J Mar Sci Technol (1999) 4:108–116
CFD simulation of 3dimensional motion of a ship in waves:application to an advancing ship in regular heading waves
Yohei Sato, Hideaki Miyata,
and
Toru Sato
Department of Environmental and Ocean Engineering, University of Tokyo, 731 Hongo, Bunkyoku, Tokyo 1138656, Japan
properties of importance are motion characteristics inwaves and maneuvering characteristics. The usefulnessof CFD simulation will be increased when these unsteady characteristics can be predicted with sufﬁcientaccuracy by improved techniques.The objective of this research was to develop a newtechnique to simulate ship motion in waves. The unsteady motion of a ship was treated by Akimoto
5
usinga boundaryﬁtted coordinate system which moves inaccordance with the ship’s motion, and is applied to asailing simulation of an IACCclass racing yacht. Otherstudies simulate maneuvering motion,
6
and the motionof underwater vehicles with controllable wings.
7
However, a realization of the 3D motion of a ship in waveshas not yet been achieved.The ship motion problem has also been treated by thetechnique of theoretical ﬂuid dynamics, with the postulation that a velocity potential exists. Since the mainpart of a ship’s motion in waves is linear, this approachhas provided a lot of useful information for the prediction of ship motion and is a commonly used, establishedmethod. However, it cannot be applied to motion whichincludes nonlinear properties, such as largeamplitudemotion, motions with a wave impact load (slamming),and capsizing.There are two ways to approach the simulation of moving bodies.1.By use of the moving grid method,
5,6,8
where themotion of the ship is represented by the deformationof a grid system ﬁtted to the ship’s surface. The gridsystem is also ﬁtted to the free surface. The disadvantage of this method is that it cannot cope with largeamplitude motion.2.Within a ﬁxed coordinate system, the motion of theship is treated as body forces introduced into theexternal force term of the Navier–Stokes equation.
7
In this study, the latter technique is employed inorder to cope with largeamplitude motion. The surface
Abstract:
A new computational ﬂuid dynamics simulationmethod has been developed for the unsteady motion of a shipadvancing in waves. The objective is to evaluate the addedresistance and predict the performance of a ship in waves. Inthis study, a ﬁnite volume method, in the framework of aboundaryﬁtted grid system, is employed. The motion of theship is solved with six degrees of freedom by using the hydrodynamic forces and moments obtained from the solution of the simulation method. The marker–density–function methodis employed to calculate the nonlinear free surface. Thismethod is applied to the coupled motion problem of heavingand pitching.
Key words:
computational ﬂuid dynamics, unsteady motion,motion in waves, density function method
Introduction
Ship hydrodynamics computations based on NS solverswere initiated in the 1980s, and since then a number of useful codes have been developed. These are theTUMMACIV code by Miyata et al.,
1
the NICE code byKodama,
2
the WISDAMV code by Zhu et al.,
3
andothers. These codes have been used by ship designers.In the last few years, computational ﬂuid dynamics(CFD) techniques have been incorporated into nonlinear optimization procedures for hull conﬁguration.
4
In this way, CFD simulation plays an important role inship design.However, these techniques are for a ship on a steadystraight course in a calm sea. The value obtained ismainly used to minimize the required horsepower of themain engine, or to increase the boat’s speed with thesame output from the engine. Other hydrodynamic
Address correspondence to:
H. MiyataReceived for publication on Nov. 15, 1999; accepted on Nov.18, 1999
of the air–water interface is moved according to theship’s motion, and the nonlinear freesurface conditionis implemented by use of the densityfunction methodwhich can simulate breaking waves.
Design by simulation
A motion simulation system was developed by combining the NS solver and the motion solution method for anequation of motion with six degrees of freedom.First, an O–Htype boundaryﬁtted coordinate system was generated, and then the NS solver was movedby forces and motions. With these hydrodynamic forcesand moments, the equations of motion are solvedand the translational and rotational accelerations areobtained. The new attitude of the ship is derived, and isthen fed back to the NS solver. This cycle is repeated,and unsteady motion is simulated (Fig. 1).The GMESH code from the Ship Research Institutewas employed for the grid generation.
Simulation method
Grid system and motion treatment
A boundaryﬁtted grid system was employed for therepresentation of the hull geometry. An O–Htypegrid system surrounds the hull, as shown in Fig. 2, andthis is not deformed or regenerated by the ship’smotion.The motion of the ship is treated in a ﬁxed coordinatesystem, which means that the grid system also movesaccording to the ship’s motion, as shown in Fig. 3. Thesrcin of the coordinate system is at the center of theship, and the trajectory and attitude of the ship aredetermined in the spaceﬁxed coordinates.
Solution method
This solution method is for the combined problem of incompressible ﬂuid motion and ship motion. The solution method used for the ﬂuid ﬂow is the ﬁnitevolumemethod based on the previously developed WISDAMV, and the densityfunction method is used for theimplementation of the freesurface condition.The governing equations for the ﬂuid ﬂow are theNavier–Stokes equation and the equation of continuity.The grid system is ﬁxed to the ship, and then the motionof the ship is expressed by the body force term
F
n
w
as
DuDt puK
w ww
=—+—+
1
2
Re
(1)
Fig. 1.
Flow chart of the motion simulation system
Fig. 2.
Overall view of the O–H grid system
Fig. 3.
Schematic sketch of the method used. The governingequation is solved in a bodyﬁxed coordinate systemY. Sato et al.: CFD simulation of ship motion109
KF
www
n
= +W
(2)
Fur ddt r dV dt
g
nwww w
wwwwww w w w
=¥¥¥
( )
¥
2
(3)where
u
w
is the velocity vector of the ﬂuid,
p
is thepressure divided by the density, Re is the Reynoldsnumber,
K
w
¯
is the body force,
W
w
¯
is the gravitationvector,
w
w
¯
is the angular velocity vector,
r
w
is theposition vector, and
V
g
w
¯
is the velocity of the srcin of the bodyﬁxed coordinates.These equations are solved in the MACtypetimemarching solution procedure, and the results areobtained continuously at each timelevel.All vector variables are deﬁned in Cartesian coordinates, and the components of velocity and pressure aredeﬁned in the staggered arrangement. Thirdorderupstream differencing is used for convective terms,and secondorder centered differencing for the otherdiscretization in space. The SOR method is used for thesolution of the pressure. The ﬁrstorder Euler explicitmethod is used for the timeintegration for simplicityand efﬁciency.The ship’s motion is given by the solution of theequation of motion, with the external forces and moments derived from integration of the pressure and thefrictional force on the hull surface. The timeintegrationis also made by the Euler explicit method.
Density function method
The density function, which is a scalar value of theporosity, is determined at the center of a cell, and isgoverned by the equation
DDtt u
r∂r ∂ r
w
=+◊—
( )
=
0
(4)The equation is discretized by the thirdorderupstream scheme for space differencing and theAdams–Bashforth scheme for time differencing.The density function takes the value 0.0 in air and 1.0in water, and the freesurface location is determined atthe point where the density function takes the value 0.5:
r rr
freesurfaceairwater

=+=
205.
(5)
Boundary conditions and numerical wavemaking
The boundary conditions for velocity, pressure, anddensity functions are listed in Table 1. The incidentwaves are assumed to be sinusoidal in inﬁnitely deepwater. The wave height
z
and the three velocity components at the spaceﬁxed position (
x
,
y
,
z
) are given by
xxw
=
( )
a
kxt
cos
(6)
Uckekxt V Wckekxt
akzakz
=
( )
==
( )
zw zw
cossin0
(7)where
c
2
=
g/k
,
z
a
is the wave amplitude,
k
is the wavenumber,
w
is the angular velocity, and
g
is the gravitational acceleration. The incident waves generated in thespaceﬁxed coordinates are introduced into the computational domain of the coordinates ﬁxed to the ship’ssurface.In order to avoid unfavorable reﬂections of the waveson the outer boundaries, the spacing is stretched in theirvicinity.
Accuracy evaluation
The accuracy of this method is examined by a simulation of a ship advancing steadily on a straight course.The wave contours in Wigley’s mathematical modeladvancing at Froude number 0.289 are calculated underthe conditions of Case A in Table 2 and compared withthe experimental results in Figs. 4 and 5. The waveproﬁle on the hull surface is also compared in Fig. 6.The wave contours in Figs. 4 and 5 show that thecomputed wave height is much smaller than the measurements taken far away from the ship. This is mainlydue to the coarse grid spacing in the far ﬁeld. However,the agreement with the wave proﬁles seems to be satisfactory in Fig. 6, except that some discrepancy is notedat the stern.
Table 1.
Boundary conditionsVelocityPressureDensityInﬂowDirichletDirichletDirichletOutﬂowExtrapolationExtrapolationExtrapolationSide boundaryExtrapolationExtrapolationExtrapolationFree surfaceExtrapolationPressure of air (
∫
0)0.5110Y. Sato et al.: CFD simulation of ship motion
The effect of grid spacing is tested with the two gridconditions listed in Table 2, and the results are compared with the wave proﬁle on the hull surface in Fig. 6.It is probable that such small differences in spacing haveonly a slight inﬂuence.
Application to pitching motion in heading waves
Conditions of simulation
This method was applied to ships advancing in regularheading waves. The pitching and heaving motions arefree, and other motions are not permitted. Two hulls,the modiﬁed Wigley model
9
and the Series 60 (
Cb
=
0.6)model were chosen, and the advance speed of eachmodel was set at Froude numbers 0.20 and 0.24, respectively. The conditions of computation are given in
Table 2.
Conditions of computationWigley modelWigley modelCase ACase BGrid points130
¥
30
¥
110130
¥
40
¥
130(
=
429000)(
=
676000)Computational domain
L
=
2.5, radius
=
0.8
L
=
2.5, radius
=
0.8Minimum grid space
z
1
2.0
¥
10

3
2.0
¥
10

3
Minimum grid space
z
2
5.0
¥
10

4
5.0
¥
10

4
Minimum grid space
z
3
9.4
¥
10

3
6.3
¥
10

3
Reynolds number1.0
¥
10
6
1.0
¥
10
6
Froude number0.2890.289Time of simulation4.04.0Step number97929640Time for acceleration1.01.0Maximum
dt
1.0
¥
10

3
1.0
¥
10

3
CFL0.50.5
z
1
, longitudinal direction of ship;
z
2
, radial direction;
z
3
, girth directionThe units are nondimensionalized by the length of the ship, the velocity of advance, and thedensity of the water
Fig. 4.
Computed wave contours of the Wigley model at
Fn
=
0.289. The contour interval is 0.001
Fig. 5.
Measured wave contours of the Wigley model at
Fn
=
0.289. The contour interval is 0.001
Fig. 6.
Comparison of wave proﬁles on the hull surface between experiment, coarse grid (case A), and ﬁne grid (case B)(Wigley model,
Fn
=
0.289)Y. Sato et al.: CFD simulation of ship motion111
Table 3, and the grid systems of the modiﬁed Wigleymodel and the Series 60 model are shown in Figs. 7 and8, respectively.
Results of simulation
A comparison is made between computation and measurement. The experimental and theoretical data of themodiﬁed Wigley model are from the results of DelftUniversity,
10
and the experiments were performed on aSeries 60 model at the University of Tokyo. The principal particulars of the two ships which were used for thecomputations and the experiments are shown in Table 4.The amplitude and phase difference of the pitchand heave are deﬁned below. The phase differenceis taken as the difference in the wave height at thecenterofgravity position with the heave and pitchamplitude.
zzt t
aezae
=◊+
( )
=◊+
( )
coscos
wez qqwez
q
(8)where
z
is the heave and
z
a
is its amplitude,
q
isthe pitch and
q
a
is its amplitude,
w
e
is the encounterfrequency, and
e
z
z
and
e
q
z
are the phase differences.
Table 3.
Conditions of computationModiﬁed Wigley modelSeries 60 (
Cb
=
0.6)Grid points115
¥
30
¥
110118
¥
30
¥
114Computational domain
L
=
2.5, radius
=
1.0
L
=
2.5, radius
=
0.8Minimum grid space
z
1
2.0
¥
10

3
2.0
¥
10

3
Minimum grid space
z
2
5.0
¥
10

4
5.0
¥
10

4
Minimum grid space
z
3
5.0
¥
10

3
5.0
¥
10

3
Reynolds number1.0
¥
10
6
1.0
¥
10
6
Froude number0.240.20Time of simulation15.015.0Time for acceleration1.01.0Start of wave making5.05.0Maximum
dt
1.0
¥
10

3
1.0
¥
10

3
CFL0.50.5Amplitude of5.00
¥
10

3
5.56
¥
10

3
incident waves
z
a
1.11
¥
10

2
z
1
, longitudinal direction of ship;
z
2
, radial direction;
z
3
, girth directionThe units are nondimensionalized by the length of the ship, the velocity of the advance, and thedensity of the water
Fig. 7.
Grid system on the horizontal plane and hull surface of the modiﬁed Wigley model
Fig. 8.
Grid system on the vertical plane and hull surface of the Series 60 (
Cb
=
0.6)112Y. Sato et al.: CFD simulation of ship motion