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t.4e4r

10 APR. 1976

ARCHIEF

1119317 FLM :3'J [-'t

/1L

/

On the interactions of

slender ships in shallow water

By RONALD W. YEUNG

Department of Ocean Engineering, Mossachu*tT Institute of Technology, Cambridge (Received 6 April 1977 and in revised form 1 August 1977)

Lab. y. Scheepsbouwki4

Technische Hogeschool

Deift

The unsteady hvdrodynamic interaction of two bodies moving in a shallow fluid is exaniined by applying slender-body theory. The bodies are assumed to be in each othei's far field and the free surface is assumed to be rigid. By matchedasymptoties, the inner and outer problems areformulated and a pair of coupled integro-differential equations for determining theunknown cross-flows is derived. The degree of coupling is shown to he related to a bottom-clearance parameter. Expressions aregiven for the unsteady sinkage force. trimming moment, sway forceand yaw moment. Numerical calculations for two weakly coupled cases are presented. One corresponds to the int.rraction of a stationary bodywith a passing one, the other to theinteraction of two

bodies moving in a steady configuration. Theoretical results are compared with

existing experimental data.

1. Introduction

The subject ofhydrodynaniic interaction between bodies moving in close proximity has been of classical interest, for there are practical situations in which interaction fortes and moments play a (lominant rote. Proximity manoeuvres of navalvessels, collision-course encounters ofships, and congested vessel traffic in harbours are a few of such situations. The interactionphenomenon is generally aggravated by the effects of shallow water. The advent of sliI)er-tankers has niade the consideration of these effects imlerative.

A briefreview ofpast analyticaiwork on hydrodynamic interactions between ships will be given. The problem of two sphei'oicls in tandem motion in a deep fluid was examined by 1-lavelock as early as 1949. Exploiting the slender-body assumptions, Newman (1965) presented closed-form results fòr a spheroid moving near a wall. Wang (1975) developed a slender-body model fòr the prediction of mooring forces on a

sta-tionarv ship due to a passing one. The effects of finite depth were also examined

under the assumption that the depth wasof the same order as the ship length. Tuck & Newman (1974) considered a similar approach, but allowed both ships to have non-zero speeds. Collatz (1963) solved the exact potential-flow problem of two elliptical cylinders iii unsteady motion. King (1977) consideredthe sanie problem Lut with the effects of

(j1( lation included. In these last two studies, the assumption of two-dimensionality is equivalent to representing the vessels by airfoils. A related mathematical model used by Dand (1976) appears to give extessivlv iare forces and moments when compared with experimental values. All studies cited are based on inviscid-fiow theory with a rigid fiee-surfuce condition.

The approach used in this paper is based on the theory of matched asymptotics. The three-dimensional probkun of two ships moving in a shallow fluid is first recast into

J. Fluid Mech. (1978), voi. . port . pp. 001

(2)

002 R. W, Yeung

U2

FIcrnaE 1. Co-ordinate systems.

two inner problems and one outer problem. Section 2 describes all essential conditions associated with the outer problem. Section 3 considers theinner problem and its outer Ihn it. The matching process is carried out in §4. The interaction hydrodynamic forces

and moments are next derived for both the vertical and the horizontal plane. The

solutions for two special cases are considered in §6. Some numerical results are pre-sented and discussed in § 7.

2. Problem formulation

Consider two vessels designated as bodies i and moving at speeds U1 and U, in an inviscid fluid of depth h. At the outset, the free surface is assumed to be rigid, which implies that the effects of waves are neglected. This is known to be a plausible assump-tion if the depth Froude number is small, i.e. U/(gh)i = o(e) (j = 1, 2), where g is the acceleration due to gravity and e a small parameter. Therefore the rigid free-surface problem formulated below may be regarded as the infinite-gravity limit of the more general problem where wave effects are important, or as the leading-order problem corresponding to a low-speed perturbation analysis. Such a free-surface condition reduces the problem to the determination of the flow about the two bodies and their images above the free surface, sandwiched between parallel walls of distances 2k apart. Let (j = 1, 2) be two moving co-ordinate systems attached to the vessels

as shown in figure 1. Let Oxyz be a third co-ordinate system fixedin space. If V(x, y, z, 1)

denotes the absolute velocity of the fluid particles due to the motion of the bodies, then

/

S >0)

the following 'exact' boundary-value problem for the velocity potential

formulated:

can be

V2ç(x, y, z, t) = O, (I)

[ç/bn](g) = U1(n)1,

[J/n](j) = U,(n,)2,

(2)

(3)

L. = 0(1), B. = O(e), = O(e) (j = 1,2),

h = O(e), s, = 0(1).

Equations (5) and (6) justify the application of slender-body theory. Assumption (7) permits the use of the so-called outer representation of one ship when the observer is near the other. The shallow-fluid assumption (6) also impliesthat, if the present zero-Froude-number theory were to be applied to situations in which Fh =o(e), the conventional Froude number based on ship length could be o(ei).

The boundary-value problem (1)(3) will now be recast into two inner problems, one for each body, and an outer problem. This procedure is similar to that used by Tuck

(1966). Hence only a brief summary will be given.

Let Y, and Z be the inner variables near the jth body stretched according to

Y = y/e and Z = z1/e. Then, to leading order, the problem for the inner potential

(j = 1, 2) can be easily derived by using (1 )-(3). inner problems:

O for (Y1,Z) near

.,

(8)

= Un(Y, Z)

(j = 1,2), (9)

[/ZL1 = 0.

(10)

Here N represents the unit two-dimensional interior normal to the section contour It. can be seen that the inner potentials satisfy Laplace's equation in the cross-plane, a flux condition on the cross-section contour and a no-flux condition on the walls. The time dependence of 1 arises from the fact that the flow incident upon a particular section of the jth ship is a function of the stagger s, which changes with time. This inner problem is illustrated in figure 2, wherein it is noteworthy that the cross-flow

V*(x, t) is unknown and its order of magnitude will depend on the blockage charac-teristics of the cross-section. This point will be addressed in a later section.

For the outer problem, x and y are both 0(1), however z is O(e). If is written in terms of an expansion of the form

=

where it is assumed that = o(()) for all n, then (1) yields = o, > = o, ). = _V,(1).(2>.

ç( and

(2>, however, cannot depend on z because of (3). If (3) is next applied to and

>, the governing equations for and can be obtained:

y, t) = O (i = 1,2). (11)

Interactions of slendir ships in shallow water 003

where (n) represents the xc'omponent of the interior normal tothe body surface The relation between the iwo moving co-ordinate systems is straightforward. viz.

XI = X9+S(t), Yi = (4)

where s is the stagger and .9, the separation, both measured with respect to the origin O. The following assumptions of slenderness are now introduced to simplify the probiem:

(4)

-V,. ix, I) ¿ c, =

Z= h

q)+ (1 =0 Y1 xi

FIGURE 2. The flow near the hull in a cross-flow plane.

Therefore the first two terms of the outer expansion satisfy Laplace's equation in the

horizontal plane. The body boundary condition (2) is not applicable to the outer

problem since to an observer 0(L) away from the bodies they would appear to have collapsed onto a line.

Only the solution of the lowest-order problem will be sought in this paper. With that as the understanding, the superscript i will henceforth be omitted. The two-dimen-sionalitv of the outer flow suggests the following representation of in terms of line-source and line-vortex distributions:

"

{

L

t) log[((x

-

+(y

(

-1)i)I1d

1 ¡'('1 (t) 1

-

1

±-J

y(,t)tan'f

- (12)

i

where the longitudinal axes of the two ships are assumed to be at y = ± and [al, aÏ] denotes the instantaneous location of the jth ship. In (12), the branch cut of the aretangent function should be chosen downstream of the translating vortices. The unknown source and vortex strengths m and y cannot be determined from the outer

problem alone. However, by matching the inner and outer solutions properly the

necessary relations can be obtained.

It is worthwhile to note that the unsteadiness of the problem gives rise to a vortex distribution in the wake of the bodies. Iii aniuch as the outer flow is two-dimensional, xisting analysis and conditions concerning the unsteady motion of a two-dimensional airfoil (see Garrick 1957) are applicable here. In particular, the linearized pressure-continuity condition across the two-dimensional wake is given by

pp

= O,i)(x,y = O,t)]/i = O

for x < a(t). (13)

On the other hand, by definition,

y(x,t) =

= 0+, t)/xb(x, y1 = Ot)/x.

(14)

Equation (13) thus implies

y(x,t)/t = O

or y(x,t) = y(x) for x < al(i). (15)

ç

00 R. W. Ycung

z,

(5)

Interactions of slender ships in shallow wate? 005 Whence any vorticitv shed in the waké remains constant in time and depends only on the wake co-ordinate in the inertial frame of reference. The rate at which vorticity is being shed by each body an be derived from Kelvin's theorem, which can be inter-preted as follows. The circulation due to any material contour that encloses each body individually should be zero at any time since the initial circulation when the bodies were far apart was identically zero. Thus any gain in the bound circulation P of the body must be compensated by shedding of vorticity of opposite sign. Whence, accord-ing to von Kármán & Sears (1938), it is possible to show that

dF1/dt = _Uy(a1(t)). (16)

Finally, to ensure uniqueness of the solution, we note the important subsidiary trailing-edge condition, which requires that the flow at the trailing trailing-edge be smooth. rllLjS Kutta

condition can be stated as

hm y1(x, t) =

um y(x, t)

for all L. (17)

- O + O

Numerical procedures for solving two-dimensional outer problems involving multiple bodies have been presented by Giesing (1968) and recently by King (1977).

3. The inner problems

It is convenient to decompose the inner problem defined by (8)( 10) into two

component potentials, one associated with the forward motion andthe other with the ja.teral flow, as follows:

cl(Y;, Z., Xj, t) = + V'(x, t) 2) +f1(x1, t)

with

11/aN

= U,n(Y,Z1) on VB

= O on

In addition, both potentials should satisfy (10). It is clear that the inhomogeneous boundary condition is satisfied by I' and that l)2) corresponds to the prohieni of unit lateral flow about a cylinder located between walls. The non-uniqueness of the inner problem asserts itself in the form of the unknown functions V and f(x, t).

By examining (9), one notes that is O(e2) with respect to outer variables. The magnitude of (l can be estimated from the behaviour of the added mass ofthe cylinder. The following asymptotic formula for a rectangular cross-section with a small bottom clearance was given by Flagg & Newman (1972):

A1(x) = 4h2

+(1_log41)_(1

_')+O()]

= O(e2/81), (21)

where A1(x) is the 'double-body' added-mass coefficint and = (h - 1)/h. Hence 2)is O( V*6t8j1). In arriving at the.3econd equality in (21). it was necessary to invoke (5) and (6). As far as completing the matching process is concerned, it is not necessary

(6)

008 R. W. Yeung

Z)

-t

FIGURE 3. Kutta condition at an abrupt trailing edge.

required. In some intermediate region where }5 = O(L), (18) can easily be shown to be

hm D5(Y,Zx,t)

)

= ( -

LS(x)/4h)

I + V[}, ± C(x1)1 +f(x, ) for a x a

O(e2) 0(f1) (22)

with the order off1 to be determined from matching. In (22), S is the area enclosed

by f and Ç is the blockage constant used frequently to characterize a lateral flow

about cascades (Sedov 1965):

-

G(x) = (A(x) + S(x))/4h. (23)

Next, consider x lying in the wake region of the jth body. Clearly, the term l» in (18) should be omitted. If the body is assumed to terminate in the form of a thin and abrupt trailing edge (figure 3) a vortex sheet will be shed downstream of this edge. By Green's theorem, the perturbation potent.ial associated with the lateral flow potential

't (j = 1, 2) can be written as 2irç(Y,Z;x)

=f

for = (24) 2ç( Y, Z; x) =

Z)ç3(0, Z)1

dZ --Ods

for x

(25)

In (25), d.s is an infinitesimal arc-length element and G is a Green function satisfying the wall conditions. In the spirit of slender-body theory, Newman & Wir (1972) have proposed a Kutta condition of a weak type to be imposed at such a trailing edge. This condition is congruent to requiring the potential be locally continuous at the juncture and .Thus it follows that the first ternis on the right-hand sides of (24) and (25) are identical. From geometrical consicleratins, the second term of both equations vanishes. Hence one arrives at the following outer behaviour of the inner potential in the wake:

(7)

/

Interaction8 of slender ships in skailou; water 007

where (is the retarded time defined by î t + (.r -afl/Uj. The retarded time arises as a result of the linearized dynamic boundary condition on the vortexsheet:

'/e y/ax

= 0, (27)

where '/at indicates differentiation with respect to time in the moving frames. It is of interest to note that the presence of the vortex sheet offers an apparent added-mass effect even though there is no physical obstruction in the fluid.

4. Matching

In order to match the outer solution with the inner solution described by (22) and (26), it will be necessary to obtain an inner expansion of (12) near each body. For

clarity of exposition, consider first the case when y1 (= y is») is small. In the

co-ordinate system of body 1,

(x1,

y,

t) = (x, 0, t) + (x1, 0,t)/ôy y + O(y)

m(x1,t)

d1y1+O(y)

=ç'(x1,O,t)-FE1(x1,t)y1±

Im+2

for aj x a, (28) where J' is the normal velocity inducedby body 2 on the axis of body 1,

1 Ca,

2+2J

x2_2 V(x1,t) = m2(0,t)

(x2-42+s

-

Y2(2 t)(x0 - 2)2+ d2, (29) sp and i a

ç(x1,0,t)

= ni1(1,t)

log

jx-+

52

Y2(2'

t) tan1 (:2) d2 ±

Ç'

t) d1. (30)

The term m1(x1,t) in (28) is inderstood to be zero when x1 <aj.

A straightforward comparison of terms of (28) with those of a similar nature in (22) and (26) yields the following four relations:

rn1(x1,t) = U1S(x1)/2h, (31) i

y1(i,t)d

(32) V(x1,t) = V2i(xjt)+_$

vc1 =

I'»»

(33) 2 XL

f' t)

=

's;(1) log

x1

d +

$2

fli2(2, t) log [(x, g2)2+

+

52

Y2(2't)tan'

(X:_2) d2.

(34)

Note that differentiation of (33) with respect to x1 yields

= 2[V(x1, t)C1(x1)i/x1

for aj ,r1 (35)

(8)

008 R. W. Yeung

A completely analogous procedure can also be carried out for body 2. The resulting equations are similar and will not be repeatedhere. In both cases, it is noteworthy that the source strength matches as in the caseof a single body considered by Tuck (I 966) whereas the vortex strength cannot. be determined until (32) is solved. The physical interpretation of (32) is as follows: the cross-flow that body I sees is that generated by the adjacent body pitis the normal velocity induced by its own vortex distribution. Tim axial flow f(x, t) aIon the body, which will determine its sinkage and trim, is generated not only by the source distributionof the body itself, but also by the singu-larit.y clistiibut ion of the adjacent body.

The following coupled integro-differential equations for V

and V can now be

derived by making use of (35), (36) and the analogous equations for body 2:

'U1 f ¡7* ('f \' /l'

i (V' C2)'(x,- e.,) (lÇ

where the primes (lenOte differentiation with respect to the space variable. The order of each term is given below the equations.

One observes that (37) and (:38) resemble a pair of coupled Prandtl lifting-line equations with V*C playing the role of bound vorticity on a large-aspect-ratio wing. Notefurther that, although these equations have to be solved for each value oft, it is necessary to determine the value ofV* in only [aj, afl and [ai, at'], since the quantity

(J'*C)' in the wake is related to Vi at the trailing edge by (36).

For two bodies of the same slenderness ratio, the degree to whichthey interact depends on the magnitude of ("i and C9. Examination of (37) and (38) shows that the

possible sit nations, may be divided into the following three eases.

Case i: = 0(1), &2 = 0(1).

This corresponds to the physical situation where the bottom clearances of both bodies are the same order as their drafts. Thus to the leading-order approximation, one notices from (.i) and (.38) that the eross-flowçdue.' to a consistent analysis is slmply that generated by the source dLstrilwtion of the adjacent body an(l that such a cross-flow is O(e).

Case 2: = O(e), 2 = 0(1).

The first equation is weakly coupled to the second. The third term of (37) can be omitted when solving foi' tile flow about body 1. V, which must be obtained by

solving (37), is seen to be O(e). On the other hand V'. which is also O(e), can be deter-mine(1 in closed form from the right-hand side of (38) and the solution fr V. These arguments, of course, apply also to the converse case 8 = O(I), 2 = O(e).

Case 3: ò1 = O(e), & = O(e).

Vr(x1, t) + - '

'

i+ -U, f O( Vr) O( Jrc81-') 1TJ_

X1-1

IT J_

(,2_2)2+8,

O(Vre&') J,,

(f _)242'

O(e) (31)

f0. ( V (9)'d f fU (Vr C1)' (xi

d1

_U1 f"i S1)s,)d1

2(a2, )+_T , 'J

i.

e \2 2

-- '2 '2

-

.i1,i) -i-8g

47Th J,,- (x1-1)9+.s'

(9)

Inleractions of slendei hips in shallow waler 009 Ño sinìplilkations are possible For this ease the coupled equations have to be solved sinìuItantously. An altcrnatie fornì of (37) and (38) which may be more amenable tu n Lirnerical solution can be obtained by using the vorticitystrength y as the unknown functions instead of the Vi':

'

-

f

-

' î

x(2) (x2_2)dc

C1(.r1)J ¡1 1' ) '1 n-J_ J (.v0E2)2+s 2 L

1'

S(9)s

- 2'rh J i ça

± f Y)

d j

-

dE Y2

2' )(2

7T J -29

j_

(x1-1)2+s

_Ir t'a

ÇUj \

' '

1'1' P

- 2mh Ja (x1-1)2+s

"

which are two coupled Volterra equations of the first kind.

Finally, it seems worthwhile to show that the inner and outer Kutta conditions

described in §2 and 3 are indeed consistent. From (16), (17) and (35),

)

yj(aj+0t)=__f(VGj)dxj

for C(afl=O, (41)

while from the inner-field condition (27) and t.he matching condition (36)

y(x»l) = _2C(afl /- V(a,Î(x))

2C(afl

V(a7,t)

t=t for

x<a1,

(42)

whiTch is clearly identical to (41) in the limit x1 = a - 0. Note further that, if the flow is steady or if the trailing edge is not fin-like y vanishes at this edge.

5. The interaction hydrodynamic force and moment

Of primary interest in the physical problem being studied are the lateral force and

moment on each body. There exists also a sinkage force and a trimming moment

acting on the wetted' half of the double body. The desired hydrodynamic force or moment can be conveniently obtained from the inner field once V' is known from solving (37) and (38). If one makes use of the inner potential (18) and the unsteady Bernoulli equation, the following expression for the fluid pressure p( Y, Z; x, t) acting on, say, body I can be derived:

the symbol K> denoting the quantity <'> + V (2)). The time differentiation is

understood to be with respect to the moving co-ordinate system. In these expressions,

p(Y, Z1 X, t) = ep1(x1, t) + 2p9(1, t) + e2P(Y1, Z1; x1, t) + 0fr3), (43)

where

p1(x1,t)/p = 'f1(x1,t)/t+Uf,

(44)

p2(x1,t)/p = (U1f)2,

= (45) (46) P0(Y1,Z1;x1,t)/p )2]

(10)

/

010 R. W. Yeung

it is worthwhile to keep in mind that = O(e) and 2) could be of an order smaller than e if the bottom clearance were not small. Note that, because of(:34), (31).aiid (35), f1(x1, t) is O(e). To the leading order, it is evident that the sinkage force - and the

trimming moment _' (about 01) are due to p1:

(s-) =

Jdxip1(x1t)

()

Nd.s =

f

°' dx1p1(x1,t)B1(x1)(1) = O(e2), (47)

where t.he contour integral in the cross-flow plane is carried out only for z < o. The expression for p, can be simplified by using (15) and by recognizing that the time dependence of the log and aIctangent function in (34) occurs via the variable x2. The

result is U2 a

s' r

U2 a s'

' )

p1(x1,t)/p =

4lTlJaXi

-

Sa (x

d2

+f [() tan-' L-2)

U2()o]d2

j \ + U2y2(a, t) tan-' (x2 f2))

where one may recognize that the first term is the usual 'sinkage pressure' that body 1 experiences while moving alone in shallow water (Tuck 1966). The second terni corre-sponds to the axial flow generated by the forward motion of body 2 and the last two terms represent the steady and unsteady effects of the adjacent vortex distribution.

If

8 is 0(1), one obtains the following simple expression for ,and .A:

j'o

U çai

j i

I

I - I - J - - I

X1 i i 5, I9J , .

\',/

47ThJa

- (x1s-52)-+s

where and .J/ are the single-body sinkage force and trimming moment

respec-tively. The additional term on the right can be used to determine the transient heave and pitch motion of ship i caused by tue motion of ship 2.

The leading-oider lateral force ', comes from J-, defined by (46). In differential form,

dYi(xi)f

F2(Y,,Z,; x,,t)Nda, (50)

A

which can actually be expressed in terms of the cross-sectional area and the added-mass characteristics of the cross-section. This was in fact carried out using momentum analysis by Newman (1975) in connexion with the swimming of a slender fish in a deep

fluid. Since the extension of his analysis to the case of finite depth is sufficiently

straight.fòrward, it does not warrant another derivation here. In the present notation, the results, after correcting for the double-body effect, can be written as

dc&1 +

= 2hC,(x,) Vr-2hU,---(VCl)A+O(e3), (51)

where represents the sway force on the 'submerged' portion of the body. Integrating (51) along the length of the body, one gets

(J ra

= 2h_J ' Vr(.r1t)C1(x1)dx1+_j

S',(x,) V,(x1)dx1+21iU1C,(aj) V'(a1,t). a1

(11)

/

Infrractions of slender ships in shallow water 011

In arriving at (52) it was assumed that body i was pointed at the bow, hence C1(afl = O. The yaw moment ..4 about O can be obtained in an analogous manner by noting that

d.ÁÇ = r1 d&'1. The 1mal expression is

=

2hk$' VrG1xjdx1+2hUJ' v*(c1+)dx1+2hUVr(ai-t)ci(aiai.

(53)

Equations (47), (52) and (53) permit t.he rapid evaluation of the instantaneous force and moment on the individual body once V' is known. Theexpediency lies in the fact that only the overall sectional characteristics of the body are needed, not a detailed knowledge of the potentials 1) and (D2>.

6. Approximate solutions for weakly coupled cases

The complete solution of (4.9) and (4.10) for bodies of arbitrary shape requires substantial numerical effort. Before embarking on such a major task it seems worth-while to test the practical usefulness of the theory for a few simple cases. Towards this end, numerical solutions have been obtained for a few situations that reflect weak coupling.

First, for case lin §4, one notes that the sway force and yaw moment can be written in closed form if and & are both 0(1). By (37) and (38), or as expected intuitively,

V'(x1,t) = V(x2),

V(x2,t) = V(x1),

(54)

where V and V° are the normal velocities induced by the source distribution, i.e. the right-hand side of (37) and (38). Next, one observes from (4) that

J7(n)

- u u ______

r

21 (r2)

-

21

- (

1 2) . . ( )

U 2 U v%1

Thus it follows from (52) and (53) that

U U

ja

=

' A(x)

(xit)dxi+-J

' S Vdx1+UA1(aj) V»(aj,t).

-

at a (56)

if

a

«(U2 - )A + LSfl .r + 4hG1(x1)} V(x1,t)dx1 +

A1(a)T1(aj,t)aj.

-

U, (57)

These are the shallow-water analogues of the equations given by Tuck & Newman (1974, §2). With the exception of the functional representation of V'f> and the values of the added-mass coefficients, the deep- and shallow-water cases are identical.

A less trivial situation would correspond to case 2 discussed in §4, in which one needs to solve tile following Prancltl lifting-line equation to obtain V(x, t):

(J7*c 't'

v'

j 1 1 1' 17(m), .

i \ '1 1 J

bi -

21. 1'

71J i51

The complexity introduced by the wake can be avoided if the following two cases are

considered: (a) body i stationary in space, which corresponds to the situation of

(12)

012 R. W. Veung

which corresponds to two vessels moving in a refuelling configuration. The integro-differential equation for both cases is

i ta (j7*fl \'

T7*I..i ;J

-

J ' ÌC - 17(m) ¡,

J

i -

21 ' 1' I'

Jaj a1-1

with the understanding that there is actuaHy no dependence on time for (b). In non-dimensional form, this can be written as

i rl (V*(:_f

V'(1,t)±- 4

"

.' d

= (59)

lIj_.j

1-where = {2x1 -(at -afl}/L1 and ?7 = 2C1/L1. Applying the Cauchy inversion

technique (luskhelishvili 1958, p. 000), one obtains

'

f' ('(''1)d?

60

i )

- iT(l-J_,

'

where i is an unknown constant which is related to the bound circulation on body 1:

r1 = f1 yj(x1,t)dx, = V(l),(!)- V(- 1)( 1).

(61)

The second term of (60) represents the homogeneous solution of (59) and must be determined by the Kutta condition.

Case (a): body i stationary

If the body ends are pointed, (61) implies that [' = 0. If the ends are square, it seems plausible to assume that the flow at the ends will be diverted by the blockage, hence V*( ± 1,t) = 0. Again, F1 = 0. Thus, in either situation, the homogeneous solution can be discarded. Integrating (60) from s to the 'leading edge' one obtains the following Fredhoim integral equation of the second kind:

-

-

ir' ,_-

-

ir'

-V'(x1, t) L1(x1) =

-

A(x,, ,) V*(1) c1

- -

I t) A (x,, ) di,, (62) 7Tj_

lTj_j

dx'

i

1

1_xg_(i_,2)t(l_x2)i

where

K(.r,)

=

f (cX')(l _'2) =

log[

t _+ (1_2) (t

(63)

For a general t), (62) has t.o be solved numerically. Once V has been deter-mined, (52) and (5) can be used to obtain the sway force and yaw moment.

Case (b): steady motion of two bodies

By (33), the appropriate end condition is (V*c1)' = O at the trailing edge = - 1.

From (60), this implies

ri

r',

= -2J

.-1

+i

[V'(,)-V,,(,)]d1.

Substituting the result into (60) and integrating with respect to r, one obtains Vfl,,) C1(x,) =

J' Ñ(,,

) [(V1)

(13)

-/

Interactions of slender ships in shallow u'ater 013

Ci ¡ .' fi / .'\.

with A(x,E) = . (I+'.,

-= K(x,) ('

)- (iT_sinx),

where to be consistent with (52), the bow lias been assumed tobe pointed, i.e. Ö1(1) = 0. It can be seen that (63) differs from (62) only by anadditional term in the definition of the kernel function. The lateral force and yaw moment can be easily obtained from (52) and (53) wit Ii the titne-11t'rivative terms discarded.

7. Numerical results and discussion

The solutions of (62) and (65) are obtained using the method of discretization, by which the original equation can be written as a system of linear equations:

(66)

2' , .v) denoting the set of grid points along the x axis and being the midpoint between two successive grid points. Most of the integrals of (68) can be evaluated analytically.

Figures 4(a) and (b) show the computed lateral force and yaw moment acting on a moored 10OK,D\VT tanker owing to the passage of a 3OK1DWT tanker. rfhe results are plotted v.s. the relative positionof the vessels, which als represents the time axis. 'rhe solid lines are speed-averaged experimental values takenfrom Remery (1975). Two sets of theoretical results are given. The dashed lines correspond to the simple formulae (.56) and (57), which do not account for the effects of blockage. Clearlythe theoretical predictions are too high. The dotted lines are obtained by solving (62) and using (52) and (53). The peak force and moment are reduced substantially but un-fortunately fall below the speed-averaged experimental values. Both sets of computa-tions use the exact values of G1(x), which are obtained by using a method discussed in Yeung & Hwang (1977), for the lateral-flow problem. These values are depicted in

an inset in figure 4(b).

The force and moment acting on a tug boat moving along the side of a cargo vessel

in shallow water are slìowii in ligures 5 (a) and (b). The experimental results aredue to Dand (1975). The theoretical predictions are obtained by solving (65) for the larger ship and determining the subsequent cross-flow incident upon the smaller ship from (38). The bow of the tug boat is assumed to be pointed. It is worthwhile to note that while the general behaviour of the experimental curves is predicted fairly well, the peak force and moment are substantially underestimated, particularly when the tug boat is at the stern region of the cargo vessel. This underestimation does not appeartobe due to the neglect of viscosity. It was found that using the computed cross-flow and a drag coefficient of 20 gives an increase in the predicted value less than 10% of the experi-mental value. One may question, of course, the validity of applying an outer theory in

-

1'

iN

Vj"(1, t) C1(1)

-

K

t) = - -

K t) (i = 1, 2, N), (67)

1Tj1 where the influence coefficients K1 are given by

K1 = or

= J"

(14)

014 I

2

10-05 0

OES l0 XI

/

R. W. Yevng (() .\ttration

075 \

051) (a) If... 025-Row- n 05 I

/

Bow-out - I,

FauaE 4. (a) The sway force and (b) the yaw moment. acting on a stationary tanker owing to the passage of another. The force coefficient C, is defined by í?//ipU B1 T1 and the moment coefficient CN by /pUBT1. The geometric parameters are LSLI = 0712, s/L1 = 239. hIT1 = 115 üd hiT2 = 172. Experimental data are speed-av+raaed values with F, ranging from 0155 to

O270. ---,theory, V = l''; , theory, V from (62); experiment, Reinery (1974). s,iL ¡5 I \ I \ /

'

10 ¡ (b) /

025

050

075

- IO))

--

Repulsion / V .vI (J s,,. L.

(15)

O-5 O-4 03 + 0-4 o o + - --Ii 2 --0-5 04 03

01

Interactions of sletder ship. in shallow water

L Attraction

-

I I

_I1-4 __O.I 41 ni i1 û-i (I-i 0-4'. 117 0-5

oli

2

+ 0! x0L2

02

0-S RptIsion

0(1 1

01

1

).-,

I I 1 I

f 02 03 04 OE50-6 + 0-7 O O-') l-O

II

-2

4. 0-2 4-+ 0-3 - 0-4 Bow-out

FLGURE 5. a) The sway force and (b) tilO y&.w moment acting on a tug moving alongside a cargo

ship. Expurimental data are for h= O-345 with s5/L1 = 0128 and h/T2 = I-38. C is defined

by I/-pU D1 T1 and CN by .,V1/pU B T1. -, theory; +, experiment, tug without screw,

Dand (19'5); C. experiment, tug with screw. Darai (1975).

such close proximity. The ratio of the lateral clearance between the two bodies t-o the length of ship 2 is o-023. It appears that the main source of error lies in the representa-tioii of the body in the outer problem by only first-order source distributio9. These

preliminary comparisons, however, show that- the theory discussed portrays the qualitative features of shiptoship interaction quite well even when the separation

between ships is not 0(1). The case of two ships in steady motion with a separation O(e) was considered by Yeung & Hwang (1977).

+ + o . + o Xo1. + + o . (a) 1-+ o 02

fo

° 03 0-6 O-5 04 O3 0-2 - BOW-In (h)

(16)

1-6

I-2

--O-8

I-8

1-4 1-0 0-0 U2

-C 1> U1

i j - IS - 1-4 - I-O 0-6-..

l-6

l-2

O-8 0-4 0-4 0-5 I -I) 0-3 ': L) -O-2 Re pulsion Bow-in 0-6 0-8 1-0 1-2 l-4 I-6 1-8

\-4 (a) (h)

-c4 O2

0-6 1-8 0 0-2 (J-4 - O-S I-O 1-2 l-4 s/L

I-0-3

- Bow-out

i - FIGURE 6. (a) The sway force and (b) the yaw moment acting on two identical vessels in head-on

E .- enetunter with/i/T = 111 sind s/L = O-5. C,

and C are non-ditnensionalized by pBTUjLT2

i and pBTL U1 IT21 respectively. . U2/U1 = -10;

. 1T2/Uj - 1.5, slowership; ---,

- - - -- - L72/L71 = - 15, faster ship.

Finally, the theory presented is applied t-o geometric configurations that are more pertinent to the stated assumptions. Two identical vessels of tanker proportions are assumecl to have parabolic a sectional area with rectangular cross-sections. Both vessels are further assumed to have pointed bows and fin-like sterns. The

length-to-beam ratio is 6667 and the length-to-beam-to-draught ratio 25. The bottom clearance is

assumed to be 10% of the vessel draught. The unsteady interaction force and moment are calculated for a ratio of separation to ship length of 05, assuming theflow to be

nib1ocked [(56) and (57)11.

Figure 6 shows the instantaneous force and moment for two tankers approaching each other. Results for two different speedratios are given. By moving along the s/L

axis from right to left, one may visualize these curves as the time history of the

interaction force and moment. One observes that during the approach each vessel experiences initially a repulsive force and a bow-out moment. Just as the lateral force becomes attractive the yaw moment becomes bow-in. This corresponds to a highly dangerous situation as far as collision is concerned. After the midships have erosed, the yaw moment changes t-o bow-out again, but the attractive force remains for some time, which may cause the st-eins ofthe vessels to collide. The several reversals of the sign of the yaw moment are of great concern tothe helmsman. Another feature observable from these curves is that the slower ship experiences a larger force and moment- than the faster one. This fact is in apparent agreement with what one may observe in the operation of a vehicle on a highway.

016 R. W. Yeung

2-5

L)-.-.2-0

(17)

/

f,

Inicraciion,s of .s1en'i.r ship.s in s1allow water 017

The author gratefully acknowledges t.he support of the National Science Founda-tion, under Grants ENG 75-70308 and GK43880X. Stimulating discussions offered

by Professor J. N. Newman and Professor E. O. Tuck during the course of this work are much appreciated.

REFERENCES

ASHLEY, H. & LANDAHL, M. 1965 Aerodynamics qf Wings and Bodies. Addison-Wesley. BECK, R. F. 1976 Forces and inouents on aship mcving in a canal. Univ. Michigan, Dept. Yaval

Archit. Mar. Engnj Rep. no. 179.

COLLATZ, G. 1 963 Potent ial theoretischeUntersuchung der HydrodynamischenWechselwirkung

zwier ScliifTskörper. Jahrbuch der Schiffbautechnischen Gesellschaft, no. 57. DANn, I. 1975 Some aspects of tug-ship interaction. 4th mt. Tug Cony, paper 5.

DANn. I. 1976 Hydrodynarnic aspects of shallow water collisions. Roy. inst. Naval. Archit. Spring Meetings, London, paper 6.

FLAGG, C. N. & NEWMAN, .J. N. 1971 Sway added-mass coefficients for rectangular profiles in shallow water. J. Ship Res. 15, 257-265.

GRIcK, J. E. 1957 Nonsteadywing characteristics. In Aerodtnarnic Components of Aircraft

at High Speeds (ed. A. F.Donovan & H. R. Lawrence), §F. Princeton Ser. High Speed Aero-dyn. ,Jet Propulsion, vol. 7. Princeton University Press.

GIESXNG, J. P. 1968 Nonlinear interaction of two lifting bodies in arbitrary unsteady motion. J. Basic Engng 90, 387-394.

HAVELOCK, T. 1949 Interaction between ships discussion on paper by A. M. Robb. Proc.

Roy. Inst. Naval Archit. 00, 336-339.

KARMAN, T. VON & SEARS, W. R. 1938 Airfoil theory for the non-uniform motion. J. Aero Sci.

5, 379-390.

KING, G. W. 1977 Unsteady hydrodynamic interactions between ships. J. Ship Res. (inPress).

MUSKHELISHVILI, N. I. 1958 Singular integral Equations.Wolters-Noordhoff.

NEWMAN, J. N. 1969 Lateral motion of a slender body between two parallel walls. J. Fluid Mech. 39, 97-115.

NEWMAN, J. N. 1975 Swimming of a slender fish in a non-uniform velocity field. J. Austr. Math. Soc. B 19, 95-111.

REMERY, G. F. M. 1974 Mooring forces induced by passing ships. Proc. Offshore Tech.Conf. pp. 349-303.

Tucx, E. 0. 1966 Shallow-water flows past slender bodies. J. Fluid Mech.26, 81-95.

TtJcK, E. O. & NEWMAN, J. N. 1974 Hydrodytiarnic interaction between ships. 10th Symp. Naval Hydrodyn. Cambridge, Mass.

WANG, S. 1975 Forces and moment on a moored vessel due to a passing ship. J. Waterways, Harbors, Coastal Engng Div. Proc. A.S.C.E.101 (WW3), 247-258.

YHUNG, R. W. & HWANG, W. Y. 1977 Nearfield Imydrodynamic interactions of ships in shallow water. J. Hydronaut. (in Press).

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