Laboratory Studies of Breaking Wave Forces Acting on Vertical Cylinders in Shallow Water

8/6/2019 Laboratory Studies of Breaking Wave Forces Acting on Vertical Cylinders in Shallow Water

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Coastal Engineering,11 (1987) 263-28 2 263E l s e v ie r S c i en c e P u b l i sh e r s B .V. , A m s t e r d a m - - P r i n t e d i n T h e N e t h e r l a n d s

L a b o r a t o r y S t u d i e s o f B r e a k i n g W a v e F o r c e sA c t i n g o n Ve r t ic a l C y l in d e r s in S h a l l o w W a t e r

C .J . A P E LT a n d J. P I O R E W I C Z

Depa rtment of Civil Engineering, University o f Queensland, St. Lucia, Q ld. 4067, Australia

(Rece ived Jun e 16, 1986; rev ised and acce pted Feb rua ry 11 , 1987)

A B S T R A C T

A p e l t , C . J . a n d P i o r e w i c z , J ., 1 9 8 7 . L a b o r a t o r y s t u d i e s o f b r e a k i n g w a v e f o r c e s c t i n g o n v e r t i c a l

c y l i n d e r s n s h a l l o w w a t e r . C o a s t a l n g . , 1 1 : 2 6 3 - 2 3 2 .

I n s o m e l o c a t i o n s , h e b e s t w a y t o c r e a t e a d e e p w a t e r p o r t i s t o c o n s t r u c t o n o f f - s h o r e b e r t h

w h i c h i s j o i n e d t o t h e l a n d b y a j e t t y . S u c h j e t t i es r e u s u a l l y s u p p o r t e d o n p i l e s a n d t h e s e a r e

o f t e n s u b j e c t e d t o b r e a k i n g w a v e s .

D i m e n s i o n a l a n a l y s i s n d i c a t e s h a t f o r s m o o t h c i r c u l a r y l i n d e r s t h e t o t a l a v e f o r c e s n d t h e

m o m e n t s f r o m d e p t h l i m i t e d w a v e s m a y b e c a l c u l a t e d a s f u n c t i o n s o f p a r a m e t e r s w h i c h , e x c e p t

f o r R e , c a n b e m e a s u r e d d ir ec tl y [ , M = f ( d / L o , H o l L o , D / H o , S , R e )] . S i n c e t h e m R x i m u m f o r c ei s e x p e r i e n c e d a t a d e f i n i t e o c a t i o n , h e p a r a m e t e r d / L oi s i m p l i e d i n t h e p a r a m e t e r s S a n d H o l L o .T h e i n v e s t i g a t i o n s e p o r t e d i n t h i s p a p e r w e r e b a s e d o n h y d r a u l i c m o d e l t e s t s i t h r e g u l a r w a v e s

a n d p r e s e n t t h e m a g n i t u d e s o f h e r e l a t i v e r e a k i n g w a v e f o r c e a n d m o m e n t i n t h e b r e a k i n g w a v e

z o n e a s f u n c t i o n s o f t h e s e p a r a m e t e r s .

F r o m t h e r e s u l t s f e x p e r i m e n t s w i t h t w o c y l i n d e r d i a m e t e r s , D - - 0 . 1 0 2 a n d 0 . 1 5 3 m , c a r r i e d

o u t i n t h r e e d i f f e r e n t a v e c h a n n e l s , e m p i r i c a l f o r m u l a e f o r f o r c e a n d m o m e n t h a v e b e e n e s t a b -

l i s h e d w i t h t h e d e e p w a t e r w a v e s t e e p n e s s , H o l L o ,a n d t h e r e l a t i v e c y l i n d e r d i a m e t e r , D / H o a si n d e p e n d e n t v a r ia b l e s . h e s e a p p l y t o s p e c i f i c o t t o m s l o p e o f S = 1 : 1 5 . T h e e x p e r i m e n t a l r a n g e

c o m p a r e s t o d r a g d o m i n a t e d c o n d i t i o n a n d a R e y n o l d s n u m b e r r a n g e 0 . 6 1 0< R e < 2 . 7 1 0w i t h i n w h i c h t h e d r a g c o ef f i c i e n t s a l m o s t c o n s t a n t .

I N T R O D U C T I O N

The widespread use of pile-supported coastal and offshore structures makesthe inte ract ion of waves on piles in the break ing zone of significant practicalimportance. The basic problem is to predict forces on a pile due to the wave-associated flow field. Wave induced flows are complex, even in the absence of

a s tructure, and predict ion of wave forces on a pile must make use o f empiricalcoefficients to a ugmen t the theoretical formulation of the problem. The mostwidely used approach to the calculation of wave forces on a rigid body is basedon the as sumption t hat the wave force can be expressed as the sum of a drag

0378-3839/87/$03 .50 1987 Else v ie r Sc ience Pub l i she rs B.V.


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f o rc e a n d a n i n e r t i a f o rc e . M o r i s o n w a s t h e f i r s t t o s u g g e s t t h e m a t h e m a t i c a lf o r m f o r t h i s a p p r o a c h . T h e r e is e x t e n s i v e l i t e r a t u r e ( H o g b e n e t a l., 19 7 7) o nw a v e f o r c e s e x p e r i e n c e d b y v e r t ic a l c i r c u l a r c y l i n d e r s w h i c h p i e r c e t h e f r e e

s u rf a ce a n d a r e i n t h e r e g io n o f n o n b r e a k i n g w a v e s . H o w e v e r, t o t h e a u t h o r s 'k n o w l e d g e , o n l y a f e w p u b l i c a t i o n s d e a l w i t h b r e a k i n g w a v e f o rc e s o n a v e r t ic a lcy l ind er (Ha l l , 1958; God a e t al ., 1966; H on d a an d M i t suyas u , 1974 ; Ap e l t an dBaddi ley, 1981 ; Wiege l , 1982 ; Och i and Chen Han Tsa i , 1984 ; Sawarag i andN o c h i n o , 19 84 ; K j e l d s e n a n d A k r e , 1 9 8 5 ) . T h i s p a p e r p r e s e n t s t h e r e s u l t s o fe x p e r i m e n t s c a r r ie d o u t w i t h c y l i n d e r s s i t u a t e d i n t h e b r e a k i n g z o n e a n d i nt h e o f f sh o r e z o n e c l o se t o t h e b r e a k i n g p o i n t . T h e e x p e r i m e n t s w e r e c a r r i e do u t f or t w o c y l i n d e r d i a m e t e r s a n d i n t h r e e d i f fe r e n t w a v e c h a n n e l s.

A P P R O A C H T O T H E F O R C E D E S C R I P T I O N

T h e w a v e f o r c e e x p e r i e n c e d b y a v e r t i c a l c y l i n d e r i s e s s e n t i a l l y o r i z o n t a l

a n d i t c a n b e c a l c u l a t e d f r o m t h e g e n e r a l f o r m o f M o r i s o n ' s e q u a t i o n :

D 2 du[: P CD D ulu] + p ~ - -~-C i - -~ (1)

w h e r e f is t h e f o r c e p e r u n i t l e n g t h o n a c y l i n d e r o f d i a m e t e r D , u i s t h e h o r i -z o n t a l c o m p o n e n t o f t h e w a t e r p a r t i c le v e l o ci ty a n d # i s t h e d e n s i t y o f w a t e r.

S e v e r al s tu d i e s h a v e s h o w n t h a t t h e m a t h e m a t i c a l f o r m o f t h i s e q u a t i o n i ss a t is f a c t o ry, b u t a p a r t i c u l a r d i f fi c u l ty i n u s i n g i t i n t h e d e s i g n o f o f fs h o r es t r u c t u r e s h a s b e e n t h e c h o i c e f r o m a w i d e r a n g e o f p u b l i s h e d v a l u e s o f t h eem pi r i ca l d rag an d in e r t i a coeff i c ien t s , CD, C~ app rop r ia te to bo th the s t ruc tu rea n d i t s d e s i g n s e a s t a te .

T h e r e i s a w i d e s c a t t e r i n p u b l i s h e d v a l u e s o f CD a n d C I a n d c o n s e q u e n t l yt h e a v e r a g e v a lu e s s u g g e s t e d b y C E R C ( 1 97 7 ) a r e s e e n as t h e b e s t e s t i m a t e sa v a i la b l e in a s i t u a t i o n o f c o n s i d e r a b l e u n c e r t a i n t y.

C E R C ( 19 7 7 ) i n it s r e c o m m e n d a t i o n fo r c a l c u l a t i o n o f t o t a l ( d e p t h i n t e -g r a t e d ) f o rc e s d u e t o d e p t h l i m i t e d b r e a k i n g w a v e s re f e r s t o H a l l ' s e x p e r i m e n t sa n d p r e s e n t s f o r m u l a e fo r t h e t o t a l m a x i m u m h o r i z o n t a l f o rc e F a n d m o m e n t ,M , as:

F ~ I . 5 p g D H~ (2 )

M~_FHb (3 )

w h e r e H b i s t h e b r e a k i n g w a v e h e i g h t .F r o m t h i s i t i s r o p o s e d t h a t M o r i s o n ' s f o r m u l a c o u l d b e u s e d f o r c a l c u l a t i o n

o f b r e a k i n g w a v e f o r c e s u s i n g t h e d r a g c o e f f i c i e n t ( C D ) b f o r b r e a k i n g c o n d i -t io ns , ( D ) 2 . 5 C D . I t s h o u l d b e n o t e d t h a t e q n . ( ) w a s d e r i v e d f r o m e x p e r -

i m e n t s o n b e d s l o p e 1 :1 0 .


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Fig. 1. Regionsof methods of b~aking wave forcecalculation (m~;erW iegel, (1982): 1=region ofM orison's formula;2 = region of impact force.

W i e g e l ( 1 98 2 ) p r o p o s e d t h e f o ll o w i n g a p p r o a c h f o r c a l c u l a t io n o f t h e f o r ced u e t o b r e a k i n g w a v e s o n a v e r t i c a l c i r c u l a r c y l i n d e r ( F i g . 1 ) :

( a ) c a l c u la t e t h e h o r i z o n t a l c o m p o n e n t o f t h e s lo w l y v a r y i n g w a v e - i n d u c e d

f o rc e u s i n g t h e M o r i s o n e q u a t i o n ( 1 ) o v e r t h e d e p t h :

O 0 .5 f o r p l u n g i n g b r ea k e r s . T h e r e c e n t i n v e s t i g a t i o n so f S a w a ra g i a n d N o c h i n o ( 19 8 4) o n t h e i m p a c t f or ce s s h o w t h a t t h e m a x i m u mc u r l i n g f a c to r, 2, c a n b e a s h i g h a s 0 . 9 fo r p l u n g i n g b r e a k e r s a t a d i s t a n c e 0 . 0 6L s h o r e w a r d s f r o m t h e b r e a k i n g p o i n t , w h e r e L i s t h e w a v e l e n g t h a t t h e d e p t hi n f r o n t o f t h e c y l i n d e r. T h e c u r l i n g f a c t o r, A, v a r i e s s i g n i f i c a n t l y w i t h b o t hb r e a k e r i n d e x a n d l o c a ti o n a n d t h e a p p r o a c h s u m m a r i s e d a b o v e h a s n o t p er -m i t t e d a n y g e n e r a l f o r m u l a t i o n t o b e d e v e l o p e d f o r p r ac t i c a l u s e i n d e s ig n .

N o n e o f t h e s t u d i e s e x a m i n e d t h e q u e s t i o n o f s ca le e ff e ct s .T h e w a v e c r e s t e l e v a t i o n , ~ /b, b a s e d o n W i e g e l 's f i e l d d a t a r a n g e s f r o m 0 .5 5t o 0 .9 5 H b. S e e li g a n d A h r e n s ( 19 8 3 ) p r e s e n t m o r e d e ta i l s a n d f o r m u l a t e a neq ua t io n to ca lcu la t e th e ~/b va lue .


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T h e C E R C r e c o m m e n d a t i o n f o r t h e i m p a c t f o r c e p e r u n i t l e n g t h o f p i l e n e a rt h e b r e a k e r c r e s t i s s i m i l a r o e q n . ( 4 ) :

/s = 0 . 5 ( C D ) bp g D H b ~ l . 2 5 CD p g D H b (5 )

w here CD var ies w i th Re yno lds N um be r (0 .7 _< CD < 1 .2 ) .T h i s t h e o r e t i c a l l y c o r r e c t a p p r o a c h c a n l e a d t o d i f f e r e n t re s u l t s i n p a r t i c u l a r

a p p l ic a t io n s b e c a u s e o f u n c e r t a i n t i e s i n e s t i m a t i o n o f t h e l e n g t h o f c y l in d e rw h i c h i s a ff e c t e d b y i m p a c t f o rc e s a n d b e c a u s e o f d if f ic u l ti e s i n e s t i m a t i o n o ft h e v e r t i c a l d i s t r ib u t i o n o f v e l o c i t y a n d a c c e l e r a t i o n i n t h e b r e a k i n g w a v e .

H o n d a a n d M i t s u y a s a ( 19 7 4) c o n s i d e r e d t h e i n f lu e n c e o f t h e d e e p w a t e rw a v e s t e e p n e ssHo/Loa s w e l l a s b e a c h s lo p e , S , o n t h e m a x i m u m v a l u e o f t h eb r e a k i n g w a v e f o r ce . T h e y p r e s e n t e d t h e f o ll o w i n g e m p i r i c a l e q u a t i o n s :

[ F / ( p g D Ho2) ] ma~ =0 .1 8(Ho/Lo)-o .s5 for S = 1:15 (6)

[ F / ( p g D H2o) ]~ x =0 .30 (Ho/Lo ) -0.55fo r S = I : 1 0 ( 7)

T h e w a t e r d e p th , d , w h e r e t h e m a x i m u m f o rc e o c cu r s c a n b e f o u n d f r o m t h ere la t ion :

d/Ho = ( 0 . 9 2 - 5 . 3 S ) ( H o / L o ) - 0 . 2 5 ( 8 )

A l l r e la t io n s p r e s e n t e d a b o v e a r e b a s e d o n e x p e r i m e n t a l w o r k a n d a p p l y tos i t u a t i o n s w h e r e d r a g f o r c es d o m i n a t e i n e r t i a l fo r ce s . T h e y i l lu s t r a t e t h e n e e df o r p r a c t i c a l f o r m u l a e f o r u s e i n d e s i g n . S u c h f o r m u l a e s h o u l d t a k e a c c o u n t o fa l l r e l e v a n t p a r a m e t e r s . S i n c e t h e r e i s n o t h e o r e t i c a l s o l u t io n a v a i l ab l e , d i m e n -s i o n a l a n a l y s i s m u s t b e u s e d t o i n d i c a t e t h e f u n c t i o n a l r e la t i o n s b e t w e e n f o r ce sa n d m o m e n t s a n d r e l e v a n t p a r a m e t e r s . S u c h a n a l y s is i n d i c a te s t h a t f o r a v er -t i c a l s m o o t h c i r c u l a r c y l i n d e r in d e p t h - l i m i t e d w a v e s :

FF = = f( d/Lo, Ho/Lo, D/Ho, S, Re) (9 )

pgDH2o

a n dM

M - p g D H2o d -f( d /L o , Ho/Lo, D/Ho, S, Re) (10)

F is d e s i g n a t e d a s " r e l a t i v e " f o rc e a n d M a s " r e l a t iv e " m o m e n t . E q u a t i o n s ( 9 )a n d ( 1 0) i n d i c a t e t h a t , i n a d d i t i o n t o t h e r e l a t i v e d e p t h ,d/Lo,d e e p w a t e r w a v es t e e p n e s s , Ho/Lo,a n d R e y n o l d s N u m b e r ,Re, w h i c h i n v o l v e h y d r o d y n a m i cc o e ff i c i e n t s , t h e b o t t o m s l o p e , S , a n d t h e r e l a t i v e c y l i n d e r d i a m e t e r,D/Ho,s h o u l d be c o n s i d e r e d as i n d e p e n d e n d p a r a m e t e r s .D/Hois a K e u l e g a n - C a r p e n -

t e r ( K C ) n u m b e r t y p e o f p a r a m e t e r. (K C - u ~ ,,T / D ,w h e r e u ~ is t h e h o r i-z o n t al m a x . p a r ti c le v e l o c it y a n d T t h e w a v e p e r i o d ) . W i t h i n a r a n g e o fRew h e r e CD i s c o n s t a n t t h e u s e o f e q n s . ( 9 ) a n d ( 10 ) a l lo w s t h e w a v e f o rc e a n dm o m e n t t o b e c a l c u l a t e d f r o m p a r a m e t e r s w h i c h c a n b e m e a s u r e d d i r e c t l y


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H/d

0.8

0.7

0.6

0..

04

0.3

0~,

01

H, L = wave parametersbreaking d : wafer depth

waves ~ wave continuity

/ I \ solitary

2 z, 6 8 10 12 1/* 16 18 20 L ..~d

F i g . 2 . A r e a s o f w a v e t h e o r y v a l i d i t y ( a f t e r D r u e t , 1 9 6 8 ) .

w i t h o u t e x p l i c it c a l c u l a t i o n o f fl u id v e l o c i t y a n d a c c e l e r a t i o n a s i s n e c e s s a r y i fM o r i s o n ' s f o r m u l a is u se d .

WAVE CONDITIONS

Wave theory

Wa v e c o n d i t i o n s f o r r e g u l a r w a v e s a r e c h a r a c t e r i s e d b y t h e r e l a t i v e w a v eh e i g h t ,H id a n d r e l a t iv e l e n g t h ,Lid. T h e v a l i d i t y o f v a r i o u s w a v e t h e o r i e s i sl i m i t e d t o d i f f e r e n t r a n g e s o f t h e s e p a r a m e t e r s a s s h o w n i n F i g . 2 , a d o p t e d f r o mD r u e t ( 19 6 8 ). T h i s i n d i c a t e s t h a t l in e a r t h e o r y c a n n o t b e u s e d i n s h a ll o w w a t e rc l o s e t o t h e b r e a k i n g z o n e . I n t h i s z o n e o t h e r t h e o r i e s , s u c h a s f o r e x a m p l eh i g h e r- o r d e r s t r e a m f u n c t i o n t h e o r y, s h o u l d b e u s e d . N e v e r t h e l e s s , s in u s o i d a lt h e o r y i s f re q u e n t l y u s e d b e y o n d t h e r a n g e o f i ts v a l i d i t y b e c a u s e o f it s

s impl ic i ty.

Wave transformation

W h e n t h e l oc a l w a v e p a r a m e t e r s a r e t o b e r e la t e d to t h e d e e p w a t e r w a v eparam ete r s , Ho,Lo an d bo t to m topography , a t r ans fo rm at ion p rocess i s r equ i red .P r e v i o u s e x p e r i m e n t s h a v e i n d i c a t e d t h a t t r a n s f o r m a t i o n b y s i n u s o i d a l w a v et h e o r y i s n o t v a l i d f o r r e l a ti v e d e p t hd/Lo< 0 .1 . In tha t case , cno ida l wa ve the -o r y i n c l u d i n g t h e i n f lu e n c e o f w a v e s t ee p n e s s ,HolLo,gives m ore rea l i s ti c r e su l t sf o r t r a n s f o r m a t i o n . I w a g a k i e t a l. ( 19 8 2 ) h a v e d e v e l o p e d s i m p l e a p p r o x i m a t ee x p r e s s i o n s w h i c h r e p l a c e t h e r e l a t i v e l y c o m p l e x c a l c u l a t i o n s a s s o c i a t e d w i t hexa c t cno ida l theo ry, a s fo llows :

H/Ho :K s =K~o+ 0 . 0 0 1 5 (d/Lo) -2.8 (Ho /Lo)l.2 (11)


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wh ere K ,o is t h e shoa l i ng coe ff i c ien t g iven by :

2k dK , = ( I ~ s i n - h 2 k d ) t a n h k d

i n w h i c h k is t h e w a v e n u m b e r= 2~ z/L.

(12)

Breaking waves

T h e b r e a k i n g p o i n t is a n i n t e r m e d i a t e p o i n t i n t h e b r e a k i n g p r oc e s s b e t w e e nthe f i rs t s t age o f i n s t ab i l it y and t he a r ea o f comp le t e b r eak ing . The re fo re , t hedep th t ha t i nd i ca t e s b r eak ing d i r ec t ly aga in s t a s t r uc tu r e i s a c tua l l y some d i s-t a n c e s e a w a rd s o f t h e s t ru c t u r e a n d n o t n e c e ss a ri ly th e d e p t h a t t h e t o e o f t h es t ruc tu r e . N o the o ry i s ava il ab le t o de sc r ibe t he p roces s o f wave -b reak ing on as lo p i n g b o t t o m n o r t h e i n i t ia t i o n o f b r e a k in g . H o w e v e r, a w a v e t h e o r y w h i c hi s app l i cab l e in t he r eg ion j u s t be fo re b r eak ing can be c om bined w i th an em p i r-i ca l b r e a k i n g c r i t er i o n to d e t e r m i n e t h e w a v e p a r a m e t e r s a t b r e a k in g . T h eem pi r i ca l f o rmu la o f l e M ~hau te a nd K oh (1967 ) was u sed i n ou r ana ly s is s incei t f it s we l l w i th ou r ex pe r im en ta l cond i t i ons . Th e wave he igh t a t b r eak ing , Hb,i s r e la t ed t o t h e b each s lope , S , and deep Wate r wave s t eep nes s by;

Hb/Ho= 0 . 7 6 S 1/v (H olL o)-1/4 (13)

w i th va l i d i ty i n t he r anges:

0.002


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269

X p ~ - X ~ = ( 4 - 9 . 2 5 S ) H b (16)

O t h e r e x p e r i m e n t s ( e.g ., N g u y e n , 1 98 3; S i n g a m s e t t i a n d W i n d , 1 9 8 0 ) m e a -

s u r e d p l u n g i n g d i s ta n c e s w h i c h w e r e m u c h l ar g e r t h a n t h o s e g i v en b y e q n .{ 1 6 ), s o m e t i m e s b y a f a c t o r o f 2.7 , w i t h w i d e s c a t t e r o f r e s u l ts .

EXPERIMENTS

A im of inves tiga tion

P r e v i o u s st u d i es r e p o r t e d by A p e l t a n d B a d d i l e y ( 19 8 1) h a d s h o w n t h a t t h ef o r c e s a n d m o m e n t s e x p e r i e n c e d b y a c y l i n d e r i n b r e a k i n g w a v e s a r e m u c h

l a rg e r t h a n h a d b e e n p r e d i c t e d b y t h e e x i s t i n g f o r m u l a e , T h e b r e a k i n g p r o c e ssi t s e lf i s c o n s i d e r a b l y v a r ia b l e f o r t h e s a m e i n c i d e n t c o n d i t i o n s a n d t h e f o rc e sa n d m o m e n t s g e n e r a t e d in b r e a k i n g w a v e s a r e v ar ia b le . T h e e x p e r i m e n t sd e s c r ib e d h e r e w e r e c a rr i e d o u t t o d e t e r m i n e t h e s t a ti s t ic a l n a t u r e o f t h i s v a r i-a t i o n a n d t o e x t e n d t h e r a n g e o f p a r a m e t e r s c o v e r e d b y t h e e a r l ie r e x p e r i m e n t s .

Exper imen ta l cond i t ions

T h e c y l i n d e r d i a m e t e r , D , w a s l i m i t e d to b e < 0.2 0 L. T h e c y l i n d e r p i e r c e d

t h e w a t e r s u r f ac e . T h e b e a c h s l o p e w a s m o o t h , u n i f o r m a n d i m p e r m e a b l e . R e g -u l a r i n c i d e n t w a v e s w e r e u s e d i n a ll c as es . T h e r a n g e s o f p a r a m e t e r s c o v e r e di n t h e e x p e r i m e n t s a re :c y l i n d e r d i a m e t e r, D = 1 0 2 a n d 1 53 r a m ;b o t t o m s lo p e , S = 1 :15 ;d e p t h a t t o e o f s lo p e ,d = 3 50 a n d 7 00 m m ;w a v e p e r i o d ,T = 0.8 to 2 .5 s ;d e e p w a t e r w a v e s t ee p n e s s ,Ho/Lo= 0.01, 0.02, 0.03, 0.05;R e y n o l d s N u m b e rR e = 0.6 to 1.5 105 for D = 102 m m

a n d R e = 1 .1 to 2 .7 105 fo r D = 153 mm ,w h e r e R e is c a l c u l a te d w i t h t h e h o r i z o n t a l w a v e v e l o c it y n e a r t h e s ti ll w a t e rlevel .

T h e m o d e l

T h e m o d e l cy l in d e r s w e r e m a d e f r o m s m o o t h c i rc u la r a l u m i n i u m t u b e. T h e yw e r e a t t a c h e d t o t w o u n b o n d e d fo r ce t r a n d u c e r s ( E t h e r L t d t y p e U F 2 w i t hr a n g e + 9 0 N ) w h i c h w e r e f i x e d t o a v e r ti c a l s t ee l ar m , s u p p o r t e d b y a h e a v y

s t ee l c h a n n e l f r a m e w h i c h s p a n n e d a c ro s s t h e w a v e c h a n n e l ( F ig . 3 ) . T h ef r a m e c o u l d b e m o v e d a lo n g t h e w a v e c h a n n e l a n d f i x e d a t s e l e c t e d p o i n t s . E n de ff ec ts a t t h e b o t t o m o f th e c y l in d e r w e re e l i m i n a t e d w i t h o u t c a u s i n g a n yr e s t r i ct i o n o n t h e m o v e m e n t o f t h e c y l in d e r. To a c h ie v e t h i s a t h i n d i sc o f t h e


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270

s f e e ( a r m F [ s t e e [ c h a n ne l

cynde~ f r a m eJ w a v e g e n e r a f o r , w a v e g a u g e s . ._ . .~ ~ [ ~ " ~ " [ I I I

Fig. 3. Diagram of m odelset-up.

s a m e d i a m e t e r a s t h e c y l i n d e r w a s f i x ed to t h e b o t t o m o f t h e v e r t i c al s u p p o r ta r m a n d t h e s m a l l g a p b e t w e e n t h e d i s c a n d t h e c y l i n d e r w a s se a l ed b y a t h i n

m e m b r a n e o f r u b b er. O n e f o rc e t r a n s d u c e r w a s l o c a t e d a b o v e S W L a n d t h eo t h e r 3 5 m m a b o v e t h e b o t t o m .B r e a k i n g w a v e fo r c es a r e i m p u l s i v e i n n a t u r e a n d t h e c o m b i n a t i o n o f s m a l l

m a s s a n d l a rg e s t i f fn e s s i n t h e f o r c e m e a s u r i n g s y s t e m i s e s s e n t i a l i f i t is t or e s p o n d a c c u r a t e l y t o t h e r a p i d l y i n c r e a s i n g w a v e f o rc e. Ty p i c a l r i se t i m e s t ot h e m a x i m u m f o rc e in b r e a k i n g w a v e s a r e o f o r d e r 1 0 m s ( s ee F ig . 5 ) . V e r yh i g h f r e q u e n c y re s p o n s e i s r e q u i r e d i n t h e m e a s u r i n g s y s t e m i n o r d e r t o m e a -s u r e t h e f o r c e s w i t h s u f f ic i e n t a c cu r ac y. T h e s y s t e m d e s c r ib e d a b o v e h a s t w od e g r ee s o f fr e e d o m . T h e n a t u r a l f r e q u e n c i e s o f v i b r a t i o n m e a s u r e d w i t h t h e

c y l i n d e r in a t y p i c a l st il l w a t e r d e p t h w e r e a p p r o x i m a t e l y 1 0 0 H z i n t h e t r a n s -l a ti o n m o d e a n d 2 0 H z i n t h e r o t a t i o n m o d e .T h e i m p u l s i v e b r e a k i n g w a v e f o r ce is a p p l i e d w i t h a f r e q u e n c y c o r r e s p o n d -

i n g t o t h e w a v e p e ri o d . T h i s f r e q u e n c y r a n g e d f r o m 0 .4 t o 1 .2 H z i n t h e s e t e s t sa n d t h e r e w a s n o d a n g e r o f r e s o n a n c e d e v e l o p i n g b e t w e e n t h e p e r i o d i c f or c ea n d t h e r e s p o n s e o f t h e f o r c e m e a s u r i n g s y s te m . R a t h e r , t h e q u e s t i o n is h o ww e ll t h e r e s p o n s e o f t h e f o rc e m e a s u r i n g s y s t e m r e p r e s e n t s t h e a p p l i e d im p u l s ea n d t o w h a t e x t e n t t h e i n e r t i a a n d f le x i bi li ty o f t h e s y s t e m m o d i f i e s t h er e s p o n s e . T h e m o d e l s u s e d i n t h e s e e x p e r i m e n t s w e r e s im i l a r i n c h a r a c t e ri s t ic s

t o t h o s e u s e d b y A p e lt a n d B a d d i l e y ( 1 98 1 ). T h e y f o r m u l a t e d a m a t h e m a t i c a lm o d e l o f t h e s y s t e m t o d e t e r m i n e t h e r e l a ti o n s h i p b e tw e e n t h e m a x i m u mi m p u l s i v e fo r ce a p p l i e d t o t h e c y l i n d e r a n d t h e i n d i c a t e d f o rc e, a s g i v en b y t h er e s p o n s e o f t h e f o rc e t ra n s d u c e r s , a n d i n t e g r a t e d t h e d i f f e re n t i a l e q u a t i o n sn u m e r i c a l ly. T h e b e s t e s t i m a t e o f r is e t i m e s f r o m t h e e x p e r i m e n t s is 1 0 m s b u ti t ap p e a r s t h a t i t c a n v a r y f r o m 8 to 12 m s . W h e n t h e r is e t i m e o f t h e i m p u l s ef u n c t i o n w a s v a r ie d o v e r t h i s r a n g e t h e m a t h e m a t i c a l m o d e l s h o w e d t h a t t h er a t i o o f p e a k r e s p o n s e t o p e a k a p p l i e d f o r ce v a r i e d f r o m 1 .1 9 t o 0 . 9 1 in t h e c a s eo f a p a r a b o l i c s h a p e o f i m p u l s e a n d f r o m 1 .2 1 t o 0 .8 8 i n t h e c a s e o f a t r i a n g u l a r

s h a p e o f i m p u ls e . T h e s e v a lu e s w e r e o b t a in e d w i t h n o d a m p i n g s i n ce o t h e rc a lc u l at io n s s h o w e d t h a t t h e s m a l l d a m p i n g p r e s e n t r e d u c e d t h e p e a k r e s p o n seb y a b o u t o n e p e r c e n t . T h i s a n a l y s i s h a s n o t b e e n d o n e f o r t h e c u r r e n t e x p er -


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TABLE 1

Wave channe ls used for experiments

271

Channel Length Widt h Depth Reflection coefficient(m) (m) (m) (%)

A 15 3.00 0.5 3.7 to 6.7B 26 0.90 0.6 5.9 to 13.2C 50 2.13 3.9 6.0 to 14.6

iments since the sy stem used has a tran sfer fu nction essentially similar to tha tused by Apelt and Baddiley.

In view of the u ncer tain ty concerning the exact shape and rise time of theimpulse function in each specific wave impact, it was decided that there waslittle to be gained by adjusting the recorded signal from the force measuringsystem for dy namic response effects. Consequently, the results present ed inthis report are the u nadjust ed values. It is noted tha t the results of Hond a andMitsu yasu (1974) are also unadjusted. The un cer tai nty in the case of the cur-ren t results is of order 15%.

The wave channels

The experiments were carried out in three different wave channels withdimens ions given in Table 1. To reduce reflection, the upper par t of the slopeswas covered with gravel absorbers. The measured reflection coefficients aregiven in Table 1.

Experimental procedure

For the experim ents in channels A and B the magnitude of forces and waveheights were measured and recorded at 1000 Hz frequency by a data acquisitionsystem based on a P DP 11/34 computer. In the case of channe l C wave heightswere recorded on a pen recorder and forces on a recording oscillograph. Theinitial wave height was measur ed in the cons tant d epth (d = 350 mm in chan-nels A and B; d= 700 mm in channel C ). For each deep water wave steepness,HolLo, five diff erent wave heights were selected in th e range of D/Ho = 0.5 to2.5. The force measur ing system was calibrated in place at the beginning andend of each set of experiments by means of hanging weights and pulleys. For

each test condition, the wave force on the cylinder was measu red at a series ofpositions beginning offshore from the breaking point and moving progressivelyshorewards through the zone of breaking waves. At each position, the forceproduced by up to 100 successive waves was recorded. From statistical analysis,


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272

the magnitudes of forces with the same probability of exceedance wereestimated.

Error in experiments

Systematic error was avoided by using two differ ent methods of recording:one with data acquisition done by computer and th e other with pen and oscil-lograph recorders, and by repeating the experiments in diff erent channels. Sincethe results obtained in all cases were similar, sys tematic error can be neglected.

To analyse the r ando m error th e following accuracy of measu rem ent wasobtained: wave height and d ept h +_ 0.5 mm; wave period + 0.01 s; force 0.5N( as a result of noise in transmissio n line to the comp uter ). T hus the deep water

wave height, Ho, and relative force, F, were calculated with max imu m probableerror of order of 5%. In the case of estimation of the relative force with 1%probability of exceedance, the stand ard deviation of the p lotted da ta poin tsabout the best fit probability distribution, a


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27 3

6 . 0

HolLo = 0.015.0 O/H o = 2 .0 .0 ~ o exp 112A

~ o i d a l t h e o ry ~ e x p 11 2 B

~. 3 .0 ~ , \ / . . xp 21/*B

2.0 o o '

1.0

J = I

0 2 '0 3 0 5 0

6"01 Ho/Lo = 0.03 )50 I ] := O /Ho = 2 .0

I~ ~ , ~ ~ o xp132A

.0 I , / . . . . . . . . . . z~ exp l :33Bl e o . x p 2 3

u.. 3.0 o~ l\ f l / " e x p 2 3 5 C

20 o ~1.0

O 0 J '0 .0 05 10 15 Z0 25 30 3 .5 0 /~5 5 .0

6 0HolLo=0.02

5 0 / cn o id a l f h e ~ D / H o = 2 . 0~.o % ~ / / ~ o ~ p lZ ~ A

a exp 122B3 .0 ~ p / / / ~ exp 225R= .

,~ e~p 22/,,E2.0 x ~ / o

x

/ ~~ " ~ ' ~ ' ' ~ , ~ - - ~ -1 . 0 . . . . . . . . . . . . . . . . . . . .

' ; J ' ' ' ' ' d , . o/H oO.u.0.. , 05 1 !5 2.0 0 3.5 /*0 5 5 .0

5.0 /mada l ~ o exp 151A~,.0 ,~ ex p 152B

e x p 2538t , . x exp 255{:

2,0

1.0

d /H o 0 .0 ' ' J ~ ~ ~ ' d / H oO 0 05 1.0 15 2.0 25 3.0 3.5 0 45 5.0

Fig. 4. W ave force vs. relative dep th. R esults of calculations and measurements.

t e n c y, t h e i r v a l u e s s h o u l d b e th o s e d e r iv e d f r o m t h e w a v e t h e o r y w i t h w h i c ht h e y a r e b e i n g u s e d . U n f o r t u n a t e l y, h o w e v e r, th e r e a r e n o g e n e r a l ly a p p l i c a b l ev a l u e s o f C D, C I d e r i v e d f o r u s e w i t h c n o i d a l t h e o r y a n d t h e v a l u e s u s e d i n a l l

o f t h e s e c a l c u l a t io n s a r e t h o s e w h i c h h a v e b e e n d e v e l o p e d f o r s i n u s o i d a l w a v et h e or y. T h i s c o u l d e x p l a i n s o m e o f th e d i f f e re n c e s b e t w e e n t h e r e s u lt s o b t a i n e df or w a v e f o r c e s u s i n g c n o i d a l a n d s i n u s o i d a l t h e o r ie s w i t h t h e s a m e v a l u e s o fCD an d CI.

W ave forces in the breaking zone

N o t h e o r y i s a v a il a b le f o r p r e d i c t in g w a v e f o r ce s i n t h e b r e a k i n g z o n e a n d

o n l y e m p i r ic a l f o r m u l a e c a n b e u s ed . F o r t h e s e e x p e r i m e n t s , t h e R e y n o l d sN u m b e r i n t h e b r e a k i n g z o n e , c a l c u la t e d w i t h t h e v e l o c i ty u = v / g d b , w a s i nt h e ra n g e ( 0 .6 t o 1 .7 ) X 1 0 5 fo r D = 1 0 2 m m a n d ( 1.2 to 2 .7 ) 1 0 b f or D = 1 5 3m m .

T h e o b j e c ti v e in t h e s e e x p e r i m e n t s w a s t o d e t e r m i n e t h e p e a k v a l u e o f t h eb r e a k i n g w a v e f o rc e . U p t o 1 0 0 s u c c e s s i v e p e a k w a v e f o r c e s w e r e r e c o rd e d i ne a c h e x p e r i m e n t .

To i ll u s tr a t e t h e d i f fe r e n c e s b e t w e e n n o n - b r e a k i n g a n d b r e a k i n g w a v e f o rc e st h e s a m p l e r e co r d s a t o n e w a v e p e ri od , T, r e co r d ed w i t h s a m p l i n g f r e q u e n c y

1 0 0 0 H z a r e p r e s e n t e d i n F ig . 5 a n d t h e i r c h a r a c t e r i s ti c s a r e s h o w n i n Ta b l e 2 .A s c a n b e s e e n f r o m Ta b l e 2 , i n e a c h e x p e r i m e n t t h e p e a k m a g n i t u d e o f t h ef o r ce m e a s u r e d a t t h e l o c a t i o n w h e r e t h e g r e a t e s t f o r c e w a s e x p e r i e n c e d v a r i e dc o n s i d e r a b ly f r o m o n e w a v e t o th e n e x t . T h i s t y p i c a l s t o c h a s t i c v a r i a ti o n


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2 7 4

FT N

1/*

12

10

8

61

(a )

2

0

-2

-4 .

- 6 , z , i ,

8 10 12f x l O O , m s

FT,N

1/+

12

10

8

6

2

0

-2

1/*

(b )

Jf x l O O , m s

I L o = 0 , 0 1= 0.066m

' ~ L , '

~ , N

-1

- 3

-5

FT,N

(a l

5

3

-1

-3

' ' ' 1 ' . ' 1 ' 6 - s

'l" x lO0 ,ms

(b )

Ho / L o = 0 ,0 5

, i I J i I

I ; 1 2 I ~ 1 6

t"xlOO,ms

F i g . 5 , R e c o r d o f t h e t o t a l a v e f o r c e f f s h o r e f r o m t h e b r e a k i n g z o n e a n d i n t h e b r e a k i n g z o n e .

r e q u i r e d u s e o f s ta t i s t ic al n a l y s i s o d e t e r m i n e w h e t h e r t h e r e s u l t s f e x p e r i -m e n t s c o u l d b e c o m b i n e d i n t o s o m e " u n i v e r s a l " p o p u l a t i o n s .

T h e s t a n d a r d d e v i a t i o n o f t h e u n i v e r s a l p o p u l a t i o n i s d e n o t e d a o a n d t h a to f t h e r e s u l t s f o n e e x p e r i m e n t i s a . I t w a s a s s u m e d t h a t t h e s t a n d a r d d e v i a -t i o n s o f t h e r e s u l t s o f e a c h e x p e r i m e n t , a , w e r e d i s t r i b u t e d a b o u t ~ o a s t h e

m e a n , w i t h s t a n d a r d d e v i a t i o n e q u a l t o ~ o / ~ / ~ - n , w h e r e n i s t h e n u m b e r o fs a m p l e s i n t h e e x p e r i m e n t . T h e n , a p p l y i n g t h e t w o - t a i l e st , h e r e s u l t s f e a c he x p e r i m e n t c a n b e t a k e n t o b e l o n g t o t h e u n i v e r s a l p o p u l a t i o n w i t h s t a n d a r d


13/20

TABLE 2

Variation of measured forces

275

Distance from he Peak orce n [N] among 100 ExampleWave parameters point of successive waves Fig. 5m a x i m u m force F~,

T Ho go /Lo Fm i, Fme~ Fm~,, [N][s] [m]

Remarks

2.11 0.066 0.01 0 12.00 13.72 18.10 15.68 Plunging10 Ho 4.27 4.57 5.09 4.66 breaker

0.94 0.066 0.05 0 4.14 5.64 7.99 6.62 Spilling5 Ho 1.87 2.18 2.68 2.15 breaker

deviation ao with a conf idence interval 0.95, i.e. 95% probability, provided asatisfies th e inequalities:

ao ( 1 - 1 . 9 6 / x / ~ ) < a < a o ( 1 + 1 . 9 6 / x / ~ ) (17)

Use of this te st indicated whethe r the results from different experiments couldbe grouped into a single distribution. It also made it possible to determine

whet her d ifferent facilities or cylinder diameters have significant influence onthe results.The results of all exper iments with the same wave steepness were assessed

by this test. T he stand ard deviation of the results of each experime nt was com-pared with those of all other ex periment s with the same wave steepness andthe experimen ts were grouped into populations on this basis. The results ineach population so obtained were the n plotted on probability paper in or der todeterm ine the magnitudes of breakin g wave forces and mome nts with 1% prob-ability of exceedance. Thi s is illustrated in Fig. 6 which con tains log-normal

probability plots for H o l L o = 0.01 in which, from a total 19 experiment s, 12 aredistributed along the line (a) , 5 along the line (b) and 2 along the line (c)indep endent ly from the cylinder diamete r and wave channel. Ther e is no cor-relation between membership of one of the populations a nd the cylinder diam-eter or the wave channel in which the ex perime nt was carried out. A similarresult was obtained for each of the othe r values of wave steepness investigated.The authors have been unable to suggest any explanation for the fo rmation ofthe three different distributions.

To express mathemat ically the relationship between values measured in

experimen ts with diffe rent para mete rs a p olynomial regression analysis wascarried out. The experimental results so obtained are compared with theempirical formulae proposed by other authors.


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2 7 6

I l l "~II I |E- -' '! ~ ! ! ' ~ F ~ = ' ' ' F i ! !~ ! ! ' '~ ' ~ i ' ~ r ~ i ~ ! ~ F ! ~ - ' ! '~ ! ! ~ ! ! ! ~ | ! ! !E | E ~ | ~ ! ~ | | | ~ | | ~ ; ~ ! ~ | | ~' = ~ ' - ' - | ~ | ~ | ~ | . ' | ~ - - | - ~ - ' ~ . ' ~ . ' ~ ' ~ ' : ' ~ : ~ . ' ~ . ' ' ~ . ' . ' | ~ ' ~ - ' | ' - ~ ' ~ | ~ ' . . ~ | ~ - ~ ' . ~ ' ~

" ~ ' ' - ~ | ~ F ' | u ~ i ~ i ~ ~ i ~ ~ M ~ u ~l l l l l l l l l l l l l l l l I I I l l l l l l l l l l l l l n l l I I I I I I I I I I I I I I l l l l n l l l l l I l l l l l U l l l l l l l l l l l l I n I l l l ~ l l , , . I ] i i ~ =~ ; i l l l l l l l l l~ u ~ ~ ~ n ~ n ~ q ' ' ~ ' ~ ` ' ~ " ~ ' r ~ I L ,a l l l= l l l l l l l l l l l l l l

~ ii;i i~ iil;i i i~ ; ; i i ; i l; i; ; i; ; ;; ;i;iil;iiii i iiiii;ii~ ;;i; i;ii; i~ ilil~ -.=- ' ! i ii ii = = . i =~ , ' , i i i id l i - -= ~-= .,-"==iiii~ii:iii= , ' - -=i i i=,~, , i~, i i i i - - ' ii i ii , l ii i i ii i i ii i i ii -=, i i=-E =~ i, i i ii i i ii ' - - - i - ---~, i ; i ; -" . - i

J ~

~- ~0 ~ ! iy i l l i iiiiiiiiiiiiiiiiiiiiiiiiiiiii!iiiiiiiiiiiiiiiiiiii~ : , , ~ i i iI- - ~ ,II ' ,IIII ' , ' , l ' , l l llll ' , ' ,I ' , I ', ', ',II',II', ', ',IIIIIIII ' ,I I I ' I I' , ', I I I I ', I I IIIII ', ' , ', I I I II ,"~II ' , I~L,~ -~--~ '~- ,l l l l l l l I l l I I I I ' , II ', I I ', I 1

1 .0 . ~ I . . l ' , i l l l l i l ] l I I l ] l l l l l l l l I

-=~ 0.60 .5

0 .0 1 0 1 0 5 1 . 0 2 . 0 5 . 0 1 0 2 0 3 0 ~ 0 6 0 8 0 9 0 9 5 9 8 9 9 9 9 .9 9 9 . 9 9C U M U L AT I V E P E R C E N T A I3 E [ % ]

F i g . 6 . L o g - n o r m a l p r o b a b i l i t y l o t f o r H o / L o= 0 . 0 1 .

~o~_o ~ . O o~ .0 - 1 Q 2 m m3 . 0 ~ ~ ' O " 1 5 3 m m

u . .- 2 . 0 I - P o o ~ , ~ , _ ~ o

1 . o f - ~ = " '~I

0 % is ~ o 1 's ~ o ~ _ s ~ o " 5 .0

o i o O : o.3 .0

, , ': zo o - oI . 0h o ~ -

~ o L = 1 i L D / H o0 . 0 0 5 1 . 0 1 5 2 . 0 2 5 3 . 05 . 0 I - I J ~ - 0 . 0 2 o 1

1oo o . , , , , ,

" v . u 0 5 ! .0 1 5 Z 0 2 .5 0 D I H oI '1 ~ _- 0 . 0 1 , ~ , , _ , , ~ . _ _ _

~.0 o ~ " ~ o o

~ 3 . 0u Z 2 . 0

1 .0I t h L L O / H

0 . 0 0 . 5 I. 0 1 . 5 2 . 0 Z 5 3 . 0

F i g . 7 . R e l a t i v e b r e a k i n g w a v e f o r ce . e s u l t s o f e x p e r i m e n t s .


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Force

The results fo r force shown in Fig. 7 show th at the relative force is not con-

stant for a fixed wave steepness, Ho/Lo, but that it varies with the relativediameter of cylinder, D/Ho. The best-fit equations which describe the magni-tude of forces with 1% probability of exceedance can be expressed as:

Ho/Lo =0.01 FI~ = 0.846D 2 + 3.462Dr +0.610 (18)

HolLo =0.02 F ~ = - 1.086D 2 + 4.031Dr +0.360 (19)

Ho/Lo =0.0 3 F ~ = 0.903D 2 + 2.914D, +0.269 (20)

Ho/Lo=O.05 (21)I~ = 0.265D~ + 1.175D~ + 0.650

where Dr = D/HoThe maxim um relative forces, F ~ , obtained from these experiments are shownin Fig. 9 as funct ions of HolLo together with the data obtained by Honda a ndMits uyas hu (1974), Hall (1958) a nd Goda et al. (1966). The data from all ofthese earlier studies are the maxi mum observed values of F and these are not,in general, the same as F ~ . Nevertheless, it is reasonable to compare overallresults and trends. As can be seen, the beach slope and.cylinder diameter influ-ence the magn itude of force. For slope 1:15 and D / H o < 1.5 the following rela-tion is proposed on the basis o f the results of our experiments:

F~ =0.41 (D/H o) '5 (Ho/Lo) - ' ~ (22)

M o m e n t s

The analyses of moments was carried out in the same manner as that offorce and produced best-fit equations describing the magnitu de of momentswith 1% probabi lity o f exceedance as follows (Fig. 8 ):

HolLo =0.01 M ~ = - 0.388D 2 + 1.967Dr +2.901 (23)

HolLo =0.02 MI~ = - 1.521D~ + 5.979Dr - 1.121 (24)

HolLo =0.03 MI ~ = - 1.207D 2 + 4.290Dr -0. 134 (25)

Ho/Lo =0.05 MI ~ =0.134D~ + 1.030Dr + 1.087 (26)

The m axim um relative moments, MI~, obtained from the experiments, are

shown in Fig. 9 as functio ns of HolLo. For slope 1:15 and D / H o < 1.5 they canbe approx imated by the expression:

MI~ =0.56 (D /H o) '~ (H o/Lo)-0.45 (27)


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2 7 8

3.0 V2.0

t o lO0 I

0.0

5.0

&O t02.01.00.0

0.0

==~ 3 0 -

2 . 0 -

0 00 0

~0.05 1102~rn

' D/Ho0.5 1~0 t5 2.0 2.5 3.0

o ~ - H olLo 0.03 o oo" D : 102ram~. D = 15]ram

0.5 1.0 1.5 2.0 Z5O/Ho

3.0

~o o HolLo= 0.02O

o D =102n~nO=153n~n I

~ ~ ~ ~ O/Ho05 10 15 Z0 2.5 3.0

7o 16.0 - z~ z~

S O L ~ ~ - ' ~ ' ~ ' -~ . . 0 t o ~ ~ ' ~ - ~

E 3 .0 - - Ho/L~0.01Z0 ~- o O=102nTn1.0 - z~ O:153mm

0.0 L ~ ~ L , L D/Ho

0.0 0.5 1.0 15 2.0 2.5 3.0F i g . 8 . R e l a t i v eb r e a k i n g w a v e m o m e n t . R es ul ts of e x p e r i m e n t s .

10 k

6

3 - I : ~ - - t - -qn.~

0 . 8

0 . 60 . 5

0.00z+ 0 0 0 6 0 01 0 002 003 O0& 0.06 0.10 H/L

10 ~ _ _ . . . .

= o s - - . ~

F i g . 9 . R e l a t i v e b r e a k i n g w a v e f o r c e a n d m o m e n t ( co m p a _ H s o n w i t h t h e r e s u lt s o f t h e o t h e r a u t h o r s ) .


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C r i t i c a ld e p t h

279

Ho/Lo C r i t i c a l d e p t h

0.01 1.22 0.080.02 1.22 0.110.03 1.00 0.090.05 1.00 0.12

I n a ll c a se s t h e l in e o f a c t io n o f t h e m a x i m u m f o rc e e x p e r i e n c e d b y t h e c y l i n d e ri s a b o v e t h e S W L , a t a n a v e r a g e l ev e l e q u a l t o 1 .3 7d . T h i s i s t h e m e a n v a l u ef o r a ll e x p e r i m e n t s .

Critical water d epth

T h e d i m e n s i o n l e s s c ri ti c a l w a t e r d e p t h ,dcr/Ho,a t t h e c y l i n d e r w h e r e t h ew a v e fo r ce h a s t h e m a x i m u m v a l u e w a s d e t e r m i n e d d u r i n g th e e x p e r i m e n t s .

T h e r e s u l t s a r e p r e s e n t e d i n Ta b l e 3 . T h e a v e r a g e c r i t i c a l d e p t h ,d J H o , iss h o w n i n F ig . 10 t o g e t h e r w i t h th e r e s u lt s o f H o n d a a n d M i t s u y a s u ( e q n . 8 ) .T h e r e la t iv e b r e a k i n g d e p t h ,db/Ho,i s t a k e n f r o m e q n s . ( 1 3) a n d ( 1 4 ) . T h er e s u l t s i n d i c a t e th a t , w i t h i n t h e s t a n d a r d d e v i a t i o n a = 0 .1 0, t h e c r i t i c a l d e p t his s h o r e w a r d s f r o m t h e b r e a k i n g p o i n t a n d c a n b e e x p r e s s e d f or t h e b o t t o mslop e 1:15 as:

dr =db -O .2 Ho =db[ 1 - 0 . 2 4 ( Ho /Lo ) '12] (28)

T h i s i n d i c a t e d th a t t h e m a x i m u m f or ce o cc u r s w i t h i n t h e d i s ta n c e d e f i n e d b y

G a l v i n a s t h e p l u n g e d i s t a n c e .

1 0

4 .

3

10 . 7 -

0 .50 . 0 0 4 Q O 0 6Q I O 0 . 0 2 ( 1 0 3 Q 0 4 0 . 0 6 (lq 6 1 0

Fig . 10 . Cr i t i c a l dep th ,dJHo, a t t h e c y l i n d e r w h e r e m a x i m u m b r e a k i n g w a v e f o r ce o c c u r re d d u r-i n g e x p e r i m e n t s .


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2 8 0

Scale effect in experim ents

F o r g r a v i t y w a v e p h e n o m e n a w i t h n e g l ig i b le v i s c o u s e f fe c ts , t h e F r o u d e L a w

of s im i la r i ty app l i es . In th e abse nce o f sca le e ffec t, the l inea r sca le ra t io , Lr, i sr e l a t e d t o t h e f o r c e r a t io b y :

L~ = ( Fp/Fm )1/3 (29)

Ho we ver, i f sca le e ffec t i s p res en t :

Fp =F ro L~ C~ (30)

w h e r e Fm d e n o t e s t h e m e a s u r e d m o d e l f or ce , F p is t h e p r o t o t y p e f o rc e a n d C ,i s t h e s c a le e f fe c t c o e f fi c ie n t . T h e t w o c y l i n d e r s u s e d i n t h e e x p e r i m e n t s g i v e

t h e g e o m e t r i c a l s c a le , L r = 1 .5 a n d t h e m e a s u r e d f o r c e f o r b o t h c y l i n d e r s s h o u l dbe in re la t io n 1 :3 .375 in the abse nce o f sca le e ffect . Cons id e r ing the re la t iveforce , the sca le e ffec t coeff i c ien t ca n be exp ressed as :

Fp Fp (31)C ~ = Fro L ~ - F ~

Ta k i n g F p a s t h e r e l a ti v e f o rc e m e a s u r e d f o r c y l i n d e r d i a m e t e r D = 0 .1 5 3 m a n dF m a s r e l a ti v e f o r c e f o r c y l i n d e r d i a m e t e r D = 0 .1 0 2 m a s h o r t a n a l y s i s b a s e do n m e a s u r e m e n t s o f F m a x i m u m h a s b e e n c a r r i e d o u t . F o r s i m i l ar v a lu e s o f

D / H o t h e r e la t iv e f o r ce s h a v e b e e n c o m p a r e d f o r d i ff e r e n t c h a n n e l s a n d w a v es teepness .

T h i r t y - t h r e e c o m p a r a b l e c a s e s w e r e f o u n d a n d , f o r t h e s e , t h e a v e r a g e s c a lee f fe c t c o e f f ic i e n t C ~ - - 1 . 0 w i t h s t a n d a r d d e v i a t i o n a - - 0 . 1 3 . T h i s r e s u l t i n d i-c a t e s t h a t t h e r e w a s n o s i g n i f i c a n t sc a l e e f fe c t b e t w e e n t h e t e s t s w i t h t h e d if -f e r e n t s iz e s o f c y l i n d e r, w h i c h i s t o b e e x p e c t e d b e c a u s e CD is a l m o s t c o n s t a n tw i t h i n t h e t e s t e d r a n g e o fR e.

DISCUSSION AND CONCLUSIONS

C o m p a r i s o n b e t w e e n t h e r e s u lt s o f o u r e x p e r i m e n t s a n d t h o s e o f e a r li e rs t u d ie s s h o w s t h a t t h e m a x i m u m b r e a k i n g w a v e f o rc e o n a v e r t ic a l c y li n d e r isi n f l u e n c e d b y t h e b o t t o m s lo p e , S , a n d b y t h e r e l a ti v e c y l i n d e r d i a m e t e r, D / H o ,a s w e l l a s b y t h e w a v e s t e e p n e s s ,HolLo. S i n c e t h e m a x i m u m f o rc e is e x p e r i-e n c e d a t a d e f i n it e l o c a ti o n , t h e p a r a m e t e rd/Lo i s e ffec t ive ly imp l ied in th ep a r a m e t e r s S a n dHo/Lo.F r o m F i g. 9 i t c a n b e s e e n t h a t o u r r e s u l t s f o r m a x i -m u m b r e a k i n g w a v e f o rc e a re g e n e r a ll y i n g o o d a g r e e m e n t w i t h t h o s e o b t a i n e dby Ho nd a and M i t suya su (1974) fo r the sa m e cond i t ions ( S = 1 :15, D/H o = 0 .5 ) .

H o w e v e r, t h o s e a u t h o r s d i d n o t i n c l u d eD / H o a s a s i g n i f ic a n t p a r a m e t e r a n di ts i m p o r t a n c e h a s b e e n d e m o n s t r a t e d b y t h e r e s u lt s o f o u r e x p e r im e n t s . T h ei n f l u e n c e o f S i s d e m o n s t r a t e d b y th e r e s u l t s o f H a l l ( 19 5 8 ) o n w h i c h t h eC E R C f o r m u l a , e q n . ( 2 ) , i s b a s e d ; t h e m a x i m u m f o rc e fo r S = 1 :1 0 is s u b s t a u -


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281

t i a l ly g r e a t e r t h a n t h a t f o r S = 1 :1 5, al l e l s e b e i n g t h e s a m e . O n t h e o t h e r h a n d ,t h e m a g n i t u d e o f t h e m o m e n t r e c o m m e n d e d b y C E R C , e q n . ( 3 ) , a p p e a rs t o b et o o s m a l l w h e n c o m p a r e d w i t h o u r r e s u l t s si n c e H b in e q n . ( 3 ) i s a l w a y s s m a l l e r

t h a n 1 .3 7 dc r, w h i c h i s t h e a v e r a g e v a l u e o f t h e e l e v a t i o n o f t h e l i n e o f a c t i o no f t h e f o rc e o b t a i n e d i n o u r e x p e r i m e n t s .

T h e i m p a c t f o r ce c o m p o n e n t o f t h e t o t a l b r e a k i n g w a v e f or ce , w h i c h m u s tb e c o n s i d e r e d i n c o n n e c t i o n w i t h t h e f a ti g u e li fe o f t h e s t r u c t u r e , c a n o n l y b ed e t e r m i n e d a c c u r a t e ly f r o m t h e e m p i r i c a l d a t a a t p r e s e n t . T h e m e t h o d f or ca l-c u l a t i n g i t w h i c h h a s b e e n p r o p o s e d b y Wi e g e l ( 1 98 2 ) r e q u i re s t h e m a g n i t u d eo f t h e c u r l i n g fa c to r , ~, a n d t h i s i s n o t k n o w n a c c u r a t e l y y e t.

W i t h t h e p r e s e n t s t a t e o f k n o w l e d g e , t h e b r e a k i n g w a v e fo r ce c a n b e d e t er -m i n e d o n l y f r o m e x p e r i m e n t . T h e e m p i r ic a l f o rm u l a e p r e s e n t e d a b o v e w h i c h

h a v e b e e n d e r i v e d f r o m t h e a u t h o r s ' e x p e r i m e n t s c a n b e u s e d f o r p r e d i c t i n gt h e b r e a k i n g w a v e fo r ce i n c o n d i t i o n s s i m i l a r t o t h o s e o f t h e e x p e r i m e n t s .H o w e v e r, t h e r e i s i n s u f f i c i e n t b a s i s f o r g e n e r a l i s i n g t h e r e s u l t s t o a p p l y t o c o n -d i t i o n s w h i c h d i f f e r s u b s t a n t i a l l y f r o m t h o s e o f t h e e x p e r i m e n t s . I t s h o u l d b en o t e d , i n p a r t ic u l a r , t h a t t h e e x p e r i m e n t s w e r e al l i n t h e s u b - c ri t ic a l o r t r a n -s i t i o n a l r e g i m e s f o rR e < 3 X 105 . La rg e - sca le cy l in de r s in r ea l s ea s w i l l be int h e s u p e r - c ri t ic a l r e g i m e w i t hR e > > 5 X 105 an d th e r e su l t s o f thes e ex per i -m e n t s c a n n o t b e e x t r a p o l a t e d t o s u c h c o n d i t i o n s w i t h o u t f u r t h e r i n v es ti g a-t i o n . I t i s v e r y d e si r ab l e t h a t e x p e r i m e n t a l s t u d i e s b e c ar r i e d o u t o n b r e a k i n g

wav e fo rces on fu l l - s ize cy l ind e r s in r ea l s eas .T h e f o l l o w i n g c o n c l u s i o n s c a n b e d r a w n f r o m t h e p r e s e n t e x p e r i m e n t s :( 1 ) T h e r e la t iv e f o rc es , F, a n d m o m e n t s , M , i n b r e a k i n g w a v e s c a n b e

e x p r e s s e d a s f u n c t i o n s o f t h e t w o p a r a m e t e r s , d e e p w a t e r w a v e s t e e p n e s s , H o/ Lo ,a n d r e la t iv e c y l i n d e r d i a m e t e r ,D / H o . G e n e r a l e x p r e s s i o n s h a v e b e e n d e v e l-o p e d w h i c h a ll ow t h e f o rc es a n d m o m e n t s e x p e c t e d in b r e a k i n g w a v es o n ab o t t o m s l o p e o f 1:1 5, w i t h p r o b a b i l i t y o f e x c e e d a n c e o f 1% , t o b e p r e d i c t e d[ e q n s . ( 2 2 ) a n d ( 2 7 ) ] f o r s m o o t h c y li n d e r s , w i t hR e < 2.7 X 105 an dD /Ho < 1 .5 .

( 2 ) I n t h e s e e x p e r i m e n t s t h e l a rg e s t b r e a k i n g w a v e fo r ce w a s f o u n d t o o c c u r

0 .2 0 H o s h o r e w a r d f r o m t h e b r e a k p o i n t .( 3 ) N o s i g n i f i c a n t s c a le e ff e c t w a s d e t e c t e d i n t h e s e e x p e r i m e n t s w h i c h c o v e ra r a n g e o f R e y n o l d s N u m b e r f r o m 0 .6 X 105 t o 2 .7 X 105 .

( 4 ) E x p e r i m e n t s o n f u l l s iz e cy l i n d e r s i n r ea l s e as a re n e e d e d to d e t e r m i n eb r e a k i n g w a v e f o r c e a n d m o m e n t c o e ff i ci e n ts in t h e s u p e r c r i t ic a l r e g i m eo f R e .

ACKNOWLEDGEMENT

T h i s w o r k w a s c a rr i e d o u t w i t h t h e s u p p o rt o f f u n d s f r o m t h e M a r i n e S ci -e n c e s a n d Te c h n ol o g i e s G r a n t s S c h e m e . T h i s s u p p o rt is g ra te fu ll ya c k n o w l e d g e d .


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REFERENCES

Apelt, C.J. and Baddiley, P., 1981. Breaking wave forces on vertical cylinders. Fifth AustralianConference on Coastal and Ocean Engineering, Perth, pp. 85-89.

Apelt, C.J. and Piorewicz, J., 1986. Breaking wave forces on vertical cylinders. Univ. Qld., CivilEng. Dept., Research Report CE71.

Chue, S.H., 1983. A Reanalysis of Nearshore Phenomena. Sixth Australian Conference on Coastaland Ocean Engineering, Gold Coast, pp. 257-263.

Druet, C., 1968. A practical method for the determination of short-period waves in hydraulicstructure foundation areas. Houille Blanche, 23 (8): 703-710.

Galvin, C.J.J., 1969. Breaker travel and choice of design wave height, ASCE J. Waterw. HarboursDiv., WW2 (96): 175-200.

Goda, Y., Haranaka, S. and Kitahata, M, 1966. Study on impulsive breaking wave forces on piles.Rept. Port and Harbour Res Inst., Vol. 6, No. 5, pp. 1-30 (in Japanese).

Hall, M.A., 1958. Laboratory study of breaking wave force on piles. Beach Erosion Board Tech.Memo, No. 106.Hogben, N., Miller, B.L., Searle, J.W. and Ward, G., 1977. Estimation of fluid loading on off-shore

structures. Proc. Inst. Civ. Eng., Part 2, 63: 515-562.Honda, T. and Mitsuyasu, H., 1974. Experimental study of breaking wave force on a vertical

cylinder. Coastal Eng. Jpn., 17: 59-70.Iwagaki, Y., Shiota, K. and Doi, H., 1982. Shoaling and refraction coefficient of finite amplitude

waves. Coastal Eng. Jpn., 25: 25-35.Kjeldsen, S.P. and Akre, A.B., 1985. Wave forces on vertical piles near the free surface caused by

2-dimensional and 3-dimensional breaking waves. MARINTEK Report No. 1.10, Trondheim,Norway.

Le M~haute, B. and Koh, R.C.Y., 1967. On the breaking waves arriving a t an angle to the shore.J. Hydraul. Res., 5 (1): 67-88.Nguyen, T.T., 1983. An evaluation of breaking wave design data. Sixth Australian Conference on

Coastal and Ocean Engineering, Gold Coast, pp. 74-79.Ochi, M.K. and Tsai Chen-Han, 1984. Prediction of Impact Pressure Induced by Breaking Waves

on Vertical Cylinders in Random Seas., Appl. Ocean Res., 6(3): 157-165.Sawaragi, T. and Nochino, M., 1984. Impact forces of nearly breaking waves on a vertical cylinder.

Coastal Eng. Jpn., 27: 249-263.Seelig, W.N. and Ahrens, J.P., 1983. The elevation and durat ion of wind crests. U.S. Army Corps

of Eng. CERC, Fort Belvoir, Misc. Rep. No. 83-1, 73 pp.Singamsetti, S.R. and Wind, H.G., 1980. Breaking wave-characteristics of shoaling and breaking

periodic waves normally incident to plane beaches of constant slope. Delft Hydraulics Lab.,Rep. M1371.

U.S. Army, Coastal Engineering Research Centre, 1977. Shore Protection Manual.Wiegel, R.L., 1982. Forces induced by breakers on piles. Proc. 18th Conf. Coastal Eng., Cape Town,

pp. 1699-1715.

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Laboratory Studies of Breaking Wave Forces Acting on Vertical Cylinders in Shallow Water

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