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Prediction of Radiation Forces
on a Catamaran at High Froude Number
M. Ohkusu (Kyushu University, Japan)
O.M. Faltinsen (Norwegian Institute of Technology, Norway)
ABSTRACT
Practical approach is investigated to predict three
dimensional hydrodynamic interaction between
two hulls of a catamaran oscillating and running
at forward speed. Chapman's approach is tried
to solve the boundary value problem for the un-
steady velocity potential around twin hulls' sec-
tion contour with retaining all the terms of the
full linear free surface condition including for-
ward speed effect. Results show that predicted
hydrodynamic forces agree generally well at high
Froude number with the ~eas~red on a anode 1
situp and it-~leraction between twill hulls is cer-
tainly weak at Ellis high speed.
INTRODUCTION
It may be a practical approach to generalize strip
theories to predict motions of a catamaran ad-
vancing in waves. Strip theories are certainly the
most successful theories for practical purpose of
predicting wave induced motions of ships at mod-
erate forward speed.
For applying strip theories two hulls of the
catamaran are considered to be close to each
other. Two dimensional theory is used to eval-
uate flow around the contour of two sections os-
cillating on the free surface. Experiments tell,
however, that hydrodynamic interaction eRects
between two hulls are not so strong as expected
from ship's lengthwise summation of the 21) ef-
fects (Ohkusu(1971~. This is more so for higher
forward speed. If we assume two hulls are located
away frown each other, theories to be applied must
not be strip theories any more. We need a prac-
tical theory to take into account correctly three
dimensionality of hydrodynamic interaction be-
tween two hulls of the catamaran.
Strip theories account for the effect of for-
war<1 speed in a simple way. The linear free sur-
face condition including forward speed term is
simplified such that the unsteady waves gener-
ated by flee body motions propagate only in the
breadthwise directions of the ship. A more com-
plicated wave system must be considered if the
three di~nensionality of the flow has to be consid-
ered.
'three di~nensinal panel method at forward
speed will be one alternative for taxis purpose.
But even will, recent progress in co~nputatio~al
scenes of the Green function (Ol~kusu and
Iwashita(1989~), it can not be so practial as strip
theories because of long computational tine. Be-
sides we have no valid approach to treat with a
line singularity appearing at the intersection o
the surface piercing body and the free surface.
Chapman's approach (Chapn~an(1976~) is a
simplified, but still satisfying the full linear free
surface condition, high speed theory for a verti-
cal surface-piercing flat plate in unsteady yaw and
sway motion. Daoud's theory (Daoud~ 1975~) for
analyzing stationary flow around the bow of the
ship is also a theory retaining all the terms of the
linear free surface condition in the region close
to the body and solving essentially 2D problem
around section's contour. Ogilvie's description
(Ogilvie (1977)) on these theories is full of in-
sight into hydrodynamics and very informative.
Furthermore one of the present author (Faltin-
sentl983~) presented a theory for solving diffrac-
tion problem at the bow of the ship along a sign
tar line. All those are attractive ideas in the point
that the -derivatives ifs tile free surface condition
is completely retained and still i~urnerical work
is almost on two dimensional problem. Recent
comprehensive analysis by Yeug and Kim (1985)
Nlak`'tc~ ()hkusu. Research Institute fair Plied \iecha~i~ i;' K~ ushu l'nix-.. Fuku`'ka Japan
()cl~-1 N1. Faltinsen. Nc:)rs~-egian Illstit~te off Technology-. Tr ~n~ll~eil:l~ .\;or~.
s
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Representative terms from entire chapter:
strip theories
gives more analytical foundation for Chapn~an's where
approach.
We present in the main text an application of
Chaprnan-Oaoud-Faltinsen's approach for com
puting hydrodynamic forces on the catamaran.
First we invesigate the far-field effect of unsteady
bow flow caused by heave and pitch motions of
the ship. There we describe a set of assumptions
leading to the full linear free surface condition
retaining the ~ derivatives in the near-field. Nu
merical solutions of the 21) boundary problems
for the velocity potentials near the body is a ver
sion of L)aoud's approach. Finally we compare
computed and measured added mass and da~np
ing of a catamaran at high Froude number and
present wave elevation generated by the ship mm
tion between two hulls of a catamaran as well as
in the far-field.
FAR-FIELD SOLUTION
Velocity potential describing the flow around
an oscillating ship at forward speed is expressed
in the form of
~ = US + +5(~ y, a) ~ Army y, Z)ei~t (1)
where the -axis directs astern from the origin at
FP, the z axis vertically upward and the By plane
coincides with the calm water surface (Fig.1~. U
is steady ambient flow velocity in the direction of
the x-axis (or the ship's speed), L the ship length,
circular frequency of motion. Is is the steady
part of the potential and Or the unsteady part.
We start with a simple expression of the sot
lution of Shr valid in the far-field of the ship's hull.
The solution Or satisfying the full linear free sur
face condition
(iw + U `' ) Or + g `' = 0
on z = 0 (2)
is expressed at My = 0(1) by the velocity pm
tential of a line source Vein distributed on
the londituidir~al axis of the ship hull since we
are concerned with the symmetrical flow field. A
Fourier transform type of expressions is given by
1 roe
assume (~-() >> y in this equation, then one
stationary point exists within the interval of each
integral, as far as the 1st and 3rd terms of equa-
tion (5) are concerned. For large T stationary
points Act for the 3rd and K2 for the 1st integral
are approximately
~c~,:~ - 11~ ~ (7)
Diary part of the righthand side of equa-
tion (5) is then evaluated as follows.
I
fir ~ - 2 J a(t,)d:
o
r.~ .h) i I(o(~ _ ()2
x At - exp [-~-(a-A)
1 ko(~-(,)2Z
4 y2
·~
- ~ 4 ]
. ~ , W ~ i(~(X _ ()2
_~ exp[-au(~-()+ 4 y
1 /(o(~ _ ()2Z I]
-
(r + 1/4)1/4 ~eXP[ik1 U (x-()]
/2~lk2~(WiUj 1 exp[7k2u(~)]}
(8)
where To = g/U2. The last two terms of this
equation come from the singularities at end points
k = kit and k2 of the interval (Erdelyi (1957~)
and are higher order than the first two. z de-
pende~ce in the last two terms is ignored because
exp(~z/U) is 1 ~ o(~il2)
If x is away from the bow region such as
~ = C)(l) and ~(~) varies smoothly there, we can
once again apply the stationary phase method
to evaluate tile parts of the integrals (8) from
= (/2) to a. We obtain
~o(~/2)
OCR for page 8
pressure of the second order on sections behind
the bow region.
FORMULATION OF THE SOLUTION
Our general assumptions are:
Al = 0(~), tI2,3 = 0~1), ~ = 0~-2)
(10)
~3 = 0(`f~ ~ ~ Or = 0(~) ( 11 )
where n~,2,3 are the x,y and z components of a
unit normal vector to the wetted part of the hull
surface whose positive direction is into the fluid. £
is the slenderness ratio of the hull geometry and ~
characterize smallness of the oscillatory motions'
amplitude. f is any flow variable caused by the
body in some region near the hull.
'the unsteady part of flow jr is rather
straightfowardly linerized based on ~ that is gen-
erally independent of the hull geometry. The
steady flow ¢, and its interaction with the un-
steady part are related closely to the hull geome-
try and their linearization is strongly dependent
Ott flow characteristics we assume. We have two
alternatives in order to retain U0/~ term in the
free surface condition:
(i) up = o(~-~12), of/ = 0~1)
(ii) U = 0~1), I/ = to-.
(i) gives ¢, = o(~3/2) and steady wave ele-
vation ¢, = 0~. The result is that we can not
transfer the free surface condition to z = 0 be-
cause of ¢~5/0Z = 0~14; the free surface condi-
tion for ¢, becomes non-linear.
(ii) may be justified with some assumptions
on the flow characteristics close to the bow as
mentioned in the previous section. Recently
Faltinsen and Zhao (1990) employed another al-
ternative no = o(~/2) to analyze the same prob-
lem. With this assumption, however, we once
again end up with a non-linear free surface condi-
tion for the steady flow ¢5. In this context jr ho
to satisfy the free surface condition not on z = 0
but on the steady free surface displaced finitely
from z = 0.
Assume (ii) as well as (10) and (11), then we
obtain ¢, = o`~2y,<~ = ote3/2) with which we
have no trouble to linearize the free surface con-
ditio~ for As. If we retain tile lowest order terries
with respect to ~ and it, tiled we reacts to tile
following linearized free surface con
free surface elevations equal to zero at x < 0. Rea-
son of taxis assurnptio~ is that no upstrea~n waves
are generated far in front of the bow because we
are concerned with the case of T >> 1/4. \Ve
assume here simply
the lowest order terms retained in (16~. Some of
results in which we keep ever the raj terns inco-
sistently will be presented.
Chapman (1975) presented a simplified high-
speed theory satisfying the full linear free surface
condition in the near field for a surface pierc-
ing flat plate in yaw and sway motion problems.
We apply a generalized and liniearized version
of his idea to solve the boundary problems (12),
(13), (14), (15) and (16~. In details we follow
the approaches proposed by Ogilvie (1977) and
Daoud (1975) in analyzing bow flow for the wave
resistance problem. This method is economical
in the computational effort because orate identical
scheme can be applied for both the steady and
the unsteady problems.
lt isconvenientin the analysts to define new ¢,r= / ~(iQLr(J:;7l~l:~°
Arm y Z) = ei(~/U)2~r(X y Z)
<,r = 0 ~ Fir
- 0 ate ~ 0 (22)
We will now introduce an expression for the
solution fir as suggested by Ogilvie (1977~.
J L(2)+R(= )
4~/~; d; J d'Er(~;71~),¢~)
o L(~)+R(~)
(18) too
x / dw exp~w2(z + ¢(, cosw2(y ~ t,))
o
Without any simplification the boundary x sin To W(X-()
value problems for both As and &6r then become
Flee similar problems with the only difference in
the body boundary conditions. For brevity we
do not state the boundary value problems for 45
hereafter.
19 tiler + t9 fir = 0 (19)
u2 jar + 9 '~¢ = 0 Off z = 0 (20)
Alar
ON
I ei(~/U)X[iW1~3-U0N ~ ]
for j = 3
ei(~/U)x[-in-L/2)
U(-n3 + (A-L/2) ON Liz )]
for j = 5
(21)
on the body surface below z = 0.
We set starting condition on fir and 0¢r/0x
at x=0. We Essence the velocity potential and the
9
where
(23)
r, r' = >/(y _ 77)2 Jr (Z IF ¢)2
L(x) and it(x) are the contours of the cata-
~naran's left and right sections at x ~ we ignore
the contribution front L(x) for a single hull ship).
We assu~ne<1 ilk tItis fortnulatio~ that each de'~i-
hull of tile catamaran locates in the near field of
each other.
Equation (23) implies that tar at ~ is repre-
sented by the potential caused by a distribution
Yr of impulsive sources on that section contours
am the effect of source distributional at all sec-
tions upstream of a. One may use Daoud's ex-
pressio~ (1975) for the velocity potential instead
of equation (23~. He distributed sources and nor-
r~al dipoles based on snore systematic derivation.
But we can transform his expression into that of
only the source distributions without difficulty by
considering another boundary value problem on
the flow interior the body. So there is no reason
that the expression (23) is not appropriate.
Contribution from the line singularities at
the intersection of the body and the free surfaces
is consi(le~re(1 to be of higher order for slender hull
form based on Daoud's arguement (1975~. We re-
~nark that if we had to include really this term,
one Galore condition such as the least singularities
of flow in addition to the body boundary condi-
tion should be imposed in order to determine the
strength of this line singularities. We do not know
what correction is physically correct.
The last term of equation (23) is the com-
plicated triple integral including the unkown Or
in its integrand. But this term gives the effect
only from the sections upstream of the section and
does not depend on the source density at ~ _ ~
where the integral equation is to be solved, be-
cause of the term sin I/ wax-Hi.
Starting from x=O, we can solve this equa-
tion step by step. Details of numerical computa-
tions are described in the following section.
When we let y ~ oo on the equation (23), a
far field approximation of the near field solution
is to be obtained as follows.
blur = e i(~/U)r A'
~ +; d(S(~;~)~Ko
X expt-it W ~,~ _ ~ To (x-(~2
1 It'o(~-(Liz
+ 4 y2
+i~ dust;
x expt-it w ~y + ~ Ifo~x _ {~2
1 I(o~x-(~2Z
+ 4
where
~ = lKo(z-() (25)
This leads to
Sin; ~) = / di Arty; 9' ()e~Z cos Fly-0)
R(~)
/ d[~r(~;77,() for y ~ oo
R(~)
(26)
Comparison of expressions (8) and (24) sug-
gests that ~) describing the far field solution is
determined from E(x) by the following relation.
U ~4 ~ /R(~) ~ ; 77' (I)
(27)
Dynamic pressure PeiWt linear to the ampli-
tude of motion os will be derived by liernoulli's
formula as follows.
P =
-ph exp(-iw/Ux)U 9~ ajar
-pt exp(-iw/Ux)t ,~ ,~ + ~9 ~ ]
(28)
in this equation we did not include the static
pressure component induced on the hull surface
by the displacement of the body in the non-
uniform steady pressure field. This component
will be measured independently of the dynamic
pressure in our experiments.
Once again there are different order terms in
the right hand side of equation (28~. The lowest
of them is the first line. The second line is ci/2
(24) higher order than the first and should not be in
cluded in the computation of forces on the body,
while we include this terns in some cases. In our
computation an identical scheme is employed to
solve the unsteady problem as well as the steady
problem and ¢, is always available when Or iS cal
culated. inclusion of the second term in the com
putation of the forces does not need more com
putational effort.
Pressure integrated on the body surface gives
the forces Fjtj = 3, 5) on the ship.
Fj = - | d
OCR for page 11
NUMERICAL COMPUTATION
'lihe source density must be determined such
that the body boundary condition (21) is satis-
fied. We rewrite the second term of equation (23)
into the discrete source density form: we divide
each section contour of the catamaran into M
pieces of segments on each of which the density
is assumed constant Er~x;j). After integrating
analytically the constant source density ore each
segment, the normal derivative is taken. Then we
have-an integral equation
ON IN /~(~)+R(~) r
~2N
7rIfoJ d.( Er(~;j)
O j=1
x Re [ - (ny + it ) exp(-id )
W(Zj+1 )
Y
~ VIZ + ¢j+1 I + i(y - q)
~ 1 1 ~ ~ · ~
+ (ny-inz ) exp(iC~; )
X (- W(Zj+l)
VIZ +¢i+ll - i(y - ~j+l)
W(Zj ) )]
VIZ + ¢j l - i(y - Rj)
(30)
where (17j,¢j) and (~j+1,¢j+1) are the coordinates
of the end points of the j-th segment, Re denotes
the real part and
Z - 1 -) (31)
J 2~1z+¢jl+i~y-0j)
CXj = tan-1 Rj+1-71: (32)
¢j+1 - ¢j
where Zj is the complex conjugate of Zj. wit)
is Error function for complex arguements defined
as
wit) = em erfc~-iz) (33)
In the derivation of (30) we assumed wall
sided hull form and the segment closest to the
free surface is vertical. This leads to
¢w('Z: )( ) ~ ( it, z, ¢j = 0
(34)
This singularity may not cause serious prob-
lem in solving the integral equation (30) numer-
ically. Efficient program to compute Liz) at
accuracy of 12 digits is available (Iwashita and
Ohkusu (1989)) in which several different expres-
sions, continued fractions, asymptotic expansions
and finite series approximations, are combined
the most efficient way.
mj, that we need to compute when we retain
the second order terms in the body boundary con-
ditions will not be so seriously singular close to
the free surface, provided the hull form is wall
sided. It is not difficult to show
0,'¢3 = 0~1),
z,,0 ¢3 = 0~1) (35)
Wave elevation Or in the region near to
the hull surface is computed with numerical
-derivatives of the solution Or onz = 0.
Or = -- (him + U ~9 ~texp~-i U x~fr]z=o
1 ,~ 4'r(X + Az)-jr(X)
--expel-Mix) Ax
Both hulls of a catamaran are considered to
be within the near field of each other, the source
density On both contours must be symmetrical
with respect to fez plane. Each section contour is
divided into a number of segments on which the
source strength is assumed constant and the inte-
gral equation (30) is solved such that the bound-
ary condition is satisfied on the midpoint of each
segment. '['his implies that upstream effect rep-
resented by the second term of (30) is evaluated
on the midpoint.
'lihe ship length is divided too into a num-
ber of strips with thicknes of Ax. Assuming the
source is concentrated at the center of /`x we step
the integral equation (30) to next section. Of
course only the sources located upstream have ef-
fect on the current section; the sources from O to
x-/\x have the effects.
11
\Ve tested our method using only the i~npul-
sive sources ageist results of Daoud (1975) and
Faltinsen ( 1983 ) on stationary waves around a
wedge. Daoud solved the integral equation on
the velocity potential on the contour of each sec-
tion instead of the source strength. laltinsen ex-
pressed the velocity potential around a section
with the fundamental sources distributed on the
free surface as well as on the section contour. The
conditions are imposed on the free surface, on the
contour and at infinity. He solved the integral
equation on the velocity potential and step the
free surface condition to next section with using
the dynamic arid the kinematic free surface con-
ditions.
Wave elevations around a wedge for a half
angle 7.5 degree are compared in Fig.2. Results
of our method agree well with their results.
In our method magnitude of /\x will have
essential effect on the accuracy of results because
we employ the concentrated sources in the x-wise
and we are assuming fast variation of the solu-
tion in the x direction. Added mass coefficient
for heave and added moment of inertia for pitch
of a hull form (one of twin hulls of a catamaran
model later described) were computed with in-
creasing the number of strips N frown 20 up to
60 as shower in Figs.3 and 4. Results of compu-
tation without raj seem to converge consistently
and N = '2() gives accurate enough results, while
results with mj, added moment of inertia in par-
ticular, are slow in convergence. This may be
due to the smallness of Ass value vulnerable to
singularity in mj close to the intersection of the
section contours and the free surface. We need a
smaller signet size there to have more consis-
tent results.
Hydrodynamic forces computed with the
present method for a mathematical hull form with
se~ni-circle section contour and water plane form
y = (0.3/L)x(L-x) are compared with those
computed by Newman's unified theory (Newman
(1978~) in Figs. 5 to 8. Results are obtained
with ignoring the terms including (, in the body
boundary condition (21) and retaining only the
first term in the pressure (28~. Actually in pitch
Node the term-Un3 in (21) is retained, though
it is of higher order. In order to evaluate the first
terra of (28) we need to dilferetiate numerically
fir ill the x direction just like in equation (36~.
Unified theory?s results we have shown here
are too without A. The present method and
unified theory predict hydrodynamic forces al
most similar in magnitude, while agreement is
not consistent. Damping of pitch computed by
the present method is much smaller in lovrer fre-
quency. We need experiments to decide which is
better in such a high I)oude number region.
We did forced heave and pitch motion test for
a catamaran whose demihull has a form shown in
Fig.9. Length of the model is lm, the breadth
0.086m, the draft 0.043m and the distance be-
tween the centers of both hulls 0.30m. Heave
or pitch of appropriate magnitude was given on
tile towed model with other Reties suppressed.
Measured force record was analyzed into several
terns of harmonics. Length of records rneasurec1
after stationary state is reached is averagely 10
cycles of motion.
In-phase force measured contain the force
that would act when the ship were displaced from
the original position at w = 0. Ilere we mea-
sured it by giving some displacement to the model
towed on calm water. Assuming magnitude of
this force is approximately linear to amount of
the ~lisplacetnent, we get a spring constant as
illustrated in Fig.14. This component was ex-
cluded frown the measured in-phase forces to ob-
tain added mass. In the frame work of the present
theory, however, difference of this component at
non-forward spee(l and at forward speed in the
non-uniform steady flow field must be considered
to be of the second order.
Results of experiments are plotted in Figs.10
to 1~3. A few of black circles depicted in Fig.11
have dash mark. At these points noise level was
so high with reasons we do not know at present
(magnitude of higher harmonics was as large as
5() percent of the fundamental one) that we are
not confident of their accuracy.
(:on~puted hydrodynamic force with the
present method are shown in those figures. Re-
sults depicted as without rat, which were ob-
tained with ignoring consistently the effects of Us
in equations (16) and (28), predict well the mea-
sured ones at high Froude number 0.5 to 0.89.
One reason of a little dicrepancy will be that the
analysis is not valid at the ends of the ship. In
the case of transom stern like our model we ex-
pect that the flow leaves the stern tangentially in
the downstream direction so that there is ato~n-
ospheric pressure at the last section. This means
that the suns of the added mass forces and hydro-
static restoring force on the last section is zero.
The same is true with the damping force on the
last section. In our analysis the hydrodynamic
2
behaviour at the last section is dependent on the
upstream effect. There is no eRect frown down-
stream. '['his means we have no information that
there should be atmospheric pressure at the last
section. 'the consequence is a wrong prediction
of hydrodynamic forces at the last section. lnclu-
sion of the interaction with steady flow at high
Froude number ~ with end ~ does not seem to i~n-
prove flee correlation between the predicted and
the measured.
Wave elevation between two hulls of a cata-
rr~aran will cause occurence of impact pressure
or the bridge structure connecting two hulls of
a catamaran. We need a practical prediction
method of it. It is well known that strip the-
ory in which we consider two hulls are in the near
field of each other predicts too strong interaction
between them (Ohkusu (1971~) and the predic-
tion of wave elevation is not realistic. Figs.15 and
16 are examples of wave elevation computed by
the present method. '['hey are for the cases of
pitch Notion at Fn = 0.5 and En = 0.89 with
w2L/g = 20. The far left of Figs.15 and 16 is the
midpoint of the distance between two hulls. It
is clear that wave elevation is almost symrnetri-
cal around each hull. This implies that there is
almost no interaction of two hulls if we are close
to the hull. Wave elevation in these examples is
certainly high at I'7n = 0.5 than at I'n = 0.89 .
We can compute the velocity potential and
wave elevation in the far-field generated with a
line source Next determined by equation (27~. In-
formation on the wave field will make it possible
to evaluate added resistance of high speed ships
in waves. L)a~nping is also computed from energy
flux in the far field. A line doublet as well as a line
source will be required to describe correctly the
wave field around a catamaran. Considering low
interaction effect between two hulls in the near
field, two line sources distributed on both hulls
may be enough to give wave elevation correctly.
Short wave components of the waves gener-
ated by a point source will be dominant at high
speed and at position close to the source. Ac-
tually waves by a point source vary dreadfully,
while waves by a line source does not because of
integration of the effects from all sources on a
line. But realizing this smoothing effect correctly
on numerical computation requires very accurate
evaluations of wave elevation generated by each
source. New scheme to evaluate the Green func-
tio~ for ship -notion at forward speed proposed by
one of the present author (Ohkusu and Iwashita
(1989~) will be usefull for this purpose.
We computed Or by solving the integral
equation at 2() strips along the ship length and
then interpolated a at 2()0 positions. Effects from
2()0 point sources are surnamed to represent the
effect of a line source. Figs.17 and 18 are corn-
puted examples of the wave elevation at t = 0 for
the heaving rno
REFEREN CES
(l)(:hapman, R.B.~1976~. Free Surface Effects for
Yawed Surface-Piercing Plate. J.Ship Research
20
(2)Daoud, N. (1975~. Potential Flow Near to a
Fine Ship's Bow, Rep. No.177, Dep. Nav. Archt.
Marine long., Univ. of Michigan.
(3)Erdelyi, A. (1956~. Asymptotic Expansions,
Dover Publications, New York.
(4)Faltinsen, 0.~1983~. Bow Flow and Added Re
sistance of Slender Ships at High Froude Number
and Low Wave Lengths. J. Ship Research 27
(5)Faltinsen, O. and Zhao, R. (1990) . Numeri
cal Predictions of a Ship Motion at High Forward
Speed, the Meeting oil the Dynamics of Ships,
the Royal Society of London . (6)Iwashita,H.
and Ohkusu, M. (1989~. Hydrodynamic Foeces
on a Ship Moving with Forward Speed in Waves,
Vol.166, J. Society Naval Architects of Japan
(7)Newman, J.N.~1978~. The theory of Ship Mo
tions, Advance in Applied Mechanics, Vol.18,
Academic Press.
(8~0gilvie, T.F. (1977~. Singular-Perturbation ' °
Problems in Ship Hydrodynamics, Advances in o.e
Applied Mechanics, Vol.17, Academic Press, New 06
York .. .
(9~0gilvie, T. F. and Tuck, E.O. (1969~. A Ratim
nal Strip Ttheory for Ship Motions, Part 1, Rep. 0.2 _
No.013, Dep. Nav. Archit. Marine Eng. Univ. o , ~' ~' ~^ '~
of Michigan
(lO)Ohkusu, M. and Iwashita, H. (1989~. Evalu
ation of the Green Function for Ship Motions at
Forward Speed and Application to Radiation and
DifEraction Problems, 4th International Work
shop on Water Waves and Floating Bodies, Nor
way
(ll)Ohkusu, M. (1971) On the MoLio~ of Twin
Hull Ship in Waves, Vol.129 J. Society Naval Ar
chitects of Japan
(12)Yeung, R.W. and Kim, S.H. (1985) A New
Development in the Theory of Oscillating and
'l'ranslating Slender Ships, 15th Symposium of
Naval Hydrodynamics
1.OI
/.6
/.4
/.2
Fig.1 Coordinate Systen~
~ / ~ :~2.0
//\W
x~v.l O
\
U. ~U.~ U.D U.0 1. V y ye~X)
H
Dooud
Fo/tinsen
· Pre s en t me tho d
F~ ' T~T ' 0 5
Xlr H F~
Fig.'2 Waves around a Wedge
FN=0.8D, ~9 L=10.0
Without mj
A55
0.1
A33
0. 5 L_ ~40.05
O
^ 3 4 5 -2
xlo
1 2
Fig.3 Convergence with Increase of Number of
Strips
4
Ass
o.os
0.5
A33
1 O j FN=O.89. ~g L=lo.o
A33
-
O 1 2 3
O
4 5
xlO
Fig.4 Convergence with Increase of Number of
Strips
1~
0.4
f, ~
0.1 _
I L I
0.4:
- Present Method(without mj)
-- Unified theory
FN=0-5
0.3h
LFN=1.O °
l l
0 5 10 15 20 2 25
wg L
Fig.5 Added Mass (Heaving) of a Single Hull Ship
~1
~1
|~ Present Method(without mj)
> - -- Unified theory
c~
1.0 _ .~ IFN=0.5
~
V 5 10
rFN=o.so
1 1 1
15 20 ,~ 25
Fig.6 Damping (Heaving) of a Single Hull Ship
0.1 °
Present Method(without mj)
--- Unified theory
1 , , ,
5 10 15
20 2 25
W T
L
9
Fig.7 Added Mt. of Inertia (Pitching) of a Single
Hull Ship
1
c~
\\~\\'
o
~ `/ FN=0~5
~_ OS / ~_4FN=1.O
_ ~~ - ___
Present Method(without mj)
--- Unified theory
Fig.8 Damping (Pitching) of a Single Hull Ship
W2 L -- Fig.9 Body Plan of a Catan~aral1
9
15
--- Present Method(without mj)
Present Method(with mj)
O Measured(FN=0.89)
~Measured(FN =0 50)
1.0
0.5
--Present Method(without mj)
-Present Method(with mj)
Measured(FN =0.89)
Measured(FN=0 50)
O O
° o orF~=0.89
1 ~O.
~ F~=O.SO
0; 5 10 15 w2I 20
ol l l I I _
5 10 15 w~L 20 9
9 Fig.13 DampiIlg (Pitchillg) of a Catamaran
Fig.10 Added Mass (Heaving) of a Catamaran
2.0H
I.0~-
Present Method(without m;)
Present Method(with mj)
O MeaSUred(FN =0~89)
· Meneured(FN=o.5o)
-W== ~ FN=0.89
~° _ 0 ~ - - _ ~==
~ - - _
L
FN=0 50
5 10 15
Fig.ll Damping (Heaving) of a Catamaran
°~1
C33
pgL B2/2
0.5 ~N=05
/'/4FN= 0.89
/ //~FN=0
~15 //; ~ 1
~/~ _
~ ~0.5
15mm
Heave
Fig.14a Variation of Restoring Force (Heave)
Cse
---- Present Method(without mj)
r Present Method(with mj) P9 L2B2/2
l!vieasured(F~ =0.89)
· Measured(FN =0.50)
0.2~
O. I _ ~ FN=0 50
o o ° ° 0_/ 0 0
FN=O
_ ~ F~=O.89
1 , , I
0 5 10 15 W2L20 ~3 /
g //,
Fig.12 Added Mt. of Inertia (Pitching) of a Cata
naran ',
~-0.2 _
~>
/ 3 deg.
/ Pitch
-FN ~0.89
-FN =0.5
Fig.14b Variation of Restoring Force (Pitch)
6
-
-
/L ~'`
oo
-0.5
5°/L
For ~ 0.5. At L .20, Pitching
,' X-0.6 _ closet "
~= ~ - A_ _ ~
- ~ ~ _ _ _ NO ~
\ 0.6
\~N
I As
~ X-0.6
0 4
~,
/ '
t
- _
_
\ /1'
1 '. /~1
-
-
J I; I ^- ~~~5
0.6 /
0.8/
Fig.15 Wave Elevation
TV
For 0.89, ~ L-20, Pitching
/'X-0.8
/
/ /0.6
/
\` X-O.B
cos ~ t 0.6 '
At'
0.~
1
0.2> ~
a\
~ 0.4
0.4
Fig.16 Wave Elevation
17
Fn = 0. 500
T = 3. 162
Y/L = 0.500
_
Y/L = 0. 4 17
Y/L = 0.333
Y/L = 0.251/'
. =/
Y/L = 0. 16 7~
Y/L = 0.084 1 \/~.
, . .
-L O L
F.P A.P
Fig.1/ Wave Elevation in the Far Field
Y/L
Y/L = 0. 333
Y/L = 0. 251
Y/L = 0. 167
-L
~ _
/
~_~y
\-,~
/~A,
_ _ ~ ~ ~ I A ~ ~ ~ ~ ~ ~ ~ ~; ~ ~ A ~ A
_ ~ ~ ~- ~V' V - ~ ~' V ~ V ~
~ ~ ~ -\f~^-J W-'`~--^~--
~ \ ~_ A~ V-~ -J~' ~ ,~J=~ _ _ ~
~ ,\ ~ ~ ~ ~ ~ ~ ~ .1 . . ~ AA~ _ . A~ _AA_ ,....
r V V ~V - ~r ] ~ V%r - ~ ~r ~
2L 3L 4L 5L
~__
,/
/
\ ~
. ,/~\,/> \ ~^J
~ ~ r ~ A ^~
~-
~ /' - ~
/ \~, ~ ~_
\ ~ - ~ ^~
- - - ~ A ~ --~V5\,~ ~ ~ r~
~ ~ \J \J ~ - v \]
;--~-^~ ,- ~,'t.,~q, `-~
2L 3L 4L 5L
Fig.18 VVave Elevation in the Far Field
18
DISCUSSION
Ronald W. Yeung
University of California at Berkeley, USA
The approach taken here is one already taken up by Yeung & Kim
(1981, 3rd Numerical Ship Hydrodynamic Conference) for a single
body. The present calculations have qualitative features, in terms of
agreement with expt., not so different from the above work which the
authors are apparently unaware of. Indeed, our experience, as I
recall, was that the terms associated with the change in waterline
width need to be accounted for. In our calculations of those days, the
mj terms involving the steady-state body potential was not included,
but improvement over strip approximation was evident. The
numerical approximation over the artificial time variables (x-O is
particularly important when (x-O is large and (yet) is small.
Accuracy and precision can be achieved if the ~ variable is integrated
analytically. The procedure is described in more detail in Yeung
(1982, J.Engrg.Math). The role played by the transverse waves was
analysed in Ref [12] but I suspect they are not important as the
Froude numbers being considered here.
AUTHORS' REPLY
I appreciate Prof. Yeung for attracting our attention to his extensive
works done before. I understand the terms associated with the change
in waterline width indicates a line integral on the intersection of the
free surface and the body surface. This term is of higher order in our
analysis, that is, of the same order as mj. Our results do not show
any essential difference from the results obtained with solving the
near field problem by distributing the fundamental sources on the free
surface as well as the body surface. This seems to show that
inclusion of the line integral has the secondary effect on the results.
19
