Nearfield hydrodynamic interactions of ships in shallow water

1.

Introduction

rHIS paper addresses the problem of hydrodynarnic

in-it teractions of ships in shallow water. Thus, in-it falls into

a category of problems brought forth by the advent of large-size

vessels, which makes the consideration of restricted-water

effects important. Such a category also is characterized by low

operation speeds, particularly in harbor or in channels,

whereby the leading-order hydrodynamic forces are those

associated with

fluid

inertia instead of th

free-surface or

viscosity. Therefore, it is hoped that the present theory, which

exploits the rigid-free-:urface condition, gives sufficiently

accurate predictions that could be used for control system

design for proximity operations.

In addition to the usual slender-body assumption, it

is

assumed that the lateral separation between the vessels

is

small. Further, the water depth is taken to be the same order

as the beams. Since only the steady-state problem is

con-sidered. this theory may be regarded as complementary to one

presented by Tuck and Newman,' which requires that the

separation distance and water depth are of the order of

a ship

length. Hence, n contrast to Tuck and Newman and Wang,

the present theory, heitig a riearlield one, requires a detailed

knowledge

of the hull geometry, and accounts for

it

ac-cordingly. The slenderness assumption allows

the

con-struCtion of the solution by solving a sequence of

two-dimensional problems.

The stated problem can, of course, be solved by a

three-dimensional singularity-distribution method (cf., Nowacki

This was, in fact, carried out in part by Norrhin,4 who found

that

the computation costs were prohibitively expensive.

Other theories related to shallow-ater interaction have been

given by Collat, s Tuck and Newman (Sec. 4), and Dand,

a!! utilizing two-dimensional theory in the horizontal plane.

From these results, it appears that

good quantitative

predictions tor realistic forms would be possible only if the

presence of the under-keel clearance is accounted for.

2.

Problem

Formulation

Consider two vessels moing at the same speed U in water

of depth h, as shown in Fig. I

- Since the flow is steady, it is

convenient to establish a common reference

system midway

between the centerplanes of the two ships. At the outset we

assume that the free surlace can be approximated by

a plane and the flow is potential. The disturbance velocity potential

kcceivcd Nov. 3. I 97(': revision received May 9. 1977.

l,ids alicsor,cs: l-fsdrodynamics; Marine Hydrody,ianics

_VecI

und Co'i i rot S u r

Ass,siant Protvssor, I)epirrrneni of(kean Ensineering. tGradu,te S, ocien,, Dcpar,_{nen, of Ovean Engir,eering}

-Lab.

_y.

_{Scheepsbouwkun.k}

J. HYDRONAI

Ii

-

- - i

VOLI !, NO.4

Nearfied Hydrodynamic Interactions of Ships

in Shal]ow VaIer

Ronald W. Yeung and Wei-Yuan Hwangt

Massachusetts Institute of Technology, cambridge, Mass.

The hydrodsainic interactiOnS Of tssO vessels mosing al the samespeedin neartieldisconsideredh apptsing the slender-body theory.li is shossnthat. for a water depth thai ¡s the same order as the beam of the vessel, ih problem reduces to a sequence nf inner problems in the cross-flo plane. This reduction to strip-theor all,oss one Io obtain the solution ssiihoui tac necessity of sotsing an otiter prohlrn. Applications '-serv nade to P.sa pairs of ship models. iheuretical predictions generally are high as compared siih available &'speriineiital measuremenls. hut offer a fairly satisfactory qualitative description of the interaction p'neflomwnon ss hen the length of the oertap of the vessels is large as compared ssith the separatioj

UOR EOEN GEBFIIJiK E

STUDIE DOELE!NDE

r

D

then satisfies the following ''exact'' conditions:

V 'çi(x,y,z) =0

ao

3et /i,

=U(nj,

--

¿in B,

=U(n ),

an

k-where B, and Th represent the underwater hull surfaces arid

their reflections about

=0. Suhscriped Cartesian variables

will be used to designate the vessel under consideration. The

problem as defined by Eqs. (t-3) represents the

three-dimensional flow about two bodies located between parallel

plates. Although this could be solved numerically by a

stir-face-singularity-distribution

method,'

we USC

iistead

matched asymptotics to obtain the solution of the problem.

Let e be a small parameter representing the body lateral

dimensions to the body length. The hull function of the jib

body therefore can be described by r, (x,O) eR, (x,O), where

R, is O(I). The lateral separation between bodies d, and the

svater depth Ji are assumed to be Ou). Furthermore, the ratio

of the under-keel clearance to the body radiu.s is assumed to he O(I). An outer expansion now can be written as follows:

ct,=e.si1' (.v,v) +e'»2' (.v,s') +...

₍₄₎

where, as indicated by Tuck,S the shallowness assumption

lciids to the fact that & i = 1,2, satisfies Laplace's equation

in the horizontal plane

/ ä2

a

--r +

--)o"

(x,y)=O

i=1,2

(5)

a.-

a-By introducing inner variables, Y=y/. Z= a/e,

etc., one

can obtain an inner expansion as follows:

(x;Y,Z) =ef, (r) +e½2' (x;Y.Z) +e

_{(e,Y,Z) -t-... (6)}

where the second-order potential sat is fies

+)2(V.Y7)_0

-(7) a /1

=U--R (-v,O)j(1±l

s ¿ix I

3=1,2

/ (R )

_----

\2

)

\

R,

/

U(t'

"

()

1980 I 2S

ARCHIEF

OCTOBER 1977

Fig. I Coordinate systems and notations.

INTERACTIONS OF SHIPS IN SHALLOW WATER 129

y

'a2

a2

3 2 + 2

)3

(X;

Y,Z) = f (.)

(IO)

(Il)

(12)

where the prime denotes differentiation with respect to x. In

the preceding equations, and in the analysis to follow, all

capitalized quantities are by assumption O(I) in the inner field.

It is worthwhile to note that neither the outer nor the inner

problems, as defined, are complete by themselves; the former lack the body boundary condition, and the latter a condition

at infinity.

3. Inner and Outer Solutions

and Matching

Consider the problem for

-'. The flux into the fluid

Q 2 is given by the rate of change of sectional areas:

=

(J,I'(x)

where ti, (x) is the sectional area of the jth ship. However, it

is not known a priori to what extent the flux will be split

unevenly. This fact can be incorporated by decomposing

into two component problems, as follows:

= Uçt,,, +

,'(2 (x)', +f2(x)

(14) with

j= 1,2

hm

)=±[(A'/4H)Y+C(,(X)J

y- ± hm

,=Y±C,(x)

Y ±'

where the subscripted 's bear no relation to the superscripted introduced earlier.

Equations (15) and (16) are of course to be satisfied in

conjunction with Eqs. (7) and (9). Thus,

,, represents a

particular solution with an evenly split flux, and

, a

homogeneous solution corresponding to unit lateral flow. C,,

and C, are blockage constants, determinable either

analytically or by the numerical solution of & and e,. The

order of magnitude of C,, and C, may be inferred by applying Green's theorem to the potentials , and respectively. In Sec. 5, it is shown that C,, is 0(2), lo the outer field, whereas

C, is O(s). Since

'

as defined by Eq. (14) is the term

associated with 2,we observe that the decomposition defined by Eq. (14)contains thetacit assumption that V'1' is O(s). The

unknown functions V,2 (x) and 11(x) can be determined

only by matching with the outer solutions.

It s convenient to regroup the terms of as even and

odd parts, which behave asymptotically as follows:

(1 6a)

(I 6b)

where sgn is the sign function. For the third-order problem a similar decomposition to that of ' _{is possible. In}

fact, can be written in terms of the previously defined 't,,

and , in the following form:

= f [o + (V'2 /U),j ½fY2 +f7ç

+ Vt3')/ ±f, (y)

(19)

where f and

V3

_{are new unknown functions, and}

corresponds to a problem similar to ,,. but subject instead to the body boundary<ondition

3N

=Y(N.)

j= 1,2

and

A(x)

Iim.=±---lYl±C(x)

with C. being another blockage constant.

Turning now to the outer problem, we note that, since d,. is assumed to be O(e), the two ships collapse onto the .v axis. Equation (5) and the unsymmetrical nature of the flow about

the xz plane suggest an outer repreentahon by distrihutin2

twoimensiona1 sources and vortices. Thus, we write

j

_n)

(E)N (v)2

+v2J

2w ,a -

-2r

Ç

Y"()tan

(21)

The fact that

'

in Eq. (6) is a function of .v only is a

consequence of =0 on S, j= 1,2. Figure 2 illustrates the boundary-value problem for 2)

_{Note that, al a given}

section, S1 and S, will not necessarily have the same shape.

Q 2

_{I Yl +f,(x)}

as lYl---co(17)

The boundary-value problem for can be shown to be

_{V-" Y+sn (Y) [UC,, + V'1'C, J}

_{J (18)} Fig. 2 is, o.hod problem in the cross-t low plane.

az

0

(9)

Here N is an interior normal to the cross-sectional curve S1.

3N 6S

=0

j=

1,2

and

äN

I= f;(u)

343) (x;Y,±H)

=0

where o and arc un knoss n source and ortex st renaths,

respectively, and [a , a J represcots the longitudinal extent

a1oti the .v axis that the t wo ships occupy. The branch cut of the arciangem fuoct ion should he chosen along the x axis with

x<

. It is evident in Eq. (21) that the first term is even and the

second odd in r. ¡:Ør mat chi ng sylt h t he inner solution, we consider first the even part. A three-term expansion in inner variables of the two-term outer solution is given by

- ½

(.V.0)]

+

['' (v.0) +

Yl]

Now, from Eqs. (6, 17, and 19), the even part of the two-term outer expansion of the three-term inner solution, sshen written in outer variables, is

(I

_{+''' h-,,}

(x) +

Ivi

- ½frv] +fff(X)

(23)

Comparing similar terms of Eqs. (22) and (23), we obtain the following immediately:

fU

c;

(X) = _uTA' (x)

f, (.v) =

_(E)CtlxEld

2,r . \

)d+

::

dE0(V')

(26)

where the second integral is to he interpreted in the Cat,chv principal-value sense. Hence the two-term inner expattsion of the two-term outer solution is

[h1

J

{±

-

y" (E)dJ

+ ±

2 - y '2'(E) dE +

L

': J

as y-O

while the two-term outer expansion of Eq. (18) takes the torni

± f2

_{UC, + V'2' C,}

I+ f V'-"v (2$)

Comparison of terms of similar nature in Eqs. (27) and (2$)

yields

½ (y." (x)

(x)) =

LIC,, +V'1'

C, I'

(29a)

(22) Equation (29b) implies that y(') is o(e), but the right-hand

side of Eq. (29a) is O(f 2). Hence, they can be consistent only

when

"

(X)= V'2'(x)

=0

(30)

This implies that

the second-order "cross-flow," which

represents the extent of the unsymmetrical flux, is identically zero. Hence, Eq. (29a) immediately yields

y(2)

(x) = - 22U(dC0/th-)

(31)

1f we now proceed to match the three-term inner expansion of

(c

''t +

) ,, svith the two-term outer expansion of (e-'

(2 _{+ ½ (3) ) we will obtain ir, closed form the following}

solution for V'3' (x)

V'3'

(x) =

_: d (32)

(24) Therefore, Eqs. (14) and (19) now are given uniquely by:

4 (2)₌

_{Uó +f,}

_(x)

=

-f;

- ½f;Y2 -t-r;. + V'3'4, +f,

(x)

For the restrictions on C,; ( ),

in order for 1'-

to exist, we

refer the reader to Nluskhelishvili.9 We poitit out however,

that sufficient conditions require that the vessels have pointed ends, and that the combined sectional area curve ,1 (y) has continuous second derivatives in [a -, a J.

The matching process is now complete. One notices that to second order the inner potential assumes a flow pattern with

equal stream velocities at both lateral itifinities. Another

significant point is that, to this order, only a strip-theory-type solution is necessary o obtain the interaction forces and

moment, since the solution of 4) at successive sections do not

interact among themselves until the third order throttgh

V'3' (x). Further, because of the symmetry of the boundary condition about the centerplanes,

only the oserlapping

portion of the vessels will contribute to interaction forces and

moment. This last point is consistent with what Tuck and Nesvman' (Sec. 3) observed when the limit of d« O of the

complementary (far-field) problem was taken.

lt is worthwhile to point out that, if the orders of the terms

itì Eqs. (27) and (28) are not kept strictly, the following in-tegral equation for V'2' results when one omits y '-' on the

left-hand side of Eq. (29a):

I

"

[UC+ V2'C, j'

,r .0

1--

.5

d=0

(35)

which is similar to one considered previously by Newman. Since the numerator in the integral is by analysis one order smaller than V'2' . this is justifiable only when C,, is (1 (t ) and

C1 is O(i) with respect ro the outer field. In fact, this would

correspond to tite case of very small under-keel clearance (cf. Taylor'') which is not considered here. For such a case, C,,'

(_7)

and C,' would become discontinuous at the stern of the

l-t. W. YEUNG AND W -Y. H WANG

J. HV[)RONALÏfI(S

J ' ' i

[b'» (.v,0)

+

Ivi

f V'(.v)=

L-

(29b)

o'2

fIxId

(25)

Here,we observe, as far as the even solution is concerned, the presence of the second body serves merely to change the

ef-fective sectional area when compared with the result of a

single body. Thus, the zero Froude-number sinkage force and trimming moment act ing on each sessel can be obtained by

Eqs. (6.12) and

(6.13) of Tuck,

with only a minor

modification ir, the definition off,. V'' in Eq. (14) and V''

in Eq. (19) still remain to be determined. These will have to come from the asymmetrical solution of the outer problem.

For the vortex-distribmion integrals, we observe that, for

OCTOBER 1977 INTERACTIONS Or: SHWS IN SHALLOW \VATER 131

kadi

ng

slnp and at he how of the trailing ship, leading to

concern rrcd loading in

Eq. (35), which must hehandled with

caution.

IuiaH ssc tinte iii passing that the conclusion of V' =0 ¡s not altered by the inclusion of first-order free-surface effects.

The torituilat ton of the inner problems " _and 2 _remain

unchanged. The First-order outer problem

still will

satisfy Laplace's equation, with the transverse variable y

replaced by (I -

-') 'v (see Tuck5), where í, is

depth-I-ronde number. The first-order vortex strength, once again,

cannot

match with the second-order discontinuity ¡n the

potential defined by Eq. (18).The first-order source strength,

hossever, will increase by a factor of (1 - J, 2)

with a

similar effect on the expression for f,

(x) .

The analysis

becomes more complicated if ° and 2I _{are included; the}

latter must satisfy Poisson's equation in the horizontal plane.

4.

Interaction Forces and Moments

Since it

¡s tiot possible co distinguish the two ships from

each other in

t he orner problem. ititeract ion forces and

niornents cati he obtained only from the inner field.

Fur-tlìernrorc, since

t ' =O, Tuck 's

inner expansion

of the

pressure field can he adopted. Thus, t follows t hat

C,,(.v;Y,Z)= (pp)/"1U=p,(.v) +[p1(x)

+P. (x,Y,7) I

(36)

w ti er e

Otd

terms up to

the second order

sviti

he considered

hereafter. Assuming centerplane symmetry, we note that p,

and p do not contribute any lateral force, alt bough i hey enter

into .sinkage and trim calculations. Hence, the leading sway

force and vass moment are 0( ) and require the knossledge of

Oht)itiCd by solving Eq. (7) with the conditions (9, 15a, and

15b). The lateral force on the jth ship

Y,

can be obtained by

integrating the proper component of pressure force over the

lì u li

Y, _ç

_far?

¡ 2

av

2 ¿.'äN

(')2}]N1

(x,c)ds (Ix (38)

where [a' ,

a'. J

is the longitudinal extent occupied by the

overlapping region, and r represents a tangential unit vector

on S,. If the stern of the leading ship does not vanish as a

point,

aH/aX

is singular at x=a'

, unless both hull functions

are stationary in ï at this point. The pressure singularity can

be interpreted as a delta function of intensity

,, or,

alter-natively, an integration by parts of the first term yields

(i N (Is₌ -\ ó,,N

1ds -

,, a N1

s'

(0)_{I d} s, x dx . s, .s, äx

ÍN

-

-1ds

(39) I ôt\' ör

Ri

Nw, trot inn that

i)ó,,iNis symmetric in Eq. (38), we finally

obtain where 1(v) = \- ,,N>ds (41) =

_{(Ns' )dO+}

(

+ --

(42)

which does not involve any numerical differentiation of che

derived quantity ,

in the longitudinal direction. In obtaining

Eq. (4f)), we have assumed that the bows are pointed; hence,

I(a

) =0. Similar analysis for the aw moment then yields

=a' ¡(a' )-

{i(x)+xY,,,(x)Jv

(43)

fr.npU

5.

Numerical Solution of the Inner Problem

A Fredholm integral equation of the second kind can be

constructed to describe the unknownn potential

_,.

What is

presented in the following also can be applied to the

per-turbation potential Ø associated with the problem for

,, with

Y. Only a change in the expression for the body

boundary-condition will be necessary. If one applies Green's

theorem to

,, and a "channel" Green function G, then for a

point Pon S1, the following identity can bederived:

where G(P,Q)

=(

Y-r1,

Zr) is the channel source

func-tion given by

G -t-- t'n[Sinh2_4

(Yn>) +sIn2

(Z)J

+

LLn{sinh2(Yo)+cos2(z+)]

₍₄₅₎

In obtaining Eq. (44), it is necessary to note that

Gir(

Y-ri)/211I',t4as

Yn>l---.

Although the blockage constant C, wa.s never used

ex-plicitly because of the absence of outer problem, we show

below that it can be calculated from integrals over the body

contours. Consider now the case in which the point Pis in the

fluid; the left-hand side of Eq. (44) is given by 2sr

(P). Let

Y ±

. Then the first integral of Eq. (44) reduces to

4

Ø,,(íVVG)ds=

_---4

r0Nds

(46)

whereas the second integral becomes

4

e,,Gds= (7r/2H) (A' I Yl

vn>ds) --A

'64

(47)

By usc of these asymptotic expressions in Eq. (44),

and comparing the result with Eq. (1 Sb), we obtain the following formula for C0:

p, (r) =2.1;

(ï)

¡U

(37a)

¡,

(ï)

=2/IU[; /U

(37h)

7r)(P)

= 4O(Q)G(P;Q)dsQ

P(x;Y,Z) =2o,, - I (i5,,

)'

+ (ç5,,)J

(37e)

v,7Gds

A'

(44)

=J(u ) - 2t'(.v)dx (40) C,,=

where X, denotesthe lateral distance of the centerplane of the

jth body from the

x

axis. A similar analysis

for o will

allow

us to recover Sedov s formula12 for C,:

C1=

H +

rl

₌

,Nyth

(49)

where p.i is now

the lateral added mass of the combined

bodies. From Eqs. (48) and (49), one notices that, to the outer

field, the constants C0 and C, are 0(f7) and 0(), respectively,

if 0,, and 0, are of the same respective order. However, as the

under-keel clearance decreases,

it

is well-known that

increases rapidly.

'

By using an asymptotic formula given

therein, one can show that C1 becomes O(I) as the

clearance-to-depth ratio changes to O(f). The magnitude of C0 also can

be inferred similarly.

The integral equation, Eq. (44), can be handled

con-veniently by the method of discretization, 1 which leads to the following system of linear equations for the discrete values of

o,,on SIUSÌ, = I, 2... 01

(1, i=j

E(irc5,, +K,1)0,,, =rD1

Ô1 = [o,

.j

j1

w here K,1=

Gds

JI . t'jI

=

[tan

' [coth-

(Y, )tan

(Z, +r)]

ir ir

--tan /

tanh--- (Y 7;)tan--- (Z -)

4H 4H

= ILs

(Q)ds0 +A 't4

(5lh)

Here (,.r,).

j= l,...M

+ 1, denotes the collection of points

that provides a polygonal approximation of the body section.

In the actual computation, the symmetry of solution about

=

O is exploited. The evaluation of K,, and D, occupies a

major portion of the computation time for a typical problem.

We recall that Eqs. (40) and (43) require the inner solution

in the overlapping region only. If the body has a fin-like

appendage at the stern, corresponding to the presence of a

rudder or skcg, it is reasonable to assume that a vortex sheet

has shed from the trailing edge on such a lin. In the context of

low-aspect-ratio lifting surfaces, such a vortex sheet has no

upstream influence,

but its

presence generates an

un-symmetrical flow about the adjacent body in the

nonoverlapping region. Newman and \Vu

showed that the

dynamic boundary condition on the sheet simply implies that

the vorticity at the trailing edge is convected downstream.

This type of inviscid wake model can be incorporated easily itr

the foregoing solution algorithm of e,,. As an illustration,

consider a typical section located downstream of a fin-like

stern of ship 2 (Fig. 3). Eq. (44) becomes:

ir, (P) =

&GSdS -

,, (x,7,)Gds

-

r',Gds,l(x)f4

for PS,

(52)

s,

Here the second integral represents a dipole distribution on

the wake surface. The strength is given by the "jump" in the

potential corresponding to an upstream point on the same

/

'/////.,,/////////J

/,,

Fig. 3 Bodysection with ssake sortes shed.

o 1.2 1.0 0.1 0.6 0.4 0.2 0,0 0.0 1.2 1.4 1.6 1.8 2 0

d/2r

Fig. 3 ntrructiitn ti,rct' nt lososph&riiids.

-

Newman's theory present c1cuIa-tions C.rg Ship

-y

A-A section Tog Boit

Iig. 5 I'uRg1III-ap()rxi,l1Iltd tittíi 1IkIflO 40 iht' &artt s11I) aliti

itiiii

streamline at the railiiig edge, j.c,,

=itçti0(

;X,,t)

(53)

with

=X,

±

v(x;X,.r)dx

(54a)

132 R. W. YEIJNG AND W.-Y.HWANG J. HYDRONAUTICS

(50)

, _/ ,*1

II.'-,,

OCTOHIIR 1977 _{INTERACTIONS OF SH PS IN SHALLOW WATER 133}

taille I Parlicutir'. ut m del,,used in ca1cuIatii,n

Mde1 A

Model 9

Fia. (i

Si furc' and

a morflent on trig in steady motion.

4,5

at the Nacional

_{Physical Laboratory, England.6"7 The}

particulars of the models were given in Table I . The first pair consists of a cargo vessel and a tug boat. The length ratio of the two ships is 0.24. The fact that the models are of rather

dissimilar size offers a stringent test on the practicability of

¡he theory. The water-depth to draft ratios are I .38 and 3.28

for the

large and small

ships, respectively. The non-dimensional separation distance based on the larger ship

length is 0.128. Polygon-approximated body plans of (he two models are shown in Fig. 5. The cargo liner has a closed stern. The skeg and large rudder of the tug are approximated by a

fin-like structure in the aftermost half-station. Computed

results are shown with experimental data in Fig. 6. Here,one

observes the rather satisfactory force prediction when the vessels are in abreast configuration. The predictions corresponding to small-overlap configurations are much less accurate, particularly when the tug is in the stern region of the cargo liner. Unquestionably, it is in such configurations that one would expect the slender-body theory to be less effective, since three-dimensional effects clearly play an important mole

in ¡he inner flow-field. The large peak in the moment curve

and the force reversal predicted by ¡he theory at x,/L, 0.8

are caused by the 'absence' of the end-effect ¡erm in Eq. (40) by virtue of the fact that the stern of the tug has moved out of

the overlapping region. Oscillations in the moment curve, although puzzling at first, arc attributable to the fact that

both the integrand and the limits of integration of Eq. (43) are functions of the stagger. The qualitative features of this curve

agree well with the inner limits of the complementary outer

theory investigated by Ycung.

Model identifier ('argo liner Tug Model A \Imrdel t I engr h. ¡ ¡ti'. meters 3.478 0.838 3.320 2.958 I.crnalr beam 6.767 3.88(1 7.023 7.836 lteamn 'me,um drab 2.700 2.570 2.848 2.3 73

0.590 0.486 97W 0.761

Waiei-depth.'drali ¡.38 3.26 1.67 1.30

L. w(x;X,,Ç)dx

(54b)

where e and w are che solutions of Eq. (52) without the S

term. Actual calculations show that such a wake vortex sheet

has rather insignificant influence on the overall force and

moment, at least for the cases investigated here.

6. Results and Discussion

A computer program based on the present theory has been

developed for the evaluation of Eqs. (40) and (43). The in-teracting vessels can be of arbitrary form. The core

sub-program consists of the solution algorithm for the inner

problem described in Sec. 5. Verifications of this

sub-program were made by applying it to obtain C, for a number of cylindrical shapes considered previously by others. The

overall programming then was checked by calculating the

sway force on a spheroid moving near a wall in an otherwise

infinite fluid. A closed-form solution of this was obtained

earlier by Newman. Figure 4 shows a comparison of the computed results with those of Newman. Each computed

point was cross-checked by repeating the computations using

Eq. (38). We note that the functional dependence of the calculated curve is in the same form as Newman's. The

discrepancy is attributable to a difference in the approximate esaluation of the slender-body force integral.

The theory presented was next applied to two pairs of

rcaLttic ship hulls, for which experimental results in shallow "ater arc available. The experiments were conducted by Dand

c

v/-f

pu2er1 CM'Mi/3-pu o.2.2,. Confiu,oliOfl O4 e/O ocerlop 02 -06 -04 -02 - o .-02 /0 -O 4 -o6 Repulsion Coro ship ) CQ Bow-in u-.-__ I 06 i-I Tuq 2 k--.. 4p -0.6 -0.4 -0.2

I-la.7 I'llRgl)n-approilnaItid body plans of rnodt.ls A and B.

CM M/ fpu2s2î -1.2 -lO -.8 -6 -.4 -'O- A U_ "i

_Ic

B -12 -(Q -.6 -.'o.- A O B _{M '} 7 06r Allrocton Q2 04 0.6.' -04E Bow -oUI C1 Y/-pU2BT CM M/_-fpU2B2T -12-1G-8 -6 -.4 -C1 loo 50 2

02

50 CM 30 o -30

Fig. $ &s a' toree and av. nionenl on model A.

Cy 00 1-IO CM 30 - 2.0 20 IO 0 -.2 1h nor y o Espt Dorid (1976) 252 36) 0/ L8 °. .6 8 lO 2 _._.0. D 2

4Ooo0

-50 20 lo ern of rr,odel A

not n one lop

'o / LB 4 6 .8 .0 2 -12-IO -B -6 -.4 -.2 . 2 .4 .6 .8 lO Z -IO - -- 3.0 -40

Fig. 9 %I% flIrte ll1II Y_1w ln,lneIIl III) mollit It.

Ecyt. o/o crnwT0cnd

'

Lept.. e screo J (075) -PreserO Theory Conf w/O Ov.rlop

-dc/Lp '0.128 h / L '0 088 Fr '0.197 F11 O345

- Iheory

o E.pt .Gond 11976) 60/L8 '.252 F5'.361 'o' LB 8 .0 12

The second pair of vessels consists of a cargo ship and a

tanker, designated by model A and B, respectively. The

models are more similar in size, compared with the first pair

considered: the length ratio being 0.839. Both are modeled by

pointed bows with fin-like sterns. Their approximated body

plans are shown in Fig. 7. Numerical results of the interaction

force and moment are shown in Figs.

8 and 9, and are

compared with the experimental data of Dand.6 The

ex-periments were conducted at two speeds; only the result; of

the lower speed case,

F11 =0.361, are cited. Theoretical

predictions for model A (Fig. 8) appear to be quantitatively

satisfactory, except when the vessel is in an ahead position

with a small overlap. In this situation, the stern terms of Eqs.

(40) and (43) play a much more dominant role than physically realized. lt

is of some interest to note that x5/L8 =0.161

corresponds to the particular stagger at which the two sterns

coincide. Here the slopes of the theoretical curves will be

discontinuous if the vessels do not have cusp ends. Figure 9

shows the corresponding comparison for the larger vessel,

model B. with the measured data. The force and moment

peaks are overestimated by a factor close to 2. Such a

discrepancy can, in fact, be traced back to the validity of

using a

fin-like

structure to

represent

the complicated

geometry at the stern region.

7.

Conclusions

Slender-body theory with a rigid free-surface assumption

has been applied to predict the nearfield interaction of two

ships. It was shown that, for the situation in which the water

depth is of the same order of the vessel draft, the solution can

be constructed, to second-order accuracy, by a stripwise

calculation in the cross-110w piane without considering the

outer problem. The inner problem, in general, requires the

numerical solution of a coupled integral equation with

specified flux conditions on both body sections.

Applications were made to two pairs of ship models for

which experimental data in shallow water are available.

Theoretical calculations appear to yied good qualitalive

agreement, particularly when the vessels have a large overlap.

For small overlaps, three-dimensional effects near the bow

and the stern regions are important, and clearly violate the

assumptions taken in the theory.

From the quantitative viewpoint. the present strip theory

tends to overpredict t he sway force and yaw moment

. Tbk is

lO he expected, since the ífctLltl badino svill he less severe

owing io the three-diinensionalit y of the flow. ihe T1lIl\ion

of a cross-flow viscous drag term, as is customarily performed

in applying slender-body theory to a single vessel at a yaw

angie. must he done with rììore Scrutiny because of the

nonuniformity of the cro;; flow ill space.

Finally, it is wort hwhile to point out t hat the com put ai ion

time for such a mathematical model of ship inleractions,

although not as prohibitive as three-dimensional calculation;,

is

suffcicntly large so thai its applications to

_{real -tilBe} sim u at ion appear to he im practical- Furt hermore, cxl reine

care also musi be exercised ill the preparation of the hull dati

to insure that the hull geometry L; sufficiently 'nooth. The

solution of the inner problem i; found to be rather sensitive to any local irregularIties.

Acknosledgment

The authors are grateful to the National Science

foun-dation for supporting this work, under grants GK43886X and

ENG75-l0308.

Re leren ces

I _{IIILk. F. O. :111(1 NC\t,lIIll .. I. N.,}

l-i\itrlld\Ilarllis ILIILraLliroImn

13cl \h /I1 Sb p .

''(Ii

(1/ (/10. /01/, SIll)/) I Ill Ill? II? \oI

I I (I,l)(IIylalnjc-ç ( )lti.e o F N.' .1? R rh .I 974 . PP- ' -7(1.

- \\ang. S.. lorcc 111111 \linlllLlll rii a \ttrored 'C''et 1)11e III .1

t'.l.Ill1 Sill)). _{.JllllIflU/ o/ fili' (I rIft'1ii/ls. /iit,/yiy,'.} 1111(1 ('(0(1(11/ IEIlcl/lCt'I7i1.0 1)11-/suoi. /',ii((eillyILs in

t.'(E, \tt.

(il . \ui. l')7.

No\s;ILkl.

ti..

"libel die \eCllsckcl)lr1cn hIaIloIrkIIIIo_'il /01./Il/Il Sdlilfsa(llI)11IlclI 111l1/Ilkt)FI)r.FlI_'' .Iii// i,,i(I Iia/c,r. 1,1111/

I 2. t I/tI t) 5p tOfnI). pp. 7(i7()),

4NrllltljIl. N. lt.. li.lIlt tIlLe)' 1)11 11 Shijr \lllLllly! I (Otu:Il .1

S Ill Il t )rL'tIoetl ( I 1,11111/I. - / j 0/ (Itt 1/Ill, Si ,,tS s(tiI// rI/f

.\ 11)1(1 f I./IfftnuII,s .( ) tI li./ u t Na'. al Re'c1I retI. i 974 mp - ' t ('(s

-'(oliai,,

G.. I'oftII)IatlI1c&1reII.e)le tllller-.UCIIIIIIO '.Iei FI vdrodv l:IIIII'./IIell WccIi'ClO ii k ii lit / I'. eier Schi II k ilF(1/F . ' lu/u ¡'Iii? (It'! .'t-h,//iou!oe/In/s(/u'n Gew/it thu/i. flaitit 57 9( ti°. 251 -329.

I

OCTOBER 1977 INTERACTIONS OF SHIPS !N SHALLOW WATER 135

I. W .

''llvdrodyuainie Aspect _{of Shallow Water} Ilk,oii. ' ' Journal of PINI, Nov. 1976, pp.323-337.

I le'.',. J. I id Smith, AMO., ''('alLIllalion ut' Potential Flow

.,hotii \rha rais Itodie'..'' l'i ,tress in .-lecwwutnu/ Siietices, Vol. X, 967. pp I-13?.

I tick. I. O., "Shallow-Water Floss Past Slender Bodies,''

loi,, i,1 v;fI/tini .tlt'cJiun,c'. Vol. 26. Sept. 1966. pp. Sl -95.

"\lu'.kl,li'.hi iii.

N. I .. .Sint,'ulur f,negral Equations, 2nd ed..

,Vtilier'.-Nntidlioff, (roiiin,zc!,, The Netherlands (Engltsh

Ura,t-.dat till t, I 95S.

Neo man, J. N.. ''Lateral Mot on ot a Slender Bod Between

I no l'a, aliel Walk,'' Journal of Fluid .tleehanicc. Vol. 39, Oct. 1969,

t'P. 9l IS.

ryl. P. J.. "The Blockage ('oeriicient for Flow About an

-\rhii rar }Jodv Iinttier,.cd in ii Channel," Journal of Sit:1, Research. Vol 17. Inne 1971. PP. 97-IllS.

- Sed iv, I - . I . Tt,t, -I)i,m',, cio,:aI Prr,hle,p,s i,, I I idrodvn,oiues citid

1 t'rodt-nau,ics, lohn WiIL'\ and Sons, New York, 1965.

''Flage. C. N. and Newman. .1. N.. "Sway Added-Mass

Coef-licielO s for Rectangii lar Prulile'. in Shallow Water.'' Journal nl S/up Rseareh, Vol. IS. Dcc. 1971. pp. 257-265.

I Hildebrand. F. B .. .i.!ethrid, o/ ,11p/ied Mai/re,na//ec, 2nd cd., Prentk'e.HalI, Englewood CuIR. 1965.

Newman. J. N. and Wo. T. Y., ''A Gencrali,cd Slcnder-Itodv rhcorv for Fish'Like Forms.'' Journal of ¡-luid %h'iiw,,cs, ViiI. 57.

March l973,pp.673-693.

" Newman. J. N.. ''The Force and Monient on a Slender Both ut Revoliit intl Moving Near a Wall ,'' David ravlor Modcl Basin, Rv'pt 2127, 1965.

7 Dand, I - W.. ''Sonic Aspects of Tug-Ship Interact on.''

Proccedinas of 4th International Tite Conven i ion, I 975. pp. 61 -75. Yeuitg. R. W.. ''(>11 tIte Interactions of Slender Ships jim Shallow \Vater,'' to appear Journuf of i-luid Mecitanu.c, 1977.

From theAIAA Progress in Astronautics andA

_{eronautics Series.}

MATERIALS SCIENCES IN SPACE WITH

APPLICATIONS TO SPACE PROCESSiNGv.

₅₂

Edited by Leo Steg

The newly acquired ability of man to project scientific instruments into space and to place himself on orbital and lunar

spacecraft to spend long periods in extraterrestrial space has brought a vastly enlarged scope to many fields of science and technology. Revolutionary advances have been made as a direct result of our new space technology in astrophysics, ecology, meleorology, communications, resource planning, etc. Another field that may well acquire new dimensions as a result of space technology is that of materials science and materials processing. The environment of space is very much different from that on Earth, a fact that raises the possibility of creating materials with novel properties and perhaps exceptionally valuable USCS.

We have had no means for performing trial experiments on Earth that would test the effects of zero gravity for extended durations, of a hard vacuum perhaps one million times liarder than the best practical working vacuum attainable on Earth, of

a vastly lower level of impurities characteristic of outer space, of sustained extra-atmospheric radiations, and of combinations of these factors. Only now, with large laboratory-style spacecraft, can serious studies be started to explore the challenging

field of materials formed in space.

This book is a pioneer collection of papers describing the first_{efforts in this new and exciting field. They were brought}

together from _{several different sources: several meetings held in 1975.76 under the auspices of the American Institute of}

Aeronautics and Astrotìautics; an internalional symposium on space processing of materials held in 1976 by the Committee on Space Research of the International Council of Scientific Unions; and a number of private company reports and specially invited papers. The book is recommended to tnaterial.s scientists who wish o consider new ideas in a novel laboratory en-viron;nent and to engineers concerned with advanced technologies of materials_processtng.

59./pp., 6x9, jI/os.,520.00 ttfemi,er 535.00 L.'st