Theoretical prediction of running attitude of a semi-displacement round bilge vessel at high speed

Applied Ocean Research41 (2013)41-47

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Applied Ocean Research

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Theoretical prediction of running attitude of a semi-displacement

round bilge vessel at high speed

Dong Jin Kim^'', Key Pyo Rheé', Young Jun You'^

'Maritime and Ocean Engineering Research Institute, Korea Institute of Ocean Science and Technology, Daejeon, Republic of Korea ^Department of Naval Architecture and Ocean Engineering, College of Engineering, Seoul National University, Seoul, Republic of Korea

Cros.sMark

A R T I C L E N F C A B S T R A C T

Article history: Received 11 July 2012

Received in revised form 14 February 2013 Accepted 14 February 2013

Keywords:

Semi-displacement round bilge vessel Running attitude

High-speed model test Added mass planing theory Plate pressure distribution method

Running attitudes of semi-displacement vessels are significantly changed at high speed and thus have an effect on resistance performance and stability of the vessel. There have been many theoretical approaches about the prediction of running attitudes of high-speed vessels in calm water. Most of them proposed theoretical formulations for the prismatic hard-chine planing hull. In this paper, running attitudes of a semi-displacement round bilge vessel are theoretically predicted and verified by high-speed model tests. Previous calculation methods for hard-chine planing vessels are extended to be applied to semi-displacement round bilge vessels. Force and moment components acting on the vessel are estimated in the present iteration program. Hydrodynamic forces are calculated by 'added mass planing theory', and near-transom correction function is modified to be suitable to a semi-displacement vessel. Next, 'plate pressure distribution method' is proposed as a new hydrodynamic force calculation method. Theoretical pressure model of the 2-dimensional flat plate is distributed on the instantaneous waterplane corresponding to the attitude of the vessel, and hy-drodynamic force and moment are estimated by integration of those pressures. Calculations by two methods show good agreements with experimental results.

® 2013 Elsevier Ltd. All rights reserved.

1. Introduction

W h e n a high-speed vessel runs i n the h i g h speed region, f l o w patterns and pressure d i s t r i b u t i o n s a l o n g the h u l l surface are d r a m a t i -cally changed, so r u n n i n g attitudes o f the vessel are v a r i e d w i t h its r u n n i n g speed. R u n n i n g attitudes o f a high-speed vessel have a s t r o n g e f f e c t o n its resistance, s t a b i l i t y and m a n o e u v r a b i l i t y . Therefore, r u n -n i -n g attitudes o f a high-speed vessel should be accurately e s t i m a t e d by m o d e l tests or t h e o r e t i c a l calculations i n its design stage.

There are some researches about v e r t i c a l plane dynamics and r u n n i n g a t t i t u d e predictions o f h i g h speed vessels. S e m i e m p i r i c a l f o r mulas based on the m o d e l tests o f p r i s m a t i c p l a n i n g hulls are d e v e l oped by Savitsky [ 1 ] and Savitsky a n d B r o w n [ 2 ] . M a r t i n [3] f o r m u -lated a m a t h e m a t i c a l m o d e l f o r t h e calculation o f forces a c t i n g o n p l a n i n g crafts, and the m o d e l is based o n linear s t r i p theory. A p l a n -ing c r a f t was m o d e l e d as a series o f strips or i m p a c t i n g wedges. A n d Zarnick [ 4 ] f o l l o w e d a n d developed M a r r i n ' s [3] w o r k s by using n o n -linear s t r i p theory. I n Akers's [5] study, Zarnick's [4] f o r m u l a s w e r e extended to predict the local d y n a m i c pressure f o r the s t r u c t u r a l de-sign o f p l a n i n g hulls. Zhao et al. [6] a p p l i e d the 2 D -i-1 t h e o r y to solve the steady p l a n i n g p r o b l e m o f a p r i s m a t i c h u l l at h i g h speed. Savander et al. [ 7 ] f o r m u l a t e d 3 D b o u n d a r y value p r o b l e m f o r steady p l a n i n g

surfaces. They m o s t l y treat h a r d chine p l a n i n g vessels, so there is l i t t l e t h e o r e t i c a l research f o r s e m i - d i s p l a c e m e n t r o u n d bilge vessels.

High-speed vessels can be categorized a c c o r d i n g to the supp o r t force or t h e range o f o supp e r a t i n g ssuppeeds. For e x a m supp l e , a s e m i -d i s p l a c e m e n t vessel operates at Frou-de n u m b e r 0.5-1.3. The w e i g h t o f a semi-displacement vessel is m a i n l y s u p p o r t e d b y buoyancy at l o w speed, b u t the h y d r o d y n a m i c l i f t becomes larger w i t h increasing speed and supports about 20-30% o f the t o t a l w e i g h t o f the vessel at h i g h speed.

I n this study, previous r u n n i n g a t t i t u d e p r e d i c t i o n m e t h o d s f o r h a r d chine p l a n i n g hulls are p a r t i a l l y m o d i f i e d and a p p l i e d to a s e m i d i s p l a c e m e n t r o u n d bilge vessel. R u n n i n g a t r i t u d e s o f a s e m i -d i s p l a c e m e n t vessel are t h e o r e t i c a l l y calculate-d, an-d m o -d e l tests are carried o u t to v e r i f y calculation results. Forces a c t i n g o n the vessel such as b u o y a n c y force, f r i c t i o n a l force and h y d r o d y n a m i c force are calculated i n a present i t e r a t i o n p r o g r a m . Buoyancy a n d f r i c t i o n a l forces are calculated w i t h reference to the change o f the w e t t e d shape o f the h u l l i n each i t e r a t i o n procedure. H y d r o d y n a m i c force is p r e d i c t e d by using 'added mass p l a n i n g t h e o r y ' and 'plate pressure d i s t r i b u d o n m e t h o d ' .

Corresponding author. Tel.: +82 42 866 3652. E-mail address: djkim@kiost.ac (D.J. Kim).

0141-1187/$ - see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.Org/10.1016/j.apor.2013.02.003

2. Theoretical approach

2.3. Force and moment equilibriums

The c o o r d i n a t e system and force c o m p o n e n t s a c t i n g on the h u l l are s h o w n i n Fig. 1. A high-speed vessel runs at the constant f o r w a r d speed U, a n d r is the t r i m angle. A b o d y - f i x e d coordinate 0-XYZ is placed at t h e center o f g r a v i t y o f a vessel. X-axis is parallel to t h e free surface and p o s i t i v e f o r w a r d , Z-axis is positive d o w n w a r d , a n d the p i t c h angle is positive b o w up.

Subscripts DS, BS, and FS respectively indicate h y d r o d y n a m i c , buoyancy, a n d f r i c t i o n a l forces at the steady state. W i s the w e i g h t o f the vessel. Heave forces and p i t c h m o m e n t s s h o u l d be i n e q u i l i b r i u m w h e n the vessel runs at constant f o r w a r d speed. Force and m o m e n t e q u i l i b r i u m equations are s h o w n i n Eq. (1).

W = FDS C O S T + FBS - FpS s i n T

M D S + MBS + MFS = Q ( 1 )

2.2. Buoyancy force

Buoyancy is obtained b y i n t e g r a t i n g all sectional w e t t e d areas a l o n g the keel, and m u l t i p l y i n g b y the mass d e n s i t y o f w a t e r a n d the g r a v i t a t i o n a l acceleration. Buoyancy force and m o m e n t are f o r m u -lated as Eq. (2). A^w is the sectional w e t t e d area, LCG and LCB are respectively the l o n g i t u d i n a l center o f g r a v i t y and buoyancy.

Aswds 0

MBS = FBS C O S T . ( L C B L C G )

(2)

The s u m m a t i o n o f g e o m e t r i c m o m e n t s o f sectional w e t t e d areas a l o n g the keel is d i v i d e d by the s u m m a t i o n o f sectional w e t t e d areas i n order to o b t a i n the l o n g i t u d i n a l center o f buoyancy, LCB.

2.3. Frictional force

W h e n sinkage and t r i m are assumed, w e t t e d surface area under the free surface can be calculated. A n d the d y n a m i c pressure 1 /2pU'^ and the f r i c t i o n a l c o e f f i c i e n t Cp are m u l t i p l i e d to o b t a i n the t o t a l f r i c t i o n a l force, as s h o w n i n Eq. (3).

FFS = IPU^CF Lswds

M f s = - F F S s i n r c o s r - ( L C f r - L C G )

(3)

Lsw is the w e t t e d l e n g t h o f each section under the free surface, the l o n g i t u d i n a l c e n t e r o f f r i c t i o n LCfr is calculated by d i v i d i n g the s u m -m a t i o n o f g e o -m e t r i c -m o -m e n t s o f Lsw by the s u -m -m a t i o n o f Lsw

2.4. Hydrodynamic force - 1st metliod: added mass planing theory

2.4.1. Added mass planing theory

'Added mass concept' or 'added mass p l a n i n g t h e o r y ' is m e n t i o n e d by some previous researchers such as W a g n e r [8], M a r t i n [ 3 ] , Payne [ 9 ] , and so on. The p l a n i n g p h e n o m e n o n o f a p l a n i n g surface was treated to be equivalent to the v e r t i c a l i m p a c t p h e n o m e n o n o f each transverse cross-section i n t h e i r studies. Under the h i g h f r e q u e n c y f r e e surface b o u n d a r y c o n d i t i o n , d a m p i n g c o m p o n e n t s vanish, t h e n the l i f t force o n the body is calculated as the t i m e rate o f change o f m o m e n t u m due to added mass c o m p o n e n t s .

H y d r o d y n a m i c forces a c t i n g o n a p r i s m a t i c p l a n i n g h u l l are cal-culated b y M a r t i n [3]. R e f e r r i n g to his research, n o t a t i o n s are s h o w n i n Fig. 2. The value s is the coordinate measured a l o n g the keel f r o m f o r e m o s t i m m e r s e d p o i n t o f the keel, and f is the c o m p o n e n t n o r -m a l to the keel. is the t o t a l w e t t e d keel length, a is the value o f s at the transverse plane t h r o u g h the center o f g r a v i t y . A n d n is the 2- d i m e n s i o n a l added mass o f the cross-section at p o i n t s.

The n o r m a l h y d r o d y n a m i c force o v e r the entire h u l l is o b t a i n e d by i n t e g r a t i n g the t i m e rate o f the m o m e n t u m o f each cross-section a l o n g the w e t t e d l e n g t h o f the h u l l and m u l t i p l y i n g by a 3 d i m e n s i o n a l e f f e c t c o e f f i c i e n t !p{X). H y d r o d y n a m i c force and m o -m e n t are f o r -m u l a t e d as Eq. (4).

MD=<PW j ^

(4) s)j^{i^t)ds

I n Eq. ( 4 ) , an i n t e g r a l t e r m o f h y d r o d y n a m i c force f o r m u l a t i o n is the s u m o f the time rate o f change o f m o m e n t u m o f a l l 2 - d i m e n s i o n a l cross-sections. Therefore, the i n t e g r a l t e r m i m p l i e s t h a t the flows are generated i n the v e r t i c a l d i r e c t i o n o f each cross-section, a n d t h e r e are no interactions b e t w e e n adjacent cross-sections. B u t the real flow is generated i n the l o n g i t u d i n a l d i r e c t i o n o f the 3 - d i m e n s i o n a l body, so the a d d i t i o n a l corrections are r e q u i r e d . I n o t h e r w o r d s , 3 - d i m e n s i o n a l e f f e c t coefficient, p(A) is the c o r r e c t i o n f a c t o r to account f o r the three-d i m e n s i o n a l i t y o f the flow. A c c o r three-d i n g t o the previous research such as Shuford's [10] report, 3 - d i m e n s i o n a l e f f e c t c o e f f i c i e n t is the f u n c t i o n o f the aspect ratio A.

E x p a n d i n g Eq. (4) i n the same w a y as M a r t i n ' s [ 3 ] report, and d r o p p i n g the second order p e r t u r b a t i o n terms, the c o n s t a n t h y d r o -d y n a m i c force an-d m o m e n t i n the stea-dy state are o b t a i n e -d as Eq. (5).

FDS--M D S ••

ip{X)U^ sin r cos r ¥)(A)U^ sin r COS z

['"'1 J a o '

ds

(5)

W h e n the sinkage and the t r i m o f a vessel are assumed d u r i n g the i t e r a t i o n procedure, added masses o f each sectional w e t t e d area are e s t i m a t e d b y Lewis's [ 1 1 ] m e t h o d . 3 - D i m e n s i o n a l e f f e c t c o e f f i c i e n t (p{X) is calculated as A / ( l + X), w h i c h is a p p l i e d to p r i s m a t i c p l a n i n g

D.J. Kim et al./Applied Ocean Research 41 (2013)41-47 43

h u i l , suggested b y S h u f o r d [ 1 0 ] . I n the case o f the calculation o f the p l a n i n g h u i l , X is the n o n - d i m e n s i o n a l i z e d value o f the mean w/etted keel l e n g t h and w e t t e d chine l e n g t h k w i t h the b r e a d t h o f the h u l l . But i n the present c a l c u l a t i o n f o r a s e m i - d i s p l a c e m e n t vessel, x is n o n - d i m e n s i o n a l i z e d value o f w e t t e d keel l e n g t h //^ w i t h the b r e a d t h of the h u l l .

2.4.2. Near-transom correction function

The t r a n s o m o f the h i g h speed vessel is d r y at h i g h speed, so the pressure o n the t r a n s o m becomes atmospheric pressure. Garme [12] proposes a n e a r - t r a n s o m pressure c o r r e c t i o n f u n c t i o n Q r f o r p l a n i n g hulls, as s h o w n i n Eq. (6). It is a r e d u c t i o n f u n c t i o n t h a t has a large g r a d i e n t a f t w h i c h decreases t o w a r d s 0, a n d approaches 1 at a distance f o r w a r d .

" (6) : t a n h • ( / k - s )

B is the breadth o f the h u l l , Cv is the Froude n u m b e r over the b r e a d t h of the h u l l , L f / V g B . i s n o n - d i m e n s i o n a l r e d u c t i o n l e n g t h as s h o w n in Fig. 3.

A f t e r the near-transom c o r r e c t i o n f u n c t i o n is a p p l i e d to b u o y a n c y and h y d r o d y n a m i c forces, force f o r m u l a t i o n s are replaced w i t h Eqs. (7) a n d ( 8 ) . Subscript 'tr.cof means the corrected terms by the near-t r a n s o m c o r r e c near-t i o n f u n c near-t i o n .

/

k

Ctr{s)-AswdS 0

^BS-tr.cor = pBSJr.cor cos r • (L C B.tr.cor - L C G )

pDS.tr.cor = 'P{^)U s i n T cos r / Ctr{s)-—ds J a

r^k Mosjr.cor = ^ (A) U s i n r cos r

r'k j

^Ctr{S)(a-(7)

(8)

Garme [12] proposes the value a' i n h i g h speed r e g i o n {Cv> 2) as 0.34, w h i c h is based on his m o d e l tests and p u b l i s h e d m o d e l test data.

2.6. Hydrodynamic force • mettiod

2nd method: plate pressure distribution

If t h e t o t a l pressure d i s t r i b u t i o n s on the h u l l b o t t o m are given, the h y d r o d y n a m i c force and m o m e n t can be d i r e c t l y obtained. There are no e x p e r i m e n t a l or n u m e r i c a l b o t t o m pressure data f o r s e m i -d i s p l a c e m e n t vessels, so t h e o r e t i c a l pressure -d i s t r i b u t i o n f o r m u l a o f 2D f l a t plate is used f o r the e s t i m a t i o n o f h y d r o d y n a m i c force and m o m e n t acting o n the vessel.

Pierson and Leshnover [13] used p o t e n t i a l t h e o r y and c o n f o r m a l t r a n s f o r m a t i o n s to f o r m u l a t e pressure d i s t r i b u t i o n o n 2-D f l a t plate as Eq. (9). P is the pressure, p is the mass d e n s i t y o f f l u i d , Lf is the h o r i z o n t a l v e l o c i t y o f the f l a t plate, and r is the t r i m angle o f a f l a t plate.

f — cos r

1/2/5U2 1 - _{1 - f cos r + sin T i / l -} (9)

Fig. 4. Longitudinal strips distribution on the waterplane.

f is the real p a r t o f t h e c o n f o r m a l t r a n s f o r m a t i o n f u n c t i o n , and is k n o w n as t h e l o n g i t u d i n a l p o s i t i o n w h i c h is n o n - d i m e n s i o n a l i z e d by the distance b e t w e e n the leading edge and the spray r o o t o f the f l a t plate, have the l i m i t s - 1 < f < 1. Both - 1 a n d 1 respectively correspond t o the t r a i l i n g edge and the spray r o o t o f the f l a t plate.

L o n g i t u d i n a l s t n p s are d i s t r i b u t e d o n the instantaneous w a t e r -plane c o r r e s p o n d i n g to the vessel's a t t i t u d e assumed, and Eq. (9) are a p p l i e d to each s t r i p as s h o w n i n Fig. 4. Then, the s t a g n a t i o n line at t h e b o w a n d the pressure d r o p near the t r a n s o m can be realistically considered.

As s h o w n i n Fig. 4, l o n g i t u d i n a l strips are d i v i d e d i n t o m a n y seg-m e n t s i n the l o n g i t u d i n a l d i r e c t i o n f o r the discretized c o seg-m p u t a t i o n . Lw is the l e n g t h o f each l o n g i t u d i n a l strip, &L and A W is the l e n g t h and the w i d t h o f each segment, and Cp_iw is the pressure c o e f f i c i e n t at each segment.

W h e n the e q u i v a l e n t t r i m angle o f a f l a t plate is veq, t h e pressure c o e f f i c i e n t at one segment, the pressure c o e f f i c i e n t a t each segment a n d the t o t a l l i f t are f o r m u l a t e d as Eqs. (10) a n d (11).

•PJ.W = 1

f - cos Teq

1 - § cos Teq + sin req\/\ - 5^

rB/2 i-Lw Fwaterplane = V^pU^ / ( C P J L W ) A L A W J -B/lJ 0 (10) (11) S h u f o r d [ 10] suggested a l i f t c o e f f i c i e n t f o r m u l a o f V - b o t t o m surfaces as Eq. (12). p is the deadrise angle, and A is the aspect r a t i o (beam over w e t t e d l e n g t h ) . S is the w e t t e d area b o u n d e d b y t r a i l i n g edge, chines, a n d spray line.

CL

Lift 0.5;rylT l / 2 p S U 2

A +

• c o s r ( l - sin/3)

sin^ T cos^ r cos fi

(12)

W h e n Pm is a m e a n deadrise angle o f the t o t a l w e t t e d v o l u m e o f the vessel. Am is the b e a m over the instantaneous w e t t e d keel length, t h e t o t a l l i f t o f the vessel can be predicted b y Eq. 03). K is a shape parameter, a n d is d e t e r m i n e d as 0.5 by present calculations.

F,essei = l / 2 p S i ; 2

O.BnAmr

Am

+

+ - s i n T c o s T cos fin

(13)

Equivalent t r i m angle o f a f l a t plate is teq is o b t a i n e d b y t h e e q u a t i o n ^vessel —F\

m e t h o d is s h o w n i n Fig. 5

geometry of eacli section / Assume t r i m Trim update Sinföge update Assume sinkage Calculate F^s, FBS. FFS No < ^ , - W < £ Calculate M^s, M^s. Mfs No

Final sinkage, trim

Fig. 6. Flow chart of an iteration program.

2.7. Computational procedure

R u n n i n g a t t i t u d e s o f a semi-displacement vessel are calculated by a present i t e r a t i o n p r o g r a m . The flow^ diagram o f the present p r o g r a m is s h o w n i n Fig. 6.

Sinkage and t r i m are updated i t e r a t i v e l y so t h a t the residues o f the force a n d m o m e n t e q u i l i b r i u m equations converge to zero. For example, the t r i m angle is assumed i n the range o f 0 - 5 ° at intervals o f 0.1°, the sinkage is calculated f r o m the force e q u i l i b r i u m e q u a t i o n i n each case, a n d assumed t r i m and calculated sinkage are s u b s t i t u t e d i n t o the m o m e n t e q u i l i b r i u m equation to check the residue o f the m o m e n t e q u a t i o n .

Figs. 7 a n d 8 s h o w one example o f calculation results o f force and m o m e n t by the present p r o g r a m . Fig. 7 shows the v e r t i c a l force c o m -ponents w i t h assumed t r i m angles. Pitching m o m e n t c o m p o n e n t s and the residues o f the m o m e n t equations are p l o t t e d a c c o r d i n g to the assumed t r i m angles i n Fig. 8.

In Fig. 7, the w e i g h t o f the h u l l is always constant regardless o f the a t t i t u d e o f the vessel. Vertical h y d r o d y n a m i c force gets larger as the t r i m angle increases. A n d the h u l l rises w i t h the increase o f t r i m , the b u o y a n c y force decreases. Therefore the total v e r t i c a l forces are in e q u i l i b r i u m . The f r i c t i o n a l force is relatively so small.

In Fig. 8, w h e n the t r i m angle increases, the h y d r o d y n a m i c m o m e n t t h a t raises the b o w up becomes larger. A n d i n t h a t case, b u o y -ancy m o m e n t lets the b o w d o w n , because the center o f b u o y a n c y moves t o w a r d the stern.

II

411

-02

-03

2 3

TMni[*dl

Fig. 7. Force components with assumed trims.

0 2 0.1

^ ao5t

3=-3 .fl.<B

-02.

1 I - ^ ^ ^

T«m[<ieg]

Fig. 8. Moment components with assumed trims.

A l t h o u g h i t is t r u e t h a t the b u o y a n c y m o m e n t becomes zero w h e n the t r i m angle is zero, calculated b u o y a n c y m o m e n t is positive w h e n the t r i m angle is zero as s h o w n i n Fig. 8. Because the near-transom pressure c o r r e c t i o n f u n c t i o n is always i n c l u d e d i n the calculation.

I n Fig. 8, the t r i m angle is 3.5° and the sinkage is 0.29% o f the l e n g t h b e t w e e n perpendiculars o f the vessel, w h e n the residues o f force a n d m o m e n t e q u i l i b r i u m equations converge to zero.

3. Model tests

H i g h speed m o d e l tests are c a r r i e d o u t to v e r i f y the calculation results. M a i n particulars o f the m o d e l ship and the h i g h speed t o w i n g carriage are s h o w n i n this chapter.

D.J. Kim et al./Applied Ocean Research 41 (2013)41-47 45

/

/

/

/

>'k

\

/

' /

1 ll

/

J

^ /

•/

' ' f ,

¥f k

j

/

/ ;

7

/ /

//:

/ /

_1//

_//

_{/ , ' / / /}

1

/ /

y

"E

Fig. 9. Body plan of a model ship.

Table 1

Main particulars of a model.

Particulars Non-dimensional value

t/B 6.21

B/T 2.70

Block coefficient 0.47

Design Froude number 1.19

Fig. 10. High speed towing system.

3.1. Modelship

The l e n g t h b e t w e e n perpendiculars o f t h e m o d e l ship is 2 m , and the d i s p l a c e m e n t o f i t is 36.975 k g . A body p l a n and m a i n particulars o f the m o d e l are s h o w n i n Fig. 9 and Table 1. The m o d e l is a w a t e r -j e t p r o p e l l e d high-speed vessel, and i t is a s e m i - d i s p l a c e m e n t r o u n d bilge t y p e h u l l .

3.2. High speed towing system

M o d e l tests are p e r f o r m e d b y a high-speed t o w i n g carriage i n a t o w i n g t a n k at Seoul National U n i v e r s i t y . The l e n g t h o f the t o w i n g t a n k is 117 m , the b e a m is 8 m , and the d e p t h is 3.5 m . The mass o f the carriage is a r o u n d 600 kg, t h e m a x i m u m t o w i n g speed is 10 m / s , and t o w i n g is accomplished by w i r e s d r a w n b y a servo m o t o r . Fig. 10 shows a schematic v i e w o f the h i g h speed t o w i n g system.

T r i m angles and t h r u s t angles o f high-speed vessels are changed considerably at h i g h speed, so t o w i n g devices t h a t can t o w the m o d e l i n the t h r u s t d i r e c t i o n are developed and used i n h i g h speed m o d e l tests.

Fig. 11. Measured and calculated sinkages according to reduction length a'.

5.0

——

— . .—_

Fig. 12. Measured and calculated trims according to reduction length a'.

4. Comparison of calculations with model test results

4. J. Comparison of measured data with calculations by added mass planing theory

A t first, r u n n i n g attitudes o f a s e m i - d i s p l a c e m e n t r o u n d bilge vessel at Froude n u m b e r 1.0-1.3 are calculated w h e n t h e r e d u c t i o n l e n g t h a' i n the n e a r - t r a n s o m pressure c o r r e c t i o n f u n c t i o n i n Eq. ( 6 ) , is 0.34, based o n Garme's [ 1 2 ] h a r d - c h i n e p l a n i n g m o d e l test data.

A t l o w speed, w a t e r m a y n o t be p e r f e c t l y separated o n the t r a n -som, so the d i f f e r e n t pressure c o r r e c t i o n f u n c t i o n is r e q u i r e d to be applied to l o w speed cases. A c t u a l l y , a' f o r t h e vessel w i t h l o w speed is m o r e t h a n 0.34, and is about 0.70-0.90 i n Garme's [ 1 2 ] h a r d - c h i n e vessel data. The o p e r a t i n g speed f o r s e m i - d i s p l a c e m e n t vessels is b e t w e e n l o w speed and h i g h p l a n i n g speed o f h a r d - c h i n e p l a n i n g vessels. So r u n n i n g a t t i t u d e s are calculated i n cases t h a t a' is changed up to 0.60. Figs. 11 and 12 s h o w t h a t 0.60 is m o s t suitable value as the r e d u c t i o n l e n g t h a' o f a present s e m i - d i s p l a c e m e n t vessel.

4.2. Comparison of measured data with calculations by plate pressure distribution method

Sinkages and t r i m s are e s t i m a t e d again b y plate pressure d i s t r i -b u t i o n m e t h o d . Figs. 13 and 14 s h o w c o m p a r i s o n of calculations -by t w o m e t h o d s w i t h e x p e r i m e n t a l results. B o t h calculations are i n g o o d agreement w i t h m e a s u r e d data, even t h o u g h sinkages b y plate pres-sure d i s t r i b u t i o n m e t h o d are a l i t t i e o v e r e s t i m a t e d .

A n e x a m p l e o f calculated force and m o m e n t c o m p o n e n t s b y t w o methods is s h o w n i n Figs. 15 and 16. Those c o m p o n e n t s are calculated w h e n t h e sinkage and the t r i m are f i x e d as measured values at design

Froude No. Moment components

Fig. 13. Measured and calculated sinkages by added mass planing theoiy and plate pressure distribution method.

Experiment

o 2D Added mass (a'=0.60) n plate pressure distribution .' i a 1 -1 +' 6 0.2 0,4 0,5 0,8 1,2 1.4 Froude No.

Fig. 14. Measur ed and calculated trims by added mass planing theoiy and plate pres-sure distribution method.

Fig. 15. Calculated force components by added mass planing theory and plate pressure distribution method,

Froude n u m b e r 1.19. Force and m o m e n t components obtained by t w o m e t i i o d s are similar.

5. Conclusions

R u n n i n g attitudes o f a semi-displacement r o u n d bilge vessel i n c a l m w a t e r are calculated and compared w i t h e x p e r i m e n t a l results i n this study.

Previous research f o r prismatic p l a n i n g vessels is extended to be applied to semi-displacement r o u n d bilge vessels. Forces a c t i n g on

Fig. 16. Calculated moment components by added mass planing theory and plate pressuie distribution method.

the vessel such as the buoyancy force, the f r i c t i o n a l force and the h y d r o d y n a m i c force are t h e o r e t i c a l l y estimated. The buoyancy a n d the f r i c t i o n a l force are calculated by using the i n f o r m a t i o n o f i n s t a n -taneous w e t t e d h u l l shapes. The h y d r o d y n a m i c force is calculated b y added mass p l a n i n g t h e o r y a n d plate pressure d i s t r i b u t i o n m e t h o d . Sinkages and t r i m s are calculated b y the present i t e r a t i o n p r o g r a m .

In calculations b y added mass p l a n i n g theory, there are some d i f -ferences b e t w e e n calculated t r i m s a n d measured t r i m s w h e n t h e n e a r - t r a n s o m c o r r e c t i o n f u n c t i o n f o r hard-chine p l a n i n g vessels is used. Reduction length i n n e a r t r a n s o m correcrion f u n c t i o n is m o d i -f i e d i n order t h a t calculated sinkages a n d t r i m s agree w i t h m e a s u r e d data.

Theoretical plate pressure d i s t r i b u t i o n s are used t o estimate t h e h y d r o d y n a m i c force and m o m e n t acting o n a semi-displacement vessel. 2 D i m e n s i o n a l plate pressure f o r m u l a t i o n s are d i s t r i b u t e d o n l o n -g i t u d i n a l strips o f the instantaneous w a t e r p l a n e c o r r e s p o n d i n -g t o t h e a t t i t u d e o f the vessel. H y d r o d y n a m i c force and m o m e n t by u s i n g plate pressure d i s t r i b u t i o n m e t h o d are s i m i l a r w i t h those b y using added mass p l a n i n g t h e o r y . A n d calculated sinkages and t r i m s are i n g o o d a g r e e m e n t w i t h e x p e r i m e n t a l results.

Acknowledgements

This research was s u p p o r t e d b y the M i n i s t r y o f L a n d , T r a n s p o r t a n d M a r i t i m e A f f a i r s o f Korea under the project, ' D e v e l o p m e n t o f m u l t i -purpose i n t e l l i g e n t u n m a n n e d surface vehicle' o f MOERI/KIOST as w e l l as M a r i n e Technology Education and Research Center o f Brain Korea 21 at Seoul N a t i o n a l U n i v e r s i t y .

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