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OCR for page 66
New Green-Function Methoc!
to Predict Wave-Induced Ship Motions and Loads
X-B Chen, L. Diebold (Bureau Veritas - DTA, France)
Y. Doutreleau (Bassin d'Essais des Carenes, France)
ABSTRACT
A three-dimensional numerical method based on the
new Green function given in Cheryl (1999) and the
higher-order description of ship hull by bi-quadratic
patches has been developed in order to predict wave-
induced ship motions and loads. The development of
this new method and its validations through a com-
prehensive comparison with semi-analytical solutions
and results of experimental measurements, are de-
scribed in the paper.
Recent work on the ship-motion Green function
in both analytical and numerical aspects provides an
accurate and efficient way to evaluate the influence
coefficients involved in integral equations. The use of
bi-quadratic patches gives a precise representation of
the ship geometry and a continuous representation of
the velocity potential over the ship hull. Furthermore,
application of the Galerkin procedure yields a square-
matrix system and improves the accuracy of solutions.
The excellent level of agreement with known semi-
analytical solutions and experimental measurements
shows that the present numerical method is reliable
and practical in a number of applications.
INTRODUCTION
Very important in practice, notably for design, several
3D panel methods making use of the Green function,
which satisfies the linearized free-surface boundary
condition corresponding to time-harmonic flows ob-
served from a translating system of coordinates, have
been developed in recent decades for computing wave-
induced ship motions and loads. However, the fairly
small number of panels used in these methods and the
large discrepancies observed among numerical predic-
tions indicate the formidable numerical burdens. In
fact, the substantial difficulties associated with the
numerical evaluation of the Green function and its
gradient, and their subsequent integrations over ship-
hull panels and waterline-segments, have been a ma-
jor stumbling block hindering the development of re-
liable and practical methods. These difficulties have
been addressed in various ways in the numerical solu-
tion procedures developed by Char~g (1977), Ir~glis ~
Price (1981), Gne'vel ~ Bongis (1982), Wit ~ Eatock
Taylor (1989) and Iwashita ~ Ohs (1992), and in
the development of methods for computing the ship-
motion Green function by Hog (1990), Jar~kowski
(1990), Bongis ~ Condray (1991) and Ba ~ Gnilband
(1995), and very recently confirmed by the work in
Cheryl (20003. It is shown formally in this work by an
asymptotic analysis that the source potential is singu-
lar and highly-oscillatory for a field point approaching
the track of the source at the free surface.
To circumvent the above-mentioned difficulties
encountered in previous studies, a new numerical
method, based on the recent results in both theo-
retical and numerical aspects of ship-motion Green
functions and the higher-order description of ship-
hull geometry and fluid kinematics, has been devel-
oped. The newly-obtained important results for ship-
motion Green functions given by Cheryl (1999) include
several innovative features. First, new formulations
of the free-surface component are developed based on
the basic decomposition of the double Fourier inte-
gral obtained by Noblesse ~ Cherl (1995) and new
expressions of the wavenumber integrals. The resul-
tant wave and local components are both expressed
by simple integrals. The asymptotic analysis of the
wave component gives analytical expressions of far-
field ship waves and reveals their direct relationship
with the dispersion relation. The singular and highly-
oscillatory properties of potential flows generated by
a source located at the free surface are analyzed and
expressed in a closed form. Analysis of the line in-
tegrals on the free surface shows that they can be
evaluated in an analytical way. Furthermore, efficient
numerical developments have been realized to evalu-
ate ship-motion Green functions in all configurations
accurately including the most critical case for which
OCR for page 67
the free-surface effects are most
source and field points are close tc
face.
important as both
or at the free sur-
In fact, this recent work gives a new lease of life
to the Green-function approach and provides the es-
sential elements to construct a sound and solid basis
to solve ship-motion problems. Further to the trea-
tise presented in Cherry (1999) on the Green functions,
a number of new developments have been achieved.
The work consists of the formulation for the gradi-
ent of ship-motion Green functions and the appro-
priate way to evaluate efficiently the influence coeffi-
cients involved in the kernel of integral equations. In
fact, the influence coefficients associated with the free-
surface effects are evaluated by quadrature integra-
tion of the wave and local components of ship-motion
Green functions over the ship's hull and along the wa-
terline. The formulation of the local component pre-
sented in Cherry (1999) is used as it is well suited for
the numerical evaluation. Indeed, the computation of
the local component expressed by the single integral
whose integrand is a smooth function is much more ef-
ficient than the double Fourier integral given in Cherry
Noblesse (1998~.
On the other hand, the single integral for the
wave component is a bit troublesome since the in-
tegrand can be highly-oscillatory when both field and
source points are close to the free surface. The
Fourier-Kochin approach presented in Noblesse
Yards (1995) and Cherry ~ Noblesse (1998) was first
adopted to evaluate this part of influence coefficients
(Cherry, 1998~. These interesting studies show that the
influence coefficients associated with a smooth distri-
bution of singularity over a higher-order panel are not
highly-oscillatory due to the cancellation of oscillatory
terms at large wavenumber. Another study in Dontre-
lean ~ Cheryl (1999) on the analytical integration of
the highly-oscillatory term leads to the same conclu-
sion. However, both the Kochin approach and that
of analytical integration need more complex calcula-
tions and are not so efficient as expected from the
experience of the first author. The option to design
a smooth function depending on the smoothness of
distribution of singularities for damping out high os-
cillations is thus retained in the computation of influ-
ence coefficients associated with the wave component.
Furthermore, the special algorithms given in Cheryl
(1999) for oscillatory integrals improve further the ef-
ficiency of the wave-term computing. This elaborated
approach allows us to evaluate accurately and effi-
ciently the influence coefficients associated with the
free-surface part of ship-motion Green functions, in
all configurations including the case of two panels at
the free surface.
Another feature of the new Green-function
method is the higher-order description of ship-hull ge-
ometry as well as singularity distributions on the hull
and along the waterline. The ship hull is subdivided
into a number of 9-nodes bi-quadratic patches which
are defined by the coordinates at 9 nodes and the
corresponding shape functions. Accordingly, the ve-
locity potential and distributions of singularities are
expressed by the classical series form associated with
the shape functions which represent the bi-quadratic
variation on each patch.
To complete the work on influence coefficients,
the Rankine term is evaluated by using the so-called
Gaussian subdivision approach, which consists of sub-
dividing the patch into a number of quadrilateral pan-
els following the coordinates defined by the Gauss-
Legendre rule. The quadrature integration of Gauss-
Legendre is performed when the field point is at a
large distance from the patch (or panels) relative to
the size of patch (or panels). For a field point close
to or on the patch, the analytical formulation is then
applied to evaluate the Rankine term on the smaller
panels.
Unlike the methods adopted in previous studies
using flat panels and constant singularities (discontin-
uous across panels) in which a large number of pan-
els are required to achieve accurate representation of
ship-hull geometry and velocity potential, the use of
the bi-quadratic patches reduces the number of un-
knowns and thus the computation time. The contin-
uous representation of the velocity potential in the
present bi-quadratic patch method makes it possible
to approximate correctly the velocity field within each
patch, which is necessary in establishing the integral
equation. Furthermore, application of the Galerkin
procedure yields a square-matrix linear system which
can be solved by either the Gauss-Elimination method
or an iterative approach, and improves the accuracy
of solutions.
The new Green-function method solves three
classes of problems including the wave diffraction-
radiation at zero forward speed, Neumann-Kelvin
steady flows and the wave diffraction-radiation with
forward speed. It can then offer a wide range of ap-
plications in ship designs and offshore industries. The
validation of this method is made through a compar-
ative study in a systematic and comprehensive way.
The numerical results obtained by using the present
method are confronted with the classic analytical so-
lution to the wave radiation around a floating hemi-
sphere given in Home (1982), the semi-analytical re-
sults of Neumann-Kelvin steady flow around an ellip-
soid studied by Farell (1973) and the semi-analytical
solutions to wave diffraction and radiation of a sphere
OCR for page 68
advancing in water waves obtained by Wn (1995~. An
excellent level of agreement is found in all configura-
tions. Furthermore, most results are presented in tab-
ulated form which may be useful to readers. Finally,
the comparison with experimental measurements on
Wigley hulls made by Jonrrte'e (1992) further confirms
the validity of the present method.
BOUNDARY VALUE PROBLEM AND
NEW GREEN FUNCTION
The reference system moving with the ship at the
mean forward speed U along the positive -axis
is defined by letting (my) plane coincide with the
mean free surface and z-axis be positive upward.
Based on the assumptions of perfect fluid, irrotational
flow and small wave steepness, the velocity potential
A, y, z, t) is written as the sum of a steady Age, y, z)
and a time-harmonic unsteady Air, y, z) potentials:
~ ~ (- ~ ~ ~ -iwt) (1) On the body surface H
Both the steady potential ~ and unsteady potential
satisfy the Laplace equation in the fluid domain and
an appropriate radiation condition at infinity. In par-
ticular, the steady potential (called Neumann-Kelvin
flow) satisfies
He + (U2/g)~ = 0 and all = ret (2)
on the mean free surface (z = 0) due to the lineariza-
tion and on the body surface H. respectively. In (2),
g iS the acceleration due to gravity and Amp mean
respectively the first derivatives of ~ with respect to z
and r', and the second derivatives of ~ with respect to
a. The normal vector n = brat, D2, Dt3) on H is defined
to point toward fluid.
The time-harmonic potential is further expressed
as the sum of various components:
6
,: = ~ ajgj + aO(¢o + ¢7) (3)
. 1
3=1
in which >i,2,...,6 are radiation potentials correspond-
ing to 6 degree of freedom oscillations of the body and
a~,2,...,6 are amplitude of corresponding motions. ¢0
is the potential of incoming waves and given by
—~g/~t, Em [z+i(~ cOs ~+y sin 9)] ¢4'
with the wavenumber k0 = w02/g associated with the
frequency we of incoming waves and ~ the wave head-
ing. ¢7 iS the potential due to the diffracted waves
associated with the amplitude So of incoming waves.
The components of the radiation and diffraction
potentials satisfy the linearized condition given by the
equation (3.23) in Newman (19783:
((UW)¢j = 0 (5)
over the deformed free surface z = ~ due to basic flow
which is steady and represented by W (Eq. 3.15 in
Newman, 19783. In (5), L: is a linear operator de-
pending on UW applied to Hi. If the uniform flow is
chosen as the basic flow, i.e. W=—en, the condition
(5) is reduced to
(0z _ ~2 + id + (u2/g)~¢j = 0 (6)
applied on z = 0. In (6), ~ = U&)/g iS the Brard number
characterizing the flow, and ~ = w2/g the wavenum-
ber associated with the encounter frequency ~ which
is defined by the frequency we and heading ~ of in-
coming wave, and the forward speed U through
O) = O)o (1 - (U~O/g) COS g] (7)
,g,> /,~ at,—iron + Umj ~ = 1, 2, ,6 <8
-~>o/0r1 ~=7
with the generalized vector
(fill, 02, 03) = n and (04, 05, 06) = X X n (9a)
and so-called mj terms defined by
ems, m2, ma) = - (n V)W (9b)
(ma, m5, me) = - (n V) (X x W) (9c)
in which X = He, y, z) is the position vector of a point
on H and the basic flow
W = V(>o - ~) (10)
Consistently, the basic potential 60 is to be chosen as
60 = ~ in the boundary condition (8) on H as well as
over the free surface (5~. However, few studies have
involved one other than the uniform flow (40 = 0) due
to the complexity of (5) and the computation of the
steady flow such as that satisfying (2) and its higher-
order derivatives needed on the free surface and in the
mj terms (9~. Only recently, the mj terms derived
from the Neumann-Kelvin flow are used in the com-
putation of time-harmonic potential by Wn (1995),
while the same condition as (6) on the free surface
was retained. By assuming small forward speed, the
consistence was respected in Cherl ~ Malerlica (1998)
since in both conditions on H and over the free sur-
face, the double-body flow satisfying
0¢O/0z = 0 and 040/0r1 = ret (11)
OCR for page 69
where k= N/~ is the wavenumber and
on z =0 and H respectively, is used as the basic flow.
Despite its limitations, the double-body potential is
used in the present study to evaluate the mj terms
while the classic condition (6) on z=0 is retained. In
fact, the same method as used in Cheryl ~ Maler~ica
/ \ ~ 1 1 1 l is the dispersion function
y1998) Is adopted In the computation of tne Tousle
derivatives of ¢0 and then mj terms.
The time-harmonic potential ~ (>j with j =
1, 2, , 7) satisfying (6) and (8) can be determined
by the integral equation
¢~/2 + X(~) = 4~) (12)
for ~ c H. and with
x<~' = JJ COG ((, ~)dS + (U2/g) .
J {~c)G~ ((, ~c)—You Aces +¢v I, ~c) }tydl
w
—i2r J ¢~c)G((, ~c~tydl (13a)
w
and
4~<, = JJ ¢~ I, kids +
H
~u2/gjJ ¢~)c0G((, ~)tydl (13b)
w
The line integral along the waterline W in (13) arises
from applying Stokes' integral theorem on the free
surface, and the following identity
¢~ = DECO + And + grlC'l (14)
is used in (13) with (cu,cv'crl) depending on the ge-
ometry of H and given by (36c). In (13), ty is the
y-component of the unit vector t tangent to the wa-
terline and oriented clockwise while (¢u,gv) are two
tangential derivatives of ~ over the ship hull H and
All is the normal derivative given by (83.
The Green function G((, ~c) involved in (13) rep-
resents the velocity potential of the flow generated at
a field point ~ = ((, 7~, () by a source of unit strength
located at the point ~ = An, y, z), is expressed by
G = Gs + OF (15)
Here Gs is the simple singularity part
41r Gs = - 1/~(—~~ + 1/~(—I'd
with A' = An, y,—z) the mirror image of ~ with respect
to the mean free surface plane z =0. The free-surface
part GF is defined by the Fourier superposition of
elementary waves (Noblesse ~ Cheryl, 1995)
Too t00 eEZ-i(~+9Y)
(47r2L)GF = 1i +0 / d/3 / Tic D + ic sign(D
it, Y. Z < 0) = ((—~7 77—y, (+z)/L
with L being the ship length. Furthermore, D in (17)
D = (For. _ f)2 _ ~ (18)
and sign(Df ~ = sign(0D/0f) =—sign(For—f ). The
dispersion function (18) is associated with the same
boundary condition as (6) on the free surface. f =
~73 is the nondimensional frequency and F =
U/~/~ the Froude number.
The free-surface part GF defined by (17) is ex-
pressed by Cheryl (1999) in the form
(47rL)GF = Gw + ON + H(r—1/4)GC (19)
where H(r—1/4) is a unit step function equal to O
for ~ 1/4. The free-
surface part GF is then decomposed into three differ-
ent components in (19) with Gw the wave component,
GN the local component and Gc the complementary
component for ~ > 1/4. The wave component Gw is
expressed as a single integral
Gw _i~, Jds (E +E )''X{'~k-)e-i(~+9Y) <20a'
along the dispersion curve defined by the dispersion
relation D = O. with E~ = sign(Df) and
~2 = ~+ erf ~ a~ + by ~ with E~ = ~ signing + gy)
along the dispersion curves k=~) given by
k~ (~) = (1/2 ~ >/1/4+r cos 0) /(F cos 0~2 (20b)
respectively. Furthermore, erf; ~ is the error function
defined in Abramowitz ~ Sternal (1967~.
The local component in (19) is defined by
ON J~-0c Ic+.~+—Ic-.~- do (21a)
-~+~c 2 >/1 /4 + ~ cos
in which the function J\r~ =
Cex(~+ (z—ih)]—i1rE~ erfc~+h~e~ (Z-ih) <21by
In (21b), we have used
h = ~ cos 0+y sin ~ and Cex~w) = ewe (w) (21c)
where erfc; ~ and Elf ~ are the complementary error
function and the exponential integral function defined
in Abramowitz ~ Sternal (1967~.
OCR for page 70
The values of be are equal to 0 for ~ < 1/4 and and finally VGC =
tic = arctan >/16~2 - 1
for ~ > 1/4. The complementary component GO in
(19) exists for ~ > 1/4 and is defined by
J7r_oc ( F c09 /J) 2
The wavenumber integral function To is given by
KC = [~ER+iLI)1(C — (ER—iLl)~C] /~2iLl) (22b)
with 1Cc = Cex~Z+~+
i1rtsign(kl) ~ sign(Z+~4H(kR)eZ (22c)
where Zip = Am I Z+ ) and the unit step function
H(ER) is equal to 0 for FRO.
Furthermore Zip in (22c) are defined by
Zip = (ER ~ ibid (z-—ih cos(0—:~] (22d)
with h = N/~ and tan fly = (y/~) and
FR = (1/2 +r cos 03/(F COS 032 (22e)
F1 = N/ - (1/4+r cos IF COS 0~2
The gradients of ship-motion Green functions
with respect to the coordinates (~ y z) of the source
point can be obtained by directly performing the
space derivatives to (16) for the simple singularity
part Gs
VG ~ + ~ ~ 23
and to (19) for the free-surface part GF:
(41rL2 JVGF = VGW + VGN + H(T _ 1 /4)VGC (24)
in which VGW =
By, J d ¢~ + ~ ~ (Q . 3. i ~ ~ he—i(c~+~y) (25)
for the wave component VGN =
icons icing 13" >~ / " do (26a)
for the local component with J\r ~ defined as
J\r'+ = W~—1/ fk~ (z—it (26b)
(l/1r) J (i cost icing 13~ my' do (27a)
for the complementary component with
Tic = [~kR+ik~) (~c—1/Z+)
(22a) —(ER—ik~32~o—1/Z-~/~2iLl) (27b)
In summary the free-surface part GF and its gradi-
ents VGF of ship-motion Green functions can be nu-
merically evaluated by using the above formulations
(20a) and (25) for Gw and VGW (21a) and (26a) for
GN and VGN and (22a) and (27a) for GO and VGC.
Once the integral equation (12) is solved the
time-harmonic pressure is given by Bernoulli s equa-
tion (neglecting second-order terms)
pj = -p(-iwfj + UW V>j) (28)
The added-mass coefficients (a~j) and damping coef-
ficients (bej) due to radiation waves are defined by
—:,;2a~j—imbue=—JJ pj ran dS (29)
for ~ j = 1 2 6. The wave exciting loads are
A A
FE =—aO JJ To + p7) 71E dS (30)
for k= 1 2 6 and then the amplitude of ship mo-
tions is evaluated by solving
6
it, [ _ w2(m~j + ahoy—imbue + comae = Flu (31)
j=1
for k= 1,2, ,6. In (31), mid and cej for ~,j =
1,2, ,6 are the inertia and stiffness matrices re-
spectively.
In the same way, solution of the Neumann-Kelvin
problem (2) gives the steady potential ~ and the hy-
drodynamic steady forces by integrating the pressure
obtained from the Bernoulli's equation
Flu =—pU2 JJ [be—(V¢~2/2] nods (32)
In particular, the wave resistance ~—Fib, the lift forces
(F3) and trim moments (F5) are important compo-
nents in practice.
BI-QUADRATIC PATCH METHOD
The hull surface of ship is subdivided into NH curved
patches. Among the NH patches, NW patches touch
the free surface so that the waterline is constituted
by NW curved segments corresponding to the upper
OCR for page 71
side of Nw patches on the hull. Each patch is repre-
sented by the coordinates at 9 nodes Ail = (I, yj, Zj)
with (j = 1, 2, , 9) 8 of which are located along the
boundary sides in the anticlockwise order, and the
ninth one is at the center of the patch (Figure 13.
The surface of patch is defined by a parametric rep-
resentation of the form
9
= ~,~cjNj(n,v) with —1 < (n,v) < 1 (33)
j=1
where Nj(n,v) are shape functions given by
shown on Figure 1, the patch is subdivided into 5 x 5
panels whose reference point (close to the centroid of
each panel) is determined by the coordinates follow-
ing the 5 x 5 Gauss-Legendre rule. Furthermore, the
coordinates in the 3 x 3 Gauss-Legendre rule are used
in the definition of panels' nodes.
The velocity potential Aid, v) as well as other flow
values is expressed over the surface by its values at 9
nodes fj (j = 1, 2, , 9) in the same way as (333:
9
f(n, v) = ~ fjNj(n, v) (35a)
j=1
(v—liven—1~n and the two tangential derivatives (>u,Qv) by
(v—loved—1) (~+ 1)
4 N1 (n, v) =
—2 Non, v) =
4 Noun, v) =
—2 Noun, v) =
4 Noun, v) =
—2 Non, v) =
4 Noun, v) =
—2 Non, v) = (v—l)(v+ 1~n—1~n
Ng(n, v) = (v—l)(v+ 1~n—1~+1)
1 - 1)V(u+ 13n
1 - 13(V+ 13~+ 13n
}+ loved+ 13n
}+ livid—1) (~+ 1)
}+ liven—13n
The basic variation of A= In, y, z) over the surface of
patch is quadratic in both ~ and v curvilinear coor-
dinates so that it is called bi-quadratic patch. Such
a description of geometry is quite common in finite-
element analysis. A number of classic results concern-
ing its fundamental properties are directly used in the
present study and not described here in detail as they
can be found in Zier~kiewicz ~ Taylor (19893.
Figure 1: Quadratic patch and its subdivision
tv
~;;r
One of particularities concerns the subdivision of the
surface into smaller sub-patches (panels) necessary for
the purpose of computing influence coefficients. As
9
Gil,, ~v) = At, f j(N?,j, Nvj) (35b)
j=1
The first derivatives of ~ in global coordinate system
are determined by the identity
Gus, Ye Act = Tale ~v ~ id (36a)
where the superscript ~ it means the transpose of the
vector and the transformation matrix is given by
_ _ -1
T= fTij, (i, j) = 1, 2, 34 = as Yv Zv (36b)
rl1 rl2 rl3
In (36b), In, y, Z ju,v are derivatives of the coordinates
(~,y,z) with respect to (n,v). Thus, the values of
(cu. cv, On ~ involved in (13) are written by
(Cur Cv7 Cal) = (Tll'T12'T13) (36C)
Now we consider the integral equation (12) at the
point All, v) on the Ith patch, the potentials on the
left side of (12) become:
9 NH 9 NW 9
2 At, Nj Aft + At, At, Cur Aft + ~ ~ Wig Aft
=1 ~=lj=1 ~=1 j=1
(37a)
and that on the right hand side:
NH 9 NW 9
At, At, Dig At + ~ ~ Big At (37b)
~=1 j=1 ~=1 j=
with the coefficients defined by
Cot = JJ Nj(~c)Gr,((,~c~dS(~c) (37c)
sk
OCR for page 72
WE (I) = (U2/g) J {Nj (~)G~ (I, A)—
wk
[cnNnj (I) + CvNvj (~)]G(67 A) }tydl(~) MIJ
—i27 J Nj (~)G((, ~)tydl(~) (37d)
wk
Dj~(~) = JO Nj(~)G((,~)dS(~) (37e)
sk
Bj~(~) = (u2lg,J cnNj(~)G(~)tydl(~) (37f)
wk
In above equations, so and we stand for the surface
of kth patch and the waterline segment of kth patch
which touches the free surface.
Now we perform an integration on both sides
(37a) and (37b) of the integral equation after mul-
tiplying the 9 shape functions within the Ith patch.
This domain collocation of Galerkin type yields
NH 9 NW 9
k.itgj~ +E ~ W~it~j~ =
~=1 j=1 ~=1 j=1
NH 9 NW 9
~it>~j+~ ~ ~~it>~j (38)
~=1 j=1 ~=1 j=1
for l= 1,2, ,NH and i= 1,2, ,9. The influence
coefficients are written as
C3i = JO Ni (I) [Cj~ (I) + 2k Nj (I)] dS(~)
wait = JO Ni(~)Wj~(~)dS(~)
Sl
Eli = JO Ni(~)Dj~(~)dS(~)
Sl
Eli = JO Ni(~)Bj~(~)dS(~)
Sl
with Elk = 1 for I = ~ otherwise Elk =0.
Finally, we note
J=(j,~) and I=(i,l)
(40)
to represent the connection between the jth node
within the kth patch (ith node within the Ith patch)
and the Jth point (Ith point) in the general number-
ing of a total of NP points on the hull surface. This
connection permits to arrange the equation (38) into
= FI for I = 1,2, , NP (41a)
I=(i,l); J=(i,~)
No 9
Coil + ~ Wail (41b)
I=(i,l); J=(i,~)
NW 9
FI = ~ { ~ ~ Iamb +E ~ ~~il¢~j } (41c)
I =(i,l ) k=1 j= 1 k= 1 j= 1
The matrix MIJ in (41a) is square and of size equal
to the number of points NP used in NH patches to
represent the ship hull. The linear system of equa-
tions (41a) can be solved by either the usual Gauss-
Elimination method or an iterative approach.
The influence coefficients defined by (39) involve
two folds of integration on the influencing (kth) patch
or waterline segment (of kth patch) and on the influ-
enced (Ith) patch. The integrand includes the jth
shape function within kth patch and the ith shape
function within Ith patch, and the Green function
as well as its normal gradient. Following (15) and
(19), the Green function is known to be composed of
a Rankine component, wave and local components,
and complementary one for ~ > 1/4 (the sum of last
three components accounting for free-surface effects).
The integration of the local and complementary com-
ponents is in general an easy task as they are not
oscillatory. However, some numerical difficulties arise
from the integration of the Rankine and wave compo-
nents.
The Rankine component is only involved in the
(39a) two-fold integration on the influencing and influenced
patches since it is nil along the waterline. The in-
(39b) tegration on the influencing patch is performed by a
compound way. It consists of using a numerical inte-
(39c) "ration of Gauss-Legendre rule when two patches are
separated at a distance larger than the patches' di-
(39d) mension. When two patches are close to each other,
the subdivision illustrated on Figure 1 is used to car-
ried out the numerical integration on smaller panels.
When the influenced point is located on the influenc-
ing patch, the analytical integration of the singular
Rankine component is performed over panels. The
second fold of integration on the influenced patch is
performed numerically by using 2 x 2, 3 x 3 and 5 x 5
rules of Gauss-Legendre depending on the distance
between two patches.
The integration of the wave component is not as
easy as expected since the integrand can be highly
oscillatory especially along the waterline. The influ-
ence coefficients involving the wave component have
been performed in several manners as described in
the introduction. Finally, the method adopted in the
present study is to multiply the integrand of (20a) and
OCR for page 73
(25) by a smoothing function of type
~ ~ jexpf—C(k:/k~m—1~2rl] ~ > hm (42)
with r' > 1, a positive real constant C satisfying
e-c << 1 and Em being determined by the size of
the patches and smoothness of distribution of singu-
larities. A number of numerical tests show the con-
vergence and stable results which are few affected by
the choice of the values of (C, Kim)
NUMERICAL RESULTS
The above-described numerical method is now tested
through a comprehensive comparison with a series of
semi-analytical results as well as experimental mea-
surements. They include the classical analytical re-
sults for a floating hemisphere, semi-analytical solu-
tions to the Neumann-Kelvin flow around an ellipsoid
and semi-analytical results for an submerged sphere
advancing in water waves. The results obtained by
the present method are also confronted with those of
experimental measurements on Wigley hulls. Further-
more, most results are presented in tabulated form
which may be useful to readers.
Hulme's hemisphere
The first set of numerical results obtained by using the
new Green function method include added-mass and
damping coefficients and wave exciting forces on the
floating hemisphere studied by Home (1982~. The
analytical solution to the radiation of waves by a float-
ing half-submerged sphere in deep water is found in
Hulme's study by constructing an expansion for the
velocity potential in terms of series of spherical har-
monics. The added-mass and damping coefficients are
then evaluated and believed to be accurate to 4 deci-
mal places. Hulme's analytical results at zero forward
speed provide naturally the first benchmark for the
present numerical method.
Figure 2: Mesh used to represent Hulme's hemisphere
A mesh composed of a total of 66 bi-quadratic
patches representing the hull of a hemisphere of radius
Table 1: Added-mass and damping coefficients for
Hulme's hemisphere
kR all tell a33 b33
0.02 0.5049 0.0000 0.8652 0.0448
0.05 0.5113
0.10 0.5234
0.20 0.5526
0.30 0.5860
0.40 0.6187
0.50 0.6452
0.60 0.6599
0.70 0.6594
0.80 0.6433
0.90 0.6137
1.00 0.5749
1.10 0.5312
1.20 0.4867
1.30 0.4438
1.40 0.4042
1.50 0.3687
1.60 0.3374
1.80 0.2867
2.00 0.2494
2.50 0.1949
3.00 0.1718
3.50 0.1634
4.50 0.1656
6.00 0.1784
8.00 0.1962
10.00 0.2081
R = 1
view (Figure 2~.
The added-mass and damping coefficients
(a~,b~) in surge and (a33,b33) in heave are eval-
uated by using this mesh. Their values divided by
(pR327r/3) for Aft and ads, and (pR3~27r/3) for bar
and b33 are given in tabulated form (Table 1) for
various values of kR varying from 0.02 to 10. By
comparing with Hulme's results which are also given
in tabulated form, an excellent precision of 2 to 4
exact figures is observed.
The wave exciting forces (not given in Helms,
1982) are also evaluated and presented in tabulated
form (Table 23. The surge forces Fit and heave
F3 are divided respectively by (pgR3kaO2:r/3) and
(pg1rR2aO) where So is the incident wave amplitude.
As for the added-mass and damping, it is believed that
an accuracy of 2 up to 4 decimal places is obtained.
0.0001 0.8768
0.0011 0.8631
0.0082 0.7942
0.0256 0.7160
0.0558 0.6455
0.0989 0.5863
0.1519 0.5383
0.2096 0.5001
0.2659 0.4700
0.3152 0.4467
0.3542 0.4287
0.3817 0.4151
0.3985 0.4050
0.4062 0.3977
0.4067 0.3927
0.4019 0.3894
0.3933 0.3875
0.3698 0.3868
0.3426 0.3888
0.2769 0.3992
0.2234 0.4115
0.1810 0.4228
0.1290 0.4407
0.0779 0.4575
0.0490 0.4703
0.0262 0.4774
is constructed and illustrated by
0.1037
0.1817
0.2794
0.3255
0.3411
0.3391
0.3271
0.3098
0.2899
0.2690
0.2483
0.2284
0.2095
0.1918
0.1755
0.1605
0.1467
0.1227
0.1028
0.0645
0.0454
0.0311
0.0158
0.0065
0.0026
0.0012
a perspective
Farell's ellipsoid
The second set of numerical results concerns wave re-
sistance associated with the Neumann-Kelvin steady
flow applied on an ellipsoid. An ellipsoid with its ma-
jor axis parallel to the free surface and in the direc-
tion of its forward speed is considered in Farell (1973~.
The steady flow around the particular ellipsoid of rev-
olution (with an identical length of two other small
axes), called slender prolate spheroid by Farell, is an-
alyzed by expanding the velocity potential in forms
OCR for page 74
Table 2: Wave loads on Hulme's hemisphere
OR JRe{F1 } ~m{F1 } JRe{F3} ~m{F3}
-0.9751
-0.9378
-0.8777
-0.7688
-0.6751
-0.5941
-0.5234
-0.4608
-0.4051
-0.3549
-0.3095
-0.2684
-0.2309
-0.1968
-0.1657
-0.1374
-0.1117
-0.0884
-0.0483
-0.0162
0.0279
0.0597
0.0592
0.0298
-0.0111
-0.0070
0.0072
0.02 1.4896
0.05 1.4732
0.10 1.4468
0.20 1.3959
0.30 1.3450
0.40 1.2898
0.50 1.2270
0.60 1.1550
0.70 1.0744
0.80 0.9880
0.90 0.8998
1.00 0.8136
1.10 0.7324
1.20 0.6578
1.30 0.5905
1.40 0.5302
1.50 0.4763
1.60 0.4280
1.80 0.3455
2.00 0.2775
2.50 0.1505
3.00 0.0649
3.50 0.0092
4.50 -0.0384
6.00 -0.0202
8.00 0.0119
10.00 0.0011
0.0000 -0.0006
0.0001 -0.0035
0.0010 -0.0121
0.0070 -0.0375
0.0201 -0.0661
0.0400 -0.0934
0.0643 -0.1174
0.0888 -0.1376
0.1091 -0.1541
0.1216 -0.1669
0.1244 -0.1763
0.1181 -0.1828
0.1043 -0.1867
0.0853 -0.1882
0.0635 -0.1877
0.0406 -0.1854
0.0180 -0.1815
0.0033 -0.1763
-0.0403 -0.1628
-0.0688 -0.1461
-0.1059 -0.0887
-0.1076 -0.0550
-0.0884 -0.0162
-0.0360 0.0249
0.0164 0.0178
0.0064 -0.0097
-0.0056 -0.0003
of source distribution of spheroidal harmonics. The
wave resistances are then evaluated for a series of el-
lipsoids by varying the ratio of length between the
major and small axes at three depths of submergence
relative to the focal distance.
A mesh composed of a total of 90 bi-quadratic
patches representing the ellipsoid's hull is constructed
and illustrated by a perspective view (Figure 33. The
lengths of major and small axes are respectively 2a =
2.3 and 2b = 0.4 so that the focal distance is c =
ia2 _ h2) = 1.132475.
The wave resistance applied on the ellipsoid is
evaluated by the present numerical method. The re-
sults are presented in tabulated form (Table 3) for
three relative depths of submergence
(d/c)~,2,3 = (0.252,0.3266,0.5)
as used by Farell. The Froude number and the wave-
Figure 3: Mesh used to represent Farell's ellipsoid
0.35
0.40
0.42
0.45
0.48
0.50
0.52
0.54
0.55
0.56
0.58
0.60
0.62
0.65
0.70
0.75
0.80
Table 3: Wave resistance coefficients on
Farell's ellipsoid
F Cw (d/c)1 Cw (d/c)2 Cw (d/c)3
0.0049
0.1222
0.2167
0.3808
0.5396
0.6306
0.7059
0.7648
0.7883
0.8081
0.8372
0.8541
0.8607
0.8557
0.8201
0.7663
0.7059
0.2907
1.2293
1.5862
2.0017
2.2685
2.3762
2.4386
2.4649
2.4673
2.4636
2.4410
2.4027
2.3534
2.2655
2.1017
1.9351
1.7766
0.0614
0.5620
0.8244
1.1799
1.4480
1.5739
1.6614
1.7163
1.7332
1.7442
1.7505
1.7401
1.7169
1.6655
1.5543
1.4318
1.3108
resistance coefficients in Table 3 are defined as
F'l = U/N/~ and Cw
-1OOOF~ /~7rpgc3)
respectively. Cw is given at three relative depths of
submergence (d/c). By reporting the values of Cw
given in Table 3 at (d/c)~ = 0.252 and (d/c)2 = 0.3266
to Fig.1 and Fig.2 in Farell (1973), any visual differ-
ence from the corresponding curves can be observed.
Furthermore, the values of Cw for a/b = 5.75 at
(d/c)3 = 0.5 given in Table 3 are well located in the
middle between the curves for a/b = 5.5 and a/b = 6
on Fig.3 in Farell (1973~. These observations show
that the present numerical method provides results
that match Farell's.
Further to above satisfactory validations for two
special cases at r=0, namely Hulme's analytical solu-
tion for a hemisphere at zero forward speed (F'2 = 0)
and Farell's semi-analytical solution for an immerged
ellipsoid advancing beneath the free surface at zero
frequency (~=0), the next logical step is to consider
the general case at ~ 7{ 0.
Wu's sphere
The first benchmark for the general case is due to
the recent work by Wn (19953. Wu used the multi-
pole expansion extended from that developed for zero
forward speed in Ursell (1949) to derive directly the
velocity potential around a submerged sphere advanc-
ing in water waves. The linearization of the condition
on the free surface leads to the decomposition of total
velocity potential into a steady part and an unsteady
part as (1~. Wu assumed further that the steady flow
is small on the free surface for a submerged body but
not on the body surface, so that the exact interaction
between steady and unsteady flows through the mj
terms (8) derived from the Neumann-Kelvin solution
is taken into account in the evaluation of unsteady
OCR for page 75
velocity potential. This original work yields accurate
1 · 1 A · ·~ ~ ~ · Figure 6: Added-mass and damping coefficients
results with ~ s~gn~ncant ngures for wave resistance . . ,
1 . ~ . In sway motion a22, b22) of Wu s sphere
anct alit forces, actcrect-mass anct clamping coemc~ents 3 0
as well as wave loads.
Figure 4: Mesh used to represent Wu's sphere
A mesh composed of 56 bi-quadratic patches rep-
resenting the sphere of radius R= 1 is constructed
and illustrated by a perspective view (Figure 4~. One
forward speed corresponding to a Froude number
Fat = U/v:~= 0.4 and two depths of submergence
(h/R) were studied. The case of h/R= 2 is one pre-
sented in Wn (1995) for which the added-mass and
damping coefficients are evaluated and presented in
Figures 5-8 as well as wave loads in Figures 9-11, and
wave resistance and lift forces in Figure 12. To note
that Wu's method gives results in water of finite depth
and to be close to those in deep water, a depth of 10
times the sphere radius is used in Wu's computations.
Figure 5: Added-mass and damping coefficients
in surge motion bait, balk of Wu's sphere
3.0 -
2.0
1.0 -
O- _
~~0
—C
/,/N
~ ~ ~ ~-~-~—.~—.~_ ~_.~_.~ r
0 0.5 1.0 1.5
The abscissae in Figures 5-8 are the reduced
wavenumber kR while the values of added-mass and
damping coefficients are non-dimensionalized by
(pR34~/3) -A and and (pR3~2~/3) -A bej
respectively. The values of added-mass coefficients
and are represented by the solid line obtained from the
present numerical method and by the circle symbols
2.0 ~
.0
O—
1 - _
it_
.~-~~~~~-~—-A—-A—-A—~—~—-A—A- - -'
0 0.5 1.0
Figure 7: Added-mass and damping coefficients
in heave motion (a33 7 baa) of Wu's sphere
3.0
2.0 ~
1.0 -
In'
o
0 0.5 1.0
for Wu's semi-analytical solution. Those of damping
coefficients bed are represented by the dot-dash line
and square symbols for the present numerical method
and Wu's semi-analytical method respectively. It is
observed that very good agreement between the two
methods is obtained, despite the difference in the eval-
uation of mj terms and in waterdepth. The present
method uses the double-body flow to derive the mj
terms, while Wu's uses the Neumann-Kelvin flow.
The difference may be considered as negligible for the
sphere at this depth of submergence.
The critical value A= 1/4 is located at kR~
0.3906 where the effect of change of flow regime is
remarkable in all figures except the added-mass and
damping coefficients in sway motion (a22,b22) illus-
trated on Figure 6, which seem not to be affected. It
is known from previous studies (Grne ~ Palm, 1983
and Lin ~ Yne, 1993) that hydrodynamic coefficients
vary sharply for ~ crossing the critical value but not
singular. The present results seem to confirm that a
finite value does exist at ~ = 1/4.
Due to the effect of forward speed, the symmetry
OCR for page 76
Figure 8: Added-mass and damping coefficients
in surge-heave motion (aid, bat ~ of Wu's sphere
0-5
o
A
i\
j \,
i O.
i
i
1
-~''n_.N
'..~,
~—-O—-O—-~._.~._. ._. ._
-it
1.0 1.5
between the coefficients in surge and sway motions is
lost (comparing Figures 5 and 6~. Furthermore, the
coupling coefficients in surge-heave motions are not
zero and far from negligible as shown on Figure 8.
Numerical results shows that the reverse flow relation
given in Timmar~ ~ Newman (1962)
1.0~
(aid, big) = - (am, bat) 0_
is satisfied accurately to 3 decimal places, so that only
(aid, bat) are illustrated on Figure 8.
Figure 9: Wave loads Fit on Wu's sphere
1.5
1.0 -
0.5
0-~
_~ ,.
JO i
x. ~
'to
,,~
1 1
0 0.5 1.0 1.5
Solutions to the diffraction problem are repre-
sented on Figures 9-11 for wave loads in surge (F~),
sway (F2) and heave (F3), respectively. The heading
of incoming waves is 3= 3:r/4. The abscissae are the
reduced wavenumber koR with k0 the wavenumber of
incoming waves. The values of forces are divided by
(pg7rR3 ~aO) ~ (~elFi7 F2), ~mIF31)
(pg1rR3kaO/10) ~ (~mlF~ ~ F2~' ~elF3~)
where ~ is the wavenumber associated with the en-
counter frequency and No the amplitude of incom-
ing waves. It may be noted that (F~,F2,F3) in the
Figure 10: Wave loads F2 on Wu's sphere
0.5
no
—0_S—
-1.0 -
.~,N,~_,~,~.~
0.
I .)
Figure 11: Wave loads F3 on Wu's sphere
2.0 -
n
a,
An,
'a..
~~-_.~,, ~
-—in-—~._~._~._~._
present study are equivalent to the complex conjugate
of tiff, iFy, iFz) in Wn (1995), following the conven-
tion used in the present method and that used by Wu.
In Figures 9-11, the real and imaginary parts of
(F~,F2,F3) are represented respectively by the solid
and dot-dash lines for the present numerical method,
and by the circles and squares for Wu's method. Ex-
cept at small wavenumber where waterdepth may
have noticeable effects, the results of both methods
are in very good agreement. In the same way as the
added-mass and damping coefficients, the change of
flow regime at koR ~ 0.2937 corresponding to the crit-
ical value A= 1/4 yields rapid variation in surge forces
Fit and heave forces F3.
Finally, the wave-resistance and lift coefficients
(Cw,C~) on Wu's sphere at this depth (h/R = 2)
of submergence are evaluated and presented on Fig-
ure 12 by the solid and dot-dash lines respectively.
The abscissae are the Froude number Fn = U/~
while the values of both coefficients here are non-
dimensionalized as Cw =—F~/(pg7rR3) and CL =
F3/(pg7rR3~. A precision of at least 2 exact figures is
found for the results of the present method by compar-
ing with Wu's represented by the circles and squares.
The case of a smaller depth of submergence h/R=
OCR for page 77
Figure 12: Wave-resistance and lift coefficients
on Wu's sphere
~ 1—
0.05 -
O-
-0.05 -
-0.1 -
-0.15 -
0 0.5 1.0 1.5 2.0
— ma_
hi,
\
lo.
'A
A.
'at,
.~
I
'en.
1.1 (h is the distance of the sphere center from the
free surface) is now considered. The same mesh (Fig-
ure 4) is used in the computation of added-mass
and damping coefficients, wave loadings and wave-
resistance and lift. Due to the small submergence,
differences between the present numerical method in-
volving mj terms derived from the double-body flow
and Wu's semi-analytical method involving mj terms
derived from the Neumann-Kelvin flow may be ex-
pected. Indeed, some discrepancy between two meth-
ods for added-mass and damping coefficients, and
wave loadings, is found and considered to be reason-
able because of differences in the two methods ex-
plained above. To confirm this, an interesting case,
to compute added-mass and damping coefficients by
setting zero values for m1,2,3 terms, is studied. The
results presented in tabulated form (Tables 4 and 5)
are in excellent agreement with Wu's (private com-
munication) with an accuracy of 2 exact figures after
the decimal point.
Table 4: Added-mass coefficients for Wu's sphere
by setting m1,2,3=0
kR all a22 a33 al3 a3
0.05 0.6709
0.10 0.7090
0.15 0.7421
0.20 0.7300
0.25 0.6026
0.30 0.4021
0.35 0.2782
0.40 0.3025
0.45 0.3616
0.50 0.3903
0.60 0.4250
0.70 0.4456
0.80 0.4580
0.90 0.4648
1.00 0.4675
1.10 0.4673
1.30 0.4614
1.50 0.4520
0.6111 0.8358
0.6320 0.9068
0.6582 0.9699
0.6895 0.9554
0.7245 0.7453
0.7589 0.4196
0.7785 0.2406
0.7021 0.3481
0.6096 0.4040
0.5599 0.4173
0.4997 0.4169
0.4619 0.4051
0.4351 0.3901
0.4153 0.3748
0.4002 0.3602
0.3886 0.3468
0.3726 0.3241
0.3633 0.3068
-0.0485
-0.1038
-0.2047
-0.3689
-0.5273
-0.5053
-0.3236
-0.0995
-0.0762
-0.0504
-0.0075
0.0261
0.0526
0.0731
0.0889
0.1006
0.1149
0.1205
0.0489
0.1043
0.2052
0.3692
0.5274
0.5052
0.3235
0.0994
0.0761
0.0504
0.0075
-0.0262
-0.0526
-0.0732
-0.0890
-0.1007
-0.1151
-0.1207
Table 5: Damping coefficients for Wu's sphere
by setting m1,2,3=0
kR tell b22 b33 bl3 b3
0.05 0.0684
0.10 0.1576
0.15 0.3178
0.20 0.5776
0.25 0.8342
0.30 0.8206
0.35 0.5636
0.40 0.2122
0.45 0.1698
0.50 0.1440
0.60 0.1226
0.70 0.1216
0.80 0.1300
0.90 0.1422
1.00 0.1552
1.10 0.1674
1.30 0.1864
1.50 0.1974
0.0024 0.1188
0.0098 0.2732
0.0276 0.5508
0.0628 0.9992
0.1254 1.4356
0.2316 1.4052
0.4078 0.9724
0.6570 0.5066
0.6460 0.5516
0.6192 0.5588
0.5680 0.5608
0.5228 0.5562
0.4822 0.5468
0.4454 0.5338
0.4116 0.5180
0.3806 0.5000
0.3256 0.4610
0.2788 0.4202
0.1760 -0.1758
0.2836 -0.2832
0.3662 -0.3654
0.3246 -0.3236
-0.0202 0.0214
-0.5502 0.5512
-0.8670 0.8676
-0.7646 0.7650
-0.6480 0.6484
-0.5996 0.6000
-0.5398 0.5402
-0.4922 0.4928
-0.4480 0.4484
-0.4052 0.4056
-0.3640 0.3644
-0.3246 0.3252
-0.2532 0.2536
-0.1918 0.1922
The results of added-mass and damping coef-
ficients with mj terms derived from the double-
body flow are presented on Tables 6 and 7 respec-
tively. Compared with the results obtained by set-
ting m1,2,3 = 0 (Tables 4 and 5), the importance of
mj terms is evident, especially in the region of small
wavenumber.
Table 6: Added-mass coefficients for Wu's sphere
with double-body mj terms at (h/R= 1.1)
kR all a22 a33 ala a3
0.05 8.3899
0.10 4.5933
0.15 3.2064
0.20 2.2659
0.25 1.3489
0.30 0.6394
0.35 0.3494
0.40 0.4361
0.45 0.5678
0.50 0.6108
0.60 0.6411
0.70 0.6459
0.80 0.6407
0.90 0.6308
1.00 0.6187
1.10 0.6057
1.30 0.5800
1.50 0.5568
5.6621 10.1824
3.2424 5.4949
2.4703 3.6898
2.1173 2.3215
1.9354 0.8796
1.8324 -0.1786
1.7358 -0.4628
1.3827 0.0009
1.0471 0.2002
0.8823 0.2606
0.7070 0.2999
0.6138 0.3093
0.5568 0.3110
0.5193 0.3107
0.4935 0.3101
0.4753 0.3098
0.4527 0.3105
0.4406 0.3126
-2.4652 2.4868
-2.0618 2.0775
-2.1359 2.1494
-2.3140 2.3257
-2.2320 2.2409
-1.6248 1.6298
-0.8836 0.8859
-0.2378 0.2389
-0.1947 0.1959
-0.1400 0.1411
-0.0606 0.0615
-0.0105 0.0111
0.0216 -0.0211
0.0421 -0.0417
0.0550 -0.0547
0.0627 -0.0625
0.0686 -0.0684
0.0674 -0.0674
The wave loads are presented in Table 8.
The forces (F1, F2, F3) are non-dimensionalized by
(oq~rR3kan) with ~ the wavenumber associated with
encounter frequency and Do the amplitude of incom-
ing waves. Finally, the wave-resistance and lift forces
(Cw, Cat ~ as well as the trim moment (CM) about the
y-axis at the sphere center (0, O. - 1. 1R) are given in
Table 9. As in Figure 12, the Froude number is de-
fined as Fn = U/,~, while the wave-resistance and
lift coefficients are defined by Cw=—Fl/(pg1rR3) and
OCR for page 78
Table 7: Damping coefficients for Wu's sphere
with double-body mj terms at (h/R= 1.1)
kR tell b22 b33 bl3 b31
0.05 2.7592 0.1186 6.3416 3.4246 -3.4454
0.10 2.6528 0.1334 5.7424 2.2746 -2.2844
0.15 2.9166 0.1804 6.0026 1.4518 -1.4544
0.20 3.2782
0.25 3.2540
0.30 2.4422
0.35 1.3998
0.40 0.4062
0.45 0.2992
0.50 0.2244
0.60 0.1484
0.70 0.1210
0.80 0.1140
0.90 0.1154
1.00 0.1196
1.10 0.1242
1.30 0.1300
1.50 0.1304
0.2672 6.4542
0.4168 6.1498
0.6700 4.4356
1.1018 2.4332
1.7212 0.9122
1.4996 0.9668
1.2914 0.8942
0.9948 0.7474
0.7968 0.6306
0.6550 0.5388
0.5482 0.4654
0.4650 0.4054
0.3982 0.3558
0.2986 0.2782
0.2286 0.2208
0.3232 -0.3190
-1.3126 1.3224
-2.6232 2.6348
-2.9276 2.9372
-2.2364 2.2420
-1.7172 1.7216
-1.4580 1.4616
-1.1398 1.1426
-0.9338 0.9360
-0.7834 0.7852
-0.6670 0.6686
-0.5740 0.5756
-0.4982 0.4996
-0.3832 0.3840
-0.3014 0.3022
Table 8: Wave loads on Wu's sphere
~oR F1
0.05 (1.2025,-0.1412)
0.10 (0.9897,-0.2357)
0.15 (0.9398, -0.3721)
0.20 (1.0862,-0.4564)
0.25 (1.2295, -0.2677)
0.30 (1.0870,-0.0292)
0.35 (0.9705,-0.0172)
0.40 (0.8909, 0.0088)
0.45 (0.8207, 0.0320)
0.50 (0.7559, 0.0512)
0.60 (0.6390, 0.0774)
0.70 (0.5380, 0.0904)
0.80 (0.4520, 0.0942)
0.90 (0.3794, 0.0923)
1.00 (0.3187, 0.0869)
1.10 (0.2679, 0.0798)
1.30 (0.1901, 0.0642)
1.50 (0.1356, 0.0497)
F2
(-1.3135 -0.0076)
(-1.2200 -0.0155)
(-1.1581 -0.0351)
(-1.1170 -0.0776)
(-1.0793 -0.1673)
(-0.9178 -0.3356)
(-0.7326 -0.2838)
(-0.6350 -0.2383)
(-0.5648 -0.2036)
(-0.5089 -0.1763)
(-0.4217 -0.1360)
(-0.3546 -0.1075)
(-0.3006 -0.0864)
(-0.2561 -0.0703)
(-0.2189 -0.0577)
(-0.1874 -0.0476)
(-0.1379 -0.0329)
(-0.1017 -0.0229)
F3
(-0.3104, 1.6410)
(-0.3887, 1.6639)
(-0.5668, 1.5282)
(-0.6814, 1.1516)
(-0.4387, 0.8159)
(-0.1865, 0.8901)
(-0.2283, 0.8666)
(-0.2310, 0.8061)
(-0.2262, 0.7416)
(-0.2179, 0.6791)
(-0.1964, 0.5661)
(-0.1726, 0.4708)
(-0.1492, 0.3917)
(-0.1276, 0.3264)
(-0.1082, 0.2724)
(-0.0913, 0.2277)
(-0.0641, 0.1595)
(-0.0444, 0.1117)
Cat, = F3/(pgfrR3), and the trim moment coefficient
by CM = F5 / ~ pg;rR4 /100 ~ . An excellent agreement is
observed between the results of (Cw, Cat ~ and those
by Wu (private communication), as at least two ex-
act figures after the decimal point are found in both
sets of results. Compared with the values presented
on Figure 12, the wave resistance and lift forces are
augmented by a factor of 5 and 7 respectively.
Journee's Wigley-hull
The second benchmark study for the general case at
{ 0 involves confronting the results of experimental
measurements on Wigley hulls. Further to the clas-
sic experimental data on vertical motions, wave loads
and added resistances of two first Wigley hullforms
in Gerritsma (1988), new tests on two other Wigley
hullforms with a midship section coefficient of 2/3 and
length-breadth ratios of 10 and 5 have been made in
Jonrr~ee (1992~. Although only the head sea is tested
Table 9: Wave-resistance, lift and moment coefficients
on Wu's sphere
Fr, CW CL CM
0.20 0.0001 0.0221 0.0001
0.30 0.0296
0.40 0.1497
0.50 0.2759
0.60 0.3270
0.70 0.3178
0.80 0.2824
0.90 0.2415
1.00 0.2035
1.10 0.1710
1.20 0.1442
1.30 0.1224
1.40 0.1047
1.50 0.0903
1.60 0.0784
1.70 0.0687
1.80 0.0605
1.90 0.0537
2.00 0.0480
0.1204 0.0026
0.2154 0.0085
0.1806 0.0130
0.0640 0.0154
-0.0601 0.0160
-0.1654 0.0157
-0.2517 0.0151
-0.3248 0.0145
-0.3905 0.0140
-0.4526 0.0135
-0.5137 0.0130
-0.5756 0.0127
-0.6392 0.0123
-0.7053 0.0120
-0.7742 0.0118
-0.8464 0.0116
-0.9219 0.0114
-1.0010 0.0112
and the surge motion is restricted (i.e. the model was
free to perform heave and pitch motions only), a large
number of reliable data were achieved. For the sake of
space, the results on the model called Wigley model
III are used for the present comparative study. Its
main dimensions (length/breadth/draught=L/B/T)
and dynamic characteristics are
L/B/T= 3/0.3/0.1875
Centre of gravity above base KG =0.1700
Gyration radius in pitch Ray =0.7500
The hull shape is defined analytically by
y = (1 - (z/T)2] (1 - (2~/L)2] (1 + (2~/L)4/5JB/2
in a coordinate system with the origin at the amid-
ship and on the waterplane. The Wigley-hull is panel-
ized with 128 bi-quadratic patches and illustrated by
a perspective view (Figure 13~.
Figure 13: Mesh representing Journee's Wigley-hull
This mesh is used, and the case of a forward
speed corresponding to the Froude number F'l =
U/~/~= 0.3 (and at zero forward speed for mo-
tions) is considered in the present numerical method.
OCR for page 79
The added-mass and damping coefficients in heave
(a33,b33), in pitch (a55,b55) and in coupling heave-
pitch (a35,b35) are presented in Figures 14, 15 and
16, respectively. The abscissae are the reduced circu-
lar frequency w :7: while the values of added-mass
(a33, a35, a55) and damping coefficients (b33, b35, b55)
are non-dimensionalized respectively by (p:L0~2)
and (p:L0~2~) with V= 0.0780 the displaced
volume, and depicted by the solid and dot-dash lines.
The results of experimental measurements are rep-
resented by the circles and downward-pointing tri-
angles (corresponding to two sets of repeated model
tests) for the added-mass coefficients, and the squares
and upward-pointing triangles for the damping coeffi-
cients. The numerical results are in very good agree-
ment with those of experimental measurements, ex-
cept for those of damping coefficients b55 in pitch.
However, the two sets of results (b55) have similar vari-
ation and close values. The critical value of a= 1/4
is located at 7=5/6, where a rapid variation is
expected due to the change of flow regime.
Figure 14: Added-mass and damping coefficients
(a33,b33) for Journee's Wigley-hull 05
5.0 -
40]
. . ~
3.0 -
2.0 -
1.0 -
r
.,
1
. .
1 \
. .
il \ ~ ~
I
. . . . . . .
0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Figure 15: Added-mass and damping coefficients
0.2 -
1 ~
(a55, b55) for Journee's Wigley-hull
n
,i1
1~ ..,-~_ ._
1
0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
The vertical wave exciting forces in heave (F3)
Figure 16: Added-mass and damping coefficients
(a35, b35) for Journee's Wigley-hull
1.0 -
0.5 -
o-
-
-05 -
-1.0 -
A
! !
, i
1 1 1 1 1 1
0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Figure 17: Wave loads on Journee's Wigley-hull
Exciting forces in heave F3
~.o~ ~
o-
A<
./
al It j
0 1.0 2.0 3.0
and moments in pitch (F5) are depicted on Figures 17
and 18. The abscissae are the ratio between the wave-
length of incoming waves and hull length (~\/L). Wave
loads (F3) and (F5) are non-dimensionalized respec-
tively by (C33aO) and (C55~0ao) where k0 and aO are
the wavenumber and amplitude of incoming waves,
C33 = 6119 and C55 = 2874 are the hydrostatic stiff-
ness in heave and pitch motions. The numerical re-
sults are depicted by the solid lines and two sets of
model tests by the circles and downward-pointing tri-
angles. Again, the numerical and experimental results
are very close to each other.
Finally, response-amplitude operators (RAOs) of
heave and pitch motions are represented in Figures
19 and 20 at forward speed (Fr' = 0.3) as well as
at zero forward speed (Fr' = 03. The abscissae in
both figures are the ratio (~\/L), while the amplitude
values of heave displacements and pitch angles are
non-dimensionalized by the wave amplitude (aO) and
(aO21r/L) respectively. The numerical and experimen-
tal results at Fr' = 0.3 are depicted respectively by the
solid lines and circles and those at zero forward speed
by the dot-dash lines and squares. The important ef-
OCR for page 80
Figure 18: Wave loads on Journee's Wigley-hull
Exciting moments in pitch F5
.0 -
0.5 -
~-
9.n
Figure 19: Heave motions of Journee's Wigley-hull
2.0
imp''
0 1.0
_ ~
2.0 3.0
feet of forward speed on heave and pitch motions is
evident by comparing the solid lines (Fin = 0.3) and
dot-dash lines (Fin = 0) in both figures. The excel-
lent agreement between numerical results (solid and
dot-dash lines) and experimental measurements (cir-
cles and squares) in both cases (F'2=0.3 and F'2=0)
shows that the present method can provide very good
prediction of wave-induced ship motions and loads.
DISCUSSIONS AND CONCLUSIONS
In this paper, we have summarized the theoretical
background and numerical development of a new 3D
method to predict the wave-induced ship motions and
loads. The excellent level of agreement with semi-
analytical results and experimental measurements is
due to several important features of the method.
First of all, the new formulation of ship-motion
Green function and its gradients expressed by vari-
ous components makes it possible to deal with differ-
ent components separately and to develop appropri-
ate algorithms in the accurate and efficient compu-
tation of influence coefficients associated with them.
The higher-order description of ship hull geometry by
Figure 20: Pitch motions of Journee's Wigley-hull
3.0 -
0 1.0 2.0 3.0
bi-quadratic patches and continuous representation of
the velocity potential lead to the solution of very good
precision and the reduction of computing time. In all
previous studies, the indirect method corresponding
to the source formulation is surprisingly used only.
The present study makes use of the direct method
corresponding to the potential formulation which is
directly derived from the Green's identity and prefer-
able although two methods are thought to be equiv-
alent. Furthermore, the Galerkin solution procedure
is adopted instead of the procedure of point colloca-
tion. This procedure of domain collocation is well
recognized to be superior in numerical stability and
convergence. Finally, the mj terms involving in the
boundary condition on the hull surface are evaluated
by using the numerical method which permits to have
accurate double derivatives of velocity potential with-
out any numerical extrapolation or interpolation. It
is understood that the mj terms are critically impor-
tant in the computation of added-mass and damping
coefficients especially for large Froude number.
The present numerical method taking advantage
of above-mentioned properties can be considered to be
sound and robust as it is able further to provide very
good results in a number of applications, although
more validations are needed especially for hulls of
more complex form as well as for computing the val-
ues of local pressures and wave patterns. It also offers
a solid basis of development for a number of perspec-
tives. Firstly, the suppression of irregular-frequency
effects can be made by applying an additional condi-
tion of zero-potential over the surface of waterplane.
The present method makes it possible since the influ-
ence coefficients between two patches located at the
waterplane can now be computed correctly. Secondly,
the linearization of free-surface boundary condition
about a basic flow like the double-body potential (in-
stead of a uniform flow) yields an extension of the in-
tegral equation by an additional zonal integral on the
free surface close to the waterline. Finally, the present
OCR for page 81
study can also be very useful in the development of a
method to couple with a near-field calculating method
accounting for nonlinear or/and viscous effects.
ACKNOWLEDGMENTS
The first author is greatly indebted to Professor Wu
Guo Xiong from University College London for pro-
viding comprehensive data of his semi-analytical re-
sults for a submerged sphere which have been criti-
cally useful in the development of computer codes.
The work on influence coefficients is partially sup-
ported by a research grant from the DGA and the Eu-
ropean Commission under the BRITE-EURAM Pro-
gram as part of the research project BE97-4406 en-
titled "Advanced Method to Predict Wave-Induced
Loads for High Speed Ships (WAVELOADS)".
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Representative terms from entire chapter:
free surface
