Hydrodynamic force and moment produced by swaying oscillation of cylinders on the surface of a fluid by F. Tasai

HYDROD!NAMXO FORCE AND MOMENT PBODUCED BY s:)AflNG OSCILLATION

OF OThINDERS ON THE SURFACE OF A FLUID.

By Fukuzo Tassi Journal of Zosen Kiokai. 110, Translation bj T. Sonoda.

Report Nr.

8.

Deift Shipbuildjn$Laborato!yt 15 Febr. 1962.

1itroduc tjon,

When a floating cylinder sways, there ¿cours a hydrodynamics]. force in water. This _{orce approximately coneta of the ád8d} ier-tial fored w)ich i _{linearly proportional to the horizontal adoelera..} tiozi, and the damping forqe _{hi.ob i.e proportional}

_to

_{the .velooity. Th.} swaying motiOn,

these toroea

dontt uauall

act to

the debter of gravi

ty of th

cylinder, becaus, of the wisymnietrical

motion of water1 Thus, there arises a rolling moment which is the coupling t of sway in rolling motion. The magnitude of those forces and moments generally is a function of the frequency of motion, as in the case of the heaving motion. Q. Grim [1 made two-dimensional caloulaticna of added mass and progressive wave height which is caused by swaying motion for three various eUipsoid cylinders, and aleo made a graph against

B

= 7

Lwhere 4)

circular frequency in swaying, Bbresdth of cylinder at waterline. And he aleo treated the added masa and mo-ruent of a cylinder of which the sections were of

the

_{Lewia'form in}

the vioinitr of Oi401

Using a model boats Motora0[3 measured

ad-ded

ZTiaØB _{in swaying and circulating motion. These above mentioned}

data are not sufficient to calculate (say, by strip method) added mase and damping forces for sway and yaw

_which

_{generally change}

ac-cording to frequency,

The author formerly calculated the added masa and progressive wave height of cylinders in heaving motion ["LI) _.

Applying the same

method,

the author

calculated

the two.dimen5ional values of added

mass, progressiva wave height and rolling moment which

are

Causad by

swaying for the

cylinder

having sections of

Lewietforma.

1 Progressive wave height

caused by swayjn

oscillation.

The y-axis is

vertical

(downward positive)

and

the x-axis lies

-2-in the still water surface (right hand positive) Assumptions are fled.

that the

depth

_{of the water is infinite and the fluid is ideal.}

Suppose the cylinder makes a swaying oscillation with email displace..

mente:

_{- Soos (&,t + L), and velocity of}

_{- -SJ.sin (dit + E).}

Sinos the length of cylinder is infinite, motion of water

rfaoe is

two dimensional and 1800 syanastrioal

to y..sxis. Therefore velocity

potential ja:

(x,y) a ₍₁₎

The linearized frei surface condition is:

(2)

assuming the amplitude of away to be email to the first order, the boundary condition at

X3 a O

is:

IX

() a U (.)

(3)

where

is outward

normal to the cylinder surface. The calculation was made for the esation of LewLs fora in this pap.r as well as in

[i1).

_{The conformal function ia expressed}

as followsi

(If)

where: B Z

* X

+ ty,

5-

iee, M

s

₂₍₁₊

a1 + 93)

Put«

0, then the circumference of Lewia'form section is;

X0 -

M{(1

.+ a) ein - a, sin 38

(5)

M(1. *i) ooeø

_{*3 008 39}

_J

7a

frem4 øB

_{and .q. (If), equation (2) is expressed}

_{as follows:}

fl

g

2

oc

-«

..3

e _*1

)

u.O(O..±)

₍₆₎

l+a1+a3

Let ¶P denote the stream function, then the boundary

condition is

expressed as follows:

U()_0a UM{(1 ...a1)sino

+ 3s,airI3O}(7)

Therefore we get:

IJMf(1

a) cose + &3

005

3é}+ c(t)

(8)

where C(t) is only a function of time.Suppoea the

following

poten-tial. which aatiefiea:

$(OÇ,Ø) - .'(°(.'G)(from equation (i)), eivatiòn

(6) and the

Laplace equations

B

Iç2m

2st

_[.(2a1)

ain2m+i)9+

i+a1+a,j

2m

-sin2m&4

-(2m+2)

looe(4t

3a

+

2m2

sin(2m+2* -e

5ifl(2m+k)Ojj,t

al,2,3 ) ₍₉₎

the stz.aa function conjugate to equation (9) is,

I

-(2m+1) oos(2m+1)9

L°

1+*1+a32m

ooa2m9

-(2s+2)cL

3a

'coeWt + 2a+2 cos(2a+2)9

(a =l,2,3,

-3-(lo)

These functions are O

attX - 0c

Take the two

dimensional horizontal Doublet at origin O which

denotee progreesive wave at

infinite distance. Let denota the

progressive wave

height caused by this Doublet at infinite djetano

the atxeam

function for the Doublet le .xpr.aeed as follows:

r

2Z1fc(x,x,7)ooewt 1Ç(X,x,ï)sinwtJ

(ii)

wheret

7t'ooe

Xx

Y5"

±7te1etnEx_Ik cos kIsthkjkx

(12)

K(x

K

the upper

5jÇ0f

compound sign in formula (12) le valid where

x>O

Then, beeide

_{the above mentioned fundamental condition, take the}

following stream function

4I,

which eatisfieB the condition of

conti-nuity with both the horizontal

and

_{vertical velocity of the fluid at} z O, _{and which produoeel, sinusoidal progressive wave at}

infinite distance,

'I'1VC(

1f(

(.0)ein)t]

(ß)

[_._(2m1)0cos(2m+l)e

f

a

"1+i+a3t2m

COB 2in+2

coe(2m+2)

3a 2m+2

-COø

+

+einWtQ231() [_e2m+1cos(m+l)e

₊ o

(

-mø

_a

e2m42

o.

2 MS+i! 2m co ( 2m+2) Ø 3a

e2

2m+k

ooe(2m+4)O}]

(13)

substituting the boundary

condition (8), (13) is expressed ae follows at the circumference,

(43)

*c)

(3O)coac.Jt +1f0

B,9)aint

(O

2mO a1cos(2m+2)e 1+a1+a3 2m + 2m+2

+

3acoe(2m+k)8

-

2a+k

}J+ainttm;

[_cosa+i

)

Y

s

a1cos(2m+2)e

3acoe(2a+k)9

+a1+a31

2m -I- 2m+2

-

2m+k

let = f'in (1k), the right hand of

(14)

ja C, again substituting

+

a,coe31+C

(1k)

this C into (14), we get the fo1io&ng formula:

and

[coB.oyoo

B.)Jooewt+Ly3o(

B1 O

)-1fo(

Ef]eincat

ajoos(2m+a)G

+coawt,P21(B)[.i.cos(2m+1)ø

_I1

a

_coe(2m+k)G

_B(41i) 1 *1 5

i+i1+a3.

+8inGJt1QC

8)[

.00e(2m+1)O

where

_{Tcyc(at*o),1Jrso_yo(ato(uIo)}

_{and fh(0)}

-6-2m+

a1ooa(2m+2)O

-+a3 21. - +

2m+

a 3a ] +a. +a3

2m+2í+'4J

oos2m a1ooe(2*+2)G 3a3ooa(2m+)S

"1+a1+a3')

2m 2nt+2 - 21

+ 1+a1 +a3

I

(19)

(18)

Assume the convergenc, of the right hand infinite eerie8 of (18)

alikewith.]1, then, P and are evalualable After that, the progreueive wave height Vis got from(17). Therefore is ex-pressed as follows:

x

r,i/2

+

UM((1-a1)eoeO

+ a,coe3}

₍₁₅₎

the right hand of (15)

ic expreaaed

ae followa:

(cIa (1..a1)coaø+a3coe3is h(0){P0cos6ft+%eiriúit}

(16)

where: (1..a1)cosO+a coe 36 -h(0) l+a1+a,

(17)

eubatit"ttng(16) into (15)MotIng

_on

each coefficient of ooet,

sinkt,

we get:

1f

°

(q130)

6-¡te for a circular cylinder for a emuli

B, PoO, 8o-1/B

Therefore:

2. Added mass.

The equation of the velocity potential which corresponds to u.

ation (13) is:

()é=

+cosiitj1P(B){e'

ain(2m+1)O+

j

1+a +a3 a

-(2m+2)(

350-(2m+k)°' I 2m+2

ein(2a+2)û-

ein(2v2+k)8}}

ein(a+1 )+

1+a+aj'2m

sin2mô 3a

(2m)0(

' 2n+2

sin(2m+8..

_{-sin(2m+'e)O j]} (22)

4,

4, are derived from the two dimenatonal doublet t the point of origin namely

4,1

1le'inKx

QQ

=

ecoeIcxTf

K cos ki+k ein

dk+

o k21K2 K(x2+y2)

The convention for the eigne is this eame.an(12). Neglecting the term of the second order, the bydrodynamic pressure acting on the surface of the cylinder is approzimatelj derived from:

p

()CC a

O

Integrating this pri.euri the hydrodynamic force in the direc-tion of x.axie is expressed as follows:

-

B()fNein,t- N0oocit

(2i)

where

T

ir

(1a

)eine +3a3sin30

.

P2

+

o

- 7s

(21)

\W

B_i1

_U

a1 3a

(1+a+a)

_LiZ

2ai2

¡

a

i 1+1»

9 (ame.2)

-.9 (2a+k) -9

Replace

to P, Q

resp.ctiv.l we get Mo, where

ç0(atX.o,

- 11(ate( - O), from (2k) the added maoa M in ewaying motion ie

M

2,2T202

N0P+M0Q0

s q o (26) o o

wber.,Oa density

o!

fluid, T - draft, E According to Landwiber'e paper [5), is expressed as follows

C

11

+(.__-uL_)2]

2 _L

therefore, atthe added maus coefficient

c'J

1e

M ()

_k

Kzsp.T

Divingb730 (Sectional area)

M (w.$)

C

Xt.,io)*513

_----h

roa squation of M (26), for anyj, Kx, 1 is as follows:

NP +MQ

k2oc

9O

JÇO

p4q

I

R NP +MQ

x

a ₅ P0

+0

S

where6 denotes oefent of sectional area, namely, , at

tA"O, that is period

, th. free surface becomes the boundary equivalent to the solid wall and we g.ti

* *17CT2C3

xx

p

(25)

E

the added mase whereC.)O0

o I Vii' ¡

l

-

-8-C (1_ai)2 + 3a32 (1_a+a,)2

(31)

3. Rolling moment.

The hydrodynamic pressure acting on the surface of the cylin... dsr has oppoelt. sign on either aide, thus it produces rolling so-ment as well as ?. This kind of moso-ment, of course,doea not appear in heaving motion. Let MR denote

the moment around

the origin O,

(positive anti-clockwise) it is derived from the following:

'[XR8iflJt!R oos(.lt] (32) where:

B,a4,*2)

f

*

¡a1(1+a)stn

2e..2a,1ain

Ô)d9+

"O

(I+a1+*,)c

. J

B(a1P2P4)

P Ç

i

m.1 2a1(1+a3)

8(1+a+s3)3

(1+a1+a3)2

(2m+i)2-4

8*

(2m+1 )-16

Replace

eo

P, sani'

and we get

MR is devidedthtßcomponente of acceleration and velocity,

namely:

2

MR

M5,

The first term means rolling moment which is caused by swaying, and the eecond,aznping moment. N8, are coupling

terms of the coupling o*ioillation of swaying and rolling, respecti-vely. Let:

M

MJ

eecp

M.K

.T

s

The ratio K5 _{of'àoupling moment arm to draft is expressed a.} follows: P XR+QQTx K _eLp/T

_2HjJQ+MQJ)

₍₃₅₎

-9--. Ç

(33)

(3')

-.9-then N5 Le expressed as:

N PQTR-QOXR

sip 2 2

p +q o o

on the other hand, wave damping N _{in swaying motion is expressed as}

N

a _(4I)

Let:

N

e (36)

and J.s expressed se followa:

0<

*

(PoTRQXR)

(37)

and

are

_{negative4n theoaee the mothent M.ienegative,}

k. Result

or caoulatton and 4iscuseSon.

The author showed the prooeee of calculation for sections of Lewieform; this can be applied to that ofa n-parameter family as was ehown in [6]. The component force F0 which is proportional to in quation(ak) is expressed as fo11oe

N Q -M P

F -A1() °

°° (Po co.t+

'

eint)

(38)

The average value ofer'-done by this force _{during one period} is:

7r2C) O O O O

This also i. equal to the energy of propagating wav, per unit time which is osused on both sides of _{the.indsr by swaying oscillation.} Therefore w. get:

NQ.MP

_{00 00}

this formula was used to verify the calculation. The convergono, of th. eerie. of the right hand in formula (18) is proved using the same method which i. used for heaving motion by T. Ureell. On the

_other

-

10

-(39)

i

10 *

hand , equation (18) has to be solved for an infinite number of

un-known quantities but the author took the expansion terme up to

a = 6

and using the same way as in [k _{made the calculation for the Lewis}

forni section.

Among the fourteen sections1 the error du. to formula (39) is le*s than 2% at H0 _{0,2 ad 1% for other sections. ?igur. 2 shows} values of and X for elvera]. elli( 0,785k) namely, H0 z 0,2 0,4, 2/3, 1,0 and 1,5 against

z

-The dotted line in the figures shows the result for a circular cylinder by 0. Grim lie also calculated for a ellipaoid cylinder at H0 _{2/3, 1,5, but there axe small errors at the maximum values.}

The maximum of K and corresponding

d increassewith the

in-creasing Be,. X increases with the increasing Ho for small values of for large values of

d' however, this trend reverses. General.

ly speaking, the difference between

_{K and}

_{X due to H0 is smaller} than that in heaving motion. X at H0 z O in the figure shows the case of a flat plate (see appendix). The influences of the area ooef. ficient _{5 are shown in figure 3, 4, 5, 6 and 7 for constant}

Judging from the igures, e

_n

that the fuller the section is, the largerje+Jsvalu, of X up to

₃

a 0,6, but this trend

rever-ses tori d)0164 The fuller the section ia,thelargerthe value

of

L

_{The result of the calculation is summarized in Table 1.} 0. Grim gives the formula for X as follows:

7C(1-a1) 2

La

₍₄₀₎ where O Replace to

3,

we get: 7(i-a )

Az

2 (41)

This formula gives a good coincidanoe up to a 0,2'O,3. 0.Grim also gives the formula for rolling moment

[23

that is, at

3d

z O

K

p>16 ai11ai+,_ala+3945,_s32

₍₄₂₎

4

11

-The values of _{derived from (42) are shown in figure}

_8.

_The

value Of l.at

d O in Tabla I is also calculated

with (42).

0. Grim aleo calculated the values of K5,, Dt for an ellipsoid

(at H0 2/3, i,, [i) . This shows a good agreement with the values in this paper. Figuree 9, 10 show the values of

K5,o(1,

for the case of full and fine sections1 Within the range of's in thie pap.r changes of

,

due to _{are generally small. K, andc(5,} are almost the same value for ellipsoid eecti _{and small H0 sections}

C9pqueion.

Two dimensional hydrodynamical force and moment are oalculatsd for Lewis' form eeotionn when a floating cylinder _{keea swaying oacf} 1-lation. The following _{is concluded within the range of treated} in this paper.

1). The change of both K and r due to H is smaller than that in heave.

2).. The fuller the section is, the larger is ¡ at constant This trend is reverse in heave.

The fuller the section is, the larger is for

d 1... than

0,6. For the valu of

d more than O,6t this trend is reversed.

The approximate formula for ¡by 0. Grim gives good approximate

valuesup to