on and de and F.G. a bed of in a in

sediment transport in rivers

flow of water in a curved open channel

with a fixed plane bed

H.J. de Vriend and F.G. Koch

report on experimental and theoretical

investigations

toegepast onderzoek

waterstaat

CONTENTS LIST OF SYMBOLS SUMMARY page 1 I n t r o d u c t i o n 1 2 E x p e r i m e n t a l s e t - u p 3 2.1 Channel geometry 3 2.2 Flow c o n d i t i o n s 3 2. 3 Measured d a t a 4 3 R e s u l t s 6 3.1 E x p e r i m e n t a l r e s u l t s 6 3.1.1 I n f l u e n c e o f t h e shape o f t h e t a i l - g a t e 6 3.1.2 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 h o r i z o n t a l v e l o c i t y components 7 3.1.3 Depth-averaged v e l o c i t y f i e l d 8 3.1.4 Water s u r f a c e c o n f i g u r a t i o n 9 3.2 Comparison w i t h t h e o r e t i c a l r e s u l t s 10 3.2.1 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 h o r i z o n t a l v e l o c i t y components 10 3.2.2 Depth-averaged v e l o c i t y f i e l d 12 3.2.3 Water s u r f a c e c o n f i g u r a t i o n 1 3 4 C o n c l u s i o n s ,,, 14 REFERENCES APPENDIX A I n s t r u m e n t a t i o n APPENDIX B M e a s u r i n g p r o c e d u r e APPENDIX C E l a b o r a t i o n o f measured d a t a APPENDIX D I n f l u e n c e o f e r r o r s i n t h e measured d a t a

LIST OF TABLES I Summary o f measurements I I Number o f c o u n t e d h o l e s d u r i n g p r o p e l l e r t e s t (TO-1) I I I D e v i a t i o n a n g l e v e r s u s o b s e r v a t i o n p e r i o d (TO--2) I V A v e r a g i n g r o u t i n e s f o r t h e d e p t h - a v e r a g e d v e l o c i t y V V e l o c i t i e s i n c r o s s - s e c t i o n EQ V I Water s u r f a c e e l e v a t i o n s i n c r o s s - s e c t i o n EQ V I I 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 m a i n v e l o c i t y i n t h e c h a n n e l a x i s V I I I 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 main v e l o c i t y i n s e c t i o n DQ I X 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 h e l i c a l v e l o c i t y i n t h e channel a x i s X 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 h e l i c a l v e l o c i t y i n s e c t i o n DQ X I Depth-averaged t o t a l v e l o c i t y v i n m/s t o t ^ X I I N o r m a l i z e d d e p t h - a v e r a g e d v e l o c i t y v ^ ^ ^ * X I I I Water s u r f a c e e l e v a t i o n i n m XIV Depth o f f l o w ( z ^ - z^) i n m XV Q u a n t i t i e s w h i c h a r e i n v a r i a n t t h r o u g h o u t t h e s i m u l a t i o n XVI Q u a n t i t i e s v a r y i n g randomly f r o m v e r t i c a l t o v e r t i c a l X V I I z-independent q u a n t i t i e s v a r y i n g f r o m p o i n t t o p o i n t X V I I I z-dependent q u a n t i t i e s v a r y i n g f r o m o b s e r v a t i o n t o o b s e r v a t i o n XIX R e s u l t s o f t h e n u m e r i c a l e r r o r a n a l y s i s

LIST OF FIGURES 1 Channel geometry 2 Bed e l e v a t i o n 3 Combined c u r r e n t - v e l o c i t y / d i r e c t i o n m e t e r 4 C a l i b r a t i o n c u r v e v e l o c i t y meter 5 T e s t s e r i e s o f v e l o c i t y measurements (TO) 6 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 n o r m a l i z e d main v e l o c i t y d u r i n g t a i l - g a t e t e s t ( T l ) 7 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 n o r m a l i z e d h e l i c a l v e l o c i t y d u r i n g t a i l - g a t e t e s t ( T l ) 8 Depth-averaged v e l o c i t y d u r i n g t a i l - g a t e t e s t ( T l ) 9 Water s u r f a c e e l e v a t i o n d u r i n g t a i l - g a t e t e s t ( T l ) 10 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 n o r m a l i z e d main v e l o c i t y i n t h e c h a n n e l a x i s (T2) 11 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 n o r m a l i z e d main v e l o c i t y i n c r o s s - s e c t i o n DQ (T2) 12 Summary o f t h e v e r t i c a l d i s t r i b u t i o n s o f t h e n o r m a l i z e d main v e l o c i t y (T2) 13 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 n o r m a l i z e d h e l i c a l v e l o c i t y i n t h e c h a n n e l a x i s (T2) 14 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 n o r m a l i z e d h e l i c a l v e l o c i t y i n c r o s s - s e c t i o n DQ (T2) 15 Summary o f n o r m a l i z e d h e l i c a l v e l o c i t i e s i n t h e c r o s s - s e c t i o n s C Q , DQ and EQ (T2) 16 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 n o r m a l i z e d h e l i c a l v e l o c i t y near t h e o u t e r w a l l i n c r o s s - s e c t i o n DQ (T2) 17 Depth-averaged v e l o c i t y f i e l d (T3)

18 Comparison between t h e depth-averaged v e l o c i t i e s (T3) 19 Water s u r f a c e e l e v a t i o n (T4)

20 Depth o f f l o w (T4)

LIST OF SYMBOLS

a Maximum e r r o r i n t h e measured f l o w a n g l e due t o t h e b a c k l a s h i n t h e keyway

A C r o s s - s e c t i o n a l a r e a B Channel w i d t h

c i Slope o f t h e c a l i b r a t i o n c u r v e o f t h e v e l o c i t y meter

C2 Value o f t h e v e l o c i t y c o r r e s p o n d i n g t o z e r o measured f r e q u e n c y o f pulses

C2^ Value o f cz a t c a l i b r a t i o n t e m p e r a t u r e C3 R a t i n g c o e f f i c i e n t o f t h e d i r e c t i o n meter Cl, C o e f f i c i e n t o f p r o p o r t i o n a l i t y i n t h e t e m p e r a t u r e c o r r e c t i o n t e r m o f C2 C Chezy's roughness f a c t o r Cl C a l i b r a t e d v a l u e o f c j d C h a r a c t e r i s t i c d e p t h o f f l o w dCj E r r o r i n Ci dZ E r r o r i n Z, f o r i > 1 1 d Z i E r r o r i n Z i

dt^l E r r o r i n t h e measured f l o w a n g l e due t o t h e b a c k l a s h i n t h e keyway

f=N/T Mean f r e q u e n c y o f p u l s e s c o u n t e d d u r i n g a v e l o c i t y measurement Fr Froude number g A c c e l e r a t i o n due t o g r a v i t y h L o c a l d e p t h o f f l o w hv Depth o f f l o w i n t h e c h a n n e l a x i s H Measured l o c a l d e p t h o f f l o w

E C E Q) Depth o f f l o w averaged over c r o s s - s e c t i o n E Q

i Number o f t h e g r i d p o i n t ( i = 1 n e a r t h e bed)

n Number o f p u l s e s t o be counted by t h e v e l o c i t y meter d u r i n g an o b s e r v a -t i o n

N Number o f p u l s e s counted by t h e v e l o c i t y meter d u r i n g an o b s e r v a t i o n np Number o f g r i d p o i n t s i n a v e r t i c a l Q D i s c h a r g e R L o c a l r a d i u s o f c u r v a t u r e R Radius o f c u r v a t u r e o f t h e c h a n n e l a x i s c Re Reynolds number T D u r a t i o n o f t h e o b s e r v a t i o n p e r i o d f o r a v e l o c i t y measurement terap^ Water t e m p e r a t u r e d u r i n g t h e c a l i b r a t i o n o f t h e v e l o c i t y meter temp^ Water t e m p e r a t u r e d u r i n g t h e measurements

LIST OF SYMBOLS ( c o n t i n u a t i o n ) V Mean v e l o c i t y V, T L o c a l time-mean h e l i c a l v e l o c i t y component h e l V,' 1 N o r m a l i z e d h e l i c a l v e l o c i t y component h e i

V . L o c a l time-mean main v e l o c i t y component

m a m v' . N o r m a l i z e d main v e l o c i t y component m a m ^ v ^ L o c a l time-mean r a d i a l v e l o c i t y component v ^ Depth-averaged v a l u e o f v ^ v ^ L o c a l time-mean t a n g e n t i a l v e l o c i t y component Depth-averaged v a l u e o f v^. ^ t V j . ^ j . L o c a l time-mean t o t a l v e l o c i t y v ^ ^ ^ Magnitude o f t h e d e p t h - a v e r a g e d v e l o c i t y v e c t o r z V e r t i c a l d i s t a n c e t o t h e bed z^ Bed l e v e l ( z ^ = 0 i n t h e s t r a i g h t s e c t i o n )

z Water s u r f a c e e l e v a t i o n (datum: bed i n t h e s t r a i g h t s e c t i o n ) s z Water s u r f a c e e l e v a t i o n averaged o v e r t h e r e l e v a n t c r o s s - s e c t i o n s Measured v e r t i c a l d i s t a n c e o f t h e g r i d p o i n t i t o t h e bed a D i r e c t i o n o f t h e d e p t h - a v e r a g e d v e l o c i t y v e c t o r n Dynamic v i s c o s i t y o f t h e f l u i d rio Dynamic v i s c o s i t y o f t h e f l u i d d u r i n g c a l i b r a t i o n o f t h e v e l o c i t y me K Von Karman's c o n s t a n t y , Mean o f dCj V K i n e m a t i c v i s c o s i t y o f t h e f l u i d aStandard d e v i a t i o n o f dCi dCj a ,„ S t a n d a r d d e v i a t i o n o f dZ dZ a ,„ S t a n d a r d d e v i a t i o n o f dZi dZi S t a n d a r d d e v i a t i o n o f t h e e r r o r i n H a., S t a n d a r d d e v i a t i o n o f t h e e r r o r i n f l (j) L o c a l time-mean f l o w a n g l e

(j)' L o c a l time-mean f l o w a n g l e t o be measured by t h e d i r e c t i o n meter <j)J True r e f e r e n c e a n g l e o f t h e d i r e c t i o n m e t e r

$' Measured l o c a l time-mean f l o v ; a n g l e

SUMMARY

I n t h i s r e p o r t t h e i l o w o f w a t e r i n a c u r v e d open c h a n n e l , w h i c h c o n s i s t s o f a 38 m l o n g s t r a i g h t s e c t i o n f o l l o w e d by a 90° bend w i t h a r a d i u s o f c u r v a t u r e o f 50 m (see F i g u r e 1 ) , has been i n v e s t i g a t e d . The c h a n n e l c r o s s - s e c t i o n was r e c t a n g u l a r w i t h a h o r i z o n t a l c o n c r e t e bed, a w i d t h o f 6 m, and a d e p t h o f f l o w o f 0.25 m, and measurements were e x e c u t e d a t two d i s c h a r g e s : 0.610m^/s and 0.305 m^/s (average v e l o c i t i e s o f about 0.4 m/s and 0.2 m/s r e s p e c t i v e l y ) .

D u r i n g these e x p e r i m e n t s t h e f o l l o w i n g phenomena were i n v e s t i g a t e d :

a. 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 t h e h o r i z o n t a l v e l o c i t y components (main f l o w and h e l i c a l f l o w ) ;

b. t h e h o r i z o n t a l d i s t r i b u t i o n o f t h e t o t a l d e p t h - a v e r a g e d v e l o c i t y ; and c. t h e h o r i z o n t a l d i s t r i b u t i o n o f t h e w a t e r s u r f a c e e l e v a t i o n .

The e x p e r i m e n t a l r e s u l t s have been compared w i t h t h e r e s u l t s o f a m a t h e m a t i c a l model o f f l o w i n c u r v e d open c h a n n e l s , d e v e l o p e d a t t h e L a b o r a t o r y o f F l u i d Mechanics o f t h e D e l f t U n i v e r s i t y o f Technology L' 2J .

The v e r t i c a l d i s t r i b u t i o n s o f t h e main v e l o c i t y t u r n e d o u t t o be h i g h l y s i m i l a r t h r o u g h o u t t h e f l o w f i e l d , t h e d i s t r i b u t i o n b e i n g w e l l d e s c r i b e d by t h e l o g a -r i t h m i c p -r o f i l e .

The h e l i c a l v e l o c i t i e s d e r i v e d f r o m t h e measured d a t a were t o o i n a c c u r a t e t o compare them more t h a n r o u g h l y w i t h t h e i r t h e o r e t i c a l d i s t r i b u t i o n s . The p o i n t i n a c r o s s s e c t i o n where t h e observed d e p t h a v e r a g e d v e l o c i t y r e a c h e s i t s m a x i -mum l i e s near t h e i n n e r w a l l i n t h e f i r s t p a r t o f t h e c u r v e and moving

downs t r e a m i t g r a d u a l l y downs h i f t downs towarddowns t h e o u t e r w a l l . T h i downs phenomenon i downs a t t r i b u t -ed t o t h e a d v e c t i v e i n f l u e n c e o f t h e h e l i c a l f l o w . As t h i s i n f l u e n c e i s n o t a c c o u n t e d f o r i n t h e p r e s e n t m a t h e m a t i c a l model, t h i s model does n o t p r e d i c t t h e phenomenon. The w a t e r s u r f a c e c o n f i g u r a t i o n agreed r e a s o n a b l y w e l l w i t h t h e computed c o n f i g u r a t i o n .

FLOW OF WATER I N A CURVED OPEN CHANNEL WITH A FIXED PLANE BED

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

W i t h i n t h e a p p l i e d r e s e a r c h group f o r sediment t r a n s p o r t i n r i v e r s , i n w h i c h R i j k s w a t e r s t a a t j D i r e c t o r a t e f o r w a t e r Management and Research, t h e 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 and t h e Department o f C i v i l E n g i n e e r i n g o f t h e D e l f t U n i v e r s i t y o f T e c h n o l o g y c o l l a b o r a t e , one o f t h e r e s e a r c h teams i s concerned w i t h w a t e r move-ment and sedimove-ment t r a n s p o r t i n r i v e r bends. The main purpose o f t h i s group i s

the development o f a p r e d i c t i o n method f o r t h e f l o w and t h e bed topography i n c u r v e d a l l u v i a l r i v e r s , w h i c h w i l l meet t h e r e q u i r e m e n t s o f r i v e r e n g i n e e r i n g

( p r e d i c t i o n o f n a v i g a b i l i t y , d e p t h o f bank p r o t e c t i o n , e t c . ) now and i n t h e n e a r f u t u r e . T h i s p r o j e c t i s i n c o r p o r a t e d i n a b a s i c r e s e a r c h programme T.O.W. (Toe-gepast Onderzoek W a t e r s t a a t ; A p p l i e d Research W a t e r s t a a t ) .

A t p r e s e n t p h y s i c a l models a r e w i d e l y used f o r t h e above-mentioned p r e d i c t i o n s , b u t t h e s e models a r e r a i s i n g a number o f problems about t h e i n t e r p r e t a t i o n o f

the model d a t a i f g r e a t a c c u r a c y i s needed. T h e r e f o r e , t h e development o f mathe-m a t i c a l mathe-models has becomathe-me a t t r a c t i v e , even though t h e s e have t h e i r own s p e c i f i c

l i m i t a t i o n s ( s c h e m a t i z a t i o n o f t h e geometry and t h e f l o w , e t c ) .

As a f i r s t s t e p towards a m a t h e m a t i c a l model f o r c u r v e d a l l u v i a l r i v e r s , a mat h e m a mat i c a l model o f mat h e f l o w i n c u r v e d s h a l l o w c h a n n e l s w i mat h v a r i a b l e c r o s s -s e c t i o n i -s b e i n g developed i n t h e L a b o r a t o r y o f F l u i d Mechanic-s o f t h e D e l f t U n i v e r s i t y o f Technology. A f i r s t v e r s i o n o f t h i s model was t e s t e d a g a i n s t a v a i l a b l e e x p e r i m e n t a l d a t a f r o m l a b o r a t o r y f l u m e s , m o s t l y w i t h f l a t beds, b u t y i e l d e d p a r t i a l l y n e g a t i v e r e s u l t s 12 . None o f t h e s e f l u m e s , however, had g e o m e t r i c a l p r o p o r t i o n s c o r r e s p o n d i n g t o those o f t h e Dutch n a t u r a l r i v e r s , w h i l e i n a d d i t i o n no s u f f i c i e n t " p r o t o t y p e " d a t a were a v a i l a b l e f r o m these r i v e r s .

I n o r d e r t o f i l l up these l a c u n a e , a s e r i e s o f e x p e r i m e n t s was executed a t t h e De V o o r s t L a b o r a t o r y o f t h e 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 (where a c u r v e d chan-n e l was a v a i l a b l e , h a v i chan-n g a f i x e d f l a t bed achan-nd g e o m e t r i c a l p r o p o r t i o chan-n s w h i c h a r e r e p r e s e n t a t i v e o f t h e Dutch r i v e r b r a n c h e s ) , and an e x t e n s i v e m e a s u r i n g programme was c a r r i e d o u t i n a bend o f t h e R i v e r I J s s e l . Moreover, a second s e r i e s o f e x p e r i m e n t s i n t h e L a b o r a t o r y channel i s planned, w i t h a n o n - f l a t , f i x e d bed h a v i n g a c o n f i g u r a t i o n o f banks and t r o u g h s as i n a n a t u r a l bend. The r e s u l t s o f these e x p e r i m e n t s w i l l be p u b l i s h e d s e p a r a t e l y i n a second p a r t o f t h i s r e -p o r t . I n t h i s P a r t I t h e measurements i n t h e f l a t bed l a b o r a t o r y channel a r e d e s c r i b e d and t h e r e s u l t s compared w i t h r e s u l t s f r o m t h e m a t h e m a t i c a l model.

2

-The e x p e r i m e n t s were e x e c u t e d by H.J. de V r i e n d o f t h e D e l f t U n i v e r s i t y o f T e c h n o l o g y and by F.G. Koch o f t h e 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 . The p r e s e n t r e p o r t i s t h e r e s u l t o f t h e i r m u t u a l e f f o r t .

-3-2 E x p e r i m e n t a l s e t - u p

D u r i n g t h e e x p e r i m e n t s p r e s e n t e d i n t h i s r e p o r t , t h e f l o w ( v e l o c i t y and d i r e c -t i o n ) and -t h e e l e v a -t i o n o f -t h e f r e e w a -t e r s u r f a c e were measured i n a c u r v e d

f i x e d bed c h a n n e l .

2.1 Channel geometry

The geometry o f t h e channel i s g i v e n i n F i g u r e 1. I t c o n s i s t e d o f a 38 m l o n g s t r a i g h t s e c t i o n , f o l l o w e d by a 90° bend w i t h a r a d i u s o f c u r v a t u r e o f 50 m

(see F i g u r e 1 ) . Throughout t h e c o n c r e t e c h a n n e l , t h e c r o s s s e c t i o n was r e c t a n -g u l a r , 6 m wide w i t h a maximum d e p t h o f 0.30 m. The c h a n n e l bed was h o r i z o n t a l i n t h e s t r a i g h t p a r t and had a l o n g i t u d i n a l s l o p e o f 3 10"** i n t h e c h a n n e l -a x i s i n t h e c u r v e d p -a r t (see F i g u r e 2 -a ) . The -a c t u -a l bed e l e v -a t i o n i s g i v e n i n F i g u r e 2b. As can be seen i n t h i s f i g u r e , t h e maximum d e v i a t i o n f r o m t h e t h e o -r e t i c a l bed l e v e l was about 0.3 * 10"^ m.

T h i s c h a n n e l , o r i g i n a l l y c o n s t r u c t e d f o r i n v e s t i g a t i o n s i n t o n a v i g a t i o n p r o b l e m s , was c o n v e n i e n t f o r t e s t i n g t h e m a t h e m a t i c a l model, because i t s most i m -p o r t a n t g e o m e t r i c -parameters ( v i z . , t h e d e -p t h - w i d t h r a t i o and t h e d e -p t h - r a d i u s o f c u r v a t u r e r a t i o ) agreed w e l l w i t h t h o s e i n t h e Dutch r i v e r branches

(d/B ^ 5 « 10"^; d/R^ < 6 ^ 1 0 " ^ ) ; l i k e w i s e , t h e y agreed b e t t e r w i t h t h e b a s i c assumptions made f o r t h e m a t h e m a t i c a l model t h a n i n most o f t h e o t h e r channels f r o m w h i c h e x p e r i m e n t a l d a t a a r e a v a i l a b l e \_9_ . B e s i d e s , t h e r e was a p o s s i b i l i -t y f o r l a -t e r e x p e r i m e n -t s , w i -t h a f i x e d bed h a v i n g a l a r g e - s c a l e c o n f i g u r a -t i o n o f banks and t r o u g h s as i n n a t u r a l r i v e r s . From t h i s t y p e o f e x p e r i m e n t s v e r y l i t t l e d a t a a r e a v a i l a b l e [_12, 13, 1 4 j , a l t h o u g h t h e y a r e even more i m p o r t a n t f o r t e s t i n g t h e m a t h e m a t i c a l model t h a n t h e f l a t bed e x p e r i m e n t s , t h e c h a n n e l c o n f i g u r a t i o n b e i n g c l o s e r t o t h e n a t u r a l one.

2.2 Flow c o n d i t i o n s

The w a t e r i n f l o w a t t h e upstream end o f t h e channel was n o t d i s t r i b u t e d u n i f o r m l y o v e r t h e channel w i d t h , as can be seen f r o m t h e measured 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 A j , F i g u r e 18a. P e r f e c t u n i f o r m i n f l o w was n o t n e c e s s a r y , s i n c e t h e measured mean i n f l o w d i s t r i b u t i o n c o u l d be imposed a t t h e u p s t r e a m boundary o f t h e m a t h e m a t i c a l model.

-4-The d i s c h a r g e was measured and r e g u l a t e d by a movable R o m i j n w e i r . Two s e r i e s o f measurements were c a r r i e d o u t , one w i t h a d i s c h a r g e o f 0.610 m'/s ( b e i n g t h e maximum d i s c h a r g e o f t h e e x i s t i n g w a t e r c i r c u l a t i o n s y s t e m ) , and t h e o t h e r one w i t h a d i s c h a r g e o f 0.305 m^/s. I n a d d i t i o n , t h e d e p t h o f f l o w a t t h e up-s t r e a m end o f t h e c h a n n e l waup-s k e p t c o n up-s t a n t (0.25 m d u r i n g b o t h e x p e r i m e n t up-s , y i e l d i n g average v e l o c i t i e s o f about 0.4 m/s and 0.2 m/s r e s p e c t i v e l y ) . The d e p t h o f f l o w was r e g u l a t e d by a movable t a i l - g a t e a t t h e downstream end o f t h e c h a n n e l . For t h e o p e r a t i o n o f t h i s t a i l - g a t e , measurements o f t h e w a t e r s u r f a c e e l e v a t i o n were e x e c u t e d i n t h e m i d d l e o f c r o s s - s e c t i o n AQ and near t h e s i d e w a l l s o f c r o s s - s e c t i o n BQ. The d i s c h a r g e and t h e w a t e r s u r f a c e e l e v a t i o n i n t h e s e t h r e e r e f e r e n c e p o i n t s were checked r e g u l a r l y d u r i n g t h e measurements, and, when n e c e s s a r y , t h e w e i r s ( R o m i j n w e i r and t a i l - g a t e ) were r e - a d j u s t e d .

The most i m p o r t a n t d a t a c o n c e r n i n g t h e c h a n n e l dimensions and t h e f l o w c o n d i -t i o n s a r e summarized i n -t h e f o l l o w i n g -t a b l e , a l o n g w i -t h -t h e Reynolds number Re = ^^-^ and t h e Froude number F r = -X- .

V /gh d i s c h a r g e d e p t h w i d t h mean v e l o c i t y Reynolds number Froude number Q d B d/B d / r a d i u s V Re Fr m V s m m -

-

The e x p e r i m e n t a l t e s t i n g o f t h e m a t h e m a t i c a l model was c o n c e n t r a t e d on t h r e e phenomena, v i z . :

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 t h e h o r i z o n t a l v e l o c i t y components (main f l o w and h e l i c a l f l o w ) (T2, f o r summary o f t h e measurement numbers see T a b l e I ) ; - t h e h o r i z o n t a l d i s t r i b u t i o n o f t h e t o t a l d e p t h - a v e r a g e d v e l o c i t y (T3) ; and

5I n o r d e r t o g a t h e r e x p e r i m e n t a l i n f o r m a t i o n a b o u t t h e s e phenomena, t h e t i m e -a v e r -a g e d m-agnitude -and d i r e c t i o n o f t h e v e l o c i t y v e c t o r were me-asured i n -a t h r e e - d i m e n s i o n a l g r i d , u s i n g a combined c u r r e n t - v e l o c i t y / d i r e c t i o n m e t e r , and t h e w a t e r s u r f a c e e l e v a t i o n was measured i n s e l e c t e d v e r t i c a l s o f t h i s g r i d ,

u s i n g s t a t i c tubes (see Appendix A ) .

The measuring p r o c e d u r e used i s d e s c r i b e d and m o t i v a t e d i n Appendix B.

The g r i d p o i n t s were d e f i n e d by 9 c r o s s - s e c t i o n s (numbered A i t o E Q; see Figure l a ) , 13 e q u i d i s t a n t v e r t i c a l s i n each c r o s s s e c t i o n (numbered 1 t o 13, see F i g u r e l b ) and 10 e q u i d i s t a n t g r i d p o i n t s per v e r t i c a l ( i n t e r v a l 0.025 m). C r o s s -s e c t i o n AQ t u r n e d o u t t o be u n s u i t a b l e f o r i n c l u s i o n i n t h e g r i d as t h e f l o w was n o t y e t f u l l y e s t a b l i s h e d i n t h i s s e c t i o n s i t u a t e d r a t h e r c l o s e t o t h e

up-s t r e a m boundary. To f i n d o u t w h e t h e r t h e t a i l - g a t e had a n e g l i g i b l e i n f l u e n c e on t h e f l o w i n s e c t i o n E Q, a s p e c i a l s e r i e s o f measurements ( T l ) was made i n t h i s s e c t i o n w i t h t h r e e d i f f e r e n t shapes o f t h e g a t e ( h o r i z o n t a l c r e s t , s t r a i g h t c r e s t i n c l i n e d o u t w a r d w i t h t h e l e v e l a t t h e o u t e r w a l l 2 cm h i g h e r t h a n a t t h e i n n e r w a l l , and s t r a i g h t c r e s t i n c l i n e d i n w a r d w i t h t h e l e v e l a t t h e i n n e r w a l l 1 cm h i g h e r t h a n a t t h e o u t e r w a l l ) . From t h e s e measurements i t was c o n -c l u d e d t h a t t h e f l o w i n s e -c t i o n EQ was n o t p e r c e p t i b l y i n f l u e n c e d by t h e shape o f t h e t a i l - g a t e (see a l s o S e c t i o n 3.1.1). Hence Eg was i n c l u d e d i n the meas-u r i n g g r i d .

-6-3 R e s u l t s 3.1 - E x p e r i m e n t a l r e s u l t s The e l a b o r a t i o n o f t h e measured d a t a i s d e s c r i b e d i n A p p e n d i x C. 3.1.1 I n f l u e n c e o f t h e shape o f t h e t a i l - g a t e ( T l ) As t h e t a i l g a t e was s i t u a t e d i n a c u r v e d s e c t i o n o f t h e c h a n n e l where a t r a n s -v e r s e s l o p e o f t h e w a t e r s u r f a c e o c c u r r e d , t h e -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 t h e g a t e c o u l d be expected t o be n o n u n i f o r m . To f i n d o u t w h e t h e r t h i s n o n u n i f o r -m i t y i n f l u e n c e d t h e f l o w i n t h e -most downstrea-m c r o s s - s e c t i o n o f t h e -m e a s u r i n g g r i d , E Q, a s e r i e s o f measurements was made i n t h i s c r o s s - s e c t i o n , w i t h t h r e e s i t u a t i o n s b e i n g examined under e q u a l f l o w c o n d i t i o n s (Q = 0.610 m^/s): a. c r e s t h o r i z o n t a l ( T l - 1 ) ; b. c r e s t i n c l i n e d w i t h t h e l e v e l a t t h e o u t e r w a l l 0.01 m l o w e r t h a n a t t h e i n n e r w a l l ( T l - 2 ) ; and c. c r e s t i n c l i n e d w i t h t h e l e v e l a t t h e o u t e r w a l l 0.02 m h i g h e r t h a n a t t h e i n n e r w a l l ( T l - 3 ) .

I n t h e s i t u a t i o n s T l - 2 and T l - 3 t h e g a t e was made o b l i q u e by f i x i n g a b a r on i t and a d j u s t i n g t h e p o s i t i o n o f t h e g a t e u n t i l t h e w a t e r s u r f a c e e l e v a t i o n i n t h e r e f e r e n c e p o i n t s i n c r o s s - s e c t i o n s AQ and Bo was t h e same as i n T e s t T l - 1 . I n a l l t h r e e s i t u a t i o n s t h e v e l o c i t i e s were measured a t 10 p o i n t s o f t h e v e r t i c a l s Eo3, EO7 , and E Q1 1 , and t h e w a t e r s u r f a c e e l e v a t i o n was measured i n t h e v e r t i -c a l s 1, 3, 5, 7, 9, 11 and 13 i n -c r o s s - s e -c t i o n EQ. T a b l e V shows t h e r e s u l t s o f t h e v e l o c i t y measurements and T a b l e V I g i v e s t h e observed w a t e r s u r f a c e e l e v a t i o n s . A d d i t i o n a l l y , F i g u r e s 6-9 g i v e a g r a p h i c a l r e s p r e s e n t a t i o n .

I t can be concluded f r o m Tables V and V I and F i g u r e s 69 t h a t 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 -t h e main v e l o c i -t y componen-t and -t h e shape o f -t h e w a -t e r s u r f a c e were h a r d l y i n f l u e n c e d by t h e shape o f t h e t a i l g a t e . 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 h e l i c a l v e l o c i t y component was t o o i n a c c u r a t e t o draw any c o n c l u s i o n s i n t h i s r e s p e c t . The o n l y i m p o r t a n t d e v i a t i o n o f t h e depthaveraged v e -l o c i t y was found i n Eo11 d u r i n g t e s t T -l - 2 ( c r e s t h i g h e r near t h e i n n e r w a -l -l ) , t h e v e l o c i t y b e i n g about 12% s m a l l e r t h a n t h e o v e r a l l average v e l o c i t y i n t h e s e c t i o n (0.383 m/s). T h i s was no r e a s o n , however, t o e x c l u d e s e c t i o n Eo f r o m t h e m e a s u r i n g g r i d .

-7-3.1.2 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 h o r i z o n t a l v e l o c i t y components (T2) The r e s u l t s o f t h e 1 0 p o i n t v e l o c i t y measurements c o n c e r n i n g 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 -t h e h o r i z o n -t a l v e l o c i -t y componen-ts a r e g i v e n i n Tables V I I . . . X and F i g u r e s 10... 16. J u d g i n g t h e r e s u l t s summarized i n t h e s e t a b l e s and f i g u r e s t h e f o l l o w i n g r e

-marks can be made:

a. F o r b o t h d i s c h a r g e s (0.610 and 0.305 m^/s) t h e v e r t i c a l d i s t r i b u t i o n s o f the n o r m a l i z e d m a i n v e l o c i t y were s i m i l a r i n t h e v a r i o u s m e a s u r i n g s t a t i o n s . Only t h e d i s t r i b u t i o n s i n t h e v e r t i c a l s Do 1 and Do 13 ( i . e . , t h e v e r t i c a l s the c l o s e s t t o t h e w a l l s ) tended t o d e v i a t e f r o m t h e o t h e r ones ( f i g u r e I I ) : i n b o t h v e r t i c a l s t h e v e l o c i t y r e d u c t i o n s near t h e w a t e r s u r f a c e were l a r g -er and extended o v e r a l a r g e r p a r t o f t h e v e r t i c a l . F o r t h e v e r t i c a l near the i n n e r w a l l (Do 13) an e x p l a n a t i o n o f t h i s phenomenon can be found i n t h e a d v e c t i v e i n f l u e n c e o f t h e h e l i c a l f l o w on t h e main v e l o c i t y d i s t r i b u t i o n

10, 1 1 , 12_ , b u t f r o m t h e same p o i n t o f v i e w t h e o p p o s i t e ( i . e . , i n c r e a s e o f t h e v e l o c i t y near t h e w a t e r s u r f a c e ) s h o u l d have been expected i n t h e v e r t i c a l near t h e o u t e r w a l l , i f a t l e a s t t h e h e l i c a l c i r c u l a t i o n had t h e same sense o f r o t a t i o n t h r o u g h o u t t h e c r o s s s e c t i o n . On c l o s e r i n v e s t i g a -t i o n o f -t h e f l o w near -t h e o u -t e r w a l l , however, a c o u n -t e r - r o -t a -t i n g h e l i x t u r n e d o u t t o o c c u r t h e r e i n t h e upper h a l f o f t h e f l o w (see F i g u r e s 14...16). Hence t h e v e l o c i t y d i s t r i b u t i o n i n DQ1 can be e x p l a i n e d f r o m a d v e c t i v e e f -f e c t s as w e l l .

b. There were d i f f e r e n c e s between t h e v e r t i c a l d i s t r i b u t i o n s o f t h e main v e l o -c i t y f o r t h e two d i s -c h a r g e s . Assuming t h e s e d i s t r i b u t i o n s t o be l o g a r i t h m i -c

(see S e c t i o n 3.2.1 Eq. 1) t h e Chêzyfactor t o be d e r i v e d f r o m s e m i l o g a -r i t h m i c p l o t s o f t h e v e l o c i t i e s a t 0.610 m^/s ( F i g u -r e 12) was about 70 m^/s, whereas t h e v a l u e d e r i v e d f r o m t h e p l o t s a t 0.305 m^/s was about 50 m^/s.

I n a d d i t i o n , t h e v e l o c i t y r e d u c t i o n near t h e w a t e r s u r f a c e was r e l a t i v e l y l a r g e r i n t h e f o r m e r case.

The d i f f e r e n c e i n t h e Chêzy-factors cannot be e x p l a i n e d by d i f f e r e n c e s i n the bed roughness, as t h e bed was c l e a n e d b e f o r e b o t h m e a s u r i n g s e r i e s . A l -so, as i n b o t h cases t h e d e p t h o f f l o w was a p p r o x i m a t e l y t h e same, t h e ex-p l a n a t i o n cannot be found i n a d i f f e r e n t d e ex-p t h o f f l o w e i t h e r .

-8"

t h e main f l o w t h r o u g h a d v e c t i v e e f f e c t s was s t r o n g e r a t t h e h i g h e r f l o w r a t e . T h i s a d v e c t i o n caused a v e l o c i t y r e d u c t i o n i n t h e upper p a r t o f t h e f l o w and an i n c r e a s e o f t h e v e l o c i t y i n t h e l o w e r p a r t . The l a t t e r can e f -f e c t a s t e e p e r s l o p e i n t h e s e m i - l o g a r i t h m i c v e l o c i t y p l o t s , s u g g e s t i n g a h i g h e r v a l u e o f t h e C h e z y - f a c t o r t h a n w o u l d o c c u r i n t h e c o r r e s p o n d i n g s t r a i g h t channel f l o w . c. The measurements o f t h e h e l i c a l v e l o c i t y p r o v e d t o be t o o i n a c c u r a t e t o draw c o n c l u s i o n s c o n c e r n i n g t h e s i m i l a r i t y o f i t s v e r t i c a l d i s t r i b u t i o n s i n t h e v a r i o u s v e r t i c a l s . I f t h i s s i m i l a r i t y i s assumed, t h e measured r e -s u l t -s f o r t h e c r o -s -s - -s e c t i o n -s Co, Do and Eo g i v e an i n d i c a t i o n o f " t h e " he-l i c a he-l v e he-l o c i t y c u r v e , and i t s p o s s i b he-l e d e v i a t i o n due t o t h e i n a c c u r a c y i n t h e measurements ( F i g u r e s 1 3 , 1 4 , 1 5 ) . d. I f t h e r e i s a h e l i c a l v e l o c i t y i n c r o s s - s e c t i o n Bo, i t i s much s m a l l e r t h a n i n t h e s e c t i o n s f u r t h e r downstream ( F i g u r e 1 3 ) . e. Near t h e o u t e r w a l l i n c r o s s s e c t i o n Do a h e l i c a l c i r c u l a t i o n i n a c l o c k -w i s e d i r e c t i o n o c c u r r e d i n t h e upper h a l f o f t h e f l o -w , p u s h i n g t h e " n o r m a l " c o u n t e r - c l o c k w i s e r o t a t i o n t o t h e l o w e r h a l f ( F i g u r e 1 6 ). Dye i n j e c t i o n s near t h e o u t e r w a l l showed t h i s c i r c u l a t i o n t o develop s h o r t l y a f t e r t h e b e g i n n i n g o f t h e bend, t o r e a c h i t s maximum w i d t h ( a b o u t 0 . 5 m) between

t h e c r o s s s e c t i o n s Co and Do and t h e n t o decrease s l o w l y u n t i l i t had v a n -i s h e d between c r o s s - s e c t -i o n s D-i and E Q .

3 . 1 . 3 Depth-averaged v e l o c i t y f i e l d ( T 3 )

The d e p t h - a v e r a g e d v e l o c i t i e s r e s u l t i n g f r o m t h e 6 - p o i n t measurements (see Appendix B.2.2) t h r o u g h o u t t h e channel a r e g i v e n i n Table X I and F i g u r e 1 7 .

I n o r d e r t o compare t h e r e s u l t s f o r t h e two d i s c h a r g e s ( 0 . 6 1 0 and 0 . 3 0 5 m^/s), t h e d e p t h - a v e r a g e d v e l o c i t i e s have been n o r m a l i z e d by t h e mean v a l u e o f t h e v e l o c i t y i n t h e r e l e v a n t c r o s s s e c t i o n , computed f r o m t h e d e p t h a v e r a g e d v a l -ues u s i n g t h e t r a p e z i u m r u l e x ^ i t h zero v e l o c i t y a t t h e w a l l s . The r e s u l t s a r e g i v e n i n T a b l e X I I and F i g u r e 18a.

-9-The c o n c l u s i o n s t o be drawn f r o m t h e s e t a b l e s and f i g u r e s a r e :

a. The 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 t h e u p s t r e a m boundary ( c r o s s - s e c t i o n A i ) i s n o t q u i t e u n i f o r m , and t h e n o n - u n i f o r m i t y has n o t y e t damped o u t when

t h e b e g i n n i n g o f t h e bend i s reached ( c r o s s - s e c t i o n B o ) .

b. I n t h e f i r s t p a r t o f t h e bend t h e v e l o c i t y maximum l i e s near t h e i n n e r w a l l , b u t g r a d u a l l y s h i f t s towards t h e o u t e r w a l l when moving f u r t h e r downstream.

c. When moving downstream t h r o u g h t h e bend, t h e r e g i o n where t h e f l o w i s appa-r e n t l y i n f l u e n c e d by t h e i n n e appa-r w a l l tends t o gappa-row l a appa-r g e appa-r , wheappa-reas i n t h e second h a l f o f t h e bend no i n f l u e n c e o f t h e o u t e r w a l l i s p e r c e p t i b l e i n t h e a d j a c e n t v e r t i c a l ( v e r t i c a l 1 ) .

d. The s h i f t i n g o f t h e v e l o c i t y maximum and t h e i n c r e a s e o f t h e i n n e r w a l l i n -f l u e n c e t e n d t o t a k e p l a c e o v e r a s h o r t e r l o n g i t u d i n a l d i s t a n c e a t t h e lower d i s c h a r g e .

3.1.4 Water s u r f a c e c o n f i g u r a t i o n (T4)

The e l e v a t i o n o f t h e w a t e r s u r f a c e above a h o r i z o n t a l datum ( v i z . , t h e bed l e v e l i n t h e s t r a i g h t s e c t i o n o f t h e f l u m e ) i s g i v e n i n T a b l e X I I I and F i g u r e

19. The l o c a l d e p t h o f f l o w , o b t a i n e d by s u b t r a c t i n g t h e t h e o r e t i c a l l o c a l bed l e v e l f r o m t h e above-mentioned w a t e r s u r f a c e e l e v a t i o n , i s g i v e n i n T a b l e XIV and F i g u r e 20. I n F i g u r e 20b o n l y t h e c u r v e f o r t h e average d e p t h o f f l o w has been drawn because o f t h e s m a l l d i f f e r e n c e s between t h e v a r i o u s v e r t i c a l s i n a c r o s s - s e c t i o n .

I t can be seen f r o m these t a b l e s and f i g u r e s t h a t :

a. A t t h e l a r g e d i s c h a r g e t h e t r a n s v e r s e s l o p e o f the w a t e r s u r f a c e was e s t a b l i s h -ed soon a f t e r t h e b e g i n n i n g o f t h e bend ( c r o s s - s e c t i o n B j i n F i g u r e 1 9 ) . At t h e s m a l l e r d i s c h a r g e t h e e s t a b l i s h i n g seemed t o go s l o w e r ( c r o s s - s e c t i o n C i ) , b u t t h i s i s n o t s u r e because t h e observed c o n f i g u r a t i o n was d i s t u r b e d by i n a c c u r a c i e s .

b. The w a t e r s u r f a c e showed a r a t h e r s t r o n g b a c k w a t e r c u r v e : h(Eo) - h(Bo) = 0.0124 ra f o r Q = 0.610 m V s and h(Eo) - h(Bo) = 0.0193 m when Q = 0.305 m V s

(see T a b l e X I V ) , w h i c h i m p l i e s t h a t t h e average l o n g i t u d i n a l s l o p e o f t h e f r e e s u r f a c e was about 1 .7 * 10""* s m a l l e r t h a n t h e l o n g i t u d i n a l s l o p e o f t h e bed i n t h a t p a r t o f t h e c h a n n e l (3.0 « 10""*) when Q = 0.610 m V s and

-10-2.7 -K 10"** s m a l l e r when Q = 0.305 m^/s. Hence an a d d i t i o n a l n o n - u n i f o r m i t y o f t h e d e p t h - a v e r a g e d v e l o c i t y was i n t r o d u c e d , a l t h o u g h t h i s n o n - u n i f o r m i t y was much s m a l l e r t h a n t h e n o n u n i f o r m i t y caused by t h e c u r v a t u r e (see F i

-g u r e 18a) .

3.2 Comparison w i t h t h e o r e t i c a l r e s u l t s

3.2.1 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 h o r i z o n t a l v e l o c i t y components

The o b s e r v e d v e r t i c a l d i s t r i b u t i o n s o f t h e main v e l o c i t y have been compared w i t h t h e l o g a r i t h m i c law:

V' . = 1 + ^ + ^ l n | - (1)

mam K C K C h

and w i t h t h e p a r a b o l i c law a p p l i e d by Engelund \_3j :

v' . = 1 + ^ { - 2 + 6 ^ - 3 ( ^ ) n (2) mam b C h n

(see F i g u r e s 10 and 11). A t t h e h i g h e r d i s c h a r g e (0.610 m^/s) t h e C h e z y - f a c t o r chosen was C = 70 m^/s, and a t t h e l o w e r d i s c h a r g e (0.305 m^/s) C = 50 m^/s

(see 3.1.2.b). I n b o t h cases Von Karman's c o n s t a n t was t a k e n 0. 4 .

I n o r d e r t o have t h e o r e t i c a l curves t h a t were comparable w i t h t h e e x p e r i m e n t a l r e s u l t s , t h e v e l o c i t i e s found f r o m (1) and (2) have been d i v i d e d by t h e i r 10-p o i n t d e 10-p t h - a v e r a g e d v a l u e , w h i c h was com10-puted u s i n g t h e same t r a 10-p e z i u m r u l e as i n t h e c o m p u t a t i o n o f t h e d e p t h - a v e r a g e d observed v e l o c i t i e s .

The c o m p a r i s o n o f t h e t h e o r e t i c a l and t h e e x p e r i m e n t a l d i s t r i b u t i o n s ( F i g u r e s 10 and 11) l e a d s t o t h e f o l l o w i n g c o n c l u s i o n s :

a. The l o g a r i t h m i c d i s t r i b u t i o n (1) tends t o agree b e t t e r w i t h t h e o b s e r v a -t i o n s -t h a n -t h e p a r a b o l i c one ( 2 ) .

b. As a consequence o f t h e a d v e c t i v e i n f l u e n c e o f t h e h e l i c a l c i r c u l a t i o n , t h e v e l o c i t i e s observed near t h e w a t e r s u r f a c e were s m a l l e r t h a n p r e d i c t e d by t h e t h e o r y , whereas i n t h e l o w e r h a l f o f t h e f l o w they tended t o be l a r g e r . These e f f e c t s were t h e s t r o n g e s t near t h e s i d e w a l l s .

The o b s e r v e d 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 h e l i c a l v e l o c i t y component has been compared w i t h t h e t h e o r e t i c a l d i s t r i b u t i o n s d e r i v e d by De V r i e n d 12' ,

Enge n Enge

-l u n d , R o z o v s k i i [s] and I k e d a , u s i n g t h e same v a -l u e s f o r t h e Chezy-f a c t o r and Von Karman's c o n s t a n t as i n t h e m a i n v e l o c i t y d i s t r i b u t i o n s . As t h e i n d i v i d u a l e x p e r i m e n t a l d i s t r i b u t i o n s were t o o i n a c c u r a t e t o draw c o n c l u -s i o n -s f r o m , t h e t h e o r e t i c a l curve-s have o n l y been p l o t t e d i n t h e -s u m m a r i z i n g F i g u r e 15, w h i c h g i v e s an i n d i c a t i o n o f " 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 t h e o b s e r v e d h e l i c a l v e l o c i t i e s , i f t h i s d i s t r i b u t i o n i s assumed t o be s i m i l a r i n a l l v e r t i c a l s .

The c o n c l u s i o n s t o be drawn f r o m t h i s c o m p a r i s o n a r e :

a. I k e d a ' s t h e o r y p r o v i d e s t h e b e s t agreement w i t h t h e measured d a t a , whereas a l l o t h e r t h e o r i e s tend t o u n d e r e s t i m a t e t h e o b s e r v e d h e l i c a l v e l o c i t y . I n I k e d a ' s t h e o r y t h e main v e l o c i t y d i s t r i b u t i o n i s d e s c r i b e d by t h e l o g a r i t h -mic l a w , j u s t as i n De V r i e n d ' s and R o z o v s k i i ' s , where t h e h e l i c a l v e l o c i t y d i s t r i b u t i o n i s s o l v e d f r o m t h e t r a n s v e r s e e q u a t i o n o f m o t i o n i n w h i c h t h e eddy v i s c o s i t y i s assumed t o be c o n s t a n t a l o n g t h e v e r t i c a l , as i n Engelund's t h e o r y . The boundary c o n d i t i o n s a r e s u p p l i e d by t h e v a n i s h i n g o f t h e t r a n s v e r s e shear s t r e s s b o t h a t t h e f r e e s u r f a c e and a t t h e bed, and by t h e i n

-t e g r a l c o n d i -t i o n o f a zero n e -t -t r a n s v e r s e d i s c h a r g e . On -t h e o r e -t i c a l grounds, however, b o t h t h e i n c o n s i s t e n c y o f t h e eddy v i s c o s i t y d i s t r i b u t i o n (parabo-l i c i n t h e main v e (parabo-l o c i t y c o m p u t a t i o n and c o n s t a n t i n t h e h e (parabo-l i c a (parabo-l v e (parabo-l o c i t y c o m p u t a t i o n ) and t h e v a n i s h i n g o f t h e t r a n s v e r s e shear s t r e s s a t t h e bed a r e d i s p u t a b l e . These o b j e c t i o n s a g a i n s t I k e d a ' s t h e o r y a r e n o t e l i m i n a t e d by t h e f a i r agreement w i t h t h e measured d a t a . The c o n c l u s i o n t o be drawn f r o m t h i s agreement i s t h a t t h e p a r a b o l i c eddy v i s c o s i t y i s n o t s u i t e d f o r m o d e l l i n g t h e p r e s e n t f l o w : t h e v e r t i c a l d i s t r i b u t i o n must be f l a t t e r i n

t h e upper p a r t o f t h e f l o w , y i e l d i n g s m a l l e r v a l u e s t h e r e i f t h e v e r t i c a l mean v a l u e remains t h e same. As t h e t h e o r e t i c a l main v e l o c i t y d i s t r i b u t i o n

i s n o t much i n f l u e n c e d by t h e d i s t r i b u t i o n o f t h e eddy v i s c o s i t y i n t h e upper p a r t o f t h e f l o w , t h e agreement between t h e o r y and e x p e r i m e n t as t o

t h e main f l o w w i l l n o t be v i o l a t e d i f t h i s d i f f e r e n t eddy v i s c o s i t y d i s t r i -b u t i o n i s a p p l i e d . The h e l i c a l v e l o c i t y d i s t r i -b u t i o n , however, depends t o a much h i g h e r e x t e n t on t h e eddy v i s c o s i t y i n t h e upper p a r t o f t h e f l o w , t h e s l o p e o f t h e h e l i c a l v e l o c i t y c u r v e (Av^^^/Az) b e i n g s t e e p e r i f t h e eddy v i s c o s i t y i s s m a l l e r . Hence t h e f l a t t e r eddy v i s c o s i t y d i s t r i b u t i o n w i l l g i v e r i s e t o a more i n t e n s e h e l i c a l f l o w .

-12-b. The h e l i c a l v e l o c i t y d i s t r i b u t i o n near t h e o u t e r w a l l i n c r o s s - s e c t i o n Do i s c o m p l e t e l y d i f f e r e n t f r o m t h e t h e o r e t i c a l c u r v e s . T h i s c o u l d be expected, because a l l t h e o r i e s a r e based on t h e e x i s t e n c e o f one s i n g l e h e l i c a l c i r

-c u l a t i o n and t h e y o n l y h o l d f a r f r o m t h e s i d e w a l l s . So a -c o u n t e r r o t a t i n g c i r c u l a t i o n near t h e o u t e r w a l l i s beyond t h e scope o f these t h e o r i e s .

3.2.2 Depth-averaged v e l o c i t y f i e l d

The d e p t h - a v e r a g e d v e l o c i t y f i e l d has been s i m u l a t e d n u m e r i c a l l y u s i n g t h e c o m p u t a t i o n a l programme d e v e l o p e d a t t h e L a b o r a t o r y o f F l u i d Mechanics o f t h e D e l f t U n i v e r s i t y o f Technology on t h e b a s i s o f t h e o r e t i c a l c o n s i d e r a t i o n s L l 2 j . The c o m p u t a t i o n a l g r i d , c o n s i s t i n g o f a c a r t e s i a n p a r t f o r t h e s t r a i g h t sect i o n and a p o l a r p a r sect f o r secthe c u r v e d s e c sect i o n , covered sect h e channel f r o m c r o s s -s e c t i o n AQ ( t h e t h e o r e t i c a l upstream b o u n d a r y ) t o c r o s s - s e c t i o n EQ ( t h e t h e o r e -t i c a l downs-tream b o u n d a r y ) . The l o n g i -t u d i n a l s -t e p s i z e was 1.00 m i n -t h e chan-n e l a x i s , achan-nd t h e t r a chan-n s v e r s e s t e p s i z e was 6.00/16 = 0.375 m. A t t h e upstream boundary a u n i f o r m v e l o c i t y d i s t r i b u t i o n and a zero v e r t i c a l v o r t i c i t y were imposed, a t t h e downstream boundary t h e c r o s s s e c t i o n a l w a t e r s u r f a c e e l e v a -t i o n was -t a k e n l i n e a r , -the -t o -t a l f a l l be-tween -t h e o u -t e r and -t h e i n n e r w a l l s b e i n g e q u a l t o t h e observed one. A d d i t i o n a l l y , t h e f o l l o w i n g v a l u e s were t a k e n f o r t h e c o n s t a n t s : K - 0,4, C = 70 (50) mz/s, Fr = 0.226 ( 0 . 1 0 8 ) , d (average d e p t h o f f l o w a t t h e down-s t r e a m b o u n d a r y ) = 0.261 ( 0 . 2 7 2 ) , t h e v a l u e down-s f o r t h e l o w e r d i down-s c h a r g e (0.305 m^/s) b e i n g i n d i c a t e d i n b r a c k e t s . The d e p t h a v e r a g e d v e l o c i t i e s r e s u l t i n g f r o m t h i s s i m u l a t i o n a r e shown i n F i -gure 18b. Only one s e t o f l i n e s has been drawn, as t h e computed v a l u e s o f t h e n o r m a l i z e d d e p t h a v e r a g e d v e l o c i t y were p r a c t i c a l l y t h e same f o r b o t h d i s -charges •

Comparing t h e t h e o r e t i c a l and t h e e x p e r i m e n t a l r e s u l t s ( F i g u r e s 18a and b ) , i t i s seen t h a t t h e r e i s a r a t h e r poor agreement^ even when the measured mean v e l o c i t y d i s t r i b u t i o n i s imposed a t t h e u p s t r e a m boundary. N e i t h e r t h e s h i f t i n g

of t h e v e l o c i t y maximum towards t h e o u t e r w a l l nor t h e growth o f t h e " i n n e r w a l l l a y e r " were r e p r e s e n t e d by t h e t h e o r y . T h i s can be e x p l a i n e d by t h e a d v e c t i v e i n f l u e n c e o f t h e h e l i c a l c i r c u l a t i o n w h i c h i s n o t a c c o u n t e d f o r i n t h e mathema-t i c a l model u n d e r l y i n g mathema-t h e compumathema-ter programme. A p p a r a n mathema-t l y mathema-t h i s e f f e c mathema-t may n o mathema-t be n e g l e c t e d , even though t h e o r e t i c a l l y i t i s o f t h e o r d e r o f magnitude

-13-3.2.3 Water s u r f a c e c o n f i g u r a t i o n

The w a t e r s u r f a c e c o n f i g u r a t i o n computed w i t h t h e a f o r e m e n t i o n e d m a t h e m a t i c a l model has been compared w i t h t h e measured d a t a i n F i g u r e s 19 and 20, and y i e l d s

t h e c o n c l u s i o n s : a. The c o n f i g u r a t i o n o f t h e w a t e r s u r f a c e i n t r a n s v e r s e d i r e c t i o n ( F i g u r e 19) i s p r e d i c t e d f a i r l y w e l l a t a h i g h e r d i s c h a r g e . A t t h e l o w e r d i s c h a r g e t h e agreement i s p o o r e r , e s p e c i a l l y i n t h e f i r s t p a r t o f t h e f l u m e , b u t i t s h o u l d be n o t e d t h a t t h e r e l a t i v e a c c u r a c y o f t h e measured d i f f e r e n c e s i n t h e w a t e r s u r f a c e i s s m a l l e r t h a n a t t h e h i g h e r d i s c h a r g e . b. The l o n g i t u d i n a l d i s t r i b u t i o n o f t h e d e p t h o f f l o w ( F i g u r e 20) i s b e t t e r p r e d i c t e d a t t h e l o w e r d i s c h a r g e . The computed b a c k w a t e r c u r v e , however, i s s t r o n g l y i n f l u e n c e d by t h e v a l u e o f t h e C h e z y - f a c t o r C a p p l i e d i n t h e ma-t h e m a ma-t i c a l model. T h i s v a l u e c o u l d be compuma-ted f r o m ma-t h e bed roughness and t h e f l o w Reynolds number, b u t i f t h e bed roughness i s n o t a c c u r a t e l y knovm t h i s y i e l d s i n a c c u r a t e r e s u l t s . For example, i f i n t h e p r e s e n t case t h e

N i k u r a d s e bed roughness l e n g t h v a r i e s between 10""* m and 10~3 m, t h e computed v a l u e o f C, computed by t h e l o g a r i t h m i c f o r m u l a f o r Q = 0.610 m^/s, v a r i e s between 79 m^/s and 61 m^/s r e s p e c t i v e l y . I n s t r a i g h t u n i f o r m shear f l o w C can be e s t i m a t e d f r o m 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 t h e v e l o c i t y and f r o m t h e t o t a l energy l o s s a l o n g t h e c h a n n e l , b u t i n t h e p r e s e n t n o n u n i f o r m c u r v e d f l o w such e s t i m a t i o n s a r e q u e s t i o n a b l e . As was s t a t e d i n S e c t i o n 3.1.2, t h e v a l u e s o f C deduced f r o m t h e v e -l o c i t y d i s t r i b u t i o n s a r e 70 m^/s f o r Q = 0.610 m^/s and 50 m^/s f o r Q = 0.305 m V s . As a consequence o f t h e i n a c c u r a t e e s t i m a t i o n o f C, c o m p a r i s o n o f t h e measured and t h e computed b a c k w a t e r c u r v e s does n o t make c l e a r w h e t h e r t h e "bend l o s s e s " a r e w e l l p r e d i c t e d by t h e m a t h e m a t i c a l model.

-14-4 C o n c l u s i o n s

1 For t h e p r e s e n t e x p e r i m e n t s t h e a c c u r a c y o f t h e f l o w d i r e c t i o n measurements i s o f p r e d o m i n a n t i m p o r t a n c e t o t h e a c c u r a c y o f t h e h e l i c a l v e l o c i t y t o be d e r i v e d f r o m t h e measured magnitude and d i r e c t i o n o f t h e v e l o c i t y v e c t o r

(see Appendix D)• T h e r e f o r e , i n a d d i t i o n t o an improvement o f t h e v e r t i c a l p o s i t i o n i n g system, a c a r e f u l c a l i b r a t i o n o f t h e m e a s u r i n g d e v i c e i s wanted f o r t h i s purpose. 2 The presence o f t h e t a i l - g a t e ( h a v i n g a h o r i z o n t a l c r e s t ) i n t h e c u r v e d s e c t i o n o f t h e c h a n n e l (where t h e w a t e r s u r f a c e has a t r a n s v e r s e s l o p e ) does n o t n o t i c e a b l y i n f l u e n c e t h e f l o w i n t h e l a s t c r o s s - s e c t i o n o f t h e m e a s u r i n g g r i d , E Q, s i t u a t e d 7 m u p s t r e a m o f i t . 3 The v e r t i c a l d i s t r i b u t i o n s o f t h e main v e l o c i t y a r e v e r y s i m i l a r t h r o u g h o u t t h e f l o w f i e l d , e x c e p t c l o s e t o t h e s i d e w a l l s i n t h e c u r v e d p a r t o f t h e c h a n n e l . There t h e main v e l o c i t y i s f u r t h e r reduced i n t h e upper p a r t o f t h e f l o w , whereas i t i s h i g h e r i n t h e l o w e r p a r t o f t h e f l o w .

4 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 main v e l o c i t y i s w e l l d e s c r i b e d by t h e l o g a r i t h m i c p r o f i l e , e x c e p t near t h e w a t e r s u r f a c e (where t h e o b s e r v a t i o n s show a v e l o c i t y r e d u c t i o n , whereas t h e t h e o r y does n o t ) and near t h e s i d e w a l l s i n t h e c u r v e d p a r t o f t h e c h a n n e l ( c f . 3 ) . These d e v i a t i o n s can be e x p l a i n e d q u a l i t a t i v e l y f r o m t h e a d v e c t i v e i n f l u e n c e o f t h e h e l i c a l f l o w on t h e main f l o w , w h i c h i s n o t a c c o u n t e d f o r i n t h e l o g a r i t h m i c law. 5 The h e l i c a l v e l o c i t i e s d e r i v e d f r o m t h e measured d a t a a r e t o o i n a c c u r a t e t o e n a b l e c o n c l u s i o n s t o be drawn as t o t h e s i m i l a r i t y o f t h e i n d i v i d u a l v e r -t i c a l d i s -t r i b u -t i o n s . I f a l l e x p e r i m e n -t a l r e s u l -t s a r e p l o -t -t e d i n one f i g u r e , however, most p o i n t s t u r n o u t t o be spread around a d i s t i n c t v e r t i c a l d i s -t r i b u -t i o n c u r v e . Only i n v e r -t i c a l s c l o s e -t o -t h e o u -t e r w a l l do -t h e h e l i c a l v e l o c i t i e s s u b s t a n t i a l l y d e v i a t e f r o m t h i s d i s t r i b u t i o n , as a consequence o f a c o u n t e r r o t a t i o n e x i s t i n g t h e r e . 6 Comparing t h e summarizing p l o t s o f t h e v e r t i c a l d i s t r i b u t i o n s o f t h e h e l i -c a l v e l o -c i t i e s w i t h v a r i o u s t h e o r e t i -c a l -c u r v e s d e r i v e d by De V r i e n d ["l 21 , R o z o v s k i i

IjTj

15agree b e s t w i t h t h e e x p e r i m e n t a l d a t a . T h i s suggests t h a t 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 -t h e eddy v i s c o s i -t y i s f l a -t -t e r -t h a n -t h e p a r a b o l i c c u r v e a p p l i e d i n De V r i e n d ' s and R o z o v s k i i ' s t h e o r i e s .

7 The o b s e r v e d depth-averaged v e l o c i t y f i e l d shows two s t r i k i n g f e a t u r e s : a. The p o i n t i n a c r o s s - s e c t i o n where t h e d e p t h - a v e r a g e d v e l o c i t y reaches

i t s maximum l i e s near t h e i n n e r w a l l i n t h e f i r s t p a r t o f t h e bend, b u t f u r t h e r downstream i t g r a d u a l l y s h i f t s towards t h e o u t e r w a l l .

b. I n t h e bend t h e " i n n e r w a l l r e g i o n " ( i . e . , t h e r e g i o n where t h e f l o w appears t o be i n f l u e n c e d by t h i s w a l l ) grows l a r g e r i n a downstream d i r e c t i o n .

8 The d e p t h - a v e r a g e d v e l o c i t y d i s t r i b u t i o n r e s u l t i n g f r o m a m a t h e m a t i c a l mod-e l basmod-ed on Dmod-e V r i mod-e n d ' s t h mod-e o r y agrmod-emod-es p o o r l y w i t h t h mod-e obsmod-ervmod-ed d i s t r i b u t i o n , e s p e c i a l l y i n t h e second p a r t o f t h e bend, where t h e s h i f t i n g o f t h e v e l o -c i t y maximum and t h e w i d t h o f t h e i n n e r w a l l r e g i o n be-come -c o n s i d e r a b l e . These phenomena, t o be a t t r i b u t e d t o t h e a d v e c t i v e i n f l u e n c e o f t h e h e l i c a l f l o w on t h e main f l o w , a r e n o t r e p r o d u c e d by t h e t h e o r y , s i n c e t h i s i n -f l u e n c e i s n o t accounted -f o r i n t h e t h e o r e t i c a l model.

9 The m a t h e m a t i c a l model g i v e s a r a t h e r good p r e d i c t i o n o f t h e t r a n s v e r s e wa-t e r s u r f a c e c o n f i g u r a wa-t i o n . Only i n wa-t h e s wa-t r a i g h wa-t p a r wa-t o f wa-t h e c h a n n e l do d i f f e r e n c e s between t h e o r y and e x p e r i m e n t occur a t t h e l o w e r d i s c h a r g e .

10 The observed l o n g i t u d i n a l d i s t r i b u t i o n o f t h e d e p t h o f f l o w i s b e t t e r p r e -d i c t e -d a t t h e l o w e r -d i s c h a r g e , b u t no c o n c l u s i o n s can be -drawn f r o m t h i s , as t h e r e s u l t s s t r o n g l y depend on t h e C h e z y - f a c t o r , t h e v a l u e o f w h i c h can-n o t be d e t e r m i can-n e d w i t h s u f f i c i e can-n t a c c u r a c y . For t h e same r e a s o can-n , i t i s can-n o t c l e a r w h e t h e r t h e "bend l o s s e s " a r e w e l l p r e s e n t e d by t h e m a t h e m a t i c a l model.

F i n a l l y , i t s h o u l d be n o t e d t h a t t h e m a t h e m a t i c a l model used h e r e was p r i m a r i l y meant t o be used as a p a r t o f a m a t h e m a t i c a l model o f t h e f l o w and t h e bed t o -pography i n c u r v e d a l l u v i a l c h a n n e l s . I n such c h a n n e l s , a f t e r a v e r a g i n g o u t t h e r i p p l e s and dunes, t h e bed shows a c o n f i g u r a t i o n o f banks and t r o u g h s t h a t may be e x p e c t e d t o e x e r t a p r e d o m i n a n t i n f l u e n c e on t h e f l o w p a t t e r n . I n t h a t case t h e s h o r t c o m i n g s o f t h e model i n p r e d i c t i n g t h e f l o w i n a c u r v e d c h a n n e l w i t h a f i x e d f l a t bed may t u r n o u t t o be o f m i n o r i m p o r t a n c e . To v e r i f y t h i s

-16-e x p -16-e r i m -16-e n t a l l y , a n o t h -16-e r s -16-e r i -16-e s o f m-16-easur-16-em-16-ents w i l l b-16-e mad-16-e i n t h -16-e p r -16-e s -16-e n t c h a n n e l , a f t e r i t has been p r o v i d e d w i t h a n o n - f l a t bed t o p o g r a p h y . These measurements w i l l be d i s c r i b e d i n P a r t I I o f t h i s R e p o r t , t o be p u b l i s h e d

REFERENCES 1 BROLSMA, P., T e m p e r a t u u r c o r r e c t i e v o o r m i c r o m o l e n m e t i n g e n . W a t e r l o o p k u n d i g L a b o r a t o r i u m , I n f o r m a t i e H 18, 1973 2 Combined c u r r e n t - v e l o c i t y / d i r e c t i o n m e t e r . 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 , T e c h n i c a l d e s c r i p t i o n , March 1975 3 ENGELUND, F.,

Flow and bed topography i n channel bends,

P r o c e e d i n g s o f t h e ASCE, J o u r n a l o f t h e H y d r a u l i c s D i v i s i o n , 100, HY 11, November 1974, p. 1631 4 GRIJSEN, J.G., R i s i c o a n a l y s e . W a t e r l o o p k u n d i g L a b o r a t o r i u m , S p e u r w e r k v e r s l a g S 164, Oktober 1974 5 IKEDA, S.,

On secondary f l o w and bed p r o f i l e i n a l l u v i a l c u r v e d open c h a n n e l ,

P r o c e e d i n g s o f t h e X V I t h Congress o f t h e lAHR, 2, paper B-14, Sao P a u l o , 1975

6 I n v e s t i g a t i o n o f t h e t o t a l e r r o r i n measurement o f f l o w by v e l o c i t y area methods,

I n t e r n a t i o n a l O r g a n i z a t i o n f o r S t a n d a r d i z a t i o n , Report ISO/TCl13/WG-l (Ne-t h e r l a n d s - 16, 17, 18) 177, 178, 179, Augus(Ne-t 1971 ( r e v . Augus(Ne-t 1974)

7 JENKINS, G.M., and WATTS, D.G.,

S p e c t r a l A n a l y s i s and i t s a p p l i c a t i o n s ,

Holden-Day s e r i e s i n t i m e s e r i e s a n a l y s i s , San F r a n s i s c o , 1968

8 ROZOVSKII, I . L . ,

Flow o f w a t e r i n bends o f open c h a n n e l s ,

I s r a e l Program f o r S c i e n t i f i c T r a n s l a t i o n s , J e r u s a l e m , 1961

9 VRIEND, H.J. de,

L i t e r a t u u r o v e r z i c h t b o c h t s t r o m i n g .

L a b o r a t o r i u m v o o r V l o e i s t o f m e c h a n i c a , A f d . Weg en Waterbouwkunde, T e c h n i -sche Hogeschool D e l f t , Rapport R 1972/1/L, 1972

REFERENCES ( c o n t i n u a t i o n ) 10 VRIEND, H.J. de. Theory o f v i s c o u s f l o w i n c u r v e d s h a l l o w c h a n n e l s , Communications on H y d r a u l i c s , Dept. o f C i v i l - . E n g i n e e r i n g , D e l f t U n i v e r s i t y o f Technology, Report 7 2 - 1 , 1973 11 VRIEND, H.J. de. Theory o f v i s c o u s f l o w i n w i d e c u r v e d open c h a n n e l s .

P r o c e e d i n g s o f t h e I n t e r n a t i o n a l Symposium on R i v e r Mechanics, Paper A17, Bangkok, T h a i l a n d , January 1973 12 VRIEND, H.J. de, A m a t h e m a t i c a l model o f s t e a d y f l o w i n c u r v e d s h a l l o w c h a n n e l s . Communications on H y d r a u l i c s , Dept. o f D i v i l E n g i n e e r i n g , D e l f t U n i v e r -s i t y o f Technology, Report 7 6 - 1 , 1976 13 YEN, C L . ,

Bed c o n f i g u r a t i o n and c h a r a c t e r i s t i c s o f s u b c r i t i c a l f l o w i n a meandering c h a n n e l ,

PH. D. T h e s i s , U n i v e r s i t y o f Iowa, Iowa C i t y , Iowa (U.S.A.), 1967

14 YEN, C.L.,

Bed topography e f f e c t on Flow i n a meander.

P r o c e e d i n g s o f t h e ASCE, J o u r n a l o f t h e h y d r a u l i c s D i v i s i o n , 96^, HY 1 , J a n u a r y 1970, p. 57

Appendix A

Appendix A

INSTRUMENTATION

1 Water s u r f a c e e l e v a t i o n

The w a t e r s u r f a c e e l e v a t i o n was measured w i t h s t a t i c tubes c o n n e c t e d t o a m e a s u r i n g p i t by p l a s t i c hoses o f about 15 m l e n g t h (see F i g u r e 1 ) . Three

c r o s s - s e c t i o n s c o u l d be reached f r o m one p i t , w h i c h i m p l i e s t h a t t h r e e p i t s were needed t o c o v e r a l l n i n e c r o s s - s e c t i o n s . I n each p i t 7 m e a s u r i n g g l a s s e s were mounted, s o t h a t f o r each c r o s s - s e c t i o n t h e w a t e r s u r f a c e e l e v a t i o n s i n a l l 7 m e a s u r i n g p o i n t s c o u l d be r e c o r d e d s i m u l t a n e o u s l y . The w a t e r l e v e l s i n t h e g l a s s e s were measured by p o i n t gauges. U s i n g a v e r n i e r f i n e s e t t i n g an a c c u r a c y i n t h e r e a d i n g s o f about 10""* m c o u l d be o b t a i n e d . The p o i n t gauge r e a d i n g s were r e l a t e d t o one datum, v i z . , t h e mean l e v e l o f t h e h o r i z o n t a l b o t t o m i n t h e s t r a i g h t p a r t o f t h e c h a n n e l .

As a consequence o f t h e r a t h e r l o n g c o n n e c t i o n s between t h e s t a t i c tubes and t h e m e a s u r i n g g l a s s e s , a c o n s i d e r a b l e t i m e ( s e v e r a l h o u r s ) had t o pass between t h e i n s t a l l a t i o n o f t h e tubes and t h e s e t t i n g o f t h e p o i n t gauges i n o r d e r t o

have o s c i l l a t i o n s o f t h e w a t e r i n t h e m e a s u r i n g system damped o u t .

2 V e l o c i t y

The m a g n i t u d e and t h e d i r e c t i o n o f t h e f l o w were measured s i m u l t a n e o u s l y by a combined c u r r e n t v e l o c i t y / d i r e c t i o n m e t e r , c o n s i s t i n g o f a m i n i a t u r e p r o -p e l l e r and a vane (see F i g u r e 3 ) . The -p r o -p e l l e r m e a s u r i n g t h e m a g n i t u d e o f t h e v e l o c i t y v e c t o r had a d i a m e t e r o f 0.011 m. The vane i n d i c a t i n g t h e d i r e c -t i o n o f -t h e v e l o c i -t y v e c -t o r was 0.020 m h i g h and 0.050 m l o n g . By means o f a s e r v o system t h e frame i n w h i c h t h e p r o p e l l e r and t h e vane a r e mounted i s t u r n e d i n t h e f l o w d i r e c t i o n . Hence t h e p r o p e l l e r measures t h e t o t a l v e l o c i t y

v.. ^. For more t e c h n i c a l i n f o r m a t i o n see fz] .

t o t I — I The a n g u l a r speed o f t h e p r o p e l l e r v a r i e d l i n e a r l y w i t h t h e c u r r e n t v e l o c i t y . A c a l i b r a t i o n c u r v e ( F i g u r e 4) o f t h e p r o p e l l e r gave t h e c o e f f i c i e n t s c i and C 2 , n e c e s s a r y t o c o n v e r t t h e observed f r e q u e n c y (number o f r e c o r d e d p u l s e s N d i v i d e d by t h e t i m e T) t o t h e v e l o c i t y : v.„. - ciN/T + C2 (m/s) ( A . l )

-2- Appendix A The c a l i b r a t i o n was e x e c u t e d i n w a t e r o f a c e r t a i n t e m p e r a t u r e , w h i c h i m p l i e d t h a t t h e c o e f f i c i e n t s c i and C2 s h o u l d be c o r r e c t e d f o r t h e a c t u a l temperature o f t h e w a t e r i n t h e c h a n n e l . I n t h i s r e s p e c t i t can be remarked t h a t o n l y C2 v a r i e d w i t h t h e w a t e r t e m p e r a t u r e t o a c o n s i d e r a b l e e x t e n t , d e p e n d i n g on t h e i n d i v i d u a l p r o p e l l e r 1 , A r o u g h i n d i c a t i o n o f t h i s dependency i s g i v e n b y : c , 1 = c . . . + 0.0234 (n - no) ( m / s ) , (A.2) z a c t u a l 2 c a l i b r a t i o n ' i n w h i c h : ri = t h e dynamic v i s c o s i t y ( i n c p ) o f t h e w a t e r a t t h e a c t u a l t e m p e r a t u r e rio = t h e dynamic v i s c o s i t y ( i n c p ) o f t h e w a t e r d u r i n g t h e c a l i b r a t i o n

The vane t u r n s i n t h e f l o w d i r e c t i o n and commands t h e frame a l s o t o t u r n i n t h e f l o w d i r e c t i o n by means o f two p o s i t i o n sensors and a s e r v o system. The p o s i t i o n o f t h e frame i s measured by means o f a p o t e n t i o m e t e r . The r e c o r d i n g equipment c o n v e r t s t h e s i g n a l t o d e g r e e s . Since t h e f l o w i s t u r b u l e n t , t h e d i r e c t i o n o f t h e vane v a r i e s i n t i m e . The average d i r e c t i o n o v e r a s e t p e r i o d i s d e t e r m i n e d by u s i n g an e l e c t r o n i c i n t e g r a t o r .

Appendix B MEASURING PROCEDURE page j Accuracy ' 1 . 1 V e l o c i t y measurements • ' 1.2 Flow d i r e c t i o n measurements 2 2 M e a s u r i n g p r o c e d u r e 4

2.1 P o s i t i o n i n g o f t h e f l o w meter and t h e s t a t i c tubes 4

2. 2 V e l o c i t y measurements 5 2.3 S u r f a c e e l e v a t i o n measurements 6

Appendix B MEASURING PROCEDURE 1 A c c u r a c y 1.1 V e l o c i t y measurements (TO-1) I n o r d e r t o draw up a m e a s u r i n g p r o c e d u r e f o r t h e v e l o c i t y measurements, t h e p r o p e l l e r o f t h e f l o w m e t e r was t e s t e d by p u t t i n g i t a t a c e r t a i n p o i n t i n t h e f l o w and t a k i n g a l o n g s e r i e s o f o b s e r v a t i o n s o f 30 s d u r a t i o n each. F i -g u r e 5 shows t h a t t h e o b s e r v e d f r e q u e n c i e s v a r i e d s l o w l y i n t i m e . T h i s pheno-menon c o u l d n o t be e x p l a i n e d f r o m v a r i a t i o n s i n t h e a c t u a l v e l o c i t y due t o

l o n g waves i n t h e c h a n n e l , s i n c e these waves s h o u l d have had an a m p l i t u d e o f 0.01 m i n o r d e r t o cause t h e o b s e r v e d v e l o c i t y v a r i a t i o n s . C o n t i n u o u s w a t e r l e v e l r e g i s t r a t i o n s by an e l e c t r o n i c "water s u r f a c e f o l l o w e r " have shown t h a t waves o f t h i s a m p l i t u d e d i d n o t e x i s t i n t h e c h a n n e l . A p o s s i b l e e x p l a n a t i o n o f t h e v a r i a t i o n s can be f o u n d i n t h e p o l l u t i o n o f t h e w a t e r i n t h e c i r c u l a t i o n system, w h i c h c o n s i s t e d m a i n l y o f open c h a n n e l s . S m a l l suspended p a r t i c l e s g r a d u a l l y f i l l e d up t h e h o l e s i n t h e p e r f o r a t e d r i n g a r o u n d t h e p r o p e l l e r , r e d u c i n g t h e number o f p e r f o r a t i o n s c o u n t e d by t h e de-t e c de-t o r . Thus de-t h e o b s e r v e d f r e q u e n c y c o u l d d e c r e a s e i n de-t i m e , whereas de-t h e a c de-t u a l f r e q u e n c y d i d n o t . I n a d d i t i o n , suspended l i t t l e h a i r s and t h r e a d s accumulated a r o u n d t h e a x i s o f t h e p r o p e l l e r , t h u s i n c r e a s i n g i t s r o t a t i o n a l r e s i s t a n c e . The f u r t h e r v e l o c i t y measurements c o n f i r m e d t h i s t h e o r y . When c h e c k i n g t h e p r o p e l l e r a f t e r a s e r i e s o f measurements, v e r y o f t e n p a r t i c l e s and t h r e a d s had t o be removed.

The above-mentioned l o n g t e s t s e r i e s p r o v i d e d t h e p o s s i b i l i t y o f Examining t h e i n f l u e n c e o f t h e o b s e r v a t i o n p e r i o d on t h e accuracy o f t h e v e l o c i t y measurements, T a k i n g t h e 30 s p e r i o d as a b a s i s , l o n g e r p e r i o d s were simulated by t a k i n g t h e average o f a number o f subsequent 30 s v a l u e s . Thus f o u r t e s t

s e r i e s were g e n e r a t e d , w i t h p e r i o d s o f 30, 60, 90 and 150 s, r e s p e c t i v e l y . The s t a n d a r d d e v i a t i o n s o f each o f t h e s e s e r i e s were t a k e n as an i n d i c a t i o n o f t h e accuracy o f t h e i n d i v i d u a l o b s e r v a t i o n s . T a b l e I I shows t h a t a l o n g e r o b s e r v a t i o n a l p e r i o d y i e l d s a r e l a t i v e l y s m a l l decrease o f t h e s t a n d a r d de-v i a t i o n . C o n s i d e r a b l y s m a l l e r s t a n d a r d d e de-v i a t i o n s were f o u n d i f t h e decreasing t r e n d i n t h e o b s e r v a t i o n s was e l i m i n a t e d by s u b t r a c t i n g f r o m t h e d a t a t h e v a l -ues a c c o r d i n g t o t h e l e a s t squares s t r a i g h t l i n e (see F i g u r e 5 ) .

Hence, assuming t h e t e s t s e r i e s t o be r e p r e s e n t a t i v e f o r a l l v e l o c i t y measure-ments , t h e measuring p r o c e d u r e s h o u l d e l e m i n a t e t h e e f f e c t o f a l i n e a r