a boundary integral equations method for computing inertial an damping characteristics

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.

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

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)

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



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


0

0.2

0.4

0.6

0.8

1

1.2

1.4

H/T=15


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

H/T=15


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

H/T=4


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



Yeung



Yeung



Yeung


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.

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.


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





-0.6

-0.4

-0.2

0


Section 02

0

0.2

0.4

0.6

0.8

1


Section 10

0

0.02

0.04

0.06

0.08

0.1


Section 10

22

22

22

24

24

24

0

0.2

0.4

0.6

0.8

1


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.

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


0

0.05

0.1H/T=5.0

Section 02



0.01

0.02

0.03

0.04

0.05

0.06

0.07


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




0

0.2

0.4

0.6


Section 02

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


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.

ARTICLE IN PRESS

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


0

0.2

0.4

0.6


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


Section 10


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


-0.4

-0.3

-0.2


0

0.04

0.08

0.12

0.16

Section 10

24

24

24


-0.16

-0.12

-0.08

-0.04

0

Section 22




ymbolsproposed method and linesHmassef.

ARTICLE IN PRESS


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.


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



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



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




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=3.0H/T=4.0H/T=5.0

bolsproposed method and linesHmassef.

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.

ARTICLE IN PRESS


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.


22

24

0 0.5 1 1.50

0.2

0.4

0.6

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


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


23

0 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


44

0

0.02

0.04

0.06

0.08

0.1


r section 10: stepped bottom.

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


2

0 0.5 1 1.50.2

0.4

0.6

0.8

1



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


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.


2

0 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


44

-0.005

0

0.005

0.01

0.015

0.02


section 10: stepped bottom.

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.


33

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


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 ;


33

0 0.5 1 1.5-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4


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.

References

Bandyk, P.J., Beck, R.F., 2008. Nonlinear ship motions in the timedomain using

a body-exact strip theory. In: Proceedings of the ASME Twenty Seventh

International Conference on Offshore Mechanics and Arctic Engineering

OMAE2008. June 1520, 2008, Estoril, Portugal, Paper OMAE2008-57069,

10pp.

Bertram, V., 2000. Practical Ship Hydrodynamics. Butterworth Heinemann, Oxford.

Bertram, V., Veelo, B., Soding, H., Graf, K., 2006. Development of a freely available

strip method for seakeeping. In: Proceedings of the Fifth Conference on

ARTICLE IN PRESS

Computer and IT Applications in the Maritime Industries (COMPIT06). Leiden,pp. 356368.

Fedyaevsky, K.K., Sobolev, G.V., 1964. Control and Stability in Ship Design, USDepartment of Commerce Translation, Washington, DC.

Frank, W., 1967. The heave damping coefcients of bulbous cylinders partiallyimmersed in deep water. J. Ship Res. 11, 151153.

Frank, Ph., Mises, R. von, 1961. Differential and Integral Equations of Mechanics andPhysics. I Mathematical Part. Dover Publications Inc. (in German).

Grim, O., 1953. Computation of hydrodynamic forces caused by oscillations ofa ship hull. In: Jahrbuch der Schiffbautechnischen Gesellschaft. Springer,pp. 277299 (In German).

ITTC, 1983. Summary of Results Obtained with Computer Programs to Predict ShipMotions in Six Degree of Freedom and Related Responses, Report on 15th and16th ITTC Seakeeping Committee Comparative Study on Ship Motion Program(19761981), 53pp.

Jaswon, M.A., 1963. Integral equation method in potential theory. Proc. R. Soc. A275/360, 2332.

Journee, J.M.J., Adegeest, L.J.M., 2003. Theoretical Manual of SEAWAY forWindows, Delft University of Technology, Report 1370, p. 336. (/www.shipmotions.nlS).

Keil, H., 1974. Hydrodynamic Forces in Periodic Motions of Two-dimensionalBodies on the Free Surface of Shallow Water, Institut fur Schiffbau derUniversitat Hamburg, Bericht no. 305 (in German).

Schmiechen, M., 1999. Estimation of spectra of truncated transient functions. ShipTechnol. Res./Schiffstechnik 46, 111127.

Soding, H., 2005. Discussion to the Paper (vol. 51, no. 2) by Serge Sutulo and CarlosGuedes Soares A Boundary Integral equation Method for Computing Inertial

and Damping Characteristics of Arbitrary Contours in Deep Fluid. ShipTechnology Research/Schiffstechnik, vol. 52, no. 1, p. 52.

Sutulo, S., Guedes Soares, C., 2004. A boundary integral equation method forcomputing inertial and damping characteristics of arbitrary contours in deepuid. Ship Technol. Res./Schiffstechnik 51, 6993.

Salvesen, N., Tuck, E.O., Faltinsen, O.M., 1970. Ship motions and sea loads. SNAMETrans. 78, 250287.

Sutulo, S., Guedes Soares, C., 2008. A Generalized Strip Theory for CurvilinearMotion in Waves. In: Proceedings of the ASME 2Twenty Seventh InternationalConference on Offshore Mechanics and Arctic Engineering OMAE2008. June1520, 2008, Estoril, Portugal, Paper OMAE2008-57936, 10pp.

Ursell, F., 1949. On the heaving motion of a circular cylinder in the surface of auid, quart. J. Mech. Appl. Math. 2, 218231.

Vantorre, M., Laforce, E., Eloot, K., Richter, J., Verwilligen, J., Lataire, E., 2008. ShipMotions in Shallow Water as the Base for a Probabilistic Approach Policy. In:Proceedings of the ASME Twenty Seventh International Conference on OffshoreMechanics and Arctic Engineering OMAE2008. June 1520, 2008, Estoril,Portugal, Paper OMAE2008-57912, 10pp.

Xia, J., Wang, Z., 1997. Time-domain hydroelasticity theory of ships responding towaves. J. Ship Res. 41, 286300.

Yeung, R.W.-C., 1973. A Singularity-Distribution Method for Free-Surface FlowProblems with an Oscillating Body. Ph.D. Dissertation, University of California,Berkeley.

Yeung, R.W.-C., 1982. Numerical methods in free-surface ows. Annu. Rev. FluidMech. 14, 395442.

Zhao, Y.-X., 1986. Computation of manoeuvring motion in shallow water. Jahrb.Schiffsbautechnischen Ges. Bd. 80, 261275 (In German).

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

a boundary integral equations method for computing inertial an damping characteristics

Documents

manoeuvring problems

d codes startedto

theoreticalring problems

applye problem

practtreatment of seakeeping

d formulation

hybrid seakeeping

strip methods