Influence of fluid density on steady ship wave characteristics

ARCHFF

ABSTRACT

On the basis of a linearized three-dimensional potential flow formulation the wavemaking resistance of a submerged prolate

spheroid and a Wigley parabolic hull form are investigated. The body moves horizontally with constant forward speed in a fluid having a vertical density profile which is either

constant or stratified with discontinuities at the interfaces. Solutions to the disturb-ance potential of the steady perturbed flow about the moving body are determined from a Neumann-Kelvin formulation, a slender body numerical approximation based on the previous approach and a conventional slender body theory. A selection of wavemaking resistance results over a wide range of Froude number is present-ed for the two bodies moving in a homogeneous fluid, a two-layer fluid and a three-layer fluid and the influence of fluid density

stratification is clearly demonstrated.

1. INTRODUCTION

The prediction of the steady state characteristics associated with ships and submerged bodies moving horizontally at con-stant forward speed in an otherwise flat calm, stationary sea has received much attention over the years [1-3]. In the linear and non-linear mathematical models developed to cal-culate the wavemaking resistance, the wave patterns generated on the free surface, the pressure fields within the fluid, etc., the fluid is assumed incompressible, inviscid, irrota-tional and the vertical fluid density profile is uniformly constant. In purely oceanographic terms this idealisation is unrealistic since viscous effects, currents, thermal gradients, density gradients, turbulent eddies, internal waves, etc. abound within the oceans. Satellite observations have produced a greater prospec-tive of these physical phenomena and of their interactions with one another but many of the fundamental mechanisms producing the observa-tions are not fully understood [4].

In this paper it is assumed that the vertical fluid density profile is not constant and discontinuities occur at the layer inter-faces. It is the influence of this fluid

INFLUENCE OF FLUID DENSITY

ON STEADY SHIP WAVE CHARACTERISTICS

W.G. Price, Y. Wang, J.J.M. Baar

Brunel University, UK

Lab. v. Scheepsbouvikundt

Technische Hogeschool

Delft

density variation on the steady state charac-teristics of a submerged prolate spheroid

[5-7] and a Wigley hull form [6-8] which is investigated. Stratified fluid models have been considered previously (9-15], but many of these studies concentrate on the description

of the wave patterns generated in the free surface and interfaces or the interaction between a wave disturbance and a wake rather than on the determination of the wavemaking resistance associated with the moving body [9].

Because of fluid stratification, internal waves

[16-19]

occur naturally in oceans and these can be generated locally by turbulence or externally by a distant storm. A body moving in a stratified fluid produces a pres-sure field within the fluid creating internal and free surface disturbances. Since waves always radiate energy away from the region where they are being formed, the stratification of the fluid provides an additional outlet for this radiation to occur.

For a ship or submerged body moving in a constant density medium, the authors

[3,6,7,20]

developed numerical procedures to determine solutions of the steady body motion problem. These were derived from a linearized three-dimensional potential flow formulation and they describe the disturbance potential of the steady perturbed flow about the moving body. The solutions were obtained by means of a Kelvin wave source distribution method. This is based on an integral identity for the Neumann-Kelvin potential involving an integral distribu-tion of Kelvin wave sources over the body's hull surface and water-line contour. The un-known source strength is determined by solving a Fredholm integral equation of the second kind which ensures that the kinematic condition at the hull surface is satisfied [21]. This theoretical approach and an iterative numerical approximation procedure [22] were applied to six different hull forms and the calculated results showed good agreement with experimental data in both a qualitative and a quantitative manner [6,7].

These approaches together with a slender body theory are again considered to solve the similar problem in astratified fluid. Through

the distribution of panel sources over the mean wetted hull surface the first two methods described previously allow the shape of the body to be defined but at the expense that solutions require much computational effort. The slender body method is a much simpler approach to use, requiring reduced computation-al resources but suffers from the disadvantage of being unable to describe the exact body shape.

These methods are used to determine the wavemaking resistance of a submerged prolate spheroid and a Wigley parabolic hull form travelling at constant forward speed in a homo-geneous fluid and a stratified fluid. This provides a simple means of demonstrating the influence of fluid density stratification on the steady state wave characteristics.

2. MATHEMATICAL MODEL Izo Yo free surface h1 0' 1 -h2 density pl I ,z E3

--

TR J y

n-hi -hi+1

ith fluid layer

Figure 1. Body moving in stratified fluid Figure 1 illustrates an arbitrary shaped rigid body travelling with constant speed U in a stratified fluid of depth H. The body is of overall length L, beam B and travels at level trim at a depth h below the mean free surface. The right hand Cartesian coordinate reference frame Oxyz moves with the body, with origin 0 situated at some convenient position in the body (i.e. centre of gravity, amidships, etc) and axes Ox, Oz point forward to the bow and to the vertical respectively. The fixed right hand axis system Oix.y.z. with origin 0' in the mean free surface z0=0 is as shown and at time t=0 the axes systems are parallel to one another and their origins lie in the same vertical plane i.e. zo=z-h.

Let P(z.) denote the density of a stable, stratified, incompressible, inviscid and irrotational fluid contained between the free surface z =0 and the flat seabed at z.. -H. In this mode i the fluid is assumed stratified into N distinct layers each of constant density Pi and thickness h. -h. for i=1,2,...,N (h=0

1+1 1 1

denotes the mean /ree surface and hN+1=H the bottom). The fluid density profile is discon-tinuous at the N-1 interfaces and when the

fluid is disturbed the layers of fluid do not mix.

The fluid flow in this idealised stratified fluid model may be described by potential functions 4).(x ,y ,z ,t) and fluid velocity

i000

functions

2i(x.,y0,zo,t) = V

yx.,y0,z0,t)

for i1,2,... ,N.

In the moving reference frame these functions can be expressed as

Tp.(x

loco

,t) ci,.(x+Ut,y,z-h)

-Ux + d'i(x,y,z) (1)

and

= -UI +V.

(2)

1

An application of Bernoulli's equation allows the pressure p.(x ,y z ,t) within the

1 o o' o layers to be expressed as p. = -p [ + ,T).2 + + i it 2 ixo lyo 1Z0 + g zo +C. (3) 1

where C. = p + -,5 p. U2 + gha for i=1,2,...,N and pa aenoth tEe Atmospheric pressure at Eo=O. Here and in subsequent formulations_

4)it = 34)./t, 324,./3x2 =

i

i

O. etc and V,72

denote the grad and Laplacian operators respectively.

2.1. Boundary conditions 2.1.1. Continuity equation

Throughout the fluid domain D(=.E, D.) the continuity equation is expressed as 1

v2i.

= o

(4)

in each domain D. provided no fluid mixing occurs between tie layers.

2.1.2. Free surface

The free surface elevation is given by

zo =n1 (xo' Yo' t)

where n denotes the upper surface disturbance

1 .

in layer 1=1. The kinematic condition exist-ing on the free surface may be expressed as

TE:

Ill

- zol = or _ nlt

+1x0

_{nxt,+ 4),Y0} n _ yo o = 0 4,lz (5) on zo = _n1

Since the position of the free surface is unknown an additional dynamic boundary condition can be deduced by letting the pres-sure at the free surface take the atmospheric value. That is at

zo

= n

p1 = pa giving

i

fT, 2 4. i

26

2 1 4. a, _ u2 = 0

lt 2 _Ixo ly, lzo ° 1 2

(6)

from equation (3) --N+1 0 1.2

2.1.3. Interface

The surface elevation in the upper inter-face of the ith layer is

z

= n.

(x , y , t) - h.

o 1 o o 1

and from equation (3) the pressure is given by

- 1 r- 2

- 2

- 2

p.+0. [O. +- 14).

+4). 1 + g

n]

= C.

1 it 2

ixo iyo lzo 1 1

The dynamical condition on this interface requires that P--1 = P. 1 1 resulting in

1 -2

2

_2

p.

_1-1

[i

_{(i-l)t 2}

+- (4)

_(i-1)x

+i(i-1)y,"(i-1)zo1

o 1 + g n(i-1)]

-

2 Pi-1

U2 =

riro _i_l c ci?

42

i

+V

i}

, 1

i

+ g nii - -2- Pi U2 "1 I t 2 ix y z o o o on z - h. 1 1

The kinematical condition is again satisfied by the expression

-{z -

n

-

h.} = 0

Dt o

i

1

and this is valid in the adjacent fluid domains such that

n. +0.

_{it lx} n.

+0.

n.

-0.

= 0

lx0 iyo iyo izo

it (i-1)x

n.oo (i-1)yo

niy

=

0o

on zo 1 1."1

2.1.4. Body surface

Because the fluid cannot permeate the wetted body surface H, it follows that in the

ith layer

(T. = 0 = on H

₍₉₎

in

where the unit normal n is positive pointing out of the hull.

2.1.5. Additional conditions

To complete the picture, additional bound-ary conditions must be imposed ensuring that the energy fluxs of the disturbances radiated by the moving body are directed outward at infinity (i.e. a radiation condition which imposes no wave disturbance a long distance ahead of the body, only waves behind the body), and on the seabed

Nz = 0 on zo = -H (= -hN,1) Or

gradN +0

as

z0+

-

0,

if the Nth layer is of infinite depth.

2.1.6. Boundary conditions in moving reference frame

The substitution of equations (1) and (2) into the previous boundary conditions allows a set of boundary conditions to be defined with respect to the moving reference frame Oxyz. The disturbance potentials Oi satisfy the following boundary conditions

continuity 724,i = 0 for i=1,2,..,N (11)

free surface

12

g 111 = U*lx (4)1X

ly

cq_z)

Up

= 4, Ti + 4) Ti

-c

(13) x lx lx ly ly lz on zo = ni or

z =

ni + h

ith layer upper interface

1 , 2 2

Pi-1 [g ni-14(i-1)x +

2'

cP(i-1)x .4'

(1)(i-1)Y +

2

4)(1-1)21]

[g ni

-

T.4ix +t;{4,2ix +

4y + 4z}]

(14)

and

4)(i-1)xlnix +

c(i-1)

Illy - 4'(i-1)z = °

(15)

= 0

calz=ni-h.or

z =

1

ni

-hi

+ h

body surface in ith layer

O. = Un on H (16) in radiation condition [23]

0(1/1x1)

x>0 O. = - as dx1=i(x4y2+,2).,w if

6(1)

x<0

where 0(.) and o(.) denote the usual Landau order symbols [24] .

seabed conditions

Nz = 0 on z=-H or z=-H+h (finite depth)o

(17)

grad0Nz = 0 as z+-0. or z+-(infinite depth)

3. LINEAR NEUMANN-KELVIN THEORY

Unfortunately the complicated non-linear nature of the free surface and interface conditions given in equations (12-15) prohibits the development of the exact solutions of the disturbance potentials. Therefore some method of approximation is required.

For a fluid of constant density Wehausen [2] introduced simplifications into the analy-sis by adopting regular or singular perturba-tion expansions. These procedures rely on the f-U cPixl

nix

(Piy niy

(1)i.z

(12)

ix = ix iy + +

--x (vi) +

(8)

ability to choose a suitable small parameter associated with the problem such that as E

decreases in value the disturbance near the free surface is gradually reduced. Specifical-ly E may be related to the beam/length ratio or draft/length ratio of a ship (i.e. the classic thin, flat and slender ship approxima-tions), the Froude number (i.e. the slow ship approximation) and the Froude number based on the immersion depth of a deeply submerged body. Each of these approaches results in the

diminishing of the free surface disturbance as is made smaller but with the penalty that they destroy the general three dimensional character of the problem.

An alternative scheme [3,6,7] may be based on linearizing only the free surface condition while the other boundary conditions are retained in their exact

form.

That is, the conditions valid on the disturbed free surface z = 0 + h may be expanded about the mean free surtace z = h (or zo = 0) using a Taylor series expansion of the form

0(x,y,n1+h)

=1(x,y,h)

+ 11 01z(x'y'h) +.. (18) (This approximation is formally inconsistent for surface ships, but some justification for its adoption may be derived from experimental evidence [25,26] which suggests that the disturbance of the free surface is relatively small except near the bow [3,7]).

The introduction of this type of expansion into equations (12) and (13) produces expres-sions of the form

n -

_(U/g)lx

= (19)

1 higher order terms

and in products of 0,

and their derivative*

lz

+ (U2/g)

0lxx

= (20)

which are valid on z=h. It can be shown [27] that the terms on the right hand side are of order

F2 0,2

or higher and these are assumed to be small in comparison with the linear terms on the left hand side of the equations. By this procedure the nonlinear boundary value problem transforms into a linear one provided that terms of 0(Fn212) are negligibly small and can be validly neglected.

If a similar approach is adopted for the stratified fluid, then at the ith interface z=n--h.+h the disturbance potential may be expressed as

0i(x,y,ni-hi+h) = 0i(x,y, h-hi) +

0.6.(x,y,h-h.)+ ."

1 1Z 1

and equation (1)4) becomes

P. [g n. - u 4) . + higher order terms in

1-1 1 (1-1)x

ni' 4)(i-1)'

etc]

=P .[gn.-U U.

+ higher order terms in

1 1 lx

n., cb., etc] (22)

i

1

on z = h-hi. By analogy with the previous discussion the higher order terms are of order

F 201' F

2 0.2.or higher and these are again

n

1-a

i

assumed sall in comparison with the remaining

linear terms.

Therefore the non-linear boundary value problem described by equations (11-17) is transformed into the linear Neumann-Kelvin

problem given by the following set of equations:

continuity

V2 0.

= 0 in each D., i11,2,...,N. 1 (23) free surface g n -U

=0

1 lx

or*

_lxx4k

_{o *lz}

=0

U nix

+lz

= 0 on z = h (2)4) where ko = g/U2.

ith layer upper surface

pi

-1[ g r.-U *(i-1)x)

= [

g ni

U 4)l x] on z = h -h. 1

ni

+

0(i_1),

p.

+ 4).

=0

1X 1Z or

15i-1

PP(i-i)xx

ko

*(i-1)z]

=

6

+k .

'ixx o 1Z where 6

= 0

on z = h -h. . 1 on z=h-l- (25) 1 *(i-1)z = *iz

i-1 = Pi-l/Pi, ko=1-71Fn-2 and i=2,3..,N.

wetted hull surface in the ith layer

0.

= U n on H (26)

in

x

radiation and bottom conditions as given previously.

On the assumption that the disturbance potentials can be derived and when the effects

of sinkage and trim are neglected, the wave-making resistance of the body can be determined by integrating the hydrodynamic pressure over

the body's hull surface. That is a non-dimensional wavemaking coefficient may be defined as [7]

(21)

Cw = Rw/p1 U2L2 =

jj

(j

x 2

-1IVj

12)nxda

(27)

where

pl

denotes the fluid density of the upper-most layer and the (non-dimensional) disturbance 01 refers to the body surface in the jth layer

(Say). This approach is adopted to evaluate the wavemaking resistance coefficient when the body moves in a homogeneous fluid. However because of numerical difficulties a method developed by Sabuncu [9] based on a three lx

=

(i)

.

-dimensional extension of Lagally's theorem is used to determine the coefficient when the body moves in a stratified fluid.

4.

ALGERBRAIC RECURSIVE RELATIONSHIPS The substitution of the transform

W

cbi(k,z)

ct'i(x,Y,Z)

k2 y) dy

(28)

into equations (23-25) gives the set of equations 4'izz k2 = °

ihallp.(29)

k 0

-k1

2 01 = 0 on z=h (30) o lz 6i-1{ ko °(i-1)z k12 k 0. - k 2 0. o iz 1 on z=h-h. (31) (45(i-1)z =

and (by way of example) for an infinitely deep Nth layer

(1)Nz _=HO on z = . (32)

In these equations i1,2,.. .,N and the horizontal wave number vector k T (kl, k2) =

(k cose, k sine), k = k12+ k22

)2.

Let there exist a unit strength source at position Ei = (Em,

ni, ci)

in the ith fluid layer. This represents an idealisation of sources distributed over the wetted surface area of the body. Equations (29-32) are satisfied by a solution of the form

+k(z-c) k,z) = -6.. e . + j -27 A.(k)ekz + B.(k) e-kz (33) J J

-where on the interface z = h-hj the upper negative sign is taken when j < i and the lower positive sign when j > i, for j=1,2,..,N. The symbol 6.. = 1 when i=j and zero otherwise.

ij

The substitution of this solution into equations (29-32) gives the following set of algebraic recursive relationships from which the constants A. B. can be determined. That

J J is -2k(h-h.) 6j-1 Aj-1 a 6j-1 Bj-1 e - B. e-2k(h-hj) { j-1d. di(j-1) -dij}(k12 -

kk

27r (k12 - k ko) -k(h-h.) j for i = 1,2,...,N; j = 1,2,...,N. (314) -2k(h-h.) -2k(h-h.) A. -B. e J - A + B. e J 3-1 3-1 j 3 - k(h-h.-C.) -k(h-h.) r," 8i(j-1)

t 6d

e

+ J e 1 _J

.

2n ₍₃₅₎ for j"2,3,... ,N.

BN=0

because an infinite deep lowest layer is assumed and

o = 0, h1=0 (i.e. free surface),

k 2+ kk k + k sec20

a = 1 o = o

k 2 -k

ko

k - ko sec2e 1

After the evaluation of the constants A., B. the fundamental solution describing the J

p8tential at a field point x(=x,y,z) in the jth layer due to a unit strength source at

i

E.(=E, n., c.)

in the ith layer has the form

-1

7

= 1{(-d../27)e;k(z-)

A. (E., k)e z + B. (E., k)e-kz}

-eik(x cose + y sine) dk de

with the radiation boundary condition still to be considered.

5. BODY MOVING IN STRATIFIED FLUID

The approach adopted to describe the free surface and internal disturbances generated by the arbitrary shaped body moving in the stratified fluid follows a generalisation of previous methods [3,6,7,201. That is, the steady flow around the body travelling in or near the free surface of a homogeneous fluid

is modelled by a three dimensional linearized potential flow theory and the wavemaking solution derived frOm a Neumann-Kelvin formu-lation. From the developed boundary value problem the associated Green function can be physically interpreted as the linearised

potential of the steady flow created by a source travelling below the free surface. Thus by

suitably distributing translating sources over the mean wetted hull surface, solutions to the disturbance potential may be derived and

cal-culations made of the wavemaking resistance, wave pattern generation, etc.

In the stratified fluid, it can be shown that the Green function G.(E., x), j=1,2,...,N associated with the Neumainlielvin _{problem is} equivalent to the form of the fundamental solution given in equation (38). Thus a boundary integral representation of the strati-fied Neumann-Kelvin potential can be developed by applying Green second formula to the disturbance potential and the stratified wave potential G.(E., x). Furthermore, the

j

-1

-'

(36) -+ -+

stratified Neumann-Kelvin potential can be represented by a distribution of translating

sources over the mean wetted hull surface such that [21]

O.(x) if G.(E.,x) Q(C.) dh (C.) +

j -

-1

-1

Fn2j Q(.) n(.) dy (E.) (39)

where Q(C.) is the unknown source strength determine

_a

from the equation

Q(.) + j G.

(c.

x) Q(E.) an(.) + jn

-1,-

-1

2

Gjn(si,x) Q(Si) nx(§i) dy(i) = n(x) (40)

on the body surface, and the normal derivative

G. (E., x) = V G.(E., x) . n(x)

jn j

In these expressions, the line integral is evaluated along the still water line con-tour c of a free surface piercing body or along an equilibrium interface contour if the body straddles an interface. Here for sim-plicity it is assumed that the submerged body moves within a single layer only so that all line integral contributions are zero.

The numerical solution of equation (40) for the unknown source strengths tends to be a time consuming task and simplifying and/or alternative approaches have been proposed. Based on the work of Noblesse [22], Baar and Price [7] investigated the merits of the "slender body approximation" which introduces a numerical refinement to ease the solution process. The assumption of length L» beam B or L» draft T is not necessary and the approach relies on an iterative procedure commencing from the hypothesis that the local normal hull velocity component nx is a reason-able first approximation to the source strength

Q. Subsequent iterations should provide an

increasing accurate prediction of the source strength.

Alternatively for a submerged slender body (L»B or L»T) of simple form (i.e. a prolate spheroid) a major simplification is

achieved by simply distributing sources along the longitudinal axis of the body rather than over the mean wetted hull surface areas. The source strength is given by

U d S(E)

=

47 dE

where S(E) is the cross-sectional area. For a prolate ellipsoid

S(E) = 0.25Y Drix { 1 - (2E/L)2}

where -L/2 < E < L/2, D denotes the maximum diameter amidships and Re source strength

Q(E) = 0.5 U (Dmax/L)2 E. (41)

6.

TBE GREEN FUNCTION

To illustrate the form of the Green func-tion G.(C.,x) let us consider a three layer

fluid iiti the source positioned in the middle layer at E2(= E2, n2, 2). After the necessary algebraic manipulations it may be shown that the Green function in the upper, middle and lower fluid layers are respectively of the

form

In these equations the coefficients are given

as a = x cose + y sine r = /{(x-E2)2+ (y-n2)2+ (z-c2)21 -k(h-h2) -kh, _kh2 A2 = a e

{(1-d2)e':()a+d1)e

' Ik((lhi611 13;2) / k(h-11 -' ` _{+ (1+02)e} }/27D kh k(h1:3()h-h2-2) -a(1-81 -kh., B2 = (1-62)e [a {(a+81)e - + -( )e kh, ' + (1-61)e kh 1 e k(h-h2-;2) (1+01 )e 2 1 e ) /27D D = a(1,52){ (a+di)e-kh 2 )ekh2 e-k(h3-h2) 1 -kh kh

+ (1+02) fa(1-1d )e 2 + (i+o1 )e 2le 3 '2

a =

(k+kosec20)/(k-kosec2e),

dl = pl/o2 62 = P2/P3 -2k(h-h3) _(e-kC2/27) A3 = A2 - B2 e -2k(h-h ) -Ya(2h-2h2-C,) 2+ce _{' /271i/} Al = a [ A2-B2 e 2kh B1 = - (A1/a)e.

Following the approach developed by Sabuncu [9] to describe the flow around a source moving in a two layer stratified fluid, it may be shown that the previous Green functions can be represented by the general expression 1 G.(E x) =

-

- 1

'

_ rij

j

r2j 1 7/2 + de

f

{ (k-kosec20) Fij(k,e) + F2j(k,e) 1 c(k,e) dk/(k-ksec2B) D(k,e)

2kh

(a+e 2,

G1 (-2E ,x)

-=

{A1 ekz+B1e-kz

_ 0

eika dkde

(E x) = - 1 _{i'{A2 ekz}

G e-kz1 eika dkde

2

+

- 2

G3(E2,x) = f

A3

j ek(ia+z) dkde

x)

1

-+

f712{(a-kosec2e) F1j(a10e) +

eb

F2J (cc'

e)} S(ai,e)

dePackosec20)Dk(ave) 7/2

+ _{F.(0) de/D0(0)}

for j=1,2,3. Here denotes a principal value integral and the upper value of the summation k depends on the number of singular roots occurring in the function D(k,e) which in turn depends on the value of the forward speed. The general expression relates to an axis system with origin transferred into the first interface, such that the source position defined with respect to the moving equilibrium axes is

.(2,n2,(=c2+h2Th)< 0). The

individual terms occurring in equation (42) are defined in the Appendix.

For 61 = 1 i.e. p =p these Green

fund-1 2

tions reduce to forms which describe the behaviour of a unit strength source moving at constant forward speed in a finite depth upper layer above an infinitely deep lower layer and for 62=1 i.e. p

i2

=p , the functions describe a

1

source moving n an infinite deep lower layer below a finite depth upper layer. Finally for 6 =1=6, the functions describe a source moving in a fluid of uniform density.

7. WAVEMAKING RESISTANCE CALCULATIONS 7.1. Homogeneous fluid (prolate spheroid)

In his pioneering investigations into the wavemaking resistance of a submerged prolate

spheroid, Havelock [28] derived a simple approximation formula by using the axial source distribution corresponding to the motion of a spheroid in an infinite fluid. However, in the presence of the free surface this distribu-tion does not produce a true spheroid but the accuracy of the approximation increases with

immersion depth. Farell [5] derived a complete analytical solution of the Neumann-Kelvin problem by representing the potential and source strength as a series of spherical har-monics. This solution encompasses the Havelock solution.

To assess the merits of the different approaches, their numerical accuracy, influence of idealisation, etc wavemaking resistance calculations were performed over a wide range of Froude number, F. = U/V(Lg), and the results compared. The spheroid is of block coefficient 0.5236, B/L = 0.1667 and the centroid is below the free surface at an immersion depth of 0.1242L when F>0.4 and 0.1633L when Fn < 0.4.

In the Neumann-Kelvin calculations the port hull wetted surface was idealised by 220,

468, 798 panel elements whilst in the slender body calculations 20 sources were evenly

distributed along the longitudinal axis. Wher-ever possible in the approximation of the hull surface by a panel distribution, panels with low aspect ratios were avoided, variations in

(42)

size between adjacent panels were kept as small as possible and panels were concentrated in areas of large curvature where the fluid flow can rapidly change. Because the source strength is assumed constant on each panel, as a general guideline, the panel dimensions were chosen to be less than one eighth of the characteristic non-dimensional wavelength A=27 F.2. This re-sults in an increasing fine mesh descretisation as the Froude number decreases.

Figure 2 illustrates the non-dimensional wavemaking resistance coefficient determined over the Froude number range 0.4 F.S 0.8 and Fn <O.4.

For the case Ft., 04 shown in figure 2(a), Farell's results [5], the full Neumann-Kelvin formulation and the slender body numerical approximation show very good agreement between themselves, but as expected, Havelock's approxi-mation or the traditional slender body theory calculation shows less favourable agreement with the other data. It is interesting to note that in this speed range the Neumann-Kelvin results derived from the different idealisations of the wetted surface area show remarkable agreement with one another indicating a convergent solu-tion not too dependent on mesh shape, size, etc.

For F.< 0.4, Farell's results, the predic-tions from slender body theory and those from a slender body numerical approximation (i.e.Q=n ) are more closely aligned as shown in figure 2( ).

In the last approach, a full body panel distribu-tion of 176 (22x8) panels was adopted and the co-efficient calculated using Lagally's method [9] rather than a body surface integration as used when F > O.L. This modified approach requires Only tRe far field disturbance solution whereas the body surface integration includes the near and far field solutions. In contrast to the ax-ial source distribution method, the modified approach allows a more general hull geometry to be modelled. The Neumann-Kelvin calculations contain the near and far field solutions and the wavemaking resistance coefficient is determined from a body surface integration. These calcula-tions were found to be sensitive to panel dist-ribution requiring denser concentrations in areas of more rapid curvature before the displayed results could be achieved.

Figure 2(b) illustrates the data derived from the different approaches. The three panel distributions adopted in the Neumann-Kelvin app-roach produce results in reasonable agreement with one another and display a similar trend to the other data, but slightly shifted. From the limited evidence presented, for F.< 0.35 and Fn>0.55 the far field solution provides the dominant contribution to the calculation, other-wise, both solutions must be considered.

7.2. Homogeneous fluid (Wigley hull) The wavemaking resistance of a Wigley parabolic hull form [8] has been extensively measured and calculated using a variety of

approaches [1-3, 25, 29]. Figure 3 illustrates a comparison between experimental data and

theoretical predictions. The measured wave pat-tern data were derived from a wake survey

analysis, whereas the measured wavemaking resistance was obtained by subtracting the estimated dominant viscous resistance from the measured total resistance [30].

For a hull of B/L = 0.1, draft T/L = 0.0625 and block coefficient 0.4444, the displayed predicted data [3,7] are derived from an

'exact' Neumann-Kelvin approach, a slender body numerical approximation model and a

slender ship theory approach. In the first two methods the mean wetted port hull surface area is discretised by 252 panels of which 36 lie in the free surface. As can be seen reasonable agreement exists between these predicted values and the experimental data but

increasing differences are observed with decreasing Froude number. As discussed previously for the prolate spheroid, this may be due to an inadequate discretisation of the

hull form, but with decreasing forward speed measurement is also prone to error and

dif-ficulties. The slender body theory results show poor agreement with the other data, clearly overestimating the magnitude of the experimental data and they display a much more oscillatory behaviour.

7.3. Stratified fluid (prolate spheroid) On the evidence presented in section 7.1 and for ease of computation, in the low Froude number range the prolate spheroid of B/L = 0.1 was modelled using an axial source distribution

in a slender body theory. For the spheroid travelling in a two layer and a three layer

stratified fluid resistance coefficients were calculated using the method proposed by Sabuncu [9] (i.e. Lagally's theorem). A

selection of results 1,3 presented in figures 4 and 5 and these are associated with a 20 axial source distribution, although the results derived using 40 (20) 80 axial source distribu-tions are in very close agreement with those shown.

For the two layer case (61=0.976), figure 4 illustrates the predicted wavemaking resistance coefficient for the spheroid moving in (a) a finite depth upper layer over an infinite deep lower layer, (b) an infinite deep lower layer under a finite depth upper layer, (c) a finite depth lower layer under an infinite deep upper layer and (d) an upper layer of finite depth sandwiched between a rigid lid and an infinite deep lower layer. In the low Froude number range considered, the predicted values are dominated by the contributions from the disturbance

created at the interface and where appropriate (i.e. figures 4(a,b)) the influence of the free surface disturbance is found to be practically non-existent. The maximum wavemaking resist-ance is associated with the generation of internal waves with the largest wave amplitudes.

As the forward speed increases the amplitude of the internal wave disturbance decreases and so does the wavemaking resistance. However for the case of the spheroid moving in the upper layer (i.e. figure 4a), as the Froude number

increases the contribution from the free sur-face Kelvin wavemaking resistance becomes dominant as shown in figure 4e. The magnitude of this component depends on the position of the spheroid relative to the free surface

though the linearized mathematical model ignores the possib4lity of wave breaking, etc which would occur when the body is in close proximity to the free surface.

For the case f = 0.5t1in figures 4(a) and (e), additional calculations were performed using the slender body numerical approximation model. A full body 176 panel distribution was used and it :as found that the calculated wave

Patterns generated in the interface and free surface, and the wavemaking resistance coeffic-ients were of similar forms and magnitudes as those determined using a 20 axial source distri-bution, but these results reauired a consider-able increase in computing effort. The wavemak-ing resistance coefficient data are in Fig.4(a).

Figure 5 illustrates results derived for the wavemaking resistance coefficient when the body moves along prescribed paths in the middle layer of a three layer fluid stratification

(61=0.9876, 60=0.9878). Internal waves are created in both interfaces and their respective contributions to the wavemaking resistance coefficient depends on the proximity of the body to the interfaces, density and layer thickness distributions.

A comparison of figures 4 and 5 shows that the wavemaking resistance coefficient curve retains its form but changes in magnitude with increasing fluid stratification.

7.4. Stratified fluid (Wigley hull)

The influence of fluid stratification on the wavemaking resistance of a surface piercing body was examined using the Wigley hull form

described in section 7.2. A double layer fluid stratification (5=0.976) was considered and two

1

separate cases examined. That is, an upper layer thickness of h2=2T and h0=4T over an infinitely deep lower layer. The results determined from a zero order slender body numerical approximation (i.e. n1Q) are dis-played in figure 6 and clearly illustrate the role of the fluid stratification on the form of the wavemaking resistance coefficient (cf. figure 3). This was calculated using Lagally's method [9], reouiring only the far field solu-tion (i.e. the single integrals in equasolu-tion (42)).

At low Froude numbers the free surface contribution to the value of the coefficient is negligibly small and the internal wave

disturbance dominates, but its magnitude decrea-ses with increasing upper layer thickness and Froude number (see figures 6(a) and 6(b)). At higher Froude numbers the free surface wave disturbance provides the dominant contribution but as previously discussed [3,29] this zero order slender body numerical approximation tends to over prediction and gives results of

a more oscillatory nature than observed experimentally. However from the evidence of section 7.2 for a more exact calculation of

the free surface influence at higher forward speed a Neumann-Kelvin approach could be deve-loped which includes contributions from the near field and far field disturbances. This, unfortunately, would require a very large increase in computing effort and computer capacity.

CONCLUSIONS

For a ship or submerged body travelling at constant forward speed in a homogeneous or stratified fluid a linearized three dimension-al model is developed to cdimension-alculate the wave-making resistance. Solutions to the

disturb-ance potential are determined from a Neumann-Kelvin theory, a slender body numerical approximation and a conventional slender body theory. From the evidence derived from studies using a prolate spheroid and a Wigley hull form in fluid of constant density it is shown that the accuracy of these approaches depends on forward speed, panel size and distribution, etc. For the spheroid, at low forward speed, the slender body theory, analytical results [5] and the slender body numerical approximation give comparable results but the Neumann-Kelvin approach is very sensitive to panel discretisation. FOrthe Wigley hull, at higher forward speeds, the Neu-mann-Kelvin theory and the slender body

numeri-cal approximation produce results and trends which better fit experimental wavemaking resistance data but predictions from slender body theory overestimate and possess a more oscillatory behaviour than is seen in the data.

A body travelling in or near the free surface of a stratified fluid excites wave disturbances at the interfaces between the layers and on the free surface. _{At low} for-ward speed the wavemaking resistance is dominated by contributions from the long wave-length, internal wave disturbances whilst in comparison, contributions from the free sur-face waves are negligibly small. As the forward speed increases the wavelengths and amplitudes of the internal waves decrease and the opposite effect occurs with the dominant contributions coming from the free surface dis-turbance. This changing role in wavemaking resistance is clearly visible in figures 4(e) and 6.

ACKNOWLEDGEMENT

We gratefully acknowledge the support of the Ministry of Defence Procurement Executive.

REFERENCES

Kostyukov, A.A. _{Theory of ship waves} and wave resistance. Iowa: Effective

Communications Inc. (English transl), 1968. Wehausen, J.V. _{The wave resistance of} ships. Adv. Appl. Mech. 13, 1973, 93-245.

Baar, J.J.M. A three dimensional linear analysis of steady ship motion in deep _water. Ph.D thesis, Brunel University, 1986.

Hughes, B.A. Surface wave wakes and internal wave wakes produced by surface ships. Sixteenth ONE Symposium, Berkeley, 1986.

Farell, C. On the wave resistance of a submerged spheroid. J. Ship. Res. 17, 1973, 1-11.

Andrew, R.N., Baar, J.J.M. and Price, W.G. Prediction of ship wavemaking resistance and other steady flow parameters using Neumann-Kelvin theory. Trans R. Istn. Maw. Arch. 130, 1988, 119-133.

Baar, J.J.M. and Price, W.G. Develop-ments in the calculation of the wavemaking resistance of ships. Proc. R. Soc. Lend. A416, 1988, 115-147.

Wigley, W.C.S. Calculated and measured wave resistance of a series of forms defined algebraically, the prismatic coefficient and angle of entrance being varied independently. Trans R. Istn. Nay. Arch. 86, 1942, 41-56.

Sabuncu, T. The theoretical wave resist-ance of a ship travelling under interfacial wave conditions. Norwegian Ship Model Exp.

Tank Publ. 63, 1961.

Hudimac, A.A. Ship waves in a stratified ocean. J. Fluid Mech.11, 1961, 229-243.

Mei, C.C. Collapse of a homogeneous fluid mass in a stratified fluid. Proc. Twelfth Intern. Congr. Appl. Mech., Springer

(Berlin), 1969, 321-330.

Miles, J.W. Internal waves generated by a horizontally moving source. Geophys. Fluid Dyn. 2, 1971, 63-87.

Keller, J.B. and Munk, W.H. Internal wave wakes of a body moving in a stratified fluid. Phys. Fluids 13, 1970, 1425-1431.

Keller, J.B., Levy, D.M. and Ahluwalia,

D.S. Internal and surface wave production in a stratified fluid. In Wave motion 3, North-Holland, 1981, 215-229.

Crapper, G.D. Ship waves in a stratified ocean. J. Fluid Mech. 29, 1967, 667-672.

Benjamin, T.B. Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29, 1967, 559-592.

Benjamin, T.B. Internal waves of finite amplitude and permanent form. J. Fluid Mech. 25, 1966, 241-270.

Lighthill, M.J. Waves in fluids. Cambridge University Press, 1978.

Ono, H. Algebraic solitary waves in stratified fluids. J. Plays. Soc. Japan 39, 1975, 1082-1091.

Baar, J.J.M. and Price, W.G. Evaluation of the wavelike disturbance in the Kelvin wave source potential. J. Ship. Res. 32, 1988,

44-53.

Brard, R. The representation of a given ship form by singularity distributions when the boundary condition on the free surface is linearized. J. Ship. Res. 16, 1972, 79-82.

Noblesse, F. A slender ship theory of wave resistance. J. Ship. Res. 27, 1983, 13-33.

Dern, J.C. Existence, uniqueness and regularity of the solution of the Neumann-Kelvin problem for two- or three-dimensional submerged bodies. In Proc. Second Int. Conf. Numerical Ship Hydrodynamics, Berkeley, 57-77.

Erdelyi, A. Asymptotic expansions. Dover Publ. (New York), 1953.

McCarthy, J. Collected experimental resistance component and flow data for three

surface ship model hulls. David W. Taylor Naval Ship Research and Development Center, Bethesda, Rept. No. DTNSRDC - 85/011, 1985. Kajitani, H., Miyata, H., Ikehata, M., Tanaka, H., Adachi, H, Namimatsu, M. and Ogiwara, S. Summary of the cooperative

experiment on Wigley parabolic model in Japan. In Proc. Second DTNSRDC Workshop Ship Wave Resistance Comp., Bethesda, 1983, 5-35.

Newman, J.N. Linearized wave resistance theory. In Proc. Int. Sen. Wave Resistance, Tokyo/Osaka, 1976, 31-43.

Havelock, T.H. The theory of wave resistance. Proc. R. Soc. Lond., A131, 1931, 275-285 also A132, 1931, 480-486.

Chen, C.Y. and Noblesse, F. Comparison between theoretical predictions of wave

resistance and experimental data for a Wigley hull. J. Ship. Res. 27, 1983, 215-226.

Ju, S. Study of total and viscous resistance for the Wigley parabolic ship form.

In Proc. Second DTNSRDC Workshop Ship Wave Resistance Comp., Bethesda, 1983, 36-49.

APPENDIX: Equation (42)

The terms occurring in equation (42) are defined as follows:

s =Ic sec20 , pl/p2 , d2 = p2/p3

C(k,e) = cos(kx cose) cos(ky sine) S(k,e) = sin(kx cose) cos(ky sine)

t = thickness of upper layer.

1

t2 = thickness of lower layer.

s(z-t1+t2+C2) Fo(e) 62 s e S(s,0), -st, Do(e) = 2 61(1-62) e _sinh(st1) -{cosh(st1 ) + (21-1) sinh(st1)}{cosh(st2) + (282-1)

sinh(st2)1

D(k,e) = {cosh(kt1)+1sinh(kt1)}{cosh(kt2) + (32

sinh(kt2)1

x d1(1-62)(k+kosec2e) e -sinh(kt ) 1 {cosh(kt1)+61sinh(kt1

)1{cosh(kt2)+(52sinh(kt2)1

- (1-d1)tanh(kt1) kosec28 1 + d1tanh(kt1) - (1-62)tanh(kt2) kosec20 1 + 62tanh(kt2)

Dk(k'e)

= d D(k,e)/dk and D(a 0=D(k,0) when

k=a.

Here a is the kth root of k in D(k,e) when

i

the term n square brackets [ lie set to zero.

For j=1, 0 < z 5 t1 1 = 0 = r r11 21 k(z-t1+t2+C) F11(k'8)=k{(1+d2)k-(1-62)sl[e -k(z-t1-t2-C)]

-k(z-t1+t2+0

+k(1-62) ((k-s)e

k(z-t1-t2-c)

-(k+s) e k(z-t1+t2+C) F21(k'e) = - 2ks [(1+52)k-(1-62y,s]e For j=2, -t2 z O. k(z+t1+t2+C) (k,0) = -k {(1+52)k -

(1-(52)0

e k(z-t1-t2-* -k(1-d2) [(k+s)e k(z+t1-t2-C)] 1

+--(k-s)(1-62)f

2 (1+6 )k -(1-61 )s}1 k(z+ti-t2-C) -k(z-ti+t2+C)] + e 1, - _{kk+s) (1-(52) {(11+d )k + (1-1}6 )sl k(z-t1-t2+C) -k(z+ti+t2-C)]

+e

+ t(k+s) (k-s) (1-(52)(1-61) [ek(z-t1-t2-) k(z+t14-t2+)

-k(z-ti+t2-c)

-k(z+ti+t2+C)]

-e

(k-s) e Le -e -e 22. 23.. -= 1 2 --kt2 k - x 1 -+

F22(1c.°) ---tOc-4-0.[(1+6 )k-41-62 2

)sT

. k(z-ti+t2+c) ji(x_E2)2 _{(y_n22} r,z702 v{(xc2)2 + (yn2)2 +

(z+04

For j=3,r13='

_{r12 7 r23 = "r22}

k(s+t1+t2+0,

F13,(k,e)= -k {(1+62k -(1-62)s} e

k(z-t1-t2-c]

-1(1-62) [{1c.+0 k(z+t1-t2-) (k-s) e + (k-s)(1-62){ (1+61)k (1-61)s 1. k(z+t

t2

4)

k(ztlt2+0

[e 1

-e

- (k+s) (1-62){ (1+61)k + t1-61)s}

k(z-t1-t2-Ft)

_e

-k(z+t1-t2-01 1 k(z+t t -+t)

{10-s)(k-s)

_(1-62)(1-61) [ e 1 2

k(z-ti-t2-0

-k(z-t1t24.e-k(z+t1-t2+C,;)t -e , k(z=t = -k.,(k+s) [(1+62) k-(l-6.2)s]e 1+t 2

Inspection of equation (42) reveals that singularities occur in the integral expressions when

k-ko sec20

= 0

and

D(k,0!) = 0:

In general, solutions k=at of this last equation can only be determined numerically.

The solution k=k sec2e represents the wavenumber producing he characteristic wave-length of free surface waves in a fluid of uniform vertical constant density.

The root k=0 is a second order singularity with D(k,e) = 0 = dD(k,e)/dk so that the second order derivative d2D/dk2 must be examin-ed to determine the critical velocities, limits of integrations and number of roots 2-, It can be shown that at k=0 the variable s=k

sec2e =

(g/U2)sec2,0 satisfies the equation

a 2 bs + 1 = 0

Where,

s = (1,-(51)t1-(52)tit, z o

b = (1-61)t + (1-62)t2 + 61.(1-62)t1 0

and two real roots exist. That is

= [b

since 0<(b2-4a)<b and for a stable, stratified, fluid 61.51, 625,1.

Thus at 0=0 we may define lower and upper critical velocities

1

1let

=.1(4/s1.)'

and U

(g/s

CU

respectively. It can now be .shown that when

0 =' 0 , DFO has always tWO roots

o

c2

u.

cucu

,

e0 o

e c D < el , D = 0 has one' root

e < 7/2 D =' 0 has two roots,

'U < U

Cu

eo <

2

s = 0 bag one root

1 + 62 tanh (kt

The first two roots remain unchanged though the root k=0 is now a first order singularity. From the last equation it may be shown that

if

U2coS20 < gt2{1-62)

a real and positive root exists within the range of integration of e. If on the other hand

U2cose gt2(1-62)

then the range of integration must, be modified to begin from

1 < < 7/2 D = 0 has two roots

where 01 =,cos-1(Uoz/U) and 02 = cos

(Veu/U1'

This determines the upper value A, of the summation occurring in equation (42)

For a two, layer fluid Z, =-1, t1 = 0 with the source moving in the upper layer,

singularities exist when

k = 01, k - ko sec20 = and k (1-62) tanh (kt2) ko sec2e = r12 = + + r22 e -e k=0 -i(b2-4a)]/2a = 82 < D , - 0 - 0 >

B. =cos- {gt-t1-26 )/U21= cos-l(t2 /1,),(1-62"-o

I/

Fn21' Thus in the two layer fluid case the limits of integration of 0 are

'0 if U2 5 :tt 2('i-&2)

if U2> gt2

and the critical velOcity is defined

U 2

gt2C1

62)

For a. Source travelling, 'with velocity U, such that

Fn<F

=

(t.2/1,r) (1-62)12

the motion is subcritical, otherwise it is supercritical and for small values of IFII-Fnc the internal waves generated are of long wavelengths.

For a body moving in an upper layer ,covered by a lid the situation is slightly different. Now two first order singularities

exist when

(1-61) thkt1)

k.

+th(k t1k.sec20

= .0t

61)

0

and it can be shown that the limits of integra-tion are

( 0 if u2,5y17,51)/61

8° =i:cos-1t1(1.(s1)/61U2if _{0,2>gt1(1-61)/61}

where t1 is the thickness of the upper layer Of density p1 and 61 =1/P2'

o= =

COS

1{gt2.(1--s2)/U2}2}

Figure 3. Non-dimenSional waveMaking resist-ance coefficient data Cw for a Wigley parabolic hull form travelling in a constant density fluid (B/L=0.1, T/L=0.0625)

Experimental, wave pattern resistance data [30]

Experimental, wavemaking resistance, data

[30]

o Neumann-Kelvin formulation using 252 (36x7) panels, near and far field solu-tions and body surface integration.

15 f.scal 3%5E-3Jacw .t cm-(Ordinate scale 10 Free Surface f /1,0.1633 Fig 2tb)

Figure 2(a) .Non-dimensional wavemaking resist-ance coefficient data C for a submerged prolate spheroid moving (Fn <

OA

in a homogeneous fluid.

Farell's theoretical predictions [5] --- Havelock's theoretical predictions [28]

o,o,x represent 220, 468, 798 port panel distri-butions respectively used in an 'exact' Neumann-Kelvin formulation with near and far

field solutions and body surface integration. 220 port panel distribution used in a slender body numerical approximation with neat and

far field solutions and body surface integ-ration.

88 port panel distribution used in a slender, body numerical approximation with a far field solution and Lagallyis method [9].

Figure 2(b): Non-dimensional wavemaking resis-tance coefficient data C for a submerged prolate spheroid moving (Fri< 0.4Y in a homogeneous fluid.

Farell's theoretical predictions [5] bol,* as in figure '2(a)

88 port panel distribution used in a, slender body numerical approximation with a far field solution and Lagallyvs

,p

method [9]

20 evenly distributed axial sourcesTanci Lagally's method. e.20 Per 9.32 130 pm retx .30

'A slender body numerical approximation using

252 panels, near and far field solutions and body surface integration.

- - Zeroth order slender body numerical approx-imation which through simplifications [29] is a generalisation of the classic thin ship

(1-62) as = -c +

-(Ordinate scale 10-4) 15E--1 y1,0.70 0.05' -Fig, 4(a) Fig 4(c) Fig. 4(e)

Figure 4: The predicted non-dimensional internal wavcmaking resistance coefficient C for a prolate spheroid moving in a two layer' fluid stratification given by

a finite depth upper layer over an infinite deep lower layer _{(81 = p1/p2 =} 0.976).

an infinite deep lower layer under a finite depth upper layer.

free &efface ((resh .ter) Interface (sea water) - fftin0.25 fit,60.60 fit in0 .75

central line source distribution op source, riyo.so central line source distribution

60 sources f/t,6050 hodysurface source distribution

176 nosh fit1w0.50 0.35 0.35 .10 (Ordinate scale 10-4) sec 0:85' 13.70` 0.75' 0.28' 0.2e 0.30' 6.35' 0.4 Fn (fresh water) Interface ti/L.0.30 _MEW (sea water)

Bed - - - f/610.25 f/tr0.50 f/t,60.7i (Ordinate scale 10-4) 0° 13 0.15 0.20 0.2 0.30 0.35' 0.10 Fn (Ordinate scale 10-4) Fig. 4(d)

a finite depth lower la:er under an infinite deep upper layer.

a finite depth upper layer with rigid surface over an infinite deep lower layer.

as in (a).

These calculations use 20 evenly distributed axial sources and Lagally's method. _{In figure 4(a) a comparison is}

shown between 40 and 60 axial _source distribu-tions, a 176 panel distribution _{over the full} ,etted surface area with a slender body

numerical approximation (Q = n.) and Lagally's method.

7igure 4(e) shows the contribution _from internal and free surface disturbances.

0.05 II.1 11.1 0.211 , 0.38 Fn ,m 0.40 Fig. 4(b) 5E-i SES 20-1 (Ordinate scale 10-4) Lid ti/Le0.13 (fresh water) Interface

I\

1'1 I (sea weter) --- fit 0.25 -f/tiw0.50 f/ti.0.75 5E-1 1E-1 3E-I 2E-s A I 1 ti/L.0.30 (freshwaur) 7.441water) -f/y0.25 - - - -f/tin3.50 Fre* Surface 3E-1

ti/Lw3.33 (fro. water) Interface 4E-4 (sea water) 7E-4 _A

- -

f/ti-0.25 I

-

f/t.,60.50 26-4

\

r,

f/t,60.75 1E-4

\\N

S. a.es 0.70 0.1 0.20 e. 0.30 0.33 0.60 Fn \ ra) =

Fig. 5(a) 7E-1 6E-4 5E-4 4E-4 3E-4 2E-4 1E-4

Li

f/t2-0.25 f/t2-0.50 f/t2.0.75 t1 /LO. 30 t2/L-0.40 0.15 8.20 Fn L

(3"

Free Surface Interface Interface 0.25 8.30 Fig. 5(b) 7E-4 8E-4 5E-4 4E-4 3E-4 2E-4 1E-4 f/t20.25 f/t20.50 f/t2.0.75 0.10 0.15 tI/L.0.30 t2/L-0.30 Fri Free Surface Interface

jf

B Interface Fiq. 5(c)

Figure (5) The predicted internal non-dimensional wavemaking resistance

co-efficient C for a prolate spheroid movinging in the middle layer of a three layer fluid stratification

(ypik2.

_0.9876,

2=P2Ab-3

0.9878).

The

upper two layers are of thick-ness t1, t2 and the lower layer is of infinite depth. These results provide an illustration of the influence of layer thickness on the coefficient and a comparison between figure 5(c) with figure 4(b) provides a measure of the influence of fluid stratification. These cal-culations involved 20 axial sources and Lagally's method

[9]. (Ordinate scale 10- )

0.25 8.30 0.20

5E-4 4E-4 11 I 3E-4 I 1 2E-4 I t, I

'\

1E-4 I

\

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Fn t1/T=4 Free Surface (fresh water) Interface (sea water) free surface -- internal 0 15 0.20 0.25 0.30 0.35 0.40 Fn

Figure 6. The nredicted internal and free surface non-dimensional wavemahin, resistance coefficient Cw for a Wigley parabolic hull form moving in a two layer stratified fluid. The upper layer is of thickness t, over an infinitely deep lower layer (d. = 0.976). These calculations use a Soo panel distribution over the mean wette6 surface area of the hull, a sender body numerical approximation and Lagally's method [9]. (Ordinate scale 10 )

Free Surface t1/T=2 (fresh water) Interface (sea water) free surface

- internal

3E-47 0.0 0.10 =