Ocean Engineering 38 (2011) 641–650
Contents lists available at ScienceDirect
Ocean Engineering
journal homepage: www.elsevier.com/locate/oceaneng
Prediction of added resistance of ships in waves
Shukui Liu, Apostolos Papanikolaou n, George Zaraphonitis
Ship Design Laboratory, National Technical University of Athens, Greece
a r t i c l e in f o
abstract
Article history:
Received 23 November 2009
Accepted 7 December 2010
Editor-in-Chief: A.I. Incecik
Available online 3 January 2011
The prediction of the added resistance of ships in waves is a demanding, quasi-second-order seakeeping
problem of high practical interest. In the present paper, a well established frequency domain 3D panel
method and a new hybrid time domain Rankine source-Green function method of NTUA-SDL are used to
solve the basic seakeeping problem and to calculate first order velocity potentials and the Kochin
functions, as necessary for the calculation of the added resistance by Maruo’s far-field method. A wide
range of case studies for different hull forms (slender and bulky) was used to validate the applicability and
accuracy of the implemented methods in practice and important conclusions regarding the efficiency of
the investigated methods are drawn.
& 2010 Elsevier Ltd. All rights reserved.
Keywords:
Added resistance
Drift forces
Far-field method
Near-field method
3D panel method
Kochin function
1. Introduction
When sailing in a seaway, the calm water resistance and
powering of a ship is increased and this is accounted for in traditional
ship design by some rough proportional increase of calm water
resistance by about 20–40%. However, the accurate prediction of
ship’s resistance in waves is nowadays of increased importance, both
for the designer and the shipowner/operator, since it greatly affects
the selection of ship’s engine/propulsion system and ship’s performance in terms of sustainable service speed and fuel consumption in
realistic sea conditions. Also, accurate and efficient predictions of the
added resistance in natural seaways are necessary for the implementation of modern onboard ship routing systems.
The added resistance is a steady force of second-order with
respect to the incident wave’s amplitude and acting opposite to
ship’s forward speed in longitudinal direction. When the ship has
zero forward speed, then the added resistance is trivially identical to
the longitudinal drift force (Papanikolaou and Zaraphonitis, 1987).
There are several alternative approaches to the added resistance
problem; they can be generally classified into two main categories,
namely far-field and near-field methods. The far-field methods are
based on considerations of the diffracted and radiated wave energy
and momentum flux at infinity, leading to the steady added
resistance force by the total rate of momentum change. The nearfield method, on the other side, leads to the added resistance as the
steady second-order force obtained by direct integration of the
hydrodynamic, steady second-order pressure acting on the wetted
n
Corresponding author.
E-mail address: papa@deslab.ntua.gr (A. Papanikolaou).
0029-8018/$ - see front matter & 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.oceaneng.2010.12.007
ship surface. The latter can be calculated exactly from first order
potential functions, and their derivatives. It is evident that though
both above methods should lead to the same numerical results for
the added resistance, this is not so in practice, because both methods
request equally accurate solutions of the basic potential, seakeeping
theory problem in the near and far field, which cannot be ensured in
practice. This is a well-known observation in the similarly posed drift
force problem. Thus, even if the same potential theory and related
numerical methods are used for the solution of the linear seakeeping
problem and the calculation of the governing potential function (and
its derivatives and of related hydrodynamic quantities, including
ship motions, as necessary) in the near- and far-field, the numerical
results by the near- and far-field method for the added resistance are
in general different, thus a basic question arises, namely which
method is more efficient for practical applications.
In this respect, several theoretical approaches of varying complexity and accuracy have been introduced in the past and
numerically implemented/verified. The first far-field approach
was introduced by Maruo (1957) in the late 50s; it was further
elaborated in the following years by Maruo (1960, 1963) and Joosen
(1966). In the early 70s, the first near field, direct pressure
integration methods appeared; however, the considered hydrodynamic pressure distribution was highly simplified (e.g. Boese,
1970). At about the same time, the radiated energy approach of
Gerritsma and Beukelman (1972) was introduced, which is basically following the far-field approach of Maruo. Strøm-Tejsen et al.
(1973) did a thorough evaluation of all the above approaches to find
large discrepancies between the numerical results form different
theoretical approaches and relevant model experimental data.
Later on, Salvesen (1974) investigated the added resistance
problem by applying Gerritsma and Beukelman’s method, but using
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S. Liu et al. / Ocean Engineering 38 (2011) 641–650
basic results of the STF seakeeping strip theory (Salvesen et al.
1970) and found quite satisfactory results for the investigated ship
hull forms. This was not surprising, considering the superiority of
the STF strip theory method for the prediction of ship motions over
other methods at that time. Also, the significance of accurate ship
motions predictions for a reliable estimation of the added resistance was acknowledged and one may have concluded that if a
more accurate calculation of the ship motions could be used
(or even ship motions from model experiments), all above alternative
approaches to the added resistance may have led to better results.
An exact in the sense of potential theory near-field, direct pressure
integration approach to the added resistance problem was introduced by Faltinsen et al. (1980), with good validation results. The
observed deficiency of the approach in short waves was addressed
by the introduction of a simplified added resistance estimation
formula, which models remarkably well the complicate interaction
between the diffraction of waves and the steady flow around the
ship. The same problem appears in exaggerated form with low
speed, full hull forms with blunt bows (bulkcarriers and tankers),
for which Ohkusu (1985) proposed an improved approach. This
method was further elaborated by Iwashita and Ohkusu (1992),
who applied Maruo’s (1963) improved far-field formulation and
the concept of Kochin functions, to obtain very good results for the
added resistance of a fully submerged spheroid by use of a 3D
pulsating and traveling source, Green function method. In his most
recent state of the art review, Naito (2008) concluded that y ‘‘good
agreement can be obtained by applying this (latter) calculation
method to all types of ship. However this method has not been
widely utilized due to the complicated calculation technique and
large computing time required’’. More recently, using an enhanced
unified theory, Kashiwagi (2009) calculated the added resistance
by a modified version of Maruo’s approach, obtaining also quite
satisfactory results.
In the present study, the well established frequency domain 3D
panel method and computer code NEWDRIFT (Papanikolaou–
Zaraphonitis, 1987, Papanikolaou–Schellin, 1992 and Papanikolaou
et al. 2000), a new time domain Green function method (Liu et al.
2007) and a hybrid time domain the Rankine source-Green
function method (Liu and Papanikolaou, 2009, 2010) of NTUA-SDL
are used to solve the basic seakeeping problem and to calculate the
first order potential and the linear ship responses, as necessary for
the added resistance calculations; then Maruo’s far-field theory
and the Kochin functions approach is adopted for the calculation of
the added resistance. For comparison, the near-field, direct
pressure integration method has been also partly investigated. Finally,
for the short waves range, the asymptotic method of Faltinsen et al.
(1980) and an improved derivative of it introduced recently by
Kuroda et al. (2008) and Tsujimoto et al. (2008) are employed.
A wide range of case studies of different hull forms (slender and
bulky) was used to validate the applicability and the accuracy of the
developed and implemented methods in practice.
2. Basic formulation
A coordinate system is introduced, as shown in Fig. 1. The x–y
plane coincides with the calm water free surface with the x-axis
pointing towards the bow and the z-axis pointing upwards.
The incident wave potential is given by
h
i
gz
ð1Þ
F0 ¼ Re i a ekzikðx cos w þ y sin wÞ eioe t ¼ Re ðf0c þ f0s Þeioe t
o
where w is the incident wave heading (w ¼ p corresponds to head
waves), oe ¼ okVcosw is the encounter frequency, k¼ o2/g ¼2p/l
is the wave number, V is ship’s advance velocity and f0c, f0s are
Fig. 1. Coordinate system.
the symmetric and anti-symmetric parts of the wave potential,
respectively,
f0c ¼ i
f0s ¼
g za
o
g za
o
ekzikx cos w cosðkysin wÞ
ekzikx cos w sinðky sin wÞ
ð2Þ
ð3Þ
The diffraction potential may be also decomposed as follows:
F7 ¼ Re½f7 eioe t ¼ Re½ðf7c þ f7s Þeioe t ð4Þ
with:
@f7c
@f
@f7s
@f
¼ 0c ,
¼ 0s
@n
@n
@n
@n
ð5Þ
The diffraction potentials f7c and f7s share the same symmetry
chrematistics with the symmetric and anti-symmetric parts of the
incident wave potential, respectively. The complex function H(kj,y),
known as Kochin function, describing the elementary waves
radiated from the ships is given by
ZZ @ @f
Gj ðyÞds
Hðkj , yÞ ¼
f ð6Þ
@n @n
S
where:
Gj ðyÞ ¼ exp½kj ðyÞz þ ikj ðyÞðx cos y þ ysin yÞ
and kj(y), j ¼1, 2 are the unsteady wave numbers
!
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
þ :j ¼ 1
K0 12O cos y 7 14O cos y
kj ðyÞ ¼
:j ¼ 2
2
cos2 y
ð7Þ
ð8Þ
In the above equations, y is the angle of elementary waves
generated by the body, O ¼ oeV/g is the Hanaoka parameter, and
K0 ¼g/V2 is the steady wave number. From Eq. (8) after some
algebraic manipulation the following expressions may be derived
for k1(y) and k2(y):
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2
17 14O cos y
2
ð9Þ
k1 ðyÞ ¼ K0 sec y
4
k2 ðyÞ ¼ k
2
2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ 14O cos y
ð10Þ
Following Maruo (1963), the added resistance may be expressed
by the above Kochin function as
(Z
Z p=2 Z 3p=2 )
a0
r
2 k1 ½k1 cos ykcos w
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dy
þ
RAW ¼
9Hðk1 , yÞ9
8p
14O cos y
p=2
a0
p=2
þ
r
8p
Z 2pa0
a0
2
9Hðk2 , yÞ9
k2 ½k2 cos ykcos w
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dy
14O cos y
ð11Þ
S. Liu et al. / Ocean Engineering 38 (2011) 641–650
643
where r is the density of sea water and a0 is the critical angle
(a0 ¼arcos(1/(4O)) for O 41/4 and a0 ¼0 for O r1/4). When V ¼0
then O ¼0, k1(y)-N and k2(y)¼k. Thus the wave systems are
reduced to the ring wave only. For the zero speed case, the drift
force may be expressed as follows (Maruo, 1960):
Z
rk2 2p
2
RAW ¼
HðyÞ9 ðcos ycos wÞdy
ð12Þ
8p 0
1+ au for the effect of forward speed. The two integrals in Eq. (17)
are calculated over the A–F section (integral I) and B–F (integral II)
section of the non-shaded part of the waterline (see Fig. 2). I1 is the
first order modified Bessel function of the first kind and K1 is the
first order modified Bessel function of the second kind. Note that in
the original proposal, the determination of CU was more complicated and required the use of experimental data.
where:
2.1. Computation of Kochin function
HðyÞ ¼
ZZ
@f
ds
exp½kz þikðx cos y þy sin yÞ fkðnz þ i cos ynx þ i sin yny Þ
@n
S
ð13Þ
For the short waves range, Faltinsen et al. (1980) derived the
following asymptotic formula for the added resistance, assuming
that the incident waves are perfectly reflected from the non-shaded
part of the ship surface that is exposed to the waves
Z
ð14Þ
F1 ¼ F n sin jdl
L
where
Fn ¼
1
2o V rg z2a sin2 ðjwÞ þ 0 1 þcos jcosðjwÞ
2
g
ð15Þ
The integration in Eq. (14) is performed over the non-shaded
part (A-F-B) of the waterline (Fig. 2).
This expression yields good results for relatively full bodies;
however, some poor results were obtained for fine hull forms like
those of containerships. In order to improve this drawback, Kuroda
et al. (2008) further investigated Fujii and Takahashi’s (1975) semiempirical original method and proposed an improved expression
for the added resistance in short waves, which takes the following
form:
1
rg z2a ad ð1 þ aU ÞBBf ðwÞ
2
RAW ¼
ð16Þ
where
Bf ðwÞ ¼
ad ¼
1
B
Z
sin2 ðjwÞsin jdl þ
I
2 2
I1 ðke dÞ
2 I2 ðk dÞ þK 2 ðk dÞ
1 e
1 e
p
Z
sin2 ðj þ wÞsin jdl
ð17Þ
II
The expression for the calculation of the Kochin functions
involves the fluid velocity on the body surface. The latter may be
calculated via the body boundary condition. For the symmetric
potentials f1, f3, f5 and the symmetric part of the diffraction
potential f7c, the Kochin function takes the following form:
ZZ
ekj ðyÞz þ ikj ðyÞx cos y cosðkj ðyÞy sin yÞ
Hm ðkj , yÞ ¼ 2
S=2
@f
kj ðyÞðfmc nz fms cos ynx Þ mc
@n
@fms
ds
þ ikj ðyÞðfms nz þ fmc nx cos yÞi
@n
ZZ
2
ekj ðyÞz þ ikj ðyÞx cos y sinðkj ðyÞy sin yÞ
S=2
ðfmc þ ifms Þkj ðyÞsin yny ds
ð22Þ
where, m¼1, 3, 5, 7c. For the anti-symmetric potentials, i.e. f2, f4,
f6 and f7s, the Kochin function takes the following form:
ZZ
Hm ðkj , yÞ ¼ 2i
ekj z þ ikj x cos y sinðkj ðyÞy sin yÞ
S=2
@f
kj ðyÞðfmc nz fms cos ynx Þ mc
@n
@fms
ds
þ ikj ðyÞðfms nz þ fmc nx cos yÞi
@n
ZZ
þ2
ekj ðyÞz þ ikj ðyÞx cos y cosðkj ðyÞy sin yÞ
S=2
ðfmc þ ifms Þkj ðyÞisin yny ds
ð23Þ
ð18Þ
where, m ¼2, 4, 6, 7s.
1 þ aU ¼ 1 þ CU Fn
ð19Þ
2.2. Behavior of the wave systems
ke ¼ kð1 þ O cos wÞ2
ð20Þ
CU ¼ max½10:0,310Bf ðwÞ þ 68
ð21Þ
In the following, we investigate the behavior of the wave
systems for the case of a submerged spheroid with a length to
breadth ratio L/B ¼5 and draught to breadth ratio d/B¼0.75 where
the draught d is measured from the free surface to the body center
with Fn ¼0.2, k¼2.5 m 1, oe E7.14 s 1 in head waves. The corresponding values of the Hanaoka parameter, the steady wave
number and the critical angle are: O E0.64, K0 E 12.7 m 1,
a0 E671. The unsteady wave numbers of the two wave systems
are plotted in Fig. 3. From this graph it may be observed that the
wave number k1 approaches infinite when y-p/2, while the wave
number k2 grows larger (although still remains bounded) as the
wave direction approaches the critical angle. It has been generally
found that the wave number k1 is much larger than k2, indicating
that the k1 waves are very small and can be neglected in the
calculation of added resistance in head waves. Fig. 4 shows the
weighting functions in the added resistance formulation. Since the
k1 wave system is neglected, only the weighting functions for the k2
wave system have been shown. The weighting functions are
defined by the following expressions:
p
B and d are the beam and draught of the ship, Bf is the bluntness
coefficient, ad accounts for the effect of draught and frequency and
Fig. 2. Coordinate system for the short waves range added resistance calculation
methods.
k2 ðyÞ2 cos y
WF2ð1Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
14O cos y
ð24Þ
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S. Liu et al. / Ocean Engineering 38 (2011) 641–650
Fig. 3. Unsteady wave numbers of the two wave systems.
Fig. 5. Kochin amplitude function of a submerged spheroid.
Fig. 4. Weighting function of the k2 wave system.
k2 ðyÞkcos w
WF2ð2Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
14O cos y
Fig. 6. Integrand in Maruo’s formula for a submerged spheroid.
ð25Þ
Fig. 5 shows the non-dimensional Kochin function amplitude for
a spheroid in the range a0 o y o p and Fig. 6 shows the integrand of
the second term in expression (11). A more detailed analysis can be
found in Naito et al. (1988).
Generated data are validated in a series of test cases in
comparison to numerical results of other authors, partly numerical
results of a near-field, direct pressure integration method
(Papanikolaou and Zaraphonitis, 1987) and experimental data, to
the extent available. Typical results from these validation studies
are presented and discussed in the following.
3. Numerical validations and discussion
3.1. Submerged spheroid
The far-field method for the calculation of the added resistance,
presented in Section 2, has been numerically implemented in a
computer code, using velocity potential and other seakeeping data
resulting from application of three different seakeeping assessment codes of NTUA-SDL, namely
1. The 3D frequency domain panel code NEWDRIFT (Papanikolaou–
Schellin, 1992); relevant results are denoted in the graphs by ND.
2. The 3D time domain Green function code (Liu et al, 2007);
relevant results denoted by LIU.
3. The 3D time domain hybrid Rankine source (near field)/Green
function (far field) code HYBRID (Liu and Papanikolaou, 2010);
relevant results denoted by HYBRID.
Iwashita and Ohkusu (1992) studied a shallowly submerged
spheroid with a length to breadth ratio L/B¼5 and draught to
breadth ratio d/B¼0.75, where the draught d is measured from the
free surface to the body center. This study case has been addressed
with two different approaches: The first one is using a hybrid time
domain Green function method presented by Liu et al. (2007) for the
calculation of the first-order potential. The second approach is based
on a frequency domain 3D panel (i.e. NEWDRIFT, Papanikolaou and
Zaraphonitis, 1987) for the solution of the first order problem and
the calculation of the potential and the body motions. The deduced
first-order results either from the time-domain or the frequency
domain approach are introduced in Eqs. (22) and (23) for the
S. Liu et al. / Ocean Engineering 38 (2011) 641–650
calculation of the Kochin functions, from which the added resistance
is calculated according to Eq. (11). In addition to the above, the drift
forces and added resistance are calculated also applying the nearfield, direct pressure integration approach, implemented in the
NEWDRIFT code (denoted as NDnear in the graphs).
Results from the first approach (denoted as LIUfar in the graphs)
for the drift force at zero speed, and for the added resistance when
the body is moving at forward speed in incident head waves with
suppressed responses (fixed body case) are presented in Figs. 7–9.
Good agreement may be observed between the present results and
the results of Iwashita and Ohkusu (1992), denoted as IWASHITA in
the graphs, both for the zero speed and nonzero speed calculations
(Figs. 7–9).
The same spheroid has been also tested at a deeper submergence of d/B ¼1.25. Comparisons have been herein made only
between the two NTUA codes, namely the time- and frequency
domain potential solvers (LIU and NEWDRIFT) at zero and at a
forward speed corresponding to Fn ¼0.2. Good agreement between
both solvers was obtained using the far-field method in the zero
speed case. For the forward speed case, though the trend and
maximum values are in general in good agreement, the herein
predicted peak value is actually slightly shifted, which might
be credited to the fact that the employed 3D panel method
corresponding to the NEWDRIFT code is essentially based on a
645
Fig. 9. Added resistance on a submerged spheroid, Fn ¼0.3, d ¼ 0.75B.
Fig. 10. Horizontal drift force on a submerged spheroid at Fn ¼0, d ¼1.25B.
Fig. 7. Horizontal drift force on a submerged spheroid at Fn ¼ 0, d¼ 0.75B.
zero speed Green function and forward speed effects are taken into
account in an approximate way by exploiting slender body theory
assumptions. These results are shown in Figs. 10 and 11.
3.2. Floating spheroid
Fig. 8. Added resistance on a submerged spheroid, Fn ¼ 0.2, d ¼ 0.75B.
The added resistance of a half-immersed (floating) spheroid
with a length to beam ratio L/B ¼5 has been studied by Kashiwagi
(1997). Fig. 12 shows the drift force (zero speed) results for the
freely moving spheroid, computed by NEWDRIFT (NDnear) and by
the far field method (NDfar) using the first-order potentials from
NEWDRIFT. A very good agreement between the near field and far
field methods may be observed, along with a reasonable agreement
with Kashiwagi’s enhanced unified theory results. Fig. 13 shows the
added resistance of the half-immersed spheroid with suppressed
motions at a forward speed corresponding to Fn ¼0.2. The results
denoted by LIN are reproduced from Lin et al. (1993). Three
different panelizations have been tested with NEWDRIFT, namely
360, 728 and 960 panels on half body. It may be observed that in
some cases 360 panels are not enough for the accurate calculation
of the added resistance, while with 728 panels, convergence has
been generally achieved. Considering that 200 panels would lead in
general to good motion results for such a simple body by the same
code, we may conclude that added resistance is much more
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S. Liu et al. / Ocean Engineering 38 (2011) 641–650
sensitive to the size of panelization, than results of first order.
A similar observation was made by Kashiwagi (2009). Nevertheless,
a significant deviation between a variety of numerical results and
experimental values is observed for this case, which is theoretically
and practically a difficult benchmark case, because of the complicated above still water level wetting of the half-immersed spheroid. Kashiwagi presented good results for this case with his
enhanced unified slender body theory, when using correct n1 term
which better accounts for the wave diffraction near the bow.
3.3. Wigley hull
A series of Wigley hulls were tested and reported by Journeé
(1992). In this section the validation of Wigley-III is presented. The
hull surface is defined by the following expression:
!
!
y
x 2
z 2
x 2
¼ 1
1 þ 0:2
1
ð26Þ
b
L=2
D
L=2
Fig. 11. Added resistance on a submerged spheroid, Fn ¼ 0.2, d ¼1.25B.
Fig. 12. Drift force on a freely moving spheroid at Fn ¼0.
Fig. 13. Added resistance on a fixed half-immersed spheroid at Fn ¼ 0.2.
where 2b/L¼0.1 and D/L¼0.0625. The hull surface was discretized
by 800 panels (port side only). For the body at zero speed, the drift
force for the fixed body case and the free floating case are predicted
by both the near field and the far field method, as shown in Fig. 14.
A very good agreement between the near field and far field drift
force results may be observed.
Calculations have been performed with non-zero forward speed
as well. Fig. 15 presents results for the added resistance by the
far-field method. Despite some scatter in the experimental data,
clearly the numerical values NDfar are over predicting the added
resistance and lead to a slight shift of the peak values towards
smaller wavelengths. This tendency is partly attributed to inaccuracies in the consideration of forward speed effects by NEWDRIFT, but mainly attributed to a common shortcoming of linear
seakeeping methods, which are based on potential theory, namely
the under-estimation of damping coefficients, leading to predictions of increased first-order motions in the vicinity of the
resonance. Another reason for the numerically predicted increased
motion amplitudes in the resonance region (thus also of added
resistance) are nonlinear effects due to the above still water level
hull form, which is not considered in linear frequency domain
methods; these effects become very significant when the motions
are of large amplitude; Papanikolaou et al. (2000) tried to overcome
this problem by including additional viscous damping to the
potential theory terms, derived with the help of ‘‘empirical’’ viscous
lift and cross-flow drag coefficients. The introduction of cross-flow
viscous damping in the first-order calculations for the Wigley hull
Fig. 14. Drift force on a Wigley-hull in head seas at Fn ¼ 0.0.
S. Liu et al. / Ocean Engineering 38 (2011) 641–650
647
available. The obtained results for Fn ¼0.266 and 0.283 are presented in Figs. 17 and 18, respectively. Once again, as in the Wigley
hull case, Fig. 15, numerical results without any viscous correction
strongly over-predict the experimental measurements. The introduction of cross-flow viscous damping in the first-order calculations (Cd ¼0.1, za/L¼1/50) resulted in a considerable reduction of
the peak values of the added resistance (and motion amplitudes),
much closer to the experimental measurements. Note here the very
good prediction by the new HYBRID time domain method of
NTUA-SDL (Liu and Papanikolaou, 2010), which, compared to
NEWDRIFT, takes accurately into account forward speed effects
and in addition accounts for the above water hull form nonlinearities. The calculation by HYBRID method has been herein done
only for the range of l/L¼0.8–1.6, where the resonance takes place
and the nonlinearity plays a significant role. In the short waves
range, added resistance calculations according to the asymptotic
formulae of Faltinsen et al. (1980) and also of Kuroda et al. (2008),
denoted as SW1 and SW2, respectively, are also presented.
3.5. S-175 container ship
Fig. 15. Added resistance on a Wigley-hull in head seas at Fn ¼ 0.3.
Fig. 16. Drift force on a S60 hull in head seas at Fn ¼0.0.
Calculations were performed also for the S175 Container ship,
a hull form that has been thoroughly investigated by many
Fig. 17. Added resistance on a Series 60 hull at Fn ¼0.266.
resulted in a reduction of the peak values for the added resistance
(denoted as NDfarCD), much closer to the experimental measurements, as shown in Fig. 15. In this calculation, the cross-flow
viscous damping coefficient is Cd ¼0.1, whereas the incident wave
amplitude is assumed as za/L¼1/60.
3.4. Series 60 hull with Cb ¼0.6
Calculations were made for a Series 60 hull with block coefficient of 0.60, which has been studied extensively by many
researchers (e.g. Strøm-Tejsen et al. (1973)). The hull surface
was discretized by 809 panels (port side). Calculations were
performed at first for the zero speed case. Fig. 16 presents the
drift force result. For the fixed body case, there is a very good
agreement between the near field and far field results. For the freefloating case however, a small discrepancy may be observed,
although calculations with both methods were based on the same
first-order potential and motion data.
Calculations have been also carried out by the far field method
for two forward speeds, for which experimental data were
Fig. 18. Added resistance on a Series 60 hull at Fn ¼0.283.
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S. Liu et al. / Ocean Engineering 38 (2011) 641–650
researchers (15th ITTC, 1978). A panelization with 885 panels was
used in the calculations. Added resistance results are presented in
Figs. 19–21. A quite satisfactory agreement between the numerical
predictions and the experimental results (reproduced from
Takahashi, 1988 and Seakeeping Committee report of ITTC 1984)
is observed. Pure potential flow calculations by the panel code
NEWDRDIFT resulted again in a small over-prediction of the added
resistance at the peak, which is eliminated when the cross-flow drag
viscous correction is introduced (Cd ¼0.1, za/L¼1/55). The results of
the new HYBRID method of NTUA-SDL are in good agreement with
the experimental data. In the short waves range, added resistance
calculations using the asymptotic formulae of Faltinsen et al. and
also of Kuroda and Tsujimoto are presented for this hull also,
denoted as SW1 and SW2, respectively. In the short waves range, the
agreement of added resistance calculations according to Kuroda and
Tsujimoto with the experiments is very satisfactory.
3.6. Bulk-carrier
Finally, results of drift force and added resistance calculations for
a Bulk-Carrier, also studied by Kadomatsu (1988), are presented.
Fig. 19. Added resistance on the S-175 container ship at Fn ¼0.25.
Fig. 21. Added resistance on the S-175 container ship at Fn ¼ 0.30.
Fig. 22. Drift Force on a Bulk-Carrier in head seas at Fn ¼ 0.0.
This is a quite full hull form, with Lpp ¼285 m, B¼ 50.0 m, D¼18.5 m,
Cb ¼0.829 and LCB ¼3.83% Lpp forward of the midship section.
A panelization consisting of 819 panels on the half body was used
for the calculations. The drift force and added resistance results are
shown in Figs. 22–25, compared against experimental data. The
numerical results agree well with the experiments, with the
exception of the short waves range at the higher speeds corresponding to Fn ¼ 0.10 and 0.15, where the added resistance is
under-predicted. However, in this range of wavelengths, calculations based on the asymptotic formulae of Faltinsen et al. and also
of Kuroda and Tsujimoto resulted in a very good agreement with
the experimental data. Note that in this case of full bodies at low
speed, no viscous correction for motion damping proves necessary,
because of the resulting small amplitude motion responses and of
related nonlinearities, along with limited forward speed effects.
4. Summary and conclusions
Fig. 20. Added resistance on the S-175 container ship at Fn ¼0.275.
The added resistance problem of ships in waves has been solved
following Maruo’s far-field theory and numerically implemented
S. Liu et al. / Ocean Engineering 38 (2011) 641–650
Fig. 23. Added resistance on a Bulk-Carrier at Fn ¼0.05.
649
using velocity potentials calculated by a Time Domain Numerical
Simulation method for submerged bodies (Liu et al. 2007), a 3D
Frequency Domain Panel method (Papanikolaou and Zaraphonitis,
1987) and a 3D time domain HYBRID method (Liu and Papanikolaou,
2010) for bodies floating on the free surface. Results of a near-field,
direct pressure integration method were used for comparison in
the zero speed case. For the short waves range, the asymptotic
formulae introduced by Faltinsen et al. (1980) and also by Kuroda
et al. (2008) were employed for comparison. A systematic validation for different hull forms (submerged and floating, slender and
bulky) and varying wavelengths and speeds were conducted.
The obtained results were compared with those of other well
established authors and experimental data and a reasonable
agreement was observed, suggesting that the implemented procedure appears a reliable and robust method for the routine
prediction of the added resistance of a ship in waves. Regarding
computational time and efficiency of the implemented 3D panel
code method, with a panelization in the order of 800 panels for the
half the body, after obtaining the potential and motion data, which
takes approximately 5 s with the NEWDRIFT code, the added
resistance computation for one frequency takes approximately
1.5 s on a PC with Intel Core 2 QUAD CPU Q8200 2.33 GHz, which
proves actually very efficient from the point of view of practical
applications.
Concluding, the original far-field method of Maruo, further
elaborated by use of a 3D panel method and the asymptotic formulae
of Faltinsen, Kuroda and Tsujimoto for the short wavelengths prove
fully satisfactory for the prediction of the added resistance of ships
(of slender and bulky hull form) in waves and may be practically
implemented within reasonable computational times.
Acknowledgements
The presented work is part of the Ph.D. work of the first author.
The continuous guidance by the second and third author, as well as
by Professor Wenyang Duan (Harbin Engineering University) is
acknowledged. Finally, the financial support by NTUA-SDL and
partly by DNV in the framework of the NTUA-DNV GIFT agreement
(2007–2010) is acknowledged.
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