GROHTH OF LONGSHORE CURRENTS DOlVNSTREAN OF A SURF-ZONE BARRIER
Peter S. Eagleson Professor of Civil Engineering
Massachusetts Institute of Technology, Cambridge, Nassachuset ts
ABSTRACT
Momentum flux considerations are used to formulate a differen-tial equation governing the growth, with distance, of the mean longshore current velocity in the surf-zone on a plane, impermeable beach due to monochromatic \.,aves. The equation is solved for the flOl., situation dOlm-stream of a surf-zone barrier and is shOlm to compare favorably \.,i th laboratory measurements.
The asymptotic (uniform flow) form of the relation is also. shOlm to be in good agreement with the field and laboratory data of other 1nves-tigators.
Conclusions are reached governing the size of laboratory models necessary to represent conditions of fully developed longshore currents.
INTRODUCTION
A thorough understanding of the mechanics of sand transportation parallel to coasts must ultimately be based upon detailed knOlvledge of :he complicated fluid motions Iolhich occur \.,ithin the surf zone. This work 1S devoted to the formulation and evaluation of a rational analytical model for the prediction of mean longshore current velocity as a functi-on of beach and \.,ave parameters.
THEORETICAL DEVELOPMENT
~lOMENTUM EQUATION
The momentum equation lVill be written for the stationarY surf-zone control volume illustrated in Figure 1. This control volurne lVas first used in the analysis of longshore currents by Putnam, Hunk a.nd Traylor (1949), and the follolVing development represents a refinement and extension of their pioneering study. The control volume is bound ed by) (1) the still-Hater surface ABED, (2) the plane beach surface CBEF, (3 a vertical plane ADFC parallel to the shoreline just offshore of
t~e
breaker line, and (4) parallel vertical planes ABC and DEF perpend1cular to the shoreline and a distance 6x apart.The longshore (i.e., x) component of the instantaneous rooment~m equation for the fluid system occupying the control volume at t :L-rne , t, 1S
F sx
+
Iff
BxPdV(1)
C.v. c.s. c.v.
/
BREAKE R LIN Ei
o I-H (— 1 o (a ) PLA N VIE W (b ) SID E ELEVATIO N Figur e 1 . Contro l Volum e fo r Momentu m Analysi si n w h i c h :
F = l o n g s h o r e component o f s u r f a c e f o r c e s sx
= l o n g s h o r e component o f body f o r c e s p e r u n i t mass p = mass d e n s i t y o f f l u i d dV = volume e l e m e n t ->-V = f l u i d v e l o c i t y w i t h r e s p e c t t o c o n t r o l volume = l o n g s h o r e component o f f l u i d v e l o c i t y dA = a r e a element r = u n i t o u t w a r d n o r m a l t o c o n t r o l s u r f a c e t = t i m e C V . s i g n i f i e s i n t e g r a t i o n t o be p e r f o r m e d over t h e c o n t r o l volume C . S . s i g n i f i e s i n t e g r a t i o n t o be p e r f o r m e d o v e r t h e c o n t r o l s u r f a c e We assume t h a t s t a r t e d f r o m r e s t , t h e f l o w w i t h i n t h e c o n t r o l volume o f F i g u r e 1 , l o c a t e d a t an a r b i t r a r y x , w i l l r e a c h a q u a s i - s t e a d y s t a t e w h i c h i s d e f i n e d ( i n p a r t ) by t h e t i m e a v e r a g e o f E q u a t i o n ( 1 ) . T h i s t i m e a v e r a g e must be t a k e n o v e r an i n t e g r a l number o f wave p e r i o d s . E q u a t i o n ( 1 ) w i l l now be expanded and t i m e a v e r a g e d t e r m by t e r m .
I n a d d i t i o n , i t i s e x p e d i e n t t o n e g l e c t a l l l o n g s h o r e v a r i a t i o n s i n t h e t e m p o r a l mean v a l u e o f t h e b a s i c p a r a m e t e r s d e s c r i b i n g t h e i n c i d e n t wave, t h e beach and t h e mean w a t e r l e v e l . Only t h e f l u i d v e l o c i t i e s i n t h e s u r f z o n e w i l l be c o n s i d e r e d dependent upon x . I t i s w o r t h w h i l e n o t -i n g t h a t t h -i s may be an a p p r o x -i m a t -i o n , even u n d e r h y p o t h e t -i c a l l a b o r a t o r y c o n d i t i o n s i n w h i c h t h e i n c i d e n t wave i s a p u r e l y t w o - d i m e n s i o n a l mono-chrome and where t h e beach i s t r u l y p l a n a r . The d i f f i c u l t y a r i s e s f r o m t h e p r o b a b l e d i f f e r e n c e between ( 1 ) t h e l o n g s h o r e p r o j e c t i o n o f t h e i n c i -d e n t wave l e n g t h an-d ( 2 ) t h e l o n g s h o r e wave l e n g t h o f t h e uprush-backwash p r o c e s s on t h e f o r e s h o r e . T h i s i n e q u a l i t y causes a l o n g s h o r e " b e a t " phe-nomenon w h i c h i s p r o m i n e n t i n t h e v i c i n i t y o f an i m p e r m e a b l e s u r f - z o n e b a r r i e r and w h i c h p r o d u c e s s p a t i a l l y p e r i o d i c v a r i a t i o n s i n such q u a n t i -t i e s as b r e a k e r p o s i -t i o n and mean v/a-ter l e v e l . The a u -t h o r b e l i e v e s -t h i s phenomenon t o p l a y an i m p o r t a n t r o l e i n t h e f o r m a t i o n o f beach cusps and r i p c u r r e n t s .
S u r f a c e f o r c e s - To e v a l u a t e t h e s u r f a c e f o r c e , F , we w i l l sx
make t h e a d d i t i o n a l a s s u m p t i o n s :
1 . The mean shear f o r c e on t h e v e r t i c a l f a c e ACDF i s s m a l l enough t o be n e g l e c t e d . T h i s a s s u m p t i o n i s a i d e d by l o c a t i n g t h i s f a c e j u s t o f f s h o r e o f t h e b r e a k e r l i n e where t h e m o t i o n i s c l o s e l y i r r o t a -t i o n a l .
490 C O A S T A L E N G I N E E R I N G
2. Wind s t r e s s e s a r e absent f r o m t h e f r e e s u r f a c e . S i n c e t h e mean w a t e r l e v e l i s b e i n g t a k e n as c o n s t a n t e v e r y w h e r e , t h e n o r m a l f o r c e s on v e r t i c a l f a c e s ABC and DEF a r e e q u a l and o p p o s i t e . The o n l y e f f e c t i v e s u r f a c e f o r c e i s t h u s t h e f r i c t i o n a l r e s i s t a n c e on f a c e CBEF, t h e t i m e a v e r a g e o f w h i c h i s w r i t t e n Ax b c. pu ^ ( x , y, t ) s i n 6 sec a d y d x d t ^ s s ( 2 ) i n w h i c h t h e s u b s c r i p t , s, i s used t o d e n o t e l o c a l c o n d i t i o n s i n t h e s u r f -zone and t h e v e l o c i t y u ^ ( x , y, t ) r e p r e s e n t s t h e l o c a l a v e r a g e , o v e r t h e d e p t h , o f u ^ ( x , y, z, t ) . We w i l l s i m p l i f y E q u a t i o n ( 2 ) b y t h e f u r t h e r a s s u m p t i o n s :
1 . The R e y n o l d s numbers o f t h e s u r f - z o n e m o t i o n a r e h i g h enough t o i n s u r e f u l l y d e v e l o p e d " r o u g h " s u r f a c e b e h a v i o r f o r a l l x , y, t and f o r a l l p r a c t i c a l s u r f a c e r o u g h n e s s e s . 2. The f r i c t i o n f o r c e i s d e f i n e d w i t h s u f f i c i e n t a c c u r a c y by n e g l e c t i n g t h e l o n g s h o r e v a r i a b i l i t y o f u and 9 . s s E q u a t i o n ( 2 ) c a n t h e n be w r i t t e n c p sec a Ax rl 2 T ( x , y, t ) s i n d y d t ( 3 )
Body f o r c e s - W i t h g r a v i t y t h e o n l y body f o r c e , B = 0 and
B pdV = 0 X
( 4 )
Momentum f l u x - The c o n v e c t i v e momentum t e r m o f E q u a t i o n ( 1 ) i n v o l v e s t h e n e t r a t e o f momentum o u t f l o w a c r o s s f a c e s DEF and ABC minus t h e i n f l o w r a t e a c r o s s ADFC. U s i n g t h e d e f i n i t i o n s o f F i g u r e 1 , t h i s t e r m c a n be w r i t t e n
<Q) V ^ ( p V T ) d A ph. Ax rT rb [ u ( x , y, t ) s i n 6 ] 2 ( 1 - ^ ) d y d t S S D 3 Ax T o J-h sb 3 , c o s e , d z d t sb sb i n w h i c h t h e s u b s c r i p t , b, s i g n i f i e s c o n d i t i o n s a t t h e b r e a k e r . ( 5 ) Unsteady t e r m S i n c e we a r e assuming t h e p r o c e s s t o be s t a -t i o n a r y i n -t h e -t i m e a v e r a g e ( i . e . , q u a s i - s -t e a d y ) , -t h e l o c a l momen-tum change v a n i s h e s . T h a t i s dt V^(pdV) = 0 (6)
Summary - E q u a t i o n s ( 3 ) , ( 4 ) , ( 5 ) and ( 6 ) a r e now s u b s t i t u t e d i n t o E q u a t i o n ( 1 ) t o o b t a i n t h e l o n g s h o r e momentum e q u a t i o n : c^ sec a rT rh 2 T ~ ( x , y, t ) s i n Ogdydt I k T ^ [ u ^ ( x , y, t ) s i n e ^ ] 2 ( l - | ) d y d t (7) - h u ^ 2 ( 2 t ) s i n 6 , cos 6 , d z d t sb ' sb sb
VELOCITY DISTRIBUTION ASSUMPTIONS
I n t e g r a t i o n o f E q u a t i o n (7) depends c r i t i c a l l y upon t h e assumed f o r m o f t h e f u n c t i o n s u ^ , u^^^, 6^ and 6^^^. As a f i r s t a p p r o x i m a t i o n , l e t us assume t h a t
492 C O A S T A L E N G I N E E R I N G 1. u ( x , y, t ) s i n 6 i s u n i f o r m l y d i s t r i b u t e d ( i . e . , con¬ s s s t a n t ) i n t h e y d i r e c t i o n f o r any x and t . 2. u ( x , t ) s i n 6 v a r i e s l i n e a r l y w i t h t i m e such t h a t o v e r s s each wave p e r i o d u ^ ( x , t ) s i n = u ^ ^ d - f ) ( 8 ) 3. The mean l o c a l l o n g s h o r e c u r r e n t v e l o c i t y i s i n d e p e n d e n t o f y and i s g i v e n by 1 s u ( x , t ) s i n 6 dA d t ( 9 ) J A s' ' s s o •'A s U s i n g E q u a t i o n ( 8 ) i n ( 9 ) u ( x ) V j x ) = (10)
Then we can c a r r y o u t t h e f i r s t momentum f l u x i n t e g r a l o f E q u a t i o n (7) t o o b t a i n T g | [ u j x , y, t ) s i n e j 2 ( l - ^ ) d y d t 2bh^ d V ^ 2 ( x ) 3 dx (11) To i n t e g r a t e t h e f r i c t i o n t e r m we w i l l make t h e a p p r o x i m a t i o n T fb [ u ( x , y, t ) s i n 6 ] ' s s d y d t (12) T s i n b Jo [u ( x , y, t ) s i n e ] 2 d y d t s s
whereupon
sec a fT
2T u 2 ( x , y, t ) s i n e d y d t s s 2bc sec a
The second momentum f l u x t e r m w i l l be i n t e g r a t e d by assuming t h e i n s t a n t a n e o u s v e l o c i t y d i s t r i b u t i o n a l o n g f a c e ADFC as shovm i n F i g -u r e 2. I n t h i s f i g -u r e t h e l o c a l E -u l e r i a n p a r t i c l e v e l o c i t y , -u^(z, t ) , due t o t h e b r e a k i n g wave i s superimposed on a mean m o t i o n w h i c h c o n s i s t s o f two p a r t s : ( 1 ) a l o n g s h o r e component, V ( x ) , n e c e s s a r y i n o r d e r f o r t h e X component o f t h e mean v e l o c i t y t o be c o n t i n u o u s a c r o s s t h e b r e a k e r l i n e and ( 2 ) an o n s h o r e component, A V ( x ) , n e c e s s a r y f o r s a t i s f a c t i o n o f c o n s e r v a t i o n o f mass i n t h e case o f a n o n - u n i f o r m l o n g s h o r e c u r r e n t . A p p l y i n g t h e c o n t i n u i t y e q u a t i o n t o t h e c o n t r o l v o l u m e , i t i s e a s i l y seen t h a t AV(x) = dV ( x ) h. Ax b b ^ ! L ^ 2 dx (14) whereupon u ^ ^ ( z , t ) s i n 8^^ = Vj^(x) + u ^ ( z , t ) s i n 6^ (15) and u ^ ^ ( z , t ) cos
b
^V!!l
2 dx + u, ( z , t ) cos 0, b b (16) U s i n g E q u a t i o n s (15) and ( 1 6 )t o Figur e 2 . Assume d Velocit y Distributio n a t Breake r
o J - h . u , 2 ( z , t ) s i n 8 , cos 8 , d z d t = sb sb sb (17) 2T u 2 ( z ^ t ) s i n 28 d z d t -- h . 4 d x Summary - S u b s t i t u t i n g E q u a t i o n s ( 1 1 ) , ( 1 3 ) and ( 1 7 ) i n t o ( 7 ) y i e l d s t h e s i m p l i f i e d l o n g s h o r e momentum e q u a t i o n : „ be sec a (18) c d V / ( x ) T2 2T - h . u, 2 ( z t ) s i n 28, d z d t b b i n w h i c h t h e i n t e g r a l on t h e r i g h t - h a n d s i d e i s r e c o g n i z e d as t h e "wave t h r u s t " ( L u n d g r e n , 1 9 6 3 ) . U s i n g s m a l l a m p l i t u d e wave t h e o r y , we have irH, c o s h k , ( h , + z ) f ^\ D D D U ^ ( Z , t ) = 3 i „ h k , h , ^ " ' ^ b D (19) f r o m w h i c h 2T gH, 2n u^^(z, t ) s i n 29j^ d z d t = - — s i n 28^ ( 2 0 ) where 1 2k h "b " 2 + s i n h 2k^h,_ b b (21)
496 C O A S T A L E N G I N E E R I N G
SOLUTION OF THE MOMENTU^I EQUATION
U s i n g (20) i n (18) t h e d i f f e r e n t i a l e q u a t i o n has t h e s o l u t i o n : V / ( x ) v / ( 0 ) = 1 - [ 1 - ^ i — - ] e-'^^ (22) i n w h i c h A = [ ] cos a s i n 6^ s i n 26^ (23) J Z D C . . D D B = ^ [ "-^ ] (24) 5 h, cos a s i n Q, b b and V (0) i s t h e v a l u e o f V ( x ) where x = 0. L L As X becomes v e r y l a r g e , E q u a t i o n (22) r e d u c e s t o t h e r e l a t i o n f o r u n i f o r m l o n g s h o r e c u r r e n t s : V 2 = A (25) J-i N o t e t h a t t h e w i d t h , b, o f t h e s u r f zone i s d i f f i c u l t t o d e f i n e due t o t h e unknown i n c l i n a t i o n o f t h e mean w a t e r l e v e l . For p r a c t i c a l p u r p o s e s t h e r e f o r e V7e w i l l assume a h o r i z o n t a l mean w a t e r l e v e l , whereupon
b = h, c o t a ( 2 6 ) b
A l s o f o r p r a c t i c a l r e a s o n s , t h e unknown r e s i s t a n c e c o e f f i c i e n t , , w i l l be e x p r e s s e d i n t e r m s o f t h e Darcy-Weisbach, f . V a l u e s o f t h e l a t t e r co-e f f i c i co-e n t a r co-e w co-e l l k n o m , a t l co-e a s t f o r s t co-e a d y u n i f o r m f l o w , as a f u n c t i o n o f r e l a t i v e b o u n d a r y r o u g h n e s s and R e y n o l d s number. The r e l a t i o n between t h e s e two c o e f f i c i e n t s i s , f o r s t e a d y u n i f o r m f l o w .
f " f = 4
U s i n g t h e f i n a l s i m p l i f i c a t i o n s ( 2 6 ) and ( 2 7 ) , t h e p a r a m e t e r s A and B become gH, n, s i n a s i n 6, s i n 2 0 , A = f [ 4 - ^ ] ^ ^ ( 2 8 ) b = ! [ h , cos I S i n 9, 1 ( 2 9 ) b b E q u a t i o n ( 2 8 ) d i f f e r s s l i g h t l y f r o m t h a t d e r i v e d e a r l i e r ( E a g l e s o n , 1964) i n t h e m a g n i t u d e o f t h e c o n s t a n t c o e f f i c i e n t . The d i f f e r e n c e a r i s e s f r o m a change i n t h e assumed s u r f z o n e v e l o c i t y d i s t r i b u -t i o n [ E q u a -t i o n ( 8 ) ] . EXPERIMENTAL PROGRAM E x p e r i m e n t s were c o n d u c t e d i n t h e H y d r o d y n a m i c s L a b o r a t o r y o f t h e D e p a r t m e n t o f C i v i l E n g i n e e r i n g a t M.I.T. The model b a s i n used was 45 f e e t by 2 2 f e e t i n i t s h o r i z o n t a l d i m e n s i o n s , and s t i l l - w a t e r d e p t h i n t h e c o n s t a n t d e p t h p o r t i o n was a p p r o x i m a t e l y 1 f o o t . The beach was c o n s t r u c t e d o u t o f smooth cement on a 1 t o 10 s l o p e ( i . e . , t a n a = 0.1) and was 30 f e e t l o n g i n t h e x d i r e c t i o n .
M o n o c h r o m a t i c waves were g e n e r a t e d by a p l u n g e r - t y p e wave-maker o f v a r i a b l e p e r i o d and s t r o k e . T r a i n i n g w a l l s w e r e i n s t a l l e d p e r p e n d i c u -l a r t o t h e wave c r e s t s f r o m t h e wave maker up o n t o t h e b e a c h . These w a -l -l s were c u r v e d t o f o l l o w t h e c a l c u l a t e d r e f r a c t i o n o f t h e wave " r a y s " i n o r -d e r t o m a i n t a i n a u n i f o r m e n e r g y -d e n s i t y a t b r e a k i n g . The " u p s t r e a m " w a l l c o m p l e t e l y o b s t r u c t e d t h e s u r f - and swash-zones w h i l e t h e " d o M S t r e a m " w a l l s t o p p e d j u s t o f f s h o r e o f t h e b r e a k e r l i n e . The e x p e r i m e n t a l t e c h n i q u e used i s r e p o r t e d i n d e t a i l by G a l v i n (1965) and w i l l o n l y be summarized h e r e .
P a r a l l e l - w i r e , r e s i s t a n c e - t y p e wave p r o f i l e gages were used t o o b t a i n t h e b r e a k i n g wave h e i g h t , H^^, t h e mean w a t e r l e v e l a t t h e b r e a k i n g p o s i t i o n (and hence t h e b r e a k i n g d e p t h , h^) and t h e b r e a k i n g wave l e n g t h , L^ (used i n c a l c u l a t i n g n ^ ) . B r e a k e r p o s i t i o n was l o c a t e d v i s u a l l y u s i n g an o v e r h e a d s i g h t i n g d e v i c e d e v e l o p e d by G a l v i n ( 1 9 6 5 ) . M a j o r changes i n t h e b r e a k e r a n g l e were o b t a i n e d t h r o u g h c h a n g i n g t h e p o s i t i o n o f t h e wave-maker. M i n o r changes o f c o u r s e accompanied each change i n wave p e r i o d .
498 C O A S T A L E N G I N E E R I N G
The mean l o c a l l o n g s h o r e c u r r e n t v e l o c i t y V ( x , y ) was d e t e r
-Li
mined by u s i n g an A r m s t r o n g M i n i f l o k e t e r AWE 1 8 3 / 1 , I s s u e A, w h i c h c o u n t s e l e c t r o n i c a l l y t h e r e v o l u t i o n s o f a j e w e l m o u n t e d p r o p e l l e r i n a 5 / 8 i n c h -d i a m e t e r h o u s i n g . The p r o b e was c a l i b r a t e -d f o r each a n g l e o f t h e wavemaker by t i m i n g t h e d r i f t o f s u r f a c e f l o a t s i n t h e r e g i o n o f u n i f o r m c u r -r e n t s . The p -r o b e was i n s e -r t e d a t m i d - d e p t h a t t h e mean y p o s i t i o n o f t h e s u r f a c e f l o a t s i n t h e s u r f - z o n e and t h e p r o p e l l e r a x i s was a l i g n e d w i t h t h e X - a x i s . I n t h i s way t h e r e l a t i o n between p r o p e l l e r c o u n t s and f l o a t v e l o c i t y was o b t a i n e d . T h i s r e l a t i o n was assumed v a l i d i n t h e r e g i o n o f n o n - u n i f o r m c u r r e n t s as w e l l s i n c e a c c u r a t e L a g r a n g i a n v e l o c i t y measure-ments a r e d i f f i c u l t i n t h e p r e s e n c e e f c o n v e c t i v e a c c e l e r a t i o n .
A b s o l u t e b o u n d a r y r o u g h n e s s was n o t measured b u t was s e l e c t e d f r o m p u b l i s h e d v a l u e s (Chow, 1959) f o r smooth cement as
k ^ = 1 X 10-3 f t . ( 3 0 )
Two s e t s o f e x p e r i m e n t s were c o n d u c t e d . T e s t S e r i e s 1 ( G a l v i n , 1965) c o n s i s t e d e f 26 i n d i v i d u a l r u n s d i v i d e d i n t o t h r e e g r o u p s a c c o r d -i n g t o t h e o f f s h o r e a n g l e o f wave -i n c -i d e n c e . T h -i s s e r -i e s was a-imed a t s t u d y o f t h e zone o f f u l l y d e v e l o p e d ( u n i f o r m ) c u r r e n t s and o n l y i n a d -v e r t e n t l y i n c l u d e d n o n - u n i f o r m -v a l u e s . T e s t S e r i e s 2 ( E a g l e s o n , 1965) c o n s i s t e d o f 7 i n d i v i d u a l r u n s a t a s i n g l e o f f s h o r e wave a n g l e and was s p e c i f i c a l l y d e s i g n e d t o i n v e s t i g a t e E q u a t i o n (22) o v e r as w i d e a r a n g e as p o s s i b l e w i t h t h e e x i s t i n g e q u i p m e n t . T a b u l a t i o n s o f t h e r e s p e c t i v e t e s t d a t a a r e g i v e n i n t h e r e f e r e n c e s c i t e d .
PRESENTATION OF RESULTS EVALUATION OF THE FRICTION FACTOR
I t has been assumed i n t h e d e r i v a t i o n o f E q u a t i o n ( 2 2 ) t h a t t h e i n s t a n t a n e o u s s u r f - z o n e f l o w i s a l w a y s h y d r a u l i c a l l y r o u g h a t a l l l o c a t i o n s . The Darcy-Weisbach c o e f f i c i e n t , f , w i l l t h u s be e v a l u a t e d f r o m t h e K a r m a n - P r a n d t l r e s i s t a n c e e q u a t i o n f o r s t e a d y , u n i f o r m , t u r b u l e n t f l o w i n r o u g h p i p e s . When w r i t t e n i n terms o f t h e h y d r a u l i c r a d i u s , R, t h i s w e l l - k n o T O e q u a t i o n becomes For t h e t r i a n g u l a r s u r f - z o n e c r o s s s e c t i o n I I \ R = — b s i n a - h t a n a =
—
( 3 2 )whereupon f i s g i v e n by
\
f = [2 l o g ^ g ^ + 1.74] 2 (33)
i n w h i c h i s t h e a b s o l u t e roughness o f t h e beach s u r f a c e . E q u a t i o n (33) i s used t o e v a l u a t e f f o r a l l t h e d a t a p r e s e n t e d i n t h i s p a p e r . The v a l u e of k^ was chosen f r o m p u b l i s h e d v a l u e s a c c o r d i n g t o t h e kno\m c h a r a c t e r o f t h e beach s u r f a c e .
IMPORTANCE OF UPSTREAiM BOUNDARY CONDITION
I n t h e d e r i v a t i o n o f E q u a t i o n (22) i t was assumed t h a t o n l y t h e f l u i d v e l o c i t i e s i n t h e s u r f z o n e v a r i e d w i t h x. T h i s d e t e r m i n e d t h e o r i -g i n o f x t o be where t h e s u r f - z o n e c r o s s s e c t i o n f i r s t r e a c h e s i t s f u l l v a l u e and was t a k e n as t h e i n t e r s e c t i o n o f t h e b a r r i e r and t h e s t i l l - w a t e r s h o r e l i n e . The mean l o n g s h o r e c u r r e n t v e l o c i t y , V^(0), a t x = 0 i s t h e r e -f o r e q u i t e s e n s i t i v e t o t h e n a t u r e o -f t h e b a r r i e r , i . e . , t o such -f a c t o r s as i t s h e i g h t , l e n g t h , a n g l e and p e r m e a b i l i t y . No a t t e m p t i s made h e r e t o p r e d i c t as a f u n c t i o n o f t h e s e v a r i a b l e s ; hov/ever, t h e p r e d i c t e d e f f e c t o f V ^ ( 0 ) upon t h e g r o w t h o f t h e l o n g s h o r e c u r r e n t can be e v a l u a t e d . T h i s i s a c c o m p l i s h e d i n F i g u r e 4 where t h e e x p e r i m e n t s ( E a g l e s o n , 1965) h a v i n g t h e l a r g e s t and s m a l l e s t average v a l u e o f t h e measured p a r a m e t e r , V (0)/A-'''2^ aj-e compared w i t h E q u a t i o n ( 2 2 ) . The agreement appears q u i t e
t l
s a t i s f a c t o r y f o r t h e s e two r u n s .
CONSOLIDATION OF AVAILABLE EXPERIMENTAL DATA
D e v e l o p i n g l o n g s h o r e c u r r e n t - I n F i g u r e 5 a l l e x p e r i m e n t s kno^m t o t h e a u t h o r a r e p r e s e n t e d i n w h i c h t h e d i s t a n c e x was r e p o r t e d as one o f t h e measured v a r i a b l e s . A l t h o u g h t h e s e two s e t s o f t e s t s were b o t h p e r f o r m e d i n n e a r l y t h e same manner and u s i n g t h e same e x p e r i m e n t a l f a c i l i t i e s , t h e d a t a o f G a l v i n (1965) appear i n somewhat w o r s e q u a n t i t a -t i v e agreemen-t w i -t h E q u a -t i o n ( 2 2 ) . The r e a s o n f o r -t h i s i s u n k n o m . Two p o s s i b l e causes a r e w o r t h y o f m e n t i o n , however. 1. For t h e s e r i e s I I I t e s t s o f G a l v i n (1965) t h e t r a i n i n g w a l l s were s t r a i g h t r a t h e r t h a n b e i n g c u r v e d t o f o l l o w t h e wave r e f r a c t i o n . T h i s d e v e l o p s a l o c a l x^ave energy d e n s i t y g r a d i e n t i n t h e l o n g s h o r e d i -r e c t i o n . 2. T h r e e y e a r s e l a p s e d between t h e p e r f o r m a n c e o f t h e two s e t s o f e x p e r i m e n t s . I n t h e i n t e r i m t h e beach s u r f a c e r o u g h n e s s may have changed.
Basi n Wall s fee t Wate r Leve l Poin t Gog e Plunge r Upstrea m Trainin g Wal l Botto m o f Beac h S W Lin e Downstrea m Trainin g Wal l Piezomete r Conduit s Piezomete r Well s @ © @ @ Figur e o. t-isn. vie w o f Laborator y Basi n @ @ @
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DEVELOPIN G LONGSHOR E CURREN T 2. 0 LEGEN D FO R EXPERIMENTA L POINT S V L(0 ) SYMBO L SOURC E Al/ 2 + 0.37 5 EAGLESO N (1965 ) A 0.09 2 EAGLESO N (1965 ) NOT E ; EXPERIMENTA L POINT S AR E PLOTTE USIN G AVERAG E VALUE S O F A AN AN D WIT H A N ASSUME D ABSOLUT ROUGHNESS , k . = I x 10" ' FT . VL(X ) A 1/ 2 1. 0 I U I M I I V|_(0 ) 0.37 5 (UNIFOR M LONGSHOR E CURREN T THEORY ) LEGEN D FO R THEORETICA L CURVE S B X V2(x ) V ^ (0) ' si n a si n 6^^ si n Zd^-I
h[ j co s a si n 0.0 4 0.0 6 0.0 8 1 0 0. 2 0. 4 0. 6 0. 8 I B X Figur e 4 . Evaluatio n o f th e Importanc e o f V, . (0 )502 C O A S T A L E N G I N E E R I N G D i s c o u n t i n g t h e s e r i e s I I I t e s t s o f G a l v i n (1965) i t may be^ c o n c l u d e d f r o m F i g u r e 5 t h a t E q u a t i o n (22) p r o v i d e s an a d e q u a t e d e s c r x p t i o n o f t h e g r o w i n g c u r r e n t , a t l e a s t a t l a b o r a t o r y s c a l e . U n i f o r m l o n g s h o r e c u r r e n t - S e v e r a l e x p e r i m e n t e r s have r e p o r t e d l o n g s h o r e c u r r e n t o b s e r v a t i o n s i n w h i c h no m e n t i o n o f p r o x i m i t y t o an up s t r e a m o b s t r u c t i o n i s made. I n F i g u r e 6^ t h e s e f i e l d and l a b o r a t o r y meas u r e m e n t s a r e compared w i t h t h e t h e o r e t i c a l uniform^ l o n g s h o r e c u r r e n g i v e n by E q u a t i o n ( 2 5 ) . B e f o r e e v a l u a t i n g t h i s c o m p a r i s o n we shou no e t h e f o l l o v 7 i n g .
1. The b r e a k i n g d e p t h , h^, was n o t measured i n any o f t h e f i e l d o b s e r v a t i o n s . I n o r d e r t o a p p l y E q u a t i o n ( 2 5 ) , t h e t h e o r e t i c a l b r e a k i n g h e i g h t t o d e p t h r a t i o o f McCowan (1894) was assumed a p p l i c a b l e ; i . e . , ( h ) b = (34) 2. The a b s o l u t e r o u g h n e s s o f t h e v a r i o u s s u r f a c e s w a s chosen a c c o r d i n g t o t h e f o l l o w i n g t a b l e : R e f e r e n c e Putnam e t a l . (1949) Putnam e t a l . (1949) Putnam e t a l . (1949) Putnam e t a l . (1949) Inman and Q u i n n (1951) L a b o r a t o r y D e s c r i p t i o n o r F i e l d o f S u r f a c e A s s u m e c M l ^ ^ o i u t e R o u g h n e s s , k^
-(fïö
( 1 mm.) L a b o r a t o r y N a t u r a l Sand 0.0033 L a b o r a t o r y 1 / 4 - i n c h Pea G r a v e l 0.0208 ( 1 / 4 - i n c h ) L a b o r a t o r y Sheet M e t a l o r Smooth Cement F i e l d F i e l d G a l v i n and Savage F i e l d (1965) N a t u r a l Sand N a t u r a l Sand N a t u r a l Sand O.OOlO 0 . 0 0 3 3 ( 1 ™"-) 0 . 0 0 3 3 ( 1 I ™ - ) 0 . 0 0 3 3 ( 1 ^ T h i s r e p r e s e n t s a m o d i f i c a t i o n o f w o r k p r e s e n t e d e l s e w h e r e C E a g l e s o n , 1 9 6 4 ) .~i i i I 1 ! I [ r — DEVELOPIN G LONGSHOR E CURREN T EXPERIMENTA L POINT S AR E PLOTTE D USIN G AVERAG E VALUE S O F A AN D B AN D WIT H A N ASSUME D ABSOLUT E ROUGHNESS , k, = I X 10" ^ FT " AL L DAT A AR E FRO M LABORATOR Y EXPERIMENT S (UNIFOR M LONGSHOR E CURRENT-THEORY)-, LEGEN D FO R EXPERIMENTA L POINT S SYMBO L RU N REFERENC E 0 m NO T MEASURE D GALVI N [ 1965 ) 9 n NO T MEASURE D GALVI N ( 1965 1 Q m NO T MEASURE D GALVI N (1965 ) 2400(0 ) 0.3 0 EAGLESO N (1965 ) A 2400(b ) 0.09 2 EAGLESO N (1965 ) O 245 0 0. 1 8 6 EAGLESO N (1965 1 + 2500(a ) 0.37 5 EAGLESO N [1965 1 V 2500(b ) 0.34 3 EAGLESO N (1965 ) • 252 0 0.35 2 EAGLESO N (1965 ) 0 1 1 255 0 0.35 0 1 1 EAGLESO N (1965 ) 1 0 10 ' B X Figur e 5 . Th e Developin g Longshor e Curren t
504 C O A S T A L E N G I N E E R I N G
3. There i s g r e a t s c a t t e r i n s u c c e s s i v e o b s e r v a t i o n s o f V r e
L l
p o r t e d b y Inman and Quinn (1951) under t h e same a p p a r e n t wave c o n d i t i o n s O n l y t h o s e mean v a l u e s o f V w h i c h exceed 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
Ll
g i v e n s e t a r e i n c l u d e d h e r e .
I t i s i n t e r e s t i n g t o n o t e i n F i g u r e 6 t h a t t h e t h e o r y p r o v i d e s an u p p e r bound f o r t h e l a b o r a t o r y v a l u e s w h i l e t h e f i e l d d a t a s c a t t e r a b o u t t h e t h e o r e t i c a l c u r v e . Perhaps t h i s d i f f e r e n c e i s due t o some v i s cous o r s u r f a c e t e n s i o n s c a l e e f f e c t a r i s i n g f r o m i n e x a c t f o r m u l a t i o n o f t h e p r o b l e m , o r p e r h a p s t h e e x p e r i m e n t a l b a s i n was n o t l a r g e enough f o r u n i f o r m c o n d i t i o n s t o be d e v e l o p e d . C o n s i d e r a b l e s c a t t e r i n t h e f i e l d d a t a i s t o be e x p e c t e d due p r i n c i p a l l y t o ( 1 ) a c t u a l n o n - u n i f o r m i t i e s i n t h e beach and wave p a r a m e t e r s and ( 2 ) d i f f i c u l t y i n e s t i m a t i n g t h e c r i t i c a l b r e a k e r p a r a m e t e r s u n d e r f i e l d c o n d i t i o n s .
I t i s c o n c l u d e d f r o m F i g u r e 6 t h a t E q u a t i o n ( 2 5 ) p r o v i d e s a good e s t i m a t e o f t h e u n i f o r m l o n g s h o r e c u r r e n t u n d e r b o t h l a b o r a t o r y and f i e l d c o n d i t i o n s .
APPLICATIONS OF THE THEORY
The r a t i o n a l p r e d i c t i o n o f l i t t o r a l t r a n s p o r t s t i l l a w a i t s de-v e l o p m e n t o f a s u i t a b l e e n t r a i n m e n t f u n c t i o n f o r t h e s u r f - z o n e f l u i d mo-t i o n and mo-t h e " m a r r y i n g " o f mo-t h i s f u n c mo-t i o n mo-t o a mean f l o w e q u a mo-t i o n s u c h as E q u a t i o n ( 2 2 ) .
M e a n w h i l e , however, E q u a t i o n ( 2 2 ) can be u s e f u l l y employed t o e s t i m a t e , among o t h e r t h i n g s , t h e s i z e o f l a b o r a t o r y b a s i n n e c e s s a r y i n o r d e r t o o b t a i n a f u l l y d e v e l o p e d l o n g s h o r e c u r r e n t f o r g i v e n wave and beach p a r a m e t e r s . I f we r e q u i r e t h a t t h e l a b o r a t o r y l o n g s h o r e c u r r e n t be a t l e a s t 95% o f t h e f u l l y d e v e l o p e d v a l u e , E q u a t i o n ( 2 2 ) r e d u c e s t o Bx = 2.32 f o r 95% development ( 3 5 ) T h i s means t h a t a m o d e l w h i c h i s b u i l t t o s t u d y a p r o t o t y p e s h o r e p r o c e s i n w h i c h a f u l l y d e v e l o p e d l o n g s h o r e c u r r e n t p l a y s an i m p o r t a n t r o l e s h o u l d have a beach w h i c h e x t e n d s a t l e a s t 2.32/B f e e t " u p s t r e a m " o f t h e a r e a o f i n t e r e s t . For t h e t y p i c a l l a b o r a t o r y s c a l e v a r i a b l e s , h, = 0.2 f t . b k^ = 1 X lQ-3 f t . cos a-1 \ = 20°
E q u a t i o n ( 2 9 ) g i v e s B = 0.14 f e e t , and t h e added model l e n g t h becomes a t l e a s t X = = 16 f t .
506 C O A S T A L E N G I N E E R I N G
SUMMARY AND CONCLUSIONS
1. An e x p r e s s i o n i s d e r i v e d f o r t h e mean l o c a l l o n g s h o r e c u r r e n t v e l o c i t y i n t h e s u r f z o n e on a p l a n e , impermeable beach. T h i s e x -p r e s s i o n i s i n t e r m s o f r e a d i l y o b t a i n e d beach and wave -p a r a m e t e r s and a r e s i s t a n c e c o e f f i c i e n t f o r h y d r a u l i c a l l y r o u g h s u r f a c e s . 2. When t h e r e s i s t a n c e c o e f f i c i e n t i s c a l c u l a t e d u s i n g an e s -t i m a -t e o f -t h e a b s o l u -t e s u r f a c e r o u g h n e s s i n c o n j u n c -t i o n w i -t h -t h e Karmön-P r a n d t l r e s i s t a n c e e q u a t i o n f o r s t e a d y , u n i f o r m f l o w i n r o u g h p i p e s , t h e v a l i d i t y o f t h e d e r i v e d r e l a t i o n s h i p i s s a t i s f a c t o r i l y d e m o n s t r a t e d . 3. V e r i f i c a t i o n o f t h e t h e o r y i s o b t a i n e d f o r t h e n o n - u n i f o r m d e v e l o p i n g c u r r e n t do\mstream o f a s u r f - z o n e b a r r i e r t h r o u g h c o m p a r i s o n w i t h l a b o r a t o r y e x p e r i m e n t s . V e r i f i c a t i o n f o r t h e u n i f o r m case i s ob-t a i n e d u s i n g b o ob-t h l a b o r a ob-t o r y and f i e l d d a ob-t a .
4. The d e r i v e d e x p r e s s i o n appears t o have p r e s e n t u t i l i t y i n t h e d e s i g n o f c o a s t a l model t e s t s and t o p r o v i d e a n e c e s s a r y component f o r t h e f u t u r e u n d e r s t a n d i n g o f l o n g s h o r e sediment t r a n s p o r t .
ACKNO;^^LEDGEMENTS
F i n a n c i a l s u p p o r t f o r t h i s w o r k was p r o v i d e d by t h e C o a s t a l E n g i n e e r i n g Research C e n t e r o f t h e Department o f t h e Army, Corps o f Eng i n e e r s u n d e r C o n t r a c t No. DA49005CIVENG629. Mr. Dara Z a r Eng a r a s -s i -s t e d w i t h t h e d a t a r e d u c t i o n .
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Chow, V. T., Open Channel H y d r a u l i c s , M c G r a w - H i l l Book Company, I n c . , New Y o r k , 1959, p. 196.
E a g l e s o n , P. S., " U n i f o r m L o n g s h o r e C u r r e n t s on a P l a n e Beach," P r o c . L a t i n Amer. Cong, on H y d r a u l i c s o f I.A.H.R., P o r t o A l e g r e , B r a z i l , A u g u s t , 1964.
E a g l e s o n , P. S., " T h e o r e t i c a l Study o f L o n g s h o r e C u r r e n t s o n a P l a n e Beach," M.I.T. D e p a r t m e n t o f C i v i l E n g i n e e r i n g . H y d r o d y n a m i c s Lab-o r a t Lab-o r y R e p Lab-o r t NLab-o. 8 2 , 1965.
G a l v i n , C. J . , J r . , and E a g l e s o n , P. S., " E x p e r i m e n t a l S t u d y o f Long-s h o r e C u r r e n t Long-s on a P l a n e Beach," Tech. Memo. Ho. 10, U. S. Army C o a s t a l E n g i n e e r i n g Research C e n t e r , W a s h i n g t o n , J a n u a r y , 1965. G a l v i n , C. J . , J r . , and Savage, R. P., "Longshore C u r r e n t s a t Nags Head,
N o r t h C a r o l i n a , " ( u n p u b l i s h e d m a n u s c r i p t r e c e i v e d i n A p r i l , 1 9 6 5 ) . Inman, D. L., and Q u i n n , W. H., " C u r r e n t s i n t h e S u r f Zone," P r o c .
Sec-ond Conf. on C o a s t a l Eng., C o u n c i l on Wave R e s e a r c h , B e r k e l e y , C a l i f o r n i a , 1951.
L u n d g r e n , H., "Wave T h r u s t and Wave Energy L e v e l , " Paper No. 1.20, I.A.H.R. Congress, London, 1963.
McCowan, J . , "On t h e H i g h e s t Wave o f Permanent Type," The London,
E d i n b u r g h and D u b l i n P h i l o s o p h i c a l Magazine and J o u r n a l o f S c i e n c e , V o l . 38, 1894.
Putnam, J . A., Munk, W. H., and T r a y l o r , M. A., "The P r e d i c t i o n o f Long-s h o r e C u r r e n t Long-s , " T r a n Long-s . A.G.ÏÏ., 3 0 , pp. 337-345, 1949.
