Added mass of two dimensional cylinders with the sections of straight frames oscillating vertically in a free surface

Added Mass of Two Dimensional. Cylinders

with the Sections of Straight Frames

Oscillating Vertically in a Free Surface

by

Jong-HeuI Hwang

Seoul National Uinversity

College of Engineering

Department of. Naval Architecture

Seoul, Korea

Reprinted from the Journal öf the Society of Naval Architects of Korea

TABLE OF CONTENTS

List of

igures

(ii)

Nomenclature

2,3

I .

Introduction

-. ..:.:

:

i

J1. General Formulation --

- 2

FormulatiOn of the Problem - 2

Boundary ConditiOns

4

Iinetic Energy of Fluid

5

.-'

D. Added Mass

--

.-

- 5

III. Added Mass for Straight-framed Séction

5

A Straight-framed Section with Raked. Side

- 5

B.. Straight-framed Section with Vertical Side.

.--

&

iV Numerical Results and Discussion

io

A.. Straight-framed Section with Vertical Side ...

- . iO

B; Straight-framed Section with Raked Side

O

Comparison of -Straight-framed Section and Lewis Form 22

V. Summary and Conclusion

23

Acknowledgments - . . References ... ., - : .24

Appendix ...

25

Evaluation of Integral (23)

. . .. . 25

Evaluation of ntegrál (24)

. .26

'LIST. OF FIGURES

Figure:

1. Mapping of polygonal boundary into a real axis

' 3

2. Characteristics of the. section

'-

4

3. Straight-framed section with raked

ide ' 5

4. Mapping of straight-framed section with raked side

6

5. -Straight-framed section with vertical side ₈

6 Mapping of straight-framed section with vertical side

₈

7. Added mass còefficients versus beam-draft ratios with parameter fi

11

8. Added mass coefficients versus fi with parameter B/H

- 12

9., Section shapes and added mass coefficients.

_12.

10. Added mass có'efficients versus sectional area coefficients with parameter 13

ji. Added thass coefficients versus sectional area coeffidents with parameter 2

- - 13

12(a)'--'(g).. 'Added mass coefficients versus beam-draft ratiò with :paameter r

14'-'17

13(a)'-'(f). Added mass coefficients versus

with parameter B/H

i7'»2O

14. Added mass coefficients versus sectional area coefficients with pàrameter 2 - 20

-

15. SectiOn shapes and ade4.ss cefficients (fi =0. 6O)....-

. ' . 21

16. The comparison of straight-framed sections with vertical side and

Lewis fórms on -added mass coefficiénts . - _{- 22}

17. The comparison of stÑight-framed sections with raked side

Vol. 5, No. 2, Nov., 1968

Added Mass of Two Dirnensoal Cylinders with the

Sections of Straight Frames Osdilating Vertically

in a Free Surface

by

J. H. Hwang*

7f]

_&IÉ1

(jc]A

Iq

-

*

f Sj

i+7i-jPj j:T;b,* g

Schwarz-hristOffel

*

MR1 +*

Lewis forms o]J

71A1 Lcwis o)1

g

OAf2] Ø1

+ 1--I,

This work is a general treatxient of added mass calculation of two.-dimensòsiat cyIindes

with straight-framed sections and chines oscillating iñ the Éree surface of

an ideal fluid

with high frequencies. Two and three parameter families in vertical oscillátions áre treated by employing Schwarz-Christoffel transformation. The results

are presented with regards to

geometrical parameters such as chine angles, sectional area coefficient and beam draft ratio.

Introduction

The hydròdynarnic added mass of two dimensional cylinders oscillating vertically at high frequencies_{in the}

f ree surface is of interest to ship vibration, and sometimes to ship motion problems. The general class of

problems dealing with cylinders of straight frames and chines has not been treated previously

After F.M. Lewis' initial work in 1929, Prohaska C2J has evaluated the inertia coefficient for a number Recieved Oct.21, 1968

* Member AssocLate Professor Department of Naval Architecture College of Cngzneerng S oui_{National Univers ty Seoul}

Korea.

-2 Journal of SNAK

of shapes by means of conformal transformations. Landweber and Macagno C33 have treated more general cases

including horizontal oscillations and derived the relations between the added mass and the sectional area

coefficient as well as beam-draft ratio. They C43 have also extended the range of available shapes by a more general classof transformations involving a third parameter. based on the moment- of inertia of the section about

the transverse axis. Above works, strictly speaking, have dealt with oscillations at high frequencies, woo.

Actually, the frequency of oscillations in the range of interest in ship motions has an effect on the added mass. Thus recently, Grim C5J and Tasai C6J- have separately considered such effects, and Porter C7J and Paulling C8J have contributed by experimental confirmátion.

For straight line sections, Lewis iJ has evaluated the inertia coefficient for rectangle and rhombus sections vibrating vertically by employing the SèhwarzChristoffel tranformation. And, Wendel .[9J has analyzed the

added masses for rectangular sections with bilge keels.

Recent'y Vugts ClOi has performed theoretical cbthpitatiôns of hydrodynamic coefficient of cylinders with

rectangle, triangle and other typical sections using the same conformal tranformation as was used for Lewis.

form His work includes checking computation with expenments at finite frequencies for all three modes of

oscillation: heave, sway, and roll.

Hwang and Kim C11J have investigated cylinders with the typical straight frames in vertical oscillation at high frequencies employing the- Schwarz-Christoffel transformation, which includes added mass coefficient calculated

for twelve typical sections with vertical side and fiat bottom using manual integration in the analysis. By comparing the results with those of the Lewis forms having same sectional area coefficient and beam-draft

ratio, it was concluded that the magnitude of chine angle seems to be a dominant factor over the sectional area coefficient on added mass coefficient for straight-framed sections.. Since then, a number of computations has now been performed and a systematic relationship between the added mass coefficient and each variable concerned can be presented for various straight frame and chine sections.

General Formulation

Formulation of the Problem

The added mass calculation for a two dimensional cylinder with straight-framed section, when it' oscillates

vertically at high frequencies in the free surface

of an infinite

invicid fluid may be accoinphshed by the

Schwarz-Christoffel transformation as stated previously 11J. Take the y-axis. in the free surface and the x-axis

normai to it as shown in Fig. 1. - ..

The transformation of the interior of a polygon PQR-VWX in the z-plane into a half plane aböve the real

axis in the t-plane may' be expressed in the form

Nomenclature

A = sectional area b = half beam of section B = full beam of section g acceleration of ravity

G,G' = designation of section and its image H = draft 'of section

= corresponding value of çs at vertices of section k2 = added mass coefficient foi vertical vibration

K = wave number = w2/g

w' added mäss for vertical vibration

n = distance measured along normal to side of section

r = modulus of complex variables in z plane = length of side of sectión near free sirf ace ri length of sHe of section near bottom s = ' arc length measured al6ng side of section

T kinetiC energy of flid

U vertical component of velocity of section

w = complex potential =

± iç

x,y = horizontal and vertical co-ordinates in plañe òf

FLOW DIRECTION

o(ii

" I . X

r'

"i

R'..

'

-¡T'

Nomenclature

z = complex variables = z ± i y

«n interior angle at vertices of polygon

ß,T = parameters controlling chine angles

C = complex variables = ¿ ± i

= horizontal and vertical co-ordinates in C plane

= points on real axis of c-plane

= angle of polar co-ordinate system in z-plane

2 = reciprocal of beam-draft ratio = H/B p = mass density ¿f fluid

2 3

"z

ç5 = velocity potential refered to the flow around the

stationary polygon : - Vço = (u,v) velocity components = velocity potential refered to the flow around the moving polygon

= stream function refered to the flow around the stationary polygon

çb' = stream function refered to the flow around the

moving polygon

w = circular frequency of oscillation ofcylinder

(o)

(b)

Fig. i

Mapping of polygonal boundary into a real axis

'where L is an arbitrary constant which can be removed by a proper selection of origin in the z-plane. The proper choice of C will fix the scale and orientation.

In applications such as this we are concerned with simple polygons with two of its sides extending tothe infinities as in the above figure. The factors corresponding to ¿

= - oo

and oo are omitted from the equation of transformation, and the angle a does not appear. The complex potential for the rectilinear flow in C-plane is expressed by

wC

(3)

for the unit velocity, where

w(C) = co + içb,

Vol. 5, No. 2, Nov., 1968 3

= C(C - ¿iY

(C - ¿2)n1

(C - ¿)

j1

(1)

ai+az+aa+

=(n-2»r,

where z and are the complex variables with z=x+iy andC=+i7ì, C may be complex or real, ¿1,e2,,

are the particular values of ¿ at the vertices of the polygon and czj,ao,a3, are the angles at corresponding vertices as shown. Integrating (1) we get

c$ [C

-

¿i)'

(C

¿)nl (C

_]

d -t-L, (2)

Ô

Journal of SNAK and ça and ç1 are the velocity potential and the stream function _{respectively.}

Hence we get

= 1,2,3,

Then, a rèctilinear 'flaw in C-plane may transfermed into a flow in z-plàne with a polygonal boundary

which is at rest in the fluid by the transformatión

= (w - çDi)' (w - ça2)1 (w '-

ç3);_1,

(5) Now we can consider the compleic potential for the flow around a stationary polygon in z-plane, say

'.v (z) = ça + içl.

₍₆₎

In Fig.1 (a), polygon QRSTth-..U'T'S'R1Q is symmetrical_{about both y-axis and ï-axis, and the part of the}

figure TUVWV'U'T' (say G) may be consideredas a' hullsection with straight-frames. The figure T'S'R'QRST is denoted as

d'

hereafter.

The rectilinear il6w around the closed polygon QRSTU'.-S'R'Q is sthmetrical about both x-axis and y-axis. Therefore it is suThcient to consider the flow around the polygon PQRSTUVWX whose boundaries consist of stream line corresponding to' çb=o.

Boundary Conditions

Equation (5) represents the flow past a stationary polygon. We, wish to obtain, the energy of the flow of' a

moving polygon in a fluid stationafy at infinity, we must accordingly add 'o thé values of ço and çbthe terms

çai=z.

cii=Y

(7)

giving

ça' = ça - z,

çli'

- y

' ₍₈₎

for the moving polygon, where ça' and çb' are velocity potential and stream function refered to the flow around

the moving polygon.

When thé body is oscillating with an angular frequency w, the libundery condition on the free surface is

'w2

where K=---, and g

G'QI

-

+Kça'.=Q onz =o,y>'-f,

' ,

is the acceleration of grvity. When w is very 'large, the bóundary condition' becomes

V

wG

- X

Fig. 2 Characteristics, of 'the section ax

-an' as ' an

where s denotes the length along the bduñdary. Hence, we get the following reÏation,

(4)

B

ça' = O

on '=0, y >---,

(10)

In the case of vertical oscillation of G the boundary

condition (10) is satisfied by supposing that the double section GG' oscillates äs a single

rigid form, so that

the bòundary conditioiòn CG' bdcomès

8ça' ax

an

-where n is the

directioñ 'df the Outwrd normal to

CG'.

Vol. 5, No. 2; Nov., 1968

(12)

We may therefore take the boundary conditio,

dy or b' y, (13)

along the polygonal oundary And it will be recalled that this same condition is

satifled by imposiiig a

uniform flow along the positive x axis as appearing in Eq. (7). 'he infinity condition thay therefore be taken as the fluid to have a uniform flow at infinity with vanishing of disturbances

Kinetic Energy of Fluid

The kinetic eergy T of a fluid at rest at infinityis given by

i

r

,

T=Tp3ç2 0dS

ot

T

f$ço' 4'

(14)

where dS dénotes an elethetary area, p is the ma deSsity of the fluid, and the integral extends over all the

boundaries of the fluid.

Since çs'=O on the free surface for the vertical oscillations when ü is very large the kinetic energy integral

vanishes over this boufldary. Therefòre tile kinetic energy of the flu below the fïee surface is. half

of that

obtained when a submerged whole poligonal cylinder

.bGr thove etlly.

-.

Added Mass

The addéd mass of the vibratiñg polygonal cylinder is then. given by

, 2T

m =

where U is the corresponding intantaileóus velocity.

If U = i,

rn =' 2T. (15)

For a prismatic body, whose cross section is a semi-ellipse, the added mass per unit length is given by

m' = __.rpb2, (16)

where bdesignates half-breadth at waterline.

For other forms. it may be written in the form

1 ₂

rn' =

k2---2rpb,

using k2 as the added mass çoecient.

From (15) and (17),we get

-2T k2

---7rpb2 or 16T k2

-where designates fùll-breadth at' waterline.

(17)

6 Journal of SNAK

Added Mass for Straight Framed Section

Straight Framed Sections with Raked Side

In the case of a two dimensional cylinder with the general straight-framed section anda single chine (Fig. 3), the flow around its section may be obtained from equation (6) wIth

B

a1

s=ßir,,

a2=a4=-Tir

a3 = 5-2(ß±T)]r

and

Soi

- 1,

Ç°2 - k, 513 = 0, 534 = k, ço

= 1

Fig. a

Straight-framed section

with raked side

2-plane w-plane

ki

()

(b)

Fig. 4 Mapping of straight-framed section with raked side

Hence the transformation corresponding to the present problem becomes

= (w2 1)ß_i (w2 - k2)7-1 W22

-

2 - k')7-' .w2(2ßffl dw ± const.,

₍₂₁₎

where

--ß<1,

l<T<,.

We are only interested in evaluating the intelgral (21) along the 'bouñ'daiiés of the sectión. Êor this,

_ç=0

and we have

z = re

₌

$ (ça2 - 1)" (ç2 - k2)T

512(2ß' _{¿ça + const;,}

(22) where r and 8 are polar coordinates.

Let r and rj be the lengths of the sides óf the polygon corresponding to _a=0,

_{ço=k) and (ça=k, ça=1)}

respectively.

Then, we have

Vol. 5, No. 2, Nov., 1968

= Zk -

2)fll

(k2 -.

2) dço (23)

(0 <

.< k <El)

lroin (13) and (14).

And y increment of the side and r1 of. the section are .expressed by dy =

drs sin (2 fi +

T)ii

and

¿y= drj simi1

ß)ir,

respectively. Hénce, (30) becomes 2T = 2p sin(2

çs2)ß(k2

2)T-1 ç552T>dç r1 ₌ I zi- - Zh

cl

is

(1

2)ß1 (2

k2)'

d1 (24)

(0<k<<1)

Where zo, z and zj are the values of z corresponding to the points at çs=0, k and 1 respectively.

The integral of (23) is apprdimated as follows by employiig the hypergeometric function (see Appendix A)

¡'(7)

¡'(1

fi ± n)

-

±

3-2ß

Ê

= k

¡'(i_fi)

r (4,

p ±

n) n!

(0<k<1)

Similarly (24) becomes (see Appendix B)

i

F(r)

_Ê

(+T_+n) r(fin)

(1)

(25) (26) (27) (28) . (29) (30) r1=

(1

r'(pr -g-)

n°

F(ß+7+n)

n!

.(oKk<1)

If we denote B and H as the full beam and dräft of the section respectively, then we get

b rs sin(2 fi ± r) r

±

r1 sin (1

ß)r

.H==rkcos(2 fi

+ r)

ir-+

r1cos(l

with b.= B/2.

And the area of the section becomes

A rs2 cos(2 r

± fi)lr

sin(2 - r +-fi)z

+

rl sin(1 fi)ir C2rAcoS (2

,7+ ß).z +

flcos (1 . ß)rj

The kinetic energy of the entrained water per únit leith of cylinder is given by 2T _{2pS6 (çs}

z) dy

8 :1

Journal of SNAK

±

sin(1 - ß)x$1(i

çoi)ß_i(ç2 k2)T_1ç5_2(ß+7) _dç

-

4-] (31)

by virtue of equations (23) ad (24).

Integrals involved in the equation (31) may be evaluated by the same approximation as employed in equations

(23) and (24). Then we have.

-

¡'(r)

_t(3Rï±n)

_k2"

2T= p [ksnß sin (2ß+T)rf(1

ß)

_{(2_ß±n)T:.ßn)}

n!

(1k2) 2ß+Tj sin (lß»r

¡'(r):

¡'(ß±n)

(1k2)"

A1 (32

- ¡'(fl±r-2)

(ß±T±ñ-1)(ß+7+n-2)

n! .1

Therefore the added mass coefficient k2 for the vertical oscillation will be deducedto

k2

=!-

_{rk2(2-ß)Sfl(2..}

¡'(r)

B2 L

r

¡'(1p)

_{(2ß±n)(1ß±n)}

_n!

¡'(r)

_-

¡'(ß ±

n)

(ik2)"

k2)ß_sjn(i

¡'(ß±T-2)

(ß+T+ni)(ß±r+n-2)

ii!

Straight-framed Section with Vèitical Side

When a is it instead of [5-2(ß+T)Jir in the preceding section

i.e. in the case of the wall sided profile, we have ß

+ T =

2.

In this case the denominator of the second teim in the bracket of the eqution (32) beoomès zero, therefore we must analyze

seapra-tely from the preceding section as a special case.

( o)

FLOW DIRECTION

Fig. 6 Mapping of straight-framed sectiòn with vertical side

-i-ko

Fig. 5 Straight-framed sectioi

with, vertical side w-plane

(b)-Vol. 5, No. 3, Nov., 1968 9

The flow around such a section may be obtained by the following transformation

dz

_{- (w2 - 1)-' (w2 - k2)i_ß}

where _fi

_<

1. Therefore we have

= 3

(1_2)ßi (k2

2) 'ß dço

ri

3 (1ço2)ß (ço2 - k2) 1-P

These integrals are performed by the same approximation as in equations (23) and (24), but they are easily

obtained by substituting

r

by 2 ß in the equations (25) and (26). Thus it follows that

Tk Ti 1 3-2ß ..

r' (1fi-t-n) F(+n)

rk ₌ k

('ß)

ï' (--_p+n)

n!

r(-- -t-n) f(ß+n)

(1k2) ¡'(2p)

2ir

,,=o

(n±1)!

where

_{o <}

k

<

1, and b =

r1 sin (1

-H

=

rh + Ti cos (1

-A =

b(rs ± H).

The kinetic energy of the entrained vater is given by

A1 2T =

2p [sin (1fi)

r$ 2ço drj

-i

=

p [2 sin (1 ß)$k (1

ta2)-1(ta2 - k2) ißdça.AJ

since dy is zero along the parallel side to the flow.

Let

I

ça( ta2)ß_i (ta2 - k2)'-ß dço.

This integral may be deduced to

I

= --

(1 - k2) B(ß, 2fi)

Hence by virtue of the property of Gamma function, we obtain

I

= --(1 -

k2)(i

-Then (39) becomes

2T=p[(1k2)(1fi)arAJ

T1 =

(1k2)"

n! ' (34) (53) (36) (37) (38) (39) (40)

10 Journal of SNAK

-.p(2(l-k?)(1-'ß)1r-B(rh±HJ

Therefore added mas coefficient k2 for vertical oscillation in this case is given by

k2 =

(1 - k2) (1

fi)

-

(rk + H)

Designate sectibnal area coefficient by a. Then

i

(

= T ±

lt follows that

k2 =

--(1 -

k2)(i -

_{) - -4- t7,}

where

2=

H

_B

Numerical Results and Discussion

Straight Framed Séction with Vertical Side

This class of problems belongs to two parameter family, containin parameters k and fi. For numerical

results, values of re, rj, B, H,

B/H, A, a, k2

eralculated by computer for the following k values, andfi

values, taken in turn for each k.

k: 0.10, 0.20, O.0, 0.40,0.45, 0.50, 0.55, 0.60, 0.0, 0.80, 0.9Ó, 0.93, 0.95, 0.98

fi: 0.50, 0.55, 0.65, Ò.7Ó, 0.75, 0.80, 0.85, 0.90, Ó.95.

The results of the computatioñ are plotted in Fig. 7 and Fig 8. The values of added mass coefficient k2 with

parameter fi on th base of B/H are plotted in Fig. 7 and k2 on the base of fi with parameter B/H are plotted

iÌi Fig. 8.

The upper limit of B/H for constant fi corresponds to tiiangular section with same p: Theíefore cúrves of

k2 for the general sectiòns must be within the two limits given by those of rectangular and triangular sections. When fi equals to 0.5, the section coriesponds to the rectangl.. The calculated values of k2

for fi=0. 5 from

equation (45) coincide well with thöse of Lewis formula for rectangular section [1J as can be seen. If B/H and fi are chosen, the sectional area coefficient is predetermined.

Thus

a=1- --tan(ß--0.5)v.

(46)

This equation shows that dì relatiòn between a and B/H(= 1/2) is linear with constant parameter fi in two

parameter family of straight-framed sections.

The curves öf h2 òn the base a with parameter fiare plotted in Fig. 10 and the same curve with pararnetter

B/H are plotted in Fig. 11. -,

From Fig. 7 and Fig. 9 it is clear that the values of the added mass coefficient k2 increase if the values of fi decrease

Straight-framed Section with RakedSide

This class of problems belongs to three parameter family, containing parameter k,

fi and r. For

umericaI

results, values of th same geometrieal and physical quantities as in the preceding section were calculated for

voI. 5, N0. 2, No 1968

0.5

2.0

Fig. 7 Addj mas5

coecjent versus beam_draft

ratIo5 wjgb _Parameter

2

Fig. 9 Section shapes and added mass coefficients

¡

-12 Journal of SNAK

0.50

.55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 A

Fig. Added mass coefficients versus ß with parameter B/H

B/N 0.5 B/I-f 1.0 B/Ha1.5

Vol. 5, No. 2, Nov., 1968 2.0 LS K2 1.0 0.5 Ka 2

o

Fig. lo Added mass coefficients versus sectional area coefficients with parameter ß

13

05

0.6

07

08

09

LO

o-Fig.Lll r Added mass coefficients versus sectional area coefficients with parameter 2

0.8 0.9 1.0

0.5 0.6 0.7

O-14 Jounal of SNAK K2 20 0.-o

Fig. 12 (a)

B/H

14 .0

Fig. 12 (b)

Fig. 12 Added mass coefficients versus sectional area coefficients with parameer ß .4

0.50

II

\\\

TRIANGLE

i

.4

0.55

Jr

EC D

ç

o

20

40

60

2

4

6

B/H

2 o

Vol. 5, No. 2, Nov., 1968 ₇₅ K2

a

0.5

0

2 3

B/H

Fig. 12 (c)

Fig. 12 (d)

s

6 P

0.6O

JR

AE

2.0 X, Q 2.0 0.

o

B/H

Fig. 12 (f)

i

TRtA1GL

O.75

'3

RECTANGLE

-'1

s 16 Journal of SNAK

0

2

B/H

Fig. 12 (e)

2 IoV., i968

Vol. 5. o.

g. iz (g)

17

R

is

'

.45

005

jo

uS

Ç'g. 13 eteT ßI

r

itb pata

1aS

ig

K K2

a

'.5

0.

Fig. 13 (e)

1_1,1

1.0

05

2

-

_

___»__

6. 4.0 aD ¡2.o TR ANGLE /

20.60

5.0

T,J4NUIII

18 Journal of SNAK

I0

It

I 2

'3

1.4

Fig. 13 (b)

1.0 12

13

r

K K2 I.10 1.05 0.50

Fig. 13 (e)

0.65

I

.6

p

0.70 1.0 D.S

TRl4?gj4

Vol. 5, No. 2, Nov., 1968 19

Ic;

'.3

1.4

Fig. 13 (d)

12

'3

1.4

2O Journal of SNAK

Fig. 14 Added mass coefficients versus sectional area coefficients with parameter 2

i

L5 LO

)

ß.L.80

v!..

9/.4

1.2 0.8 t

0.60

W

.7 .6

-1.15 1.20

Fig. 13 (f)

05

06

o

K2t.

o

o

o

1. t.

Vol. 5, No. 2, Nov., 1958 21

the relation 1.5 <fi+T<2 were taken.

0.70, 0.80, 0.90

0.90

T: appropriate values between 1.05 and 1.45 satisfying the relation 1.5 <p +r <2 for each fi value above.

The results of the computation are plotted in Fig. 12 and Fig. 13. Figure 12 shows the values of k2 with parameters _fiand

r

on the base of B/H, and Fig. 13 shows the values of R2 on the base of r with parameters

fi and B/H.

The whole points lie between those of rectanglular and triangular section in Fig. 14.

It is obvious that the lower limit of B/H for constant fi and r corresponds to the triangular section with a

bottom angle (fi2r) taking r value.

If B/H,

fi and r are chosen, the sectional area coefficient is predetermined.

B/HeI B/Me 2

B/He 5

B/H 3

Fig. 15 Section sbapes and added mass coefficient (p 0.60)

k: 0.10, 0.20, 0.30, 0.40, 0.45, 0.50, 0.55, 0.60,

fi: 0.50, 0.55, 0.60, 0.65, 0.70, 0.75, 0.80, 0.85,

Thus

.__tañ(2_ß±T)x} {1___-tanft--'o.5)}..

1tan(

-- O.5)r tan(2 - ß + T),r

It is sèen from above equation tt. the contours of ç9nstat ß and r induce the hyperbolic relation between

¿r and B/H(= i/) in

three parameter family of straight-framed section. The curves of k2 on the base of

parameter at constant fi value of 0.60, for example, are plotted in Fig. 14.

Several sections and corresponding added mass coefficients k2 when fi = 0.6 are shown in Fig. 15 to show the. effect of r value. From Fig. 12, and Fig 15 it i.seen that the values of added mass coefficient k2, at

contant fi values, increase with the r values in considerable range of beam draft ratio and. decrease while r

increase comparatively from certain high beam draft ratios It is interesting to note that the every k2 curve in Fig

12 shows that its values are closer to that of trianglulár section and are fàr from that of rectagWar section. Comparisoü of Straight-framed Section and Lewis Form

It is seen from the results of computation in the preceding sections that in straight-framed section the chine

angles are also basic parameters having marked effects. on added mass coefficients as well as beam-draft ratio and sectional ârea coefficient. Thus the comparison between straight-framed sectiòns and Lewis forms on added mass coefficient is of little merit if it is done only at the same beam-draft ratio and sectional areá coefficient,

neglecting the chine angles.

-05.

06

0.1 .

ó.ä

r

69

-.

Fig. 16 The comparison of straight-framed sections with vertical side

and Lewis forms on added mass coefficients

For the two parameter family of stráight-framed section, however, angle is predetermined if beam-draft ratio nnd ectiônàl area còeffieient are chosen. Therefore; for this tlass of straight-framed sections it

is clear that

the comparison of the straight-frarned section and Lewis form is possible provided a and B/H re used as basic

parameters. .. . .

For B/H = 2(2 '=- 0.5)änd 333(2'=' 0-9)-- for eïamples, thestraightrframed sections of vertical side give

Ereater values of added '-mass coefficient than those of Lewis foi-m in almost entire region of sectional area

Vol; 5, No. 2, Nov., 1968 23

coefficieñts. The maximum difference of about 20 percent over Lewis form. are attained near. the sectional- area coefficient 0. 8 and it is maintained in the considerably broad band as shown in Fig; :16.

The differences decrease as a values approach to the rectangles (à = I)' or triangles (a = 0.5).

K1

1.2

0.5

Fig. 17 The comparison oÉ straight-framed sections with raked side (ß=0. 60) and Lewis

froms on added mass coefficients

Hwang and Kim Cil] have shòwn, from their computation on added mass coefficients for twelve typicaL stra-ight-framed sections having vertical side and flat bottom and covering fur groups of chine angles, that the added mass coefficient of those sections had considerably higher values than. those of Lewis forms háving same sectional

area coefficients and beam" 'draft ratiòs and that

the increments ranged fróm 3% 'to 30% in whole being

significantly controlled by the chine angle as stated at the early part of this paper.

Nevertheless, from the results of 'systematic computation on the straight-framed iection with taked side and bottom, as examplified for the constant angle ß of 0.60, added mass oefficieüts associated with vertical Oscillation at high fiequency give greater valùe thañ those of Lewis forms beyond 'the range of _{= 0:63 for B/H =3. 33}

(1 = 0.3) and beyond the range of over

0.73 for B/H'= 2:0(2 = O:'5)as shón' in "ETg. 17

And it is interesting to note' that the straight-framed sections for fi '' 0 60' give sthallèr added'"thass coefficients than those of Lewis forms at those c values smaller than 0.63 for. B/H = 3.33 ad 6 'ùlues smaller than 0.73 for

B/H = .2.0.

It appears that these results are màinly dependent ùpon the slope' of.the. sides of the section.

Summar.y

and'.Conc1usion-The foregoing analyses have, demonstrated a general echthque of employing Schwartz-Christoffel transformation for the added mass calculation of two-dimensional cylinders with straight framed sections and chines oscillating

vertically in the free surface of an ideal fluid at high frequencies...

Specifically, two and three' parameter families, including sections found in practical chine, ship forms such

as with raked sides and deadrise bottoms were analyzed and the results were found to be well within the

expected range of values compared to Lewis forms and other previous works.

The study shows that the parameter controlling chine angle is a significant one as are other parameters such as beam draft ratios and sectional area coefficients It is significant to note that the method such as employed

-

CHINE FORM LEWIS FORM

07;». 08

'

2. Journal' of SNAK here raises no dffficulty with regard to raked sides and deadrise; and 'thus such èffects may be. studied fürther in detail by employing this échnique. .'

In real flow, due to the sprays 'on the aidés ad eddies .t the knuckles, if any, certain differences in added

mass coefficients are expected And the effect of frequencies of oscillation is of interest in practical application An experimental study, therefore, to complement the above would show further interesting results, and such is

recommended.

Acknowledgments

The author expresses sinc&ely his graditude to the President of the Seoul National University Dr. Mun Whan

Çhoe who has supported this work. He is also grateful to Professors. Hokee 'Minn and Keuck Chun Kim of

the Seoul National University who were always willing to discuss the details and to help him in solving many

aspects' of the problem. .

The author is especially indebted to Professor Masatoshi Bessho of the Defence Academy, Japan, for the

encouragement and helpful suggestions given to him and furnishing several references. He is also indebted to

Professor C.W Prohaska for furnishing the copy of his earliest paper on ship vertical vibration.

He also takes this opportunity tò acknowledge Dr. Hun Chol Kim of the Korea Institute of Science and

Technology for his' excellent help in preparing this English text.

The efforts of Mr. YK. Hin, who pgr

ed the'rnpiitation, is also grcatfully acknòwledged. Lastly, he

wishes to express his thanks to Mr. Y. J. Kwon and the students who has rendered assistance in preparing

the graphs.' .

References

,

ï. FM. Lewis; «The Inertia of the Water Surrounding a Vibrating Ship," 'Trans. SNAME, vol. 37, 1929.

C.W. Prohaska, «Vibrations verticales ie naviré," Bulletin 'de L'Association Technique Maritine,. et

Aero-nautique, 1947.

L. Lanciweber and M. Macagno, "Added mass of Two-Dimensional Forms Oscillating in a Free Surface," Journal of.Ehip Reseazch, vol. 1. No. 3, 1957. . .

L. Landweber and M. Macagno, "Added Mass of a Three-Parameter Family of Two-Dimensional 'Forms Oscillating in a Free Surface," Journal of Ship Research, vol. 2, No. 4, 1959e

5 0 Grim

Die Schwingungen von Schwimmenden Zweidimensionalen Kdrpern" Hamsburgische Schiffbau

Versuchsanstalt, Gesell.cckaft Report 1171, 1959. . .

F'. Tasai, "On the Damping Force and Added Mass of Ships Heaving and Pitching," Journal of Zosen

Kiokai, vol. 105, 1959.

W.R. Porter, «Pressure Distributiòns, Added Mass, and Damping Coefficients for Cylin'ders Oscillating in a Free Surface," University of California Report, Series 82, Issue 16, 1960.

8 J.R. Paulling, "Measurèmentôf Préssures, Forcesand Radiating Waves for Cylinders Oscilláting in a Free Surface," University of California Repoit, Series 82, 'Issue 23, 1962.

9. K. Wendel, «Hydrodynamische Massen und hydrodynamische Massenträgheitsmomente," STG, vol 44,

1950. ' , '

10.. J. H. Vugts, "The Hydródyúthic Coefficients for Swaying, Heaving añd Rolling Cylinders in a Free Surface",

ISP, vol. 15 No. 167, 1968 . .

li. JH; 'Hwang 'aiid K.C. Kim, "A Study on the' Added 'Mass Äsioãiated with the Vertical Oscillation of the

rk = S:(

2)ß-1(ki 2)T_14_2iß+7) dçs

Let (ço/k)2 t, then (23) bcomes

rk _

1k32ß5' t3J2-(i

t)1

(i -

k2t)1dt

In evaluating the above iitegral we employ the following relation (12J

2F1(a, b; c; x) - Î(b).T(c

_{- 1,5} S

tb-i(1 -

t)C_6-l(i

-

tx)dt

( Re(c) > Re(b) > Ô, 1x141 J

Then (23) will be. deduced to

- ß

-r) 1(r)

= k3'-2P _{2F1(1 - fi,}

since

kl <land,

-'ß >

.ß±

7

> o.'

Hypergeomêtrical function iFi(a, b; c; x)iS also expressed as follows,

.1(c)

1(a

+

n) 1(b

+

n) z"

2Fi(a, b; C; z)

_{¡'(a) .1(b)}

_,,o

₁₍

_{± n)}

Then

(lxl<)

(23), (48) (49)

fi; k2). '(50)

r(. 1-1g)

1(1 - fi

+ñ)1(-

-

p4.'r+n)

From (49) and (51) we have .

-= .-f3--2ß

¿___ . " ... (25)

2

1(1 - fi)

_p

₊

_n) n!

Sirce I k <i in the: present problem, the seriés appeared in the right sidé of (25) convergés. It is noted

that

p<i, 1<T<--and4<ß±T<2.

_:.

The integral appeared in the first term of equation (31) may be readily evaluated similarly.

2F1(1 -

'F- p;

k2)

=

-'--"Q

1(1 - ß) r'(4_p+r)

(jkl<1)

VoL 5 No. 2, Nov., 1968 25

National University vol 3 No i (written in Korean) 1968

12 E T Whittaker and G N Watson A Course of Modern Analys2s Cambridge University Press 1927

APPENDIX A

26 _{JoUrnal of SNAK}

APPENDIX B'

Evaluation of Integral (24)

rl =

2)ß-1 (2

k2)'l

2(2j)

d ₍₂₄₎

(o <k< <i)

Let (p2. k2)/(1 k2) = t, then (24) becomes

r1

--(1

k2)ß+T_iSl (1 .-t)ß-1 tT1(k2 +

(i

2)tJ3J2.-dt

₍₅₃₎ Since

_{k < i and the above integral does not seem to}

_{be approximated to any convergent series as the}

previous one we cóndesider a new transformation of variable, say

i

t = u.

Then (53) bècomës

r1 = -(i - k2)ß1.T1S' ufr1(1 ---ù)T'Ci (i

k2)uJ3i2_. _dii. By the saiñê technique applied in equatión (-23) we have

rl

3 .

. \rf

-

Pf \

¼P

-

5 ) '.J-

i (i...2\n

-r =_(i -

i _.2

k2ß-1

/

.

Y'

\--

r'

r

"

"°

1

+ T +

flj fl.

-Since- kl < i,

there follows o

< i

k2

< land ft ±

T > _ft

>

O, the series in (26) converges. The.