Approximate evaluation of added mass and damping coefficients of Two-dimensional SWATH sections

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DAVID W. TAYLOR NAVAL SHIP

E S E A R C H AND DEVELOPMENT CENTER

Bethesda, Md. 20084

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APPROXIMATE EVALUATION OF ADDED MASS AND DAMPING C O E F F I C I E N T S OF

TWO-DIMENSIONAL SWATH SECTIONS

by Choung M. Lee A P P R O V E D F O R P U B L I C R E L E A S E : D I S T R I B U T I O N U N L I M I T E D

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SHIP PERFORMANCE DEPARTMENT R E S E A R C H AND DEVELOPMENT REPORT

October 1978

MAJOR DTNSRDC ORGANIZATIONAL COMPONENTS D T N S R D C C O M M A N D E R 0 0 T E C H N I C A L D I R E C T O R 01 O F F I C E R - I N - C H A R G E C A R D E R O C K 0 5 S Y S T E M S D E V E L O P M E N T D E P A R T M E N T I I S H I P P E R F O R M A N C E D E P A R T M E N T 15 S T R U C T U R E S D E P A R T M E N T 17 S H I P A C O U S T I C S D E P A R T M E N T 19 S H I P M A T E R I A L S E N G I N E E R I N G D E P A R T M E N T 23 O F F I C E R - I N - C H A R G E A N N A P O L I S 0 4 A V I A T I O N A N D S U R F A C E E F F E C T S D E P A R T M E N T C O M P U T A T I O N , M A T H E M A T I C S A N D L O G I S T I C S D E P A R T M E N T 118 P R O P U L S I O N A N D A U X I L I A R Y S Y S T E M S D E P A R T M E N T _{2 7} C E N T R A L I N S T R U M E N T A T I O N D E P A R T M E N T 2 9

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REPORT DOCUMENTATION PAGE B E F O R E C O M P L E T I N G FORM R E A D I N S T R U C T I O N S

1. R E P O R T N U M B E R

DTNSRDC-78/084

2 . G O V T A C C E S S I O N N O . 3 . R E C I P I E N T ' S C A T A L O G N U M B E R

4. T I T L E f a n d S u b ( l ( / e ;

APPROXIMATE EVALUATION OF ADDED MASS AND DAMPING COEFFICIENTS OF

TWO-DIMENSIONAL SWATH SECTIONS

5. T Y P E O F R E P O R T 4 P E R I O D C O V E R E D

F i n a l

4. T I T L E f a n d S u b ( l ( / e ;

APPROXIMATE EVALUATION OF ADDED MASS AND DAMPING COEFFICIENTS OF

TWO-DIMENSIONAL SWATH SECTIONS 6 . P E R F O R M I N G O R G . R E P O R T N U M B E R 7. A U T H O R C » ;

Choung M. Lee

B. C O N T R A C T O R G R A N T N U M B E R f » ;

9 . P E R F O R M I N G O R G A N I Z A T I O N N A M E A N D A D D R E S S

David W. T a y l o r Naval Ship Research and Development Center Bethesda, Maryland 20084 10. P R O G R A M E L E M E N T , P R O J E C T , T A I S K A R E A a W O R K U N I T N U M B E R S (See reverse s i d e ) I I . C O N T R O L L I N G O F F I C E N A M E A N D A D D R E S S 12. R E P O R T D A T E October 1978 I I . C O N T R O L L I N G O F F I C E N A M E A N D A D D R E S S 13. N U M B E R O F P A G E S 41

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1 5 « . D E C L A S S I F I C A T I O N / D O W N G R A D I N G S C H E D U L E

16. D I S T R I B U T I O N S T A T E M E N T C o / l / i l s R e p o r O

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I B . S U P P L E M E N T A R Y N O T E S

19. K E Y W O R D S (Continue on reverse aide II neceaeary and Idenllly by black n u m b e r ;

Small-Waterplane-Area, T^<rin-Hull Ships Ship M o t i o n i n Waves

Two-Dimensional Added Mass and Damping

A B S T R A C T (Continue on reverse aide II naceasary and Identity by 6 / o c * n u m b e r ;

D e r i v a t i o n o f approximate formulas f o r d e t e r m i n i n g the added mass and damping c o e f f i c i e n t s o f two-dimensional, small-waterplane-area, t w i n - h u l l

(SWATH) s e c t i o n s i s d e s c r i b e d . The added mass and damping c o e f f i c i e n t s o f i n t e r e s t a r e those a s s o c i a t e d w i t h a f o r c e d o s c i l l a t i o n of SWATH s e c t i o n s m heave, sway, or r o l l mode i n a f r e e s u r f a c e . The o b j e c t i v e o f d e r i v i n g t h e

S E C U R I T Y C L A S S I F I C A T I O N O F T H I S P A G E (When Dele Entered) (Block 10) Program Element 62543N P r o j e c t ZF-43421 Task Area ZF-43421001 Work U n i t 1-1500-102 (Block 20 continued)

approximate formulas f o r t h e hydrodynamic c o e f f i c i e n t s i s t o s i m p l i f y the computation of motion of SWATH ships i n waves w i t h o u t s a c r i f i c i n g i t s accu-racy s i g n i f i c a n t l y .

The approximate formulas are d e r i v e d based on the p o t e n t i a l - f l o w

t h e o r y . The damping c o e f f i c i e n t s are obtained i n terms of the o u t g o i n g wave amplitudes, and the added mass c o e f f i c i e n t s are o b t a i n e d by using the damp-i n g c o e f f damp-i c damp-i e n t s through the s o - c a l l e d Kramers-Krondamp-ig r e l a t damp-i o n s . W damp-i t h damp-i n t h e frequency range of p r a c t i c a l i n t e r e s t , the approximate formulas p r o v i d e s a t i s f a c t o r y r e s u l t s .

TABLE OF CONTENTS : . Page LIST OF FIGURES „ i i i NOTATION. . . . V ABSTRACT 1 ADMINISTRATIVE INFORMATION 1 INTRODUCTION 1 ANALYTICAL METHODS 3 BACKGROUND . . . . . . . 3 ANALYSIS 4 Heave Damping A p p r o x i m a t i o n 4 Sway Damping A p p r o x i m a t i o n 12 R o l l Damping A p p r o x i m a t i o n 14 Sway-Roll C o u p l i n g Damping A p p r o x i m a t i o n 17 Added Mass A p p r o x i m a t i o n • • • 18 DISCUSSION OF RESULTS 20 SUMMARY AND CONCLUSIONS 29

ACKNOWLEDGMENTS 30 REFERENCES 31

LIST OF FIGURES

1 - D e m i h u l l Cross S e c t i o n o f a SWATH Ship 6 2 - Heave Damping C o e f f i c i e n t o f a SWATH S e c t i o n 21

3 - Heave Added-Mass C o e f f i c i e n t o f a SWATH S e c t i o n 2 1 4 - Heave Added-Mass and Damping C o e f f i c i e n t s

o f Submerged Twin C i r c l e s 23 5 - Sway Damping C o e f f i c i e n t o f a SWATH S e c t i o n 25

Page 7 - Sway Added-Mass and Damping C o e f f i c i e n t s

of Submerged Twin C i r c l e s 26 8 - R o l l Damping C o e f f i c i e n t o f a SWATH S e c t i o n . 28

NOTATION

A = A^/h^ R a t i o o f wave a m p l i t u d e t o o s c i l l a t i o n a m p l i t u d e

A^ A m p l i t u d e o f o u t g o i n g wave

a.. T w o - d i m e n s i o n a l added mass i n t h e i t h mode due t o t h e m o t i o n i n t h e j t h mode

(a„„,a„„) = ^ N o n d i m e n s l o n a l (sway, heave) added mass 22' 33' pS.

c o e f f i c i e n t

^24

= , T_o ^ N o n d i m e n s l o n a l s w a y - r o l l c o u p l i n g added mass

24 (pbS^) ^44 a, , = = N o n d i m e n s l o n a l r o l l added i n e r t i a ^ (pb^S^) b One-half o f t h e d i s t a n c e between t h e c e n t e r l i n e o f each h u l l b. . T w o - d i m e n s i o n a l wavemaklng damping i n t h e i t h mode due t o t h e m o t i o n i n t h e j t h mode

b S t r u t t h i c k n e s s a t t h e mean w a t e r l i n e o

- - ^^22'^33^

^^22'^33^ = — N o n d i m e n s l o n a l (sway, heave) damping c o e f f i c i e n t

b. '24 N o n d i m e n s l o n a l s w a y - r o l l c o u p l i n g damping 24 (pwbS.) A c o e f f i c i e n t ^ 4 b, , = ^ N o n d i m e n s l o n a l r o l l damping c o e f f i c i e n t (pwb^S^) C o r r e c t i o n f a c t o r ( 1 = 1,2,3) C C e n t e r l i n e o f t h e t w i n b o d i e s

D r a f t o f c r o s s s e c t i o n Depth t o t h e c e n t e r o f c i r c u l a r h u l l Depth o f s t r u t G r a v i t a t i o n a l a c c e l e r a t i o n A m p l i t u d e o f o s c i l l a t i o n o f body I m a g i n a r y u n i t Wave number y- and z-component o f u n i t n o r m a l v e c t o r p o i n t i n g i n t o body Radius o f c i r c u l a r p o r t i o n o f h u l l C y l i n d r i c a l c o o r d i n a t e s as d e f i n e d on page 5 Submerged c r o s s s e c t i o n a l a r e a o f one h u l l Sway v e l o c i t y Heave v e l o c i t y Right-handed C a r t e s i a n c o o r d i n a t e system; Oz i s d i r e c t e d v e r t i c a l l y upward and Oy c o i n c i d e s w i t h calm w a t e r l i n e

One-half o f sway added mass o f t w i n c i r c l e s submerged under a r i g i d s u r f a c e

Heave added mass o f SWATH d e m i h u l l c r o s s s e c t i o n a t zero f r e q u e n c y

Water d e n s i t y

V e l o c i t y p o t e n t i a l r e p r e s e n t i n g f l o w f i e l d d i s t u r b e d by (heave, sway, r o l l ) o s c i l l a t i o n

ABSTRACT

D e r i v a t i o n o f a p p r o x i m a t e f o r m u l a s f o r d e t e r m i n i n g t h e added mass and damping c o e f f i c i e n t s o f

two-d i m e n s i o n a l , s m a l l - w a t e r p l a n e - a r e a , t w i n - h u l l (SWATH) s e c t i o n s i s d e s c r i b e d . The added mass and damping c o -e f f i c i -e n t s o f i n t -e r -e s t a r -e t h o s -e a s s o c i a t -e d w i t h a f o r c e d o s c i l l a t i o n o f SWATH s e c t i o n s i n heave, sway, o r r o l l mode i n a f r e e s u r f a c e . The o b j e c t i v e o f d e r i v i n g t h e a p p r o x i m a t e f o r m u l a s f o r t h e hydrodynamic c o e f f i -c i e n t s i s t o s i m p l i f y t h e -c o m p u t a t i o n o f m o t i o n o f SWATH s h i p s i n waves w i t h o u t s a c r i f i c i n g i t s a c c u r a c y s i g n i f i c a n t l y . The a p p r o x i m a t e f o r m u l a s a r e d e r i v e d based on t h e p o t e n t i a l - f l o w t h e o r y . The damping c o e f f i c i e n t s a r e o b t a i n e d i n terms o f t h e o u t g o i n g wave a m p l i t u d e s , and t h e added mass c o e f f i c i e n t s a r e o b t a i n e d by u s i n g t h e damping c o e f f i c i e n t s t h r o u g h t h e s o - c a l l e d KramersK r o n i g r e l a t i o n s . W i t h i n t h e f r e q u e n c y range o f p r a c t i c a l I n t e r e s t , t h e a p p r o x i m a t e f o r m u l a s p r o v i d e s a t i s f a c -t o r y r e s u l -t s . ADMINISTRATIVE INFORMATION

T h i s work was sponsored by t h e N a v a l M a t e r i a l Command as p a r t o f t h e High Performance V e h i c l e Hydrodynamic Program o f t h e Ship Performance Department, D a v i d W. T a y l o r Naval Ship R&D Center (DTNSRDC). Funding was p r o v i d e d under t h e S h i p s , Subs and Boats Program, Element 62543N, Task Area ZF-43421001, P r o j e c t Number ZF-43421, Work U n i t 1-1500-102.

INTRODUCTION

One o f t h e advantages e x p e c t e d f r o m a s m a l l w a t e r p l a n e a r e a , t w i n -h u l l (SWATH) c o n f i g u r a t i o n i s t -h e improved seakeeping q u a l i t i e s compared t o m o n o h u l l s h i p s i n moderate sea c o n d i t i o n s . To a s s i s t i n t h e development o f t h e SWATH c o n c e p t , an a n a l y t i c a l p r e d i c t i o n o f m o t i o n o f SWATH s h i p s i n waves has been d e v e l o p e d a t t h e D a v i d W. T a y l o r N a v a l Ship R&D Center

1 * 2

(DTNSRDC). ' The a n a l y t i c a l method can p r e d i c t t h e m o t i o n o f a SWATH s h i p i n f i v e d e g r e e s - o f - f r e e d o m i n r e g u l a r and i r r e g u l a r waves. The s u r g e m o t i o n i s e x c l u d e d due t o i t s minor p r a c t i c a l s i g n i f i c a n c e .

A l t h o u g h t h e e x i s t i n g a n a l y t i c a l method has i t s own m e r i t s and s e r v e s i m p o r t a n t p u r p o s e s , i t has been f e l t t h a t i t i s t o o cumbersome t o be used i n t h e c o n c e p t u a l d e s i g n s t a g e a t w h i c h numerous c a n d i d a t e h u l l forms a r e

3

examined. D a l z e l l has addressed t h i s p r o b l e m by d e v e l o p i n g a s i m p l i f i e d method o f computing t h e hydrodynamic c o e f f i c i e n t s a s s o c i a t e d w i t h heave-p i t c h c o u heave-p l e d m o t i o n o f SWATH s h i heave-p s a t zero sheave-peed i n head waves. Since

2

t h e m a j o r p o r t i o n o f t h e computer program f o r p r e d i c t i o n o f m o t i o n i s comp r i s e d o f comcomputing t h e s e c t i o n a l added mass and damcomping, t h e a c c o m comp l i s h -ment o f D a l z e l l i n r e d u c i n g t h e s i z e and c o m p u t a t i o n t i m e o f t h e computer program i s r e g a r d e d t o be v e r y w o r t h w h i l e .

I n t h i s r e p o r t , an e x t e n s i o n o f t h e D a l z e l l e f f o r t t o a l l hydrodynamic c o e f f i c i e n t s a s s o c i a t e d w i t h t h e f i v e d e g r e e s - o f - f r e e d o m o f m o t i o n o f SWATH s h i p s i s d e s c r i b e d . The approach employed h e r e i s s i m i l a r t o t h e D a l z e l l approach. That i s , t h e s e c t i o n a l wavemaklng damping c o e f f i c i e n t s o f SWATH c r o s s s e c t i o n s a r e o b t a i n e d u s i n g t h e f a r - f i e l d p o t e n t i a l - f l o w t h e o r y . By

4

a p p l i c a t i o n o f t h e K r a m e r s - K r o n i g r e l a t i o n s t o t h e damping c o e f f i c i e n t s , t h e added-mass c o e f f i c i e n t , w h i c h can be r e g a r d e d as t h e c o n j u g a t e p a i r o f t h e damping, i s d e t e r m i n e d .

The advantage o f t h i s p r o c e d u r e over t h e c o n v e n t i o n a l m u l t i p l e -e x p a n s i o n m-ethod^ o r s o u r c -e - d i s t r i b u t i o n m-ethod^ i s t h a t t h -e wav-emaklng damping can be o b t a i n e d by d e t e r m i n i n g t h e o u t g o i n g wave a m p l i t u d e a t a f a r f i e l d r a t h e r t h a n by d e t e r m i n i n g t h e p r e s s u r e d i s t r i b u t i o n on t h e body. The o u t g o i n g wave a m p l i t u d e can be c l o s e l y a p p r o x i m a t e d by r e p r e s e n t i n g t h e f l o w d i s t u r b a n c e s by a p u l s a t i n g s i n g l e s o u r c e , a d i p o l e , o r a c o m b i n a t i o n of b o t h . F u r t h e r m o r e , t h e K r a m e r s - K r o n i g r e l a t i o n s can be expressed i n terms o f a h a l f - r a n g e d o u b l e F o u r i e r t r a n s f o r m . Because an e f f i c i e n t n u m e r i c a l p r o c e d u r e such as f a s t F o u r i e r t r a n s f o r m (FFT)'' i s a commonly a v a i l a b l e computer r o u t i n e , t h e p r o c e s s o f o b t a i n i n g t h e addedmass c o e f f i -c i e n t s -can be q u i -c k l y p e r f o r m e d .

Once t h e sèctional added mass and damping c o e f f i c i e n t s i n heave, sway, r o l l , and r o l l s w a y c o u p l e d modes a r e o b t a i n e d , t h e r e s t o f t h e motiön p r e -d i c t i o n p r o c e s s f o l l o w s i -d e n t i c a l l y t o t h a t -d e s c r i b e -d i n Reference 1 .

Comparison o f t h e r e s u l t s o b t a i n e d by e x p e r i m e n t s , t h e method o f source d i s t r i b u t i o n , and t h e p r e s e n t a p p r o x i m a t e method a r e p r e s e n t e d i n graphs. Agreements a r e , i n g e n e r a l , f a i r . The a p p r o x i m a t e method y i e l d s s a v i n g s i n t h e c o m p u t a t i o n t i m e by an o r d e r o f magnitude compared t o t h e c o n v e n t i o n a l method o f Reference 8. The r e a l m e r i t o f t h e a p p r o x i m a t e method w i l l be e v a l u a t e d i n t h e f u t u r e when an a n a l y t i c a l method o f p r e -d i c t i o n o f SWATH m o t i o n s i n waves i s -develope-d.

ANALYTICAL METHODS BACKGROUND

W i t h i n a l i n e a r a n a l y s i s , t h e added mass and damping c o e f f i c i e n t s o f o s c i l l a t i n g t w i n - c y l i n d r i c a l b o d i e s o f a r b i t r a r y c r o s s s e c t i o n have been

Q

d e t e r m i n e d by t h e method o f p u l s a t i n g s o u r c e d i s t r i b u t i o n . T h i s method has shown s a t i s f a c t o r y agreement w i t h t h e e x p e r i m e n t a l r e s u l t s o f v a r i o u s c r o s s s e c t i o n a l shapes such as t w i n s e m i c i r c l e s , r e c t a n g l e s , t r i a n g l e s , ^

9 and SWATH s e c t i o n s .

To o b t a i n r e a s o n a b l y a c c u r a t e added mass and damping c o e f f i c i e n t s o f a SWATH c r o s s s e c t i o n , a t l e a s t 15 t o 20 p o i n t s o f source d i s t r i b u t i o n on the submerged c o n t o u r o f one h u l l s h o u l d be t a k e n f o r each f r e q u e n c y o f o s c i l l a t i o n . To o b t a i n a l l t h e hydrodynamic c o e f f i c i e n t s i n t h e l i n e a r e q u a t i o n s o f m o t i o n c o v e r i n g a l l d e g r e e s - o f - f r e e d o m , t h e s e c t i o n a l added mass and damping c o e f f i c i e n t s s h o u l d be computed f o r t h e sway, heave, and r o l l modes f o r about 20 c r o s s s e c t i o n s a l o n g t h e l e n g t h o f a s h i p . I f t h e m o t i o n o f a SWATH s h i p i n i r r e g u l a r waves f o r d i f f e r e n t wave headings and s h i p speeds i s t o be computed, t h e f o r e g o i n g number o f c a l c u l a t i o n s s h o u l d be r e p e a t e d f o r each wave h e a d i n g , s h i p speed, and f r e q u e n c y o f o s c i l l a t i o n . Thus, one can e a s i l y e x p e c t t h a t a s i g n i f i c a n t amount o f computer t i m e w i l l be r e q u i r e d t o e v a l u a t e t h e seakeeping q u a l i t i e s o f a s h i p .

I n s p i t e o f t h i s l a r g e expense o f c o m p u t a t i o n t i m e , what we a r e ob-t a i n i n g , a ob-t b e s ob-t , a r e m o ob-t i o n s based on ob-t h e hydrodynamic c o e f f i c i e n ob-t s w h i c h a r e o b t a i n e d under t h e a s s u m p t i o n o f t w o - d i m e n s i o n a l f l o w c o n d i t i o n s a t each c r o s s s e c t i o n . F u r t h e r m o r e , u n l i k e m o n o h u l l s h i p s , t h e m o t i o n o f SWATH s h i p s cannot be p r e d i c t e d a c c u r a t e l y by u s i n g o n l y t h e c o e f f i c i e n t s

o b t a i n e d f r o m t h e p o t e n t i a l - f l o w t h e o r y . There a r e v i s c o u s e f f e c t s and f i n e f f e c t s t o be i n c l u d e d i n t h e hydrodynamic c o e f f i c i e n t s . I t seems r e a s o n a b l e , t h e r e f o r e , t o seek a s i m p l i f i e d method o f o b t a i n i n g t h e sec-t i o n a l hydrodynamic c o e f f i c i e n sec-t s , i f sec-t h e economic s a v i n g s i n sec-t h e l a b o r a t i m e can j u s t i f y t h e l o s s o f a c c u r a c y o f t h e r e s u l t s w i t h i n a c e r t a i n m a r g i n .

ANALYSIS

From t h e p r i n c i p l e o f c o n s e r v a t i o n o f energy, t h e wavemaklng damping of an o s c i l l a t i n g t w o - d i m e n s i o n a l body can be d e t e r m i n e d by t h e a m p l i t u d e of t h e r a d i a t i n g waves a t a f a r d i s t a n c e f r o m t h e body. T h i s i m p l i e s t h a t the damping c o e f f i c i e n t can be d e t e r m i n e d by examining t h e f a r - f i e l d be-h a v i o r o f t be-h e d i s t u r b a n c e s g e n e r a t e d by t be-h e o s c i l l a t i n g body. Once t be-h e wavemaklng damping i s known, t h e s o - c a l l e d Kramers-Kronig r e l a t i o n s can be

i n v o k e d t o o b t a i n t h e added mass o f t h e body. Thus, b o t h c o e f f i c i e n t s can be o b t a i n e d by examining t h e f a r - f i e l d b e h a v i o r o f t h e o u t g o i n g waves. The advantage o f examining t h e f a r - f i e l d b e h a v i o r o f t h e f l u i d d i s t u r b a n c e s l i e s i n t h e f a c t t h a t t h e s o l u t i o n o f t h e p o t e n t i a l f u n c t i o n can be obt a i n e d by e x a m i n i n g obt h e a s y m p obt o obt i c b e h a v i o r o f obt h e f u n c obt i o n as obt h e h o r i -z o n t a l c o o r d i n a t e , f o r example y, goes t o i n f i n i t y . The a n a l y s i s w i l l be c a r r i e d o u t i n t h e f o l l o w i n g manner. F i r s t , t h e p o t e n t i a l f u n c t i o n r e p r e s e n t i n g t h e o u t g o i n g wave a t t h e f a r f i e l d , w h i c h i s g e n e r a t e d by o s c i l l a t i n g t h e d e m i h u l l o f a t w o - d i m e n s i o n a l SWATH f o r m , w i l l be d e t e r m i n e d . Then, t h e i n t e r f e r e n c e and b l o c k i n g e f f e c t s o f t h e o t h e r h u l l on t h e o u t g o i n g waves a t t h e f a r f i e l d w i l l be d e t e r m i n e d . The n e x t s t e p i s t o d e t e r m i n e t h e damping c o e f f i c i e n t i n terms o f t h e wave a m p l i t u d e and t h e n , t h e added-mass c o e f f i c i e n t by t h e K r a m e r s - K r o n i g r e l a t i o n s .

Heave Damping A p p r o x i m a t i o n

Assume t h a t t h e c i r c u l a r p o r t i o n o f a c r o s s s e c t i o n o f t h e d e m i h u l l of a SWATH s h i p can be r e p r e s e n t e d by a v e r t i c a l d i p o l e , w i t h t h e p u l s a t i n g

v e l o c i t y V l o c a t e d a t a c e r t a i n p o i n t on t h e submerged area o f t h e d e m i h u l l c r o s s s e c t i o n , w h i c h i s shown i n F i g u r e 1 . The n o t a t i o n s d e n o t i n g t h e dimensions o f t h e c r o s s s e c t i o n a r e a l s o g i v e n i n F i g u r e 1. The v e l o c i t y p o t e n t i a l r e p r e s e n t i n g t h e v e r t i c a l d i p o l e i n an un-bounded f l u i d i s g i v e n by (1) where p = w a t e r d e n s i t y = submerged c r o s s s e c t i o n a l a r e a o f t h e d e m i h u l l ~ heave added mass o f t h e d e m i h u l l c r o s s s e c t i o n i n an

unbounded f l u i d

( r, e ) = c y l i n d r i c a l c o o r d i n a t e s such t h a t y = r s i n 6 and z = r cos 6 - d , whe

o t h e d i p o l e i s l o c a t e d

z = r cos 6 - d , where d i s t h e d e p t h o f t h e p o i n t where o' o ^ ^

The heave added mass y ^ ^ w i l l be a p p r o x i m a t e d by

y ^ ^ = pTTR^ \ 1 - ( 2 )

Due t o t h e p r e s e n c e o f t h e s u r f a c e p i e r c i n g v e r t i c a l s t r u t , t h e d i s p l a c e d a r e a o f t h e c r o s s s e c t i o n changes a c c o r d i n g t o t h e v e r t i c a l o s c i l l a -t i o n o f -t h e body. The change i n -t h e d i s p l a c e d a r e a i s e q u i v a l e n -t -t o -t h e f l u x t h r o u g h t h e w i d t h o f t h e s t r u t w h i c h i s Vb^. Thus, an e q u i v a l e n t s o u r c e s t r e n g t h t o g e n e r a t e t h i s f l u x can be o b t a i n e d by V b ^ / ( 2 7 T ) , and t h e c o r r e s p o n d i n g s o u r c e l o c a t e d a t t h e p o i n t ( 0 , - d ) i s expressed by s Vb 27T , /' £n | / y 2 + ( , + d ^ ) 2 ( 3 )

Thus, t h e v e l o c i t y p o t e n t i a l r e p r e s e n t i n g a h e a v i n g d e m i h u l l SWATH c r o s s s e c t i o n i s o b t a i n e d a p p r o x i m a t e l y by V 2TT V -33 o Vb cos 6 o „ H -7,— x,n r ZT T | / y ^ + ( z + d ^ ) ^ ( 4 ) The f a r - f i e l d b e h a v i o r o f t h e s i n g u l a r i t i e s r e p r e s e n t e d by E q u a t i o n ( 4 ) , where t h e z = 0 i s r e g a r d e d as t h e calm-water l i n e , can be o b t a i n e d by * H ~ - i S . + ^ l K e ' ^ ^ - V ^ y ) A p , , K ( z - d + i y ) + b e s ^ o , . , - i w t . (-lOJh^e ) (5) where o n l y t h e r e a l p a r t o f t h e r i g h t - h a n d s i d e i s t o be r e a l i z e d , and . , - i w t V = - i w h e o 2 K = CO / g , t h e wave number CO = c i r c u l a r f r e q u e n c y o f o s c i l l a t i o n g = g r a v i t a t i o n a l a c c e l e r a t i o n 1 = /=ï

H e r e a f t e r , when a r e a l f u n c t i o n i s expressed i n a complex f o r m i n v o l v i n g e '^^^, i t w i l l be t a c i t l y assumed t h a t o n l y t h e r e a l p a r t o f t h e complex e x p r e s s i o n i s meant.

Because t h e wave a m p l i t u d e , w h i c h i s denoted by A^, i s o b t a i n e d by

z y

= 0 — 00

A = -K _^A+ '33

-Kd -Kd

+b Ke

o (6)

From t h e c o n s e r v a t i o n o f energy, t h e r a t e o f work i m p a r t e d on t h e f l u i d by t h e body d u r i n g one c y c l e o f o s c i l l a t i o n s h o u l d be e q u a l t o t h e r a t e o f energy c a r r i e d by t h e r a d i a t i n g s u r f a c e waves d u r i n g t h e same p e r i o d . T h i s means t h a t where T F, a F^ V d t = - 2 p T 0 d t (})^ d) dz (7) p e r i o d o f o s c i l l a t i o n o

[-00 (M+a..) cos OJt-üJb.. s i n a3t+c cos ( O t ] h ^

-wh s i n (jOt o

gA^ cos (Ky-oat+a)e

to

Kz

phase a n g l e w i t h r e s p e c t t o t h e v e r t i c a l m o t i o n o f t h e body

( a . . , b . . , c . . ) = (added mass, damping, r e s t o r i n g ) i n t h e j t h mode 11 11 -11 due t o t h e m o t i o n i n t h e j t h mode M mass o f t h e body From E q u a t i o n ( 7 )

^j =Pn^

(8) S u b s t i t u t i n g E q u a t i o n ( 6 ) i n t o ( 8 ) / - \ 2 (9) where t h e s u p e r s c r i p t 1 i s used t o i n d i c a t e t h e q u a n t i t i e s c o r r e s p o n d i n g t o t h e d e m i h u l l o n l y .

I f b (1) i s n o n d i m e n s i o n a l i z e d by poaS , t h e n 33

(

-Kd -Kd 2 -K 7. °+b Ke s o ( 1 0 ) A To t a k e i n t o account t h e i n t e r f e r e n c e o r b l o c k a g e e f f e c t s f r o m t h e o t h e r h u l l , t h e r e f l e c t e d images a t y = 2b o f t h e d i p o l e and source i n -t r o d u c e d i n -t h e f o r e g o i n g a n a l y s i s s h o u l d be c o n s i d e r e d . F u r -t h e r , assume t h a t t h e d e m i h u l l i s r e p l a c e d by a v e r t i c a l b a r r i e r h a v i n g t h e d e p t h D; hence, t h e waves g e n e r a t e d by one h u l l cannot be t r a n s m i t t e d c o m p l e t e l y beyond t h e o t h e r h u l l .

I f we l e t e denote t h e r a t i o o f t h e wave a m p l i t u d e o f t h e i n c i d e n t wave b e f o r e t h e b a r r i e r t o t h e a m p l i t u d e o f t h e t r a n s m i t t e d wave b e h i n d t h e b a r r i e r , t h e waves g e n e r a t e d by t h e s i n g u l a r i t i e s can be expressed i n

where t h e i n c i d e n t wave a m p l i t u d e i s assumed t o be u n i t y , and a,^, i s t h e change o f t h e phase a n g l e o f t h e t r a n s m i t t e d wave.

E q u a t i o n ( 1 1 ) can be expressed as t h e f o r m

C = cos ( K y- ( j O t ) + e cos (Ky+2b-6Jt+a ) _{( 1 1 )}

C = 3 cos (Ky-oot+Ó) ( 1 2 )

O 111 = [1+e +2e cos (2Kb+a„)]

o J-(13) -e s i n (2Kb+a^) 6 = t a n l + e cos (2Kb+a,j,) , 12 The t r a n s m i s s i o n c o e f f i c i e n t e and t h e phase a n g l e a r e g i v e n by

K, (KD) e = ^ 1 (14) j/ V l ^ ( K D ) + K ^ ^ ( K D ) a„ = t a n _^ /Trl^(KD) T - "-^^ K, (KD) (15)

where I ^ and K^ a r e t h e m o d i f i e d B e s s e l f u n c t i o n s . For X < L i t can be shown t h a t 3 5 7 T rv^ - X + ^ + ^ + + .. . (16) \ W - 2 16 384 18432 - 3 2 ~ ^ " 768 ^ 36864 ^ where m Y = £im ^ - 5,n m ^ n = l = 0.57721566...

Because 3^ r e p r e s e n t s t h e i n t e r f e r e n c e and b l o c k a g e e f f e c t s on t h e wave a m p l i t u d e a t t h e f a r f i e l d , t h e t w i n - h u l l damping c o e f f i c i e n t can be d e t e r m i n e d by

S 3 = ^ 1 ^ 3 ' ^ ^ '

where i s i n t r o d u c e d h e r e as a c o r r e c t i o n f a c t o r i f t h e need a r i s e s . When t h e s t r u t t h i c k n e s s i s z e r o , t h e source p o t e n t i a l g i v e n by E q u a t i o n ( 3 ) can be d i s r e g a r d e d because b^ = 0.

A l s o , due t o t h e absence o f t h e v e r t i c a l b a r r i e r , i t i s assumed t h a t e = 1 and a,j, = 0. Thus, t h e heave damping o f c o m p l e t e l y submerged t w i n c i r c l e s can be o b t a i n e d f r o m E q u a t i o n s ( 9 ) and (18) as or (19) where 1TR' 2 d t h e d e p t h t o t h e c e n t e r o f t h e c i r c l e c

The l a s t two terms i n t h e b r a c k e t o f t h e e x p r e s s i o n o f y^^ r e p r e s e n t , r e s p e c t i v e l y , t h e t w i n h u l l e f f e c t and t h e f r e e s u r f a c e e f f e c t w h i c h i s a s

-13 sumed t o be a r i g i d w a l l .

Sway Damping A p p r o x i m a t i o n

From a f a r - f i e l d v i e w , t h e d e m i h u l l o f a SWATH s e c t i o n i n t h e sway m o t i o n can be r e g a r d e d as a v e r t i c a l p l a t e o f t h e same d r a f t as t h e body d r a f t . For s m a l l f r e q u e n c i e s o f o s c i l l a t i o n w h i c h a r e o f p r a c t i c a l

i n t e r e s t t o SWATH m o t i o n , i t can be f u r t h e r assumed t h a t , near t h e body, t h e f l u i d d i s t u r b a n c e s w o u l d be c l o s e t o t h o s e made by a v e r t i c a l p l a t e o f t w i c e t h e l e n g t h o f t h e body d r a f t i n sway m o t i o n i n an unbounded f l u i d .

I f t h e sway v e l o c i t y o f t h e body m o t i o n i s denoted by U, t h e f l o w f i e l d can be r e p r e s e n t e d by t h e v e l o c i t y p o t e n t i a l w h i c h can be expressed i n terms o f t h e d i p o l e d i s t r i b u t i o n as t i o n o f a f l a t p l a t e s u b j e c t t o n o r m a l u n i f o r m f l o w by t a k i n g t h e p o t e n t i a l jump a c r o s s t h e p l a t e . The c o r r e s p o n d i n g f a r - f i e l d e x p r e s s i o n w h i c h r e p r e s e n t s t h e r a d i a t i n g f r e e - s u r f a c e waves i s o b t a i n e d f r o m E q u a t i o n (13.29) o f Reference 1 1 as 0 (20) where t h e d i p o l e d e n s i t y can be o b t a i n e d f r o m t h e w e l l k n o w n s o l u -0 = i2KUe K ( z + i y ) B(K) (21) where

O

B(K) = f e^^ |/D2-?2 dC -D

= 2^ ( I j ^ ( K D ) - L ^ ( K D ) ) (22)

i n w h i c h i s t h e m o d i f i e d S t r u v e f u n c t i o n .

The a m p l i t u d e o f r a d i a t i n g wave a t |y| = °° i s o b t a i n e d by

. . 2aj|u|KB(K) 1 g and I f U = -ia)h e - i ' ^ ^ o t h e n A, A E - j ^ = 2 r B ( K ) (23) O

Thus, t h e sway damping f o r d e m i h u l l o f a SWATH c r o s s s e c t i o n can be ob-t a i n e d f r o m E q u a ob-t i o n s ( 8 ) and (23) as

'22 - pa3S^ [^2 )

= ^ ^ ^ (24)

^A

The i n t e r f e r e n c e e f f e c t f r o m t h e o t h e r h u l l w i l l be t r e a t e d t h e s _ame as i n t h e p r e v i o u s case. That i s , t h e f a r - f i e l d wave a m p l i t u d e A^ w i l l be m o d i f i e d as 3^A^ where 3^ i s g i v e n by E q u a t i o n ( 1 3 ) . C o n s e q u e n t l y , t h e sway-damping c o e f f i c i e n t f o r t h e t w i n h u l l s i s o b t a i n e d by

where i s a c o r r e c t i o n f a c t o r w h i c h s h o u l d be d e t e r m i n e d as t h e need a r i s e s .

For t h e c r o s s s e c t i o n w i t h o u t t h e s t r u t s , t h e sway damping i s assumed t o be s i m i l a r t o t h e heave damping g i v e n by E q u a t i o n ( 1 9 ) , i . e . . , -2Kd 2 / R2 2 = 2C K e

S

R

2- +

1

(1+ cos 2Kb) (26) 2b 2d ^ c where S, = TTR^ A y 22

o n e - h a l f o f t h e sway added mass o f t w i n c i r c l e s under a r i g i d s u r f a c e

9 2 2b'^ 2d ^

c

R o l l Damping A p p r o x i m a t i o n

The r o l l moment e x e r t e d on a SWATH c r o s s s e c t i o n due t o hydrodynamic p r e s s u r e s p can be expressed by

MR = j p(yn^-zny)d£, (27)

where n^ and n^ a r e , r e s p e c t i v e l y , t h e y-component and t h e z-component o f t h e u n i t n o r m a l v e c t o r p o i n t i n g i n t o t h e body on t h e submerged c o n t o u r o f t h e c r o s s s e c t i o n , and (I d£ i s t h e i n t e g r a l over t h e submerged c r o s s

-J s e c t i o n c o n t o u r .

I f (f)^ d e n o t e s t h e v e l o c i t y p o t e n t i a l a s s o c i a t e d w i t h a f o r c e d r o l l o s c i l l a t i o n o f t h e c r o s s s e c t i o n , t h e n , f r o m t h e l i n e a r i z e d B e r n o u l l i e q u a t i o n , P = ipoj(j)j^ hence = ipo) j ) (j)^(yn^-zny)d£ ( 2 8 ) Assume t h a t \ = - "^s^ (29) Then, s u b s t i t u t i n g E q u a t i o n ( 2 9 ) i n t o E q u a t i o n ( 2 8 ) , d l y = ipu) y^ j (})j^n^d£+z^ j ^^n^ - (I yz(())gn^+(})^ny)d£ ( 3 0 )

where t h e mean-value theorem i s used by d e f i n i n g

-1

f y'Vz^^

y = /= (I (j) n d £ il z -1 f ^ ^ ^ ^ ^ / ^ •7 = *- T it (f)^n d£ S y

The l a s t i n t e g r a l i n E q u a t i o n ( 3 0 ) can be a p p r o x i m a t e d by

(I yz ((t)gn^-(|)yny)d£ = 2b j) z((})gn^-(l)^ny)d£ DH

where ƒ means t h e i n t e g r a l a l o n g t h e submerged c o n t o u r o f t h e d e m i h u l l . DH

From t h e f a r - f i e l d p o i n t o f v i e w , I t can be a p p r o x i m a t e d t h a t (f)g i s odd and é i s even w i t h r e s p e c t t o t h e v e r t i c a l c e n t e r l i n e o f t h e d e m i h u l l .

H

Then, due t o t h e f a c t t h a t z and n a r e even and n i s odd, t h e i n t e g r a n d z y

o f t h e f o r e g o i n g i n t e g r a l becomes odd, hence t h e i n t e g r a l v a n i s h e s . Because t h e i m a g i n a r y p a r t o f E q u a t i o n ( 3 0 ) c o r r e s p o n d s t o t h e r o l l damping, t h e r o l l damping can be o b t a i n e d by

^ 4 = y ^ 3 ' ^22 or - = 44 ' 4 4 " 2 pa)b^S^ iy' b33+z2 b^^) . ^33+^22 ^ (31)

2 2 2 where I t i s assumed t h a t y = b = An a p p r o x i m a t e z i s g i v e n by u -D B(K) (32) For t h e c r o s s s e c t i o n w i t h o u t t h e s t r u t s , t h e r o l l damping i s a p p r o x i -mated by ^ 4 = ^ 3 1^+ , 2 (33)

where b^^ corresponds t o t h e one g i v e n by E q u a t i o n ( 1 9 )

Sway-Roll C o u p l i n g Damping A p p r o x i m a t i o n

The same assumptions made i n t h e a p p r o x i m a t i o n o f t h e r o l l damping w i l l y i e l d ^^22 ^22" ^24 ^ ^42^ pojbS^ b (34) where C d g B(K) (35) For t h e c r o s s s e c t i o n w i t h o u t t h e s t r u t s . b„„d 22 c '24 " b (36) where g i v e n by E q u a t i o n ( 2 6 )

Added Mass A p p r o x i m a t i o n Once t h e damping c o e f f i c i e n t s a r e d e t e r m i n e d , t h e n t h e c o r r e s p o n d i n g c o n j u g a t e p a i r s , i . e . , t h e added-mass c o e f f i c i e n t s can be o b t a i n e d by t h e Kramers-Kronig r e l a t i o n s . A c o n v e n i e n t f o r m f o r n u m e r i c a l m a n i p u l a t i o n s i s g i v e n i n Reference 4 as 2 a., (cj) a., (°°) = -j k ' -j k TT r f b.k(^')-b..(°°)

cos üJt ^, J*^ s i n co't dco'dt ( 3 7 ) 0

f o r j , k = 2, 3, and 4

where a., i s t h e added-mass c o e f f i c i e n t i n t h e i t h mode due t o t h e m o t i o n i n t h e k t h mode, and b., (°°), i n g e n e r a l , i s zero because no s u r f a c e wave

j 1^

can be g e n e r a t e d a t 00 = °°. The n o n d i m e n s i o n a l added-mass c o e f f i c i e n t s w i l l be denoted by t h e b a r s i g n , and a r e d e f i n e d by (-22'^33) ^ (^^^ ^ 4 = ^ 4 2 = ^ (^5) ^44 " The added-mass c o e f f i c i e n t s a t t h e i n f i n i t e f r e q u e n c y a r e a p p r o x i m a t e d by a^^C-) = ( 4 1 ) w h i c h i s based on t h e f l a t p l a t e r e s u l t s .

(42)

j / b

2 2 —

where TT D /32 i s o b t a i n e d by a^^ (°°)/a22 (°°) f o r a v e r t i c a l p l a t e h a v i n g t h e d r a f t D / ' ^ ^42^") = ^ 2 ^ ^ ) l b ^''^ For t h e c r o s s s e c t i o n w i t h o u t t h e s t r u t s , assume t h a t 2b^ 2d ^ ' c = (46) 2b^ 2d ^ ' c .2 / D^ 2 a ^ ^ ( . ) = a 3 3( « ^ ) H- 3 ^ m (47)

where C3 i s a c o r r e c t i o n f a c t o r w h i c h s h o u l d be d e t e r m i n e d when t h e need a r i s e s . The e x p r e s s i o n s g i v e n by E q u a t i o n s (45) and ( 4 6 ) c o r r e s p o n d t o t h e cases t h a t t w i n c i r c l e s , s e p a r a t e d by 2b between t h e c e n t e r s , move w i t h a c o n s t a n t v e l o c i t y i n t h e unbounded f l u i d i n t h e d i r e c t i o n o p p o s i t e t o each o t h e r , and i n t h e same d i r e c t i o n n o r m a l t o t h e l i n e c o n n e c t i n g t h e c e n t e r s o f t h e c i r c l e s , r e s p e c t i v e l y .

DISCUSSION OF RESULTS

The n u m e r i c a l r e s u l t s o b t a i n e d by t h e a p p r o x i m a t e methods d e s c r i b e d i n the p r e c e d i n g s e c t i o n s a r e p r e s e n t e d i n F i g u r e s 2 t h r o u g h 9. The r e s u l t s of t h e a p p r o x i m a t e methods a r e compared w i t h t h e r e s u l t s o b t a i n e d by t h e a c c u r a t e t h e o r y , ^ and w i t h t h e a v a i l a b l e e x p e r i m e n t a l r e s u l t s . H e r e a f t e r , the method d e s c r i b e d i n Reference 8 s h a l l be r e f e r r e d t o as " a c c u r a t e t h e o r y . "

The damping c o e f f i c i e n t s o f a h e a v i n g , t w o - d i m e n s i o n a l SWATH c r o s s s e c t i o n a r e p r e s e n t e d i n F i g u r e 2. The heave damping b^^ i s made nond i m e n s i o n a l by t h e p r o nond u c t o f t h e nond i s p l a c e nond f l u i nond mass (pS^) annond t h e c i r -c u l a r f r e q u e n -c y o f o s -c i l l a t i o n ((JO) . The r e l a t i v e dimensions o f t h e s e c t i o n a r e as shown i n t h e f i g u r e . The a p p r o x i m a t e heave damping c o e f f i c i e n t s a r e o b t a i n e d by E q u a t i o n (18) w i t h C^ = 1 and = 0. I t means t h a t t h e phase change o f t h e t r a n s m i t t e d wave beyond t h e v e r t i c a l b a r r i e r i s n e g l e c t e d . I n c l u s i o n o f t h e phase change i n E q u a t i o n (18) r e s u l t e d i n p o o r e r a g r e e -ment w i t h t h e a c c u r a t e t h e o r e t i c a l r e s u l t s . As can be seen, t h e t h r e e

2

r e s u l t s agree r e a s o n a b l y w e l l f o r t h e f r e q u e n c y number (o) R/g) l e s s t h a n 0.5.

Because t h e s h o r t e s t w a v e l e n g t h o f p r a c t i c a l i n t e r e s t i n SWATH m o t i o n a t z e r o f o r w a r d speed i s about o n e - h a l f o f t h e s h i p l e n g t h , t h e h i g h e s t f r e q u e n c y number c o r r e s p o n d i n g t o t h i s w a v e l e n g t h f o r a SWATH s h i p h a v i n g the l e n g t h - t o - d i a m e t e r r a t i o o f 15 i s about 0.8. Beyond t h i s h i g h e s t f r e q u e n c y l i m i t , SWATH s h i p s would b a r e l y respond t o waves, w h i c h means t h a t t h e r e i s no need o f computing t h e t r a n s f e r f u n c t i o n o f t h e m o t i o n response beyond t h i s f r e q u e n c y l i m i t . Thus, t h e a p p r o x i m a t e r e s u l t s shown i n F i g u r e 2 would be q u i t e r e a s o n a b l e f o r use i n t h e c o m p u t a t i o n o f m o t i o n of SWATH s h i p s i n waves.

The heave added-mass c o e f f i c i e n t f o r t h e SWATH s e c t i o n shown i n

F i g u r e 2 i s p r e s e n t e d i n F i g u r e 3. The r e s u l t s f r o m t h e a p p r o x i m a t e method a r e o b t a i n e d f r o m E q u a t i o n s (37) and ( 4 2 ) . The i n t e g r a l s on t h e r i g h t - h a n d s i d e o f E q u a t i o n ( 3 7 ) , i n f a c t , r e p r e s e n t d o u b l e F o u r i e r t r a n s f o r m s . A f a s t F o u r i e r t r a n s f o r m s u b r o u t i n e o f t h e system l i b r a r y o f t h e computer

(Mi 0.5 h 0.2 h 0.1 1 r 1 r 1.25-A E X P E R I M E N T A C C U R A T E T H E O R Y A P P R O X . T H E O R Y J L 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 W-^R/g

F i g u r e 2 - Heave Damping C o e f f i c i e n t o f a SWATH S e c t i o n

A ' ' ^ i l l l 1 1.2 • A E X P E R I M E N T A C C U R A T E T H E O R Y •=• = - A P P R O X . T H E O R Y 1.0 ~ A — A 0.8

-^ n c

1

0.6

1

t ^

0.4 I • 0.2

1 \y

— n 1 1 1 I l l l 1 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 w'^R/g

f a c i l i t y a t t h e Center was used t o e v a l u a t e t h e double F o u r i e r t r a n s f o r m s . The s u b r o u t i n e i s c a l l e d "FFT" w h i c h was o r i g i n a l l y developed a t t h e Los Alamos S c i e n t i f i c L a b o r a t o r y i n C a l i f o r n i a .

As can be observed i n F i g u r e 3, an a p p a r e n t d i s c r e p a n c y between t h e e x a c t and a p p r o x i m a t e methods i s a t those f r e q u e n c i e s a t w h i c h t h e e x a c t method shows a b r u p t d i s c o n t i n u i t i e s . So f a r as t h e problem i s c o n f i n e d t o a t w o - d i m e n s i o n a l f l o w c o n d i t i o n , t h e d i s c o n t i n u i t i e s e x h i b i t e d by t h e e x a c t method can be r e a l p h y s i c a l phenomena*; a l t h o u g h , t h e e x p e r i m e n t a l r e s u l t s , u n f o r t u n a t e l y , were n o t t a k e n a t t h o s e f r e q u e n c i e s . However, f o r t h r e e -d i m e n s i o n a l t w i n b o -d i e s such as SWATH, t h e r e cannot be a complete t r a p p i n g o f waves between t h e two h u l l s ; hence, no a b r u p t d i s c o n t i n u i t i e s o f t h e hydrodynamic c o e f f i c i e n t s a t c e r t a i n f r e q u e n c i e s s h o u l d e x i s t . I n t h i s r e s p e c t , t h e a p p r o x i m a t e method, w h i c h does n o t c r e a t e t h e d i s c o n t i n u i t i e s , may cause l e s s e r r o r s a t those f r e q u e n c i e s a t w h i c h t h e d i s c o n t i n u i t y i n

t h e hydrodynamic c o e f f i c i e n t o c c u r s .

The SWATH c r o s s s e c t i o n s a t f a r f o r w a r d o r a f t p o r t i o n o f t h e body do n o t have v e r t i c a l s t r u t s . For t h e s e s e c t i o n s , t h e c r o s s - s e c t i o n v i e w i s c o m p l e t e l y submerged t w i n c i r c l e s . The heave added mass and and damping c o e f f i c i e n t s o f such a c r o s s s e c t i o n a r e shown i n F i g u r e 4. For t h e

— 2KR

damping. E q u a t i o n (19) was used w i t h = e , and f o r t h e added mass. E q u a t i o n s (37) and (46) w i t h C^ = 1.1 were used. Agreement between t h e two methods appears good f r o m a p r a c t i c a l v i e w p o i n t f o r t h e p r e d i c t i o n o f SWATH m o t i o n . T h i s i s because a s i g n i f i c a n t l y l a r g e r magnitude o f heave damping c o n t r i b u t e d by v i s c o s i t y and s t a b i l i z i n g f i n s s h o u l d be added t o the wavemaklng damping i n t h e a n a l y t i c a l p r e d i c t i o n o f SWATH m o t i o n ; t h e r e -f o r e , a 50 p e r c e n t e r r o r , as can be observed i n t h e heave damping i n F i g u r e 4, i s n o t g o i n g t o a f f e c t , s i g n i f i c a n t l y , t h e f i n a l r e s u l t s o f t h e m o t i o n . There appears t o be a maximum o f about 5 p e r c e n t e r r o r i n t h e heave added mass c a l c u l a t e d by t h e a p p r o x i m a t e method. Because t h e SWATH m o t i o n i s f a r

l e s s s e n s i t i v e t o t h e added mass t h a n t o t h e damping, a 5 p e r c e n t e r r o r i n the addedmass c o e f f i c i e n t i s n o t g o i n g t o a f f e c t , s i g n i f i c a n t l y , t h e p r e -d i c t e -d r e s u l t s o f SWATH m o t i o n .

F i g u r e 4 - Heave Added-Mass and Damping C o e f f i c i e n t s o f Submerged Twin C i r c l e s

The sway damping o f a SWATH c r o s s s e c t i o n i s p r e s e n t e d i n F i g u r e 5, t o g e t h e r w i t h t h e r e l a t i v e d i m e n s i o n o f t h e c r o s s s e c t i o n . The damping co-e f f i c i co-e n t s o b t a i n co-e d by t h co-e a p p r o x i m a t co-e mco-ethod a r co-e basco-ed on E q u a t i o n (25) (d /(2R)-KD) w i t h t h e m o d i f i c a t i o n f a c t o r = e ° where d^, t h e d e p t h o f t h e c e n t r o i d o f t h e submerged c r o s s s e c t i o n a l a r e a , can be o b t a i n e d by A t p r e s e n t , t h e r e i s no r a t i o n a l e x p l a n a t i o n f o r t h e d e r i v a t i o n o f t h e m o d i f i c a t i o n f a c t o r as i t i s e x c e p t t h a t such a f a c t o r seems t o r e p r e -s e n t t h e b e h a v i o r o f t h e -sway damping i n t h e low and h i g h f r e q u e n c y range-s r e a s o n a b l y w e l l . A l s o , i n t h e v i c i n i t y o f t h e f r e q u e n c y where an a b r u p t d i s c o n t i n u i t y o c c u r s , t h e m o d i f i c a t i o n f a c t o r appears t o make t h e a p p r o x i -mate method behave smoothly a c r o s s t h e d i s c o n t i n u i t y p r e d i c t e d by t h e a c c u r a t e t h e o r y . The c h a i n e d c u r v e shown i n F i g u r e 5 r e p r e s e n t s t h e r e s u l t s o f t h e a p p r o x i m a t e method u s i n g =

1-The sway added mass o f t h e SWATH c r o s s s e c t i o n shown i n F i g u r e 5 i s p r e s e n t e d i n F i g u r e 6. The a p p r o x i m a t e r e s u l t s were o b t a i n e d by E q u a t i o n s

(37) and ( 4 1 ) . I n t h e f r e q u e n c y range o f 0.2 < A / g < 0.5, t h e a p p r o x i -mate r e s u l t s show a l a r g e d i s c r e p a n c y . I t i s n o t o b v i o u s a t t h e p r e s e n t s t a g e w h e t h e r t h e sway added mass computed by t h e a p p r o x i m a t e method would n e c e s s a r i l y r e s u l t i n p o o r e r p r e d i c t i o n o f SWATH m o t i o n t h a n t h a t o b t a i n e d by t h e a c c u r a t e t h e o r y .

The sway added mass and damping c o e f f i c i e n t s f o r submerged t w i n c i r c l e s a r e p r e s e n t e d i n F i g u r e 7. E q u a t i o n (26) was used t o o b t a i n t h e damping, and E q u a t i o n s (37) and (45) w i t h C3 = 1.1 were used t o o b t a i n t h e added mass. The t r e n d o f t h e r e s u l t s i s v e r y c l o s e t o t h a t o f t h e heave added mass shown i n F i g u r e 4.

The r o l l damping o f a SWATH c r o s s s e c t i o n i s p r e s e n t e d i n F i g u r e 8. The dimensions o f t h e c r o s s s e c t i o n a r e a l s o shown i n t h e f i g u r e . Note t h a t t h e c e n t e r o f r o l l i s l o c a t e d above t h e calm w a t e r l e v e l by t h e

3.8 1 r SWAY DAMPING A E X P E R I M E N T A C C U R A T E T H E O R Y APPROX. T H E O R Y APPROX. T H E O R Y WITH -1.25 H 1.8 tJ^R/g

F i g u r e 5 - Sway Damping C o e f f i c i e n t o f a SWATH S e c t i o n

A E X P E R I M E N T A C C U R A T E T H E O R Y A P P R O X . T H E O R Y - 1 . 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 W^R/g

0.9 h 0.8'

-i

L 0.2 0.4 0.6 J L 0.8 1.0 A C C U R A T E T H E O R Y APPROX. T H E O R Y 2.5 W^R/g

F i g u r e 7 - Sway Added-Mass and Damping C o e f f i c i e n t s o f Submerged Twin C i r c l e s

d i s t a n c e o f 3.1 t i m e s t h e r a d i u s o f t h e c r o s s s e c t i o n . The damping c o e f f i -c i e n t s were o b t a i n e d by E q u a t i o n ( 3 1 ) . Both t h e e x p e r i m e n t a l and t h e a c c u r a t e t h e o r e t i c a l r e s u l t s show d i s c o n t i n u i t y a t about W^R/g = 0.65. However, as i n t h e case o f heave and sway, t h e a p p r o x i m a t e r e s u l t s do n o t e x h i b i t any d i s c o n t i n u i t y . The r e l a t i v e t r e n d s o f t h e t h r e e d i f f e r e n t r e s u l t s a r e q u i t e s i m i l a r t o t h e sway damping sho\m i n F i g u r e 5.

The r o l l added i n e r t i a c o e f f i c i e n t o f t h e c r o s s s e c t i o n shown i n F i g u r e 8 i s p r e s e n t e d i n F i g u r e 9. The a p p r o x i m a t e r e s u l t s show a l a r g e

2

d i s c r e p a n c y i n 0.2 < co R/g < 0.5 as was a l r e a d y observed i n F i g u r e 6 f o r the sway added mass. The same remarks as g i v e n e a r l i e r on t h e sway added mass a p p l y t o t h e r o l l added i n e r t i a o b t a i n e d by t h e a p p r o x i m a t e method.

The a c c u r a t e t h e o r y , r e f e r r e d t o f r e q u e n t l y i n t h i s r e p o r t , i s based on t h e Green f u n c t i o n method w h i c h l e a d s t o an i n t e g r a l e q u a t i o n i n v o l v i n g the unknown s t r e n g t h s o f t h e source d i s t r i b u t i o n over t h e submerged c o n t o u r s of t h e c r o s s s e c t i o n o f t h e t w i n h u l l s . The n u m e r i c a l e v a l u a t i o n o f t h e i n t e g r a l e q u a t i o n r e q u i r e s a s e g m e n t a t i o n o f t h e i n t e g r a l a l o n g t h e c r o s s -s e c t i o n c o n t o u r i n t o a f i n i t e number o f i n t e g r a l -s . I n c r e a -s i n g t h e number of segments o f t h e c o n t o u r i n c r e a s e s t h e accuracy o f t h e r e s u l t s . For SWATH s e c t i o n s , i t t a k e s about f i f t e e n t o t w e n t y segments on t h e c o n t o u r o f the d e m i h u l l c r o s s s e c t i o n t o o b t a i n a c c u r a t e r e s u l t s . A t y p i c a l computa-t i o n by computa-t h e CDC 6 0 0 0 - s e r i e s compucomputa-ter a computa-t computa-t h e Cencomputa-ter f o r computa-t h e sway and heave hydrodynamic c o e f f i c i e n t s f o r submerged t w i n c i r c l e s f o r t w e n t y f r e q u e n c i e s t o o k about 220 seconds o f e x e c u t i n g t i m e .

On t h e o t h e r hand, t h e p r e s e n t a p p r o x i m a t e method t o o k o n l y about 16 seconds t o o b t a i n t h e sway, heave, r o l l , and s w a y - r o l l c o u p l i n g c o e f f i c i e n t s f o r 256 f r e q u e n c i e s . * Even i f a d i s c o u n t i s made f o r t h e l a r g e number o f f r e q u e n c i e s w h i c h a r e d i c t a t e d by t h e usage o f t h e f a s t F o u r i e r t r a n s f o r m t e c h n i q u e b u t n o t by t h e p r a c t i c a l n e c e s s i t y , t h e a p p r o x i m a t e t h e o r y seems to y i e l d a t i m e s a v i n g i n t h e c o m p u t a t i o n by an o r d e r o f magnitude compared to t h e a c c u r a t e t h e o r y . *Most o f t h e f a s t F o u r i e r t r a n s f o r m s u b r o u t i n e r e q u i r e s t h a t t h e f u n c -t i o n -t o be -t r a n s f o r m e d s h o u l d be g i v e n i n numbers o f 2^^ where n i s an

i n t e g e r . For t h e p r e s e n t work, i t was found t h a t n = 8 y i e l d s s a t i s f a c t o r y r e s u l t s .

A A "I 1 1 1 1 ^ R O L L DAMPING i A E X P E R I M E N T A P P R O X . T H E O R Y 3.1

t

1 I A ; F i g u r e 8 = R o l l Damping C o e f f i c i e n t o f a SWATH S e c t i o n

T r

A E X P E R I M E N T A C C U R A T E T H E O R Y A P P R O X . T H E O R Y

J L A

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 U J ^ R / g F i g u r e 9 - R o l l A d d e d - I n e r t i a C o e f f i c i e n t o f a SWATH S e c t i o n

SUMMARY AND CONCLUSIONS

To develop a s i m p l i f i e d a n a l y t i c a l p r e d i c t i o n o f SWATH m o t i o n s i n waves, a p p r o x i m a t e methods o f computing t h e hydrodynamic c o e f f i c i e n t s o f o s c i l l a t i n g c r o s s s e c t i o n s o f a SWATH c o n f i g u r a t i o n a r e d e v e l o p e d . The hydrodynamic c o e f f i c i e n t s s t u d i e d a r e t h e added mass and t h e wavemaking damping c o e f f i c i e n t s i n sway, heave, r o l l , and s w a y - r o l l c o u p l i n g modes.

The a p p r o x i m a t e methods a r e based on t h e concept t h a t t h e wavemaking damping c o e f f i c i e n t s can be o b t a i n e d by t h e a m p l i t u d e s o f t h e o u t g o i n g waves a t t h e f a r f i e l d . The o u t g o i n g waves a t t h e f a r f i e l d can be r e p r e -sented by a p a i r o f sources and o f d i p o l e s f o r t h e heave o s c i l l a t i o n and by a d i p o l e d i s t r i b u t i o n on t h e v e r t i c a l c e n t e r l i n e o f each h u l l f o r t h e sway and r o l l o s c i l l a t i o n s . The s t r e n g t h o f t h e s i n g u l a r i t i e s a r e o b t a i n e d under t h e assumption t h a t t h e c r o s s s e c t i o n i s o s c i l l a t i n g i n an unbounded f l u i d .

The hydrodynamic i n t e r a c t i o n s between t h e two h u l l s a r e t r e a t e d as i f one h u l l i s r e p l a c e d by a v e r t i c a l b a r r i e r h a v i n g t h e d r a f t as t h e

o r i g i n a l f o r m and, t h e r e f o r e , i t b l o c k s t h e passage o f t h e waves g e n e r a t e d by t h e o t h e r h u l l .

The added-mass c o e f f i c i e n t s a r e o b t a i n e d by i n v o k i n g t h e Kramers-K r o n i g r e l a t i o n s w i t h t h e known damping c o e f f i c i e n t s and t h e added-mass c o e f f i c i e n t s a t t h e i n f i n i t e f r e q u e n c y , as i n d i c a t e d by E q u a t i o n ( 3 7 ) . The e v a l u a t i o n o f E q u a t i o n (37) i s c a r r i e d o u t by a w e l l - k n o w n n u m e r i c a l

a l g o r i t h m c a l l e d f a s t F o u r i e r t r a n s f o r m t e c h n i q u e . When needed, t h e ap-p r o x i m a t e methods a r e augmented by t h e m o d i f i c a t i o n f a c t o r s w h i c h a r e de-t e r m i n e d by a n u m e r i c a l de-t r i a l - a n d - e r r o r approach.

The r e s u l t s f r o m t h e a p p r o x i m a t e methods, t h e a c c u r a t e t h e o r y based on t h e source d i s t r i b u t i o n on t h e submerged c o n t o u r s , and t h e e x p e r i m e n t s , i f a v a i l a b l e , a r e compared i n F i g u r e s 2 t o 9. The agreements o f t h e r e s u l t s i n t h e f r e q u e n c y range o f p r a c t i c a l i n t e r e s t f o r SWATH m o t i o n a r e , i n g e n e r a l , f a i r ; however, t h e a b r u p t d i s c o n t i n u i t i e s o f t h e hydrodynamic c o e f f i c i e n t s a t c e r t a i n f r e q u e n c i e s e x h i b i t e d by t h e a c c u r a t e t h e o r y and t h e e x p e r i m e n t s a r e n o t r e p r o d u c e d by t h e a p p r o x i m a t e methods. The a b r u p t

d i s c o n t i n u i t i e s a r e m e r e l y caused by t h e t w o - d i m e n s i o n a l f l o w c o n d i t i o n , and i n r e a l i t y , f o r t h r e e - d i m e n s i o n a l b o d i e s , t h e y do n o t e x i s t ; hence, the c o n t i n u i t y o f t h e hydrodynamic c o e f f i c i e n t s a t a l l f r e q u e n c i e s g i v e n by the a p p r o x i m a t e methods c o u l d be more r e a l i s t i c .

The t i m e s a v i n g i n t h e c o m p u t a t i o n o f t h e hydrodynamic c o e f f i c i e n t s a c h i e v e d by use o f t h e a p p r o x i m a t e method i s r e m a r k a b l e ; t h e a p p r o x i m a t e method can p r o v i d e t h e r e s u l t s i n l e s s t h a n one-hundredth o f t h e t i m e w h i c h would be r e q u i r e d by t h e a c c u r a t e t h e o r y .

I t i s t o o e a r l y t o j u d g e whether t h e a p p r o x i m a t e methods developed h e r e w i l l p r o v i d e r e a s o n a b l e p r e d i c t i o n o f SWATH m o t i o n s i n waves. The u s e f u l n e s s o f t h e a p p r o x i m a t e methods w i l l be r e a l i z e d o n l y when t h e y a r e sho\«/n t o p r o v i d e adequate p r e d i c t i o n s o f t h e m o t i o n s f o r use d u r i n g t h e s t a g e o f c o n c e p t u a l h u l l d e s i g n o f SWATH c o n f i g u r a t i o n s . However, i t i s b e l i e v e d t h a t a f i n a l u s e f u l method o f p r e d i c t i n g SWATH m o t i o n can be de-v e l o p e d on t h e b a s i s o f t h e a p p r o x i m a t e methods p r e s e n t e d i n t h i s r e p o r t by an i n t e l l i g e n t usage o f t h e m o d i f i c a t i o n f a c t o r s a s s o c i a t e d w i t h t h e ap-p r o x i m a t e methods.

ACKNOWLEDGMENTS

The a u t h o r would l i k e t o express h i s a p p r e c i a t i o n f o r t h e f u n d i n g s u p p o r t g i v e n by t h e H i g h Performance V e h i c l e s Program O f f i c e o f t h e Ship Performance Department a t t h e Center. He a l s o would l i k e t o e x t e n d h i s g r a t i t u d e t o Ms. M a r g a r e t D. Ochi and Mr. Grant R. Hagen f o r t h e i r en-couragement d u r i n g t h e c o u r s e o f t h i s work.

REFERENCES

1 . Lee, CM., " T h e o r e t i c a l P r e d i c t i o n o f M o t i o n o f S m a l l W a t e r p l a n e -A r e a , T w i n - H u l l (SW-ATH) Ships i n Waves," DTNSRDC Report 76-0046 ( 1 9 7 6 ) .

2. M c C r e i g h t , K.K. and CM. Lee, "Manual f o r M o n o h u l l o r T w i n - H u l l Ship M o t i o n P r e d i c t i o n Computer Program," DTNSRDC Report SPD-686-02 ( 1 9 7 6 ) .

3. D a l z e l l , J.F., "Seakeeping E v a l u a t i o n o f t h e MARAD SWATH F e a s i -b i l i t y Designs," Davidson L a -b o r a t o r y o f Stevens I n s t i t u t e o f Technology, L e t t e r R e p o r t , SIT-DL-77-1970 ( 1 9 7 7 ) .

4. Landau, L.D. and E.M. L i f s h i t z , " S t a t i s t i c a l P h y s i c s , " Addison Wesley P u b l . Co., I n c . , pp. 392-398 ( 1 9 5 8 ) .

5. U r s e l l , F., "On t h e Heaving M o t i o n o f a C i r c u l a r C y l i n d e r on t h e S u r f a c e o f a F l u i d , " Q u a r t . J . , Mech. and A p p l . Math., V o l . 2, pp. 218-231

( 1 9 4 8 ) .

6. Frank, W., " O s c i l l a t i o n o f C y l i n d e r s I n o r Below t h e Free S u r f a c e o f Deep F l u i d s , " DTNSRDC R e p o r t 2375 ( 1 9 6 7 ) .

7. S i n g l e t o n , R.C, "On Computing t h e F a s t F o u r i e r T r a n s f o r m , " Communication o f A s s o c i a t i o n f o r Computing M a c h i n e r y , V o l . 10, pp. 647-654

(Oct 1967).

8. Lee, CM. e t a l . , "Added Mass and Damping C o e f f i c i e n t s o f Heaving Twin C y l i n d e r s i n a Free S u r f a c e , " DTNSRDC Report 3695 ( 1 9 7 1 ) .

9. Lee, CM. and R.M. Curphey, " P r e d i c t i o n o f M o t i o n , S t a b i l i t y , and Wave Load o f S m a l l - W a t e r p l a n e - A r e a , T w i n - H u l l S h i p s , " Trans. Soc. Naval A r c h i t e c t s and M a r i n e E n g i n e e r s , V o l . 85, pp. 94-130 ( 1 9 7 7 ) .

10. K o t i k , J . and V. M a n g u l i s , "On t h e K r a m e r s - K r o n i g R e l a t i o n s f o r Ship M o t i o n s , " I n t . Ship B l d g . P r o g r e s s , V o l . 9, No. 97, pp. 361-368 ( 1 9 6 2 ) .

1 1 . Wehausen, J.V. and E.V. L a i t o n e , " S u r f a c e Waves," E n c y c l o p e d i a o f P h y s i c s , V o l . 9, S p r i n g e r - V e r l a g , B e r l i n ( 1 9 6 0 ) . (See p a r t i c u l a r l y E q u a t i o n s (13.28) and ( 1 3 . 2 9 ) . )

12. U r s e l l , F., "The E f f e c t o f a F i x e d V e r t i c a l B a r r i e r on S u r f a c e Waves i n Deep Water," Proc. Cambridge P h i l . S o c , V o l . 43, pp. 374-382

13. Kennard, E.H., " I r r o t a t i o n a l Flow o f F r i c t l o n l e s s F l u i d s , M o s t l y o f I n v a r i a b l e Depth," DTNSRDC R e p o r t 2299, pp. 235-239 ( 1 9 6 7 ) .

14. K o t i k , J . , "Damping and I n e r t i a C o e f f i c i e n t s f o r a R o l l i n g o r Swaying V e r t i c a l S t r i p , " J. Ship Res., V o l . 7, No. 2, pp. 19-23 ( 1 9 6 3 ) .

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