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OCR for page 499
Side-Wall 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 slender-ship theory is presented
for predicting the side-wall effects on the
added-mass 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 side-wall boundary condition in the
far-field solution is acheived by the method
of mirror images, with a closed-form expres-
sion obtained of the resultant infinite se-
ries. The inner expansion of the far-field
solution dictates the asymptotic behavior of
the near-field solution, thereby determining
the near-field homogeneous component. This
component accounts for the side-wall inter-
ference in the inner region as well as the
hydrodynamic interactions along the ship's
length.
Computed added-mass and damping coeffici-
ents are presented for a half-immersed prolate
spheroid. Validity of the proposed theory is
confirmed by the comparison with the 3-D 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 side-wall 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 side-wall 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 thin-ship
theory is a rational theory of the side-wall
effects, but did not give reliable information
on the effects. Hosoda ~3,4] and Takaki t5]
are based on the strip-theory 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 slender-ship 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 side-wall 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 open-sea problem. Namely it can be const-
ructed by the superposition of the particular
solution given by the strip theory plus a
homogeneous solution giving three-dimensional
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 side-wall 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 3-D wave sources with unknown
strength along the ship's length. The veloci-
ty potential of 3-D wave source satisfying the
side-wall boundary condition is derived by
considering an infinite number of image singu-
larities and a closed-form 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
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Representative terms from entire chapter:
forward speed
integral equation for the strength of 3-D wave
sources, the solution of which then settles
the coefficient of the near-field homogeneous
solution; thereby completing the velocity po-
tential necessary for the calculations of
added-mass and damping coefficients.
Solving the integral equation obtained re-
quires the numerical evaluation of Cauchy's
principal-value integrals involved in the ker-
nel function representing the side-wall 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 non-singular 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 10-5.
Computational results are presented of the
heave and pitch added-mass and damping coef-
ficients, for a half-immersed prolate spheroid
of length-beam ratio 8.0 advancing at a Froude
number 0.1 in the waterway of width twice the
spheroid's length.
The appearance of side-wall 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 diverging-wave component, as
the parameter ~ increases across the critical
value 1/4. Corresponding to this complicated
variation of the wave pattern, the added-mass
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 3-D
panel method, for example, by Kashiwagi [73.
The results of the present theory agree excel-
lently with the 3-D panel-method 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-
low-water effect in the water-wave phenomena.
A coordinate system used is shown in Fig.1.
The x-axis is coincident with the centerline
of the waterway and positive in the direction
of the ship's forward velocity, the y-axis is
Side Wall
BT
1' L 'I
—B: ~ X
.............................................................
Side Wall
~ AX
z
Fig.1 Coordinate system and notations
horizontal, and the z-axis 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 xj-axis with
extended definition of ns=znl-xn3. mj is the
so-called m-terms 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
three-dimensional boundary-value problem, we
exploit a slender-ship 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
x-axis, and thus the flow field is insensitive
to the details of ship's hull geometry. There-
fore the outer problem is defined by the 3-D
Laplace equation [L], subject to the free-
surface iF], radiation ~R], sea-bottom [B],
and side-wall tW] boundary conditions, that
is, (3)-(6~. Since these boundary conditions
are homogeneous, the outer solution can be
described in terms of the 3-D Green function
with unknown wave sources distributed on the
x-axis, in the form
501
~j~x,y,z) = JLqj(~)G~X-t,y,Z)d; (9)
Here G~x,y,z) denotes the Green function or
the potential of a "translating-pulsating"
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 e-KIy-pBT|V~dn
p=_=
-~sgn~ 1+2kl) e
·£ e-iSgn~l+kl)~|y-pBT|/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 right-hand side of
each equation vanishes and only the first term
remains. Thus the side-wall 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 open-sea
Green function Go~x,y,z) plus the side-wall-
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 3-D
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 side-wall 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
open-sea 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) + ~ (l-Kz~fo~k)
where + O(Kr~l-v),K2r2) (19)
G2D(y,z) = Go(°;Y'Z)
- ~ {(l-Kz)(logKr+y)
+ Kr~cosO+OsinO)}-i~l-Kz) (20)
502
fork) = logl 21 + Hi
~ ~ (cosh~~(lv~+nisgn: l +k~ )}
L ~k2 / 2 { COS-\ ~ I V I )_~}
(21)
In (19) and (20), two-dimensional 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 side-wall-effect part
of the Green function can be obtained with
comparatively simple reduction, in the form
iE 1 *
GT(k;y,z) = ~ (l-KZ)fT(k)
where + O(Kr~l-v),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/V2-l ~)
_ (23)
As in (10), the upper and lower expressions in
brackets in (21) and (23) correspond to | k |
where
The inverse Fourier transform of the side-
wall-effect part (22) can be expressed as
GT(x,y,z) ~ rr (l-KZ)dd FT(KX) (28)
FT(X) = 2~ ,: e ikX dk/k
·: 2n 2 { -l+coth( 2T~)} ins +kZ
+ ) Z · Z m - sGn(i+kT) e-ikX
2 KBT p-1 m_O [kldv/dk-k/vl ~k=kpm
2 [ ~k2+:k4]
IT
· cot( 2 1//l-k2/v2)dk/k
i ~ {k2 (key -ikX 1
+ 2 L Jkl Jk3J ,/k2/V2-l
. coth(—2ik2/v2-1 )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 right-hand side of (29)
represents the contribution of non-radiating
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 (m-O,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 hydrodynamic-force 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
side-wall effects, this term will contribute
to the inertia force. The integrals in the
third term must be treated as Cauchy's princi-
pal-value 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 open-sea 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)
- ~ ( 1-Kz )lLqj (A )dE,: [Fo{ K(x-: )}
+ FT{K(x-E,)} id: (31)
2.2 The inner problem
Since the ship hull is assumed slender,
changes of the flow in the x-direction 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 2-D Laplace equation subject
to the free surface condition which is inde-
pendent of forward velocity and applicable to
the 2-D problem in the y-z 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 side-wall
boundary condition can not be specified, be-
cause the side walls are absent in the inner
region. With these taken into account, the
boundary-value 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 2-D transverse plane,
and Nj and M denote the slender-body approxi-
mation of tie 3-D quantities in (7) and (8),
which are explicitly given as
N5 = -xN3 ~
M3 = -N2Xzy~N3Xzz ~ (35)
M5 = N3-XM3 J
The inner boundary-value problem defined
from (32) to (35) is the same as the conven-
tional 2-D formulation except that the radia-
tion condition is absent. Thus the inner
solution can be identified with Newman's uni-
fied-theory solution t6], composed of a par-
ticular solution commonly used in the strip
theory plus a homogeneous solution multiplied
by a three-dimensional-effect 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
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 2-D effective
source strengths; these can be given by sol-
ving the 9~- and -problems respectively.
G2D(y,z) is the 2-D 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 3-D 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 Kfx-t )} +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 3-D source
strength q jinx) of the form
q jinx) - 2~ t(>j~x)/oj~x)-l]J.Lqi(~)
· d<; ~Fo{Kfx-E~}+FT{Kfx-E~ 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 side-wall 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 slender-ship theory in the
open-sea 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 multi-body-type 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 3-D source strength qj~x), which includes
the side-wall effects as a solution of (42)
the coefficient of homogeneous solution Cj (x ~
accomodates not only the 3-D interaction ef-
fects between transverse cross-sections but
also the side-wall effects of the waterway.
3. Added-mass 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 i-th direction due to the j-th
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 added-mass (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 2-D added-mass and damping coefficients
respectively, involving the 3-D interaction
effects and the side-wall 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 open-sea
case the relative importance of the contribu-
tions frog the M3-term and related velocity
potential 95, by comparing the numerical re-
sults with experiments. His results reveal
that the inclusion of the M3-term leads to a
substantial overprediction of the damping co-
efficients. This overprediction may be at-
tributed to the inaccuracy of the m-terms near
the ship ends, which have been evaluated with
slender-body approximation. Therefore the m-
terms should be evaluated from the 3-D 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 m-terms in the
unified theory. Thus in the numerical calcu-
lations with side-wall 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
unified-theory corrections in the open-sea
case and for the effects of side-wall 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
3-D source strength q (x). For this purpose,
after dividing the swipes longitudinal axis
into NX segments of equal length, the 2-D
boundary-value 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 2-D 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 3-D 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
~ (x-xk 1 ~ / (Xk-Xk- 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
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 side-wall effects, computed by the slender-
ship theory described in this paper. In order
to show the magnitude of side-wall effects,
the values in open sea are shown by short-
dashed lines, which were obtained with the
side-wall-effect part of the kernel function
Fly Kfx-; )} set to be zero in the integral
equation (42), and therefore must be identical
to the unified-theory solutions t94. Also
shown in the open-sea case are the strip-the-
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/g-FniKL 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 non-negative 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 damping-force 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 tank-resonant mode in the transverse di-
rection can be given by KBT=2nm (m=1 , 2, · · ~ and
thus in the present case by KL-nm ; 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 tank-resonant frequency is shifted to
the lower frequency than the zero-speed 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 added-mass 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 added-mass 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 added-mass and damping coefficients
reduce to the corresponding values in open
sea shown by short-dashed lines around the
nondimensional wavenumber KL=14. O. In this
range, i.e. between KL=8. 0 and 14 .0, the di-
verging-wave component may be dominant in the
side-wall effects on hydrodynamic forces on a
ship.
We have a conventional diagram [1] which
can be used to judge whether the side-wall
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 side-wall 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. 2-5, 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 2-D 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 2-D boundary-value 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
Numerical integrations in (25~-~27) and (29)
are, as described earlier, performed using
Clenshaw & Curtis quadrature, with an abso-
lute convergence requirement of 10-s applied.
Therefore the remaining thing to be checked is
the accuracy of the solution of integral equa-
tion (42~.
Fig.6 presents the added-mass and damping
coefficients when the number of divisions in
the x-direction NX was changed from 10 to 70,
under the same computational conditions as in
Figs. 2-5. 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
heave-mode 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 x-direction
(NX) vs. added-mass 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 Chebyshev-polynomial 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, 3-D "exact" calculations based on the
integral-equation method may be available,
with the Green function modified to satisfy
the side-wall boundary condition. A 3-D 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 fore-and-aft symmetry. If we
compare the zero-speed results computed by the
present theory with the corresponding ones by
the 3-D integral-equation method, the valida-
tion for less complicated case can be accomp-
lished.
Fig.7 and Fig.8 present respectively the
heave added-mass (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 added-moment 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 forward-speed case shown from
Fig.2 to Fig.5. Also included in Figs. 7-10
are the results of 3-D integral-equation
(panel) method, which are presented by plus
symbols for the case of open sea and by open
circles for the case of side-wall effects
present. A certain amount of inaccuracy should
be expected in the results of 3-D 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
slender-ship theory and of the 3-D 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-
1
o -
-1
s
Prolate Spheroid L/B=8 at U=0
in Open Sea
- St rip Theory
---- Slender Ship Theory
+ 3-D Panel Method
with Side-Wall Effect ( BT/B=16 )
i? - Slender Ship Theory
'TV I
~~-
~ 1
0 3-D 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 3-D panel-method
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 + 3-D Panel Method
\ with Side-Wall Effect (BT/B=16)
i. Slender Ship Theory
0 3-D 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
3-D panel-method 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 3-D 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 3-D
panel-method predictions.
509
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 side-wall effects
are prominent. Furthermore the theory is cor-
rect even for the case of a narrow waterway,
because the side-wall 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 3-D panel method. In the case of non-zero
forward velocity, however, the present theory
may be the first one which is able to give
precise predictions of the side-wall effects
on hydrodynamic forces. Therefore with this
theory, we are ready to make quantitative
discussions on the effects of tank-wall inter-
ference included in the results of experiments
for a ship model.
Computations were performed for the heave
and pitch added-mass and damping coefficients
of a prolate spheroid of length-beam 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.189-206 (1960~.
2. Hanaoka, T., "On the side-wall effects on
the ship motions among waves in a canal",
J. Soc. Nav. Arch. Japan No.102, pp.l-5
(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.23-30 (1976~.
4. Hosoda, R., "Effect of side-wall inter-
ference of towing tank on the results of
experiments in waves (2~", J. Soc. Nav.
Arch. Japan No.143, pp.52-60 (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.173-184 (1979~.
6. Newman, J.N., "The theory of ship motions"
Adv. Appl. Mech. Vol.18, pp.221-283 (1978~.
7. Kashiwagi, M., "3-D integral-equation 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-
ti-body-type floating breakwater", J. Soc.
Nav. Arch. Japan No.149, pp.54-64 (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.1-22
(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.29-47 (1984~.
12. Timman, R. and Newman, J.N., "The coupled
damping coefficients of symmetric ships",
J. S. R. Vol.5, No.4, pp.34-55 (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.1-92 (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.l-10 (1953~.
15. Haraguchi, T. and Ohmatsu, S., "On an im-
proved solution of the oscillation problem
on non-wall-sided floating bodies and a new
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DISCUSSION
by C.M. Lee
I think that it is the first paper which
presented a 3-D theory for an oscillating
ship, including the side-wall effects. The
authors should be congratulated for their
excellent work.
Similar to the case of twin-hull ships,
this paper shows negative added mass at
certain frequencies. This kind of phenomenon
does not occur in the open-sea 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 open-sea 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 side-wall effects.
If the calculations of wave-exciting force and
moment are completed, the ship motion in waves
can be readily computed from them, using the
added-mass 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
