Some recent advances in the prediction of ship motions and ship resistance in waves

1

International Jubilee ]S4eeting

on the Occasion of the

40th Anniversary of the

Netherlands Ship Model Basin

August 30 - September 1, 1972

The NSMB - 40 years of scientific industrial service in marine

technology

and

synopsis of papers

1972

Netherlands Ship Model Basin

Wageningen the Netherlands

Contents

Preface

D r R . J. H . K r u i s i n g a , Secretary o f State

The NSMB-40 years o f scientific i n d u s t r i a l service i n m a r i n e technology

Prof. Dr Ir J. D. van Manen

C o n t r i b u t i o n s o n some c u r r e n t p r o b l e m s o f ship resistance

Prof. Dr L. Landweber

O n w i n d resistance

Prof. Dr Ing. K. Wieghardt

Recent developments i n marine p r o p e l l e r h y d r o d y n a m i c s Dr Ir M. W. C. Oosterveld and Ir P. van Oossanen 35 35 36

Some developments i n the area o f strength and

v i b r a t i o n s o f ships 40

Dr E. Abrahamsen

Propeller v i b r a t o r y s h a f t forces affected by design

and e n v i r o n m e n t a l c o n d i t i o n s . 41

Prof Dr Ir R. Wereldsma

C o m p u t e r aided ship p r o d u c t i o n , management and

c o n t r o l 41

Prof E. G. Frankel

D e s i g n and operations 42

Ir J. HoUrop and Ir A. Koops

C a v i t a t i o n and its d e t r i m e n t a l effects

Ir J. H. J. van der Meulen

36 A p p l i e d mathematics i n ship h y d r o d y n a m i c s Prof. Dr R. Timman 37 F i s h p r o p u l s i o n Prof Th. Y. Wu 37 M a n e u v e r a b i l i t y , State o f the a r t Prof. Dr S. Motora 38

Some recent advances i n the p r e d i c t i o n o f ship m o t i o n s

a n d ship resistance i n waves 38

Prof. Ir J. Gerritsma

R e t r o s p e c t i o n o n 15 years NSMB seakeeping activities 39

M. F. van Sluijs and Ir S. G. Tan

Ocean T e c h n o l o g y

Dr Ir J. P. Hooft

Some recent advances in the prediction of ship motions and ship resistance in waves

P r o f . I r J. G e r r i t s m a

U n i v e r s i t y o f T e c h n o l o g y , S h i p b u i l d i n g D e p a r t m e n t , D e l f t

Synopsis

Since the p u b l i c a t i o n o f the strip theory m e t h o d , as f o r m u l a t e d by K o r v i n - K r o u k o v s k y and Jacobs a n d m o d i f i e d by others, the c a l c u l a t i o n o f heaving a n d p i t c h i n g m o t i o n s i n a given seaway c o u l d be carried o u t w i t h sufficient accuracy f o r m a n y p r a c t i c a l purposes. Substantial experimental evidence to c o n f i r m the theory is available and the n u m e r i c a l methods based o n the strip theory m e t h o d are n o w used f o r design purposes a n d strength calculations. The use o f ship m o t i o n theory i n ship design to o b t a i n o p t i m u m seakeeping quahties under given wave c o n d i t i o n s is slowly expanding. T i m e r e d u c t i o n o f the calculations used f o r this purpose is essential and some interesting developments i n this d i r e c t i o n liave been r e p o r t e d .

There is an increasing d e m a n d f o r the analysis o f six degrees o f f r e e d o m m o t i o n s a n d l o a d i n g . F o r this m o r e general case, the s i t u a t i o n is somewhat d i f f e r e n t f r o m the ship m o t i o n p r o b l e m i n head waves: there is n o c o m -parable v e r i f i c a t i o n o f the theory available. Despite this f a c t considerable progress has been r e p o r t e d i n the analysis o f lateral m o t i o n s d u r i n g the last f e w years.

I t s h o u l d be m e n t i o n e d that i m p l i c i t l y the s t r i p theory methods are restricted to certain s h i p f o r m s because parts o f the three-dimensional effects are i g n o r e d .

A systematic series o f seakeeping experiments is p l a n n e d at D e l f t S h i p b u i l d i n g L a b o r a t o r y to investigate the l i m i t s o f the a p p l i c a b i l i t y o f the strip t h e o r y methods. The tests i n c l u d e a very w i d e v a r i a t i o n o f the l e n g t h / beam r a t i o to include a very t h i n a n d a very f a t m o d e l .

A d d e d resistance i n waves is another subject w h i c h draws the a t t e n t i o n . A recent treatment o f this p r o b l e m employs the o u t g o i n g energy o f the d a m p i n g waves, caused by the o s c i l l a t i n g ship. The results are c o n f i r m e d by experiments w h i c h , i n a d d i t i o n , show t h a t the resistance

Some recent advances in the prediction of ship motions and ship

resistance in waves

P r o f . I r J. G e r r i t s m a / U n i v e r s i t y o f T e c i i n o l o g y , S h i p b u i l d i n g D e p a r t m e n t , D e l f t

Introduction

I n the last f e w years some interesting i m p r o v e m e n t s have been i n t r o d u c e d i n the n u m e r i c a l methods f o r p r e d i c t i n g the seagoing qualities o f ships and other f l o a t i n g objects. Since the p u b l i c a t i o n o f t h e s t r i p theory m e t h o d , as f o r m u l a t e d by K o r v i n - K r o u k o v s k y and Jacobs and m o d i f i e d by others, tlie c a l c u l a t i o n o f heaving and p i t c h -i n g m o t -i o n s -i n a g-iven seaway c o u l d be carr-ied o u t w i t h sufficient accuracy f o r m a n y practical purposes, w i t h o u t using e m p i r i c a l or experimental data. F o r this p a r t i c u l a r case there is substantial experimental evidence to c o n f i r m the theory, and the numerical methods based on the strip theory m e t h o d are n o w used f o r instance f o r design purposes and strength calculations. The wave load c a l c u l a t i o n p r o v e d to be a valuable t o o l f o r the extra-p o l a t i o n o f strength rcqtiirements f o r very large sliiextra-ps, a l t h o u g h the statistics, w h i c h are used f o r the l o n g t e r m predictions are sometimes questionable. Wetness calcula-tions are i n use f o r the d e t e r m i n a t i o n o f m i n i m u m free-b o a r d and p r e d i c t i o n o f s l a m m i n g and s p r i n g i n g is carried out w i t h a view o f the local load o n the construc-t i o n . The use o f ship m o construc-t i o n construc-theory i n ship design construc-to o b t a i n o p t i m u m seakeeping qualities under given wave c o n d i t i o n s is s l o w l y e x p a n d i n g . The advantage o f a t o o l to compare a large n u m b e r o f alternatives w i t h regard t o seakeeping behaviour is o b v i o u s f o r certain designs, b u t m o s t o f the existing c o m p u t e r p r o g r a m s are n o t adapted to this purpose because o f the relatively large t i m e c o n -s u m p t i o n . T i m e r e d u c t i o n f o r -such calculation-s i-s e-s-sen- essen-t i a l and some inessen-teresessen-ting developmenessen-ts i n essen-this d i r e c essen-t i o n have been r e p o r t e d .

There is an increasing demand f o r the analysis o f six degrees o f f r e e d o m m o t i o n s a n d l o a d i n g . F o r this general case, represented by the ship sailing i n o b l i q u e waves, the s i t u a t i o n is somewhat d i f f e r e n t f r o m the ship m o t i o n p r o b l e m i n head waves: there is no c o m p a r a b l e v e r i f i c a -t i o n o f -the -t h e o r y available. Despi-te -this f a c -t consider-able progress has been reported i n the analysis o f lateral m o t i o n s d u r i n g the last f e w years.

A p a r t i c u l a r d i f f i c u l t y i n the linear strip theory presents

the r o l l i n g m o t i o n , w h i c h can be very non-linear due to n o n - l i n e a r d a m p i n g and restoring moments. Significant scale effect i n r o l l i n g o n ship models can be expected because o f f r i c t i o n a l a n d e d d y i n g effects and dependable scaling methods to extrapolate m o d e l values to the ship are n o t available. C l e a r l y there is a need f o r systematic research i n this area. I t s h o u l d be m e n t i o n e d that i m -p l i c i t l y the stri-p theory methods are restricted to certain ship f o r m s because parts o f the three-dimensional effects are i g n o r e d . O n the other h a n d an astonishing c o r r e l a t i o n w i t h experimental results is sometimes f o u n d where the ship f o r m c o u l d be described as f a t or b l u n t .

A systematic series o f seakeeping experiments is p l a n n e d at D e l f t S l i i p b u i l d i n g L a b o r a t o r y to investigate the l i m i t s o f the a p p l i c a b i l i t y o f the strip t h e o r y methods. The tests include a very w i d e v a r i a t i o n o f t h e length/beam r a t i o to include a very t h i n and a very f a t m o d e l . Some o f the results are available and w i l l be presented i n the f o l l o w -ing chapter.

A d d e d resistance i n waves is another subject w h i c h draws the a t t e n t i o n . H a v e l o c k ' s w e l l - k n o w n m e t h o d to calculate the average net resistance force i n head waves, by using the u n d i s t u r b e d wave pressure, is not c o m p l e t e l y satisfactory and can o n l y be regarded as a first a p p r o x i -m a t i o n .

A recent treatment o f this p r o b l e m employs the o u t g o i n g energy o f the d a m p i n g waves, caused by the o s c i l l a t i n g ship.

The results are c o n f i r m e d by experiments w h i c h , i n a d d i -t i o n , show -tha-t -the resis-tance increase c o m p o n e n -t varies as the wave height squared.

I n the f o l l o w i n g , some o f the subjects m e n t i o n e d i n the i n t r o d u c t i o n , w i l l be reviewed i n m o r e d e t a i l , i n p a r t i c u l a r w i t h respect to their practical a p p l i c a t i o n i n naval a r c h i -tecture.

Improvements of fhe strip theory method

Detenninatioii of Iw o-dimensioiial added mass and damping

The first refinement o f t h e s t r i p t h e o r y m e t h o d , as i n -t r o d u c e d by Korvin-Kroul&l-t;ovsl&l-t;y and Jacobs was a m o r e accurate d e t e r m i n a t i o n o f the h y d r o d y n a m i c mass and d a m p i n g f o r actual ship cross sections. The p r o b l e m to cope w i t h was equivalent to that o f the l i a r m o n i c m o t i o n o f a cylinder w i t h a r b i t r a r y cross section i n the free surface o f a fluid.

UrselTs s o l u t i o n f o r the vertical o s c i l l a t i o n o f a circular cylinder [1] was generalized to ship cross sections by using a 2-parameter c o n f o r m a l t r a n s f o r m a t i o n (Lewis f o r m s ) by G r i m [2] and Tasai [ 3 ] . P o r t e r f u r t h e r extended this w o r k to a r b i t r a r i l y shaped sections by using m u l t i -coefiicient t r a n s f o r m a t i o n s a n d he also treated the case o f finite water depth [4]. D e Jong [5] a n d Tasai [6] also developed Ursell's m e t h o d f o r swaying and r o l l i n g m o t i o n s .

The use o f the m u l t i - c o e f f i c i e n t t r a n s f o r m a t i o n or the so-called close fit methods, avoids the restrictions imposed by the Lewis t r a n s f o r m a t i o n . The resulting shapes show indeed a very good fit f o r actual ship sections, i n c l u d i n g extreme b u l b o u s f o r m s .

I n the analysis o f t h e h o r i z o n t a l cylinder o s c i l l a t i o n the s o l u t i o n o f t h e p o t e n t i a l is given i n the f o r m o f a d i p o l e i n the o r i g i n and a hnear c o m b i n a t i o n o f a s y m m e t r i c m u l t i p o l e potentials, instead o f an analogous s y m m e t r i c s o l u t i o n f o r the heaving m o t i o n .

There is an extensive experimental v e r i f i c a t i o n o f the calculated h y d r o d y n a m i c properties o f o s c i l l a t i n g cy-linders, n o t a b l y by Tasai [ 7 ] , Porter [4] a n d V u g t s [ 8 ] . I n general the agreement is satisfactory as s h o w n f o r instance i n Figs. 1, 2 and 3. T h e a s s u m p t i o n o f l i n e a r i t y and o f a non-viscous fluid seems to be acceptable f o r the heave a n d p i t c h m o t i o n s , b u t f o r r o l l viscous effects are t o o i m p o r t a n t to be neglected.

A n alternative s o l u t i o n f o r the d e t e r m i n a t i o n o f added mass a n d d a m p i n g is given by F r a n k [9],

H e uses a p u l s a t i n g source d i s t r i b u t i o n o n the surface o f

1 K 3 | PA ^2g 100

\

1

tb

\ •

\

— - • • \ O 0 —^—• > 0 §•0 1é> ° °^ 0 2 5 0 5 0 0 7 5 1,00 125 1.50 1.75 2 0 0 >-B/T = 2 A 8 0 <> • 2a = 0 0 » • D • 0 0 1 m 0.02 m 0.03 m / « / " / • t V ^ / • / Cl 1 / • / 0 n 0 0 2 5 0 5 O 0 7 5 100 125 150 1.75 2 0 0

C L O S E FIT APPROXIMATION O F S E C T I O N CONTOUR L E W I S - F O R M APPROXIMATION O F S E C T I O N CONTOUR O O O M E A S U R E M E N T S

Fig. 1 Added mass and damping coefficient i n heaving [8].

C L O S E F I T A P P R O X I M A T I O N O F S E C T I O N C O N T O U R L E W I S - F O R M A P P R O X I M A T I O N O F S E C T I O N C O N T O U R o o o M E A S U R E M E N T S ° yy PA O 9y O Ó

•

•

• 0 0.25 0.50 0,75 1 0 0 , 1.25 1 5 0 175 2 0 0 P A 2 g ° o 0 0 B / T O

•

= 2 4 8

°/l

1 0 . O \ s ° OS

-VJ-—

- S ^ Ö c J / /

_^^^^^

0 2 5 0 5 0 0 7 5 100 1,25 1,50 175 Z O O

i k P A B / / — _ / / s ° ° 0 — — -^li / / 1/ 1 u O A lo \ Ir / / u O \ / / ' / ° O 0 ' 0 2 5 0 5 0 0 7 5 1.00 , 1.25 1 5 0 175 2 0 0 C L O S E F I T A P P R O X I M A T I O N O F S E C T I O N CONTOUR L E W I S - F O R M A P P R O X I M A T I O N O F S E C T I O N CONTOUR O o o M E A S U R E M E N T S 0 1 0 0 P A B ^ o . o s d

-

_{•1 1}

0^ O « O O 0 O V O O 0

lis

O O 0 O V O O 0

lis

jg S 8 6 ** • ° o • , • • t r 0 5 0 0.75 1 0 0 125 150 175 2 0 0 2g p A B ^ 2 g - 0 2 O l O 0 ° \\ \ \ 0 \ O o ° O 0 O 0 O O 0 2 5 Q 5 0 Q 7 S 1.00 , 1.25 1,50 175 2.00

Fig. 2b Coupling coefficients o f sway into roll [8].

P A B ^ 2g 0 * a = 0 • 0 0 5 0 1 0 0 2 0 • n • a D ' • • o • • < T T t \ _{a 8 8} 3a - 0 1 ol p A B - Q I 51 0 2 5 0 5 0 0 7 5 1.00 155 150 175 2 0 0

1 1

/

/

l l

/

\ \ ^ \ 1 1 II 1 ' f " o \ _ 0 / / / 0 0 Q25 0.50 0.75 1.00 125 1 5 0 175 2 0 0

Fig. 3a Added mass moment of inertia and damping coefficient in roli [8].

Fig. 3b Coupling coefficients of roll into sway [8].

- 0 . 1 0 pAB 2g - Q 1 5 - 0 2 0

^

0 \ \ o O 0 0 2 5 0 5 0 0,75 1.00 125 150 175 2 0 0 3b

the cylinder in the mean p o s i t i o n . The source strength f o l l o w s f r o m the b o u n d a r y c o n d i t i o n o n the c y l i n d e r surface, by means o f an integral e q u a t i o n .

The results o f b o t h methods are very close to each other except at certain frequencies where irregularities are observed i n the values according to the F r a n k m e t h o d . T h i s d i t f i c u i i y c o u l d be removed by r e s t r i c t i n g the m o t i o n o f the "fluid" inside the considered c y l i n d r i c a l b o d y . A t h i r d m e t h o d employs finite element techniques. Opsteegh [10] f o u n d a complete agreement w i t h the results o f the close fit methods f o r a rectangular cross section o f w h i c h the experimental values were also given, see F i e . 4.

T H E O R E T I C A L ( 2 P / a = 5 ) M E A S U R E D ( „ ) y 2 " x A ( O N E C Y L I N D E R )

Fig. 5 The amplitude ratio (wave amplitude/lieaving amplitude) [12]. p A P A " 2 9 - C A L C U L A T E D B Y U S I N G C O N F O R M A L T R A N S F O R M A T I O N . C A L C U L A T E D WITH F I N I T E E L E M E N T M E T H O D . E X P E R I M E N T A L R E S U L T S ,

Fig. 4 Added mass and damping coefficient, calculated with a finite element method f o r a rectangular cross section

BIT = 2 [10].

A f u r t h e r development o f the t w o - d i m e n s i o n a l t h e o r y concerns cross sections i n use w i t h catamarans a n d other m u l t i h u U c o n f i g u r a t i o n s . De Jong gives the a n a l y t i c a l solutions f o r heave, sway a n d r o l l o f t w o i d e n t i c a l sym-m e t r i c a l cylinders o f a r b i t r a r y f o r sym-m w h i c h oscillate i n the free surface [11].

O l i k u s u [12] c o m p a r e d calculated values f o r t w o c i r c u l a r cylinders w i t h the results o f experiments f o r several distance/diameter ratios 2P/a. F i g . 5 shows the results f o r 2P/a = 5.

The close fit m e t h o d , using c o n f o r m a l t r a n s f o r m a t i o n to derive the actual ship's cross-section f r o m a circle, gives

a very satisfactory s o l u t i o n w i t h regard to the goodness o f t i t .

H o w e v e r , the n u m e r i c a l procedure consumes too m u c h t i m e w h e n f o r instance a great n u m b e r o f ship designs have to be c o m p a r e d . The t r a n s f o r m a t i o n f o r m u l a : Z = C + Z " 2 „ + l c 11 = 0 - ( 2 ; i + 1) (1) w o r k s s a t i s f a c t o r i l y f o r instance w i t h A'^ = 10 — 20 f o r ship crosssections. F o r extreme bulbous f o r m s a p p r o x i -mately 20 — 30 coeflücients c o u l d be necessary.

The accuracy o b t a i n e d w i t h the close fit m e t h o d is seldom necessary and f o r m o s t cases the Lewis t r a n s f o r m a t i o n (A^ = 1) is sufficient, except f o r bulbous sections. A n impression o f t h e differences between the actual a n d the derived f o r m s a n d the calculated wave b e n d i n g m o m e n t s w i t h A' = 1 and A' = 9 is given i n Figs. 6 a n d 7 f o r a Series 60 b l o c k .70 s h i p f o r m [13]. O n l y at the highest speed there is a s m a l l , n o t significant diflference i n the calculated values o f t h e wave l o a d .

L o u k a k i s [14] p r o p o s e d a simple 2parameter t r a n s f o r m a -t i o n f o r b u l b o u s sec-tions:

z = C +

C + A (2)

w h i c h gives sufficient accuracy, also f o r extreme cases, see Fig. 8.

Thus a c o m b i n a t i o n o f t w o simple t r a n s f o r m a t i o n s can be used f o r m a n y purposes to save c o m p u t e r t i m e . A n a d d i t i o n a l advantage o f o n l y tvvo parameters is the d e f i n i t i o n o f a cross section by o n l y the b r e a d t h / d r a f t r a t i o a n d the area coefficient, w h i c h determine and

A C T U A L S E C T I O N L E W I S F O R M C L O S E F I T

X / L = r 2 5

Fig. 6 Two and ten coefficient fit to sections of series 60,

Cb = .70 [13]. 1 X= 0.025 1 a =4.1 5

;

/ — J NO LEWIS-FORM FITS T H I S STATION • STATION OFFSETS I X = 0 2 2 2 2 0 \ CT.1.815 \ . STATION V 2 0 1 0 NO LEWIS-FORM FITS 0 T H I S STATION 0

Fig. 8 MIT b u l b f o r m , Lewis f o r m [14].

o r A and B f r o m (1) and (2). T h i s is a useful s i m p l i f i c a -t i o n i n -the p r e l i m i n a r y design s-tage,

A^eu' developments of the strip theory

A s already m e n t i o n e d , there is progress i n the develop-m e n t o f the 6 degrees o f f r e e d o develop-m p r o b l e develop-m , b u t an exten-sive and detailed c o m p a r i s o n , i n c l u d i n g the effects o f f o r w a r d speed is only available f o r heave and p i t c h i n g m o t i o n s i n head waves. We shall restrict ourselves to this p a r t i c u l a r case. The new methods w h i c h were i n t r o d u c e d i n the last few years show some small b u t interesting differences w i t h the o r i g i n a l K o r v i n K r o u k o v s k y f o r m u -l a t i o n [15], and those w h i c h were derived f r o m it [13]. These were criticized f o r n o t h a v i n g the s y m m e t r y

rela-o 2 0 E X P E R I M E N T CALCULATION Fn=,30 O 2 C a / L = l / 4 8 • 2 C a / L = V ' ' 0 LEWIS FORM CLOSE FIT

tions. as l o r m u l a t e d f o r instance by T i m m a n and N e w m a n [16]. The o m i s s i o n o n l y concerns tlie mass cross-coupling coefficients and n o t the m o r e i m p o r t a n t d a m p i n g cross-c o u p l i n g cross-coeflicross-cients, w h i cross-c h were already i n t r o d u cross-c e d by K o r v i n - K r o u k o v s k y . The s y m m e t r y r e l a t i o n and the absolute values o f the latter terms were c o n f i r m e d by forced o s c i l l a t i n g tests at f o r w a r d speed w i t h a ship m o d e l [17]. It was also s h o w n t h a t these terms are essen-tial to get agreement between calculated and measured amplitudes and phases o f the ship m o t i o n s i n waves. I n the new \ ersion o f the strip theory methods as derived by Shintani [18] S ö d i n g [19] S e m e n o f - T j a n - T s a n s k i j , Blago-wetsjenskij, G o l o d i l i n [20] Tasai [21] a n d others, the expressions f o r the h y d r o d y n a m i c mass c o n t a i n a d d i t i o n -al f o r w a r d speed dependent terms and the mass cross c o u p l i n g terms have the desired s y m m e t r y .

A c o m p a r i s o n o f t h e calculated and experimental values o f t h e pitch mass cross-coupling t e r m itself by Beukelman [22] shows the i m p r o v e m e n t at h i g h f o r w a r d speeds f o r the D a v i d s o n T y p e A destroyer, a rather radical ship f o r m h a v i n g a p r o n o u n c e d b u l b o u s f o r e b o d y , see Figs. 9 a n d 10.

H o w e v e r , the p i t c h and heave amplitudes calculated w i t h the i n c l u s i o n o f f o r w a r d speed eflFect i n the p i t c h mass c o u p l i n g term show differences w i t h the results o f the earlier m e t h o d over a f a i r l y wide frequency range. T h i s is also the case f o r less extreme ship f o r m s as s h o w n by V u g t s [ 8 ] , see F i g . 11. I n p a r t i c u l a r the p i t c h amplitudes do n o t tend to the m a x i m u m wave slope at l o w f r e q u e n -cies o f encounter i n head waves, w h i c h c o u l d have been expected.

T o analyse the new expressions f o r the h y d r o d y n a m i c coefficients o f the equations o f m o t i o n i n m o r e d e t a i l , a c o m p a r i s o n o f calculated and e x p e r i m e n t a l values was p l a n n e d f o r a systematic series o f ship models, w h i c h have a wide v a r i a t i o n o f l e n g t h / b e a m r a t i o . T h e models are derived f r o m the T o d d 60 Series .70, by m u l t i p l y i n g the offsets w i t h constant f a c t o r s . I n t o t a l five models w i t h

LjB respectively 4, 5.5, 7, 1 0 a n d 2 0 w i l l be force oscillated

w i t h three f o r w a r d speeds {Fn = 0.15, 0.20, 0.30) and a wide range o f frequencies to o b t a i n the h y d r o d y n a m i c mass and d a m p i n g . Some p r e l i m i n a r y results o f the i n v e s t i g a t i o n are given here and a c o m p a r i s o n is made w i t h c o r r e s p o n d i n g calctilated values, a c c o r d i n g to the earlier and new strip theory methods f o r three l e n g t h / beam ratios [23]. F o r easy reference the expressions o f the various coefticients are given as w e l l .

T h e equations o f m o t i o n i n s t i l l water are given b y :

{pV + a)z -^ bz -t- cz ^d'Ó - eb — gO = 0 (heave)

(3)

{ly, + A)'Ó + BÓ + CO — Dz — Ez - Gz = 0 ( p i t c h )

Fig. 9 Davidson type A destroyer.

0 2 4 E X P E R I M E N T C A L C U L A T I O N o r . 0 . 0 1 0 m • r = 0 0 2 0 m N E W S T R I P T H E O R Y E A R L I E R .

125 15Q S H I P P I N G OF WATER 0 0 3 Q s F n ^ O 0 2 0 \ \

\ \

\ \ \ \

\ ^

0 2 5 0 5 0 0 7 5 1 0 0 0 2 0 J

H

k . V \ 0 2 0 \ 0 2 0 \ > \ " \ F n = 0 / ^ 125 150 0 2 5 0 5 0 0 7 5 1 0 0 1 2 5 1 5 0 S H I P P I N G O F WATER

\

1 / \ \ \

u

l'J \ 2 0 \ ^ l = O y ' • /

/

/ 0 3 0 / rn=oX T l ' ' 1 • \ 0 2 5 0 5 0 0 7 5 1 0 0 125 ).50 0 2 5 0 5 0 0 7 5 1 0 0 125 U p y B / 2 g

Fig. 11 Vertical motions in liead waves for a model of the Todd 60 series, Q = .70, L/B = 1 [8].

The expressions f o r the various coefficients are f o u n d f r o m : pyz I j F'dx, F'x.dxu A c c o r d i n g to K o r v i n - K r o u k o v s k y : f ' = - 2 p g v,„ (= - x,0} - N ' { z - x,Ó + VO) - - {m' ( i - .v„() + I/O)} dt F'= 2pg3^„ (z A,0) N ' i z X, Ó + ]0) -- m' (z -- x,Ö + 2V0) + (z -- x,i) + I'0) (5) dx,

w h e r e : A ' ' and m' are the sectional d a m p i n g coefficient and the sectional added mass and is the h a l f w i d t h o f the load waterline.

The new f o r m u l a t i o n can be f o u n d easily f o r instance f r o m [19] a n d [ 2 0 ] : F ' = - 2pgy,, (z - x,0) ct dXk. m - — o r : F' = - 2 p g y „ , (z - x,0) - m' (z - x,Ö + 2VÓ)-- A2VÓ)--' (z 2VÓ)-- x.Ó + 2VQ) + F — (z 2VÓ)-- x,Ó + VO) + dx„ + V dN' dx (6) w h e r e :

(o^, is the frequency o f o s c i l l a t i o n .

F r o m equations (3), (4), (5) and (6) it f o l l o w s t h a t : lu'dx, VE ( V B m'xi,^dx, + N'dx N'x,\lx, + N'xulx, V' N'dx c = 2pg C = 2pg Vs^dx, y,,xldxi, d = D = m'x.dx. + Vb CO,. E = g = 2pg G = 2pg ,Vb w x^dx, - I —^ N'x,d.x, - Va N'x,dx, + Va y,,xi,dxi, y,,Xi,dx. (7) T h e coelficients a c c o r d i n g to K o r v i n - K r o u l c o v s k y are f o u n d by replacing the terms i n brackets i n (7) by zero. I n the i n t e g r a t i o n o f the sectional values over the l e n g t h o f the ship, it is assinued that b o t h A ' ' and in' are zero at the ends. W h e n this is n o t the case, f o r instance w i t h a t r a n s o m stern, the c o r r e s p o n d i n g a d d i t i o n a l terms are easily f o u n d f r o m (3), (4), (5), and (6) by t a k i n g i n t o account the p r o p e r l i m i t s o f i n t e g r a t i o n . I t s h o u l d be r e m a r k e d that the expressions f o r d and A, w h i c h c o n t a i n

Vb VE

respectively — - and — a r e not equivalent w i t h the

CO, CO,

o r i g i n a l K o r v i n K r o u k o v s k y version. T h e y were i n t r o -duced by G e r r i t s m a and B e u k e l m a n [24].

The expression f o r the sectional h y d r o d y n a m i c f o r c e as given by (6) not o n l y contains 'l'"-, as already k n o w n ,

dN'

but also I n p r i n c i p l e G r i m m e n t i o n e d this result i n

dx,

1963 [25].

F o r the considered ship m o d e l f a m i l y the results o f t h e c o m p a r i s o n are given i n Figs. 12 and 13. T h e n u m e r i c a l results s h o w t h a t the new a d d i t i o n a l t e r m o f A is t o o small to be o f any practical i m p o r t a n c e . T h e new a d d i -tions t o the p i t c h mass cross-coupling coefficient D and to the p i t c h d a m p i n g B are the o n l y terms w h i c h are o f interest. T h e y c o n t a i n the squared frequency o f encounter i n the d e n o m i n a t o r a n d consequently the large values result at l o w frequencies.

T h e e x p e r i m e n t a l values f o r added mass and d a m p i n g agree f a i r l y w e l l w i t h the predictions, except f o r the p i t c h d a m p i n g , where the experimental values at the l o w e r frequencies do n o t correlate so well w i t h b o t h o f the calculations. T h e d a m p i n g cross-coupling coefficients show a g o o d agreement w i t h the p r e d i c t i o n , i n p a r t i c u l a r w h e n the absolute m a g n i t u d e o f the c o r r e s p o n d i n g terms i n the equations o f m o t i o n is taken i n t o account.

T h e e x p e r i m e n t a l p i t c h mass cross-coupling coefficients do n o t reveal a preference f o r one o f the t w o methods, b u t also i n this case the absolute value o f the c o u p l i n g terms has t o be considered i n the c o m p a r i s o n . F o r h i g h speeds the p r e d i c t e d values a c c o r d i n g t o the new m e t h o d give a better c o r r e l a t i o n w i t h the experiment, as s h o w n already i n F i g . 10, b u t a c c o r d i n g to [22] the a d d i t i o n a l t e r m D is the cause f o r erratic behaviour o f the p i t c h / wave slope r a t i o over an i m p o r t a n t p a r t o f the frequency range, as i l l u s t r a t e d i n F i g . 11. T h i s aspect s h o u l d be investigated m o r e closely.

The e x c i t i n g forces a n d m o m e n t s due to waves are cal-culated i n tvvo p a r t s : one w h i c h uses the u n d i s t u r b e d

wave pressure i n tlie e v a l u a t i o n o f the vertical forces o n a s t r i p , w h i c h is sometimes called the F r o u d e - K r y l o f f p a r t , and the other is the estimation o f the d i f f r a c t i o n effects. The first p a r t is i m p o r t a n t in magnitude and due care must be given to account f o r the actual s h i p f o r m i n the n u m e r i c a l process.

T h e d i f f r a c t i o n part o f the e x c i t i n g force and m o m e n t is calculated by using the now well k n o w n H a s k i n d relations [26]. These relations require the s o l u t i o n o f the oscilla-t i o n p r o b l e m i n c a l m waoscilla-ter o n l y and oscilla-therefore eliminaoscilla-te the d i f f i c u l t p r o b l e m o f the d i f f r a c t i o n o f the waves due t o the presence o f the ship at zero speed o f advance.

N e w m a n [27] removed the l i m i t a t i o n o f zero speed and showed that the extension to a f o r w a r d m o t i o n f o r t h i n or slender ships and f o r deeply submerged bodies is possible, p r o v i d e d that the disturbance o f t h e free sur-face is s m a l l . A p p l i c a t i o n o f the H a s k i n d relations is s h o w n i n the w o r k by Salvesen et al [28] a n d by Vugts [ 8 ] , w h o used N e w m a n ' s extension to the f o r w a r d speed case. F o r zero speed o f advance no difference exists w i t h the w i d e l y used relative m o t i o n concept [12], [14].

T h e difference between the various methods is extremely small and not o f m u c h practical significance f o r p i t c h i n g and heaving m o t i o n s , as shown f o r instance i n F i g . 14.

Resistance increase of a ship in head waves

A n o t h e r subject where corrections to the F r o u d e - K r y l o f f hypothesis are necessary to arrive at an acceptable pre-d i c t i o n , is the apre-dpre-depre-d resistance i n waves.

H a v e l o c k ' s r o u g h estimate is f o u n d by i n t e g r a t i o n o f the l o n g i t u d i n a l c o m p o n e n t o f the u n d i s t u r b e d wave pressure forces over the wetted p a r t o f t h e h u l l . This e s t i m a t i o n avoids the d i f f i c u l t p r o b l e m o f the e v a l u a t i o n o f t h e d i f f r a c t e d waves. A s p o i n t e d o u t by FirsofF [29] the F r o u d e - K r y l o f f hypothesis is clearly n o t applicable i n this case, b u t i t s h o u l d be realized that i n the c a l c u l a t i o n o f the m o t i o n s , w h i c h is a basic p a r t i n H a v e l o c k ' s m e t h o d , h y d r o d y n a m i c corrections are i n c l u d e d . There-f o r e the result is m o r e o r less acceptable as a There-first estimate. Havelock's expression f o r the mean added resistance i n waves is as f o l l o w s :

k

(F„z„ sin + M„0„ sin £„„) (8)

w h e r e : F„ and M„ are the amplitudes o f t h e e x c i t a t i o n force and m o m e n t i n heave and p i t c h , r„ and 0„ are the c o r r e s p o n d i n g m o t i o n amplitudes w i t h phase lags £.,• and

C A L C U L A T I O N S ACCORDING TO PRESENT E Q U A T I O N S • C A L C U L A T I O N S A C C O R D I N G TO PREVIOUS E Q U A T I O N S E X P E R I M E N T S 0 2 5 0 5 0 0 7 5 , l O O 1 2 5 I S O 0 7 5

!

\ \ F n . 0 1 0 Pn.QS Ü Ï Ü 0 \ \ *

^^^^^

0 2 5 0 5 0 0 7 5 1 0 0 1 2 5 1.50 9 0 = MC F n . 0 I V JL \ \ \ p 2 0 \ • 010_

J

a Fn.Q 010_ \ ( g O \ \ 0 3 0; S 1 \ 0 2 5 0 . 5 0 0 7 5 f— 1 0 0

A n alternative m e t l i o d to find this expression is to equal-ize the w o r k done by the e x c i t i n g force and m o m e n t to the w o r k done by the f o r c e w h i c h is necessary to t o w the ship t h r o u g h the given wave field [30].

Havelock's p r e d i c t i o n fails i n t w o respects: as s h o w n by experiments the added resistance i n waves does n o t vanish when a ship model is restrained i n heave and p i t c h and secondly the p r e d i c t i o n overestimates the added resis-tance at or below resonance c o n d i t i o n s . I n order to get a m o r e satisfactory agreement the concept o f the relative vertical m o t i o n o f t h e ship w i t h respect to the water has to be used. Joosen calculated the added resistance o f a ship i n short waves by e x p a n d i n g M a r u o ' s expression i n t o an asymptotic series w i t h respect to the slenderness parameter [31]. T a k i n g i n t o account o n l y the first order terms, he f o u n d a reasonable agreement w i t h the experi-ment, a l t h o u g h this s i m p l i f i e d treatment results i n a speed independent added resistance.

O f p a r t i c u l a r interest is his expression f o r the added resistance:

, , 3

(9)

w h i c h is completely equivalent to H a v e l o c k ' s e q u a t i o n (8). However, e q u a t i o n (9) shows that the added resis-tance can be regarded as a result o f t h e radiated d a m p i n g waves. A s in the case o f e q u a t i o n (8) this expression does not take i n t o account the relative vertical n r o t i o n o f the ship, w i t h respect to the water.

T h e r e f o r e the f o l l o w i n g procedure is adopted f o r the c a l c u l a t i o n o f t h e radiated energy ƒ" o f the oscillating ship d u r i n g one p e r i o d o f encounter. W e consider l o n g i t u d i n a l reeular head waves.

P = where: 'Te f L 0 . 0 b' • vl clxull dm' (10)

b' = N' — V , the sectional d a m p i n g coeflRcient at dx,

f o r w a r d speed;

K- = i — x,0 + VO — ( * , the vertical relative water v e l o c i t y ;

a n d :

y / ' ' d x , ) ,

F o r this concept reference is made to [13]. As K , is a h a r m o n i c f u n c t i o n w i t h a m p l i t u d e V^„ and a frequency equal to the f r e q u e n c y o f encounter co,, we find:

P = b' Vldxu ₍₁₁₎

F o l l o w i n g the reasoning given i n [30] the w o r k being done by the t o w i n g f o r c e R^„, is also given b y :

P = RA^y {V + c)T, = R^,y • X (12)

where: c is the wave celerity and X is the wave length. F r o m (11) and (12) i t f o l l o w s t h a t :

k

2ü7 b' Kldx, (13)

F r o m (13) i t f o l l o w s that the added resistance varies as the wave height squared, because K,„ is p r o p o r t i o n a l t o the wave height. A ship w i t h o u t o s c i l l a t o r y m o t i o n i n waves can be represented b y : V^ = — ( * and the cor-r e s p o n d i n g avecor-rage cor-resistance f o l l o w s f cor-r o m : w h e r e : ko/ b'e-T * = ^ i n f l -k dx. (14)

F r o m the fact that the speed V. appears as a q u a d r a t i c f o r m i n the integrand i n the expression f o r R,,»-, it f o l l o w s that the resistance increase i n waves is n o t merely the sum o f the resistance increase o f a ship o s c i l l a t i n g i n c a l m water (C* = 0 i n e q u a t i o n (10)) and the resistance increase o f a motionless ship i n waves.

T h e resistance increase i n waves was calculated a c c o r d i n g to the equations (13) and (14) f o r a fast cargo ship, see F i g . 15.

The results are c o m p a r e d w i t h experimental values i n F i g . 16, w h i c h shows a very satisfactory agreement f o r the case o f a heaving and p i t c h i n g ship. F o r short waves (A/^ < 0,8) the experimental values are somewhat higher f o r speeds exceeding Fn = 0.15. A possible e x p l a n a t i o n c o u l d be the u n d e r e s t i m a t i o n o f the d a m p i n g at h i g h frequencies w h i c h is o f t e n observed (see also F i g . 1). T o show the i m p r o v e m e n t w i t h regard to H a v e l o c k ' s f o r -m u l a , the values o f R^^y c o r r e s p o n d i n g t o e q u a t i o n (9) are also given i n F i g . 16.

the effective vertical wave displacement f o r a cross section.

The resistanee increase o f the motionless ship is small but should n o t be neglected.

There is a f a i r agreement w i t h the experimental values at the lower speeds, b u t at the highest speed the d i f f r a c t i o n resistance is underestimated. W i t h regard to the t o t a l added resistance i n waves, the differences are o f m i n o r importance.

The model experiment, w h i c h was carried o u t t o check the calculated added resistance, was also used to i n

E X P E R I M E N T WITHOUT S U R G E . C A L C U L A T I O N - D E L F T .

Fig. 17 Influence of surge on the added resistance in waves and calculated values.

vestigate the relation between resistance and wave height. There is some discussion w i t h regard to the assumed linear r e l a t i o n between added resistance and wave height squared as used f o r s u p e r p o s i t i o n purposes, p a r t i c u l a r l y i n the case o f slender ship f o r m s [32].

H o w e v e r , the present tests, carried o u t w i t h a three metre m o d e l o f a fast cargo ship h a v i n g a b l o c k coeffi-cient = 0.56, c o n f i r m e d the square l a w to a large extent.

T h e experiments include an extensive series o f added resistance and m o t i o n measarements i n a large range o f wave lengths, f o r w a r d speeds and three to f o u r wave heights. I n one p a r t i c u l a r case the infiuence o f surge was also investigated s h o w i n g a negligible influence on m o t i o n s and the added resistance i n head waves see F i g . 17.

T h e various experimental results are summarized i n F i g . 18.

References

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2 O. G r i m . Berechnung der durch Schwingungen eines Schilïskörpers erzeugten hydrodynamischen Krafte. Jahrb. der Schiff bau Technische Gesellschaft 1953.

3 F. Tasai. On the damping force and added mass of ships heaving and pitching. Report of Research Institute f o r A p -plied Mechanics, Kyushu University, Vol. V I I no. 26, 1959. 4 W. R. Porter. Pressure distribution, added mass and damping coefficients for cylinders oscillating in a free surface. Institute of Engineering Research, University of California Report, 1960.

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23 J. Gerritsma, C. C. Glansdorp and J. G. L . Pijfers. A note on damping and added mass f o r vertical motions. D e l f t Shipbuilding Laboratory, Report no. 343, 1972. 24 J. Gerritsma and W. Beukelman. The distribution of the hydrodynamic forces o n a heaving and pitching shipmodel in still water, I S P , 1964.

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