Prediction of hydrodynamic forces and moments acting on ships in heaving and pitching oscillations by means of an improvement of the slender ship theory

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Prediction of Hydrodynamic Forces and Moments Acting on

Ships in Heaving and Pitching Oscillations by Means of an

'Improvement of the Slender Ship Theory

by Hajime MaruO, Member

Jun Tokura** Member

Summary

A method of prediction of hydrodynamic forces and moments acting on a ship in heaving and pitching oscillations is presented. It is based on a revision of the slender

ship theory, which can interpolate between the original slender ship theory at low frequencies and the strip theory at high frequencies. Numerical computations are carried out for Series 60 models and results are compared with experiments by

Ger-ritsma and Beukelman. Much better agreement between computed and measured

results is observed when compared with results of the strip theory.

1... Introduction

The present practice of the theoretical predic-tion of Hydrodynamic forces and moments acting on oscillating ships depends generally upon the

strip theory, and it is regarded that the

calcula-tion according to the strip theory shows a

plausi-ble agreement with measured results in many

cases". However a detailed comparison of

computed and measured results reviels some

discrepancies which become appreciable

espe-cially at lower frequencies. As is well known,

the strip theory assumes that the fluid motion

around the hull at each transverse section

is identical with the two-dimensional motion around an infinitely long oscillating cylinder having the same sectional shape as that of the section of the hull. This becomes a valid

ap-proximation only when the frequency is sufficient-.

ly large, so that errors appear if the frequency

becomes lower. Purely numerical methods*

may be available to the calculation of hydro-dynamic forces on an oscillating

three-dimen-sional body at least in case of zero forward speed.

However they need much computer time and

axe by no means suitable to the routine practice. A simplification of the three-dimensional theory can be achieved by the application of the slender

body theory". It assumes the smallness of the

ratio of the beam and draft to the length of the

ship. Unfortunately numerical results according

Faculty of Engineering, Yokohama

Na-tional University

Mitsui Shipbuilding Co, Ltd.

to the original formulation of the slender ship theory have been much disappointing". They

show only poor agreement with actual phenomena

except in very low frequency range. The

diffi-culty involved is the slender ship approximation takes quite different formulations either at high

frequency or at low frequency,

In order to

remove this difficulty one of the author" has

proposed a revision of the original slender ship

formulation which has been valid only at low

frequencies, so as to give a valid approximation

at both low and high frequency limits The

present work gives results of numerical computa-tion which will be compared with experimental data.

2'. Outline of the slender ship formulation

In the present analysis, we consider the case.

bf zero forward speed. Take the cartesian

co-ordinates with the axes x and y in the undisturbed free surface and the axis z vertically upward, and consider a slender ship with its longitudinal axis

along the x-axis. The slender body technique

divides the flow field around a body into near

field and far field.

In the near field, y and z

coordinates are stretched by the ratio e-1, where e is a small parameter indicating the slenderness, of the ship. In the present case, the beam length

ratio is taken as the small parameter and the draft is assumed to be of the same order as the

beam.

On rewriting the Laplace equation

satisfied by the velocity potential 15 by the new

coordinates and omitting terms of 0(0), one

obtains the two-dimensional Laplace equation in

y z-plane . 2 4 JUlt 1978

ARCHIEF

1-13

* **

J-4.:111,441400014q5F,s:

!ou;47.,

_{112Atilgt=11,t% '45 143 S.'}

spol-/ Irbzindls1

a2A 141.13

ayz az'

=°

If we consider a periodical motion of circular

frequency co with small amplitude, the boundary condition to be satisfied at the free surface is the

linearized free surface condition such as

K0=0

at z=0

( 2 )

Oz

where K=&.2/g, the wave number. Here we consider a velocity potential 0(2D, which

re-presents a two-dimensional fluid motion which accompanies the oscillation of a two-dimensional cylinder corresponding to the transverse section

of the ship. Then a general solution of (1) in

three-dimensions can be expressed by the form

like

0=0,2D, +g(x)+ zh(s) ( 3 )

because of the condition of port and starboard

symmetry. Since 0,2D, satisfies the free surface

condition, eq. (2) is satisfied by when there

is a relation

h(x) = Kg (x) ( 4 )

g (x) is an arbitrary function of x, but is deter-mined in such a way that the outer expansion

of the near field potential matches the inner

expansion of the far field potential. The fluid

motion at a great distance from the ship is

expressed asymptotically by the velocity poten-tial of a line distribution of periodical wave sour-ces along the longitudinal axis. The two-dimen-sional motion due to an oscillating cylinder can be expressed by

kz cos k y dk_n.jek. cos Ky

c5(2D) = ao[y

_{e kK}

aun, z (2m 1) ! L az=n zz+yz ) 02M-2( \

+K

_{a .e+} )J

Then the matching procedure with the

field potential gives the relation

g (x)= ao(y

ao'(x')sgn(xx')

xln (21Clx + .5to(x')

4

x [Ho(Klxx'1)+ Yo(KixX')

+2i J o(Kix x'))]dx' ( 6 )

where Ho(x) is the Struve function, Yo(x) and

Jo(x) are Bessel functions, and y is Euler's con-stant. ao(x) is the coefficient of source term in

the two-dimensional solution and ao' (x) is its

derivative. If we write

( 5 )

far

N (u)= 7-1n2u+i- S:Ho(u')du'

Yo(u')du'-7Ti-F7Ti10 J o(u')du'

2 o

( 7 )

and assume that ac(s) vanishes at both ends of

the ship, the near field potential is expressed as

cb= 0(2D)+-1-2(l+ Kz)ao' (x')N (Kix el)

x sgn(xx')dx'

( 8 )

Now let us consider the boundary condition

on the hull surface. Designate the vertical

dis-placement of the hull in heave by ZQ and the

angle of rotation about y-axis by 0, and the

boundary condition on the hull surface in small oscillations takes the form

00 Oz .

= (zg xcb)

On On

where n is the outward normal to the hull surface,

and dots on g and 0 mean the differentiation with respect to time. Since the ship is very slender, one can replace the normal to the hull by the normal drawn to the section of the hull

Then we can write (10)

where nv and nz are direction cosines of the

normal in the transverse plane with respect to

axes y and z respectively. If we put

-xsb= V(x) e"t (11)

15(2.7,)= cy2D) etwt

and differentiate (8) by n, we obtain

O(2D)

On _[v(x)_-K5 ao' (x')N (Kix2 x'1)

x sgn (x-x')dn'inz (13) Now we consider a two-dimensional problem of

a cylinder making heaving oscillation of unit

velocity at a free surface. The boundary condi-tion on the body is

,30(w)

On

The solution is obtained by the method of Ursell and Tasaio, and the source term in the expansion

of the form of (5) can be determined at each

section. It is a function of x and let us designate

it by A(s). From the equation (13) we have the

relation

ao(x)=[V (x) 1K5ao' (x')N (Kix

x sgn(xx')dx121(x) (15) within the transverse plane.

a

_{a+flz a}

fl2i On _Oy Or ( 9 ) (12) (14) -z + (

One can solve the above integro-differential equation by numerical methods, but we take a successive approximation in the present case.

If we put

ao(x) = V (x)A(x) (16) the solution becomes exact at both limits of high

and low frequencies. Then we can adopt (16)

as the first approximation. The approximation

at the intermediate frequencies, we can employ the expression as

ao(x)=[V (x) (V ' (x')A (x')+ V (x')A' (x')) x N (Klx xi) sgn (x x')dx'1A (x) (17) Since the function A (x) is determined

numerical-ly, the above equation is not convenient for

numerical work because it involves the derivative

of A (x). However (17) can be rewritten in the form which does not include A' (x). If A (x)

vanishes at bow and stern ends, we can write

ao(x)= V(x)A(x)[1--liK A (x)(N (Kila+ xi)

1

+N (Kilf xl))1 210A (x)

xc if [V (x')A (x') V (x)A(x)] _ -la

xN' (Kix s'i)dx' (18)

where if, la denote the ordinates of bow and

stern ends respectively, and

N' (Kix x'I)= 1 110(KIXX'D IX 2 ± 71170(K IX el)±7Tijo(KIXel) 2 If we put W (x)= 1KV (x)A (x)(N (KV x I) 2 +N(Kllf x1)) +1K211 f EV (x')A(x') V (x)A (x)]N ' (Kix x'1)ds' (20) the boundary condition (13) can be expressed as

ads(2D)

On (x) W (x)]nz (21)

The strip theory is based on the solution for the boundary condition

O(2D)

On V (x)nz

Prediction of Hydrodynamic Forces and Moments Acting on Ships 113

(19)

(22) Therefore the three-dimensional effect appears in the term of W (x) which gives the correction to

the relative velocity V(x) of the section. This

circumstance resembles the down wash on an aerofoil of finite span.

The solution of the

boundary condition (21) can be easily found by

the method

for a two-dimensional heaving

cylinder.

3. Calculation of forces and moments

The vertical component of the hydrodynamic force on the hull is

F z= 55tpdS

(23)

and the moment about y-axis is my=

(tx--8,:zz)pdS

Now we define the following integral along the

contour of each section

nzds= el" f (x) (25) a at

By virtue of the small oscillation and the slender

ship approximation, the force and moment are

expressed by the integral

ei"f (x)d x (26)

Ma= f (x)x d x (27)

Therefore f (x) means the hydrodynamic force per unit length at each section. The velocity

potential 0 given by (8) consists of the

two-dimensional potential 0"E) and a term which re-presents the three-dimensional effect. The latter

may be expressed by the form e't (1+ Kz)0(x). Then one can divide the function f (x) into the

portion related to 0(21') and that due to the

three-dimensional effect. The former is identical with the vertical force acting on a cylinder which makes a heaving oscillation with velocity [V (x)

W (x)] ei". This can be expressed in terms of the added mass mz and the damping coefficient

INT: of the section. Thus we can write

f (X)= (i(0111z+Nz): V(X) W(x) ] (28)

The latter portion is easily evaluated by Gauss'

theorem.

f 2(x)= i po.)W (x)EB(x) KS(x)] (29)

where B(x) and S(x) are the waterplane width

and the area of the section respectively. From

(18) and (20), we have

ao(x)jV (x) W(x)JA (x) (30) On comparing (8) with (15), we can put

W (x)=KVi (x)

Therefore the force at each section is given by

f (x)= (iannz+N z)E V (x)W (x)]

i ixoS(x)W B (x)W (x) (32)

The function W (x) is defined in (20) and is

deter-mined by the functions V (x) and A (x) is defined

by (11) and determined from the ship's motion and A (x) is determined by the solution of the

(24) (31) + . Fz .

114 '

flil

g

'two-dimensional problem of heaving motion of

each section. The latter is related to the

ampli-tude and phase of the radiating wave generated

by the heaving cylinder. The amplitude and -phase of the periodical source is given in

Ap-pendix 1

In pure heaving, we put V (x)=iwzg and we

can write

f (x)e"=-(mz-Hns')(--(oio)'-(Nz-FIVZ).(itozg)

-where In/ and No' indicates the three-dimen-sional effect to the added mass and damping

coefficient respectively.

In the case of pure

pitching, we put V (x)-= -iwx0 and we can write

f (x), e" = - (11 x -Fm.")00 - (Is ?V/')(ieocb) _ Here we put

a* =Inz-Emz' b*=Is

d* =mtx+1112" e* N 7,x-EN z" (35)

and define the following integrals,

I f ,b= b*dx

' a

SrE e.= e*dX -la If B= e* xdx: -la

These constants give hydrodynamic coefficients in

-the coupled equation of heaving and pitching.

.(a+ pV)29-1-b2o-Fczo-d0-esi;-gcb=F

m(A +Pm, pf2') +Bsb-F - Fig -.Gz-=III}

(37)1

where 17 is the displacement volume and kyv is

the radius of gyration. The formulae for the calculation of above coefficients are given in

Appendix 2. On account of Haskind's relation,

we have relations

D=d, E=e - (38Y

4. Numerical examples

Series 60 models with 0.7 block coefficient is

employed for numerical examples, because

numer-ical results by means of the strip theory and

experimental data by Gerritsma and Beukelman

are available for comparison. The distribution

of periodical sources along the x-axis expressed

by ao(x) is given in Figs. 1-3. Results by the

present theory are compared with the

two-dimensional calculation of the strip theory and

results by the original slender ship theory which.

assumes that the source density is given by

(36)1

II'a" x ilL'I

PRESENT METHOD ORIGINAL SLENDER' SHIP THEORY STRIP THEORY -10.0° -,150° ,arg a. ( 180°

Fig. 1 Distribution:of Sou rce (KL 12. = 0.2094)

=

-,- V (x)B(x)lir . As far a§ the Source intensity is

concerned, the agreement between the strip

theory and the present method is

ncyticiabl

throughout the whole frequency range. The

original slender ship theory shows considerable

deviations at higher frequencies. Figs. 4,-,9

show the lengthwise distribution of hydrodynamic

forces defined by eq. (35). The results by the

present method and those of the strip theory

are compared with results of experiments by

Gerritsma and Beukelman. A remarkable

im-provement by the present theory is observed in

lower frequencies. Figs. 10,42 show, the

hydro-dynamic coefficients of coupled equations of

heave and pitch. The present method improves

the agreement between computed and measured

results to a great extent especially at lower

frequencies. -The result by the original slender

ship theory is shown in Fig. 10.. It is found that

this method is

almost useless for prediction

purposes, 143 (33) 4.0 - 2.0

--

-(34) A!P a= a* d x d= d*dx -d*xdx a*xdx -1b*xdx

AP F.P I a. (WI_ I XI (Y2 _4.0. arg_._cisk(x),

\

1800 1-4 AP

)/L

_{arg aq(x)} -'IL 4.0 -" 4.0 21..Q '01 - --- PRESENT METHOD

STRIP THEORY EXPERIMENT

Pig. 2 Distribution of Sources (KL/25=-1.1342) Fig, 3 Distibution I Sources (KLI2=7.368) 'Fig.

4 Distribution of a* and b* (1-4.0 rad/sec)

0 i I - - 150' - -150° 2 3 4 5 6

'7

-

-2.0

-

-100° (x I a)

-20

20 E 0

a) 4 0 - 2.0 0

20-(4 a> 10 -2.0 10 _{-1.0 -2.0} 10 'N 0.5 0 wa) -0.5 Fig. 5

Distribution of a* and b* (f -=6.0 rad/sec)

Fig. 6

Distribution of a* and b* (11=8.0 rad/sec)

Fig. 7

Distribution of d* and e* (er=4.0 rad/sec)

t1/4) 1 2 3 4 5 6

1 7/

IPI

\

_lilt

----.--zt ,r-s,

A/

-

-/

\

---A V V

--

-/

201

CP. 10 * 4.0 a) Q 2 3

-0 2.0 1.0 -2.0 2 3 4 5 6 10 - 1.0

Prediction of Hydrodynamic Forces and Moments Acting on Ships

a/pL..° EXPERIMENT PRESENT METHOD - ORIGINAL SLENDER SHIP THEORY STRIP THEORY o 0 -1.0

2.0

7:3

-2.0

1.0 0.5 * -0.5 XI0-2 2.0 -1.0 1.0 -10 51.0 510

Fig. 10 Added Mass and Damping Coefficient in Heave

117

\

2 3 4 6

Fig. 8 Distribution of d* and e* (w=6.0 rad/sec) Fig. 9 Distribution of d* and e* rad/sec)

b/pg05L2' KL/2 l KL /2 x10-' 20-1.0 -

/

4 7

118 7i011c$ V.; 143 -g" 0.0 -0.5-x10, A/ PL! K L/2 5. Conclusion

Numerical calculations of hydrodynamic forces and moments of Series 60 models are carried out

by means of a method which is based on a revision

of the slender body theory for oscillating ships.

A remarkable improvement in the agreement

between computed and measured results is ob-served when compared with results of the strip theory, and the three-dimensional effect is shown to be important at lower frequencies.

The authors express their thanks to Mrs.

Kazuko Hirayama for her cooperation in

com-puter programming. The numerical computation

has been carried out by the aid of computer

HITAC 8700/8800 of Tokyo University Computer Center.

0.5-_

KL/2

Fig. 11 Added Mass and Damping Coefficient in Pitch

xio-3

hg. 12 Coupling Coefficients Between Heave and Pitch

References

Gerritsma, J., Beukelman, W.,

Distribu-tion of damping and added mass along the length of a shipmodel. I.S.P. 10 (1963).

Chang, M. S., Three-dimensional

ship-motion computation. Second International

Conference on Numerical Ship

Hydro-dynamics (1977).

Newman, J. N., Tuck, E.

0., Current

progress in the slender body theory for

ship motions. Fifth Symposium on Naval

Hydro. (1964).

Joosen, W., Slender body theory for an oscillating ship at forward speed. Fifth Symp. on Naval Hydro. (1964).

Maruo, H., An improvement of the slender body theory for oscillating ships with zero,

forward speed. Bulletin Faculty of

Engi-d/PL e/Pg"L35 KL/2 so KL/2 0.0 -0.5-x10-3 / Pg°5L45 x10-3 0.5-51.0 50 .1) .

Prediction of Hydrodynamic Forces and Moments Actg on Ships 119 neering Yokohama National University 19

(1970).

Tasai, F., Takagi,. M., Symposium on

Seakeeping, Society of Naval Architects of Japan, (1969),_.,t

Appendix 1

Consider a two-dimensional cylinder heaving at the free surface with instantaneous velocity V.

It generates radiating waves of amplitude CA

which correspond to waves generated by a

periodical source at the free surface. We assume that the strength of the source is m sin wt and that the corresponding motion of the cylinder shows the velocity

2g CA.

V =

77,618(x)(Ao cos tot + Bo sin at)

The coefficients A0 and Bo are determined by the method described in Reference 6. If the motion

of the section is given by the vertical

displace-ment

Zgo=ZAe"''+4

the vertical velocity becomes V =toz,1

Then the corresponding isource pressed by B(x)1V1 i/A02 Bo2 tan a = BoIA Therefore weobtain B(x) LA (x) _{= 2s7A02 + B} arg A (x)=B ol A oi Appendix 2

Express the kernel function N (u) and the source

strength A (x) by complex numbers

N (u)=N 3(1)+ iNa(u) A (x) =A s(x) a(x)

Then the hydrodynamic coefficients are 'expressed5 as follows.

pgB(x)As(x):- pS (x) A .(X)

-=msAs(x)- (x)

X_{[Ns(Klx±1.1)±N3,(Kix -1 fl)]}

1 ₁

-F-2K[msA a(x)d-p.S.:(x)Aa(x),toN.A i(Xj PgB(x)Aa(x)] X_{[Na(Klx +74) N a(Klx} 1 1 +_2A2[ -7pgg(x),-pS(x).=md_'Or' (A-31 strength_ is ex7 (A-6) X fN s' (KI.X x1)12 Mx')'.- A s(x)]dx ta 1

--

K2[ 1 pgB(x)"-pS(x)-ms] 2

co-N a' (Kix - s'IKAa(e), - A a(x)ide

I = '

Ix2iY

N s."(K -2 co -1 f X [A a(x') - A a(x)]dx4 Na'(Klx- x'1,1 2 co -la x [As(x')-As(x)]clx' (A-9) a(x)± pcoS(x)Aa(x), 1 pg.B(x)A a(x)1 X [N s(Klx +14-FN s(Klx -1 f 1 2K[NsAa(x)+ poS(x)4i(x) com.A 3(x)-1 -6 pxB(x)As(x)1 x[Na(Klx +14± N a(Klx --F-2K2[coms- F pco,S(x)--1pgB(x)11 If X Ns' (Klx - x'pr[Aa(x')- Aa(X)id.e. -t a ±-2IC2PoPrz.± pcoS(X)--pg111(x)] )4z/t.Navcix-xti[As(e)'-4,(x)id.x,

1.roN2ifN/(Kix-xl

_{2 -t a} x [As(x')-As(x)]dx'

+.I'K2Ns2 -laNa'(Kjxx'',1)

X [A a(x4)- A a(x)]dx' d* m.x_{±--1 K [1}x pg B (x)A 3(x), 2 co' N. pS(x)A3,(x),-m.As(x)-7-0-Aa(x):1 x.[Ns(Klx +WI+ N s(Klx -143 +-2Kx[msAa(x)± pS(x)Aa(xf (x1- pgfp(X)Aii(x,)] X EN a(Klx +1a1)± N a(Klx -1 fl)]

.11,2[4pgmx)

_{2 to} - pS (x)r-m.1 f 'x _{N (Kjx- x'j),[Asfx')x' A 3(x)x]dxr} . -la =_{-1 K2[-L2-pg Th(x))- pS(x) .,m.]} 2 (/) lf

x Na'.(Kix --X1) [A a(xle - A a(x)x]dx'

z N.

--K2

Ns, .(Kjx I) 2 co. TA-10) 'Ii .0) (A-1) (A-2) 4 a* =m.-F+K[ x b* -

N.As(x)

-120 " '0 143 p< [A a(x')x' Aa(x)x]dx' 1 N.

- -

r

N a(K

-2 w -1. x [A s(x')x' -A s(x)x]dx' _ (A-1117 1 4*=Nzx+-2Kx[wnizAa(s)+ peo,S(x)Aa(x) N.A.(x)- pgB(x)Aa(x)1

tNs(Klx +/al) +Ns(KIs -IA)] 1

+-Kx[NzAa(x)- paLS(x)A (x,)

+ pgB(x)AAx)]

)( [Na(Klx +led) Nz(Klx _lfI)]

+

-2 K2[con: -1- pcoS(x)- pgB (x.)1

f

x N.' (IfIx&- x'11,),[Aa(xle - A a(x)x]dx' --ta

+-ix

1 [(Am.+ pwS(x,),- 1 pgB(x)11 Na' (Klx x'1)[A.(x')x' - A s(x,)x]d x' f '±Kz./Vz 51 j"N.' (Kix -) 2

-I.

x [As (e)x' - A s(x)x]c/ x' 1 z ±-2K2N aN a' (Kix -xlAa(xl)x'- A a(x)x]dx'

*12)

x'l)