp 193ndash215 1932

[13] W S Vorus ldquoA flat cylinder theory for vessel impact and steadyplaning resistancerdquo Journal of Ship Research vol 40 no 2 pp89ndash106 1996

[14] T Miloh ldquoOn the initial-stage slamming of a rigid sphere in avertical water entryrdquo Applied Ocean Research vol 13 no 1 pp13ndash43 1991

[15] J M Aristoff and JW Bush ldquoWater entry of small hydrophobicspheresrdquo Journal of Fluid Mechanics vol 619 pp 45ndash78 2009

[16] P Ghadimi A Dashtimanesh and S R Djeddi ldquoStudy of waterentry of circular cylinder by using analytical and numericalsolutionsrdquo Journal of the Brazilian Society ofMechanical Sciencesamp Engineering vol 34 no 3 pp 225ndash232 2012

[17] A TassinD J PiroAAKorobkinK JMaki andM J CookerldquoTwo-dimensional water entry and exit of a body whose shapevaries in timerdquo Journal of Fluids and Structures vol 40 pp 317ndash336 2013

[18] E Yao H Wang L Pan X Wang and R Woding ldquoVerticalwater-entry of bullet-shaped projectilesrdquo Journal of AppliedMathematics and Physics vol 02 no 06 pp 323ndash334 2014

[19] M-S Park Y-R Jung and W-G Park ldquoNumerical study ofimpact force and ricochet behavior of high speed water-entrybodiesrdquo Computers amp Fluids vol 32 no 7 pp 939ndash951 2003

[20] D Battistin and A Iafrati ldquoHydrodynamic loads during waterenergy of two-dimensional and axisymmetric bodiesrdquo Journalof Fluids and Structures vol 17 no 5 pp 643ndash664 2003

[21] A A Korobkin and M Ohkusu ldquoImpact of two circular platesone of which is floating on a thin layer of liquidrdquo Journal ofEngineering Mathematics vol 50 no 4 pp 343ndash358 2004

[22] K M Kleefsman G Fekken A E Veldman B Iwanowski andB Buchner ldquoA volume-of-fluid based simulation method forwave impact problemsrdquo Journal of Computational Physics vol206 no 1 pp 363ndash393 2005

[23] Y Kim Y Kim Y Liu and D K P Yue ldquoOn the water-entryimpact problem of asymmetric bodiesrdquo in Proceedings of the9th International Conference on Numerical Ship Hydrodynamics(NSH rsquo07) August 2007

[24] Q Yang and W Qiu ldquoNumerical solution of 2D slammingproblem with a CIPmethodrdquo in Proceedings of the InternationalConference on Violent Flows Research Institute for AppliedMechanics Kyushu University Japan 2007

[25] A C Fairlie-Clarke and T Tveitnes ldquoMomentum and gravityeffects during the constant velocitywater entry ofwedge-shapedsectionsrdquo Ocean Engineering vol 35 no 7 pp 706ndash716 2007

[26] Q Yang andW Qiu ldquoNumerical simulation of water impact for2D and 3D bodiesrdquo Ocean Engineering vol 43 pp 82ndash89 2012

[27] G Wu ldquoNumerical simulation for water entry of a wedge atvarying speed by a high order boundary element methodrdquoJournal of Marine Science and Application vol 11 no 2 pp 143ndash149 2012

[28] M Ahmadzadeh B Saranjam A Hoseini Fard and A RBinesh ldquoNumerical simulation of sphere water entry prob-lem using Eulerian-Lagrangian methodrdquo Applied MathematicalModelling vol 38 no 5-6 pp 1673ndash1684 2014

[29] V-T Nguyen D-T Vu W-G Park and Y-R Jung ldquoNumericalanalysis of water impact forces using a dual-time pseudo-compressibility method and volume-of-fluid interface trackingalgorithmrdquo Computers amp Fluids vol 103 pp 18ndash33 2014

[30] S Kim and N Kim ldquoIntegrated dynamics modeling forsupercavitating vehicle systemsrdquo International Journal of NavalArchitecture and Ocean Engineering vol 7 no 2 pp 346ndash3632015

18 Geofluids

[31] V-T Nguyen D-T Vu W-G Park and C-M Jung ldquoNavier-Stokes solver for water entry bodies with moving Chimera gridmethod in 6DOF motionsrdquo Computers amp Fluids vol 140 pp19ndash38 2016

[32] M Mirzaei M Eghtesad and M Alishahi ldquoPlaning forceidentification in high-speed underwater vehiclesrdquo Journal ofVibration and Control vol 22 no 20 pp 4176ndash4191 2016

[33] G Qiu S Henke and J Grabe ldquoApplication of a CoupledEulerian-Lagrangian approach on geomechanical problemsinvolving large deformationsrdquoComputersampGeosciences vol 38no 1 pp 30ndash39 2011

[34] M R Erfanian M Anbarsooz N Rahimi M Zare andM Moghiman ldquoNumerical and experimental investigationof a three dimensional spherical-nose projectile water entryproblemrdquo Ocean Engineering vol 104 pp 397ndash404 2015

[35] W M Isbell F H Shipman and A H Jones ldquoHugoniotequation of state measurements for eleven material to fiveMegabars Materials and Structural Laboratory ManufacturingDevelopments General Motors Technical centerrdquo Tech RepMichigan 1968

[36] M A Jandron R C Hurd J L Belden A F Bower W Fennelland T T Truscott ldquoModeling of hyperelastic water-skippingspheres using AbaqusExplicitrdquo in Proceedings of the SIMULIACommunity Conference 2014

[37] H Forouzani B Saranjam R Kamali and A Abdollahi-far ldquoElasto-plastic time dependent impact analysis of highspeed projectile on water surfacerdquo Journal of Solid and FluidMechanics vol 3 pp 281ndash298 2016

[38] M R Erfanian andMMoghiman ldquoNumerical and experimen-tal investigation of a projectile water entry problem and studyof velocity effect on time and depth of pinch-offrdquo Journal ofModares Mechanical Engineering vol 15 pp 53ndash60 2015

[39] A Sillem Feasibility study of a tire hydroplaning simulation in afinite element code using a coupled Eulerian-Lagrangian method[MSc thesis] 2008

[40] M Lee R G Longoria and D E Wilson ldquoCavity dynamics inhigh-speed water entryrdquo Physics of Fluids vol 9 no 3 pp 540ndash550 1997

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Submit your manuscripts atwwwhindawicom

Geofluids 11

Table 2 Moment of inertia of pounders with different geometric shapes

Geometric shape of the pounder Cube Cylinder Sphere Pyramid ConeMoment of inertia (cm4) 685 5137 411 3082 411

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

01 02 03 040Time (s)

2 ＝Ｇ

175 ＝Ｇ

15 ＝Ｇ

Figure 7 Comparison of the effect of the Eulerian mesh size ondisplacement-time graph of spherical pounder

Eulerian domain was modeled using quarter symmetry witha mesh size of 15 cm and contains 1538943 elements

63 Lagrangian Model Definition Figure 8 illustratespounder dimensions for different geometric shapes Thesepounders are identical in terms of material mass densityYoungrsquos modulus Poissonrsquos ratio volume mass and dropheight from the water surface The only difference is theirgeometric shapeThese parameters were regarded as identicalin all pounders to investigate the effect of pounder shapeon deepwater displacement and velocity air cavity shapepinch-off time and depth and the effect of water drag forceon velocity depreciation when impacting the model bed

Pounder dimensions were selected so that they had avolume of 523 cm3 and equal mass due to their identicalmaterial and mass density Table 2 illustrates the moment ofinertia for pounders with different geometric shapes

Table 3 illustrates thematerial properties of the poundersThese parameters are regarded as identical for all pounderswith the mentioned geometric shapes

64 Lagrangian Mesh Size Due to the negligible effect ofLagrangian mesh size on the results and to improve analysistime the Lagrangian mesh size for all pounders was set to05 cm

Pyramid Cone

Cube Cylinder Sphere

10 ＝Ｇ10 ＝Ｇ10 ＝Ｇ 10 ＝Ｇ

10 ＝Ｇ10 ＝Ｇ10 ＝Ｇ

10 ＝Ｇ5236 ＝Ｇ

20 ＝Ｇ15708 ＝Ｇ

666 ＝Ｇ

Figure 8 The dimensions of pounders with different geometricshapes

Table 3 Material properties of pounders

Materialtype Mass (kg) Density (120588)

(kgm3)

Youngrsquosmodulus (119864)(kgm2)

Poissonrsquosratio (120592)

Steel 411 7850 21 times 1010 03

65 Boundary Condition Wall boundary conditions wereapplied to the sides and bottom of the Eulerian domain tosimulate a water tank Pounder displacement was not con-strained in any direction and the pounders could freely movein all directions The sides and bottom boundaries of thethree-dimensional Eulerian domain were set as nonreflectingboundaries so that the waves released due to pounder impacton water surface did not return to the computational domainafter reaching these boundaries and being absorbed by themThis boundary condition is written as [37]

119889119901 minus 119901119888119889119906 = 0119889119909119889119905 = minus119888 (41)

where 120588 is density 119888 is the speed of sound in the fluid 119901 ispressure 119906 is velocity perpendicular to the wave and 119909 is thedirection perpendicular to the boundary

66 Drop Height and Impact Velocity of Pounders Afterdropping the pounders from a height (ℎ) above the watersurface the velocity of the pounders at themoment of impactwas determined using

1198812 minus 11988120 = 2119892ℎ (42)

where1198810 is the pounder initial velocity before being droppedfrom height ℎ above the waterrsquos surface 119881 is pounder

12 Geofluids

01 02 03 04 050Time (s)

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

CubeCylinderSphere

PyramidCone

Figure 9 Displacement-time graph of pounders with differentgeometric shapes

secondary velocity when impacting the water surface and 119892is the gravitational acceleration of the Earth

From (42) which expresses a time independent equationfor the free fall of objects the initial velocity and the terminalvelocity of the object drop height and the gravitationalacceleration of the Earth are interdependent in the absenceof time In this numerical simulation the pounders weredropped from a height of 1m above thewaterrsquos surfacewith aninitial velocity (1198810) equal to 0ms Therefore the secondaryvelocity of the pounders when impacting the water surfacewas obtained using

119881 = radic2119892ℎ (43)

Instead of simulating a full dropping event from theinitial position the pounders were placed exactly tangent towaterrsquos surface and their secondary velocity at the momentof impact was determined according to (43) A gravitationalacceleration (gravity) of minus98ms2 was applied in the normaldirection of the pounders Therefore a secondary velocityof 443ms in the normal direction of the waterrsquos surfacecorresponds to the speed that would be obtained for thepounders

7 Results and Discussion

71 Effect of the Geometric Shape of the Pounder on Deep-water Displacement Figure 9 illustrates the displacement-time graph of pounders with different geometric shapes Thegraph continues for the cubical pounder from the momentof water surface impact at time zero up to the depth of 1mat 415ms The displacement-time graph of the cylindrical

pounder indicates that after entering the water it reachesthe model bed in a shorter amount of time than the cubicalpounder The graph for this pounder continues from themoment of water surface impact at time zero up to the depthof 1m at 355ms The linear displacement-time graph forthe spherical and pyramidal pounders indicates the fasterdeepwater movement of these pounders compared to thecubical and cylindrical pounders The deepwater movementtime of the spherical and pyramidal pounders after enteringthe water to themodel bed is 285ms and 265ms respectivelyThe linear displacement-time graph of the conical pounderindicates that it has a faster deepwater movement comparedto other pounders as it takes 235ms to reach the model bedafter entering the water

72 Mechanism of Air Cavity Formation and Pinch-Off dueto Deepwater Movement of Pounders with Different Geomet-ric Shapes Figure 10 illustrates the movement trajectory ofpounders with different geometric shapes from the momentof water impact towhen the pounders reached themodel bedAfter entering the water a symmetric air cavity is formedbehind the pounders Air cavity formation involves severalsteps including air cavity creation and expansion behind thepounder air cavity contraction and air cavity collapse [34]As the pounder moves downward in water depth it imposesa force onto surrounding fluid in its radial directions andtransfers its momentum to the fluid This extension is facedwith the hydrostatic pressure resistance of fluidThe directionof the radial flow is reversed and eventually leads to aircavity contraction and collapse In the other words the aircavity contraction is accelerated until pinch-off occurs andthe air cavity is divided into two distinct parts The upperpart of the air cavity continues its contraction and movestowards the waterrsquos surface whereas the lower part of the aircavity clings to the pounder and moves along with it [34]After impacting the water surface the cubical and cylindricalpounders due to their flat contact surfaces lose some of theirvelocityThis velocity depreciation slows down and continuesuntil pinch-offThe spherical pounder with its curved contactsurface has less velocity depreciation compared to the cubicaland cylindrical pounders However some pounder velocityis always depreciated due to impacting the water surfaceThe velocity depreciation process for the spherical pounderduring deepwater movement continues until pinch-off Afterimpacting the waterrsquos surface the pyramidal pounder with itspointy contact surface has a slight velocity depreciation butcontinues to move deeper into the water with approximatelythe same secondary velocity until pinch-off By imposing thedrag force of water onto the pounders surface its velocityis gradually decreased until it reaches the model bed Afterimpacting the water surface the conical pounder with itspointy contact surface continues to move deeper into thewater with an increasing velocity The increasing velocity ofthe conical pounder during deepwater movement continueseven after pinch-off When the pounder moves towards themodel bed the waters drag force is imposed onto poundersurface causing velocity depreciation The pinch-off time forthe pounders with the above-mentioned shapes is between225 and 250ms

Geofluids 13

t = 0 25ＧＭ 50ＧＭ 100ＧＭ 200ＧＭ 225ＧＭ 250ＧＭ 275ＧＭ 300ＧＭ 325ＧＭ 350ＧＭ150ＧＭCu

beCy

linde

rSp

here

Pyra

mid

Con

e

Figure 10 Air cavity formation due to deepwater movement of pounders with different geometric shapes and pinch-off time and depth

73 Effect of the Pounder Geometric Shape onDeepwater Veloc-ity Figure 11 illustrates the velocity-time graph for pounderswith different geometric shapes The cubical and cylindricalpounders lose some of their secondary velocity at the earlymoments of water entry Hence the velocity-time graph ofthese pounders illustrates great velocity depreciation up to15ms The velocity reduction of the cubical and cylindricalpounders gradually continues until pinch-off The velocityof the spherical pounder decreases up to 45ms after waterentry but its velocity depreciation is less than the cubicaland cylindrical pounders This velocity reduction continuesat a slower rate until pinch-off In the initial moments ofwater entry the pyramidal pounder has a slight reduction insecondary velocity up to 55ms Then velocity depreciationcontinues at a faster rate due to the drag force imposed onto

pounders surface and the velocity-time graph of the pyrami-dal pounder shows a gradual velocity reduction until pinch-off After impacting the water surface the conical poundercontinues its deepwater movement with an increasing veloc-ity rate Due to its very low slope the velocity-time graphindicates an increase in velocity for the conical pounder butdue to the waters drag force its velocity begins to decrease at145ms The velocity-time graph indicates a gradual decreasein velocity for conical pounder and velocity depreciationcontinues until pinch-off After pinch-off pounder velocitiesgradually decrease until they reach themodel bed Eventuallywith a velocity of 388ms and 157ms at the moment ofmodel bed impact the conical and cubical pounders have thehighest and lowest velocity respectively Water drag force isimposed in the opposite direction of poundermovement and

14 Geofluids

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Velo

city

(ms

)

01 02 03 040Time (s)

CubeCylinderSphere

PyramidCone

Figure 11 Velocity-time graph of pounders with different geometricshapes

it has the most and least influence on the cubical and conicalpounders respectively

8 Critical Drop Height of the Pounder

In the dynamic compaction of the seabed determining thecritical drop height of the pounder from the freewater surfaceis an effective parameter for the pounder to reach maximumdeepwater velocity After dropping the pounder from height119867 above the free water surface with an initial velocity (1198810)of 0ms its velocity gradually increases until it reaches thesecondary velocity (119881) at the moment of water impact Afterentering the water pounder velocity is decreased due to waterdrag force According to the free fall equation for objectsincreasing the drop height from the waterrsquos surface leads toincreased pounder velocity at the moment of water surfaceimpact but it does not necessarily lead to an increase inpounder velocity during seabed impact because the criticaldrop height of the pounder is a determining factor ofits maximum deepwater velocity According to Figure 12 aconical pounder with a mass of 411 kg was dropped fromdifferent heights (119867) and its velocity when impacting thewaters was measured according to the time independentequation for the free fall of objects Table 4 illustrates thevelocity of the conical pounder with different drop heightswhen impacting the waterrsquos surface

81 Effect of the Drop Height of the Pounder from WaterSurface on Deepwater Displacement Figure 13 illustrates thedisplacement-time graph of the conical pounder with dif-ferent drop heights from the waterrsquos surface As can beseen increasing the drop height of the pounder leads to

Water levelℎ

Figure 12 Schematic of drop height of the conical pounder fromwater surface

Table 4 Velocity of the conical pounder with different drop heightsat the moment of impacting the water surface

Drop height of the pounderfrom water surface (m)

Pounder velocity at themoment of impacting water

surface (ms)025 22105 313075 3831 443125 49515 542175 5852 626225 66425 7

displacement-time graphs that are closer to each other Infact an increase in the drop height of the pounder from thewaterrsquos surface leads to an increase in its secondary velocitywhen impacting the waterrsquos surface Hence the pounderenters the water with a higher velocity and reaches the modelbed in less time

82 Effect of the Drop Height of the Pounder from WaterSurface on Its Deepwater Velocity Increasing the drop heightof the pounder from the waterrsquos surface leads to an increasein its velocity before impacting the water but the poundervelocity gradually decreases after entering the water due tothe its drag force Figure 14 illustrates the velocity-time graphfor the conical pounder with different drop heights from thewaterrsquos surface When the pounder is dropped from a lowheight due to its low velocity it moves deepwater due to itsweight Moreover because of the low velocity of the pounderwhen entering the water pinch-off occurs in less time and atlower depths and its velocity increases after pinch-off until itreaches terminal velocity Increases in pounder drop heightsincrease water entry velocity Hence it will have greatervelocity depreciation in the first moments of water entry

Figure 15 illustrates the terminal velocity-time graph forthe conical pounderwith different drop heightswhen impact-ing the model bed As can be seen increasing the drop height

Geofluids 15

01 02 03 040Time (s)

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

H = 1Ｇ

H = 075Ｇ

H = 05 Ｇ

H = 025 Ｇ

H = 175Ｇ

H = 15 Ｇ

H = 125 Ｇ

H = 25 Ｇ

H = 225 Ｇ

H = 2Ｇ

Figure 13 Displacement-time graph of the conical pounder withdifferent drop heights

of the pounder from the waterrsquos surface leads to an increasein its terminal velocity when impacting the model bedTheseconditions are dominant as long as the pounder reaches acritical drop height from the water surface but increasingthe pounder drop height higher than the critical drop heightleads to decreased terminal velocity when impacting themodel bed In this numerical simulation after dropping thepounder from a height of 2m above the waterrsquos surface itssecondary velocity was 626ms when impacting the watersurface After entering the water the secondary velocity ofthe pounder decreased until it impacts the model bed witha terminal velocity of 454ms Therefore a drop height of2m from above the waterrsquos surface has the highest terminalvelocity when the pounder impacts themodel bed Increasingthe drop height bymore than 2m leads to decreased pounderterminal velocity when impacting the model bed The heightwhich has the greatest effect on pounder terminal velocitywhen impacting the model bed is called the critical dropheight Increasing the drop height of the pounder to greaterthan its critical drop height leads to decreased terminalvelocity when impacting the model bed

Figure 16 illustrates the secondary and terminal velocityof the conical pounder with different drop heights As canbe observed increasing the drop height of the pounder fromwaterrsquos surface leads to increased differences between thesecondary and terminal velocities of the pounder Increasingthe drop height to more than the critical drop height leadsto increased pounder secondary velocity when impactingthe waterrsquos surface and decreased terminal velocity whenimpacting the model bed Hence the difference betweenthe secondary and terminal velocities of the pounder is

01 02 03 040Time (s)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)H = 175Ｇ

H = 125 Ｇ

H = 075Ｇ

H = 025 Ｇ H = 225 Ｇ

H = 1Ｇ

H = 05 Ｇ

H = 25 Ｇ

H = 2Ｇ

H = 15 Ｇ

Figure 14 Velocity-time graph of the conical pounder with differentdrop heights

minus430

minus447 minus454minus415

minus426 minus380

minus367minus355

minus332minus310

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Term

inal

velo

city

(ms

)

01 02 03 040Time (s)

Figure 15 Terminal velocity of conical pounder with different dropheights versus time

not reasonable and a significant amount of the pounderssecondary velocity is depreciated when entering the waterThe great difference between the secondary and terminalvelocities of the pounder reveals the necessity of determiningthe critical drop height of the pounder before dynamic seabedcompaction operations Increasing the pounder drop heightto be greater than the critical drop height not only hasa reverse result on pounder velocity when impacting theseabed but it also requires special equipment such as lengthy

16 Geofluids

Secondary velocityTerminal velocity

025 05 075 1 125 15 175 2 225 250Drop height (m)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)

Figure 16 Secondary and terminal velocities of conical pounderversus drop heights

1 2 3 4 5 6 7 80Impact velocity (ms)

0

005

01

015

02

025

03

Pinc

h-off

tim

e (s)

Figure 17 Pinch-off time versus impact velocity of conical pounder

cranes and huge barges leading to increased implementationcosts

Figure 17 illustrates pinch-off time based on the sec-ondary velocity of the pounder at the moment of watersurface impact As can be seen pinch-off time is a weakfunction of pounder impact velocity and increasing impactvelocity leads to a slight reduction in pinch-off time Thisbehaviour was also observed in the analytical and numericalresults found by Lee et al [40]

Figure 18 illustrates pinch-off depth based on the sec-ondary velocity of pounder at the moment of water surfaceimpact As can be seen the pinch-off depth is a linear

1 2 3 4 5 6 7 80Impact velocity (ms)

minus12

minus1

minus08

minus06

minus04

minus02

0

Pinc

h-off

dep

th (m

)

Figure 18 Pinch-off depth versus impact velocity of conicalpounder

function of pounder impact velocity These results are in linewith the analytical equations of Lee et al [40]

9 Conclusions

In this study the water entry problem of three-dimensionalpounders with different geometric shapes of cube cylindersphere pyramid and cone was numerically simulated bycommercial finite element code Abaqus 614-2 An explicitdynamic analysis method was employed to model the fluid-structure interaction using a Coupled Eulerian-Lagrangian(CEL) method Before applying the model to water entryof the pounders a three-dimensional numerical model wasdeveloped to simulate a transient free-falling wedge impact-ing the free surface of the water in order to calibrate theadopted computationalmodelThe accuracy of the numericalmodel was examined by comparing numerical results withthe experimental data of Aristoff et al The good agreementbetween numerical simulation results with the theoreticaland experimental data of Aristoff et al demonstrates thecapability of the employed numerical methodology to sim-ulate the water entry problem for three-dimensional geome-tries with acceptable accuracy After water entry by a pounderdropped fromaheight of 1m a symmetric air cavity is formedbehind the pounder As the pounder moves downward inwater depth it imposes a force onto surrounding fluid inthe radial directions and transfers its momentum to thefluid This extension is faced with the hydrostatic pressureresistance of the fluid causing the direction of the radialflow to be reversed leading to air cavity contraction andcollapse The results show that the water drag force had thehighest and lowest effect on the cubical and conical poundersrespectively and increasing the drop height to greater thanthe critical drop height leads to increased pounder velocitywhen impacting the waterrsquos surface and decreased velocitywhen impacting the model bed Pinch-off time is a very weak

Geofluids 17

function of pounder velocity when impacting the waterrsquossurface but the pinch-off depth increases linearly along withincreased pounder impact velocities It is recommended thatthe critical drop height for the pounder is determined beforedynamic seabed compaction operations and the adjustmentof crane heights to this height In addition it is also recom-mended that perforated pounders be used in order to reducevelocity depreciation when impacting the waterrsquos surface andto reduce the effect of water drag force

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors warmly appreciate the cooperation and guidanceof Dr B Hamidi and Dr A Mehrabadi in this study

References

[1] B Hamidi J-M Debats H Nikraz and S Varaksin ldquoOffshoreground improvement recordsrdquo Australian Geomechanics Jour-nal vol 48 no 4 pp 111ndash122 2013

[2] SWatanabe ldquoResistance of impact onwater surface part I-cone Institute of Physical andChemical ResearchrdquoTokyo vol 12 pp251ndash267 1930

[3] S Watanabe ldquoResistance of impact on water surface part II-cone (continued)rdquo Institute of Physical and Chemical Researchvol 14 pp 153ndash168 1930

[4] A May and J C Woodhull ldquoDrag coefficients of steel spheresentering water verticallyrdquo Journal of Applied Physics vol 19 no12 pp 1109ndash1121 1948

[5] A May ldquoEffect of surface condition of a sphere on its water-entry cavityrdquo Journal of Applied Physics vol 22 no 10 pp 1219ndash1222 1951

[6] AMay ldquoVertical entry ofmissiles into waterrdquo Journal of AppliedPhysics vol 23 no 12 pp 1362ndash1372 1952

[7] A P New T S Lee and H T Low ldquoImpact loading and waterentrance characteristics of prismatic bodiesrdquo in Proceedings ofthe third international offshore and polar engineering conferencepp 282ndash287 1993

[8] M Anghileri and A Spizzica ldquoExperimental validation of finiteelement models for water impactsrdquo in Proceedings of the secondinternational crash usersrsquo seminar Cranfield Impact Centre LtdEngland UK 1995 second international crash users seminar

[9] A Engle andR Lewis ldquoA comparison of hydrodynamic impactsprediction methods with two dimensional drop test datardquoMarine Structures vol 16 no 2 pp 175ndash182 2003

[10] J M Aristoff T T Truscott A H Techet and J W M BushldquoThe water entry of decelerating spheresrdquo Physics of Fluids vol22 no 3 Article ID 009002PHF pp 1ndash8 2010

[11] T Von Karman The impact of seaplane floats during landingNational Advisory Committee for Aeronautics NACA TN no321 1929

[12] HWagner ldquoPhenomena associatedwith impacts and sliding onliquid surfacesrdquo Journal of Applied Mathematics and Mechanicsvol 12 pp 193ndash215 1932

[13] W S Vorus ldquoA flat cylinder theory for vessel impact and steadyplaning resistancerdquo Journal of Ship Research vol 40 no 2 pp89ndash106 1996

[14] T Miloh ldquoOn the initial-stage slamming of a rigid sphere in avertical water entryrdquo Applied Ocean Research vol 13 no 1 pp13ndash43 1991

[15] J M Aristoff and JW Bush ldquoWater entry of small hydrophobicspheresrdquo Journal of Fluid Mechanics vol 619 pp 45ndash78 2009

[16] P Ghadimi A Dashtimanesh and S R Djeddi ldquoStudy of waterentry of circular cylinder by using analytical and numericalsolutionsrdquo Journal of the Brazilian Society ofMechanical Sciencesamp Engineering vol 34 no 3 pp 225ndash232 2012

[17] A TassinD J PiroAAKorobkinK JMaki andM J CookerldquoTwo-dimensional water entry and exit of a body whose shapevaries in timerdquo Journal of Fluids and Structures vol 40 pp 317ndash336 2013

[18] E Yao H Wang L Pan X Wang and R Woding ldquoVerticalwater-entry of bullet-shaped projectilesrdquo Journal of AppliedMathematics and Physics vol 02 no 06 pp 323ndash334 2014

[19] M-S Park Y-R Jung and W-G Park ldquoNumerical study ofimpact force and ricochet behavior of high speed water-entrybodiesrdquo Computers amp Fluids vol 32 no 7 pp 939ndash951 2003

[20] D Battistin and A Iafrati ldquoHydrodynamic loads during waterenergy of two-dimensional and axisymmetric bodiesrdquo Journalof Fluids and Structures vol 17 no 5 pp 643ndash664 2003

[21] A A Korobkin and M Ohkusu ldquoImpact of two circular platesone of which is floating on a thin layer of liquidrdquo Journal ofEngineering Mathematics vol 50 no 4 pp 343ndash358 2004

[22] K M Kleefsman G Fekken A E Veldman B Iwanowski andB Buchner ldquoA volume-of-fluid based simulation method forwave impact problemsrdquo Journal of Computational Physics vol206 no 1 pp 363ndash393 2005

[23] Y Kim Y Kim Y Liu and D K P Yue ldquoOn the water-entryimpact problem of asymmetric bodiesrdquo in Proceedings of the9th International Conference on Numerical Ship Hydrodynamics(NSH rsquo07) August 2007

[24] Q Yang and W Qiu ldquoNumerical solution of 2D slammingproblem with a CIPmethodrdquo in Proceedings of the InternationalConference on Violent Flows Research Institute for AppliedMechanics Kyushu University Japan 2007

[25] A C Fairlie-Clarke and T Tveitnes ldquoMomentum and gravityeffects during the constant velocitywater entry ofwedge-shapedsectionsrdquo Ocean Engineering vol 35 no 7 pp 706ndash716 2007

[26] Q Yang andW Qiu ldquoNumerical simulation of water impact for2D and 3D bodiesrdquo Ocean Engineering vol 43 pp 82ndash89 2012

[27] G Wu ldquoNumerical simulation for water entry of a wedge atvarying speed by a high order boundary element methodrdquoJournal of Marine Science and Application vol 11 no 2 pp 143ndash149 2012

[28] M Ahmadzadeh B Saranjam A Hoseini Fard and A RBinesh ldquoNumerical simulation of sphere water entry prob-lem using Eulerian-Lagrangian methodrdquo Applied MathematicalModelling vol 38 no 5-6 pp 1673ndash1684 2014

[29] V-T Nguyen D-T Vu W-G Park and Y-R Jung ldquoNumericalanalysis of water impact forces using a dual-time pseudo-compressibility method and volume-of-fluid interface trackingalgorithmrdquo Computers amp Fluids vol 103 pp 18ndash33 2014

[30] S Kim and N Kim ldquoIntegrated dynamics modeling forsupercavitating vehicle systemsrdquo International Journal of NavalArchitecture and Ocean Engineering vol 7 no 2 pp 346ndash3632015

18 Geofluids

[31] V-T Nguyen D-T Vu W-G Park and C-M Jung ldquoNavier-Stokes solver for water entry bodies with moving Chimera gridmethod in 6DOF motionsrdquo Computers amp Fluids vol 140 pp19ndash38 2016

[32] M Mirzaei M Eghtesad and M Alishahi ldquoPlaning forceidentification in high-speed underwater vehiclesrdquo Journal ofVibration and Control vol 22 no 20 pp 4176ndash4191 2016

[33] G Qiu S Henke and J Grabe ldquoApplication of a CoupledEulerian-Lagrangian approach on geomechanical problemsinvolving large deformationsrdquoComputersampGeosciences vol 38no 1 pp 30ndash39 2011

[34] M R Erfanian M Anbarsooz N Rahimi M Zare andM Moghiman ldquoNumerical and experimental investigationof a three dimensional spherical-nose projectile water entryproblemrdquo Ocean Engineering vol 104 pp 397ndash404 2015

[35] W M Isbell F H Shipman and A H Jones ldquoHugoniotequation of state measurements for eleven material to fiveMegabars Materials and Structural Laboratory ManufacturingDevelopments General Motors Technical centerrdquo Tech RepMichigan 1968

[36] M A Jandron R C Hurd J L Belden A F Bower W Fennelland T T Truscott ldquoModeling of hyperelastic water-skippingspheres using AbaqusExplicitrdquo in Proceedings of the SIMULIACommunity Conference 2014

[37] H Forouzani B Saranjam R Kamali and A Abdollahi-far ldquoElasto-plastic time dependent impact analysis of highspeed projectile on water surfacerdquo Journal of Solid and FluidMechanics vol 3 pp 281ndash298 2016

[38] M R Erfanian andMMoghiman ldquoNumerical and experimen-tal investigation of a projectile water entry problem and studyof velocity effect on time and depth of pinch-offrdquo Journal ofModares Mechanical Engineering vol 15 pp 53ndash60 2015

[39] A Sillem Feasibility study of a tire hydroplaning simulation in afinite element code using a coupled Eulerian-Lagrangian method[MSc thesis] 2008

[40] M Lee R G Longoria and D E Wilson ldquoCavity dynamics inhigh-speed water entryrdquo Physics of Fluids vol 9 no 3 pp 540ndash550 1997

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Submit your manuscripts atwwwhindawicom

12 Geofluids

01 02 03 04 050Time (s)

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

CubeCylinderSphere

PyramidCone

Figure 9 Displacement-time graph of pounders with differentgeometric shapes

secondary velocity when impacting the water surface and 119892is the gravitational acceleration of the Earth

From (42) which expresses a time independent equationfor the free fall of objects the initial velocity and the terminalvelocity of the object drop height and the gravitationalacceleration of the Earth are interdependent in the absenceof time In this numerical simulation the pounders weredropped from a height of 1m above thewaterrsquos surfacewith aninitial velocity (1198810) equal to 0ms Therefore the secondaryvelocity of the pounders when impacting the water surfacewas obtained using

119881 = radic2119892ℎ (43)

Instead of simulating a full dropping event from theinitial position the pounders were placed exactly tangent towaterrsquos surface and their secondary velocity at the momentof impact was determined according to (43) A gravitationalacceleration (gravity) of minus98ms2 was applied in the normaldirection of the pounders Therefore a secondary velocityof 443ms in the normal direction of the waterrsquos surfacecorresponds to the speed that would be obtained for thepounders

7 Results and Discussion

71 Effect of the Geometric Shape of the Pounder on Deep-water Displacement Figure 9 illustrates the displacement-time graph of pounders with different geometric shapes Thegraph continues for the cubical pounder from the momentof water surface impact at time zero up to the depth of 1mat 415ms The displacement-time graph of the cylindrical

pounder indicates that after entering the water it reachesthe model bed in a shorter amount of time than the cubicalpounder The graph for this pounder continues from themoment of water surface impact at time zero up to the depthof 1m at 355ms The linear displacement-time graph forthe spherical and pyramidal pounders indicates the fasterdeepwater movement of these pounders compared to thecubical and cylindrical pounders The deepwater movementtime of the spherical and pyramidal pounders after enteringthe water to themodel bed is 285ms and 265ms respectivelyThe linear displacement-time graph of the conical pounderindicates that it has a faster deepwater movement comparedto other pounders as it takes 235ms to reach the model bedafter entering the water

72 Mechanism of Air Cavity Formation and Pinch-Off dueto Deepwater Movement of Pounders with Different Geomet-ric Shapes Figure 10 illustrates the movement trajectory ofpounders with different geometric shapes from the momentof water impact towhen the pounders reached themodel bedAfter entering the water a symmetric air cavity is formedbehind the pounders Air cavity formation involves severalsteps including air cavity creation and expansion behind thepounder air cavity contraction and air cavity collapse [34]As the pounder moves downward in water depth it imposesa force onto surrounding fluid in its radial directions andtransfers its momentum to the fluid This extension is facedwith the hydrostatic pressure resistance of fluidThe directionof the radial flow is reversed and eventually leads to aircavity contraction and collapse In the other words the aircavity contraction is accelerated until pinch-off occurs andthe air cavity is divided into two distinct parts The upperpart of the air cavity continues its contraction and movestowards the waterrsquos surface whereas the lower part of the aircavity clings to the pounder and moves along with it [34]After impacting the water surface the cubical and cylindricalpounders due to their flat contact surfaces lose some of theirvelocityThis velocity depreciation slows down and continuesuntil pinch-offThe spherical pounder with its curved contactsurface has less velocity depreciation compared to the cubicaland cylindrical pounders However some pounder velocityis always depreciated due to impacting the water surfaceThe velocity depreciation process for the spherical pounderduring deepwater movement continues until pinch-off Afterimpacting the waterrsquos surface the pyramidal pounder with itspointy contact surface has a slight velocity depreciation butcontinues to move deeper into the water with approximatelythe same secondary velocity until pinch-off By imposing thedrag force of water onto the pounders surface its velocityis gradually decreased until it reaches the model bed Afterimpacting the water surface the conical pounder with itspointy contact surface continues to move deeper into thewater with an increasing velocity The increasing velocity ofthe conical pounder during deepwater movement continueseven after pinch-off When the pounder moves towards themodel bed the waters drag force is imposed onto poundersurface causing velocity depreciation The pinch-off time forthe pounders with the above-mentioned shapes is between225 and 250ms

Geofluids 13

t = 0 25ＧＭ 50ＧＭ 100ＧＭ 200ＧＭ 225ＧＭ 250ＧＭ 275ＧＭ 300ＧＭ 325ＧＭ 350ＧＭ150ＧＭCu

beCy

linde

rSp

here

Pyra

mid

Con

e

Figure 10 Air cavity formation due to deepwater movement of pounders with different geometric shapes and pinch-off time and depth

73 Effect of the Pounder Geometric Shape onDeepwater Veloc-ity Figure 11 illustrates the velocity-time graph for pounderswith different geometric shapes The cubical and cylindricalpounders lose some of their secondary velocity at the earlymoments of water entry Hence the velocity-time graph ofthese pounders illustrates great velocity depreciation up to15ms The velocity reduction of the cubical and cylindricalpounders gradually continues until pinch-off The velocityof the spherical pounder decreases up to 45ms after waterentry but its velocity depreciation is less than the cubicaland cylindrical pounders This velocity reduction continuesat a slower rate until pinch-off In the initial moments ofwater entry the pyramidal pounder has a slight reduction insecondary velocity up to 55ms Then velocity depreciationcontinues at a faster rate due to the drag force imposed onto

pounders surface and the velocity-time graph of the pyrami-dal pounder shows a gradual velocity reduction until pinch-off After impacting the water surface the conical poundercontinues its deepwater movement with an increasing veloc-ity rate Due to its very low slope the velocity-time graphindicates an increase in velocity for the conical pounder butdue to the waters drag force its velocity begins to decrease at145ms The velocity-time graph indicates a gradual decreasein velocity for conical pounder and velocity depreciationcontinues until pinch-off After pinch-off pounder velocitiesgradually decrease until they reach themodel bed Eventuallywith a velocity of 388ms and 157ms at the moment ofmodel bed impact the conical and cubical pounders have thehighest and lowest velocity respectively Water drag force isimposed in the opposite direction of poundermovement and

14 Geofluids

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Velo

city

(ms

)

01 02 03 040Time (s)

CubeCylinderSphere

PyramidCone

Figure 11 Velocity-time graph of pounders with different geometricshapes

it has the most and least influence on the cubical and conicalpounders respectively

8 Critical Drop Height of the Pounder

In the dynamic compaction of the seabed determining thecritical drop height of the pounder from the freewater surfaceis an effective parameter for the pounder to reach maximumdeepwater velocity After dropping the pounder from height119867 above the free water surface with an initial velocity (1198810)of 0ms its velocity gradually increases until it reaches thesecondary velocity (119881) at the moment of water impact Afterentering the water pounder velocity is decreased due to waterdrag force According to the free fall equation for objectsincreasing the drop height from the waterrsquos surface leads toincreased pounder velocity at the moment of water surfaceimpact but it does not necessarily lead to an increase inpounder velocity during seabed impact because the criticaldrop height of the pounder is a determining factor ofits maximum deepwater velocity According to Figure 12 aconical pounder with a mass of 411 kg was dropped fromdifferent heights (119867) and its velocity when impacting thewaters was measured according to the time independentequation for the free fall of objects Table 4 illustrates thevelocity of the conical pounder with different drop heightswhen impacting the waterrsquos surface

81 Effect of the Drop Height of the Pounder from WaterSurface on Deepwater Displacement Figure 13 illustrates thedisplacement-time graph of the conical pounder with dif-ferent drop heights from the waterrsquos surface As can beseen increasing the drop height of the pounder leads to

Water levelℎ

Figure 12 Schematic of drop height of the conical pounder fromwater surface

Table 4 Velocity of the conical pounder with different drop heightsat the moment of impacting the water surface

Drop height of the pounderfrom water surface (m)

Pounder velocity at themoment of impacting water

surface (ms)025 22105 313075 3831 443125 49515 542175 5852 626225 66425 7

displacement-time graphs that are closer to each other Infact an increase in the drop height of the pounder from thewaterrsquos surface leads to an increase in its secondary velocitywhen impacting the waterrsquos surface Hence the pounderenters the water with a higher velocity and reaches the modelbed in less time

82 Effect of the Drop Height of the Pounder from WaterSurface on Its Deepwater Velocity Increasing the drop heightof the pounder from the waterrsquos surface leads to an increasein its velocity before impacting the water but the poundervelocity gradually decreases after entering the water due tothe its drag force Figure 14 illustrates the velocity-time graphfor the conical pounder with different drop heights from thewaterrsquos surface When the pounder is dropped from a lowheight due to its low velocity it moves deepwater due to itsweight Moreover because of the low velocity of the pounderwhen entering the water pinch-off occurs in less time and atlower depths and its velocity increases after pinch-off until itreaches terminal velocity Increases in pounder drop heightsincrease water entry velocity Hence it will have greatervelocity depreciation in the first moments of water entry

Figure 15 illustrates the terminal velocity-time graph forthe conical pounderwith different drop heightswhen impact-ing the model bed As can be seen increasing the drop height

Geofluids 15

01 02 03 040Time (s)

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

H = 1Ｇ

H = 075Ｇ

H = 05 Ｇ

H = 025 Ｇ

H = 175Ｇ

H = 15 Ｇ

H = 125 Ｇ

H = 25 Ｇ

H = 225 Ｇ

H = 2Ｇ

Figure 13 Displacement-time graph of the conical pounder withdifferent drop heights

of the pounder from the waterrsquos surface leads to an increasein its terminal velocity when impacting the model bedTheseconditions are dominant as long as the pounder reaches acritical drop height from the water surface but increasingthe pounder drop height higher than the critical drop heightleads to decreased terminal velocity when impacting themodel bed In this numerical simulation after dropping thepounder from a height of 2m above the waterrsquos surface itssecondary velocity was 626ms when impacting the watersurface After entering the water the secondary velocity ofthe pounder decreased until it impacts the model bed witha terminal velocity of 454ms Therefore a drop height of2m from above the waterrsquos surface has the highest terminalvelocity when the pounder impacts themodel bed Increasingthe drop height bymore than 2m leads to decreased pounderterminal velocity when impacting the model bed The heightwhich has the greatest effect on pounder terminal velocitywhen impacting the model bed is called the critical dropheight Increasing the drop height of the pounder to greaterthan its critical drop height leads to decreased terminalvelocity when impacting the model bed

Figure 16 illustrates the secondary and terminal velocityof the conical pounder with different drop heights As canbe observed increasing the drop height of the pounder fromwaterrsquos surface leads to increased differences between thesecondary and terminal velocities of the pounder Increasingthe drop height to more than the critical drop height leadsto increased pounder secondary velocity when impactingthe waterrsquos surface and decreased terminal velocity whenimpacting the model bed Hence the difference betweenthe secondary and terminal velocities of the pounder is

01 02 03 040Time (s)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)H = 175Ｇ

H = 125 Ｇ

H = 075Ｇ

H = 025 Ｇ H = 225 Ｇ

H = 1Ｇ

H = 05 Ｇ

H = 25 Ｇ

H = 2Ｇ

H = 15 Ｇ

Figure 14 Velocity-time graph of the conical pounder with differentdrop heights

minus430

minus447 minus454minus415

minus426 minus380

minus367minus355

minus332minus310

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Term

inal

velo

city

(ms

)

01 02 03 040Time (s)

Figure 15 Terminal velocity of conical pounder with different dropheights versus time

not reasonable and a significant amount of the pounderssecondary velocity is depreciated when entering the waterThe great difference between the secondary and terminalvelocities of the pounder reveals the necessity of determiningthe critical drop height of the pounder before dynamic seabedcompaction operations Increasing the pounder drop heightto be greater than the critical drop height not only hasa reverse result on pounder velocity when impacting theseabed but it also requires special equipment such as lengthy

16 Geofluids

Secondary velocityTerminal velocity

025 05 075 1 125 15 175 2 225 250Drop height (m)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)

Figure 16 Secondary and terminal velocities of conical pounderversus drop heights

1 2 3 4 5 6 7 80Impact velocity (ms)

0

005

01

015

02

025

03

Pinc

h-off

tim

e (s)

Figure 17 Pinch-off time versus impact velocity of conical pounder

cranes and huge barges leading to increased implementationcosts

Figure 17 illustrates pinch-off time based on the sec-ondary velocity of the pounder at the moment of watersurface impact As can be seen pinch-off time is a weakfunction of pounder impact velocity and increasing impactvelocity leads to a slight reduction in pinch-off time Thisbehaviour was also observed in the analytical and numericalresults found by Lee et al [40]

Figure 18 illustrates pinch-off depth based on the sec-ondary velocity of pounder at the moment of water surfaceimpact As can be seen the pinch-off depth is a linear

1 2 3 4 5 6 7 80Impact velocity (ms)

minus12

minus1

minus08

minus06

minus04

minus02

0

Pinc

h-off

dep

th (m

)

Figure 18 Pinch-off depth versus impact velocity of conicalpounder

function of pounder impact velocity These results are in linewith the analytical equations of Lee et al [40]

9 Conclusions

In this study the water entry problem of three-dimensionalpounders with different geometric shapes of cube cylindersphere pyramid and cone was numerically simulated bycommercial finite element code Abaqus 614-2 An explicitdynamic analysis method was employed to model the fluid-structure interaction using a Coupled Eulerian-Lagrangian(CEL) method Before applying the model to water entryof the pounders a three-dimensional numerical model wasdeveloped to simulate a transient free-falling wedge impact-ing the free surface of the water in order to calibrate theadopted computationalmodelThe accuracy of the numericalmodel was examined by comparing numerical results withthe experimental data of Aristoff et al The good agreementbetween numerical simulation results with the theoreticaland experimental data of Aristoff et al demonstrates thecapability of the employed numerical methodology to sim-ulate the water entry problem for three-dimensional geome-tries with acceptable accuracy After water entry by a pounderdropped fromaheight of 1m a symmetric air cavity is formedbehind the pounder As the pounder moves downward inwater depth it imposes a force onto surrounding fluid inthe radial directions and transfers its momentum to thefluid This extension is faced with the hydrostatic pressureresistance of the fluid causing the direction of the radialflow to be reversed leading to air cavity contraction andcollapse The results show that the water drag force had thehighest and lowest effect on the cubical and conical poundersrespectively and increasing the drop height to greater thanthe critical drop height leads to increased pounder velocitywhen impacting the waterrsquos surface and decreased velocitywhen impacting the model bed Pinch-off time is a very weak

Geofluids 17

function of pounder velocity when impacting the waterrsquossurface but the pinch-off depth increases linearly along withincreased pounder impact velocities It is recommended thatthe critical drop height for the pounder is determined beforedynamic seabed compaction operations and the adjustmentof crane heights to this height In addition it is also recom-mended that perforated pounders be used in order to reducevelocity depreciation when impacting the waterrsquos surface andto reduce the effect of water drag force

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors warmly appreciate the cooperation and guidanceof Dr B Hamidi and Dr A Mehrabadi in this study

References

[1] B Hamidi J-M Debats H Nikraz and S Varaksin ldquoOffshoreground improvement recordsrdquo Australian Geomechanics Jour-nal vol 48 no 4 pp 111ndash122 2013

[2] SWatanabe ldquoResistance of impact onwater surface part I-cone Institute of Physical andChemical ResearchrdquoTokyo vol 12 pp251ndash267 1930

[3] S Watanabe ldquoResistance of impact on water surface part II-cone (continued)rdquo Institute of Physical and Chemical Researchvol 14 pp 153ndash168 1930

[4] A May and J C Woodhull ldquoDrag coefficients of steel spheresentering water verticallyrdquo Journal of Applied Physics vol 19 no12 pp 1109ndash1121 1948

[5] A May ldquoEffect of surface condition of a sphere on its water-entry cavityrdquo Journal of Applied Physics vol 22 no 10 pp 1219ndash1222 1951

[6] AMay ldquoVertical entry ofmissiles into waterrdquo Journal of AppliedPhysics vol 23 no 12 pp 1362ndash1372 1952

[7] A P New T S Lee and H T Low ldquoImpact loading and waterentrance characteristics of prismatic bodiesrdquo in Proceedings ofthe third international offshore and polar engineering conferencepp 282ndash287 1993

[8] M Anghileri and A Spizzica ldquoExperimental validation of finiteelement models for water impactsrdquo in Proceedings of the secondinternational crash usersrsquo seminar Cranfield Impact Centre LtdEngland UK 1995 second international crash users seminar

[9] A Engle andR Lewis ldquoA comparison of hydrodynamic impactsprediction methods with two dimensional drop test datardquoMarine Structures vol 16 no 2 pp 175ndash182 2003

[10] J M Aristoff T T Truscott A H Techet and J W M BushldquoThe water entry of decelerating spheresrdquo Physics of Fluids vol22 no 3 Article ID 009002PHF pp 1ndash8 2010

[11] T Von Karman The impact of seaplane floats during landingNational Advisory Committee for Aeronautics NACA TN no321 1929

[12] HWagner ldquoPhenomena associatedwith impacts and sliding onliquid surfacesrdquo Journal of Applied Mathematics and Mechanicsvol 12 pp 193ndash215 1932

[13] W S Vorus ldquoA flat cylinder theory for vessel impact and steadyplaning resistancerdquo Journal of Ship Research vol 40 no 2 pp89ndash106 1996

[14] T Miloh ldquoOn the initial-stage slamming of a rigid sphere in avertical water entryrdquo Applied Ocean Research vol 13 no 1 pp13ndash43 1991

[15] J M Aristoff and JW Bush ldquoWater entry of small hydrophobicspheresrdquo Journal of Fluid Mechanics vol 619 pp 45ndash78 2009

[16] P Ghadimi A Dashtimanesh and S R Djeddi ldquoStudy of waterentry of circular cylinder by using analytical and numericalsolutionsrdquo Journal of the Brazilian Society ofMechanical Sciencesamp Engineering vol 34 no 3 pp 225ndash232 2012

[17] A TassinD J PiroAAKorobkinK JMaki andM J CookerldquoTwo-dimensional water entry and exit of a body whose shapevaries in timerdquo Journal of Fluids and Structures vol 40 pp 317ndash336 2013

[18] E Yao H Wang L Pan X Wang and R Woding ldquoVerticalwater-entry of bullet-shaped projectilesrdquo Journal of AppliedMathematics and Physics vol 02 no 06 pp 323ndash334 2014

[19] M-S Park Y-R Jung and W-G Park ldquoNumerical study ofimpact force and ricochet behavior of high speed water-entrybodiesrdquo Computers amp Fluids vol 32 no 7 pp 939ndash951 2003

[20] D Battistin and A Iafrati ldquoHydrodynamic loads during waterenergy of two-dimensional and axisymmetric bodiesrdquo Journalof Fluids and Structures vol 17 no 5 pp 643ndash664 2003

[21] A A Korobkin and M Ohkusu ldquoImpact of two circular platesone of which is floating on a thin layer of liquidrdquo Journal ofEngineering Mathematics vol 50 no 4 pp 343ndash358 2004

[22] K M Kleefsman G Fekken A E Veldman B Iwanowski andB Buchner ldquoA volume-of-fluid based simulation method forwave impact problemsrdquo Journal of Computational Physics vol206 no 1 pp 363ndash393 2005

[23] Y Kim Y Kim Y Liu and D K P Yue ldquoOn the water-entryimpact problem of asymmetric bodiesrdquo in Proceedings of the9th International Conference on Numerical Ship Hydrodynamics(NSH rsquo07) August 2007

[24] Q Yang and W Qiu ldquoNumerical solution of 2D slammingproblem with a CIPmethodrdquo in Proceedings of the InternationalConference on Violent Flows Research Institute for AppliedMechanics Kyushu University Japan 2007

[25] A C Fairlie-Clarke and T Tveitnes ldquoMomentum and gravityeffects during the constant velocitywater entry ofwedge-shapedsectionsrdquo Ocean Engineering vol 35 no 7 pp 706ndash716 2007

[26] Q Yang andW Qiu ldquoNumerical simulation of water impact for2D and 3D bodiesrdquo Ocean Engineering vol 43 pp 82ndash89 2012

[27] G Wu ldquoNumerical simulation for water entry of a wedge atvarying speed by a high order boundary element methodrdquoJournal of Marine Science and Application vol 11 no 2 pp 143ndash149 2012

[28] M Ahmadzadeh B Saranjam A Hoseini Fard and A RBinesh ldquoNumerical simulation of sphere water entry prob-lem using Eulerian-Lagrangian methodrdquo Applied MathematicalModelling vol 38 no 5-6 pp 1673ndash1684 2014

[29] V-T Nguyen D-T Vu W-G Park and Y-R Jung ldquoNumericalanalysis of water impact forces using a dual-time pseudo-compressibility method and volume-of-fluid interface trackingalgorithmrdquo Computers amp Fluids vol 103 pp 18ndash33 2014

[30] S Kim and N Kim ldquoIntegrated dynamics modeling forsupercavitating vehicle systemsrdquo International Journal of NavalArchitecture and Ocean Engineering vol 7 no 2 pp 346ndash3632015

18 Geofluids

[31] V-T Nguyen D-T Vu W-G Park and C-M Jung ldquoNavier-Stokes solver for water entry bodies with moving Chimera gridmethod in 6DOF motionsrdquo Computers amp Fluids vol 140 pp19ndash38 2016

[32] M Mirzaei M Eghtesad and M Alishahi ldquoPlaning forceidentification in high-speed underwater vehiclesrdquo Journal ofVibration and Control vol 22 no 20 pp 4176ndash4191 2016

[33] G Qiu S Henke and J Grabe ldquoApplication of a CoupledEulerian-Lagrangian approach on geomechanical problemsinvolving large deformationsrdquoComputersampGeosciences vol 38no 1 pp 30ndash39 2011

[34] M R Erfanian M Anbarsooz N Rahimi M Zare andM Moghiman ldquoNumerical and experimental investigationof a three dimensional spherical-nose projectile water entryproblemrdquo Ocean Engineering vol 104 pp 397ndash404 2015

[35] W M Isbell F H Shipman and A H Jones ldquoHugoniotequation of state measurements for eleven material to fiveMegabars Materials and Structural Laboratory ManufacturingDevelopments General Motors Technical centerrdquo Tech RepMichigan 1968

[36] M A Jandron R C Hurd J L Belden A F Bower W Fennelland T T Truscott ldquoModeling of hyperelastic water-skippingspheres using AbaqusExplicitrdquo in Proceedings of the SIMULIACommunity Conference 2014

[37] H Forouzani B Saranjam R Kamali and A Abdollahi-far ldquoElasto-plastic time dependent impact analysis of highspeed projectile on water surfacerdquo Journal of Solid and FluidMechanics vol 3 pp 281ndash298 2016

[38] M R Erfanian andMMoghiman ldquoNumerical and experimen-tal investigation of a projectile water entry problem and studyof velocity effect on time and depth of pinch-offrdquo Journal ofModares Mechanical Engineering vol 15 pp 53ndash60 2015

[39] A Sillem Feasibility study of a tire hydroplaning simulation in afinite element code using a coupled Eulerian-Lagrangian method[MSc thesis] 2008

[40] M Lee R G Longoria and D E Wilson ldquoCavity dynamics inhigh-speed water entryrdquo Physics of Fluids vol 9 no 3 pp 540ndash550 1997

Hindawiwwwhindawicom Volume 2018

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ChemistryArchaeaHindawiwwwhindawicom Volume 2018

Marine BiologyJournal of

Hindawiwwwhindawicom Volume 2018

BiodiversityInternational Journal of

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EcologyInternational Journal of

Hindawiwwwhindawicom Volume 2018

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Applied ampEnvironmentalSoil Science

Volume 2018

Forestry ResearchInternational Journal of

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International Journal of

Geophysics

Environmental and Public Health

Journal of

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Hindawiwwwhindawicom Volume 2018

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Volume 2018Hindawiwwwhindawicom Volume 2018

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Geological ResearchJournal of

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

Submit your manuscripts atwwwhindawicom

Geofluids 13

t = 0 25ＧＭ 50ＧＭ 100ＧＭ 200ＧＭ 225ＧＭ 250ＧＭ 275ＧＭ 300ＧＭ 325ＧＭ 350ＧＭ150ＧＭCu

beCy

linde

rSp

here

Pyra

mid

Con

e

Figure 10 Air cavity formation due to deepwater movement of pounders with different geometric shapes and pinch-off time and depth

73 Effect of the Pounder Geometric Shape onDeepwater Veloc-ity Figure 11 illustrates the velocity-time graph for pounderswith different geometric shapes The cubical and cylindricalpounders lose some of their secondary velocity at the earlymoments of water entry Hence the velocity-time graph ofthese pounders illustrates great velocity depreciation up to15ms The velocity reduction of the cubical and cylindricalpounders gradually continues until pinch-off The velocityof the spherical pounder decreases up to 45ms after waterentry but its velocity depreciation is less than the cubicaland cylindrical pounders This velocity reduction continuesat a slower rate until pinch-off In the initial moments ofwater entry the pyramidal pounder has a slight reduction insecondary velocity up to 55ms Then velocity depreciationcontinues at a faster rate due to the drag force imposed onto

pounders surface and the velocity-time graph of the pyrami-dal pounder shows a gradual velocity reduction until pinch-off After impacting the water surface the conical poundercontinues its deepwater movement with an increasing veloc-ity rate Due to its very low slope the velocity-time graphindicates an increase in velocity for the conical pounder butdue to the waters drag force its velocity begins to decrease at145ms The velocity-time graph indicates a gradual decreasein velocity for conical pounder and velocity depreciationcontinues until pinch-off After pinch-off pounder velocitiesgradually decrease until they reach themodel bed Eventuallywith a velocity of 388ms and 157ms at the moment ofmodel bed impact the conical and cubical pounders have thehighest and lowest velocity respectively Water drag force isimposed in the opposite direction of poundermovement and

14 Geofluids

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Velo

city

(ms

)

01 02 03 040Time (s)

CubeCylinderSphere

PyramidCone

Figure 11 Velocity-time graph of pounders with different geometricshapes

it has the most and least influence on the cubical and conicalpounders respectively

8 Critical Drop Height of the Pounder

In the dynamic compaction of the seabed determining thecritical drop height of the pounder from the freewater surfaceis an effective parameter for the pounder to reach maximumdeepwater velocity After dropping the pounder from height119867 above the free water surface with an initial velocity (1198810)of 0ms its velocity gradually increases until it reaches thesecondary velocity (119881) at the moment of water impact Afterentering the water pounder velocity is decreased due to waterdrag force According to the free fall equation for objectsincreasing the drop height from the waterrsquos surface leads toincreased pounder velocity at the moment of water surfaceimpact but it does not necessarily lead to an increase inpounder velocity during seabed impact because the criticaldrop height of the pounder is a determining factor ofits maximum deepwater velocity According to Figure 12 aconical pounder with a mass of 411 kg was dropped fromdifferent heights (119867) and its velocity when impacting thewaters was measured according to the time independentequation for the free fall of objects Table 4 illustrates thevelocity of the conical pounder with different drop heightswhen impacting the waterrsquos surface

81 Effect of the Drop Height of the Pounder from WaterSurface on Deepwater Displacement Figure 13 illustrates thedisplacement-time graph of the conical pounder with dif-ferent drop heights from the waterrsquos surface As can beseen increasing the drop height of the pounder leads to

Water levelℎ

Figure 12 Schematic of drop height of the conical pounder fromwater surface

Table 4 Velocity of the conical pounder with different drop heightsat the moment of impacting the water surface

Drop height of the pounderfrom water surface (m)

Pounder velocity at themoment of impacting water

surface (ms)025 22105 313075 3831 443125 49515 542175 5852 626225 66425 7

displacement-time graphs that are closer to each other Infact an increase in the drop height of the pounder from thewaterrsquos surface leads to an increase in its secondary velocitywhen impacting the waterrsquos surface Hence the pounderenters the water with a higher velocity and reaches the modelbed in less time

82 Effect of the Drop Height of the Pounder from WaterSurface on Its Deepwater Velocity Increasing the drop heightof the pounder from the waterrsquos surface leads to an increasein its velocity before impacting the water but the poundervelocity gradually decreases after entering the water due tothe its drag force Figure 14 illustrates the velocity-time graphfor the conical pounder with different drop heights from thewaterrsquos surface When the pounder is dropped from a lowheight due to its low velocity it moves deepwater due to itsweight Moreover because of the low velocity of the pounderwhen entering the water pinch-off occurs in less time and atlower depths and its velocity increases after pinch-off until itreaches terminal velocity Increases in pounder drop heightsincrease water entry velocity Hence it will have greatervelocity depreciation in the first moments of water entry

Figure 15 illustrates the terminal velocity-time graph forthe conical pounderwith different drop heightswhen impact-ing the model bed As can be seen increasing the drop height

Geofluids 15

01 02 03 040Time (s)

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

H = 1Ｇ

H = 075Ｇ

H = 05 Ｇ

H = 025 Ｇ

H = 175Ｇ

H = 15 Ｇ

H = 125 Ｇ

H = 25 Ｇ

H = 225 Ｇ

H = 2Ｇ

Figure 13 Displacement-time graph of the conical pounder withdifferent drop heights

of the pounder from the waterrsquos surface leads to an increasein its terminal velocity when impacting the model bedTheseconditions are dominant as long as the pounder reaches acritical drop height from the water surface but increasingthe pounder drop height higher than the critical drop heightleads to decreased terminal velocity when impacting themodel bed In this numerical simulation after dropping thepounder from a height of 2m above the waterrsquos surface itssecondary velocity was 626ms when impacting the watersurface After entering the water the secondary velocity ofthe pounder decreased until it impacts the model bed witha terminal velocity of 454ms Therefore a drop height of2m from above the waterrsquos surface has the highest terminalvelocity when the pounder impacts themodel bed Increasingthe drop height bymore than 2m leads to decreased pounderterminal velocity when impacting the model bed The heightwhich has the greatest effect on pounder terminal velocitywhen impacting the model bed is called the critical dropheight Increasing the drop height of the pounder to greaterthan its critical drop height leads to decreased terminalvelocity when impacting the model bed

Figure 16 illustrates the secondary and terminal velocityof the conical pounder with different drop heights As canbe observed increasing the drop height of the pounder fromwaterrsquos surface leads to increased differences between thesecondary and terminal velocities of the pounder Increasingthe drop height to more than the critical drop height leadsto increased pounder secondary velocity when impactingthe waterrsquos surface and decreased terminal velocity whenimpacting the model bed Hence the difference betweenthe secondary and terminal velocities of the pounder is

01 02 03 040Time (s)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)H = 175Ｇ

H = 125 Ｇ

H = 075Ｇ

H = 025 Ｇ H = 225 Ｇ

H = 1Ｇ

H = 05 Ｇ

H = 25 Ｇ

H = 2Ｇ

H = 15 Ｇ

Figure 14 Velocity-time graph of the conical pounder with differentdrop heights

minus430

minus447 minus454minus415

minus426 minus380

minus367minus355

minus332minus310

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Term

inal

velo

city

(ms

)

01 02 03 040Time (s)

Figure 15 Terminal velocity of conical pounder with different dropheights versus time

not reasonable and a significant amount of the pounderssecondary velocity is depreciated when entering the waterThe great difference between the secondary and terminalvelocities of the pounder reveals the necessity of determiningthe critical drop height of the pounder before dynamic seabedcompaction operations Increasing the pounder drop heightto be greater than the critical drop height not only hasa reverse result on pounder velocity when impacting theseabed but it also requires special equipment such as lengthy

16 Geofluids

Secondary velocityTerminal velocity

025 05 075 1 125 15 175 2 225 250Drop height (m)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)

Figure 16 Secondary and terminal velocities of conical pounderversus drop heights

1 2 3 4 5 6 7 80Impact velocity (ms)

0

005

01

015

02

025

03

Pinc

h-off

tim

e (s)

Figure 17 Pinch-off time versus impact velocity of conical pounder

cranes and huge barges leading to increased implementationcosts

Figure 17 illustrates pinch-off time based on the sec-ondary velocity of the pounder at the moment of watersurface impact As can be seen pinch-off time is a weakfunction of pounder impact velocity and increasing impactvelocity leads to a slight reduction in pinch-off time Thisbehaviour was also observed in the analytical and numericalresults found by Lee et al [40]

Figure 18 illustrates pinch-off depth based on the sec-ondary velocity of pounder at the moment of water surfaceimpact As can be seen the pinch-off depth is a linear

1 2 3 4 5 6 7 80Impact velocity (ms)

minus12

minus1

minus08

minus06

minus04

minus02

0

Pinc

h-off

dep

th (m

)

Figure 18 Pinch-off depth versus impact velocity of conicalpounder

function of pounder impact velocity These results are in linewith the analytical equations of Lee et al [40]

9 Conclusions

In this study the water entry problem of three-dimensionalpounders with different geometric shapes of cube cylindersphere pyramid and cone was numerically simulated bycommercial finite element code Abaqus 614-2 An explicitdynamic analysis method was employed to model the fluid-structure interaction using a Coupled Eulerian-Lagrangian(CEL) method Before applying the model to water entryof the pounders a three-dimensional numerical model wasdeveloped to simulate a transient free-falling wedge impact-ing the free surface of the water in order to calibrate theadopted computationalmodelThe accuracy of the numericalmodel was examined by comparing numerical results withthe experimental data of Aristoff et al The good agreementbetween numerical simulation results with the theoreticaland experimental data of Aristoff et al demonstrates thecapability of the employed numerical methodology to sim-ulate the water entry problem for three-dimensional geome-tries with acceptable accuracy After water entry by a pounderdropped fromaheight of 1m a symmetric air cavity is formedbehind the pounder As the pounder moves downward inwater depth it imposes a force onto surrounding fluid inthe radial directions and transfers its momentum to thefluid This extension is faced with the hydrostatic pressureresistance of the fluid causing the direction of the radialflow to be reversed leading to air cavity contraction andcollapse The results show that the water drag force had thehighest and lowest effect on the cubical and conical poundersrespectively and increasing the drop height to greater thanthe critical drop height leads to increased pounder velocitywhen impacting the waterrsquos surface and decreased velocitywhen impacting the model bed Pinch-off time is a very weak

Geofluids 17

function of pounder velocity when impacting the waterrsquossurface but the pinch-off depth increases linearly along withincreased pounder impact velocities It is recommended thatthe critical drop height for the pounder is determined beforedynamic seabed compaction operations and the adjustmentof crane heights to this height In addition it is also recom-mended that perforated pounders be used in order to reducevelocity depreciation when impacting the waterrsquos surface andto reduce the effect of water drag force

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors warmly appreciate the cooperation and guidanceof Dr B Hamidi and Dr A Mehrabadi in this study

References

[1] B Hamidi J-M Debats H Nikraz and S Varaksin ldquoOffshoreground improvement recordsrdquo Australian Geomechanics Jour-nal vol 48 no 4 pp 111ndash122 2013

[2] SWatanabe ldquoResistance of impact onwater surface part I-cone Institute of Physical andChemical ResearchrdquoTokyo vol 12 pp251ndash267 1930

[3] S Watanabe ldquoResistance of impact on water surface part II-cone (continued)rdquo Institute of Physical and Chemical Researchvol 14 pp 153ndash168 1930

[4] A May and J C Woodhull ldquoDrag coefficients of steel spheresentering water verticallyrdquo Journal of Applied Physics vol 19 no12 pp 1109ndash1121 1948

[5] A May ldquoEffect of surface condition of a sphere on its water-entry cavityrdquo Journal of Applied Physics vol 22 no 10 pp 1219ndash1222 1951

[6] AMay ldquoVertical entry ofmissiles into waterrdquo Journal of AppliedPhysics vol 23 no 12 pp 1362ndash1372 1952

[7] A P New T S Lee and H T Low ldquoImpact loading and waterentrance characteristics of prismatic bodiesrdquo in Proceedings ofthe third international offshore and polar engineering conferencepp 282ndash287 1993

[8] M Anghileri and A Spizzica ldquoExperimental validation of finiteelement models for water impactsrdquo in Proceedings of the secondinternational crash usersrsquo seminar Cranfield Impact Centre LtdEngland UK 1995 second international crash users seminar

[9] A Engle andR Lewis ldquoA comparison of hydrodynamic impactsprediction methods with two dimensional drop test datardquoMarine Structures vol 16 no 2 pp 175ndash182 2003

[10] J M Aristoff T T Truscott A H Techet and J W M BushldquoThe water entry of decelerating spheresrdquo Physics of Fluids vol22 no 3 Article ID 009002PHF pp 1ndash8 2010

[11] T Von Karman The impact of seaplane floats during landingNational Advisory Committee for Aeronautics NACA TN no321 1929

[12] HWagner ldquoPhenomena associatedwith impacts and sliding onliquid surfacesrdquo Journal of Applied Mathematics and Mechanicsvol 12 pp 193ndash215 1932

[13] W S Vorus ldquoA flat cylinder theory for vessel impact and steadyplaning resistancerdquo Journal of Ship Research vol 40 no 2 pp89ndash106 1996

[14] T Miloh ldquoOn the initial-stage slamming of a rigid sphere in avertical water entryrdquo Applied Ocean Research vol 13 no 1 pp13ndash43 1991

[15] J M Aristoff and JW Bush ldquoWater entry of small hydrophobicspheresrdquo Journal of Fluid Mechanics vol 619 pp 45ndash78 2009

[16] P Ghadimi A Dashtimanesh and S R Djeddi ldquoStudy of waterentry of circular cylinder by using analytical and numericalsolutionsrdquo Journal of the Brazilian Society ofMechanical Sciencesamp Engineering vol 34 no 3 pp 225ndash232 2012

[17] A TassinD J PiroAAKorobkinK JMaki andM J CookerldquoTwo-dimensional water entry and exit of a body whose shapevaries in timerdquo Journal of Fluids and Structures vol 40 pp 317ndash336 2013

[18] E Yao H Wang L Pan X Wang and R Woding ldquoVerticalwater-entry of bullet-shaped projectilesrdquo Journal of AppliedMathematics and Physics vol 02 no 06 pp 323ndash334 2014

[19] M-S Park Y-R Jung and W-G Park ldquoNumerical study ofimpact force and ricochet behavior of high speed water-entrybodiesrdquo Computers amp Fluids vol 32 no 7 pp 939ndash951 2003

[20] D Battistin and A Iafrati ldquoHydrodynamic loads during waterenergy of two-dimensional and axisymmetric bodiesrdquo Journalof Fluids and Structures vol 17 no 5 pp 643ndash664 2003

[21] A A Korobkin and M Ohkusu ldquoImpact of two circular platesone of which is floating on a thin layer of liquidrdquo Journal ofEngineering Mathematics vol 50 no 4 pp 343ndash358 2004

[22] K M Kleefsman G Fekken A E Veldman B Iwanowski andB Buchner ldquoA volume-of-fluid based simulation method forwave impact problemsrdquo Journal of Computational Physics vol206 no 1 pp 363ndash393 2005

[23] Y Kim Y Kim Y Liu and D K P Yue ldquoOn the water-entryimpact problem of asymmetric bodiesrdquo in Proceedings of the9th International Conference on Numerical Ship Hydrodynamics(NSH rsquo07) August 2007

[24] Q Yang and W Qiu ldquoNumerical solution of 2D slammingproblem with a CIPmethodrdquo in Proceedings of the InternationalConference on Violent Flows Research Institute for AppliedMechanics Kyushu University Japan 2007

[25] A C Fairlie-Clarke and T Tveitnes ldquoMomentum and gravityeffects during the constant velocitywater entry ofwedge-shapedsectionsrdquo Ocean Engineering vol 35 no 7 pp 706ndash716 2007

[26] Q Yang andW Qiu ldquoNumerical simulation of water impact for2D and 3D bodiesrdquo Ocean Engineering vol 43 pp 82ndash89 2012

[27] G Wu ldquoNumerical simulation for water entry of a wedge atvarying speed by a high order boundary element methodrdquoJournal of Marine Science and Application vol 11 no 2 pp 143ndash149 2012

[28] M Ahmadzadeh B Saranjam A Hoseini Fard and A RBinesh ldquoNumerical simulation of sphere water entry prob-lem using Eulerian-Lagrangian methodrdquo Applied MathematicalModelling vol 38 no 5-6 pp 1673ndash1684 2014

[29] V-T Nguyen D-T Vu W-G Park and Y-R Jung ldquoNumericalanalysis of water impact forces using a dual-time pseudo-compressibility method and volume-of-fluid interface trackingalgorithmrdquo Computers amp Fluids vol 103 pp 18ndash33 2014

[30] S Kim and N Kim ldquoIntegrated dynamics modeling forsupercavitating vehicle systemsrdquo International Journal of NavalArchitecture and Ocean Engineering vol 7 no 2 pp 346ndash3632015

18 Geofluids

[31] V-T Nguyen D-T Vu W-G Park and C-M Jung ldquoNavier-Stokes solver for water entry bodies with moving Chimera gridmethod in 6DOF motionsrdquo Computers amp Fluids vol 140 pp19ndash38 2016

[32] M Mirzaei M Eghtesad and M Alishahi ldquoPlaning forceidentification in high-speed underwater vehiclesrdquo Journal ofVibration and Control vol 22 no 20 pp 4176ndash4191 2016

[33] G Qiu S Henke and J Grabe ldquoApplication of a CoupledEulerian-Lagrangian approach on geomechanical problemsinvolving large deformationsrdquoComputersampGeosciences vol 38no 1 pp 30ndash39 2011

[34] M R Erfanian M Anbarsooz N Rahimi M Zare andM Moghiman ldquoNumerical and experimental investigationof a three dimensional spherical-nose projectile water entryproblemrdquo Ocean Engineering vol 104 pp 397ndash404 2015

[35] W M Isbell F H Shipman and A H Jones ldquoHugoniotequation of state measurements for eleven material to fiveMegabars Materials and Structural Laboratory ManufacturingDevelopments General Motors Technical centerrdquo Tech RepMichigan 1968

[36] M A Jandron R C Hurd J L Belden A F Bower W Fennelland T T Truscott ldquoModeling of hyperelastic water-skippingspheres using AbaqusExplicitrdquo in Proceedings of the SIMULIACommunity Conference 2014

[37] H Forouzani B Saranjam R Kamali and A Abdollahi-far ldquoElasto-plastic time dependent impact analysis of highspeed projectile on water surfacerdquo Journal of Solid and FluidMechanics vol 3 pp 281ndash298 2016

[38] M R Erfanian andMMoghiman ldquoNumerical and experimen-tal investigation of a projectile water entry problem and studyof velocity effect on time and depth of pinch-offrdquo Journal ofModares Mechanical Engineering vol 15 pp 53ndash60 2015

[39] A Sillem Feasibility study of a tire hydroplaning simulation in afinite element code using a coupled Eulerian-Lagrangian method[MSc thesis] 2008

[40] M Lee R G Longoria and D E Wilson ldquoCavity dynamics inhigh-speed water entryrdquo Physics of Fluids vol 9 no 3 pp 540ndash550 1997

Hindawiwwwhindawicom Volume 2018

Journal of

ChemistryArchaeaHindawiwwwhindawicom Volume 2018

Marine BiologyJournal of

Hindawiwwwhindawicom Volume 2018

BiodiversityInternational Journal of

Hindawiwwwhindawicom Volume 2018

EcologyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2018

Forestry ResearchInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Environmental and Public Health

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Microbiology

Hindawiwwwhindawicom Volume 2018

Public Health Advances in

AgricultureAdvances in

Hindawiwwwhindawicom Volume 2018

Agronomy

Hindawiwwwhindawicom Volume 2018

International Journal of

Hindawiwwwhindawicom Volume 2018

MeteorologyAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

ScienticaHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Geological ResearchJournal of

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

Submit your manuscripts atwwwhindawicom

14 Geofluids

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Velo

city

(ms

)

01 02 03 040Time (s)

CubeCylinderSphere

PyramidCone

Figure 11 Velocity-time graph of pounders with different geometricshapes

it has the most and least influence on the cubical and conicalpounders respectively

8 Critical Drop Height of the Pounder

In the dynamic compaction of the seabed determining thecritical drop height of the pounder from the freewater surfaceis an effective parameter for the pounder to reach maximumdeepwater velocity After dropping the pounder from height119867 above the free water surface with an initial velocity (1198810)of 0ms its velocity gradually increases until it reaches thesecondary velocity (119881) at the moment of water impact Afterentering the water pounder velocity is decreased due to waterdrag force According to the free fall equation for objectsincreasing the drop height from the waterrsquos surface leads toincreased pounder velocity at the moment of water surfaceimpact but it does not necessarily lead to an increase inpounder velocity during seabed impact because the criticaldrop height of the pounder is a determining factor ofits maximum deepwater velocity According to Figure 12 aconical pounder with a mass of 411 kg was dropped fromdifferent heights (119867) and its velocity when impacting thewaters was measured according to the time independentequation for the free fall of objects Table 4 illustrates thevelocity of the conical pounder with different drop heightswhen impacting the waterrsquos surface

81 Effect of the Drop Height of the Pounder from WaterSurface on Deepwater Displacement Figure 13 illustrates thedisplacement-time graph of the conical pounder with dif-ferent drop heights from the waterrsquos surface As can beseen increasing the drop height of the pounder leads to

Water levelℎ

Figure 12 Schematic of drop height of the conical pounder fromwater surface

Table 4 Velocity of the conical pounder with different drop heightsat the moment of impacting the water surface

Drop height of the pounderfrom water surface (m)

Pounder velocity at themoment of impacting water

surface (ms)025 22105 313075 3831 443125 49515 542175 5852 626225 66425 7

displacement-time graphs that are closer to each other Infact an increase in the drop height of the pounder from thewaterrsquos surface leads to an increase in its secondary velocitywhen impacting the waterrsquos surface Hence the pounderenters the water with a higher velocity and reaches the modelbed in less time

82 Effect of the Drop Height of the Pounder from WaterSurface on Its Deepwater Velocity Increasing the drop heightof the pounder from the waterrsquos surface leads to an increasein its velocity before impacting the water but the poundervelocity gradually decreases after entering the water due tothe its drag force Figure 14 illustrates the velocity-time graphfor the conical pounder with different drop heights from thewaterrsquos surface When the pounder is dropped from a lowheight due to its low velocity it moves deepwater due to itsweight Moreover because of the low velocity of the pounderwhen entering the water pinch-off occurs in less time and atlower depths and its velocity increases after pinch-off until itreaches terminal velocity Increases in pounder drop heightsincrease water entry velocity Hence it will have greatervelocity depreciation in the first moments of water entry

Figure 15 illustrates the terminal velocity-time graph forthe conical pounderwith different drop heightswhen impact-ing the model bed As can be seen increasing the drop height

Geofluids 15

01 02 03 040Time (s)

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

H = 1Ｇ

H = 075Ｇ

H = 05 Ｇ

H = 025 Ｇ

H = 175Ｇ

H = 15 Ｇ

H = 125 Ｇ

H = 25 Ｇ

H = 225 Ｇ

H = 2Ｇ

Figure 13 Displacement-time graph of the conical pounder withdifferent drop heights

of the pounder from the waterrsquos surface leads to an increasein its terminal velocity when impacting the model bedTheseconditions are dominant as long as the pounder reaches acritical drop height from the water surface but increasingthe pounder drop height higher than the critical drop heightleads to decreased terminal velocity when impacting themodel bed In this numerical simulation after dropping thepounder from a height of 2m above the waterrsquos surface itssecondary velocity was 626ms when impacting the watersurface After entering the water the secondary velocity ofthe pounder decreased until it impacts the model bed witha terminal velocity of 454ms Therefore a drop height of2m from above the waterrsquos surface has the highest terminalvelocity when the pounder impacts themodel bed Increasingthe drop height bymore than 2m leads to decreased pounderterminal velocity when impacting the model bed The heightwhich has the greatest effect on pounder terminal velocitywhen impacting the model bed is called the critical dropheight Increasing the drop height of the pounder to greaterthan its critical drop height leads to decreased terminalvelocity when impacting the model bed

Figure 16 illustrates the secondary and terminal velocityof the conical pounder with different drop heights As canbe observed increasing the drop height of the pounder fromwaterrsquos surface leads to increased differences between thesecondary and terminal velocities of the pounder Increasingthe drop height to more than the critical drop height leadsto increased pounder secondary velocity when impactingthe waterrsquos surface and decreased terminal velocity whenimpacting the model bed Hence the difference betweenthe secondary and terminal velocities of the pounder is

01 02 03 040Time (s)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)H = 175Ｇ

H = 125 Ｇ

H = 075Ｇ

H = 025 Ｇ H = 225 Ｇ

H = 1Ｇ

H = 05 Ｇ

H = 25 Ｇ

H = 2Ｇ

H = 15 Ｇ

Figure 14 Velocity-time graph of the conical pounder with differentdrop heights

minus430

minus447 minus454minus415

minus426 minus380

minus367minus355

minus332minus310

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Term

inal

velo

city

(ms

)

01 02 03 040Time (s)

Figure 15 Terminal velocity of conical pounder with different dropheights versus time

not reasonable and a significant amount of the pounderssecondary velocity is depreciated when entering the waterThe great difference between the secondary and terminalvelocities of the pounder reveals the necessity of determiningthe critical drop height of the pounder before dynamic seabedcompaction operations Increasing the pounder drop heightto be greater than the critical drop height not only hasa reverse result on pounder velocity when impacting theseabed but it also requires special equipment such as lengthy

16 Geofluids

Secondary velocityTerminal velocity

025 05 075 1 125 15 175 2 225 250Drop height (m)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)

Figure 16 Secondary and terminal velocities of conical pounderversus drop heights

1 2 3 4 5 6 7 80Impact velocity (ms)

0

005

01

015

02

025

03

Pinc

h-off

tim

e (s)

Figure 17 Pinch-off time versus impact velocity of conical pounder

cranes and huge barges leading to increased implementationcosts

Figure 17 illustrates pinch-off time based on the sec-ondary velocity of the pounder at the moment of watersurface impact As can be seen pinch-off time is a weakfunction of pounder impact velocity and increasing impactvelocity leads to a slight reduction in pinch-off time Thisbehaviour was also observed in the analytical and numericalresults found by Lee et al [40]

Figure 18 illustrates pinch-off depth based on the sec-ondary velocity of pounder at the moment of water surfaceimpact As can be seen the pinch-off depth is a linear

1 2 3 4 5 6 7 80Impact velocity (ms)

minus12

minus1

minus08

minus06

minus04

minus02

0

Pinc

h-off

dep

th (m

)

Figure 18 Pinch-off depth versus impact velocity of conicalpounder

function of pounder impact velocity These results are in linewith the analytical equations of Lee et al [40]

9 Conclusions

In this study the water entry problem of three-dimensionalpounders with different geometric shapes of cube cylindersphere pyramid and cone was numerically simulated bycommercial finite element code Abaqus 614-2 An explicitdynamic analysis method was employed to model the fluid-structure interaction using a Coupled Eulerian-Lagrangian(CEL) method Before applying the model to water entryof the pounders a three-dimensional numerical model wasdeveloped to simulate a transient free-falling wedge impact-ing the free surface of the water in order to calibrate theadopted computationalmodelThe accuracy of the numericalmodel was examined by comparing numerical results withthe experimental data of Aristoff et al The good agreementbetween numerical simulation results with the theoreticaland experimental data of Aristoff et al demonstrates thecapability of the employed numerical methodology to sim-ulate the water entry problem for three-dimensional geome-tries with acceptable accuracy After water entry by a pounderdropped fromaheight of 1m a symmetric air cavity is formedbehind the pounder As the pounder moves downward inwater depth it imposes a force onto surrounding fluid inthe radial directions and transfers its momentum to thefluid This extension is faced with the hydrostatic pressureresistance of the fluid causing the direction of the radialflow to be reversed leading to air cavity contraction andcollapse The results show that the water drag force had thehighest and lowest effect on the cubical and conical poundersrespectively and increasing the drop height to greater thanthe critical drop height leads to increased pounder velocitywhen impacting the waterrsquos surface and decreased velocitywhen impacting the model bed Pinch-off time is a very weak

Geofluids 17

function of pounder velocity when impacting the waterrsquossurface but the pinch-off depth increases linearly along withincreased pounder impact velocities It is recommended thatthe critical drop height for the pounder is determined beforedynamic seabed compaction operations and the adjustmentof crane heights to this height In addition it is also recom-mended that perforated pounders be used in order to reducevelocity depreciation when impacting the waterrsquos surface andto reduce the effect of water drag force

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors warmly appreciate the cooperation and guidanceof Dr B Hamidi and Dr A Mehrabadi in this study

References

[1] B Hamidi J-M Debats H Nikraz and S Varaksin ldquoOffshoreground improvement recordsrdquo Australian Geomechanics Jour-nal vol 48 no 4 pp 111ndash122 2013

[2] SWatanabe ldquoResistance of impact onwater surface part I-cone Institute of Physical andChemical ResearchrdquoTokyo vol 12 pp251ndash267 1930

[3] S Watanabe ldquoResistance of impact on water surface part II-cone (continued)rdquo Institute of Physical and Chemical Researchvol 14 pp 153ndash168 1930

[4] A May and J C Woodhull ldquoDrag coefficients of steel spheresentering water verticallyrdquo Journal of Applied Physics vol 19 no12 pp 1109ndash1121 1948

[5] A May ldquoEffect of surface condition of a sphere on its water-entry cavityrdquo Journal of Applied Physics vol 22 no 10 pp 1219ndash1222 1951

[6] AMay ldquoVertical entry ofmissiles into waterrdquo Journal of AppliedPhysics vol 23 no 12 pp 1362ndash1372 1952

[7] A P New T S Lee and H T Low ldquoImpact loading and waterentrance characteristics of prismatic bodiesrdquo in Proceedings ofthe third international offshore and polar engineering conferencepp 282ndash287 1993

[8] M Anghileri and A Spizzica ldquoExperimental validation of finiteelement models for water impactsrdquo in Proceedings of the secondinternational crash usersrsquo seminar Cranfield Impact Centre LtdEngland UK 1995 second international crash users seminar

[9] A Engle andR Lewis ldquoA comparison of hydrodynamic impactsprediction methods with two dimensional drop test datardquoMarine Structures vol 16 no 2 pp 175ndash182 2003

[10] J M Aristoff T T Truscott A H Techet and J W M BushldquoThe water entry of decelerating spheresrdquo Physics of Fluids vol22 no 3 Article ID 009002PHF pp 1ndash8 2010

[11] T Von Karman The impact of seaplane floats during landingNational Advisory Committee for Aeronautics NACA TN no321 1929

[12] HWagner ldquoPhenomena associatedwith impacts and sliding onliquid surfacesrdquo Journal of Applied Mathematics and Mechanicsvol 12 pp 193ndash215 1932

[13] W S Vorus ldquoA flat cylinder theory for vessel impact and steadyplaning resistancerdquo Journal of Ship Research vol 40 no 2 pp89ndash106 1996

[14] T Miloh ldquoOn the initial-stage slamming of a rigid sphere in avertical water entryrdquo Applied Ocean Research vol 13 no 1 pp13ndash43 1991

[15] J M Aristoff and JW Bush ldquoWater entry of small hydrophobicspheresrdquo Journal of Fluid Mechanics vol 619 pp 45ndash78 2009

[16] P Ghadimi A Dashtimanesh and S R Djeddi ldquoStudy of waterentry of circular cylinder by using analytical and numericalsolutionsrdquo Journal of the Brazilian Society ofMechanical Sciencesamp Engineering vol 34 no 3 pp 225ndash232 2012

[17] A TassinD J PiroAAKorobkinK JMaki andM J CookerldquoTwo-dimensional water entry and exit of a body whose shapevaries in timerdquo Journal of Fluids and Structures vol 40 pp 317ndash336 2013

[18] E Yao H Wang L Pan X Wang and R Woding ldquoVerticalwater-entry of bullet-shaped projectilesrdquo Journal of AppliedMathematics and Physics vol 02 no 06 pp 323ndash334 2014

[19] M-S Park Y-R Jung and W-G Park ldquoNumerical study ofimpact force and ricochet behavior of high speed water-entrybodiesrdquo Computers amp Fluids vol 32 no 7 pp 939ndash951 2003

[20] D Battistin and A Iafrati ldquoHydrodynamic loads during waterenergy of two-dimensional and axisymmetric bodiesrdquo Journalof Fluids and Structures vol 17 no 5 pp 643ndash664 2003

[21] A A Korobkin and M Ohkusu ldquoImpact of two circular platesone of which is floating on a thin layer of liquidrdquo Journal ofEngineering Mathematics vol 50 no 4 pp 343ndash358 2004

[22] K M Kleefsman G Fekken A E Veldman B Iwanowski andB Buchner ldquoA volume-of-fluid based simulation method forwave impact problemsrdquo Journal of Computational Physics vol206 no 1 pp 363ndash393 2005

[23] Y Kim Y Kim Y Liu and D K P Yue ldquoOn the water-entryimpact problem of asymmetric bodiesrdquo in Proceedings of the9th International Conference on Numerical Ship Hydrodynamics(NSH rsquo07) August 2007

[24] Q Yang and W Qiu ldquoNumerical solution of 2D slammingproblem with a CIPmethodrdquo in Proceedings of the InternationalConference on Violent Flows Research Institute for AppliedMechanics Kyushu University Japan 2007

[25] A C Fairlie-Clarke and T Tveitnes ldquoMomentum and gravityeffects during the constant velocitywater entry ofwedge-shapedsectionsrdquo Ocean Engineering vol 35 no 7 pp 706ndash716 2007

[26] Q Yang andW Qiu ldquoNumerical simulation of water impact for2D and 3D bodiesrdquo Ocean Engineering vol 43 pp 82ndash89 2012

[27] G Wu ldquoNumerical simulation for water entry of a wedge atvarying speed by a high order boundary element methodrdquoJournal of Marine Science and Application vol 11 no 2 pp 143ndash149 2012

[28] M Ahmadzadeh B Saranjam A Hoseini Fard and A RBinesh ldquoNumerical simulation of sphere water entry prob-lem using Eulerian-Lagrangian methodrdquo Applied MathematicalModelling vol 38 no 5-6 pp 1673ndash1684 2014

[29] V-T Nguyen D-T Vu W-G Park and Y-R Jung ldquoNumericalanalysis of water impact forces using a dual-time pseudo-compressibility method and volume-of-fluid interface trackingalgorithmrdquo Computers amp Fluids vol 103 pp 18ndash33 2014

[30] S Kim and N Kim ldquoIntegrated dynamics modeling forsupercavitating vehicle systemsrdquo International Journal of NavalArchitecture and Ocean Engineering vol 7 no 2 pp 346ndash3632015

18 Geofluids

[31] V-T Nguyen D-T Vu W-G Park and C-M Jung ldquoNavier-Stokes solver for water entry bodies with moving Chimera gridmethod in 6DOF motionsrdquo Computers amp Fluids vol 140 pp19ndash38 2016

[32] M Mirzaei M Eghtesad and M Alishahi ldquoPlaning forceidentification in high-speed underwater vehiclesrdquo Journal ofVibration and Control vol 22 no 20 pp 4176ndash4191 2016

[33] G Qiu S Henke and J Grabe ldquoApplication of a CoupledEulerian-Lagrangian approach on geomechanical problemsinvolving large deformationsrdquoComputersampGeosciences vol 38no 1 pp 30ndash39 2011

[34] M R Erfanian M Anbarsooz N Rahimi M Zare andM Moghiman ldquoNumerical and experimental investigationof a three dimensional spherical-nose projectile water entryproblemrdquo Ocean Engineering vol 104 pp 397ndash404 2015

[35] W M Isbell F H Shipman and A H Jones ldquoHugoniotequation of state measurements for eleven material to fiveMegabars Materials and Structural Laboratory ManufacturingDevelopments General Motors Technical centerrdquo Tech RepMichigan 1968

[36] M A Jandron R C Hurd J L Belden A F Bower W Fennelland T T Truscott ldquoModeling of hyperelastic water-skippingspheres using AbaqusExplicitrdquo in Proceedings of the SIMULIACommunity Conference 2014

[37] H Forouzani B Saranjam R Kamali and A Abdollahi-far ldquoElasto-plastic time dependent impact analysis of highspeed projectile on water surfacerdquo Journal of Solid and FluidMechanics vol 3 pp 281ndash298 2016

[38] M R Erfanian andMMoghiman ldquoNumerical and experimen-tal investigation of a projectile water entry problem and studyof velocity effect on time and depth of pinch-offrdquo Journal ofModares Mechanical Engineering vol 15 pp 53ndash60 2015

[39] A Sillem Feasibility study of a tire hydroplaning simulation in afinite element code using a coupled Eulerian-Lagrangian method[MSc thesis] 2008

[40] M Lee R G Longoria and D E Wilson ldquoCavity dynamics inhigh-speed water entryrdquo Physics of Fluids vol 9 no 3 pp 540ndash550 1997

Hindawiwwwhindawicom Volume 2018

Journal of

ChemistryArchaeaHindawiwwwhindawicom Volume 2018

Marine BiologyJournal of

Hindawiwwwhindawicom Volume 2018

BiodiversityInternational Journal of

Hindawiwwwhindawicom Volume 2018

EcologyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2018

Forestry ResearchInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Environmental and Public Health

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Microbiology

Hindawiwwwhindawicom Volume 2018

Public Health Advances in

AgricultureAdvances in

Hindawiwwwhindawicom Volume 2018

Agronomy

Hindawiwwwhindawicom Volume 2018

International Journal of

Hindawiwwwhindawicom Volume 2018

MeteorologyAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

ScienticaHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Geological ResearchJournal of

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

Submit your manuscripts atwwwhindawicom

Geofluids 15

01 02 03 040Time (s)

minus12

minus1

minus08

minus06

minus04

minus02

0

Disp

lace

men

t (m

)

H = 1Ｇ

H = 075Ｇ

H = 05 Ｇ

H = 025 Ｇ

H = 175Ｇ

H = 15 Ｇ

H = 125 Ｇ

H = 25 Ｇ

H = 225 Ｇ

H = 2Ｇ

Figure 13 Displacement-time graph of the conical pounder withdifferent drop heights

of the pounder from the waterrsquos surface leads to an increasein its terminal velocity when impacting the model bedTheseconditions are dominant as long as the pounder reaches acritical drop height from the water surface but increasingthe pounder drop height higher than the critical drop heightleads to decreased terminal velocity when impacting themodel bed In this numerical simulation after dropping thepounder from a height of 2m above the waterrsquos surface itssecondary velocity was 626ms when impacting the watersurface After entering the water the secondary velocity ofthe pounder decreased until it impacts the model bed witha terminal velocity of 454ms Therefore a drop height of2m from above the waterrsquos surface has the highest terminalvelocity when the pounder impacts themodel bed Increasingthe drop height bymore than 2m leads to decreased pounderterminal velocity when impacting the model bed The heightwhich has the greatest effect on pounder terminal velocitywhen impacting the model bed is called the critical dropheight Increasing the drop height of the pounder to greaterthan its critical drop height leads to decreased terminalvelocity when impacting the model bed

Figure 16 illustrates the secondary and terminal velocityof the conical pounder with different drop heights As canbe observed increasing the drop height of the pounder fromwaterrsquos surface leads to increased differences between thesecondary and terminal velocities of the pounder Increasingthe drop height to more than the critical drop height leadsto increased pounder secondary velocity when impactingthe waterrsquos surface and decreased terminal velocity whenimpacting the model bed Hence the difference betweenthe secondary and terminal velocities of the pounder is

01 02 03 040Time (s)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)H = 175Ｇ

H = 125 Ｇ

H = 075Ｇ

H = 025 Ｇ H = 225 Ｇ

H = 1Ｇ

H = 05 Ｇ

H = 25 Ｇ

H = 2Ｇ

H = 15 Ｇ

Figure 14 Velocity-time graph of the conical pounder with differentdrop heights

minus430

minus447 minus454minus415

minus426 minus380

minus367minus355

minus332minus310

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Term

inal

velo

city

(ms

)

01 02 03 040Time (s)

Figure 15 Terminal velocity of conical pounder with different dropheights versus time

not reasonable and a significant amount of the pounderssecondary velocity is depreciated when entering the waterThe great difference between the secondary and terminalvelocities of the pounder reveals the necessity of determiningthe critical drop height of the pounder before dynamic seabedcompaction operations Increasing the pounder drop heightto be greater than the critical drop height not only hasa reverse result on pounder velocity when impacting theseabed but it also requires special equipment such as lengthy

16 Geofluids

Secondary velocityTerminal velocity

025 05 075 1 125 15 175 2 225 250Drop height (m)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)

Figure 16 Secondary and terminal velocities of conical pounderversus drop heights

1 2 3 4 5 6 7 80Impact velocity (ms)

0

005

01

015

02

025

03

Pinc

h-off

tim

e (s)

Figure 17 Pinch-off time versus impact velocity of conical pounder

cranes and huge barges leading to increased implementationcosts

Figure 17 illustrates pinch-off time based on the sec-ondary velocity of the pounder at the moment of watersurface impact As can be seen pinch-off time is a weakfunction of pounder impact velocity and increasing impactvelocity leads to a slight reduction in pinch-off time Thisbehaviour was also observed in the analytical and numericalresults found by Lee et al [40]

Figure 18 illustrates pinch-off depth based on the sec-ondary velocity of pounder at the moment of water surfaceimpact As can be seen the pinch-off depth is a linear

1 2 3 4 5 6 7 80Impact velocity (ms)

minus12

minus1

minus08

minus06

minus04

minus02

0

Pinc

h-off

dep

th (m

)

Figure 18 Pinch-off depth versus impact velocity of conicalpounder

function of pounder impact velocity These results are in linewith the analytical equations of Lee et al [40]

9 Conclusions

In this study the water entry problem of three-dimensionalpounders with different geometric shapes of cube cylindersphere pyramid and cone was numerically simulated bycommercial finite element code Abaqus 614-2 An explicitdynamic analysis method was employed to model the fluid-structure interaction using a Coupled Eulerian-Lagrangian(CEL) method Before applying the model to water entryof the pounders a three-dimensional numerical model wasdeveloped to simulate a transient free-falling wedge impact-ing the free surface of the water in order to calibrate theadopted computationalmodelThe accuracy of the numericalmodel was examined by comparing numerical results withthe experimental data of Aristoff et al The good agreementbetween numerical simulation results with the theoreticaland experimental data of Aristoff et al demonstrates thecapability of the employed numerical methodology to sim-ulate the water entry problem for three-dimensional geome-tries with acceptable accuracy After water entry by a pounderdropped fromaheight of 1m a symmetric air cavity is formedbehind the pounder As the pounder moves downward inwater depth it imposes a force onto surrounding fluid inthe radial directions and transfers its momentum to thefluid This extension is faced with the hydrostatic pressureresistance of the fluid causing the direction of the radialflow to be reversed leading to air cavity contraction andcollapse The results show that the water drag force had thehighest and lowest effect on the cubical and conical poundersrespectively and increasing the drop height to greater thanthe critical drop height leads to increased pounder velocitywhen impacting the waterrsquos surface and decreased velocitywhen impacting the model bed Pinch-off time is a very weak

Geofluids 17

function of pounder velocity when impacting the waterrsquossurface but the pinch-off depth increases linearly along withincreased pounder impact velocities It is recommended thatthe critical drop height for the pounder is determined beforedynamic seabed compaction operations and the adjustmentof crane heights to this height In addition it is also recom-mended that perforated pounders be used in order to reducevelocity depreciation when impacting the waterrsquos surface andto reduce the effect of water drag force

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors warmly appreciate the cooperation and guidanceof Dr B Hamidi and Dr A Mehrabadi in this study

References

[1] B Hamidi J-M Debats H Nikraz and S Varaksin ldquoOffshoreground improvement recordsrdquo Australian Geomechanics Jour-nal vol 48 no 4 pp 111ndash122 2013

[2] SWatanabe ldquoResistance of impact onwater surface part I-cone Institute of Physical andChemical ResearchrdquoTokyo vol 12 pp251ndash267 1930

[3] S Watanabe ldquoResistance of impact on water surface part II-cone (continued)rdquo Institute of Physical and Chemical Researchvol 14 pp 153ndash168 1930

[4] A May and J C Woodhull ldquoDrag coefficients of steel spheresentering water verticallyrdquo Journal of Applied Physics vol 19 no12 pp 1109ndash1121 1948

[5] A May ldquoEffect of surface condition of a sphere on its water-entry cavityrdquo Journal of Applied Physics vol 22 no 10 pp 1219ndash1222 1951

[6] AMay ldquoVertical entry ofmissiles into waterrdquo Journal of AppliedPhysics vol 23 no 12 pp 1362ndash1372 1952

[7] A P New T S Lee and H T Low ldquoImpact loading and waterentrance characteristics of prismatic bodiesrdquo in Proceedings ofthe third international offshore and polar engineering conferencepp 282ndash287 1993

[8] M Anghileri and A Spizzica ldquoExperimental validation of finiteelement models for water impactsrdquo in Proceedings of the secondinternational crash usersrsquo seminar Cranfield Impact Centre LtdEngland UK 1995 second international crash users seminar

[9] A Engle andR Lewis ldquoA comparison of hydrodynamic impactsprediction methods with two dimensional drop test datardquoMarine Structures vol 16 no 2 pp 175ndash182 2003

[10] J M Aristoff T T Truscott A H Techet and J W M BushldquoThe water entry of decelerating spheresrdquo Physics of Fluids vol22 no 3 Article ID 009002PHF pp 1ndash8 2010

[11] T Von Karman The impact of seaplane floats during landingNational Advisory Committee for Aeronautics NACA TN no321 1929

[12] HWagner ldquoPhenomena associatedwith impacts and sliding onliquid surfacesrdquo Journal of Applied Mathematics and Mechanicsvol 12 pp 193ndash215 1932

[13] W S Vorus ldquoA flat cylinder theory for vessel impact and steadyplaning resistancerdquo Journal of Ship Research vol 40 no 2 pp89ndash106 1996

[14] T Miloh ldquoOn the initial-stage slamming of a rigid sphere in avertical water entryrdquo Applied Ocean Research vol 13 no 1 pp13ndash43 1991

[15] J M Aristoff and JW Bush ldquoWater entry of small hydrophobicspheresrdquo Journal of Fluid Mechanics vol 619 pp 45ndash78 2009

[16] P Ghadimi A Dashtimanesh and S R Djeddi ldquoStudy of waterentry of circular cylinder by using analytical and numericalsolutionsrdquo Journal of the Brazilian Society ofMechanical Sciencesamp Engineering vol 34 no 3 pp 225ndash232 2012

[17] A TassinD J PiroAAKorobkinK JMaki andM J CookerldquoTwo-dimensional water entry and exit of a body whose shapevaries in timerdquo Journal of Fluids and Structures vol 40 pp 317ndash336 2013

[18] E Yao H Wang L Pan X Wang and R Woding ldquoVerticalwater-entry of bullet-shaped projectilesrdquo Journal of AppliedMathematics and Physics vol 02 no 06 pp 323ndash334 2014

[19] M-S Park Y-R Jung and W-G Park ldquoNumerical study ofimpact force and ricochet behavior of high speed water-entrybodiesrdquo Computers amp Fluids vol 32 no 7 pp 939ndash951 2003

[20] D Battistin and A Iafrati ldquoHydrodynamic loads during waterenergy of two-dimensional and axisymmetric bodiesrdquo Journalof Fluids and Structures vol 17 no 5 pp 643ndash664 2003

[21] A A Korobkin and M Ohkusu ldquoImpact of two circular platesone of which is floating on a thin layer of liquidrdquo Journal ofEngineering Mathematics vol 50 no 4 pp 343ndash358 2004

[22] K M Kleefsman G Fekken A E Veldman B Iwanowski andB Buchner ldquoA volume-of-fluid based simulation method forwave impact problemsrdquo Journal of Computational Physics vol206 no 1 pp 363ndash393 2005

[23] Y Kim Y Kim Y Liu and D K P Yue ldquoOn the water-entryimpact problem of asymmetric bodiesrdquo in Proceedings of the9th International Conference on Numerical Ship Hydrodynamics(NSH rsquo07) August 2007

[24] Q Yang and W Qiu ldquoNumerical solution of 2D slammingproblem with a CIPmethodrdquo in Proceedings of the InternationalConference on Violent Flows Research Institute for AppliedMechanics Kyushu University Japan 2007

[25] A C Fairlie-Clarke and T Tveitnes ldquoMomentum and gravityeffects during the constant velocitywater entry ofwedge-shapedsectionsrdquo Ocean Engineering vol 35 no 7 pp 706ndash716 2007

[26] Q Yang andW Qiu ldquoNumerical simulation of water impact for2D and 3D bodiesrdquo Ocean Engineering vol 43 pp 82ndash89 2012

[27] G Wu ldquoNumerical simulation for water entry of a wedge atvarying speed by a high order boundary element methodrdquoJournal of Marine Science and Application vol 11 no 2 pp 143ndash149 2012

[28] M Ahmadzadeh B Saranjam A Hoseini Fard and A RBinesh ldquoNumerical simulation of sphere water entry prob-lem using Eulerian-Lagrangian methodrdquo Applied MathematicalModelling vol 38 no 5-6 pp 1673ndash1684 2014

[29] V-T Nguyen D-T Vu W-G Park and Y-R Jung ldquoNumericalanalysis of water impact forces using a dual-time pseudo-compressibility method and volume-of-fluid interface trackingalgorithmrdquo Computers amp Fluids vol 103 pp 18ndash33 2014

[30] S Kim and N Kim ldquoIntegrated dynamics modeling forsupercavitating vehicle systemsrdquo International Journal of NavalArchitecture and Ocean Engineering vol 7 no 2 pp 346ndash3632015

18 Geofluids

[31] V-T Nguyen D-T Vu W-G Park and C-M Jung ldquoNavier-Stokes solver for water entry bodies with moving Chimera gridmethod in 6DOF motionsrdquo Computers amp Fluids vol 140 pp19ndash38 2016

[32] M Mirzaei M Eghtesad and M Alishahi ldquoPlaning forceidentification in high-speed underwater vehiclesrdquo Journal ofVibration and Control vol 22 no 20 pp 4176ndash4191 2016

[33] G Qiu S Henke and J Grabe ldquoApplication of a CoupledEulerian-Lagrangian approach on geomechanical problemsinvolving large deformationsrdquoComputersampGeosciences vol 38no 1 pp 30ndash39 2011

[34] M R Erfanian M Anbarsooz N Rahimi M Zare andM Moghiman ldquoNumerical and experimental investigationof a three dimensional spherical-nose projectile water entryproblemrdquo Ocean Engineering vol 104 pp 397ndash404 2015

[35] W M Isbell F H Shipman and A H Jones ldquoHugoniotequation of state measurements for eleven material to fiveMegabars Materials and Structural Laboratory ManufacturingDevelopments General Motors Technical centerrdquo Tech RepMichigan 1968

[36] M A Jandron R C Hurd J L Belden A F Bower W Fennelland T T Truscott ldquoModeling of hyperelastic water-skippingspheres using AbaqusExplicitrdquo in Proceedings of the SIMULIACommunity Conference 2014

[37] H Forouzani B Saranjam R Kamali and A Abdollahi-far ldquoElasto-plastic time dependent impact analysis of highspeed projectile on water surfacerdquo Journal of Solid and FluidMechanics vol 3 pp 281ndash298 2016

[38] M R Erfanian andMMoghiman ldquoNumerical and experimen-tal investigation of a projectile water entry problem and studyof velocity effect on time and depth of pinch-offrdquo Journal ofModares Mechanical Engineering vol 15 pp 53ndash60 2015

[39] A Sillem Feasibility study of a tire hydroplaning simulation in afinite element code using a coupled Eulerian-Lagrangian method[MSc thesis] 2008

[40] M Lee R G Longoria and D E Wilson ldquoCavity dynamics inhigh-speed water entryrdquo Physics of Fluids vol 9 no 3 pp 540ndash550 1997

Hindawiwwwhindawicom Volume 2018

Journal of

ChemistryArchaeaHindawiwwwhindawicom Volume 2018

Marine BiologyJournal of

Hindawiwwwhindawicom Volume 2018

BiodiversityInternational Journal of

Hindawiwwwhindawicom Volume 2018

EcologyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2018

Forestry ResearchInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Environmental and Public Health

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Microbiology

Hindawiwwwhindawicom Volume 2018

Public Health Advances in

AgricultureAdvances in

Hindawiwwwhindawicom Volume 2018

Agronomy

Hindawiwwwhindawicom Volume 2018

International Journal of

Hindawiwwwhindawicom Volume 2018

MeteorologyAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

ScienticaHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Geological ResearchJournal of

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

Submit your manuscripts atwwwhindawicom

16 Geofluids

Secondary velocityTerminal velocity

025 05 075 1 125 15 175 2 225 250Drop height (m)

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Velo

city

(ms

)

Figure 16 Secondary and terminal velocities of conical pounderversus drop heights

1 2 3 4 5 6 7 80Impact velocity (ms)

0

005

01

015

02

025

03

Pinc

h-off

tim

e (s)

Figure 17 Pinch-off time versus impact velocity of conical pounder

cranes and huge barges leading to increased implementationcosts

Figure 17 illustrates pinch-off time based on the sec-ondary velocity of the pounder at the moment of watersurface impact As can be seen pinch-off time is a weakfunction of pounder impact velocity and increasing impactvelocity leads to a slight reduction in pinch-off time Thisbehaviour was also observed in the analytical and numericalresults found by Lee et al [40]

Figure 18 illustrates pinch-off depth based on the sec-ondary velocity of pounder at the moment of water surfaceimpact As can be seen the pinch-off depth is a linear

1 2 3 4 5 6 7 80Impact velocity (ms)

minus12

minus1

minus08

minus06

minus04

minus02

0

Pinc

h-off

dep

th (m

)

Figure 18 Pinch-off depth versus impact velocity of conicalpounder

function of pounder impact velocity These results are in linewith the analytical equations of Lee et al [40]

9 Conclusions

In this study the water entry problem of three-dimensionalpounders with different geometric shapes of cube cylindersphere pyramid and cone was numerically simulated bycommercial finite element code Abaqus 614-2 An explicitdynamic analysis method was employed to model the fluid-structure interaction using a Coupled Eulerian-Lagrangian(CEL) method Before applying the model to water entryof the pounders a three-dimensional numerical model wasdeveloped to simulate a transient free-falling wedge impact-ing the free surface of the water in order to calibrate theadopted computationalmodelThe accuracy of the numericalmodel was examined by comparing numerical results withthe experimental data of Aristoff et al The good agreementbetween numerical simulation results with the theoreticaland experimental data of Aristoff et al demonstrates thecapability of the employed numerical methodology to sim-ulate the water entry problem for three-dimensional geome-tries with acceptable accuracy After water entry by a pounderdropped fromaheight of 1m a symmetric air cavity is formedbehind the pounder As the pounder moves downward inwater depth it imposes a force onto surrounding fluid inthe radial directions and transfers its momentum to thefluid This extension is faced with the hydrostatic pressureresistance of the fluid causing the direction of the radialflow to be reversed leading to air cavity contraction andcollapse The results show that the water drag force had thehighest and lowest effect on the cubical and conical poundersrespectively and increasing the drop height to greater thanthe critical drop height leads to increased pounder velocitywhen impacting the waterrsquos surface and decreased velocitywhen impacting the model bed Pinch-off time is a very weak

Geofluids 17

function of pounder velocity when impacting the waterrsquossurface but the pinch-off depth increases linearly along withincreased pounder impact velocities It is recommended thatthe critical drop height for the pounder is determined beforedynamic seabed compaction operations and the adjustmentof crane heights to this height In addition it is also recom-mended that perforated pounders be used in order to reducevelocity depreciation when impacting the waterrsquos surface andto reduce the effect of water drag force

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors warmly appreciate the cooperation and guidanceof Dr B Hamidi and Dr A Mehrabadi in this study

References

[1] B Hamidi J-M Debats H Nikraz and S Varaksin ldquoOffshoreground improvement recordsrdquo Australian Geomechanics Jour-nal vol 48 no 4 pp 111ndash122 2013

[2] SWatanabe ldquoResistance of impact onwater surface part I-cone Institute of Physical andChemical ResearchrdquoTokyo vol 12 pp251ndash267 1930

[3] S Watanabe ldquoResistance of impact on water surface part II-cone (continued)rdquo Institute of Physical and Chemical Researchvol 14 pp 153ndash168 1930

[4] A May and J C Woodhull ldquoDrag coefficients of steel spheresentering water verticallyrdquo Journal of Applied Physics vol 19 no12 pp 1109ndash1121 1948

[5] A May ldquoEffect of surface condition of a sphere on its water-entry cavityrdquo Journal of Applied Physics vol 22 no 10 pp 1219ndash1222 1951

[6] AMay ldquoVertical entry ofmissiles into waterrdquo Journal of AppliedPhysics vol 23 no 12 pp 1362ndash1372 1952

[7] A P New T S Lee and H T Low ldquoImpact loading and waterentrance characteristics of prismatic bodiesrdquo in Proceedings ofthe third international offshore and polar engineering conferencepp 282ndash287 1993

[8] M Anghileri and A Spizzica ldquoExperimental validation of finiteelement models for water impactsrdquo in Proceedings of the secondinternational crash usersrsquo seminar Cranfield Impact Centre LtdEngland UK 1995 second international crash users seminar

[9] A Engle andR Lewis ldquoA comparison of hydrodynamic impactsprediction methods with two dimensional drop test datardquoMarine Structures vol 16 no 2 pp 175ndash182 2003

[10] J M Aristoff T T Truscott A H Techet and J W M BushldquoThe water entry of decelerating spheresrdquo Physics of Fluids vol22 no 3 Article ID 009002PHF pp 1ndash8 2010

[11] T Von Karman The impact of seaplane floats during landingNational Advisory Committee for Aeronautics NACA TN no321 1929

[12] HWagner ldquoPhenomena associatedwith impacts and sliding onliquid surfacesrdquo Journal of Applied Mathematics and Mechanicsvol 12 pp 193ndash215 1932

[13] W S Vorus ldquoA flat cylinder theory for vessel impact and steadyplaning resistancerdquo Journal of Ship Research vol 40 no 2 pp89ndash106 1996

[14] T Miloh ldquoOn the initial-stage slamming of a rigid sphere in avertical water entryrdquo Applied Ocean Research vol 13 no 1 pp13ndash43 1991

[15] J M Aristoff and JW Bush ldquoWater entry of small hydrophobicspheresrdquo Journal of Fluid Mechanics vol 619 pp 45ndash78 2009

[16] P Ghadimi A Dashtimanesh and S R Djeddi ldquoStudy of waterentry of circular cylinder by using analytical and numericalsolutionsrdquo Journal of the Brazilian Society ofMechanical Sciencesamp Engineering vol 34 no 3 pp 225ndash232 2012

[17] A TassinD J PiroAAKorobkinK JMaki andM J CookerldquoTwo-dimensional water entry and exit of a body whose shapevaries in timerdquo Journal of Fluids and Structures vol 40 pp 317ndash336 2013

[18] E Yao H Wang L Pan X Wang and R Woding ldquoVerticalwater-entry of bullet-shaped projectilesrdquo Journal of AppliedMathematics and Physics vol 02 no 06 pp 323ndash334 2014

[19] M-S Park Y-R Jung and W-G Park ldquoNumerical study ofimpact force and ricochet behavior of high speed water-entrybodiesrdquo Computers amp Fluids vol 32 no 7 pp 939ndash951 2003

[20] D Battistin and A Iafrati ldquoHydrodynamic loads during waterenergy of two-dimensional and axisymmetric bodiesrdquo Journalof Fluids and Structures vol 17 no 5 pp 643ndash664 2003

[21] A A Korobkin and M Ohkusu ldquoImpact of two circular platesone of which is floating on a thin layer of liquidrdquo Journal ofEngineering Mathematics vol 50 no 4 pp 343ndash358 2004

[22] K M Kleefsman G Fekken A E Veldman B Iwanowski andB Buchner ldquoA volume-of-fluid based simulation method forwave impact problemsrdquo Journal of Computational Physics vol206 no 1 pp 363ndash393 2005

[23] Y Kim Y Kim Y Liu and D K P Yue ldquoOn the water-entryimpact problem of asymmetric bodiesrdquo in Proceedings of the9th International Conference on Numerical Ship Hydrodynamics(NSH rsquo07) August 2007

[24] Q Yang and W Qiu ldquoNumerical solution of 2D slammingproblem with a CIPmethodrdquo in Proceedings of the InternationalConference on Violent Flows Research Institute for AppliedMechanics Kyushu University Japan 2007

[25] A C Fairlie-Clarke and T Tveitnes ldquoMomentum and gravityeffects during the constant velocitywater entry ofwedge-shapedsectionsrdquo Ocean Engineering vol 35 no 7 pp 706ndash716 2007

[26] Q Yang andW Qiu ldquoNumerical simulation of water impact for2D and 3D bodiesrdquo Ocean Engineering vol 43 pp 82ndash89 2012

[27] G Wu ldquoNumerical simulation for water entry of a wedge atvarying speed by a high order boundary element methodrdquoJournal of Marine Science and Application vol 11 no 2 pp 143ndash149 2012

[28] M Ahmadzadeh B Saranjam A Hoseini Fard and A RBinesh ldquoNumerical simulation of sphere water entry prob-lem using Eulerian-Lagrangian methodrdquo Applied MathematicalModelling vol 38 no 5-6 pp 1673ndash1684 2014

[29] V-T Nguyen D-T Vu W-G Park and Y-R Jung ldquoNumericalanalysis of water impact forces using a dual-time pseudo-compressibility method and volume-of-fluid interface trackingalgorithmrdquo Computers amp Fluids vol 103 pp 18ndash33 2014

[30] S Kim and N Kim ldquoIntegrated dynamics modeling forsupercavitating vehicle systemsrdquo International Journal of NavalArchitecture and Ocean Engineering vol 7 no 2 pp 346ndash3632015

18 Geofluids

[31] V-T Nguyen D-T Vu W-G Park and C-M Jung ldquoNavier-Stokes solver for water entry bodies with moving Chimera gridmethod in 6DOF motionsrdquo Computers amp Fluids vol 140 pp19ndash38 2016

[32] M Mirzaei M Eghtesad and M Alishahi ldquoPlaning forceidentification in high-speed underwater vehiclesrdquo Journal ofVibration and Control vol 22 no 20 pp 4176ndash4191 2016

[33] G Qiu S Henke and J Grabe ldquoApplication of a CoupledEulerian-Lagrangian approach on geomechanical problemsinvolving large deformationsrdquoComputersampGeosciences vol 38no 1 pp 30ndash39 2011

[34] M R Erfanian M Anbarsooz N Rahimi M Zare andM Moghiman ldquoNumerical and experimental investigationof a three dimensional spherical-nose projectile water entryproblemrdquo Ocean Engineering vol 104 pp 397ndash404 2015

[35] W M Isbell F H Shipman and A H Jones ldquoHugoniotequation of state measurements for eleven material to fiveMegabars Materials and Structural Laboratory ManufacturingDevelopments General Motors Technical centerrdquo Tech RepMichigan 1968

[36] M A Jandron R C Hurd J L Belden A F Bower W Fennelland T T Truscott ldquoModeling of hyperelastic water-skippingspheres using AbaqusExplicitrdquo in Proceedings of the SIMULIACommunity Conference 2014

[37] H Forouzani B Saranjam R Kamali and A Abdollahi-far ldquoElasto-plastic time dependent impact analysis of highspeed projectile on water surfacerdquo Journal of Solid and FluidMechanics vol 3 pp 281ndash298 2016

[38] M R Erfanian andMMoghiman ldquoNumerical and experimen-tal investigation of a projectile water entry problem and studyof velocity effect on time and depth of pinch-offrdquo Journal ofModares Mechanical Engineering vol 15 pp 53ndash60 2015

[39] A Sillem Feasibility study of a tire hydroplaning simulation in afinite element code using a coupled Eulerian-Lagrangian method[MSc thesis] 2008

[40] M Lee R G Longoria and D E Wilson ldquoCavity dynamics inhigh-speed water entryrdquo Physics of Fluids vol 9 no 3 pp 540ndash550 1997

Hindawiwwwhindawicom Volume 2018

Journal of

ChemistryArchaeaHindawiwwwhindawicom Volume 2018

Marine BiologyJournal of

Hindawiwwwhindawicom Volume 2018

BiodiversityInternational Journal of

Hindawiwwwhindawicom Volume 2018

EcologyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2018

Forestry ResearchInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Environmental and Public Health

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Microbiology

Hindawiwwwhindawicom Volume 2018

Public Health Advances in

AgricultureAdvances in

Hindawiwwwhindawicom Volume 2018

Agronomy

Hindawiwwwhindawicom Volume 2018

International Journal of

Hindawiwwwhindawicom Volume 2018

MeteorologyAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

ScienticaHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Geological ResearchJournal of

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

Submit your manuscripts atwwwhindawicom

Geofluids 17

function of pounder velocity when impacting the waterrsquossurface but the pinch-off depth increases linearly along withincreased pounder impact velocities It is recommended thatthe critical drop height for the pounder is determined beforedynamic seabed compaction operations and the adjustmentof crane heights to this height In addition it is also recom-mended that perforated pounders be used in order to reducevelocity depreciation when impacting the waterrsquos surface andto reduce the effect of water drag force

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors warmly appreciate the cooperation and guidanceof Dr B Hamidi and Dr A Mehrabadi in this study

References

[1] B Hamidi J-M Debats H Nikraz and S Varaksin ldquoOffshoreground improvement recordsrdquo Australian Geomechanics Jour-nal vol 48 no 4 pp 111ndash122 2013

[2] SWatanabe ldquoResistance of impact onwater surface part I-cone Institute of Physical andChemical ResearchrdquoTokyo vol 12 pp251ndash267 1930

[3] S Watanabe ldquoResistance of impact on water surface part II-cone (continued)rdquo Institute of Physical and Chemical Researchvol 14 pp 153ndash168 1930

[4] A May and J C Woodhull ldquoDrag coefficients of steel spheresentering water verticallyrdquo Journal of Applied Physics vol 19 no12 pp 1109ndash1121 1948

[5] A May ldquoEffect of surface condition of a sphere on its water-entry cavityrdquo Journal of Applied Physics vol 22 no 10 pp 1219ndash1222 1951

[6] AMay ldquoVertical entry ofmissiles into waterrdquo Journal of AppliedPhysics vol 23 no 12 pp 1362ndash1372 1952

[7] A P New T S Lee and H T Low ldquoImpact loading and waterentrance characteristics of prismatic bodiesrdquo in Proceedings ofthe third international offshore and polar engineering conferencepp 282ndash287 1993

[8] M Anghileri and A Spizzica ldquoExperimental validation of finiteelement models for water impactsrdquo in Proceedings of the secondinternational crash usersrsquo seminar Cranfield Impact Centre LtdEngland UK 1995 second international crash users seminar

[9] A Engle andR Lewis ldquoA comparison of hydrodynamic impactsprediction methods with two dimensional drop test datardquoMarine Structures vol 16 no 2 pp 175ndash182 2003

[10] J M Aristoff T T Truscott A H Techet and J W M BushldquoThe water entry of decelerating spheresrdquo Physics of Fluids vol22 no 3 Article ID 009002PHF pp 1ndash8 2010

[11] T Von Karman The impact of seaplane floats during landingNational Advisory Committee for Aeronautics NACA TN no321 1929

[12] HWagner ldquoPhenomena associatedwith impacts and sliding onliquid surfacesrdquo Journal of Applied Mathematics and Mechanicsvol 12 pp 193ndash215 1932

[13] W S Vorus ldquoA flat cylinder theory for vessel impact and steadyplaning resistancerdquo Journal of Ship Research vol 40 no 2 pp89ndash106 1996

[14] T Miloh ldquoOn the initial-stage slamming of a rigid sphere in avertical water entryrdquo Applied Ocean Research vol 13 no 1 pp13ndash43 1991

[15] J M Aristoff and JW Bush ldquoWater entry of small hydrophobicspheresrdquo Journal of Fluid Mechanics vol 619 pp 45ndash78 2009

[16] P Ghadimi A Dashtimanesh and S R Djeddi ldquoStudy of waterentry of circular cylinder by using analytical and numericalsolutionsrdquo Journal of the Brazilian Society ofMechanical Sciencesamp Engineering vol 34 no 3 pp 225ndash232 2012

[17] A TassinD J PiroAAKorobkinK JMaki andM J CookerldquoTwo-dimensional water entry and exit of a body whose shapevaries in timerdquo Journal of Fluids and Structures vol 40 pp 317ndash336 2013

[18] E Yao H Wang L Pan X Wang and R Woding ldquoVerticalwater-entry of bullet-shaped projectilesrdquo Journal of AppliedMathematics and Physics vol 02 no 06 pp 323ndash334 2014

[19] M-S Park Y-R Jung and W-G Park ldquoNumerical study ofimpact force and ricochet behavior of high speed water-entrybodiesrdquo Computers amp Fluids vol 32 no 7 pp 939ndash951 2003

[20] D Battistin and A Iafrati ldquoHydrodynamic loads during waterenergy of two-dimensional and axisymmetric bodiesrdquo Journalof Fluids and Structures vol 17 no 5 pp 643ndash664 2003

[21] A A Korobkin and M Ohkusu ldquoImpact of two circular platesone of which is floating on a thin layer of liquidrdquo Journal ofEngineering Mathematics vol 50 no 4 pp 343ndash358 2004

[22] K M Kleefsman G Fekken A E Veldman B Iwanowski andB Buchner ldquoA volume-of-fluid based simulation method forwave impact problemsrdquo Journal of Computational Physics vol206 no 1 pp 363ndash393 2005

[23] Y Kim Y Kim Y Liu and D K P Yue ldquoOn the water-entryimpact problem of asymmetric bodiesrdquo in Proceedings of the9th International Conference on Numerical Ship Hydrodynamics(NSH rsquo07) August 2007

[24] Q Yang and W Qiu ldquoNumerical solution of 2D slammingproblem with a CIPmethodrdquo in Proceedings of the InternationalConference on Violent Flows Research Institute for AppliedMechanics Kyushu University Japan 2007

[25] A C Fairlie-Clarke and T Tveitnes ldquoMomentum and gravityeffects during the constant velocitywater entry ofwedge-shapedsectionsrdquo Ocean Engineering vol 35 no 7 pp 706ndash716 2007

[26] Q Yang andW Qiu ldquoNumerical simulation of water impact for2D and 3D bodiesrdquo Ocean Engineering vol 43 pp 82ndash89 2012

[27] G Wu ldquoNumerical simulation for water entry of a wedge atvarying speed by a high order boundary element methodrdquoJournal of Marine Science and Application vol 11 no 2 pp 143ndash149 2012

[28] M Ahmadzadeh B Saranjam A Hoseini Fard and A RBinesh ldquoNumerical simulation of sphere water entry prob-lem using Eulerian-Lagrangian methodrdquo Applied MathematicalModelling vol 38 no 5-6 pp 1673ndash1684 2014

[29] V-T Nguyen D-T Vu W-G Park and Y-R Jung ldquoNumericalanalysis of water impact forces using a dual-time pseudo-compressibility method and volume-of-fluid interface trackingalgorithmrdquo Computers amp Fluids vol 103 pp 18ndash33 2014

[30] S Kim and N Kim ldquoIntegrated dynamics modeling forsupercavitating vehicle systemsrdquo International Journal of NavalArchitecture and Ocean Engineering vol 7 no 2 pp 346ndash3632015

18 Geofluids

[31] V-T Nguyen D-T Vu W-G Park and C-M Jung ldquoNavier-Stokes solver for water entry bodies with moving Chimera gridmethod in 6DOF motionsrdquo Computers amp Fluids vol 140 pp19ndash38 2016

[32] M Mirzaei M Eghtesad and M Alishahi ldquoPlaning forceidentification in high-speed underwater vehiclesrdquo Journal ofVibration and Control vol 22 no 20 pp 4176ndash4191 2016

[33] G Qiu S Henke and J Grabe ldquoApplication of a CoupledEulerian-Lagrangian approach on geomechanical problemsinvolving large deformationsrdquoComputersampGeosciences vol 38no 1 pp 30ndash39 2011

[34] M R Erfanian M Anbarsooz N Rahimi M Zare andM Moghiman ldquoNumerical and experimental investigationof a three dimensional spherical-nose projectile water entryproblemrdquo Ocean Engineering vol 104 pp 397ndash404 2015

[35] W M Isbell F H Shipman and A H Jones ldquoHugoniotequation of state measurements for eleven material to fiveMegabars Materials and Structural Laboratory ManufacturingDevelopments General Motors Technical centerrdquo Tech RepMichigan 1968

[36] M A Jandron R C Hurd J L Belden A F Bower W Fennelland T T Truscott ldquoModeling of hyperelastic water-skippingspheres using AbaqusExplicitrdquo in Proceedings of the SIMULIACommunity Conference 2014

[37] H Forouzani B Saranjam R Kamali and A Abdollahi-far ldquoElasto-plastic time dependent impact analysis of highspeed projectile on water surfacerdquo Journal of Solid and FluidMechanics vol 3 pp 281ndash298 2016

[38] M R Erfanian andMMoghiman ldquoNumerical and experimen-tal investigation of a projectile water entry problem and studyof velocity effect on time and depth of pinch-offrdquo Journal ofModares Mechanical Engineering vol 15 pp 53ndash60 2015

[39] A Sillem Feasibility study of a tire hydroplaning simulation in afinite element code using a coupled Eulerian-Lagrangian method[MSc thesis] 2008

[40] M Lee R G Longoria and D E Wilson ldquoCavity dynamics inhigh-speed water entryrdquo Physics of Fluids vol 9 no 3 pp 540ndash550 1997

Hindawiwwwhindawicom Volume 2018

Journal of

ChemistryArchaeaHindawiwwwhindawicom Volume 2018

Marine BiologyJournal of

Hindawiwwwhindawicom Volume 2018

BiodiversityInternational Journal of

Hindawiwwwhindawicom Volume 2018

EcologyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2018

Forestry ResearchInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Environmental and Public Health

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Microbiology

Hindawiwwwhindawicom Volume 2018

Public Health Advances in

AgricultureAdvances in

Hindawiwwwhindawicom Volume 2018

Agronomy

Hindawiwwwhindawicom Volume 2018

International Journal of

Hindawiwwwhindawicom Volume 2018

MeteorologyAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

ScienticaHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Geological ResearchJournal of

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

Submit your manuscripts atwwwhindawicom

18 Geofluids

[31] V-T Nguyen D-T Vu W-G Park and C-M Jung ldquoNavier-Stokes solver for water entry bodies with moving Chimera gridmethod in 6DOF motionsrdquo Computers amp Fluids vol 140 pp19ndash38 2016

[32] M Mirzaei M Eghtesad and M Alishahi ldquoPlaning forceidentification in high-speed underwater vehiclesrdquo Journal ofVibration and Control vol 22 no 20 pp 4176ndash4191 2016

[33] G Qiu S Henke and J Grabe ldquoApplication of a CoupledEulerian-Lagrangian approach on geomechanical problemsinvolving large deformationsrdquoComputersampGeosciences vol 38no 1 pp 30ndash39 2011

[34] M R Erfanian M Anbarsooz N Rahimi M Zare andM Moghiman ldquoNumerical and experimental investigationof a three dimensional spherical-nose projectile water entryproblemrdquo Ocean Engineering vol 104 pp 397ndash404 2015

[35] W M Isbell F H Shipman and A H Jones ldquoHugoniotequation of state measurements for eleven material to fiveMegabars Materials and Structural Laboratory ManufacturingDevelopments General Motors Technical centerrdquo Tech RepMichigan 1968

[36] M A Jandron R C Hurd J L Belden A F Bower W Fennelland T T Truscott ldquoModeling of hyperelastic water-skippingspheres using AbaqusExplicitrdquo in Proceedings of the SIMULIACommunity Conference 2014

[37] H Forouzani B Saranjam R Kamali and A Abdollahi-far ldquoElasto-plastic time dependent impact analysis of highspeed projectile on water surfacerdquo Journal of Solid and FluidMechanics vol 3 pp 281ndash298 2016

[38] M R Erfanian andMMoghiman ldquoNumerical and experimen-tal investigation of a projectile water entry problem and studyof velocity effect on time and depth of pinch-offrdquo Journal ofModares Mechanical Engineering vol 15 pp 53ndash60 2015

[39] A Sillem Feasibility study of a tire hydroplaning simulation in afinite element code using a coupled Eulerian-Lagrangian method[MSc thesis] 2008

[40] M Lee R G Longoria and D E Wilson ldquoCavity dynamics inhigh-speed water entryrdquo Physics of Fluids vol 9 no 3 pp 540ndash550 1997

Hindawiwwwhindawicom Volume 2018

Journal of

ChemistryArchaeaHindawiwwwhindawicom Volume 2018

Marine BiologyJournal of

Hindawiwwwhindawicom Volume 2018

BiodiversityInternational Journal of

Hindawiwwwhindawicom Volume 2018

EcologyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2018

Forestry ResearchInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Environmental and Public Health

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Microbiology

Hindawiwwwhindawicom Volume 2018

Public Health Advances in

AgricultureAdvances in

Hindawiwwwhindawicom Volume 2018

Agronomy

Hindawiwwwhindawicom Volume 2018

International Journal of

Hindawiwwwhindawicom Volume 2018

MeteorologyAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

ScienticaHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Geological ResearchJournal of

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

Journal of

ChemistryArchaeaHindawiwwwhindawicom Volume 2018

Marine BiologyJournal of

Hindawiwwwhindawicom Volume 2018

BiodiversityInternational Journal of

Hindawiwwwhindawicom Volume 2018

EcologyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2018

Forestry ResearchInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Environmental and Public Health

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Microbiology

Hindawiwwwhindawicom Volume 2018

Public Health Advances in

AgricultureAdvances in

Hindawiwwwhindawicom Volume 2018

Agronomy

Hindawiwwwhindawicom Volume 2018

International Journal of

Hindawiwwwhindawicom Volume 2018

MeteorologyAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

ScienticaHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Geological ResearchJournal of

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

Submit your manuscripts atwwwhindawicom