Contents Journal of Ship Research 1958 (Mag-8)

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By Johts P. C@msü@€k^

A GREAT deal of w o r k is b e i n g done or is p l a n n e d l o o k i n g t o a b e t t e r k n o w l e d g e of t h e loads a n d stresses a c t u a l l j ^ experienced b y ships a t sea. Several p r o j e c t s a i m e d a t a m o r e " r e a h s t i c " or " l o g i c a l " design of h u l l s t r u c t u r e are r a i d e r w a y . T h e F i v e A g e n c y H u l l S t r u c t u r e C o m m i t t e e a n d i t s w o r k i n g s u b c o m m i t t e e , t h r o u g h i t s a d -v i s o r y C o m m i t t e e o n S h i p S t r u c t u r a l D e s i g n , are t a k i n g a l o n g l o o k at t h e m a t t e r . T h e C l a s s i f i c a t i o n Societies are deeply interested, p a r t i c u l a r l y as regards t h e large ships. T h e Society's H 3 r d r o d j a i a m i c s C o m m i t t e e a n d H u l l S t r u c t u r e s C o m m i t t e e h a v e several panels en-gaged a l o n g t h i s line. O u r S-10 P a n e l is m e a s u r i n g h u l l g i r d e r stresses o n ships a t sea, a n d is s t u d y i n g t h e r -m a l stresses i n h u l l s t r u c t u r e s .

S e a w a y Stresses

I n some of t h i s , t h e r e seems t o be a t e n d e n c y t o over-l o o k , or a t over-least t o u n d e r e s t i m a t e , t h e c o m p o s i t e n a t u r e of t h e stresses i n ships' h u l l s . F o r example, t h e c o m -m e n t was -m a d e r e c e n t l y r e g a r d i n g seaway stresses measured o n ships i n h e a v y weather, t h a t i t seems strange stresses n o higher t h a n those recorded s h o u l d h a v e r e s u l t e d occasionally i n f a i l u r e s . T h e i m p l i c a t i o n was t h a t t h e seaway stress is t h e o n l y stress of consequence, a n d t h a t i f we k n e w accuratelj^ t h e w o r s t b e n d i n g m o -m e n t t h a t s t o r -m seas w o u l d ever i-mpose o n a ship, w e c o u l d s i m p l y design f o r t h a t b e n d i n g m o m e n t alone a n d a l l w o u l d be w e l l .

A c t u a l l j ^ , of course, seaway stresses are o n l y one of several k i n d s of h u l l - g i r d e r stress, a l l of w h i c h m a y h a v e t o be resisted s i m u l t a n e o u s l y .

Composite Nature of Hull-Girder Stress

I t seems t i m e l y , t h e r e f o r e , t o r e v i e w t h i s c o m p o s i t e aspect of h u l l - g i r d e r stress. I t a f f e c t s b a s i c a l l y t h e concept of " r a t i o n a l " design, a n d i t applies particularl3^ t o stresses measured o n f u l l - s c a l e h u l l s t r u c t u r e s as is b e i n g done b y our S-10 Pa:nel.

I n a n y f u l l - s c a l e h u l l - s t r a i n measurements, i t c a n n o t be overemphasized t h a t t h e stresses a n d strains e x i s t i n g i n t h e ship w h e n t h e s t r a i n gages are a t t a c h e d w i l l be o n l y a n a p p a r e n t z é r o , a n d t h a t t h e s t r a i n ( a n d stress) ranges observed t h e r e a f t e r , w h e t h e r seaway, t h e r m a l , or other, w i l l be changes o n l y , a n d w i l l be no i n d i c a t i o n w h a t e v e r of a c t u a l stress.

These i n i t i a l or "base-line" stresses w i l l i n c l u d e a t least t h r e e categories of u n m e a s u r e d stresses; n a m e l y .

1 N a v a l A r c h i t e c t , N e w p o r t News S h i p b u i l d i n g & D r y pool;; C o m p a n y , N e w p o r t News, V a . , a n d C h a i r m a n of the Societ3''s S-10 Panel a n d M e m b e r of the H u l l S t r u c t u r e C o m m i t t e e .

b u i l t - i n r e s i d u a l stresses f r o m several causes, i n i t i a l t h e r m a l stresses, a n d s t i l l - w a t e r b e n d i n g stresses.

Bui!f-ln Stresses

B u i l t - i n r e s i d u a l stresses are u s u a l l y considered as due p r i m a r i l y t o w e l d i n g , b u t a c t u a l l y t h e y i n c l u d e stresses f r o m m a n y o t h e r sources. F o r example, d u r -i n g t h e b u -i l d -i n g , t h e w a y s u s u a l l y w -i l l settle u n e v e n l y as t h e w e i g h t of t h e s h i p increases, causing stress i n t h a t p a r t of t h e s t r u c t u r e a l r e a d y b u i l t , b u t n o t i n s t r u c t u r e a d d e d t h e r e a f t e r . I n c i d e n t a l l y , steel as r e -ceived f r o m t h e m i l l , b o t h plates a n d r o l l e d shapes, is b y no means stress-free. W e l d i n g , of course, results i n large r e s i d u a l stresses, f r e q u e n t l y large e n o u g h t o l i f t t h e b o w a n d s t e r n c o m p l e t e l y off t h e keel b l o c k s . U n f a i r a n d d i s t o r t e d p l a t i n g l i k e l y w i l l h a v e been s t r a i g h t -ened b y h e a t i n g a n d s h r i i r k i n g ; t h e stresses r e s u l t i n g f r o m t h i s alone have caused f r a c t u r e s p r i o r t o l a u n c h i n g . A f t e r l a u n c h i n g t h e d i s t r i b u t i o n of s u p p o r t b y b u o y a n c y d i f f e r s s i g n i f i c a n t l y f r o m t h a t of t h e keel blocks, shores, a n d c r i b b i n g b e f o r e l a u n c h i n g , causing s t i l l - w a t e r stress i n t h a t p a r t of t h e s t r u c t u r e c o m p l e t e d b e f o r e l a u n c h i n g . T h e r e m a i n d e r of t h e s t r u c t u r e , such as f o r e x a m p l e u p p e r deck p l a t i n g l e f t o f f f o r s h i p p i n g m a c h i n e r y , w i l l be a d d e d , p r e s u m a b l y i n a n unstressed c o n d i t i o n , t o t h e stressed h u l l .

T h e r e h a v e been a f e w studies of t h e r e s i d u a l stresses b u i l t i n t o a ship, b u t , as m a y be expected f r o m t h e r a n -d o m n a t u r e of b u i l -d i n g c o n -d i t i o n s , t h e results s h o w n o consistency, a n d these b u i l t - i n stresses i n general w i l l be c o m p l e t e l y u n k n o w n , b u t b y no means n e g l i g i b l e . Thermal Stresses I n i t i a l t h e r m a l stress, p r i o r t o a n y w h i c h m a j ' ' be d e t e r m i n e d e x p e r i m e n t a l l y , w i l l iirclude b o t h t h e p e r -m a n e n t r e s i d u a l t h e r -m a l stresses r e s u l t i n g f r o -m t h e v a r y i n g t e m p e r a t t t r e s a t w h i c h t h e v a r i o u s p a r t s w e r e i n i t i a l l y j o i n e d t o g e t h e r a n d t h e t r a n s i e n t t h e r m a l stresses r e s u l t i n g f r o m t h e t e m p e r a t u r e g r a d i e n t s exist-i n g t h r o u g h o u t t h e shexist-ip ( n o t j u s t a t t h e gages) w h e n t h e gages were a p p l i e d . These i n i t i a l t h e r m a l stresses t o o i n general w i l l be c o m p l e t e l y u n k n o w n .

Bending Stresses in Still Water

T h e s t i l l - w a t e r b e n d i n g stresses result f r o m t h e l o a d i n g c o n d i t i o n e x i s t i n g w h e n t h e gages were a p p l i e d . These stresses, alone a m o n g those m e n t i o n e d here, can be c a l c u l a t e d as average stresses w i t h some degree of ac-c u r a ac-c y , b u t t h e y ac-c a n n o t be measured b y themselves.

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ly James A. F©iy^

A n analysis o f t h e f o r c e d i s t r i b u t i o n o n a vessel m o v i n g i n t o a r e g u l a r w a v e system is p r e s e n t e d , s t a r t i n g w i t h a " c r o s s - f l o w " h y p o t h e s i s s i m i l a r t o t h a t p r o p o s e d by M u n k ( 1 5 ) ^ f o r a i r s h i p s a n d a p p l i e d t o v a r i o u s aspects o f vessel h y d r o d y n a m i c s by W e i n b l u m a n d St. D e n i s ( 9 ) , G r i m ( 3 ) , a n d K o r v i n - K r o u k o v s k y ( 1 1 ) . E x p r e s s i o n s f o r t h e vessel m o t i o n ( p i t c h a n d heave) a n d i n t e r n a l reac-t i o n s ( b e n d i n g m o m e n reac-t a n d s h e a r ) are o b reac-t a i n e d a n d c o m p a r e d w i t h e x p e r i m e n t a l measurements by L e w i s ( 1 2 ) . T h e a g r e e m e n t b e t w e e n t h e o r y a n d e x p e r i m e n t f o r t h e m o t i o n is q u i t e g o o d , b u t f o r the b e n d i n g m o m e n t is p o o r . R e m a r k s c o n c e r n i n g t h e j u s t i f i c a t i o n o f the c r o s s f l o w h y -p o t h e s i s are a -p -p e n d e d . Nomenclafure T H E f o l l o w i n g n o m e n c l a t u r e is used i n t h e paper: a

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surface-wave a m p l i t u d e B

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vessel b e a m a m i d s h i p s c w a v e velocits^ d

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d a m p i n g f o r c e per u n i t l e n g t h per u n i t v e l o c i t y = i n t e g r a l d e f i n e d b y E q u a t i o n [20] i n t e g r a l d e f i n e d b y E q u a t i o i r [20]

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acceleration of g r a v i t y h

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h y d r o d y n a m i c i n e r t i a l f o r c e per u n i t l e n g t h per u n i t acceleration H

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vessel d r a f t F , i n t e g r a l d e f i n e d b y E q u a t i o n [20] L

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vessel l e n g t h

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vessel mass per u n i t l e n g t h

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1 Associate Professor, D e p a r t m e n t of M e c h a n i c a l Engineering, Massachusetts I n s t i t u t e of Technology, Cambridge, Mass.

' N u m b e r i n parentheses refer t o t h e B i b l i o g r a p h y at the end of the paper.

N o t e : T h i s w o r k has been sponsored i n p a r t b y the H u l l Struc-t u r e C o m m i Struc-t Struc-t e e of T h e SocieStruc-ty of N a v a l A r c h i Struc-t e c Struc-t s and M a r i n e Engineers t h r o u g h a g r a n t - i n - a i d t o t h e D e p a r t m e n t of M e c h a n i c a l E n g i n e e r i n g , Massachusetts I n s t i t u t e of Technology.

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dis-p l a c e m e n t S

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v e r t i c a l shear t t i m e V vessel v e l o c i t j ^ ( p o s i t i v e i n -(-.'E-direction) w

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local w a t e i i i n e w i d t h X , y h o r i z o n t a l a n d v e r t i c a l ( d o w n w a r d ) c o o r d i -nates fixed i n space

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co-ordinates of fluid element i n x, 3'-system

yo d i s p l a c e m e n t of p o i n t Z = 0, T = 0 meas-u r e d i n X, y - s y s t e m d

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p i t c h angle w

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e n c o u n t e r c i r c u l a r f r e q u e n c y wo

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w a v e c i r c u l a r f r e q u e n c y

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n a t u r a l c i r c u l a r f r e q u e n c y of p i t c h '1 d e f l e c t i o n of free surface of w a v e X w a v e l e n g t h

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p i t c h a n g l e / m a x w a v e slope f

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h e a v e / w a v e a m p l i t u d e X , Y

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h o r i z o n t a l a n d v e r t i c a l ( d o w n w a r d )

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co-ordinates of p o i n t i n vessel m e a s u r e d i n X , F - s y s t e m

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s y m m e t r i c vessel ( s u b s c r i p t ) Introduclion

I n recent years t h e r e has arisen a renewed i n t e r e s t i n t h e p r o b l e m of vessel m o t i o n i n a seaway a n d t h e d y -n a m i c i -n t e r -n a l loads e-nge-ndered t h e r e b y . T h e questio-ns of p i t c h i n g a n d h e a v i n g , a n d l o n g i t u d i n a l s t r e n g t h , were m u c h discussed i n t h e l a t t e r q u a r t e r of t h e n i n e t e e n t h c e n t u r y , a n d t h e solutions proposed were universall}^ accepted a n d h a v e f o r t h e m o s t p a r t p r o v e d s a t i s f a c t o r y i n o r d i n a r y a p p l i c a t i o n s . Nevertheless, f r o m a s c i e n t i f i c p o i n t of v i e w progress i n t h e subsequent 50 years has rendered t h i s e a r l y w o r k obsolete w h i l e f r o m a n e n g i -n e e r i -n g v i e w p o i -n t m o r e r e f i -n e d a-nalyses are r e q u i r e d i f s t r e n g t h a n d seaworthiness r e q u i r e m e n t s are t o be m e t w i t h a f a i r degree of assurance. I t is precisely these s t i n i u H w h i c h h a v e regenerated i n t e r e s t i n a t t a c k i n g t h i s o l d p r o b l e m , an endeavor w h i c h u l t i m a t e l y w i l l l e a d t o new a n d m o r e s a t i s f a c t o r y f o r m u l a t i o n s t o a i d t h e prac-t i c i n g n a v a l a r c h i prac-t e c prac-t .

T h e general p r o b l e m has been a p p r o a c h e d f r o m d i f -f e r e n t p o i n t s o-f v i e w . F i r s t o-f a l l , t h e r e h a v e been i n ¬ , v e s t i g a t i o n s of some of t h e h y d r o d y n a m i c f o r c e s i n v o l v e d .

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Fig. 2 General view o f torsion-arm loading system n n L n

By J©seplii Peti^ieri'

g ] m y ) l l f f 5 r o ) [ l ( i T H E s t r u c t u r a l m o d e l , F i g . 1, tested i n t h i s i n v e s t i g a -t i o n Avas designed -t o s i m u l a -t e -t h e essen-tial f e a -t u r e s of a M a r i n e r t y p e ship. T h i s m o d e l was of a p p r o x i m a t e l y 1 : 4 0 scale i n o v e r - a l l cross-sectional dimensions. T h e thickness of p l a t i n g a n d transverse s t i f f e n e r a r r a n g e m e n t h o w e v e r were n o t scaled as i t w o u l d h a v e been i m p r a c t i c a l t o scale such f e a t u r e s w i t h t h i s s m a l l a m o d e l . A l s o i t was f e l t t h e general stress p a t t e r n s a n d r e l a t i v e r i g i d i t i e s w o u l d n o t be i n f l u e n c e d g r e a t l y b y such a change of scale.

P r a c t i c a l considerations set t h e spacing of transverse s t i f f e n e r s a t 4 i n . T h e thickness of p l a t i n g was t h e n selected of s u f f i c i e n t thickness t o p r e v e n t b u c k l i n g of p l a t i n g between stiffeners. B u c k l i n g of e n t i r e panels was p r e v e n t e d b y t h e transverse stiffeners a n d b y c o n t i n u o u s l o n g i t u d i n a l h a t c h coamings. These h a t c h coamings h o w e v e r were scaled a l o n g w i t h t h e cross section since t h e i r presence affects t h e l o n g i t u d i n a l stiffness of t h e m o d e l .

R i g i d ends were p r o v i d e d o n t h e m o d e l so t h a t t h e ap-p l i e d t o r q u e s w o u l d be d i s t r i b u t e d ap-p r o ap-p e r l y i n t o t h e b o x T h e i n v e s t i g a t i o n r e p o r t e d h e r e i n is a p a r t o f a m o r e g e n e r a l i n v e s t i g a t i o n w h i c h w a s i n i t i a t e d f o r t h e p u r p o s e o f e v a l u a t i n g the e f f e c t o f h a t c h o p e n i n g sizes o n the s t r u c t u r a l b e h a v i o r o f s h i p g i r d e r s . T h i s g e n e r a l i n v e s t i g a t i o n has i n c l u d e d b o t h a study o f stress d i s t r i b u t i o n a n d a study o f r i g i d i t y o f a m o d e l s h i p g i r d e r h a v i n g h a t c h o p e n i n g s o f v a r i o u s sizes. L o a d i n g s c o n s i d e r e d w e r e h o g g i n g , s a g g i n g , a t h w a r t s h i p s b e n d i n g a n d t o r s i o n . O n l y t h a t p o r t i o n o f t h e p r o g r a m d e a l i n g w i t h t o r s i o n a l r i g i d i t y w i l l be p r e s e n t e d i n t h i s p a p e r .

' Associate Professor of C i v i l Engineering, U n i v e r s i t y o f C a l i -foi'nia, Berkeley, Cal.

g i r d e r . A h e a v y l o a d i n g a r m was p r o v i d e d a t t h e e n d of t h e m o d e l as s h o w n i n F i g . 2 t h r o u g h w h i c h t o r q u e was a p p l i e d . A n h j ^ d r a u l i c j a c k was p l a c e d u n d e r one e n d of t h e a r m a n d h o l d - d o w n bars were p l a c e d a t t h e o t h e r end. A similar- t o r s i o n - a r m a r r a n g e m e n t was p l a c e d a t t h e

M A R C H , 1958

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I N a recent paper (1)* i t was i n d i c a t e d t h a t t i i e r e a p -peared t o be a need f o r g e n e r a l i z i n g t h e w e l l - k n o w n l o g a r i t h m i c l a w of t u r b u l e n t b o u n d a r y l a y e r s a n d such a g e n e r a l i z a t i o n was d e r i v e d o n t h e basis of a suggestion due t o T o w n s e n d ( 2 ) . I t was f o u n d t h a t t h e l o g a r i t h m i c f o r m u l a s c o n s t i t u t e o n l y one m e m b e r of a f a m i l y of possi-ble f o r m u l a s , a m o n g w h i c h t h a t one w h i c h best f i t s t h e b o u n d a r y - l a y e r d a t a should be selected.

A p r e v i o u s s t u d y of d a t a o n f l a t - p l a t e b o u n d a r y l a y e r s (3) o n t h e basis of t h e l o g a r i t h m i c l a w h a d r e s u l t e d i n m e a n curves Mdiich i t was s t a t e d c o u l d be considered o n l y as t e n t a t i v e because of t h e w i d e s c a t t e r of t h e d a t a . I t was t h e n suggested t h a t t h e n i e t h o d o f analysis t h e r e i l -l u s t r a t e d be r e a p p h e d t o m o r e precise d a t a w h e n t h e y become a v a i l a b l e .

T h e characteristics of t h e b o u n d a r y l a y e r o n a f l a t p l a t e i n zero pressure g r a d i e n t were s u b s e q u e n t l y measured i n t h e w i n d t u n n e l a t t h e I o w a I n s t i t u t e of H y d r a u l i c R e search. A n i m p o r t a n t f e a t u r e of t h e n e w d a t a is t h e i n -d e p e n -d e n t -d e t e r m i n a t i o n of t h e shear stress a t t h e w a l l b y means of P r e s t o n ' s m e t h o d (4) e m p l o y i n g a c a l i b r a t e d s t a g n a t i o n t u b e . W h i l e t h e e x p e r i m e n t a l v a l u e of t h e m o m e n t u m t h i c k n e s s r e h a b l y gives t h e t o t a l surface shear, i t s d e r i v a t i v e can be considered as o n l y a p o o r m e a s u r e m e n t of t h e l o c a l surface-shear stress. T h e reason is t h a t t h e m e a s u r i n g o f t h e l o c a l slopes of a c u r v e d r a w n t h r o u g h e x p e r i m e n t a l p o i n t s c a n n o t be v e r y accurate. Because of t h e p r e c a u t i o n s a n d care t a k e n i n o b -t a i n i n g -t h i s se-t of d a -t a , -t h e a u -t h o r s believe -t h a -t i -t is s u f S c i e n t l y precise t o serve as a basis f o r t h e c r i t i c a l analysis of b o u n d a r y - l a y e r laws. O t h e r sets of flat-p l a t e b o u n d a r y - l a y e r d a t a h a v e n o t been i n c l u d e d i n t h i s analysis because i t was f o u n d i n such a p r e v i o u s s t u d y (3) t h a t t h e dispersion of these d a t a is large a n d conse-q u e n t l y m i g h t confuse t h e desired c o m p a r i s o n b e t w e e n t h e t w o laws. C o n s e q u e n t l y o n l y t h e n e w d a t a w i l l be presented a n d a n a l y z e d i n accordance w i t h b o t h t h e l o g a r i t h m i c l a w a n d i t s g e n e r a l i z a t i o n .

T h e analysis based u p o n t h e l o g a r i t h m i c l a w s has been r e f i n e d i n one p r a c t i c a l aspect. T h e necessity f o r

select-1 P r o j e c t sponsored b y Office of N a v a l Research, C o n t r a c t N8onr-500(03).

2 Research Engineer, I o w a I n s t i t u t e of H y d r a u l i c Research, a n d Professor, D e p a r t m e n t of Mechanics a n d H y d r a u l i c s , State U n i -v e r s i t y of I o w a , I o w a C i t y , I o w a , M e m b e r S N A M E .

= F o r m e r l y Research Engineer, I o w a I n s t i t u t e of H y d r a u l i c R e search, State U n i v e r s i t y of I o w a ; presently a t I n s t i t u t e of H } ^ -draulic Research, Academia Sinica, Peking, C h i n a .

N u m b e r s i n parentheses refer t o t h e B i b l i o g r a p h y a t the end of the paper.

i n g a v a l u e f o r the b o u n d a r y - l a y e r thickness a t each v e l o c i t y p r o f i l e , a v a l u e w h i c h is i n h e r e n t l y i l l d e f i n e d be-cause of i t s as3aiiptotic n a t u r e , has been a v o i d e d b y f o l l o w i n g a suggestion m a d e i n t h e A p p e n d i x of t h e p r e -v i o u s w o r k (3) t o use a n a t u r a l l e n g t h scale d e f i n e d b y t h e b o u n d a r y - l a y e r l a w s themselves. I t w i l l be seen t h a t t h i s m o d i f i c a t i o n does n o t a f f e c t t h e f o r m a l n a t u r e of t h e r e s u l t i n g expressions f o r t h e b o u n d a r y - l a y e r characteristics, a l t h o u g h t h e n u m e r i c a l values of some of t h e c o n -s t a n t -s a p p e a r i n g i n t h e e q u a t i o n -s are g r e a t l y a l t e r e d .

T h e analyses a c c o r d i n g t o t h e l o g a r i t h m i c l a w a n d i t s g e n e r a l i z a t i o n , w h i c h w i l l h e r e a f t e r be called t h e p o w e r law, appear t o fit t h e d a t a e q u a l l y w e l l i n t h e n a r r o w range of R e y n o l d s n u m b e r s encompassed b y t h e tests. T h e r e s u l t i n g curves f o r t h e coefficients of shear stress a n d d r a g versus R e y n o l d s n u m b e r d e v i a t e a p p r e c i a b l y f r o m each other a t h i g h e r R e y n o l d s n u m b e r s , t h e v a l u e p r e -d i c t e -d b y t h e l o g a r i t h m i c l a w excee-ding t h a t f r o m t h e p o w e r l a w b y a b o ü t 10 per cent a t a R e y n o l d s n u m b e r of 101 T h u s c o m p a r i s o n w i t h t w o - d i m e n s i o n a l flat-plate d a t a a t h i g h e r R e y n o l d s n u m b e r s s h o u l d i n d i c a t e w h i c h of t h e b o u n d a r y - l a y e r laws is p r e f e r a b l e . Nomenclature a, b, B, C = c o n s t a n t s Co, Cl, C2, Cs = c o n s t a n t s d e f i n e d b y d e f i n i t e i n t e g r a l s Cf = f r i c t i o n a l - r e s i s t a n c e c o e f f i c i e n t Cr = shear-stress c o e f f i c i e n t d = a c o n s t a n t D = p i t o t-t u b e d i a m e t e r ƒ, fo = i n n e r - l a w f u n c t i o n ƒ(?/*); fo = f(]/o*) F, Fl = o u t e r - l a w f u n c t i o n / ( f ) ; = F(^i) (7 = a f u n c t i o n , g{a) H = shape p a r a m e t e r , 5i/52 k, K = c o n s t a n t s L = l e n g t h scale of b o u n d a r y l a y e r n = a n e x p o n e n t ?j = pressure p = pressure outside b o u n d a r y l a y e r Pt = t o t a l pressure, p + ^pu^ r, Tt = o u t e r a n d i n n e r r a d i i of p i t o t t u b e Rx, Rs, Rsi, Rsi = R e j a i o l d s n u m b e r s based o n i n d i c a t e d

l i n e a r d i m e n s i o n s ; e.g.,Rx = Ux/v u = .^-component of m e a n v e l o c i t y

u' = .^•-conlponent of fluctuating p a r t of v e l o c i t j ' '

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o f n i d i i F i f e i / - l ! ^ i ( o ! i f ( i [ H J y d r @ f @ i l i '

fn Noncavihafing and Fuil-(2avify Flov/^

I n t r o d u c t i o n T H E p r i m a r y a d v a n t a g e t o be gained f r o m h y d r o f o i l boats is t h e p o s s i b i l i t j ' of a t t a i n i n g v e r y h i g h o p e r a t i n g speeds w i t h o u t i n o r d i n a t e l j ' - h i g h p o w e r requirements. H o w e v e r , the f u l l e s t e x p l o i t a t i o n of t h i s i m p o r t a n t a d -v a n t a g e m a y m a k e the occurrence of c a -v i t a t i o n on t h e h y d r o f o i l system u n a v o i d a b l e w i t h a r e s u l t i n g increase m d r a g a n d a decrease i n l i f t . F o r some a p p l i c a t i o n s where v e r y l i i g h speeds are essential, i t m a y be possible t o relax t h e r e q u i r e m e n t f o r a h i g h l i f t coefficient and, i f t h e c a v i t a t i n g h y d r o f o i l s operate i n a stable manner, t h e onset of c a v i t a t i o n need pose no i n s u r m o u n t a b l e p r o b l e m . I n f a c t i t m a y be desirable t o operate t h e h y d r o f o i l s over t h e greatest possible speed range i n t h e f u l l y c a v i t a t i n g regime i n order t o t a k e f u l l e s t a d v a n t a g e of t h i s stable f l o w c o n f i g u r a t i o n . T h u s , we are led n a t u r a l l y t o profiles w i t h sharp leading a n d t r a i l i n g edges because t h e v e r y l o w m i n i m u m pressures p r o v i d e d b y t h e i r sharp edges m a k e such profiles c a v i t a t e a t r e l a t i v e l y l o w speeds. M o r e o v e r , f o r f u l l - c a v i t . y flows t h e c a v i t y springs f r o m t h e sharp edges, so f o r a large range of a t t a c k angles t h e g e o m e t r y of t h e flow p a s t t h e h y d r o f o i l remains es-s e n t i a l l y the es-same, t h e r e b y aes-ses-suring es-steady h j ' - d r o f o i l operation. O n t h e o t h e r ha:nd, there are s i t u a t i o n s i n w h i c h t h e onset of f u l l - c a v i t y flow c o m p l e t e l y changes t h e flow g e o m e t r y so t h a t t h e forces o n t h e h y d r o f o i l sufi'er drastic changes as t h e f u l l c a v i t y develops. T h e n a t u r e of such f o r c e v a r i a t i o n s a n d t h e range of flow geometries f o r w h i c h thej^ occur should be of considerable interest f o r t h e a p p h c a t i o n of sharp-edged' profiles t o de-sign situations.

I n v i e w of t h e f a c t t h a t t h e f u l l c a v i t y generally spruigs f r o m t h e sharp leading a n d t r a i l i n g edges, t h e t w o -duiiensional c a v i t y flow p a s t such sharp-edged profiles of simple g e o m e t r y has considerable t h e o r e t i c a l interest. These fixed separation p o i n t s enable f r e e - s t r e a m l i n e t h e o r y t o be a p p l i e d t o a c a v i t y flow f o r w h i c h t h e separation p o i n t s are k n o w n i n advance, guch a t h e o r y of forces on f u l l y c a v i t a t i n g profiles has been developed

' T h i s research was carried o u t under the B u r e a u of Ships F u n d a m e n t a l H y d r o m e c h a n i c s Research P r o g r a m P r o j e c t N S 7 1 5 - 1 0 2 , D a v i d T a y l o r M o d e l Basin. T h e w o r k was p e r f o r m e d at the C a h f o r n i a I n s t i t u t e of Technology under the d i r e c t i o n of Prof. M . S. Plesset, O f f i c i a l I n v e s t i g a t o r . R e p r o d u c t i o n i n whole or i n p a r t is p e r m i t t e d f o r a n y purpose of t h e t l n i t e d States G o v e r n -m e n t . ^ T h e R a n d C o r p o r a t i o n , Santa M o n i c a , C a l i f . A n i n v e s t i g a t i o n i n t h e H i g h - S p e e d W a t e r T u n n e l o f t h e t w o d i m e n s i o n a l h y d r o d y n a m i c c h a r a c t e r i s -tics o f s h a r p - e d g e d h y d r o f o i l s is d e s c r i b e d . T h e l i f t , d r a g , a n d p i t c h i n g m o m e n t w e r e m e a s u r e d i n c a v i t a t i n g a n d n o n c a v i t a t i n g f l o w s f o r f l a t - p l a t e a n d c i r c u l a r - a r c p r o f i l e s . T h e t h e o r y o f W u f o r the f o r c e s o n sharp-edged p r o f i l e s i n f u l l - c a v i t y flow a n d t h e e x p e r i m e n t a l results s h o w e d g o o d a g r e e m e n t o v e r a w i d e r a n g e o f a t t a c k angles. b y T . Y . W u '(1).3 I t is of i n t e r e s t t o s u b m i t t h i s t h e o r y , w h i c h makes no l i n e a r i z i n g a p p r o x i m a t i o n s , t o e x p e r i m e n t a l v e r i f i c a t i o n . T h e present r e p o r t includes a p r e s e n t a t i o n of t h e t w o -d i m e n s i o n a l f o r c e a n -d m o m e n t characteristics w h i c h were o b t a i n e d i n H i g h - S p e e d W a t e r T u n n e l (2) experiments. These d a t a are presented f o r sharp-edged h y d r o f o i l s of flat p l a t e , a n d concave circular-arc p r o f i l e s f o r b o t h c a v i t a t i n g a n d n o n c a v i t a t i n g flow. Those forces w h i c h were measured i n t h e f u l l y c a v i t a t i n g - f l o w regime are c o m p a r e d w i t h W u ' s exactHlieory.

Experimental Procedures^

T h e m e t h o d s of measurement a n d of d a t a r e d u c t i o n w i l l be described i n t h i s section. B r i e f d e s c r i p t i o n s of t h e e x p e r i m e n t a l e q u i p m e n t used i n t h e e x p e r i m e n t s are also g i v e n .

T h e present experiments, w h i c h were m a d e t o determ i n e t h e t w o d i determ e n s i o n a l forces o n flatplate a n d c i r c u lararc p r o f i l e s i n f u l l c a v i t y fiow, r e q u i r e d t h a t a t w o -d i m e n s i o n a l channel be i n s t a l l e -d i n t h e H i g h - S p e e -d W a t e r T u n n e l w o r k i n g section. W h e n t h i s m o d i f i c a t i o n t o t f i e

' N u m b e r s i n parentheses refer t o t h e B i b l i o g r a p h y a t t h e end of the paper.

* A d d i t i o n a l details are given i n P a r k i n and l i e r m e e n , " W a t e r T i m n e l Techniques f o r Force Measurements on C a v i t a t i n g H y d r o

-f o i l s , " J O U R N A L OF S H I P R E S E A H O H , v o l . 1 , no. 1 , A p r i l 1 9 5 7 , p . 3 6 . See also " E x p e r i m e n t s on C i r c u l a r A r c a n d F l a t P l a t e H y d r o f o i l s i n N o n c a v i t a t i n g a n d F u l l - C a v i t y F l o w s , " C a l i f o r n i a I n s t i t u t e of Technology H y d r o d y n a m i c s L a b o r a t o r y R e p o r t N o . 4 7 6 , F e b r u -a r y 1 9 5 6 . 3 4 _{J O U R N A L O F SHIP R E S E A R C H}

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ir

in V. Bii'esoiiri A b r i e f s u m m a r y is g i v e n o f e l ï o r t s i n the f i e l d o f naval a r c h i t e c t u r e o n t h e p r o b l e m o f d e t e r m i n i n g the v i b r a t o r y f o r c e s a n d m o m e n t s p r o d u c e d by a s h i p p r o p e l l e r , a n d an a c c o u n t o f t h e p r i n c i p a l c o n t r i b u t i o n s m a d e b y a e r o n a u t i c a l researchers o n t h e p r o b l e m o f c o m p u t i n g t h e fluctuating pressure f i e l d near a p r o p e l l e r . T h i s pressure is d e r i v e d f r o m a l i f t i n g - l i n e r e p r e s e n t a t i o n a n d t h e e x p r e s s i o n s are t h e n i n t e r p r e t e d i n t e r m s o f the f i e l d s o f t w o d o u b l e t d i s t r i b u t i o n s . I t is s h o w n t h a t t h i s r e s u l t can be v e r i f i e d by specializ-i n g t h e e x p r e s s specializ-i o n s f o r t h e s o u n d - p r e s s u r e f specializ-i e l d o f a p r o p e l l e r . Some c h a r a c t e r i s t i c s o f the pres-sure field are discussed b r i e f l y . D e t a i l e d evalua-t i o n s a n d c o m p a r i s o n w i evalua-t h e x p e r i m e n evalua-t w i l l be p r e s e n t e d i n a subsequent p a p e r .

Nomenclature

T H E f o l l o w i n g n o m e n c l a t u r e is used i n t h e p a p e r : A R = amplitudes of the cosine and sine

constitu-ents of

A,

m e a n v a l u e of fluctuating pressure h = propeller r a d i u s d = propeller d i a m e t e r J

=

advance r a t i o ( = U/nd) J'

=

TV/J VI

=

n u m b e r of blades

n = propeller r e v o l u t i o n s per second

V = difterence b e t w e e n pressure a t a n y p o i n t a n d t h a t a t a p o i n t f a r ahead of a n y d i s -t u r b a n c e

v'

=

pressure difference d e t e r m i n e d w i t h respect t o m o v i n g axes

pressure change due t o b o u n d v o r t e x Vc

=

c o n v e c t i v e pressure [ = — pC(ö<^/ö,x)] Ph pressure change due t o helical v o r t e x

Vi i m p u l s i v e pressure difference [ = p{b4>/bt)] Vm = b l a d e - f r e q u e n c y pressure of an ???-bladed p r o p e l l e r \Vm\ = a m p l i t u d e of p,„ ' T e c h n i c a l D i r e c t o r , E x p e r i m e n t a l T o w i n g T a n k , Steveirs I n -s t i t u t e of Technology, H o b o k e n , N . J .

N O T E : T h i s s t u d y was carried o u t a t the E x p e r i m e n t a l T o w i n g T a n k , Stevens I n s t i t u t e of T e c h n o l o g y , under B u r e a u of Ships C o n t r a c t N o . NObs47790, T . O. 17, sponsored b y the Ship D i v i s i o n of t h e D a v i d W . T a y l o r M o d e l B a s i n .

Vl, qi, »'i = y, 2-components of a n g u l a r - v e l o c i t y vec-t o r

' R = distance between a p o i n t o n v o r t e x a n d a p o i n t i n space

r = r a d i a l distance f r o m propeller axis t o p o i n t i n space; u s u a l l y expressed i n m u l t i p l e s of p r o p e l l e r radius

T = t o t a l t h r u s t of propeller

T' = p r o p e l l e r - d i s k l o a d i n g ( = T/7r6^)

t = t i m e (t is also used f o r t i p clearance d e f i n e d b e l o w )

t/d = tip-clearance r a t i o w h i c h is r e l a t e d t o r b y . r = 1 - I - 2{i/d)

U = speed of f r e e s t r e a m or speed of a d v a n c e u, V, IV = X, y, ^-components of p e r t u r b a t i o n v e l o c i t i e s u', v', w' = x', y', ^'components of p e r t u r b a t i o n v e l o c i

-ties measured w i t h respect t o m o v i n g co-ordinates x', y', z'

Ub, Vi, Wh = c o m p o n e n t s of v e l o c i t y f r o m b o u n d v o r t e x

u,,, v,„ iu„ = c o m p o n e n t s of v e l o c i t y f r o m h e l i c a l v o r t e x

Uo, Vo, lOo = X, y, ^-components of v e l o c i t y of m o v i n g

axes

V = r e s u l t a n t v e l o c i t y a t a p o i n t

v'y = t a n g e n t i a l - v e l o c i t y c o m p o n e n t i n moving-f r a m e omoving-f removing-fereirce

X, y, z = fixed r e c t a n g u l a r co-ordinates x', y', z' = co-ordinates of helix

a = 0 — J To = c i r c u l a t i o n of line v o r t e x 7 = angle f o r m e d b y r a n d l i n e p a r a l l e l t o +y-axis e = phase angle of p„j e = i n s t a n t a n e o u s angle of p r o p e l l e r - b l a d e axis (also b o u n d v o r t e x line) m e a s u r e d f r o m l i n e p a r a l l e l t o -|-?/-axis

do = phase angle r e f e r r i n g t o a single b l a d e

posi-t i o n

p = fluid-mass d e n s i t y 0 = v e l o c i t y p o t e n t i a l

w angular v e l o c i t y of p r o p e l l e r

Introduction

T h i s paper presents some of t h e results of a t h e o r e t i c a l s t u d y of t h e transient-pressure field near an o p e r a t i n g ship p r o p e l l e r i n t h e absence, of o t h e r bodies. W i t h t h e present emphasis o n increased ship speeds a n d conseq u e n t l y m u c h h i g h e r propeller t h r u s t , h j ^ d r o d y n a m i c -i n d u c e d v -i b r a t -i o n has become a v e r y -i m p o r t a n t p r o b l e m i n n a v a l a r c h i t e c t u r e . A l t h o u g h t h e t r a n s i e n t pres-sures are r e l a t i v e l y s m a l l (peak prespres-sures b e i n g of t h e

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R e s e a r c h T r e n d s

I y y If

Q

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l y

PFriillip

Eisen

T l

o c o o u u T H E R E are f e w disciplines w h i c h h a v e t h e f a r - r e a c h i n g a n d i n t i m a t e consequences i n t h e w h o l e s p e c t r u m of N a v j ' operations, vehicles, a n d i m p l e m e n t s t h a t H y d r o -d y n a m i c s has. M o r e o v e r , i t is a c o n t e m p o r a r y charact e r i s charact i c of charact h i s f i e l d charact h a charact charact h e basic p h y s i c a l a n d m a charact h e -m a t i c a l p r o b l e -m s n o w recognized as t h e c e n t r a l ones are also a m o n g t h e v e r y ones e n c o u n t e r e d b y t h e designer of t h e m o d e r n F l e e t a n d i t s m y r i a d a c c o u t e r m e n t s . T h e recent years h a v e been p a r t i c u l a r l y n o t e w o r t h y f o r r a p i d e x p l o i t a t i o n of s c i e n t i f i c a n d t e c h n o l o g i c a l b r e a k t h r o u g h s i n t h e developmeirt o f Fleet c o m p o n e n t s a n d N a v a l strategic a n d t a c t i c a l concepts. I t m a y be f a i r l y s t a t e d t h a t advances i n H j ^ l r o d y n a m i c s h a v e p l a y e d a s i g n i f i c a n t p a r t i n t h e e v o l u t i o n o f t h e n e w N a v j ' ; o n t h e o t h e r h a n d , t h e need f o r i n t e n s i f i c a t i o n of research i n t h i s d i s c i p l i n e has been emphasized b y t h e h y d r o d y n a m i c a l p r o b l e m s w h i c h m u s t be solved if t h e m o s t e f f e c t i v e N a v a l s y s t e m c o m p a t i b l e w i t h t h e u n p r e c e d e n t e d d e v e l o p m e n t s i n o t h e r fields is t o be p r o v i d e d . T o h i g h l i g h t t h e i r significance i n N a v a l a p p l i c a -t i o n s , -t h e m a n y aspec-ts ( a n d -these m a k e u p m o s -t of -t h e field) t h a t are of i n c r e a s i n g i m p o r t a n c e have been c h a r -acterized as " N a v a l F l y d r o d y n a m i c s . " ^

Scope, Orientation, a n d i\Aotivation

D u r i n g t h e past 4 years o r so, t h e O f i i c e of N a v a l Research p r o g r a m i n H y d r o d y n a m i c s has been s u b j e c t t o r a t h e r sweeping r e a p p r a i s a l a n d r e o r i e n t a t i o n . T h e changes t h a t have been a n d are c o n t i n u a l l y b e i n g m a d e reflect t h e N a v y ' s i n c r e a s i n g l y m o r e s o p h i s t i c a t e d t e c h n o l o g i c a l r e q u i r e m e n t s . I n t h i s respect, our goal is t o ensure t h a t research i n H y d r o d y n a m i c s is i n t e l -l i g e n t -l y f o c u s e d t o y i e -l d n e w d e v e -l o p m e n t s c o m p a t i b -l e w i t h , a n d designed t o t a k e f u l l a d v a n t a g e of, t h e dis-coveries a n d advances i n o t h e r areas of t e c h n o l o g y ; e.g., nuclear power a n d h i g h - e n e r g y f u e l s , i m p r o v e d radar, a n d so o n . O n t h e o t h e r h a n d , we m u s t endeavor

1 H e a d , Mechanics B r a n c h , Office of N a v a l Research, D e p a r t -m e n t of t h e N a v j ' , W a s h i n g t o n , D . G.

2 A cross section of t h e m a j o r areas of interest was reviewed d u r -i n g t h e f-irst Sympos-ium on Naval Hydrodynam-ics sponsored b y t h e Office of N a v a l Research and t h e N a t i o n a l Academjr of Sciences-N a t i o n a l Research Council, Sept. 24-28, 1956. Proceedings of t h i s s y m p o s i u m h a v e been p\iblished i n N A S - N R C P u b l i c a t i o n 515, 1957, W a s h i n g t o n , D . C.

t o e x p l o i t a t t h e earliest possible m o m e n t a n d i n a b o l d b u t nevertheless hard-headed a n d s y s t e m a t i c Avay those new results of h y d r o d y n a m i c s research w h i c h show promise f o r m a t e r i a l l y i m p r o v i n g t h e effectiveness of t h e Fleet.

T o achieve these goals, t h e o r i e n t a t i o n developed i n t h e O N R p r o g r a m is e x a m i n e d f r o n r t\YO vie-\\ p o i n t s :

(1) A d e q u a c y of research coverage of basic u n s o l v e d p r o b l e m s i n h y d r o d y n a m i c s , i.e., s t o c k i n g of t h e c u p -iDoard of basic k n o w l e d g e u p o n w h i c h n e w . d e v e l o p m e n t s m u s t u l t i m a t e l y d r a w , aird, (2) a d e q u a c y of t h e p r o g r a m i n p r o v i d i n g b a c k g r o u n d i n f o r m a t i o n f o r d e v e l o p m e n t s w h i c h are a l r e a d y recognized t o be, or are speculated t o be, e x p l o i t a b l e f o r N a v y use. W h i l e t h i s b r e a k d o w n i m p l i e s " b a s i c " research o n t h e one h a n d a n d specific N a v y categories o n t h e o t h e r , i t is m o r e a c c u r a t e (since, i n t h e largest p a r t , .all of t h e research s u p p o r t e d i n t h i s p r o g r a m is of a more-or-less f u n d a m e n t a l c h a r a c t e r ) t o d i s t i n g u i s h b e t w e e n these t w o g r o u p i n g s as " f o u n -d a t i o n a l " a n -d " m o t i v a t e -d " research. I n other w o r -d s , t h e w o r k u n d e r so-called f o u n d a t i o n a l research is a p r o g r a m of b r o a d scope c o v e r i n g those aspects o f t h e f o u n d a t i o n s of H y d r o d y n a m i c s of i n t e r e s t t o t h e N a v y b u t w i t h t h e emphasis d e t e r m i n e d i n i t i a l l y b y t h e s c i e n t i f i c u r g e n c y or i n t e r e s t r a t h e r t h a n b y s p e c i f i c a l l y i d e n t i f i e d t e c h n o l o g i c a l a p p l i c a t i o n o r r e q u i r e m e n t . T h e r e m a i n d e r o f t h e p r o g r a m has been m o t i v a t e d d i r e c t l y b y r a t h e r c l e a r l y d e f i n e d N a v a l r e q u i r e m e n t s or b y p o s s i b i l i t i e s of i m p o r t a n t a p p l i c a t i o n s as de-t e r m i n e d a c c o r d i n g de-t o o u r besde-t j u d g m e n de-t a n d esde-timade-tes.

I t w i l l be recognized, o f course, t h a t i t has n o t been possible w i t h i n t h e research budgets of recent y e a r s t o p r o v i d e c o m p l e t e l y adequate s u p p o r t f o r a l l of t h e areas o f H y d r o d y n a m i c s of N a v y i n t e r e s t . T h u s , w i t h i n t h e c r i t e r i a a n d estimates m e n t i o n e d , i t is neces-s a r y t o aneces-sneces-sign neces-some k i n d o f p r i o r i t j ^ f o r emphaneces-sineces-s o n t h e basis of our best forecasts f o r s c i e n t i f i c a n d t e c h -n o l o g i c a l p a y o f f . T o bias t h e cha-nces o f success, t h e p r o g r a m is closely c o - o r d i n a t e d w i t h t h e " i n - h o u s e " a n d c o n t r a c t research p r o g r a m s of t h e N a v y ' s design B u r e a u s a n d t h e r e l e v a n t p r o g r a m s of t h e i r respective l a b o r a t o r i e s w h i c h s u p p o r t each B u r e a u ' s m i s s i o n . I n t h i s w a y , a n a t t e m p t is m a d e t o p r o v i d e t h e flexi-b i l i t y a n d coverage essential t o r a p i d research progress a n d e x p l o i t a t i o n .

F i n a l l y , space does n o t a l l o w d e t a f l e d discussion or

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@ ] y © ] ü D ( ö ) [ n ] i O O O O

„ . f o r

Stress and Disi

l d A t i m r i M j

By Joseph Kempner

Energy expressions and tlie related equilibrium equations and natural boundary conditions fer the determination of the stresses in and displacements of uniform, thin-walled cylinders of arbitrary cross section loaded in an arbitrary manner by surface and edge forces and moments are presented. The derivations are based upon the KirchhofT-Love assumptions of the classical theory of shells and are per-formed to within a degree of accuracy employed by Flügge in his derivation of the equilibrium equations applicable to circular cylindrical shells; hence, in terms of stress resultants, the exact, small-deflection equilibrium equations are obtained. Methods of simplification of the relations denved and of solution of the differential equations presented are indicated.

\aiyses

Cylifidfkxil ShAk

I N c o m p a r i s o n t o t h e a m o u n t of l i t e r a t u r e concerned w i t h equations a p p l i c a b l e t o t h e solutions of m a n y p r o b -lems i n t h e f i e l d of c i r c u l a r c y l i n d r i c a l shells, l i t t l e is a v a i l a b l e o n t h e c o r r e s p o n d i n g , p r o b l e m s of n o n c i r c u l a r c y l i n d r i c a l shells. Such shells find mde i n d u s t r i a l a p p l i c a t i o n , p a r t i c u l a r l y i n s u b m a r i n e a n d a i r c r a f t f a b r i c a -t i o n .

T h i s paper represents t h e first of several r e p o r t s w h i c h g i v e t h e basic r e l a t i o n s a p p r o p r i a t e t o s o l u t i o n s of stress a n d s t a b i l i t y p r o b l e m s r e l a t e d t o a r b i t r a r y , elastic c y -l i n d r i c a -l she-l-ls [1 t o 3 ] . ' M o r e s p e c i f i c a -l -l y , i t presents t h e d e r i v a t i o n of energy expressions a n d t h e correspond-i n g d correspond-i f f e r e n t correspond-i a l equatcorrespond-ions a n d n a t u r a l b o u n d a r y c o n d correspond-i - . t i o n s s u i t a b l e f o r t h e analysis of t h e states of stress a n d d i s t o r t i o n i n a r b i t r a r i l y loaded, u n i f o r m w a l l - t h i c k n e s s c y l i n d e r s . F o r t h e m o s t p a r t , t h e r e l a t i o n s d e r i v e d are c o m p l e t e l y analogous t o those of F l ü g g e [4, 5 ] , a n d reduce t o his corresponding expressions f o r c i r c u l a r c y l i n -ders. I n t h e present d e r i v a t i o n second-order ( b u c k l i n g ) effects are excluded, a n d heirce t h e d e r i v a t i o n is based u p o n t h e s t r a i n - d i s p l a c e m e n t r e l a t i o n s of classical elast i c i elast y elast h e o r y . A d d i elast i o n a l m a elast e r i a l o n a r b i elast r a r y c y l i n -ders p r e s e n t i n g r e l a t i o n s s u i t a b l e f o r t h e analysis of b u c k l i n g a n d p o s t b u c k l i n g b e h a v i o r is g i v e n i n references [2, 3 ] . Basic A s s u m p t i o n s

T h e present w o r k is concerned w i t h a n open or closed c y l i n d r i c a l shell whose cross section is characterized b y

1 T h e results were o b t a i n e d i n t h e course of research j o i n t l y spon-sored b y t h e Office of N a v a l Research a n d t h e B u r e a u of Ships.

2 Professor of A e r o n a u t i c a l E n g i n e e r i n g , P o l y t e c h n i c I n s t i t u t e of B r o o k l y n , B r o o k l y n , N . Y .

' N u m b e r s i n brackets indicate References at end of paper.

t h e plane c u r v e r e s u l t i n g f r o m t h e i n t e r s e c t i o n of t h e m e d i a n surface a n d a plane n o r m a l t o t h e axis of t h e c y l i n d e r . F i g . 1. I t is f u r t h e r assumed t h a t t h e l a t e r a l boundaries of t h e shell are a t distances z = ( / i / 2 ) a n d 2 = — {h/2) f r o m the m e d i a n surface, i n w h i c h z is meas-u r e d f r o m t h e m e d i a n smeas-urface along a n o r m a l t o t h i s surface a n d h is t h e u n i f o r m w a l l t h i c k n e s s of t h e s h e l l ; the c u r v a t u r e ( 1 / r ) of t h e cross section of t h e m e d i a n surface is considered p o s i t i v e w h e n z is d i r e c t e d t o w a r d t h e center of c u r v a t u r e . T h e thickness is assumed t o be e v e r y w h e r e v e r y s m a l l c o m p a r e d t o t h e r a d i u s of cur-v a t u r e , l e n g t h , a n d w i d t h of t h e shell. T h e m a t e r i a l of t h e shell is considered t o be i s o t r o p i c , homogeneous, a n d elastic.

I n t h e analysis t h e l o c a t i o n of a n y p o i n t i n t h e Avail of t h e shell is specified i n t e r m s of a n a x i a l c o - o r d i n a t e .T fixed t o t h e m e d i a n surface, a c i r c u m f e r e n t i a l coo r d i n a t e s w h i c h fcoollcooAVS t h e m e d i a n l i n e coof t h e t r a n s -verse cross section, a n d t h e r a d i a l c o - o r d i n a t e z. T h e r e s u l t i n g o r t h o g o n a l c o - o r d i n a t e s y s t e m is chosen t o be r i g h t - h a n d e d . F i g . 1.

T h e p r i n c i p a l k i n e m a t i c a s s u m p t i o n i n a d e i n t h e f o l -loAving d e v e l o p m e n t is t h a t t h e s t r a i n - d i s p l a c e m e n t rela-t i o n s are rela-those of rela-t h e classical rela-t h e o r y of e l a s rela-t i c i rela-t y , a n d hence t h e change i n g e o m e t r y of t h e shell due t o loads p l a y s no role i n t h e d e v e l o p m e n t of t h e e q u a t i o n s of e q u i l i b r i u m . T h i s a s s u m p t i o n i m p l i e s t h a t t h e strains i n a n d t h e r o t a t i o n s of a n element of t h e shell are of t h e same s m a l l order of m a g n i t u d e c o m p a r e d t o u i f i t y a n d t h a t t h e a x i a l , c i r c u m f e r e n t i a l , and r a d i a l displacements are s m a l l c o m p a r e d t o t h e Avail t h i c k n e s s of t h e sheh. F u r t h e r m o r e , t h e u s u a l K i r c h h o f f - L o v e a s s u m p t i o n s of thin-Avalled shell t h e o r y t h a t n o r m a l s t o t h e m e d i a n

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mm

aenaing Moments in Regular W a v e s

i y Win mitred R. Jacobs^

The longitudinal distribution of forces acting on a T2-SE-A1 tanl<er moving in regular head seas is found by on analysis based on the "strip method" used by Korvin-Kroukovsky in the development of his theory of ship motions, bending moments at the midship section are calculated and compared with bending moments measured during model tests. The theoretical values of bending moments are of the same order of magnitude as those found in the expedments. However, some discrepancies between them appear. These may be due to experimental error as well as to deficiencies in the method of calculation. Certain important facts are brought out by the theoretical calculations: 1) The midsh.p bending moment is virtually a second-order effect dependent on local variations in the longitudinal distribution of the loads. 2) Dynamic bending moments at reasonable speeds are less than those obtained by the conventional static calculation with Smith correchon. This confirms experimental findings.

T H I S paper is a r e p o r t o n a p r o j e c t c a r r i e d o u t a t t h e E x p e r i m e n t a l T o w i n g T a n k a t Stevens I n s t i t u t e of T e c h n o l o g y u n d e r t h e j o i n t sponsorship of P a n e l S-3 of t h e H u l l S t r u c t u r e C o m m i t t e e a n d P a n e l H - 7 o f t h e H y d r o d y n a m i c s C o m m i t t e e of T h e S o c i e t y of N a v a l A r c h i t e c t s a n d M a r i n e Engineers. T h e o b j e c t o f t h e research was t h e a n a l y t i c a l c a l c u l a t i o n of s h i p b e n d i n g m o m e n t s i n r e g u l a r waves. C a l c u l a t i o n of t h e l o n g i t u d i n a l b e n d i n g m o m e n t s m waves is based o n a " s t r i p " m e t h o d of a n a l y z i n g t h e 1 E x p e r i m e n t a l T o w i n g T a n k , ' S t e v e n s I n s t i t u t e - o f Technology, H o b o k e n , N . J.

^ N u m b e r s i n brackets indicate References at end of paper.

forces a c t i n g o n a s h i p i n a r e g u l a r h e a d sea a n d deter-m i n i n g t h e r e s u l t i n g deter-m o t i o n s . T h i s is t h e l i n e a r i z e d m e t h o d developed b y K o r v i n - K r o u k o v s k y [ 1 , 2 ] , ^ as o u t l i n e d i n t h e A p p e n d i x of t h i s paper. B y e m p l o y i n g t h e s t r i p m e t h o d , t h e d i s t r i b u t i o n of b o t h t h e s t a t i c a n d d y n a m i c forces a l o n g t h e l e n g t h of t h e s h i p , i.e., t h e l o a d c u r v e , can be o b t a i n e d a f t e r t h e m o t i o n s h a v e b e e n c a l -c u l a t e d . A d o u b l e i n t e g r a t i o n of t h e l o a d -c u r v e deter-m i n e s t h e shear forces a n d b e n d i n g deter-m o deter-m e n t s . T h e first step t h e n i n c a l c u l a t i n g b e n d i n g m o m e n t s i n r e g u l a r h e a d seas is t o c o m p u t e t h e m o t i o n s i n heave a n d p i t c h f r o m t h e i n t e g r a t e d forces a n d m o m e n t s . T h e c o m p u t e d values of m o t i o n a m p l i t u d e s a n d phase rela-t i o n s h i p s f o r rela-t h e m o d e l discussed i n rela-t h i s paper, rela-t o g e rela-t h e r A, B, C, D, E, G, a, h, c, e, g, ll, coefficients of miscellaneous terms of d i f f e r e n t i a l equa-t i o n s of m o equa-t i o n A = r a t i o of a m p l i t u d e of waves m a d e b y a ship t o a m p l i -t u d e of heaving m o -t i o n B * = beam (local) cw = wave celerity F = h y d r o d y n a m i c h e a v i n g force due t o waves ƒ = t o t a l force due t o m o t i o n s of waves a n d b o d y g = acceleration óf g r a v i t y h = wave a m p l i t u d e J = l o n g i t u d i n a l m o m e n t of i n -e r t i a of a ship i n mass u n i t s Kl, Kl = coefficients of e q u a t i o n (25) of reference [2]

= added mass coefficient i n t w o -dimensional v e r t i c a l flow about a ship section

ki = correction coefficient f o r eft'ect

of f r e e w a t e r surface L = l e n g t h of ship

M = h y d r o d y n a m i c p i t c h i n g m o -m e n t

mass of a ship mass of a ship segment v e r t i c a l ds^mping force per

^ u n i t of b o d y l e n g t h per f o o t per second r = local radius of s e m i c y l i n d r i -cal b o d y S = sectional area ( t o s t i l l W L ) m. Sm

mo

t = t i m e V = ship speed X = l o n g i t u d i n a l co-ordinate w i t h respect t o w a v e n o d a l p o i n t mean d e p t h of a section, area

d i v i d e d b y b e a m heaving displacement angle between l o n g i t u d i n a l t a n g e n t t o b o d y surface a n d s-axis local w a v e - h e i g h t co-ordinate angle of p i t c h wave l e n g t h l o n g i t u d i n a l co-ordinate w i t h respect t o C G water d e n s i t y

phase angles of exciting forces

01 = f r e q u e n c y of w a v e encounter y = z = (3 = > = ê = P =

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es Associateu vvirn

emergence and Ship Slamming in Irregu

By Lee J. Tieft^

series

havP n , n f ? ' ^^^^S"' ^« h i g h l y desirable t o son e s e i r " ' ^ ' ^ ^ P°^^^ble t h e c a l c u l a t i o n of

ome seakeepmg p r o p e r t i e s of a g i v e n design. T o do t h i s I t h e m a t i l ^ ^ r ' T , *r ^ ' ^ " ^ ^ - a l i s t i c \ n d t r a c t a b f e n r f d i n t b

""""^'^ 1 ^

""'^ "^^^^ ^e able t o P i e d i c t t h e response of a ship t o a sea; finally one m u s t

^SkTT'

1 seaworthiness w h i c h are b^otT m ^ ! m a n n l t i o , ^ ' ' ^

'T't"^

^"^"""'^^^ ' ° m a t h e m a t i c a l

The S e a w o y

T h e m a t h e m a t i c a l m o d e l d e r i v e d b y Pierson [8] ^ f o r a

s e n t l l l I observed t o occur. E s -r a d o i ^ n ' "'""^'^u' ^ ^ ^ ^ ^ ' ^ ^ i ^ " ' s t a t i o n a r y , ^ a n d o n i process, s u c h t h a t each r e a l i z a t i o n satisfies t h e MoMVX''und::ko^^^^^^^^^ D - i d T a y l o r

E n g i n e e r m g , l e T y o ^ k t e ^ ^ ^ ^ ^ ^ ^ « « ^ ' ^ ^ ^ °f N u m b e r s m brackets indicate References at end paper.

p o t e n t i a l e q u a t i o n , w i t h i n f i n i t e w a t e r d e p t h , a n d t h e l i n e a r i z e d f r e e surface c o n d i t i o n of h y d r o d y n a m i c s . ^ Some of these t e r m s m a y n o t be f a m i l i a r , hence a b r i e f Z n r t h ° ;

l^'"""^- ""^'^ ^

^-^"^^"^ P ™ - - . t h i s means t h a t there is a c o l l e c t i o n of f u n c t i o n s say X,(t)

o l t T i d " " " " " ^ ' " ^ * ° ^ ^ ^ ' - y f^»^«tion i n t h i collection) or as i t is u s u a l l y t e r m e d , t h e ensemble, a n d each m e m b e r of t h e ensemble is called a r . a f e ^ t ó o » H m t h e r , each r e a l i z a t i o n has a t t a c h e d t o i t a n u m b e r be-t w e e n zero a n d 1, called i be-t s probabilibe-ty.^ P r o b a b i l i be-t y is rene {"t^^'P^'^t^d as t h e r e l a t i v e f r e q u e n c y of o c c u i -b l T ; 1 W t , " ' ' ^ ' ' " ' " f ' *^hen (these m a y -be sums or

1 egiaIs) t h e y are r e f e r r e d t o t h i s ensemble. T h e p r o c

-i^ J A h ^kl^'k'"^ s t a t i o n a r y i f t h e ensemble

IS n o t changed b y t r a n s l a t i o n s i n t i m e , t. A n o t h e r w a y of l o o k i n g a t s t a t i o n a r i t y is t o say, t h a t i f w e take one

ea i z a t i o n a n d s h i f t i t m t i m e , t h e r e w i l l be a n o t h e r l e a h z a t i o n m t h e ensemble w h i c h is i d e n t i c a l w i t h t h i s k ! ^ i ^ i ^ ^ - ' ' ' ' ° ' ^ ^ ° - - discussion of t h e a, b, d e A,B, D E - parameters of d i f f e r e n t i a l equations of m o t i o n = force

= expected n u m b e r o f slams per second = m o m e n t = f u n c t i o n s of parameters, forces a n d moments E = p r o b a b i l i t y averaging opera-tor G F S TJ Re, I m L s ( u ) h, k t V - real a n d i m a g i n a r y parts = distance f r o m center of g r a v -ity t o b o w = spectrum of sea = ship f r e e b o a r d at bow = ship d r a f t at bow = t i m e = ship speed = c r i t i c a l v e r t i c a l v e l o c i t y a t bow f o r s l a m m i n g = distance a l o n g h o r i z o n t a l axis = displacement i n heave w = circular f r e q u e n c y = f r e q u e n c y of encounter 1, Vb = sea-surface elevation 0 = p i t c h angle

P = difference of bow height a n d

sea height at bow

I o-,-,- J = covariance m a t r i x

$ ( s ) = error f u n c t i o n i n t e g r a l

<P = angle between wave a n d keel

line

S(x) = D i r a c d e l t a f u n c t i o n

f = AVeiner r a n d o m process

<r' = variance of a process

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In this paper the motion of a liquid produced by an ellipsoid of revolution under vibration is considered. It is assumed that each point of the ellipsoid performs a linear oscillation at a given common frequency and of small relative amplitude. General expressions for the potential function and the kinetic energy are presented. Assuming that half the ellipsoid is a first approximation of a ship hull, some applications to ship vibrations are then made. The influence of different distributions of the velocity on the ellipsoid, differ-ent positions of the nodes, and differdiffer-ent forms of the vibration-amplitude curve is studied for some cases. A general formula comparing "strip" theory v/ith three-dimensional theory is derived for vertical, pure-shear vibrations. The results are presented graphi-cally as ratios of the kinetic energy for the three-dimensional case to that computed from "stnp" theory, by means of which the added mass of a vibrating ship, computed by the latter theory, can be corrected approximately.

a a,, bi, Ci, di c k t U, V, 10

X, y, z

Ti s n s ri

N

V T f , A;, e p r a./27r p o l a r r a d i u s of ellipsoid coefficients e q u a t o r i a l r a d i u s of ellipsoid f o c a l distance t i m e c o m p o n e n t s of v e l o c i t y C a r t e s i a n co-ordinates coefficients Legendre associated f u n c t i o n s n o r m a l v e c t o r v e l o c i t y v e c t o r k i n e t i c energy of l i q u i d e l l i p s o i d a l co-ordinates p o t e n t i a l f u n c t i o n s d e n s i t y of l i q u i d r a t i o of k i n e t i c energies f r e q u e n c y of v i b r a t i o n s I N t h e present s t a t e of t h e a r t , t h e d i s t r i b u t i o n of t h e a d d e d mass a l o n g t h e l o n g i t u d i n a l axis o f a v i b r a t i n g s h i p is c o m p u t e d b y e m p l o y i n g t h e t w o - d i m e n s i o n a l addedmass coefficient f o r each section a n d t h e n c o r r e c t -i n g t h e ord-inates of t h -i s m a s s - d -i s t r -i b u t -i o n c u r v e b y a c o n s t a n t f a c t o r o b t a i n e d f r o m a n exact p o t e n t i a l - f i o w

1 T h i s w o r k was p e r f o r m e d a t t h e I o w a I n s t i t u t e of H y d r a u l i c Research, State U n i v d r s i t y of I o w a , on behalf of the H - 1 1 F l o w Studies Panel of the H y d r o d y n a m i c s C o m m i t t e e of T H E S O C I E T Y

O F N A V A L A R C H I T E C T S A N D M A R I N E E N G I N E E R S .

2 V i s i t i n g Research Engineer, I o w a I n s t i t u t e of H y d r a u l i c Re-search, State U n i v e r s i t y of I o w a .

I' ^ Research Engineer, I o w a I n s t i t u t e of H y d r a u l i c Research, a n d Professor, D e p a r t m e n t of Mechanics a n d H y d r a u l i c s , State U n i -v e r s i t y of I o w a . c a l c u l a t i o n f o r a v i b r a t i n g ellipsoid of r e v o l u t i o n . T h i s p r o c e d u r e was developed a l m o s t s i m u l t a n e o u s l y b y F . M . L e w i s [1]* ( N o v e m b e r 1929) a n d J . L . T a y l o r [2] ( J a n u -a r y 1930). I n t h e discussion f o l l o w i n g t h e r e -a d i n g of t h e L e w i s paper, T a y l o r p o i n t e d o u t t h a t t h e t y p e of m o t i o n (shearing or flexural) is of some i m p o r t a n c e ; b u t , perhaps because of a t y p o g r a p h i c a l e r r o r i n t h e p r i n t i n g of T a y l o r ' s discussion w h i c h , as L e w i s p o i n t e d o u t i n h i s p r i n t e d r e p l y , i n d i c a t e d t h e p o s s i b i l i t y of n e g a t i v e k i n e t i c energy, T a y l o r ' s c o n t r i b u t i o n appears n o t t o h a v e r e -ceived t h e same a t t e n t i o n as t h a t of L e w i s . A l t h o u g h some use is m a d e of T a y l o r ' s results (see c u r v e i n F i g . 35 of reference [ 1 1 ] ) , a n d t h e i r use has r e c e n t l y been r e c o m -m e n d e d [3] i n several papers [4, 5, 6 ] , t h e L e w i s s o l u t i o n has been a d o p t e d a n d n o significance a t t r i b u t e d t o t h e t y p e of m o t i o n . F u r t h e r m o r e , t h e efiects of t h e p o s i t i o n of t h e nodes a n d of t h e f o r m of t h e a m p l i t u d e c u r v e a l o n g t h e ship's axis have n o t been i n v e s t i g a t e d .

T h e present w o r k g r e w o u t of a n i n i t i a l a t t e m p t t o r e -solve t h e c o n t r o v e r s y i n t h e discussion of L e w i s ' s p a p e r . W h e n i t became clear t h a t t h e results of b o t h L e w i s [ 1 ] a n d T a y l o r [ 2 ] were correct, a n d t h a t t h e d i f f e r e n c e s were due t o t h e p a r t i c u l a r t y p e of v i b r a t i o n t h a t each h a d assumed, i t seemed desirable t o e x t e n d t h e i r s t u d y b y c o n s i d e r i n g t h e efl'ects of o t h e r possible t y p e s of v i b r a -t i o n .

I n t h e present paper, t h e general expression f o r t h e v e l o c i t y p o t e n t i a l f o r a submerged as w e l l as f o r a h a l f -s u b m e r g e d e l l i p -s o i d w i l l be pre-sented a n d -s o l u t i o n -s w i l l be w o r k e d o u t c o m p l e t e l y i n some special cases. T h e general expression f o r t h e k i n e t i c energy also w i l l be de-r i v e d a n d a p p l i e d t o difi'ede-rent cases.

T o show t h e e f f e c t of t h e v e l o c i t y d i s t r i b u t i o n , a q u i t e ^ N u m b e r s i n brackets i n d i c a t e References a t end of paper.

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om

é Wok® Fraction for Potential Fl@wgf

W a k e fraction and tlirust deduction are determined in closed-form expressions for a potential-flow case. The afterbody of a ship hull is represented by a sink distribution and the propeller by two superposed singularity systems. The first of these singularities simulates the uniform part (7i) of the thrust distnbution and the second represents the non-uniform part {Ti). A functional relationship between thrust deduction (f) and wake frac-tion (w) is obtained, which, when compared with the corresponding Dickmann expression, includes a correction term. The general expressions for t and w are reduced to a simpler form for the case of a body of revolution. Numencal results obtained from these reduced forms are in agreement with previous results obtained for the air-ship Akron. Additional computations are made for a hypothetical case where the body singularity distribution has a pronounced asymmetry. This computation demonstrates the important role of the nonuniform thrust distribution on the thrust-deduction fraction and confirms the belief that the nonuniform thrust could be the most decisive factor in the evaluation of the thrust de-duction for the case of a very pronounced asymmetry. Since the hull is represented by discrete sources and sinks of known strength, which is a general method applicable to any form bf a body, and since the thrust-loading vanations, defined by T = Ti + Ta cos u, represents well the important features of the propeller action, therefore, the developed analysis can be considered of general importance and applicable to most practical prob-lems.

K N O W L E D G E of w a k e f r a c t i o n a n d t h r u s t d e d u c t i o n is necessary f o r t h e selection of a ship p r o p e l l e r . B o t h f a c t o r s , k n o w n i h t h e field of n a v a l a r c h i t e c t u r e as " h u l l p r o p e l l e r i n t e r a c t i o n , " h a v e a t t r a c t e d m a n y f a m o u s i n v e s t i g a t o r s . Because of t h e n u m b e r of s i g n i f i c a n t v a r i -ables i n v o l v e d i n t h i s p h e n o m e n o n , a n u m b e r of s i m p l i f i e d a s s u m p t i o n s h a v e t o be e m p l o y e d i n order t o s t u d y i t s general characteristics. T h e m o s t i m p o r t a n t assump-t i o n , i m assump-t i a l l y f o r m u l a assump-t e d b y D i c k m a n n [1],^ is assump-t h a assump-t assump-t h e p o t e n t i a l , w a v e a n d viscous c o n t r i b u t i o n s t o t h e w a k e f r a c t i o n a n d t h r u s t d e d u c t i o n are assumed t o be separate a n d superposable effects, i.e.,

10 = lUp + io„ + w, a n d

i = + i., +

where w a n d t are t h e w a k e f r a c t i o n a n d t h r u s t deduc-t i o n s , respecdeduc-tively, a n d subscripdeduc-ts p , lo, a n d ƒ represendeduc-t t h e p o t e n t i a l w a v e a n d viscous c o n t r i b u t i o n s .

1 T h i s paper is t a k e n f r o m t h e s t u d y " P r o p e l l e r H u l l I n t e r a c t i o n , " conducted at the E x p e r i m e n t a l T o w i n g T a n k , Stevens I n -s t i t u t e of Technology, under Office of N a v a l Re-search C o n t r a c t N o . N6onr-24705, sponsored b y the B u r e a u of Ships N o . NS715-102 and technically administered b y the D a v i d T a y l o r M o d e l B a s u i .

2 E x p e r i n i e i i t a l T o w i f i g T a n k , Stevens I n s t i t u t e of Technology, H o b o k e n , N . J.

' N u m b e r s i n brackets indicate References a t end of paper.

T h e present w o r k deals w i t h t h e e v a l u a t i o n of w a k e . f r a c t i o n a n d t h r u s t d e d u c t i o n i n a p o t e n t i a l - f i o w case.

I n ' a n e f f o r t t o d e t e r m i n e t h e v e l o c i t y field i n t h e h u l l -p r o -p e l l e r i n t e r a c t i o n m o s t i n v e s t i g a t o r s d r a w f h e i r con-clusions m o r e or less o n t h e same basic assumptions, T h e y d i f f e r i n t h e i r a t t e m p t t o s i m u l a t e t h e flow a r o u n d t h e h u l l a n d p r o p e h e r b y v a r i o u s t y p e s of s i n g u l a r i t y systems. T h e s i m p l e s t m e t h o d of representing a p o t e n t i a l flow a b o u t a r i g i d b o d y is t o assume a c e r t a i n s i n g u l a r i t y d i s t r i b u t i o n (sources, sinks or d o u b l e t s ) . I ^ o r v i n I C r o u k o v -s k y [ 2 ] , who-se analy-si-s i-s u-sed a-s t h e f r a m e w o r k of t h e present i n v e s t i g a t i o n , represented t h e h u l l b y source d i s t r i b u t i o n s over t h e f o r e b o d y l e n g t h a n d b y s i n k dis-t r i b u dis-t i o n s over dis-t h e a f dis-t e r b o d y l e n g dis-t h . T h e m e dis-t h o d of e v a l u a t i n g t h e s t r e n g t h of t h i s s i n g u l a r i t y s y s t e m is based o n t h e f a c t t h a t t h e fluid-velocity v e c t o r is t a n g e n t t o t h e surface of t h e h u l l . A general expression f o r t h e e v a l u a t i o n of t h e s t r e n g t h of a s i n g u l a r i t y s y s t e m , w i t h a n a p p l i c a t i o n t o t h e A'^ictory ship, w i l l be presented s h o r t l y i n a subsequent r e p o r t [ 3 ] .

A ship p r o p e l l e r can be represented [ 4 , 5 ] b y a s i n g u -l a r i t y s y s t e m w h i c h s i m u -l a t e s t h e t h r u s t d i s t r i b u t i o n over t h e d i s k area a c c o r d i n g t o t h e f o l l o w i n g r e l a t i o n -s h i p :

T = T l 4- Ta cos CO