J Mar Sci Technol (2003) 8:47–60

DOI 10.1007/s00773-003-0163-5

Original articles

Evaluation of added resistance in regular incident waves by

computational fluid dynamics motion simulation using
an overlapping grid system
Hideo Orihara1 and Hideaki Miyata2
1
Hydrodynamic Engineering Laboratory, Technical Research Center, Universal Shipbuilding Corporation, 1-3 Kumozu-Kokan-cho,
Tsu 514-0398, Japan
2
Department of Environmental and Ocean Engineering, School of Engineering, The University of Tokyo, Tokyo, Japan

Abstract A computational fluid dynamics simulation method the added resistance, and to apply these methods for
called WISDAM-X was developed to evaluate the added re- an evaluation of added resistance in the ship design
sistance of ships in waves. The Reynolds-averaged Navier– process.
Stokes (RANS) equation was solved by the finite-volume Traditionally, the prediction of added resistance has
method and a MAC-type solution algorithm. An overlapping been done by analytical methods based on the potential
grid system was employed to implement rigorous wave gen-
flow theory. Since these methods are appropriate for
eration, the interactions of ships with incident waves, and the
the gross estimation of the added resistance, they have
resultant ship motions. The motion of the ship is simulta-
neously solved by combining the solution of the motion of the been widely used as a practical design tool.1–4 However,
ship with the solution of the flow about the ship. The free a clear disadvantage of these methods is that they can-
surface is captured by treatment by the density-function not account for nonlinear flow features such as a free-
method. The accuracy of WISDAM-X is examined by a com- surface shock wave or a wave breaking in the near field
parison with experimental data from a container carrier hull of a ship. Because these nonlinear features contribute
form, and shows a fairly good agreement with respect to ship considerably to the added resistance, analytical meth-
motion and added resistance. Simulations were also con- ods are not appropriate for the quantitative prediction
ducted for a bow-form series of a medium-speed tanker to of added resistance.
examine the effectiveness of the WISDAM-X method as a With the recent development of computer techno-
design tool for a hull form with a smaller resistance in waves.
logy, computational fluid dynamics (CFD) has been
It was confirmed that the WISDAM-X method can evaluate
applied to a variety of problems in ship hydrody-
the added resistance with sufficient relative accuracy and can
be used as a design tool for ships. namics. CFD simulation methods are advantageous
because they can deal directly with the nonlinear flow
Key words Computational fluid dynamics · Added resistance phenomena without explicit approximations. There-
in waves · Ship motion in waves · Overlapping grid fore, they are believed to be suitable for problems with
strong nonlinearity, such as the prediction of added
resistance.
In recent years, CFD simulations have been applied
Introduction to flows about a ship in waves.5–11 However, in most of
these studies, attention is focused on the ship’s motions,
The reduction of added resistance in waves is of crucial and the prediction of added resistance is not discussed
importance for the design of the hull form for better in detail.
performance in a seaway. To achieve this, it is very In this study, a new simulation method called
important to develop accurate prediction methods for WISDAM-X is developed for the accurate evaluation
of added resistance of practical hull forms. The
Reynolds-averaged Navier–Stokes (RANS) equation
and the continuity equation are solved using the finite-
volume method in the framework of an overlapping grid
system consisting of two different types of computa-
Address correspondence to: H. Orihara tional grids. A curvilinear body-fitted grid is used for
(e-mail: orihara-hideo@u-zosen.co.jp) computation in the vicinity of the hull, and a rectangular
Received: July 31, 2003 / Accepted: August 22, 2003 grid is used for computation in the far field, and extend-
48 H. Orihara and H. Miyata: Added resistance in regular incident waves

ing the computational domain several ship’s lengths

away from the hull surface. The computations in the
overlapping grid systems are performed by solving the
governing equations iteratively in each grid. The motion
of the ship is simultaneously solved by combining the
equation of motion of the ship’s body with the flow
computation.
A brief description of WISDAM-X is given in the
next section. Then the accuracy of WISDAM-X is ex-
amined through the grid-dependency test and a com-
parison with experimental data. The SR-108 container Fig. 1. Definition of the coordinate systems
ship was selected for the examination of accuracy be-
cause of the availability of extensive experimental re-
sults. In the subsequent section, WISDAM-X is applied
vertically upwards. The body-fixed coordinate system
to bow-form improvements, where two hull forms of a
O-XYZ has its origin at the center of gravity (CG) of
medium-speed tanker were selected, and its effective-
the ship, the X-axis is oriented from the bow to the
ness as a design tool for ships of smaller resistance is
stern, the Y-axis is oriented to the starboard side, and
examined. Some brief conclusions are given in the final
the Z-axis is normal to the other two. The translation
section.
coordinate system O-X0Y0Z0 shares its directions with
o-xyz, and the origin with O-XYZ. For the representa-
tion of the relative orientation among the coordinate
Computational method
systems, the Euler angles W = (F, Q, Y) are used. The
relations among the three coordinate systems are de-
Coordinate system
scribed as follows:
Three sets of orthogonal coordinate systems are used in
T T
the WISDAM-X method, as shown in Fig. 1. The space- (X , Y , Z ) = (x, y, z) - r (t )
0 0 0 c
(1)
fixed coordinate system o-xyz is defined with the o-xy
T T
plane on the still free surface, and the z-axis directed (X , Y , Z) = E (F, Q, Y) ◊ (X , Y , Z )
0 0 0
(2)

where
Ê cos Q cos Y cos Q sin Y - sin Q ˆ
E (F, Q, Y) = Á sin F sin Q cos Y - cos F sin Y sin F sin Q sin Y + cos F cos Y sin F cos Q ˜ (3)
ÁÁ ˜˜
Ë cos F sin Q cos Y + sin F sin Y cos F sin Q sin Y - sin F cos Y cos F cos Q¯

and rc(t) is the position vector of the center of gravity of ping grid system is employed in the WISDAM-X
the ship defined in o-xyz. method. Computation in the overlapping grid system is
The relation between the angular velocities is de- performed by the overlapping grid method.12 By em-
scribed as ploying the overlapping grid system, the computational
dW domain can be divided into two overlapping domains
w = H (F, Q) ◊ (4) for which a grid system can easily be generated. In the
dt
WISDAM-X method, the overall computational do-
where w = (P, Q, R)T is the angular velocity vector about main is divided into two solution domains, as shown in
the the X, Y, Z axes, and dW/dt is the angular velocity Fig. 2. The inner solution domain covers the region in
about the x, y, z axes. H(F, Q) is written as the vicinity of the hull. The outer solution domain ex-
È1 0 - sin Q ˘ tends to the outer boundary, which is located several
Í ˙ ship’s lengths away from the hull surface. In each solu-
( )
H F, Q = Í0 cos F sin F cos Q ˙ (5)
tion domain, the computational grid is generated inde-
Í0 - sin F cos F cos Q˙
Î ˚ pendently. In the inner solution domain, an O–H-type
body-fitted grid is generated. In the outer solution do-
main, a simple rectangular grid system is generated. By
Domain decomposition and overlapping grid system
using a rectangular grid, the computational time and
To implement both the interaction of a ship with inci- memory requirements can be significantly reduced com-
dent waves and the resultant ship motions, an overlap- pared with the case of the body-fitted grid system. In
H. Orihara and H. Miyata: Added resistance in regular incident waves 49

STO U T tion) of the inner grid at the overlapping outer grid

Inner solution domain LA OUT locations; (6) compute and update flows in the outer
grid; (7) return to (1) to proceed to the next time-step.
LH
SOO U T

LFO U T Governing equations

The governing equations which must be computed are
HO U T SSO U T the three-dimensional, time-dependent, incompressible
RANS equations, and the continuity equation for fluid
DO U T velocity and pressure. The RANS and the continuity
SBO U T equations are expressed in integral form for a control
S CO U T
Outer solution domain volume Wc as follows:
BO U T SIO U T ∂u
a ÚWc ∂t
dV + Ú c TdS = Ú c KdV
∂W W
(6)

S TI N
LAIN
Ú∂W u ◊ dS = 0
c
(7)
LH SOIN
where u is the fluid velocity vector, T is the fluid stress
tensor, and K is the body-force vector accounting for
LFIN
the inertial effect due to the motion of the coordinate
system. All the fluid variables are made dimensionless
SSIN with respect to the mean advancing velocity of the
HIN ship U0, the ship length L, and the fluid density r. The
C R IN
L dimensionless parameters, the Reynolds number (Re)
SCIN
SIIN
and the Froude number (Fn), are defined respectively
as
b
U0L U0
Fig. 2. Definition of the overlapping solution domain. a Over- Re = , Fn = (8)
all view of the overlapping solution domains. b Close-up view v gL
of the inner solution domain
where v and g are the kinematic viscosity of the fluid
and the gravitational acceleration, respectively.
The fluid stress tensor T is expressed as
addition, the numerical accuracy of the computation of
Ê 1 ˆ
wave propagation to the far field can be enhanced, be- + vt ˜ È—u + —u ˘
T

cause the grid spacing can be kept small near the free
T = uu + fI - Á
Ë Re ¯ ÎÍ
( )
˚˙
(9)
surface. For simplicity, the grids of the inner and outer where I is the identity tensor, — is the gradient operator,
solution domains will be denoted as inner and outer (·)T denotes the transpose operator, and f is the piezo-
grids, respectively, in the following descriptions. metric pressure excluding the hydrostatic pressure,
In the overlapping grid system, the flow computations which is defined as
are performed iteratively between the two grid systems,
and the flow information is exchanged between the z
f ∫ p+ (10)
grids by interpolating the flow variables at each bound- Fn 2
ary. In computing the ship’s motion, only the inner grid where p is the static pressure. The kinematic viscosity vt
is moved in accordance with the ship’s motion. The is evaluated by the Baldwin–Lomax algebraic turbu-
generation of the inner grid is made by the GMESH lence model.15
grid generation code developed at the National Mari- The body force vector K is given as
time Research Institute.13,14
dw dV
The numerical calculation procedure for the overlap- (
K = -2w ¥ u - w ¥ w ¥ r - ) dt
¥r-
dt
(11)
ping grid system in WISDAM-X is as follows: (1) move
the inner grid in accordance with the ship’s motion; (2) where w is the angular velocity about the body-fixed
search for grid points located in the overlapping region coordinate system O-XYZ, r is the position vector de-
of the two grids; (3) interpolate the flow variables of the fined in O-XYZ, and V is the translation velocity vector
outer grid at the overlapping inner grid locations; (4) of the ship in the directions of X, Y, and Z.
compute and update flows in the inner grid; (5) interpo- The discretization of the governing equations is
late the flow variables (velocity, pressure, density func- carried out by the finite-volume methods. The spatial
50 H. Orihara and H. Miyata: Added resistance in regular incident waves

discretization of the convective flux is carried out by the (

Ê uw ˆ Ê w wz A e kz cos kx - w w t )ˆ
QUICK scheme.16 The spatial discretizations of other Á ˜
u w = Á vw ˜ = Á 0 ˜ (15)
fluxes are carried out with the 2nd–order centered ÁÁ ˜˜
scheme. w Á
Ë w¯ Ë w Aw z e (
kz sin kx - w t
w ) ˜
¯

where VA is the wave amplitude, k = w w2/g is the wave

Free-surface boundary condition number, and ww is the angular frequency.
Equation 14 is implemented by giving the values of
The free-surface boundary condition consists of the ki-
the density function so that the vertical location of the
nematic condition for mass conservation on the free
isosurface of rm = 0.5 coincides with the wave height
surface, and the dynamic condition for stress balance on
given by Eq. 14.
the free surface.
The kinematic condition is treated using the density-
function method.17,18 In the density-function method, Other boundary conditions
the density function rm, which is a scalar variable, is
defined in the entire computational region as On the body boundary, the no-slip condition is imposed
for fluid velocity. To reduce the requirement for
Ï1.0, in the fluid region minimum grid spacing in the direction normal to the
rm = Ì (12) body surface, a wall function is employed. The body-
Ó0.0, otherwise boundary condition for pressure f is derived by incor-
This means that rm is unity at any point occupied by the porating the no-slip condition for the velocity into the
fluid, and changes from unity to zero at the free surface. RANS equation (Eq. 6), and assuming that the inner
In the control volume, including the free surface, the product of the diffusion term and the normal vector to
value of rm is approximated by the fractional volume of the body surface is zero. The gradient of the density
the fluid that occupies the cell. Then the location of the function normal to the body surface is assumed to be
free surface is defined as the isosurface of rm = 0.5. The zero.
time-dependent evolution of the density function is de- Except for the case of zero advance velocity, a uni-
termined by solving the following transport equation of form flow is given at the inflow boundary of the outer
rm, which is written as solution domain (SOUTI in Fig. 2). At the outflow and side
boundaries of the outer solution domain (SOUT O , and

∂r m SOUT
S , respectively, in Fig. 2), the open-boundary condi-
ÚWc ∂t
dV = - Ú c r m u ◊ dS
∂W
(13) tion is imposed for all the flow variables. At the center
plane boundaries (SCIN and SCOUT in Fig. 2), the symmetric
where u is the fluid velocity. The right-hand side of Eq. condition is imposed for all the flow variables.
13 is discretized using the QUICK sheme,13 and tempo- At the inflow, outflow, and side boundaries of the
ral discretization is made by the second-order Adams– inner solution domain (SIN I , SO , and SS , respectively, in
IN IN

Bashforth method. Fig. 2), the Drichlet boundary conditions are obtained
The dynamic condition is treated by extrapolating the by interpolating the flow variables of the outer solution
velocity and the pressure above the free surface in a domain as described earlier.
similar way to that used by Orihara and Miyata,19 in
which the surface tension and external stress on the free
surface are ignored, and the zero-stress condition is ap- Pressure solution procedure
proximately satisfied on the free surface. For a time-accurate solution of the incompressible flow,
the divergence-free condition must be satisfied at each
Wave-making condition at the inflow boundary time-level. To achieve this, a MAC-type pressure solu-
tion algorithm is employed. The pressure is obtained by
In the WISDAM-X method, incident waves are gener- solving the Poisson equations using the SOR method,
ated in a manner based on linear wave theory. The and the velocity components are obtained by correcting
generation of incident waves is approximated by giving the velocity predictor with the implicitly evaluated
the fluid velocity and wave height explicitly at the inflow pressure.
boundary of the outer grid (SOUTI in Fig. 2). The wave
height (zw) and the fluid velocity due to the wave par-
ticle motion uw are given in the space-fixed coordinate Ship motion treatment
system (o-xyz) as follows: The ship’s motion is determined by solving the equa-
tions for the ship’s motion. The equations for transla-
z W (t ) = z A cos(kx - w W t ) (14) tional and rotational motions are written as follows:
H. Orihara and H. Miyata: Added resistance in regular incident waves 51

Ê d* ˆ Flow un , p n u p, p p uc , pc 5 un+ 1 , pn+1

mÁ V + w ¥ V ˜ = F (16)
Ë dt ¯ F p, G p F c, G c
d* 2 4
h +w ¥ h = G (17) 1 3
dt K p
Kc

where h (= I0 · w) is the angular momentum, m is the Motion V n, w n V p, w p V,w

c c
V n+1, w n+ 1
mass of the ship, I0 is the tensor of inertia of the ship
about the body-fixed coordinate system, and d*/dt
stands for the time differencing in the noninertial coor- tn (prediction) (correction) t n+1
dinate system fixed to the body. Fig. 3. Coupling scheme for the computation of flow and ship
The hydrodynamic force F and the moment G are motion
obtained by integrating the fluid stresses over the wet-
ted surface of the ship. By solving Eq. 16 for V, the
acceleration of the ship in the body-fixed coordinate SR108
system is obtained as
d*V F

FP
= -w ¥V (18)

/2
91
dt m

B
AP

9
1/
DWL DWL
For the rotating motion, the angular acceleration d*w/dt

2
1

8
is obtained by incorporating the relation h = I0 · w into

7
Eq. 17, and is written as

6
4

5
d*w 4 7/8

dt
[
= I 0-1G - I 0-1 w ¥ ( I 0 ◊ w ) ] (19)
Fig. 4. Body plan of the SR108 container ship
-1
where I denotes the inverse of I0.
0
The velocity V and the angular velocity w are ob-
tained by integrating Eqs. 18 and 19 in time as the flow computation (Fig. 3). (1) The predictors of the
ÊF ˆ hydrodynamic force, Fp, and the moment, Gp, are calcu-
d*V
V =Ú dt = Ú Á - w ¥ V ˜ dt (20) lated using the flow solution at the present time-level tn.
dt Ëm ¯ (2) The predictors of the ship’s motion, Vpc , and the
d*w angular velocity, w p, are calculated using Fp and Gp, and
w=Ú
dt { [ ]}
dt = Ú I 0-1G - I 0-1 w ¥ ( I 0 ◊ w ) dt (21) the predictor of the body force, Kp, is then evaluated.
(3) The predictors of the flow field, up, and the static
V and w are transformed into the coordinate system pressure, pp, are calculated using Kp, and the correctors
O-X0Y0Z0 using Eqs. 2 and 4. Then, in the O-X0Y0Z0 of hydrodynamic force, Fc and the moment, Gc, are then
system, the velocity Vc and the angular velocity dW/dt calculated. (4) The correctors of the ship’s motion, Vcc,
are obtained as follows: and the angular velocity, w c, are calculated using Fc and
Gc, and the corrector of the body force, Kc, is then
Vc = E -1 (F, Q, Y) ◊ V (22) evaluated. (5) The correctors of the flow field, uc, and
dW dt = H -1 (F, Q, Y) ◊ w (23) the static pressure, pc, are calculated, and the flow field
at a new time-level tn+1 is obtained.
Finally, the trajectory of the CG, xc, and the attitude
of the ship, W, are obtained by integrating Vc and dW/dt
in time. Validation of WISDAM-X

The accuracy of the WISDAM-X method is examined

Flow updating procedure
by the grid-dependency test and a comparison with ex-
In the WISDAM-X method, the motion of the ship is perimental data. The hull form of the container carrier
simultaneously solved by combining the solution of the SR-108, shown in Fig. 4, was selected for the validation
motion of the ship with the solution of the flow compu- study because of the availability of extensive experi-
tation about the ship. To take into account the effects of mental results.
ship motion on the flow computation at each time level,
the following predictor–corrector scheme is employed
to combine the solution of the motion of the ship with
52 H. Orihara and H. Miyata: Added resistance in regular incident waves

Table 1. Size of the computational grids

Grid points

Case Inner Outer

1 67 ¥ 21 ¥ 41 95 ¥ 27 ¥ 67
2 99 ¥ 31 ¥ 61 141 ¥ 41 ¥ 101
3 148 ¥ 46 ¥ 91 211 ¥ 61 ¥ 151

Table 2. Size of the overlapping solution domain

Inner (1/L) Outer (1/L)

Length Length
LFIN 0.25 LOUT
F 1.00
LH 1.00 LH 1.00
LIN
A 0.25 LOUT
A 2.00
Radius Width
RIN 0.30 BOUT 1.00 Fig. 5. Close-up view of the overlapping grids for the SR108
Depth Depth container ship
HIN 0.06 HOUT 0.12
DOUT 1.00
z

Y x

Grid-dependency test
To check the grid dependency of the WISDAM-X
method, a numerical test was conducted using a set of
three grids. The numbers of grid points are given in
Table 1. The number of grid points in each direction is
increased by a factor of 1.5 between each grid. A partial
view of the overlapping grid system (case 2) is shown in z
Fig. 5. The three grids used in the test are shown in Fig.
6. The size of the solution domain of these grids is given Y x

in Table 2. Figure 2 shows the notation used in Table 2,

where LH is the length between the FP and AP of the
ship, LF is the length between the inflow boundary and
the FP of the ship, LA is the length between the AP of
the ship and the outflow boundary, H is the height of the
solution domain above the still free surface, and D is the
depth of the solution domain below the still free surface.
The simulations were conducted in regular heading z
waves at Fn = 0.275 and Re = 1.0 ¥ 106. Two examples of
the wave-length to ship-length ratio (l/L), 0.7 and 1.0, Y x

were chosen, with a constant wave-amplitude to ship-

length ratio (zA/L) of 0.01. The ship was set free to
heave and pitch, while the surging motion was fixed.
The time increment was controlled automatically so
that the CFL number did not exceed 0.5. The flow was
accelerated to a steadily advancing condition during the
time from T = 0.0 to T = 6.0, where T is time made
dimensionless with respect to (L/U0). The wave compu-
tations started at T = 12.0 and continued until T = 30.0. Fig. 6. Comparison of the inner grids used for the examina-
Figure 7 shows some comparisons of the pressure tion of grid-dependency
distributions on the hull surface at two instants in the
case of l/L = 0.7, where the pressure, excluding the
H. Orihara and H. Miyata: Added resistance in regular incident waves 53

hydrostatic component, is made dimensionless with re- pitching motions were made dimensionless with respect
spect to rU 20 · Te in the encounter period. The contours to the wave amplitude VA and the mean wave slope kVA,
are very similar in the three grid cases. This result indi- respectively. The phase difference was defined as that
cates that the difference in grid size has only a minor relative to the wave elevation at the CG of the ship. The
effect on the pressure distribution. added resistance (RAW) was obtained by subtracting the
Table 3 shows a comparison of the response ampli- still-water resistance from the mean resistance in waves.
tude operators (RAOs) of the ship’s motion (motion The added resistance was made dimensionless with re-
amplitudes zj and phase difference ej, where the sub- spect to r gVA2 (B2/L), where B is the breadth of the ship.
scripts j = 3, 5 stand for heaving and pitching motions, Table 3 shows that the variations in the motion ampli-
respectively) and the added resistance in waves. The tudes and phase differences are small in the three grid
amplitudes of the ship’s motions were obtained by the cases, and that the computed results almost converge in
harmonic analysis of the computed time histories of case 2. Since the pressure component is dominant in the
ship’s motion, where the first harmonic is taken as the hydrodynamic force acting on the ship in waves, the
motion amplitude. The amplitudes of the heaving and convergence of the ship’s motion accords with the con-
vergence of pressure distributions shown in Fig. 7.
For the added resistance, however, the variation
among the three grid cases is greater than that of the
ship’s motion. The difference between cases 2 and 3 is
–0
.11
0.3

0.2
5
about 12% of the computed value of case 3. This may
indicate that the size of the grid point used in the test
–0.05

0.2

.11
–0
15
0.
5

was not sufficient for the convergence of the added

1
0.

5
0.05 0.0
0

0 5
.0
–0
0
5
–0.05
–0
.1
–0
.1
0
resistance, and that some grid dependency remains in
– –

this range of grid size. However, the computed results

0.1

show monotonic convergence, and this magnitude of

quantitative difference can be regarded as being within
–0
.1 0.3 the allowable range for all practical purposes.
The results of the grid-dependency test confirm that
–0

0.2
.0
5

15
0.

the difference in grid size examined in this study has

1
0.

0.05
0
0

0
5
–0.05
5
–0.05
–0
.1
–0
.1
0
only a small influence on the ship’s motion and added
–0.1 –0.05
resistance, and that the numerical method has sufficient
0.1

0.05

convergence to predict these variables. Thus, in the fol-

lowing computations, the grid in case 2 (inner grid size
–0
.1 0.3
5
99 ¥ 31 ¥ 61; outer grid size 141 ¥ 41 ¥ 101) is used.
0.2
5
–0.0

0.2

15
0.
1
0.

0.05
0
Comparison with experimental data
0

0 – .1
5
–0.05 –0
.1
–0 0
–0.1 –0.05
5
To verify the accuracy of the WISDAM-X method, the
0.1

0.05

computed ship’s motion and added resistance were

a b compared with experimental data.20,21 The computations
Fig. 7. Comparisons of the pressure distributions on the hull were performed in regular heading waves at three
surface of the SR108 at Fn = 0.275 and l/L = 1.0. a t = 2/4 Te. Froude numbers: Fn = 0.250, 0.275, 0.300. The Reynolds
b t = 4/4 Te number was set at 1.0 ¥ 106 for all computations. The

Table 3. Grid-dependence of ship’s motions and added resistance

Heave Pitch Added resistance

l/L Case x3/zA e3 (°) x5/kzA e5 (°) dAWa

0.7 1 0.049 -21.4 0.050 140.7 0.92

2 0.053 -18.5 0.052 138.9 1.41
3 0.055 -19.0 0.051 138.4 1.61
1.0 1 0.636 -95.4 0.532 158.6 5.01
2 0.684 -96.9 0.538 159.4 6.93
3 0.691 -97.5 0.541 159.9 7.20
a
dAW = RAW / rgzA2(B2/L)
54 H. Orihara and H. Miyata: Added resistance in regular incident waves

Fig. 8. Time-evolution of wave-height contour maps about

the SR108 container ship advancing in head waves at Fn =
0.275 in a wave of l/L = 1.0 and zA/L = 0.01

ship was set free to heave and pitch, while the surging Fig. 9. Comparison of the RAOs of ship’s motion for the
SR108 container ship at Fn = 0.275 in waves of zA/L = 0.01.
motion was fixed. The conditions of the incident waves RSM, Rankine source method
were 0.5 £ l./L £ 1.6 with zA/L = 0.010. The conditions of
the solution domain and the grid size were the same as
those used in case 2 in the grid-dependency test (see
Tables 1 and 2). The time-increment was controlled made dimensionless with respect to L/103. Contour
automatically so that the CFL number did not exceed maps are shown at an interval of one-quarter of the
0.5. The flow was accelerated to a steadily advancing encounter period (Te) in heading waves of l/L = 1.0.
condition during the time from T = 0 to T = 6.0. The The periodical wave formation due to the interaction of
wave computations started at T = 12.0 and continued the ship’s motion with the incident wave is clearly seen
until T = 30.0. in Fig. 8.
Figure 8 shows time-evolution of the wave-height Comparisons of the ROAs of the ship’s motion and
contour maps about SR108, where the wave height was the added resistance are shown in Figs. 9 and 10. Figure
H. Orihara and H. Miyata: Added resistance in regular incident waves 55

: M58F0
: M58F1

X
D
FP
AP

9 1/2
DWL DWL
9

1/2 8
7

2
3
4 5, 6

DW L

: M58F0
: M58F1
FP X

Fig. 11. Body plans and bow profiles of the M58 series models

dent wave (zA) for the amplitude of the heaving motion,

and within 1% of the mean slope of the incident wave
(kzA) for the amplitude of the pitching motion. Our
computed results agree well with the experimental data.
It is noted that the degree of agreement is better than
the agreement with the linear methods. In our calcula-
tions, the motion peak is predicted quite well for both
heaving and pitching motions. The reason for the accu-
rate prediction may be the exact treatment of the hull
surface and the free-surface boundary conditions in the
WISDAM-X method. Comparisons of the added resis-
tance at Fn = 0.250, 0.275, and 0.300 are shown in Fig.
10. The computed results were compared with the ex-
perimental data of Takahashi.21 Although some discrep-
ancies can be seen in Fig. 10, the calculated results agree
reasonably well with the experimental data in all cases.
Fig. 10. Comparison of the RAOs of added resistance for the
SR108 container ship at Fn = 0.250, 0.275, and 0.300 in waves From the results of the comparisons with experimen-
of zA/L = 0.01 tal data, it has been demonstrated that the WISDAM-X
method has sufficient accuracy to predict ship’s motions
and the added resistance.
9 shows comparisons of the amplitudes and phase dif-
ferences of heaving and pitching motions at Fn = 0.275.
The computed results were compared with experimen- Application to the bow-form improvement
tal data from the ITTC20 and with linear methods
(results calculated by the strip method1 and the To examine its effectiveness as a design tool for the hull
frequency-domain Rankine source method (RSM)22). form to give less resistance in waves, the WISDAM-X
Since the uncertainty of the experimental data is not method was applied to improvements of the bow form
given in the reference,20 the magnitude of the uncer- above the still waterline. Two hull forms, M58F0 and
tainty of the experiment cannot be evaluated. However, M58F1, of a medium-speed tanker were chosen. The
in general, it may be said that the uncertainty of the body plans and bow profiles are shown in Fig. 11. The
experiment is within 1% of the amplitude of the inci- parent hull form, M58F0, has a conventional rounded
56 H. Orihara and H. Miyata: Added resistance in regular incident waves

bow. The modified hull form, M58F1, has a long, pro- the mean incident wave slope kVA. The computed results
truding bow above the still waterline. It was newly agree well with the experimental data. It is noted that
designed by the authors to reduce the added resistance. the motion amplitudes of the two hull forms are almost
The experiment on the two hull forms was conducted the same in the experiment and the computation. This
in the experimental tank at the University of Tokyo
using 2.5-m models. In the experiment, the ships’ mo-
tions and total resistance were measured in regular
heading waves of 0.5 £ l/L £ 1.5 at Fn = 0.224. The
motion response amplitude operators and the added
resistance were obtained by analyzing the measured
time histories.
The conditions of these simulations were almost the
same as those of the experiment. Fn and Rn were set at
0.224 and 1.0 ¥ 106, respectively. The solution domain
and grid sizes were the same as those used in case 2 of
the grid-dependency test (see Tables 1 and 2). The time-
increment was controlled automatically so that the CFL
number did not exceed 0.5. The flow was accelerated to
a steadily advancing condition during the time from T =
0 to T = 6.0. The wave computations were started at T =
12.0 and continued until T = 30.0. Close-up views of the
grid system are shown in Fig. 12, where the difference in
bow shape between the two hull forms can clearly be
seen.
Comparisons of the amplitudes of the ships’ motions
are shown in Figs. 13 and 14. In both figures, the calcu-
lated and measured results for M58F0 and M58F1 are
compared in waves of l/L = 0.7, 1.0, 1.2, and 1.5, at a
constant wave height of VA/L = 0.012. Figure 13 shows
comparisons of the amplitudes of heaving motion made
dimensionless with respect to the incident wave ampli-
tude zA. Figure 14 shows comparisons of the amplitudes Fig. 12. Close-up views of the M58 series bow configuration
of pitching motion made dimensionless with respect to and grid system

Fig. 13. Comparison of the ampli-

tudes of heaving motion for the M58
model series at Fn = 0.224
H. Orihara and H. Miyata: Added resistance in regular incident waves 57

Fig. 14. Comparison of the ampli-

tudes of pitching motion for the M58
model series at Fn = 0.224

Fig. 15. Comparison of the added

resistance of the M58 model series at
Fn = 0.224

implies that the small difference in bow shape has a The superiority of a hull form, i.e., the smaller added
negligible effect on the ship’s motion. resistance, is more directly indicated by the comparison
Figure 15 shows comparisons of the added resistance of the pressure distribution on the hull surface shown in
in the same manner as in Figs. 13 and 14. In the figure, Figs. 16–19. Fig. 16 shows a comparison of the hull
the added resistance is made dimensionless with respect surface pressure (Cf) distributions in still water condi-
to rgV2A(B2/L). Although the computation overestimates tions at Fn = 0.224, where Cf is the pressure excluding
the added resistance, it correctly indicates the relative the hydrostatic component and made dimensionless
magnitude of the added resistance between the two hull with respect to rU02 . The difference in the pressure dis-
forms. It may be safe to say that the computations can tributions between the two hull forms is very small.
predict the difference in the magnitude due to the Since the measured resistances of the two hull forms in
modification of the bow shape. still water conditions are almost the same, it is consid-
58 H. Orihara and H. Miyata: Added resistance in regular incident waves

Fig. 16. Comparison of the hull surface pressure (Cf) distribu- Fig. 17. Comparison of the time-averaged pressure (C̄f,WV)
tion of the M58 model series at Fn = 0.224 in still water distribution on the hull surface of the M58 model series at
conditions Fn = 0.224 in a wave of l/L = 0.7

ered that the computations correctly predict the flow in the high-pressure region. Thus, the surface pressure
field about a ship in this condition. distribution can be used to choose a superior hull form
Figure 17 shows a comparison of the distributions of with less resistance in waves. It is also noted that the
the time-averaged pressure (C̄f,WV) in waves of l/L = 0.7, magnitude of Cf,AW below the wave profile in still water
where C̄f,WV is obtained by averaging Cf over the dura- conditions is quite small, and that the higher-pressure
tion of several encounter periods with incident waves. region is confined to the relatively small area near the
In general, the C̄f,WV distributions of the two hull forms bow. This implies that the modification of the bow
are very similar, but some difference is noted near the shape above the free surface has a remarkable effect on
bow. It is shown that the area of the high pressure the added resistance.
region, for instance C̄f,WV ≥ 0.2, is smaller for M58F1. Since the WISDAM-X method predicts the added
The difference between the two hull forms is indi- resistance and pressure distribution resulting from a
cated more clearly in Figs. 18 and 19, which show a small difference in the bow shape with satisfactory accu-
comparison of the distribution of the pressure compo- racy, as shown here, it is considered that the WISDAM-
nent associated with the added resistance, Cf,WV, in X method can be used effectively for hull-form
waves of l/L = 0.7, obtained by subtracting the pressure improvement purposes. Hull form improvement can be
in still water conditions, Cf,S, from C̄f,WV. Since the inte- achieved by a succession of comparisons of the com-
gration of Cf,AW over the wetted hull surface produces puted added resistances and pressure distributions of a
added resistance, the distributions of C̄f,WV show details series of modified hull forms. A quantitative estimation
of the added resistance. would be better made by experimentation with the final
It is noted that the extension of the high-pressure hull form realized after a series of simulations.
region is considerably reduced by the new long, pro-
truding bow shape of M58F1. The reduction of the high-
pressure region for M58F1 obviously indicates the Conclusions
superiority of M58F1 with the small added resistance
shown in Fig. 15. As shown in Figs. 18 and 19, a hull A new CFD simulation method, WISDAM-X, was
form with a smaller added resistance shows a reduction developed to predict the added resistance of a ship in
H. Orihara and H. Miyata: Added resistance in regular incident waves 59

Fig. 18. Comparison of the distribution of the pressure com-

ponent associated with the added resistance (Cf,AW) of the
M58 model series at Fn = 0.224 in a wave of l/L = 0.7. Dot–
dash line, wave profile in still water Fig. 19. Close-up view of the distribution of the pressure com-
ponent associated with the added resistance (Cf,AW) of the
M58 model series at Fn = 0.224 in a wave of l/L = 0.7. Dot–
dash line, wave profile in still water
regular heading waves. The accuracy of this method was
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