Increase of bottom shearstress of a current due to wave motion

WATERLOOPKUNDIG LABORATORIUM. I n c r e a s e o f b o t t o m s h e a r s t r e e s o f c u r r e n t due t o wave m o t i o n . by E.W. B i j k e r . P r e l i m i n a r y R e p o r t . D e l f t H y d r a u l i c s L a b o r a t o r y A p r i l 1965. ,.

CONTENTS. 1. I n t r o d u c t i o n page 1 2 . Equipment o f t h e t e s t s 2 3. T h e o r e t i c a l approach ₅ 4. Keasurements 8 ^. F u t u r e i n v e s t i g a t i o n s

₉

FIGURES. 1. Layout o f model b a s i n . 2. V/ave h e i g h t d i s t r i b u t i o n i n o r o a s - s e c t i a n 0. 3 . Mean v e l o c i t y d i s t r i b u t i o n i n c r o s s - s e c t i o n 0 . 4 . Measured v e l o c i t y p r o f i l e s i n T306 i n c r o s a - e e c t i o n 0. 5. Comparison between l o g a r i t h m i c v e l o c i t y d i s t r i b u t i o n o f u n i f o r m f l o w and o r b i t a l v e l o c i t y d i s t r i b u t i o n o f waves. 6. - 1) v e r s u s U 7 . |i v e r s u s k. — ) f o r smooth aand b o t t o m . ~ ) f o r sand b o t t o m w i t h r i p p l e s . ) f o r b o t t o m w i t h s t o n e s . TABLES. Measuring r e s u l t s f o r I . B o t t o m w i t h s t o n e s . I I . Sand b o t t o m w i t h r i p p l e s , I I I , Smooth sand b o t t o m .

SYMBOLS. r e s i s t a n c e c o e f f i c i e n t o f Chezy. base o f n a t u r a l l o g a r i t h m e . a c c e l e r a t i o n o f e a r t h g r a v i t y . . w a t e r d e p t h w a v e - h e i g h t . s l o p e o f energy l e v e l . bottom roughness, m i x i n g l e n g t h , wave l e n g t h , t i m e wave p e r i o d . o r b i t a l v e l o c i t y a t t h e b o t t o m . a m p l i t u d e o f o r b i t a l V e l o c i t y a t t h e b o t t o m . mean v e l o c i t y . v e l o c i t y a t h e i g h t y, r e s p . y' above t h e b o t t o m s h e a r s t r e s s v e l o c i t y . r e s u l t a n t v e l o c i t y . , h e i g h t above t h e b o t t o m , c o n s t a n t o f von Karman, c o n s t a n t . c o n s t a n t . b o t t o m s h e a r s t r e s s , r e s u l t a n t bottom s h e a r s t r e s s i n t h e d i r e c t i o n o f t h e main c u r r e n t a t t i m e t .

component o f t h e bottom s h e a r a t r e s e Irï t h e d i r e c t i o n o f t h e c u r r e n t a t t i m e t .

. 1 '

mean v a l u e o f t ' ^ ^ ^

b o t t o m s h e a r s t r e s s o f t h e c u r r e n t . coëfficiënt o f k i n e m a t i c v i s c o s i t y .

Summary.

Due t o t h e i n c r e a s e d v e l o c i t y v e c t o r i n t h e case o f c u r r e n t combined w i t h waves t h e b o t t o m s h e a r s t r e s s i n t h e d i r e c t i o n o f the c u r r e n t w i l l i n c r e a s e . T h i s i n c r e a s e has been d e r i v e d and the d e r i v a t i o n has been checked by measurements.

oominaire.

A cause du v e c t e u r de l a V i t e s s e augmentee dans l e cas d'un courant combine aveo l a h o u l e , l e f r o t t e m e n t du f o n d dans l a d i r e c t i o n du c o u r a n t augmentera. C e t t e a u g m e n t a t i o n e s t derive'' et c e t t e d e r i v a t i o n e s t v e r i f i e p a r l e s mesures.

I n c r e a s e o f b o t t o m s h e a r e t r e s R nr « c u r r e n t due t o wave m o t l n n .

1. I n t r o d u c t i o n .

From d a t a i n n a t u r e as w e l l as i n models i t i s Icnown t h a t t h e bedload movement caused by a c u r r e n t i n c r e a s e s when a wave m o t i o n i s superimposed upon t h i s c u r r e n t . T h i s e f f e c t i s n o t m e r e l y t h e r e s u l t o f t h e combined t r a n s p o r t c a p a c i t y o f t h e waves and t h a t o f the c u r r e n t , because t h e i n c r e a s e o f t h e bedload movement o c c u r s a l s o when t h e d i r e c t i o n o f p r o p a g a t i o n o f t h e waves i s p e r p e n d i c u l a r upon t h e c u r r e n t .

Since t h e r e e x i s t s , a t a^y r a t e w i t h c u r r e n t o n l y , a r e l a t i o n between b o t t o m s h e a r s t r e s s and bedload movement i t seems w o r t h w h i l e to t r y t o f i n d t h i s r e l a t i o n a l s o f o r t h e case o f t r a n s p o r t under the combined i n f l u e n c e o f waves and c u r r e n t . T h e r e f o r e i t w i l l be f i r s t l y n e c e s s a r y t o s t u d y t h e i n c r e a s e o f t h e b o t t o m s h e a r s t r e s s of a c u r r e n t due t o a wave m o t i o n .

I n t h i s paper an a t t e m p t i s made t o s t u d y t h e i n c r e a s e o f t h e bottom s h e a r s t r e s s o f a c u r r e n t due t o a wavJ m o t i o n w i t h a d i r e c t i o n o f p r o p a g a t i o n p e r p e n d i c u l a r upon t h i s c u r r e n t .

As i t i s n o t w e l l p o s s i b l e t o measure t h e b o t t o m s h e a r s t r e s s d i r e c t l y an i n d i r e c t method had t o be chosen. The method o f d e t e r -n i i -n a t i o -n o f t h e b o t t o m s h e a r s t r e s s f r o m t h e v e l o c i t y p r o f i l e i -n

the v i c i n i t y o f t h e b o t t o m i s n o t f e a s i b l e i n t h i s case as t h e ' combined v e l o c i t y p r o f i l e i s o f a r a t h e r c o m p l i c a t e d n a t u r e . The s h e a r s t r e s s o v e r t h e b o t t o m w i l l t h e r e f o r e be d e t e r m i n e d f r o m t h e energy s l o p e .

- 2 ~

2, Equipment f o r t h e t e s t a .

The t e s t s have been executed i n a b a s i n w h i c h was 27 m l o n g and 17 m v/ide ( f i g u r e I ) . At one o f t h e l o n g s i d e s a wave g e n e r a t o r was i n s t a l l e d , and a t t h e o p p o s i t e s i d e a t a l u s w i t h a s l o p e o f 1:7 was c o n s t r u c t e d i n o r d e r t o absorb t h e waves. The r e f l e c t i o n was reduced t o an a c c e p t a b l e degree. I n f i g u r e 2 t h e wave h e i g h t d i s t r i b u t i o n over a c r o s s - s e c t i o n p e r p e n d i c u l a r upon t h e t a l u s and t h e wave-c r e s t s i s g i v e n f o r some s i t u a t i o n s .

The wave h e i g h t s have been measured by means o f a r e s i s t a n c e

wave h e i g h t - m e t e r . . i^"^ (ilj/'d * A maximal d i s c h a r g e o f O.7 m^/sec c o u l d be a d j u s t e d w i t h an

accuracy o f 3 ^ by an a u t o m a t i c a l l y governed i n l e t s l u i c e . T h i s d i s c h a r g e was d i s t r i b u t e d e v e n l y o v e r t h a t p a r t o f t h e m o d e l w h i c h has a c o n s t a n t d e p t h o f t h e m o d e l by means o f an o v e r f l o w w e i r and a g r i d . I n f i g u r e J t h e v e l o c i t y d i s t r i b u t i o n o f soide s i t u a t i o n s i s given,and i n f i g u r e 4 t h e v e l o c i t y d i s t r i b u t i o n o v e r t h e v e r t i c a l i s shown. As t h e f l o w i s w i t h s u f f i c i e n t a p p r o x i m a t i o n u n i f o r m t h e energy l e v e l can be d e t e r m i n e d by m e a s u r i n g t h e e l o p e o f t h e w a t e r l e v e l . T h i s i s done by m e a s u r i n g t h e d i f f e r e n c e o f t h e water l e v e l a t two p o i n t s a t a d i s t a n c e o f 10 m i n t h e c e n t e r -l i n e o f t h e mode-l. The w a t e r -l e v e -l s were r e c o r d e d by means o f f l o a t s p l a c e d i n drums b e s i d e s t h e model. The drums were connec-t e d b y a p i p e w i connec-t h m e a s u r i n g p o i n connec-t s i n connec-t h e boconnec-tconnec-tom o f connec-t h e model. S p e c i a l p r e c a u t i o n s were t a k e n i n o r d e r t o g u a r a n t e e t h a t t h e water l e v e l was r e c o r d e d w i t h o u t any v e l o c i t y e f f e c t . By p o t e n t i o m e t e r s a t t a c h e d t o t h e f l o a t s t h e d i f f e r e n c e i n w a t e r l e v e l a t t h e two p o i n t s was r e c o r d e d w i t h an a c c u r a c y o f 0 , 0 5 nun.

3 -T h e o r e t i c a l approach. The b o t t o m s h e a r s t r e s s o f t h e c u r r e n t t o g e t h e r w i t h wave m o t i o n can be o b t a i n e d by c a l c u l a t i n g t h e g r a d i e n t o f t h e r e s u l t a n t v e l o c i t y v e c t o r o f t h e main c u r r e n t and t h e o r b i t a l m o t i o n . A c c o r d i n g t o P r a n d l t h e s h e a r s t r e s s o v e r t h e b o t t o m i n a t u r -b u l e n t c u r r e n t may -be w r i t t e n as ^ - ( a / ) b o t t o m (see R o t t a l i t t . 1 ) . , i n which 1 - m i x i n g l e n g t h

V = v e l o c i t y a t h e i g h t y above the bottom' Q = d e n s i t y

T = s h e a r s t r e s s a t t h e b o t t o m . • y " d i s t a n c e f r o m t h e b o t t o m .

According t o t h e t h e o r y o f P r a n d t l 1 i s f o r a r o u g h b o t t o m d e t e r -mined by t h e roughness o f t h i s b o t t o m and the' d i s t a n c e t o t h e b o t t o m , so . r , 1 = K y y Oj . '-' ''^"^^ ( 2 ) i n which K i s a u n i v e r s a l c o n s t a n t w i t h v a l u e 0 . 4 » For a n o r m a l f u l l y t u r b u l e n t c u r r e n t t h e d i f f e r e n t i a l q u o t i e n t o f the v e l o c i t y d i s t r i b u t i o n ( t h e v e l o c i t y g r a d i e n t ) o u t s i d e t h e v i s c o u s sublayer c l o s e t o t h e b o t t o m can be w r i t t e n as 3v V whara v , . / i - \ J s t a - ^ ^ , . ( 4 )

i . 4 -i n v/h-ich vj» = s h e a r s t r e s s v e l o c -i t y h =. w a t e r d e p t h I • s l o p e o f energy l e v e l V = mean v e l o c i t y C = r e s i s t a n c e c o e f f i c i e n t o f Chezy g • a c c e l e r a t i o n o f e a r t h g r a v i t a t i o n I n t e g r a t i o n o f e q u a t i o n 5 g i v e s t h e v e r t i c a l d i s t r i b u t i o n o f the v e l o c i t y , . . • v i z . ^ i n ^ i n which k i s a v a l u e f o r t h e b o t t o m r o u g h n e s s . For t h e c a l c u l a t i o n o f t h e s h e a r s t r e s s a t t h e b o t t o m , s h o u l d be known. A c c o r d i n g t o f i g u r e 5 i t w i l l be assumed t h a t ^dy '^bottom b o t t o m y' So i n t h i s case ( 5 ) ( 6 ) A f t e r s u b s t i t u t i o n t h i s v a l u e i n e q u a t i o n 5 o f the' v e r t i c a l d i s t r i -b u t i o n o f t h e v e l o c i t y one f i n d s ke y' • i n w h i c h e^= base o f t h e n a t u r a l l o g a r i t h m e ,

The v e c t o r o f t h e r e s u l t a n t v e l o c i t y o f c u r r e n t and wave motion i s

I n t h i s case i t i s assumed t h a t t h e v i s c o u s boundary l a y e r s f o r the o r b i t a l m o t i o n and f o r t h e main c u r r e n t a r e e q u a l . I n o r d e r to achieve t h e c o n n e c t i o n between t h e l i n e a i r v e l o c i t y p r o f i l e i n t h e v i s c o u s s u b l a y e r and t h e o r b i t a l v e l o c i t y a t t h e b o t t o m u^, a f a c t o r fi i s i n t r o d u c e d . So \i w i l l be s m a l l e r t h a n 1 . The o r b i t a l v e l o c i t y a t t h e b o t t o m u^ i s a f u n c t i o n o f t h e time a c c o r d i n g t o "b - % ^ » ' ( 8 ) i n which T - wave p e r i o d , w h i c h has been d u r i n g these t e s t s 1.57 sec

i i n which H - wave h e i g h t L = wave l e n g t h So t h e r e s u l t a n t b o t t o m s h e a r s t r e s s a t any moment t w i l l be • o 9v 2 ^ V 2

M t )

" - 51

(7?)

( 1 0 ) T h i s r e s u l t a n t s h e a r s t r e s s , a c t i n g i n t h e d i r e c t i o n o f t h e r e s u l t a n t c u r r e n t w i l l make an angle DC w i t h t h e main c u r r e n t ;

C O S C X / . v O ,1 /^^N

The component o f t h e r e s u l t a n t s h e a r s t r e s s w o r k i n g i n t h e d i r e c t i o n o f t h e main c u r r e n t can be w r i t t e n as

6

-IHj.

T h i s can be w r i t t e n as • ^

" ^ ' ( t ) " 5 v ^ ^ \ / 1 + ^ ( j i u ^ ) ^ , s i n c e 1 = K y.

and a f t e r r e p l a c i n g under t h e r o o t s i g n by ^ ,

2 \ / U^^C^ ,UbN^ /.iN

27tt When f o r u ^ i s w r i t t e n u ^ s i n » 2 \ / u^K^C^ /"ON^ , 2 2 u t /.,s T ' ^ ^ ^ - Qv,, \|^ + ^ — ( ~ ) s i n -Y". , ( 1 4 ) 2 2 2 For ^ w i l l be w r i t t e n C . C i s t h e r e s i s t a n c e v a l u e o b t a i n e d f r o m t h e g e n t i r e p r o f i l e , so w i t h H i n s t e a d of. h. The s h e a r s t r e s s due t o t h e c u r r e n t o n l y i s . • ' -^c - QV^^ »

So t h e r a t i o between t h e s h e a r s t r e s s o f c u r r e n t and waves and /

o n l y c u r r e n t a t any moment t can be w r i t t e n as ^>

. 4 ^ . \ / i . K'(^)\i^' ^ ( 1 5 )

The mean v a l u e can be o b t a i n e d v i a i n t e g r a t i o n o v e r j T and w i l l be T ^ ^ 1 . C = ( ^ ) % i n 2 a t 1 • 4 The i n t e g r a l i n t e r v a l f r o m 0 t o T / 4 i s s u f f i c i e n t as s i n i s v a r i a t i n g between 0 and + 1 . .

7

-T h i s i n t e g r a l , w h i c h i s o f t h e • l l i p t i o t y p e has been computed on an e l e c t r o n i c computer and t h e r e s u l t l a shown on f i g u r e 6.

T h i s f u n c t i o n oan be w r i t t e n I n t h e r e a c h o f I n t e r e s t as

f -

1 0 . 2 2 ( 1 7 )

T h i s d e r i v a t i o n has been made f o r a h y d r a u l i c r o u g h b o t t o m .

The same d e r i v a t i o n can be made f o r a h y d r a u l i c smooth b o t t o m or f o r t h e r e g i o n between h y d r a u l i c r o u g h and h y d r a u l i c smooth.

I n t h i s case however y' w i l l be d i f f e r e n t f o r t h e mere c u r r e n t and f o r t h e c o m b i n a t i o n o f c u r r e n t and waves, as i n t h i s case y' i s a l s o d e t e r m i n e d by t h e v i s c o u s s u b l a y e r w h i c h i s dependent on t h e v e l o c i t y ( s e e a l s o R o t t a , l i t t . 1 ) . T h i s works o u t r a t h e r c o m p l i c a t e d i n t h e f i n a l f o r m u l a e as y' i n t h i s case I s a f u n c t i o n o f \ ^ v ^ +(^* "b^^* AO y' w i l l be s m a l l e r w i t h i n c r e a s i n g v e l o c i t y ' ^ ' ( ^ j w i l l i n c r e a s e . For t h e r e d u c t i o n o f v , t o r a h i g h e r v a l u e o f C has t o be i n t r o d u c e d K G 80 t h e f a c t o r — — w i l l i n c r e a s e . These two e f f e c t s w i l l be i n t r o d u c e d by a n o t h e r v a l u e o f ^ w h i c h w i l l be t h e r e f o r e g r e a t e r .

The v a l u e o f w i l l be c a l c u l a t e d f r o m t h e measurements and p l o t t e d a g a i n s t k ( f i g u r e 7 ) .

No a t t e m p t w i l l be made f o r t h e moment t o d e t e r m i n e t h e r e l a t i o n between ji and k f o r c o n d i t i o n s smoother t h a n r o u g h , s i n c e t h i s r e l a -t i o n w i l l be a r a -t h e r c o m p l i c a -t e d f u n c -t i o n o f -t h e v e l o c i -t i e s . Moreover u n d e r n o r m a l c o n d i t i o n s t h e b o t t o m w i l l be h y d r a u l i c r o u g h or almost h y d r a u l i c r o u g h s o t h a t t h e miniaum v a l u e o f j i ' c a n be t a k e n . As f o r l a m i n a r m o t i o n t h e t h i c k n e s s o f t h e boundary l a y e r w i l l be p r o p o r t i o n a l w i t h •> i n w h i c h v i s t h e coëfficiënt o f k i n e -matic v i s c o s i t y (Lamb a r t . 5 4 5 , l i t t .2 ) i t i s n o t i m p o s s i b l e t h a t t h e r e w i l l be some i n f l u e n c e o f T on t h e f a c t o r ^. ïïirther t e s t s v / i l l loave t o be e x e c u t e d t o d e t e r m i n e t h i s e f f e c t . .

8

-4» lleasurementa. • • ,

The measurements were e x e c u t e d f o r t h r e e d i f f e r e n t b o t t o m c o n d i t i o n s , v i z , a b o t t o m covered w i t h s t o n e s w i t h a mean d i a -meter o f 3 t o 4 cm, a sand b o t t o m covered w i t h r i p p l e s o f some em's h e i g h t and a smooth sand b o t t o m . The mean d i a m e t e r o f t h e sand was 0 . 2 5 mm and t h e d^Q^ was 0 , 5 4 mm. F o r t h e t e s t s w i t h the smooth sand b o t t o m t h a roughness v a l u e o f t h e sand was i n

some cases a p p r e c i a b l e l o w e r as t h e b o t t o m was sometimes smoothered by some s i l t .

For t h e t e s t s w i t h t h e sand b o t t o m w i t h r i p p l e s s p e c i a l p r e -c a u t i o n s were t a k e n so t h a t t h e r i p p l e f o r m a t i o n d u r i n g t h e t e s t s w i t h o n l y c u r r e n t and w i t h waves and c u r r e n t were as much a l i k e as p o s s i b l e . T h i s was r e a l i s e d by measuring t h e s l o p e f o r o n l y c u r r e n t i m m e d i a t e l y a f t e r a t e s t w i t h waves and c u r r e n t . I n t h a t case t h e r i p p l e p a t t e r n changed so s l o w l y t h a t t h e same roughness value c o u l d be assumed. The r e s u l t s o f t h e measurements a r e g i v e h on t a b l e s 1 t h r o u g h 3 , and t h e r e s u l t s a r e p l o t t e d on f i g u r e s 8 t h r o u g h 1 0 , The r e s u l t s f o r t h e t h r e e i n v e s t i g a t e d b o t t o m c o n d i t i o n s can be g i v e n i n t h e f o l l o w i n g f o r m u l a e , 1 5 Bottom w i t h s t o n e s - I ) - 0 , 0 ? ~ ) ' ( 1 8 ) Sand b o t t o m w i t h r i p p l e s ( — - 1 ) - O.O85 ( ^ - J l ^ ) * ( 1 9 ) TQ Yg V F l a t sand b o t t o m (^^ - 1 ) - 0 , 1 2 5 { ^ - ~ . ^ ) ^ ' ^ ( 2 0 ) From c o m p a r i s o n between t h e measurements ( e q , 18 t h r o u g h 2 0 )

9

-The v a l u e s a r e For b o t t o m w i t h s t o n e s O.47 For sand b o t t o m w i t h r i p p l e s 0 , 5 5 For f l a t sand b o t t o m 0 , 6 8 ~p=^ C o n s i d e r i n g t h e a c c u r a c y o f t h e measurements t h e f o r m u l a f o r a rough b o t t o m - t h e n o r m a l case - oan be w r i t t e n as

( f -

1 ) . 0 . 2 2

ïs/-'

( 2 1 )

5. F u t u r e i n v e s t i g a t i o n .

Other t e s t s are executed i n o r d e r t o d e t e r m i n e t h e r e l a t i o n b e t w e e n u^/v and t h e i n c r e a s e o f s h e a r s t r e s s f o r o b l i q u e waves. The t h e o r e t i c a l r e l a t i o n has been a l r e a d y d e r i v e d and has been checked w i t h some measurements.

The same t e s t program has t o be e x e c u t e d f o r o t h e r p e r i o d s , s i n c e i t may be expected t h a t \i w i l l be a l s o a f u n c t i o n o f the p e r i o d of t h e waves.

Moreover these t e s t s have t o be completed w i t h d a t a about t h e t r a n s p o r t a t i o n o f b o t t o m m a t e r i a l .

10 -'TABLE I , B o t t o m w i t h s t o n e s , Test h V _{H I h i c} m m/ s 10-2 m 10-4 10~5 m mV2/, 121 0 . 2 0 • 0 . 1 0 _{0 . 3 9 0 . 7 9} 36 121g 0 . 2 0 ' 0 . 1 0 5 . 6 0 . 6 4 1 . 2 8 122 0.21 0 . 1 9 1 . 4 5 3 . 0 5 34 i22g 0.21 0 . 1 9 5 . 6 1.90 4 . 0 0 124*^ 0 . 3 5 0.21 1 . 0 0 5 . 5 0 36 124**g 0 . 3 5 0.21 6 . 8 1 . 5 8 5 . 5 0 124 _{0 . 3 5} 0 . 5 0 2 . 0 4 6 . 7 0 36 124g 0 . 3 5 0 . 5 0 6 . 6 2 . 7 3 9 . 0 0 126 0 . 1 9 0.21 2 . 2 4 _{4 . 2 5 32} i26g 0 . 1 9 0.21 2 . 2 2 . 6 6 5 . 0 5 101 0 . 1 9 0 . 2 2 2 . 5 6 4 . 8 5 31 lOlg 0 . 1 9 0 . 2 2 2 . 6 2 . 9 6 5 . 6 0 101 0 . 2 0 0.21 2.91 5 . 8 2 28 •i 01 "g 0 . 2 0 0.21 • 2 . 6 _{5 . 2 4} 6 . 4 8 102 0.21 0 . 2 4 2 . 5 7 5 . 4 0 33 1 02g 0.21 0 . 2 4 2 . 8 5 . 0 2 6 . 5 2 103 0 . 3 2 0.21 _{0 . 8 9 2 . 8 5 39} i05g 0 . 3 2 0.21 _{3 . 4} 1.06 _{5 . 3 9} 104 0 . 3 2 0 . 2 5 _1.54 4 . 2 8 ₅₈ 104g 0 . 3 2 0 , 2 5 3 . 8 1.56 4 . 9 8 105 0 . 3 6 0 . 2 4 _{0 . 8 4 3 . 0 5 44} i05g 0 . 3 6 0 . 2 4 _{4 . 7} 1.06 3.81 :06 0 . 3 6 0 . 2 7 _1.17 4.21 42 106g 0 , 3 6 0 . 2 7 _{4 . 4} 1 . 4 0 5 . 0 2 114 0.31 0 . 2 5 _{1 . 9 5} 6 . 0 5 32 0.31 0 . 2 5 6 . 0 _{2.41 7 . 4 9} ',15 _{0 . 3 5} 0 . 1 4 0 . 5 6 1 . 2 6 ₃₉ 0 . 3 5 0 . 1 4 6 . 6 0 . 6 4 2 . 2 4 116 _{0 . 3 5} 0 . 2 7 1 . 6 0 5 . 6 0 36 116g _{0 . 3 5 0 . 2 7 6 . 6 2 . 1 8 7 . 6 6}

k UQ T'/T(O) UQ/V ( u y v ) 2 c_K C)c ^o.

r 2 m ra/e Yg" /g v - 0 . 1 1 4 1.62 1.14 1.50 4.6 5 . 2 5.3 0 . 1 1 0 1.51 0 . 5 8 0 . 3 4 4 . 5 2 . 5 U 2 0 . 1 4 9 1.57 0 . 7 0 0 . 4 9 4.6' 3 . 2 0 . 1 5 0 1.34 0 . 5 0 0 . 2 5 4 . 6 2 . 3 1.8 0 . 0 7 2 1.18 0 . 3 5 0 . 1 2 4i,1 1.44 •4 0 . 0 8 5 1 . 1 5 0 . 3 9 0 . 1 5 4 . 0 1.56 0 . 0 8 2 1.11 0 . 3 9 . 7 0 . 0 8 6 1.17 0 . 3 5 . 8 0 . 0 7 9 1 . 1 9 0 . 5 8 0 . 1 4 5 . 0 1 . 9 0 . 8

I

0 . 0 8 8 1.16 0 . 3 5 0 . 1 2 4 , 9 1.72 . 8 0.101 1 . 2 9 0 . 4 2 0 . 1 8 5^6 2 . 3 0 . 0 9 4 1 . 1 9 0 . 5 4 0 . 1 2 5.4 1.84 2 ^' 0 . 1 4 3 1 . 2 4 0 . 5 7 0 , 3 3 4.1 2 . 5 . 9 ' ! 0 . 1 4 4 1 . 7 8 1.05 1 . 0 6 5^0 5 . 2 •2 0 . 1 4 4 1.36 0 . 5 3 0 . 2 8 4.6 2 . 4 0 . 1 5 3'.6 1.40 • I 0 , 1 2 4 . 2 1,46

11 -TABLE I I . Sandbottom w i t h r i p p l e s . T e s t H h i m m/s 10"^ m 10-4 10-5m 312 **• 0 . 2 0 0 . 1 5 0 . 2 2 _{0 . 4 4} 31 2 •'g 0 . 2 0 0 . 1 5 3 . 7 0 . 6 9 1 . 5 8 314 0.21 0.28 1 .28 2 . 6 9 514s 0,21 0.28 _{4 . 5} 2 . 1 7 _{4 . 5 5} 0.21 0 . 4 0 _{4 . 2 4} 8 . 9 0 315g 0.21 0 . 4 0 4 . 5 5 . 2 9 1 1 . 1 0 516 0 . 3 0 0 . 1 2 0 . 1 2 0 . 5 6 316g 0 . 3 0 0.12 5 . 8 0 . 5 9 1.17 516 0 . 3 0 0 . 1 2 0 . 1 2 0 . 3 6 516 "-g 0 . 3 0 0 . 1 2 5 . 8 0 . 5 4 1 . 0 2 317 0 . 3 0 0.21 0 . 4 5 1 . 2 9 5l7g 0 . 3 0 0.21 5 . 6 0.62 2 . 4 6 5I8 0 . 3 0 0 . 5 0 1 . 0 5 3 . 0 9 518g 0 . 3 0 0 . 5 0 _{5 . 5} 1.66 4 . 9 8 519 0 . 3 0 0 . 4 0 2 . 6 6 9.00 519g 0 . 3 0 0 . 4 0 6 . 4 5 . 6 5 1 0 . 9 6 320 0 . 3 8 0 . 1 5 0 . 1 2 0 . 4 6 520g 0 . 3 8 0 . 1 5 _{7 . 3 0 . 5 7 1.41} 322 0 . 3 8 0.51 0.91 3 . 4 6 522g 0 . 5 8 '0.51 7 . 5 1 . 4 8 5 . 6 3 300 0 . 2 0 0 . 1 5 0 . 2 7 _{0 . 5 4} 500g 0 . 2 0 0 . 1 5 2 . 5 0 . 4 5 0 . 8 6 502 0 . 2 0 0 . 5 0 1 . 5 3 . 0 0 502g 0 . 2 0 0 . 5 0 2 . 2 _{1 . 7 5} 3 . 5 0 502 * 0 . 2 0 0.51 1 .64 3.28 502 *g 0 . 2 0 0.51 2 . 3 2 . 0 2 4 . 0 4 303 0 . 2 0 _{0 . 5 7 4 . 2 7 8 . 5 4} 303g 0 . 2 0 _{0.57 2 . 5} 4 . 5 8 9.16 504 0 . 5 0 0 . 1 5 0. M 0 . 4 2 304g 0 . 5 0 0 . 1 5 2 . 6 0 . 2 0 0 . 6 0 62 54 45 63 63 59 54 50 70 53 56 55 54 40 64 0 . 9 2 . 5 7 . 9 0 . 8 0 . 8 1 . 9 3 . 6 6 . 0 0 . 6 5 . 2 1 . 8 2.1 2 . 4 1 4 . 5 1 . 0 m/s 9 0 . 1 1 6 3 . 1 4 ' 0 . 8 9 0 . 1 3 0 1 . 6 9 0 . 4 6 0 . 1 3 0 1.25 0 . 3 2 0 . 1 3 8 3 . 2 5 1 . 1 5 0 . 1 3 8 2 . 8 3 1 . 1 5 0 . 1 3 5 1.91 0 . 6 4 0.151 1.61 0 . 4 4 0 . 1 5 2 1.37 0 . 3 8 0 . 1 4 6 3 . 0 8 0 . 9 7 0 . 1 5 0 1,62 0 . 4 8 0 . 0 7 8 1 . 5 9 0 . 6 0 0 . 0 6 9 1.17 0 , 2 5 0 , 0 7 5 1.23 0 . 2 4 0 , 0 7 3 1.07 0 , 2 0 0 . 0 6 2 1,43 0 , 4 8 (uo/v)2 C K C K "o Vg \fs V 0 . 7 9 0,21 1 . 3 2 1.52 0.41 0 , 1 9 0 . 1 4 0 . 9 4 0 . 2 5 0 . 5 6 0 . 0 5 0 . 0 6 7 . 9 6. 8.0 8.0

I

I 7 . 5 I I 6 . 9 6 . 4 8.9 ,i 6 . 8 7 . 2 I] 7 . 0 •i 6 . 9 j J 7 . 0 ? 5 . 2 0 . 1 0 5»'7 1.82 9 . 2 9 . 2 4 . 8 3 . 0 2 . 4 8 . 6 3 . 3 4 . 3 1.61 1 . 6 5 0 . 0 4 5.1 1.02 0 . 2 3 8 . 2 3 . 8

12 -TABLE I I ( c o n t i n u e d ) Test h y H I h i C m m/8 10-2 m 10-4 1 0 - 5 m ml A . 506 0 . 3 0 0 . 3 3 0 . 7 9 2 . 3 7 68 306g 0 . 3 0 0 . 3 3 2 . 8 0 . 9 2 2.76 306 0 . 3 0 0 . 3 3 0 . 8 9 2 . 6 7 64 506 "g 0 . 3 0 0 . 3 3 2 . 8 1 .21 3 . 6 3 507 0 . 3 0 0 . 3 9 2 . 3 3 6*99 47 507 g 0 . 3 0 0 . 3 9 3 . 4 2 . 7 0 8 . 1 0 310 0 . 3 8 0 . 3 2 0 . 5 2 1.98 72 51 Og 0 . 3 8 0 . 3 2 _{4 . 5} 0 . 8 7 3 . 3 0 0 . 6 1 . 0 8 . 8 0 . 5 8.7 O.O67 1 . 1 6 0 . 2 0 0 . 0 4 0 . 0 6 7 1 . 3 6 0 . 2 0 0 . 0 4 8. 0*081 1.16 0.21 0 . 0 4 6 . 0 t 0 . 0 9 0 1.67 0.28 0 . 0 8 9 . 2 2 . 6 1.74 1.64 1.26 1 ^ M ^

13 -TABLE I I I . Tes-t} h V H I h i m m/s 10*^ m 10-4 1 0 - 5 a 9 0 . 4 0 0.21 0 . 1 9 0 . 7 6 9g 0 . 4 0 0.21 3 . 4 0 . 3 2 1.28 15 0 . 3 0 0.22 0 . 2 5 0 . 7 5 I5g 0 . 3 0 0.22 2. 6 0 . 3 8 _1.14 21 0 . 3 0 0.21 _{0 . 2 3 0 . 6 9} 21g 0 . 3 0 0.21 2 . 6 0 . 4 0 1 , 2 0 25 0 . 3 0 0.22 _{0 . 1 9 0 . 5 8} .25g 0 . 3 0 0.22 2 . 6 0 . 4 0 1 , 2 0 34 0 . 3 0 0 . 2 4 0 . 1 2 0 . 5 6 34g 0 . 3 0 0 . 2 4 5 . 2 0 . 4 2 1.26 53 0 . 4 0 0.22 _{0 . 4 2 1 , 6 8} 53g 0 . 4 0 0 . 2 2 6 . 6 1 . 0 0 4 . 0 0 66 0 . 2 0 0 . 2 0 _{0 . 3 3} 0 , 6 6 66 g 0 . 2 0 0;20 3 . 8 0 . 8 3 1.66 67 0 . 2 0 0 . 2 0 0 . 4 8 0 . 9 6 67g 0 . 2 0 0 . 2 0 3 . 8 0 . 8 8 1.76 69 0 . 2 5 0.21 _{0 . 5 7} 0 , 9 2 69g 0 . 2 5 0.21 v 3 . 8 0.82 2 , 0 5 70 0 . 2 5 0.21 0.28 0 , 7 0 70g 0 . 2 5 0.21 3 . 8 0 . 7 8 _{1 . 9 5} 79 0 . 3 0 • 0 . 2 4 0.28 0,82 79g 0 . 3 0 0 . 2 4 5 . 2 0 . 9 2 2 . 7 6 m^y^/a l O- 3 m m/s ^ 76 0 , 3 80 0,1 80 0.1 92 0 126 0 54 4 . 8 78 0,1 65 0. 6 69 0 , 4 80 0,1 84 0,1 0.067 1.69 0 , 3 2 0 , 1 0 9.7 5.1 0 , 0 6 3 1.52 0 . 2 9 0.08 10.2 5 , 0 0 , 0 6 5 1-74 0 . 5 0 0 , 0 9 . 1 0 ; . 2 5,1 0 , 0 6 3 2 , 0 7 0 , 2 9 0,08 11.7 5 . 4 0 , 1 2 6 5 . 5 0 0 . 5 2 , 0 . 2 7 16 _{'.1 8 . 4} 0 , 1 5 0 2 , 5 8 0 , 5 9 0 , 3 5 6I9 4.1 0 , 1 2 0 2 . 5 2 0 . 6 0 0 . 5 6 9w8 5 , 9 0, 1 2 0 1 , 8 5 0,60 0 , 5 6 8 3 5 . 0 0 , 1 0 5 2 . 2 3 0 , 5 0 0 , 2 5 8 1 8 4 . 4

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H Y D R A U L I C S L A B O R A T O R Y * D E L F T

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