-
SLENDER SHIPS WITH FORWARD SPEED
A NEW APPROACH AND A NEW THEORY
by
Sea Heon Kim
B.S., Seoul National Uhiversity, Seoul, Korea
Department of Naval Architecture and Marine Engineering
(1976)
M.S., Massachusetts Institute of Technology, Cambridge, Mass.
Department of Ocean Engineering
(1980)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
December 1982
Massachusetts Institute of Technology 1982
Signature Redacted
Signature of Author
Department of Ocean Engineering
December 1982
Signature R(edacted
Certified by
Roniald W. Yeung, Supervisor
Signature Redacted
Accepted by
Chairman, Departmental Coafi.ttei--nraduate
MAS;t3L'upr,
INSTiTUTE
OF TECHNOLOGY
JUN 0 2 1%J3
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Archives
Students
SLENDER SHIPS WITH FORWARD SPEEDA NEW APPROACH AND A NEW THEORY
by
Sea Heon Kim
Submitted to the Department of Ocean Engineering
in February, 1983, in partial fulfillment of the requirements
for the Degree of Doctor of Philosophy
ABSTRACT
A linear theory is developed for the heave and pitch
motions of a slender ship moving with constant speed in calm
water. The ship beam-to-length and draft-to-length ratios
are assumed to be small. The method of matched asymptotic
expansions is applied to obtain a new slender-ship formulation.
Novelty lies in the fact that a complete linearized free-surface
condition is used in the formulation of the inner problem, which
contains forward-speed effects not heretofore accounted for in
other theories. Genuine three-dimensional effects are also
incorporated by introducing a generalized inner Green function
that matches with the inner limit of the outer Green function.
It is shown in the course of development of this theory that
the inner problem, interpreted in a pseudo-time sense, is completely compatible with the three-dimensional outer problem.
The new theory is expected to be more valid for general
frequencies of oscillation of the heave and pitch motions.
Several special cases of simplification of this theory are
In particular, the present formulation yields a
considered.
rather promising approach to the steady forward motion problem
of a slender ship, which is simply a limiting case of the more
comprehensive theory presented here.
Thesis Supervisor:
Title:
Ronald W. Yeung
Professor of Naval Architecture,
Visiting (Univ. of Calif., Berkeley)
-3-
ACKNOWLEDGEMENT S
I would like to express my sincere gratitude to my
advisor Professor Ronald W. Yeung not only for his advice
and guidance during the course of this work but also for
his continuous support and encouragement during my stay at
M.I.T.. Without his constant support my graduate studies
would not have been completed.
I am also indebted to Professors P. D. SclavounOs,
T. F. Ogilvie, J. N. Newman, and M. S. Triantafyllou for
having served on my thesis committee.
Special. thanks go to
Professor Sclavounos for his valuable discussions.
Financial support during this study was provided
primarily by the Office of Naval Research under Task
# NR62-611, Contract # N00014-78-C-0390 and in part by the
National Science Foundation under Grant ENG 77-17817.
Both
supports are gratefully acknowledged here.
Thanks are also due to Miss Beth Germanotta for her
proficient and patient typing of this manuscript.
Finally, I would like to thank my wife Kyong-Ho for her
endless encouragement and assistance.
I would also like to
thank my parents and parents-in-law for their continuous
interest and support throughout my education.
-3a-
TABLE OF CONTENTS
Page
ABSTRACT
.......................
2
ACKNOWLEDGEMENTS .............
3
,................
TABLE OF CONTENTS
......
I.
INTRODUCTION
.,
II.
THE BOUNDARY-VALUE PROBLEM ......................
................................
...
....
...
...
.........
..
..
....
111.2
111.3
IV.
IV.1
IV.2
V.
VI.
30
Forward-Speed Transformation and
Resulting Representations ..
*.............
Inner Behavior of the Modified
Green Function ..
........................
Simplification of the Surface Integrals
THE INNER PROBLEM
..
4
20
III. THE OUTER PROBLEM ............
III.1
3a
.
The Pseudo-Time Inner Problem ... ..........
Integral-E uation Formulation in
terms of Gi)
...........................
MATCHING OF INNER AND OUTER SOLUTIONS ...........
....................
The Matched Solution
V.2
Behavior of Interaction Functions
F and F .................................
35
38
44
*.*.......
V.1
33
47
49
57
57
61
V.2.1
Zero-Speed Forced Oscillation .....
64
V.2.2
Steady Forward Motion ...
70
*......
V.3
The High-Frequency Approximation ......... ...
75
V.4
Alternative Representation of the Solution
in terms of a Line Distribution of
Wave Sources
.....
......
......
. . .
ADDED MASS AND DAMPING
.........................
VII. DISCUSSIONS AND CONCLUSIONS
.............
78
83
90
-3b-
Page
.-
REFERENCES
...
..
..
e.
..
.
..
L
.....
t
.
*
..
.... S
a
99
a
.
APPENDIX 1 :
Derivation of Modified Greenks Theorem
103
APPENDIX 2 :.
Derivation of the Pseudo-time-dependent
.........
Green Function ..........
105
APPENDIX 3:
111
APPENDIX 4:
Alternative Expressions and Asymptotic
Approximations of the Pseudo-time.dependent Green Function
The Inner Expansion of h (x,y,z) .....
APPENDIX 5:
Reduction of the Kernel
APPENDIX 6:
Inner Expansion of the Outer Solution
O(x,y..Z)
APPENDIX 7:
..
.
.
.
.
.
116
................
.
.
.
.
.
-
.
..
Relationship between a Line Distribution
and a Surface Distribution of the
.......
Generalized Inner Green Functions
.
-------a
120
128
130
-4-
I.
INTRODUCTION
One of the primary: goals of the naval architect is to
be able -to design ships which operate satisfactorily in a
wave environment that is frequently uncomfortable and quite
irregular in nature.
The accurate prediction of the ship
motions and the dynamic sea loads is of unquestionable
importance for a safe and economical operation.
However
this is such a complex problem that reliable methods of
making these predictions have only been available for the
last thirty years.
One of the major difficulties in develop-
ing a suitable ship motion theory was the lack of any
technique which could describe the complex nature of the
random ocean waves.
Since the well-known paper of St. Denis and Pierson
(1953) on the application of spectral analysis to ship
motions, there has been much development in both experimental
and theoretical efforts for the ship-motion problem in
regular waves, which, by linear superposition, is ultimately
applicable to an irregular seaway.
With the assumption of
small unsteady motions of the ship and of the surrounding
fluid, the ship motion problem can be decomposed linearly
into two problems; the radiation problem where the ship
undergoes prescribed oscillatory motions in otherwise calm
water, and the diffraction problem where the ship is kept
fixed at its mean advancing position but is otherwise free
-5-
to interact with the incident wave train.
The present
study is concerned only with the former problem., and is
restricted to the solution of the forced heave and pitch
motions of the ship which is also advancing at a constant
forward speed.
The first theoretical analysis of the ship motion
problem is connected with the works of Froude
Krylov (1896).
(1861) and
Froude and Krylov derived differential
equations for the motion of a ship in waves.
The inertial
and hydrostatic restoring forces for the radiation problem
were evaluated without attempting to analyze the hydrodynamic
disturbances associated with the ship's oscillatory motion.
Furthermore, the exciting force for the diffraction problem
was evaluated using only the pressure field of the undisturbed
incident wave under the hypothesis that the presence of the
ship did not alter its pressure field.
The resulting
exciting force has become known as the Froude-Krylov exciting
force.
Michell
(1898) made the first
significant step to account
for the hydrodynamic disturbance due to a realistic
ship
hull with his steady-state wave-resistance theory.
He
developed a so-called thin-ship theory of wave-resistance
under the assumption that the ship has a small beam compared
to its length and wavelength.
Although Michell recognized
the possibility of extending his theory to include unsteady
-6-
motins, a promised paper was never published.
workers, however,
Subsequent
did take up his ideas for bQth. wave-
resistance and ship motion studies.
A comprehensive analysis of the linearized ship motion
problem was made by Haskind (1946).
Green's theorem was
used to construct the velocity potential due to the presence
of a ship hull and the necessary Green's function or source
potential was derived.
The thin-ship approximation was then
invoked to solve the resulting integral equation.
He also
introduced the now widely adopted procedure of decomposing
the velocity potential into a canonical form so that thp
radiation problem and the diffraction problem could be
solved independently.
Throughout the development of ship motion theory,
certain geometrical characteristics of a typical ship has
been taken advantage of.
Thin-ship theory is the result of
assuming the beam is small compared to the length, with the
draft being taken as arbitrary.
Slender-body theory, how-
ever, assumes that the beam and draft are the same small
order of magnitude by comparison to the length.
Since
Michell's pioneering effort, the thin-ship theory has been
extended and refined by many investigators, Haskind (1946),
Peters and Stoker (1957), Havelock (1958), and Newman (1961).
In spite of the fact that real ships are hardly thin, study
of the thin ship prospered, and the theory provided some
-7-
.guidance in the reduction of wave resistance, see Wehausen
(1973).
But corresponding results for ship motion problem
have not been so useful.
A breakthrough was made by Korvin-Kroukovsky and Jacobs
(1957).
Using the concepts of the slender-body theory of
aerodynamics supplemented by substantial physical insight,
they derived a "three-dimensional" strip theory for the
approximation of ship motions in head waves.
They assumed
that the hydrodynamics associated with the ship could be
represented by a series of two-dimensional transverse ship
elements or 'strips'.
They solved two-dimensional boundary-
value problem for each cross-section of the ship by neglecting hydrodynamic interactions between adjacent ship sections.
The two-dimensional solutions were then adjusted to include
certain three-dimensional forward-speed effects based on
intuitive, physical arguments.
Even though their theory was
derived mainly from 'physical intuition' rather than rational
mathematics, there is no doubt that this original strip theory
deserves its recognition as one of the most significant
contributions in the field of seakeeping.
The Korvin-
Kroukovsky and Jacobs theory has since been modified and
extended.
For example, Smith (1967) has shown that a modified
strip theory by Gerritsma and Buekelman (1967) predicts the
head-seas motions for a high-speed destroyer hull which
agree quite well with experiments.
Here the word 'modified'
-8-
is used to represent a modification leading to a speeddependence of some of the hydrodynamic coefficients which
was not found with. a 'pure' strip theory calculation.
Even
though the agreement between experiments and the KorvinKroukovsky and Jacobs strip theory has usually been satisfactory, a major objection to this theory has been that the
forward-speed dependent terms in the cross-coupling coefficients do not satisfy the symmetry relationships proved by
Timman and Newman
(1962)'.
Subsequent work was concentrated on providing a rigorous
derivation of the ship motion theory based on the slenderbody approach of aerodynamics.
This has resulted in a so-
called long-wavelength ordinary slender-body theory of ship
motions.
This was done initially for the steady-state wave-
resistance problems by Cummins
(1956).
Later, Tuck
(1963)
reformulated this theory by using rigorous singular perturbation and matched asymptotic expansions techniques, with the
assumptions that the beam and draft are small compared to the
wavelength scale as well as the ship length.
Using a similar
slender-body assumption, Newman and Tuck (1964) derived a
long-wavelength slender-body theory .for unsteady motions of
the ship.
In this case, however, most nontrivial hydrodynamic
effects as well as the inertial force due to body mass are
higher order by comparison to the hydrostatic restoring force
and the Froude-Krylov exciting force, which depend primarily
-9-
on waterplane area.
This implies that the resulting
leading-order equationa of motion are nonresonant.
Thus,
their unsteady solutions only gave adequate results for low
forward-speed and low frequencies of oscillation.
In order to improve the predictions of the slender-body
formulation the inertia terms must be retrieved into the
lowest order theory in such a way that added mass and damping
also appear.
This can be done under the assumption that the
wavelength is the same small order of magnitude as the
transverse dimensions of the ship.
A systematic analysis of
the short-wavelength slender-body theory for the radiation
problem was carried out by Ogilvie and Tuck (1969), where a
strip theory approximation was shown to be an adequate
representation of the flow field adjacent to the ship hull.
By consistently retaining the higher order terms of relative
magnitude
e1/2
in their perturbation analysis, e
being
the slenderness parameter defined as the ratio of transverse
dimension to ship length, Ogilvie and Tuck provided a
rational approximation for the effects of the ship's forward
speed.
Their analysis gave some integral terms over the
free surface as being an additional contribution to the
cross-coupling coefficients.
These additional contributions
are essentially due to the forward-speed effects on the free
surface condition.
Faltinsen (1974) noted that better
agreement with experiments was achieved by using the Ogilvie
and Tuck's cross-coupling coefficients.
A cQmprehensive derivation of a strip theory, valid for
short waves, was carried out by Salvesen, Tuck and Faltinsen
(1970) in the context of the prediction of ship motions and
sea loads.
Even though this formulation was derived less
rigorously compared with Ogilvie and Tuck's (1969), the
results were in good agreement with experiments.
theory of Salvesen, et al.
The strip
(S.T.F.) agrees in the zero-speed
case with Ogilvie and Tuck's short-wavelength slender-body
theory, which is known as the 'rational strip theory'.
However, the SFT'theory included in the forward-speed case
the forward-speed squared terms that were discarded as
higher-order effects in the particular perturbation scheme
of Ogilvie and Tuck.
In spite of the work of Ogilvie and
Tuck (1969), all strip theory formulations have deficiencies
in one way or another.
The principal questions are the
validity of the solution at lower frequencies, and the
validity of treating forward-speed effects as simple
corrections to the zero-speed solution or only as higherorder corrections.
Newman (1978) derived a unified slender-body theory,
which embraces the ordinary slender-body theory and the
strip theory, under the single assumption of small beam
and draft to length ratio.
The ordinary slender-body theory
and the strip theory were shown to be the long- and the
-11-
short-wavelength liMit of this theory respectively,
Computa-
tions for the radiation prob.lem wi.th. ?k unified 4lendernbody
theory were presented by Mays
Sclavounos'
1980).
CI9781
Newman and
The theory has been extended to the
diffraction problem by Sclavounos
19811,
Throughout the analysis of the above slender-kbody
theories, the technique of matched asymptotic expansions,
described by Ogilvie (1977), was used to split the problem
into an inner problem and an outer problem,
The inner
problem, which describes the flow field in an inner region
close to the ship hull, involves two-dimensional flow about
a long slender body with some unknown three-dimensional
interactions.
This inner problem is governed by the two-
dimensional Laplace equation subject to the linearized
time-harmonic free-surface boundary condition and the hull
boundary condition.
The outer problem which applies far
from the hull surface is fully three-dimensional, and is
governed by the three-dimensional Laplace equation subject
to the complete linearized free-surface boundary condition
and the proper radiation condition at infinity.
The unknown
three-dimensional interactions in the 'inner problem are
determined by requiring the two solutions to be compatible*
in a suitably defined overlap region.
It is worthwhile to point out that the free-surface
boundary condition in the inner problem has be'en customarily
-12-
taken as time-harmonic so that it
was free of forward-speed
dependence,' while thle hull boundary conditiQn did include
forward-speed effects.
However, one will not expect the
time-harmonic free-'surface houndary condition in two
dimensions to remain physically valid if
the characteristic
time associated with the ship speed is comparable to the
period of oscillation.
In the rational strip theory of
Ogilvie and Tuck (1969), the forward-speed effects were
taken into account in the free-surface boundary condition
as higher order effects, which provided the cross-coupling
coefficients with the additional contributions represented
by some integral terms over the free surface.
The speed-
squared terms were, however, discarded as being higher order,
while most of the intuitive versions of the theory retained
these terms.
The conventional strip theory is deficient not only for
low frequencies but also for high speeds.
A complementary
approach, which includes the forward-speed effects on the
free surface condition, was initiated by Chapman (1976, 1977).
The flow at each section along the ship was analyzed in a
quasi-three-dimensional manner such that a part of longitudinal interactions could be incorporated through the free
surface condition.
Here the interactions were assumed to
propagate downstream only.
Chapmanls computations for the
sway and yaw response of a vertical surface-piercing flat
-13-
plate were suppQrted by an impressive agreement with
experiment,
But the 'roude number was extraordinarily
Using a similar idea, Yeung and Kim (1981) derived a
high..
formulation for the radiation problem of a slender ship.
By letting the ship pierce through a sequence of control
planes defined in a fixed frame of reference and by solving
the resulting initial
boundary-value problem on a control
plane, the forward-speed effects on the free-surface boundary
condition could be easily included in a quasi-three-dimensional
manner.
The computational results from the resulting
intlegral equation of Volterra type showed some improvement
over the strip theory predictions at low and moderate
frequencies.
Since the solution satisfied a two-dimensional
Laplace equation and no outer matching was used in the above
approach, there could be no representation of transverse
waves.
A direct numerical solution of the linearized threedimensional ship motion problem was carried out by Chang
(1977)
and Inglis
(1980).
It was found from the computa-
tional results that the simple speed-correction normally
used in strip theory does not adequately characterize the
actual speed effects observed in the measurements, and that
the speed terms in the free-surface boundary condition have
to be included in order to obtain reasonable quantitative
and qualitative 'predictions.
Even though this three-
-14-
dimensional approAch is derived in. the Most 5reneral way
within linear theory, one 'MAy be faced with the need to
assume that the ship is either thin, slender or completely
submerged to justify the linearization of the free-surface
boundary condition, for the disturbances due to the steady
forward motion of the ship are not necessarily small
compared to the unsteady ones unless the geometrical
restrictions are imposed.
-15In the present study a new slender-body formulation of
the ship motion is derived by using the technique of matched
asymptotic expansions under the single assumption of small
beam and draft to length ratio.
It is a well-known fact
that the hydrodynamic disturbances due to ship motions are
represented by a velocity potential which satisfies the
three-dimensional Laplace equation, the complete linearized
free-surface condition, the hull boundary condition, and
the radiation condition of outgoing waves at infinity.
It
is also known that the velocity potential can be constructed
by using Green's theorem and appropriate Green functions.
Conventional ship hulls are slender in the sense that
the geometry variation in the longitudinal direction is small
compared to the corresponding variation in the transverse
directions.
This geometric characteristic of the slender
ship has been used for the simplification of the fully
three-dimensional boundary-value problem.
In what follows,
the flow pattern and rationals behind the present theory
are briefly described from the physical standpoint.
The
theoretical framework of the present approach is built from
mathematical analysis presented in the subsequent chapters.
Far away from the ship, at distances comparable to
ship length, the flow is fully three-dimensional and depends
mainly on the elongated nature of the ship hull, being
relatively insensitive to its local geometric details.
The
velocity potential in this outer region, therefore, satisfies
-16the three-dimensional Laplace equation, the complete
linearized free-surface condition, and the radiation condition
at infinity.
The outer solution is either obtained by using
Green's theorem, or approximated by a distribution of freesurface wave sources and dipoles along the ship's centerSince the hull boundary condition is not imposed on
line.
the outer problem, the precise quantitative information is
not available until the flow close to the ship hull is
analyzed.
Near the ship, at transverse distances of the order of
the ship's lateral dimension, the flow gradients in the
longitudinal direction are small by comparison to those in
the transverse directions due to the slenderness of the ship,
which justifies the reduction of the three-dimensional
Laplace equation to the two-dimensional one in the inner
region.
It is common also to simplify the free-surface
boundary condition by relating the wave parameters and the
forward speed to the slenderness parameter - e .
In the
zero forward-speed case of forced oscillations of a slender
ship, there is only one wave system, whose wavelength depends
on the oscillation frequency
w , far away from the ship.
In the unsteady forward-speed case, however, Faltinsen (1981)
showed that there may be up to 5 wave systems with different
wavelengths behind a translatory harmonic oscillating source.
Furthermore, the wavelengths depend on the forward speed and
the wave propagation direction as well as on the oscillation
-17frequency, which makes the unsteady forward-speed problem
more complicated than the zero-speed case.
The forward-
speed terms in the free surface condition have been usually
either neglected completely or included as higher order
effects under the assumption that
U = 0(1)
and
w = 0(C-1/2)
However, it is difficult to know how to relate the different
wavelengths to the slenderness parameter in a way that leads
to meaningful results.
The three-dimensional approaches of
Chang (1977) and Inglis (1980) also suggested the importance
of the forward-speed terms in the free-surface condition for
the reasonable quantitative and qualitative predictions.
From the above reasoning the present formulation employs the
complete linearized free-surface condition which includes
the forward-speed terms under the simple assumption that
the forward speed is appropriately limited so as to justify
the linearization.
Even though these forward-speed terms
may be considered higher order effects, including terms of
higher order should not significantly affect the results in
a strict sense.
In reality, the present approach provides
physically meaningful results.
Thus, in view of the above
arguments, we adopt the following formulation in the inner
region.
The velocity potential satisfies the two-dimensional
Laplace equation, the complete linearized free-surface
condition, and the hull boundary condition.
Since a
radiation condition cannot be imposed, the inner solution
is not complete, unless the qualitative information is
-18-
provided from the outer solution, appropriately approximated
in an overlap region.
The mathematical analysis of the
present study is largely devoted to the derivation of this
approximation, which is known as the inner expansion of the
outer solution, and to its subsequent effects on the inner
problem.
As it turned out, the general solution of the inner
problem requires the knowledge of a Green function which
cannot be determined uniquely from the inner problem alone.
It is determined uniquely and completely by examining the
outer expansion of the inner solution and requiring it to
be compatible with the inner expansion of the outer solution.
Owing to the forward-speed dependence of the free surface
condition in the inner problem a line distribution of Green
functions along the ship's centerline cannot produce the
same wavemaking effectiveness even in the far field as that
of a surface distribution.
In other words, the line
distribution should be extended over an infinite length in
order to produce the same far-field wave effects as that of
the surface distribution.
The outer solution, therefore,
is also represented in terms of the surface distribution of
Green functions in the present approach.
In Chapter II, the exact linearized boundary-value
problem for the unknown radiation potential is formulated,
and the fundamental assumptions of the present approach are
explicitly stated.
The method of matched asymptotic
-19-
expansions is used for the solution of the problem.
Thus,
the outer solution is firstly derived in Chapter III together
with its inner expansion.
The inner problem is formulated
in Chapter IV and then the matching is carried out in
Chapter V.
In Chapter VI, the pressure on the hull and the
resulting added mass and damping coefficients are determined.
Finally, the present formulation is compared with the
existing theories, and conclusions and future directions of
work are described in Chapter VII.
-20-
II.
THE BOUNDARY-VALUE PROBLEM
We consider a ship which moves in the positive x-
direction with constant forward speed U, while performing
small harmonic oscillations of frequency
pitch.
It
w
in heave and
is - convenient to introduce three Cartesian
coordinate systems, with x =(x , y , z ) fixed in space,
x =(x,', y', z')
fixed with respect to the ship, and
x=(x,
y, z)
moving in steady translation with the mean forward-speed of
the ship.
These coordinate systems are illustrated in
Figure 1.
The
z0=0
plane is taken as the undisturbed
free surface, the positive x,-axis in the direction of the
ship's forward speed, and the positive zo-axis upward.
The steady-moving coordinate system
X, which is
defined by (2.1), is an inertial frame of reference in which
the motions of the ship are periodic.
coordinate system x',
The ship-fixed
which is defined such that x'=x
in steady-state equilibrium, is the best to derive the hull
boundary condition on the ship's wetted surface.
x=
(xo-Ut, y 0, z )
(2.1)
-21-
,
z
Z
X
.x.
Figure 1.
Coordinate System
The fluid is assumed to be ideal, incompressible with
constant density
p, and its motion to be irrotational.
Surface tension effects are neglected.
With these
assumptions, the fluid velocity vector
V (X
,t)
is
represented as the gradient of the velocity potential ND(xo ,t)
which satisfies the three-dimensional Laplace equation,
V20 (
,t)
= 0
(2.2)
The fluid pressure
throughout the fluid domain.
p(x.,,
t)
is determined by Bernoulli's equation,
p (xO, t) =
Here
p
-
p (t
+
V
+ gz0 ) +
a
is the fluid density, g is the gravitational
(2.3)
-22-
acceleration, and pa is the atmospheric pressure which is
In (2.3) and hereafter, when the
assumed constant.
independent variables
(x, t) appear as subscripts, partial
differentiation is indicated, i.e., tt = at/at, etc.
On the submerged portion of the ship's wetted surface
S, the normal velocity of the ship is equal to that of the
adjacent fluid such that
on S,
V-n = Vson
where
Vs
(2.4)
is the local velocity of the ship's wetted
surface, and n
is the unit normal vector pointing out of
the fluid domain.
On the free surface, whose elevation is given by
z
(x,
the kinematic boundary condition is
y , t),
expressed by means of the substantial derivative
D/Dt E/at + V-V ,
(D/Dt)(
in the form
-
zo) = 0
on z
=
(2.5)
An additional dynamic boundary condition is obtained by
requiring that the pressure on the free surface is
atmospheric.
t +
Since
Bernoulli's equation (2.3) then gives
V2 + gz
0
on z0
=
(2.6) holds on the free surface for all time, its
(2.6)
-23substantial derivative can be set equal to zero.
This leads
to an exact nonlinear free-surface boundary condition,
1 vo-V(Vo-vo)
+ go 0 = 0
on z
=
,
0tt + 270-VOt +
(2.7)
with
Z
12
(1/g) [P +1 V2
(2.8)
0
The fluid velocity V is assumed to vanish at z
+
-
and a
radiation condition has to be imposed such that the energy
flux associated with the disturbance of the ship is directed
away from the ship at infinity.
The set of equations (2.1)-(2.8) formulates the exact
boundary-value problem within the limitation of an ideal
incompressible fluid.
However, the nonlinear free-surface
condition precludes any solution of the unsteady motion
problem without further simplifications.
In the theory of ship motions it is customary to
assume the amplitude of the incident wave system to be small
by comparison to its wavelength, which justifies the smallness of the oscillatory motions of the ship and the surrounding fluid.
However, the steady disturbances associated with
the ship's forward motion are not necessarily small compared
to the unsteady ones unless geometrical restrictions are
imposed.
Under the previous assumptions the total velocity
potential can be decomposed linearly into a steady and an
-24unsteady part.
Thus, following Newman(1978), we write
(t)
(Nt)
+
)=U(x)
UT(N) is the velocity potential due to the steady
where
i(N,t) is the unsteady one
forward motion of the ship, and
resulting from ship's oscillatory motions.
i
(2.9)
( x, t)
and
Both potentials
n'1satisfy.Laplace's equation (2.2) subject to the
boundary conditions that are appropriate to them.
The velocity
vector of the steady flow relative to the moving reference frame is
= UV(T - x)
(2.10)
The boundary condition on the hull surface in its steadystate position
S
takes the form
n = 0
q
(2.11)
on
In the moving reference frame the nonlinear free-surface
condition for
becomes
-
(W
+
gUoz = 0
on z = I
i
where the steady free-surface elevation
(2.12)
,
is given by
(g/2) (W2 - U2
(2.13)
Neglecting second order terms in
T
and using a
Taylor-series expansion, Newman (1978) showed that the
unsteady velocity potential
T
satisfies the following
boundary condition on the steady-state free surface
z=i.
-25-
-V)[I
-V
+ Ttt + 2A.V'ft +
vT-vW 2 + g z = 0
+
]-,z
+
l/[g
(Vw2) + gt
-
+
on z
=
(2.14)
This boundary condition is a combination of steady and
unsteady effects, and it is impossible at present to derive
a velocity potential which satisfies (2.14).
Using a similar analysis, Newman obtained the
linearized boundary condition on the ship's wetted surface,
n=
(
+
(t-v)
(a-.)%
AL
n on S, S
(2.15)
where S denotes the instantaneous position of the hull
surface, S the steady-state mean position, and the overdot
signifies time differentiation in the reference frame of
the ship.
Here
a = x
-
x
is the local oscillatory
4.
displacement of the ship's surface.
a
Since
is a small
oscillatory quantity and each member of (2.15) is 0(a),
this boundary condition can be applied either on S or S
with the difference 0 (a2 .
From a vector identity the
equation (2.15) is expressed in a more compact form
4.
"I
'n
=
E
onF
+ Vx(a x W)]-n
(2.16)
If the perturbation of the flow due to steady forward
motion of the ship is neglected,
reduces to
=
-
Ut,
and (2.14)
-26-
tt -
+ U2 x
2Ux
+ g
on z = 0
= 0
(2.17)
This assumption is justified for a slender ship at moderate
speeds.
Since the unsteady motions are assumed small, the vector
can be expressed as
a
displacement
=
where
I
1
and
denote the unsteady translation and rotation
of the ship relative to the origin
P
X' = 0.
The potential
in (2.9) can thus be decomposed linearly into separate
components due to each of the six rigid body motions of the
ship.
With the restriction that the unsteady motions are
sinusoidal in time with the frequency w, the motions of the
ship are denoted by
+
& = Re{(%1,
= Re{(
1
,
2'
3
e
iwt
22' a 3
S Re{(&4 ,41'
6)it
(2.19)
1
t
(2.20)
Here the Re stands for the real part of the expression.
accordance with this notation, the unsteady component of
the velocity potential can be expressed as
S(X, t) = Re{
Tx)eit} (
jJJ
(2.21)
In
-27Under the assumptions stated above the linear boundaryvalue problem satisfied by
T(N)
and
P. (') in the moving
J
reference frame is summarized as
V2= 0
(i
V?
22
+ V) 2
+ 0
on z = 0, w
= 0
on
+0
at z =
and
f
(2.22)
+
0
(2.23)
(2.24)
= f
*
where
U~
-
in the fluid domain
-c
(2.25)
denote the corresponding velocity potential
and the normal velocity, respectively, which are defined in
(2.9), (2. 21),
(2.11) , and (2.16) .
Here we introduce an
artificial Rayleigh viscosity parameter
P
for the purpose of
specifying an appropriate radiation condition at infinity.
From (2.11), (2.16), and (2.18) the hull boundary
conditions for (2.24)
S=
mio
take the forms,
n
+Um.)
V.Fjn == iwn.
1w
Here the components
n
(n4 , n 5 , n 6 )
(x
(2.26)
on~
(2.27)
are defined as
n.
(nl, n 2 , n 3 )
on
(2.28)
x
n)
(2.29)
and, following Ogilvie and Tuck (1969),
U(m1 , m 2 , m 3 )
UM = -(n-V)W
U(m4 , m 5 , M 6 ) = -(,-V) (,
x
')
(2.30)
(2.31)
-28In a strict sense, a theoretical model which satisfies
(2.23)
and (2.24), where
f
is given by (2.26)-(2.31),
is inconsistent since the influence of the perturbation of
the steady flow has been excluded from the free-surface
boundary condition but retained in the hull boundary
condition.
For a slender ship at moderate speeds, however,
the disturbances due to the steady forward motion of the
ship can be assumed small compared to the unsteady ones.
Thus this model is certainly valid for a slender ship at
moderate speeds since retaining higher order quantities that
are mathematically of a higher order in the formulation
could possibly improve the results.
For an arbitrary geometry the solution of the above
boundary-value problem can be obtained using Green's
theorem, which results in an integral equation for the unknown
potential on the wetted surface.
Such a Green function should
satisfy the three-dimensional Laplace ecuation, the
linearized free-surface condition, and the radiation condition
at infinity.
The solution by this approach involves, in
general, lengthy computations of the kernel associated with
this fully three-dimensional Green function as well as the
inversion of a big matrix.
Conventional ship hulls are slender and this characteristic will be used in the present approach to reduce the
complex three-dimensional problem into the simpler one through
the method of matched asymptotic expansions.
Before
-29proceeding any mathematical manipulations we state our
The beam and draft are assumed small
basic assumptions.
compared to the length such that
B/L, T/L = 0(),
(2.32)
E. .<<
where B, T, and L are the beam, the draft, and the length
of the ship respectively.
The ship's forward speed U is
restricted only to the extent that interactions between the
steady and unsteady disturbances remain relatively unimporThus we set
tant.
U
(2.33)
5 0 (1)
The frequency of oscillation w varies independently of the
ship dimensions.
W
We will introduce the parametrization of
for the ease of analysis in the application of the method
of matched asymptotic expansions.
O= O(j)
,0
Thus we set
S y. S 1/
2
(2.34)
Since only the radiation problem for heave and pitch
is analyzed in the present study, the subscript
used hereafter with the understanding that
j
j
will be
= 3 or 5.
-30-
III.
THE OUTER PROBLEM
r =(x2 +Y2 +z 2 1/2
The outer solution, which applies for
0 (1)
,
satisfies the following set of equations,
in the fluid domain
V24= 0
(iw
V
0
U-ax +1)2) + g 3
-
+
-0
0
on
z =0,
at
z =
+
0+
(3.1)
(3.2)
(3.3)
--
Here the radiation condition of outgoing waves at infinity
is incorporated by introducing an artificial Rayleigh viscosity
parameter
p
.
An appropriate outer solution can be either
exactly obtained by using Green's theorem, or approximately
by a distribution of wave sources and dipoles along the
ship's centerline.
The former representation is used in the
present study for reasons that will be readily apparent.
Applying Green' s second identity to
4)(Q)
and
G (0) (PQ)
in Q-space, which is bounded by the ship's wetted surface
a large surface
surface
1,
S, , and the portion of the undisturbed free
SF between
S
and
S.
as shown in Figure 2, we
can express the outer solution as
4[f
(P)
ds(Q) a) -
S+S +S F
G(0) (PQ)
)n
an
(3.4)
-31-
Sao
Figure 2.
where
nQ
Q-space.
(o)
Domain of application of Green's theorem
is the unit normal vector out of the fluid in
The Green function
G
(o) (P,Q), with superscript
denoting "outer", is defined as
G(O)(P,Q)
=
-
l/r + 1/r1 + h(P,Q)
1 lim
h (P,Q)= .- ~ 1+o+
dkj
x
d
exp[k2+2 2 1/2
2 2 1/2_
exp[-ik(x-E)-it(y-n)J
(3.5)
2
(3.6)
where
r =(x-)
2 + (yn) 2 + (Z-0 2 1/2
(3.7)
-32-
r1=
with
(x-02+
P = (x,y,z)
and
2+
Q =
(+
2 1/2
(,n, I)
.
(3.8)
,
The function
h
is
defined such that it is harmonic in the fluid domain and
satisfies conditions (3.1)-(3.3) with respect to P-variable.
The details of the derivation can be found in Wehausen and
Laitone (1960).
The integral over large surface
vanish if
G(O
and
can be shown to
Since both
tend to infinity.
S,
we let
S.
satisfy the free surface condition, the
4
bracketed quantity in (3.4) evaluated on
=0
can be written
as
)
[-
1
1-
{2iwU[$hJ+U2[4 a
3h}
C=O
;=O
(3.9)
Making use of (3.9) and Stokes's theorem, we obtain
47r4 (P)
dS(Q)[4,(Q)
dn
where
C
G
-
Q
()(P(,Q)
Q
[2iwU4(Q)+U2 (0
)h(P
-
,Q)
(3.10)
represents the intersection of the hull surface
and the undisturbed free surface.
Since the hull boundary
condition is not imposed on the outer problem, the precise
quantitative informations for
4
and
an
will be determined
-33from the matching with the inner solution.
III.1
Forward-Speed Transformation and Resulting
Representations
It is convenient in the matching procedure to introduce
the following "forward-speed transformation"
=
*(x,y,z)
-0
=
0
(U
-
)2
VO
(3.11)
ei(ox/U 0 (x,y,z)
then becomes
*
The boundary-value problem for
2+
defined by
in the fluid domain
+ g a
+
on
z=0, P
at
z= -w
0+
(3.13)
(3.14)
represent generalized gradient
v+
where the special symbols
+
(3.12)
(vector) operators which are defined as
S
i(+( a= 0
) +
+
k
(3.15)
in rectangular coordinates. The modified Laplacian operators
2
V+
are, therefore, of the forms,
2
2+
,
2
- iw
+
a2
Dy
a2
z
Dz
(3.16)
Following a similar analysis to what we did a little
earlier, the solution
0
can be obtained from a "modified
Green theorem" proved in Appendix 1.
More specifically,
-34-
if
and
0
are any differentiable scalar functions,
*0
then
J JJ
S
=
-
-
0 TdS
3n03n
S
(3.17)
where the normal-derivative operators in (3.17) are defined
as
(n-V +) = n(
+ n 2-
-
+ n3az
(3.18)
If the function
*0
Green function
G 0 ()
in (3.17) is chosen as a 'modified'
then the following expression for
,
results
47r
()
n 0 -- 0(G ) (P,-)
@nQ+
(Q)
dS(Q) [
fJ
=
DnQ-
T
-
U2
g
ra
Cdn[f
0
C
(Q)
~
-
Here the particular Green function
V+GO
a 2aG
0h
(3.19)
(P,Q)
p.
G (0)
has the properties,
in the fluid domain
(PQ) = 6 (P-Q)
-
(Uax _
3
(0)
2 G 0-(o)+ g G0z
and is in fact related to (3.5)
0
on z=0, P
such that
+
(a)
G(
0+
is given
(3.20a)
(3.2 Ob)
-35by the following expressions,
(o)
Go ((PQ)
=
(-/r + 1/r 1 )e
h (PQ) = e U
1 tim
eO((k'-*/U)+2 2- (Uk-i )2
(x-0)
U
+ h 0 (PQ)
(3.21)
h(PQ)
dk'
-*++
dt exp[(k'-w/U) 2
[
exp [-ik'
2 1/2
(x- )-iZ(y-n)
(3.22)
It is of interest to notice that the line integral in
(3.10) has been reduced to a simpler form in (3.19),
and that the expression in (3.19) will be seen to be easily
matchable to the inner solution later
on.
Inner Behavior of the Modified Green Function
111.2
If we are later going to match the outer solution in
(3.19)
with an inner solution, which will be derived in
Chapter IV, it will be necessary to obtain an inner expansion
We shall first examine the expansion
of this outer solution.
of
ho
for small values of
R(
y2
+ z2)
.
In the next
section, we shall derive the corresponding inner approximation
.
of the integrals in (3.19)
The expansion of
h
will be derived in the Fourier
space because of the relative simplicity of the algebraic
-36manipulations that are needed in the transform space as
opposed to the more cumbersome manipulations in the physical
We define a Fourier-transform pair as follows
space.
Odx e
F{P (x)}
p*(k)
F
P(x)
1
(3.23)
P(x)
(3.24)
Cdk e-ikxp*(k)
{P (k)}
The Fourier transform of (3.22) then gives
h 0 (y*, z;k)
Jt
k1 exp(k-w/U) 2 1/2z costy
d0 [ (k-w/U) 2 +t2 11/2- (Uk-ip
)2
im
-&
-4
0+
/
=
(3.25)
The expansion of
for small values of
h
k0 R0 (=k-w/UIR)
is derived in Appendix 4, and is given by
h
O(k R ,
(y,z;k) = 2[H (y,z;k)+ F (y,z-k)]+
0
K
R2 ) (3.26)
00
where
-2
H00 (y,z;k)
*1
FF*-(y,z;k)
0 (y2z7k
=--
+
0
dZ e costy
t- (Uk-ip)
*-*
(3.27)
/g
*
~- f *0 (kK0 ) [H*0 (y, z;k)-H 0 (y,z;k) ]
(3.28)
-37-
f (k,
0
0
)
ag sn (k) + cosh" (K /k )22 1/2
-sgn (k)
-
+ sgn(k) Zn(2c /k ) +
1
(3.29)
i
*
is the complex conjugate of
H
Here
I
(/k
0o 0
+
*
H
,
and the upper or
lower expression in (3.29) is applicable according as
K 0
/k0 Z 1, with
K
0
U 2k2 /g
k0
and
(k-w/U).
Transforming
2R , R )
(3.30)
back to a physical space, we obtain
h (x,y,z)=2[H (y, z;x)+F0 (XyZ)+0
where
F
1
(x,y,z)=
*
*
F1
F-1 {f9 (k, K )[H* (y,z;k)-Hf0 (y,z;k)]}
*
/K..
-l-k
It will be seen later in Chapter IV that if
(3.31)
we interpret
x/U as time t,- H (y,z;x) can be considered as a two-dimensional
time-dependent source potential which satisfies a twodimensional Laplace equation
condition
K
0
H oxx + H oZ = 0
V22D H 0 = 0, the free surface
on
z=0, and the initial (i.e.,
upstream) conditions of vanishing motion
x > 0
on z=O.
H *(y,z;k)0
,
H = H ox= 0
for
Furthermore, the inverse Fourier transform of
designated as
H0
(y,z;x) , will represent the
corresponding source potential that satisfies a reverse flow
problem such that
=
H(c)
.
H(C)
Ox =
0
for
x S 0
on
z=0.
-38-
Simplification of the Surface Integrals
111.3
On account of slender-body approximations of the ship
geometry, we have
= 0( e)
n
and
0(1)
n2 , n 3
(3.32)
Thus, to leading order, the normal derivatives in (3.19)
reduce to
a
anQ+
4.
a
nQ
n
(3.33)
aNQ
a
anQ
1 + O(weR 0 , eR0 ) ,
with an error factor
where
NQ
is
the two-dimensional unit normal vector out of the fluid and
the vector operators
V
are defined in (3.15).
From (3.33) we obtain the slender-body approximation
of (3.19) as
rL/2
(
da
d'
L/2
+
Q) 3N..
ds
S( W
-L/2
dsQ[I
(h)(PrQ
0 N0
/2
U
I
g C
dnli
0II-....
T
-o
1
(
d
U
N os[--Ni+
a.
+
(P)
(p/ =
)
4wo
''
-
~
-
a
h
MQ0
0(PQ)
(3.34)
-39with an error factor
,
0)
1 + O(weRQ .R
where
.
the transverse section of the hull surface at
B(e) is
Now we consider the first term of (3.34) defined by
L/2
OA (P)
s
df
f/
-L/2
ds
N
[0(Q)
-
-1W.(x-0U
-
r
+
] (-
e
8 U)
(3.35)
It is seen that the kernels in (3.35) for small values of
are approximately
1/r , 1/r
Ix- I
except for
small
R
=
I
1/x-
(3.36)
+ 0(R
)
R0
= 0(R,).
Therefore, we recast (3.35) for
as follows
AL/2 d9
-L/2
ds
0 (x
[
8(x)
+0)
(
-
lnC)
Q
Q
L/2
Q
8(x)Q
1 )e
r
1
+
L)
(-l +
ar (-
- f ds [0 (x,n,t)
1
)
aN Q
-
B(
-
)
-L/2
jri,)
-
d{J dsQH[ 0 (
(x-)
I
(337
(3.37)
Now we can apply the approximations (3.36) to the second term
of (3.37) because the curly bracketed quantity in (3.37)
tends to zero for
-+
x.
Thus
A
becomes approximately
-40-
OA (
L/ d9
ds
[ (x,n,)dQoa(r.-NQ
- - -J-Q
-L/2 .8(x)
with an error factor
Completing the integration
1 + 0(R ).
9
j ~~ jjI
dsQ[00(x
B(x)
we obtain
,
x.
- - aN.]
)3*
x
2.
x
(3.38
Q1
in (3.38) with respect to
OA(P) =
] (-QL + 1-)
2
2
1/2
tn (x-L/2)+[*(x-L/2) +(y-n) +(z
2+(Z-4 2 11/2
(x+L/2) + [ (.x+L/2 )2+Yn2
+ t
2 1/2
2
(x+L/2)+ [ (x+L/2) 2(-)
2+ N-n 2+(Z+4) 2 11/2
(X-L/2)+[ (x-L/2)2
(3.39)
For small values of R, and
-L/2 < x < L/2, it is also
seen that
/2.2
(x+L/2) +
(x+L/2)+R
(x+L/2)+ (x+L/2) [1+
2
-
+...]
2 (x+L/2)
=2(x+L/2)
(x-L/2) +
(x-L/2) 2 +R 2
+ 0(R 2
22
(3.40a)
- (L/2-x)+ (L/2-x) [l+
]
2 (L/2-x)
2
+ O(R )
2(L/2-x)
(3.40b)
-41From (3.39)
M
Q
8(x)
(3,41)
]nR/R
-
ds0(xn)
= 2
A(p)
and (3.40) we obtain
where
R =
[
(y-n) 2+
R = ( (y-n) 2+
(z-)2 11/
2
(3.42a)
(z+;)2 1/
2
(3.42b.)
with the cumulative error factor
collecting all
1 + 0 (R )
Therefore,
the terms in (3.34), the inner expansion of
the outer solution in (3.19) takes the form
2ir
(p) =f
ds
0
0
0 (x,n,,)-
]Q N nR/R1
8(x)
+
L/2
d
ds10(
(Q)
-
0t
][H (PQ)+F (PQ)]
-L/2 8(E)
- -
[H (PQ)+F (PQ)J
with a cumulative error factor
1 + 0(w2R 2, weR 0 ,
-
(Q)
ldn[
(3.43)
IR, 2
We include in (3.43) the higher-order terms represented
by a line integral over the free surface.
The justification
for this comes from the same arguments we made in Chapter I
for the inclusion of speed-dependent terms into the free
surface condition.
Actually,
the contribution from this
-42line integral turned out to be important in the computational
results of Yeung and Kim (1981).
The main,result so far is the inner expansion of the
outer solution defined in (3.19).
approximation for
0
The corresponding
may be obtained using a similar procedure
to what we did above (see Appendix 6 for details).
2-rr 0 (P) =
T, )
ds 1s0[(XI Q
5(x)Q
f
QL
+ +L d
ds QN
[O(E,
-L/2 S(E)
-nR/R
a
3N Q1
T,)
Q-
1N
NQ
[H (PIQ)+F(PQ)]I
dn[2iwU(E,n,O) + U 2(0,.-
g
The results are
0)]
C
[H(P,Q) + F(PQ)J
with an error factor
1 + O(w 4 R,
(3.44)
R , R
)
x
The same result will be obtained if we substitute into
(3.43) the following relations,
O(P)
=
eiwx/U(P)
H(P,Q)= e U
HO (PQ)
(3.11)
(3.45)
-W(x-0)
F(P,Q)= eU
Fi(PQ)
(3.46)
which are consequences of the forward-speed transformation.
It
is obvious that (3.11) should not be used in the case of
-43zero forward-speed forced oscillation prohlem.
Instead
(3.44) will be used without the line integral over the free
surface.
It is seen that the forward-speed transformation
in (3.11) simplifies this line integral in the non-zero
forward-speed case.
The inner expansions of the outer
solution in (3.43) or (3.44) will be eventually matched
with the corresponding expressions of the inner solution in
a suitably defined overlap region.
-44-
IV.
THE INNER PROBLEM
The inner problem is defined in an appropriately
restricted region close to the ship hull, at transverse
distances of the order of the ship's lateral dimension.
The
basic assumption in the inner region is that the flow
gradients in the longitudinal direction are small by
comparison to those in the transverse direction due to the
slenderness of the ship.
of
0(1)
y
and that
Assuming that the x coordinate is
and
z
coordinates are both of
0(R0 ), a coordinate stretching suggests that
-z 0 ,
ax,
ay0(1)
=0(R7
1)
(4.1)
Exploiting the slenderness of the ship, it has also been
assumed that
),n2,n
3=0(1)
n
(4.2)
= 0(
n
e is the slenderness parameter.
where
Now the flow equations (2.22)-(2.25) will be approximated
using (4.1) and (4.2).
The three-dimensional Laplace
equation is approximated as the two-dimensional one,
+
ay
0
in the fluid domain
(4.3)
az
with an error factor
1 + O(R 0 ).
The body boundary condition
-45-
reduces to
(n2
2ay + n 3 az
with an error factor
onn
N(y,z;x)
Nyzx
aN
1 + O(eR), where
VN
(4.4)
is the inner
approximation of the normal velocity of the ship's wetted
Thus,
surface.
N
=
N5 =
(2.28)-(2.31)
(N2 ,N3 )
(4.5a)
-xN3
(4.5b)
= -(N- + N 3a7
M 3 32ay
M5
(4.5c)
z
(4.5d)
-xM3 + N3
with an error factor
2
1 + O(e ),
the inner approximations of
T
reduces to
nj
N.
where
and
J
m.
and
M.
are
v)
respectively, and
is the velocity potential due to the steady forward
motion of the ship.
The reduction of the free-surface boundary condition
needs further comments.
Neglecting influences of the
perturbation of the steady flow, we obtain the linearized
,
free-surface condition in the form
(iso-U)2
ax
I + g a
= 0
on z=O
If we were to further neglect the forward-speed terms in
(4.6) , it
would reduce to
(4.6)
-46-
a-
az
vV0
on
z;=
, v 2/g
(4.7)
with a resulting error factor
1 + 0 (wR 0).
Then, the inner
problem becomes a truly two-dimensional one, being free
from any explicit
U
dependence.
In the strip theory of Salvesen et al.(1970),
(4.7) is
used for the .free surface condition so that the forwardspeed effects are involved only through hull boundary
condition.
However, the three-dimensional approaches of
Chang(1977)
and Inglis(1980) pointed out the importance of
the forward-speed terms in the free surface condition for
reasonable quantitative and qualitative predictions.
In a
systematic perturbation analysis of Ogilvie and Tuck(1969),
these terms are included as higher order effects of relative
order
e /2
For reasons described in Chapter I, we simply use (4.6)
for the free surface condition in the present work under
the assumption that the forward speed is appropriately
limited so as to justify the relations
(4.6).
Even though
these forward-speed terms may be considered higher order
effects, it is possible that their inclusion can significantly
improve the results.
In summary, the boundary-value problem satisfied by
the inner solution takes the form,
-47-
a2
+
0
az
= 0
U I) 2 ++ ga
ax
(iW
with the cumulative error factor
(4.9)
in z=0
97
on
VN (yZ;X)
=N
IV.1
(4.8)
in the fluid domain
(4.10)
1 + 0 (eR, R
)
ay
2
0
0
The Pseudo-Time Inner Problem
In accordance with the formulation of the outer solution,
we set
(y,z;x) = eiwx/U ( (y'Z;X)
(4.11)
The corresponding boundary-value problem for
40
then
takes the form,
a2
2
2
4
1 +
U2 2
0
aN
=
a -w/
2
+g
=
in the fluid domain
(4.12)
on z=0
(4.13)
on
(4.14)
0
---= 0
z
NYzx
with the cumulative error factor
F
1 + 0( eR, R2
0
0
-48-
A similar boundary-value problem to (4.11)
-
(4.14)
has
been studied by Yeung and Kim(1981) based on a "transient
This is readily apparent
formulation" of radiation problem.
if one simply replaces the x/U by time t.
For more details
Yeung and Kim(1981) should be referred to.
Since no outer
matching was used in the analysis of Yeung and Kim, initial
(i.e., upstream) conditions of vanishing motion were imposed
for the uniqueness of the solution, which could be justified
for
T = wU/g > 1/4.
We will use the method of Green function to solve the
above inner problem.
(y,z,n,rC;x,5), with super-
G
Let
script (i) denoting "inner", be an appropriately defined
Green function for (4.12)-(4.14), of the form,
G0
G
Here
"initial"
=
(4.15)
G 00(y,z,n,C;x,F) + E (yznI;xr)
is defined such that it satisfies the upstream
conditions of
G0 = G
The boundary-value problem for
=
G0
0
for
and its
x > E
on
z=0.
solution are
derived in Appendix 2, of the form,
G=
H
6(x-&)tnR/R1 + H (y,z,n,r ;xI)
u (E -x)
d
(4.16)
e(z+r) cost (y-n) sin//7 a (E-x)
(407
(4.17)
-49-
where
2
(y-n) 2 + (Z
R =
1/2
(4.18)
,
R= [(y-n) 2+ (z+;) ]l/2
(4.19)
a = U 2 /g
(4.20)
6(x-)
u(E-x)
and
are the Dirac delta function and the
Heaviside step function respectively.
Since the inner problem defined by (4.12)-(4.14)
has no
E
corresponding initial conditions, the harmonic function
cannot be determined until we proceed the matching with the
outer solution.
However, it is expected that
and
H0
some modulated combination of
E
will be
(c)
for both of
Hc,
them satisfy the free surface condition similar to (4.13).
The boundary-value problem for
H(c)
0
and its
solution are
also derived in Appendix 2, of the form,
H c
0
=
-
afp
(x-
e(Z+,) cost (y-n) sin/Z7a
/ dZ
0
Here
Hc)
0
(4.21)
is defined such that it satisfies the downstream
initial conditions of
IV.2.
0(x-E)
H(c)= H(c)
ox
0
-
0
for
x < E
on
z=0.
Integral-Equation Formulation in terms of G
In order to solve the problem at hand, we start out by
applying Green's second identity to
t (Q;E)
0
and
(PIQ;X.
G(i)
0
-50-
S (c), a large
in the region bounded by the body contour
semicircle
CR, and the portion of the undisturbed free
F(E)
surface
as shown in Figure
CR
and
B(E)
between
Thus we obtain
3.
2n6 (x-E) 00 (P; E)= f ds Q [ 0o3N
(Q;E)
8 (E) +F W)
Q
-
(4.22)
normal vector out of the fluid in Q-space.
over
CR
CR
vanishes if we let
(PQ;x-)
0-4G
9
is the two-dimensional unit
N
P=(y,z), Q=(rj,,), and
where
-
The integral
tend to infinity.
E
ting both.sides of (4.22) with respect to
Integra-
from -L/2 to
L/2, we obtain
ds [
(P;x)=
(Q;x)
--
0Q
+
JL/2
-L/2
ds
d
I[
4,DG
Q
B(E)
x[H
fL/2
-L/2
0- nR/R
Q
-Q
0] x
00
(P,Q;x- ) + E (P,Q;x-F)]
dn[ $ (Q;.)
0
-
1; ]G 0i)
(P,Q;x-
)
27r
F(E)
(4.23)
-51-
AL
.a
FW
N
FC
Figure 3
Domain of application of Green' s theorem
Since both
and
G(1)
0
0
satisfy the free surface
condition, it follows that
[(
a
0 a=O
0 )G (i)
-
2
9~G5E
-)
0E
3E
(H +E)
00r
(4.24)
Making use of (4.24)
takes the form,
and Stokes's theorem, equation
(4.23)
-52-
ds0 [% (Q;x)
27r 0 (P;x)=
0
nR/R
-
8 (x)
L/2
+
as
[ 1 (Q;
ds Qs[t
d()
0
8 (W
-L/2
x
Q
[H 0(P,Q;x-E) + E (P,Q;x-)]
a
2
-U
g
0
0
H
[ (Do (Q; 0
Cd
x
N
a NQ
(P,Q;x-F)+E (PQ;x- )I
0
(4.25)
where
C
is the intersection of the hull surface and the
undisturbed free surface.
An integral equation for
5 (x)
on
00
can be readily
obtained by letting
P
that the factor
in (4.25) is now replaced by
2w
approach
The net effect is
.
B(x)
7.
Thus
we obtain
7 0 (P;x) -
ds [
(Q ;x)
0]ZnR/R
-
5
8(x)0
f L/2
d- f
-L/2
ds0 I
B (9)
Q
(Q;)
0
aN Q
as
2
g
x
+ E (PQ;x-V)]
x[H 0Qx-V
U
as
-0]
3NQ
-0]
Cdo [ D (Q;
[H0 (PQ;x-C)+E
(PFQ;x-0) J
C
for
P(y,z) e 8(x)
(4.26)
-53-
t, which satisfies the boundary-
The inner solution
value problem defined in (4.8)-(4.10),
can be obtained from
(4.25) and (4.11), or alternatively it can be expressed as
21r t(P;x) =
' x
ds It(Q; x)'--- a ao InR/R 1
Q
Q
5(x)
ds [(Q;1)
+(L/2d4
-L/2
-
+ E(PQ;x-)]
x[H(P,Q;x-E)
id
[2iwU
a
(Q;E)+U2
x[H(P,Q;x-)
where
H and
E
Q
Q
S(E)
+ E(PQ;x-)]
(4.27)
are defined such that
H(P,Q;x-E) = e U
-(x-o)
H (PQ;x-)
(4.28a)
E (PQ;x-)
(4. 28b)
-(x-)
E(P,Q;x-E)
= e U
which are consequences of the forward-speed transformation.
Thus, an integral equation for
form,
t
on
8(x)
now takes the
-54-
7 t (P;x)
i
-
dsQ [
8 (x)
f 2d
-L/2
(g)
x
aQ
Q
I [H (PQ;x-)+E (P Q;X-0)I
ds [ + (Q; ) a
NQ
aNQ
x
[2iwU(DQ;)+U2 (0 a
[H (P, Q; x-g
)
Cdn
S
D]ZnR/R
aNQ
-
Q;x)-
+ E(PQ;x-)]
for P (y, z) e 8 (x)
(4. 29)
D
is obtained from the integral equation for
t (P;x)
-
f
8(x)
-
MQ
{
L/2
ds
8(
-L/2
I [tlnR/R
[
(Q;)
0a
+ H2D(PIQ)I
QN0
-
aD]E(PQ;x-)
I
)
=
a
ds
Q
0
,
For the zero speed case, where (4.11) is not applicable,
for P(y,z)
e 8(x)
(4.30)
where
H 2 D (PQ)
-
e (z+)cost(y-1)
m
i-2
2+O
0f
at
t-(W-iv) 2/g
(4.31)
-55-
Here the limiting behavior,
SL/2d
-L/2
=
f
dsQ[V(Q;H)
,Q;x-E)
-
Q
8(t)
dsQ[O(Q;x)
-
0H
(PQ)
(4.32)
Q
8(x)
has been used to obtain (4.30) from (4.29).
It is worthwhile to discuss the character of the motion
furnished by (4.25) in more detail.
The first term in (4.25)
is evidently the fluid motion generated by an impulsive body
motidn characterized by the hull boundary condition at
x,
assuming no previous disturbances existed in the fluid.
However, the ship moves with forward speed U, and therefore,
each cross section of the ship will pass through the flow
field disturbed by the rest of the ship.
The second term
represents the memory effects due to hull boundary conditions
on the rest of ship sections.
Since the inner solution satisfies the two-dimensional
Laplace equation, effects due to the presence of any transverse waves are absent unless the three-dimensional interactions characterized by
E0
is included.
The third term
of (4.25) has been resulted from the inclusion of the
forward-speed effects in the free surface condition.
If
the ship's forward speed approaches zero, the fluid motion
-56-
becomes time-harmonic so that it is free from any explicit
dependence on U.
Owing to the transient disturbances
generated by a finite forward speed, however, we can hardly
expect the two-dimensional time-harmonic representation of
the solution to always remain physically valid.
-57-
V.
V.1
MATCHING OF INNER AND OUTER SOLUTIONS
The Matched Solution
We are now in a position to match our inner and outer
solutions in a suitably defined overlap region
e << R
<< 1
to determine the unknown source and normal dipole strengths
of the outer solution, and the three-dimensional interaction
function in the inner solution.
The process of matching
involves equating terms of similar form from the inner
expansion of the outer solution with those from the outer
expansion of the inner solution.
The outer and inner
problems were analyzed in Chapter III and IV respectively,
and the corresponding solutions were expressed by (3.19) and
(4.25) respectively.
Equating the inner expansion of the outer solution in
(3.43) with the corresponding expansion of the inner solution,
we obtain
4
(x,y,z) =
0 (y,z;x)
(5.1)
N
,z;x)
ON (yD
(5.2)
E 0 (y,z;x) = F (x,y,z)
(5.3)
where the cumulative error factor in (5.1)-(5.3) takes the
form,
Err = 1 + 0(W R2, weR ,
0
0
R , eR 0 )
0
(5.4)
-58-
The approximations (3.43) and (4.25) are valid in a
region where the corresponding error factors approach one
as
Let
-i+-O.
R
=
O(eB)
0 < B <
,
in the matching region.
(5.5)
Thenthe error factor in (5.4)
takes the form,
Err = 1 + 0(e 2 $-2y I
w
where
1+$-y
1+a)
20
is parameterized in (5.6) as
W = O(J T )
(5.7)
Hence, the matching conditions
only if
(5.6)
2U-2y
>
0
and
(5.1)-(5.3) are legitimate
1+0-y > 0.
Therefore,
y < a < 1(5.8)
defines the matching region.
Alternatively, if we do not apply the forward-speed
transformation, we will obtain from (3.44) and the
corresponding expression of (4.26) the matching conditions,
=
.(x,y,z)
ON (x,y,Z)=
0(y,z;x)
PN (y,Z;X)
(5.9)
(5.10)
-59-
E(y,z;x)
= F(x,y,z)
(5.11)
with the cumulative error factor,
42
2
Err=l + o(w R ,R ,
0
0
eR )
(5.12)
0
Thus, the direct matching provides the matching region,
2y < 0 < 1
(5.13)
For sufficiently large values of
w
such that
+
(e-1/2
a separate matching must be conducted from the high-frequency
inner approximation of the outer solution.
This will be
discussed in more detail in section V.3.
By comparing (3.43) and (4.25), it is now clear that
the wave kernel
H
in the pseudo-time inner problem, i.e.,
the portion associated with homogeneous "initial conditions",
forms only a part of the inner expansion of the outer Green
function.
of
E0
The remaining part is accounted for by the presence
in the inner Green function.
It is precisely due to
this reason that we need to introduce a generalized Green
function in formulating the inner problem in Chapter IV.
The matching procedure thus defines
E 0 , which contains the
relevant three-dimensional characteristics of the problem.
In terms of the inner variables, we have therefore the
following integral equation for
t (P;x)
to solve:
-60-
Tr
0f
-
(?;x)
ds [P
Q
8 (x)
Q
0
Cd
-~
(Q;x)
-
aNQ a
-
o (;
ZnR/R1
[H0 (P,Q;x-)+F0 (P,Q;x-)]
)(a
BC
-L/2
0
Q(
[H (P,Q;x-()+F(P,Q;x-E)],
-
for P(y,z)
e B(x)
(5.14)
which represents the final results of the matching analysis.
0
must be solved numerically from this
integral equation.
The behavior of the kernal function
The potential
F0
is of obvious importance and will be examined in the next
section.
For completeness, we will also provide the corresponding
integral equation for
which is obtained from a
4D(P;x)
formulation without the forward-speed transformation:
, 4 (P;x)
- -]nR/R 1
ds [((Q;x)
-
8(x)
=L/2d
-L/2
-1
f
ds [(Q;-)
-
[H(P,Q;x-E)+F(P,,Q;x-E) I
8(E)
dn[2iwU (Q; )+U2 (,
-
for P(y,z)
f][H(PQ;x-E)+F(PQ;x-)],,
e 8(x)
(5.15)
-61-
F.
Behavior of Interaction Functions
V.2
The kernel function
and.
F
in (5.14) essentially contributes
F
a "three-dimensional correction" to the pseudo-time inner
problem, which would otherwise have no upstream effects
because of the step-function property of
(4.17).
H
as given by
Considerable amount of analyses were necessary to
extract the essential behavior of
out in Appendix 5.
This was carried
The final results may be summarized as
1
F
F (x,y,z)
o0
~
Fo.
2
) (x,y,z) + F
(
(x,y,z)
,
x < 0
,
x > 0
(5.16)
0
(x,y,z)
F
where
F
(x,,Z)
kldk e-ikxeKozcosK y {l-[l-k2 K2 -1/2
=
+ 2i[
fk
-
2i
k2dk e-ikxeKoxcosK y
0
2
W/Udk e-ikxe Kozcos K y {l-[l-k /K 2 -l/2
k2
(5.17)
F 0((x,y,z)
k3 +
=-i[
ici/U
r-ikx
0dk
k
KozCOS KY
0
x{l-[l-k2
0
-i
4
k
dk e
eK 0
X
cOsK y
2 -1/2
0
{l-ik 2 /K 2
0
0
-
1
/2}
3
0 < -T < 1/4
(5.18)
-62-
Zcos K y {1- [1-k /K
dk e-kXe
0
f W/U
T
where
K
=
U2 k2/g
,
root singularities of
(k
2 -1/2
-
2
k
(2
(xyz)
F
>
k0 = |k-w/U|, and
(5.19)
1/4
are the square-
k.
F 0 (y,z;k) defined as
W/U)2 =.U k /g2
i=1, . . ., 4
(5.20)
T = oU/g > 0
(5.21)
Thus,
k
i2 {1 ~l-+4r}
=
,F
2U
0 <
k
=
T < 1/4
(5.22)
9 1
2U, 1 + i
T > 1/4
F (x,y,z)
The previous expression indicates that
regular, being finite for all
totic behavior of
x.
for large
F
is
The leading-order asymp-
lxI
can be obtained from
the theorem on Fourier integrals (see Copson,1976), and
the results are of the following forms,
21r 1
7X-
/2 (1+4-c1
4
x
ipix/6-i32/4
2
P2 e
2 e-i2x+i
3
/ 4ep
)}x~ + 0(1x|)
e 22/6~cos(p 22y/M
Ix I-*
,
C > 0
,
/
F 0 )x~~)[
(5.23)
-63-
F(2)
2
4x-i3 w/4, 42/
-i
-1/4
1/2
2
-1
cos(p 4 y/6) + O(IxI
X+-co,
0
<
< 1/4
T
(5.24)
1/2
y Z/
F o( ()21r
&/4 2 x
1 /4
-
x
3e~ P 3x/d+i37r/4 e
z/6 x
cos(p 2y/6) + O(IxKI')
x+
0 < T < 1/4
+00,
(5.25)
F (2)(x,y,z)=(e- V4T-lx|/6)
6= U2/g
where
and
p
,
xf+o
,
T
> 1/4
(5.26)
= k 6.
The corresponding expressions for
F(x,y,z)
in (5.15)
can be easily derived from (5.16)-(5.26) by using the
following relations,
F(x,y,z) = e1wx/U F0 (x,y,z)
(5.27)
F (y,z;k)= F 0 (y,z;k+w/U)
(5.28)
and we will not repeat the expressions here.
for
T
We note that
> 1/4, the interaction function (5.26) is exponentially
negligible upstream.
We will now discuss some special cases
of the parameters
and
w
U
in some detail.
-64-
V.2.1
Zero-Speed Forced Oscillation;
U=0
, w#O
This limiting case corresponds to the physical problem
of a ship with
U=O
performing forced oscillations at a
w
on the otherwise calm free surface.
finite frequency
Since the forward-speed transformation is not applicable
in this special case, we will analyze the kernel function
F(x,y,z) instead of F (x,y,z).
The corresponding expression
for F(x,y,z) can be obtained from (5.16)-(5.22) by using
(5.27) and (5.28).
f"dk e-ikx F (y,z;k+/U)
F(x,y,z) =1
-
Here
K =
Thus, we have
(W+Uk) 2 /g
dk e-ikxD(k)eKz cos Ky
and
D (k)
is defined such that
D(k)=Zn[2K/k]+7ri sgn(w+Uk)-7
x
(5.29)
1 - k2
i sgn (w+Uk) -cosh -(K
_
+ cos
2
(K/JkJ)
-1/ 2
/Ikl)
k
(5.30)
where the upper or lower expression in the bracket is
applicable according as
K/|kj
'< 1
(5.31)
-65-
F
The corresponding square-root singularities of
are obtained from (5.21), (5.22), and (5.28).
,
k 1,2
=
k 1,2 - w/U
=
(y,x;k)
Thus, we have
- 2U22 {1+2T + /l+4T }, T=wU/g
0
>
W/U 1
= k
4
111-2T
2U 21-2T
1-41
+
-i/4-r-1
0
<
T
1/4
<
T > 1/4
,
k
,
(5.32)
(5.33)
Since
T+O
U-0, the upper expression in
for
applicable in this limiting case.
timk
(5.33)
is
It is easily proved that
(5.34)
-O
=
- v
imk 23
U
2,3
2
v W /g
(5.35)
Replacing the integrand and the limits of integral in
(5.17)-(5.18) by the corresponding ones in (5.29)-(5.35),
k = k/v
and making use of a coordinate transformation
we obtain
F
F(x,y,z)
=
(1) (x,y,z) + F (2) (x,y,z)
(2) (xyz)
x < 0
(5.36)
x > 0
where
F(1) (x,y,z) = -2iv eVz Cos vy {dk e
-2iv e VZ Cos vy
J dk
e
-ikvx
ikvx
1
1-k 2 1-.1/27
(5.37)
-66-
F (2) (x,,z) = -iv eVzcos vy
1 dk e-ikvx{l-l-k 2 -l/ 2
Q
-iv e'Vzcos
vy
dk e-ikvx{1-i[k2_1]-1/ 2
(5.38)
By using integral representations of the Bessel and Struve
functions defined in Abramowitz and Stegun(1964), the
x)r--x)
F (x,y,z)=e V z cosvy{-l/ lx|+(7rv/2) [H (v
)
expressions in (5.36)-(5.38) can be reduced to
(5.39)
+ 2iJ 0 (vIxj)]}
where
J0 , Y , and H
of zeroth order.
+e
are the Bessel and Struve functions
The resulting expression (5.39) shows a
symmetric disturbance in x.
Now the integral equation for
O (P;x)
corresponding to
this limiting case can be obtained from (5.15) and (5.39) as
1
aD(Q;X)
-
-
[tnR/R1
+ H2D
8(x)
-L/2
dsQ[ 1(Q;)
]F(PQ;x- )
-
Q
8(e)
for P(y,z)
e
(x)
(5.40)
(
=L/2 d
,
ift (P;x)
-67-
where
2
Zim H2
+(Z+c)cost.Zy-n) H
2 D (PQ) = -2 V.+
(5.41)
2/g
Here the limiting behavior,
L/2 dE
-L/2
dsQ[tO(Q;)
-
1H(P,Q;x-E)
8(E)
(5.42)
,r
H2(PQ)
-
fxdsQ[O(Q;x)
has been used to obtain (5.40) from (5.15).
However, the integral equation (5.40) is not complete,
defined in (5.39) is singular at
F(x,y,z)
for the kernel
If
x=0, and furthermore the singularity is not integrable.
we integrate the right-hand side of (5.40) by parts, we
will obtain
Wi (P;x) -
f
ds
[
(Q;x)
MQ
Q
-
a] [ZnR/R +H 2 D(PQ)I
12
NQ
B (x)
-L/2
B(
-
ev (z+r:) cosv(y-n),
)
ds [ (Q; )0 a -
L/2dFA (X-)
for P(y,z)
e 8(x)
(5.43)
FA(x)
= sgn(x){[Zn(2vjx
+ y + 7il
-
where
7r IXIdt [IH(t)+Y (t)+2iJ( t) ]}
I
(5.44)
-68-
Here
y
" 0.577...
that the kernel
is Euler's constant.
FA (x)
It is observed
in (5.44) is identical to the
corresponding one in the unified slender-body theory
(Newman and Sclavounos (1980)).
In this special limiting case, the three-dimensional
interaction function
F(x,y,z)
in (5.39) is represented
simply as the product of a homogeneous Green function
times a function of
x
such that
G M= 6(x-E)G 2 D(PQ) + F(PQ;x-)
= 6 (x-") G2 D
-
[(X-[G
G2 D1
(5.45)
where
G 2 D(P,Q) = ZnR/R1 + H
(PQ)
(5.46)
Far from the ship in the inner region, the inner
solution
0
can be written as (see Appendix 7)
L/2
O(y,z;x)
i)
d~a(E)G
=
(y,z,0,0;x-E)
L/2
= a(x)G 2 D(y,z,O,O)-
[G2
2D]
L/2 da(E)f(x-E)
-L/2
(5.47)
Here
a (x)
is an effective source strength which is defined
-69-
It is Qf interest to compare (5.47) with the
in (A7.13).
corresponding far-field expression of the unified slenderbody theory,
c(y,z;x){ (s)
-
x) + C(x) [a (s)x)
C (x) a
(x)
+ a (S)
(5.48)
[G 2 D~ U 2 D]
where
CCX)
1 L/2
d( o( )f(x-)
=
(5.49a)
,
-L/2
a (x) = a()
and
a (S)
+ C x) [
(S (X) +
(5.49b)
(s)(x) 1
is an effective source strength which is derived
from the strip theory solution
ds
a (S)(X)
t (s)
such that
[t (S) (N,(;x)
TIF I
6(x)
a-s) -ev( +in)
aN 0
NQ
(5.50)
From (5.47) and (5.48) we can draw a conclusion that
the present formulation and that of the unified slenderbody theory have the same far-field behavior of the inner
solution.
In principle, both theories are identical in
this special limiting case, even though their approaches
to the solution are quite different.
More specifically,
-70-
the former approach includes the three-dimensional interactions through the construction of a generalized inner
Green function, while the latter approach does through the
construction of a general inner solution.
Both approaches
involve unknown interaction functions to be determined by
outer matching.
In a computational standpoint, the unified slender-body
theory approach (Newman and Sclavounos,1980) is more
economical than the present one especially for the zero-speed
problem, in which the two-dimensional time-harmonic free
surface condition appears to be a good approximation.
How-
ever, it should be noted that the present work is directed
to the development of a new approach to the more general
non-zero speed problem, and that the expression in (5.43)
represents simply a limiting form of the present more general
results.
V.2.2
Steady Forward Motion;
w=0
t0
T,
This limiting problem is of particular interest in the
sense that the present formulation may provide a promising
approach to the steady forward-motion problem of waveresistance.
It has long been known that quantitative
predictions from the ordinary slender-body theory are poor
(see Newman,1970).
This has generally been attributed to
-71-
the fact that the flow caused by a line distribution of
sources has wavemaking effectivenesses that are different
from those of a surface distribution.
In addition to the
above fact, Ogilvie (1977) pointed out that ordinary slenderbody theory at the lowest order is deficient in the sense
that it does not include any effects of the diffraction by
the ship of its own waves.
Waves are generated at each cross section of the ship,
and these waves must undergo diffraction by the rest of
the ship.
In other words, these waves have an associated
velocity field which violates the body boundary condition.
Since the interference among cross sections appears in the
form of a function of
x
alone, these diffraction effects
cannot be accounted for in the ordinary slender-body theory.
Daoud' s (1975)
and Chapman' s (1976)
analyses,
in which the
speed-dependent terms in the free surface condition were
retained, both included the diffraction of the diverging
waves by the body itself.
However, since their solutions
satisfied a two-dimensional Laplace equation and no useful
outer matching was used, there can be no representation of
transverse waves, which, on the other hand, did appear in
the ordinary slender-body theory as a function of
x
alone.
Recently, Maruo(1982) derived a new slender-body
formulation by approximating the kernel function of the
Neumann-Kelvin problem near the ship axis.
It is of interest
-72-
to notice that Maruo' s analysis turns out to be a rather
crude approximation of the present more comprehensive results.
First, it can be noticed from (5.27) that
F(x,y,z) = FO(x,y,z)
for w=0
(5.51)
The corresponding square-root singularities of
F (y,z;k)
are, therefore, obtained from (5.21) and (5.22) as
=
k2
3
k1 ,4 -
im
i
m
k
=
2
g
(5.52)
0
(5.53)
Replacing the integrand and the limits of integral in (5.17)(5.18) by the corresponding ones, and making use of a
coordinate transformation
k
(xyZ)
F (x,y,z)
kd
=
+
6 = U 2 /g,
,
F(2) (x,y,z)
we obtain
,
x < 0
,
x > 0
(2)
F
(0
(5.54)
(x,y,z)
where
i
F~1 )(xyz)
dk e-ikx/6e k 2 z/6cos(k2y/ 6 [lI-k
f4dk ei
+~
(2)
-ikx/S6
=
-
(1 dk e
0
e
cos~ky/S)
7k2
(5.55)
k 2Z/6cos(k2y/)
ek2 Z/6cos (k2y/6 ) [1-ik k2
(5.56)
-73-
These integrals in (5.55) and (5.56) are greatly simplified
compared to those which we started with.
If we simply set
y
and
equal to zero (i.e.,
z
evaluated at the x-axis), the resulting expression reduces
F (x,0,0)=-2+sgn(-X)J/x +
{H(-x/6)
+
to the one derived by Tuck (1963):
(5.57)
+ 2(2+sgn(-x)]Y 0 (x|/6)}
which shows that Tuck's three-dimensional interaction function
is a special limit of the present one.
was a very much simplified one,
Tuck's inner problem
viz. the rigid free-surface
condition was assumed, in contrast to our more elaborate
development.
Replacing
T, we obtain the inner
in (3.43) by
cO
expansion of the outer solution for this limiting problem as
ds [T(xn, )
27rT (6)=
InR/R
-
6(x)
L/2
d
+
dsQ[T(
-L/2
x
-
B(E)
x
U
F,)
Cdn [T(
[H 0(PQ;x-E)+FO(Q0x-E
In ,)'
-E
I
[H (P,Q;x-E)+F0 (PQ;x-9)]
(5.58)
with an error factor
1 + 0 (eR , R2), where
i
represents
the velocity potential due to the steady forward motion of
the ship, and
F
is defined in
(5.54).
Similarly, the
-74inner solution can be obtained from (4.25)
ds [T(Q;x)
Q
5(x)
L/ l
+
.aNQ ]nR/R1
-
-L/2
f dsQ
5(eM
x
-
a[
I(Q;)
-
-
-f
21r(P;x)
as
[H 0(P,.Q; X-)
(Q;()
a-
9 C
[H
x
QN
+E (,0
I
(P,Q;x-E)+E, (PQ;X-M)
(5.59)
Therefore, the matching procedure described in section
V.1 will provide the matching conditions,
N(x,y,z) =
T(y,z;x)
(5.60)
TN(y,Z;X)
(5.61)
(x,y,z)
(5.62)
E0 (y,z;x) =
F
with the cumulative error factor
integral equation for
=
-
DQ
B(x)
L/2dt
-L/2
g
j(P;x) takes the form,
ds [ET(Q;x)
f
B(
C
Now the
as [I (Q; 0)
-nR/R
aNQ
-
Q
9NQ
1
I (H0 (P,Q;x-E.)+F0
(PFQ;X-M~
0
)
WiT(P;x)
1 + 0 (eR ,R ).
0 (PQ;x-E)+Fo( . ,0X-0
for P (y, z)
S
W(x)
,
=
T(x,y,z)
(5.63)
-75-
We should note that the combination of
and
F
(x-g,O,O)
H (P,Q;x-g)
in (5.58) turns out to provide the similar
expression as Maruo's(1982), which is, strictly speaking,
an inconsistent approximation in the present context.
V.3
>,>l
The High-Frequency Approximation;
The high frequency approximation of the inner expansion
of
presents a particular interest because of the
h(x,y,z)
simplicity of the final expressions.
cR 0
and
kR , where
0
For small values of
K=(w+Uk) 2 /g, we obtain from the
results of Appendix 6
2 2 2 2
*
*
*
h (y,z;k) = 2[H (y,z;k)+F (y,z;k)] + O(k R 2,K R ) (5.64)
where
,,,~
(5.65) F (y,z;k)
=
1
-
*
f kK H (y,z;k)J
H (y,z-k)
= 2(1+Kz)[enKR 0 + y + ni sgn(w+Uk)-KZ+KyO]
+ 0 (K2
* (
)
=
)
(5.66)
,
-sgn~w+Ukfi sgn(w+Uk)+cosh
f -)f + cos
1
11-k2 /K2
(K/k)
+ sgn(w+Uk)Zn(2K/Ikj) + x i
and
(K/lkI)
y = 0.577... is Euler's constant.
,
(5.67)
-1/2
-76-
For high frequencies of oscillation such that
and
K>[kj,
f
is
f*(k,K)
(k,K)
0
=
(k2
w+Uk>O
approximated as
(5.68)
K2)
Here the upper expression in the bracket of (5.67) is only
applicable and its limiting behavior,
1-k2
I
-1/2cosh~I(K/|kI) = Zn(2K/jkJ)+0(k2 /K2
(5.68) from (5.67).
is used to obtain
KR
and large
0
K,
(5.69)
Therefore, for small
h (y,z;k) reduces to
**
2 2
2
h (y,z;k) =.2H (y,z;k) + O(k /K , k R /K,
2 2
R 2)
K
(5.70)
The inverse Fourier transform of (5.70) then gives
= 2H(y,z;x)
+ 0(w~
,
R-2R, w R
(5.71)
)
h(x,y,z)
Since the approximation (5.71) has been derived based
on the assumption that
KR0
<< 1, a complementary analysis
is required for sufficiently large values of
+
of
(F- -1/2
h
and
for large
integration of
h*(y,z;k)
>> 1.
KR
h
= -4
K
w
such that
The precise form of the expansion
will be obtained by using a contour
defined by
+J.
I+>0 fo
dZ exp[k 2 + 2 1/2z cos y
[k2 + 2 1 2 - (w+Uk-ip ) 2/g
(5.72)
-77-
of
and
KR
for large values
h
The asymptotic approximation of
has been derived in Newman(1978), and is
K
given by
h
2 -1 /
2
(x,y,z) = 47i(1-k
rexp[Kz-i ly(K 2 - k 2)1/23
2
cose
0
+
(5." 73)
0 (K-2R2
0
From the results of Appendix 3, we also obtain the
asymptotic approximation of
H
H
for large
(y,z;k) = 27i exp[Kz-iK|yl ]
+
cOs 6
0
KR
as
0
(5.74)
+ 0 (K-2 R-2
0
Comparison of (5.73) and (5.74) then gives the result,
h
(y,z;k) = 2H (y,z;k) + 0(k2
K 2,
k2 R/K,
Thu0
K- 2R-
)
(5.75)
Thus,
h(x,y,z)
= 2H(y,z;x)
+ O(w
,
-R 0 ,
4R-2)
(5.76)
Therefore, in this special limiting case, we obtain
=
$N(xryz) =
E(y,z;x)
D(y,z;k)
D (y ,z;x)
= F(x,y,z)
= 0
(
t(x,y,z)
5.77)
(
from (5.9)-(5.11) and (5.76) the matching conditions,
5.:78)
(5.79)
-78-
with the cumulative error factor,
-
Err = 1 + '0 (W
-4R-
,
R
,
R )
(5.80)
Thus, in the matching region, we will have
(5.81)
0 < $ < 2y
Since
F(x,y,z)
vanishes for
w>>l
in accordance
with (5.71), the results (5.9)-(5.11) are consistent with
the high frequency results.
For this reason (5.1)-(5.3)
are valid, in general, for all frequencies of oscillation.
V.4
Alternative Representation of the Solution in terms
of a Line Distribution of Wave Sources
In Chapter III, an appropriate outer solution has been
obtained by using Green's theorem.
In other words, the
outer solution is expressed in terms of a surface distribution of wave sources and normal dipoles over the ship's
wetted surface.
It is possible to show that the matching
procedure is not fundamentally changed if
of wave sources is used instead.
a line distribution
To do this,
we consider
the outer solution being represented as follows:
(x,y,z) =
d
q( ) G (
(x-F,y,z)
(5.82)
-79-
Here
q(x)
is the unknown source strength which will be
determined by matching with the inner solution, and
G
source located on the x-axis at
From (3.5),
x=
.
is the potential of a translatory harmonic oscillating
is expressed as
G
h(x-_,yz)
G(O) (x- ,y,z)
(5.83)
In order to match the inner expansion of (5.82) with the
outer expansion of the inner solution, an asymptotic expression of
G (0)
for small
R (=
y +z 2)is required.
From
the results of Chapter III, the inner approximation of
G(O)
in (5.83) can be expressed as
G(O) (x-E,y,z)
2(H(y,z;x-E)+F(x-Ey,z)]
=
R<<l
,
(5.84)
The inner solution has been derived in Chapter IV and
given in (4.27) as
2Trr(P;x)
ds Q(IQ;x)D- -9
=
..nR/R
B(x)
[
L/2 d
+
-L/2
-
)
(-]
d
D
x
B(E)
Cdn[2iwU('(Q; )+U2 ((D
x
X
[H(P(4;X-2)+E(P,7Q;x-)
(4.27)
-80-
E
Here the harmonic function
is unknown and will be
determined by matching with the outer solution.
Far from the ship in the inner region, the inner
solution in (4.27) can be expressed in terms of a line
distribution of effective wave sources,
OdE a (E)
(y,z;x) =
G
(y,z;x-)
(5.85)
R 0 >>
,
where the exact relation between the effective source
strength
derivative
a(x)
DN
and the body potential
will be addressed later.
D
and its normal
In (5.85)
G
is the generalized inner Green function which is defined as
G(i) (y,z;x-E) = H(y,z;x- ) + E(y,z;x-E)
(5.86)
Here the relations (4.28a,b) are used to obtain (5.86) from
(4.15).
Equating the inner expansion of the outer solution in
(5.82) with the outer expansion of the inner solution in
(5.85), we obtain
(5.87)
q(x) = a(x)/2
E(y,z;x)
= F(x,y,z)
(5.88)
Now the matching procedure defines the unknown function
by (5.88).
Thus, the integral equation for
(P;x)
E
can be
-81-
obtained from (4.27) in terms of inner variables,
WO
(P;x)
ds [QD(Q;x)
.
-
-1nR/R
-
a(x)
f L/
aI[H(P ,Q;x-)+F(P ,Q;X-)
2 df
-L/2
ds [4(Q;)
Q ~~~
-
awQ
N
8(C)
dn [2iwMUcNQ; )+U 2 (H(pQ
-1
X )
F(pL
C
for P (y,z) e B (x)
The unknown source strength
q(x)
(5.89)
in (5.82) is also'
determined from this matching procedure, provided
a(x)
is known.
The effective source strength
a (x)
has been derived
in terms of inner variables in Appendix 7,
L/2
ds [
. d
-L/2
(D , ;) a;xQ
B(t)
C.dn [2iwMD (n,0;g)+U2(
M0
-
-W
<
) M(T ,0 ;x-g)
x
<
,
2ra (x) =
(5.90)
W
where
S(-4U 2
/
(~n,,C 04U
g
4U
2
(+in)]exp[
+)
2
in+
U
(5.91)
-82-
It is of some interest to point out that for the special
case of zero forward-speed,
U=O, .
ds [c(Dn, ;x)
2ra (x)
(x)
simplifies to
-
3 e.v (C+in),
|xl<
L/2
lx>
L/2
(x)
0
,
(5.92)
where
v
2 /g
The representation of the outer solution in terms of
a line distribution of wave sources turns out to provide
the same results as those of surface distribution of wave
sources and normal dipoles.
However, it is worthwhile to
note that this line distribution, in general, should be
extended over an infinite length in order for the outer
solution to be completely compatible with the outer
expansion of the inner solution.
From a computational
standpoint, the representation (5.82) would probably be
more economical than that of surface distribution if farfield behavior of the outer solution is desired for a given
q(x).
-83-
VI.
ADDED MASS AND DAMPING
We consider the unsteady component of the hydrodynamic
pressure force, with the usual assumption that the oscillatory
motions of the ship and the fluid are small.
Neglecting
second-order terms in Bernoulli's equation (2.3), the unsteady
hydrodynamic pressure is given by
-p7iWV+-Vi+l/2$-'VW 2 ]
P(x)=
on
(6.1)
The last term in (6.1) gives a force proportional to the
unsteady displacement of the ship, and hence an additional
buoyancy force to the hydrostatic restoring force.
Since
we are now interested in the linearized unsteady pressure
force associated with the added mass and damping, we will
use (6.1) without the last term.
Following the conventional definition of added mass
and damping coefficients, the unsteady hydrodynamic pressure
force can be written in a form,
F
=
dSpn=
p
dS(iw.+W'.-V?
j
-w 2 1(a
+b t/io)
)n
J
,
i=3,5
j=3,5
(6.2)
Cb
-84-
where the quantities
a
are the added mass and
b
and
damping coefficients associated with the force (moment) in
the i-th direction due to the j-th mode of motion.
The term in (6.2) proportional to the steady velocity
can be transformed by means of a theorem due to
field
(Ogilvie and Tuck, 1969)
,
Tuck
dS(n-V)T 1 -
=
dS( -vVT)i
- (6.3a)
(dxp*)
C
N
fdS(v') (Zxn) =
(dZx I)x x
dS(-V) (nxW) -
(6.3b)
C
K
where the line integral is over the intersection of the ship
hull with the undisturbed free surface.
It is noted that
the line integral in (6.3) may be ignored for a slender
ship, for it is of higher order than the remaining surface
integral by the relative order
of
in (2.30)
m
F=
and (2.31),
.
From the definition
(6.2)
can be transformed to
e
(6.4)
dS(iwn. - Um )T.]
p E. E [
S..
3
and
a.
+ b
1)
/iw =
13
(P/ 2) fdS(iwn
-
1
)T
(6.5)
-85-
Let
* (yz;x)
and
$ (y,z;x)
be the numerically
determined solutions of the Inner problem defined in (4.8)(4.10), with the following hull boundary conditions on
S(x)
=n
(6.6a)
=m
(6.6b)
aN
aN
i
From the boundary condition for
T.
J
in (2.27), the unsteady
velocity potential can now be expressed in terms of
and
'.
V
such that
=
From (6.5)
and (6.7) the added mass and damping
coefficients can be obtained in the form,
a
+ b 1)i/iw13 )1= f)
+ f!I)
+ f
i)
(6.8)
where
f
=
p
fdS n *.
(6.9)
J
ij
W
j
i
-86-
U )2
f II)
(6.11)
i=3,5
dS m
j3,5
=f
Now the expressions in (6.8)-(6.11) will be discussed
From Green's theorem and the boundary
in more detail.
conditions (6.6), we can show that
dS[n 4i
0=
where
ES
dS[*
=
i
ff
ES
]
- ni
rsJ
-
DNaN
represents a closed surface bounded by the
ship's wetted surface
F,
a closure surface
portion of the undisturbed free surface
and
S,
as shown in Figure 2.
be shown to vanish if we let
*
both
(6.12)
-]
and
SF
S.,
and the
between
The integral over
S,
tend to infinity.
So
S
can
Since
satisfy the free surface condition (4.9),
ii
it follows that
)- fI)
FfdS[n i
=
(p/g)
-n
dy[2iwU*
]
+ U2 (4i
a
-
f
C
(6.13)
It is of interest to notice that the line integral in
(6.13) has the similar form as that of (4.27).
Actually,
this line integral results from the inclusion of the forward
-87-
speed effects in the free surface condition.
It is also seen
that the line integral in (6.13) is of higher order than the
surface integral by the relative order -ew.
If we further
neglect the contribution due to this line integral, we would have
)
f
=
(6.14)
P
By using a similar analysis to what we did above, we can also
show that
(II)
f(II)
(6.15)
J.
1)
(6.16)
!111)
f III)= f ji
:LJ
with the resulting error terms represented by a line integral
similar to the one in (6.13).
Since the velocity potential
in the present formulation retains the similar line integral
(see equation 4.27), we do not expect that the relations
(6.14)-(6.16) will generally hold.
In the derivation of the reverse-flow theorem of Timman
and Newman (1962), it was assumed that
= G(O
+(+
where the superscripts
) x
(6.17)
( ) denote directions of the forward
speed, and the Green functions
G
~
the linearized free surface conditions
g 3G ()
/ax] = 0
conditions
aG
on
/an
z=0
are defined to satisfy
[(iw UB/3x) 2G ()
+
G
and the homogeneous body boundary
on the body surface.
Strictly speaking,
-88-
the reciprocity relation (6.17) was obtained from the
approximation which is equivalent to a neglect of the
forward-speed effects on the free surface condition (see
Since these
more details in Timman and Newman, 1962).
forward-speed effects are, however, included in the present
formulation, the applicability of the reverse-flow theorem
to the present formulation is questionable.
In an intuitive approach, if the gradients of the
steady-state disturbance velocity field are considered
negligible, then the only nonzero element of
m5
which is identified to
3*
5
and
n3 .
m (j=3,5) is
Thus it leads to
*3 = 0
With these simplifications in (6.8)-(6.11),
we obtain
a
+
3 3 /i
a53/iA
a
=
(I)
b
=
-(U/ii)
+ b3 5
a5 5 + b/iW
=
(6.18a)
f
(6.18b)
M
+ (U/iw) f(I)
M
-
35
(6.18c)
53 I
(6.18d)
-(U/iw)2 fI)
These equations are 'structurally' very similar to the strip
theory results for heave and pitch derived by Salvesen et
al.(1970).
But the coefficients
f1J) in (6.18) are now
three-dimensional ones which include the three-dimensional
interactions, while those of the traditional theories are
-89-
their corresponding two-dimensional values.
In the next
chapter we will compare the results in (6.8)-(6.11) with
those of the existing theories.
-90-
VII.
DISCUSSIONS AND CONCLUSIONS
A linear theory has been presented for the heave and
pitch motions of a slender ship, moving with forward speed
in calm water.
Using the complete linearized free-surface
condition in the formulation of the inner problem, the
velocity potential includes the forward-speed effects in a
physically meaningful manner.
Three-dimensional interactions
are also incorporated through the appropriate outer matching.
The resulting slender-body formulation is quite different
from the existing theories of the slender ship, and the new
results are valid for more general frequencies of oscillation.
Furthermore, the present approach has provided a quite
promising slender-body formulation for the steady forward
motion problemwhich is simply a limiting case of the more
general problem being examined.
The resulting expressions
for the added mass and damping coefficients are 'structually'
very similar to those of the existing strip theories even
though the origin of the corresponding physical quantities
is quite different.
In the process of deriving the present formulation we
introduce a forward-speed transformation by which we can
show very clearly how the inner problem reduces to an
initial-boundary value problem.
From a physical standpoint
we can interpret the present work as a transient formulation
-91-
of the time-harmonic motion problem.
With the restriction that the unsteady motions of the
ship are sinusoidal in time, the unsteady components of the
velocity potential can be expressed as time-harmonic solutions.
Due to the "transient" nature of the free surface condition
in the forward-speed case, however, the resulting spacedependent velocity potential, which is obtained by factorizing out the time-dependent component of the unsteady velocity
potential, has a "transient" character along the
ate.
x
coordin-
These transient or memory effects are adequately
considered in this work.
Before making any further comments
on the present work, we shall describe briefly other slenderbody approaches together with their corresponding results.
Strip Theory of Salvesen et al. (1970) :
This formulation assumes that there are no longitudinal
interactions regardless of frequency.
This assumption is
generally valid for high frequencies of oscillation.
The
added mass and damping coefficients of the ship are given
by integrating corresponding sectional coefficients with
an appropriate weighting factor,
a333+
3
3 + b 333/w
a + b5 3 i
viz.
()
(7.la)
(0 . (U/iw) f (0
(7.lb)
3 33-/f
-92-
a 3 5 + h35)
+ (U/i)
a5 5 + b 5 55
5
f
(7.lc)
(U/i) 2 (O)
=f
55"w
55
(7.ld)
33
where
f
dx[A 3 3 (x) + B 3 3 (x)/iwJyi (x)
,
i=3,5
(7.2)
1=3,5
1
i
i=j=3
-x
i or j=5
x2
Here
A33
and
(7.3)
i=j=5
B33
represent the sectional added mass and
damping coefficients respectively, and the superscript (o)
is used in (7.1) to denote that the corresponding quantities
are zero speed results.
Rational Strip Theory of Ogilvie and Tuck (1969):
This formulation uses a systematic perturbation analysis
under the assumption that the frequency of oscillation
S= O(C-1/2)
and the forward speed
U=0(1).
By consistently
retaining the higher order terms of relative magnitude
e 1/2
this formulation provides the results,
a 3s3 +33/w
+ b 3 3 /io
a
3 +
b5 3 /iW
=
= f 3(3
=
f 0
(7.4a)
Ui)f(0)
l(2pwg)
s
(7.4b)
-93-
a 3 5 + b3
() +(U/iW)f (-(2ipwU/g)
a5 5 +55b 555/iw
5/
where
f !0?
ds
3
(7. 4c)
f555
(7.4d)
are defined in (7.2)-(7.3).
Here the integral
over the free surface in (7.4b)-(7.4c) has a special interpretation which was described in more detail in Ogilvie and
Tuck(1969).
This integral actually results from the inclu-
sion of the forward-speed effects on the free surface
condition of the inner problem.
Since the forward-speed effects are accounted for only
as higher-order corrections, the term proportional to
has been discarded in (7.4d).
(U/W) 2
Comparative merits of this
formulation to the strip theory of Salvesen et al. (1970)
have not been resolved yet, although Faltinsen(1974) has
shown that the comparison with experiments is improved by
using the Ogilvie-Tuck cross-coupling coefficients.
Unified Slender-Body Theory of Newman(1978):
In this formulation the velocity potential for the
inner problem is represented as the superposition of a
particular solution similar to that of the high-frequency
strip theory, plus a homogeneous solution which accounts
for interactions along the length in an analogous manner
to the low-frequency ordinary slender-body theory.
The
-94-
resulting expressions are, therefore, valid for more general
frequencies of oscillation.
The homogeneous solution is
constructed such that
[ ts) 1
Dhomo = C(x)D
where the superscript
(7.4)
(s)
is used to denote that the
corresponding quantity is a strip theory solution, and
C(x)
represents the longitudinal interaction function to be
determined from outer matching.
The resulting expressions for the added mass and damping
coefficients take the forms,
a
= f.f(O)
+ b. ./i
()
(2) + f..
(1) + f..
+ f..
(7.5)
where
s
(7.6b)
f~o
n
=
(s)
)
(
5
=
(p(-fm. -f~
C (x)[n
f ()= p
s
(7.6c)
(U/iw)m 1](s)+
(s))ds, i=3,5
j=3, 5
(7.6d)
-95(S)
where
J
and
theory
;(s) are the corresponding strip
solutions which satig fy the hull boundary conditions
(6.6a)
and (6.6b) respectively.
is of interest to compare the results in (6.8)-(6.11)
It
with those of the foregoing existing theories.
If the
gradients of the steady-state disturbance velocity field are
neglected, the resulting simplified expressions in
(6.18)
will be 'structurally' very similar to the corresponding
expressions of Salvesen et al. (1970).
f!)
in (6.18)
But the coefficients
are now three-dimensional ones which do
include the effect of longitudinal interactions, while the
corresponding strip theory coefficients .f!
do not.
At high frequencies of oscillation the longitudinal
interaction function
(5.79).
F(x,y,z)
vanishes in accordance with
In such a case, longitudinal interactions still
occur via the free surface condition and are of a downstream type, i.e., only the upstream sections have effects
on a given section.
This is precisely the approach adopted
by Yeung and Kim(1981), who in fact devoted considerable
effort to give a physical interpretation of the formulation
by using a fixed-frame of reference.
To a certain extent,
the strip theory formulation of Salvesen et al. (1970)
represents a further approximation of Yeung and Kim(1981),
since neither upstream nor downstream influence is accounted
for in constructing the local solution at a given ship
-96-
section.
it is important.to note that forward-speed effects on
the free surface condition are not accounted for in the
derivation of the strip-theory potentials whereas it is here.
The rational strip theory of Ogilvie and Tuck(1969) do include
the forward-speed effects in a more systematic way than the
present analysis.
Due to the high-frequency assumption
which is made in the systematic derivation of the rational
strip theory, however, the solutions in (7.4) do not have
any longitudinal interactions either.
Furthermore, the term
proportional to (U/w)2 has been discarded as being higher
order, while the other theories including the present
approach keep this.
The present formulation and the unified slender-body
theory formulation have many common viewpoints.
The unified
slender-body theory includes the longitudinal interactions
through the construction of a general solution for the inner
problem based on the two-dimensional time-harmonic solution.
The present formulation includes such interactions through
the construction of a generalized Green function for the
inner problem.
Thus both theories are expected to be valid
for more general frequencies.
However, the general solution
of the unified slender-body theory is so constructed that
it satisfies the two-dimensional time-harmonic free surface
condition, with which no direct forward-speed effects on
-97-
the free surface condition can be accounted for.
In the
present formulation the complete linearized free-surface
condition is used for the inner problem so that the forwardspeed effects can be included in both body and free surface
conditions.
It is also worthwhile to note that the general solution
of the unified slender-body theory is not matchable with
the outer solution unless some further approximations of
the inner expansion of the outer solution are made.
More
specifically, it is assumed in the unified slender-body
theory that the longitudinal interactions are simply some
modulations of the two-dimensional time-harmonic solution
However, the inner expansion of the
along the x-axis.
outer Green function clearly shows that this is not the
case, and that the longitudinal interactions of the unified
slender-body theory are simply approximations of the real
interactions.
The principal complication of the present approach is
that the factor
m3
must be determined, and that the
solution involves the inversion of a big matrix and lengthy
computations for the kernel even though it is greatly
simplified compared with that of the fully three-dimensional
approach.
From a computational standpoint the present
formulation is not expected to be more economical than the
existing theories.
However, the present study has provided
-98-
a new insight on the slender-body formulation and would
likely lead to the reasonable quantitative and qualitative
predictions of ship motions and wave resistance in the future.
-99-
REFERENCES
"Handbook of
(1964).
Abramowitz, M., and Stegun, I., eds.
Mathematical Functions." 'U.S. Gov. Print. Off.,
Washington, D.C.
(1977). Computations of three-dimensional ship
Chang, M.-S.
Proc. Int..Conf. Numer.
motions with forward speed.
Ship Hydrodyn., 2nd, 124-135. Univ. of California,
Berkeley.
(1976). Free-surface effects for yawed
Chapman, R. B.
surface-piercing plate. J. Ship Res., 20, 125-136.
(1977). Survey of numerical solutions for
Chapman, R. B.
Proc. Int. Conf. Numer. Ship
free-surface problems.
Hydrodyn., 2nd, 5-16. Univ. of California, Berkeley.
(1965). "Asymptotic Expansions." Cambridge
Copson, E. T.
Univ. Press, Cambridge.
(1956). The wave resistance of a floating
Cummins, W. E.
slender body. Ph.D. Thesis, American University,
Washington, D.C.
(1975). Potential flow near to a fine ship's bow.
Daoud, N.
Rept. No. 177. Dept. Nav. Archit. Mar. Eng., University
of Michigan, Ann Arbor.
(1974). A numerical evaluation of the OgilvieFaltinsen, 0.
Tuck formulas for added mass and damping coefficients.
J. Ship Res. 18, 73-85.
(1981). Bow flow and added resistance of
Faltinsen, 0.
slender ships at high Froude number and low wave lengths.
(to be published).
Froude, W.
(1861).
On the rolling of ships.
Inst. Nav.
Archit., Trans. 2, 180-229.
(1967). Analysis of the
Gerritsma, J., and Beukelman, W.
modified strip theory for the calculation of ship motions
Int. Shipbuild. Prog. 14: (156),
and wave bending moments.
319-337.
The hydrodynamic theory of ship
(1946a).
Haskind, M. D.
Prikl. Mat. Mekh.
oscillations in rolling and pitching.
(Engl. transl., Tech. Res-. Bull. No. 1-12,
10, 33-66.
Soc. Nav. Archit. Mar. Eng., New York, 1953.)
pp. 3-43.
-100-
Haskind, M. D.
*(1946b.). The oscillation of a ship in still
water.
Izv. Akad. Nauk. SSSR, Otd. Tekh. -.Nauk 1, 23-34.
(Engl. transl.,
Tech. Res. Bull. No. 1-12, pp. 45-60.
Soc. Nav. Archit. Mar. Eng., New York, 1953.)
Havelock, T. H.
(1958). The effect of speed of advance
upon the damping of heave and pitch.
Inst. Nav. Archit.,
Trans. 100, 131-135.
Inglis, R. B.
(1980). A three dimensional analysis of the
motion of a rigid ship in waves.
Ph.D. Thesis, University of London.
Korvin-Kroukovsky, B. V., and Jacobs, W. R.
(1957). Pitching and heaving motions of a ship in regular waves.
Soc. Nav. Archit. Mar. Eng., Trans. 65, 590-632.
Kriloff, A.
(1896). A new theory of the pitching motion of
ships, and of the stresses produced by this motion.
Inst. Nav. Archit., Trans. 37, 326-368.
Lighthill, M. J. (1958).
"An introduction to Fourier
analysis and generalized functions. " Cambridge Univ.
Press, Cambridge.
Maruo, H.
(1982). New approach to the theory of slender
ships with forward velocity.
Bull. Faculty Eng.,
Yokohama National University 31, Yokohama, Japan.
Mays, J. H.
(1978). Wave radiation and diffraction by a
floating slender body. Ph.D. Thesis, Massachusetts
Institute of Technology, Cambridge, Massachusetts.
Michell, J. H.
(1898). The wave resistance of a ship.
Philos. Mag. [5] 45, 106-123.
Newman, J. N.
(1961). A linearized theory for the motion
of a thin ship in regular waves.
J. Ship Res. 3:(1),
1-19.
Newman, J. N., and Tuck, E. 0.
(1964). Current progress in
the slender-body theory of. ship motions.
Proc.- Symr>.
Nav. Hydrodyn., 5th ACR-112, 129-167. Off. Nav. Res.,
Washington, D.C.
Newman, J. N.
(1970). Application of slender-body theory in
ship hydrodynamics.
Ann. Rev. Fluid Mech. 2, 67-94.
-101-
(1978). The theory of ship motions.
Newman, J. N.
Appl. Mech. 18,. 221-283.
Adv.
(1980). The unified
Newman, J. N., and Sclavounos, P. D.
Proc. Symp. Nav. Hydrodyn.,
theory of ship motions.
13th, Tokyo, Japan.
(1969). A rational strip
Ogilvie, T. F., and Tuck, E. 0.
theory for ship motions. Part 1, Rept. No. 013.
Dept. Nav. Archit. Mar. Eng., University of Michigan,
Ann Arbor.
(1977). Singular-perturbation problems in
Ogilvie, T. F.
ship hydrodynamics. Adv. Appl. Mech. 17, 91-188.
Peters, A. S., and Stoker, J. J. (1957). The motion of a
Commun.
ship, as a floating rigid body in a seaway.
Pure Appl. Math. 10, 399-490.
(1970). Ship
Salvesen, N., Tuck, E. 0., and Faltinsen, 0.
Soc. Nav. Archit. Mar. Eng.,
motions and sea loads.
Trans. 78, 250-287.
Sclavounos, P. D. (1981). On the diffraction of free surface
Ph.D. Thesis, Massachusetts
waves by a slender ship.
Institute of. Technology, Cambridge, Massachusetts.
(1967). Computation of pitch and heaving
Smith, W. E.
Int. Shipbuild.
motions for arbitrary ship forms.
Prog. 14: (155),
(1953). On the motion of
St. Denis, M., and Pierson, W. J.
ships in confused seas. Soc. Nav. Archit. Mar. Eng.,
Trans. 61, 280-354.
(1962). The coupled damping
Timman, R., and Newman, J. N.
coefficients of symmetric ships. J. Ship Res. 5:(4),
34-55.
(1963). The steady motion of a slender ship.
Tuck, E. 0.
Ph.D. Thesis, University of Cambridge.
(1962).. - Slender oscillating ships at zero forUrsell, F.
ward speed. J. Fluid Mech. 19,' 496-516.
Surface waves.
(1960).
Wehausen, J. V., and Laitone, E. V.
In "Handbuch der Physik" (S. Flugge, ed. ), Vol. 9,
pp. 446-778. Springer-Verlag, Berlin and New York.
-102-
(1973). The wave resistance of ships.
Wehausen, J. V.
Adv. Appl. Mech. 13, 9-3-245.
(1981). Radiation forces on
Yeung, R. W. and Kim, S. H.
Proc. Int. Conf. Numer.
ships with forward speed.
Ship Hydrodyn., 3rd, (in press).
-103-
APPENDIX 1:
0
If
Derivation of Modified Green's Theorem
are any differentiable scalar
0
and
functions of position, the following is true
-
0
a 0 11
2
-2.
JJI I
.eyIdV
V
-
-
0
.1JdS
(Al.1)
S
Here the operators in (Al.1) are defined by
+
-)
+L
)
i(3x
V Vnijn-3
where
+
+
+ k
(Al.2)
) + a
ay
n3
+
n
(Al.3)
4
3aza
an
n- _= n lax + -n)
23y. + n
an
a
= n(a
ne~V -l
2aY + n 3az
1ax + a) + na
to the surface
Proof.
theorem,
S
and
(l5
(l6
represents the unit vector normal
(n1 , n 2 , n 3 )
n
(Al.4)
a
is an arbitrary constant.
Making use of Green's theorem and Stokes's
(Al.1) can be proved as follows:
-104-
a
I,
S
11
an+
iiD- 1-dS
I
~
go, i
ff
go,
an
+ 2a(yoi
n 1JdS
S
=1~ f 1~
=1 fr
I
[
i V20
-
Oilv 20)
+ 2ca
( 0
)tdV
(
V
(1 (V
+ct2
'1 +2ct
V
a
ax
+
0o 2il
-
=111
*2
2
a 20 1 ) dV
- 20
dV
(Al. 7)
V
If
Jf
-
V +11
S ian +
V = 0
1~-d
in
V, there follows from (Al. 1)
(Al.8)
-105-
Derivation of the Pseudo-time-dependent
APPENDIX 2:
Green Function
We construct a Green's function which satisfies the
following set of equations,
2
2
+ -z-2)G (y,z,n,;x,() = 6 (x- ) 6 (y-n)S(z-c)
(
ay
az
(U ax
VG
0
a
G0 + g a
)
-
=
on z=O
0
,
x+0+
(A2.2)
at z= -c
0
+
(A2.1)
(A2.3)
where the initial conditions of vanishing motion upstream
are incorporated introducing an arbitrary Rayleigh viscosity
parameter
V
such that
aG
G0
for x
>
t, z=0
(A2.4)
It is of some interest to notice the similarity between
G
and the time-dependent Green function corresponding to
an unsteady two-dimensional free surface flow.
we replace
x
Green function.
by
Ut,
G0
Actually, if
will reduce to the time-dependent
The solution for
G0
can be found in
Wehausen and Laitone(1960) in the form,
G0 (y,z,n,ic;x,t) = 6(x-C)ZnR/R1 + H (y,z,n,C;xE)
H
= -
u(E-x)
Jf
(A2.5)
e(Z+c) cost(y-n )sin /
0
( -x)
0
(A2.6)
-106-
where
R =
(y-n) 2 + (z-)2 1 1 / 2
R
(y-n) 2 + (z+0)2
&=
1/2
(A2.8)
(A2.9)
2/g
are the Dirac delta function and the
u(g-x)
and
6(x-)
(A2.7)
,
Heaviside step function respectively.
Here the first term
in (A2.5) represents an impulse at the point
when
E=x,
while
Q =
(n,)
describes the disturbance at
H
due to this impulse as it propagates away from
course of pseudo-time
Q
(y,z)
P
in the
x/U.
the flow direction is reversed, the corresponding
If
conjugate Green function will satisfy the following set of
equations,
(
+
ay
az
)
(c) (y,z,n,;x,) =
(x-
(y-
(z(A2.10)
3G (c)
0
(ax + P)2G(c)
VG 0(c)
0
+ g a 0Gc
=
0
on z=0,
(+0+
A2.ll)
at z=
(A2.12)
--
where the initial conditions of vanishing motion are
reserved such that
9G(c)
G0
Q
-107is easily derived in the form,
G(c)
0
The solution for
G(c) (y,z,n,?;x,g) = 6(x- )nR/R
1
+ H c) (y,z,r,C;x,5)
.
0(
(A2.l14)
where
H (c)= 00
-
dZ
u(x-)
(Z+C) cost(y-n)sinF7-C)
o V/a0 %J
(A2.15)
It is instructive to derive the Green function which
satisfies the following set of equations,
(-L
2
2
+
1y
)G(yz,r, ;x,)
77
-
2
+
(A2.16)
= 6(x-)6(y-n)6(z-)
G
on z=O, P+0+
VG + 0
at
z= -=
(A2.17)
(A2.18)
If we set
is to satisfy the following set of equations,
H
then
(A2.19)
,
G = 6(x-E)ZnR/R1 + H(y,z,n,i;x,)
a2
2
(
+
ay
(iW
-
3 )H(y,z,n,4;x,) = 0
az
UaAx + -P)2H + gz
-6 (X-0) g[nR/R
a2z.
(A2.20)
] on z=0,
-yO+
(A2. 21)
VH +- 0
at z= -m
(A.
22)
-108-
We define the double Fourier transform of
respect to
H
**
and
x
y
H
with
as
+ ity
(Z,z,ri,iyk,5)
dy H(y,z,n,;x,)e
dx
(A2.23)
The double Fourier transform of (A2.20)- (A2.22) gives
d2 H
t 2 H **
= 0
(A2.24)
dz
**
dH
(c+Uk-iu2 )H
=
Ij I c+ikE+tn
2wg
on z=0, 1 +0+
(A2. 25)
**
dli
at
+ 0
z =
-w
(A2. 26)
From (A2.24) -(A2.26)
H
we obtain
I (z+c)+ik +itn]
(Z,z,n,?;k,&) = (-2r) exp[ It
I I- (w+Uk-ipi) 2/g
(A2.27)
and
)0 = 2
,C;x,
4 7r
l_ im
-
df
U .+0+
dt
2 --
0
(t ,z,
T,.c; k,
e-ikx-ity
0
dk
exp [Z (z+c) -ik (x- ) -iZ (y-n ) I
Z-(w+Uk-iu) 2
iw
a
T
dk H
)
H(y,z,
**
o
1 0
dZ
x
o
Z/a0
et(z+)cost(y-n)sin
Z (E-x)
0
A2..28)
-109-
Here the contour integral in the complex k-plane, which is
shown in Figure A2-l, is used to obtain (A2.28).
Im k
when
x- E< 0
Re k
U
U
0
when
when
Figure A2-l.
x-E>
x-E>00
Integral paths for the k-integral
We consider a Green function
G(c)
which satisfies a
,
reverse-time and reverse-flow problem of (A2.16)-(A2.18)
a2
2
+
ay
)G(c) (y,z,n, ;x,) =
a z2
(iW-Ua -
VG (c
)2 G (c) +
(c)
=
0
(x-E) 6 (y-n) 6 (z-c)
(A2.29)
on z=0,; y+40+
(A2.30)
at z= -w
(A2. 31)
-110-
If we set
G(c)
= 6(x-C)ZnR/R1
+ H(c) (y,z,,;x,)
,
(A2.32)
the similar analysis as above will provide the solution for
H(c)
in the form,
H (c) =
(X-C)
00t(+
-2-u_(x-o)e U
ds
ao
(X-)
cost(y-n)sin
a
fo
/1
(A2.33)
It
is worthwhile to note that
G(y,=z,n,;x,C)
= eU
G
(A2.34)
iuJ
G(c) (y,z,n,4;x,C) = eU
G(c)
0
(A2.35)
-111-
APPENDIX 3:
Alternative Expressions and Asymptotic
Approximations of the Pseudo-time-dependent
Green Function
The Fourier transforms of the homogeneous terms of the
pseudo-time-dependent Green functions derived in Appendix 2
take the forms,
H 0 (y,z;k)
-
=
1-+
+O
Zim
H(c)
* (y,z;k)=
0
.l et(e
Jo
ez
r
a+0
e 'ly)
ty+
-(Uk-ip) 2
ieZy+ e
ZIyI)
Z-(Uk+iu ) /g
(A3. lb)
*
= H
H (y,z;k)
=-
(y,z;k)
fd
'+
+
(A3.la)
e
0
(eZItY+
2
t-(w+Uk-ip) /g
.
*
= H~(y,z;k+w/U)
H c)*(y,z;k)= H 0(c) *-* (y,z;k+w/U) = H (y,z;k)
(A3.lc)
(A3.ld)
where the overhead bar denotes the complex conjugate of
the expression involved.
; = z - ilyl
,
,
, = z + ilyl
Making a change of variables,
(A3.2a)
WA. 2b)
-112-
to the form,
(A3.la)
H 0 (y,z;k) =
[eKOC
-
Ko(
0
(I
ds
+ eK
dse
1
)
we may reduce
(A3.3)
where
K
= U 2 k 2 /g.
Here the integral paths are sketched
easily shown that the integral over
as
R+
e, and that (A3.3)
H (y,z;k) =
-
[eoc
It can be
k < 0.
in Figures (A3-1)-(A3-2) according as
r = R0 e
vanishes
reduces to
ds
K0
ds
+ e oC
-
{
K0
e 0
+ 27ri sgn(k)
(A3. 4)
eK o
where the upper and lower terms in brackets are applicable
according as
k.
0.
The equation (A3.4) can be expressed in terms of the
exponential integral function,
CO
E l(z) = fzds
-s
s
,
(jargzf
defined in Abramowitz and Stegun(1964).
< 7r)
(A3.5)
-113-
K
K
0
pnr
0
s-plane
.- plane
j
2
r
K0
Figure A3-1
Integral paths for
k>O
-fexp (K0c)
(y, z;k)-e-
0 E
(0c)+e
CE
-2Re {e'< (z+2.y)E 1(K
(KO)
+27Ti
sgn(k)
exp(K 0
0Z+i K y)
}+27ri sgn(k)
x(0
exp
(K0 c
(A3.6)
The asymptotic properties of (A3.6) can be obtained
from the corresponding approximations of the exponential
integral.
For small values of
K
the sourcelike
logarithmic singularity is displayed in the approximation
-114-
K0C
0
snr
s-plane
t-plane
I2r
0
t
Figure A3-2
Integral paths for k<0
H 0 (y,z;k) = 2(1+K 0 z) [ZnK 0 R 0 + y + wi sgn(k) +
Here
0 (K 2 R 2
(A3. 7)
0 0
y = 0.577... is Euler's constant, and
polar coordinates defined such that
z = -R 0cosG..
Koz + K 0 ye]
For large values of
(R ,9)
y = Rosine
K0 R
are
and
(A3. 6) is
approximately
exp iK0 (z-i jyf)
I
+ 0(-)
H (y,z;k) = 2ni sgn(k)
exptrO (z+iIy
00
(A3. 8)
The corresponding expressions for
H.
can be obtAined froM (A3,61-CA3,81
(c*
Q
*
-115-
rI~ and
conbined with
(A3.1) as
exp
H 0*
(y,z;kY=-c2 Re{eg Z+0y) E C,0 z+iK 0
(K 0)
}y1-21"'1 sgn (k)
exp (K
)}
(A3. 9)
exp (Kr)
H (y,z;k)
=-2 Re{e K(Z+iy) E (KZ+iKy) }+2wi sgn (w+Uk)
)
exp (Oc
(A3.10)
exp (Kr)
(y,z;k)=-2 Re{eK (z+iy)E (KZ+iKy)1-2ti sgn-(w+Uk)
(A3.ll)
where
K =
(w+Uk) 2/g
exp
(KcI
-116-
APPENDIX 4:
h (xyfz)
The Inner Expansion of
In this appendix we shall derive the inner expansion
of the modified Green function defined in (3.22).
h
in (3.25)
.
with the Fourier transform of
2
i-4"+fd 2 11exp[(k-w/U)
(k-w/U) 2 +Z /2-(Uk-ii) 2g
=
h*(yz;k)
0 ykO
=0 d
We start
21
2z
cosy
(A4 .1)
Making a change of variable,
(A4.2)
t = jk-w/Ulsinh v
we may reduce (A4.1) to the form,
coshx
+imf0dv
h (y,z-k) = -4
0
Vt).+
o coshv -(Uk-i ) 2/g |k-w/U|
x
exp(zIk-w/Ulcosh v)cos(yjk-w/Ujsinh v)
(A4.3)
Ursell(1962) derived a series representation of the
expression similar to that of (A4.3).
Ursell's analysis to (A4.3) provides
The application of
-117-
coshm
0
h 0 (y,z;k) = 4 [1 0 (k0 R ) + 2 M m=. (-1) mm
(k R ) cosme{
1
(X i sgn(k)
I
+ a)
-
4
4K (kR)
sinhma cotha
+ 8m
~a 1
m
v
(k 0R 0)cos
v e)]
v =m
sin ma* cota*
(A4. 4)
In' K
where
are the modified Bessel functions defined in
Abramowitz and Stegun(1964),
y = R sinO
,
z = -R 0 cose
and the upper and lower terms in curly brackets are applicable
cosh a >
K
/k
a*
0Cos
where
.
according as
K0
U 2 k 2 /g
<1
and
I
(A4. 5)
k0
Ik-w/U|
Using the ascending series expansions for
and neglecting terms of
In
K0
0 (k2R2), we obtain
-l
0~
h* y~zk) (lK Z)4 &iisnkcohlI/,O
i
sgn
(k)+cosh
h *(y,z;k)=4(1+K
-W
+ cos
(K /o)
11-k /K 21/2
(K 0 /k)
+ 4[Zn(k 0 R 0 /2)+ y] + 4K 0 z[Zn(k0 R0 /2)
+ 4K ye + O(k2R2
where
and
, Kc2R)
y = 0.577... is Euler's constant.
+ y-1]
(A4.6)
-118-
The Fourier transform of a pseudo-time-dependent Green
ing asymptotic expansion for small values of
KOR
,
function has .been derived in Appendix 3 with the correspond-
of the form
H 0 (y,z;k) = 2(1+K 0 z) [n
K
R
+ y + 7i sgn(k)
-
+
z
(A447)
2
O(K 2R
0 0
+
K 0 yel
and
47ri sgn(k)
H
where
(1+K
denotes the complex conjugate of
the expansion of
h (yz;k) = 2 H (y,z;k)
Thus,
KOR0
and
0
0
-
.
H
and
H
can be expressed in terms of
H
k0 R0
for small values of
h
(A4.8)
z)
4 sgn(k) (1+K0 z)f 0
(k,K
)
H (y,z;k) - H (y,z;k)
0
0
[H-
(f /2ini) (H0
-
H0 ) ]
(A4.9)
where
0
-sgn (k)
n)
sgn(k)+cosh
-7r+cos
1 (K
(i,/k
0
/k)
+ sgn(k)tn(2K /k0 ) + in
with an error factor
1 + O(k2R2
, K2R ).
21-k 2/K 2-/
)
*i
)=
f 0 (k,r
)=
(
(A4.10)
-119-
If we define
= -2Tri
f 0 (k,K )[H
(y, z;k)
*
-2
h0
0
(y,z;k)
sgn(k)f (k,K )e0zcos
-
0 (y, z;k)]
Koy
(A4.11)
,
S*
F
in (A4.9) reduces to
*
22
h (y, z;k) = 2[H (y, z;k) + F 0 (y, z;k) ] + 0 (k2R
0 0
2 2
0 0
(A4.12)
Therefore, the inverse Fourier transform of (A4.12) yields
an inner expansion of
ho (x,y,z)
as
h (X,y Z) = 2[H (y,z;x) + F (x,y,z)J
1+ 0
1 + (m
22
2
e
,
with an error factor
0
-120-
Reduction of the Kernel
APPENDIX 5:
In this appendix, we shall make a reduction of the
kernel defined in (3.31) following a similar analysis as
Sclavounos(1981)
Using (3.27)-(3.29)
did.
and the results
of Appendix 3 followed by a coordinate transformation,
k=k6
,
6= U2/g, we obtain
F (x,y,z) =
J
dk e -ikx/6D
--
(k)ek2Z/6 cos(k 2 y/6)
(A5.1)
+ in
sgn(k)
-
i sgn(k)+cosh 1 (k2 / |k-Ti)
%. -7+cos
xl
-
x
1(k 2/|jk-Tj
)
D (k) = Zn(2k 2 /jk-Tj)
+
where
(k--t) 2/k41-1/ 2
(A5.2)
where the upper or lower expression in bracket is applicable
according as
k2 /jk-T
Since the kernel
1 ,
D 0 (k)
(A5.3)
T= WU/g
defined in (A5.2)
similar to that of Sclavounos(1981),
analysis for the present work.
is structurally
we will repeat his
-121-
cosh- (x) = Zn[x + (x2 _l) 1/2
,
We use the definition
and start by analyzing the function
x 5 1
(1-z
f(z)
1/2
2
- 2 -1/2
)
n[z+(z -1)
]
on the complex z-plane, where
Znz
and
(A5.4)
z1 / 2
are said to
be the principal branches of the corresponding multivalued
functions with
-7
< arg (z) < i.
The function
f (z)
is
analytic on the finite complex z-plane with a branch cut
along (--,-l].
The values of
shown in Figure (A5-l), where
0 <lxi
<
f(z)
along the x-axis are
0< cos 1 |x
1, and the previous definition of
understood for
<
7r/2
cosh
LFr
is
l-
~v--
-II-
/1-2
0
Figure A5-1
1 x1
JxJ> 1.
Cx,
~,r
for
1.
Icsxiii--76
The values of f(z) along the x-axis
COSh
-122-
f(k 2/ (k-r)]
The analytical structure of the function
on the complex k-plane can be determined if we first consider
the mapping,
z = x + iy = k 2 /(k-T)
T>O
,
(A5.5)
k
Solving the resulting quadratic equation for
gives
[z-(z 2 -4tz) 1/2]
k = u + iv =
(A5.6)
Here the branch of the square root is chosen so that the
The
complex z-plane is mapped onto the lower k-plane.
branch point at
equation
z=-l
corresponds to the roots of the
k2/(k-t)=-l, defined by
,
-[l+(1+4t)
1 /2
] ,
(A5.7)
z =.-liO
The corresponding roots for the
which are real for all
T.
singular point at
are defined, according as
z=1
4T 5 1, by
1 ;(1-4)
1 /2]
,
z = l iO
,
4t< 1
(A5.8)
P3 , 4
[1-i (4T-1)
I
,
z = 1 iO
,
4T> 1
It is not difficult to show that the mapping function
z (k)
is a singlevalued analytic function of
lower k-plane.
The inverse mapping function
k
in the
k(z)
is also
-123-
7 .,P
t\
A
c
A
Ir
l
E
D
. T <1/4
&
gKDu
N
T L
4%+
K
0 ------
A
1
EB
X.
--
PB
F
V.
L/4
Figures Al -2, A5-3
Mappings between z-plane and k-plane
-124-
a singlevalued analytic function in the finite z-plane
except branch cuts along (-w,O] and t4T,<o).
Thus the
mappings between the corresponding domains in the complex
k-plane -and z-plane are one-to-one.
The previous considera-
tions are schematically shown in Figures (A5-2) and (A5-3)
according as
4T
1.
Next we define
R(k) = [l-(k-r) 2 /k
2 /(k-r)+[k 4 /(k-t) 2_
1
-1/2Ztn{k
1/2
(A5.9)
The function
Dku
and
R(k)
is analytic in each of the domains
DkL, its values being determined from the
corresponding values of
where
f(z)
is analytic.
f(z)
in Dzu
and
DzL
Since the values of
respectively,
along
R(k)
the curves CD and IJ correspond, in the limit, to the values
of
f(z)
along the segment (0,4T) on the x-axis where
is analytic,
R(k)
f(z)
can be analytically continued across
the dotted semicircle from the domain
Dku
to
DkL
and
vice versa, using definition (A5.9) uniformly in the lower
half k-plane.
Furthermore, R(k)
takes real values along
the real axis except the branch cuts, one located along
the segment (-o ,plJ
and the other along
be analytically continued for
Im k > 0+
[p 2 ,T],
and it can
using Schwartz's
-125-
reflection principle,
R
(k)
=
(A5.10)
where the overhead bar denotes the complex conjugate of the
expression involved.
Consequently
R(k)
is analytic in
the finite k-plane apart from the two branch cuts.
Now we define
W(k) = Zn[2k 2/(k-r).
Since the function
-
R(k)
(A5.ll)
Zn[2k 2 /(k-T)]
is analytic in the finite
k-plane except for a branch cut along (-ct],
W(k)
is an
analytic function in the finite k-plane with a branch cut
along
(-c, T] .
Thus the values of
D (k)
along the real
k-axis can be expressed in terms of the values of
W(k iO),
where the upper or lower sign is applicable according as
x 5 0
Figures
in (A5.1).
Using (A5.4), (A5.8), (A5.10),, and
(A5-1) - (A5-3) , we obtain
Do (k)
=
W(k iO) + iri sgn(k)
7ri[u(T-a)-2u(-a)]
+ (|1-(k-T)2 /k 4 -1/ 2 )g (k)
where
u(k)
(A5.12)
is the Heaviside step function and
g+(k)
= 2ni ,
g+(k)
= 0
,
k
<
p1
(A5.13a)
p. < k
<
p2
(A5.13b)
--
<
-126-
g_ (k)= 0
-00
<
g (k) = -'ri
T
g (k) = -ni
k. <
(A5.13d)
*
k < p3
,
,
(A5.13c)
< t
p3
< k
p4
<
-
< k
k
T. <
1/4
(A5.13e)
< p4
T
< 1/4
(A5.13f)
<
T
< 1/4
(A5.13g)
T
>
(A5.13h)
0w
< co
,
g (k) = -ni
,
< k
=
,
-27i
p2
g+(k)
,
1/4
It can be seen that
W(k)
= 0(nlkl)
)
W(k) = 0(tnlk|/k 2
as k
0
(A5.14a)
as k
00
(A5.14b)
Applying Jordan's lemma, we obtain
dk
-ikX/6 W(k iO) = 0
,
x
>0
(A5.15)
-00
Using (A5.15) and the convolution theorem, we can also
show that
f
00
2
Ok eik/ W(k iO)e kz6cos(k 2Y/6) = 0,
x
> 0
(A5.16)
Combining (A5.1), (A5.12), and (A5.13) we obtain
IF~
F(1) (X
,Z
0
F (x,y,z) =
1
F
(2)
+ F (2) (x,y,z)
x < 0
(x,y,z)
(x,y,z)
(A5. 17)
,
x
> 0
-127-
F
0
(x,y, Z)
2i
P
di e-ikx/6ek2Z/6 cos(k 2y/6) {1-[l-(k-t) 2/k 4
J
+
:2
e-ikx/5 ek 2 z/6 cos (k2 y16)
- 1
2
4
dk e-ikx/6ek 2 z/6 cos (k2 y/ )'U1- [ 1- (k-r) /k ] -1/2k
2i
6
f P
(A5.18)
F (2) (XIyTrZ)
+
f43
i (P
f
2
4 &c e -ikx/Sek Z/
cos (k2 y/6)'{1-[l-(k-T) 2 /k 4 ]-1/2}
2
dk e -ikx/6 ek z/6 cos(k 2y/c)x
4
J P3
-
x {1-i[
(k-t) 2/k 4-1]-1/2
. < 1/4
(A5.19)
and
F 0(2)xyz)
dk e-ikx/e k2Z/6 cos (k2Y/S)
=-o
x
{1- [l-(k--c) 2 /k 4 1- 1/2
T
where
6 = U2/g.
>
1/4
Th a corresponding expressions for
(A5.20)
F (x,y,z)
can be obtained from (A5.17)-(A5.20) by using the relations,
F(xyz) = e x/'
F (x,y,z)
F (y,z;k)= F 0 (y,z;k+w/U)
(A5.21a)
(A5.21b)
-128-
Inner Expansion of the Outer Solution
APPENDIX 6:
4(x,y,z)
In this appendix we shall derive the inner expansion
of the outer solution
$
The outer solution
4 ur$(p)
for the purpose of comparison.
0
has been derived in (3.10) as
ds(Q) [(Q)
=
Q
-
C
g
dn[2iwUt
r[(-+
) + h(pQ)j
a
Q
(Q) + U2 W
n
-
(A6 .1)
")I)h(p,Q)
C
The Fourier transform of
4m(m
h+0
h(yk)=
h
0
is obtained from (3.6) as
2 + 21/2z cos ty
exp[k
[k 2+2212(w+Uk-iz) 2/g
A6.2)
Appedix4 fr te drivtionof he nne exanson
*
Making use of a similar analysis to what we did in
Appendix 4 for the derivation of the inner expansion
h,0
we will obtain
**
*
2 2
2 2
h (y,z;k)= 2[H (y,z;k)+F (y,z;k)]+O(k R ,K R 2)
(A6.3)
where
H (y,z;k)= H (y,z;k+w/U)
= 2(1+KZ) [Zn KR0 + y + 'ri sgn(w+Uk)-Kz+Ky61
+
0(K2R )
(A6.4)
*
-129-
*
F (y, z;k) = F (y,Z;k+w/U)
*
1
f (k,K)
*
-*
(A6.5)
f (k,i) [H (y,z;k) -H (y,z;k)]
-
=
= -sgn(w+Uk)
t
sgn(w+Uk)+cosh 1 1(K/Ik|)
_-1 (< I
in
-V
+
Cos
I
11-k 2
K2
-1/2
(A6. 6)
+ sgn(w+Uk)tn(2K/fkl) + 7i
Here the upper or lower term in bracket is applicable
according as
ic/|kI
>
1 I
K
=
(A6.7)
(w+Uk) 2/g
The inverse Fourier transform of (A6.3), therefore, provides
(A6.8)
h(x,y,z) = 2[H(y,z;k) + F(x,y,z)1
1 + O(w 4,R )
.
with an error factor
Thus, an application of the similar analyses in section
111.3 to (A6.1) will give rise to the inner expansion of
(A6.1) as
2n$(p) =
ds [$(Xn,)
t ]ZnR/R
-
a
3Q1
B (x)
ds Q[
d [2iwU (
-1
(,
, )
-
Q
DNQ
O'n,0)+U 2
9C
with a resulting error factor
1+0(W4R 2,
-
1 [H(PQ)+F(PQ)]
)- [H (PQ)+F(PQ) I
(A6.9)
R0 ,R
)
-L/2
f
(
j
)
+
L/2
/2
-130-
APPENDIX 7:
Relationship between a Line Distribution and a
Surface Distribution of the Generalized Inner
Green Functions
In this appendix we shall derive an effective line
distribution of wave sources which produces far-field wave
effects comparable to those of a surface distribution.
The
solution of the inner problem in (4.8)-(4.10) takes the form,
2ff(P;x) =
- IG
(n-&)
L/2
ds
-L/2
Q
8(e)
- dn[2iwU0(n,;)+U 2
(u-
-
)]G
(yzrn,;x--)
(A7.1)
where
G ()= 6(x-E)ZnR/R1 + H(y,z,rn,C;x-E) + F(y,z,n,r;x-E)
(A7.2)
Here
H
and
F
are defined in Chapter III.
The Fourier
transform of (A7.1) gives
L/2
-L/2
ds [
B(
G
x
-
;)
(,
a-x
NQ aN Q
(ylzjn,C;k)eik
Cdn[2iwU((n,0;E)+U2 (x
C
Y
a
0;k
-
(y, z;k)
)
2*
aE
k
(A7.3)
-131-
(y,z,n, C;k) = e K
G (W
(;C+in)
G
(1)*
(y,z,O,O;k) + 01(
)
From the results of Appendix 3 we know that
(A7.4)
where
R0 = (y2+z2)1/2
and
K
=
(w+Uk) 2 /g
The velocity potential due to a free-surface line
distribution of wave sources can be expressed as
t(y,z;x)
Here
a(x)
=f
da(&)G(i) (y,z,,O;x-&),
R>>1
is the effective source strength.
transform of (A7.5)
(A7.5)
The Fourier
gives
$*(y,z;k) = a (k)G ('(y,z,0,0;k)
(A7.6)
Far from the ship in the inner region, therefore, the
effective source strength
of inner variables.
2Tra
-1
can be expressed in terms
From (A7.3)-(A7.6) we obtain
ai,;l) K(c+in)+ik&
ds, [
(k) =(L/2d
-L/2
a(x)
8(&)
Q
Cdn [2iUt (n , 0 ; &) +U2 (t
Q
ol
ign+ik t
(A7.7)
Thus, we can immediately write down the effective source
strength as
-132-
f
-L/2
B(E)
ds
[4(
Q
[2iwo(n,0;
-T
-O;
- NQ
)
fL/2
Jd
2ra (x) =
DQ
(n,C;x-)
)+U2
Cd
-0 < X
< 0
(A7.8)
where
M(-n,, ;x-V
7~1
dk e-ikx e (C+in)+ik
-47rU2 (+in)
9
F2
I -1/2
I4U
( +in)
U
(A7.9)
For the special case of zeror forward-speed,
U=O, we
recast (A7.9) as
7(a/1/2 exp[-a (x-+a2 ) 2]exp(aa 2
)
M(n,c;X-)
(A7.10)
where
a2
4U (+in)
2iwU (c+in)
(A7.llb)
.
1
(A7.lla)
-
=
a
From the definition of a generalized function
Lighthill(1958), we know that
6 (x)
(see
j
-133-
tim
Thus,
*dx e-ax 2 (a/Tr) 1 /2F(x) = F (0)
a (x)
(A7.12)
simplifies to
ds [I(
x
e
,x
. L/2
(x)
21ra(x) =
Ix|> L/2
0
(A7.13)
where
v =
obtain
2 /g.
(A7.13)
Here (A7.10)-(A7.12) have been used to
from (A7.8).
From (A7.8) and (A7.13) we can draw a conclusion that
a free-surface line distribution of sources, in general,
should be extended over an infinite length in order to
produce far-field wave effects comparable to those of a
surface distribution, while the line, distribution over a
ship length sufficies for the case of zero forward-speed.