Research Article Numerical Investigation of Wave Slamming of...

Research ArticleNumerical Investigation of Wave Slamming of Flat Bottom Bodyduring Water Entry Process

Xiaozhou Hu and Shaojun Liu

School of Mechanical and Electrical Engineering Central South University Changsha 410083 China

Correspondence should be addressed to Xiaozhou Hu smallboathu163com

Received 23 December 2013 Revised 4 April 2014 Accepted 8 April 2014 Published 29 April 2014

Academic Editor Benchawan Wiwatanapataphee

Copyright copy 2014 X Hu and S Liu This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A numerical wave load model based on two-phase (water-air) Reynolds-averaged Navier-Stokes (RANS) type equations is used toevaluate hydrodynamic forces exerted on flat bottom body while entering ocean waves of deploying process The discretization ofthe RANS equations is achieved by a finite volume (FV) approach The volume of fluid (VOF) method is employed to track thecomplicated free surface A numerical wave tank is built to generate the ocean waves which are suitable for deploying offshorestructures A typical deploying condition is employed to reflect the process of flat bottom body impacting waves and the pressuredistribution of bottom is also presented Four different lowering velocities are applied to obtain the relationship between slammingforce and wave parameters The numerical results clearly demonstrated the characteristics of flat bottom body impacting oceanwaves

1 Introduction

At the beginning of deployment of offshore structures thewave impact load due to slamming between the deployedbodies and free surface of ocean wave is a dangerousphenomenon which attracts more and more attention fromocean engineering field Firstly wave impact is a strongnonlinear phenomenon and a random process which is verysensitive to relative motion between body and free surfaceand the deployed bodies undergo hydrodynamic loads thatmay induce localized plastic deformation even catastrophicdamage in hostile sea conditions Secondly during the pro-cess of water entering ocean waves the nonlinear interactionbetween deployed bodies and waves may result in dynamictension of deploying cable which connects the deployedbodies and crane equipment even leading to fracture of cableand missing of deployed bodies Therefore it is significant topredict the whole water entry process of flat bottom bodyas well as the pressure distribution and maximum impactpressure on the body which is the practical motivation forthe theoretical and numerical studies in this paper

Slamming phenomena have been studied over sev-eral decades especially in naval hydrodynamics Pioneering

research had been carried out by Von Karman [1] Wanger[2] Chuang [3] and Zhao and Faltinsen [4] These studiesmainly focused on the calculation of slamming force withoutconsidering the effect of ocean waves

Concerning the wave impact of flat bottom body indeploying process the present solution is empirical equations[5] Although available empirical equations from laboratoryprototype experiments can be used to get approximateestimations of wave impact forces on deployed flat bottombody the nonlinear influence of ocean waves on body isdifficult to evaluate Also most empirical equations provideonly maximum values of wave forces rather than the forcetime history

In this regard numerical modeling of wave loads ondeployed flat bottom body is a very useful supplementaryapproach for estimating wave loads With the developmentof computer hardware and computational fluid dynamicsnumerical methods have been extensively used to solve thewater entry of rigid body under fully nonlinear free surfaceconditions

Recently there are many significant studies concerningrelated problems which are introduced as follows

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 821689 10 pageshttpdxdoiorg1011552014821689

2 Mathematical Problems in Engineering

Baarholm and Faltinsen [6] used a boundary elementmodel to study water impacts on a fixed horizontal platformdeck from regular incident waves They simplified this prob-lem to a two-dimensional potential flow problem and solvedthe resulting boundary value problem by three differentnumerical methods simplified Wagner-based method andtwo boundary element methods solving the perturbationvelocity potential due to the impact and the total velocitypotential respectively Chen and Xiao [7] used commercialsoftware MSC Dytran to study the water entry of flat-bot-tom structure with constant velocity and air cushion andsplash were observed Bunnik and Buchner [8 9] applied theimproved volume of fluid (iVOF) method for better numer-ical prediction of the behavior of a subsea structure in thesplash zone dedicated model tests verified simulations Dinget al [10] presented the experimental investigation of unidi-rectional random wave slamming on the three-dimensionalstructure in the splash zone Sarkar and Gudmestad [11]discussed the hydrodynamic coefficients (drag coefficientsand added mass coefficients) and analysis methodology forthe splash zone deploying analysis Zhang et al [12] built amodel of fluid with CFD software to analyze the dynamiccharacter of the deployment and retrieval system when itwas deployed into water and retrieved out of water withoutconsidering the effect of ocean waves Chen and Yu [13]implemented CFD simulation of wet deck slamming withinterface-preserving level-set method W-H Wang and Y-YWang [14 15] used a CFD method to numerically simulatethe physical process of cylinder vertically entering water inregular waves and the effect of wave parameters on thehydrodynamic force was analyzed

Accordingly in this work regular waves of ocean condi-tions (4ndash6 degrees) which are for the deployment of offshorestructures are generated in a numerical wave tank andtheir properties and sensitivity to governing parameters areinvestigated

This paper is motivated by developing a practical methodin simulating the wave slamming of flat bottom body basedon the CFD solver Fluent It is organized as follows the wavemodel equations and numerical methods implemented in thesimulation are described in Section 2 including governingequations boundary conditions generation of waves andnumerical wave-absorbing beach as well as parameters andmesh of numerical wave tank numerical experiments andresults together with their physical interpretation are pre-sented in Section 3

2 The Numerical Method

21 Governing Equations For simulations of water surfacewaves the Navier-Stokes equation-based model is knownas one of the numerical models capable of overcoming thelimitations of nonlinearity dispersion and wave breakingwhich most of the conventional wave models cannot doTherefore the CFD calculations of this paper are based on the

RANS equations The flow of a homogeneous incompress-ible viscous fluid is described by theNavier-Stokes equationsrespectively they are given by

120597120588

120597119905+

120597 (120588119906119894)

120597119909119894

= 0 (119894 = 1 2)

120597 (120588119906119894)

120597119905+ 119906119895

120597 (120588119906119894)

120597119909119894

=120597

120597119909119895

[120583 (120597119906119894

120597119909119895

+120597119906119895

120597119909119894

)] minus120597119901

120597119909119894

+ 120588119891119894

(1)

where 119894 119895 = 1 2 for two-dimensional flows 119906119894

= 119894th com-ponent of the velocity vector 120588 is the density of fluid 119901 is thepressure 120583 is the dynamic viscous coefficient and 119891

119894is the

body forceBy implementing the Reynolds decomposition in the N-

S equations and averaging the Reynolds-averaged Navier-Stokes (RANS) equations governing the average flow field arederived

120597 (120588119906119894)

120597119905+

120597 (120588119906119894119906119895)

120597119909119894

= minus120597119901

120597119909119894

+120597

120597119909119895

[120583 (120597119906119894

120597119909119895

+120597119906119895

120597119909119894

)]

+120597

120597119909119895

(minus1205881199061015840

1198941199061015840

119895) + 120588119891

119894

(2)

where the superscripts ldquo rdquo denote ensemble average of thephysical variables and minus120588119906

1015840

1198941199061015840

119894is the Reynolds stress

In this paper the standard 119896minus120576model where theReynoldsstress is approximated by a nonlinear algebraic stress modelis employed for turbulence closure The governing equationsfor 119896 and 120576 model are given by

120597 (120588119896)

120597119905+

120597 (120588119896119906119895)

120597119909119894

=120597

120597119909119895

[(120583 +120583119905

120590119896

)120597119896

120597119909119895

] + 120588119866119896

minus 120588120576

(3)

120597 (120588120576)

120597119905+

120597 (120588120576119906119895)

120597119909119894

=120597

120597119909119895

[(120583 +120583119905

120590120576

)120597119896

120597119909119895

]

+120576

119896(1198621205761

120588119866119896

minus 1198621205762

120588120576)

(4)

120583119905

=1198621205831205881198962

120576 (5)

119866119896

=120583119905

120588(

120597119906119894

120597119909119895

+120597119906119895

120597119909119894

)120597119906119894

120597119909119895

(6)

where 119896 is the turbulence kinetic energy 120576 is the dissipationrate of turbulence kinetic energy 120583

119905is the eddy viscosity and

119866119896is the turbulent energy productionThe empirical coefficients in (3)ndash(5) have been deter-

mined by performing many simple experiments and enforc-ing the physical realizability condition the values are given asfollows

119862120583

= 009 1198621205761

= 144 1198621205762

= 192

120590119896

= 10 120590120576

= 13

(7)

Mathematical Problems in Engineering 3

z

x

Ω(t)

Free surface Γf

Dampingzone

Bottom Γb

Inputboundary

ΓI

Wall Γw

Ld

L

Wavegeneration

zone

Workingzone

H

Moving body

h

L0

Figure 1 Numerical wave tank and moving body

The VOF method is employed to track the complicatedfree surface with a volume of fluid function F to define thewater region The physical meaning of the F function is thefractional volume of a cell occupied by water with unity torepresent a cell full of water and zero for a cell with no waterCells with F values between zero and unitymust then containthe free surface

If the volume fraction of water and air in each cellis denoted as 119886

119908and 119886

119886 respectively the tracking of the

interface between the phases is accomplished by the solutionof a continuity equation for the volume fraction of water

This equation of 119886119908has the following form

120597119886119908

120597119905+ 119906119894

120597 (119886119908

)

120597119909119894

= 0 (8)

The volume fraction equation will not be solved for airthe volume fraction of air will be computed based on thefollowing constraint

119886119908

+ 119886119886

= 1 (9)

For (9) 119886119908

= 0 means air phase and 119886119886

= 0 means waterphase while 119886

119886= 0sim1 means the mixture phase

22 Boundary Conditions

221 Free Surface Boundary Conditions Dynamic free sur-face condition can be written as

120597120601

120597119905= minus119892120578 minus

1

2

1003816100381610038161003816nabla12060110038161003816100381610038162

minus119875119886

120588 (10)

where 119875119886is the pressure on the free surface and is assumed to

be zero from now onAnd kinematic free surface condition can be written as

120597120578

120597119905= minusnabla120601 sdot nabla120578 +

120597120601

120597119911 (11)

Both the dynamic and kinematic free surface boundaryconditions are satisfied on the exact free surface

222 Bottom Boundary Condition No normal-flux condi-tion is applied on the bottom of numerical wave tankConsider

120597120601

120597119899= 0 (12)

223 Body Boundary Condition The body boundary condi-tion is written as

120597120601

120597119899= 119881 sdot 119899 (13)

whereV and n are body velocity with respect to gravity centerand surface normal vector of body respectively

224 Wall Boundary Condition The solution domain isbounded by a wave-maker on the left wall boundary At thewave-maker boundary the horizontal velocity of motion ofthe boundary is imposed on the water particle velocities atthe boundary as shown in Figure 1

No-slip condition is applied on the right vertical wall ofnumerical wave tank Consider

120597120601

120597119899= 0 (14)

The no-slip condition ensures that the fluid moving overa solid surface does not have velocity relative to the surface atthe point of contact

23 Generation of Waves and Numerical Wave-AbsorbingBeach Thepistonwave-maker is employed to generatewavesin this paper A schematic diagram of a numerical wave tankwith piston wave-maker is shown in Figure 1

The waves are generated by a piston type wave-makerlocated at the left boundary of the solution domain If theplunger moves sinusoidally with the function

119883 (119905) =1198830

2sin120596119905 (15)


(a) General diagram of mesh

(b) Mesh surrounding moving body (c) Mesh near wave-maker

Figure 2 Schematic diagrams of mesh

where 1198830

is the maximum horizontal displacement ofplunger and 120596 is the angular frequency of motion then thewave free surface equation is as follows

120578 (119909 119905) =1198830

2(

4 sinh2119896119889

2119896119889 + sinh 2119896119889cos (119896119909 minus 120596119905)

+

infin

sum

119899=1

4 sin2 (119896119899119889)

2119896119899119889 + sin (2119896

119899119889)

119890minus119896119899119909 sin120596119905)

(16)

where 119889 is the water depth and 119896 is the wave numberThe first part of (16) is the incoming wave with wave

number 119896 and the angular frequency is 120596 and the secondpart is attenuating standing wave induced by the piston If thesecond part of (16) can be eliminated the surface elevationequation can be given

120578 (119909 119905) =1198830

2

4 sinh2119896119889

2119896119889 + sinh 2119896119889cos (119896119909 minus 120596119905)

=119867

2cos (119896119909 minus 120596119905)

(17)

where 119867can be obtained

119867 =4 sinh2 (119896119889)

2119896119889 + sinh (2119896119889)1198830 (18)

Nonreflecting boundaries have to be used or a damp-ingdissipation zone is added to the solution domain fordamping the waves In this work towards the end of the com-putational domain the porous media technology is appliedto form an artificial damping zone so that the wave energy isgradually dissipated in the direction of wave propagation

The porous media are modeled by the addition of amomentum source term to the standard fluid flow equationsthe momentum source term of simple homogeneous porousmedia can be written as follows

119878119894= minus (

120583

120572V119894+ 1198622

1

2120588

1003816100381610038161003816V1198941003816100381610038161003816 V119894) (19)

where 119878119894is the source term for the 119894th (x y or z) momentum

equation |V119894| is themagnitude of the velocity120583 is the viscosity

of the fluid 1198622is the inertial resistance factor and 120572 is the

permeability which is a measure of the ability of porousmaterial to transmit fluids The determination methods ofpermeability for different waves can refer to Hu et alrsquos work[16] Besides the length of the damping zone is determinedto be at least 2 wavelengths

24 Parameters and Mesh of Numerical Wave Tank The 2Dnumerical tank is set as a rectangle with 300m length and30m height and the width (B) and height of deployed bodyare 10m and 3m The still water depth (d) is set as 20m Thenumerical grid exported by Gambit is shown in Figure 2 thegrid sizes are nonuniform and change case by case accordingto the wavelength and the water depth The general rule forconstructing grids is to make them denser for the shoalingand wave breaking region near the water surface

3 Numerical Experiments

31 Verification of Numerical Wave Tank A regular wavewith wave height 25m and period 6 s which is suitable fordeployment of offshore structures is used to verify the effec-tiveness and correctness of numerical wave tank introduced


Time (s)

Wav

e ele

vatio

n (m

)

100806040200

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

SimulationTheory

(a)

Time (s)

Wav

e ele

vatio

n (m

)

100806040200

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

SimulationTheory

(b)

Time (s)

Wav

e ele

vatio

n (m

)

100806040200

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

SimulationTheory

(c)

Time (s)

Wav

e ele

vatio

n (m

)

100806040200

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

SimulationTheory

(d)

Figure 3 Wave elevation at four different locations (a) x = 50m (b) x = 100m (c) x = 200m (d) x = 300m

in Section 2 A user defined function (UDF) implementsthe numerical wave generation and absorbing approach Thesimulation parameters are as follows permeability of porouszone 120572 = 1 times 10

minus6 the length of porous zone is determinedas 150m Wave elevations of locations of x = 50m 100m200m and 300m are determined as wave elevation probeswave elevations of each ocean condition are compared withthe corresponding analytical solutions

The simulated waves are shown in Figure 3 FromFigure 3 it can be seen that wave elevations of working zone

of wave tank (x = 50m 100m) are almost identical withthe analytical solutions which means that the error betweensimulation results and analytical solutions is very small andthat the reflected waves are eliminated Besides Figure 3 alsoreflects that the wave energy in damping zone of wave tank(x = 200m and 300m) has been absorbed that is to saythe amplitude attenuation rate is 100 Therefore it can beconcluded that the performance of numerical wave tank isperfect and can satisfy the demand of studying water-entryof flat bottom body in ocean waves numerically


32 Numerical Example of Flat BottomBody ImpactingWavesA numerical example is performed to show the whole waterentry process of flat bottom body in wave The main infor-mation of numerical example is as follows the wave heightand period of aimed wave are 25m and 8 s respectively theconstant vertical velocity of body is V

119889= 2ms the initial time

of simulation is 1199050

= 0 s and the motion of wave-maker willmake aimed waves from the initial time the initial movingtime of flat bottom body is 119879

0= 1931 s

The phase diagrams and pressure distribution diagramsof bottom of T = 20412 s 20462 s 20612 s and 20658 sare shown in Figure 4 These figures can clearly reflect thewhole water entry process of flat bottom body in wave AtT = 20412 s the left part of flat bottom body impacts thewave surface and with wave continuing to propagate to theright the contact length betweenwave and body is increasingAccording to Hu and Liu [17] when flat bottom body enterscalm water some phenomena can be observed firstly theair cushioning effect exists when the air is trapped betweenthe falling flat bottom body and the fluid which is alsomentioned by Faltinsen [18] secondly the impact pressuredistribution of bottom is axis-symmetric and the maximumpressure appears at the center of bottom Compared with theclam water entry case the case of entering water in waveis totally different as shown in Figure 4 impact pressuredistribution of bottom is nonsymmetric and the maximumpressure appears at the interface between rigid body (theflatted bottom body) and two-phase fluid (water and air)besides the pressure of bottom getting in contact with wateris much bigger than that getting in contact with air andthe trapped air between body bottom and fluid is relativelylimited (it is determined by the length of bottom) which willbe discussed in Section 33

33 Slamming Force Analysis During the transit of the struc-ture through the wave surface high slamming force mayoccur thus it is important to evaluate impact loadThe slam-ming coefficient of flat bottom body can be defined whichis nondimensional impact force similar to that of circularcylinder

The slamming coefficient 119862119904is defined as

119862119904

=119865119904

120588119908V2119889119861

(20)

where 119865119904is the maximum slamming force during water entry

process 120588119908is the density of ocean water V

119889is the deploying

velocity and B is the width of flat bottom bodyIn the numerical experiments four different velocities

of body are 1 2 3 and 4ms To investigate the influencethat wave height and period exert on slamming force thecoefficient Hd which is the ratio between wave height (H)and water depth (d) is employed as well as BL which is theratio between length (B) of deployed body and wave length(L) Waves with different heights (H) and wave lengths (L)are selected so as to study the relationship between slammingcoefficient and coefficients BL and Hd which are shown inTable 1

Table 1 Parameters of numerical experiments

Parameters Values119867 (m) 1 2 3 4 5 6119867119889 005 01 015 02 025 03119879 (s) 3 4 6 8 10 12119871 (m) 14 25 55 887 1212 1532119861119871 0714 04 0182 0113 0083 0065

Related researches reflect that the wave height plays animportant role in affecting wave impact force [10] Figure 5shows the relationship between Hd and slamming coeffi-cients with different deploying velocities and BL

As shown in Figure 5 when the body impacts oceanwaves the slamming coefficient increases with Hd increas-ingwhile it decreases rapidlywith deploying velocity increas-ing which is similar to the case in which the body impactscalm water however the slamming coefficient of the lattercase is explicitly smaller than the former one

Most pioneer experimental and numerical researchesfavor the suppose that the maximum impact pressure isapproximately proportional to the square of the relativeimpact velocity 119881

0[3 7] thus (20) can be rewritten as

119862119904

=119865119904

120588119908V2119889119861

=119865119904(1198812

0)

120588119908V2119889119861

1198810

= V119889

+ V119908

(21)

where V119908

is instantaneous velocity of water particles invertical direction For regular waves V

119908is determined by H

and T and it is clear that slamming coefficient 119862119904increases

with the ratio V119908

V119889increasing

BL is also an important factor which determines theslamming force As shown in Figure 4 when the bottom ofthe deployed body impacts wave the length of the bottomis the displacement that the wave travels and it also affectsthe compression level of trapped air between body bottomand fluid Figure 6 shows the relationship between BL andslamming coefficients with different deploying velocities andHd

As can be seen in Figure 6 slamming coefficient increaseswith BL increasing which is due to air cushioning effectwhich is also mentioned by Chuang [3] and Chen et al [19]For different deploying velocities the same law can be seenwhen BL is small 119862

119904increases slowly comparatively when

BL is bigger than 02 119862119904changes rapidly

4 Conclusion

The numerical wave-load model based on Navier-Stokestype equations and the VOF method has been applied toinvestigate the dynamic impact of wave forces on flat bottombody during its deploying process Based on the statisticalanalysis of the simulation results the following conclusionscan be listed


X (m)

Pres

sure

(kpa

)

807876747270

15

10

5

0

100e + 00950e minus 01900e minus 01850e minus 01800e minus 01750e minus 01700e minus 01650e minus 01600e minus 01550e minus 01500e minus 01450e minus 01400e minus 01350e minus 01300e minus 01250e minus 01200e minus 01150e minus 01100e minus 01500e minus 02000e + 00

(a)


X (m)807876747270

60

50

40

30

20

10

0

Pres

sure

(kpa

)

(b)


X (m)807876747270

Pres

sure

(kpa

)

300

250

200

150

100

50

0

(c)


X (m)807876747270

Pres

sure

(kpa

)

500

400

300

200

100

0

(d)

Figure 4 Phase diagram of the flow field and pressure profile at the body bottom at four different times (a) T = 20412 s (b) T = 20462 s (c)T = 20612 s (d) T = 20658 s


Hd

0 005 01 015 02 025 03 035

Cs

1500

1000

500

0

d = 1ms

(a)

Hd

0 005 01 015 02 025 03 035

Cs

500

400

300

200

100

0

d = 2ms

(b)

Hd

0 005 01 015 02 025 03 035

Cs

BL = 0714

BL = 04

BL = 0182

BL = 0113

BL = 0083

BL = 0065

300

200

100

0

d = 3ms

(c)

Hd

0 005 01 015 02 025 03 035

Cs

BL = 0714

BL = 04

BL = 0182

BL = 0113

BL = 0083

BL = 0065

200

100

0

d = 4ms

(d)

Figure 5 Relationship between slamming coefficient and Hd

(1) Waves which are generated based on piston typewave-maker method and porous media wave absorp-tion method clearly demonstrate the flexibility andaccuracy of this type of numerical wave tank forsimulating kinds of waves of different ocean condi-tions thus it can be used for studying the impactload between flat bottom body and ocean waves in itsdeployment

(2) The impact pressure distribution of bottom is non-symmetric and themaximum pressure appears at the

interface between rigid body and two-phase fluid thepressure of bottom getting in contact with water ismuch bigger than that getting in contact with air

(3) The slamming coefficient increases with Hd or BLincreasing while it decreases with deploying velocityincreasing which means that Hd BL and thedeploying velocity can exert great influence on thewave impact force

(4) This study can be a reference for studying the inter-action between flat bottom body and ocean waves

Mathematical Problems in Engineering 9Cs

1500

1000

500

0080604020

BL

d = 1ms

(a)

Cs

500

400

300

200

100

0080604020

BL

d = 2ms

(b)

Cs

300

200

100

0

Hd = 005

Hd = 01

Hd = 015

Hd = 02

Hd = 025

Hd = 03

080604020

BL

d = 3ms

(c)

Cs

200

100

0

Hd = 005

Hd = 01

Hd = 015

Hd = 02

Hd = 025

Hd = 03

080604020

BL

d = 4ms

(d)

Figure 6 Relationship between slamming coefficient and BL

and also provide the foundation for investigatingdynamic characteristics of deploying system of oceanstructures during deployed body lowering throughsplash zone

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors gratefully acknowledge the support of theNational Natural Science Foundation of China (no51305463)

References

[1] T Von Karman ldquoThe impact of seaplane floats during landingrdquoTN 321 National Advisory Committee for Aeronautics 1929


[2] H Wanger ldquoTrans phenomena associated with impacts andsliding on liquid surfacesrdquo Math Mechanics vol 12 no 4 pp193ndash215 1932

[3] S L Chuang ldquoExperiments on flat-bottom slammingrdquo Journalof Ship Research vol 10 pp 10ndash17 1966

[4] R Zhao and O Faltinsen ldquoWater entry of two-dimensionalbodiesrdquo Journal of Fluid Mechanics vol 246 pp 593ndash612 1993

[5] DNV-RP-H103 Recommended Practice Modelling and Analysisof Marine Operations DET NORSKE VERITAS Oslo Norway2011

[6] R Baarholm and O M Faltinsen ldquoWave impact underneathhorizontal decksrdquo Journal ofMarine Science andTechnology vol9 no 1 pp 1ndash13 2004

[7] Z Chen and X Xiao ldquoSimulation analysis on the role of aircushion in the slamming of a flat-bottom structurerdquo Journal ofShanghai Jiaotong University vol 39 no 5 pp 670ndash673 2005

[8] T Bunnik and B Buchner ldquoNumerical prediction of wave loadson subsea structures in the splash zonerdquo in Proceedings of the14th International Offshore and Polar Engineering Conference(ISOPE rsquo04) pp 284ndash290 Toulon France May 2004

[9] T Bunnik B Buchner andA Veldman ldquoThe use of a Volume ofFluid (VOF) method coupled to a time domain motion simula-tion to calculate themotions of a subsea structure lifted throughthe splash zonerdquo in Proceedings of the 25th International Confer-ence on Offshore Mechanics and Arctic Engineering (OMAE rsquo06)vol 4 pp 839ndash846 Hamburg Germany June 2006

[10] Z Ding B Ren Y Wang and X Ren ldquoExperimental study ofunidirectional irregular wave slamming on the three-dimen-sional structure in the splash zonerdquo Ocean Engineering vol 35no 16 pp 1637ndash1646 2008

[11] A Sarkar and O T Gudmestad ldquoSplash zone lifting analysis ofsubsea structuresrdquo in Proceedings of the 29th International Con-ference on Ocean Offshore and Arctic Engineering (OMAE rsquo10)vol 1 pp 303ndash312 Shanghai China June 2010

[12] J Y Zhang D J Li X Y Gao Q C Meng C J Yang andY Chen ldquoSimulation analysis of deployment and retrieval ofjunction box used in cabled seafloor observatoryrdquo Ship EngI-neering vol 32 no 6 pp 53ndash59 2010 (Chinese)

[13] H-C Chen and K Yu ldquoCFD simulations of wave-current-bodyinteractions including greenwater and wet deck slammingrdquoComputers amp Fluids vol 38 no 5 pp 970ndash980 2009

[14] W-H Wang and Y-Y Wang ldquoNumerical study on cylinderentering water in waverdquo Journal of Shanghai Jiaotong Universityvol 44 no 10 pp 1393ndash1399 2010 (Chinese)

[15] W-H Wang and Y-Y Wang ldquoAn essential solution of waterentry problems and its engineering applicationsrdquo Journal ofMarine Science and Application vol 9 no 3 pp 268ndash273 2010

[16] X-Z Hu S-J Liu and Y Li ldquo2D numerical wave tank simu-lation for deployment of seafloor mining systemrdquo Journal ofCentral South University vol 42 no 2 pp 246ndash251 2011

[17] X-Z Hu and S-J Liu ldquoNumerical simulation of calm waterentry of flatted-bottom seafloor mining toolrdquo Journal of CentralSouth University of Technology vol 18 no 3 pp 658ndash665 2011

[18] O M Faltinsen Sea loads on ships and offshore structure Cam-bridge University Press 1993

[19] Z Chen X Xiao and D-Y Wang ldquoSimulation of flat-bottomstructure slammingrdquoChinaOcean Engineering vol 19 no 3 pp385ndash394 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Baarholm and Faltinsen [6] used a boundary elementmodel to study water impacts on a fixed horizontal platformdeck from regular incident waves They simplified this prob-lem to a two-dimensional potential flow problem and solvedthe resulting boundary value problem by three differentnumerical methods simplified Wagner-based method andtwo boundary element methods solving the perturbationvelocity potential due to the impact and the total velocitypotential respectively Chen and Xiao [7] used commercialsoftware MSC Dytran to study the water entry of flat-bot-tom structure with constant velocity and air cushion andsplash were observed Bunnik and Buchner [8 9] applied theimproved volume of fluid (iVOF) method for better numer-ical prediction of the behavior of a subsea structure in thesplash zone dedicated model tests verified simulations Dinget al [10] presented the experimental investigation of unidi-rectional random wave slamming on the three-dimensionalstructure in the splash zone Sarkar and Gudmestad [11]discussed the hydrodynamic coefficients (drag coefficientsand added mass coefficients) and analysis methodology forthe splash zone deploying analysis Zhang et al [12] built amodel of fluid with CFD software to analyze the dynamiccharacter of the deployment and retrieval system when itwas deployed into water and retrieved out of water withoutconsidering the effect of ocean waves Chen and Yu [13]implemented CFD simulation of wet deck slamming withinterface-preserving level-set method W-H Wang and Y-YWang [14 15] used a CFD method to numerically simulatethe physical process of cylinder vertically entering water inregular waves and the effect of wave parameters on thehydrodynamic force was analyzed

Accordingly in this work regular waves of ocean condi-tions (4ndash6 degrees) which are for the deployment of offshorestructures are generated in a numerical wave tank andtheir properties and sensitivity to governing parameters areinvestigated

This paper is motivated by developing a practical methodin simulating the wave slamming of flat bottom body basedon the CFD solver Fluent It is organized as follows the wavemodel equations and numerical methods implemented in thesimulation are described in Section 2 including governingequations boundary conditions generation of waves andnumerical wave-absorbing beach as well as parameters andmesh of numerical wave tank numerical experiments andresults together with their physical interpretation are pre-sented in Section 3

2 The Numerical Method

21 Governing Equations For simulations of water surfacewaves the Navier-Stokes equation-based model is knownas one of the numerical models capable of overcoming thelimitations of nonlinearity dispersion and wave breakingwhich most of the conventional wave models cannot doTherefore the CFD calculations of this paper are based on the

RANS equations The flow of a homogeneous incompress-ible viscous fluid is described by theNavier-Stokes equationsrespectively they are given by

120597120588

120597119905+

120597 (120588119906119894)

120597119909119894

= 0 (119894 = 1 2)

120597 (120588119906119894)

120597119905+ 119906119895

120597 (120588119906119894)

120597119909119894

=120597

120597119909119895

[120583 (120597119906119894

120597119909119895

+120597119906119895

120597119909119894

)] minus120597119901

120597119909119894

+ 120588119891119894

(1)

where 119894 119895 = 1 2 for two-dimensional flows 119906119894

= 119894th com-ponent of the velocity vector 120588 is the density of fluid 119901 is thepressure 120583 is the dynamic viscous coefficient and 119891

119894is the

body forceBy implementing the Reynolds decomposition in the N-

S equations and averaging the Reynolds-averaged Navier-Stokes (RANS) equations governing the average flow field arederived

120597 (120588119906119894)

120597119905+

120597 (120588119906119894119906119895)

120597119909119894

= minus120597119901

120597119909119894

+120597

120597119909119895

[120583 (120597119906119894

120597119909119895

+120597119906119895

120597119909119894

)]

+120597

120597119909119895

(minus1205881199061015840

1198941199061015840

119895) + 120588119891

119894

(2)

where the superscripts ldquo rdquo denote ensemble average of thephysical variables and minus120588119906

1015840

1198941199061015840

119894is the Reynolds stress

In this paper the standard 119896minus120576model where theReynoldsstress is approximated by a nonlinear algebraic stress modelis employed for turbulence closure The governing equationsfor 119896 and 120576 model are given by

120597 (120588119896)

120597119905+

120597 (120588119896119906119895)

120597119909119894

=120597

120597119909119895

[(120583 +120583119905

120590119896

)120597119896

120597119909119895

] + 120588119866119896

minus 120588120576

(3)

120597 (120588120576)

120597119905+

120597 (120588120576119906119895)

120597119909119894

=120597

120597119909119895

[(120583 +120583119905

120590120576

)120597119896

120597119909119895

]

+120576

119896(1198621205761

120588119866119896

minus 1198621205762

120588120576)

(4)

120583119905

=1198621205831205881198962

120576 (5)

119866119896

=120583119905

120588(

120597119906119894

120597119909119895

+120597119906119895

120597119909119894

)120597119906119894

120597119909119895

(6)

where 119896 is the turbulence kinetic energy 120576 is the dissipationrate of turbulence kinetic energy 120583

119905is the eddy viscosity and

119866119896is the turbulent energy productionThe empirical coefficients in (3)ndash(5) have been deter-

mined by performing many simple experiments and enforc-ing the physical realizability condition the values are given asfollows

119862120583

= 009 1198621205761

= 144 1198621205762

= 192

120590119896

= 10 120590120576

= 13

(7)


z

x

Ω(t)

Free surface Γf

Dampingzone

Bottom Γb

Inputboundary

ΓI

Wall Γw

Ld

L

Wavegeneration

zone

Workingzone

H

Moving body

h

L0




119908and 119886




120597119886119908

120597119905+ 119906119894

120597 (119886119908

)

120597119909119894

= 0 (8)


119886119908

+ 119886119886

= 1 (9)

For (9) 119886119908






120597120601

120597119905= minus119892120578 minus

1

2

1003816100381610038161003816nabla12060110038161003816100381610038162

minus119875119886

120588 (10)



120597120578


120597120601

120597119911 (11)



120597120601

120597119899= 0 (12)


120597120601

120597119899= 119881 sdot 119899 (13)




120597120601

120597119899= 0 (14)




119883 (119905) =1198830

2sin120596119905 (15)





where 1198830


120578 (119909 119905) =1198830

2(

4 sinh2119896119889

2119896119889 + sinh 2119896119889cos (119896119909 minus 120596119905)

+

infin

sum

119899=1

4 sin2 (119896119899119889)

2119896119899119889 + sin (2119896

119899119889)

119890minus119896119899119909 sin120596119905)

(16)



120578 (119909 119905) =1198830

2

4 sinh2119896119889

2119896119889 + sinh 2119896119889cos (119896119909 minus 120596119905)

=119867

2cos (119896119909 minus 120596119905)

(17)


119867 =4 sinh2 (119896119889)

2119896119889 + sinh (2119896119889)1198830 (18)



119878119894= minus (

120583

120572V119894+ 1198622

1

2120588

1003816100381610038161003816V1198941003816100381610038161003816 V119894) (19)









Time (s)

Wav

e ele

vatio

n (m

)

100806040200

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

SimulationTheory

(a)

Time (s)

Wav

e ele

vatio

n (m

)

100806040200

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

SimulationTheory

(b)

Time (s)

Wav

e ele

vatio

n (m

)

100806040200

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

SimulationTheory

(c)

Time (s)

Wav

e ele

vatio

n (m

)

100806040200

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

SimulationTheory

(d)











0= 1931 s




119862119904

=119865119904

120588119908V2119889119861

(20)












119862119904

=119865119904

120588119908V2119889119861

=119865119904(1198812

0)

120588119908V2119889119861

1198810

= V119889

+ V119908

(21)

where V119908





V119889increasing





4 Conclusion



X (m)

Pres

sure

(kpa

)

807876747270

15

10

5

0


(a)


X (m)807876747270

60

50

40

30

20

10

0

Pres

sure

(kpa

)

(b)


X (m)807876747270

Pres

sure

(kpa

)

300

250

200

150

100

50

0

(c)


X (m)807876747270

Pres

sure

(kpa

)

500

400

300

200

100

0

(d)



Hd

0 005 01 015 02 025 03 035

Cs

1500

1000

500

0

d = 1ms

(a)

Hd

0 005 01 015 02 025 03 035

Cs

500

400

300

200

100

0

d = 2ms

(b)

Hd

0 005 01 015 02 025 03 035

Cs

BL = 0714

BL = 04

BL = 0182

BL = 0113

BL = 0083

BL = 0065

300

200

100

0

d = 3ms

(c)

Hd

0 005 01 015 02 025 03 035

Cs

BL = 0714

BL = 04

BL = 0182

BL = 0113

BL = 0083

BL = 0065

200

100

0

d = 4ms

(d)








1500

1000

500

0080604020

BL

d = 1ms

(a)

Cs

500

400

300

200

100

0080604020

BL

d = 2ms

(b)

Cs

300

200

100

0

Hd = 005

Hd = 01

Hd = 015

Hd = 02

Hd = 025

Hd = 03

080604020

BL

d = 3ms

(c)

Cs

200

100

0

Hd = 005

Hd = 01

Hd = 015

Hd = 02

Hd = 025

Hd = 03

080604020

BL

d = 4ms

(d)





Acknowledgment


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










z

x

Ω(t)

Free surface Γf

Dampingzone

Bottom Γb

Inputboundary

ΓI

Wall Γw

Ld

L

Wavegeneration

zone

Workingzone

H

Moving body

h

L0




119908and 119886




120597119886119908

120597119905+ 119906119894

120597 (119886119908

)

120597119909119894

= 0 (8)


119886119908

+ 119886119886

= 1 (9)

For (9) 119886119908






120597120601

120597119905= minus119892120578 minus

1

2

1003816100381610038161003816nabla12060110038161003816100381610038162

minus119875119886

120588 (10)



120597120578


120597120601

120597119911 (11)



120597120601

120597119899= 0 (12)


120597120601

120597119899= 119881 sdot 119899 (13)




120597120601

120597119899= 0 (14)




119883 (119905) =1198830

2sin120596119905 (15)





where 1198830


120578 (119909 119905) =1198830

2(

4 sinh2119896119889

2119896119889 + sinh 2119896119889cos (119896119909 minus 120596119905)

+

infin

sum

119899=1

4 sin2 (119896119899119889)

2119896119899119889 + sin (2119896

119899119889)

119890minus119896119899119909 sin120596119905)

(16)



120578 (119909 119905) =1198830

2

4 sinh2119896119889

2119896119889 + sinh 2119896119889cos (119896119909 minus 120596119905)

=119867

2cos (119896119909 minus 120596119905)

(17)


119867 =4 sinh2 (119896119889)

2119896119889 + sinh (2119896119889)1198830 (18)



119878119894= minus (

120583

120572V119894+ 1198622

1

2120588

1003816100381610038161003816V1198941003816100381610038161003816 V119894) (19)









Time (s)

Wav

e ele

vatio

n (m

)

100806040200

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

SimulationTheory

(a)

Time (s)

Wav

e ele

vatio

n (m

)

100806040200

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

SimulationTheory

(b)

Time (s)

Wav

e ele

vatio

n (m

)

100806040200

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

SimulationTheory

(c)

Time (s)

Wav

e ele

vatio

n (m

)

100806040200

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

SimulationTheory

(d)











0= 1931 s




119862119904

=119865119904

120588119908V2119889119861

(20)












119862119904

=119865119904

120588119908V2119889119861

=119865119904(1198812

0)

120588119908V2119889119861

1198810

= V119889

+ V119908

(21)

where V119908





V119889increasing





4 Conclusion



X (m)

Pres

sure

(kpa

)

807876747270

15

10

5

0


(a)


X (m)807876747270

60

50

40

30

20

10

0

Pres

sure

(kpa

)

(b)


X (m)807876747270

Pres

sure

(kpa

)

300

250

200

150

100

50

0

(c)


X (m)807876747270

Pres

sure

(kpa

)

500

400

300

200

100

0

(d)



Hd

0 005 01 015 02 025 03 035

Cs

1500

1000

500

0

d = 1ms

(a)

Hd

0 005 01 015 02 025 03 035

Cs

500

400

300

200

100

0

d = 2ms

(b)

Hd

0 005 01 015 02 025 03 035

Cs

BL = 0714

BL = 04

BL = 0182

BL = 0113

BL = 0083

BL = 0065

300

200

100

0

d = 3ms

(c)

Hd

0 005 01 015 02 025 03 035

Cs

BL = 0714

BL = 04

BL = 0182

BL = 0113

BL = 0083

BL = 0065

200

100

0

d = 4ms

(d)








1500

1000

500

0080604020

BL

d = 1ms

(a)

Cs

500

400

300

200

100

0080604020

BL

d = 2ms

(b)

Cs

300

200

100

0

Hd = 005

Hd = 01

Hd = 015

Hd = 02

Hd = 025

Hd = 03

080604020

BL

d = 3ms

(c)

Cs

200

100

0

Hd = 005

Hd = 01

Hd = 015

Hd = 02

Hd = 025

Hd = 03

080604020

BL

d = 4ms

(d)





Acknowledgment


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













where 1198830


120578 (119909 119905) =1198830

2(

4 sinh2119896119889

2119896119889 + sinh 2119896119889cos (119896119909 minus 120596119905)

+

infin

sum

119899=1

4 sin2 (119896119899119889)

2119896119899119889 + sin (2119896

119899119889)

119890minus119896119899119909 sin120596119905)

(16)



120578 (119909 119905) =1198830

2

4 sinh2119896119889

2119896119889 + sinh 2119896119889cos (119896119909 minus 120596119905)

=119867

2cos (119896119909 minus 120596119905)

(17)


119867 =4 sinh2 (119896119889)

2119896119889 + sinh (2119896119889)1198830 (18)



119878119894= minus (

120583

120572V119894+ 1198622

1

2120588

1003816100381610038161003816V1198941003816100381610038161003816 V119894) (19)









Time (s)

Wav

e ele

vatio

n (m

)

100806040200

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

SimulationTheory

(a)

Time (s)

Wav

e ele

vatio

n (m

)

100806040200

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

SimulationTheory

(b)

Time (s)

Wav

e ele

vatio

n (m

)

100806040200

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

SimulationTheory

(c)

Time (s)

Wav

e ele

vatio

n (m

)

100806040200

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

SimulationTheory

(d)











0= 1931 s




119862119904

=119865119904

120588119908V2119889119861

(20)












119862119904

=119865119904

120588119908V2119889119861

=119865119904(1198812

0)

120588119908V2119889119861

1198810

= V119889

+ V119908

(21)

where V119908





V119889increasing





4 Conclusion



X (m)

Pres

sure

(kpa

)

807876747270

15

10

5

0


(a)


X (m)807876747270

60

50

40

30

20

10

0

Pres

sure

(kpa

)

(b)


X (m)807876747270

Pres

sure

(kpa

)

300

250

200

150

100

50

0

(c)


X (m)807876747270

Pres

sure

(kpa

)

500

400

300

200

100

0

(d)



Hd

0 005 01 015 02 025 03 035

Cs

1500

1000

500

0

d = 1ms

(a)

Hd

0 005 01 015 02 025 03 035

Cs

500

400

300

200

100

0

d = 2ms

(b)

Hd

0 005 01 015 02 025 03 035

Cs

BL = 0714

BL = 04

BL = 0182

BL = 0113

BL = 0083

BL = 0065

300

200

100

0

d = 3ms

(c)

Hd

0 005 01 015 02 025 03 035

Cs

BL = 0714

BL = 04

BL = 0182

BL = 0113

BL = 0083

BL = 0065

200

100

0

d = 4ms

(d)








1500

1000

500

0080604020

BL

d = 1ms

(a)

Cs

500

400

300

200

100

0080604020

BL

d = 2ms

(b)

Cs

300

200

100

0

Hd = 005

Hd = 01

Hd = 015

Hd = 02

Hd = 025

Hd = 03

080604020

BL

d = 3ms

(c)

Cs

200

100

0

Hd = 005

Hd = 01

Hd = 015

Hd = 02

Hd = 025

Hd = 03

080604020

BL

d = 4ms

(d)





Acknowledgment


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Time (s)

Wav

e ele

vatio

n (m

)

100806040200

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

SimulationTheory

(a)

Time (s)

Wav

e ele

vatio

n (m

)

100806040200

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

SimulationTheory

(b)

Time (s)

Wav

e ele

vatio

n (m

)

100806040200

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

SimulationTheory

(c)

Time (s)

Wav

e ele

vatio

n (m

)

100806040200

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

SimulationTheory

(d)











0= 1931 s




119862119904

=119865119904

120588119908V2119889119861

(20)












119862119904

=119865119904

120588119908V2119889119861

=119865119904(1198812

0)

120588119908V2119889119861

1198810

= V119889

+ V119908

(21)

where V119908





V119889increasing





4 Conclusion



X (m)

Pres

sure

(kpa

)

807876747270

15

10

5

0


(a)


X (m)807876747270

60

50

40

30

20

10

0

Pres

sure

(kpa

)

(b)


X (m)807876747270

Pres

sure

(kpa

)

300

250

200

150

100

50

0

(c)


X (m)807876747270

Pres

sure

(kpa

)

500

400

300

200

100

0

(d)



Hd

0 005 01 015 02 025 03 035

Cs

1500

1000

500

0

d = 1ms

(a)

Hd

0 005 01 015 02 025 03 035

Cs

500

400

300

200

100

0

d = 2ms

(b)

Hd

0 005 01 015 02 025 03 035

Cs

BL = 0714

BL = 04

BL = 0182

BL = 0113

BL = 0083

BL = 0065

300

200

100

0

d = 3ms

(c)

Hd

0 005 01 015 02 025 03 035

Cs

BL = 0714

BL = 04

BL = 0182

BL = 0113

BL = 0083

BL = 0065

200

100

0

d = 4ms

(d)








1500

1000

500

0080604020

BL

d = 1ms

(a)

Cs

500

400

300

200

100

0080604020

BL

d = 2ms

(b)

Cs

300

200

100

0

Hd = 005

Hd = 01

Hd = 015

Hd = 02

Hd = 025

Hd = 03

080604020

BL

d = 3ms

(c)

Cs

200

100

0

Hd = 005

Hd = 01

Hd = 015

Hd = 02

Hd = 025

Hd = 03

080604020

BL

d = 4ms

(d)





Acknowledgment


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














0= 1931 s




119862119904

=119865119904

120588119908V2119889119861

(20)












119862119904

=119865119904

120588119908V2119889119861

=119865119904(1198812

0)

120588119908V2119889119861

1198810

= V119889

+ V119908

(21)

where V119908





V119889increasing





4 Conclusion



X (m)

Pres

sure

(kpa

)

807876747270

15

10

5

0


(a)


X (m)807876747270

60

50

40

30

20

10

0

Pres

sure

(kpa

)

(b)


X (m)807876747270

Pres

sure

(kpa

)

300

250

200

150

100

50

0

(c)


X (m)807876747270

Pres

sure

(kpa

)

500

400

300

200

100

0

(d)



Hd

0 005 01 015 02 025 03 035

Cs

1500

1000

500

0

d = 1ms

(a)

Hd

0 005 01 015 02 025 03 035

Cs

500

400

300

200

100

0

d = 2ms

(b)

Hd

0 005 01 015 02 025 03 035

Cs

BL = 0714

BL = 04

BL = 0182

BL = 0113

BL = 0083

BL = 0065

300

200

100

0

d = 3ms

(c)

Hd

0 005 01 015 02 025 03 035

Cs

BL = 0714

BL = 04

BL = 0182

BL = 0113

BL = 0083

BL = 0065

200

100

0

d = 4ms

(d)








1500

1000

500

0080604020

BL

d = 1ms

(a)

Cs

500

400

300

200

100

0080604020

BL

d = 2ms

(b)

Cs

300

200

100

0

Hd = 005

Hd = 01

Hd = 015

Hd = 02

Hd = 025

Hd = 03

080604020

BL

d = 3ms

(c)

Cs

200

100

0

Hd = 005

Hd = 01

Hd = 015

Hd = 02

Hd = 025

Hd = 03

080604020

BL

d = 4ms

(d)





Acknowledgment


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










X (m)

Pres

sure

(kpa

)

807876747270

15

10

5

0


(a)


X (m)807876747270

60

50

40

30

20

10

0

Pres

sure

(kpa

)

(b)


X (m)807876747270

Pres

sure

(kpa

)

300

250

200

150

100

50

0

(c)


X (m)807876747270

Pres

sure

(kpa

)

500

400

300

200

100

0

(d)



Hd

0 005 01 015 02 025 03 035

Cs

1500

1000

500

0

d = 1ms

(a)

Hd

0 005 01 015 02 025 03 035

Cs

500

400

300

200

100

0

d = 2ms

(b)

Hd

0 005 01 015 02 025 03 035

Cs

BL = 0714

BL = 04

BL = 0182

BL = 0113

BL = 0083

BL = 0065

300

200

100

0

d = 3ms

(c)

Hd

0 005 01 015 02 025 03 035

Cs

BL = 0714

BL = 04

BL = 0182

BL = 0113

BL = 0083

BL = 0065

200

100

0

d = 4ms

(d)








1500

1000

500

0080604020

BL

d = 1ms

(a)

Cs

500

400

300

200

100

0080604020

BL

d = 2ms

(b)

Cs

300

200

100

0

Hd = 005

Hd = 01

Hd = 015

Hd = 02

Hd = 025

Hd = 03

080604020

BL

d = 3ms

(c)

Cs

200

100

0

Hd = 005

Hd = 01

Hd = 015

Hd = 02

Hd = 025

Hd = 03

080604020

BL

d = 4ms

(d)





Acknowledgment


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Hd

0 005 01 015 02 025 03 035

Cs

1500

1000

500

0

d = 1ms

(a)

Hd

0 005 01 015 02 025 03 035

Cs

500

400

300

200

100

0

d = 2ms

(b)

Hd

0 005 01 015 02 025 03 035

Cs

BL = 0714

BL = 04

BL = 0182

BL = 0113

BL = 0083

BL = 0065

300

200

100

0

d = 3ms

(c)

Hd

0 005 01 015 02 025 03 035

Cs

BL = 0714

BL = 04

BL = 0182

BL = 0113

BL = 0083

BL = 0065

200

100

0

d = 4ms

(d)








1500

1000

500

0080604020

BL

d = 1ms

(a)

Cs

500

400

300

200

100

0080604020

BL

d = 2ms

(b)

Cs

300

200

100

0

Hd = 005

Hd = 01

Hd = 015

Hd = 02

Hd = 025

Hd = 03

080604020

BL

d = 3ms

(c)

Cs

200

100

0

Hd = 005

Hd = 01

Hd = 015

Hd = 02

Hd = 025

Hd = 03

080604020

BL

d = 4ms

(d)





Acknowledgment


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1500

1000

500

0080604020

BL

d = 1ms

(a)

Cs

500

400

300

200

100

0080604020

BL

d = 2ms

(b)

Cs

300

200

100

0

Hd = 005

Hd = 01

Hd = 015

Hd = 02

Hd = 025

Hd = 03

080604020

BL

d = 3ms

(c)

Cs

200

100

0

Hd = 005

Hd = 01

Hd = 015

Hd = 02

Hd = 025

Hd = 03

080604020

BL

d = 4ms

(d)





Acknowledgment


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Numerical Investigation of Wave Slamming of...

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