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OCR for page 499
SideWall Effects on Hydrodynamic Forces Acting on
a Ship with Forward and Oscillatory Motions
M. Kashiwagi and M. Ohkusu
Kyushu University
Fukuoka, Japan
Abstract
A rational slendership theory is presented
for predicting the sidewall effects on the
addedmass and damping coefficients of a ship,
moving with forward velocity and performing
heave and pitch motions in a waterway with
vertical and parallel side walls. Satisfaction
of the sidewall boundary condition in the
farfield solution is acheived by the method
of mirror images, with a closedform expres
sion obtained of the resultant infinite se
ries. The inner expansion of the farfield
solution dictates the asymptotic behavior of
the nearfield solution, thereby determining
the nearfield homogeneous component. This
component accounts for the sidewall inter
ference in the inner region as well as the
hydrodynamic interactions along the ship's
length.
Computed addedmass and damping coeffici
ents are presented for a halfimmersed prolate
spheroid. Validity of the proposed theory is
confirmed by the comparison with the 3D panel
method for the special case of zero forward
speed.
Hydrodynamic forces acting on a ship model
with forward and oscillatory motions are in
most cases measured in a towing tank with
limited width. When the forward speed and os
cillation frequency of a ship model are rela
tively small, reflection waves from the side
walls of a towing tank affect the measured
hydrodynamic forces: they must be different
from what we expect for a ship model in open
water.
A diagram [1] is prepared for predicting
whether the sidewall effect is expected or
not, which gives the critical value of para
meter ~=Um/g (where U is the forward speed,
the oscillation frequency, and g the gravi
tational acceleration) as a function of the
ratio of tank width to ship length. This di
agram is, however, derived with heuristic ways
and thus not entirely precise.
There have been a few theoretical studies
concerning the sidewall effects on hydro
499
dynamic forces acting on a ship moving at a
finite forward speed in waves. The pioneering
work of Hanaoka [23 based on the thinship
theory is a rational theory of the sidewall
effects, but did not give reliable information
on the effects. Hosoda ~3,4] and Takaki t5]
are based on the striptheory approach with a
number of approximations in the mathematical
evaluation of reflection waves from the side
walls. Accuracy of their numerical results
are therefore questionable despite their math
ematically complicated expressions.
In the present paper, the slendership the
ory is applied to develop a rational method
which is able to predict the effects of side
wall interference, particularly when a ship
has a finite forward speed. The theory desc
ribed in this paper may be regarded as an
tension of Newman's unified theory [6] to
case of sidewall effects present.
In the inner region close to the ship hull,
since the side walls and the radiation condi
tion are absent, the inner solution can be
identified with that in the unified theory for
the opensea problem. Namely it can be const
ructed by the superposition of the particular
solution given by the strip theory plus a
homogeneous solution giving threedimensional
effects. The latter component plays an impor
tant role in accounting not only for longitu
dinal flow interactions along the ship hull
but also for the sidewall interference in the
inner region.
In the outer region far from the ship, the
ship may be seen as a segment on the longitu
dinal axis, but the side walls are present.
Thus the solution is represented by a line
distribution of 3D wave sources with unknown
strength along the ship's length. The veloci
ty potential of 3D wave source satisfying the
sidewall boundary condition is derived by
considering an infinite number of image singu
larities and a closedform expression of the
resultant infinite series.
The source strength in the outer solution
and the coefficient of a homogeneous component
in the inner solution are to be determined
from the asymptotic matching procedure. The
implementation of the asymptotic matching bet
ween the inner and outer solutions leads to an
ex
the
OCR for page 499
integral equation for the strength of 3D wave
sources, the solution of which then settles
the coefficient of the nearfield homogeneous
solution; thereby completing the velocity po
tential necessary for the calculations of
addedmass and damping coefficients.
Solving the integral equation obtained re
quires the numerical evaluation of Cauchy's
principalvalue integrals involved in the ker
nel function representing the sidewall ef
fects. Numerical implementation of these in
tegrals is performed by firstly subtracting
the singular behavior from the integrand, sec
ondly integrating analytically the subtracted
singular part, and finally integrating numeri
cally the resultant nonsingular part by means
of an appropriate numerical technique. Clen
shaw & Curtis quadrature is employed in this
paper with a tolerance of absolute error less
than 105.
Computational results are presented of the
heave and pitch addedmass and damping coef
ficients, for a halfimmersed prolate spheroid
of lengthbeam ratio 8.0 advancing at a Froude
number 0.1 in the waterway of width twice the
spheroid's length.
The appearance of sidewall effects is
closely related to the wave pattern generated
by an oscillating and translating ship and to
its reflection from side walls of the water
way. Starting from the ring wave at ~=0, the
wave pattern changes to the complicated one
dominated by the divergingwave component, as
the parameter ~ increases across the critical
value 1/4. Corresponding to this complicated
variation of the wave pattern, the addedmass
and damping coefficients including the side
wall effects show complex variations.
In order to check the validity of the
theory, numerical results for the special case
of zero forward speed are compared with inde
pendent "exact" calculations based on the 3D
panel method, for example, by Kashiwagi [73.
The results of the present theory agree excel
lently with the 3D panelmethod predictions.
The present paper is restricted to the
radiation problem of heave and pitch oscilla
tions, but the diffraction problem may be
analyzed in a similar manner with the know
ledge of Sclavounos' diffraction theory t11]
for the case of open sea, which is left for
future work.
2. Formulation of problem
As shown in Fig.l, we consider a ship in a
waterway with vertical and parallel side
walls. Let L, B. and d denote the length,
breadth, and draft of a ship respectively, and
BT the breadth of a waterway. The ship is as
sumed to move at constant forward velocity U
and to oscillate sinusoidally with circular
frequency ~ in heave and pitch; the depth of
waterway is assumed deep enough, with no shal
lowwater effect in the waterwave phenomena.
A coordinate system used is shown in Fig.1.
The xaxis is coincident with the centerline
of the waterway and positive in the direction
of the ship's forward velocity, the yaxis is
Side Wall
BT
1' L 'I
—B: ~ X
.............................................................
Side Wall
~ AX
z
Fig.1 Coordinate system and notations
horizontal, and the zaxis is vertical and
positive downward, with the origin placed at
midships and on the undisturbed free surface.
Assuming the flow to be inviscid with irrota
tional motion, the flow field can be described
in terms of the velocity potential ¢(x,y,z,t)
satisfying the Laplace equation of the form
~ L ~ ~xx + tyy + fizz = 0 (1)
in the fluid domain Z>O, IYI
[F] ~jz + K~j + i2l~jx  K tjxx
[R]  ip( Ktj + il~jX ) = 0 (3)
where K=w2/g, I=Um/g, Ko=g/U2 (4)
[B] fjz ~ O
as z ~ ~ (5)
[A] fjy = 0 on y = +BT/2 (6)
[H] Jan = nj + ~~ mj on ship hull (7)
Here ~ in (3) is Rayleigh's artificial visco
sity coefficient to ensure the appropriate
radiation condition [R] being satisfied. The
subscript n in (7) denotes normal differentia
tion, with the unit normal vector defined
positive when pointing into the fluid domain
(see Fig.l), and nj is the components of the
normal vector parallel to the xjaxis with
extended definition of ns=znlxn3. mj is the
socalled mterms representing the forward
speed effect due to the oscillatory motions in
the steady flow, which has been originally
derived in Timman and Newman [12] and can be
explicitly written as
(ml, m2, m3) = (n V)VX
(m4, m5, m6) = (n V)(rxVX)
(8)
In order to obtain a solution of the above
threedimensional boundaryvalue problem, we
exploit a slendership theory. In this theory,
the flow field to be analyzed will be divided
into the inner and outer regions, and in each
region the governing equation and boundary
conditions may be simplified, making it pos
sible to obtain the inner and outer solutions
respectively with relative ease. However, both
of these solutions include indeterminate coef
ficients, since nothing has been prescribed
about the respective asymptotic behavior far
away in the inner problem and close to the
ship in the outer problem. These indetermi
nate coefficients can be settled by requiring
the two solutions to be compatible in an over
lap region between the inner and outer fields.
2.1 The outer problem
In the outer field far from the ship hull,
the effects of three dimensionality and of
side walls of the waterway must be accounted
for. When the ship is seen from the outer
field, it may be viewed as a segment on the
xaxis, and thus the flow field is insensitive
to the details of ship's hull geometry. There
fore the outer problem is defined by the 3D
Laplace equation [L], subject to the free
surface iF], radiation ~R], seabottom [B],
and sidewall tW] boundary conditions, that
is, (3)(6~. Since these boundary conditions
are homogeneous, the outer solution can be
described in terms of the 3D Green function
with unknown wave sources distributed on the
xaxis, in the form
501
~j~x,y,z) = JLqj(~)G~Xt,y,Z)d; (9)
Here G~x,y,z) denotes the Green function or
the potential of a "translatingpulsating"
source with unit strength, and q j(X) is the
strength of the source to be determined. The
Green function appropriate to the present
problem can be derived by applying Hanaoka's
approach t2], in which the method of mirror
images is utilized.
Considering the Fourier transform with
respect to x of a unit source located at the
origin and its mirror images with respect to
both of the side walls of waterway, the Green
function satisfying homogeneous boundary con
ditions (3~~6) is given in the Fourier space
as follows:
G35(k;y z) = J. Gfx,y,z)eikKXdx
_00
1 j~ ncostnKz)vein~nKz) n
.Z eKIypBTV~dn
p=_=
~sgn~ 1+2kl) e
·£ eiSgn~l+kl)~ypBT/l_k2/v2
p=_=
1 _
The function appearing as the first term in
brackets on the right side of (15) denotes the
infinite series of Dirac's delta function,
defined by
~ ' KBT) m m6(Q KBT ~ (16)
and therefore contributes only when Q=2nm/KBT.
If we consider the limit of BT ~ ~ in (14)
and (15), it is relatively easy to confirm
that the second line on the righthand side of
each equation vanishes and only the first term
remains. Thus the sidewall effects are repre
sented by the second line in (14) and (15~;
this suggests that the Green function can be
expressed in an addition form of the opensea
Green function Go~x,y,z) plus the sidewall
effect part GT(x,y,z). The final expression
of the Green function can be given by consi
dering the inverse Fourier transform of the
above expressions, with the following formula:
Gfx,y,z) = Go~x,y,z) + GT(X,y~z)
K lo {G*(k;y Z)+GT(k;y~z~}e~ikKXdk (17~
In the outer solution given by (9), the 3D
source strength q jinx) is unknown but will be
determined by requiring the inner expansion of
(9) to be compatible with the outer expansion
of an appropriate inner solution. For this
matching procedure, the inner expansion of the
Green function must be sought. Following the
method of matched asymptotic expansions, we
seek the inner expansion with the following
order of magnitude:
Ky, Kz=O(~), k=0~1), ~<0~1) (18)
Considering that the sidewall effects will be
expected when the forward velocity and oscil
lation frequency of a ship are relatively
small, the assumption on the order of para
meter ~ in (18) seems reasonable. It should
be noted, however, that this assumption does
not necessarily mean the applicability of the
present theory is restricted to the range
under the critical value given by ~=1/4. The
transitional value of the parameter ~ where
the theory becomes invalid may be determined
from numerical computations and comparison of
those with experiments.
The analysis for the inner expansion of the
opensea Green function is identical to that
in the unified theory devised by Newman [6],
and hence with the present notations the
desired result can be expressed in the form
Go~k;Y,Z) = G2D(Y,Z) + ~ (lKz~fo~k)
where + O(Kr~lv),K2r2) (19)
G2D(y,z) = Go(°;Y'Z)
 ~ {(lKz)(logKr+y)
+ Kr~cosO+OsinO)}i~lKz) (20)
502
fork) = logl 21 + Hi
~ ~ (cosh~~(lv~+nisgn: l +k~ )}
L ~k2 / 2 { COS\ ~ I V I )_~}
(21)
In (19) and (20), twodimensional polar co
ordinates (r,6) are used with the relation
(y,z) = (rsinG,rcosO), and y is Euler's cons
tant equal to 0.5772~.
The expansion of the sidewalleffect part
of the Green function can be obtained with
comparatively simple reduction, in the form
iE 1 *
GT(k;y,z) = ~ (lKZ)fT(k)
where + O(Kr~lv),K2r2) (22)
fT(k) = J 2 2 (l+coth; T ~ '
dn
a,,
nisen(~+~1)( 2n ~ ~ ~ KBT) 1
+ i sgn ~ l+kl ~ cot ~—<_k2 /V2 ~ }
~2/V2l ~)
_ (23)
As in (10), the upper and lower expressions in
brackets in (21) and (23) correspond to  k 
OCR for page 499
where
The inverse Fourier transform of the side
walleffect part (22) can be expressed as
GT(x,y,z) ~ rr (lKZ)dd FT(KX) (28)
FT(X) = 2~ ,: e ikX dk/k
·: 2n 2 { l+coth( 2T~)} ins +kZ
+ ) Z · Z m  sGn(i+kT) eikX
2 KBT p1 m_O [kldv/dkk/vl ~k=kpm
2 [ ~k2+:k4]
IT
· cot( 2 1//lk2/v2)dk/k
i ~ {k2 (key ikX 1
+ 2 L Jkl Jk3J ,/k2/V2l
. coth(—2ik2/v21 )dk/k
[ ; Jk2 Jk4] vS~ dk/k
~ [ Jk2 Jk4] ikX 2/V2_l / (29)
and ~ 0 = 2 ~ Em = 1 (miO) (30)
The first term on the righthand side of (29)
represents the contribution of nonradiating
local waves. The second term is obtained from
the infinite series of delta function in (23)
and physically the contribution of the out
going waves at infinity. In this second term
k m denotes the values satisfying KBT~
=5Trm (mO,1,2,··), which exist at most four in
number and when m=0 coincide with kj ( j=1~4)
given in (12) and (13).
A part of the radiated waves are reflected
on the side walls, and changed in phase by the
factor rr/2 and represented as the third term
in (29). In other words from the standpoint
of hydrodynamicforce calculation, the third
term is originally to be contributed to the
damping force, as is the same as the second
term, but by the phase shift of ~r/2 due to the
sidewall effects, this term will contribute
to the inertia force. The integrals in the
third term must be treated as Cauchy's princi
palvalue integral at the points of k sat
isfying sin(KB7~/2)=0, namely at k=kpm
defined in the second term.
The fifth and sixth terms in (29) are
independent of BT and thus to be cancelled out
by some terms given in the opensea expres
sions (25)(27). However as discussed previ
ously in connection with the infinite series
of (l4) and (15), these two terms play a role
in cancelling out respectively the third and
fourth terms of (29) in the limit of BT ~ Go.
When ~>1/4, the expressions from third to
sixth terms should be understood with k3=k4.
Since we got the desired inner expansion of
the Green function, by substituting it in (9)
we have readily the inner expansion of the
outer solution in the following form:
~j(x,y~z' ~ qj (X)G2D(Y'Z)
 ~ ( 1Kz )lLqj (A )dE,: [Fo{ K(x: )}
+ FT{K(xE,)} id: (31)
2.2 The inner problem
Since the ship hull is assumed slender,
changes of the flow in the xdirection are
small in the region close to the ship hull, by
comparison to changes in the transverse plane.
Thus the flow in the inner region may be
described by the 2D Laplace equation subject
to the free surface condition which is inde
pendent of forward velocity and applicable to
the 2D problem in the yz plane; this can be
mathematically justified by the coordinate
stretching argument with the assumption of
x=O(1), y=O(~), and z=O(~). In the inner prob
lem, the radiation condition and the sidewall
boundary condition can not be specified, be
cause the side walls are absent in the inner
region. With these taken into account, the
boundaryvalue problem can be written as
[L] tjyy + Jazz = 0 for z>0 (32)
[F] 4jz + Kjj = 0 on z=0 (33)
tH] IN = Nj + 7.U Mj on ship hull (34)
Here we note that the subscript N in (34) de
notes the normal differentiation on the sec
tional contour in the 2D transverse plane,
and Nj and M denote the slenderbody approxi
mation of tie 3D quantities in (7) and (8),
which are explicitly given as
N5 = xN3 ~
M3 = N2Xzy~N3Xzz ~ (35)
M5 = N3XM3 J
The inner boundaryvalue problem defined
from (32) to (35) is the same as the conven
tional 2D formulation except that the radia
tion condition is absent. Thus the inner
solution can be identified with Newman's uni
fiedtheory solution t6], composed of a par
ticular solution commonly used in the strip
theory plus a homogeneous solution multiplied
by a threedimensionaleffect coefficient. To
be more specific,
(X;y~z) =ijP(y z) + U9P( )
+ Cj(x) ~jH(y,z) (36)
4J(Y,z) = ~J(Y'Z) ~j(Y,Z)
(37)
where the overbar~denotes the complex conju
gate, and Liz and 98 are the particular solu
503
OCR for page 499
Lions determined to satisfy the following
boundary conditions on the body profile at
station x:
P ED
4jN = Nj ~jN = Mj (38)
The coefficient of the homogeneous solution
Cj (x ~ in (36) is indeterminate at this stage,
but may be settled by matching the outer
expansion of (36) with the inner expansion of
the outer solution already given by (31~.
Far from the ship hull in the inner region,
(36) reduces to
~x;y,z) ~ [Oj(X)+1,U ~j(X)
+ Cj~x)~i~x)~j(X)) iG2D(Y,Z)
 e cos(Ky) 2iCj (x)oj (x) (39)
Here ~j(X) and Dj(X) denote the 2D effective
source strengths; these can be given by sol
ving the 9~ and problems respectively.
G2D(y,z) is the 2D Green function and iden
tical to the one shown in (20) or (31~.
2.3 Matching
In the analysis described above, the un
knowns are the 3D source strength q j (x ~ in
the outer solution and the coefficient Cj~x)
of a homogeneous component in the inner solu
tion. These will be determined by the match
ing of the inner and outer solutions. Compar
ing (31) with (39) and equating the factors of
G2D, the following relation can be found:
Ajax) = ~j(X)+'U ~j(X)+Cj~x){oj~x)oj~x)l ~
Equating the remaining terms in (31) and (39)
gives
i2rrCj~x)Oj~x) = ~Lqi(~)
· d: [For Kfxt )} +FT[ Kfx` )} id: (41 )
with the error of order O(K2r2 ).
Eliminating Cj (x) from (40) and (41 ) we
have an integral equation for the 3D source
strength q jinx) of the form
q jinx)  2~ t(>j~x)/oj~x)l]J.Lqi(~)
· d<; ~Fo{KfxE~}+FT{KfxE~ ids
= ~j(X)+~U Ajax) (42)
Once q jinx) is determined by solving (42 ~ with
an appropriate numerical method, the coeffi
cient Cj~x) can be readily determined from
(40) and thus the inner solution will be
completed.
In the case of no sidewall effects, i.e.
BT ~ ~ the function F  Kfx~} becomes zero
as already mentioned, and the integral equa
tion (42) reduces to the corresponding one in
Newman's unified slendership theory in the
opensea case [63. In the special case of zero
504
forward speed, Kinoshita and Saijo [8] derived
an analogous equation to (42 ~ in the study on
a multibodytype floating breakwater, consis
ting of an infinite array of slender bodies.
The inner solution (36) appears formally to
be invariable regardless of whether the side
walls are present or not. However, through
the 3D source strength qj~x), which includes
the sidewall effects as a solution of (42)
the coefficient of homogeneous solution Cj (x ~
accomodates not only the 3D interaction ef
fects between transverse crosssections but
also the sidewall effects of the waterway.
3. Addedmass and damping coefficients
Since the inner solution has been deter
mined, we proceed to the calculation of hydro
dynamic pressure force and moment acting on a
ship with forced heave and pitch motions. The
linearized hydrodynamic pressure is given from
Bernoulli's equation. Then the hydrodynamic
force in the ith direction due to the jth
mode of motion can be provided by integrating
the pressure over the mean wetted surface of
the ship hull, and can be expressed in terms
of the addedmass (Aij) and damping (Bij)
coefficients, in the form
Pi =  3 5~(iw)2Aij + (i~)Bij}`j
' (i=3,5) (43)
~..~, ~ ,
Aij+Bij/itl) = J Ltaij(X)+bij~x)/im~dx (44)
a i j +b i j l it) = ~ P :CNi ~id'
+ ip ~ ~ ~ (Ni(j)j~i~j Do  p (`—~ ~ ~ Ni~jdQ
 pCj~x)l (Ni~ ~i)( $j~ $J)di ~
Here p is the fluid density, and in deriving
the above, Tuck's theorem t13] has been used.
aij and bid defined in (44) and (45) denote
the 2D addedmass and damping coefficients
respectively, involving the 3D interaction
effects and the sidewall effects, and c to
the integral sign in (45) denotes the submer
ged portion of the contour of the transverse
section.
In order to perform the calculations of
(45), the term M3 defined TAX (35) and the
related velocity potential 95 must be known,
besides the velocity potential 93 commonly
calculated in the strip theory. If 95 and
95 are obtained, the remaining velocity poten
tials for pitch (j=5) follow from (35)
95P=x93P
~ (46)
95 = $3 X93
Sclavounos [lO] studied in the opensea
case the relative importance of the contribu
tions frog the M3term and related velocity
potential 95, by comparing the numerical re
sults with experiments. His results reveal
that the inclusion of the M3term leads to a
OCR for page 499
substantial overprediction of the damping co
efficients. This overprediction may be at
tributed to the inaccuracy of the mterms near
the ship ends, which have been evaluated with
slenderbody approximation. Therefore the m
terms should be evaluated from the 3D precise
calculation for the steady perturbation poten
tial. Fortunately, according to his numerical
study, a better agreement with experiments is
provided by simply omitting the mterms in the
unified theory. Thus in the numerical calcu
lations with sidewall effects presented here
too, it was decided to neglect the M3^term
and consequently the velocity potential 95 in
(45~.
It should be noted that the last term in
(45), multiplied by the coefficient Cj~x),
plays an important role in accounting for the
unifiedtheory corrections in the opensea
case and for the effects of sidewall inter
ference in the presence of waterway. Without
this last term, the remaining expressions in
(45) are identical to those in the strip
theory.
4. Numerical calculation method
An improtant task in the present theory is
to solve the integral equation (42) for the
3D source strength q (x). For this purpose,
after dividing the swipes longitudinal axis
into NX segments of equal length, the 2D
boundaryvalue problem for heave (j=3) in each
divided transverse plane must be solved; which
gives 03(X) necessary in calculating the right
side of (42~. Since we neglect the contribu
tion of steady perturbation potential, 93(X)
becomes zero, and the 2D effective source
strength for pitch (j=5) can be evaluated
directly from o3(X), with the result of o5(X)
=xc3 (x) and oryx)= 03(x).
The 3D source strength q jinx) , which is
to be determined, has been assumed to vary
linearly in x between adjacent nodal points,
with the value of fix) at station x=xk denoted
by qk
where
~ (xxk 1 ~ / (XkXk 1 ~ Xk 1
40E
2.0
1 n
^3s~V
_
in 0~ See
_
_   S ~ ~ ~
i~ ELI ~ (e=le} G.0
Slender Ship Theory
4.0
n
, . ~ ~ O
KL 15
Fig.2 Heave sided mass of a prolate spheroid
(~/B=8) at Fn=O.1 in waterway of B~/B=16
^~V~
0.10
0.05
0.05 _
Elate Sphe~ld ~8 = e 81 Fn =0d
In Open See
~ ~ 0 4
 Slender Shlp Theory
( ~ ~ )
Slender Shlp Theory
O . , , , , I
5
0.3
0.2[
Fig.4 Pitch added moment of inertia of a
prolate spheroid (~/B=8) at F~=O.1 in
waterway of B~/B=16
506
F Ba3/~V/~7[
Fig.3 Heave damping coefficient of a prolate
spheroid (~/B=8) at Fn=O.1 in waterway
of B~/B=16
s~V~
. ~ ~ `,
Fig.5 Fitch damping coefficient of a prolate
spheroid (L/B=8) at Fn=O 1 in waterway
of B~/B=16
OCR for page 499
form of these coefficients are displayed in
the ordinate of each figure.
In all of these figures, thick solid lines
indicate the numerical results in the presence
of sidewall effects, computed by the slender
ship theory described in this paper. In order
to show the magnitude of sidewall effects,
the values in open sea are shown by short
dashed lines, which were obtained with the
sidewalleffect part of the kernel function
Fly Kfx; )} set to be zero in the integral
equation (42), and therefore must be identical
to the unifiedtheory solutions t94. Also
shown in the opensea case are the stripthe
ory predictions, which are indicated by dash
dotted lines. Comparing the predictions of the
strip theory with those of the unified slender
ship theory, we can understand that the ef
fects of three dimensionality are prominent
only in the low frequencies.
Since the forward velocity is present, with
the incensing wavenumber KL, the parameter
=Um/gFniKL increases and takes the critical
value ~=l/4 at KL=6. 25. The position of this
critical wavenumber is shown by the vertical
thin solid line with a downward arrow. In the
frequencies less than ~=1/4, the effects of
side walls are considerable not only in heave
but also in pitch modes. In particular, Ass
and Bss change drastically in the frequency
range slightly less than the critical frequen
cy ~=l/4, and Ass takes a negative value. It
should be noted, however, that the damping
coefficients B33 and Bss predicted by the
present theory are definitely positive, al
though they vary greatly in magnitude and
become nearly equal to zero at some frequen
cies. This nonnegative damping force seems
quite reasonable, judging from the considera
tion on the energy flux radiating in the
longitudinal direction of the waterway. In
some published results by a heuristic method
[5], negative dampingforce coefficients are
predicted in the low frequencies; this is not
the case.
It is known in the case of zero forward
velocity that the wavenumbers corresponding to
the tankresonant mode in the transverse di
rection can be given by KBT=2nm (m=1 , 2, · · ~ and
thus in the present case by KLnm ; at which
the ratio of wavelength to tank width is equal
to the inverse of an integer. When the forward
velocity is present, the wavelength of the
wave component radiating in the transverse
direction is diminished in comparison to the
wavelength at U=O, due to the effects of
forward speed. With this knowledge, we can
observe particularly in the range of ~<1/4
that the tankresonant frequency is shifted to
the lower frequency than the zerospeed tank
resonant frequency given by KL=nm.
The wave pattern generated by a ship with
forward and oscillatory motions in open sea is
known to change drastically, dependent on the
value of ~ [143. In particular for ~ close to
but larger than 1/4, the angle of the sector
in which no radiating waves exist increases
rapidly from zero to more than 90 degrees.
This leads to the conjecture that, in the
range of ~1/4, there exist the short waves
which originate from the cusp part of the wave
pattern and propagate in the transverse direc
tion of the waterway. These waves reflect on
the side walls and may exert a complicated
influence on hydrodynamic forces on a ship. In
the numerical results of the addedmass and
damping coefficients shown from Fig.2 to
Fig.5, we can observe fast variations in the
short range of the wavenumber approximately
between KL=7.3 and 7.8. The parameter ~ cor
responding to these wavenumbers takes the
values ranging from 0. 27 to 0. 28. Therefore
the fast variations in the addedmass and
damping coefficients might be attributed phys
ically to a contribution of short waves origi
nating from the cusp part of the wave pattern.
As the motion frequency increases across
the range where the fast variations occure,
the effects of side walls gradually decrease,
and the addedmass and damping coefficients
reduce to the corresponding values in open
sea shown by shortdashed lines around the
nondimensional wavenumber KL=14. O. In this
range, i.e. between KL=8. 0 and 14 .0, the di
vergingwave component may be dominant in the
sidewall effects on hydrodynamic forces on a
ship.
We have a conventional diagram [1] which
can be used to judge whether the sidewall
effects are expected or not, by means of the
parameter ~ and the ratio of tank width to
ship length BT/L. In the present calculations
the ratio of BT/L is 2.0 and thus the critical
angle of a sector, Fc' is given as
Oc = tan~: (BT/L) = 63.4 deg.,
where the critical angle Fc is determined
geometrically such that the wave emitted from
the ship bow will strike the afterbody of the
ship by the reflection from tank walls. This
critical sector angle, on the other hand, is
estimated from the calculations of the wave
pattern generated by an oscillating and trans
lating source [14], and is depicted in the
diagram as a function of ~. Using this diag
ram with the critical angle Fc as the input,
we get ~=0.365 as the predicted critical fre
quency. In the frequency range lower than this
point the sidewall effects will be expected.
For Fn=O.l, the value ~= 0.365 gives the crit
ical wavenumber of KL= 13.3. Looking at the
computed values shown in Figs. 25, this crit
ical wavenumber turns out to be a good approx
imation.
5.2 Accuracy check and validation
The items to be checked for the accuracy of
the present calculations are the 2D solution
in the transverse plane, numerical evaluation
of the kernel function (25~~27) and (29), and
the solution of the integral equation (42~.
Since the 2D boundaryvalue problem is well
posed, no discussion is needed. Haraguchi &
Ohmatsu's method [15] is utilized in the
present work, which easily get rid of irregu
lar frequencies and give an accurate solution.
507
OCR for page 499
Numerical integrations in (25~~27) and (29)
are, as described earlier, performed using
Clenshaw & Curtis quadrature, with an abso
lute convergence requirement of 10s applied.
Therefore the remaining thing to be checked is
the accuracy of the solution of integral equa
tion (42~.
Fig.6 presents the addedmass and damping
coefficients when the number of divisions in
the xdirection NX was changed from 10 to 70,
under the same computational conditions as in
Figs. 25. Computed results are plotted with
the values of NX=70 set to 1.0. The upper
results in Fig.6 are for KL=5.0 (~=0.224<1/4),
and the lower ones are for KL=10.0 (~=0.316
>1/4~.
For ~=0.224, all computed coefficients ap
pear to converge as the number of division
increases, although the coefficients associ
ated with the pitch mode dictate a finer dis
cretization relative to that necessary for the
heavemode calculations. (Here we note that
the relative error in Bss might be noticeable
but its absolute error is not so large, be
cause the value itself is small at KL=5.0 as
seen in Fig.5.)
Prolate Spheroid of L/B=8
in waterway of BT/B=16
1 If
_ ~
r
1.0
0.95
1.05
1.0

0.95
B55 ,' "`
$,, "
: B33 "`
\
, ~ ~
 ~ ~
/
. . . .
10 30
K L = 5.0
Fn = 0.1
~ = 0.224
Values of NX=70
are set to 1 .0
l
50 NX 7 0
"~x B55
B33 .+ ~
_
A55
1_ . , , ,
10 30
K L = 1 0.0
En =0.1
~ =0.316
Values of NX=70
are set to 1 .0
A 70
Fig.6 Number of division in the xdirection
(NX) vs. addedmass and damping coef
ficients of a prolate spheroid, at KL=
5.0 (upper) and KL=10. 0 (lower).
Results are plotted with the values at
NX=70 set to 1.0.
508
For ~=0.316, the results at NX=60 are
slightly different from those at NX=50 or
NX=70 with relative error of approximately
1.0 %. This suggests that the solution of the
integral equation (42) tends to be unstable as
the value of ~ increases beyond 1/4. Sclavou
nos t10] has found this kind of instability
occurs in the unified theory for large values
of I, and proposed an alternative scheme,
using the Chebyshevpolynomial expansion for
the unknown source strength. However in the
range of ~ calculated here the instability
seems not so serious, and Fig.6 reveals that
40 segments along the ship's length are suffi
cient to give a solution with relative error
less than 2.0 %. On this basis, all of the
computations shown in this paper have been
carried out with NX=40.
In the special case of zero forward veloci
ty, 3D "exact" calculations based on the
integralequation method may be available,
with the Green function modified to satisfy
the sidewall boundary condition. A 3D cal
culation method of this kind has been develop
ed by Kashiwagi ~7], in which almost perfect
agreement is shown between the calculated and
experimental values for a hemisphere and a
ship model with foreandaft symmetry. If we
compare the zerospeed results computed by the
present theory with the corresponding ones by
the 3D integralequation method, the valida
tion for less complicated case can be accomp
lished.
Fig.7 and Fig.8 present respectively the
heave addedmass (A33) and damping (B33)
coefficients, for a prolate spheroid of L/B=8
floating at zero forward speed in the waterway
of BT/B=16. Similarly Fig.9 and Fig.10 show
the pitch addedmoment of inertia (Ass) and
damping (Bss) coefficients under the same
conditions. In these four figures, the same
scale of the ordinates and the same line sym
bols are used as those in the corresponding
figure for the forwardspeed case shown from
Fig.2 to Fig.5. Also included in Figs. 710
are the results of 3D integralequation
(panel) method, which are presented by plus
symbols for the case of open sea and by open
circles for the case of sidewall effects
present. A certain amount of inaccuracy should
be expected in the results of 3D panel meth
od, too. However numerical accuracy is be
lieved to be fairly good, because the hull
surface of spheroid and the normal vector on
it can be mathematically given and some ana
lytical manipulations are thus used to improve
t'ne numerical accuracy. We see that very good
agreement exists between the results of the
slendership theory and of the 3D panel
method, showing the validity of the present
theory.
We proposed a new rational theory for
predicting the hydrodynamic forces on a ship,
moving at constant forward speed and oscilla
ting in heave and pitch in a restricted water
OCR for page 499
1
o 
1
s
Prolate Spheroid L/B=8 at U=0
in Open Sea
 St rip Theory
 Slender Ship Theory
+ 3D Panel Method
with SideWall Effect ( BT/B=16 )
i?  Slender Ship Theory
'TV I
~~
~ 1
0 3D Panel Method
7_ .
C~ BI ~
/ KL
Fig.7 Heave added mass of a prolate spheroid,
the same as Fig. 2 except for Fn=O.O.
Comparison with the 3D panelmethod
predictions.
0.05
6
)0 2
~ J
All 15
Ass/PVL Prolate Spheroid L/B=8 at U=0 B55/pVL
0.1 5 . in Open Sea
  Strip Theory
~ ~ Slender Ship Theory
0.10
j + 3D Panel Method
\ with SideWall Effect (BT/B=16)
i. Slender Ship Theory
0 3D Panel Method 0 2
~'~ I;__
O _
0.3
10 KL 15
Fig.9 Pitch added moment of inertia of a
prolate spheroid, the same as Fig. 4
except for Fn=O. O. Comparison with the
3D panelmethod predictions.
10 KL
Fig.8 Heave damping coefficient of a prolate
spheroid, the same as Fig. 3 except for
Fn=O. O. Comparison with the 3D panel
method predictions.
Hi
1 1
10
KL
15
Fig.10 Pitch damping coefficient of a prolate
spheroid, the same as Fig. 5 except
for Fn=O. O. Comparison with the 3D
panelmethod predictions.
509
OCR for page 499
way. Only the slenderness of the ship hull is
assumed, and thus the proposed theory is valid
for all frequencies and forward velocities of
practical interest where the sidewall effects
are prominent. Furthermore the theory is cor
rect even for the case of a narrow waterway,
because the sidewall effects are taken into
account not only on outgoing waves but also on
evanescent local waves.
Validity of the proposed theory is confirm
ed for the special case of zero forward velo
city by comparison with the numerical results
of 3D panel method. In the case of nonzero
forward velocity, however, the present theory
may be the first one which is able to give
precise predictions of the sidewall effects
on hydrodynamic forces. Therefore with this
theory, we are ready to make quantitative
discussions on the effects of tankwall inter
ference included in the results of experiments
for a ship model.
Computations were performed for the heave
and pitch addedmass and damping coefficients
of a prolate spheroid of lengthbeam ratio
8.0, moving at the Froude number 0.1 in the
waterway of width twice the spheroid's length.
The computed hydrodynamic forces show complex
variations as the frequency increases. It is
noted that these variations correspond to the
appearance of complicated wave pattern, which
starts from the pattern dominated by the ring
wave and changes to the markedly different one
dominated by the diverging wave, as the param
eter l=mU/g increases across the critical val
ue 1/4.
References
1. Vossers, G. and Swaan, W.A., "Some sea
keeping tests with a victory model", I. S.
P. Vol.7, No.69, pp.189206 (1960~.
2. Hanaoka, T., "On the sidewall effects on
the ship motions among waves in a canal",
J. Soc. Nav. Arch. Japan No.102, pp.l5
(1958~.
3. Hosoda, R., "Side wall effects of towing
tank on the results of experiments in waves
(1~", J. Soc. Nav. Arch. Japan No.139,
pp.2330 (1976~.
4. Hosoda, R., "Effect of sidewall inter
ference of towing tank on the results of
experiments in waves (2~", J. Soc. Nav.
Arch. Japan No.143, pp.5260 (1978~.
5. Takaki, M., "Effects of breadth and depth
of restricted waters on longitudinal mo
tions in waves", J. Soc. Nav. Arch. Japan
No.143, pp.173184 (1979~.
6. Newman, J.N., "The theory of ship motions"
Adv. Appl. Mech. Vol.18, pp.221283 (1978~.
7. Kashiwagi, M., "3D integralequation me
thod for calculating the effects of tank
wall interference on hydrodynamic forces
acting on a ship", to be published in J.
Kansai Soc. Nav. Arch. Japan No.212 (1989~.
S. Kinoshita, T. and Saijo, K., "On the mul
tibodytype floating breakwater", J. Soc.
Nav. Arch. Japan No.149, pp.5464 (1981~.
12 .
510
9. Newman, J.N. and Sclavounos, P.D., "The
unified theory of ship motions", Proc. 13th
Symp. on Nav. Hydrodyn. Vol.4, pp.122
(1980~.
10. Sclavounos,
body theory:
15th Symp.
(1984~.
11. Sclavounos, P.D., "The diffraction of free
surface waves by a slender ship", J. S. R.
Vol.28, No.1, pp.2947 (1984~.
12. Timman, R. and Newman, J.N., "The coupled
damping coefficients of symmetric ships",
J. S. R. Vol.5, No.4, pp.3455 (1962~.
. Ogilvie, T.F. and Tuck, E.O., "A rational
strip theory for ship motions", Dept. Nav.
Arch. Mar. Eng., Univ. Michigan, Rep.No.13,
pp.192 (1969~.
14. Hanaoka, T., "On the velocity potential in
Michell's system and the configuration of
the wave ridges due to a moving ship", J.
Zosen Kiokai, No.93, pp.l10 (1953~.
15. Haraguchi, T. and Ohmatsu, S., "On an im
proved solution of the oscillation problem
on nonwallsided floating bodies and a new
method for eliminating the irregular frequ
encies", Trans. WestJapan Soc. Nav. Arch.,
No.66, pp.923 (1983~.
P.D., "The unified slender
ship motions in waves", Proc.
On Nav. Hydrodyn., pp.177192
OCR for page 499
DISCUSSION
by C.M. Lee
I think that it is the first paper which
presented a 3D theory for an oscillating
ship, including the sidewall effects. The
authors should be congratulated for their
excellent work.
Similar to the case of twinhull ships,
this paper shows negative added mass at
certain frequencies. This kind of phenomenon
does not occur in the opensea case, and,
therefore, induces puzzlement to those who
cannot accept the motion of "negative added
mass". My advice to those people has been
that one should not get too excited by just
observing unusual hydrodynamic coefficients
alone but should wait until the computed
results of ship motion in waves are shown.
My prediction is that although the
hydrodynamic coefficients may look quite
different from those of the opensea case, the
motion results may not show significant
differences, particularly for the tank width
being twice the ship length as chosen in the
sample calculations in this paper. I would
like to encourage the authors to compute the
ship motion to check if my prediction is
correct.
Author's Reply
Thank you for your comment. We are now
applying the proposed theory to the
diffraction problem with sidewall effects.
If the calculations of waveexciting force and
moment are completed, the ship motion in waves
can be readily computed from them, using the
addedmass and damping coefficients predicted
by the present theory. Therefore, I think
that the computed results of ship motions in
waves can be shown in the foreseeable near
future.
511
OCR for page 499