FORCE AND MOMENT ON A SLENDER BODY OF
REVOLUTION MOVING IN WATER OF FINITE DEPTH
by
William R. McCreight
B.S., Webb Institute of Naval Architecture
(1967)
SUBMITTED INPARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF
SCIENCE
at the
MASSACHUSETTS INSTITUTE OF
TECHNOLOGY
June 1970
Signature of Author ....................................................
Department of Naval Architecture and
Marine Engineering, June 1970
Certified by ............ . ...
..
u..
..
..........
...
.. .......
Thesis Supervisor
Accepted by
......... ,
hairman, Departmental Committee
on Graduate Students
Archives
JAN 2N 1i71
,trap'a T
Nau XTziý
1
-
ii
-
FORCE AND MOMENT ON A SLENDER BODY OF
REVOLUTION MOVING IN WATER OF FINITE DEPTH
by
William R. McCreight
Submitted to the Department of Naval Architecture
and Marine Engineering on June 3, 1970 in partial
fulfillment of the requirement for the degree of
Master of Science.
ABSTRACT
A consistent first-order theory is derived for the heave force and
pitching moment on an arbitrary slender body of revolution moving
horizontally with constant velocity in water of finite depth. A computer
program is given for evaluating the resulting expression. Numerical
results obtained using this program are presented and compared with
previous theory and with published experimental results for the case of a
Rankine ovoid.
Thesis Supervisor:
Title:
John Nicholas Newman
Associate Professor of Naval Architecture
-
iii -
ACKNOWLEDGMENTS
The author is indebted to his advisor, Professor J. Nicholas Newman,
for his advice and encouragement throughout the course of this thesis.
Thanks are due to Miss J. A. Davis for her excellent typing.
All computations were performed at the Computation Center of the
Massachusetts Institute of Technology.
-
iv
-
TABLE OF CONTENTS
Page
1.
INTRODUCTION
1
2.
FORMULATION OF THE PROBLEM FOR THE POTENTIAL
3
3.
SOLUTION FOR THE POTENTIAL
8
4.
APPROXIMATE FAR-FIELD SOLUTION USING GREEN'S THEOREM
16
5.
FORCE AND MOMENT
24
6.
NUMERICAL RESULTS
30
7.
CONCLUSION
4o
8.
REFERENCES
41
APPENDIX:
PROGRAM LISTING
- V
-
LIST OF FIGURES AND TABLES
Figures
Page
1
The Coordinate System and Flow Geometry
2
The Surface
3
Shape of Rankine Ovoid Used in Calculations
32
4
Heave Force on Rankine Ovoid at 0.144L Submergence
33
5
Pitching Moment on Rankine Ovoid at 0.144L Submergence
34
6
Heave Force on Rankine Ovoid at 0.266L Submergence
35
7
Pitching Moment on Rankine Ovoid at 0.266L Submergence
36
8
Heave Force on Rankine Ovoid as a Function of Submergence
and Depth of Water
37
Pitching Moment on Rankine Ovoid as a Function of
Submergence and Depth of Water
38
9
S
for Applying Green's Theorem
Tables
1
EFFECT OF SUBMERGENCE AND DEPTH OF WATER ON FORCE AND
D
MDMENT
ON
A
RANKINE
4
17
1. INTRODUCTION
When a submerged body moves horizontally near a free surface, the
body experiences a vertical force and a pitching moment due to the
presence of the free surface.
If the water is shallow, there will also be
a force and moment due to the presence of the bottom.
In addition, in
this case the free surface waves, and hence the force and moment due to
the presence of the free surface, will be altered.
The magnitude and sign
of the force and moment will depend in a complicated manner on the
geometry, location and Froude number of the body, and on the depth of
water.
In order to obtain a tractable problem, some simplifying assumptions
must be made.
The flow is considered to be inviscid, incompressible,
homogeneous, and irrotational.
The analysis is carried out for a slender
body of revolution, which can be represented by a line distribution of
sources along the body axis, as is usual for a slender body, and a similar
distribution of vertical dipoles to correct for the crossflow induced by
the presence of the free surface and bottom.
Although the source distribution is of lower order than the dipole
distribution, both result in contributions to the moment of equal order.
The expression for the potential is also obtained directly for the far
field by applying Green's theorem, which justifies the assumed form of
the potential.
Numerical results are presented for the case of a Rankine
ovoid to demonstrate the effect of depth of submergence, depth of water,
and Froude number.
These results are compared with published experimental
results for the case of infinite depth of water.
- 2 This problem has been studied previously by Havelock [1] for the
special case of a prolate spheroid in deep water.
He justifies the second
(dipole) term in the expression for the moment on order of magnitude
grounds and suggests its applicability to other bodies.
Pond [2] , who
has studied the case of a Rankine ovoid moving in deep water, derives the
dipole term by a heuristic argument and justifies its use by comparison
with experimental results.
The other limiting case has been studied by
Newman [3], who calculated the force and moment on a slender body of
revolution moving laterally near a wall, but with no free surface.
-32.
FORMULATION OF THE PROBLEM FOR THE POTENTIAL
We assume a slender body of revolution to be moving horizontally with
a constant velocity
U
parallel to its axis at a depth
free surface of a body of water of uniform depth
h
.
b
Let
beneath the
Oxyz
be a
Cartesian coordinate system with the z-axis upward moving with velocity
in the direction
Ox
parallel to the body axis.
U
The origin is located on
the free surface directly above the midpoint of the body axis.
This is
shown in Figure 1. We nondimensionalize all quantities with respect to
body length
L
, the velocity
U
, and the fluid density
p
.
With the further assumption of irrotational flow in an inviscid,
incompressible, and homogeneous fluid, there exists a potential
(u,v,w)
that the velocity components
such
are given by
From the continuity equation and the assumption of incompressibility, we
obtain Laplace's equation
+
..
= O
(2)
On the free surface, which is at an unknown location which is to be found
as a part of the solution of the problem, the dynamic boundary condition
that the pressure be constant must be satisfied.
equation,
2.
2IN7ý1
-P +
Thus from Bernoulli's
- 4ý
z
K:
Figure 1
The Coordinate System and Flow Geometry
-5where
C(t)
p
is the pressure,
g
is the gravitational acceleration, and
is a function of time only, which we may take as identically zero.
On the free surface we must also satisfy the kinematic boundary condition,
which requires that a particle of fluid on the surface remains on the
surface; that is, has the same velocity normal to the surface as the
surface itself has.
Thus if the free surface is given by
we must have
D
O=
(5)
Dt
on the surface, where
D
Z
D/Dt
is the substantial derivative
+b
a
(6)
On any solid boundary we require that the velocity be tangent to the
surface, or
- = O(7)
on the surface, where
)/An
is the derivative normal to the surface.
In
the case of a horizontal bottom, this is just
O
0-
on
z = -h
(8)
The fluid velocities must be bounded everywhere in the fluid.
Finally, to insure uniqueness of the solution, we must impose a radiation
condition at infinity.
If
the problem were formulated as the limit for
large time of an initial value problem, this difficulty would not arise
-6(Stoker, [4]).
However, it is simpler to avoid the time dependence by
solving a steady-state problem with the appropriate additional condition.
Thus we require that ahead of the body the motion must vanish, except for
the uniform stream,
(-LU ,oo )
±
(9)
Since we want the potential for a body in a uniform stream, it is
natural to write
=
where
$
is the disturbance potential due to the presence of the body.
, Twe obtain the problem
Then in terms of
S•
on
(10)
z = S(
-
from equations (3) and (5)
D•v,
on the body surface,
Sn
(12)
for the time-independent case,
(13)
-7-
on
z = -h, and the radiation condition
(15)
)=
$
de also require that
be bounded at infinity.
The bottom condition for
h - c is
7k= O
(16)
The free surface condition (12) is nonlinear and must be evaluated at
an unknown location.
By expanding the potential in a Taylor series about
z = 0 and taking the lowest order terms in 5, we obtain the linearized
free surface condition
D
+
WID
:0
to be satisfied on the undisturbed free surface
and Laitone, [5J).
z = 0
(cf. Weyhause
The wave height is, to lowest order,
and thus we have neglected terms of order
condition.
(17)
0( 2)
in the boundary
-83. SOLUTION FOR THE POTENTIAL
We now seek an approximate solution to the linearized free surface
problem formulated in the preceding section.
The solution will be obtained
ye first consider a line distribution
using a slender-body approximation.
of sources of arbitrary strength cO(x)
on the body axis over the interval
i
J/
((19)
2
wrhere
2
p
= O
2(½)
= y
2
+ (z+b)
2
.
Integrating by parts and assuming c(-)
=
[8],
, we obtain, as in Newman
(20)
Then, with the body shape defined by
here the parameter
slenderness
4here the slenderness parameter
is defined as the ratio of the maximum(21)
is defined as the ratio of the maximum
-9p = O(C)
radius to the body length, the limit of (20) for
is
(22)
-2c (/) 47 P - o (Cer)
On the body surface we must satisfy the boundary condition (13), which
can be written
S---where
ao(
denotes the derivative normal to the curve
the plane
x = constant
(23)
plrX.
r =
R(x)
in
The normal derivative of (22) is
IL
:B-
z 2(z - )
(24)
2o
Since
0:)
o = O(e) , the second term is
0(d/E )
, while the first is
Thus the source strength is found from (23) and (24) to be
C:-64)where
.Oi(E
(25)
24Q(5L
L4 Tra/?(
S(x) =yR 2 (x) is the nondimensional sectional area of the body,
and the body boundary condition is satisfied to
O(c-)
. We see from
(25) that there are certain restrictions on the geometry of the body near
- 10 -
the ends if the assumption of d(--)
that the derivative of
S(x)
=c(½) = 0
is to be correct; namely,
must disappear at the ends.
If
F
is the
distance from the end of the body and
for small
,
then
and thus we must have
below that we must have
c,>
for
S = 0
S' = 0
at the end.
at the ends of the body.
It will be shown
Tuck [7]
presents a detailed discussion of this point.
The boundary conditions (14) and (17) on the bottom and the free
surface are not satisfied.
By using the source-like Green's function
which satisfies these conditions instead of the free-space Green's
function, these conditions will be met, but only at the expense of
introducing an additional error in the body boundary condition.
The
required Green's functions satisfy
,0OK)
"G-____
%
=
(>
(26)
- 11 as appropriate, and the radiation condition
+
-a
CO
17.",,OZ,
I/.z
(27)
The required Green's functions are given by Wehausen and Laitone, [5].
For finite depth
GYF
I
S0
-TTT zC
cw
cOjt( 2 4A ) [CAeAJ3f~) (I
~f
Lc~e~
V(:O< U(IX7 T) 10 O
-ITI'2
-aR
e
IH+
Co42O
0zý,
where
and
PV
k
o
[,o(-x)
LLB~t
Vj
-P)l
2/)
~P~R.-~a~c~R-hec
co ·
91.1401
CU4
v) -
ý(4 FI(I
)
(28)
(L
eI
Ac~t
^ z2R
COIC-oc EAo (40•) 4re] de
denotes the principle value,
is the real positive root of
a
ee(
ts-
a
e :=O/
(29)
- 12 -
2.j= CA/U2ý1
a.Y~cr..o3Siz7IF
O6,
where
oC
= O
1P
-r1 >
In the case of water of infinite depth, the Green's function is
+±LH
r
ro
P
F,~7KY
TT
_~~-r/2
C"O Eý (I -i)
(/X
C4S)
,
A11j
+':
4.prO1
4(
c~IL/-lA:~
4~
j
(30)
where
t-2 -
ý(7_ý
(-,-1
+
The functions
H
4~-s2·
~+)
-- MZd
r~~l=~
(t- S
are the part of the Green's function which are analytic
in the fluid domain and which must be added to the free-space Green's
function
1/r
to satisfy the boundary conditions on the free surface and
the bottom.
Using the Green's function above, the potential can be written
( Y-)
J
= I,, +I,
~~
J
3/
0,)-b) J
- 13 -
)
H('6,
+ c7-
(31)
The error in meeting the boundary condition on the body is now
OQ(6)
O(621,Q)=
+
(32)
This error is of lower order than the error in the free surface condition,
which is
0(02 ) = 0(6 )
.
It will be shown in section 5 that this error
leads to an error in the pitching moment which is of the same order as the
leading term in the moment obtained using the potential (19).
The error
in the heave force is of higher order.
To obtain the next term in the potential, we consider a line
distribution of vertical dipoles of arbitrary strength
axis over the interval
-1
x <
.j(x)
on the body
.
2
(33)
aS L13~'/-~at
PV~l/X- 4
Zpi-)
~e
2 P2pljd
'2 -
- 14 4(-)
by the procedure used above, for
tan
-l
z+b!y
.
In terms of
I12
and
= 4()
I2
= 0
, where
, the condition (13) becomes
-"- +
(34)
since (23) is satisfied using
series in
y
and
Ill
as
.
Expanding
112
in a Taylor
z about the body axis and taking account of transverse
symmetry, we obtain
(35)
Now
(36)
p2
Thus
(37)
It is seen that we have assumed
result.
to order
S(-) = S(-½) = 0
in deriving this
Since from (35) the boundary condition on the body is satisfied
0(/6
E ) = O( E)
, there is no inconsistency in including a
term
O(6 )
in the potential.
also
O( C )
, we may use just the free-space dipole Green's function.
Since the error in the free surface is
- 15 -
The error in force and moment resulting from neglecting the free surface
condition for the dipole distribution will be shown to be of higher order.
In the usual slender-body approach (Tuck, [7]), it is necessary to
find an inner expansion for the potential and match the outer limit of this
solution with the inner limit of the outer solution, which is the line
singularity distribution.
We have been able to avoid this because of the
particularly simple form of the boundary condition to be met on a body of
revolution, and hence have been able to use the inner limit of the outer
solution to satisfy the boundary condition directly.
- 16 4.
APPROXIMATE FAR-FIELD SOLUTION USING GREEN'S THEOREM
The results obtained in the preceding section can be obtained, in the
far-field, by applying Green's theorem.
This more direct approach, in
which it is not necessary to assume a form of the solution a priori, is of
considerable interest.
It is necessary only to apply the slender-body
assumption in the approximate evaluation of the surface integral in Green's
theorem, which is an exact result.
Green's theorem has been used to find
the near-field potential in various slender-body problems involving a free
surface by Vossers [81, Newman [61, [9J,
If
S
and Joosen [10].
is a surface enclosing a volume
V
in which the disturbance
potential satisfies Laplace's equation (11), then the potential inside
V
is given in terms of the values of the potential and its normal derivative
on the surface
S
by
5
where
;/~n
(38)
denotes the inward normal derivative, and
function given in the preceding section.
body surface
SS
, the free surface
vertical cylindrical surface
SR
, the bottom
SFS
of radius
The surface is
is the Green's
We take as the surface
surrounding the body, and a tube connecting
down to a vanishing line.
G
R
SS
S
the
, and a
SB
centered at the origin
and
SFS
which we shrink
shown in Figure 2.
- 17 -
U
Figure 2
The Surface S for Applying Green's Theorem
- 18The integral over
(14).
S3
is zero due to the bottom boundary condition
On the free surface
(39)
where the integral is taken around the intersection of
In the limit as
R - c
, this integral is zero since both
satisfy the radiation condition.
the surface
SR
SFS
and
•
and
SR
G
For the same reason the integral over
also vanishes in *the limit
R-
with the integral over the surface of the body.
, and we are left
Thus we want to solve
the integral equation
(4o)
c •3) [, -Vd,
-d
We will find an approximate solution valid in the far-field for small
values of the slenderness parameter
E
.
We split the integral into two parts
where
A ds
(42)
- 19 -
and
I
J,=
21T
I:Lpm '3ZVI%~
-
4Tw
Ln
(43)
jej
From the body boundary condition (23), we have
-6Y1
D&
In the integral
and
I" about
31
J1
f = 0
" r/
we expand
and
/Y-V, ,-z
a"
-It
0(3)
G(x,y,z;g,j,C)
(44)
in a Taylor series in
C = -b and obtain
-[_
, 6)
,;
j
(45)
CE
O?;I
,;D
C-E~t ;
RMli
Since
R
is constant on a body of revolution for
66:
0 4)qb
T
constant, the
second and third terms integrate out, and we have the result
(1(46)
This potential, a distribution of sources on the axis proportional to the
derivatives of the sectional area curve, is the usual result of slenderbody theory (Tuck, [7]).
Since in the above, we have expanded the
1/r
- 20 -
term in a Taylor series, these results are valid only for small
It
S(E)
is sometimes more convenient to have the integral in terms of
. Thus, integrating by parts, we obtain
I
J-1(/,-)
=
S~r)
(/X7
C), )
dl
iY
2
This represents an axial distribution of doublets (cf. Batchelor, [11]).
We now consider the integral
, equation (43).
J2
Using
Jl
for
in this integral, we see that
O(e J'> O (3)
=C2
and thus we need use only the
order term of
surface
S
J2
.
Jl
term of
(48)
to evaluate the lowest
Since Green's theorem for a point
(x,y,z)
on the
is
(49)
-C
C)T
we must take
2J;
~,
(50)
in the evaluation of equation (43).
and
J12
function.
, corresponding to the
Then we write
J2
2
We divide
1/r
and
as the sum of
H
J
into two parts,
Jll
terms of the Green's
J21
21. and
J22
22
corresponding
- 21 -
to
Ill
and
112
respectively.
Thus
( 51
Then
aL
J.aL
c
DcTG&J
4
D33 4
(52)
-
Expanding these derivatives in Taylor series about
= O,
01
= -b
, w
obtain
(53)
Glrx~rr;i~t;S
Jll
is a function of
x
4
and
-b)]
p
sn? de Br +O0(64
only.
Hence it is a constant on each
section on a body of revolution, and the first two terms in
integrate to zero leaving
J21
J11
= O(E4)
Finally, we consider
e
-2
2n
(54)
- 22 -
Expanding
112
and
AG/ýn
in a Taylor series about the body axis, we
have
(25
(55)
1-3
44
C(oczr;l~,q-b
m;ej
Then integrating and noting
12 = 0
--
+ (6i
R(nYd~d
due to transverse symmetry
of the H-term of the Green's function, we obtain the result
2
21
Since
J12
=
0(C2)
discarding terms
also.
(56)
O(
, we have
J
= 0(
)
.
Because we have been
4) all along, to be consistent we should drop
However, it will be shown in the next section that
J22
J22
leads to
a contribution to the moment of leading order, while the discarded terms
will lead to higher order force and moment terms.
The part of
J22
due
to the H-term of the Green's function in equation (42) also leads to a
higher order force and moment and hence need not be included.
In
addition, in the previous section it was shown that in the near-field the
- 23 dipole term is needed to satisfy the boundary condition on the body to the
same order as the free surface condition.
- 24 5. FORCE AND MOMENT
The force and moment on the body may be calculated by several methods.
We shall simply integrate the pressure over the body.
The vertical force
and pitching moment are given by
(57)
where
n
is the outward unit normal on the body.
For a slender body of revolution whose shape is defined by (21),
e . 0 (C)
rl, =
4-3 0)
40,+OC
where
e
S2Tr
_
F,•,
is defined as above.
f)
MA;
-
Thus the integral can be written
0ki
,)
•(
+-
+ 0 (6
,.)
•
(58)
2
· i)
+
kc
001
0
&ý
C)
r = O(C)
since
on the body.
- 25 From Bernoulli's equation the pressure is
<)Pg )7~
The
pgz
- P
2
(59)
term yields the hydrostatic force and moment
8
F6 V,-/0o where
*
buoyancy.
,-(n
-P I-
is the body volume, and
(60)
x
o
is the x-coordinate of the center of
We will neglect this hydrostatic force and moment in the follow-
ing.
In terms of the disturbance potential
-p·-P
ý
, the pressure is
2
22
I
-/y+.,
2U
CUZ
Oý4
(61)
ýA
The U2 term isa constant and hence isintegrated out. The terms
/y, and
y
2
z
are higher order and will be neglected.
2,
Thus the dynamic
pressure to order is
SPUT~l 4 ~
(62)
The disturbance potential near the body has been found to be
-VI
Z.
-)
2- :5-
47
I'~cr)Wr··,L·,b;i46)~I
L
(63)
4
-2,(
I-~O
(E
- 26 where now
2
r
respectively.
= y
2
2.
and 0J
and
are given by (25) and (37)
Then
(/9(c,A:;1,2
t
+ (z+b)
2
-2
3
a~L~x)
a~c
',,:e(f
c'(t D2 H(/?ý
d~cC(~19-b;
(64)
r (1) 4;,
o,-~lq-)
r
On a body of revolution, the first and second terms of (64) are functions
of
x
only and hence integrate out.
_~(
-P•~
(
I,
The third term yields
VGc) A ekt dLWJ,
(65)
-6 '44-_
J2 S
jý-a3 H(/x,
2.3
2.
J,,-bd~x
The contribution to the moment from this term is
5 D o0
2
(66)
_ -_ 4p
51~~3
·r
5-i~~3
I -l5V)~o)
-r~
~~-)
- 27 -
From the last term of
, we have the force contribution
x
22·n
(67)
according to the assumptions made in the derivation in section 3. The
corresponding contribution to the moment is
222
These expressions are all
.
O( C)
lead to force and moment terms
O(
The
5)
2
and
,
and higher.
terms resulting from the dipole distribution are
z
terms in (61)
The free surface
0(4 )
and clearly will
result in higher order contributions to the force and moment and hence can
be neglected.
Combining (66) and (68), we have the result
I
C r-r
I
_
(69)
It is of some interest to obtain these results by applying Lagally's
theorem (Cummins, [12j).
The force on a source is equal to
-4 Treco" ,
- 28 where
c
et is
is the flow velocity due to external singularities, and
the source strength.
the moment is given by
To lowest order, the force on a dipole is zero, and
-4vrpcp
, where now
c
is the external flow
velocity component normal to the dipole.axis, in this case the stream
velocity
U
, to lowest order, and
p
is the dipole strength.
The
"Lagally" force and moment on the source distribution are easily found to
be, to lowest order,
I
0
(70)
MSL2w~ma~
Z-a
ldd
Similarly, the force and moment due to the dipole distribution are
FDL-
O
The expressions for the force are the same as obtained by integrating the
linearized pressure, (65) and (67), but the moments due to the source and
dipole distributions are not those obtained previously.
ML = MS + MDL
However, the sum
- 29 -
is that given by (69) and thus the final results are identical.
This
discrepancy in the intermediate results for the moment is a direct
consequence of the fact that the source distribution (19) alone does not
satisfy the body boundary condition.
In the first calculation we found
the force and moment on the surface which was to be represented by the
source distribution, while the application of Lagally's theorem has given
the force and moment acting on the surface which is actually represented
by the source distribution, which clearly is significantly different from
what is wanted.
Similarly, the integral of the pressure over the intended
body surface gives the contribution to the moment due to the dipoles,
while the Lagally moment on the dipoles does not correspond to the moment
on any given body, since a line distribution of vertical dipoles in a
uniform horizontal stream by itself does not by itself represent a body,
but rather represents the effect of a correction to the shape of the body
represented by the source distribution.
- 30 6.
NUMERICAL RESULTS
A program has been written to evaluate the expressions for the force
and moment on a body of revolution moving horizontally in water of finite
depth, equations (65) and (69).
The program is listed in the appendix.
Figures 4, 5, 6, and 7 show the force and moment on a Rankine ovoid
of length-diameter ratio 10.5 for two different submergences in infinitely
deep water, along with the experimental results obtained by KorvinKroukovsky, et al.,
[13] for comparison.
The shape of the body is shown in
Figure 3. The agreement between calculated and measured results for the
moment is very good.
The agreement for the force is not as good.
Figures
5 and 7 also show the results of computations by Pond [2ý who used the
exact uniform flow potential (a source and a sink) instead of the source
distribution obtained by using the slender-body approximation.
The dipole
terms were calculated in the same manner as was used above, based on a
heuristic argument.
Both the moment on the source-sink combination alone,
calculated by Lagally's theorem (labeled "Pond's First Result"), and the
moment on the source, sink, and dipoles are given.
It is clear from these
curves that the source potential alone is insufficient from a practical
point of view.
The calculations using equation (69) agree very closely
with Pond's results in spite of the difference in the source potential
and the bluntness of the body.
Figures 8 and 9 and Table 1 present the results of calculations for
Froude number 0.55, depth of submergence varying from 0.144L to 2.5L, and
depth of water varying from 0.5L to infinite depth.
It is clear that for
a body near the surface the effect of the bottom is not felt if the water
- 31 is deeper than about two body lengths.
Furthermore, from Table 1 it is
clear that if the body submergence is more than about one or two body
lengths the force and moment due to the free surface are very small,
unless the body is close to the bottom, in which case the force and moment
are due to the wall effect of the bottom.
- 32 -
Surface of water corresponding to submergence b = 0.266L
Surface of water corresponding to submergence b = 0.144L
Ir
max
-I-
T
r-
Only one quarter of body is shown
length/diameter = 10.5
Figure 3
Shape of Rankine Ovoid used in Calculations
- 33 -
1.0
0.0
0
(C
-2.0
00
S-3.0
0
-4.0
-5.0
0.2
0.3
0.4
Froude Number F
Figure 4
0.6
0.5
0.7
V/1
Heave Force on Rankine Ovoid at 0.144L Submergence
- 34 -
/
/
/l
3.0
2.0
0
V
1.0
N
//
4•-
K
I
I
0.0
C-H
0l
II
0,
-1.0
Program (equation 69)
-
Model test results (Korvin-Kroukovsky, et. al. 131)
Pond's first result
-2.0
(Pond [21)
""
Pond's second result
0.3
I·
I·
0.4
0.5
(Pond [2J)
I·
0.6
1
0.7
Froude Number Fn = Vi1L
Figure 5
Pitching Moment on Rankine Ovoid at 0.144L Submergence
- 35 -
2.
31)
1.
IN>
0.
4cA
.
O
C-2.
0II
-3.
0.3
o.4
0.5
o.6
0.7
Froude Number F n = V/
n
Figure 6 Heave Force on Rankine Ovoid at 0.266L Submergence
- 36 -
Program
(equation 69)
Model test results (Korvin-Kroukovsky, et. al. [131)
O
M
1.0
c4
0.0
II
E*
el2 -2.0
O
-p
-3.0
0.2
0.3
0.4
0.5
0.6
Froude Number Fn = V/!
Figure 7
Pitching Moment on Rankine Ovoid at 0.266L Submergence
- 37
-
-tO
-3.0 r.
.rCU
b = 0.144
a,
o
0
.r
-2.0
L.
II
4,
r4
0
b = 0.266
1.0
k-
b = 0.500
0.0
0.0
1.0
2.0
3.0
4.o
Depth of water/body length
Froude Number = 0.55
Figure 8
Heave Force on Rankine Ovoid as a
Function of Submergence and Depth of Fluid
- 38 -
b = 0. 144
3.0
O
2.0
b = 0.266
0a,
o
4-
1.0
E
b = 0.500
0
0.0
0.0
I
1.0
2.0
Depth of water
3.0
4.o
/ Body length
Froude Number = 0.55
Figure 9
Pitching Moment on Rankine Ovoid as
a function of Submergence and Depth of Water
- 39 -
TABLE 1.
EFFECT OF SUBMERGENCE AND DEPTH OF WATER
ON FORCE AND MOMENT ON RANKINE OVOID
Froude Number = 0.55
Submergence
1.OL
Depth of Water
Force
Coefficient
Moment
Coefficient
2.OL
- 1.22 x 106
5.11 x 108
3.OL
- 2.59 x 10-6
3.86 x 10-7
- 2.86 x 10- 6
4.10 x 10-7
Deep Water
1.5L
9.48 x 10o
3.0
- 1.29 x 10- 7
Deep dater
2.OL
2.5L
-6b
2.0
3.0
-7
- 4.67 x 10 -7
9.54 x 10-7
Deep dater
3.0
- 1.26 x 10 - 7
9.65 x 10 - 6
Deep dater
- 4.72 x 10 -
8
- 3.97 x 106
- 6.24 x lO-8
1.20 x 10 -- 8
- 3.47 x 10 - 7
3.76 x 10-10
- 3.98 x 10
1.22 x 10- 11
- 40 7.
CONCLUSION
The theory of Havelock and Pond for the force and moment on a body of
revolution moving under a free surface has been shown to be a consistent
first-order theory, while theories derived ignoring the effect of the body
generated waves on the flow near the body are not.
The theory has been
extended to the case of an arbitrary slender body of revolution and to
include the effect of finite depth of water.
This more general theory
compares very well with the theory of Pond and with experimental results
for the special case of a Rankine ovoid in infinite depth water.
- 41 8.
1
REFERENCES
HAVELOCK, T. H., Quart. Journ. Mech. and Applied Math., Vol. V,
pt. 2 (1952).
2
POND, H. L., Journal of Ship Research, Vol. 2, No. 4 (1959).
3
NEWMAN, J. N., David Taylor Model Basin Report 2127 (1965).
4
STOKER, J. J., Comm. Pure and Appl. Math. IX, pp. 577-595 (1965).
5
WEHAUSEN, J. V., and E. V. LAITONE, "Surface Waves", Vol. IX, Part III
of Handbook of Physics, S. Flugge, Ed., Berlin, Springer-Verlag
(1960).
6
NEWMAN, J. N., J. Fluid Mech., Vol. 18, pt. 4, pp. 602-618 (1964).
7
TUCK, E. 0., "The Steady Motion of a Slender Ship", Ph.D. Thesis,
Cambridge (1963).
8
VOSSERS, G., "Some Applications of the Slender Body Theory in Ship
Hydrodynamics", Thesis, Delft (1962).
9
NEWMAN, J. N., "Lectures on the Theory of Slender Ships", Berkeley,
California (1963).
10
JOOSEN, W., "Oscillating Slender Ships at Forward Speed", Publication
268, Netherlands Ship Model Basin, Wageningen, Holland.
11
BATCHELOR, G. K., An Introduction to Fluid Dynamics, Cambridge (1967).
12
CUMMINS, W., Journal of Ship Research, Vol. 1, No. 1, pp. 7-18 (1957).
13
KORVIN-KROUKOVSKY, B. V., F. R. CHABROW, P. M. BELOUS, and W. H.
SUTHERLAND, "Theoretical and Experimental Investigation of the
Interaction of a Free Fluid Surface and a Body of Revolution
Moving under It', Experimental Towing Tank, Stevens Institute
of Technology, Report 390 (1950).
- 42 -
APPENDIX
PROGRAM LISTING
- 43 -
A-4P- P-4
ZZZ
V-4 P-4 P--0-4 . NNNN
o N NN
ZZZZZZZZZZZZZ
ZZ
-
0Ev
-0
ON
N *
II
<
tJ
-j
a1
CI
-0
'4CIL
u
O
- E-
-E
EX
-
- 44 -
a, o P-4
m %r
un
ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
N %0 *__ -W CV_-
N4
M
*
ýC
f-
W
C %,
0
SW
4
v-
N
M
-
-t
1-
f-
M
r--4
-
C'
Z
U.
o
t
S4
%
w
v0.
0.
II.,,
- ::
.n+
-U')
-"
c *O
4
0
00
C:12
W
U.*
0,-AZ
it3:
C)
0
•
3
-
•-"
00
*".Z
@
0
V)
:Du
11 F
a
.
CI
Z0\J
0
W,-.. 6-sW•
L'UJ mLL
ZeuI.w x
,Z
4
I DM
Cm
(.
L
*no
Z
C
n
Q4
(InV
in-
0te)
0OZ
U.
it z
0 -0'C
Q
u a,..
C
W
$
m)
N
C
ZM
a
*
(X
0
x.
zL
Zcc
*.U0
0 •*I
Ca
-,
<
A..)
u
4U
XL)
OZ
>
tn
.
U
M
+-
U, CC
boZo
4
0
C0 I
W
< a
4
)
C-
C
a
c.
II Q
, ,"
mHI•I=Z
z x QN1-12LL
2
Ct cC
Z
'
- 45 -
p!"-!
aO
go
aO O,
p ao
Q0 00 0•%3 ONa', - 0 - 0
o
0-
0
-00
0
ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
SX,.
= -LU
U.
V4 %C S.
f- uj 0'x
2
Z. 9 I
,
Z
*
•1a
2
S
II
If ,.4
*
c
+
W+.
It if M
I-
Ii .0
Wx
I-*
LL x
*
V) V) 0
*
C)
c
w
Z
ZL
*
bm"
*
1
0I
f
" 0
V) V) 0
U.
" 64 )
W) 4A
OIV
01 v
u
j
*%%
a
W)
a 4NIf*
U
=
I :I t ' ý
Id
u
o4
ZZZ
Q 04I.Y Ian
U.I(.I=
-4
bo
0 0 0
W 4A V) kn V)
-
(n
kA %-, -
a-
*
q9
O
CICLLU-
w
(J
:x w
*
b
't
0
U.
I1
<
< ,
-
"
9
O
*QUO4*
%a
WW
9
f 11 - WZ d 6U
Q4
tD#=
-N
-
r 00 k-. ,,.4I
co
0Z ~X
X*Z•.
L-
'%..Z
b.-4
n0WW%.
V)
z1
ft
1V
W 0aW N%"* i
Z
-
,-
0"U..L
X. LL
cEm
*
L,%%, - -- C
l
-
.*'u.
0,w
4I
*C
*e+-O
' bo
0
•,U
Z
000
ICY
o"Y
IN0
0
wl
I0
P- "0 lo
S+
m W+
•C -. .
0 - x X-"
04b
u
0
14n 4 Z -
- .
+
*
S-
Q
T
Q x
N 0-x
UJ
-,wX 4
,x
•
S\
Z*
-ru. Q
I
-
,.,x
9.
zl.
.
0
P-4
4w
O Xj
w
11
-
x
1 m<
II -4
-4-ýZ-tZ
a
S Uj *
LL U- U0
IfLn\C~
- 46 -
o -
--
-
z
-
o
·
%t,
0
V-4 l.
a.
M
I.t LA
to tn
-
- 47 -
--,NM -to1 o r- awa, or h4M-4-0 %or-w CPO
s-4 - 4V-4a
-
-4 V- "
a
ww
a ww ww w m m ww
mmwa am(
-4 N
ui
WW
W
z
-
.
wn
z
*
a
N
I
I
I
at
N
.
N
0
"MNt
U3 cc
NNrfff
a
r--
+Z
X
S.mm
^j
NM
yZv
X o
Ir
SIWW
0 +w xftw~w*
*
*
0
*
* -- ~ c. w 6o=
<C-4(AcxxM
M
+
I-*(D
+ 2: M u''
mW r. O
* 1S" N
+ N
li-CW.
a l )W
c+r Z
=)
N
NT
M
CL C
1acif " >- x 11 Is I*.LCV
=
c
W)
xe LL1
cCic
q It
V xti
M=rn-W
.jitCL
x
3E11I
2M= 0w
Z
Ca M
U. C) C
M >LC(a%
cc X cr.
zc 0NN0.
o,
Q .tj WN
=NZNO
M
X
cnCOLLI
Oz
CaCaOWN<Nr'"I
Sf
lW
- 48 -
No I*- wN
,--,
N,.,..,t
mV-ýr ,- ,-,
-- u-I
I,
o V-wN m1n
%01-
0
N
*
.N*
**0
*
"t
V**
N
4-..
N Nc
CV 444444r
4.
r %t "V ("1
-ft(N n t
FI
S-
0"
Z X**
*3
• ,*
*444
N,
.
N N
zOft Z
*R-"it
4.
*
at c.
4 a
If** ,.0
-,
1S
N atw
A
.
***44
U-
.. NNNNNNN
-
*4
P-4 P • . ,
-4
" aN
"4 N e *.
* 0
N d
**0
si4
.*
e****
rs444.4444
a
Z
ZN4.
**
***
e * eN
*NNNNNNN
CIM
*
s
Cy
NNNNNNN
*rC
3d~~
> * * ** *
M
m0
z
j
*0
*
*0w*
'
.eN'
,
; It
UU..Lcw..
*
Z
N
0
**Io .ZW
.
*j~7aP
*
-
*OLL
'
Q
.
. UC.
o.
WL
.-
Z
I
'
N
w
..)
*-
ON-*
0
12
II
4
>4 nag MJ
t-.c Nw
0
C
ZI
_
a
ZZ
**lI
t
I I
t I I
ZZAR...ZQ
t-NrfM0e
22r22
U
WZ
I
U
0
NIJ5 z
t-
j
Z
O
C
-
a
e
*
*
0
C
I
*
00
- -
Uu
**.
4*
e
W*U.
.J
*
e.
,
MuEI
00
v a
NN
t
Z
*.
.
*******
*N
0
Z
ZZ
V-cC
4
'
t c
rM
,U
e'. 024
+N4v>
li
)(*
'CI
)(
II
21
Y
UIa F
.J.a
-
L=
0
Nr. O
ull
- 49 -
o o
cmatw
ZZ
l.
-A
W.
Nla
j it
M 0
•
1
O
V-4
44
NN
0"
.I.ZZ
SU.
xZZ
I.- I--
**u...
u0
N rfn
•=
*
".-
N
•tz
0 .0.
.ZZ
poCatata:
I% a
NC.•
"
ZZ
N=
at
o
I.
)m.Po
p...
4
- it
•
ai
o)
*"
- lo-
c-0i
I lw
It
PC4
.IS w
a uCcc cc 0
a
w 0- 1.- I
*) 4N1mn- lw
ml
(fl
I•
.. I.Wtr• a a
t*j.J
to I- t
I'.-
--
.
=
cP-
0
en
u
,a V1 I•
0
49
of
- 50 -
O4 m.4
P-mA
-4 P4 v49- v-
NL(
N
O
I
0,
ow
z*
w
SL
.
o
cc
z
4
4a
c
C)c 0.Z
Sci!
-
W0
C >"r
C
X
4
L
a
I-
Q
Z
I-,
a 0oiw0
cr
a a
~
63 m a
N
x
a
:3
IIZ
rI
)
ZC
0
110d
xu
0W
)
V M )
) L I
N
=" t.
tLL N P
v~
I
a
-4W
ItW
-
C
ui
N
1 C
U
,
!1z
QN
*- U
Ittr
c -,eF
rrCUr3O *C -· V)
ul.
V
<
U
-F
I
UV)Y U
.
-51
mm4 4 4 4 4 4 4 4
Y
C3
)-vtn
Y
c3
~
~sh ~ul: ~L
tl
C3 t3
)I· tle
uv,
u,
Y
C3
)e
u,
r:
e3
t·
m
yi
C3
)e
vr
y
C3
t~
v,
-
nmtn inLn'%\
ts 0tnuA
A %o o
y
y
e3 c3
CI ~
v,
v,
pm-Ig- m
U0
o
.o
>
z
r
C <
8
%UJ
N
'
',
*
W
*
2
C
I-.
<
L
-O -
-II
c
Z
uC
00zw
-j
N
o
.
Iw*i
LO
It
Z4
QP
C.-
woo,
o'o
"
o
o 'o
4
'o r- -I.
- 52 -
mao
-
aa
- ---000m0000
o W
0
.0
cc-.
D
"oP4 v-4 -4-"
0
(
o.
cn*C:N
14
4
S.
eZJ
zce
cc
"4
i --U 4o-Z 1" Q3
I a=
W
.0
0 *"4
oe
*
4,
1
U>
0
SU.
+
,f
.
-C0*.4-.-.,
Om-.
w. ZZC
*W It-
j
J
l0
V8
N'•
.
*
a
.V o-"1
WUO..J
ZUt
k
ZN
II t
- 0'o0
t P-4
I.-a
j
Uj
Z
0
4
Ii
a
't X - U.Ww
(i L16
-,,
Z.U,.
#-4.tW
a
C
N
Z
* . io lz
It
V4
Z::N
O
0
Q0Z
0
LLl
3*00fUJU)W)WWQ
Z0
"
0
:
P-403
j
m
N
Aw M-GC0
zUW0
^-.. "
1
0m
0
m+N
0P"
vi
- 53-
U) SO r M
M
t
CS N
P"40 4P-4P-4OW4P-4V.4r.4P-4'.
N MJ MM%
MN
aP4
(\
t
4-M
Unl-p
v-4
UA
D
ON
I--
'-'U
0
I
ct
I:
V),,V),
I-I--u
*
u,~)
a'-4
u,
S'
44
ow
Il-i.. *)
ZZU
zu".10I-Q
NZ
I, I"-
If
LZ
01.* Ni
- IU. -U. .
W
uUl.
i4
U.
p(II,.t.A
001
I-.I
o ULL
uJ
•,,
15,e(%
,w
(r.
a nun m
i,1, I,,I,
-
4-%
. 4
I+
23
a
M
3
*
*,
e*
Nm
u: U
II II
i i
I,, I.
II
V
Ome1Qm
V)
U,'-
-i
,i,,
,U,
e
NM
00
S0O0
ct re
t
cC
3CU.
00w
Coo
- 54-
'-te Ume r
ZZZZZZZZZZZZZZZ
XE
4M
~
00
0
00
5
;
0%
V-4 v-4
g
L
4g
0
a
; 00
e
r-4P-4
r-4 r-i
a
0
o0
P-4 N
M ,t
FC
-4 NM
N NN N ( *N NV N m m m m
ZZzzzz
mcr. cr. (XW.w
u j
I
A
0.
z
u.
UN
a
C
I-
enen
N
N
ON
0.
ccm-
Y. li
O
NN
*, I
+
o
e
p-Z
0,.-
1
.,jZ
*3 4,
O
w* up.
a e
WW
I-.I-
t,. J tJ
u,-
-I4 ,,
_1--1
z.
O,.
1 ,.[ --
b- cc u w
00
z z
•ey
Zi.U. Uw
r,
m m
- 55 -
Z
Z ZZ
ZZZ
Z2
2
X
N
"t
CC
N
N
O
-. USZ0
ww
)-
-
NO
C,
NZ
NN
N
-
4-
w
Nm
-
- Zc
It 11 ý- Z
0
UU " =.w=M 1UIWc
LL.L
LU~. :L
*Id L
· LLL.. .W
C,
- 56 -
ZZZZZZZZZZZZZZZZZZzZZZZZZZZZZZZZZZZZ
N
*
N
-
v•
**
*
o
N
*0
N
I.N
U.1
S.
=
N
e
*
*IcN N N
"
N
"l
EN
N
,
N . N SNN
.
o,.
**0
* * *** . *
**
*
c*
*CS S
o,
*
N* r4
*
*
*+***.,.
aS.
*
•
9
Z
** , o l
S.
S
0
0.000
* *r
P-4 4s
cm
"N
S. S. 0*
,,
* N
* S.
***
NN
NNMNNN
*.
* *..
*
S.N(
.iN
(S
Z
Z4
*•*
*
OU1
-Z
Z-
S.S.*
S.
0 .
S.
-.
,- -
--.
r, --
-
S.
'o.
**
*.
SS.
S
. * S
SS.n*
S* S
.r - - -- I - - --
0
0
*
I.
NN
m. " w. W
?
I
Z
4
S.
**. S.
S
o
S e S.
S
-. -.
,-,
oI-
N
*
w
II
S.
Qt
-.. a
**
IZ
!
e
I(*.Z.l!
.-
,,
-•C 44N, 4
'n ..
I4 ZN
*-eZ
-C z
"
*n
* S.
S
..
**
S
*
S.·
S.
S*
S.
S
,NNNNNNNNNNNNNNN.t
C **j •l
S * S ** S * S. * S.
S.c,9 d **.
S *•
S.
S
a
0 000*@
S*
**
*
,S.*
S,
c,
S.
*r
S. -4
S -S*4-4 *
* .
.
r-4 "4 P-4 -4 r-4 v-4 w-4P-4*-*P-4*p.4* *"4 v-4
**
** *
SNNNNNNNNNNNNNNN Vtt1M
z
l'*
NN
N
oS. ,.S. S.
**o
NNNNN
N W * S*o**
*, S
*
e, *
I.-.4. a
0
0 0 . •
S.N
%'
*
Ne N
,
NNNNNNNNNNNNNN
.N0 N•*
Z•I S
Z,
*******
**o***
N'i
N v eS N N N C4N
u
S
r 0 No N
t I -
.ZZZ.4
0
0U
Z
(YMZ. ZI
•.
H"INZ
S.
I
Z
Wa
l
- 57 -
SOM
N
oZZ
-u00
m
-4Z
ft
0
zP- r , Z iza
00
Z LL
Z
C)
-- *
(
"W"1
N
.
.
W
.
w
0u Li0UJ
lfn
0
f
I-.ZZ I. -
-z
J
-u-
.
cc to
z
00
Ob
ZL
(D Iwpp 4.. .
.
"1
LU
a0-
N
N
W
yN
*V n
%. %w
c<<i I
0.,,
**L
I.-
•
a
C0
O
I-
M
UiN
4t
-
Pw % .1%%
%.-
U-U.
+
Z
-,,N
I I
M-ifIzff
_j*
ONMNQ
tU - W
II
+
y II n C
)C
,IL'
WI.-I m
.ww =W * * N
.JW
m-0.
w E.ri f"
*cnE/)ZZN"Nj1-0-**
W * 0 oý-'UJLU
W
OI-WX
O
W
Qr
W
W
U - Ncw.u, Y
QU-00f )
d 0 · 0W
")
SL
t
,. Z m M
U
•r
Z•
w NWU
*-N*
.mN
JZ
- 58 -
v.4r-
vV) W
")
%0 co
M
P- 0.4 0-4 V-4
P4-
W WI,
ý16
Nd
V) V) 4)
0
I
Z
*
Z
o
3Cc
0010
z
C 0z.
. O.
-LI
cOS
-
0
0
*
I
*
•
O0 v0
...
tZ
li
+
O
*. *
I
C.
x X"
W U
u. --ur
o*
aNoNr
N
N"
M
-- co7 0
(..I cN (\IN N
Nn
W
W
-4Nm 4
,
m mm m m
mm
V)W
V) WW
Uj
V
- 59 -
(/
x0
N
I
C0w
u.
x
u.
.-
0
*
*
It
I-
- 60 -
l',10-,,--
II
•'-"
• • •
I"too•- )L,,
"
10I w, I
•
•
C
• t
• • t
*,1
N
0
*
d
N N N0**(
X
4
* N L NN
N CQ
9"
Z
00 iwNm
aite..
Z
o. N
* * .
* .NNN
rU
*. *
'' x
w"
0
0.
U.
(D
,
9',** 0
. *%Nm N
1
iI.
Z N
+"
0
to0
=::
*' goo
•
• *N
•
*N
SZ1
,4r
*, *
o
N,..NN *
1,
4'O,
-, " "
a ALU In
**NNN
* -0 *0 WM
•
NU.
V) UJ
Z--.
- •
Ur
·
W
Z
o,
*NNN
.. *• ..
.
..
*
•
=-
*•
eM
0
-
Q""
. **.
00
On"
C'0Zw* 4~
* NNNU" •,
4*
I-
,
* I .L,=I,=.
l
nN4-LWOL
1=I N
Z
I. C
,-0 .N
.U.++
%V
,.
W.
C
O
It
l
w4
II Z
! =
Z
+ +
a
Z
W Zi.
1'
n
r
I(U)0
•J
0N.
.ZZ'
!.
u
Z
Z Z
1V') Z 1
0N
ZZ
.=-I--0
Z Z
U.
IZ,
4_j I--0X
ZZ
LJ
0N 0
-r, l -..
-
"*Z
'.UD(3.eL.
.
I w.
.4N
o
i
I
*I
-
LI--
ZU.Za
0u
..
0.0.
uI I, uZ-U
0U
. 0
a
I" I=V,
- '
I
C0C
-
0
-Z
+*
wqe
t---
v ) - 1: L
0 +C •N
N
•
-
=l,
oCtn
.-•
0
- 61 -
F: Po Oi Z: Zr Ri n
I: :. :E I: P. CF O,
(V rr
tn 'D r.- Wa0, 0
m 't
-
0-
of
Uw
tz.Uz
ZZ
0 a
WW
* <
* U. =. ,
I,--
-.LL
f
Mifn
I-
ZZ
"m-I00W
NN
f
z
r• %W*
)U
a %W
JI
-.
.-..
W
'
S*-.
I
*
1"
.I
tNNWW00
ZZ
I z
t If
. Nl
%WCI.
. %NoW
..
"Wo
cZ
(N
Z
,LU*
!
aJUJ
0%"t
UV
oe.
tiNN
UV
*w
N'
* *0
,,cv
0%
,r~e!,..!.NNAOOI
I
...0
*Ow-N
W
.**
WA
go
CA
V
•
0
W
1
U0
W- e
WooWCe
-W'W
it
I.
l 3
=U . . ,
mawXx.
04
•
W4
sft
O-
ZUw
r
J
t.ZZNNjt-i'' I
toIt
N14
NNOO
cu
wN J
rJ
,
-- iN
)
.4
I/W
W00
CW"It
I
4
-414
: W<
=
"Z Z
a
MW
WgL. WOQ
OinOs
in
1us
s o.'u
62 -
0>V-m Mt-t 0 %0
r-- w a%a V-4
Ci.m%t0 "Dt-. w
4v44
V4o
-
.v4-
C44M
.t W,-D
N
M- M
M4
2:
0
xL
(DON
. N'*
Z
a-*W
0i
4J Ip-0.4ZJ 43 UJ Lin
li4
z•-@W-•
V)
Sa LU
M QZ c
C
Q0o Z to0 V-4*=
.
04
O4
0
*"-')
Suit
a
OWZ
W a0
--w
0
w2
*Z
c l4
U
=0
@
(%
aLL
Om
uII
Q Q.
*0
o
m00
W
r"
Z (A
*L.)y
a
.0%
LU,
0c
11Z+
-
QZ)WO
! ,,,N W
"a
1,,,
*
0 WW
U-0 0ý
N
b
-
0
= W U.
le
I-O
IItI u
U CD ,,Q
4
n
U
f
004s -
0
x
0
UQ
"4
C U
hI
(X
zI>
u
Lm0
?-
uj
QI
cc
63 -
-
0
*
((
MO
4
O-
,m-
roei
0
*
S3
<Z n
-,NO
CI0
m.O
I-
-
<N
x/
o
*O
. 0
r.
.0
C
00DI,-'j--(3
,4
S(3
•
•0
0WC
"-".
*
0C.•w.U
.
LWNm
.
*
I.
CL
0.
,"0
0*O
u
0-4
*
Z
.,,
ftA
c
:.
L~-me
h
" x
0
** .
00
g..
Z
c-
-.
LU .UJ4
ýld '4c
0ýe ýld (V
w x
X-Z
42
uZ,> Q
F~
a , a:
tX
N•)
0
/)(I)
*
"
0
-
*C. W-.m4,-•,
M
--
Z
Z-.W
0
(-
0
Y IIQ
CZ ~ 10
X:
0X
.
0.)
W ltZXi
4QWW - W W U
lP Z--
C.'
. t
UJ
00
o'Y.
c YA
XwZ
w
%
C
0
W
-
CO
--
0
r
=
X
...
-0
=IC
m-
0
O
-- a
.lI
*0
o
(WJ
..
- 64 -
-i
O
N
X
u1
x
Uul
u-
x
x
w
t
0
Q
C• Q
Ce
NcWZWMNEM
I--
4
mV
0
II,
Z
oO a
*Lt%.
. *L Z OOZ
O Z-
z
9 -1
II
1
LC
N
*