747
1 INTRODUCTION
Theship‐bankinteraction,whichisalsoreferredtoas
bankeffects,oftenleadstoasuctionforcetowardsthe
bankandabow‐outorbow‐inmoment.Thereforeit
significantlyincreasestheriskofcollisionagainstthe
bankorpassingships.Fornavigationsafetyconcern,
investigating
themanoeuvrabilityofashipinfluenced
by ship‐bank hydrodynamic interaction is of
importance.Duringthepastdecadesresearchershave
been focusing on providing suitable formulations to
predict bank‐induced lateral force and yaw moment
accordingto extensiveexperimentalresults (Norrbin
1974, Ch’ng et al. 1993, Li et al. 2003,
Lataire and
Vantorre 2008). On the other hand, the role of
hydrodynamic derivatives in evaluating the
manoeuvrability of the ships close to banks was
discussed in recent papers. Sano et al. (2014)
presentedthevariationof hydrodynamicderivatives
with the ship speed, water depth and off‐centreline
displacement and the consequent
change of course
stabilitybasedontheircaptivemodeltests.Liu etal.
(2016) carried out a series of planar motion
mechanism (PMM) tests in a Circulating Water
Channel (CWC) to study the impact of ship‐bank
interaction on the manoeuvring performance of a
VLCCshipmodel.
Nowadays researchers resort
to using
computationalfluiddynamics(CFD)todealwiththe
problem of ship‐bank interaction. Zou and Larsson
(2013) utilized a Reynold‐averaged Navier Stokes
(RANS)solver to studythe bank effectson atanker
hullthatproceedsindifferentcanals.Hoydoncketal.
(2015) applied several CFD methods to predict
the
loads on the KVLCC2 Moeri tanker for varying
depths and positions in a channel. In order to
determine the hydrodynamic derivatives through
virtual test technology, Mucha and el Moctar (2013)
conductedthesimulationsofPMMtestsforKVLCC2
tanker with various distances to a vertical bank. To
generatethe
time‐varyingcomputationalgridsforthe
Numerical Study of Hydrodynamic Derivatives and
Course Stability under Ship-Bank Interaction
H.Liu,N.Ma&X.C.Gu
ShanghaiJiaoTongUniversity,Shanghai,China
ABSTRACT:Sinceship‐bankinteractionaffectsthemanoeuvrabilityofashipnavigatingclosetoabank,the
determinationofhydrodynamicderivativesisofgreatimportancetoassesstheshipmanoeuvrability.Toobtain
thehydrodynamicderivativesoftheKVLCC2modelshipwithdifferentwaterdepthsand
ship‐bankdistances,
the simulation of PMM tests are carried out using an unsteady Reynolds‐Averaged Navier–Stokes (RANS)
based solver. Hybrid dynamic mesh technique is proposed to realize the simulation of pure yaw tests in
confined water. Studies on the grid convergence and time‐step‐size convergence are firstly performed.
Hydrodynamicderivativesfortheshipindifferentwaterdepthsandship‐bankdistancesarecompared.The
coursestabilityisinvestigatedbasedontime‐domainsimulationsandeigenvalueanalysis,andtheresultsshow
thattheship‐bankinteractionandshallowwatereffecthavearemarkableinfluenceonthecoursestability.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 12
Number 4
December 2018
DOI:10.12716/1001.12.04.14
748
dynamic motion of PMM tests, methods such as
dynamic layering combined with sliding interface
(Yang2011)andtetrahedralgridremeshing(Panetal.
2012) have been used in the past. However, these
methodsrequireenoughspacebetweentheshipand
the boundaries of the computational domain for
remeshing,thusthey
areincapableofsimulatingthe
motioninconfinedwater.
Inthispaper,thehybriddynamicmeshtechnique
is proposed to solve the problem of remeshing in
confined computational domain. The uncertainties
relating to grid and time‐step discretization are
quantified. A series of calculations is conducted for
thePMMtests
withdifferentwaterdepthsandship‐
bank distances. The first‐order derivatives are
obtainedandthechangeofthederivativeswithwater
depth and ship‐bank distance are presented. The
course keeping performanceunder ship‐bank
interaction is simulated in time domain and the
coursestabilityisevaluatedbyeigenvalueanalysis.
2 MANOEUVRINGFORMULATION
2.1 Maneuveringtheory
The coordinate system used for all numerical
simulationsinthis paper is shownin Figure1.Both
the earth‐fixed coordinate system O
0ξηζ and ship
fixed coordinate system Oxyz are a right‐handed
coordinate system with the positive ζ and z axis
pointingintothepageandtheoriginOsetatthemid‐
ship point. The ship is expected to move in the
direction of the ξ axis at speed U. Affected
by the
suctionforceandyawmomentfromthebankeffect,
the ship’s velocity vector is [u, v, r] and a heading
angle ψ appears between ξ axis and x axis.
Consequentlyarudderdeflectionδisneededtokeep
theship’soriginalcourse,whichcanbealsodescribed
by
thedistancebetweenshipandbank,ybank.
Figure1.Coordinatesystems.
Theproblemofshipmanoeuvring closetoabank
features a small deviation in sway and yaw motion
andthe ship speed is assumedconstant, i.e.
uU
.
Therefore the equation of surge motion is neglected
andtheequationsofshipmotionaregivenas

G
mv ur xr Y

(1)

zG
I
rmxvur N

(2)
wheremistheshipmass,I
zisthemomentofinertia
aboutthe z axisand Y andN are the externalsway
force and yaw moment acting on the ship,
respectively. All variables are transferred to non‐
dimensionalvalueswithrespecttotheshiplengthL,
draftT,shipspeedUandwaterdensity
ρthroughthe
equationsasfollows:
, , ,
0.5 0.5
, , , = ,
,
0.5 0.5
z
z
G
G
I
mvLrL
mI vr
LT LT U U
x
uvrL
uvr x
UUUL L
YN
YN
LTU L TU










2
2422
222
(3)
Assuming small deviations in sway and yaw
motion,thetotalswayforceandyawmomentacting
ontheshipareexpressedasthepolynomialsoffirst‐
order hydrodynamic derivatives and bank induced
interactionforces.
+
vrvrbank
Y YvYrYvYr YY
   



(4)
+
vrvrbank
N NvNrNvNr NN
   



(5)
InEquations4and5,thelinearvelocity(
v
Y
,
v
N
)
andacceleration(
v
Y
,
v
N
)deriva tivesaredetermined
byapuresway testand therotational (
r
Y
,
r
N
)and
angular acceleration (
r
Y
,
r
N
) derivatives are
determined by a pure yaw test. The derivatives for
bank induced forces (
bank
Y
,
bank
N
) and the rudder
control derivatives (
Y
,
N
) are measured through
the rudderangle tests, where the value of the fitted
curveof Y’and N’atδ=0 referto
bank
Y
and
bank
N
.
For a detailed explanation of the approaches to
identify the derivatives, the readers are referred to
Yoonetal.(2015).SubstitutingEquations4and5into
the non‐dimensional form of Equations 1 and 2
resultsinthefollowingequations:


vv Gr r
bank
mYvYv mx Yr mYr
Y
Y
 
 



(6)

Gv v zr Gr
bank
mx N v Nv I N r mx N r
N
N
  




(7)
2.2 Coursestabilityanalysis
The course (directional) stability is particularly
important for a ship sailing under bank effect. In
practice, a ship is stable if a disturbance applied to
deflecttheshipfromitscurrentcourse,resultsinthe
shipeventuallyresuminganewstraightlineafterthe
disturbance
disappears (Kobylinski 2003). For ships
nearbanks,rudderforcesarefirstusedtocounteract
the interaction force. After a ship reaches motion
equilibrium, the course stability can be judged
according to the eigenvalues of the homogeneous
formofEquations6and7.
Inthisstudy,thepoleplacementmethod(Kautsky
et
al. 1985) is adopted to control the rudder angle
749
before the motion equilibrium is achieved. Firstly
Equations 6 and 7 are rearranged with respect to [v
r]
T
,yieldingthestateequationsasfollows:
RB
vv
rr
 

 

 



M NF F
(8)
where
=
v
RB
v
r
G
v
Grz
rv
r
G
bank
bank
Ym Ymx
Nmx
NI
Ym
Y
Nmx
N
Y
Y
NN









 









M
N
FF

(9)
Then,thestatevector[vr]
T
isexpandedbyadding
thecrosstrackerroryandtheheadingangleψwhich
satisfy:
=cos+sin +yv u v


(10)
r

(11)
Thefinalstate‐spaceequationsaregivenas:
1
1
00
00
00
01 00
00
10 10
RB
vv
rr
yy




















FF
MN
M
(12)
TheinputvariableinthelinearsystemofEquation
12istherudderangle.Inordertocompleteaclosed
loop,theinputinEquation12isexpectedtobegiven
as
uKx
. Here the pole placement is used to
computethegainmatrixKthatcanensurethepoles
oftheclosed‐loopsystematthedesiredlocations.The
time‐domaincomputationofEquation12willendup
withasteadyruderangleδandaheadingangleψ.At
this phase
the right‐hand side of Equations 6 and 7
becomes zero. Then, the ship’s motion variables v’
andr’duetoasmalldisturbancearedefinedas:
12
,
tt
vCe rCe


(13)
where C
1 and C2 are arbitrary variables. By
substituting Equation13into Equations6 and7, the
characteristicequationiswrittenas:
2
0ABC

 (14)
where



vz r G v Gr
vz r v G r
vGr Gv ur
vGr v ur
AmYIN mx NmxY
BYIN NmxY
mYmx N mx N m X Y
CYmxN NmXY











(15)
Toensuretheship’scoursestability,thefollowing
Hurwitzstabilitycriterion(Bergmann1964)shouldbe
satisfied, meaning that Equation 9 has negative real
rootsonly:
0, 0, 0 or 0, 0, 0ABC ABC (16)
Since
,
vzr
mY I N



and
v
Y
are always
largerpositivevaluesthanothertermsinEquations6
and 7, it can be concluded that
0, 0AB
is
ensuredformostships.Therefore,thecriterioncanbe
reducedto
0C
.
3 NUMERICALSIMULATIONMETHOD
3.1 Numericalmethod
ThesimulationsofPMMtestsareperformedwiththe
Reynolds‐averaged Navier‐Stokes (RANS) equations
in the CFD software ANSYS FLUENT. The flow is
defined as single‐phase, 3‐D and incompressible
viscous flow. The flow model is implicit unsteady
with a segregated
predictor‐corrector solver. Semi‐
Implicit Method for Pressure‐Linked Equations‐
Corrected (SIMPLEC) algorithm is applied to solve
the velocity‐pressure coupling. The RNG k‐ε
turbulence model by Yakhot and Orszag (1986) is
introducedtoclosetheRANSequations.
Figure2.Computationaldomainandboundaries.
Thecomputational domainand boundary
conditionsforPMMsimulationissetasillustratedin
Figure2.Itextends4.0L
PP(LPPisthelengthbetween
perpendiculars)fromtheafttothepressureoutlet,1.0
L
PPfromthebowtothevelocityinlet.Thebreadthof
the domain is 4.0 L
PP, while the length of ybank from
starboardtotherightsidewallisvaried.Thedepthh
between the free surface and the bottom is also
alterable. Due to the low Froude number (F
r=0.066
basedon L
PP),the freesurface isconsideredrigid as
symmetryboundary.
Ahybriddynamicmeshtechniqueisproposedto
simulatethepureyawmotionnearthebank.AsFig.3
shows, the whole computational domain is divided
into three parts. The internal region is meshed with
750
tetrahedral cells, and the external and stationary
regionareswept withprismatic cells.Thepure yaw
motion is decomposed into transverse movement
(puresway)andhorizontalrotation.Thepureswayis
realized in the external region by layering method,
andtherotationiscompletedintheinternalregionby
local
remeshing.Thedistributionofgridsfollowsthe
rulethatcellslocatedintheinternalregionisaround
60‐70percentand10‐15percentintheexternalregion
and 20‐25 percent in the stationary region. The
dynamicandstationaryregionsareseparatedbygrid
interfaces. All the mesh motion
at each time step is
programmedbyUserDefinedFunctions(UDF).
Figure3.Hybridgridregionsforpureyawtestsimulation.
3.2 Convergencestudyongridandtimestepsize
Thespecificcaseforthepresentedstudyisknownas
the KVLCC2 Moeri tanker. The model data are
published via (SIMMAN 2008).The principal
particulars of the KVLCC2 test model together with
thefullscaleshiparelistedinTable1.
Table1.PrincipledimensionsoftheKVLCC2
_______________________________________________
ParametersFullscale Model
_______________________________________________
LengthbetweenperpendicularsLm 320.00 2.4850
BreadthBm58.00  0.4504
DesigndraftTm20.80  0.1615
Displacement Δ m3312540 0.1464
LCBfromMid‐shipxBm11.04  0.086
Scale128.77:1
_______________________________________________
The convergence study follows the methodology
adoptedintheInternationalTowingTankConference
(ITTC) recommended procedures forthe uncertainty
analysis in CFD (ITTC 2002). This paper focuses on
the verification work in the recommended
procedures,whichisappliedtoassessthenumerical
uncertaintyinthesimulationswithgraduallyrefined
grids and
time steps. Firstly, the grid‐convergence
study focuses on the simulation of pure sway test.
Three sets of grid are generated with the grid
refinementconformingtoauniformrefinementratio
4
2
G
r
. The grid volume of the three sets, namely
Grid 1, Grid 2 and Grid3 is respectively 2656360,
1681260 and 1016970. The values of the derivatives
frompureswaymotion(symbolizedasS
1,S2andS3in
termsof grid volume)are selected as theobjects for
uncertainty estimation and shown in Table 2 The
changes between S
1, S2 and S3are given by
21 2 1
SS
and
32 3 2
SS
, and the grid
convergence ratio R
G are obtained by
21 32G
R
.
The numerical uncertainty of grid size U
G can be
calculated through a complex derivation, for which
the readers refer to (ITTC 2002). In Table 2, the
monotonicconvergenceappearsin
v
Y
and
v
N
with
0＜R
G＜1 while oscillatory convergence appears in
v
Y
and
v
N
with RG＜0.The uncertaintyUG canbe
estimated in both conditions and the results
(presented as the percentage of the derivatives for
Grid1)showthatthenumericaluncertaintyinterms
ofgridnumberisfairlysmall.Grid2ischoseninthe
subsequentsimulationforcomputationalefficiency.
Table2.Resultsofgridconvergencestudy
_______________________________________________

v
Y

v
Y

v
N

v
N
_______________________________________________
Grid1‐0.3275 ‐0.1611 ‐0.5064 ‐0.0209
Grid2 ‐0.3296 ‐0.1620 ‐0.5064 ‐0.0206
Grid3 ‐0.3342 ‐0.1606 ‐0.5062 ‐0.0210
R
G0.4785 ‐0.6272 0.2753 ‐0.7785
UG 0.00848 0.00069 0.00021 0.00020
U
G(%Grid1) 2.59% 0.43% 0.04% 0.96%
_______________________________________________
Secondly,thetimestepconvergenceischeckedby
usingthreetimestepsΔt=0.007s,0.01sand0.014sand
theGrid2tosimulatethesame pureswaycaseasthe
gridconvergencestudy.Theresultsofthederivatives
with the corresponding convergence ratio R
T and
uncertainty U
T are listed in Table 3. The numerical
uncertainties in terms of time step are smaller and
t
=0.01s is chosento simulate further cases of pure
sway. The time steps for pure yaw test is set
t
=0.003sbecausethesmallerstepisneededtoavoid
negative‐volume cells’ appearance during the
remeshing.
Table3.Resultsoftimestepconvergencestudy
_______________________________________________

v
Y

v
Y

v
N

v
N
_______________________________________________
Δt=0.007s‐0.3299 ‐0.1618 ‐0.5063 ‐0.0205
Δt=0.01s‐0.3296 ‐0.1620 ‐0.5064 ‐0.0206
Δt=0.014s ‐0.3292 ‐0.1622 ‐0.5079 ‐0.0207
R
T0.5118 0.7472 0.0543 0.6316
UT0.00022 0.00052 0.00016 0.00017
U
T0.07% 0.32% 0.03% 0.81%
(%Δt=0.007s)
_______________________________________________
4 RESULTSANDDISCUSSION
4.1 Hydrodynamicderivatives
ThetypesofPMM test includepureswaytest, pure
yaw test and rudder angle test, covering a series of
ship‐bank distance to breadth ratios y
bank
/B =2.8, 1.7
and1.35,andwaterdepthtodraftratiosh /T=1.2,1.5
and10.ThevelocityinletspeedissetatU=0.326m/s
(F
r=0.066)thatis35percentoftheservicespeed.
Figure3and4showthederivativesversush/Tat
y
bank
=1.7B as well as the derivatives versus y
bank
/B at
h=1.2T.Tocomparethederivativesunderbankeffect
with respect to the derivatives in open water with
infinitedepth,theconditionofy
bank
=10Band h=10T
wassimulatedadditionally, ofwhich thederivatives
aremarkedas“Deepopenwater”.Itcanbeseenthe
magnitudes of most derivatives except
r
Y
and
r
N
751
are larger in confined water than the derivatives in
deepopenwater.
The absolute values of
v
Y
,
r
N
and
r
Y
increase at
shallowwaterdepth.Theycorrespondtoaddedmass
andaddedmomentof inertiawith achangeofsign.
Therefore the ship sailing in shallow water needs
longertimetochangeitsstateofmotionthanindeep
water. An exceptional case is the oscillation of
v
N
thatismainlyduetothechangeofgriddistributionin
the case of shallow water. Since the value of
v
N
is
relativelysmall,theaccuracyof
v
N
byCFDisnotas
good as other derivatives. For velocity derivatives
androtationalderivatives,themagnitudesof
v
Y
and
v
N
risestrikinglywiththedecreasingh/Twhile
r
N
appears an opposite trend. And the value of
r
Y
oscillateswithh/T.
Figure3.Accelerationderivativesversusybank/Bandh/T.
Figure4.Velocityderivativesversusybank/Bandh/T.
As to the results versus ybank/B at h=1.2T, the
magnitudes of acceleration derivatives increase with
the decreasing ship‐bank distance except
v
N
which
showsaslightdowntrend.Thesimilartrendappears
inthevelocityandrotationalderivatives.Thevalues
of
v
Y
,
r
Y
,
v
N
and
r
N
refer to a push‐out force
andabow‐outmomentonthemovingshipcausedby
v and r respectively. It indicates that the
hydrodynamicforcesareenhancedby theship‐bank
interaction.
Thefittingcurvesoftherudderforceandmoment
versus the rudder angle with different ship‐
bank
distancesath/T=1.2areshowninFigure5.Thecurves
haveapparentasymmetrieswithrespecttotheorigin
point,whichmeanstheship‐bankinteractionexertsa
suctionforceandabow‐outmomentontheship.The
rudder derivatives
Y
and
N
plotted in Figure 6
do not show significant changes in different ship‐
bankdistances.
Figure5.Rudderforceandmomentversusδwithvariations
iny
bank
/B.
Figure6.Rudderderivativesversus y
bank
/B.
4.2 Coursestability
The course keeping performance using the pole
placement control is simulated for different water
depthsandship‐bankdistances.Thepolesarechosen
to‐0.5,‐16.1and‐2.4±0.16iandfoundfromrepeated
tests. The simulation is performed for the speed
corresponding to F
r=0.066 and the rudder deflection
and steering velocity followed by the control law is
controlledwith35°and7.6°/s.
Figure7 showsthe resultsof coursecontrol with
variations inwater depth. Theinitial position of the
ship is at y
bank
=2.5B, and the positive y/B means the
shipisclosertothebank.Theshipmovestothebank
at the start phase. Then, the rudder angle grows
drastically to change the ship’s direction to “away
from the bank”. A negative heading angle is
generatedtogive a Munk
momentto counteractthe
rudder forces. Finally, the ship reaches the motion
equilibrium before t=20s. The rudder angle and the
1.2 1.5 10
1.35 1.7 2.8
-1.0
-0.8
-0.6
-0.4
-0.2
h /T
y
bank
/B
Results versus h/T
at y
bank
/B=1.7
Results versus y
bank
/B
at h/T=1.2
Deep open water
v
Y
1.2 1.5 10
1.35 1.7 2.8
-0.09
-0.06
-0.03
0.00
h /T
y
bank
/B
Results versus h/T
at y
bank
/B=1.7
Results versus y
bank
/B
at h/T=1.2
Deep open water
r
Y
1.2 1.5 10
1.35 1.7 2.8
-0.06
-0.04
-0.02
0.00
h /T
y
bank
/B
Results versus h/T
at y
bank
/B=1.7
Results versus y
bank
/B
at h/T=1.2
Deep open water
v
N
1.2 1.5 10
1.35 1.7 2.8
-0.04
-0.03
-0.02
-0.01
h /T
y
bank
/B
Results versus h/T
at y
bank
/B=1.7
Results versus y
bank
/B
at h/T=1.2
Deep open water
r
N
1.2 1.5 10
1.35 1.7 2.8
-1.2
-0.9
-0.6
-0.3
0.0
0.3
h /T
y
bank
/B
Results versus h/T
at y
bank
/B=1.7
Results versus y
bank
/B
at h/T=1.2
Deep open water
v
Y
1.2 1.5 10
1.35 1.7 2.8
0.00
0.02
0.04
0.06
h /T
y
bank
/B
Results versus h/T
at y
bank
/B=1.7
Results versus y
bank
/B
at h/T=1.2
Deep open water
r
Y
1.2 1.5 10
1.35 1.7 2.8
-0.3
-0.2
-0.1
0.0
h /T
y
bank
/B
Results versus h/T
at y
bank
/B=1.7
Results versus y
bank
/B
at h/T=1.2
Deep open water
v
N
1.2 1.5 10
1.35 1.7 2.8
-0.09
-0.06
-0.03
0.00
h /T
y
bank
/B
Results versus h/T
at y
bank
/B=1.7
Results versus y
bank
/B
at h/T=1.2
Deep open water
r
N
-30 -20 -10 0 10 20 30
-0.005
0.005
0.010
0.015


Y'
Exp Fit
y
bank
/B=2.8
y
bank
/B=1.7
y
bank
/B=1.35
-30-20-10 1020300
-0.004
-0.003
-0.002
-0.001
0.001

N'
Exp Fit
y
bank
/B=2.8
y
bank
/B=1.7
y
bank
/B=1.35
1.35 1.7 2.8
-0.020
-0.015
-0.010
-0.005
y
bank
/B
Y'
1.35 1.7 2.8
0.000
0.002
0.004
0.006
y
bank
/B
N'
752
overshoot of y/B increase as the water becomes
shallower in general. The largest overshoot of y/B
appears at h=1.5T. It shows that the shallow water
effectenhancestheship‐bankinteractionandalarger
rudderdeflectionisneededtokeepcourse.However,
ath=1.2Tthe
situationmaynotbeevenworse.
Figure7.Timetracesofcoursecontrolwithvarioush/T.
Figure8.Timetracesofcoursecontrolwithvariousy
bank
/B.
Figure 8 shows the ship‐bank distance has the
significant effect on the results of course keeping
control. The simulation is conducted at h=1.2T with
theshipinitiallyplaced aty
bank
=2.8B,1.7B and1.35B.
Dueto the increasing suctionforce by the bank, the
crosstrackerrortendstobeclosertothebankwhen
y
bank
reduces to 1.7B. Consequently, larger rudder
deflection is required for theequilibrium of motion
whentheship isveryclose tothe bank.The rudder
angleneededtoachievetheequilibriumath=1.2Tand
y
bank
=1.35B is nearly 30°.Also, the reverse heading
angle grows from‐0.1° to‐0.2°. Since the ship’s
deviation from the initial path is quite small in
Figures 7 and 8, the control system following pole
placementmethodmanifestitsadvantageofoffering
large control inputs to counteract the ship‐bank
interaction.
Based on the sign of parameters of A, B and C
calculatedbyEquation15,thecoursestabilityofthe
shipafterthedisturbancedisappearsinproximitytoa
bankcanbediscussed.Figure9presentsthevaluesof
A,BandCversush/Tandy
bank
/B.Forallthecases,A
andBarepositivebutCisnegative.Thisimpliesthat
theshipisunabletokeepastraightlineifnocontrol
action is taken after the disturbance. C for shallow
waterconditionsismuchmorenegativethanitisfor
h=10T
,whichindicatestheshallowwatereffectmakes
thisproblemevenworse.
Figure9.ParametersA,BandCversush/Tandybank/B.
5 CONCLUSIONS
The present study applies CFD method to simulate
thePMMtestsofKVLCC2modelshipwithvariations
in water depth and ship‐bank distance, and
investigatesthechangeofhydrodynamicderivatives
andcoursestabilityfortheship.Thehybriddynamic
mesh method is developed to solve the problem of
mesh configuration and remeshing in limited space.
Validation study is performed through the
convergence study of grid and time step. Further
calculation results show that shallow water effect
enlargesthevalueofhydrodynamicderivatives,and
the magnitudes of derivatives mostly increases with
the decrease of ship‐bank distance. The pole
placement method is capable of controlling the
KVLCC2 to achieve the equilibrium of motion in
differentwater depthsandship‐bankdistances.Large
rudder deflections are necessary to achieve the
motionequilibriuminshallowwaterand“veryclose
to bank” conditions e.g. h=1.2T and y
bank
=1.35B. The
course stability analysis indicates that the ship is
unstable without control and the shallow water
conditioncanworsenthisproblem.
ACKNOWLEDGEMENT
The research is supported by the China Ministry of
Education Key Research Project “KSHIP‐II Project”
(Knowledge‐based Ship Design Hyper‐Integrated
Platform):No.GKZY010004.
02468101214
-0.010
-0.005
0.000
0.005
y/B
x/L
h/T=10.0
h/T=1.5
h/T=1.2
0 20406080100
-0.2
-0.1
0.0
0.1
(°)
t (s)
h/T=10.0
h/T=1.5
h/T=1.2
0 20406080100
0
10
20
30
(°)
t (s)
h/T=10.0
h/T=1.5
h/T=1.2
02468101214
-0.008
0.000
0.008
0.016
y/B
x/L
y
bank
/B=2.8
y
bank
/B=1.7
y
bank
/B=1.35
0 20406080100
-0.2
-0.1
0.0
0.1
(°)
t (s)
y
bank
/B=2.8
y
bank
/B=1.7
y
bank
/B=1.35
020406080100
0
10
20
30
(°)
t (s)
y
bank
/B=2.8
y
bank
/B=1.7
y
bank
/B=1.35
0246810
h/
T
-0.06
-0.03
0
0.03
0.0
6
A
B
C
753
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