Some predictions of the values of the wave resistance and moment concerning the rankine solid under interfacial wave conditions

N OR WE G IA N SHIP M ODE L

EXP ER ¡ME N T

TA NK

THE TECHNIcAL

UNIVERSITY OF

NORWAY

SOME. PREDICTIONS OF

THE

VALUES OF

THE WAVE RESISTANCE AND MOMENT

CONCERNING THE RANK/NE SOLID UNDER

I N TERFA CIA L

WA VE C ONDI TION S

BY

T. SABUÑCU

NORWEGIAN SHIP MODEL EXPERIMENT

TANK

PUBLICATIOÑ N0 65

SOME PREDICTIONS OF THE VALUES OF THE WAVE RESISTANCE MTh MO ENT CONCERNING THE 1WIKINE SOLID UNDER INERFACIAL WAVE CONDITIONS

by

T. Sabuncu

I

Norwegian Ship Nadel experiment Tank Publication No, 63 ABSTRACT

In this report, as a supplementary to thé

previous one*, uzder interfacia]. wave conditions _a

linearized theory of the moments acting on slender bodies is developed. The theoretical results

obtained for wave resistance and momenta are applied to a Rankine solid and solutions are given in the form of curves. (pp. 28 -38)

Although numerical calculations of the wave resistance rest upon linearized first order theory it has been found necessary to compute the momenta through two approximations.

REFERENC ES

24 26

LIST OF FIGURES

Figure 3. The shape and offsets

of

a Rankine solid . . . 27

Figure 4. J values for Rankine olïd for (h-t)/D 0.5 . . 28

Figure 5. © values for Rankine solid for (h-t)/D = 1.0 . 29

Figure 6. © values for Rankine solid for (h-t)/D = 1.5 30

Figure 7. © values for Rankine solid for infinite depth

of submergence 3].

Figure8. Added wave resistance,kg. per ton, for different'

depths of submergence 32

Figure 9. Added © values for different depthaoof submergence 33

Figure 10. Noment caused by free surface waves for .(h-t)/D l ₃₄

Figure 11. Noment caused by internal waves for (h-t)/D 1 - 35

Figure l2..J1ornent caused by internal waves for infinite

depth of submergence 36

Figure 13. oment coefficients for Rankine solid against

various type of Froude humbers . 37

Figure 14. Nonint coefficients for Razilcine solid against

inverse FrOude numbers .

J

CONTENTS

₁₁ PAGE Introduction i T TT ..

TEE ¡O'!ENT ACTIiG OiJ A SUB'1ERGED BODY UNDER INPERFACIAL WAVE

CONDITIONS

The moment acting on a slender body moving in the light fluid layer, bounded by the free water surface and interface .

The moment acting on a slender body moving in a two-layer fluid system bounded by horizontal plane boundaries at the

top and bottom 4

Iv

NUMERICAL CALCULATIONS CONCERNING THE RANKINE SOLID

1. The wave resista.nce of a submerged Rankne solid moving

in the light fluid layer bounded by the free water surface

and interface . . 9

ave.resistance expression of a Rankine,so].id 9

Integration of the wave resistance when 'h'

becomes infinitely large 10

Asymptotic evaluationof the wave resistnce . 11

The snape and dimensions of the Rankine solid. 12

Evaluation of the wave resistance . 13

2. The moment acting on a Rankine solid moving in the light

fluid layer bounded with the free water surface and interface 14

The first order approximation 14

The second approximation 15

Integration of the moment expression when 'h'

becomes infinitely large 20

Asymptotic evaluation of the moment expression . 20

.Evaluation of the moment expression 21

CONCLUSIONS ... . .

...-...-... ...

23

1. Introduction

Generally the ocean has been accepted as a vertical density stÑoture# the occurrence of a layer of water of smaller density ón the top of denser water is quite common. Such layera may be caused by the spreading out. of fresh-water from rivers, by the melting of ice, by the heating of the top layem 'and by other oceanographic processes.

A comprehensive amount 'of work has been done to study the forces -and

moments acting both on ships and submerged bodies theoretically by various approxi-mations; and to control the theoretical results many model experiments have al5o been carried out. In practically all cases the ocean is treated as if it were

homogeneous. Very few works exist which deal with' the problem in a stratified

ocean; but f the present, an attempt has been made to solve the problem for._a

different purpose.

The transfer of ênerr from the ship easily raises waves of considerable height on the common boundary (interface) at the expense of its increasing resistance to its motion. But still higher wave resistañce augmentations and .maneeuvrin

moments may be expected for submarines when they are moving on or neár the

inter-face. These forces and moments may even be high enough to cause the aubmrine _to

lose control. '

Therefore, the aim of this work is, as a supplementary to the previou3 one, to formulate moment expressions under interfacial wave conditions, to apply

the theoretical results obtained for resistances -and-moments to a'Rakine _solid,,

and to give the numerical values

in

_{-the form of curves. Although, in- the present}

"work, numerical calculations of wave resistance for a

Rankine.

_{aolid. rest upon}

first order theory, it has been found necessary to compute momemts antig _{n the}

same solid through two

approximations.

R

III. THE MONENT ACTING ON A SUBMERGED BODY UNDER INTERFACIAL

WAVE CONDITIONS.

1. The moment acting on a slender body moving in the light fluid layer, bounded by the free water surface and interface.

It is assumed that there is a slender body symmetric with respect to the midship section, and moving with a uniform speed along its principal axis in the light fluid layer. It is further assumed that the shape of the body generated by a line distribution of sources and sinks moving close to either one of the

surfaces is identical with the shape of the corresponding body of revalution

generated by the same singularities in an unbounded fluid. With the usual

assumptions of the linearized theory (discussed in the previous work in some

detail) the velocity potential of the light fluid motion due to a body

repre-sented by such a distribution is (cf. previous work page ¶O Equation 64)

e

¿

1172

[°°4(.-21?*2t)

-

/üdJ[S2262'9/

e

-o

(k-k0)

8

+

_{(i:&/a.a'i/a'g f}

9

Jo J

kh-k44-

(i-4h

,

/cdf/do2/

g

k0 set.)c[k

_-1)c

Jc *s

_{) c/k}

11

,iii,,ir,

')JkJ,

2&7 -/.-,

'O

'V -

1'

j

r

f'

kih-t)

- (-t) -k(#t)

!!._/crdf/a't9/

{(k-k0.cec9)e

-

(k0)&

Je

(kÇ,%)j

('t)kh

k

_â etf9 -'e'

É0-2h*2,)29

2 ' z e

[-.'(1-x,)e

2SJ

17/2

é

Ç(oq3-3/42t) 2 2 2 2

-(i

-)/a-df/l e (+ k08&) (+k0s1c)(oc-k0 sc6')+e

¡.cc)3

(«-k0&)[(c4A)-

(1-x),4ksJ

2

-i-,.i4'«,) .çs4?[o<

(r- f)c]cti (

ç'n )

Q'.

(1)

where

rectangularcoordinates, z positive upwards and

xy-plane

ooinoidea

with the horizontal

symmetry

plane of the body (figure 1),

h depth of the interface below the undisturbéd free surface,

t

height

of the horizontal

symmetry plane

of the body above the

undis-turbed interface,

c

=

uniforn velocity of the submerged body,

r2

((-il'232J

- the ròot of the equation

-

!l$k0ec

-

l#ao

8Ö ô ¡

c2<fr).6

8 -

C'f(7-x)0qhj1"2

'f

C2>

(1-xh

Three-dimensional extension of Lagally's theorem may be applied to obtain the

moménts

acting on slender bodies.: According to

this

theorem the moment acting

on a body whóse surface is a closed

stream surface

of the fluid motion is

given

by thé sum of' the vector equations of thé type

/

(2)

where

01

strength of a source internal to the stream surface,

=

the vector from the origin

to the source placed on the generatitg

line

of the body atP(!,OO1

=

resultant fluid velocity vector at the lOcation of the source due to

all other sources (its components in the x,yy,-z directions are

_{'4;- %))}

Hence, the whole moment acting on the slender body about ïts centre of

buoyancy is

fe

_4llJ'/Jcdf

J_e

fo?'

g::

-

()

Since

0

source distribution along the longitudinal axis of the slender body

(symmetric with respect to midship

section) is an odd function of, therefore

when the first four integral

terms

of (i)

are

substituted into moment

ex-pression these vanish.

Hence, the remaining integral terms of the velocity

potential contribute their parts

to the moment

2

i

-2(h-t)k0sCs

M

f

'-e

Jo

(

(fC)e_2k0h'&J

8 8

_{r(3,4-2) 2 cc (h-14)}

-

fff

(f-4

ií

'Q-"d'fCdff ¡[e ( k0s&e (kstò)J('c4th

hcff-j)ce

j

J

j

(°

ka

)T(c.hxM«h (i-)h442]

tfith

4/a_a]

2(h40seI9.

1

7

+

,6ft(i_9)40/[/

V- v2i

f

c_

tjt

°'czoct3 °c

(a'-. t-t-x) e

(i-.4,, &ecJ

(U,-.t.

:e

-4--

¿0zsec

P+Q

/o-c.x)e c'.x,

ic.xcr8

-

r

/

az)

e

cz'.X

[u

/

vi

= /oC.z e

le

Ç4)

2. The moment acting on a slender body moving in a two-layer

fluid

_system

bounded by horizontal plane boundaries at the top and bottom.

It is supposed that there is à slender bo&y symmetric

_with

_{reapeot tO the}

midship section

_and

moving with

uniform

speed along its principal

axis

in the

light fluid layer, with the usual assumptions of the linearized theory discussed

previously in

some

detail, the velocity potential óf the light

fluid

_{motion due}

the

root

o±

th

equatïon

(it4

auch

,-D

j$+

j/cdf

o d

t

((-Y

323

I

ae-ca J

the

root

of the

a<

(P2-F)/

t2h1

-5-to a body represented by a line distribution of sources

and

sinks, is

given

below

(for the säke of brevity intermediate long oeloulations are omitted

here)..

e 1.1/2

6(k)

h(-J)]cÇfr?14'k

f

_{J [}

(4).t(da')ac#4

(kd)kczg

7

D o

e.4 (k

/?)

M

Ckd)

J

odf/6(t)[4(t4 (t

(tX2J,&tiz fr(x-j)

c'&J

c47

(t&)

c/&

(M

(èh)

.4 (ra')

a'cd (½) s-4 (a')J

2

ix)[a'

Z()

C td)J

k&9

(5)

where

x,y,z rectangular coordinates, z positive upwards

and xy-plane

cotnoidee

with the horizontal

synmetry

plane of the body (figure 2),

h height of the horizontal plane boundary above the interface,

t

depth of

the interface measurod from the horizontal symmetry plane of

the body,

depth of the solid bottom below the undisturbed interface, uniform velocity of the body,.

equation

(i-x)

4

r(-Jrr2)1

-t4

2td

2

-'-2 e

t)1-?C)(tk0Ce9)e ((k&)+f c)e J

2

-ra'

e

[(i-)(t-keec9)e

[(kc)--

(t-k6ssio,)e

8"=

cc[(1°Pi)hd

c2/"2

/ c'

(P:)kd

o

4hc

/ i///'l////.

SO/d

_8ottam

Using the threedimensional application of Lagally's theozn to the

_mÓmnt,

thus the moment acting on the slender body about its centre of buoyancy is

for

x=j

41=O

C

Inserting into (3) the zderivative of the velocity potential(..z,J.) taken _at

the point

P(1'oo)

, the first integral term of the 'quation (5) vanieheB

since source distribution along the longitudinal axis of the slender body

(supposed symmetric with respect to the midship section) is an odd function

either of s'or Hence, the remaining integral term oÇ, the velocity potential

contributes its part to the moment

g

8

"i'

J'l= 2F

ff/df/t)[oc«4(td)

t4»4(td)]44 L

r(Ñ)Jy'

2 where -t, II

(3)

')(tr

E

C t-k0s'c&) r( tk2c)]

d1

2e

i-c(t k0

)etd((t*kò

o +x(-

ecj î

2

[(f-x)(r-kecec1)e

_{-x(t-k0c&)]é}

_J

(6)

Using the relation o

(Y)th(t4kUCand

_{after some extensive}

reductions

('h)

_c4(»-d)

the moment acting on a body of revolution may be written as follows

2 f%dl=

8ffJ'X(i-ê)1é/

[uy-jk228

(7)

L,

itXc4Q

[t'iv).

/o(.x)& c1z

-t

_¿e

t

the root of the

Q9uatiOn ?.L.

(1)/4t4(y)

:=

if

c2(f-P)hd

)

e"=

271/2

_c7>

(I-f)kd

F2h4fc

w

here _e

One may

draw from the

above sorne relatively simpler resulto with the help

of various limiting processes.

a. If d

and

h both tend to

infinitr

Que obtains moment

expression

of a slender body tzavelliñg in the light fluid above the interface an

light

and dén

fluids extend to infinity in their rospective regions

/

4

-2k0tsec29

f1

1611f

x(1-x)2d4 / (u'v- vu] seû &

(i.tx)3.

_J

C where -

where

[uí

-c

b. If the height of the horizontal

plane boundary measured from

the

interface becomes infinitely large, one finds the moment expression of a slender

body moving in the

light

fluid lying on top of a shallow dense e

M= 1611.Çx(i-a'9k

e

r

[u

-

iv]

c-cx) e

,.

,

(u

e

j

e

'k0z

dx

(a)

/['v-vvJc.cec

-2

e

z(d)]

(1-d)k0a'sC2t9

tu

/

V] = /o.x) e

I ol 6

a4cte'((1-?()

h]'2

c. tf the depth 'd of shallow dense water increases to an infinite

depth, One gets the moment expression of a slender body moving in the light

fluid between the interface and a flat horizontal beindry

1½

M = BiT jj(i-) k0

f [LI"v Vu] fcE2/9h-t)J-

ijp £eIt9 d&

j(Mo9h)aec4h)J- (1-de)hk.V429

E u-tv

c2, g.h(i-3e)

e doc

(9)

-e

(fo)

wheré

the root of the equation

C2(

(.f-ì)íd

Jhtf1d

I

e.

r

[u'v]= o-c.z) d

-E

(i-w)

th't4)4 (ta'

_{k ,secc}

ô

'

M[(Jt4)hd

c'

¿P

c>

(ñPiMc9

f2h.id

_ihf4ci

the root of:the equation

/=

(1-X)t.4t3h428

t9'O if G2h (

_,)

6,

cCOZ1[

(Ç_i)] " ij

C QT1)

In

the

above we have developed the monient expression for

slender body

moving in thé light fluid layer of a twofluid system, bounded by

horontal

_flat

boundaries at the top

and

bottom;

if

the body moves in the lower dense fluid

_and

the whole boundary conditions remain the saine, we can obtain the following result

-

_&flP(1e)k

[

(L/'V- v'UJ[%J2C2t(dt)1Jtec&d8

IV.

WUYRICJIL CALCULATIONS CONCERJiING THE RANKIN SOLID

ii. The wave resistance of a submerged

Rankinà

solid moving in the

light fluid

layer bounded by the free water surface and interface.

a. Wave resistance expression ofa

Rankine

solid.

By

following

numerous comparisons between the theorétical

añd experimental

results in a homogeneous fluid referring to submérged bodies, it

has

been found

reasonable (though by no

means

entirely satisfactory) agreement between the wave

resistance calculatiOns performed by using first order theory

and

experiments.

Therefore, it would be reasonable to draw the same.00nclusion in the case of

first

order linearized interfacial wave conditions. Although the closer the body is

either to the free water surface or interface, the

larger

are the deformations

with respect to the real shape of the body generated in the unbounded fluid; this

fortunately does not effect the

wave

resistance by

any

considerable amount.

In the subsequent analysis the assumption has been. made that the shape

of the distorted body generated by a source

and

sink moving close to either one

of the

boundary

surfaces, is identical with the shape of the

_Rankine

solid

generated by the

saine source and.sink

in an unbounded fluid.

The

linearized wave resistance expression of a submerged slender body

moving in the light fluid layer bounded with the free water surfáce

_and

interface

has been

given in.

the previous work (page 94 Equation 74). Concentrating the line

distribution of the sources

and sinks to a

single source and equal

sink

of the

strength 'in1 placed at a

distance '' apart,

on a line parallel. to the uniform

stream we ôbtain iiinnìècïiately the first order wave resistance expression of a

Rankine solid

2

2(1, -t)k ,sX9

R

G4

fT4z;.

,,/ LTi (k.a..e8)]

secs e

d&

r,

2°'h

.o.ct

2

641?f(1-X)40

2/[(J

[

(c.t

()-

ze

¿4h)J

,ecg d&

J[.(

1-r)

(1-X)i4k0(S .oh)2p)

2]

where

the root

of the equation .

(1-e) I(°&

.i +f1(o,4)

eoo ¡f

c2

(1-X)h

Sco,[(í..x)b)''2

,,.

O>(i)

b. Integration of the wave resistance when h becomes infinitely large.

If h the height of the free water surface measured from the interface

tends to, infinity the expression for 'R takes its simple form

2

1

2

R=

NJ

_{(s(ec)J.gc&.e.d&}

where 2 2

M

6417J

(1-ar)

3e kern

(ia93

2Sk,cz

j

f2!kdt

_--

_..i-i

_2??

Since series

c4:1t4s.e)

-

.j:

2()(l»

are uniformly convergent, by expanding the integrand of thé above Jquation (13)

in an infinite series and integrating term by term we obtain

k=

where

A/ e.

r

-('5)

It

has been shown by J. K. .Lunde (Refereoce 4 page90) that integrals of the above

type

can

be expressed in terms of modified Bessel functions of the second

_kind

of orders zero and one, provided that '?2is. a positive integer

_and

o . bne

procedure for doing thIs depends on the reduction formula

A

VH'f

2fl-1

A

(16)'

17 i Cf

11

"

fl1

the above equation (16) may be derived by integrating the erpression

r

r'z

/

e

's

û

by part. For fl1 reduction formula gives

and oontìnuing the procedure

A= i°A .!..ì

_{3 z z'?}

_I

A31

_'f

n Cf Adfl.4

-(13)

Hence, from

and A1 all

other _A may be determined. The transformation

sc20=

ppIi

to

the integrals for

_{A6and A}

_{transforms them into}

own expressions

for modified Bess1

funótions of

the second

kind

1<

snd K

fte result is

ç. 2 e _{A0 =} co

4A =

_I

where

-il-Wi

..

e

@12/

=

Ee.

Pi)

e

nd a

few

others

obtained from reduction formulas

Ço

4'feA2=L2=L'fe

(<'/)±(1-I-%')e

_/<(w2)J

4P2f4

23

°/2

4 're

L

_{}<(4')+( 6+4}

2cp.-p)e

(17b)

Using the- results o'otained above, the expression for R, when h tends to infinity,

becomes

4

2"°°°

-77-1

an-!

R.3217P(1)e.k4a2meZo(cz)

_1(w2)

k

o = 2

r-r

o

c. Asymptotic evaluation of the wave resistance.

Asymptotic formulas may be applied to the wàve resistance.. Under the

restrictions of k0

a> 9

and

h 2t , the wave resistance expression

given by equation (12) can be reduced aproximate1y to

11/2

R'

where

\/3211I

(-efaek,n2

-(iac)3

--

74

O)

cp

12iA.

(17a)

t.t

has

already been shown that integrai of the type

-17/a

/s

A'=

sre

can b

expres8ed in terme öf the modified Bessel functiona of the second

--Id.nd

and first -and. zero Orders

-

The result is

-On the other hand, by ising the principle of stationary phase

17/z 2 _/

r

77

J

Cec9)se

c'

-'

(-) e c,i(o-)

finally from the above written results we obtain asymptotic evaluation of the

wave rsistance, when 12_a' and / 2 are

271f dC

(k2m2et

1

(i)3

°

1

L1<f2)1

-12-+

kJ-}/

e4i2ka+..]

j

(19)

d. The shape and dimensions of the Rankine solid.

If we combine the source and equal sink of strength 'ni' placed at a

distance of ' ' apart, with a uniform flow c in an inbounded fluid, we obtain

a Rankine solid whose surface is now a closed stream surface of the fluid motion.

The following relations exist between body dimensions, generating source strength and velocity of the uniform flow

¿C

(I

4?i

wherè

b = half of the maximum diameter,

a = hail of the distance between source and sink,

1 = half of the length of the Rankine solid,

m generating source strength,

c = velocity of the uniform flow.

With sufficiently.large length-diameter ratios, the strength of the source can be written approximately as

?n

4L

2 -

₄

(20)

For a length-diameter ratio of 10.5 or greater the error made by taking the above approximation is.smaller than one per cent.

D = 2b. 0.21

L = 21 =2.20

2a =2.095 in. n It

-13-L/D = 10.5

b/a

0.2.

g j = 1000 (Kg./m3) 1040 °

a2 =0.9615

r2.

P.C. = Prismatic. coefficient -

A_=

0.926

0.07056 (rn3)

¡Tb2L-The longitudinal cross-section and offsets of such a Rnkine's solid with length-diameter ratio of 10.5 are shown in figure (3).

e. Evaluation of wave resistance.

There appears to be no possibility of the direct integration of the

wave resistance expression given by Equation (12). Therefore, integrals computed

numerically; namely numerical integration was performed by using Simpson*s rule and corresponding ordinates of the integrand were calculated at suitable intervals

of the variable 'e' (usually 1, 2 or 4 degree intervals have been selected

-according to the frequency of the integrand). In addition to this, calculated ordinates of the integrand plotted as a function of 'G' and graphical integrations were also made by using the planirneter- for checking purposes. Numerical

evaluation of the part of the wave resistance caused by internal waves is rather

time-consuming, and requires a great deal of labour, espésially when is

large; i.e., the smaller the speed of the body, the greater is the oscillation frequency of the integrand. Therefore fOr this part of the calculations the

asymptotic formula, given by Equation (19) was used. If h the height of the

free water surface tends to infinity, the effect of the free water surface dis-appears and the expression for wave resistance takes its simplest form. In this casé, resistance integral, can be evaluated in a empÌratively easy manner, either by uing Sirnpson's rule or by expanding the integrand in an infinite asries and integrating term by term. Th'e series expansion formula of the wave resistance is developed under subparagraph c and given by Equation (18). Expansion formula

iS rapidly convergent for t &

fer

The calculations of wave resiStance were carried outrfour different

depths of submergencé below the free surface, and in, eac case it is supposed

that the Ran]cLñe aolidtouches the interface; thus the following depth and height-diameter ratios were taken for the numerical calculations

-14-(h-t)=o.5'

t/D=o.5

D (h - t)

-1.0

t/D = 0.5 D (h - t) 1.5 t/D 0.5 (h - t) t/D = 0.5 i)

Computàtions for a wide range of speeds are.surnmerized in figures from (4) to (9.).

In figure (4) nondimensional resistance coefficIents, namely caused

''

by the surface and internal waves, are plotted as a function of' sped and at

submergence of 0.5 dIameter. The curves in this figure relateto a rather extreme and artificial case, where the solid touches both the free water surface and the interface, and exäites waves of mximum heights on both fluid surfaces. As a

consequence of this assumption, curves of values belong to a rather distorted,

Rankine solid. Probably, in this case, it is plausible to expect a poor agreement

with reality, especially for the part of the resistance caused by the free water surface waves. In figures (s), (6) and (7) plots are given of© values against

speéd and F at three different depths of submergence below the free water

surface. In all these-three cases, it has been assumed that the solid touches interface. As it may be observed troni figure (7), owing to the infinite depth of submergence below the free surface, the parts of' the © values caused by the free surface waves have disappeared. In order to be able to examine thoroughly the effect of submergence on that part of the wave resistance caused by internal

waves, figures (8) and (G) _{were prepared. The former gives the values of}

on a base of various Froude' numbers related1 at submergences of.0.5, 1.0,

1,5 and o@ diameters. The latter gives the values of _{again on the:.aame base}

and at the same submergence-diameter ratios.

2. The

moment acting on a Rankine solid moving in the light fluid layer. bounded with the free water surface and interface.

a. The

first order approximation

15-approximation, that for a given body the deep.-imîneraion di5tribution of singularities can be used, instead of the actual distribution valid for near surface conditions appears to be a serious one when the body is close to the surface. It has been proved by various comparisons between theoretical and

experimental results that:Ìt leads to inconsistent results with respect to momento

acting on the bodies in a homogeneous liquid (Reference 7 ). Therefore it would

be reasonable to draw the saiñe conclusion for submerged bodies travelling Índer interfaôiál wave conditions. That is, by using related first order linearized

theory for moments where we might expect poor agreement between calculated and real va]uès. Therefore in such cases, it is neeessary to find an additional second

approximation.

-The first order linearized moment expressïon of a submerged slender body moving in the light fluid layerbounded with the free water surface and interface

has been given by Equation (4). Concentrating the line distribution of the sources

and sinks to a single source and equal sink of strength 'rn' placed at a distance

of (

2a)

apart on a line parallel to the uniform stream we get immediately

the first order aproximation of the moment acting on a distorted Rankine's solid

about theorigin 4

_2(h-t)kO8.

-- 'k0

(2k00)/

s

(2k0a.e)SJCO e d

+

[x-i. (()

-2k0h

Sc9J

tf(1-X)bC%

f(2a) ae

IØ'h

-

- C

2'

iû)

S4

+

I

()2kO/.I

o

₍₂₁₎

wheré we have made use of. in

4

(En the above expression a positive moment acts to turn the nose..of the body

down)

b. The second approximation

One procedure for obtaining second order approximation frôm the general formulas might be the application of the perturbation method, in spite of its plausibility it leads to great complications in formulating the case. A'relative1y simple and direct approximation has been suggested by H. L. Pond for homogeneous

fluids, (Reference 7 ), and very good agreement has bean obtained betwean

calculated and experimental results. Therefore, in thè'subsequent devølopment the same method of approximation has beén adapted, owing to its simpler applica-bility to the present problem. To obtain a closer approximation to the moment

-16-wave eonditiöús, the 8imple distribution of a source and equal sink on the axis

of the closed stream surface representing the Rankinet solid must be modified.

This might be done rigorously by 8uccessive applications Of the correcting potentials

( Reference i ). However, instead of attempting the aforementioned procedure a simple approximation to the problem will be sought.

It has been shOwn by von Karman (Reference 3 ) that for a body of revolution with its axis parallel to a uniform stream, the effect of superimposing a flow

perpendicular to the axis may be obtained approximately by a suitable distribution

of doublets (Reference 3 ) along the axis of the body between the limits of the

source-sink distribution which defines the body in the uniform stream. The doublets are oriented so that their axes are opposite in direction to the transverse flow and their strength per unit distance along the axis of the body is

&=-'2r2w-

(22)

where 'r' is the radius of the body at the position of the doublet, and 'w' is

the superimposed transverse velocity. Therefore, in the case of the Rankine solid moving under interfacial wave conditions, the effect of the velocity

induced by the free water surface and interface may be represented approximately by a suitable distribution of doublets along the axis of the stream surface between

the source and sink. It

will

be apparent that for the calculation of moments, only

the vertical component of the induced velocity need be considered. If the velocity

potential of the distorted Rankine solid is denoted by (x.Y,) this vertical

velocity is obtained by evabuating at points àlong the line between the

source and sink. Since the vertical velocity does not change very rapidly with depth, even under interfacial wave conditions, this calculation of the vertical velocity is probably satisfactory, especially for a body whose diameter is

suf-ficiently small in comparison with its depths of submergence measured from either

of the fluid surfaces. For Rakine solid.with fairly large length-diameter

ratios the diameter of the body is nearly constant in the region between the source and sink. Thus, as a further simplification the radius 'r' in Equation (22)

will be considered constant and

equa].

to the maximum radius of the body (cf.

figure 3). The desired doublet distribution is now given by ¿(/

(23)

where 'b' is the maximum radius of the undistorted Rankine solid.

From the extension of Lagally's theorem to doublets the moment acting on a body whose surface is a closed stream surface of the fluid motion is given by the sum of the vector equations of the type

(24)

where

t4

-

47lPI6A

_9i

Ç

mass density of the light fluid

±

8k0

p

TT

resultant fluidvelocity vector at the location of the doublet dueto

ali

other singularities. .

If the

axis

of the interna], doublet is normal to the direction.of the

auperimposéd uniform stream (-c), we obtain

4.17'c

(25)

Hence, from Equation (23). and (25.), the moment per

unit

length for the

Ramkine'e

solid due to the uniform motion is

given

by

(26)

and

the total moment on the body due to the doublet distribution is

2cb2/dx

427)

The veioòity potential of the fluid motion about the distorted Rankina sOlid

with source at

(a,O,O )

and sink

at

(-t2O.û )

may be obtained from the Equation

.(].) The result i . .

c.z+

2/z

,,g

_a'O/é

sn ( kxC4z1) si.,n

(k

c9) e

(k ysto) «k

o

i

ff/ 1004(3..3)14.2t)

.2

( °.-'49)S «k ZC45&)S61 (

C49)C4«k 's)o4

J, J,

Cc.4kJi

sIhKk- 0.eh)[k

(t4&iqçc2J

-T::I.

+

+2

rnPII e

17

J J

. . o . .

i-äet4h d

.1

tW2 Ç°

k(h-t)

-AOz-t) -k(3t)

p,,'

j

((k_k0.ec)e

-

k+)Q0sec2e) e

I

sn(k.x c.o)s4" (ka

18)c#2 (i..t)o

77.

j

f

i&k/i)1h-

,)Q2O]

.

-ko

f e °

S.C8

d

J

Ci)

e_2J

. luz . _ocg+k) -2 (1-?71

fíe

2f_

(3-h)

_(c-k0UC)

+e

e

(°

¿uì9)

J

.

(c°"

(a6)M8)'9

Using the Equation given above the veiocìty 'w'

is

W'.=

for

o

°-4(h-t)

.1

-

empf,er!dfe

- Jo

k-ko

2

1(h-2t)

2

2(i...3e)mpJcIgf fe íssth)-e (-o8)J$n

(i.xe&).iin

ac8)Auz'

J

(ckh+a'M*6)(k-k0sec)[k-

ke'J

'Ti'2 -2(/i- )k0

/

gk02m f

(cl_x)e_2k0e267J

o z _oczt-ì,) 2

+

2(1-)1fl/

f e (c+k uc28)- e (_k0

g)J(ch

+XøA)ç47

(o.x

d:,9)Si;i (oea

co(

J

(pc-k0ec79)

((chMh)i)Akec267J

o

The first two integral terms on the right of Equation (29) give that part of 'w'

which is antisymmetric about

x=O, and

the last. two give the symmetric part.

Since the integrals of the antisymmetric terms vanish after integrating with repect. to 'x' Equation (27) now becomes

1hZ

. 2

2Ch-t)kò

,1

=+32I1fZcbZbfl/ C'-k4uec)]

.2

_{[(le)e_ZhohSßc'61J}

-

8)7Ç(i-?cb2rn

Hence, using the relation

Equation (30) may be written as

û

li'2_(3h_2t)

2..

ci(2_i,)

_2? 2

(e (ok0sec9)- e

_k0

3(

hç/s*(«aq)]

77/2 2

r

, 2

-2(h-i)10.C9

/En(k0a0ceco)Jee&e

a'

+

J

tx -

(i-x)

_2J

gzj

on

r

-t

_

-

,,

z

-

vi,(i-a')b4c2

_/

(re

g-,',oh- e

M«h4 e

c.cih3f

io<ac)J.eSg ciD

J

80

t'i =+

ß17Prbc2

2

-18-+

(30)

ø-

(1-)tha(h.kj.$&

14

C cc.

h

and

putting Qn (3')

-19-where ce.

and 8

have the

usual meanings.

The second approximation to the total momént acting on the

undiaturtéd

Rnnfrmne solid about its centre ofbnóyancy. is nov

given as the sum of the

moments

given by equations (2.1) and (31). The result may be written fas

2

r

. .

.

-2(h-t))!0gc9

M

_1'2

_u1Ç0I

((2k0a9)S(20$

&)--4C6(2Jeasec8)-4.

e

I

((l)e2h03

. .

O

..C.(t

oh

..h

.

t(2«a8)£42

(2ae

9)4ceg2a

ë Mc h - e e cc h- e c/icc.3¿W9 c/6

[xC1-e)

where

= o.

if

U-X)h

,

8,

a'tCC4.'

i-)

c2>

(iX)t,4cth

_k0sec9

1iJ/ì°th

-j-(in the above expression.a positive moment acts to turn the nose of the body down) (of. figure f).

In

the limiting procêas when J/ or

e=i

moment expression given by

Equation

(32)

is simplified tothe moment expression of the infinite homogeneous

fluid,

Thus

ll/

r

3 -Zth-t)Ie0.CC

N

= -

17Pbc2k0

c(zk seA9_4}s.ec9 e o(9

gw.s

/

û

(33)

The

above

found

result is identical

with

that obtained by-H.L. Pond for homogeneous

fluid

(Reference 7 ,

page 5). On the other hand, if ve take

the

limiting

value

of the Equation (32) when h becomes infinitely large, the effect. of the free water

surface vanishes

and

ve get the moment expression of a deep1

immersed RnHñe

solid moving above the interface

4= +ff

_of

av4.)si?(z

kase)41C2

)]-2kòt s.e,

c. Integration of the moment expression when 'h' becomes infinitely large.

If the height of the free water surface measured from the interface

tends to infinity the moment expression given by Equation (34)

can

be re-written

as foliors

where

N =

174 _k0b4 2 2. 3 ¡'liN.v/. c

h-' °

'7/z o -20-e ₂

By substituting the series expansion of the sine and cosine functions, Equation

(35)

becomes 2flr2

f

2fl?5 cfSec8

t1= NZ(-')

(2n-2)

e 'o. n=C

(zn+2)I e

or C1

W(_1(217_2)

OA ('f)

77O _(2n2)Í ° fl#2

r

where C& e - functions have been dealt with under section IV.-lb.

C

Using the same type of notations the result is

Ç

:)r7L

'z) .

(7)

where L,(f/z) functions are given with Equation (rib).

The

above given formula

is rapidly convergent when O Z. I

(Note that when 7Z I thé second term oï (7) vanishes).

d. ÀsymptQtiC evaluation of the moment expression.

For larger -2i.k0c values moment expression given by (32)

can

be

approximated to '7/2 2

11/2

-

f

3 - cPSC

M

Nf((o0s)

(°e°)

-4-4 c (o seco)] .SCO e dO -4 e

. do

(38')

ø( _{2L: k0}

ea

1-#dC

(35)

(36)

'o dû

Using the principle of stationary phase, first integral term of (36) may be

written approximately as

N

e)1[.smn(-i-

)

A coc,+iE)]

On the other hand, second integral term of (36) is equal to A1-function which

purpose.

-21-first and zero orders. Namély

-A1

f

Ek0('P/2)*K1("/z)J

from the results given above we can obtain asymptotic evaluation of the moment

expression

Mrs.'

N

Á(o+..)j_(t'2KO( W2)+

()

where

The above given approximation is sufficiently accurate for our purpose when

*

-k

> a

and i, :- 2t

e. Evaluation of the moment expression.

The moment expression given by equation (32) is so complex that a direct

integration of the integrals doea not appear to be possible. Therefore, integration

was performed numerically by using Simpson's. rule,in the same manner: as

explained in the evaluation ol' the resistance integrals. Por larger

values, that is for low speed range, asymptotic formula in Equatìort (39) vas

used. If the height of the free surface increases infinitely large, the effect

Of the free surface vanishes, and the moment expression takes its relatively

simple form. In this case, for

0023,cz1

values, that is for high speed

range, the series expansion formula in (37) gives a quick solution for the

Calculations were made for two different depths of submergence below

the free water surface, and in each case it is supposed that Rankine solid

touòhes the interface; namely the following depth and height-diameter ratios

were taken for numerical solutions

(h - t)/D = i. , t/D = 0.5 h = 0.315 w.

, t 0.105m.

(h

t)/D ,

t/D = 05

h-.

t-0.105m.

Rankine solid for which the moment values computed, has the same

dimension and shape as used in the resistance calculations. The results are

summarized in figures 10 to 14. Figure 10 illustrateS a comparison of the values

of the moments obtained from the first and second approximations and caused

-22-to make a clearer comparison, the parts of the -22-total moments caused by the internal waves have been calculated according to the first and second

approxi-mations; and the results for (h - t)/D = i and (h - t)/D = are shown in

figures 11 and 12 respectively. One may deduce from all these figures the big difference between first and second approximations. The curves shown in. figure

10 are similar in character to those given in figure 4 of Reference () in

_which

a good agreement has been obtained between observed and predicted moments by using second approximation.

In figure 13 nondimensional moment coefficients, namely

c-1/2TrLbc'

values caused by the free surface and internal waves are plotted seperately as a

function of speed and various type Froude numbers at submergences _{(h - t)/D} ::l

and (h - t)/D = . In figure 14 the saine nioment coefficients plotted on a base

of inverse Froude numbers. It is interesting to note from the examinations of

ligures 13 and 14 that the values of C caused by internal waves have nearly the

same variations for all submergences below the free surface and for constant heights above the interface; only for a small speed range moment coefficients

make a pronounced increase with respect to those values' of _{obtained for}

infinite depth of submergence belOw the free surface. In the present case, at a submergence of 1 diameter, the maximum.of the mqment coefficient occurs at that speed, the ratio of which to the çritical internal wave speed is equal to 0.843,

i.e.

-

__E_....= ú.S43

-Conclusions:

From

the

theory developed and numerical predictions made in this report

the

following conclusions may be dràwn:

The wave resistances of bodies operating in the stratified óceane are composed of two types of components; the first component excited by

the. free surface waves has nearly the same magnitude as _{occure in a}

homogeneous sea; the second component caused by internal waves is similar in. character to that of shallow water wave resistance,. but charactèristic humps and hollows occur at a very low and narrow speed range,where the wave resistance caused by the free surface waves may be àonsidered to be practically zero.

An examination of the figures show that the maximum added resistance

induced byinternal waves occur when the speed of the body attains a.

certain fraction of the critical speed, and the fraction decreases with the increasing submergences below the free water surface. In the present case where the Rnkine àolid toucheè bóth free sürface ánd intérface, the ratio of the subject speed to the critical value is approximately

equal to _{C/Ok =} 0.85 and decreases at infinite depth of submergence

to zero. (cf. figure 8)

s

stated

_{in the first part of the previous}

report;

_{when the densityof}

the

IeRvier fluid becomes infinitely large the resistance exoression of

tne internal waves. reducea'to the wave, resistance expression öf the shallow water. Therefore, added wave resistance increases in magnitude

with the increase of the, density of the heavier fluid, from zero tò the value of the r1ave resistance in shallow water. Hnòe, for practical density ratios the added resistance would not be so great in magnitude

for surface ships even for submarines. Thïs fact may be seen more clearly

from the examination of figure 8.

In contrast to the added wave resistance, which is decreased for _the

speeds even a small amount larger than the critical one, figures 8 and 9, the added moment increases with the speed up to certain considerable values. On the other hand, the maximum of the dimensionless moment

coefficient occurs again beforethe critical speed and has a pronounced

high value. Even for submergences where the free surface effect may be considered practically nil, this maximum reaches to appreciable values when typical Froude number attain. a value in close vicinity tO

F=C//2ag

C/J/

_O.60

(cf. figure 13).

Submarines, which are fitted with fins and control surfaces

designed for

_2lhjgher

_{operational speeds and for comparatively moderate}

hydrodynamic moment and force coefficients, may come close to such a deep interface with the result that at a certain low speed range moment

and vertical force coefficients are high.enoughtocause. the _submarines

to lose control even under automatic control due toiùsufficency

of the areas of fins and control 'surfaces needed

_under

_{such coùditions.}

-23-

-24-LIST OF SYrIBOLS

one half of the distancé between source and sink of Rankine solid

A a function defined in text relating to Equations 15 and 16

b maximum radius of the Rankine solid

C velocity of the body in the + x-direction

critical speed =

/ 213 -.9ì resistance coefficientrnk' // C

C1.,= moment coefficient

=

f%v1/Lv2.fîï.L.b2.clJ

depth of the solid bottom below the undisturbed interface maximum diameter of the .Rnkine solid

/=

Froude number

9

acceleration of gravity

h

the height of the free surface measured from the undisturbed interface

k

=

variable of integration

't=

V1

modified Bessel functions of second kind

P = one half of the length of the Rankine solid

over-all length of the Rankine solid

a function defined in text relating to Equations 17 a and 17 b generating source strength

M,M1Mmoments

_'f z resistance function resistance function moment function moment function

resultant fiuid-velodty vector

radial distances from the source and sinks

wave resistance

-the height of -the body from undisturbed interface

resistance function. resistánce function moment fuction moment function rectangular co-ordinates

rr

12

v=

v

LIST OF SYOLS

(Cont'd)

= the root of the equation

J

o

J3

the root of the equation ¡3=

(i-)«e

se/rzt t?'4)J

%

the root of the equation

Aweight

displacement

9= wave

direction angle and variable of integration

lower limit of the resistanàe and moment integrals

¿9 lower limit of the moment integral

=

fi

o strength of doüblet

the coordinate of the source along the longitudinal axis of the Blender body

mass density of the light fluid mass density of the heavy fluid

o-=

söurce distribution along the longitudinal

axis

of the slender body

ço=oç

-26-REEERENC ES

Havel

00k,

T. H. 'dave resistance theory and its application to ship

problems. Trans. Soc. Naval Arch. farine Engra., 1961, 12-24.

Hudimac, A. A. Ship waves in a stratified ocean. J. Fluid Mech.

Vol. 11, Part 2, September, 1961, 229-24.

voñ Karman, P. Calcûlation of pressure distribution on airship hulls.

National Advisory Committee for Aeronautics, Tèchnical Memorandum

No. 574, 1930.

Lunde, J. K. A note on the linearized deep water theory of wave

profile and wave resistance calculation. University of California

Institute Engineering Research Berkeley,

_1957.

Mime-Thomson, L. M. Theoretical Hydrodynamics. Macmillan and. Co.

London.

Nilne-thomson, L. N. and Cornrie, L. J. Standard four-figure

mathematical tableso Edition B.

Pond, E. L. The moment acting on a Rankine ovoid moving under free.

surface. J. ShipResearch, Vol. 2, No. 4, March, l95, l-9.

Sabuzcu, T. The theoretical wave resistañce of a ship travelling

under interfacial wave conditions. Norwegian Ship Model Experiment Tank Publication, No. 63, Nay, 1961.

Watson, G. N. A treatise on the theory of Bessel functions. The.

RANKINE'S SOLID

L- 2.200 rn

O-2b-- 0.210

2= 2.095

_{/:C.= -4--0.926}

f

02L

A= 0.07056 'n3

f0,5

x/t

rib

0.00 0.10

0.0998

0.20

0,30

0.9992 0.60 0.9979 0.50 O.99G5 0.60

0.9933

0,70 0.9862 0.B0 09 654 0.90

0.8765

0.6644 0.4985 0.99 0.3507 1.00 D.oOoo

L = 2E

sink

source

2 a=q95228L.

I -I I 0.1 0.2 0,3 O.&ft

The ±/?ape and of/se/s

of a Ran.kine

co/i'

with

len2ih-a'/ameter ra/l'o

of

10.5

Figure 3.

0,S 0.6 0.7 0.3 0.9

zig

qj b (J (n

2

, w

, (JV1%IIV .$ULIU

Ill

Pik

Iu/-u I

0FSW.

L 1m p

pI v__

0g

71

Free

surface waves. o7ily

er na

wa

s, on

y

L

2.20

D'= 0.21

ft=

10.5

to.t ai

F $. W. I.,

ltÉ=a5

PC.2o.g26

o

0.07056

m3.

Free water surface

¿nkr face

0.5 D t=0.û4g

.L0O4

L 0.3 0.4 05 06

o.i

as 09

lo

11 f2. 1.3 14 15 .16 1.7 18 1.9 20 21 2.2 2.3 24 2.5 26' 27 2.9 j .

i.

i -I n

-Fa'1"

I 2 3 i 4 S 6 i 'i. 7 . i I 'i i 'i i i -I 1.1 I i I I I 'I i i i i i I i 'I i I i' i i .i i 'i. i i i .1.. i 'i' i i i, I i S I i i i 0.1 0.2 0.3

0.5

F

...-Scale of Froude

Number

F'ure 4.

C (771./sec,]

h 0.210 ,n.

t'=.O.105 i, 1V.

0.1 oa 0.3 0.5 L I I I I j

RANKINE'S SOLID

©

Im W. =

®totat

97 08 P9

i0

1.1 1 Z I.3 1 4 1 5 .1 6 1.7e 1 1 1.9 L 2.20

D_02I

10.5 P.C.= o.g26. 04 O5 0.6 I i . . i . . . . . . . . i . . a 0.1 0.2

a3

0.4

Scale

of

Froude.. Number

'Piiire 5.

®totai

Free water surface g

O3f5 t = O. 105

_J)*=.1.O

¿AtLQ.095

-.a048

Inter face 2.0 2.1 2.2 23

FiO.3S F3

C/j(f-)9h2

6 0.8.

Q, 0' u U)

--"-T V. O5. I n n n n . n n I . I I I I I 0.1

0.2

-n 'p I J

03

0.4

ScaLe

of Froude Number

1=

--riqure 6.

2.0 2.1 5 2

=0.926

1LL4.__0143

---=ao48

DL

L L

0.07056

2.2 2.3

hy=O.42Om. F=C//i9h5

I I I I I I t n I I

0.5

0.6

®In.:

ota1

©

©

FW.s

total

o

nA ,

-

..

-c(Lm./scc.j

RANK!NE'S SOLID

cree water surce

'1 _-c X i be i-l'ace L

0=

Lio.5

D

2.20

0.21 "17. 'i

h= 0.420 'n.

t= 0.105 ii

=

7'

15 I

RANKI'NE'S SOLID

C p I I -I I I

j

I

13 2.

Scale of

F;:

A,re Z

2.5

-)

inerPace

2

io n n n

L= 2.20 m.

h-.

0=

O.21 fr O.105

û

10.5 D

05

0.926

f;

0.048 4 o 0.5

ecj

z

1.5 1.0 0.5 0.110 0.160 0.090 R, 0.070 I

h=4315rn. f,OJ05n

0.060

irgji;.

0.040 0.050 0.0I0

,r%,v7rrI

¿;-I-Ei 0.0 0,000

-0,1 0.2 0,5 k I 0.5 4.0 05 1,0 I I I I Oi 03 0,4 05 o, 0.7

F'/gare

8. 1.0 03 0.210 in., tI O.105m. 0.4 05 1(j 1.5 a.o O,420?n. t.. 0.105 ?Y1 0.6 0,8 O9 1.0

Scales

òf.

r

h-,o t:0.1DS

09

3.0 2',S

t'

1.2 1.3 1.4

1f,çancF4

07 2,5 2.0 .8

L= 2.20 in.

OO.2s

10.5

=0.926

-D2L

4

t

0.0785cm3

w

3.5 3.0 2,5 s i t f5 i, i,7 1.8

Free waLer surface 1.2 3,5 _3.0

_Ji..

-interface

C

('zvk.J

h,o.a:o m hzO315-'. F,CI11(i-x$h tL i 0.420'.

C//1-a13

h-,..oc'

= 0.105

h- 0.315 m. t= 0.105

h O.420m t=0.105

-

+'- -'---i i i C L'ri/sec]

f

di 0. 0.3 0

i5

106 07 0.81 09 10

II

1.2 ' 1.8 1.9 .0 2.1 b d.5 2 i 6 i I /?fO.315 rl o I i 2 i i i 5 6

h3OÀ20ô =//_j3

O

05

f 2 3 I

f

I h-boo

F=cÏf2a

I 4 4 1.# 0.5 1 1.5 2

F1g.

.9.

Scale

k

= 10.5 D _RC.

--0.92

D2L

007056

? L

2.20 7,.

'I

-2

M o

ne4

Ica

use

db

yf

ree

sur

lac e

a ve

Jvl seconc? app.

0,315 m

O.1C5 9.

t'i

¡rst app.

i

1. 2.

Veloci

y

3. ¡n

ne/er

4.

pe'r

seco-cid

5.

C oìnparison

of the

first

and

second

approximations

Figure

lo.

Ve/octy

_{I,i meter per second}

C omparison of the first and second ápproxímations

F/ìare ft.

3

2.

M

f _PC1)pE/

_j,p.

ien

cajsed1

wa'e

'ni

h=01rn.

-43 ¼)

it

0.105in.

qj

-4 '-, _'s

's,q

'I

_b

. f I

--ti

-

-

T i --

--

i i i i - 02 0.3 04 05 06 07

os

_{0.9 1.0} 11 1.2

31.4 -

_1.5

IS

1.7 14

-41

-Q f -0.1 -01 -0.1

-01

-0.1

-01

-0.1 -0. -0.0 -o -0.0 -LQ -0.0

-00

-0.0 -0.0 o 0 02 0,03 0,04 Cas

-

15

-

14

-013

- 12

-Q 1f -0.10 -Q 0g -Qox

-007

-0.0 -0.05 -Ô 04 -0.03 -a 02

-

0-1 02 0.3 Q4 05 0.6 07

09

0.9 1 0 1.1 1.2 1.3 1.4

Velocty

ii

neter per second

comparison of the first and second appro%irncxtíns

Figure 12.

F = C / YfiL 0.02 0.03 0.04 0.03 0 06 0.07 O6

omet coe ic/eni

ir

nk,

e's sohi

-I

IjHorrrens

-are tak n 'boul C

)

Free waler surface

1;

(A positive mom ni a is

t

tur

1703 dOwn)

,

--T J

ô

m

h1 0,31:5 in. 1

!?

___

77T1.rna ways

1"

D;) 2 1

h2-' oc

P.C.

42=0.926

.hl_t....fc

L-o5

IIIUI1

. : ..

'liii'

. LI?

na

aves fJfJ

___lÍ.._

I h1= 'LJhZ /

sur ace

aves

f

. iS

ji

02 03 4 O °C 0? 08 9 O

II

. .. 2 13 Velocity 4 1

¡17 meter per

. 4. 16 17 18 19 20 SCCO7)

,.

1

.I. ...

6. 23

.

7. 24 0.5 p 1 i 2. 3.

F= C/Y(1-)9fl

.: i ' ' '5 ' ' 2 o4 05 -0.1 1 - o. i o - 0.00 - 0.09 - 0.07

s

"3 -0.06

z

-J - 0.05

-0.04

'

Q

- 0.03 -14

- 0.02 3

- 0.01 -I.' O qj E

00f Z

ÇV) w

L

k

- Qil

-0.10

-- Q08 -. q07 - 0.06 Ñ -0.05, L)

-Q -0;04

t.

t::

-003

-0.02

-0.01. o

0M

L) 0.02 0,03 E 0.04 0.0 0.06 ,0Q7 0;0 (1q s ii i ,posEi

wr

e me oo

¿" ¿orn nose

's

111

I

¡

ri

.

____

t

0 105 ?n. - own)

t

I

¡

p!,

111

e sr/ace

I,

I

Fr

w-tvr,iiiA

. . . O 4 6 8 .10 12 14 1g 20 22 , .

1/,rï'ç.

-,

24

,

26 28 30 ,

,,

34 3 31

-'-h o 4 6 8 10 12 .14 16 20 22 I. .

'Ir

-24 26 28 30 3Z

3

. 1 2 3 .

S,.

L./±2ag/c