The force and moment on a twin-hull ship in a steady potential flow

- - JUN I 1976

ARCHEF

ABSTRACT

The linearized problem of potential flow around a twin-hull ship is solved by two sheet distributions of sources and

normal doublets. The thickness problem

is solved by the thin-ship approximation, while the lifting problem is solved by a

slender-ship approximation. General

formulas for the steady hydrodynamic force and moment acting on a hull are obtained as an extended form of Lagally's theorem for two floating lifting bodies. This theory is applied to the prediction of wave resistance of

Small-Waterplane-Area Twin-Hull (SWATH) ships. Computed

and experimental results are compared and presented.

I. INTRODUCTION

Recently, there has been a consider-able amount of interest in developing a particular type of twin-hull configura-tion, namely, the Small-Waterplane-Area Twin-Hull (SWATH) ship, as a new design

concept for a number of specific Navy

applications. As part of the initial

effort to investigate the hydrodynamic performance of SWATH ships, an analytical tool for predicting wave resistance was

developed. This paper presents the

results of subsequent efforts to extend the scope of analysis to include an investigation of the steady hydrodynamic force and moment on a twin-hull ship of

THE FORCE AND MOMENT ON A TWIN-HULL SHIP IN A STEADY POTENTIAL FLOW

by

WEN-CHIN LIN

Naval Ship Research and Development Center Bethesda, Maryland

a more general shape.

The wave resistance problem of catamarans has been investigated by

Lunde (1951) and Eggers (1955). In both

papers, a catamaran was treated as two

separate but identical thin ships, and the velocity potential of the problem was obtained from two sheet distributions

of simple sources (or, more precisely,

Havelock sources). However, since each

hull is in proximity to the oth2r, and since each will generally experience an asymmetrical flow field around it, dis-tributions of simple sources alone are insufficient for solving the problem. To account for the asymmetric flow field around each hull, distributions of

doublets normal to the sheets should be

simultaneously considered. Fortunately,

by virtue of linearization, the effects of doublet distributions may always be superimposed to refine the results initially obtained from source distri-butions without jeopardizing the initial results.

In this paper, the twin hulls are represented by two distributions of sources and normal doublets on their planforms (which, in the case of a symmetric ship, correspond to the ship

centerplane). The normal-doublet

distri-bution is also extended into the infinite downstream in the wake regions which

Lab.

v. Scheepsbouyikunde

Technische Hogeschool

trail each4u11. Hence, the so-called "lifting" effect is also included in

the present analysis. The mathematical

problem is formulated within the context of the linearized thin-ship theory. Therefore, the "thickness problem", i.e., the determination of the source density, may be solved immediately upon

applica-tion of the boundary condiapplica-tions. For

the "lifting problem", i.e., the deter-mination of the doublet density, an integral equation is initially derived with the full expression of the Havelock doublet as its kernel.

As a first approximation, the

kernel function of the original integral equation is replaced by the so-called "zero-Froude-number" Green's function.

mean camber surface of a demihull half thickness of a demihull

2b

hull-separation distance

= total velocity potential = distribution potential

total fluid velocity relative to the ship

disturbance velocity planform of Hull I

wake sheet trailing Hull 1 disturbance potential due to source distribution

disturbance potential due to normal doublet distribution = local density of the source

distribution

= local density of the normal-doublet distribution

H

= ship draft

L = ship length

B

= ship beam

This equation is further reduced to a singular integral equation of one varia-ble by introducing low draft-to-length ratio and small hull separation

distance-to-length ratio approximations. For this

reduced problem, both the source density and doublet density of the distributions

have been obtained. General formulas

for force and moment expressed in terms of these singularity-distribution

den-sities have been derived. The result,

as might be expected, is an extended form of Lagally's theorem for thin lifting bodies making a uniform motion in the

free surface. In addition to the

expressions obtained by Cummins (1957) for the steady case, terms involving integrations along the waterline and the trailing edge of a hull now appear in

NOMENCLATURE

(Additional nomenclature are defined as they appear in the text)

A673 =

scalar product of two vectors = Cauchy principal value integral

A/03 =

vector product of two vectors

2:

= bounding surface of a control

volume excluding ship wetted surface

control volume bounded by and the wetted surface of Hull 1, S1

tt =

I'Ve plus the displacement of

Hull 1

= (0,1,0) = the unit normal to $(:',)

- tangential component of

q

with

respect to

= normal component of F with

respect to

= = the difference

in at both sides of Sri,'

averaged value of the fluid velocities at both sides of the singularity distribution

F

= hydrodynamic force acting on Hull 1

= hydrodynamic moment acting on Hull 1 with reference to the origin of the Oxyz-frame

7Rw = total wave resistance

_I C

t(-)

= = =

these formulas. The significance of these additional terms for making better predictions of force and moment on a floating lifting body is being assessed by actual computations and by comparison

of the computed results with experi-mental data.

As an example of the application of the theory presented in this paper, the wave resistance of two SWATH ships has

been investigated. The results of

com-putations together with their comparisons with experimental results are presented. Unfortunately, the entire scope of the

investigation of force and moment on a twin-hull ship is not covered in this paper since this work is still in progress and further computations are being made. We shall continue to report the results of such investigations as they become available.

II. FORMULATION OF THE PROBLEM

Consider a twin-hull ship moving with constant forward speed U into

other-wise calm water of infinite depth. The

two hulls will be referred to as Hull 1

and Hull 2, respectively. Let Oxyz be a

right-hand Cartesian coordinate system moving together with the ship, with Oz

directed upward (against gravity), Ox in the direction of motion, Oy to port, and Oxy coinciding with the undisturbed free surface.

As usual, we shall assume that each hull is "thin" and has small camber, and that the two hulls are mirror images of

each other. We shall further assume that

Hull 1 is situated approximately in the Oxz plane, and Hull 2 is situated

approx-imately in the plane y = 2b. Thus, the

hull separation distance is 2b, and the plane y = b is a plane of symmetry for the whole flow field,

(2.3)

ship geometry may be described as follows: _{velocity relative to the ship as follows:}

Using this coordinate system, the _{so that its gradient gives the total fluid}

Hull I:

[V,

PC.

a)

_(2.1a)

Hu112: (2.1b)

cLs = 71'46, a-) 4-.26

where the subscript S and P indicate the starboard and port sides of the hull, respectively.

For convenience, let us define the mean camber surface and the half-thick-ness function, respectively, as follows:

6,4)1

(2.2a)

(2.2b)

Then, 1 -16,,,z-)= C6,-, )+ I 6,,,z) , ,1-6,,z)= C 6,,j) t6,4)

Finally, we shall assume that the draft of the ship is H and the ship length is L, extending from -L/2 to L/2.

We shall now develop a potential flow of an inviscid fluid for a twin-hull ship. This development will be based upon the linearized theory of ship waves using the thin-ship approximation.

Let the total velocity potential be given by

?0

?x.

which represents the disturbance velocity

due to the ship. We shall use the

nota-tion et , i = 1,2,3 to represent the

three base vectors of the Oxyz coordinate

system. Then, by a systematic

perturba-tion expansion based on the thin-ship assumption, we may obtain the following boundary-value problem appropriate for

the first-order approximation of the

dis-turbance potential

0

.

0 ( x>o,

o(1), ior x<o,

In addition,

0

is required to satisfy

the following approximate kinematic con-ditions at the ship:

a.sX1411--pcs, to) 24(X, _{ufac (..o.t)} ?), JJ, and 441,4) u ui 4re,

_ ?.1

a ,

for

co)

,(o)

where Sib and 326 represent projections

of the wetted surfaces of Hull 1 and Hull 2 onto the vertical planes y = 0 and y = 2b, respectively, when the ship is at

rest. In wing theory terminology,

(2.4) _(o)

and 526 are the planforms of Hulls 1 and

(2.5) 2, respectively.

(2.6)

(2.7)

Before proceeding further, let

us

observe the flow situation around the

ship. Because each hull is now cambered

and is situated in proximity to the other, each will experience a uniform flow coming

from an angle. Hence, in addition to the

usual thickness effect of a symmetric monohull ship, the so-called cross-flow

effect must also be considered. This is

true even if the camber is reduced to zero since, for two identical monohull ships sailing abreast, the flow around each ship will still be asymmetric with respect to its centerplane.

Since both the differential equation and the boundary conditions (2.6) and (2,7) are linear, we may consider the effect of

thickness and the effect of the position of the mean camber line (which now includes

the incidence) separately. This can be

done by assuming C(x,y) = 0 and t(x,z) = 0 in (2.7), respectively, and superimposing

the results. Thus, for example, the flow

around Hull I may be decomposed into two

problems as follows: (1) the flow around

a symmetric hull (with respect to the

plane y = 0) without the presence of Hull 2, and (2) the flow around a very thin hull (zero thickness), with or with-out camber, in the presence of Hull 2, Subsequently, these two problems will be referred to as the "thickness" and the

"lifting" problems, respectively. The flow around Hull 2 may be similarly

decomposed. This suggests that the

dis-turbance potential

0

may have the

following decomposition: such that -F I,/..__

,t.

(2.8)

=

where = 61,4/, 4v-) v ,

-2b,Z) = =

I

=

to, a) ?

?A,

Tx- , ?t,(x,.26,±0, a) 1.)?i" 7) 4.- 2x ,

or

f oc,0.i)

(2.9)

tm, tjaC 0.4 6, 24 *)

a-(2.10)

;Or fx, zb, a)t S2b

In view of these boundary conditions and the corresponding flow problem each compo-nent potential represents, we may expect

the potentials Om and

0

to be

symme-trical about the planes y = 0 and y = 2b,

respectively, while Olp.

and 4. should

be antisymmetrical about these two planes. Hence, and ;,t) t,.(-,

g, )=

-;,

= t,(x,

(At,. 1-) =

Our next task is to find solutions to the boundary-value problems given by

(2.6), (2.9), and (2.10). We shall try

to find the respective solutions by the method of Green's functions.

Solution to the Thickness Problem Let us first find the solution to

Ow

which is required to satisfy the

requirements in (2.6) and the first

equation in (2.9). Since Oicr is expected

to be symmetrical with respect to the plane y = 0 (see 2.11), it suggests that

Oicr may be obtained by a distribution

of sources and sinks over the planform of

(0)

Hull 1,

S. Thus

a.) g

a,..)./x.de,,,

(2.13)

sr

where(r(x

0 z0) is the local surface density of the distribution, and G (x,y,z,

x_OP,y0 ,z0) is known as a

Green's function. This function

satis-fies all the requirements in (2.6),

except at (x,y,z) = (x01y01z0), and has

the form

In this expression

A..

and H is harmonic everywhere in the lower

half space. The explicit expression of

the function G is well known (see, for example, p. 149 of Wehausen (1973)), and may be written in the following form:

= 71--, (2.11)

4 f%

,

X e...s[io,n)e-req. ( 0] (2.15)

(2.12)

_X

A

R.

where TL,R 0)t(1-1,,f)2 I

= 3/U2,

and the integral with respect to k is a Cauchy principal-value integral.

The function G for this specific case is called a "Havelock source" and,

in addition to having the properties of a source potential, satisfies the

linear-ized free-surface condition in (2,6), Thus, it is obvious from (2,13) and (2,14)

that 010- has the properties of a

single-layer potential, and, in particular, has the following jump property at the

sur-face of distribution, _S16 '

(2.16)

6 (x.1,e: e.)

-k

H

1-x.,a (2.14) -44.4,

-

(x. =

-tr(x.,

From (2.16) and the first equation of (2.9), the local surface density,cr(xz,), of the distribution is found to be

related to the half-thickness function of a hull as follows:

_II

0(;

I) =

.4

In a similar manner, we may conclude that

tr(''1.4) = (2.18)

(')

where is the planform of Hull 2.

Since we assume that Hull 2 is a mirror

image of Hull 1, the local surface

density of the source distribution for

Hull

2 in (2,18) will still be given by

(2.17).

Thus, both

OiT

and

020-

are now

completely determined. Our next task

is to determine and 942fr,, .

Solution to the Lifting Problem Suppose that Hull 1 is of zero thickness, but cambered, and is situated in close proximity to Hull 2 which has

finite thickness. As was previously

discussed, describes this portion of

the flow field around

Hull 1,

_{By (2.11),}

4s,

is antisymmetrical with respect to

the plane y = O. This suggests that _Oto.

can be represented by a distribution of doublets of surface densityiu in the

x,z-plane. To satisfy the

antisymmetri-cal property of (2,11), the orientation of the doublet distribution should be

normal to the x,z-plane. Analogous to

the situation of an incident flow past a

finite thin wing, we shall assume that there is now a steady vortex wake trailing

the stern of

Hull

1 and remaining in the

x,z-plane. We shall also assume a

similar physical picture for

_Hull

2.

Furthermore, since the wake is now a sheet of discontinuity, the doublet

dis-, (0)

tribution extends over the wake

S.

as

well as over the planform 5 of

Hull 1,

Thus,

S.

(c.)

where the suffix 3, + J

indicates

that the integral is taken over the

plan-form of

Hull 1

_{and the wake trailing}

behind it.

Similarly, for

Hull

2, we have

.1) = 5?- a6,,,.e;x,,,2, .),,&../z, (2.20)

+

where the distribution is now extending

over the planform of

_Hull

2 and its wake.

The minus sign in (2.20) is due to the

assumption that Hull 2 is a mirror image of

Hull 1;

_{hence, the doublet density for} Hull 1 is equal in magnitude and opposite

in orientation from that of Hull 2. From

(2,14), (2,19), and (2.20), it is clear

that both

0,,

_{and 02/, have the}

proper-ties of a double-layer potential. _In

particular, we have:

f.) --

96 (x,-ai.)-= Cx,0, z-,)z S74),-and tm< .26 -a, R-.)--=+47r/-4-(x., t,), .S:r

Since, by (2.11) 451).., is an odd function

of y, (2.21) gives (2.17)

(2.19)

9bip..06

t

0, i.) (2.21) (2.22) +o,

(x,,.

A few more observations can be made

for the doublet density . The

expres-sion for the pressure appropriate for the

first-order approximation may be obtained

from Bernoulli's law as

= f1j-

= f U.

I O.'s+

(2.25)

a Oa,/

Since, in (2,25), A is the only

function which suffers a jump

discontin-uity across the sheet

5

, (a)

,

the pressure difference on both sides of this sheet is

401

u

(2.26)

=

-47rf U

The assumption that the pressure is con-tinuous across the wake implies that

or

ptcx.,-E0)=-}41-1.,?-0) (2.27)

x

(G)

in the wake . That is, ),A is

con-stant along lines which are parallel to the direction of flow in the wake. Similarly, we may also set "6/"'" 21x. =

along the keel if we assume continuity of

pressure. At the keel we shall further

assume that the function is

continuous. This means that

it (s, ) 0,

(2.28)

everywhere.

Since JA

(X00

is no longer related

directly to the local geometry of the ship, it is more difficult to determine 951/... .

,However, the second equation of (2.9) may

now be used. If we substitute expressions

in (2.15), (2.18), and (2.20) into (2.9), we may write ,) /6(,' ; t,) e SZ+SI:

(2.29)

-

?1

This is an integral equation to be solved for the unknown doublet density if.". , The

right-hand side of (2.29) involves only known quantities, since the source density

q-(xi0)

is already related to the local

slope by (2.17). However, this equation

becomes extremely complex if the complete expression of the kernel function G is to

be substituted from (2,15). A closed-form

solution of this complete integral equation appears to be beyond our resources.

Although we may try to solve this equation numerically, it is not advisable to take a

brute-force approach to evaluate the function G in the form given by (2.15)

without further reduction. Since, in this

expression, G contains a double-integral term with a highly oscillating integrand, a direct numerical evaluation of this term

is not only time-consuming but prone to inaccuracy.

Hu (1961) has shown that for a simpler case, i.e., for y = 0 and yo = 0, the

k-integral of the double-integral term in (2.15) may be expressed in terms of the

exponential-integral function. Since the

exponential-integral function is well behaved and is also extensively tabulated, the degree of numerical difficulty appears to be considerably reduced.

Z.) =

(4

1A4(xe, # )

(2.23)

.2 7r"

Similarly, for Hull 2, we have from (2.22)

j"-(x,g0

=

_r

(x.,26+0 .-E0. .27

(2.24)

,,

0

=

-To fill the need of the problem here, Flu's approach has been extended to

a general case of G. It is shown in

Appendix A that (2.15) may be written in the following form,

G x., _IL + 12.

-n- ?ct.

IdA e

tuo

_va- ,

e

LvPc-,(4-,8)]

a.= (1-)'

the rest of the variables are already

defined in (2.15). The special function

El is the "exponential-integral function" as defined by equation (A.4) of Appendix

A.

In (2.30), the double-integral term has been reduced to a single integral. Since the function E1 is extensively tabulated and its method of computation well studied, an efficient numerical

scheme for evaluating G may be developed from (2,30).

As an alternate approach, we may also

try to solve the integral equation (2.29)

by an iterative perturbation technique. In fact, for a yawed-ship problem, Hu

(1961) assumed perturbation expansions for both G and the doublet densityik and solved the integral equation of the problem by an iterative procedure.

In terms of (2.30), the expansion for G used in Hu may be written as follows,

(2.31)

(2.30)

The remaining terms of (2.31),

G(n)

, n 1,2 3 are obtained by

making use of the asymptotic expansion

of the E1 function. Their explicit

expressions are given by equations (A,18)

and (A,19) of Appendix A. Since the

asymptotic expansion of the El function is valid only for large arguments, and since the argument of the El function in (2.30) is now v (a.,. L) which is related to the inverse of the Froude number squared, the expansion (2.31) is valid only for low Froude numbers.

G(C)

i

In fact, n (2.31) corresponds

to the limiting case of setting Froude number equal to zero, and is referred to as the "zero-Froude-number" approximation

of G, Hu further reasoned that Cic° and

represent the lowest- and highest-order terms in the expansion, respectively,

and pursued the iterative approximations accordingly,

In Flu's report, the computational results so obtained compared favorably with corresponding experimental data. Therefore, it may be inferred that a similar technique may be applied here to

solve the integral equation (2.29). On

the other hand, some investigators have experienced difficulty in using the

iterative technique based on a perturba-tion expansion which has the zero-Froude-number approximation as its first term

in the series, and have warned that such a technique may not be valid, In any case, the information available in this regard

seems inconclusive in proving or

dis-proving the validity of such a technique. Continuing efforts are being made to

solve the integral equation (2.29) both by a direct numerical scheme and by the

iterative perturbation technique. It is

felt that, short of rigorous mathematical proof, one can draw a conclusion about the validity of the iterative perturbation

technique only after a careful examination

(2.32)

2' a d

e

4s09-73)7. _(2.33) (0,

4

a_d (11)

= 4

a;

E,

, 11 ,,,,,

-of the computed results obtained by

different methods, such as those mentioned above, and by comparison between the

computed results and experimental data.

Zero-Order Solution

In the following we shall attempt to derive an approximate solution to the integral equation (2.29) by replacing the complete expression of G by Ci("°. Such a solution is equivalent to the zero-order approximation in the asymptotic

expansion (2.31). The problem will be

further simplified by adding three more assumptions concerning ship geometry: (1) that each hull has a rectangular

plan-form, (2) that the draft-to-length ratio

is small, and (3) that the ratio between the hull separation distance and ship length is also small and is of a com-parable order of magnitude as the

draft-to-length ratio.

Thus, (2.29) shall now be written as

U3T-C,.4.

/7

Cr (X. Y'r, ; N.).1x,

if we assume

,c;.°11

(2.34)

In the above equation, \--T is to be

substituted from (2.32). After such a

substitution, however, each term in (2.34) must be further manipulated into a form suitable for taking the limit as indicated. Thus, the first term on the left side of

(2.34) becomes (x, 1, z) I ; 4.a N.)

?4t-St.

2 I' _{z. ( -gat ,16,-,f--,1'4(i."-Nt/} , #

= I

.1,1d Itc.64.g.)

r

f -

(2.35)

/It

g. ) - i.)

(2.36)

Since

satisfies Laplace's equation, (2,35) may be replaced by

#

(--L) + 2L. (-L )1

747,(*, 1. ) = ' Dx: /Z 4: /1

(2.37)

Next, we adapt the usual assumptions made for a rectangular wing (for example, Robinson and Laurmann, p. 232) that:

(1) i-t(,(,z.) = 0 along the finite

boun-S

(o)

daries of _{1 b} I v./ ; (2)

along the bow profile and its reflection,

x = L/2; and, (3)

0

along the

keel and its reflection, zo = +H . Then,

by making use of these assumed boundary conditions, of integration by parts, and

of the fact that = 0 in the

wake, the following results may be established: ti 1.4 fro g.)

a)

-and

(2.38)

?/4)

N

_(xr

4) -% _iy

?x E. Lg+

(E-Z02:11Z-

(2.39)

The algebraic details for deriving these results may be found in Robinson and

Laurmann (1956, pp. 232-237). After

sub-stitution from (2.38) and (2,39) into (2,37), we obtain 4/, /

)

/

1 4 `4. Cx- ) - f.)t (xr, =

a-'

I,

=

:,:/;4e, x.) aN e-J

'at

(11.

)

The integrals in (2.40) are singular

for y = 0. It can be shown that in this

case the correct limiting value is obtained by taking the principal values of the

integrals. Thus, upon taking the limit,

% H

= idqlf°'"*)/211L)

g

?

_2t1L)

z6 £/t_ ?2/1 --m -e.) .1 (Y-.26)%,. N- N, (7-.26)'..(1-141

where we have assumed

N.) cr 6r. , - i-,)

(2.46)

4..24 dy, S4' -N 44t-i. ti.,4)7) cr(x., a 6

"

2i

_(2.45)

-N [fr-xdN4 44.4

(2.42)

where we have again made use of (2.36) and

the fact that 1/A.. satisfies Laplace's

equation.

(2.43)

Since there is no longer any singularity

at y = 0 in (2.43), (A,c,-E) may be

obtained immediately from (2,43) as

= _{0.111:4 ?x?,'4.}

(7f((:2ii';(4(°-4.:*():417

_(2.44)

1- N. 4 (4- i.).` 4 L

The remaining term to be similarly manipulated is the last term on the right

side of the equal sign in (2.34). This

term represents the componert velocity

normal to SV+ 51(ci) contributed by the

source distribution over the planform of

Hull 2, Hence, let us write

i/LL-.N)

(2.40)

-4N

?i.

(-te)r-Following a similar manipulation as

in the case of Art), , we obtain

1,7 (,

t.4

(x-..)(N-e.) -,1d1.71 D'Ac Ji(--N)'4-(1-,z)71 N 11

+

?A(- -"N`')

(2.41)

?Z.

N-=

A/4)c

.

)

)

Z-Note that upon taking the limit y 0, the

second double integral in (2.40) tends to

zero. Hence, only the principal values

of the first double integral and the last single integral of (2.40) remain

Next, let us derive a similar result for the second term on the left of (2,34), Clearly, this term represents the component

(0)

(0)

velocity normal to + due to

the doublet distribution over the planform

and its wake of Hull 2, Hence, we shall

write )6,4)1') y, =

?yi

sT,s,Z

e

=

?'

1 iv-1 ),16,-,)'+ (N-4) -A (N+)0

=if./CC (x,

i.)

(

1-.26)4) z..)4 -N (z-. (4ae#0 -,?1

4X-Because of the relationship between the source-distribution density and the local

ship-hull slope as given by (2,17), the assumption (2.46) means that the ship geometry is also an even function of Z, i.e., symmetric with respect to the

Oxy-plane, so that

.6(10,,,L.,)/x

The integral equation for the doublet-distribution density appropriate for the flow around a ship with two thin hulls may now be obtained by substituting the results obtained in (2.41), (2.44), and

(2.45) into (2.34). Hence,

-N

N

{(1(7-1:."24.L

)

(.-x.)a+ ei4 (I-4).

.44'4

2C(.,2) r% 2

,,,

1,Ja

b.",1"-,4e,

r.)

This is an integral equation of the

first kind for . Although

we may try to solve this equation numeri-cally, it appears that, for the problem

considered here, further simplifying

assumptions may be introduced. To do

this, we observe that the draft of a ship is generally much smaller than its length. Furthermore, for the steady forward motion of a twin-hull ship, the cross-flow effect becomes a problem of concern only when

the two hulls are sufficiently close to

each other. Hence, the two assumptions

introduced earlier concerning the draft-to-length ratio and separation

distance-to-length ratio will now be used. They

may be stated as

(2.48)

Thus the radicals Lc%-.0'.(7_ zOT2 and

(z-1.)'rz

may both be

replaced by IX-X- in (2.47). For

example, two of the terms in (2,47) will be approximated as follows;

(2.49)

lz-i.) (xx,) zo)

After making substitutions from (2.49) and (2,50) and carrying out the integration with respect to xo, the left side of the equal sign in (2.47) becomes:

r*

I

di,

2/4("'?') f

(2.47) "

E 4

(2-2,)')

(2.51)

To treat the last double-integral term in (2.47) similarly, we integrate by parts with respect to xo first before

replacing the radical

E(A-x)2.46,2

with X - . Thus, this term becomes:

= ° -1/

-jdX.

_,x

/(x-x.)L + °

=

I' I

_{41'7-4;4::::)):}

-#

That is, H/L and 2b/L are both small and

their orders of magnitude are comparable, Substituting (2.51) and (2.52) into

so that the usual approximations made for (2.47), we obtain an approximate integral

low-aspect ratio wings may be applied here, equation for 1/"111. as follows:

(2.52) (2.50)

-.'W:(xuTg..)/ax.

44 0-4)1) (x'x.)(4-4,)

t

(x-x.)

/x-x."

' d 0-Eat- /

1°)

(

j

46'4. (a-i-.)z

-#

DC /

f"

0,-(x,4)

_dz.

_ _{+ 4 b}

?),

-H

This equation, aside from being simpler than the original one in (2,47), has an interesting physical significance

which shall be discussed here. It can be

shown, by integrating by parts and by making use of the fact that log

(/-ri

satisfies Laplace's equation in the plane perpendicular to the x-axis (i.e., in y- and z-variables), that

(2.53) is precisely the same equation as

2_{7 _N/} (x, 10)

i(g-g°f+ /1"t)%...0

_

(?- io)L 1 di,

=

0-8(x, (./111( Z.,' ? -N (2.53)

In the cross-sectional plane, x = constant, the left side of the equal sign in (2.53a) represents the net y-component velocity contributed by the two line-distributions of horizontal doublets, located at yo = 0 and yo.= 2b and extending from zo = -H to

zo = H. On the other hand, the right side

of this equation represents the cross-flow velocity due to both the camber of Hull 1

and the source distribution representing

Hull 2. Thus (2.53a) states that the

doublet density may be determined by

equating these two velocities when eval-uated at y = 0 (which corresponds to the

planform of Hull 1). Hence, ).k is now

to be determined two-dimensionally at

each cross-section. This is in agreement

with one of the familiar conclusions obtained from the slender-body approxima-tion.

To find the doublet density , we

note first that (2.53) is a complete

singular integral equation. Except for

some particular types (see, for example, Gakhov, §51) for which closed-form

solutions may be obtained, the solution of such an equation is generally carried out by a reduction to a Fredholm integral

equation. A number of techniques are

available. Here, we shall use the method

of Carleman-Vekua. Namely, we shall

eliminate the singular integral by solving the corresponding dominant equation (for example, Gakhov, pp, 186-194),

Morgan (1962) derived an integral equation of this type for a ducted pro-peller problem, and used the same method

to solve the equation. A similar approach

will be applied here to solve (2.53). First, let us rewrite (2.53) in the following form: 2.)

₌

_{0e, z )} (2.54) --i where

F_

I De

x-rr ,/

4(/ 1 )

X (1*..Z ),1-1 -N Z., 4 h."-+

Note that in (2.55) we have made a

sub-stitution for cr from (2.17).

By making use of a known inversion formula, (2.54) may be solved in a closed form if Fo is regarded temporarily as a

known function, Thus, according to this

formula, (2,54) may yield the following result:

e),zft

F Fc,-1

4

?,t (x,.)

d (2.56) -H E0 But '))0, 2.0 /1-1(x, iv) --/C(.. -NJ =o , (2.57)

,

Z. (2.55) at y = O. (2.53a) ?)140c.

i-ie

tCy-y'..)1+

/A. .

I ?Z -17-2110-Zt.,/ 1 /-17-

F

a.)

di'

z where rm k,,( a'

=

_{(4-Z)Z 40,}

-H N

)t, a) = _

_

I?t(x.

-z ?), -rr-

?x

(2.60)

H

462+(z-Note that (2,58) is a Fredholm

equation of the second kind for

l',P'hz

.

However, it may not be solved in this form since there are singularities at z . +H. Instead, let us define

where

(f0.2'),

-7' (ki)[.4W.e,Z!)1.1 ,#

= -÷

F,(x'Z.)

dz °

The kernel function K(z,z') and the known function F2(x'z) are both defined

by Cauchy principal-value integrals. In

general, for a given ship geometry, (2.62)

may be solved for

*Ala

by a numerical

Then, in terms of this new unknown function, (2.58) may be written as

*(x,i)

I

k

(2.62)

(2.61)

III. HYDRODYNAMIC FORCE AND MOMENT

Thus far, both the thickness and lifting problems previously formulated

have been solved. That is, both the

corresponding source and doublet densi-ties may now be determined explicitly for

a given ship geometry. The logical next

step is to find the hydrodynamic force and moment acting on the ship in terms of

these known singularity densities.

Formu-las derived for such purposes are general-ly referred to as Lagalgeneral-ly's theorem.

Recall that we have obtained the flow field of our problem by two sheet-distributions of sources and normal (or

transverse) doublets. Across such

sur-face distributions of singularities, two important properties of the fluid velo-city are of particular interest to our

investigation, namely: (1) that part of

the velocity generated by a source dis-tribution which has a jump discontinuity in its component normal to the surface, and (2) that part generated by a normal doublet distribution which has a jump discontinuity in its component tangential

to the surface. These two properties

shall now be used to derive general expressions for force and moment.

In the subsequent development, we shall derive formulas for force and

moment acting on Hull 1 alone. Let

St

and

S2

be the wetted surface of Hull 1

and Hull 2, respectively. Let $,, and

S,w

represent the plus and minus sides

of the wake sheet of Hull 1, respectively. Next, consider a volume of fluid,

which is bounded by the wetted surface

SI

and a control surface

Z

enclosing

Hull 1 and its wake sheet. Thus, part

since we have assumed that tA is even in scheme without much difficulty. Finally,

z and also vanishes along the keel, Hence, the doublet density /4.,,.(,,,$) may be

com-upon substitution from (2,55) for Fo, pletely determined from the solution for

(2.56) becomes:

_11-kxh,i

.

(2.58)

(2.59)

(2.63)

(2.64)

1

of consists of the free surface and

both sides of the wake sheet. Hence,

the combined surface

S

and /

bounds

only fluid. As before, we shall define

the positive direction of the unit

normal to this bounding surface

Si

to be pointing away from the fluid

volume -V-c . We shall assume that

contains no singularities, and therefore

Hull 2 lies outside of

Va

. For a

steady-state problem we shall assume that

the control surface

E

moves with the

ship.

The total momentum of the fluid in

the volume tf, is

where io is the mass density of the

fluid, and i is the total fluid

velo-city relative to the ship as defined by

(2.4). Then, by making use of the

momentum theorem (see, for example, Wehausen (1973), p. 102), we may obtain an expression for the force acting on Hull 1 as

F

(3.2)

(Sn

(3.1)

in which 1, is the pressure, and

n

is the unit normal to . Next,

recall that the generating singularities representing Hull 1 are distributed only

on its planform

S,

Let (S)and

-represent the plus and minus sides (0)

of

Si6

, respectively. To simplify our

derivation, we shall assume that the planform of Hull 1 lies entirely within

Si

, the wetted surface of Hull 1.

Then, the volume bounded by the combined

(o)

surface (

,S,V)

and

kS,, )

also

con-tains no singularities. Let this volume

be denoted by Atc . Note that the

difference between -V-, and -V-, in this

case is the displacement of Hull 1.

Let us now rewrite (3.2) as follows.

"L'i6L7

es`:) .-tan...

.

(3.3)

Applying Gauss'Gauss' theorem to the first sur-face integral in (3.3) results in

[V

f(V)9 ]

(3.4)

where we have used a variant of Gauss' theorem,

5t

(g n Lis

_III[(B.v)A

A (i7,9 )

and the fact that

V.

= 0 for an

incompressible fluid. Note that Euler's

equation of motion may be written with reference to the Oxyz frame as follows:

- -

T

If we neglect the external body force

IXI , then for steady-flow problems we

have

-74-f(iG9

= o

(3.5)

From (3.5) and (3.4) it is clear that the first surface-integral term in (3.3)

vanishes identically. Hence, the

expres-,

-sion for the force, (3.3),-becomes simply-:' _{In view of the decomposition (2:8),}

' Let us write

in Which

a

substitution has been made

from Bernoulli's equation for the

pres-sure

1,

Let us now define the unit normal

vector to the planform S,(r7) to be 1,

such that it is pointing in the positive

'-direction. Then, according to our

convention, the nOrmal vector m in (3.6)

is related to 1r; AS folloWs:

1

(ST14. n, =I -I! ,I,.. (s,71-.

,(0),

Recognizing the planform Jib to be a

sheet

pf

discontinuity.,, let

us

also

define

c /1(c, _{F(i..no,.4) (SZ1}

Furthermore, We shall decompose the

fluid velocity

I.

on either side

of

s to)

into the normal and tangential

components as follows.

9,

Substituting (3.9), (3.8), and (3.7) into (3,6) and working out the necessary algebraic manipulations, we

may obtain an expression of

F

_as

F

-fiRcricr)

÷i(c.c(f...-,LC)s-sT

-

(4.4 n.(4-1-1,..-2-y]ds.

( SZ1)+ ,

as,

in which we have defined

C41 ,-,

qt =

(t

(977'4- <7)

On the other hand, the discontinuity in the tangential component will bewritten

oo

(,-)

1-t -

9'4

r:

(3.13)

In terms of (3.12) and (3.13), the three terms In the integrand of (3.10) beoome;

CZ" V)(Ct

4' 2( gv") g7',)

it 4 4 ircr

6"4" -;z1- ,( C4 0 (fL."-- _417r

e-(3.9)

_)*(

_g'-')

_{/. = (lt}

0.10) With substitUtions of the results

obtained in (i), (.40, and the

Thus, 9.e _{represents the uniform stream,}

and the 4sturbance velocity due to

Hull 2, On the other hand, y.,

and

VIC

represent the disturbance

velo-cities due to the normal doublet and

(o),

source distributions on Sib

respec-tively; If we now decompose %fir _and

yir

into the normal and tangential

components according to (3.9)', then fran (2.16) and (3.11) we have, 6.;

Cil

- 4 rz--

(3.12)

F

_{f(l.f1)115 f.} (3.,6) here (3. ,7.1)i

I- f

4 f(pni #.-11W2.14,,

t

V UX OL,4 959,' )

(c).4an-

_{e, 4 + VI,.} t V (3171

(3.8)

= = ,

expression for the force in (3.10) becomes,

F

=-4f0- (-) cis

IL

,

j)-v]is

47r //a.

is

,

(3.15)

s`ft

where 'I- = q-t+ qv- represents the

average value of the velocities on both

(0)

sides of 316 . The first term of

(3.15) is exactly the familiar form of Lagally's theorem for the force acting

on a source distribution. However, (3.15)

states that for a continuous sheet dis-tribution, across which the velocity suffers a jump discontinuity, the average value of the velocities on both sides of

the sheet should be used when applying Lagally's theorem to compute the force.

Since (3.15) represents the force due to a simultaneous distribution of sources and normal doublets, the second integral must represent the force on

the doublet distribution. Nevertheless,

it is not in the familiar form of Lagally's theorem since it is now expressed in terms of vorticity rather than explicitly related to the density of the doublet distribution.

It will be shown next that the second integral in (3.15) can indeed be brought into the familiar form of

Lagally's theorem for doublets. However,

because the singularity distribution now represents a lifting body, additional terms must be added to the usual form of

Lagally's theorem for doublets. To

derive this result, we shall make use of the properties of the potential of

normal-doublet distribution. Since the

subsequent analysis involves integration by parts, in order to simplify our deri-vation we shall assume that Hull 1 has a

rectangular planform. Hence, the

boun-dary conditions imposed, following (2.37), on the doublet density and its deriva-tives for a rectangular planform will

still be used here. Extension to a

plan-form of a more ship-like shape is

straightforward but adds additional com-plications to the final result.

To begin with, let us write the second integral of (3.15) as

Recall that in (3.13) we defined as

the jump discontinuity in that component

of <1. which is tangential to

Thus from (3.13), (3.11), and (2.21) we have

Let us now examine each component

of in (3.18). After integration

by parts, the x-component gives

o

fi I;q-s

<ix!. 4 7rf

J7 ?

4,r,111 A 2.de. But if we define (3.16)

-4-n-C?,"),

0 ?i4 ) _(3.17)

Substituting (3.17) into (3.16) and

noting that

-r

= (0,1,0) and i

the expression for Fr, becomes

F..=

4

irark-

)

(3.18)

ET

I

(x,/.1)/

/

{_L (2956,,,,t) (3.19)

ax

that is, the limit y 0 is to be

taken after the partial differentiation with respect to y, then

2),

With substitution from (3.20), the last

expression for becomes

Fi1=:-476r/a&de

S:

-47rd

1,g41

IE

(3.21) (3.20)

The first term of (3.21) is the standard form of Lagally's theorem for a distribution of doublets with its

axis in the positive y-direction. The

second term of (3.21), on the other hand, can be identified with the induced

drag of a wing. As is indicated in

(3.21), the doublet density and the

(0) "averaged" velocity normal to

_S

_ib are both to be evaluated at x = -L/2, which is the trailing edge (i.e., the stern)

of the planform of Hull 1. Note also

that (3.21) appears to provide an inter-esting insight into the difference

between a lifting and a non-lifting body.

For simplicity, let us consider the case of a thin asymmetric body making a uniform motion in an infinite fluid (which is incompressible and inviscid). So long as the flow field is to be generated by a distribution of

singu-larities on the planform of the body, both sources and normal doublets will

be required. Since in the wake the

doublet density remains constant along the direction of the uniform stream,

i.e., /...()(,z) AA (- L,2) , if we

assume /./. (x.,

0=

0

for any xo< -L/2,

then there will be no wake trailing the

thin asymmetric body. In this case the

induced-drag term in (3.21) vanishes. Thus the flow problem is solved as a non-lifting potential problem which has its velocity field continuous

every-where outside the body. On the other

hand, if

/A

is non-zero at the

trail-ing edge, then there will be both a wake trailing behind the body and induced

drag due to the wake. Clearly, in this

case, the flow problem is solved as a lifting problem.

Applying a similar analysis to the

two remaining components of F,M. in

(3.18), we may obtain the following results, ----

477-f

#11'

chci,

o

-& 47r!

(/

A( 9

I4t -M I

+471-f j (0,

, )1 _(3.22)

2..

in which the second term is to be inte-grated along the trailing edge, and the third term is to be integrated along the waterline.

In summary, let us write the force acting on Hull 1 as

where

-_{477-1 II}a dx.1,-4

(3.24)

and F

_{has already been given by (3.22).}

(3.23)

an

Moment

To obtain formulas for the hydro-dynamic moment acting on Hull 1, we may use an analysis similar tO that uSed for'

deriving the force expression. However,

moment of momentum will be used rather

than momentum. We shall use the origin

of the Oxyz frame as the point of reference

for the moment expression. Thus, in place

of (3.1), the momentum, we shall consider the moment of momentum with respect to the origin 0 = .(0,0,0) in the control

volume as

follows:,

ih which

X =

(x,y,z) is the position

vector of a point P

(x,y,z)

in the

control volume.

Then, by a Similar Analysis applied

to YC.

,

we may obtain the expression

for the moment acting on Hull 1 with reference to the origin of the Oxyz frame, as follows

1 z,

s,

cr

x

),z;

s:4

f

f,

= ivocr

Noi,

(3.26)

where

Pio,-

and A4A- represent the

first and second integrals in. (3,26), respectively.

ty a similar analysis following (3.16) we may obtain the final expression

for AV

,

as follows (3.25) e IA

4 f

I 14-2.( g

sX

--4rf

/Pt

_Wx4

sC,, -1-47:

17,142c, (-1

g,o)ji

'ft

+47rfj

fit2..(n (0,

4' E)11

. (3.27) lmo

Again, in this expression, the first two integrals represent the well known

result of Lagally's theorem. The last

two integrals in l3.27) are required only if the flow problem is being solved as a

lifting-surface problem. In particular,

the last integrals in (3.22) and (3.27) appear only for surface ship problems since they are to be evaluated along the

undisturbed waterline, z = O. These

integrals are obtained as the result of

integration by parts. Their physical

Meaning and numerical importance to a better prediction of the force and moment

are not yet Well understood- Hehte,

further investigations are required to assess their contributions,

IV. APPLICATION TO 'SWATH SHIPS

With the source density 0-6c,zey

and doublet density )1U6cZ0 obtained

in Chapter II, and the general formulas for force and moment derived in Chapter. III, we may now proceed to compute all three components of force and moment. However, since this work is still in progress and further computations are. being made, we shall not be able to covet' the entire scope of this investigation here, but shall continue to report the results as they become available... In this paper, we shall present the results ,of Our specific application of the theory,

namely: the prediction of wave

resis-tance of the Small-Waterplane-Area Twin, Hull (SWATH) Ships_

=

-N

A SWATH demihull is characterized by having a combination of an elongated, slender, round body which is totally sub-merged, and a thin strut which joins the submerged main body and extends above

the free surface. There has been a

con-siderable amount of interest in develop-ing such configurations as a new design

concept. As a part of the effort to

investigate the hydrodynamic performance of SWATH ships, an analytical tool for predicting the wave resistance was

developed. Such a tool may be obtained

by applying the theory presented in this paper.

To begin with, observe that the main body of a SWATH demihull has a round shape, and its transverse dimen-sion is wider than that of the strut. Such a configuration tends to prevent the flow from going around the keel. Hence, a SWATH demihull is less likely to behave as a lifting surface as

com-pared to a conventional catamaran. Thus,

in applying the theory to SWATH ships, it was assumed that wake may be neglected

in the initial investigation. Such an

approach does not represent a serious compromise in accuracy since, as a con-sequence of linearization, the effect of the wake may be superimposed on the results of the initial investigation as a later refinement, if necessary.

To obtain the wave resistance formu-la, we only have to be concerned with the x-component of equation (3.23). Since we have assumed that Hull 2 is the mirror image of Hull 1, the resultant wave resistance of the twin-hull ship is

twice that experienced by Hull 1. Thus,

with the effect of wake neglected, we may obtain from (3.23):

=

F

(

Fr+

_F,)

= z (R

(4.1)

where

R.. = 16frpk:

fa.

sic'ett.c...(2bk. sees *me)

°

_(4.2)

R. R.,. = /de5ec'esin(2bk.s.c'es,ne)

X[R. , (4.3)

Re?. = 16

frfk: f

de see8[1- cos(2 bk. sec2E3 son

X [P;

(4.4)

k $ee0

cr(.x.z)e cosl (kx sece),

(a' S' sm

= k. secle _{fidx,i)....(v.s.)}

kZsec'e_{{c5c7s-1(k/Sete)}

o.

k= 3/u2

, and

S

is the

projec-tion of et! and

Sft

onto the Oxz

plane.

The quantity represents the

wave resistance experienced by one hull contributed by the source distribution

alone.

Similarly,-Rpiu

is that due to

the transverse dipole distribution alone,

and Pcp.4-Ritvo- is due to the

inter-action between the sources and the trans-verse doublets.

No-Cross-Flow Assumption and Its Consequences

Here we shall discuss a special

con-dition on camber. Under such a condition,

each demihull will experience no

cross-flow effect. Recall the integral equation

(2.34) which is to be solved for the

unknown doublet density )4 . Suppose

now that a demihull is so designed that its camber satisfies the following relationship:

.?D .24

(21-.7x.)

(4.7)

S'

Then, since the right side of equation (2.34) vanishes, we may conclude that

}-1(xe,?..)=0,

(4.81 (4.5) (4.6) ,X e ! ) 32

if the integral equation has a unique

solution. Thus, if the condition (4.7)

is satisfied, then the force and moment

associated with Ak vanish. In

particu-lar (4.1), the wave resistance formula, simplifies to merely

2 (4.9)

Thus, the relationship (4.7) represents

an interesting design application. If

we apply the low draft-to-length ratio and the small separation-distance-to-length ratio approximations used in obtaining (2.52), the relationship (4.7) may be further simplified as follows:

N

Z) = - dzo

7r 46'4 (4.10)

-N

with the boundary condition

i)=0

and C(_1-/z,z)=0 . According to

(4.10), the no-cross-flow camber varies

with z as well as with x. In practice,

however, it would be more convenient and sufficiently accurate to use an averaged value of C(x,z) so that it would be

con-stant along the draft and would vary

only along the length. If we let -C. be

such an averaged value, then from (4.10) we have:

Ftw-i(xZ-) d

-(4.11)

1

/ 46")[4'210-1-:T )/d.

Z,H7r

Thus, the averaged no-cross-flow camber may now be determined for each station by using (4.11).

Results of Computations and Comparison with Experimental Data

In making wave resistance computa-tions for SWATH ships, we have assumed that the ship either has camber which

satisfies the no-cross-flow requirement of (4.11), or a low

beam-to-separation-distance ratio. Under such an

assump-tion, the simplified formula (4.9) may be considered an acceptable

approxima-tion. Thus, with this simplification,

only the source distributions are now

involved. Nevertheless, as can be seen

from (4.2) and (4.5), the actual compu-tation is still quite involved for an arbitrarily given ship geometry.

To take advantage of SWATH geometry, the source distribution is further simpli-fied to the sum of a sheet distribution representing the thin strut, and a line

distribution representing the body. With

this simplification, only the thickness of the strut and the cross-sectional area of the body appear explicitly in the

wave-resistance formula. However,

carry-ing out the computation still requires the strut thickness and body sectional

area to be described analytically. This

may present problems since hull geometry is usually given in terms of offsets.

To solve this problem, a special curve-fitting technique has been developed. This technique employs the Chebyshev

series, utilizes the offsets of strut thickness and area curve of the body, and provides analytical expressions for the strut-thickness function and the

body area curve. Analytical development

of this technique is presented in Appendix B.

With two Chebyshev-series represen-tations, one for the strut thickness function and one for the body area curve, the expression for wave resistance may be put in the following final form:

Rw = 2 (Rs+ R. Rs.) (4.12)

where R' RB' and RSB represent the

S

wave resistance of one strut, one body, w =

i(--vt,

and the interaction between strut and

body, respectively. Furthermore, each

of the three components is now expressed as a finite sum of the products of

Chebyshev coefficients and auxiliary

functions. For example, Rs is given as

follows;

Rs = (41-es-ra Lsros)

x

{A,ATsmn+B.,BW,,,,n1

, (4.13)

nI

where T and L are the maximum thickness

and length of the strut, respectively,

and

4

is a dimensionless number related

to ship speed. The equations for Rs and

RSB have a similar structure and are

given by equations (B.9) through (B.17) in Appendix B.

A computational program has been developed on the basis of those final

sets of equations. Since the three

principal equations for Rs, Rss, and Rs have a similar structure, we shall use

the one for R as an example for

discuss-ing some of the computational aspects of

the program. In the above equation for

R the Chebyshev coefficients' Asm and

S'

Bsm are determined solely by

strut-thickness offsets, while the auxiliary functions, Tsmn and Wsmn' are determined by the draft-to-length ratio and the speed (or, to be precise, the Froude number); that is, the auxiliary functions do not vary with changes in strut shape. Thus, in case there are several alter-nate designs to be investigated, the

auxiliary functions need to be computed only once provided those designs have

the same draft-to-length ratio.

In making the wave resistance com-putations, the major portion of computer time is taken up by the auxiliary func-tions, while the Chebyshev coefficients require only a minor amount of effort.

Hence, the present development is especially suitable for investigating

the effect of hull-form variations on

wave resistance. In particular, it would

be a straightforward extension to formu-late an optimization problem to investi-gate the theoretical hull form of

"minimum" resistance. The procedure developed in Lin, Webster, and Wehausen (1963) may be easily adapted to the

present formulation. The computer

pro-gram developed on the basis of the present analytical work has been applied

to most of the SWATH models investigated

at NSRDC. Only the results of two

SWATH models will be reported in this

paper as typical examples. Since no

experiment was conducted to measure wave resistance directly, comparison of theo-retical predictions can be made only

against residuary resistance. Despite

the fact that residuary resistance con-sists of the so-called "form drag" as well as wave resistance, good

qualita-tive agreement between residuary resis-tance and theoretical wave resisresis-tance was observed.

Schematic diagrams and geometric characteristics of the two SWATH models are given in Figures 1 and 2 and Table I,

respectively. The demihulls of both

models have similar cross-sectional shapes, the major difference being that SWATH III has two "straight" (i.e., zero camber) demihulls, while SWATH IV has a slight camber on each demihull intended to minimize the cross-flow effect.

Another distinction is in the waterplane shapes, as can be seen from the diagrams.

A number of test cases has been devised to investigate the merits of this theoretical tool in making

resis-tance predictions. First, to test its

ability to reflect the effect of a small change in hull geometry, the demi-hull of SWATH III with different strut sm

AWNIII .12.1AL t

-

d.)

I I 11551SL IN F. 20 19 19 17 lb 15 14 13 IN ill ç IS 6 5 4 3 2 1 0 STAT

FIGURE 1 - REDUCED' SCALE AWINGOF WTH ill FORESENTED BY WI 5275

CMOIER

11/10402I L100

20 19 II 17 16 15 14 la 12 11 10 9 11 7 6 5 4 1 2 I 0

,}47 lows

FIGURE 2 - RIM], SCALE DRAWlit OF NAN IV KRUM) BY MIL 576?

shapes was used. Results of theoretical

predictions and the corresponding residuary resistance coefficients are

presented in Figure 3. In this figure,

the same main body was used for both

cases, Although the only difference

is in the strut shape, experimental data show marked differences in the residuary-resistance characteristics

at lower speeds. This trend is very

well predicted by the theory. However,

at the higher speed range, the compari-son between theory and experiment is not as favorable for this specific

example. Although both theory and

experiment indicate that differences in the resistance characteristics will be small at higher speeds, theory predicts

a different trend. One thing that may

help explain this reversed trend is the fact that the "original" strut possessing a "Coke-bottle" shape may have a higher-than-usual percentage of "form drag" and therefore a higher residuary resistance,

TABLE I

Ship Dimensions and Coefficientz For SWATH Demihulls

Note: For a Twin Hull Configuration the

Separation Distance Between Demihull

Center-line is 3.68 feet, model scale. This

corresponds to a beam/separation distance ratio of 0.106.

The theory, on the other hand, is intended to predict only wave resistance and does not account for the viscous form drag of the Coke-bottle effect.

Figure 4 shows a comparison between

single- and twin-hull ships. Since the

demihulls of SWATH III are "straight", each is exactly one half of a twin-hull

ship. In this figure, the solid- and

dotted-line curves show the predicted wave resistance characteristics of the twin-hull (i.e., SWATH III) form and its

demihull, respectively. For most speeds,

the twin-hull interference effect amounts Wetted Surface (sq ft) (S) 17350 17540 LCB/LOA _0.479 Length/Diameter of Body 16.55 16.02 Cp (Body) _0.758 0.758 C (Strut) 0.709 0.740 T/Ls(Strut) depth to E body 0.035 1.12 0.035 1.06 diameter of body

Design Speed in Knots 32.0 _32.0

Scale Ratio _{20.4 20.4}

SWATH III SWATH IV

NSRDC Model Number 5276 ₅₂₈₇

Length of Body (ft)(Ls) 267 288

Length of Strut (ft)(L,) 226 227

Diameter of Body (ft)(D) 17.3 18.o

Beam of Strut (ft) (t) 8.0 8.0

Depth of Submergence

to g Body (ft) 19.4 19.0

Total Draft of Demihull

(ft) 28.0 28.0 Displacement (tons, s.w.) 3760. 3960. -,

08

0.2 025 03

SWATH III DEMIHULL

SWATH (CAMBERED DEMIHUL LS)

28 FT DRAFT 8 75 FT SPACING FOR PROTOTYPE RESIDUARY RESISTANCE COEFFICIENTS FROM EXPERIMENT

, MODEL FREE TO TRIM 8 HEAVE

.^, MODEL CAPTIVE

10 12 14 16 18 20 22 24 26 28 30 3; 34 36 PROTOTYPE SPEED IN KNOTS (226 FT WL LENGTH)

STRUT SHAPE

SPEED-LENGTH RATIO V/VLSTRUT

FIG 3 -EFFECT OF STRUT VARIATIONS-COMPARISON BETWEEN

THEORY AND EXPERIMENT

---2 0 7

114 16 113 20 2.2 24

SPEED-LENGTH RATIO V/VLSTRUT

FIG 4 -EFFECT OF TWIN - HULL INTERFERENCE ON RESISTANCE CHARACTERISTICS

THEORETICAL PREDICTION OF WAVE RESISTANCE COEFFICIENT

1 10 1.2 4 16 18 20 22 24 SPEED-LENGTH RATIO V/VL T 04 0.5 0.6 FROUDE NUMBER _v/V91 STRUT

FIG 5 -COMPARISON BETWEEN THEORETICAL

PREDICTION AND DATA FROM TWO EXPERIMENTAL TECHNIQUES

to an increase in resistance. However,

at certain speed ranges (for example, the speed-length ratio between 1.3 and 1.5), the interference effect becomes favorable. In this case, the twin-hull ship would have less resistance than would two

demi-hulls travelling individually. The

experimental data appear to substantiate this predicted trend very well.

Figure 5 shows the theoretical pre-diction and its comparison with

experi-mental data for SWATH IV. Data from two

model-experiment techniques, one with the model captive and the other with the model free to trim and heave, are

pre-sented. In contrast with SWATH III,

SWATH IV has camber which makes the flow field more closely satisfy the no-cross-flow assumption of the present theory. Thus, the agreement between theory and experiment in this case is the most striking among all the cases

investi-gated. It is also evident from this

figure that the theory agrees better with the data from the captive-model experi-ments.

Comparison of the computed and experimental data shows that, for SWATH ships, we may expect generally good correlation between theoretical wave

resistance and residuary resistance. A

number of reasons which are rather obvious may explain such good agreement. First, the SWATH ships have thin and slender demihulls which well satisfy the

assumptions of the theory. Second, the

traditional method of estimating viscous drag (i.e., Froude's method) must be quite satisfactory, at least qualita-tively, for SWATH ships so that residuary resistance consists mostly of wavemaking

resistance. Third, the speeds of

inter-est for SWATH ships fall in a higher Froude number range for which the theory

is known to be more reliable. As was

previously mentioned, lifting effect seems to be less important to SWATH

con-THEORY (WAVE) EXPERIMENT (RESIDUARY) SWATH 10 L\ DEMIHULL OF SWATH LI

0

z 1.0 0 w 0 cr 0 6 08 iO 2 2.0 .0 06 0.8 "> 3.0 al4.0 z _3.0 -0 .0. I-0

SHAPE THEORY IEXPERIPNT

(OWE) tRESIDUARy)

0

0 0 2 0 1.0 k2 1.4 .1.6 2.4 . STRUT 0 0 0 0,7

figurations. For a conventional cata-maran, however, lifting effect may become

rather significant. In this case, the

dipole distributions as well as the source distributions must be considered simultaneously.

ACKNOWLEDGMENT

The author is indebted to Professors R. Timman and J. N. Newman for many help-ful ideas and discussions; to Mr. V. J. Monacella for continued encouragement and guidance through the course of this

investigation; and to Miss Claire E. Wright for her expert assistance in editing and preparing the manuscript.

This work was supported under the Naval Ship Research and Development Center's General Hydromechanics Research Program and was funded by the Naval Ship Systems Command under Task Area

SR 023 01 01.

REFERENCES

Cummins, W.E. The force and

moment on a body in a time-varying

potential flow. J. Ship Res., Vol. 1,

No. 1 (1957) PP. 7-18 & 54.

Eggers, K. 'Ober

Widerstands-verhalt nisse von Zweikorperschiffen. Jahrb. Schiffbautech. Ges. Band 49

(1955), pp. 516-537; ErOrt. 538-539. Gakhoy, F.D. "Boundary Value Problems" (1963) English translation by I.N. Sneddon, Pergamon Press.

Havelock, T.H. Wave resistance:

the mutual action of two bodies. Proc.

Roy. Soc., 2er. A, Vol. 55 (1936), pp. 460-472.

Hu, P.N. Forward speed effect on

lateral stability derivatives of a

ship. Davidson Lab. Report 6-829

(Aug 1961), Stevens Inst. of Tech.,

38 pp + 2 figs.

Kellog, O.D. "Foundations of Potential Theory," Dover Publications, N.Y. (1929) ix + 384 pp.

KOchin, N.E. On the wave-making

resistance and lift of bodies submerged

in water. (Russian Org., 1936)

Trans-lated in Soc. Nay. Archs, and Mar. Engrs. Tech. Res. Bull. No. 1-8 (1951), 126 pp.

Landweber, L.; Yih, C.S. Forces,

moments, and added masses for Rankine

bodies. J. Fluid Mech. Vol. 1 (Sep

1956), pp. 319-336.

Landweber, L. Lagally's Theorem

for multipoles. Schiffstechnik Bd. 14

(1967), pp. 19-21.

Lin, W.C.; Day, W.G., Jr. The

still-water resistance and propulsion characteristics of

Small-Waterplane-Area Twin-Hull (SWATH) Ships. AIAA/

SNAME Adv. Marine Vehicles Conf.(1974), San Diego, Calif., 14 pp.

Lin, Wen-Chin; Webster, W.C.; Wehausen, J.C. Ships of "minimum"

total resistance. Int. Sen. Theoret.

Wave Resistance, Ann Arbor (1963), pp. 907-948; disc. 949-953.

Lunde, J.K. On the linearized

theory of wave resistance for displace-ment ships in steady and accelerated

motion. Trans. Soc. Nay. Archs. and

Mar. Engrs. Vol. 59 (1951), pp. 25-76; disc. 76-85.

Magnus, W.; Oberhettinger, F.; Soni, R.P. "Formulas and Theorems for the Special Functions of Mathe-matical Physics." Springer-Verlag, New York, Inc. (1966), VIII + 508 pp.

Morgan, W.B. Theory of the

annular airfoil and ducted propeller. Sym. Nay. Hydrodyn., Washington, D.C.

(1962), pp. 151-197.

Muskhelishvili, N.I. "Singular Integral Equations." English Trans-lation by J.R.M. Radok, F. Noordhoff N.V.

Fien, P.C.; Lee, C.M. Motion and

resistance of low-waterplane-area

catamaran. Sym. Nay. Hydrodyn., Faris

(1972).

Robinson, A.; Laurmann, J.A. "Wing Theory." Cambridge University Press (1956), ix + 569 pp.

Wehausen, J.V. The wave resistance

of ships. "Advanced in Applied Mechnn-ics," Vol. 13, pp. 93-245, Academic Press, New York (1973).

Yokoo, K.; Tasaki, R. On the

twin-hull ships. Rept. Transportation

Tech. Res. Inst. of Japan, No. 1, 20 pp.

(1951); No. 2, 9 pp. (1953). Translated

In U. of Michigan Dept. of Nov. Arch. and Mar. Eng., Rept. No. 33 and 34

(1969).

Zucker, R.D. Lagally's Theorem

and the lifting body problem. J. of

Ship Res. Vol. 14, No. 2 (1970), pp. 135-140.

EXPONENTIAL-INTEGRAL FUNCTION AND HAVELOCK SOURCE POTENTIAL The Havelock source potential given

in (2.15) may be expressed in terms of the well tabulated exponential-integral

function. To this end, we first rewrite

(2.15) in the following form:

(-;6, 0,2, IT

-

db/ d

_eAZI

-4-id,

(4.4..49) eGE,....4oci 0 )(Atz-f-i.6,-,)4,7 6,414. (w--.7.) a -4-5 9

_

(A.1)

I

IT"

_e

04.-x .)

t.jC.4

tvt;-4-where a ----

z + z< o,1) = ko sec20 >

0,

and = (x-x. ) cos() + (y-ye)

sine.

The third term in (A.1) involves a double integration and thus presents the

major computational difficulty. Hence,

our primary concern is to express the k-integral in terms of a well-known

func-tion for which computafunc-tions may be

per-formed efficiently.

Let us first introduce the follow-ing change of variable:

= - eiv)(a-÷-:ez)

(A.2)

Then, by making use of the definitions of exponential-integral functions, the Cauchy integral theorem, and a suitable choice of the paths of integration in a complex plane, the following identity may be established:

APPENDIX A

vo,,z)

IF;(7.4,,,,,2-4±: 7r

I

,

0;

(A.3)

The special functionsEt and Ei in

(A.3) are known as exponential-integral functions and are defined as follows:

,...

-t

E, 6414 =

,..,

(a)= -7-11fr >°

Further information concerning these functions may be found in Abramowitz and Stegun (1964, pp. 228-252).

Next, we substitute (A.3) into the

double-integral term of (A.1). However,

since the

e

integral ranges from -7r to

+tr, and variable 2Z changes sign in this interval, care is required in making the

substitution. Let us introduce the plane

polar coordinates /9 and such that

x-xo .(ocosis

Then, the variable Z., may be written as

-.(A-,..),cs#9

- _/e9cc-4e74)

(A.5)

Thus, the double-integral term in (A.1) becomes

F Z:6*1:4

_6.44(7,1z)

=Idoi

di L.-, pt.to'l =

e

-z; (A.4) (A.6)