Research Article Influences of Vorticity to Vertical ... · Faltinsen et al. [ ] investigated the piston mode occur-ring in a square-shaped catamaran hull form under forced heave

Research ArticleInfluences of Vorticity to Vertical Motion of Two-DimensionalMoonpool under Forced Heave Motion

Jae-Kyung Heo1 Jong-Chun Park2 Weon-Cheol Koo3 and Moo-Hyun Kim4

1 Software DNV Korea Busan Republic of Korea2Naval Architecture and Ocean Engineering Pusan National University Busan Republic of Korea3 Naval Architecture amp Ocean Engineering University of Ulsan Ulsan Republic of Korea4Civil Engineering Texas AampM University College Station TX USA

Correspondence should be addressed to Jong-Chun Park jcparkpnuedu

Received 3 October 2013 Accepted 6 February 2014 Published 30 March 2014

Academic Editor Shaoyong Lai

Copyright copy 2014 Jae-Kyung Heo et alThis 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

Potential (inviscid-irrotational) andNavier-Stokes equation-based (viscous) flowswere numerically simulated around a 2D floatingrectangular body with a moonpool particular emphasis was placed on the piston mode through the use of finite volume method(FVM) and boundary element method (BEM) solvers The resultant wave height and phase shift inside and outside the moonpoolwere comparedwith experimental results by Faltinsen et al (2007) for various heaving frequencies Hydrodynamic coefficients werecompared for the viscous and potential solvers and sway and heave forces were discussed The effects of the viscosity and vortexshedding were investigated by changing the gap size corner shape and viscosityThe viscous flow fields were thoroughly discussedto better understand the relevant physics and shed light on the detail flow structure at resonant frequency Vortex shedding wasfound to account for the most of the dampingThe viscous flow simulations agreed well with the experimental results showing theactual role and contribution of viscosity compared to potential flow simulations

1 Introduction

A moonpool is a vertical well directly connecting the freesurface to the deck of offshore vessels for drilling divinginstallation or other offshore support works The discon-tinuous openings on the bottom of ships not only increaseresistance but also cause harmful wave motions that degradethe safety and working ability of crews The behavior of thefree surface inside the moonpool has two types of modespiston and sloshing The piston mode involves the verticalmovement of water and in this mode a violent motionat resonant frequency is generated Horizontal motions areobserved in the sloshing mode which results in increasedresistance

Most previous research can be divided into two groupsresistance increase and violent water motions Studies onresistance increase mainly considered drillships English[1] measured the resistance and wave height for different

moonpool shapes of a drillship Yoo and Choi [2] examineddetailedmoonpool shapes with appendages Veer andTholen[3] surveyed the principal dimensions of moonpools forseveral drillships and discussed the mode of the free surfacerelated to the increase in resistance

Fukuta [4] and Aalbers [5] studied the water motion andits resonant frequency for drillships They both employed aspring-mass-damper system Fukuta [4] proposed an empir-ical formula to estimate the resonant frequency Aalbers [5]measured the damping coefficient from free-decay tests forthe spring-mass-damper model Molin [6] derived analyticformulas of the resonant frequency for 2D and 3Dmoonpoolsbased on the potential theory

Two-dimensional moonpoolsmdashfor example catamaransand side-by-side moored vesselsmdashhave the same resonantbehavior Faltinsen [7] presented a summary on the piston-mode resonance for twin hulls As new concepts for floatingstructures are developed such as liquefied natural gas floating

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 424927 13 pageshttpdxdoiorg1011552014424927

2 Mathematical Problems in Engineering

production storage offloading (LNG-FPSO) and liquefiednatural gas floating storage and regasification unit (LNG-FSRU) with a side-by-side mooring arrangement pistonmode resonance has drawn more attention The free-surfaceelevation inside the gap and the motion of the two float-ing bodies are important design factors In general linearpotential theory in the frequency domain is widely usedin the design process However this approach inherentlyoverestimates the wave elevation near resonance Koo andKim [8] and Hong et al [9] studied side-by-side mooredsystems and addressed the discrepancy in the piston modeThe overestimation of the free-surface elevation can besuppressed by employing a damping lid or artificial damping[10ndash12] However the exact value of the artificial damping isnot available prior to model tests and improper use of the lidor artificial damping coefficientmay decrease reliability in thenumerical estimation

Faltinsen et al [13] investigated the piston mode occur-ring in a square-shaped catamaran hull form under forcedheave motion by using experiments and linear numericalanalysis Kristiansen and Faltinsen [14] compared their sim-ulation results by using the vortex-tracking method withexperimental results They recently used a finite volumemethod-based domain decomposition method in which theviscous domain for the body region communicates with theoutside potential region [15]

The aim of this study is to determine the effects of eddydamping and viscosity on the piston mode by varying thegap size corner shape and numerical solver Fully nonlinearpotential flow solver and Navier-Stokes equation solver forviscous flowswere employed for computations using differentgap sizes and corner shapes to determine the contributionsof viscosity and vortices to the gap flows In this paper wepresent the governing equations and numerical methods ofeach flow solverThe computational and boundary conditionsalong with the grid convergence test results are describedNumerical results are presented and compared with theexperimental results of Faltinsen et al [13] Vortical flowfields according to computational fluid dynamics (CFD)are discussed in detail to better understand the relevantphysics The force response amplitude operator (RAO) andhydrodynamic coefficients are also compared for differentflow solvers Finally conclusions are drawn based on thecomputations along with remarks on the eddy damping

2 Numerical Modeling

A fully nonlinear potential code [16] and a commercialCFD software (FLUENT ver 63) were used The governingequations and numerical schemes are summarized as follows

21 Potential Flows Assuming the computational domain isfilled with homogenous inviscid incompressible and irro-tational fluid the Laplace equation is applied as a governingequation using the velocity potential 120601

nabla2120601 = 0 (1)

On the free surface the fully nonlinear dynamic andkinematic boundary conditions are respectively applied asfollows

120597120601

120597119905= minus119892120578 minus

1

2

1003816100381610038161003816nabla12060110038161003816100381610038162

minus119875119886

120588

120597120578

120597119905= minusnabla120601 sdot nabla120578 +

120597120601

120597119911

(2)

where 119905 is the time 120578 is the free surface elevation 119892 is thegravitational acceleration 120588 is the water density 119875

119886is the

pressure on the free surface assumed to be zero and 119911 is thevertical coordinate

No normal-flux condition is applied to rigid boundaries

120597120601

120597119899= 0 (3)

To solve the Laplace equation with all boundary condi-tions the Green function is applied and the governing equa-tion can be transformed to the boundary integral equation

120572120601119894= intintΩ

(119866120597120601119895

120597119899minus 120601119895

120597119866

120597119899)119889119904 (4)

where 119866 is the basic Green function satisfying the Laplaceequation and 120572 is a solid angle (120572 = 05 when singularitiesare on the boundary)

By discretizing all boundary surfaces and placing a nodeon each panelmdashwhich is called the constant panel methodmdashand integrating the discretized boundary integral equationthe velocity potential and its normal derivative on eachsegment can be obtained at each time step For 2D problemsthe Green function (119866

119894119895) as a simple source is given by

119866119894119895(119909119894 119911119894 119909119895 119911119895) = minus

1

2120587ln1198771 (5)

where R1is the distance between source and field points

The velocity potential and free-surface elevation at eachtime step are obtained by using the Runge-Kutta fourth-ordertime integration scheme and the mixed Eulerian-Lagrangianmethod to solve (2) The total derivative (120575120575119905) = (120597120597119905) +V sdot nabla can be used to modify the fully nonlinear free-surfaceconditions in the Lagrangian frame

The nonlinear body pressure can be calculated fromBernoullirsquos equation

119901 = minus120588119892119911 minus 120588120597120601

120597119905minus1

21205881003816100381610038161003816nabla120601

10038161003816100381610038162

(6)

Finally the nonlinear wave forces on a body can beobtained by integrating the nonlinear pressure over theinstantaneous wetted body surface 119878

119887at each time step

rarr

119865= intint119878119887

119901rarr

119899 119889119904 (7)

whererarr

119899 is the unit normal vector of the instantaneous bodysurface

Mathematical Problems in Engineering 3

An artificial damping zone is used in front of the end wallto dissipate the outgoing wave energy as much as possiblethat is thewave energy gradually dissipates in the direction ofwave propagation This is necessary to avoid wave reflectionfrom the rigid end wall Both 120601

119899and 120578-type damping terms

are added to the fully nonlinear dynamic and kinematic free-surface conditions

A Chebyshev five-point smoothing scheme is appliedalong the free surface during time marching to avoid theso-called saw-tooth numerical problem which is not aphysical phenomenon but a numerical instability Detailedmathematical formulations for the numerical schemes usedhere are given by Koo and Kim [16 17]

22 Viscous Flows The governing equations for viscousincompressible and rotational fluids are the continuity andNavier-Stokes equations

nabla sdotrarr

119881= 0

120597 (120588rarr

119881)

120597119905+ nabla sdot (120588

rarr

119881rarr

119881) = minusnabla119901 + nabla sdot (120591) + 120588rarr

119892

(8)

where 120591 is the stress tensor which is defined as follows

120591 = 120583 [(nablararr

119881 +nablararr

119881119879

) minus2

3nabla sdotrarr

119881119868] (9)

where 120583 is the molecular viscosity and 119868 is the unit tensorThe governing equations can be written in the following

integral form for a general scalar 120595 on an arbitrary controlvolume 119881 whose boundary is moving at the mesh velocity ofthe moving mesh

rarr

119881119892

int119881

120597 (120588120595)

120597119905119889119881 + ∮

119878

120588120595(rarr

119881 minusrarr

119881119892) sdot 119889rarr

119860

= ∮119878

120583nabla120595 sdot 119889rarr

119860 +int119881

119878120595119889119881

(10)

where 119878 is the control surfacerarr

119860 is the area vector normal to119878 and 119878

120595is the source term

The integral equation is discretized by the FVM into alinear algebraic systemThe second-order upwind and centraldifference schemes are used for the convection and diffusionterms respectively The pressure implicit with splitting ofoperators (PISO) algorithm is used to couple the pressure andvelocity for the continuity equation

The free surface is represented by an explicit volumeof fluid (VOF) method with a modified high-resolutioninterface capturing schemeThe range of the volume fractionof the air and water varies from 0 to 1 where the free surfaceis defined at 05

3 Boundary and Computational Conditions

The computational domain was set up based on experimentsby Faltinsen et al [13] who performed tests in a 2D wave

B

D x

z

L1

Figure 1 Schematic definition of moonpool

Table 1 Principal particulars of computational setup

Narrow case Wide caseBeam 119861 (m) 036 036Draft119863 (m) 018 018Heave amplitude 120578

3119886(m) 00025 00025

Gap 1198711(m) 018 036

Corner shape Squarerounded Square

flume tank and measured the wave height inside and outsidethe moonpool Figure 1 shows the schematic definition of theexperimental setup

In this research we carried out numerical simulationsfor two moonpool geometries narrow and wide gaps Thenarrow moonpool had square and rounded corner shapesPrincipal particulars are summarized in Table 1 Details onthe boundary conditions are explained in the subsection foreach flow solver

31 Potential Flow Computations The water depth was setto 103m and the width of the entire computational domainwas 135m these are very similar to the wave tank conditionsin the experiments The bottom and the two bodies weretreated as solid walls that is no flux through the boundaryA numerical damping zone of 2m was placed on both sidesof the free-surface area to prevent reflection from the sideboundary walls

In total there were 265 elements and about 20 panelsdistributed over one wavelength 120582 The number of panels perwavelength was chosen from a set of panel convergence teststhis is discussed in the next section

The time step was T128 where T is the period of pre-scribed body motion Because the simulation was fully non-linear the instantaneous wetted body surface was updatedand regridded at each time step The fully nonlinear free-surface condition was also satisfied at the instantaneous free-surface positions

32 Viscous Flow Computations The CFD computationswere carried out with half of the numerical wave tank asymmetry boundary condition was applied to the center lineof themoonpoolThewidth of the computational domainwas150m consisting of a 70m wave zone and 80m artificialdamping zoneThehorizontal size of themesh in the dampingzone was exponentially increased for artificial damping toprevent reflected waves [18] On the bottom boundary 103mdeep no-slip wall was applied for the viscous computationsHydrostatic pressure was imposed and a gradient conditionfor the velocity and volume fraction was applied at the outlet


03 04 05 06 07 08 090

2

4

6

8

10

12

14

Outside

Inside

10 panels12058220 panels12058230 panels120582

40 panels120582Linear [13]

1205783120578

3a

1205962L1g

(a) Panel convergence

Time (s)

Wav

e hei

ght (

m)

0 20 40 60 80

0

001

002

003

004

005

CoarseMediumFine

69 70 71

00050000

001000150020

minus001

minus002

minus003

(b) Grid convergence

Figure 2 Wave height in the narrow gap for grid convergence

The entire computational domain with the grid systemperiodically moved up and down during the computationaccording to the prescribed bodymotion as shown in (11) thesize of the cells remained unchanged except for the top andbottom boundary cells Only the closest rows of cells to thetop and bottom boundaries changed their height to absorbthe imposed heaving motion at every time step

V3= minus1205783119886120596 cos (120596119905) (11)

where V3 1205783119886

of 00025m and 120596 represent the verticalvelocity heave motion amplitude and angular frequencyrespectively

The viscous flow was assumed to be laminar becausethe prescribed oscillating velocity was low with a Reynoldsnumber at the approximate order of 103 A total of 176000rectangular cells which is a highly dense grid for 2D compu-tation were used after a grid dependency test Such a high-resolution grid may be capable of capturing flow structuresincluding shedding vortices

The minimum height and width of the cell were 120578311988620

and 00042 B respectively Equally spaced regular rectangularcells were allocated inside the moonpool and on the freesurface to capture the precise vortex structure and waveelevation

The time step was set to 0005 s which is around T250A linear ramp duration was employed during the initial 10 sto prevent sudden transient fluctuations

4 Numerical Results

41 Grid Convergence The number of panels per wavelengthis an important variable for potential flow computations

Table 2 Principal particulars for CFD grid convergence tests

Grid Total number of cells Min 119889119909 Min 119889119911

Coarse 84375 00020m(00056119861)

00010m(041205783119886)

Medium 116450 00015m(00042119861)

00010m(041205783119886)

Fine 176000 00015m(00042119861)

00005m(021205783119886)

Tests were performed on the narrow moonpool for 10 2030 and 40 panels per wavelength Figure 2(a) shows the waveheight RAOs inside and outside the moonpoolThe case of 10panels per wavelength showed a discrepancy from the othersat least 20 panels per wavelength were required

For the CFD simulations three grid sets were comparedcoarse medium and fine grids Table 2 lists the details ofeach grid The medium grid was only denser than the coarseone in the horizontal direction near the free surface The fineone was also twice as dense vertically over the height of thebody Figure 2(b) shows the time history of the wave elevationfor the narrow moonpool at the peak frequency The threegrid sets showed fully converged agreement with each otherthe fine grid set was chosen to capture vortex shedding moreprecisely

42 Wave Elevation The free-surface elevations reached aperiodically steady state within 60 s for all runs Compu-tations were performed which took longer than 60 s datasampling and analysis were carried out on elevations obtainedafter 60 s Figure 3 shows the free-surface elevation of viscous


Time (s)0 2010 4030 6050 8070

Wav

e hei

ght (

m) 001

002

003

004

minus001

minus002

minus003

minus004

0

wout

win

(a) 12059621198711119892 = 05180 narrow

Time (s)0 2010 4030 6050 8070

wout

win

Wav

e hei

ght (

m) 001

002

003

004

minus001

minus002

minus003

minus004

0

(b) 12059621198711119892 = 07719 wide

Figure 3 Sequential wave elevation probed inside and outside the moonpool

flows around the peak frequency The black-dashed dot linerepresents the forcing amplitude for reference and 119908in (redline) and 119908out (blue dotted line) are the wave elevation atthe symmetry line inside the moonpool and 700mm outsidethe body respectively The tendency of the time series ofwave elevations observed in the CFD computations was verysimilar to that of the experiment results of Faltinsen et al [13]

The wave height inside the moonpoolmdasha major concernin moonpool operations and side-by-side mooringsmdashwascompared with the experimental results shown in Figure 4In the narrow moonpool the fully nonlinear viscous resultspredicted the resonance frequency and its amplitude wellwhereas the fully nonlinear potential flow solver predictedhigher wave elevations than the experiments which is thesame as the linear results of Faltinsen et al [13] In thepresent example the small forcing amplitude did not bringabout appreciable nonlinearity in the potential flow solversTherefore the discrepancy between the potential resultsand experiments can be attributed to the viscous effectsThe viscous flow computations actually produced excellentcorrelation against the measurement and did not show anyoverestimation at the peak frequency

Interestingly such overestimation by the potential flowsolvers almost disappeared in the wide-gap case The poten-tial and viscous results are very similar which means thatviscous effects in this case are not significantThis will furtherbe discussed in the next section Compared to the narrowgap case the free-surface elevation became lower and theresonance occurred at a higher frequency The difference inpeak frequency relative to the experiment results was becauseof the reflected waves of the experiments as pointed out byKristiansen and Faltinsen for their experiments [14]

43 Viscous Flow Fields CFD is an expensive computationaltool compared to panel methods Nevertheless it can providevaluable clues to understanding the relevant physics Thevelocity vorticity and pressure fields were investigated tobetter understand the flow fields

Figure 5 shows the velocity vectors and vorticity contoursat two frequencies off the resonant frequency low and highWeak vortices were generated at both sides Vortices wereattached around the corners and the flow fields were notdisturbed at the low frequency In contrast the vortices atthe high frequency were shed into a circular trail and slightlydisturbed the vertical watermotion like awater column insidethe moonpool The difference in inner and outer velocitymagnitudes was not significant thus a small amount of swayforce was induced

Stronger vortices were generated at the resonant fre-quency as shown in Figure 6 Because the water velocityinside themoonpool was faster than that on the outsidemorevigorous vortex shedding was observed in the vicinity of theinner corner The vortex changed the direction of rotation atthe corner as the body changed the direction of its verticalmotion The shed vortices trailed into a circular motion forwhich the diameter reached approximately half the size of themoonpool The vortices interrupted the vertical motion anddisturbed most of the water column leaving only the upperpart undisturbed Shear-stress-driven vortices were observedon the sidewall of themoonpool they become another sourceof viscous damping

The resonant vertical movement of the water columnaffected the pressure field as shown in Figure 7 The innerpressure was periodically higher or lower than the outerpressure corresponding to the free surface elevation The


03 04 05 06 07 08 09

2

0

4

6

8

10

12

141205783120578

3a

win potentialwout potentialwin viscouswout viscous

win Exp [13]wout Exp [13]Linear [13]

1205962L1g

(a) Narrow

06 07 08 09 1 11 12

2

0

4

6

8

10

12

14

1205783120578

3a



1205962L1g

(b) Wide

Figure 4 Wave height RAOs inside and outside the moonpool

t = 719 s t = 719 s

050045040035030025020015010005000

100806040200minus20minus40minus60minus80minus100

(a) 12059621198711119892 = 03975

t = 719 st = 719 s

050045040035030025020015010005000


(b) 12059621198711119892 = 07244

Figure 5 Velocity vectors and vorticity contours in the narrow moonpool


t = 715 s t = 715 s

t = 719 s t = 719 s

t = 723 s t = 723 s

050045040035030025020015010005000


050045040035030025020015010005000

050045040035030025020015010005000



Figure 6 Velocity vectors and vorticity contours in the narrow moonpool near resonance frequency 12059621198711119892 = 05180

t = 715 s t = 721 s

120100200

80400minus40minus80

minus100minus200

minus120

120100200

80400minus40minus80

minus100minus200

minus120

Figure 7 Pressure contours in the narrow moonpool near resonance frequency 12059621198711119892 = 05180


t = 721 s t = 721 s

t = 725 s t = 725 s

t = 729 s t = 729 s

050045040035030025020015010005000

050045040035030025020015010005000

050045040035030025020015010005000




Figure 8 Velocity vectors and vorticity contours in the wide moonpool near resonance frequency 12059621198711119892 = 08252

difference in pressure between inside and outside the moon-pool induced a sway force When the free surface reachedits lowest level the inner pressure became minimum thus asway force inward the moonpool and heave force downwardwere generated and vice versa

Figure 8 shows the velocity vectors and vorticity contoursin the wide gap at the peak frequency A similar vortexbehavior as that in the narrow gap was observed Vorticesgenerated at the inner corner showed lower strength thanthose at the narrowmoonpool owing to its weaker resonanceThe shed vortices moved along a big circular trace howeverthey could not block the movement of the water column asmuch as in the narrow gap This is one reason for the smalldiscrepancy between the potential and viscous flows in thewave height RAO shown in Figure 4

44 Corner Shapes In the previous section the shed vorticesfrom the square corner were shown to disturb the verticalwater motion We changed the corner shape to be roundedto investigate its effect on vortex shedding A bilge radiusof 5 of the draft was applied Although the vortices didnot completely vanish the strength was weaker than for thesquare corners as shown in Figure 9 The vortices in theboundary layer along the wall by shear stress still remainedThe circular track of the shed vortices from the square cornerwas no longer observed Instead the water columnmoved upand down very freely like a solid column owing to the weakervortex strength and the disappearance of the circular trace ofthe shed vortices

The effect of weakened vortex shedding due to roundedcorner is shown in Figure 10(a) with the wave height RAOs


t = 715 s t = 715 s

t = 719 s t = 719 s

t = 723 s t = 723 s

050045040035030025020015010005000


050045040035030025020015010005000


050045040035030025020015010005000


Figure 9 Velocity vectors and vorticity contours in the narrowmoonpool with rounded corner near resonance frequency 12059621198711119892 = 05180

In the fully nonlinear potential flows the change in cor-ner shapes little affected the maximum wave amplitude asexpected and they showed similar RAOs However applyingthe bilge radius significantly increased the wave height in theviscous flows This fact may be very important in the designof oscillating water column (OWC) wave energy converter(WEC) At low and high frequencies the wave height was notaffected by the corner shapes As the frequency approachedthe resonant frequency the wave height increased by up to32 This significant increase in the wave height explainsthe important role of vortex shedding associated with theinteraction with free-surface motion Vortex shedding fromthe hull or appendages such as damping platesmay contributeto the suppression of harmful free-surface behavior It is alsonotable that the damping effects are dominant around the

peak frequency thus artificial treatments to suppress thewave elevation might be taken at the resonance

The phase shift was not related to the corner shapesand vortex shedding as shown in Figure 10(b) The phaseshift of the two corner shapes in potential and viscous flowscoincided with each other and showed good agreement withthe experimental results

5 Hydrodynamic Forces

51 Force RAOs As discussed in the previous section themovement of water inside themoonpool generates a periodicpressure change that causes hydrodynamic forces Figure 11shows the sway andheave forceRAOs respectively calculatedusing viscous CFD solvers The wave height RAO is included


03 04 05 06 07 08 09

2

0

4

6

8

10

12

141205783120578

3a

win potential (square)wout potential (square)win potential (rounded)wout potential (rounded)win viscous (square)wout viscous (square)

win viscous (rounded)wout viscous (rounded)win Exp [13]wout Exp [13]Linear [13]

1205962L1g

(a) RAOs

Phas

e shi

f (ra

d)

03 04 05 06 07 08 09

2

0

4

6

8

10

12

win potential (square)wout potential (square)win potential (rounded)wout potential (rounded)win viscous (square)

wout viscous (square)win viscous (rounded)wout viscous (rounded)win Exp [13]wout Exp [13]

1205962L1g

(b) Phase shift

Figure 10 Wave height RAOs and phase shift in the narrow moonpool

Table 3 Effects of the corner shape of the narrow moonpool at12059621198711119892 = 05180

Corner shape Square Rounded Wave elevation (119908in1205783119886) 887 1171 1320Sway force (119865

2120578311988610minus3) 1135 1517 1337

Heave force (1198653120578311988610minus3) 751 941 1253

for reference It is definitely observed that the wave heightRAO is in the great agreement with the force RAOs showinga similarity The phase of the two forces also became close tothat of the wave elevation and approached 120587

Table 3 shows the effects of the corner shape whichresulted in the vortex damping near the peak frequencyIn the narrow gap case the corner shape can reduce thewave elevation and sway and heave forces by about 30This remarkable amount of damping should be accountedfor while designing offshore structures Taking the similaritybetween the wave elevation and the forces into accountaccurate estimation of the wave height is highly requiredto estimate hydrodynamic forces acting on the bodies ormooring lines if they are moored

52HydrodynamicCoefficients The linear equation for heavemotion is given in (12) With Newtonrsquos second law in (13) we

can use the least square method to calculate hydrodynamiccoefficients [16]

(11986033+119872) 120578 + 119861

33120578 + 11986233120578 = 119891 (119905) (12)

119872 120578 = 119865 (119905) (13)

where A33 M B

33 C33 f (t) F(t) and 120578 are the added

mass body mass linear damping coefficient restoring coef-ficient wave exciting force total force on the body andheave displacement respectively The quadratic dampingterm 119861

33qd| 120578| 120578may be added to the equation of motionBy subtracting the restoring force from F(t) the added

mass and linear damping can be derived as

11986033= minus

int119865 (119905) 120578 119889119905

int 1205782 119889119905

11986133= minus

int119865 (119905) 120578 119889119905

int 1205782 119889119905

(14)

If we consider the quadratic damping term another termwith the quadratic damping coefficient emerges The linearand quadratic damping terms are coupled to each other Inthe present study we considered only linear damping

Figure 12 shows the hydrodynamic coefficients for thenarrowmoonpoolThe present results showed typical charac-teristics of the piston mode in the moonpool well such as the


0

2

4

6

8

10

12

14

16

18

20

03 04 05 06 07 08 09

2

0

4

6

8

10

12

14

16

1205783120578

3a

F2120578

3a

times103

SquareRound12057831205783a square

1205962L1g

(a) Sway force RAOs

0

2

4

6

8

03 04 05 06 07 08 09

10

12

Phas

e shi

f (ra

d)

SquareRoundwin square

1205962L1g

(b) Phase shift of sway force

024681012141618202224

03 04 05 06 07 08 09

1205783120578

3a

0

2

4

6

8

10

12

14

16

18

20

F3120578

3a

times103


1205962L1g

(c) Heave force RAOs

03 04 05 06 07 08 090

2

4

6

8

10

12

Phas

e shi

f (ra

d)


1205962L1g

(d) Phase shift of heave force

Figure 11 Force RAOs and phase shift in the narrow gap of viscous flows

negative added mass and linear damping approaching zeroaround the peak frequency

The hydrodynamic coefficients from the potential-flowsimulations were appreciably higher than those from theviscous flows especially for sharp corners This result isrelated to the overestimation of the free-surface elevation bythe potential-flow solvers Since forced-motion amplitude issmall in the present example the fully nonlinear potential-flow simulation is not significantly different from that ofFaltinsenrsquos linear calculation The effects of the corner shape

are small as intuitively expected in the potential flows incontrast it has a great effect on the viscous flow which cantake the rotational fluid motions into account

The hydrodynamic coefficients were found to be very sen-sitive to the hull forms only the viscous flow solver providedreliable results It is difficult to derive a general formula thatconsists of the main particulars of the moonpools Furtherstudy on the eddy damping and its quantification is neededfor practical purposes We carefully propose to quantifythe difference in hydrodynamic coefficients between the

12 Mathematical Problems in EngineeringAd

ded

mas

s and

dam

ping

coe

ffici

ent

0302 04 05 06 07 08 09 1

0

2

4

6

8

A33 potential (square)B33 potential (square)A33 potential (rounded)B33 potential (rounded)A33 viscous (square)

B33 viscous (square)A33 viscous (rounded)B33 viscous (rounded)A33 linear [13]B33 linear [13]

minus2

minus4

minus6

1205962L1g

Figure 12 Hydrodynamic coefficients in the narrow moonpool

viscous and potential results This might be implemented asnonlinear viscous damping in time domain motion analysisto account for the eddy and viscous damping

6 Conclusion

A two-dimensionalmoonpool by a rectangular section underforced heavemotion including pistonmode was successfullysimulated by fully nonlinear solvers for potential and viscousflows

All numerical methods of the present study predicted theresonant frequency for different gap sizes well The viscousflow CFD solver estimated the most reliable and acceptableresonance amplitudes compared to the experimental resultsThe potential flow solver tends to overestimate the free-surface elevation near the resonance frequency Outside theresonance region potential solutions are also reliable Inthe wide-gap case the moonpool resonance is relativelyweak and their differences are not appreciable The viscoussolver produces significant differences inmoonpool elevationbetween the narrow-gap cases of sharp and rounded cornerswhile the phenomenon cannot be reasonably predicted by thepotential simulations

The free-surface elevation inside the moonpool wasfound to be related to the hydrodynamic forces To accu-rately estimate the hydrodynamic forces and hydrodynamiccoefficients the free-surface elevation should be correctlypredicted first

Vortex shedding and surface viscosity are the mainsources of the damping that differentiated the viscous andpotential flow solvers The vortical flow fields at the resonant

frequency clearly showed the mechanism and contributionof vortex shedding and it should be taken into account formoonpool problemsThe amount of eddy dampingwas foundto depend on the sharpness of corners therefore free-surfaceelevation may be controlled by changing the corner shape orappended devices

Conflict of Interests

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

References

[1] J W English ldquoA means of reducing the oscillations in drillwellscaused by the vesselrsquos forward speedrdquoTheNaval Architect no 3pp 88ndash90 1976

[2] J Yoo and S-H Choi ldquoPerformance of drillship withmoonpooland azimuth thrustersrdquo Journal of the Society of Naval Architectsof Korea vol 41 no 6 pp 33ndash39 2004 (Korean)

[3] R V Veer and H J Tholen ldquoAdded resistance of moonpools incalm waterrdquo in Proceedings of the 27th International Conferenceon Offshore Mechanics and Arctic Engineering (OMAE rsquo08)Paper no 57246 pp 153ndash162 Estoril Portugal June 2008

[4] K Fukuta ldquoBehavior of water in vertical well with bottomopening of ship and its effects on ship motionrdquo Journal of theSociety of Naval Architects of Japan vol 141 pp 107ndash122 1977(Japanese)

[5] A B Aalbers ldquoThe water motions in a moonpoolrdquo OceanEngineering vol 11 no 6 pp 557ndash579 1984

[6] B Molin ldquoOn the piston and sloshing modes in moonpoolsrdquoJournal of Fluid Mechanics vol 430 pp 27ndash50 2001

[7] O M FaltinsenHydrodynamics of High-Speed Marine VehiclesCambridge University Press 2005

[8] B J Koo and M H Kim ldquoHydrodynamic interactions andrelative motions of two floating platforms with mooring linesin side-by-side offloading operationrdquo Applied Ocean Researchvol 27 no 6 pp 292ndash310 2005

[9] S Y Hong J H Kim S K Cho Y R Choi and Y SKim ldquoNumerical and experimental study on hydrodynamicinteraction of side-by-side moored multiple vesselsrdquo OceanEngineering vol 32 no 7 pp 783ndash801 2005

[10] R H M Huijsmans J A Pinkster and J J De Wilde ldquoDiffrac-tion and radiation of waves around side-by-side moored ves-selsrdquo in Proceedings of the 11th International Offshore and PolarEngineering Conference pp 406ndash412 Oslo Norway June 2001

[11] X-B Chen ldquoHydrodynamics in offshore and navalApplicationsmdashpart Irdquo in Proceedings of the 6th InternationalConference on Hydrodynamics (ICHD rsquo04) Perth AustraliaNovember 2004

[12] T Bunnik W Pauw and A Voogt ldquoHydrodynamic analysis forside-by-side offloadingrdquo in Proceedings of the 19th InternationalOffshore and Polar Engineering Conference pp 648ndash653 OsakaJapan June 2009

[13] O M Faltinsen O F Rognebakke and A N Timokha ldquoTwo-dimensional resonant piston-like sloshing in a moonpoolrdquoJournal of Fluid Mechanics vol 575 pp 359ndash397 2007

[14] T Kristiansen and O M Faltinsen ldquoApplication of a vortextracking method to the piston-like behaviour in a semi-entrained vertical gaprdquo Applied Ocean Research vol 30 no 1pp 1ndash16 2008


[15] T Kristiansen and OM Faltinsen ldquoGap resonance analyzed bya newdomain-decompositionmethod combining potential andviscous flowrdquoAppliedOcean Research vol 34 pp 198ndash208 2012

[16] W C Koo and M H Kim ldquoNumerical simulation of nonlinearwave and force generated by a wedge-shape wavemakerrdquoOceanEngineering vol 33 no 8-9 pp 983ndash1006 2006

[17] W Koo and M-H Kim ldquoFreely floating-body simulation by a2D fully nonlinear numerical wave tankrdquo Ocean Engineeringvol 31 no 16 pp 2011ndash2046 2004

[18] J-C Park M-H Kim and H Miyata ldquoFully nonlinear free-surface simulations by a 3D viscous numerical wave tankrdquoInternational Journal for Numerical Methods in Fluids vol 29pp 685ndash703 1999

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


production storage offloading (LNG-FPSO) and liquefiednatural gas floating storage and regasification unit (LNG-FSRU) with a side-by-side mooring arrangement pistonmode resonance has drawn more attention The free-surfaceelevation inside the gap and the motion of the two float-ing bodies are important design factors In general linearpotential theory in the frequency domain is widely usedin the design process However this approach inherentlyoverestimates the wave elevation near resonance Koo andKim [8] and Hong et al [9] studied side-by-side mooredsystems and addressed the discrepancy in the piston modeThe overestimation of the free-surface elevation can besuppressed by employing a damping lid or artificial damping[10ndash12] However the exact value of the artificial damping isnot available prior to model tests and improper use of the lidor artificial damping coefficientmay decrease reliability in thenumerical estimation

Faltinsen et al [13] investigated the piston mode occur-ring in a square-shaped catamaran hull form under forcedheave motion by using experiments and linear numericalanalysis Kristiansen and Faltinsen [14] compared their sim-ulation results by using the vortex-tracking method withexperimental results They recently used a finite volumemethod-based domain decomposition method in which theviscous domain for the body region communicates with theoutside potential region [15]

The aim of this study is to determine the effects of eddydamping and viscosity on the piston mode by varying thegap size corner shape and numerical solver Fully nonlinearpotential flow solver and Navier-Stokes equation solver forviscous flowswere employed for computations using differentgap sizes and corner shapes to determine the contributionsof viscosity and vortices to the gap flows In this paper wepresent the governing equations and numerical methods ofeach flow solverThe computational and boundary conditionsalong with the grid convergence test results are describedNumerical results are presented and compared with theexperimental results of Faltinsen et al [13] Vortical flowfields according to computational fluid dynamics (CFD)are discussed in detail to better understand the relevantphysics The force response amplitude operator (RAO) andhydrodynamic coefficients are also compared for differentflow solvers Finally conclusions are drawn based on thecomputations along with remarks on the eddy damping

2 Numerical Modeling

A fully nonlinear potential code [16] and a commercialCFD software (FLUENT ver 63) were used The governingequations and numerical schemes are summarized as follows

21 Potential Flows Assuming the computational domain isfilled with homogenous inviscid incompressible and irro-tational fluid the Laplace equation is applied as a governingequation using the velocity potential 120601

nabla2120601 = 0 (1)

On the free surface the fully nonlinear dynamic andkinematic boundary conditions are respectively applied asfollows

120597120601

120597119905= minus119892120578 minus

1

2

1003816100381610038161003816nabla12060110038161003816100381610038162

minus119875119886

120588

120597120578

120597119905= minusnabla120601 sdot nabla120578 +

120597120601

120597119911

(2)

where 119905 is the time 120578 is the free surface elevation 119892 is thegravitational acceleration 120588 is the water density 119875

119886is the

pressure on the free surface assumed to be zero and 119911 is thevertical coordinate

No normal-flux condition is applied to rigid boundaries

120597120601

120597119899= 0 (3)

To solve the Laplace equation with all boundary condi-tions the Green function is applied and the governing equa-tion can be transformed to the boundary integral equation

120572120601119894= intintΩ

(119866120597120601119895

120597119899minus 120601119895

120597119866

120597119899)119889119904 (4)

where 119866 is the basic Green function satisfying the Laplaceequation and 120572 is a solid angle (120572 = 05 when singularitiesare on the boundary)

By discretizing all boundary surfaces and placing a nodeon each panelmdashwhich is called the constant panel methodmdashand integrating the discretized boundary integral equationthe velocity potential and its normal derivative on eachsegment can be obtained at each time step For 2D problemsthe Green function (119866

119894119895) as a simple source is given by

119866119894119895(119909119894 119911119894 119909119895 119911119895) = minus

1

2120587ln1198771 (5)

where R1is the distance between source and field points

The velocity potential and free-surface elevation at eachtime step are obtained by using the Runge-Kutta fourth-ordertime integration scheme and the mixed Eulerian-Lagrangianmethod to solve (2) The total derivative (120575120575119905) = (120597120597119905) +V sdot nabla can be used to modify the fully nonlinear free-surfaceconditions in the Lagrangian frame

The nonlinear body pressure can be calculated fromBernoullirsquos equation

119901 = minus120588119892119911 minus 120588120597120601

120597119905minus1

21205881003816100381610038161003816nabla120601

10038161003816100381610038162

(6)

Finally the nonlinear wave forces on a body can beobtained by integrating the nonlinear pressure over theinstantaneous wetted body surface 119878

119887at each time step

rarr

119865= intint119878119887

119901rarr

119899 119889119904 (7)

whererarr

119899 is the unit normal vector of the instantaneous bodysurface







nabla sdotrarr

119881= 0

120597 (120588rarr

119881)

120597119905+ nabla sdot (120588

rarr

119881rarr


119892

(8)


120591 = 120583 [(nablararr

119881 +nablararr

119881119879

) minus2

3nabla sdotrarr

119881119868] (9)



rarr

119881119892

int119881

120597 (120588120595)

120597119905119889119881 + ∮

119878

120588120595(rarr

119881 minusrarr

119881119892) sdot 119889rarr

119860

= ∮119878

120583nabla120595 sdot 119889rarr

119860 +int119881

119878120595119889119881

(10)








B

D x

z

L1




3119886(m) 00025 00025

Gap 1198711(m) 018 036









03 04 05 06 07 08 090

2

4

6

8

10

12

14

Outside

Inside



1205783120578

3a

1205962L1g


Time (s)

Wav

e hei

ght (

m)

0 20 40 60 80

0

001

002

003

004

005

CoarseMediumFine

69 70 71

00050000

001000150020

minus001

minus002

minus003




V3= minus1205783119886120596 cos (120596119905) (11)

where V3 1205783119886






4 Numerical Results




Coarse 84375 00020m(00056119861)

00010m(041205783119886)

Medium 116450 00015m(00042119861)

00010m(041205783119886)

Fine 176000 00015m(00042119861)

00005m(021205783119886)





Time (s)0 2010 4030 6050 8070

Wav

e hei

ght (

m) 001

002

003

004

minus001

minus002

minus003

minus004

0

wout

win

(a) 12059621198711119892 = 05180 narrow

Time (s)0 2010 4030 6050 8070

wout

win

Wav

e hei

ght (

m) 001

002

003

004

minus001

minus002

minus003

minus004

0

(b) 12059621198711119892 = 07719 wide










03 04 05 06 07 08 09

2

0

4

6

8

10

12

141205783120578

3a



1205962L1g

(a) Narrow

06 07 08 09 1 11 12

2

0

4

6

8

10

12

14

1205783120578

3a



1205962L1g

(b) Wide


t = 719 s t = 719 s

050045040035030025020015010005000


(a) 12059621198711119892 = 03975

t = 719 st = 719 s

050045040035030025020015010005000


(b) 12059621198711119892 = 07244



t = 715 s t = 715 s

t = 719 s t = 719 s

t = 723 s t = 723 s

050045040035030025020015010005000


050045040035030025020015010005000

050045040035030025020015010005000




t = 715 s t = 721 s

120100200

80400minus40minus80

minus100minus200

minus120

120100200

80400minus40minus80

minus100minus200

minus120



t = 721 s t = 721 s

t = 725 s t = 725 s

t = 729 s t = 729 s

050045040035030025020015010005000

050045040035030025020015010005000

050045040035030025020015010005000










t = 715 s t = 715 s

t = 719 s t = 719 s

t = 723 s t = 723 s

050045040035030025020015010005000


050045040035030025020015010005000


050045040035030025020015010005000









03 04 05 06 07 08 09

2

0

4

6

8

10

12

141205783120578

3a



1205962L1g

(a) RAOs

Phas

e shi

f (ra

d)

03 04 05 06 07 08 09

2

0

4

6

8

10

12



1205962L1g

(b) Phase shift




2120578311988610minus3) 1135 1517 1337






(11986033+119872) 120578 + 119861

33120578 + 11986233120578 = 119891 (119905) (12)

119872 120578 = 119865 (119905) (13)

where A33 M B





11986033= minus

int119865 (119905) 120578 119889119905

int 1205782 119889119905

11986133= minus

int119865 (119905) 120578 119889119905

int 1205782 119889119905

(14)




0

2

4

6

8

10

12

14

16

18

20

03 04 05 06 07 08 09

2

0

4

6

8

10

12

14

16

1205783120578

3a

F2120578

3a

times103


1205962L1g

(a) Sway force RAOs

0

2

4

6

8

03 04 05 06 07 08 09

10

12

Phas

e shi

f (ra

d)


1205962L1g


024681012141618202224

03 04 05 06 07 08 09

1205783120578

3a

0

2

4

6

8

10

12

14

16

18

20

F3120578

3a

times103


1205962L1g


03 04 05 06 07 08 090

2

4

6

8

10

12

Phas

e shi

f (ra

d)


1205962L1g








ded

mas

s and

dam

ping

coe

ffici

ent

0302 04 05 06 07 08 09 1

0

2

4

6

8



minus2

minus4

minus6

1205962L1g



6 Conclusion








References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















nabla sdotrarr

119881= 0

120597 (120588rarr

119881)

120597119905+ nabla sdot (120588

rarr

119881rarr


119892

(8)


120591 = 120583 [(nablararr

119881 +nablararr

119881119879

) minus2

3nabla sdotrarr

119881119868] (9)



rarr

119881119892

int119881

120597 (120588120595)

120597119905119889119881 + ∮

119878

120588120595(rarr

119881 minusrarr

119881119892) sdot 119889rarr

119860

= ∮119878

120583nabla120595 sdot 119889rarr

119860 +int119881

119878120595119889119881

(10)








B

D x

z

L1




3119886(m) 00025 00025

Gap 1198711(m) 018 036









03 04 05 06 07 08 090

2

4

6

8

10

12

14

Outside

Inside



1205783120578

3a

1205962L1g


Time (s)

Wav

e hei

ght (

m)

0 20 40 60 80

0

001

002

003

004

005

CoarseMediumFine

69 70 71

00050000

001000150020

minus001

minus002

minus003




V3= minus1205783119886120596 cos (120596119905) (11)

where V3 1205783119886






4 Numerical Results




Coarse 84375 00020m(00056119861)

00010m(041205783119886)

Medium 116450 00015m(00042119861)

00010m(041205783119886)

Fine 176000 00015m(00042119861)

00005m(021205783119886)





Time (s)0 2010 4030 6050 8070

Wav

e hei

ght (

m) 001

002

003

004

minus001

minus002

minus003

minus004

0

wout

win

(a) 12059621198711119892 = 05180 narrow

Time (s)0 2010 4030 6050 8070

wout

win

Wav

e hei

ght (

m) 001

002

003

004

minus001

minus002

minus003

minus004

0

(b) 12059621198711119892 = 07719 wide










03 04 05 06 07 08 09

2

0

4

6

8

10

12

141205783120578

3a



1205962L1g

(a) Narrow

06 07 08 09 1 11 12

2

0

4

6

8

10

12

14

1205783120578

3a



1205962L1g

(b) Wide


t = 719 s t = 719 s

050045040035030025020015010005000


(a) 12059621198711119892 = 03975

t = 719 st = 719 s

050045040035030025020015010005000


(b) 12059621198711119892 = 07244



t = 715 s t = 715 s

t = 719 s t = 719 s

t = 723 s t = 723 s

050045040035030025020015010005000


050045040035030025020015010005000

050045040035030025020015010005000




t = 715 s t = 721 s

120100200

80400minus40minus80

minus100minus200

minus120

120100200

80400minus40minus80

minus100minus200

minus120



t = 721 s t = 721 s

t = 725 s t = 725 s

t = 729 s t = 729 s

050045040035030025020015010005000

050045040035030025020015010005000

050045040035030025020015010005000










t = 715 s t = 715 s

t = 719 s t = 719 s

t = 723 s t = 723 s

050045040035030025020015010005000


050045040035030025020015010005000


050045040035030025020015010005000









03 04 05 06 07 08 09

2

0

4

6

8

10

12

141205783120578

3a



1205962L1g

(a) RAOs

Phas

e shi

f (ra

d)

03 04 05 06 07 08 09

2

0

4

6

8

10

12



1205962L1g

(b) Phase shift




2120578311988610minus3) 1135 1517 1337






(11986033+119872) 120578 + 119861

33120578 + 11986233120578 = 119891 (119905) (12)

119872 120578 = 119865 (119905) (13)

where A33 M B





11986033= minus

int119865 (119905) 120578 119889119905

int 1205782 119889119905

11986133= minus

int119865 (119905) 120578 119889119905

int 1205782 119889119905

(14)




0

2

4

6

8

10

12

14

16

18

20

03 04 05 06 07 08 09

2

0

4

6

8

10

12

14

16

1205783120578

3a

F2120578

3a

times103


1205962L1g

(a) Sway force RAOs

0

2

4

6

8

03 04 05 06 07 08 09

10

12

Phas

e shi

f (ra

d)


1205962L1g


024681012141618202224

03 04 05 06 07 08 09

1205783120578

3a

0

2

4

6

8

10

12

14

16

18

20

F3120578

3a

times103


1205962L1g


03 04 05 06 07 08 090

2

4

6

8

10

12

Phas

e shi

f (ra

d)


1205962L1g








ded

mas

s and

dam

ping

coe

ffici

ent

0302 04 05 06 07 08 09 1

0

2

4

6

8



minus2

minus4

minus6

1205962L1g



6 Conclusion








References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










03 04 05 06 07 08 090

2

4

6

8

10

12

14

Outside

Inside



1205783120578

3a

1205962L1g


Time (s)

Wav

e hei

ght (

m)

0 20 40 60 80

0

001

002

003

004

005

CoarseMediumFine

69 70 71

00050000

001000150020

minus001

minus002

minus003




V3= minus1205783119886120596 cos (120596119905) (11)

where V3 1205783119886






4 Numerical Results




Coarse 84375 00020m(00056119861)

00010m(041205783119886)

Medium 116450 00015m(00042119861)

00010m(041205783119886)

Fine 176000 00015m(00042119861)

00005m(021205783119886)





Time (s)0 2010 4030 6050 8070

Wav

e hei

ght (

m) 001

002

003

004

minus001

minus002

minus003

minus004

0

wout

win

(a) 12059621198711119892 = 05180 narrow

Time (s)0 2010 4030 6050 8070

wout

win

Wav

e hei

ght (

m) 001

002

003

004

minus001

minus002

minus003

minus004

0

(b) 12059621198711119892 = 07719 wide










03 04 05 06 07 08 09

2

0

4

6

8

10

12

141205783120578

3a



1205962L1g

(a) Narrow

06 07 08 09 1 11 12

2

0

4

6

8

10

12

14

1205783120578

3a



1205962L1g

(b) Wide


t = 719 s t = 719 s

050045040035030025020015010005000


(a) 12059621198711119892 = 03975

t = 719 st = 719 s

050045040035030025020015010005000


(b) 12059621198711119892 = 07244



t = 715 s t = 715 s

t = 719 s t = 719 s

t = 723 s t = 723 s

050045040035030025020015010005000


050045040035030025020015010005000

050045040035030025020015010005000




t = 715 s t = 721 s

120100200

80400minus40minus80

minus100minus200

minus120

120100200

80400minus40minus80

minus100minus200

minus120



t = 721 s t = 721 s

t = 725 s t = 725 s

t = 729 s t = 729 s

050045040035030025020015010005000

050045040035030025020015010005000

050045040035030025020015010005000










t = 715 s t = 715 s

t = 719 s t = 719 s

t = 723 s t = 723 s

050045040035030025020015010005000


050045040035030025020015010005000


050045040035030025020015010005000









03 04 05 06 07 08 09

2

0

4

6

8

10

12

141205783120578

3a



1205962L1g

(a) RAOs

Phas

e shi

f (ra

d)

03 04 05 06 07 08 09

2

0

4

6

8

10

12



1205962L1g

(b) Phase shift




2120578311988610minus3) 1135 1517 1337






(11986033+119872) 120578 + 119861

33120578 + 11986233120578 = 119891 (119905) (12)

119872 120578 = 119865 (119905) (13)

where A33 M B





11986033= minus

int119865 (119905) 120578 119889119905

int 1205782 119889119905

11986133= minus

int119865 (119905) 120578 119889119905

int 1205782 119889119905

(14)




0

2

4

6

8

10

12

14

16

18

20

03 04 05 06 07 08 09

2

0

4

6

8

10

12

14

16

1205783120578

3a

F2120578

3a

times103


1205962L1g

(a) Sway force RAOs

0

2

4

6

8

03 04 05 06 07 08 09

10

12

Phas

e shi

f (ra

d)


1205962L1g


024681012141618202224

03 04 05 06 07 08 09

1205783120578

3a

0

2

4

6

8

10

12

14

16

18

20

F3120578

3a

times103


1205962L1g


03 04 05 06 07 08 090

2

4

6

8

10

12

Phas

e shi

f (ra

d)


1205962L1g








ded

mas

s and

dam

ping

coe

ffici

ent

0302 04 05 06 07 08 09 1

0

2

4

6

8



minus2

minus4

minus6

1205962L1g



6 Conclusion








References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Time (s)0 2010 4030 6050 8070

Wav

e hei

ght (

m) 001

002

003

004

minus001

minus002

minus003

minus004

0

wout

win

(a) 12059621198711119892 = 05180 narrow

Time (s)0 2010 4030 6050 8070

wout

win

Wav

e hei

ght (

m) 001

002

003

004

minus001

minus002

minus003

minus004

0

(b) 12059621198711119892 = 07719 wide










03 04 05 06 07 08 09

2

0

4

6

8

10

12

141205783120578

3a



1205962L1g

(a) Narrow

06 07 08 09 1 11 12

2

0

4

6

8

10

12

14

1205783120578

3a



1205962L1g

(b) Wide


t = 719 s t = 719 s

050045040035030025020015010005000


(a) 12059621198711119892 = 03975

t = 719 st = 719 s

050045040035030025020015010005000


(b) 12059621198711119892 = 07244



t = 715 s t = 715 s

t = 719 s t = 719 s

t = 723 s t = 723 s

050045040035030025020015010005000


050045040035030025020015010005000

050045040035030025020015010005000




t = 715 s t = 721 s

120100200

80400minus40minus80

minus100minus200

minus120

120100200

80400minus40minus80

minus100minus200

minus120



t = 721 s t = 721 s

t = 725 s t = 725 s

t = 729 s t = 729 s

050045040035030025020015010005000

050045040035030025020015010005000

050045040035030025020015010005000










t = 715 s t = 715 s

t = 719 s t = 719 s

t = 723 s t = 723 s

050045040035030025020015010005000


050045040035030025020015010005000


050045040035030025020015010005000









03 04 05 06 07 08 09

2

0

4

6

8

10

12

141205783120578

3a



1205962L1g

(a) RAOs

Phas

e shi

f (ra

d)

03 04 05 06 07 08 09

2

0

4

6

8

10

12



1205962L1g

(b) Phase shift




2120578311988610minus3) 1135 1517 1337






(11986033+119872) 120578 + 119861

33120578 + 11986233120578 = 119891 (119905) (12)

119872 120578 = 119865 (119905) (13)

where A33 M B





11986033= minus

int119865 (119905) 120578 119889119905

int 1205782 119889119905

11986133= minus

int119865 (119905) 120578 119889119905

int 1205782 119889119905

(14)




0

2

4

6

8

10

12

14

16

18

20

03 04 05 06 07 08 09

2

0

4

6

8

10

12

14

16

1205783120578

3a

F2120578

3a

times103


1205962L1g

(a) Sway force RAOs

0

2

4

6

8

03 04 05 06 07 08 09

10

12

Phas

e shi

f (ra

d)


1205962L1g


024681012141618202224

03 04 05 06 07 08 09

1205783120578

3a

0

2

4

6

8

10

12

14

16

18

20

F3120578

3a

times103


1205962L1g


03 04 05 06 07 08 090

2

4

6

8

10

12

Phas

e shi

f (ra

d)


1205962L1g








ded

mas

s and

dam

ping

coe

ffici

ent

0302 04 05 06 07 08 09 1

0

2

4

6

8



minus2

minus4

minus6

1205962L1g



6 Conclusion








References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










03 04 05 06 07 08 09

2

0

4

6

8

10

12

141205783120578

3a



1205962L1g

(a) Narrow

06 07 08 09 1 11 12

2

0

4

6

8

10

12

14

1205783120578

3a



1205962L1g

(b) Wide


t = 719 s t = 719 s

050045040035030025020015010005000


(a) 12059621198711119892 = 03975

t = 719 st = 719 s

050045040035030025020015010005000


(b) 12059621198711119892 = 07244



t = 715 s t = 715 s

t = 719 s t = 719 s

t = 723 s t = 723 s

050045040035030025020015010005000


050045040035030025020015010005000

050045040035030025020015010005000




t = 715 s t = 721 s

120100200

80400minus40minus80

minus100minus200

minus120

120100200

80400minus40minus80

minus100minus200

minus120



t = 721 s t = 721 s

t = 725 s t = 725 s

t = 729 s t = 729 s

050045040035030025020015010005000

050045040035030025020015010005000

050045040035030025020015010005000










t = 715 s t = 715 s

t = 719 s t = 719 s

t = 723 s t = 723 s

050045040035030025020015010005000


050045040035030025020015010005000


050045040035030025020015010005000









03 04 05 06 07 08 09

2

0

4

6

8

10

12

141205783120578

3a



1205962L1g

(a) RAOs

Phas

e shi

f (ra

d)

03 04 05 06 07 08 09

2

0

4

6

8

10

12



1205962L1g

(b) Phase shift




2120578311988610minus3) 1135 1517 1337






(11986033+119872) 120578 + 119861

33120578 + 11986233120578 = 119891 (119905) (12)

119872 120578 = 119865 (119905) (13)

where A33 M B





11986033= minus

int119865 (119905) 120578 119889119905

int 1205782 119889119905

11986133= minus

int119865 (119905) 120578 119889119905

int 1205782 119889119905

(14)




0

2

4

6

8

10

12

14

16

18

20

03 04 05 06 07 08 09

2

0

4

6

8

10

12

14

16

1205783120578

3a

F2120578

3a

times103


1205962L1g

(a) Sway force RAOs

0

2

4

6

8

03 04 05 06 07 08 09

10

12

Phas

e shi

f (ra

d)


1205962L1g


024681012141618202224

03 04 05 06 07 08 09

1205783120578

3a

0

2

4

6

8

10

12

14

16

18

20

F3120578

3a

times103


1205962L1g


03 04 05 06 07 08 090

2

4

6

8

10

12

Phas

e shi

f (ra

d)


1205962L1g








ded

mas

s and

dam

ping

coe

ffici

ent

0302 04 05 06 07 08 09 1

0

2

4

6

8



minus2

minus4

minus6

1205962L1g



6 Conclusion








References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










t = 715 s t = 715 s

t = 719 s t = 719 s

t = 723 s t = 723 s

050045040035030025020015010005000


050045040035030025020015010005000

050045040035030025020015010005000




t = 715 s t = 721 s

120100200

80400minus40minus80

minus100minus200

minus120

120100200

80400minus40minus80

minus100minus200

minus120



t = 721 s t = 721 s

t = 725 s t = 725 s

t = 729 s t = 729 s

050045040035030025020015010005000

050045040035030025020015010005000

050045040035030025020015010005000










t = 715 s t = 715 s

t = 719 s t = 719 s

t = 723 s t = 723 s

050045040035030025020015010005000


050045040035030025020015010005000


050045040035030025020015010005000









03 04 05 06 07 08 09

2

0

4

6

8

10

12

141205783120578

3a



1205962L1g

(a) RAOs

Phas

e shi

f (ra

d)

03 04 05 06 07 08 09

2

0

4

6

8

10

12



1205962L1g

(b) Phase shift




2120578311988610minus3) 1135 1517 1337






(11986033+119872) 120578 + 119861

33120578 + 11986233120578 = 119891 (119905) (12)

119872 120578 = 119865 (119905) (13)

where A33 M B





11986033= minus

int119865 (119905) 120578 119889119905

int 1205782 119889119905

11986133= minus

int119865 (119905) 120578 119889119905

int 1205782 119889119905

(14)




0

2

4

6

8

10

12

14

16

18

20

03 04 05 06 07 08 09

2

0

4

6

8

10

12

14

16

1205783120578

3a

F2120578

3a

times103


1205962L1g

(a) Sway force RAOs

0

2

4

6

8

03 04 05 06 07 08 09

10

12

Phas

e shi

f (ra

d)


1205962L1g


024681012141618202224

03 04 05 06 07 08 09

1205783120578

3a

0

2

4

6

8

10

12

14

16

18

20

F3120578

3a

times103


1205962L1g


03 04 05 06 07 08 090

2

4

6

8

10

12

Phas

e shi

f (ra

d)


1205962L1g








ded

mas

s and

dam

ping

coe

ffici

ent

0302 04 05 06 07 08 09 1

0

2

4

6

8



minus2

minus4

minus6

1205962L1g



6 Conclusion








References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










t = 721 s t = 721 s

t = 725 s t = 725 s

t = 729 s t = 729 s

050045040035030025020015010005000

050045040035030025020015010005000

050045040035030025020015010005000










t = 715 s t = 715 s

t = 719 s t = 719 s

t = 723 s t = 723 s

050045040035030025020015010005000


050045040035030025020015010005000


050045040035030025020015010005000









03 04 05 06 07 08 09

2

0

4

6

8

10

12

141205783120578

3a



1205962L1g

(a) RAOs

Phas

e shi

f (ra

d)

03 04 05 06 07 08 09

2

0

4

6

8

10

12



1205962L1g

(b) Phase shift




2120578311988610minus3) 1135 1517 1337






(11986033+119872) 120578 + 119861

33120578 + 11986233120578 = 119891 (119905) (12)

119872 120578 = 119865 (119905) (13)

where A33 M B





11986033= minus

int119865 (119905) 120578 119889119905

int 1205782 119889119905

11986133= minus

int119865 (119905) 120578 119889119905

int 1205782 119889119905

(14)




0

2

4

6

8

10

12

14

16

18

20

03 04 05 06 07 08 09

2

0

4

6

8

10

12

14

16

1205783120578

3a

F2120578

3a

times103


1205962L1g

(a) Sway force RAOs

0

2

4

6

8

03 04 05 06 07 08 09

10

12

Phas

e shi

f (ra

d)


1205962L1g


024681012141618202224

03 04 05 06 07 08 09

1205783120578

3a

0

2

4

6

8

10

12

14

16

18

20

F3120578

3a

times103


1205962L1g


03 04 05 06 07 08 090

2

4

6

8

10

12

Phas

e shi

f (ra

d)


1205962L1g








ded

mas

s and

dam

ping

coe

ffici

ent

0302 04 05 06 07 08 09 1

0

2

4

6

8



minus2

minus4

minus6

1205962L1g



6 Conclusion








References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










t = 715 s t = 715 s

t = 719 s t = 719 s

t = 723 s t = 723 s

050045040035030025020015010005000


050045040035030025020015010005000


050045040035030025020015010005000









03 04 05 06 07 08 09

2

0

4

6

8

10

12

141205783120578

3a



1205962L1g

(a) RAOs

Phas

e shi

f (ra

d)

03 04 05 06 07 08 09

2

0

4

6

8

10

12



1205962L1g

(b) Phase shift




2120578311988610minus3) 1135 1517 1337






(11986033+119872) 120578 + 119861

33120578 + 11986233120578 = 119891 (119905) (12)

119872 120578 = 119865 (119905) (13)

where A33 M B





11986033= minus

int119865 (119905) 120578 119889119905

int 1205782 119889119905

11986133= minus

int119865 (119905) 120578 119889119905

int 1205782 119889119905

(14)




0

2

4

6

8

10

12

14

16

18

20

03 04 05 06 07 08 09

2

0

4

6

8

10

12

14

16

1205783120578

3a

F2120578

3a

times103


1205962L1g

(a) Sway force RAOs

0

2

4

6

8

03 04 05 06 07 08 09

10

12

Phas

e shi

f (ra

d)


1205962L1g


024681012141618202224

03 04 05 06 07 08 09

1205783120578

3a

0

2

4

6

8

10

12

14

16

18

20

F3120578

3a

times103


1205962L1g


03 04 05 06 07 08 090

2

4

6

8

10

12

Phas

e shi

f (ra

d)


1205962L1g








ded

mas

s and

dam

ping

coe

ffici

ent

0302 04 05 06 07 08 09 1

0

2

4

6

8



minus2

minus4

minus6

1205962L1g



6 Conclusion








References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










03 04 05 06 07 08 09

2

0

4

6

8

10

12

141205783120578

3a



1205962L1g

(a) RAOs

Phas

e shi

f (ra

d)

03 04 05 06 07 08 09

2

0

4

6

8

10

12



1205962L1g

(b) Phase shift




2120578311988610minus3) 1135 1517 1337






(11986033+119872) 120578 + 119861

33120578 + 11986233120578 = 119891 (119905) (12)

119872 120578 = 119865 (119905) (13)

where A33 M B





11986033= minus

int119865 (119905) 120578 119889119905

int 1205782 119889119905

11986133= minus

int119865 (119905) 120578 119889119905

int 1205782 119889119905

(14)




0

2

4

6

8

10

12

14

16

18

20

03 04 05 06 07 08 09

2

0

4

6

8

10

12

14

16

1205783120578

3a

F2120578

3a

times103


1205962L1g

(a) Sway force RAOs

0

2

4

6

8

03 04 05 06 07 08 09

10

12

Phas

e shi

f (ra

d)


1205962L1g


024681012141618202224

03 04 05 06 07 08 09

1205783120578

3a

0

2

4

6

8

10

12

14

16

18

20

F3120578

3a

times103


1205962L1g


03 04 05 06 07 08 090

2

4

6

8

10

12

Phas

e shi

f (ra

d)


1205962L1g








ded

mas

s and

dam

ping

coe

ffici

ent

0302 04 05 06 07 08 09 1

0

2

4

6

8



minus2

minus4

minus6

1205962L1g



6 Conclusion








References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

2

4

6

8

10

12

14

16

18

20

03 04 05 06 07 08 09

2

0

4

6

8

10

12

14

16

1205783120578

3a

F2120578

3a

times103


1205962L1g

(a) Sway force RAOs

0

2

4

6

8

03 04 05 06 07 08 09

10

12

Phas

e shi

f (ra

d)


1205962L1g


024681012141618202224

03 04 05 06 07 08 09

1205783120578

3a

0

2

4

6

8

10

12

14

16

18

20

F3120578

3a

times103


1205962L1g


03 04 05 06 07 08 090

2

4

6

8

10

12

Phas

e shi

f (ra

d)


1205962L1g








ded

mas

s and

dam

ping

coe

ffici

ent

0302 04 05 06 07 08 09 1

0

2

4

6

8



minus2

minus4

minus6

1205962L1g



6 Conclusion








References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










ded

mas

s and

dam

ping

coe

ffici

ent

0302 04 05 06 07 08 09 1

0

2

4

6

8



minus2

minus4

minus6

1205962L1g



6 Conclusion








References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Influences of Vorticity to Vertical ... · Faltinsen et al. [ ] investigated the piston mode occur-ring in a square-shaped catamaran hull form under forced heave