International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 3
Number 1
March 2009
31
CFD Based Hull Hydrodynamic Forces for
Simulation of Ship Manoeuvres
T. Tabaczek, T. Gornicz & J. Kulczyk
Wrocław University of Technology, Wrocław, Poland
1 INTRODUCTION
During the process of designing a new ship the de-
signer has to answer a lot of questions. Some of
them refer to the manoeuvrability of a ship. Moreo-
ver, the IMO regulations define precisely the mini-
mum manoeuvring requirements. The possibility to
determine the manoeuvring properties in early stage
of design results in significant reduction of cost and
time. There have been developed numerous mathe-
matical models describing a motion of a ship. The
authors of those methods usually report common
problems like poor accuracy, limited range of appli-
cation, or need of model tests to determine charac-
teristics and coefficients. Recent advances in IT and
CFD are promising in solving problems referring to
the need of model tests. In the opinion of the present
authors the CFD is mature enough to determine most
of hydrodynamic characteristics necessary to simu-
late ship manoeuvres. The characteristics of hull,
propeller and rudder and interactions between hull,
propeller and rudder can be determined separately
with confidence. In this paper the authors present the
attempt to determine the hull hydrodynamic forces
using the results of CFD computations of ship flow.
2 EQUATIONS OF SHIP MOTION
Usually the equations of ship motion are written in
the co-ordinate system with the origin at the centre
of gravity of a ship. The left-hand sides of equations
describe the dynamics of rigid body, and the right
hand sides represent the external forces:
zz
mu mvr X
mv mur Y
Ir N
−=
+=
=
(1)
m denotes the mass of a ship, u, v, r - forward speed,
transverse speed and yaw rate,
randv,u
- accelera-
tions in respective directions, I
zz
- the moment of in-
ertia of a ship, X, Y and N - the external forces: surge
force, sway force and yaw moment, measured at
ship’s centre of gravity.
The same equations can be written in a co-
ordinate system with the origin at midship:
2
()
()
()
G
G
zz G
m u vr x r X
m v x r ur Y
I r mx v ur N
−− =
++=
+ +=
(2)
ABSTRACT: There have been developed numerous mathematical models describing the motion of a ship. In
opinion of present authors the CFD is mature enough to determine with confidence the hydrodynamic charac-
teristics necessary to simulate ship manoeuvres. In this paper the authors present the attempt to determine the
hull hydrodynamic forces using the results of CFD computations of ship flow. Results show qualitative
agreement with reference data and reveal shortcomings due to simplifying assumptions applied in CFD com-
putations.
32
In this case u, v, r, X, Y, N denote rates and forces
measured at midship, and x
G
- the distance from the
midship to the centre of gravity.
Sometimes it is convenient to solve equations
written in co-ordinate system with origin at the cen-
tre of gravity when forces are determined at mid-
ship:
zz G
mu mvr X
mv mur Y
I r N xY
−=
+=
= −
(3)
Equations (3) are also used in the following for
simulation of ship motion.
3 EXTERNAL FORCES
In order to verify the idea of determination of hy-
drodynamic forces using CFD the present authors
chosen the modular model of MMG to represent the
external forces acting on manoeuvring ship:
HPR
HR
HR
XX X X
YY Y
NN N
= ++
= +
= +
(4)
The subscripts "H" "P" and "R" denote the hull
hydrodynamic forces and forces from propeller and
rudder respectively. This modular model is suitable
for testing the individual mathematical models one
by one.
3.1 Hull hydrodynamic forces
The mathematical model described in [1] was used
to represent hull hydrodynamic forces for its sim-
plicity and availability of reference data. Model is
based on the quasi-steady approach and forces de-
pend only on rates and accelerations:
2 24
0
3 2 23
3 2 23
()
()
H vv vr y rr vvvv
H v r x vvv vvr vrr rrr
H v r vvv vvr vrr rrr
X X X v X m vr X r X v
Y Yv Y mrYv YvrYvr Yr
N NvNrNv NvrNvr Nr
′ ′ ′ ′ ′ ′ ′′ ′ ′ ′ ′
=+ +− + +
′ ′ ′ ′ ′ ′ ′ ′ ′ ′′ ′ ′
=+− + + + +
′ ′′′′′′ ′′′′′′ ′′
=++ + + +
u', v', r' denote the non-dimensional rates, X
0
=
-R
T
(u) - ship resistance in considered co-ordinates,
and X’
vv
, X’
vr
, ..., Y’
v
, Y’
r
, ..., N’
v
, N’
r
, ... - hydrody-
namic coefficients.
The non-dimensional forms of forces are defined
as follows:
22
22
32
···
2
···
2
···
2
X X LU
Y Y LU
N N LU
ρ
ρ
ρ
′
=
′
=
′
=
3.2 Propeller force
The model described in [2] was adopted to represent
the longitudinal force generated by propeller, includ-
ing the effects of propeller-hull interaction:
24
0
(1 ) ( )
P tP p P T P
X C t nDK J= −
(6)
2
12 3
2
0
() · ·
1
cos
·exp( 4.0 )
·
TP P P
P
P
P
PP
P
K J C CJ CJ
w
JU
nD
ww
xr
β
β
ββ
=++
−
=
′
= −
′ ′′
= −
t
P0
denotes thrust deduction factor in straight ahead
ship motion, n - rotational speed of propeller, D
P
-
propeller diameter, K
T
- thrust coefficient, J
P
- ad-
vance coefficient, C
1
, C
2
, C
3
- coefficients for evalu-
ation of K
T
from open water characteristics, U - ship
speed, β - drift angle, w
P0
- effective wake fraction in
straight ahead ship motion, x'
P
- non-dimensional x-
ordinate of propeller.
3.3 Rudder forces
Forces from the rudder, including the interaction be-
tween hull, propeller and rudder, are calculated us-
ing the mathematical model described in [2] for rec-
tangular spade rudder:
(1 ) sin
(1 ) cos
( · ) sin
R RN
R HN
R R HH N
X tF
Y aF
N x ax F
δ
δ
δ
′′
=−−
′′
=−+
′ ′ ′′
=−+
(7)
0
0
1
0.6
1
cos
1 (1 )
·
2·
P
R
P
R
P
P
RR
P
RR
RR
D
h
w
K
w
s wU
nP
w
ww
w
xr
η
β
α δ γβ
ββ
=
−
=
−
=−−
=
′
= −
′ ′′
= −
( )
( )
( )
2
2
2
2
1 (1 · ( ))
sin
·
6.13·
2.25
2 (2 )·
()
1
RR
R
R
H
H
R
N NR R
R
N
R
U w Cg s
x
x
L
x
x
L
A
F CU
Ld
K
C
K
K ss
gs K
s
α
η
=−+
′
=
′
=
′
=
=
+
−−
=
−
t
R
- denotes the coefficient for additional drag, F
N
-
normal force acting on rudder, δ - rudder angle (pos-
33
itive to starboard), a
H
- ratio of additional lateral
force, x’
R
- non-dimensional x-ordinate of applica-
tion point of F
N
, x’
H
- non-dimensional x-ordinate of
application point of additional lateral force, h
R
-
height of rudder, s - propeller slip coefficient, P -
propeller pitch, w
R0
- effective wake fraction at loca-
tion of rudder, in straight ahead ship motion, α
R
- ef-
fective rudder inflow angle, γ - flow straightening
coefficient, U
R
- effective rudder inflow velocity, A
R
- rudder area, K
R
- aspect ratio of rudder.
4 HYDRODYNAMIC COEFFICIENTS
The clue of the present paper is the approximation of
hull hydrodynamic forces using the results of CFD
computations. To this end a series of ship flow com-
putations was carried out for a couple of combina-
tions of drift angle and yaw rate. The scope of drift
angle and yaw rate was predetermined based on re-
sults of free running model tests of basic manoeu-
vres, i.e. the turning manoeuvre and the 15/15deg
zig-zag manoeuvre [1]. It was estimated that drift
angle varies in the range -10<β<20deg and yaw rate
in the range 0<r’<1.0.
Computations of ship flow were carried out with
the assumption of low Froude number (negligible
heel and effect of free surface). The commercial
Fluent software was used to compute single phase,
turbulent steady flow in moving reference frame.
Same assumptions were applied when computing
the flow around the accelerating ship, in order to de-
termine the components of added mass: m
x
and m
y
.
In this case the accelerated flow with constant accel-
eration was computed around ship in rest.
Computed forces, moment and components of
added mass were subsequently used to determine all
hydrodynamic coefficients in the mathematical
model (5). The coefficients were estimated using
standard statistics procedure of fitting the user de-
fined function to the set of data.
Reported computations and simulations described
in next section were carried out for the Esso Osaka
model ship of length L
PP
=6.0m. Hydrodynamic
forces approximated using coefficients given in [1]
and coefficients based on CFD computations are
compared in Fig.1. If one takes the hydrodynamic
forces approximated using coefficients from [1] as
reference, surge force X’
H
seams to be predicted sat-
isfactory. Sway force Y’
H
is predicted well except
for drift angles above 10deg. Yaw moment N’
H
is
overpredicted at drift angles above 10deg and at
high yaw rate r’>0.4. The effect of differences in
hydrodynamic forces on the manoeuvring perfor-
mance of model ship is shown in the next section.
5 SIMULATION OF STANDARD
MANOEUVRES
The turning manoeuvre and the 10/10deg zig-zag
manoeuvre of model ship were simulated using
equations (3), nodular model (4) of external forces,
and mathematical models of hull, propeller and rud-
der forces described in previous sections. Data for
simulation collected from [3] and [4] are listed in ta-
ble 1. Model ship resistance was estimated accord-
ing to the idea of form factor:
C
TM
= (1+k)C
F0M
+ C
RM
There were applied the ITTC-57 model-ship correla-
tion line to evaluate frictional resistance C
F0M
, the
assumption of low Froude number (negligible wave
resistance C
RM
=0), and the form factor k=0.27. Open
water propeller characteristics K
T
(J) was approxi-
mated using the characteristics of corresponding
propeller from B-Wageningen screw series.
The differential equations of motion (3) were
solved using 4-th order Runge-Kutta method with
adaptive time step. However, the examinations
shown that this equation can be solved even precise-
ly with simpler methods but with the time step re-
striction.
Table 1 Data for simulation of motion of the Esso Osaka mod-
el ship
L
PP
6.0 m
B
0.978 m
T
0.402 m
C
B
0.83
x
G
0.190 m
m’= m/½ρL
3
0.01813
I’
zz
= I
zz
/½ρL
5
0.00110
m’
x
= m
x
/½ρL
3
0.00138
m’
x
(computed)
0.00133
m’
y
= m
y
/½ρL
3
0.01580
m’
y
(computed)
0.01703
J’
zz
= J
zz
/½ρL
5
0.00069
D
P
0.168 m
P/D
P
0.715
t
P0
0.27
w
P0
0.365
A
R
0.0408 m
2
h
R
0.256 m
K
R
2.49
U
0
0.699 m/s
The results of simulation of turning manoeuvre
with δ=35deg are shown in figures 2 and the results
of 10/10deg zig-zag manoeuvre are shown in figure
3. The differences in estimation of hydrodynamic
forces seen in figure 1 are reflected also in results of
both simulations.
34
6 CONCLUSIONS
The authors used the results of CFD computations of
ship flow to approximate hydrodynamic forces and
moment for simulation of ship manoeuvres. The
comparison of hydrodynamic forces approximated
using the reference hydrodynamic coefficients and
CFD based coefficients, shown in Fig.1, revealed
that sway force Y’
H
estimated using CFD based coef-
ficients is evidently underestimated at drift angles
above 10deg. Yaw moment N’
H
is overpredicted at
drift angles above 10deg and at high yaw rates
r’>0.4. That differences in estimation of hydrody-
namic forces are reflected also in results of simula-
tions shown in figures 2 and 3.
Taking into account that discrepancies in force
estimation and in simulated turning circle appear at
higher values of drift angle and yaw rate, one may
suspect that the assumption of low Froude number
applied to computations of ship flow is valid only at
low drift angle and yaw rate. Then at higher values
of drift angle and yaw rate the ship heel, trim, sink-
age, and especially the effect of free water surface
around the ship cannot be neglected in CFD compu-
tations.
-0,08
-0,06
-0,04
-0,02
0
0,02
0,04
0 0,2 0,4 0,6 0,8 1
r'
X'H
coeff. [1]
CFD, β=-10deg
CFD, β= 0deg
CFD, β= 10deg
CFD, β= 20deg
coeff. CFD
-0,1
0
0,1
0,2
0,3
0,4
0,5
0 0,2 0,4 0,6 0,8 1
r'
Y'H
coeff. [1]
CFD, β=-10deg
CFD, β= 0deg
CFD, β= 10deg
CFD, β= 20deg
coeff. CFD
-0,12
-0,1
-0,08
-0,06
-0,04
-0,02
0
0,02
0,04
0,06
0,08
0 0,2 0,4 0,6 0,8 1
r'
N'H
coeff. [1]
CFD, β=-10deg
CFD, β= 0deg
CFD, β= 10deg
CFD, β= 20deg
coeff. CFD
Fig.1. Surge force X’
H
, sway force Y’
H
, and yaw moment N’
H
computed with CFD and approximated using coefficients given
in [1] (dashed line) and CFD estimated (solid line).
0
0,5
1
1,5
2
2,5
3
3,5
0 1 2 3
y'
x'
CFD
coeff.[1]
test[1]
0
0,2
0,4
0,6
0,8
1
1,2
0 5 10 15 20 25 30
t'
U/U0
CFD
coeff.[1]
test[1]
35
0
5
10
15
20
25
0 5 10 15 20 25
t'
β
[deg]
CFD
coeff.[1]
test[1]
0
0,1
0,2
0,3
0,4
0,5
0,6
0 10 20 30 40
t'
r*L /U0
CFD
coeff.[1]
test[1]
Fig.2. Turning circle of model ship with δ=35deg simulated us-
ing hydrodynamic coefficients from [1] and coefficients based
on CFD computations
(a)
coeff.[1]
-30
-20
-10
0
10
20
30
0 5 10 15 20
t'
δ, ψ
[deg]
δ
ψ
(b)
CFD
-40
-30
-20
-10
0
10
20
30
40
0 5 10 15 20
t'
δ, ψ
[deg]
δ
ψ
Fig.3. 10/10deg zig-zag manoeuvre of model ship simulated
using hydrodynamic coefficients from [1] (a) and coefficients
based on CFD computations (b)
ACKNOWLEDGEMENT
The research reported in this paper was financially
supported by the Minister of Science and Higher
Education under grant No. N509 02932/2113.
BIBLIOGRAPHY
[1] The Specialist Committee on Esso Osaka Final Report and
Recommendations, Conference Proceedings of the 23rd
ITTC, Volume II, Venice, 2002
[2] Kijima, K., Tanaka, S., Furukawa, Y., Hori, T. : On predic-
tion Method of Ship Maneuvering Characteristics, MAR-
SIM'93, St. John's, Newfoundland, Canada, 1993
[3] Crane, C.L. : Maneuvering Trials of the 278 000 DWT Esso
Osaka in Shallow and Deep Water, Transactions of the
SNAME, Vol. 87, 1979
[4] Prediction of manoeuvrability of a ship, Bulletin of the
SNAJ, No.668, February 1985
