10 APR. 1978
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SYNPOS lUN ON ASPECTS OF NAVIGABILITY
APPLICATIONS OF SLENDERBODY THEORY TO SHIPS MOVING IN RESTRICTED SHALLOW WATER
BY Ronald W. Yeung, Ph.D.
Professor of Naval Architecture, Department of Ocean Engineering
Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A.
SYNOPSES
This paper examines the hydrodynamic forces and moments acting cn ships
moving in shallow water through the application of slenderbody theory. The
free surface is considered to be rigid, and the clearance between the keel
and fluid bottom s assumed small. The paper consists of two parts. In
part I, a general theory for a single ship moving in a nonuniform incident
stream is developed. Expressions are derived for all six components of forces
and moments acting on the ship in terms of the shifl characteristics. In part
II of the paper, the slenderbody theory is utilized to predict the transient sway force and yaw moment exerted on a ship as she moves along an irregular
coastline, or certain verticalside objects. Numerical results are presented
for the case of a ship approaching a breakwater or a wedgeshaped pier. It
was found that in all cases considered, the ship experiences an attractive force and a bowin moment during the approach towards the object.
INTRODUC I ION
The advent of largesize tankers has brought forth a class of interesting and challenging hydrodynamics problems, in which the consideration of shallowwater effects as well as the complexities of the physical environment is often
very important. For instance. it has been observed that the maneuvering
characteristics of a vessel are highly sensitive to the size of the underkeel clearance'*, and such characteristics are further altered by bank effects2. From the viewpoint of theoretical modelling, it is fortunate that a number of these problems can be described reasonably well by inviscidflow theory with
the free surface being treated as a rigid wall. The rigid freesurface
condition can be justified on the ground that the Froude number based on the
water depth is small and that the dominant hydrodynaxnic forces are those
associated ;itìA i1e fluid inertia. Even with such a simplifying assumption,
the resulting exact (but timedependent) boundaryvalue problem still requires extensive numerical computations, the implementation of which on a realtime basis, say on a simulator, is still beyond the capability of the present
geiieration of computers. Furthermore, such complicated coinDutations generally
provide relatively little insight into the understanding of the underlying
physical phenomena.
*
Supr5cript denotes references given at
the end of die
pafler.Lab. y.
Scheepsbouwkun
Technische Hogeschool
If one exploits the fact that most shiplike bodies tend to be slender, the
threedimensional problem in shallow water can be recast as two twodimensional
problems, an inner problem in the crossflow piane of the ship and an outer
problem in the horizontal plane, each of which is relatively easy to solve.
The two
solutions can be combined together by the technique of matched asymptotics.
The
classical work of such an approach in shallowwater ship hydrodynamics is
due to
Tuck3, who obtained the sinkage and trim of a slender ship moving steadily in
water of constant depth. Extension of this theory to the case of a dredged
canal
was carried out by Beck, Nenan, and Tuck. ore recently, the steadystate
banksuction problem of a ship in a canal was studied by Beck5, who noted a
generally
good agreement between his theoretical results and existing experimental
measurements. All of the steadystate problems described above
contained the
leadingorder effects of the freesurface. However, the inclusion of such effects in
the case of an unsteady flow will make the problem much less tract.ble.
Utilizing
a rigid freesurface condition, Tuck and Newman6 presented a
simple slenderbody
theory for the prediction of hydrodynamic interaction of ships in deep water.
The shallowwater theory of the same problem was examined by Yeung7.
Both works
confirm the extreme usefuliness of these "0(1) lateralseparation" slenderbody
theory in providing an excellent qualitative description of the physical
phenomena.
In this paper, two additional aspects of slender ships in shallow water are
examined theoretically. In part I, the general theory of a single ship
undergoing rectilinear motion in a spatially, and possible timewise nonuniform
incident stream is developed. A onedimensional integral equation can be
derived
to determine the local crossflow as observed at each ship section. Various
components that contribute to the forces and moment in both the vertical and
horizontal planes are noted. The results of such an analysis is useful to the
common realistic situation of a deepdraft vessel crossing a current that
has
spatial gradients, as in the case of seasonal or tidal currents near entrances
of navigation channels.
In part II of this paper, we investigate the effects of certain
verticalsided physical obstructions on a flo field generated by a moving ship in
otherwise calm water. The theory is developed with the assumption that a
general Green function associated with the obstacle geometry is available and is
illustrated with numerical results for the geometrically simple cases of a
breakwater and a wedgeshaped pier. A host of interesting features of the
transient sway force and yaw moment acting on the ship are pointed out and discussed.
PART I
Slender Ship in a NonUniform Stream
1. GOVERNING EQUATIONS
The problem to be considered is that of determining the hydrodynamic forces
and moments acting on a slowmoving ship in an arbitrary streaming flow.
Throughout, the fluid will be assumed ìnviscid and the flow irrotational except in a
wakevortex region behind the ship. Let h be the water depth and O'x'y'z'
a coordinate system fixed in space, with the O'x'y' plane being the undisturbed
free surface and the zaxis pointing upwards. A twodimensional streaming flow
is assumed given and described by the velocity potential (x',y',t), where
t
denotes time. it is convenient to establish a second ccorinate system
Oxyz,
fixed on the ship with the origin at midship and the xaxis pointing towards the
bow (see Figure 1). e shall assume that the ship has lateral symmetry
and
moves in the positive xdirection with velocity U3 (t). if we now further
assume that the freesurace can be treated as a 1ane, the original problem
is
equivalent to one of studying the nonuniE low flow about the ship plus its
image about y = O (the double body) translating between two planes at z = ±h.
2*
VN
y ¿ (x)2. INNER AND OUTER EXPANSIONS
We assume the length of the body L to be 0(1), and its tranverse and
virtl dimensions 0(c). The water depth h is assumed to be such that the
underkeel clearance is also 0(c). Further, if the incident stream varies slowly over
Ii
a distance of
t,
we can write  = 0(1) and = O  for (y,z) near th±dy oz
body. In this
inner
region, we will express the total potential x;y,z,t as*
4(x;y,z,t) (x,0,t) + V y + _{2 y} + (4)
The first three terms are part of a Taylor expansion representing the incident
field as seen at a body section whereas ,, represents the innerfield body
disturbances. Allowing V*, and ct be unknowns at this point makes it
possible to account for an9 interaction between the body and
incidentstream potentials.
If we make use of (4) and that
/x «(/y
, /z) in (2), the boundarycondition for reduces to
c; ' o N
+0(c3)

1n y j (5)Figure 1: Slender Ship in a NonUniform Stream
Let
Then the boundaryvalue
be the potential associated
problem for
with disturbances generated by the body. is: (x,y,z,t) = 0 (1) B 2 2 2 +
=U n
_{lx}
(2) 13=0
(3) z = ±h4where 13 is the surface of the double body and n the exterior unit normai
to the fluid. An approximate solution of this threedimensional problem will now
where is. the twodimensional normal in the crossflow plane on the body
section E(x). We observe that since the body is assumed to be slender N =
0(1). n = O(E). Next, by substituting into (1) and (5) an inner expansion
of the following form for
(l) + (2) +
B B B
where = we immediately obtain
y2
(l)_
y,z B
2 (2)
y,z B
E
/x.
Now, in order to keep quantities involving the bodyFor a body with lateral symmetry, and are even in y. By examining the
ne t flux emitted from the body boundary, we can easily show that
s, 4h for _{¡yJ}
»
C=4h
_{_V*N}
(7)  y B (U _U*)n YN
, (8)  1 x ywhere S(x) is the
doublebod
sectional area at the location x. The prime in(12) is. used to indicate differentiation with respect to the space variable. The
potential
2 is associated with the lateral motion of a cylinder of geometry
defined by E and is odd in y. Its asymptotic behavior for large y can be
shown to be S+A .4. (6) (12) (13) (14)
where C is a rather wall known blockage constant introduced by Sedov8. In (14),
oA renresents
the
ateral added mass of the doublebody section, where p is thedensity of the fluid. We note in passing that C is 0(1) if the bottom clearance
with U*(x,t)
geometry separate °from the various unknown coefficients in the inner problems,
it is useful to define the following cannonical potentials which depend only on the body geometry at a given section:
y,z
_{z±h}
=0
(9) V2_{ji1=O}
=n
E X V2=0
E y y,z_{z±h}
(10) y,z V2=0
=0
z= ±h (li) E = yNis 0(c), which is what we assumed. In terms of
, 2
and ', it is obvious
that we can now write
± = (x,0,t) +
rV(Y_2)
+ f1(x,t) +o B o
[ (u1U*) + (y2  2;) + f2(x,t)] +
0(c3)
(15YThe functions f and f7 are introduced to represent the homogeneous solutions cf
(7) and (S). Tie unkn6wns , V, f1, c, and f7 can only be determined by
matching with the outer soluion.
Sufficiently far away from the body, (i.e. (x,y) = 0(1)) , we anticipate
the flow to be essentially twodimensional in the horizontal plane because of
the shallowdepth assumption. This can be verified by substituting an outer
expansion of the form
B,Y,z,t)
) + + (16)into (1) and applying the condition (3). Hence, the outer solution can be
represented by twodimensional vortex and source distribution as follows:
(x,y,z,t) 0(x,y,t) =+ y(,t) tan 1 ) d +
L/2 C(t)
W(t)
a(,t)
log ((x_)2 _{+} y2)1'2 d (17)L/2
where y and
a
are unknown vortex and source strengths in the intervalx = [L/2, L/2]. Because of Kelvin's theorem of conservation of circulation,
a wake vortex line will be present and convected downstream by the zerothorder
potential after the vorticity is shed at the stern. At a given time, we may
assume that 0the strength y in the wake and the location of the wake zit) are
both known. (t) as used in (1» is defined by (x[L/2, L/2j). .
3. MATCHING
We now match the inner limit of the outer solution with the outer limit
of the inner solution. First, a twoterm outer expansion of the twoterm inner
solution can be obtained from (14) and (15):
hm
+ (x,0,t) + {V*(y ± C(x)) + f1] (18)whereas, according to (17), the twoterm inner expansion of tue twoterm outer solution for x in [L/2,L/2] is given by:
L,12 L/2 him ( + (x,Q,t) + [V y ± I
y(,t)d
if
_{(t)d}
y] o xy±O
(19)where V0 E (x,o,t), which represents the normal velocity of the incident
stream at thxaxis. Clearly, equations (18) and (19) together imply that
V0(x,t)
ly(,t)d
(L / 2
VC =
Iix
Thus, the longitudinal flow is not altered, as one may have expected intuitively;
accordingly, UU E ç(x,0,t).
However, equations (21) and (22) can be combinedto yield to following integrodifferential equation for V*:
V*(x,t)
+
J
[v*(x,t)CJtd _{= V0(x,t)}
(23)
which is identical to one examined by Yeung7 in shipinteraction problems. We note the interesting special case that if C is 0(c), the cross flow can be
approximated, with good accuracy, simply by V (see also Yeung and Hwang'°).
Proceeding to the next order of matching, we obtain the threeterm outer expansion
of the threeterm inner solution as
hm
+ = (x,0,t) + [V*(y ± C) + f1
»1
S'
c2
S+
Nu.

U1)iII
+ (
 i!) +
_{f2]}which can be compared with the following term inner expansion of the
threeterre outer solution
1(L/2
i
Jum (
+ ) = (x,0,t) ± [Vy ±zj (,t)d + o B oy±O
Xu,
_{o2}
y + J,t)logxd]
+ H
2 y t(x) j (x,t) ] (5)to yield the equalities:
(26) o = [(U  U )S'  aS]/4h (27) o I ( L/2 f = J
a(,t)
logx
:Id
(28)2Tr
The unmatched term f7 in (24) and yjyterm in (25) can easily be shown to match
with higherorder teems in the inner expansion. We note that (26) is consistent
with (20) obtained earlier. In comparison with the constantspeed case considered by Tuck3, (27) states that the source strength has an additional contribution,
U'S/2h, due to the spatial gradient of the longitudinal flow. As will be seen
lter, the function f1 will determine the leadingorder sinkage force and trimming moment due to the body potential.
I
(24)
6(30)
IIIpp
k
i
It is worthwhile to recaoituìate all results below. From (15), the
matched inner solution can be written as:
(x,y,z) = (x,0,t) + V*(y
 + f1 + (U1 U)1
(2

2) ±
0(t3)
where V is the solution of
(23)
and f1 is given by (28).4. HYDRODYNAMIC PRESSURE
In order to obtain the differential force and moment acting at a ship
section, we need to examine the hydrodynamic pressure p in the inner field.
In the Oxyzsystem, Bernoulli's theorem is given by
a
= 
C(t)+
(pu
_{1x}
)'
i(2
2)
±0(t3)
2 y z
where C(t) is a given constant that depends only on the incident flow at infinity. This constant is unimportant for the determination of the longitudinal
and transverse forces, which vanish because the doublebody is a closed body and
that the underwater contribution is identical to the image contribution. For
the sinkage force, since the integration is carried out only over the "wetted"
portion of the body, C(t) will give rise to a net vertical force proportional
to the waterplane area and acting at the center of flotation. With the
understanding that this componenet has already been properly accounted for, we shall
omit C(t) in the ensuing calculations. If equation
(29)
is now substituted into(30) and terms that are even or odd in y are gathered separately, the leading
order results may be written as:
(p/p) = [p(x,t) + P(x;y,z,t)] + p1(x,t) + P1(x;y,z,t) +
0(t2)
where, for terms that are even in y,
=
+uu uI
o o10
o2
po =
v2[(l
_{}
2y
+ 2z_p
= f
+ (U
 U
)f' 1 1 o11
pl
= Owhile for terms that are odd in y
p =P
0:
= *(y _{ U}f
_{V*(y}

+ v*(U1  u0)(l
 2yly
+ v*u'[_(l 
)(Y
_{}
_{2zz}
In the above expressions, a dot indicates partial derivative with respect to time whereas a prime indicates partial derivative with respect to x.
4.1 Surge and Heave Forces and Trimming Moment
The forces and moment for verticalplane motion are determined by a
pressure field which is even in y, viz. Equation (31). The components of the
pressure that depend only on the axial coordinate can be integrated rather
easily. First, we recall the following identity, valid within the context of
the slenderbody theory:
L/2
fi
_{} _{I}_{S'(x)}
p(x) n dS  dxp(x)[ . ± B(x) k]
j 2
L/2
where indicates that the integration is to be performed only on the wetted
portionf the hull and B(x) is the local beam at the location x. Let F and
F be the surge and heave forces, respectively, and M the trimming moment. If
we denote the contributions due to po, Po, and P1 by Tthe superscript, a, b, and c respectively, then
dF(a) _{=} 4JS(x)[11_{o} + (U_{o}  U )U I dx 1 o dF(a) _{=} pB(x)[ + (U /2  U )U I dx z o o 1 o dM(a) = dF(a)X y z
which are effect0caused by the longitudinal component of the incident stream and are all O(c2. The contribution due to the "thicknesst' effects of the body
is due to Pi' which is one order higher. Upon substituting the definition of
f1 into (31) and performing the necessary integration by parts, we obtain
F(c) _{'S'(x)} (33) L/2 L/2
= 
Idx Jd z 8Trh J L/2 L/2 M(c) yThe evaluation of the integrals associated with P requires the knowledge of
certain hydrodynamic properties of the body sectio. If we define
k(x)
1 2 2 n X =ii[(i) + ] .ds (35)k(x)
2)
2y 2z n Z E Zwhich can be readily evaluated once the cannonical potential
2 is obtained,
the zerothorder forces and moment due to the modified cross flow V' is given by
(36) I Q F x y = p L/2 L/2
J
V*2 kxl
kzI
xk zII dx 2B(x)+ (U(x,t)U[(U(,t)U1)S'
X(34) 2B(x)x4.2 Differential lateral force
It is evident from (32) that the leading order lateral force F is O(e2).
The various terms of P1(x;y,z) may be regrouped in a more orderly fashion as
follows: O yNds  O N ds y
2y
f
L E ]N_{lds}}
 lz 2z_{Y)}
+ tf ¿
[YNy_{ [(l_2)}
+ IN ids oL
y2zz
y) (37) Ft(x,t) = O P N ds = _[*_(U1_U )V*'  °  (U U )v* i o O x ( N ) +12yly
whereby, one recognizes that
O yNds = S ,
Nds
= A (38)Accordingly, the first two terms yield:
+ A) with  (U1  U) (39)
The third term can be written as
ds = (J Ó2N _{ds} _{}
 dz
_{(40)}E E
where y = fl(x,z) represents the body surface. Although it is not entirely
obvious, Lighthill'' showed that the last term of (40) cancel with the fourth
term in (37). The fifth term in (37) is straightforward, but it does not appear
to be possible to simplify the last term any further. In any event, if we define
a new coefficient (x), where
k() =
[(l_2Y)Y + y2 
21
N ds (41)The final result for the differential lateral force dF
is given simply by
(lD ' D _{i}
F; =  ._V*JS +  (V*A)  _{V' U'(S + k)} _{(42)}
Like k
and k earlier, is a function only of the section geometry and thepotential 2
and . Equation (42) says that the lateral force consists of
three components: an effective buoyancy force on the body section due to the
accelerating cross flow, a force due to the change of momentum of the fluid
as a consequence of the movement of the body section in the incident field, and a force representing the interaction of the cross flow and longitudinal velocity
J
 rvfl. .t.tc  S;
5.
SMARY
A simple theoretical model based on the slenderbody theory in a
shallowfluid has been developed for the rectilinear motion of a single ship
in a
nonuniform stream. The hydrodynamic forces and moments are
given by Equations
(33)(36), and (42). Evaluation of these forces is possible if the
hydrodynamic
characteristics of the sections
A, k, k,
, are known.These coefficients
are nontimedependent. At a given instant, the actual cross flow V*(x,t)
an be determiaed by solving a relatively s'nple integral equation
(23). Note
that although the ship is assumed to have a singledegree of freedom in
the above
analysis, lateral and rotational motion can be incorporated in the theory
with little additional complications.
PART II
Interaction of Ships with a Coastline or Fixed Obstacles
1. THE PHYSICAL PROBLEM
In this part of the paper, we consider a problem which is complementary
to the one studied in the earlier part. We consider a ship moving
in calm
shallow water near a coastline or some obstacles. We are interested in examining
how the vessel interacts with such obstacles, resulting in a lateral force
and
yaw moment acting on the ship. The nomenclature is defined in Figure 2, in
which we noted the obstacle geometry is best described in a fixed frame of
reference.
10N
Figure 2: Slender Ship in Shallow Water near Obstacles
In addition to all the assumptions stated in paragraph 1 of the earlier
part of this paper, we would require the minimum separation Sp to be 0(1).
Note that this does not necessarily mean the theory developed below will
breakdown when
Sp IS
small, which is perhaps the case of more practiced interest,but instead, the theory would likely provide less accurate quantitiative
predictions in such a circumstance.
2. SOLUTION BY MATCHED ASYMPTOTICS
We can proceed to solve this problem in precisely the same manner as
earlier, viz, using inner and outer expansions. In fact, considerable
simplifications occur because the "incident't flow or is now by assumption
identically zero. Near the body, since Sp = 0(1), the representation given by
x,y,z,t) =
1(x,t)
+ [V*(y + U
11
+ f (x,t)12 + O(E3)(43)
where and
2 are defined by (9) and (10).
We observe that V* is now
O(E) rather than 0(1) since its occurence must be due to the effects of the
forward motion of the ship in a laterally unsymmetrical physical environment. We now need a representation of the solution in the outer region, which, to leading order, is twodimensional in the horizontal plane as before.
Let
and G'
be a source and vortex function, respectively,satisfying the noflux boundarycondition on the obstacles. In other words, we assume
the existence of two functions and such that
G°(P;Q)
= logf(x' T)2 + (y' T)21l/2+ H(P;Q)
J
G(P;Q) =:tan
x'
+ H(P;Q)
'J
where H(0) and are functions harmonic in (',n') and are so constructed that
(cx)
= O , and O
(46)
p
Here, Q E (',n') is the source or vortex point, and P E (x',y') is a field point.
The construction of the G's are discussed in a later section. Meanwhile, it is
not difficult to verify that by the application of Green's Theorem, the following representation of the leadingorder outer solutions can be obtained
li=
_{(s,t) G(x',y';',n') ds}
_{+}(t)
2n
_{J}
y(s,t) G(x',y';',n') ds
(47)
(t) zy
where ds in an infinitesimal arclength element along the xaxis; ',fl') are
parametric functions of s as seen in the O'x'y' system.
In
Equation (47),(t) is used to designate the axis of the ship, x = [L/2, L/2], as seen in the
fixed coordinate system. Note that in acaordance with linearized theory, the
wake
position ¿AY
reamins on the xaxis, since any lateral or longitudinalmovement must be induced by the fluid disturbances generated by the body. Such
movements are of a secondorder nature. The source arid vortex strengths in
(47) have to be determined from a matching of
(47)
with the inner solution,whichis given by (43).
/
While all details concerning the matching process may be found in Yeung and Tan'2, it is possible to deduce the findings given, therein from the results
we have obtained earlier. First we note that Equation (26), with U = = 0,
remains valid because the function H(G). is continuous across the axis of the ship
y = 0. Whence,
U 1L/2 _{aH(a)} L/2 I
S'()
_{d} +(t)[_F
ay
F =2(v*)
S y 2 Dt + (v*A) 12d = V*(x,t) (50) 3 i 0(s ) (53)Using a similar argument for H(1), we can show that the cross flow, V', must
be related to y by (22) as before:
y = 2(VC)' (49)
Finally, the cross flow must be determined from (21) with (x) replaced by
aG'Iay and
V by H y/ay Therefore,L/2 i
which states the physically obvious fact that the net cross flow that one sees at a section is the algebraic sum of the vortex distribution and source
distribution. An alternate view is that the vorticity y in (50) is driven
by the first term on the lefthand side, which represents the "reflection"
effects of the obstacle due to the forward motion of the ship.
o
Equation (5) should be augmented by the additional conditions stating that the pressure across the wake tY is continuous and that Kelvin's theorem
is satisfied. The linearized version of the former yields the fact that:
y(x',t) = y(x') for x' GtÚ (51)
whereas the latter can be shown to give
y(x = t) = (52)
where F is the circulation about the body. If Equation (50) with the
auxiliary conditions is now solved for all t during a transit motion of the ship,
then V* is known, and at any given instance of time, the differential lateral
force F' is given accordingly by (42):
where D/Dt is now simply a/at  u1a/ax . Note that the last term in (42) is
discardéd since it is of a higher order.
3. THE GREEN FUNCTION
The Green functions defined by (44), (45), and (46) for a few obstacles of
simple geometry can be obtaired rather easily. The analysis below is
considerably simplier if one introduces the complex variables z' x' + iy'
Lets consider a simple wedge obstacle with one side of the wedge coinciding
with the O'x' axis. Let 3 be the interior angle formed by the two sides of
the wedge. With the introduction of a cew complex variable
The wedge is now conformally mapped onto the real axis of the cplane, in whicui we obtain easily the following expressions for the complex velocity
w(z) E
u  iv of a unit source and a unit vortex located at z' = z'_{o}(a)
i
_{W' ()}
+w (c)
o o to
where a bar denotes the complex conjugate and = z'. Transforming this
velocity back to the physical piane, we obtain °
(a),,
n ri
i
I
w. (z ,z = , ln ,n nn ,n
(z ) L z z' z z o or
1 1 ln I ,n ,n + (z ) I z z z o owith the principal value of z in {O,2ii] and in {2Tr,O]. Thus, the quantities
G0), C'
to be used in the solution of (50) can be calculated very easilyusing (56) and (57). We note that for n =  , (56) and (57) correspond physically
tó an infinite breakwater, while a rightangle pier corresponds to the case of
n = 4/3.
For the general case where one has an arbitrary contour, a seminumerical
representation of G can be obtained via the use of SchwartzChristoffel
transformation.
4. NUMERICAL RESULTS
A numerical procedure has been developed to solve the integral equation
(50). Details pertaining to its treatment is described in Reference 1112. Here
certain interesting aspects of a ship moving past a wedge are presented and
discussed.
The vessel used in the calculations is a 280K DWT ore/oil carrier used
earier by Norrbin13to study bankeffects on a ship in a channel. For convenience,
the quantitity C(x) in (14) is obtained with the assumption that the section
shape is a rectangle of width equal to the local beam. The actual value would
be slightly lower than the approximated value. Figures 3 and 4 show the
lateral force and yaw moment acting on this vessel for three wedge angles,
as she approaches the side of the wedge. The separation between the body axis
and the sida is taken to be one half of the ship length. The bottom clearance
to water depth ratio is chosen as 1/40. cte that when the bow is just about
at the tip of the wedge, the vessel experiences a maximum suction force and a
moment (bowin) pulling her towards the wedge. Since the force and moment are
in phase with each other in this part of the transients, they symbolize a
rather local sort of interaction of the bow geonietry and the wedge tip. A
memory effect starts to develop when the midship point coincides with the tip.
During this time, the force quickly achieves a maximum value of a repulsive
nature while the moment continues to grow in the bowout direction. Thus,
the overall pattern of the transient force and moment indicates that the vessel
has a dangerous tendency to move tewaids the wedge during the approach while
her stern may collide with the wedge after the transit. It is of interest to
0.09 o. oc 0.04 0.02 0.00 0.02 0.04 0.06  U2.5IO?t CF

_{./L0, 5}U t.  .025 oo. 040 ADTICT1Otl t 0.08 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5Figure 3: Sway Force induced by a
wedgeshaped Obstacle
during transit, CF _{=}
V / U2BT
y 2 1
remains the saine with the case of the breakwater offering the strongest
disturbance. When the bow of the vessel is about one ship length from the tip,
the. transients start to decay and the usual banksuction force and bowout
moment are approached gradually.
Figures 5 and 6 illustrate the effect of the bottom clearance on the force
and moment for a vessel moving past a breakwater at right angles. The pattern
of the transient is similar to that considered earlier. The peak of the
attractive force increases montonically as the bottom clearances is reduced.
The twodimeniona1 results are recovered i we set O.
Figures 7 and 8 show the lateral force and yaw moment acting on the
vessel as she approaches or leaves a rightangle pier. The results are plotted
against the midship position relative to the corner of the pier. That the two
curves are not identical is indicative of the fact that the two situations give
rise to two different wake composition behind the vessel and that the vessel is
not entirely symmetrical about midship. It is of interest to note that the
vessel leaving the pier experiences a 30% increase in the attraction force compared with the steady banksuction value, and a strong bowin moment before
it passes the corner.
It is worthwhile to recall that all the above results were obtained with
the assumption that the free surface is rigid. In the realistic situation, one
would expect an oscillatory pattern .be superposed onto our results
be.cause of transient effects of surface waves. Nevertheless, we hope that these
calculations will provide a better understanding of the nature of the hydrodynamic forces that slowly moving tankers experience in restricted shallow
water. Similar calculations involving more complicated interaction phenomenon
are now in progress and will be reported in the near future.
1425E 20 is 10 5 towour z:: 20 owxo 25E 3 2.0 2.0 CN o o o  o. 1.5 1.0 0.5 UT/L. I I 0.0 0.5 1.0 1.5 20
Figure 4: Yaw Moment about Midship
induced by a wedgeshaped
Obstacle during Transit,
k7TRCrIO 0.08 0.08 2.0 1.5 1.0 0.04 i ¡ I I I I 20E2 EPULS COW APP CIIIÑO ATTRAC?IOW LEAVING LEAVING L t 1.0 i_5 'Pli__o_s 6.0.025 APPROACHING
Figure 7: Effect of direction of
motion on Sway Force for motion near a rightangle
Pier 15E 10 5 io C?s
o.
jSE 2.0 20 ACKNOWLEDGMENTThe author wishes to thank Professors J. N. Newman, E. O. Tuck and C. C. Mei for many interesting discussions during the course of this work. The valuable assistance of Mr. W. T. Tan in obtaining the numerical results
is greatly appreciated. The research reported here was supported by the U.S.
National Science Foundation, under Grdnds GKi38S6X and ENG7717187.
BON007? 60.1 d0.05 d0.025 60. 0125 NOW IN 1.5 1.0 0.5 0.0 UTtL 0.5 d0.025 01.0125 i..O j_5 2.0 I I I t 0.5 0.0 0.5 1.0 1.5 2 0 X'/L
Figure 8: Effect of direction of
motion on Yaw Moment for motion near a rightangle
Pier
Figure 5: Effect of Bottom Clearance Figure 6: Effect of Bottom Clearance
on Sway Force for a Transit on Yaw Moment for a
Tranperpendicular to an Infinite sit perpendicular to an
Breakwater Infinite Breakwater
0.0 0.5 0.5 uTil_ i_9  NOVOU? io Lf.AVONG APPROACO1ING 10  NOWIN is 20E 2.0 2.0 1.5 LO 0.02 0.00 F 0.02 0.04 0.06 0.08 0.10 0.12 2.0 1.5 1.0 0.5 0.0
xt
0.5 LO i_5REFERENCE S
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