-
ha
, In
tour
ati
s o
er o
on
ase
f th
Slenderness of most ship hulls encouraged creation ofsimplied
hydrodynamic models exploiting this property. Namely,
two-diical mnoeuvenouger, somsistancajor incould
higher. However, latest progress in computing power pre-
keeps advancing especially when it goes about hybrid
seakeeping-and-manoeuvring problems or nonlinear formulations
(Sutulo
h areintoload964).more
oscillating contours are much less numerous. Probably, the
rst
ARTICLE IN PRESS
Contents lists available at ScienceDirect
lse
Ocean Eng
Ocean Engineering 36 (2009) 109811111982) developed a
boundary-integral-equation method, suitableE-mail address:
[email protected] (C. Guedes Soares).determined a slight
natural drift towards 3D codes. At the same systematic study of the
nite-depth radiation problem belongs toKeil (1974). However, his
solution, as presented also by Journeeand Adegeest (2003), was
based on the Lewis conformal mappingand is only valid for a limited
family of sections. Yeung (1973,
0029-8018/$ - see front matter & 2009 Elsevier Ltd. All
rights reserved.
doi:10.1016/j.oceaneng.2009.06.013
Corresponding author.slender vessels, but improvements in
accuracy in many cases wereat best uncertain, while the CPU-time
requirements were much
extensively, enough to mention classic works by Ursell
(1949),Grim (1953), Frank (1967), publications on the
shallow-waterstimulated its extensions allowing for partial account
for 3Deffects (Bertram, 2000). At the same time, purely 3D codes
startedto develop primarily with applications to non-slender
maritimestructures, where strip methods could not be expected to
bringsatisfactory results. Three-dimensional codes were also
applied to
maneuvering problems (Zhao, 1986). Viscous effects,
whicespecially important for the horizontal modes, are then
takenaccount just by means of articial reduction of the
transverseacting on the aft part of the body (Fedyaevsky and
Sobolev, 1
While the similar deep-uid problem was investigatedstrip method
in seakeeping-and-manoeuvring problems, wherethey were usually
estimated empirically or semi-empirically.
One of the most known example of a matured seakeeping
striptheory was presented by Salvesen et al. (1970). Certain
limitationsof the strip method, especially at higher Froude
numbers,
characteristics of the ship sections, usually related to
theirsmall-amplitude oscillations within a certain frequency
interval,and this predetermines the problem considered in the
presentpaper. It is also important to bear in mind that
low-frequency datacorresponding to the horizontal motions can also
be useful forin many cases it was possible to ngradients and to
apply the strip metthe natural 3D formulation to thedecades, this
was the only practtreatment of seakeeping and matransverse loads
could be reliablyof the 2D hydrodynamics. Howevsimilar approaches
to the wave relongitudinal resistance force is of mfailed.
Similarly, longitudinal forcesich effectively reducesmensional one.
Duringethod for theoreticalring problems, whereh estimated by
meanse attempts to apply
e problem, where theterest, have practicallynot be predicted by
the
formulations (Xia and Wang, 1997). Besides that, the strip
methodis nowadays regarded as a quite adequate tool for
educationalpurposes, which encouraged development of an open source
stripcode PDSTRIP (Bertram et al., 2006). One of the
possibleextensions of existing strip codes is their adaptation to
theshallow-water situation which is extremely important from
theviewpoint of determining tidal and weather windows for
largeships approaching harbours (Vantorre et al., 2008).
The keystone of every strip method are hydrodynamiceglect the
longitudinal owhod wh
and Guedes Soares, 2008; Bandyk and Beck, 2008) or
hydroelasticComputation of inertial and damping cshallow water
S. Sutulo, J.M. Rodrigues, C. Guedes Soares
Centre for Marine Technology and Engineering (CENTEC), Technical
University of Lisbon
a r t i c l e i n f o
Article history:
Received 11 November 2008
Accepted 22 June 2009Available online 14 July 2009
Keywords:
Shallow water hydrodynamics
Vibrating ship sections
Boundary integral equation
Stepped bottom
a b s t r a c t
Hydrodynamics of 2D con
oscillations with a modic
deep-uid case. Alteration
present study and a numb
These include comparis
calculations made for the c
can be used for estimation o
1. Introduction
journal homepage: www.eracteristics of ship sections in
stituto Superior Tecnico, Av. Rovisco Pais, 1049-001 Lisbon,
Portugal
s representing ship sections is considered for the case of small
harmonic
on of a boundary-integral-equation method implemented earlier
for the
f the algorithm required by the nite-depth case are described in
the
f numerical results are given.
with another code for the case of at horizontal bottom and
comparative
of the abrupt change of depth near the ship (stepped bottom).
The results
e bottoms inuence on the manoeuvring and seakeeping qualities of
ships.
& 2009 Elsevier Ltd. All rights reserved.
time, it is denitely premature to consider as obsolete the
stripmethod which is often much more efcient. Its development
still
vier.com/locate/oceaneng
ineering
-
ARTICLE IN PRESS
for arbitrary sections, primarily just for the nite depth,
althoughthe innite-depth generalization was also provided: all
thecalculations were performed for the nite depth and the
limitingdeep-water case was treated as a very large nite depth. As
theYeung method seemed to be very promising as free of
irregularfrequencies and potentially applicable to domains of
arbitraryshape, it was further modied by Sutulo and Guedes
Soares(2004) aiming at better fulllment of the body
boundarycondition. At that time, this latter method was only
implementedand veried for the case of innite depth. Now, the code
wasextended to the shallow-water case with arbitrary shape of
thebottom. Results of its verication and application to the
poorlyexplored case of the stepped bottom modeling the situation
thatcan be encountered when the ship is moving along or near
adredged channel are described and discussed in the presentarticle.
This is preceded with a rather detailed statement ofproblem and
some comments on the solution method are given.Analytic formulae
for the inuence functions are mostly omittedas they are the same as
used by the deep-uid code and aredescribed in full by Sutulo and
Guedes Soares (2004).
unity normal n nxex+nyey is supposed to be dened almost
everywhere on the domains boundary qG which is
@G S SF [ SC [ SR [ SL [ SB: 1
In the general case, the motion of a slightly deformable
contouris described by the time-dependent velocity distribution on
thecontours boundary V(P, t), where t is time and P(x, y)ASC a
pointon the contour. The motions of the contour are exciting the
uidwhose motions potential is F(M, t) where M(x, y)AG. Thepotential
must satisfy the following relations constituting aninitial- and
boundary-value problem:
the Laplace equationDF 0 in G; 2
the free-surface boundary condition@2F@t
g @F@y
at y 0; 3
where g is the acceleration of gravity;
hatim
re
V
S. Sutulo et al. / Ocean Engineering 36 (2009) 10981111 10992.
Formulation of problem and boundary integral equation
2.1. Problem statement
Considered is a two-dimensional problem of determining
thehydrodynamic characteristics of a smooth contour SC
oscillatingnear the free surface SF of an incompressible uid which
isconstrained by the rigid bottom SB which, in general case, can
beof any shape (Fig. 1).
The amplitude of any form of oscillations is assumed to
beinnitesimally small, so that the solution uid domain G could
beconsidered steady that is having xed boundaries what is
typicalfor linear formulations. The contour can intersect the free
surfaceor be completely submerged beneath it (as some bulb
sections),but the rst case is of greater interest and it will be
furtherconsidered as the main one. Finally, considered are two
articialboundaries formed by half-innite vertical straight lines SL
and SRto make the uid domain technically nite in the
horizontaldirection. The origin of the principal co-ordinate system
Oxylies on the free surface and, as a rule, inside the contour.
Thex-axis is directed to the right and the y-axisdownwards. TheFig.
1. Global frame of reference apresented as
P; t VPeiot ; 7excIn the following, it is assumed that the
contour is oscillatingrmonically with the frequency o and this
motion started a longe ago, so that the initial conditions inuence
vanishes. Theitation velocity distribution along the contour can
then be the boundary condition on the contour@F@n
P VP nP at P 2 SC; 4
the bottom boundary condition@F@n
P 0 at P 2 SB; 5
the initial conditions
FM; 0 F0M;@F@t
M; 0 C0M; M 2 G; 6
where the functions in the right-hand sides describe
initialdistributions of the potential and of its time derivative.nd
domain boundaries.
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ARTICLE IN PRESS
S. Sutulo et al. / Ocean Engineering 36 (2009) 109811111100where
the real part only is supposed to be retained in the right-hand
side and V is the complex shape function that can berepresented as
a superposition of certain simple modal shapes.Exclusively rigid
contours will be further considered, for whichthree modal shapes
(heave, sway and roll) are sufcient todescribe any motion. In this
particular case,
VP V0 X rOP ; 8
where VO* u*ex+v*ey and X* p*ez are complex amplitudes of
the linear and angular velocities, respectively, and rOP xex+yey
isthe radius vector from the origin to the point P.
The velocity potential is then described with the helpof
time-independent complex amplitude F*(M), so thatF(M, t) F*(M)eiot.
The complex potential F*(M) also satisesthe Laplace equation in G
and the boundary condition on thebottom remains the same as dened
by Eq. (5). The boundarycondition on the free surface takes the
form
@F
@y k0F 0; 9
where k0 o2/g.The initial conditions are absent in the
time-independent
problem, but the radiation conditions are required toguarantee
the solutions uniqueness. These are dened as,(Yeung, 1973)
@F
@nxR;L; y ikR;LFxR;L; y; at y 2 0; hR;L; 10
where xR and xL are the abscissae of the vertical boundaries SR
andSL, respectively, hR and hL the water depth values at
thoseboundaries, and kR,L the wave numbers of the outcoming
wavesdened by the equation kR,L tanh kR,LhR,L k0
The complex amplitude of the potential is usually decomposedas
F* u*j2+v*j3+p*j4, where j2, j3, j4 are the radiationpotentials
which satisfy all the boundary conditions formulatedabove for the
complex potential amplitude and the followingconditions on the
contour:
@j2@n
f2 nx;@j3@n
f3 ny;@j4@n
f4 xny ynx: 11
The formulated problem for each of the radiation functions is
amixed boundary-value problem for the elliptic equation.
Thisproblem is solved here by means of a boundary integral
equationas proposed by Jaswon (1963) and Yeung (1973).
2.2. Boundary integral equation
As all the following considerations are valid for all
theradiation functions, the indices i will be dropped. It will be
alsoassumed that the point P(x, y) is the observation point in G
G[S,while Q(x, Z) is the current (integration) point in the
same
domain. The distance between these two points is r jxP xQ j x x2
y Z2:
qApplying the second Greens formula, (Frank
and Mises, 1961), to a radiation function j which is supposed
tobe harmonic in G and to the fundamental solution of the
two-dimensional Laplace equation log r yields:
ZjD log r Dj log rdG
Z@j
log r j @ log r
dS: 12
G S @n @nAs Dj0 and D log|x| 2pd(x), where d(x) is the
Diracfunction, one can obtain for the point PAG
npjP Z
S
@j@n
log r j @ log r@n
dS; 13
where n 2 if PAG, and n 1 if PAS and the surface S is
smooth(more precisely: it must be a Lyapunov contour) in the
neighbourhoodof P. Then, assuming PAS, decomposing the integrals in
Eq. (13),according to Eq. (1) and applying the boundary
conditionsformulated above obtained is the following boundary
integralequation with respect to the distribution j(P):
pjP Z
SjQ KP; Q dSQ
ZSC
f Q log r dSQ ; 14
where the kernel K is
KP; Q
@ log r
@nQif Q 2 SC [ SB;
@ log r
@nQ k0 log r if Q 2 SF ;
@ log r@nQ
ikR;L log r if Q 2 SR;L:
8>>>>>>>>>>>>>>>:
15
2.3. Discretisation of the boundary-value problem
The discretisation procedure was described in detail in
Sutuloand Guedes Soares (2004) for the case of the innite-depth
uid.As the procedure remains practically the same in the
nite-depthcase, it will be just outlined here.
First, the whole boundary S is subdivided into a reasonably
largenumber of segments: S [iSi, where every two segments have
notmore than one common boundary point. Then, each
curvilinearsegment Si is approximated with the rectilinear segment
Si havingthe same end points and the radiation function j is
approximatedover each Si with the constant valueji. After this
step, the boundaryequation does no longer contain any
integrals:
pjP XN1j0
KjPjj XNC1j0
FjPjj; P 2 S; 16
where N is the overall number of segments, NC is the number
ofsegments on the contour, S [iSi, and
KjP Z
Sj
KP; Q dSQ ; FjP Z
Sj
f Q log r dSQ : 17
To nalize the discretisation, this equation must also besatised
on a nite discrete set which would result in linearalgebraic
equations for the radiation function values ji. This canbe
performed at least in two ways: either the equation is satisedat
the center Pi of each rectilinear segment (simple collocation) orit
is satised in the integral sense over each rectilinear segment
Si(integral collocation). The latter method was proposed by
Sutuloand Guedes Soares (2004) and it results in better accuracy at
agiven number of panels at the expense of a somewhat
morecomplicated algebra. The nal algebraic equations look
indenti-cally in both the cases:
pSiji XN1j0
Kijjj Fi XNC1j0
Fij; i 0; . . . ; N 1; 18
where the inuence functions in the case of the simple
collocation
are just Kij Kj(Pi) and the excitation functions Fij Fj(Pi)
while
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ARTICLE IN PRESS
computed as
accuracy, each half of the free surface must cover at least
two
S. Sutulo et al. / Ocean Engineering 36 (2009) 10981111 1101mkl
rXNC1i0
jli fkiSi: 22
Numerical results in the present paper are represented in
thenon-dimensional form. The non-dimensional frequency waso0
o2Lref/g, where the reference length Lref is taken asmaximum of the
draught T and waterline half-breadth B/2 foreach contour. The
dimensionless added masses are denedas follows:
m022 m22=rpT2; m023 m23=rpL2ref ; m024 m24=rpL3ref ;m034
m34=rpL3ref ; m044 m44=rpL4ref ; m033 4m33=rpB2:
23
The damping coefcients are additionally divided by thefor the
integral collocation
Kij Z
Si
KjPdSP; Fij Z
Si
FjPdSP: 19
Analytic expressions for all inuence and excitation
functionswere obtained by Yeung (1973) for the simple collocation
case andby Sutulo and Guedes Soares (2004) for the integral
collocation.They are rather cumbersome and are not reproduced here,
exceptfor one component which was corrected as it sometimes failed
inthe nite-depth case. This component is described in Appendix.
The algebraic set (18) can be typically of order
80200.Unfortunately, this set is not diagonal dominant and the
iterativeGaussSeidel method is not applicable. The GaussJordan and
LU-decomposition methods both work reliably and are of equal
valuefrom the viewpoint of speed.
3. Added masses and damping coefcients
After the potential distribution on the contour and
theboundaries are dened, the potential can be calculated at
everypoint inside the uid domain. Then, the velocity eld can
berestored by means of the numerical differentiation as well as
thepressure distribution, etc. However, when the mentioned
quan-tities are required, it is more convenient to use another
methodfor solving the boundary-value problemwhich would produce
thevelocity eld directly. At the same time, many manoeuvring
andseakeeping problems can be solved when used is only thepotential
distribution j(P) on the contour and the complex addedmass
coefcients m*kl dened as
mkl rZ
SC
jl fk dS; 20
where r is the uid density, jl, l 2, 3, 4 the radiation
functionscorresponding to the sway, heave and roll motions,
respectively,and the functions fk, k 2, 3, 4 dened by Eq. (11).
Traditionally,the complex added masses are represented as
mkl mkl i
onkl; 21
where mkl are the usual real added masses and nkl the
dampingcoefcients. These two real quantities are normally displayed
forevaluation and comparisons.
In the discretized form, the complex added masses
arefrequency.lengths of the radiating waves and at least 23
segments arerequired for each wavelength l. The latter depends on
theoscillations frequency, but in the nite-depth case it also
dependson the water depth. In any case, the actual repartition must
bedynamic with respect to the frequency. As at low frequencies,
therequired free-surface panel length can become too large
ascompared to the contour panel length, a transition interval
wasorganized in the vicinity of the contours corner. Within
thisinterval, the segment length was varying linearly from
theminimum contour panel length to the panel length requested bythe
oscillation frequency.
In the nite-depth case, the paneling problem is treatedsomewhat
differently: the far-eld sidewalls are to be discretizedmore or
less in the same way as the free surface and the bottom ofthe
domain are forming a closed box. As larger absolute values ofthe
potentials gradient are expected to happen near the boxcorners, the
panels lengths must be reduced near the angularpoints. The
so-called cosine distribution is typically used in thesecases.
While in the deep uid it was applied to rectangularcontours, in the
nite depth it becomes also convenient for thesidewalls, bottom and
the free surface. The vertices should becondensed near the corners,
but also on the bottom near thecontour when the clearance is small.
Usually, there is no sense toconsider gaps which are smaller than
the corresponding panellength. Hence, when possible, the
repartition on the contourshould depend on the relative depth and
when the contourvertices are xed, restrictions on the minimum water
depth mustbe imposed.
4.2. Added masses and damping coefcients for semi-circle over
at
bottom
The semi-circle is a traditional benchmark shape. Due tocentral
symmetry, all coupled coefcients are zero in the case ofunlimited
at bottom. Calculations were carried out for 50 panelson the circle
and around 150 panels on the remaining boundaryand results were
compared with those obtained by Yeung (1973)for H/T 5, 15 and by
Kim (as presented by Yeung) for H/T 4, 10.It must be noted that the
relative depth 15 and even 10 practicallycorresponds to the deep
uid.
The agreement with Yeungs data (Fig. 2) may be considered
asgood, although the present data are likely somewhat moreaccurate;
as they were obtained with 50 panels against only 18panels were
used by Yeung. Kims data show disagreement for theadded masses at
lower frequencies. This, however, does not showany deciency of the
present method as Kims data obviously do4. Verication of the method
and numerical examples
4.1. Peculiarities of the boundarys discretisation
Although the method is almost the same for the innite-depthand
nite-depth cases, the repartition of panels over the boundaryhas
some specics. First, in the deep uid (Sutulo and GuedesSoares,
2004) each of the far-eld vertical boundaries contains,but a single
semi-innite panel with the exponential potentialdistribution. This
reduces greatly the total number of panels,although it requires the
introduction and evaluation of additionalinuence functions, which
are more complex than the functionsoriginating from Eqs. (15) to
(19). If the number of panels on thecontour is pre-determined by
the available hull shape database,which is a typical situation, the
only remaining repartitionuncertainty for the deep uid is related
to the representation ofeach part of the free surface. It was found
that to achieve goodnot meet the KramersKronig relations
(Schmiechen, 1999)
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ARTICLE IN PRESS
1.2
1.4
1.6
1.8
2
H/T=533
33, present method33,
present method33, Yeung33,
Yeung
S. Sutulo et al. / Ocean Engineering 36 (2009) 10981111110222
0.8
1
1.2
1.4
H/T=5
22, present method22,
present method22, Yeung22,
Yeunglinking values of the added mass and the damping
coefcientwhich are automatically satised in the proposed
method.
4.3. Added masses and damping coefcients for ship sections
over
at bottom
The developed code was rst applied to the at-bottom case.The
three characteristic sections of the container ship S175 (ITTC,
Dimensionless Frequency
22
,
0 0.5 1 1.5 2 2.50
0.2
0.4
0.6
Dimensionless Frequency0 0.5 1 1.5 2 2.5
0
0.2
0.4
0.6
0.8
1
Dimensionless Frequency0 0.5 1 1.5 2 2.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
H/T=15
Dimensionless Frequency0 0.5 1 1.5 2 2.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
H/T=15
Dimensionless Frequency0 0.5 1 1.5 2 2.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
H/T=4
Dimensionless Frequency0 0.5 1 1.5 2 2.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
H/T=10
22
,22
33
,33
33
,33
33
,
33,
33
22, present method22,
present method22, Yeung22,
Yeung
33, present method33,
present method33, Yeung33,
Yeung
33, present method33,
present method33, Yeung33,
Yeung
33, present method33,
present method33, Kim33,
Kim
Fig. 2. Added mass and damping coefcients for a semi-circle in
sway and heave.
Fig. 3. Panels distribution for section 2 at H/T 1.3:
leftgeneral view (verticallystretched); centercentral part; and
rightarticial vertical wall.
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ARTICLE IN PRESS
1983) i.e. the bulbous bow section #02, the midship section
#10and the stern section #22 were used for test computations.
Thenumber of panels on the contour was relatively small,
thuscorresponding to the common practice of seakeeping
calculations.Some examples of the distribution of panels
representing also thecontours shapes are shown on Figs. 3 and 4,
and the results forthe hydrodynamic coefcientson Figs. 58.
As the case is symmetric, all the coefcients with
differentindices, except for 2 and 4, are zero. The results are
compared with
Fig. 4. Panels distribution for sections 10 (left) and 22
(right) at H/T 1.3: centralpart.
Dimensionless Frequency0 0.5 1 1.5 2 2.5
0
0.2
0.4
0.6
0.8
1
H/T=10.0H/T=15.0H/T=1.1H/T=1.3H/T=1.5H/T=1.7H/T=2.0H/T=3.0H/T=4.0H/T=5.0
Section 02
Dimensionless Frequency0 0.5 1 1.5 2 2.5
Dimensionless Frequency0 0.5 1 1.5 2 2.5
Dimensionless Frequency0 0.5 1 1.5 2 2.5
Dimensionless Frequency0 0.5 1 1.5 2 2.5
-0.6
-0.4
-0.2
0
H/T=10.0H/T=15.0H/T=1.1H/T=1.3H/T=1.5H/T=1.7H/T=2.0H/T=3.0H/T=4.0H/T=5.0
Section 02
0
0.2
0.4
0.6
0.8
1
H/T=10.0H/T=15.0H/T=1.1H/T=1.3H/T=1.5H/T=1.7H/T=2.0H/T=3.0H/T=4.0H/T=5.0
Section 10
0
0.02
0.04
0.06
0.08
0.1
H/T=10.0H/T=15.0H/T=1.1H/T=1.3H/T=1.5H/T=1.7H/T=2.0H/T=3.0H/T=4.0H/T=5.0
Section 10
22
22
22
24
24
24
0
0.2
0.4
0.6
0.8
1
H/T=10.0H/T=15.0H/T=1.1H/T=1.3H/T=1.5H/T=1.7H/T=2.0H/T=3.0H/T=4.0H/T=5.0
Section 22
-0.15
-0.1
-0.05
H/T=10.0H/T=15.0
Fig. 5. Sway and swayroll added mass coefcients:
S. Sutulo et al. / Ocean Engineering 36 (2009) 10981111
1103Dimensionless Frequency0 0.5 1 1.5 2 2.5
-0.3
-0.25
-0.2
H/T=1.1H/T=1.3H/T=1.5H/T=1.7H/T=2.0H/T=3.0H/T=4.0H/T=5.0
Section 22symbolsproposed method and linesHmassef.
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ARTICLE IN PRESS
S. Sutulo et al. / Ocean Engineering 36 (2009)
1098111111040.15
0.2H/T=10.0H/T=15.0H/T=1.1H/T=1.3H/T=1.5H/T=1.7H/T=2.0H/T=3.0H/T=4.0those
obtained with the code Hmassef developed by Soding asextension and
improvement of his own code Hmasse described byBertram (2000), see
also (Soding, 2005). That code accounts forthe seabed by means of
mirror reection which means that itcan only be used with the at
bottom. Although, the two codeswere developed absolutely
independently and are very different,the agreement is qualitatively
good in all the cases. As to the
Dimensionless Frequency0 0.5 1 1.5 2 2.5
0
0.05
0.1H/T=5.0
Section 02
Dimensionless Frequency0 0.5 1 1.5 2 2.5
Dimensionless Frequency0 0.5 1 1.5 2 2.5
0.01
0.02
0.03
0.04
0.05
0.06
0.07
H/T=10.0H/T=15.0H/T=1.1H/T=1.3H/T=1.5H/T=1.7H/T=2.0H/T=3.0H/T=4.0H/T=5.0
Section 10
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12H/T=10.0H/T=15.0H/T=1.1H/T=1.3H/T=1.5H/T=1.7H/T=2.0H/T=3.0H/T=4.0H/T=5.0
Section 22
44
44
44
Fig. 6. Roll and heave added mass coefcients: sym0.8
1quantitative match, it is always acceptable, i.e. the
observeddifferences are not essential for applications, and often
good andeven excellent. Larger differences are observed at smaller
waterdepths, but these differences remain insignicant from
theviewpoint of practical applications.
In some cases, mainly at smaller water depths, the proposedcode
shows somewhat wavy character of the curves. This is an
Dimensionless Frequency0 0.5 1 1.5 2 2.5
Dimensionless Frequency0 0.5 1 1.5 2 2.5
Dimensionless Frequency0 0.5 1 1.5 2 2.5
0
0.2
0.4
0.6
H/T=10.0H/T=15.0H/T=1.1H/T=1.3H/T=1.5H/T=1.7H/T=2.0H/T=3.0H/T=4.0H/T=5.0
Section 02
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
H/T=10.0H/T=15.0H/T=1.1H/T=1.3H/T=1.5H/T=1.7H/T=2.0H/T=3.0H/T=4.0H/T=5.0
Section 10
0
0.2
0.4
0.6
0.8
1H/T=10.0H/T=15.0H/T=1.1H/T=1.3H/T=1.5H/T=1.7H/T=2.0H/T=3.0H/T=4.0H/T=5.0
Section 22
33
33
33
bolsproposed method and linesHmassef.
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S. Sutulo et al. / Ocean Engineering 36 (2009) 10981111
11050.8
1
Section 02H/T=10.0H/T=15.0H/T=1.1H/T=1.3H/T=1.5H/T=1.7artifact
stemming from satisfying the free-surface boundarycondition on the
nite boundary and known also for the innitedepth, where it is,
however, less pronounced (Sutulo and GuedesSoares, 2004). This
effect could be reduced by double computationwith the far-eld
boundary shifted by half-wavelength as done inthe code Hmassef, but
this was not done here, as it would also
Dimensionless Frequency0 0.5 1 1.5 2 2.5
0
0.2
0.4
0.6
Dimensionless Frequency0 0.5 1 1.5 2 2.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
H/T=10.0H/T=15.0H/T=1.1H/T=1.3H/T=1.5H/T=1.7H/T=2.0H/T=3.0H/T=4.0H/T=5.0
Section 10
Dimensionless Frequency
22
22
22
0 0.5 1 1.5 2 2.50
0.1
0.2
0.3
0.4
0.5
0.6H/T=10.0H/T=15.0H/T=1.1H/T=1.3H/T=1.5H/T=1.7H/T=2.0H/T=3.0H/T=4.0H/T=5.0
Section 22
H/T=2.0H/T=3.0H/T=4.0H/T=5.0
Fig. 7. Sway and swayroll damping coefcients: s-0.1
0
Section 02require the double computation time and the obtained
improve-ment of accuracy is of no practical value.
All observed differences are due to uncertainties containedin
both methods which are related to different number ofpanels,
especially on the free surface and to the distance atwhich the
radiation condition is imposed. However, somewhat
Dimensionless Frequency0 0.5 1 1.5 2 2.5
-0.4
-0.3
-0.2
Dimensionless Frequency0 0.5 1 1.5 2 2.5
0
0.04
0.08
0.12
0.16
Section 10
24
24
24
Dimensionless Frequency0 0.5 1 1.5 2 2.5
-0.16
-0.12
-0.08
-0.04
0
Section 22
H/T=10.0H/T=15.0H/T=1.1H/T=1.3H/T=1.5H/T=1.7H/T=2.0H/T=3.0H/T=4.0H/T=5.0
H/T=10.0H/T=15.0H/T=1.1H/T=1.3H/T=1.5H/T=1.7H/T=2.0H/T=3.0H/T=4.0H/T=5.0
H/T=10.0H/T=15.0H/T=1.1H/T=1.3H/T=1.5H/T=1.7H/T=2.0H/T=3.0H/T=4.0H/T=5.0
ymbolsproposed method and linesHmassef.
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ARTICLE IN PRESS
S. Sutulo et al. / Ocean Engineering 36 (2009)
1098111111060.15
0.2
H/T=10.0H/T=15.0H/T=1.1H/T=1.3H/T=1.5H/T=1.7
Section 02irregular dependence of the agreement on the case
(somecoefcients are better for one section, other coefcients are
betterfor another; sometimes agreement is better for low
frequencies,sometimesat high frequencies; even when the agreement
forthe added mass is not so good, it is much better for
thecorresponding damping coefcient) indicates that no one of
thecodes contains systematic errors and all results are viable.
Dimensionless Frequency
44
44
44
0 0.5 1 1.5 2 2.50
0.05
0.1H/T=2.0H/T=3.0H/T=4.0H/T=5.0
Dimensionless Frequency
Dimensionless Frequency
0 0.5 1 1.5 2 2.50
0.005
0.01
0.015
0.02
Section 10
0 0.5 1 1.5 2 2.50
0.01
0.02
0.03
0.04
0.05
Section 22
H/T=10.0H/T=15.0H/T=1.1H/T=1.3H/T=1.5H/T=1.7H/T=2.0H/T=3.0H/T=4.0H/T=5.0
H/T=10.0H/T=15.0H/T=1.1H/T=1.3H/T=1.5H/T=1.7H/T=2.0H/T=3.0H/T=4.0H/T=5.0
Fig. 8. Roll and heave damping coefcients: sym1
1.2
1.4
Section 02
H/T=10.0H/T=15.0H/T=1.1H/T=1.3H/T=1.5H/T=1.7H/T=2.0It can be
noticed that depending on the sections shape andrelative water
depth, the dependency of the added masses anddamping coefcients on
the frequency can be very different andsometimes even unexpected.
This proves that using deep-waterdata for seakeeping calculations
in shallow water can lead tosignicant errors of undetermined sign.
The theoretical symmetryrelation m24m42 is only met approximately,
although the
33
33
33
Dimensionless Frequency
Dimensionless Frequency
Dimensionless Frequency
0 0.5 1 1.5 2 2.50
0.2
0.4
0.6
0.8
0 0.5 1 1.5 2 2.50
0.4
0.8
1.2
1.6
2
2.4
Section 10
0 0.5 1 1.5 2 2.50
0.4
0.8
1.2
1.6
2
Section 22
H/T=10.0H/T=15.0H/T=1.1H/T=1.3H/T=1.5H/T=1.7H/T=2.0H/T=3.0H/T=4.0H/T=5.0
H/T=10.0H/T=15.0H/T=1.1H/T=1.3H/T=1.5H/T=1.7H/T=2.0H/T=3.0H/T=4.0H/T=5.0
H/T=3.0H/T=4.0H/T=5.0
bolsproposed method and linesHmassef.
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ARTICLE IN PRESS
differences are small and even hardly perceptible at larger
depthsof uid.
4.4. Ship sections over stepped bottom
The next explored case was that of a ship section over
thestepped bottom. This case is modeling the situation typical
forharbour approach channels. In this case, the results will
depend,besides the frequency, on three parameters: (1) depth of
theshallower part, (2) depth of the deeper part, and (3)
lateraldisplacement of the section with respect to the step. As
thesymmetry with respect to the centerplane does not exist
anymore,more coupling effects will be observed. Namely, the
couplingsbetween the heave and sway (indices 23) and between heave
androll (34) will take place.
The test computations were carried out for the maximum-contrast
cases i.e. when the relative depth of the shallower partwas 1.1 and
for the deeper part15.0, which is practicallyequivalent to the
unlimited depth. Of course, such a depthcontrast is not likely in
the real situations but all step effectsare supposed to be more
pronounced in the studied case.The calculations were performed for
the same three sections,but only results for section 10 are
presented here in full. A typicalgeneral distribution of panels is
shown on Fig. 9 for theshallow part located at the left, but most
calculations werecarried out for its inversed location and for ve
different positionsof the section with respect to the step shown on
Fig. 10. Resultsof these calculations for the section 10 are
presented on Figs. 11and 12.
The results differ from those obtained with the at bottom
notonly quantitatively, but also qualitatively: most dependencies
ofthe added mass and damping coefcients look highly
oscillatorywhich is no longer an artifact, but indicates to the
presence ofsome interference. Its details are still not clear, but
apparently thisis due to the fact that the same oscillation
frequency o results inat least two different wavelengths 2p/kR and
2p/kL, dened by twopresent depths according to Eq. (10). Although
the at-bottomresults for the relative depths 1.1. and 15 tend to
serve asenvelopes for the stepped-bottom data, they hardly can be
used asa viable approximation which conrms the necessity to
performestimations for any actual bottom shape.
The asymmetric coupling characterized by coefcients with
-100 -50 0 50
0
20
40
60
80
100
120
140
160
180
Fig. 9. General view of the computational domain for a
left-stepped bottom.
Position 1
S. Sutulo et al. / Ocean Engineering 36 (2009) 10981111
1107Position 3 Position 4Fig. 10. Ship section 10indices 23 and 34
is signicant enough in most cases, but usuallytends to diminish at
higher frequencies. This, and the oscillatorybehaviour of the
corresponding dependencies point out thatthe asymmetry is mainly
governed by the wave effects and thenear-eld asymmetry stemming
from the presence of the step isless important.
For all the coefcients, the character of the dependencies onthe
frequency can vary with the contours shape. Sometimes, thebehaviour
can be similar for different shapes but sometimes not.To illustrate
this second possibility, plots for the heave added
Position 2
Position 5over a right step.
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ARTICLE IN PRESS
S. Sutulo et al. / Ocean Engineering 36 (2009)
1098111111080.8
1
1.2
Position1Position2Position3Position4Position5mass of the
sections 2 and 22 are shown on Fig. 13. Comparisonwith similar data
for the section 10 on Fig. 11 indicates an unusualbehavior in the
latter case when the positions inuence is muchgreater than that
observed for sections 02 and 22, especially athigher frequencies.
Likely, this happens due to proximity of twoat bottoms which does
not happen in the case of the bow andstern ship sections.
Dimensionless Frequency
22
24
0 0.5 1 1.50
0.2
0.4
0.6
Dimensionless Frequency0 0.5 1 1.5
Dimensionless Frequency0 0.5 1 1.5
0
0.02
0.04
0.06
0.08
0.1
Position1Position2Position3Position4Position5
34
-0.12
-0.08
-0.04
0
0.04
0.08
Position1Position2Position3Position4Position5
Fig. 11. Added mass coefcients fo0.6
0.8
1
Position1Position2Position3Position4Position55. Conclusion
A exible implementation of the boundary integral equationmethod
for two-dimensional contours intersecting the free sur-face of the
nite-depth uid has been developed. The salientfeature of the method
is that the non-penetration boundarycondition can be satised in the
integral sense over each of the
Dimensionless Frequency
23
0 0.5 1 1.5
Dimensionless Frequency0 0.5 1 1.5
Dimensionless Frequency0 0.5 1 1.5
-0.4
-0.2
0
0.2
0.4
33
0
0.4
0.8
1.2
1.6
2
2.4
2.8
Position1Position2Position3Position4Position5
44
0
0.02
0.04
0.06
0.08
0.1
Position1Position2Position3Position4Position5
r section 10: stepped bottom.
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ARTICLE IN PRESS
S. Sutulo et al. / Ocean Engineering 36 (2009) 10981111
11092
1.2
1.4
1.6
1.8
Position 1Position 2Position 3Position 4Position 5approximating
at panels. The calculation domain is naturallylimited by the wetted
part of the contour, free surface, bottom(seabed) and is also
articially limited with two far-placed verticalboundaries onwhich
the radiation condition is fullled. The shapeof the bottom can be
arbitrary.
The method was successfully veried for at bottom, in whichcase
some published results and an independent code were available.Then,
computations for the stepped bottom with an extremely high
Dimensionless Frequency
2
0 0.5 1 1.50.2
0.4
0.6
0.8
1
Dimensionless Frequency0 0.5 1 1.5
Dimensionless Frequency0 0.5 1 1.5
24
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Position 1Position 2Position 3Position 4Position 5
34
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
Position 1Position 2Position 3Position 4Position 5
Fig. 12. Damping coefcients for3
0
0.4
0.8depth contrast were carried out. The obtained results showed
that thepresence of a step heavily affects the hydrodynamic
characteristics ofthe contour. The dependency on the oscillation
frequency becomeshighly oscillatory and cannot be approximated with
any at-bottomresults. The bottom asymmetry results in a signicant
hydrodynamicasymmetry even on geometrically symmetric contours and
certaincoupling effects, which are usually absent or negligible,
like sway-heave and heave-roll, can become important.
Dimensionless Frequency
2
0 0.5 1 1.5
Dimensionless Frequency0 0.5 1 1.5
Dimensionless Frequency0 0.5 1 1.5
-1.2
-0.8
-0.4Position 1Position 2Position 3Position 4Position 5
33
0
0.4
0.8
1.2
1.6
2
Position 1Position 2Position 3Position 4Position 5
44
-0.005
0
0.005
0.01
0.015
0.02
Position 1Position 2Position 3Position 4Position 5
section 10: stepped bottom.
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ARTICLE IN PRESS
ions
ineering 36 (2009) 10981111Appendix
Corrected explicit formulae for the excitation function
Fij(4)
According to Sutulo and Guedes Soares (2004)
F4ij 12 nyjxOj nxjyOj I0 12I1; A1
where nxj, nyj are projections of the unity normal on the
jthelement and xOj ; yOj are co-ordinates of the same element; I0
andAcknowledgments
The study was carried out within the framework of theresearch
Project PTDC/ECM/65806/2006 Dynamics and Hydro-dynamics of Ships in
Approaching Fairways nanced by Funda-c- ~ao para a Ciencia e a
Tecnologia (FCT), Portugal, The rst authorwas supported by the FCT
Grant SFRH/BPD/26722/2006. Theauthors appreciate Mr. Antonio Pac-os
considerable aid at editingnumerous plots.
Dimensionless Frequency
33
0 0.5 1 1.50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Position 1Position 2Position 3Position 4Position 5
Fig. 13. Sway added mass coefcients for sect
S. Sutulo et al. / Ocean Eng1110I1 are the auxiliary functions
dened as
I 0
1
Z Si=2Si=2
dx
Z Sj=2Sj=2
logbij
sijx2 aij cijx x21
x
( )dx; A2
where aij, bij, cij and sij are auxiliary geometric parameters
denedin Sutulo and Guedes Soares (2004).
The integrals I0 and I1 were evaluated in Sutulo and
GuedesSoares (2004) analytically for the general case of arbitrary
mutualorientation of the ith and jth elements and also for the
specialcases of parallel and co-planar elements. All formulae are
correctexcept for the case of I1 calculated for parallel elements
when
I1 H0Si=2; Sj=2 H0Si=2; Sj=2 H0Si=2; Sj=2
H0Si=2; Sj=2; A3
where
H0x; y H00x; y H01x; y H02x; y H03x; y; A4and where the rst two
auxiliary functions are computed as
H00x; y logjbijjxy2 b2ij; H01x; y 2aijbijH010x; y
2cijbijH011x; y; H010x; y x atanaij y cijx
bij
cijaij y atan2xbij; aij y cijx12cijbij logb2ij aij y cijx2;
A5
H011x; y 1
2cijbijx x2 atan
aij y cijxbij
aij y2 b2ij atan2xbij; aij y cijx
bijaij y logb2ij aij y cijx2
A6
and
atan2xx; y ( atan2x; y at yZ0p sign x atan x
yat yo0 ;
Dimensionless Frequency
33
0 0.5 1 1.5-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Position 1Position 2Position 3Position 4Position 5
2 (left) and 22 (right) over stepped bottom.atan2x; y ( atan
x
yat ya0
p2sign x at y 0
A7
The corresponding formulae in Sutulo and Guedes Soares(2004)
break on elements whose normals are directed to eachother. This
situation did not happen in deep-water calculations.All the
remaining auxiliary functions from Eq. (A4) are correctlygiven in
Sutulo and Guedes Soares (2004). Their structure anddegree of
complexity are similar to those of the formulaepresented here.
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S. Sutulo et al. / Ocean Engineering 36 (2009) 10981111 1111
Computation of inertial and damping characteristics of ship
sections in shallow waterIntroductionFormulation of problem and
boundary integral equationProblem statementBoundary integral
equationDiscretisation of the boundary-value problem
Added masses and damping coefficientsVerification of the method
and numerical examplesPeculiarities of the boundarys
discretisationAdded masses and damping coefficients for semi-circle
over flat bottomAdded masses and damping coefficients for ship
sections over flat bottomShip sections over stepped bottom
ConclusionAcknowledgmentsCorrected explicit formulae for the
excitation function Fij(4)
References