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Applications of slender body theory to ships moving in restricted shallow water


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10 APR. 1978










BY Ronald W. Yeung, Ph.D.

Professor of Naval Architecture, Department of Ocean Engineering

Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A.


This paper examines the hydrodynamic forces and moments acting cn ships

moving in shallow water through the application of slender-body 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 non-uniform 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 slender-body 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 vertical-side objects. Numerical results are presented

for the case of a ship approaching a breakwater or a wedge-shaped pier. It

was found that in all cases considered, the ship experiences an attractive force and a bow-in moment during the approach towards the object.


The advent of large-size tankers has brought forth a class of interesting and challenging hydrodynamics problems, in which the consideration of shallow-water 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 inviscid-flow theory with

the free surface being treated as a rigid wall. The rigid free-surface

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 time-dependent) boundary-value problem still requires extensive numerical computations, the implementation of which on a real-time basis, say on a simulator, is still beyond the capability of the present

geiier-ation of computers. Furthermore, such complicated coinDutations generally

pro-vide relatively little insight into the understanding of the underlying

physical phenomena.


Supr5cript denotes references given at

the end of die


Lab. y.


Technische Hogeschool


If one exploits the fact that most ship-like bodies tend to be slender, the

three-dimensional problem in shallow water can be recast as two two-dimensional

problems, an inner problem in the cross-flow piane of the ship and an outer

prob-lem 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.


classical work of such an approach in shallow-water 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


was carried out by Beck, Nenan, and Tuck. ore recently, the steady-state

bank-suction problem of a ship in a canal was studied by Beck5, who noted a


good agreement between his theoretical results and existing experimental

measure-ments. All of the steady-state problems described above

contained the

leading-order effects of the free-surface. However, the inclusion of such effects in

the case of an unsteady flow will make the problem much less tract.ble.


a rigid free-surface condition, Tuck and Newman6 presented a

simple slender-body

theory for the prediction of hydrodynamic interaction of ships in deep water.

The shallow-water theory of the same problem was examined by Yeung7.

Both works

confirm the extreme usefuliness of these "0(1) lateral-separation" slender-body

theory in providing an excellent qualitative description of the physical


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

under-going rectilinear motion in a spatially, and possible time-wise non-uniform

incident stream is developed. A one-dimensional integral equation can be


to determine the local cross-flow 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 deep-draft vessel crossing a current that


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

vertical--sided physical obstructions on a flo field generated by a moving ship in

otherwise calm water. The theory is developed with the assumption that a

gen-eral 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 wedge-shaped pier. A host of interesting features of the

trans-ient sway force and yaw moment acting on the ship are pointed out and discussed.


Slender Ship in a Non-Uniform Stream


The problem to be considered is that of determining the hydrodynamic forces

and moments acting on a slow-moving ship in an arbitrary streaming flow.

Through-out, the fluid will be assumed ìnviscid and the flow irrotational except in a

wake-vortex 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 z-axis pointing upwards. A two-dimensional streaming flow

is assumed given and described by the velocity potential (x',y',t), where


denotes time. it is convenient to establish a second ccorinate system


fixed on the ship with the origin at midship and the x-axis pointing towards the

bow (see Figure 1). e shall assume that the ship has lateral symmetry


moves in the positive x-direction with velocity U3 (t). if we now further

assume that the free--surace can be treated as a 1ane, the original problem


equivalent to one of studying the non-uniE low flow about the ship plus its

image about y = O (the double body) translating between two planes at z = ±h.




y ¿ (x)


We assume the length of the body L to be 0(1), and its tranverse and

vir-tl dimensions 0(c). The water depth h is assumed to be such that the

under-keel clearance is also 0(c). Further, if the incident stream varies slowly over


a distance of


we can write --- = 0(1) and = O - for (y,z) near th±

dy oz

body. In this


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 inner-field body

disturbances. Allowing V*, and ct be unknowns at this point makes it

possible to account for an9 interaction between the body and

incident-stream potentials.

If we make use of (4) and that

/x «(/y

, /z) in (2), the boundary

condition for reduces to

c; ' o N



1n y j (5)

Figure 1: Slender Ship in a Non-Uniform Stream


Then the boundary-value

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


(2) 13


(3) z = ±h

4-where 13 is the surface of the double body and n the exterior unit normai

to the fluid. An approximate solution of this three-dimensional problem will now


where is. the two-dimensional normal in the cross-flow 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) +


where = we immediately obtain



y,z B

2 (2)

y,z B



Now, in order to keep quantities involving the body

For 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





(7) - y B (U _U*)n

- YN

, (8) - 1 x y

where S(x) is the


sectional area at the location x. The prime in

(12) is. used to indicate differentiation with respect to the space variable. The


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


ateral added mass of the double-body section, where p is the

density 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:




(9) V2



E X V2


E y y,z


(10) y,z V2



z= ±h (li) E = yN


is 0(c), which is what we assumed. In terms of

, 2

and ', it is obvious

that we can now write

± = (x,0,t) +


+ f1(x,t) +

o B o

[ (u1U*) + (y2 - 2;) + f2(x,t)] +



The 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 two-dimensional in the horizontal plane because of

the shallow-depth assumption. This can be verified by substituting an outer

expansion of the form


) + + (16)

into (1) and applying the condition (3). Hence, the outer solution can be

re-presented by two-dimensional vortex and source distribution as follows:

(x,y,z,t) 0(x,y,t) =+- y(,t) tan -1 ) d +

L/2 C(t)



log ((x_)2 + y2)1'2 d (17)


where y and


are unknown vortex and source strengths in the interval

x = [-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 zeroth-order

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). -.


We now match the inner limit of the outer solution with the outer limit

of the inner solution. First, a two-term outer expansion of the two-term inner

solution can be obtained from (14) and (15):


+ (x,0,t) + {V*(y ± C(x)) + f1] (18)

whereas, according to (17), the two-term inner expansion of tue two-term outer solution for x in [-L/2,L/2] is given by:

L,12 L/2 him ( + (x,Q,t) + [V y ± I




y] o x



where V0 E (x,o,t), which represents the normal velocity of the incident

stream at thx-axis. Clearly, equations (18) and (19) together imply that




(L / 2

VC =



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 combined

to yield to following integro-differential equation for V*:




[v*(x,t)CJtd = V0(x,t)


which is identical to one examined by Yeung7 in ship-interaction problems. We note the interesting special case that if C is 0(c), the cross flow can be

approx-imated, with good accuracy, simply by V (see also Yeung and Hwang'°).

Pro-ceeding to the next order of matching, we obtain the three-term outer expansion

of the three-term inner solution as


+ = (x,0,t) + [V*(y ± C) + f1










+ -(

- -i!) +


which can be compared with the following term inner expansion of the

three-terre outer solution




um (

+ ) = (x,0,t) ± [Vy ±zj (,t)d + -o B o





y + J


+ 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






The unmatched term f7 in (24) and yjy-term in (25) can easily be shown to match

with higher-order teems in the inner expansion. We note that (26) is consistent

with (20) obtained earlier. In comparison with the constant-speed 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 leading-order sinkage force and trimming moment due to the body potential.








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) ±


where V is the solution of


and f1 is given by (28).


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 Oxyz-system, Bernoulli's theorem is given by


= -










2 y z

where C(t) is a given constant that depends only on the incident flow at in-finity. This constant is unimportant for the determination of the longitudinal

and transverse forces, which vanish because the double-body 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

under-standing that this componenet has already been properly accounted for, we shall

omit C(t) in the ensuing calculations. If equation


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) +


where, for terms that are even in y,


+uu -uI

o o



po =



+ 2z


= f

+ (U

- U

)f' 1 1 o



= O

while for terms that are odd in y

p =P



= *(y - U




+ v*(U1 - u0)(l

- 2yly

+ v*u'[_(l -




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 vertical-plane 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 slender-body theory:



- I


p(x) n dS - dxp(x)[ . ± B(x) k]

j 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)[11o + (Uo - 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) y

The evaluation of the integrals associated with P requires the knowledge of

certain hydrodynamic properties of the body sectio. If we define


1 2 2 n X

=ii[(i-) + ] .ds (35)



2y 2z n Z E Z

which can be readily evaluated once the cannonical potential

2 is obtained,

the zeroth-order forces and moment due to the modified cross flow V' is given by

(36) I Q F x y = -p -L/2 L/2


V*2 k




xk zII dx 2B(x)

+ (U(x,t)-U[(U(,t)-U1)S'

X-(34) 2B(x)x


4.2 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



L E ]N


- lz 2z


+ t

f ¿


- [(l_2)

+ -IN ids o




y) (37) Ft(x,t) = O P N ds = _[*_(U1_U )V*' - ° - (U -U )v* i o O x ( N ) +


whereby, one recognizes that

O yNds = -S ,


= 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



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 -


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 the

potential 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



- rvfl. .t.tc - S;



A simple theoretical model based on the slender-body theory in a

shallow-fluid has been developed for the rectilinear motion of a single ship

in a

non-uniform stream. The hydrodynamic forces and moments are

given by Equations

(33)-(36), and (42). Evaluation of these forces is possible if the


characteristics of the sections

A, k, k,

, are known.

These coefficients

are non-time-dependent. 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 single-degree of freedom in

the above

analysis, lateral and rotational motion can be incorporated in the theory

with little additional complications.


Interaction of Ships with a Coastline or Fixed Obstacles


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


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



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

break-down when


small, which is perhaps the case of more practiced interest,

but instead, the theory would likely provide less accurate quantitiative

pre-dictions in such a circumstance.


We can proceed to solve this problem in precisely the same manner as

earlier, viz, using inner and outer expansions. In fact, considerable

simpli-fications 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) =


+ [V*(y

- + U


+ f (x,t)12 + O(E3)


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 two-dimensional in the horizontal plane as before.


and G'

be a source and vortex function, respectively,

satis-fying the no-flux boundary-condition on the obstacles. In other words, we assume

the existence of two functions and such that


= logf(x' T)2 + (y' T)21l/2

+ H(P;Q)


G(P;Q) =:tan



+ H(P;Q)


where H(0) and are functions harmonic in (',n') and are so constructed that


= O , and O



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 leading-order outer solutions can be obtained


(s,t) G(x',y';',n') ds





y(s,t) G(x',y';',n') ds


(t) zy

where ds in an infinitesimal arc-length element along the x-axis; ',fl') are

parametric functions of s as seen in the O'x'y' system.


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


position ¿AY

reamins on the x-axis, since any lateral or longitudinal

movement must be induced by the fluid disturbances generated by the body. Such

movements are of a second-order nature. The source arid vortex strengths in

(47) have to be determined from a matching of


with the inner solution,which

is 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


d +



F =


S y 2 Dt + (v*A) -12--d = 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 left-hand side, which represents the "reflection"

effects of the obstacle due to the forward motion of the ship.


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

auxil-iary 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

dis-cardéd since it is of a higher order.


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

consider-ably 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 c-plane, 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



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 °


n r




w. (z ,z = , l-n ,n n

n ,n

(z ) L z -z' z -z o o


-1 1 l-n I ,n ,n + (z ) I z -z z o o

with 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 easily

using (56) and (57). We note that for n = - , (56) and (57) correspond physically

tó an infinite breakwater, while a right-angle pier corresponds to the case of

n = 4/3.

-For the general case where one has an arbitrary contour, a semi-numerical

representation of G can be obtained via the use of Schwartz-Christoffel



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


The vessel used in the calculations is a 280K DWT ore/oil carrier used

earier by Norrbin13to study bank-effects 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 (bow-in) 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 bow-out 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, 5U t-. - .025 o-o. 0-40 ADTICT1Otl t -0.08 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Figure 3: Sway Force induced by a

wedge-shaped 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 bank-suction force and bow-out

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 two-dimeniona1 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 right-angle 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 bank-suction value, and a strong bow-in 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 hydro-dynamic 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.

-14-25E 20 is 10 5 tow-our -z:: -20 ow-xo -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 wedge-shaped

Obstacle during Transit,


k7TRCrIO -0.08 -0.08 -2.0 -1.5 -1.0 0.04 i ¡ I I I I 20E-2 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 right-angle

Pier 15E 10 5 -io C?s



The 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 ENG-77-17187.

BON-007? 60.1 -d-0.05 d-0.025 6-0. 0125 NOW- IN -1.5 -1.0 -0.5 0.0 UTtL 0.5 d-0.025 0-1.0125 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 right-angle


Figure 5: Effect of Bottom Clearance Figure 6: Effect of Bottom Clearance

on Sway Force for a Transit on Yaw Moment for a

Tran-perpendicular to an Infinite sit perpendicular to an

Breakwater Infinite Breakwater

0.0 0.5 -0.5 uTil_ i_9 - NOV-OU? io -Lf.AVONG APPROACO1ING -10 - NOW-IN -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


0.5 LO i_5



1Dand, I. W., "Hydrodynamic Aspects of Shallow Water Collisions," Journal


RINA, pp. 323-337, November, 1976.

2Norrbin, N. 'i., "Maneuvering in Confined vaters: Interaction Phenomenona due

to Side Banks or Other Ships," Report of the Maneuverability Committee, 14th

ITTC, 1975.

3Tuck, E. O., "Shallow-Water Flows Past Slender Bodies," Journal



Mechanics, Vol. 26, pp. 81-95, September, 1966.

Beck, R. F., Newman, J. N., and Tuck, E. O., "Hydrodynamic Forces for Ships in Dredged Channels," Journal


Ship Research, Vol. 19, No. 3,

pp. 166-171, 1975.

5Beck, R. F., "Forces and Moments on a Ship Moving in a Shallow Channel,"



Ship Research, Vol. 21, No. 2, pp. 107-119, 1977.

6Tuck, E. O., and Newman, J. N., "Hydrodynamic Interactions -Between Ships,"



the 10th Symposium on Naval Hydrodynamics, Office of Naval

Research, pp. 35-70, 1974.

7Yeung, R. W., "On the Interactions of Slender Ships in Shallow Water," to

appear in the Journal


Fluid Mechanics, 1978.

8Sedov, L. I., Two-Dimensional Problems in Hydrodynamics and Aerodynamics, John Wiley and Sons, New York, 1965.

9Newman, J. N., "Lateral Motion of a Slender Body Between Two Parallel Walls,"



Fluid Mechanics, Vol. 39, pp. 97-115, October, 1969.

'Yeung, R. W., and Hwang, W. Y., "Nearfield Hydrodynamic Interactions of Ships

in Shallow Water," Journal


Hydronautics, Vol. 11, No. 4,

pp. 122-135, October, 1977.

Lighthi1l, M. J., "Note on the Swimming of Slender Fish," Journal



Mechanics, Vol. 9, pp. 305-317, 1960.

12Yeung, R. W., and Tan, W. T., "Shallow-Water Interaction of Slender Ships

With Fixed Obstacles," to be submitted to the Journal


Ship Research, 1973.

13Norrbin, N. H., "Bank Effects on a Ship Moving Through a Short Dredged Channel," Proceedings of the 10th Symposium on Naval Hydrodynamics, Office

of Naval Research, pp. 71-88, 1974.

1'Newman, J. N., USwimining of a Slender Fish in a Non-uniform Velocity Field,"



Australian Math. Society, XIX(B), Part 1, pp. 95-111, 1975.


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