Seakeeping prediction of KVLCC2 in head waves with RANS.pdf

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Applied Ocean Research 35 (2012) 5667

Contents lists available at SciVerse ScienceDirect

Applied Ocean Research

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Seakeeping prediction ofKVLCC2 in head waves with RANS

B.J. Guo a,, S. Steen a, G.B. Deng b

a Department of MarineTechnology, NTNU, Trondheim, Norwayb E.M.N., L.M.F-UMR6598, Ecole Centrale de Nantes, 1, Ruede la Noe, France

a r t i c l e i n f o

Article history:

Received 16 August 2011

Received in revised form

22 November 2011Accepted 20 December 2011

Available online 3 February 2012

Keywords:

Added resistance

Ship motions

KVLCC2

Head waves

a b s t r a c t

The present work is devoted to the prediction of added resistance and ship motion ofKVLCC2 in headwaves. Systematic validation and verification of the numerical computation demonstrate that reliable

numerical results can be obtained in calm water as well as in head waves. The numerical results areanalyzed in terms of added resistance, ship motions and wake flow. Both free to heave and pitch and

fixed model are studied to investigate the contribution to added resistance from ship motion at differentwavelengths, and the results show that ship motion induced added resistance is negligible when the

wavelength < 0.6 Lpp. The comparison with theoretical calculation based on strip theory and experi-mental results shows that RANS predicts the added resistance better in all wavelengths. Ship pitch and

heave motion in regular head waves can be estimated accurately by both CFD and strip theory. Finally,wake flow at the propeller plane in waves is discussed, and the numerical results show that the changeofaxial velocity due to one studied incoming head waves is at most over 30% ofthe ship forward speed.

2011 Elsevier Ltd. All rights reserved.

1. Introduction

CFD approach relying on the resolution of the Reynolds Aver-aged Navier-Stokes equation (RANS) overcomes the limitation ofthe potential flow with respect to effects of water viscosity, wave

dispersion, nonlinearity and wave breaking [1]. Moreover, effectsof turbulence are taken into account with turbulence model. Con-sequently, the application of CFD in the ship industry is increasing.In addition to traditional ship motions and resistance informa-

tion,RANS calculations can easily provide comprehensive flow fieldinformation, which is helpful for ship design optimization [2].

Prediction of flow around ship hulls using RANS started in the1980s, and a majority of the studies were initially devoted to ship

resistance and flow field prediction in steady state. Even in theGothenburg CFD workshop 2000, all benchmark test cases con-cerned steady RANS computation [3]. Before this workshop, a fewseakeepingcomputationshadbeenperformed.Wilsonetal. [4] pre-

sented an unsteady flow study in head waves, studying the forceand diffraction wave pattern, for the Wigley hull and the DTMBModel 5415. However, only the diffraction problem with forwardspeed at a few wavelengths was discussed. Sato et al. [5] sim-

ulated pitch and heave motions in head waves with a modifiedWigley model and Series 60. The agreement between the CFD and

Corresponding author at: Department of Marine Technology, Norwegian Uni-

versity of Science andTechnology, OttoNielsens v10, NO-7491Trondheim, Norway.

Tel.: +47 7355 1198.

E-mail address: [email protected](B.J. Guo).

experimentalresults was good,but some discrepancies were noted,

especially for the heave motion of the Series 60 model. Rhee andStern [6] performed a validation work on ship resistance, wave fieldand wake in head waves for a modified Wigley hull. Their calcula-tionsrestrainedship motionto studythe diffractionproblem.Hideo

et al. [7] studied added resistance in regular incident waves withthe SR108 container ship. A further study of unsteady surface pres-sure, added resistance and ship motions of the SR221C model wasperformed by Hideo [8]. These works lack a systematic verification

study of the numerical computation.A forward speed diffraction benchmark test case is proposed

for the Tokyo CFD workshop in 2005. It is the first time that theunsteady RANS for ship hydrodynamics was benchmarked [9]. Four

groupswere involved in this calculation.The standard deviationforthe 0th-ordertotal resistanceand heave force are4.5% and6.7%, butit is about 70% for the pitch moment [10]. Such high level of uncer-tainty clearly suggested that CFD approach was not yet a reliable

tool for seakeeping application at that time. Since then, more andmore efforts are devoted to improve the accuracy of CFD compu-tation for seakeeping application. Weymouth et al. [11] extendedthework by Wilson et al. [4], and Rhee and Stern [6] to predict ship

pitch and heave motion of a modified Wigley hull in head waves.Their simulation shows very good agreement with experimentaldata, and extensive uncertainty analysis was performed followingthe procedure given by Stern et al. [12]. Their studies focus mainly

on long waves. Only a few short wave cases were studied. Denget al. [13] studied several short wave cases for a container ship,but the discrepancy for added resistance estimation in short wavesreached 50%.

0141-1187/$ see front matter 2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.apor.2011.12.003
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B.J. Guo et al. / AppliedOcean Research35 (2012) 5667 57

To prepare for the Gothenburg 2010 CFD workshop, Deng et al.

[10] further studied the added resistance of KVLCC2 in three differ-ent wavelengths, but they lacked experimental validation. KVLCC2are selected as a benchmark model in the Gothenburg 2010 CFDworkshop [14]. 5 groups from 4 countries performed the relevant

calculation, and the seakeeping prediction for six wavelengths wascompared. The shortest wavelength is Lpp/= 1.57. The standarddeviation for the mean resistance is around 11.3%, while the maxi-mum deviation of 22.0% happens at the longest wave. The standard

deviations of the 0th-order heave and pitch motions are 30.9% and30.6%, and that for the 1st-order cases are 14.1% and 14.6%.

While the number of participants contributing to seakeepingbenchmarksduring the Gothenburg workshop conveys a clearmes-

sage that CFD becomes a feasible tool for seakeeping application,but the high value of standard deviation and comparison error ofthe submitted results do not allow to conclude that CFD is a reliabletool for such kind of application. One of the main reasons lead-

ing to such high level of uncertainty is that nearly all participantshave performed their computation for each test case with a sin-gle grid using a single time step. Almost no numerical verificationwas performed. In order to assess the reliability of CFD approach

for seakeeping application, this paper extends the work of Denget al. [10] about added resistance ship motion analysis for KVLCC2

at different wavelengths, especially in short waves. Due to datascattering, an uncertainty analysis based on the least squares root

method is followed here, as proposed by Eca and Hoekstra [15]. Arecently revised version can be found in Eca et al. [26]. The short-est wavelength is Lpp/=2.453. Both free heave and pitch motionand restrained motion in head waves were performed to study the

contribution from ship motions to added resistance. The ship resis-tance components in waves are also analyzed to study the viscouseffect on the added resistance coefficient. Another objective is tostudy the nominal wake field at the propeller plane in waves, and

to determine how it changes due to the incident waves and shipmotions. Such a study is generally not possible with any linearizedor inviscid method, and it is quite difficult to measure in a modeltest.

2. Ship geometry and test conditions

The KVLCC2 model with a scale 58 is adopted for the numeri-

cal calculation. The ship length and main parameters are given inTable 1. For the sea-keeping test, thedesigned depth is too small toavoid wateron deck.Thus,the depth ofthe hull isincreasedthroughthe extrapolation of the bodyplanto a depth of0.7 m (modelscale).

Fig. 1 shows the body plan of model used in both model tests andnumerical calculations. With Fn=0.142, the ship speed at modelscale is 1.047m/s.

The waves considered in this study are given in Table 2. The

waves are divided into two groups here. The first group, containingtheshortandintermediatewaves,isthewavesappliedintheexper-

iments performed by thefirsttwo authors in the towing tank at theMarine Technology Centre in Trondheim, Norway [18]. The sec-

ond group contains longer waves applied in experiments at Osaka

Table 1

Main particulars of KVLCC2.

Symbol Unit Ship Model

Scale 1 58

Lpp m 320.0 5.5172

B m 58.0 1.0000

T m 20.8 0.3586

D m 30 0.5172/0.7Cw 0.9077 0.9077

CB 0.8098 0.8098 m3 312,622 1.6023

Fig. 1. Body plan (dimensions in mm full scale).

Table 2

Waves used in calculations and experiments.

Lpp/ (m) H(m) T(s)

2.453 2.249 0.05 1.200

1.571 3.511 0.05 1.499

1.571 3.511 0.15 1.4991.224 4.507 0.15 1.699

1.090 5.060 0.15 1.800

1.667 3.310 0.10 1.456

0.909 6.069 0.10 1.972

0.625 8.827 0.10 2.378

University. Part of these experimentaldata canbe found in Gothen-

burg 2010 CFD workshop [14]. The model scale in the experimentsperformed by the authors is 58, consistent with the numerical cal-culation in this study. The model scale is 100 in the experiment byOsaka University, and those results are scaled to model scale 58 in

Table 2.In order to analyze the numerical and experimental results, a

right-handed coordinate system (x, y, z) used in the Gothenburg2010 workshop (Larsson et al. [14]) is adopted here (see Fig. 2).

The numericaland experimental results areall transferred into thiscoordinate for comparison and analysis.

3. Computation method

There are three distinct physical problems modeled in this

paper:free trim andsinkage motionin calm water,wave diffractionwith forward speed and the free heave and pitch motion responsedue to incident waves. In the forward speed diffraction problem,the model is fixed in the design condition but move with the for-

ward speed. However, a regular wave comes into the fluid domainand introduces an unsteady flow around the ship hull. In the free

motion response problem, the hull, moving forward with constant

Fig. 2. The coordinate system.


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58 B.J. Guo et al. / AppliedOcean Research35 (2012) 5667

speed in waves, responds to the force and moment with coupled

pitch and heave motion.

3.1. Computational implementation

The ISIS-CFD flow solver developed by EMN (Equipe Modlisa-tion Numrique) is used for the study here. The turbulent flow is

simulated by solving the incompressible RANS witha finite volume

method. The solver can process multi-phase flows and dynamicgrids. Details refer to FINE/Marine V2.2 [16]. In thepresent numer-ical simulation, a second order implicit scheme is applied for the

time discretization. The SIMPLE-like algorithm is used for couplingthe velocity and pressure. An analytic weighting mesh deforma-tion approach is adopted to simulate the ship motion. EASM kmodel, which is believed to inherit some advantage of the orig-

inal RSM model, while maintaining the numerical robustness ofa two-equation model, is used for turbulence modelization. Thefree surface is handled with a VOF-type free-surface capturingapproach, which has been proved to be suitable for flow involving

hull shape with section flare and breaking waves [17].

3.2. Calculation domain and boundary condition

The calculation domain is1.75Lpp


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Fig. 4. Mesh used for numerical simulation.

our measurement, the sinkage is measured with respect to a refer-ence position at the carriage. The variation in elevation of towingtank railsabovewateris estimated tobe around1 mm.Although the

difference between numerical prediction and measurement data issmaller than 1 mm, the comparison error is still as high as 12%.

5.1.1. Verification of numerical calculation

The numerical resistance and ship motions versus refinementratio hi/h are shown in Figs. 5 and 6. The fitted curves based onthe least squares root approach as well as fitted curves based on

theoretical accuracy Ptheory = 2 are also plotted, where hi is a meshsize calculated with cell number.

The observed order of accuracy PG =2.02 for the resistance isvery close to the theoretical value. The PGof sinkage and trim are

also close to the theoretical value Ptheory =2. The observed order ofaccuracy PG, extrapolated solution 0, standard deviation US%D,and uncertaintyUGare all listed in Table 4. The standard deviation

US%D is less than 1% foreach physical parameter, which shows that

the numerical results distribute smoothly along the fitted curve.

Table 3

Ship resistance andmotions in calm water.

Coarse Medium Fine Finest Exp.

Mesh 351,018 775,031 1,365,729 2,284,161

Rt 19.886 19.166 18.942 18.674 18.200

e 9.26% 5.31% 4.08% 2.60%

3 5.938 5.753 5.671 5.642 6.419

e 7.49% 10.38% 11.65% 12.10%5 0.1328 0.1297 0.1282 0.1277 0.126

e 5.40% 2.94% 1.75% 1.35%

This could also be seen in Figs. 5 and 6. The uncertainty is less than6% for all the predicted parameters.

5.1.2. Validation of numerical calculation

The validation of the numerical model in calm water is given inTable 5. Steady flowis adopted forthe numericalsimulation in calm

water,so time step uncertainty (UT) does not exist in the validation.Since iterative errors are much smaller than the relative change of

Fig. 5. Grid convergenceof total resistance in calm water.


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Fig. 6. Grid convergenceof ship motions in calm water.

Table 4

Uncertainty of ship resistance and motions.

PG 0 US%D UG UG% D

Rt 2.02 18.232 0.36% 0.219 3.39%

3 2.78 5.5073 0.13% 0.37 5.76%5 2.57 0.1263 0.15% 0.0064 5.07%

Table 5

Validation of numericalmodel in calm water.

E UG%D UD% D UVal%D

Rt 2.60% 3.39% 1.0% 3.53%

3 12.10% 5.76% 1.26% 5.90%5 1.35% 5.07% 0.88% 5.15%

calculation parameter with mesh size, iterative uncertainty (UI) isalso neglected. Then numerical simulation uncertaintyUSNin calmwater is equal to grid uncertainty (UG) [12]. The validation uncer-

tainty is calculated according to U2Val = U2G + U2D, where UD is theuncertainty of model test. The validation comparison error Eis thedifference between the numerical solution from finest case (S) andexperimental data (D).

Table 5 shows that the validation comparison error E of theresistance and trim is smaller than the validation uncertainty.Beside, both the comparison error and the validation uncer-tainty are relatively small, from which it can be concluded that

calculated resistance and trim are in good agreement with exper-imental results. For sinkage (3) this is not the case the relativeerror between calculation result and experimental result is large(12.1%), however, the absolute error is relatively small (0.78 mm),

since the sinkage is very small at the low speed studied. The lowvalue of the sinkage means that even small bias errors might have

large influence on the relative error. We believe that a variation ofthe elevation of the rails in the towing tank is a major contributor

to the deviation in sinkage. The validation uncertainty of sinkagewould be 15.5% in this case, if a variation of rail elevation of 1 mmis included in the experimental uncertainty.

5.2. Unsteady flow

5.2.1. Uncertainty estimation for spatial discretization

Considering the large discrepancy near the ship motion peakarea (22.0%) reported at the Gothenburg 2010 workshop, wave-length /Lpp = 0.9171 is selected here for the uncertainty analysis.To evaluate the spatial discretization error, 4 different meshes are

used. For those computations, the same time step with 140 time

stepsper wave encounterperiod isused.Numerical results inwaves

Table 6

Ship resistance andmotions at=0.9171Lpp predicted withT= Te/140.

Coarse Medium Fine Finest Exp

Mesh 748,748 1,346,253 2,147,731 3,626,036

Caw 5.035 4.827 4.664 4.610 4.601

e 9.43% 4.91% 1.37% 0.20%

0th3 6.135 6.258 6.369 6.385 6.516

e 5.84% 3.96% 2.25% 2.01%

0th5 0.1301 0.1327 0.1331 0.1335 0.137

e 5.03% 3.14% 2.85% 2.56%1st3 12.005 11.873 10.963 11.170 11.631

e 3.22% 2.08% 5.74% 3.96%

1st5 1.373 1.360 1.295 1.298 1.357

e 1.16% 0.199% 4.58% 4.38%

from different meshes are compared with experimental data inTable 6. Added resistance coefficient Caw is calculated and non-dimensionalized according to:

Caw =Raw

g2aB2/Lpp

(1)

where Raw = Rwave Rcalmis added resistance,Rwaveis mean resis-

tance in waves and Rcalmis resistance in calm water. is the waterdensity, ais the wave amplitude.B and Lpp are given in Table 1.

Fig. 7. Grid convergence of added resistance in waves.


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Fig. 8. Grid convergenceof ship motions in waves.

Table 6 indicates that the numerical added resistance

approaches theexperimental data as themesh is refined. Theaddedresistance is almost constant and the error is reduced to 1.37%,when the mesh number is increased to fine case. It is observedthat the comparison error for added resistance in waves is smaller

than that for the calm water resistance. The possible reason for thehigher accuracy of added resistance is that the error in both experi-mental and numerical results is canceled by subtracting resistancein calm water from that in waves.

The numerical resistance and ship motions in waves versus

refinement ratiohi/h, together with fittedcurves based on theleastsquares root approach as well as fitted curves based on theoreticalaccuracy Ptheory =2 areshownin Figs. 7 and 8. Fig. 7 shows the grid

convergenceof addedresistance.That for shipmotions is illustratedin Fig. 8.

Fig. 7 shows that the PG =2.43 of added resistance is very closeto the theoretical value.The pointsare distributed aroundthe fitted

curve with standard deviation US less than 1% of the experimentaldata, which clearly indicates that the numerical added resistanceis very reliable.

Fig. 8 indicates that the observed order of accuracy (PG = 2.74,

PG =1.44 for the 0th and 1st order separately) of heave motion isclose to 2, while the PG of the pitch motion is a bit farther fromthe theoretical value. The 1st order of pitch converges very slowly.

The possible reason forthe poor convergence behavior is that other

sources of numerical errors, such as nonlinear iteration error and

data sampling error, can not be neglected compared with spatialdiscretization error when the mesh becomes very fine. In addition,time discretization error with the same time step may not be thesame for different meshes. The PG for the 0th order pitch is very

large. The 0th order pitch from different meshes is almost con-stant after the mesh is increased to themedium size.The 0th orderamplitude for motion is a second order quantity in a seakeepingapplication. It is quite normal that poor convergence behavior is

observed for those quantities.

The observed order of accuracy PG, the extrapolated solution0, standard deviation US%D, and uncertainty UG %D are all listedin Table 7.

The uncertainty of numerical results in waves is larger than thatin calm water. The large uncertainty of added resistance and the

Table 7

Spatial uncertainty of ship resistance and motions in waves.

PG 0 US%D UG%D

Caw 2.43 4.4304 0.72% 7.62%

0th3 2.74 6.5443 0.49% 4.86%

0th5 6.97 0.1357 1.11% 3.14%

1st3 1.44 10.141 3.59% 13.63%

1st5 0.70 1.096 2.15% 6.60%


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Table 8

Shipresistance andmotions at=0.9171Lpppredictedwith fine meshand different

time step sizes.

Large Middle Small Extrapolated Exp

T/t 140 200 280

Caw 4.664 4.7250 4.7451 4.756/3.034 4.601

e 1.37% 2.70% 3.13% 3.37%

0th3 6.369 6.3299 6.3131 6.298/2.261 6.419

e 0.77% 1.39% 1.65% 1.89%0th5 0.1331 0.1334 0.1333 0.1320/ 0.137

e 2.85% 2.63% 2.70% 3.65%

1st3 10.963 12.043 12.259 12.320/4.4 11.631

e 5.74% 3.54% 5.40% 5.92%1st5 1.295 1.399 1.377 1.396/ 1.357

e 4.58% 3.10% 1.47% 2.87%

1st heave motion is due to the discrepancy of the result from thecoarse mesh. We believe that numerical results from the present

numericalmodelare accurate enough. Thehigh level of uncertaintyis due to the design of uncertainty estimation procedure using datarange based on very coarse grid solution.

5.2.2. Uncertainty estimation for time discretizationIn Table 6, only a very small change of added resistance is

observed between the fine and the finest mesh. In order to saveCPU time, the fine mesh is used for the time convergence study.The time convergence study is given in Table 8.

The extrapolated values in Table 8 are estimated with theoreti-

calorder of convergence PG = 2.The reasonfor theuse oftheoreticalorder of convergence is that it is not always possible to obtainmonotonic convergent for all cases in the time convergence study.The observed order of convergence is also given in Table 8, for

resistance, as well as for 0th-order and 1st-order heave motion.Monotonic convergence is not observed for other quantities;hence,order of convergence cannot be given. For those quantities, thechange of numerical results with time step is very small. Thus, it

might be concluded that time discretization error for those quan-tities is small compared with other sources of error, such as datasampling error and non-linear iteration error.

5.2.3. Validation of numerical model

For this unsteady simulation, the simulation time is chosen as

60 s, which includes about 45 wave encounter periods (Te). In orderto reduce the effect of sampling error on the numericaladded resis-tance and ship motions in waves, the data in the last 16s (about12Te) is used for analysis. In order to show the effect of sampling

error, the numerical results at three different time windows aregiven in Table 9. These three different time windows are all chosenin the last 16s of simulation. All of them have the same start point.Table9 showsthattheresultfromeachtimewindowisverysimilar,

which indicates that the effect of sampling error can be neglected.The validation of the numerical model in waves is given in

Table 10. The validation uncertainty UVal is calculated accordingto U2Val= U2D + U

2T+ U

2G

, and UT is time step uncertainty.

Table 9

Numerical results at three different time windows.

Time window 1 2 3 Exp

t/T 8 10 12

Caw 4.7493 4.7453 4.7451 4.601

0th3 6.3110 6.3092 6.3131 6.419

0th5 0.1330 0.1327 0.1333 0.137

1st3 12.2629 12.2523 12.259 11.631

1st5 1.377 1.377 1.377 1.357

Table 10

Validation of numerical model in waves(/Lpp=0.9171).

E%D UD%D UT%D UG%D UVal%D

Caw 3.13% 0.48% 0.89% 7.62% 7.69%

0th3 1.65% 1.26% 0.40% 4.86% 5.04%

0th5 2.70% 0.88% 0.17% 3.14% 3.27%

1st3 5.40% 1.26% 6.99% 13.63% 15.37%

1st5 3.38% 0.88% 2.86% 6.60% 7.25%

It is seen from Table 10 that the validation comparison error

E is smaller than the validation uncertainty Uval for all studiedvariables, from which it can be concluded that the numericaladdedresistance and ship motions in waves are in agreement with the

experimental data. The results presented in Table 10 indicate thatcomparison errors for all quantities are relatively small. Four ofthem are smaller than 4%, which can be considered as accurateenoughfor engineeringapplication. However, the validation uncer-

tainty is relatively high, ranging from 3% to 15% with three of themhigherthan7%. We believe that such a high levelof uncertaintydoesnot imply that the numerical prediction is unreliable. It indicates

that reliable uncertainty estimation cannot be obtained due to thelimitation of computational resources for such kind of application.

A typical run with a grid containing 2 M cells over one encounterperiods using 200 time steps per encounter periods and 10 non-

linear iterations per time step takes about 400 min with 16 cores,corresponding to about one-third of a clam water resistance com-putation. More reliable uncertainty estimation certainly requiresfiner mesh, smaller time step, better nonlinear convergence, and

longer simulation time.With the validation of the numericalmodel,additional seakeep-

ing simulations are performed with different wavelengths. In orderto reduce the numerical error, the finest mesh case is refined

near free surface for simulation in short waves < 0.9171Lpp, withtime step t=Te/140. For the simulations of the longer waves(> 0.9171Lpp),the fine meshis changed a bit nearthe freesurfacefor simulation with time stept=Te/200. The number of grid cells

increase vertically and decrease longitudinally, and the total num-ber of cells is almost the same. All these mesh cases ensure thereare at least 70 cells per wavelength and 11 cells per wave heightnear the free surface for the all waves studied here.

5.3. Shipmotion inwaves

Comparison of numerically predicted ship motions with exper-

imental data are given in Fig. 9. Ship heave and pitch arenon-dimensionalized according to

Y3 =3a, Y5 =

5ka

(2)

where a is wave amplitude, k is wave number. Experimentalresult by the authors are marked with triangle ( ) in Fig. 9,

whereas experimental results from Osaka University are markedwith square ( ) [14].

Strip theory proposed by Salvesen et al. [21] has been provedto be able to predict ship motions with satisfactory accuracy. Thenumerical results from CFD are also compared with the results

given by strip theory in Fig. 9. The calculation with strip theoryrefers to Guo and Steen [18].

Fig. 9(a) compares heave motion amplitude at the center ofgravity predicted by CFD and strip theory with the experimen-

tal data measured by ourselves and Osaka University. The heavemotion amplitude is almost zero at high wave frequency, wherewavelength is less than 0.4 Lpp. At low wave frequency, wherewavelength is longer than 1.5 Lpp, the heave amplitude is close to

the incident wave amplitude. The heave motion changes very fast,

when wavelength is between Lppand 1.5 Lpp. Both CFD simulation


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Fig. 9. Comparison of RAO of ship motions amongst experiment, strip theoryand CFDcalculation.

and strip theory could estimate heave motion of KVLCC2 with high

accuracy. Pitch motion is depicted in Fig. 9(b), which shows that

both experiment and CFD result falls on the pitch curve given bystrip theory. In details, the CFD results match experimental datawell except at low frequency. The discrepancy between CFD and

experiment may be due to a slight difference in pitch radius of gyration (r55) between CFD simulation and experiment.

Thecomparison shows that strip theoryalso canpredict the ver-tical motion of KVLCC2 with high degree of accuracy. The reason is

that large hydrostatic coefficients C33, C55and linear exciting forcegive high accuracyof strip theoryin thepresent study.Ship motionsfrom CFDsimulation show good agreementwith experimentaldatain relatively short waves. In long waves, numerically and experi-

mentally predicted heave also agree well, but the numerical pitchmotion does not match the experimental result well. It should benoticed that surge motion is free in the measurement performed inOsaka University, while it is fixed in the CFD computation.

The 0th order harmonic pitch and heave predicted with CFDsimulation arecomparedwith measured datain Table 11. The com-parison shows that numerical results match the experimental datawell, and that the ship sinkage and trim in waves are independent

of wavelength.

5.4. Added resistance

Reliable evaluation of added resistance is important for shipdesign aiming at sustaining ship service speed and saving propul-sion power. The added resistance calculated by CFD simulationis compared with that from the radiated energy method (REM)

proposed by Gerritsma and Beukelman [22]. Knowledge of shipmotions and hydrodynamic characteristics required to calculateadded resistance with the REM, is provided by strip theory in thisstudy. The detailed calculation procedure of REM is given in Guo

and Steen [18].

Table 11

The0th harmonic pitch and heave in waves.

Lpp/ 3 (CFD) 3 (EFD) 5 (CFD) 5 (EFD)

2.453 6.322 6.223 0.125 0.122

1.571 6.359 6.409 0.121 0.123

1.224 6.333 6.375 0.121 0.125

1.090 6.385 6.419 0.134 0.137

1.667 6.293 5.431 0.120 0.124

0.909 6.201 5.517 0.128 0.137

0.625 5.865 6.051 0.127 0.145

Added resistance comparison amongst CFD calculation, experi-

mental measurement and theoretical calculation REM is shown

in Fig. 10. Comparison shows that the CFD results have betteragreement with experimental data than that from REM, espe-cially in short waves. REM could predict added resistance in long

waves with acceptable accuracy. However, it underestimates theship added resistance in short waves. The reason for small addedresistance in short waves is that the wave diffraction effect isnot well modeled by REM, and the diffraction dominates the

added resistance in short waves. The largest added resistance isobserved around =1.1 L, and CFD results near this wavelengthshows good agreement with measurement. REM underestimatesadded resistance near this region. The discrepancy of REM predic-

tion is probably due to the nonlineareffects with large ship motion,which cannot be modeled by REM.

The results show that CFD calculation predicts the added resis-tance with higher accuracy than REM. However, the cost of CFD

simulation in small waves is very high, due to the small meshsize and time step (/x=70, Te/t =140) to capture the waveswell. Fujii and Takahashi [23] divided added resistance into twoparts: one part is due to ship motions and the other part is due

to wave reflection. Since the ship motions are very small in shortwaves, it will be economical to fix the hull in predicting addedresistance, if the contribution of ship motions to added resistanceis also negligible. The contribution from ship motions to added

Fig. 10. Comparison of added resistance from REM and CFD calculation.


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Fig. 11. Added resistances from free and fixed models.

resistance is studied here. The numerical model is fixed at the

design load condition in short waves, and added resistances fromfree and fixed model are compared with experimental results in

Fig. 11. The comparison shows that numerical added resistancefrom fixed and free hull are almost equal in very short waves, i.e.

Lpp/>1.57 (


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Fig. 14. Thenominal wake flow in calm water.

Fig. 15. Mean wake flow in waves.

wave height H=0.138m, which is the measured wave amplitudein a model test with nominal wave height H=0.15m.

5.6.1. The 0th order harmonic amplitude ofwake at propeller disc

The 0th order harmonic amplitude of wake flow is the meanvalue in waves, which is illustrated in Fig. 15. The wake velocityin three different directions is non-dimensionalized with ship for-ward speed in the same way as in Fig. 14. Comparison with the

wakeflow incalmwater(Fig.14) indicates that the area of lowaxialvelocity becomes smaller in waves and the variation of axial veloc-ity at the top part becomes large. The mean vertical wake velocityin waves becomes negative over the whole disc.

In order to show the change of wake flow induced by wavesquantitatively, the difference of mean wake velocity along threedirections in waves (Ui, i= 1, 2, 3) and that in calm water (Ui-steady,

i= 1, 2, 3) as a fraction of ship forward speed U (U=1.047m/s) is

shown in Fig. 16. In Fig. 16(a), thelargest increase of themeanaxial

velocity in waves relative to that in calm wateris about 35% of shipforward speed. The change in axial velocity is mainlydue to the fact

that thebilge vortex moves up and down in waves. The region withlow axial velocity in calm water indicating the location of the bilge

vortex disappears in waves, resulting in a wake with higher meanaxial velocity. Fig. 16(b) shows the change of the mean transver-

sal velocity in waves relative to that in calm water. The transversalvelocity is zero at the center plane due to the symmetry condi-tion adopted in the CFD simulation. The mean transversal velocityshows that the largest changeis about 8% of theship forward speed.

In Fig.16(c), thechangeof themean vertical velocity relative to thatin calm water is negative at thewholepropeller disc. Themaximummean velocity increase is about 25% of ship forwards speed.

5.6.2. The 1st order harmonic amplitude of wake at propeller disc

In Fig. 17(a), the 1st order axial velocity shows very large ampli-

tude at the lower part of the propeller disc, where the maximum

Fig. 16. Difference of the0th harmonic velocities andsteady flow velocities (UiUi-steady)/Ualong threedirections at propeller disc.


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Fig. 17. The1st harmonic amplitudeof wake flow at propeller disc.

value is about 35% of ship forward speed. This value superimposedonthe 0thharmonicamplitudewillinduce a large axialvelocityandlow pressure at the propeller disc. In Fig. 17(b), the 1st order har-

monic amplitude of transversal velocity is zero at the center planedue to the symmetry boundary conditon. The value increases awayfrom the center plane. Compared with 1st order axial and verticalvelocity, the 1st transversal velocity is smaller. This is not surpris-

ing, since purely head sea is considered here. Fig. 17(c) shows thatthemaximum vertical first harmonic amplitudeis about 22%of shipforward speed.

6. Conclusions

Ship motions andadded resistanceof KVLCC2advancingin headwaves have beencalculatedwith unsteady RANS code. A systematicverification and validation analysis of numericalresult in bothcalm

water and waves is performed to show the reliability and accuracyof the numerical model in terms of the studied sea-keeping prob-lems. Numerical added resistance and ship motions in waves areverified to be convergentthrough time step andgrid spacing refine-

ment studies. The uncertainty is less than 6% for all the studiedvariables in calm water.

CFD prediction of ship motions and added resistance matchesthe experimental data well for different wavelengths, down to

the shortest wavelength ofLpp/=2.453. The comparison of shipmotions show that CFD has no advantage over strip theory in pre-dicting ship motion. However CFD predicts added resistance withhigher accuracy than the theoretical calculation REM based on

strip theory, especially in short waves. Although CFD could accu-rately predict added resistance in short waves, the computationalcost is still very high due to the requirement of very small meshsize. To save CPU time and also ensure the numerical accuracy, it is

recommended to estimate the added resistance by fixing the shipmodel when the wavelength is less than 0.63Lpp.

Pressure resistance dominates the first order ship resistance inwaves, and the first order frictional resistance is small. The addedresistance mainlycomes from the change of pressure force, and theadded frictional resistance is small and almost constant in shortwaves.

Both the0th and the 1st order axial velocities inducedby wavesat the propeller disc predicted by numerical simulation are verylarge, and the maximum values are approximately 35% of ship for-ward speed.Wave-induced mean transversalvelocity is quitesmall

except for thelocal region, where maximum velocity is about 8% ofthe ship forward speed. The1st order transversal velocity increasesaway from the center plane, but is still small. The mean verticalvelocity in waves is negative in the whole propeller disc. Wave

induced maximum 1st-order vertical velocity is about 22% of the

ship forward speed. The numerical simulation indicates that the

wake flow is strongly affected by the incoming waves. The effect ofwaves on the wake is surprisingly strong, and more studies shouldbe performed to verifyour findings. Also, theeffect of the change of

wake due to waves on the propeller performance should be inves-tigated.

Acknowledgements

The work is part of the research project SeaPro, sponsored byRolls-Royce Marine and the Research Council of Norway.

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