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OCR for page 239
Numerical Appraisal of the
New Slender Ship Formulation In Steady Motion
H. Marco (University of California, Santa Barbara, USA)
W.S. Song (Shanghai liao Tong University, China)
ABSTRACT
A new formulation for the fluid motion
around a slender ship is developed, on the
basis of an asymptotic expression of the
Kelvinsource around its track. The boundary
value problem is expressed by an integral eq
ation which is much more simplified than
the solution of the NeumannKelvin approxima
tion. In order to examine the validity of
this theory, numerical computations are carri
ed out with respect to the pressure distribu
tion, wave pattern and wave resistance of
several types of hull forms, i.e. the Wigley
hull, a sailing yacht hull, and a Series
60 (Cg= 0.60) hull. The results are compared
with experimental data.
1. INTRODUCTION
The final goal of ship hydrodynamics
is the theoretical determination of hydrodyna
mic forces acting on the ship hull within
the accuracy of practical allowance. One
of the most important in this respect is
the computation of wave resistance in the
steady forward motion. The pressure distri
bution over the hull surface becomes important
when the boundary layer calculation is intend
ed. Because of the complex geometry of the
ship hull, the solution of fully or partially
nonlinear boundary value problem by means
of the computational approach of numerical
simulation has not achieved the practical
feasibility yet. The analytical solution,
on the other hand, has to depend on the per
turbation technique which leads to the linear
ization of the problem as the first approxi
mation. It is well known, that the classical
thin ship perturbation has not provided result
which shows a satisfactory agreement with
measured data. Several versions of linearized
theory have been proposed, such as the Neumann
Kelvin approximation (1~. However most of
them are rather inconsistent approach, lacking
the rational basis in the sense of the pertur
bation analysis. The slender body theory
is another possibility of rational approach
of this problem. The first attempt of the
application of the slender body theory to
ships in steady forward motion appeared in
1962  1963 (2~3~4~5~. However the formul
239
ation of the wave resistance by this theory
was found quite unsatisfactory, because the
values computed according to this theory
showed a remarkable deviation from measured
values, and the agreement was even poorer
than the result of the classical Michell
theory (6~. No progress in this problem
has been observed for as long as 20 years
since that time. In 1982' one of the present
authors developed a new approach to the slen
der ship in steady forward motion (7~. The
difference of this theory from the former
theory lies in the treatment of the singular
ity which represents the body. The original
formulation has followed the method of pertur
bation analysis, which is employed in the
slender body in the unbounded fluid. Then
it assumes that the slender body is represent
ed by the source distribution along the longi
tudinal axis. The new theory, on the other
hand~begins with the expansion of the Kelvin
source aroundits track. It is disclosed that
it is not possible to represent the slender
ship floating on the free surface by the
source distribution along the longitudinal
axis considered in the free surface.
The boundary value problem is expressed
by an integral equation on the hull surface,
because the singularity representing the
hull must be distributed over the surface.
Then the solution is more complex than the
original slender body theory. However the
solution of the integral equation is much
more simplified than the solution in the
Neumannkelvin approximation. The reason
is first that the integral equation is of
the Volterra type, so that the boundary value
problem becomes parabolic. That means there
is no contribution from the disturbance in
the downstream to the boundary condition
at the upstream section of the body. The
marching procedure starting from the bow
end is possible to solve the boundary value
problem in each section. Secondly, the kernel
function of the integral equation can be
expressed by known functions, so that the
high accuracy of the numerical work is achiev
ed. An analytical method of solution by
means of the conformal mappi ng has been i ntend
ed, and several numeri Cal resul ts have been
obtai ned i n 1983. However i t i s found that
the accuracy of the computati on i s not sati s
OCR for page 239
factory (8~. Furthermore, the analytical
method is not suitable for the numerical
work, because the mapping of the transverse
section to a unit circle needs much computer
time. Then a numerical method of solution
is developed. This method employs the source
distribution to represent the hull, and the
density of sources is determined numerically
by the panel method. The program library
is prepared for the computation of kernel
functions. Three kinds of hull forms are
employed for the numerical example. They
are the Wigley hull, a sailing yacht hull,
and the Series 60 (CB= 0.60) model. Items
of the numerical work are the pressure distri
bution on the hull surface, wave resistance,
the lateral force when moving obliquely,
the wave profile alongside the hull, and
the wave pattern around the hull. Some of
the numerical results are compared with meas
ured data.
2. LINEARIZATION OF THE VELOCITY POTENTIAL
It is assumed that the fluid is inviscid
and incompressible, and the depth of water
is infinite. Take the Cartesian coordinate
system with the origin on the undisturbed
free surface, x and yaxes on the horizontal
plane, and zaxis directing vertically down
wards. Consider a slender ship fixed in
a uniform flow of velocity U in the direction
of positive x. Assume the irrotational motion
and write the velocity potential in the form
like Ux+~. The field equation is the Laplace
equation.
[L] V2¢ = 0
(1)
The boundary condition on the hull surface is
~ An= U3x/3n = Unx (2)
where n is the normal drawn outwards on the
hull surface, and ~ = al/3n. The kinematic
condition on the free surface at z=` is
[K] (U ~ ~x)~x+ ~y~y~ Liz= 0 (3)
The dynamic condition on the free surface,
that the pressure is constant, is
[D] U¢x+ 2~¢X2+ ~y2+ ~Z2) _ 9` _ o (4)
Since the depth of water is infinite, ~ = 0 at
Zen=, and ¢~0 at x,y+~. One can eliminate
between (3) and (4) such as
[F] [(U+$x~aa +$yaay +¢za33tu~x+ 2~¢x2+$y2+lz2)
_9z] = 0 (5)
In order to express the velocity potential
of the fluid motion around the hull, we assume
Green's function G(P,Q) with P=(x,y,z), Q=(x',
y',z'), and apply Green's theorem in the
x',y'z'space bounded by the hull surface
S bellow the still waterline, a large surface
Son surrounding S in the lower half space,
and the horizontal plane SO between S and
00.
: S+Sm+so ~ ~ P. Q ) ~$ ~ Q )~ ] d SQ
(6)
We have assumed the analytic continuation of ~
to the entire space below the still water
surface in the above equation. We will employ
the Kelvinsource as Green's function, which
satisfies the boundary condition,
U2¢  gaG = 0
(7)
on the horizontal plane z=0. If the surface S.
is taken at infinite distance, the integral
on S vanishes. On the horizontal surface
SO, we have the relation derived from the
free surface condition (5) such as
)=0= U ~ )Z=0+ ~ (X,y) (8)
where
t~ x, y ) = [ 2u($xTxx+$y~xy+¢zfxz )+2 ~¢x~y~xy+
+$xtz~xZ+¢y~z~yz )  y ~ 2~¢ ~ 2 ~
+/ (U2¢xxzg~zz)dz (9)
Making use of relations (7) (8) in the inte
gral on SO, and integrating by parts with
respect to x', we obtain
4~lSt ~ (P'Q)~(Q) ~ idSQ
4nglL [G(P'Q) ~  ¢(Q) ~ ~z'=odY'
o
+ 4~9i; t~x',y')G(P,Q)dx'dy' (10)
SO
where Lo is the curve of intersection of
the hull surface with the still water plane.
Now let us assume that the ship is very
slender and the beam to length ratio is a
small fraction £<< 1. Then the slope of the
hull surface to the longitudinal axis is
small in the order of £, i.e. nx= 0~£). From
the hull boundary condition (2), we have
An ~x~x+ny~y+nztzc Uo(£) (11)
where n , n , n are direction cosines of
the outXwardY norZmal to the hull surface.
Because of the slender body, n =O(1), nz=0~1).
If U=O(1), we have ~ =0~£), ~ YO(£). Since
the hull surface are] is regaZrded as 0~£),
the first term on the right hand side of (10)
is O(£2). t~x,y) is at most 0~£2) in the near
field in the area of 0~£) within S=, so that
the third term on the right hand side is at
most 0~£3 ). Omitting this term, we have
240
OCR for page 239
ff [a¢(Q)G(p Q'_~(Q)aG(P'0)]d
~ " S vilQ Q
_ 4U J [G(P,Q) ~ _ ¢(Q)DG(P Q)]zlOd
0 (12)
Next assume a velocity potential ¢' in the
lower half space, which is harmonic inside S.
and satisfies the boundary conditions,
¢'= ~ on S
and
U257T2 gang= 0 on z=0
Applying Green's theorem to $' and G(P,Q) in
the domain bounded by S and the plane z=0,
0 = 44~; [a~niQ)G(P'Q)~¢'(Q)3G(P Q)]dSQ
+ 4nglL [G(P'Q) ~ ~¢ (Q.) aX : ) ] Z , Body
o
..... (13)
where n' is the normal of S drawn inwards.
Adding (12) and (13), and putting
~ than + an')
we obtain
~ = ffS0(Q)G(P'0)ds + 9 fL o(Q)G(P'Q)nx~s ds
(14)
..... (15)
This is the basic equation of the Neumann
Kelvin approximation. If the slender body is
assumed, nx and dy/ds are 0(£~. Then the inte
gral along the waterline Lo is 0(£3 ), and can
be omitted. In consequence, the velocity pot
ential is given by
= iiS0(Q~G(P,QjdS (16)
Thus the fluid motion around the hull is exp
ressed by the distribution of Kelvinsources
over the hull surface. The Kelvinsource is
given by the formula,
G(P,Q) = G(x,y,z;x'y'z')
= ~ r + r'+ G'(x,y~z) (17)
G'(x,y,z) =
Ko lode ~ exp(kz+3kxcosK6+ikysin6) dk (18)
where r =[(xx' )2+(y_y')2+(Z_z' )2 ]2
r'=[(XX' )2+(y_y' )2+(z+z' '2 ]2
x=xx', y=yy', z=z+z', Ko= g/U2.
The integral with respect to k is taken along
the real axis indented by a small semicircle
in the lower side of the pole at k=KOsec20.
Then the velocity potential is written in the
form like
(19)
¢1 ifSo~x my LIZ )(r~ r,jdS (20,
$2 ffS0( x , Y , Z JIG (x,y,z~dS (21)
3. ASYMPTOTIC EXPRESSION FOR THE KELVINSOURCE
Let us consider the asymptotic behavior of
the Kelvinsource near the xaxis. In order
to find out the asymptotic expression of
G'(x,y,z), we consider the following integral
in the complex uplane.
I = ~ exp[uz+iu(xcosO+ysinO)] du (22,
c 'C u  KOsec'd
along a closed circuit C composed of the
positive real axis indented by a small semi
circle in the lower side around the pole at
u=K sec2D, and the positive or negative part
of The imaginary axis together with a large
quadrant arc connecting the ends of the axes.
In the case of xcosO +ysinO~ 0, the closed
circuit is taken in the first quadrant.
Since the pole is inside the contour, Cauchy's
theorem gives
Ic= 2ni(Residue at u=K0sec20)
If the radius of the large circle tends to
infinity, the integral along it vanishes,
so that
~ 0
I +r = 2niexp[K0zsec20+iK0sec20(xcosO+ysinO)]
and accordingly
exp[uZ+iu(xcKoSO+~inG)] du
~=tCostzKosec Using t(XcosO+ysin0)
J t + Ko sec ~ dt
 2ne KOzSeC ~sin[K0sec20(xcosO+ysinO)]
In the case of xcos0+ysinG<0 on the other hand,
the closed circuit is taken in the fourth quad
rant. Since the pole is outside the closed
circuit, Cauchy's theorem gives I = 0, and we
have the result, c
r exp[uz+iu(xcos0+ysinG)]
)0 u  K0sec20 du
tcostzKOsec2~sintz 
t2+ K02sec40 etlxcos~+ysin~ldt
241
OCR for page 239
Therefore the function 0'~x,y,z) is given by
2Ko 2 loos zKOseC2 S intz
see d  Liz+ K ~ sect ~
x et~xcosO+ysinO~dt
4Kof~ e~KOSec ~Zsin(KOxsecO)cos(KOysecOtanG)
x sec2OdO (23,
where 01 is an angle between ~/2 and n/2 such
that tanOl= x/~y;.
The double integral is bounded, and expressed
on the xaxis by known functions such as
11 sec2OdOi e~t~x~cosO ~ tdt
~ :tHl(Ko~x~)  Yl(Ko~x~) _ 1] (24)
where H is the Struve function and Y is the
Bessel Junction of the second kind `93. Next
we consider the single integral of the second
term on the right hand side of (23~.
Changing the integration variable by
secO = u'
e KOzSeC ~sin(KOxsecO)cos(KOysecOtanO)
~ ) ~ e KoZU sin(KOxu~cosKOyu~u21 )
x sec2OdO
_>DmJ~ e Kozu ~ U cos(KOyu~cos(KOyu
z x sin(KOxu~du
so
KodXJlCoS(KoXU)(~  u~du
2 Yl(KOX)  c°S(Kox)/(KOx)
The second integral is finite.
(27)
yiom~oe KoZU sin(KOxu~cosKOyu Mu = 0
z  O
Kox
(28)
Then the integral (25) is expressed asymptoti
cally as
+ 2 ~ e~v Zcos(v2y~sin(v ~ ~dv when x>O,
= 0 when x<0 (29)
Summarizing the above results, the asymptotic
expression for G(x,y,z) near the xaxis is
given by
G(x,y,z) ~ 87o: e v Zcos(v2y~sin(viOx~dv
x U dU when x>O, + ~KotHl(Kox)+3Yl(KoX)] + x ~ 2Ko
,~
rm
e KoZU sin(KOxu~cos(KOyui5~) u du
when x<0 (25)
It has an essential singularity along xaxis.
In order to isolate the singularity, let us
consider the identity,
~ e KoZU sin(KOxu~cos(KOyui;~) u du
= ~ e KoZU sin(KOxu~cos(KOyu2)du
0 1
 ~ e KoZ U s i n( Koxu ~ cos ( Kosu2 Mu
+ ~ e Kozu [ U cos(KOyui~cos(KOyu2 ~ ]
x sin(KOxu~du (26)
The singularity is condensed in the first
term on the right hand side. It is readily
shown that the last integral is bounded and
uniformly convergent on the xaxis.
~  ~KotHl(Kox)Yl(KoX)] + x 2Ko
when x>0 (30)
when x<0 (31)
The integral in (30) is expressed by the Fres
nel function of complex argument such as
E(x,y,z) = ~ e v Zcos(v2y~sin(vi~0x~dv
= ]m e~i KoX /4Z: F[XiKo/(2nZ)] (32)
where Z = y+iz, and
F(x) = C(x) + iS(x) = ~ ei~U2/2du (33)
o
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4. BOUNDARY VALUE PROBLEM FOR THE SLENDER SHIP
The velocity potential near the slender
ship is simplified by the asymptotic expres
sion of the Kelvinsource given in the preced
ing section. We have divided the velocity
potential into two parts ¢1 and ¢2 in (19~.
1 is expanded with respect to E . Omitting
higher order terms, it is expressed near
the hull by
x ln'Y Y.) + 'Z Z.' ds
¢1 is(x) ~ ~ (y_y )2+ (z+z )2
(34)
The expression for $2 near the hull is obtain
ed from the expression (30) and (31~.
2= idx iC`x ~ (x ,y ,z JIG (xx ,yy ,z+z ids
..... (35)
where
G (x,y,z) = 47oE(x,y,z)~1 + sgn x)
+ ~KoH1(Ko~x~) + [nKoY1(Ko~xj)+ ?/~x ~ ~
x (1 + 2sgn x)  2Ko (36)
Then we can write
¢2= 4iro/~1 + son x~dx ic~x,'0(x ,y ,z ~
X E(x,y,z~ds + JH(x~dx Jc~x''0(x ,y ,z ids
where
H(x) = KoH1(Ko~x~) + [nKoYl(Ko~x~+2/~x~]
(37)
x (1 + 2sgn x)  2Ko (38)
Let us consider first, the case that the
longitudinal axis of the ship is along the
xaxis. The ship is moveing at zero drift
angle, and the fluid motion is symmetric
on both sides of the ship. The hull surface
is given by the equation,
y = +f~x,z)
Then the boundary condition on the hull surface
is written as
3¢ = Ufx/il+fx2+fz2 (40)
Because of the slender body assumption f  0~£)
f = C(l). Omitting terms of o(~2), the ~irec
t~on cosine of the normal can be written as
axe fX/i:fz = VX
nay +1/~ = v
nor fz/ ~ = v
Then the hull surface condition is expressed as
a;; ~ it;= Unix
where 3/3v = vy3/Dy + v 3/3z.
Taking the representation ~ = ¢1+ $2 we can
write
3$l 3~2
 = us  ~
When the ship is placed obliquely to the
uniform flow, with a drift angle A, the bound
ary condition on the hull surface becomes
(43)
= (Ucos~nx+ Usin~ny) (44)
Then the velocity potential can be divided
into the symmetric part ~ and the antisymmet
ric part Ma such as s
= Viscose + Casing (45)
If ~ = 0~), the boundary condition on the hull
surface for Us is given by (42), and that for
is
a
atlas = Uvy
From the definition of 11 12
=200(x,y,z)
(46)
+ J ~ ,~[ln'yyyy, '2+ ~z+zl '20`x y z ids
..... (47)
a 2= 8j~oJ dx ~ Ev~x,y,z)·~(x ,y ,z ids
..... (48)
where x=xO gives the bow end, and
Ev~x,Y,Z) = vy3E + vzaz '
Equations (47) (48) are substituted in (43),
giving an integral equation for o~x,y,z).
The expression (48) suggests that the integral
`39' equation with respect to x is of the Volterra
type, so that the boundary value problem
is parabolic. This fact facilitates the
solution to a great extent. The integral
of (48) is determined by the source density
in cross sections upstream. Then a¢~/3v
is regarded as a known function at the section
where the integral equation along the hull
contour is solved. The solution begins at the
bow end, and marches downstream.
5. WAVE PATTERN AND WAVE RESI STANCE
The pressure on the hull is given by the Ber
(41) noulli equation.
p = p _ pO
= p[utcos~$x+sin~¢y)  2~¢y2+lz2 ~ + 9Z] (50)
(42)
243
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is omitted because of higher order compared
with ~ 2, ~ 2. The elevation of the free sur
face if obtained by
~ = z
=  [U(cos~¢x+sin~ly) + 2(¢y2+¢z2)]Z O (51)
The quadratic term ~ 2+¢ 2 may be omitted in
the formula for the iaveZpattern, but it is
better to include in the pressure distribution
below the waterline.
The wave resistance, the lateral force
and the yaw moment are calculated bv the ore~
sure integral.
F = fisPn dS ~ fdxfc( )Pv ds
F = JfsPn dS ~ fdxfc( )Pv ds
Nz= fisP(nxy+nyx)dS ~ fxdxic(x'Pvyds (54)
Substituting the expression for ~ in
f50;, the pressure distribution is determined.
The free surface elevation, forces and moment
are calculated therefrom. The velocity poten
tial ¢2 includes a term which does not depend
on z. It gives the pressure distribution ir
respective of the depth. However the numeric
al example for pressure distribution indicates
that better agreement with measured results
is obtained by taking account of the attenua
tion of this term by depth of water. The
draftwise variation is related to DG/az. There
is the relation by the free surface condition,
(52)
(53)
DG = ~ a2G at z=0 (55)
Therefore the variation of the kernel H(x),
defined by (38), is given by (z/Ko)02H(x)/3x2.
This problem appears when Koz is not small, or
Kox is large. Then the asymptotic expression
for H(x) may be employed. At large Kox, the
asymptotic expression is determined by Y1(Kox).
and we have the asymptotic relation
32Yl(Kox)/8x2~ Ko2Y1(Kox)
Therefore we can express the value of H(x) at
depth z in the form like
(1  Koz)H(x)
We make further simplification by taking the
average of the attenuation factor through z.
Then the factor 1  2K z is multiplied to the
corresponding term in ~he free surface eleva
tion, and (1  2K Z)2 iS multiplied in the
calculation of hy~rodynamic forces.
6. NUMERICAL METHOD
The solution of the integral equation for
the distribution of sources is calculated
by means of the panel method. Since the
hull is symmetric, one side of the hull sur
face under still waterline is divided in
IXJ = M panel elements AS. , with I divisions
in x and J divisions in ziJ The source density
is defined at the center of the panel, over
which the density is assumed uniform.
The integral equation is discretized as
J i1 J
k1 ij( ) vxijQelki o(Qk)Mjj(Qk)
(j = 1,2, J) for symmetric part (56)
J i1 J
k1 ij( ) vyjjQilki o(Qk)Mjj(Qk)
(j = 1,2,...J) for antisymmetric part. (57)
L (k) and Mj.(Qk) denote the normal velocity
oiJ$~ and $2 Jby unit source at the control
point respectively. The left hand side is the
source density which is to be determined. The
summation on the right hand side is determined
by the source density along the cross section
upstream. It is regarded as a known quantity,
because the solution is carried out from the
foremost section and proceeds backwards.
Lj (k) is calculated analytically. The
co~putation of the kernel matrix Mj (Qk) is
more timeconsuming. J
Mjj(Qk) = vyjjEy(x,y,z) + vzjjEz(x,y,z) (58)
We have to calculate the derivatives of ~ and
¢2 for the determineation of various quantiti
es. 3~1/3x is replaces by the finite differ
ence,
a~l (¢l )i ($l )i
Dx Ax
(59)
Analytical expressions are employed for other
derivatives. The most timeconsuming is
the computation of E, E , E , E . In order
to facilitate this, theXfol~owinzg transform
ation is employed. Consider the integrals
ECC = J e Ctcos(2a2t2)cos(bt)dt
ECS = f e Ctcos(2a2t2)sin(bt)dt
QSC = f e~ctt2sin(2a2t2)cos(bt)dt
QSS = J e Ctt2sin(2a2t2)sin(bt)dt
EQC = J e ctt 2sin(2a2t2)cos(bt)dt
EQS = I e ctt 2sin(2a2t2)sin(bt)dt
Put p = c+ib, and define
A = ECC  iECS = J e Ptcos(2a 2 t 2) dt ~
B = QSC  iQSS = J e Ptt2sin(2a2t2)dt ~ (61)
C = EQC  iEQS = J e~Ptt~2sin(2a2t2)dt J
(~ (60)
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OCR for page 239
Then we have
ECC = ~A, ECS = ~A
QSC = ;~8, QSS = SUB} (62)
EQC = Ji:C, EQS = _aac
Applying the Laplace transform,
A = p1+ jrr2a2p~3/2e~a/Perf~ia2p2y(63)
B 2p2 j~2p5/2~2payea/perf~ia2p 2) (64)
C = i rr2p~5/2e~a/Perf~ ja2p2'(65)
where erf(Z) is the error function defined by
erf(Z) =  Erf(Z) = J. e dt
~ ~0
There is the expansion for the error function
about Z=0 as
n_O41 e~Z2> 2nZ2n+](67)
where
(2n+1~!! = (2n+1~2n1~2n3 ~5.3.1
(1~!! = 1
On the other hand, the error function has an
asymptotic expansion at CZAR such as
Erf(Z) = v; e~Z2) (mu) (22~
Now we put
Erf(Z) ~ e~Z ~ (a) (2n1~!! + ~' ~z' (69)
(68)
The first term on the right hand side is the
asymptotic expansion. If an denotes the nth
term of the asymptotic expansion, and£gives
the accuracy, we have
N(Z) 2 for ~an~_£ (70)
J)N(Z) = EN ntO ni(2n2N+l)
for ~an~>£ (71)
where N is the integral part of ~z2 Me When
N=0, the first term of (69) becomes zero, and
Erf(Z) = $0(Z). Then (69) coincides with (67~.
If Z is pure imaginary, Z = iy,
Erf~iy) = iiYe~tdt = jfyet2dt
O O
Then it is pure imaginary. Put
Erf~iy) = i[eY ~ (2nn+~2n+l] + ~N(iy) (73)
(iy) is given by
(72)
A)N(iY) = 0 for an £ (74)
N(iY) it EN nt0 n!~2n2N+~]
for an>£ (75)
Sample calculations of the Fresnal integral,
to which the existing program library is avail
able, confirm excellent accuracy of the above
method. Putting
a = Kox2/2, b = y, C = z > 0, (76)
we obtain
E (x y z) = 2~oECC = 2;140^A~
Ey~x,y,z) = 2QSS = 2 ~ B `77'
(66) Ez~x,y,z) = 2QSC = CAB
E(x,y,z) = 2EQC = ARC
In order to check the computer time, the wave
Pattern of a point source is calculated. The
computation by means of the above expression
takes 2.04sec.CPU by HITAC M240H, while it
takes 2min.7sec.CPU, if E is calculated by
Simpson's rule. x
7. RESULTS OF THE WIGLEY HULL
equation
The hull surface geometry is given by the
y = bt1 _ (Q)23~1 _ (Z'2 ~ (78)
where Q = L/2 is half length, b = B/2 is half
breadth, and d is the draft at still waterline
A model of dimensions L = 2.000m, B = 0.200m,
d = 0.125m is employed for experiments in the
towing tank of Yokohama National University.
The panel division for the numerical
work is 40~1ength) x 10(draft) = 400, in
equal intervals. The hull form with panel
division is illustrated in Fig.1.
First of all, the source distribution
over the hull surface is determined. A few
sample results are illustrated in Figs.2,3.
Comparison is made with the source distribu
tion of the double model. Remarkable differ
ence is observed near the free surface.
This fact suggests the inadequacy of the origi
nal form of the slender ship theory, which
employs the double body potential as the
near field solution. Fig.4 shows the pressure
distribution on the hull surface at Froude
number 0.267. The result of computation
is compared with measured results with 6m
model published by Namimatsu et al (10~.
Generally speaking, good agreement is observ
ed between computed and measured results.
Slight deviation at the stern region may
be attributed to the boundary layer displace
ment effect.
243
OCR for page 239
Present Flethod
X U x lo2 ~Doub le Hodel
Fig. 1 Coordinate system (Wigley hull)
Present Metl~od SC'~E
Dolible Model u 1 2 3 vXl02
X/1O  975
i:
x/~ o. B75 x/~=O. 825
X/, 0. 1 75 X/~O. 1 25 0  ~
47/
X/~ 0.825 X/1 0. 675
X/1= 0. 92S X/~ 0. 975
Fig. 2 Source distribution at each
cross section F = 0.267
' n
246
o. ,
o.o
o.,
,
F.P.
Fig. 4 Pressure distribution on the
hull surface at F= 0.267
n
.n's
~=~~_~`Q3363~\ 1'
.P.
r2
o s ~,Z/~:0000.0125 ~
~=~t 2
AP.
11\ / W;~,
r2
0;~'' U.S~' \_ Z/L:.JO00.0l2s 1,
I,
F" = 0 . 3 1 6
F.P.
Fi g. 3 Long i tud i na 1 d i stri buti on
of sources at uppermost
pane 1 s
c  P ~ P°
P ~ u 2
\e
\~
O O
F =0 267 Present Hethod
n . ° Measured (6m Model)
, ~Z/~0. 015 0 9/
~ ~:~
NNbW _ ~/eo,~,~
_ , Z/l=0 . 08S ~
v''° ° °~° ~ ,
'~s:
0),% . z/~ 0 . I 05
~ 1
OCR for page 239
Samples of computation of the wave pro
file alongside the model are illustrated
in Fig.5. The result at Froude number 0.267
is compared with the experiment of 2mmodel
at YNU towing tank, and the result at Froude
number 0.316 is compared with the measurement
of 6mmodel mentioned before. Slight discre
pancies are observed at the wave trough.
They may be attributed to the nonlinear effect
because the phase of the wave is in good
agreement. The computed wave pattern around
the hull at Froude number 0.267 is illustrated
in Fig.6, and the corresponding measurement
is illustrated in Fig.7. Similar configura
tions in the crest and trough of waves are
observed between computeation and measurement.
Fig.8 shows the result of computation of
the wave resistance coefficient. The computed
values given by white spots are compared
with the values obtained by the longitudinal
cut wave survey given by black spots. Good
agreement in the position of humps and hollows
is observed. The computed wave resistance
is slightly higher than the wavepattern
resistance. It may be a common trend, that
the wavepattern resistance shows a little
lower value. The dotted line gives the wave
resistance obtained by the subtraction of
viscous resistance defined by CF(1+K), where
OF is the Schoenherr friction coefficient
and K = 0.15, the form factor, from the total
resistance coefficient. This curve fits
well with the computed values. The full
line gives the Michell resistance, which
shows a great deviation from the measurement.
Computations are also carried out with
respect to the Wigley hull at finite drift
angles, ~ = 5°, 10°, 15°, 20°, as the example
of the asymmetric flow. Fig.9 shows the
computed wave profile alongside the model
at ~ =10°, Froude number 0.267. The result
of measurement with 2mmodel at Yokohama
National University is also shown. Fairly
good agreement except near the bow on the
back side (leeway side), at which the leading
edge separation may be present. The computed
and measured wave pattern around the model
is illustrated in Fig.10 and Fig.ll respect
ively. In spite of good agreement in the
wave profile' some deviation is observed
in the diverging wave pattern. It may be
attributed to the distortion of the base
flow due to the hull, while the computation
does not take account of it.
Longitudinal and lateral components
of the force and the moment about the vertical
axis are given in Fig.12, Fig.13 and Fig.14
respectively. The nonlinearity of curves at
large angle may be due to the inclusion of the
term of velocity squared. There are no data
available for comparison with measurement.
The computation does not include the viscous
force which must be present, though the flow
separation at the leading edge is observed
clearly. Therefore the computation under
the condition of continuous flow of a perfect
fluid may not hold in the actual condition.
O.04: 2~/L
~o
0.02 :\
J. ~
O.00 OjI on ~`
n no
Present Method
pleasured (Y..~.U.)
 Pleasured (em Node
j
Fig. 5 Wave profile alongside the
model at En= 0.267, 0.316
0.6
).5
0.4
03
0a
O.'
Michell
° Present Method
Towing  t (Krause
· Wave Analysis
, ~
O ~ ~ 1<' ~ ,:
0.15 a23 n ,~ 0~0 n ~ EN
Fig.8 Wave resistance coefficient
of Wigley hull
Fig. 6 Computed wave pattern at
F = 0.267
Fig. 7 Measured wave pattern at
F = 0.267
n
247
OCR for page 239
0.06r 2~/L
0.04 _
0.0_
_n n:
n n
n n:
o.n
O.0~
~.
Fi 9. 9 Computed and measured wave
profi le alongside the hul l
o~ = 10°, Fn= 0.267
Fi 9. 10 Computed wave pattern ~ .6
o ~= 10°, Fn= 0.267
o.~
PRESENT METHOD
: · MEASURED
.CX~to3
,__ t~ __
Rx fno.3le
ro u L
 ~e~
o . I
~ ~_ ~ _ ~
fn0 . 204
O O I .' ~
. ~° s° ~o° a 15° 20°
Fn0 . 2nd
F i 9. 1 2 Computed l ong i tud i na l force
coeff i c i ent
1.8 Ci
~_~In'
side
~a9c~
Fig. 11 Measured wave pattern
~= 10°, Fn=0.267
/~
0.0( ~' ~1 1
'0 50 10° a ls° 20°
248
`~,,~
Fig. 13 Computed Lateral force
coeffi ci ent
OCR for page 239
0.6
n 5~
  
0.4
0.3
0.2
O . 1
 CURIO:
//
o.~( i, ,
or so I Go
if/
Fig. 14 Computed yaw moment
coefficient
8. RESULTS OF THE SAILING YACHT HULL
Cu=< 1~
, ,
a 15° 20°
The hull form has been designed by NCAC
following the rule of 12 metre class yacht.
The hull form with panel division is illust
rated in Fig.15. The length is divided into
40 segments in equal intervals. The radial
cut is employed in the draftwise division.
The angle of the radial cut from the waterline
j 0° 8° 1 6° 24° 32°,40°,50°,70°, 80°, 90° .
The panel division on the solid keel is
11X8, but small panels are combined, resulting
70 panels. A model with a detachable solid
keel and a rudder is made for towing tank
experiment. The resistance test at zero
drift angle is carried out at Yokohama National
University tank. Experiments at the finite
drift angle are conducted by Akishima Labora
tory of Mitsui Shipbuilding Co.
_~
FP
Fig. 15 Sailing yacht hull form
Fig.16 gives the longitudinal distribu
tion of the symmetric part of sources on
the canoe body with and without keel, and
Fig.17 gives that of the antisymmetric
part of sources at Froude number 0.269.
A remarkable effect of the vertical keel
to the source distribution on the canoe body
at finite drift angle is observed. Fig.18
gives the wave profile alongside the hull, and
Fig.19 illustrates the wave pattern around the
hull by the wave contour, at the drift angle
0°, 4°, 8°, at Froude number 0.269. Because
of small length to beam ratio and the flat
stern, the wave pattern behind the stern
becomes very complex. No measured data are
available for comparison, but a particular
feature of this hull form is observed in
these figures.
4 O,xIO 4
8_ on_ B°
2 ~2
Of ~O
2
PRtSENT HtTHO0
lUltH Kttt )
 PRESIFT HEtHOD
4 'al THOUT KEEL )
 DOUBLE HOtEL
IHITH KtEL)
2~_O.S
FP
9= 9°  16°
L2
~4
~ 2
UP
4 O,x10
§=16°  24°
·RISE'r HETHOØ
INtTH KEEL,
___ FRESEtll HETHO8
4' tH1 tHOUt KEEL )
ount HOTEL
IN l TH KEEL )
l_2
4 O,XIO
A_
PRESENT HETHe0
`~11H KEEL )
___ PRESENT HETHO0
IRITHOUT KEtL)
 DOUI'E NOtEL
tH1 TH KEEL ~
=50°  60°
=60°  70°
L_2
UP
~4
L 2
O
2
2
FP
249
Fig. 16 Longitudinal distribution
of symmetric part of sources
Fn= 0.269
OCR for page 239
 onto
8 ~  ~
' ~
r ~~
v o.s o.o ~ e.s \ .` 1
re'3'Hr NETHER
tHl TH HEEL )
 FR`SEH' HtTH00
lu I TNOUt REEL )
 lOUlLE ttOt'L
(~1 TH HEEL )
2
O
2
PREsEHr NETHOD
IH1 TH HIEt )
 "ESEHT NETHOD
'~ 1 You r KEEL )
 lOUJ~£ HOVEL
tH 1 TH KEEL ) 6 =2 (a3
O
. O2x 1 0
2
4
 2
\~2
~4
l2
UP
~ =5~6~
~2
FR6SEH! METRO:
IH 1 TH KEEL ~
_ PRESEH! nETHOD
. IN 1 THOU r KEEL I
~ lOUlLE HOVEL
tH I TN KEEL )
UP
PtESEHr NETHOD
IUlTH KEEL ~
 rtE5EH' NEtHUD
4_ t~1 THOUr KEEL )
 GUILE noDEL
INtTH KEEL
2
~' O
0 ~=in 2
 2
FP
L_2
UP
Fig. 17 Longitudinal distribution
of antisymmetric part of
sources En= 0.269
Fig.20 shows the wave resistance coeffi
cient of the canoe body without keel. Towing
test results are given by the total resistance
coefficient CT, the residual resistance coef
ficient CR based on the Schoenherr friction
coefficient, and the wave resistance coeffi
cient derived by the assumption of the form
factor K = 0.29. The curve of residual resis
tance fits the computed values approximately.
The deviation at higher Froude numbers may be
attributed to the change of wetted hull geo
metry due to sinkage and trim, together with
the bow wave elevation. Fig.21 shows the
wave resistance coefficient of the hull with
keel. The effect of the keel to the computed
wave resistance is mainly due to the differ
ence in the source distribution near the
stern. The difference in CT between results
with and without keel is remarkable. The
conventional method of the form factor for
viscous resistance gives 0.54, which seems to
be too large. The computed points are not
parallel to the experimental curve. This
deviation may be partly due to the change
of trim, which is more remarkable than in
the case without keel.
The axial and lateral forces when the
model is moving at finite drift angle are
calculated. The axial force coefficient
C is shown in Figs.22,,23. The computation
does not include the induced drag of the
solid keel. Therefore the difference between
the results with and without keel is not
remarkable. Experimental data obtained by the
Akishima laboratory are shown in white spots.
Since the model is free to heel and trim
in the experiment, there is a considerable
difference between experiment and computation.
Then the comparison is only for reference.
The lateral force coefficient Cy is
shown in Figs. 24, 25. The computation does
not include the lift of the solid keel.
In order to calculate the lift of the keel,
the lifting surface computation such as the
vortex lattice method is necessary. However
the result of computation by the theory of
large aspect ratio is added for simplicity.
The result is shown in dotted lines. The
measured data are shown by white spots.
However comparison is difficult, because
of the difference in conditions mentioned
before. The computed yaw moment coefficient
C is shown in Figs.26, 27. Different from
the lateral force, difference between the
moment with and without keel is remarkable.
This means that the theory includes the moment
of the keel. One may conclude from the above
results, that the present theory is applicable
to the wave resistance of the canoe body
of the yacht, in spite of the small length
to beam ratio, while other theories such as
the vortex lattice computation is necessary
for the prediction of hydrodynamic forces
acting on the hull with vertical keel at
finite drift angle.
250
OCR for page 239
 2 t/L
_ . _
0.02
O. 00 U.1
fP
0 . 02 _ Fn=0 .269
F
ATTACK ANCLE=ee
 AITACK ANCLE=4°
A I TACK AIdGLE _0°
FAtE 5 I DE ,,
, =3 I /~\ ,~t ,~/r~oA/P
o.o2t
O.OC
O. O~
~ ~/ 2X/L
_ f~ 
Fn_O. 269
F i ~ . 1 8a Computed wave prof i 1 e
alongside the hul 1
without keel Fn= 0.269
.~
Q~ lIlifK A'lr.l E=4° [n=0.269
~ ~r.l .F =R~ Fn :0 . 269
Fi 9. 1 9a Computed wave contour
wi shout keel Fn= 0 269
0.06F
0.04L
0. 02~
0.0c
1
2 (/L
A 1 T ACK AllCLE = e°
 AITACH ANCLE=4°
Al TACH AtICLE=0°
FACE SIDE
Fn=0.269
0.0 _
0.04 2t/L
lACK SIDE ~0.02~1 ' ~,~
F i 9. 1 8b Computed wave prof i 1 e
alongside the hul 1
with keel Fn= 0.269
A'TArx AN0.t E=48 Fn=0.269
~? ~AT TACK AllGLE1~° Fn ~0. 269
lr I I I
Fi 9. 1 9b Computed wave contour
with keel Fn= 0.269
251
OCR for page 239
X 1 O  3
6 _
5 _
4
C R
 rOU2L2
TOHINC TEST C'
. C~(K=0.29)
~CALCULI`TED CU
2 I
~CI
~ 74 CFO
_
0.3 0.4 Fn 0.5
Fig.20 Computed wave resistance
coefficient compared with
test results of yacht model
without keel
s.o~
4.0 _
3.0 _
2.0 _
1.0 _
c,
Fig. 21 Computed axial force
coefficient of yacht model
F = 0.269
n
C= ~
reNl'JG JEST Cl. CR.
CuIK=0.54 ~
~ CALCULATED CU
4~t
_CXXl03 5 0
Cx= ~ r Fn =0 . 269
C' MEASURED B). n I TSU I
(WITH KEEL ~ RUDDER )
 CALCULATED
{HITHOUT ItEEL ~ RUDDER)
CALCULA TED
t)J I TH KEEL )
(~AYEMAKING FORCE ONLT )
c,
1 n
252
/CI
/ CR
, /
///cH
/ //
~ ~ /,/
Fig. 21 Computed wave resistance
coefficient compared with
test results of yacht model
with keel
Cxxlo3
4.0 _
3.O _
2.0 _
CX= ~ Fn =0 .359
nEAsuRED BT n I TSU I
(UITI.I KEEL ~ RUDDER)
~ CALCULA TED
{HITHIJUT KEEL ~ RUDDER)
CALCULA TED
(HITH KEEL )
(II4VE_MAK ING FORCE ONLY )
cFn I
Fig. 23 Computed axial force
coefficient of yacht model
Fn= 0 359
OCR for page 239
15.0
10.0~
s.o~
4.0 1
3.0 _
2.0 _
1.D
:) . U

Crxlo3
lS.0 crXlo3
Fn=~.269
EASURED B] H 1 TSU I
(HItH KEEP ~ RUDD£R )
CALCULA TED
(HItHCUT KEEL ~ RUDDER)
CACCULA TED
{~1 TH KEEP )
(~AVEMAKING fCRCE ONLY'
CALCULAtED
HlTH KEEL )
(HAVEdAK ING fORCE*
LlfT OF KEEL )
~10.0
C7=~ fn=~.359
MEASURED BY H 1 TSU I
(N I tH KEEL ~ RUDDER )
 CALCULA TED
~ ~ l rHcu r KEEL ~ RUD DER )
 CALCULA TED
(~1 TH KEEL )
(~AVEMAKING fCRCE ONLY )
 CALCULATED
(H I TH KEEL ~
(HAVE MAK I NG fURCE.
L I FT 3F KEEP )
~,,
~oO 20 4° 6° 8a
a
Fi 9. 24 Computed 1 ateral force
coefficient of yacht model
F = 0.269
_CMX 104
CM= .~'U: L] Fn =~.263
 CALCULATED
(HITHCUt KEEL 4 RUDDER) 3.o
CALCULAtED
(~1 TH KEEP )
(HAVEMAKING FORCE 3NLT
24
40
Fi 9. 25 Computed 1 ateral force
coefficient of yacht model
Fn= 0.359
4 ~ CMX10
__.
, ,
6° 80
a
CM=  fn =~ .359
 CALCULAtED
{HITHOUT KEEP 4 RUDDER )
CALCULA TED
(HITH KEEP ) /
{~4VEMAK ING fORCE CNcT )
60 80
Fig.26 Computed yaw moment
n n
a
coefficient of yacht model Fig. 27 Computed yaw moment
__
,_
a
coef f i c i ent of yacht mode 1
Fn= 0 359
253
,_
OCR for page 239
9. RESULTS OF SERIES 60 HULL
As an example of the conventionalhull
form, the wellknown Series 60 model (C =0.60)
is employed for computation. The bodily plan
is shown in Fig.28. The panel division is
40~1ength~x 8(draft) in equal interval. The
towing test of 3mmodel is conducted at Yoko
hama national University tank. The wave
profile alongside the model at Froude number
0.28, 0.30, 0.32, 0.34 are illustrated in
Figs.29 ~ 32. Full lines give the measured
results and dotted lines give the computation.
Fairly good agreement between computed and
measured wave profiles is observed throughout
the results. The slight difference may be
attributed to effects of the nonlinearity and
the boundary layer displacement. The wave
patern around the model at Froude number 0.30
is illustrated by contour courves in Fig.33.
Similarity between computed and measured
wave patterns is observed.
In Fig.34, the computed wave resistance
coefficient is compared with the residual
resistance coefficient based on the Schoenherr
friction coefficient and the wave pattern
resistance by the longitudinal cut method.
The computed value is slightly higher than
the residual resistance. This may correspond
to the tendency of lower wave height at the
stern in the computed wave profile. Since
the curvature of the hull surface at the
stern is much higher than the case of other
models, the number of panel division in this
area may not be sufficient for good accuracy.
The position of humps and hollows is in good
agreement in theoretical and experimental
curves.
L~38 Bso.lOOs Cb=.8
1
.. _
l\ V\'\\\\\<
In\\ \\:L~\
Ah
~\1
~1
id.
Fig. 28 Body plan of Series 60
CB= 0.60
0.03~
:N 0.0;
~ of f
1.0 <.l 0.~/ O.] ~ J^" 0 '
n at I
~.ml
Fig. 29 Computed and measured wave profile
alongside Series 60 model
Fn= 0.28
A ivy
.
0.0
odor
~_._.
~0 01 1 ~
 a  a  flu {2~, ,~o.:e
 x  ~   .
06t ^4 0.2 t ~ ~2 ~0;`'
4.05
^04
Fig. 30 Computed and measured wave profile
alongside Series 60 model
Fn= 0 30
To:
.' '`\
0.03 j
o ml
0.01:
dry.
1.0 0.8 0.6 ~.',0. ~ 0.2 0 0
~ 0.01 ~
~ {.,
0.03
O. O.
 X  X  Con
1 ~
; ~Q;5, 8~ 1,0~
Fig. 31 Computed and measured wave profile
alongside Series 60 model
Fn= 0.32
2h'L
1 ~
I; \
6 i: o.m
,. `:
;~
1.0 ^e ^l \,(0~` ~L2 0.
V. Mel
~` ^02
^04 ,
_._
0.03
n hi
°  .o ~ <2.) r_~.~`
x. x  con it.
my;
~ :
?~^ a
0.2 tar. ,,~'o.e 1.0
f.~'
/.''
,~,i
/,,
Fig. 32 Computed and measured wave profile
alongside Series 60 model
Fn= 0~34
254
OCR for page 239
o
,4 ~_
844:
~ Tar
10.
Fig. 33 Computed and measured wave contour
of Series 60 model En= 0 30
9.0' ,
B.04 i
7.~. ,
6.Q.
5.04
4~0t
~ no
2 D
 Cw T11£0RETICAL RESISTANCE COEFF.
Cr IESIOUAI" lESISTA#CE COEFF. nAr.l7~089 (Schoenherr)
V.T.I4.O'C
VAVE FATTERN RESISTANCE COEFF.
x CUP
Fig. 34 Computed wave resistance coefficient
compared with test results
Series 60 model
10. CONCLUSIONS
The validity of the new slender ship
formulation is examined by the computation
of the wave pattern and wave resistance of
the Wigley hull, the sailing yacht hull and
the Series 60 model.
Satisfactory agreement is obtained in
pressure distribution on the hull surface,
the wave profile alongside the hull, and
the wave resistance, between computed and
measured values with respect to the Wigley
hull. The computed wave resistance of the
canoe body of the sailing yacht hull without
keel shows good agreement with towing test
results, in spite of small length to beam
ratio. The effect of the solid keel is not
fully accounted for by the present theory.
The hydrodynamic forces, when the hull is
at a finite drift angle, may be calculated
by the present theory supplemented by the
lifting surface computation. Although a
good agreement between the computed wave
profile alongside the Series 60 hull and
the measurement is obtained, a slight devia
tion is observed in the computed wave resis
tance from the towing test result. This
fact may suggest that fine panel division
is required for the prediction of wave resis
tance of conventional hull forms such as
Series 60 model, because of the large curva
ture of the hull surface, especially at the
stern area.
As the conclusion, the new formulation
of slender ship approximation has achieved
a remarkable improvement in the theoretical
computation of the wave pattern and wave
resistance. The result seems to confirm
the usefulness of this theory in the predic
tion of hydrodynamic characteristics of prac
tical hull forms in steady forward motion.
ACKNOWLEDGMENTS
The authors express their thanks to
Prof. M.Ikehata and staffs of Marine Hydro
dynamic Laboratory of Yokohama National Uni
versity for their cooperation in the experi
mental work. Their thanks are also to NCAC
and the Akishima Laboratory of Mitsui Ship
building Co. for their generous permission
for publishing the data concerning the sailing
yacht hull,
It is noted that the numerical work
has been carried out by the use of HITAC
M28D computer of Yokohama National University
Information Processing Center.
REFERENCES:
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Ship Form by Singularity Distributions when
the Boundary Condition on the Free Surface is
Linearized." Journ. Ship Res. Vol.16 (1972)
. Vossers, G., "Some Applications of the
Slender Body Theory in Ship Hydrodynamics."
Thesis, Delft Technological University, (1962)
. Maruo, H., "Calculation of the Wave Resis
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small as the Beam." Journ. Soc. Naval Arch.
Japan, Vol.112 (1962)
4. Tuck, E.O., "The Steady Motion of a Slender
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5. Joosen, W.P.A., "Velocity Potential and
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Slender Ship." Internationnal Seminar on Theo
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6. Lewison, G.R.G., "Determination of the
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~;~3 u~v~Resistance' Ann Arbor (1963)
255
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7. Maruo, H., "New Approach to the Theory of
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Vol.31 (1982)
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10. Namimatsu, M., Ogiwara, S., Tanaka, H.,
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. . .
11.i Maruo, H., Ikehata, M., Takizawa, Y.,
Masuya, T., "Computation of Ship Wave Pattern
by the Slender Body Approximation." Journ. Soc.
Naval Arch. Japan, Vol.154 (1983)
12.i Song, W.S., Ikehata, M., Suzuki, K.,
Computation of Wave Resistance and Ship Wave
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(1988)
13.i Song, W.S., "Wavemaking Hydrodynamic
Forces Acting on a Ship with Drift Angle and
Wave Pattern in her Neighborhood."
Journ. Kansai Soc. Naval Arch. Japan No.211
(1989)
14.~ Song, W.S., Ikehata, M., Suzuki, K.,
"On Wavemaking Hydrodynamic Forces and Wave
Pattern of a Sailing Yacht." Journ. Soc. Naval
Arch. Japan, Vol.166 (1989)
15.2 Maruo, H., "Evolution of the Theory of
Slender Ships." Ship Technology Research, Vol.
36, No.3 (1989)
Results in this paper are in part reported
in these articles.
2 The theory is discussed from the perturba
tion point of view.
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DISCUSSION
Ronald Yeung
University of California at Berkeley, USA
In spite of the nonlinear conditions stated in the beginning of section
2, it seems clear that the starting point of this work remains the same
as the NeumannKelvin problem. This is evident from the
representation, Eq. (12) of the paper. What followed from there is
essentially an approximation to the NK solution, and one should not
expect the present calculations can do any better than the 3D
NeumannKelvin solution. In terms of the slenderbody
anoroximation carried out here. or in Prof. Maruo's 1982 work, I
don't feel that it is rationally based, at least not completely. I will
point out 2 objections. (1) The neglect of the line integral in Eq.
(15) cannot be justified simply on the basis of traditional infinitefluid
slenderbody theory. Sources on the free surface exert much stronger
influence than submerged distribution. It is well established these
days that the waterline integral in the NK problem yields a
significant contribution. This contribution is taken into account in
the matched asymptotic theory of Yeung & Kim (1984, 15th ONR
Symposium). This leads to the 2nd point. (2) In our work, which
Prof. Marno might not be aware of, we showed that the near field
approximation, Eq. (25), of paper is more elaborate than an impulsive
2D source and a function, say, F. that depends only on the axial
distance 'x'. Using matched asymptotics, we showed that the
transverse wavefield is contained in F(x,y,z), with explicit DISCUSSION
expressions given in Yeung & Kim (1984). Prof. Marno's Ffunction
corresponds to setting yz0 in ours. It is clear that your analysis
eventually lead to a rather adhoc afire in the paragraph following
Eq. (55). It appears this deficiency can be corrected in the manner
that we have derived from the matched field. I don't think such
development should be done as a matter of convenience, rather, it
should be rationally based. I would like Prof. Marno to comment on
these two issues.
err ~
AUTHORS' REPLY
The slender body theory is based on the rational perturbation
analysis. The fundamental technique is the series expansion of the
complete solution of the fully nonlinear boundary value problem, its
existence being assumed, with respect to the slenderness ratio a, as a
small perturbation parameter. The lowest order of the expansion
gives the linearized solution, which is discussed in this paper. The
rigorous derivation of the result by the perturbation technique is not
employed in this paper, because it has been Riven in another

literature. The general form of the linearized solution in the near
field is given by the velocity potential of the form ~ = HAD' + g(x)
where Cited' is the solution of the twodimensional Laplace equation
q}~D',,~, + q}~2D'zZ = 0 and satisfies the boundary conditions on the
body surface and on the free surface. g(x) is a function of x only,
which is determined by matching with the far field solution. Both of
these functions are o(~2~. It is readily proved that the line distribution
of sources along the waterline b is 0(63), SO that it must be deleted
from the linearized scheme. The function F(x,y,z) referred in the
discussion is derived from the linearized far field potential, which is
not correct in the near field. In order to obtain the consistent
approximation in the near field, one must expand it with respect to
the transverse coordinates y, z, and detain only the term of the lowest
order. It is reduced to the limit at y = z = 0, if the function is
bounded. It corresponds to the function g(x). It should be
emphasized that the higher order terms with respect to ~ should not
be detained, because they are subject to the nonlinear portion of the
boundary conditions which is not taken into account in the theory.
The present discussion is concerning the higher order terms only, and
does not make sense accordingly. It seems that the argument
presented by Prof. Yeung is noting but the consequence of the lack
of knowledge about the rational perturbation analysis of the complex
nonlinear problem.
DISCUSSION
Hongbo Xu
Massachusetts Institute of Technology, USA (China)
Prof. Marno, the results you have are impressive. My question is
about the theoretical results for Wigley hull at an angle of attack.
Have you compared the lateral force and yaw moment coefficients
with experimental data? The angle of attack or used in your
computation appears to be rather large (up to 20°). As we know, the
stall angle for a wing is about 12° to 15°. It may be important to
find the approximate range for ~ in which the slender ship theory is
valid.
AUTHORS' REPI,Y
The example for the Wigley hull in finite drift angle is rather an
academic aspect because the leading edge separation must be present
by the form with a sharp edge, though the theory does not take
account of it. Therefore, the forces and moment computed by the
theory may not represent the actual value except at small angles, less
than 5° say. However, the comparison of the wave profile shown in
Fig. 9 indicates that the theory can predict the behavior of the free
surface flow fairly well, even at 10°, which is not so small.
Kazuhiro Mori
Hiroshima University, Japan
You explained that your results agree with the measured Fairly
wells I don't think so, but the agreement is strikingly well! In your
introductory remarks, you disclosed your negative opinion to the
direct numerical method. Although it takes much computing time, it
has a potentiality; e.g., the viscosity can be taken into account. The
methods may be complimentary. I hope you may not have such a
negative opinion to the numerical method, for you are so influential.
AUTHORS' REPLY
The computational fluid dynamics depends on the capacity of
computers. It has achieved a great success in various fields in
hydrodynamics, such as in compressible aerodynamics.
Unfortunately, the present stage of the application of CFD to the
freesurface flow around the hull does not seem to reach the level of
feasibility as a useful tool to resolve problems in the practical field
of shipbuilding. However, the recent progress of the computer
capacity is remarkable, so that there is much prospect that CFD will
become a powerful tool with practical feasibility in the field of the
full form research in future. Please, never be afraid.
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