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OCR for page 529

The Numerical Solution of the Molions
of a Ship Advancing in Waves
G. X. Wu and R. Eatock Taylor
University College London
London, UK
Abstract
The hydrodynamic problem of a surface
ship advancing in regular waves at
constant forward speed is analysed using
a three dimens tonal theory based on the
linearized velocity potential. The
potential is represented by a
distribution of sources over the surface
of the ship and its waterline. Various
numerical schemes are introduced to
overcome some of the maj or difficulties
in this problem. Calculation is made for
a submerged sphere. Results are compared
with the analytical solution and very
good agreement is found. Some preliminary
calculations have also been made for a
series 60 ship with block coefficient
0.7.
1. Introduction
The wave induced motions of a ship have
several implications for ship
performance, increased resistance, deck
wetting, slamming, vertical acceleration
and propeller emergence, etc. While all
of these aspects are important subj ects
in ship hydrodynamics, the fundamental
problem remains that of estimating the
overall motion of the ship in waves.
In order to predict ship motions in
waves, the ship is usually regarded as a
rigid floating body having six degrees of
freedom, and the fluid loading is
estimated from linearized potential flow
theory. This theory assumes that the
fluid is inviscid and incompressible, the
flow is irrotational, and both incoming
wave elevation and body oscillation are
small. The velocity potential therefore
satisfies the Laplace equation, and the
corresponding boundary condition is
imposed on the mean pos ition of the fluid
boundary .
Even after such drastic assumptions
have been introduced, the solution of the
resulting equation is still not easy to
obtain. One of the maj or difficulties
529
arises from the complicated free surface
condition. Further difficulty is
associated with the fact that for a
practical ship its shape is usually
described by coordinates of discrete
points rather than by a simple
mathematical function. As a result, the
solution can be only obtained
numerically .
Attempts to predict ship motions in
waves can be traced much earlier, but a
significant breakthrough was the work by
Korvin-Kroukovsky and Jacobs (1957).
Based on phys ical intuition rather than
rigorous mathematics, they provided the
early version of strip theory. Even
though their theory was later found to be
mathematically inconsistent, ( in
particular it does not satisfy the
Timman-Newman relation(l962) ),
experimental data have shown that it
nevertheless provides very good results
in many cases. A number of modified
versions of this strip theory have since
been developed, of which, that proposed
by Salvesen, Tuck and Faltinsen (1970) is
widely used in ship des ign. Another very
significant step was the work of Newman
(1978). He overcame the limitation of the
conventional theory to the region of high
frequency, and proposed a "unified strip
theory" which is valid throughout the
whole frequency region. In particular
this theory takes some account of wave
interactions between different cross
sections of the ship. Numerical results
for heave and pitch (Sclavounos 1985) in
infinite water depth have shown that the
unified theory is superior to the
conventional strip theory in such a case.
Even though the s trip theory can
provide satisfactory results in many
cases, and has had a very important role
in ship des ign, it has its inherent
limitations. It requires the ship to be
slender, and the magnitudes of forward
speed and encounter frequency to be in
appropriate ranges. Furthermore, while it

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may provide good results for the total
force on the ship, it usually gives a
very poor prediction for the detailed
hydrodynamic pressure distribution around
the hull. Thus attempts to remove some of
the limitations of strip theory, by using
a three dimensional approach have been
initiated by Chang (1977) and others
(Inglis & Price 1981, Kobayashi 1981,
Guevel & Bougis 1982). These
investigations have all adopted the
constant panel method (Hess & Smith
1964): the ship hull is represented by
small panels on which the sources or
dipoles are assumed to be constant. It
has been found that these frequency
domain three dimensional theories in
general improve the results and provide
better agreement with experimental data.
However, it has been observed that the
numerical solution is sensitive to the
size of the panels, and high accuracy is
not easy to obtain. This was also
noticed in recent work by King, Beck &
Magee (1988) using a three dimensional
method in the time domain.
The present work is part of an
investigation which aims to obtain a
stable and accurate solution of the
linearized three dimensional problem,
using the source distribution approach.
Attention is focused here on certain
numerical aspects. Firstly, we use
quadratic isoparametric boundary elements
instead of plane constant panels. As the
associated wave resistance is known to be
sensitive to the shape of the ship, it
seems likely that isoparametric elements
should enable us to model the ship hull
with a higher degree of accuracy. They
also provide a more convenient means of
calculating the velocity of the fluid on
the ship surface. Secondly, we impose the
body surface condition by averaging over
the body surface using the Galerkin
method, rather than at discrete nodes.
Experience has shown that this method
usually gives more accurate results. In
this particular problem, as the body
surface condition on the waterline is
averaged over the body surface, we can
avoid the difficulty of both the source
and field points being on the free
surface when solving the integral
equation. The use of the Galerkin method
also avoids another serious numerical
difficulty: the second order derivatives
of the steady potential due to forward
speed (which appear in the body surface
condition on the unsteady potential due
to the ship oscillation) can be reduced
to first order derivatives, as in the
coupled finite element method (Wu &
Eatock Taylor 1987a)
In the integral equation, we express
the Green function in terms of the
exponential integral (Wu & Eatock Taylor
1987b). Extensive tests have been carried
out to try to achieve accurate evaluation
of the Green function, and a technique
has been introduced to remove the
singularity in its integrand. We use a
similar technique to that of Noblesse
(1983) to reduce the order of the dipole
singularity 1/r (where r is the distance
between the source and the field points).
We do not however need to evaluate the
constant by integrating the Green
function over the waterplane of the ship.
Finally, to remove the singularity due to
the source 1/r, we adopt triangular
polar coordinates when calculating the
contribution of an element to itself (Li,
Han & Mang 1985). An alternative method
for achieving this, by subdivision of the
element, has also been investigated.
These numerical procedures are found to
be very effective for a submerged sphere.
Compared with the analytical solution (Wu
& Eatock Taylor 1988), the numerical
method provides very accurate results
when 12 elements for half of the sphere
are used. Calculation are also made for a
series 60 hull of block coefficient 0.7
at Froude number Fn~0.2.
2. Mathematical Formulation
We define the right-handed coordinate
system O-xyz so that x points in the
direction of steady forward speed U of
the ship and z upwards; the origin of the
system is located on the undisturbed free
surface and the middle section of the
ship. The whole system is moving with the
ship at the same forward speed. For a
time-periodic incoming wave at a
frequency w0, the total potential can be
written as
=-Ux+Ui (x , y , z )
+Re[j[0~7j jj (x,y,z)e ] (1)
where ~ is the steady potential due to
unit forward speed, ¢. ( j51, . . . ~ 6) are
radiation potentials corresponding to the
six degree of freedom oscillations of the
body and ~.(j=1,...,6) are corresponding
motion amplitudes; ¢0 and ¢7 are the
potentials of the inch dent and diffracted
waves respectively; and ~=~77 is the
incoming wave amplitude. The encounter
frequency ~ is given by
~> no ~ (~>O/g)U cost (2)
where g is the gravitational acceleration
and ,`3 is the incident angle of the
incoming wave and ,B=0 indicates a
following sea.
Based on the assumptions of the
linearized theory, we have for the steady
potential
v2¢ =o
530
(3)

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in the whole fluid domain R;
Adz+ RIO
(4)
on the undisturbed free surface S. , where
~g/U ; F
3~/3n n
(5)
on the body surface SO, where n is the
inward normal of the body surface and n
is its component in the x direction; andX
3~/3n =0
(6)
on the bottom SB of the fluid or z,-- in
the present case of infinite water depth.
To complete the boundary-value problem,
we also need to include the radiation
condition at infinity: it is usually
assumed that there is no wave due to
far in front of the ship but there are
waves far behind the ship.
The components of the radiation and
diffraction potentials are assumed to
satisfy the following equations (Newman
1978)
in R; (7)
~jz+(, /~)¢jxx~2ir~ix~~¢i~o on SF (8)
where r=wU/g and ~c~2/g; and
where G is the Green function for a
pulsating translating source, which is
taken in the form derived by Wu & Eatock
Taylor (1987b). From the above equation,
we obtain
~n(P) 4~ HIP)+ It |S an~pjQ)a(Q)ds
g JL an ~ P j ~ (Q) nX (Q) dY] ( 12 )
on S0, where a(P) is the inner subtended
angle of the ship surface at point P and
the integration excludes the point Q P.
Substituting equation (9) into the above
equation, we can obtain the corresponding
boundary-integral equation for the source
distribution.
One difficulty in dealing with
equation (12) is caused by the normal
derivative of the Green function. It
contains a second order singularity of
the dipole when CAP. To avoid that we
define
F(P Q.) ~ 1 + l (13)
where r is the distance between P and Q.
and r is that between P and the mirror
image 1of Q about the undisturbed free
surface. Applying Green's second identity
in the domain enclosed by the ship and
its water plane where ~F/3n O. we have
Bjj/3n -i~nj+Umj (j-1, ,6) (9a) akp) _ | ~F(P,Q) dS (14)
3¢j/3n .-~¢O/8n j=7 (9b) SO ~n(Q)
on SO, where
(n1,n2 ,n3) = (nX,ny,nZ)
(n4,nS,n6) = X n
U(ml,m2,m3) ~ -(n.V)W
U(m4,m5,m6) - -(n.V)(X W)
W = UV(~-x) .
(lea)
(lob)
(lOc)
(led)
(lee)
X is the position vector of a point on SO
relative to the origin of the
coordinates. The potentials ¢. alto
satisfy the same condition on theJbott~m
of the fluid as ¢. The radiation
condition on ¢. states that the outgoing
wave with itsJgroup velocity larger than
forward speed travels far in front of the
body; otherwise the waves propagate
behind.
Following the derivation of Brard
(1972), the unknown potential can be
represented by a source distribution
over the body surface SO and water line
L We have
¢(P) 4~ [ JS G(P,Q)a(Q)dS
g JL (P Q)a(Q)nX(Q)dY] (11)
By substitution of equation (14) into
(12), the latter becomes
n(P) 4~|Sot ~n`Pj ~(Q)~n`QjQ)a(p)]ds
4~ g JL ~n(Pj ~(Q)nx(Q)dy (15)
It is easy to confirm that the order of
the singularity in this equation has been
reduced.
3. Numerical Discretisation
We now discretise equation (15) using
the shape function Nj. We write
=.§ a.N.
J=1 J J
where n is the number of nodes. By use of
the Galerkin method, equation (IS) can be
written as
[A][~]=[B]
531
(16)
(17)
where [A] is the square matrix with the
coefficients
ij 4~ iSOISo[ ~n`Pj ;(Q)
~F'QPjQ) Nj(P)]Ni(P)dSQdSp
g iSo[iL ~ntP) Nj(Q)nx(Q)dy]Ni(p)ds };
(18a)

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and [B] contains the body surface
boundary condition and has the
coefficients
bij ~ JS ~ N (P) dS (lab)
To obtain an accurate solution of the
above equation, careful consideration
must be given to several factors which
have most significant effects. Firstly,
the Green fucntion is expressed in an
integral form with a complicated and
highly oscillatory integrand. The
numerical evaluation of such an integral
requires a very small step and takes most
of the computer time. In our analysis, we
have perfomed certain transformations of
variable to reduce the oscillation. To
deal with the following singularity in
the Green function (Wu & Eatock Taylor
1987b)
I ~ Jo 7(4rcose 1)
at ~ ~~acos(l/4~) when r>0.25, we
introduce the following scheme
I --1 {) sin~f(~)-sin~f(-r) do
sin, J O J(4,cos8-1)
sin, 2, 7(47-1) (19)
A similar scheme is adopted in the range
(,,~/2) and is found to be effective.
The second important factor which
significantly affects the accuracy is
the method of discretisation of the ship.
Initially, a coarse mesh can be refined
by using more of the coordinates of the
ship hull provided by the offsets; but
this process is limited by the number of
coordinates available. When a still finer
mesh is needed, the commonly adopted
procedure is to interpolate using the
shape functions. Consequently this may
refine the representation of the source
distribution but it does not improve the
representation of the ship hull. This
leads to the problem that different shape
functions will give different hulls. It
may not be important when these ship
hulls are close to each other; but a
problem can arise when even then the
results do not converge. Ultimately,
different shape functions may lead to
different converged solutions when the
above subdivision procedure is adopted.
When this happens, subdivision of
elements must be based on measuring the
nodal coordinates on the lines drawing of
the ship.
The third factor is the integration
over the body surface in equation (18).
After numerous tests and careful
consideration of accuracy and efficiency,
we have chosen the four point Gaussian
scheme. To avoid the singularity when
UP, we have investigated two methods:
that proposed by Li, Han & Mang (l9,85)
using a triangular polar local coordinate
system; and a method based on subdividing
the element when the integration is
performed. We have found that both
schemes give very similar results and the
latter has been chosen in the main
computer program. To improve efficiency,
we have also used the fact that
components of G(P,Q) are either symmetric
or antisymmetric. This reduces the
computer time by almost a half.
Finally, to avoid the difficulty of
calculating the second order derivatives
in equations (lOc) and (led), we can
perform the integration in equation (lab)
by parts. This reduces the derivatives to
first order (Wu & Eatock Taylor 1987a).
After the solution has been found, the
added masses A.. and damping coefficients
Aij can be obtained from (Newman 1978)
2
- 1~> j
P |so(i° fj+ w.v¢; )nidS
P iso(iu)ni~Umi)¢j dS+PUtL¢j~znidL(20)
where the second term has been
transformed using the relation derived
by Ogilvie and Tuck (1969). In general,
the second form of this equation has no
apparent advantage over the first. In
fact the second order derivative in mi
makes the calculation even more
difficult. However, when the steady
potential ~ can be neglected, such as for
sufficiently slender ships, the latter
form has the advantage of not requiring
calculation of the derivatives of the
unsteady potential. Thus for a slender
ship we have
2
~ . . ~ ~ . . - 1~> . .
1J 1J 1J
P |So(i~)ni Umi)4q~ [ |sG(P,Q)a(Q)ds
g |LG(P,Q)a(Q)nx(Q)dy] dS
P 4~ k~lak k
where a corresponds to +. and
k |SO(i(~)ni-umi) [ |S G(P,Q)Nk(Q)dS
(21)
g |LG(P,Q)Nk(Q)nx(Q)dy] dS (22)
Ck can of course be calculated when
matrix [A] is assembled rather than after
the solution has been found; otherwise
the computer time would almost be
doubled.
532

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4. Numerical Results
In the following calculated example
we have taken ¢~0. This is in fact
consistent with the linearized free
surface condition given in equation (8).
The presence of the steady disturbance
potential in the body surface condition
alone does not appear to provide a
consistent improvement to the accuracy of
the boundary-value problem for a surface
piercing body. In general, when ~ can not
be regarded as a small quantity, its
contribution should be included in the
free surface condition (Newman 1978) as
well as in the body surface condition. If
the incoming wave is of large amplitude,
a fully nonlinear mathematical model
should be used.
Tables la and lb give the added mass
and damping coefficients for a sphere of
radius a undergoing forced oscillations.
Table 2 gives the exciting force on the
sphere in an incoming wave with incident
angle p~0.75~. The sphere is submerged at
hula (h being distance between the centre
of the sphere and the mean free surface)
and translates at a Froude number
Fn=U/~(ga)~0.4. The hydrodynamic
coefficients are nondimensionalized as
~ij/p~a ~ and the exciting forcees as
F./pg~a uq0. In the tables, ~ and u0
correspond to the values of ~ and w0 al
the critical point r,0.25. We have
omitted r3 from table 1, since it is
Observed That the Timman-Newman (1962)
relation is very well satisfied at this
forward speed. The results from the
numerical method (designated N in the
tables) are based on an idealization
using only 12 elements (45 nodes) on one
half of the surface of the sphere. These
are seen to agree reasonably with results
from the analytical solution (Wu & Eatock
Taylor 1988), designated A in the tables.
It should be noted, however, that these
results are not the same as those in Wu &
Eatock Taylor (1988), since the latter
included the effect of the steady
disturbance potential on the body surface
boundary condition. We have observed that
excluding this term has one marked
effect: namely it leads to a non-zero
rotational moment about the centre of the
sphere. This should not occur, and it
highlights the importance of including
in this case. Nevertheless, the
comparisons shown in the table provide
evidence of the reliability of the
numerical procedures adopted to solve the
boundary value problem by equations (7)
and (9).
~11 ~22 ~33 ~13
ma A N A N A N A N
0.1 0.7009 0.6948 0.6939 0.6996 0.7291 0.7251 -0.0028 -0.0029
0.2 0.7200 0.7141 0.7055 0.7116 0.7607 0.7577 -0.0259 -0.0264
0.3 0.7128 0.7064 0.7075 0.7137 0.7529 0.7501 -0.0758 -0.0773
0.4 0.6077 0.5987 0.6936 0.6996 0.6259 0.6153 -0.0799 -0.0828
0.5 0.6290 0.6224 0.6722 0.6779 0.6315 0.6244 -0.0198 -0.0211
0.6 0.6374 0.6311 0.6579 0.6629 0.6271 0.6201 -0.0007 -0.0011
0.7 0.6420 0.6356 0.6482 0.6532 0.6226 0.6156 0.0102 0.0103
0.8 0.6443 0.6379 0.6416 0.6465 0.6186 0.6115 0.0170 0.0174
0.9 0.6452 0.6387 0.6371 0.6423 0.6153 0.6081 0.0212 0.0215
1.0 0.6452 0.6388 0.6341 0.6388 0.6125 0.6051 0.0236 0.0240
Table la. Comparison of added mass coefficients for
sphere (h=2a, Fn=U/~(ga)=0.4, ~ a=0.3906)
>11
a submerged
>22
N A N A N A N
0.1 0.0035 0.0035 0.0025 0.0023 0.0062 0.0063 0.0181 0.0184
0.2 0.0275 0.0279 0.0155 0.0153 0.0450 0.0460 0.0360 0.0367
0.3 0.0722 0.0781 0.0376 0.0380 0.1205 0.1237 0.0249 0.0253
0.4 0.0871 0.0884 0.0595 0.0605 0.1454 0.1513 -0.0891 -0.0937
0.5 0.0371 0.0380 0.0631 0.0644 0.0975 0.1013 -0.0695 -0.0712
0.6 0.0275 0.0280 0.0599 0.0609 0.0850 0.0879 -0.0592 -0.0604
0.7 0.0242 0.0245 0.0548 0.0557 0.0769 0.0793 -0.0502 -0.0511
0.8 0.0230 0.0233 0.0492 0.5000 0.0703 0.0725 -0.0420 -0.0428
0.9 0.0225 0.0228 0.0438 0.0444 0.0645 0.0665 -0.0348 -0.0355
1.0 0.0222 0.0225 0.0388 0.0393 0.0593 0.0612 -0.0285 -0.0291
Table lb. Comparison of damping coefficients for a
sphere (h=2a, Fn=U/~(ga)~0.4,~ a=0.3906)
533
submerged

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I F1 l
A N A N A N
1F31
0.1 1.0669 1.0633 1.0789 1.0859 1.5441 1.5411
0.2 0.8434 0.8402 0.8601 0.8626 1.2345 1.2342
0.3 0.6957 0.6950 0.6828 0.6846 0.9640 0.9610
0.4 0.5560 0.5548 0.5395 0.5407 0.7678 0.7661
0.5 0.4449 0.4437 0.4299 0.4306 0.6119 0.6107
0.6 0.3559 0.3548 0.3444 0.3449 0.4889 0.4881
0.7 0.2851 0.2840 0.2768 0.2772 0.3916 0.3911
0.8 0.2286 0.2278 0.2232 0.2233 0.3145 0.3143
0.9 0.1837 0.1828 0.1801 0.1802 0.2530 0.2530
1.0 0.1478 0.1470 0.1457 0.1456 0.2039 0.2042
Table 2. Comparison of exciting forces on a submerged sphere
(h'2a, Fn.U/~(ga)~0.4,~0ca~0.2937,§~0.75~)
As an application of this analysis to a
surface ship, we have calculated results
for a series 60 hull with block
coefficient 0.7. We first investigated
convergenece by assuming a rigid free
surface condition,and using two meshes
on one half of the ship hull: with 168
elements and 567 nodes and with 280
elements and 935 nodes, as shown in
figure 1. The second of these meshes is
substantially finer than those used by
others in earlier published work. he
found that the former provides results
within 3.1% and 5.3% of the finer mesh
results for added mass in heave and pitch
respectively, while these two meshes give
virtually identical area and volume fqr
the ship. Next we calculated the
hydrodynamic coefficients by using the
translating pulsating source Green's
function in equation (22), with the
source strength in equation (21) based on
the rigid free surface calculation. This
has the advantage of providing a much
more rapid calculation of source
strength, and is related to the
approximate method used by Newman (1961)
for a submerged ellipsoid. The results
from the coarse mesh are shown in figures
2 and 3.
(a) Coarse mesh (567 nodes)
(b) Fine mesh (935 nodes)
Figure 1. Mesh for Series 60 hull
534

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xt~1
~1
20.
to.
Q
10
S.
, ,
2 3 t
~ Legal)
(a) heave
x,~2
~1
—5
-10.
-t5
—20-
-25'
t 5
2
6J /~)
(c) heave/pitch
xlr2
1K
to
to
to.
8.
~ ._
x'0-2
16.
11. .
12.
10.
a
Q
- ; ~ 5
~ '~q~L)
(b) pitch/heave
. . . .
3 t S
Legal)
(d) pitch
Figure 2. The added masses of the series 60 (Cb=0.7) at Fn=0.2
535

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xt~l X18-l
8.
25. ~ ~ _' ~
J
tS.
to,
Is
~ Ethyl)
, .
2 3
~ Mogul)
(a) heave (b) pitch/heave
xt.-l Xt.
j, 6,
~ t1.
3.
7
1~
_ ~ ~
to.
~ ,
0-
—1 .
o.
~ 8
>3 6.
1, ,
, , , ~ ~ 2. . J
1 2 3 t S 2 3
~ AL) ~ ', GEL)
(c) heave/pitch (d) pitch
Figure 3. The damping coefficients of the series 60 (Cb=0.7) at Fn=0.2
536
~ _z
J
_
4 5
7 - J
~ 5

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5. ConcIusions
The hydrodynamic problem of a surface
ship advancing in regular waves at
constant forward speed is analysed based
on the linearized velocity potential
theory and using the boundary integral
technique. The numerical techniques
introduced have been found to be
effective in overcoming some difficulties
encountered previously. A major remaining
difficulty is the evaluation of the Green
function when both source and field
points are near the free surface, which
appears to be the direction towards
which further work in this area should be
directed.
References
Brard, R. "The representation of a given
ship form by singularity distributions
when the boundary condition on the free
surface is linearized", J. Ship Res.,
Vol. 16, pp.79-92, (1972)
Chang, M.S. "Computation of
three-dimensional ship-motions with
forward speech, 2nd Int. Conf. on Num.
Ship Hydrodyn., pp.l24-135, University
of California, Berkeley, (1977)
Guevel, P. and Bougis, J. "Ship-motions
with forward speed in infinite depth",
Int. Shipbuilding Prog., Vol. 29,
pp.1-3-117, (1982)
Inglis, R.B. and Price, W.G. nA three
dimensional ship motion theory-comparison
between theoretical predictions and
experimental data of the hydrodynamic
coefficients with forward speed", Trans.
R.I.N.A., Vol. 124, pp.l41-157, (1981)
King, B.K., Beck, R.F. and Magee, A.R.
"Seakeeping calculations with forward
speed using time-domain analysis", 17th
Symp. on Naval Hydrodyn., The Hague, The
Netherlands, (1988?
Kobayashi, ~ God the hydrodynamic forces
and moments acting on an arbitrary body
with a constant forward speed", J.S.N.A.
Japan, Vol. 150, pp.61-72, (1981)
Korvin-Kroukovsky, B.V., and Jacobs, W.R.
"Pitching and heaving motions of a ship
in regular waves n, Trans. SNAME, Vol. 65,
pp. 590-632, (1952)
Li, H.B., Han, G.M. and Mang, H.A. "A new
method for evaluating singular integral
in stress analsysis of solids by the
direct boundary element method", Int. J.
Num. Meth. Eng., Vol. 21, pp.2071-2098,
(1985)
Newman, J.N. "The damping of an
oscillating ellipsoid near a free
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OCR for page 529

DISCUSSION
by R. Huijsmans
The authors are to be congratulated on
their treatment of the full forward speed
diffraction problem. We at MARIN have the
experience that using the exact Green's
function for the translating oscillating
source as was studied by eg Bougis, Inglis
needs a very careful treatment of the panel
sizes. In the above mentioned studies only
very coarse grids were used. In recent
calculations at our Institute, we calculated
added mass and damping of a series 60 ship
with the number of panels increasing up to
856. Apart from the very large computational
burden the results for the added mass and
damping did not seem to converge really with
increasing panel sizes. This was especially
the case when the pressure distribution was
examined. Our opinion is that there is
conflicting requirement regarding the
stationary and the oscillating part of the
Green's function with respect to the panel
sizes, especially when the forward speed is
not very large.
1) Does the authors have the some
experience regarding the statement made above?
2) Can the authors give some indication of
the type of computer they used and computer
time they have used for the non-zero speed
case? Especially when comparing with the zero
speed case.
3) Does the authors have some
idea/indication how the local hydrodynamic
quantities, like pressure and velocities
behave when increasing the number of panels?
Author's Reply
We thank Dr. Huijsmans for his interesting
comments. From our experience using quadratic
boundary elements for the problem, we are
certainly not surprised that using 850 or so
constant elements to represent the series 60
hull did not always provide satisfactory
results. Our own coarse mesh used 1085 nodes
for the submerged hull (the numbers in Fig.1
referring to one half of the hull), and the
finer mesh 1789 nodes. Like Dr. Huijsmans we
have been looking at various ways of
overcoming the conflicting requirements at
small forward speed, and some of our thoughts
on this are to be published elsewhere[A1].
We do not yet have an efficient algorithm
for the Green function, and the computing time
therefore still quite long. For the series 60
results given in Figs.2 and 3, they range from
about 2000 to 7000 seconds per frequency on a
CRY 1, with the longer runs corresponding to
results at higher frequencies. We also run
these programs on a Microvax II, and have to
wait a few days for results at one frequency!
We have not evaluated pressures and local
kinematics for the case of the body with
forward speed. But we would expect to draw
similar conclusions to those given in Eatock
Taylor and Sincock[A2].
[Al] Wu, G.X. and Eatock Taylor, R.: The
Hydrodynamic Force on an Oscillatory ship
with Low Forward Speed, J. of Fluid
Mech. to appear 1990).
[A2] Eatock Taylor, R. and Sincock, P.: Wave
Upwelling Effects in TLP andSemi-
submersibleStructures,Ocean Engineering
16, pp.281-306 (1989).
DISCUSSION
by G. Jensen
Isoparametric elements are associated with
numerical integration. Could you please give
some more details about the computation of the
velocities and may be higher derivatives on
the body?
Author's Reply
The first order derivatives of the
velocity potential on the body surface (and
hence the fluid velocities at any point on the
surface) can be obtained from the nodal
solutions, using the shape functions, together
with the known normal derivatives. Thus uses
ax = acax + away + adz
ad axar cyan azar
an apex away adz
_ _ ~ _ +
an- Dxarl ayarl azar,
an an an an
On axnX + Dyne + aznZ
to solve for the derivatives of ~ in the x,y
and z directions.
This approach can not be used directly to
obtain the higher order derivatives. As
discussed in the paper, however, it may only
be the integrated effect of such derivatives
that is required on the body surface, and in
some circumstances this can be obtained by
alternative means.
538