Discussion of bedload movement formulas of Kalinske, Einstein and Meyer-Peter and Müller and their application to recent measurements of beload in the rivers in Holland

DISCUSSION DES POmTULES DE DEBIT SOLIDE DE KALINSICE, EINSTEIN ET MEYER-PETER ET OTLLER COMPTE T.E^TUE DES MESURES RECE^iTES •

DE TRAl>TSrOET D M S L E S R I V I E R E S NEERLA3TDAISES DISCUSSION OF BEDLOAD MOVMENT FORMULAS OP

KAJ.IESKE, EINSTEIN, AND MEYER-PETER m) MUT.LER AND THEIR APPLICATION TO RECENT

MEASTIREMEIvTS OP BEDLOAD MOVEMENT IH THE RIVERS OP HOLLAIJD

Cy^yy-^U, r^-^yl^y -j.:^^-...^ 7y^-.^.-'^ •A''^/^'^,

By ' H.C. F r i j l i n k E n g i n e e r a t t h e D e l f t H y d r a u l i c L a b o r a t o r y • H o l l a n d D e l f t , June 1952 ( T r a n s l a t i o n by C h a r l e s W. Thomas, H y d r a u l i c E n g i n e e r , Bureau o f R e c l a m a t i o n , Denver, C o l o r a d o , August 1953)

Note; T h i s i s a w o r k i n g t r a n s l a t i o n p r e p a r e d p r i m a r i l y f o r u s e w i t h i n the Bureau of R e c l a m a t i o n . I t s h o u l d n o t he c o n s i d e r e d f i n a l , hut p r a c t i c a l l y a l l the i d e a s e x p r e s s e d "by the o r i g i n a l a u t h o r can be o b t a i n e d from i t i n i t s p r e s e n t form. A l l i l l u s t r a t i o n s have

been d u p l i c a t e d from the o r i g i n a l a r t i c l e w i t h o u t complete t r a n s l a t i o n of t h e d e s c r i p t i v e m a t e r i a l o r r e d u c t i o n of a l l v a l u e s t o E n g l i s h u n i t s .

L I S T OP SYli/IBOLS USED Syrnljol Dimension D e s c r i p t i o n A B 1, 2, 3 l 1» 2 D i m e n s i o n l e s s c o n s t a n t s B 1 Normal w i d t h of t h e r i v e r "b 1 E f f e c t i v e w i d t h of t h e r i v e r i -1 C 1'"^ t Chezy c o e f f i c i e n t i -1 C' 1'^ t Chezy c o e f f i c i e n t r e l a t e d t o t h e r o u g h n e s s of t h e g r a i n s <i 1 D i a m e t e r o f g r a i n s e Base o f n a t u r a l l o g a r i t h i n s _2 S I t A c c o l o r a t i o n o f g r a v i t y h 1 Depth o f w a t e r Ö h 1 V a r i a t i o n o f depth d u r i n g t h e o b s e r v a t i o n I Slope o f t h e energy g r a d i e n t Slope i n r e l a t i o n t o r o u g h n e s s K F r a c t i o n of m a t e r i a l composing t h e bed w i t h a f i x e d g r a i n s i z e 1 E q u i v a l e n t rougltness o f t h e w a t e r c o u r s e

1 Rougkness of t h e gi'^ins i n t h e bed p F r a c t i o n o f t h e m i x t u r e o f g r a i n s 2 -1 q. . I t D i s c h a r g e p e r u n i t o f w i d t h E 1 H y d r a u l i c r a d i u s 1 H y d r a u l i c r a d i u s i n r e g a r d to t h e r o u g h n e s s o f g r a i n s k ^B F r a c t i o n o f t h e t r a n s p o r t e d m a t e r i a l i n movement w i t h a f i x e d g r a i n s i z e .

GOKTENTS

Page

1. I n t r o d u c t i o n 1 2. Comparison o f the F o r m u l a s of Kalinske» E i n s t e i n and

M e y e r - P e t e r and. M u l l e r 2 a. K a l i n s k e ' s f o r m u l a . . . 2 h. E i n s t e i n ' s f o r m u l a 3 c. F o r m u l a o f M e y e r - P e t e r and M u l l e r 4 d. Comparison o f t h e t h r e e f o r m u l a s 3 3. C h a r a c t e r i s t i c D i a m e t e r s o f G r a i n s o 4. R i p p l e C o e f f i c i e n t s 7 a. T r a n s p o r t o f bed m a t e r i a l and l o s s of energy 7

b. C o e f f i c i e n t s o f r i p p l e s a c c o r d i n g t o E i n s t e i n and a c c o r d i n g t o M e y e r - P e t e r and M u l l e r ü O. C o n s i d e r a t i o n i n d e t a i l o f t h e r i p p l e c o e f f i c i e n t [x . . . 10 d. I n f l u e n c e o f r i p p l e s i n K a l i n s k e ' s f o r m u l a 12 5. C r i t i c a l V e l o c i t y o f C r i t i c a l S h e a r f o r t h e B e g i n n i n g o f T r a n s p o r t 13 6. E t a p i r i c a l S i m p l i f i c a t i o n Proposed f o r t h e Formulae D i s c u s s e d 1'4 7. C o n c l u s i o n s Ij Appendix A '17 D e s c r i p t i o n o f o b s e r v a t i o n s i n t h e r i v e r s of H o l l a n d . . . I'i Appendix B I 0 D e s c r i p t i o n o f measurements made a t t h e H y d r a u l i c L a b o r a t o r y a t D.ELFT l o T a b l e Measurements of T r a n s p o r t i n t h e V/aal r i v e r n e a r Nimegue 1 Measurements o f T r a n s p o r t i n t h e Lower Rhine R i v e r

n e a r Arnhem 2 Measurements of T r a n s p o r t of Pumice Stone i n a

G l a s s - s i d e d Plume 3 Measurements o f T r a n s p o r t o f B a k e l i t e i n a G l a s s -s i d e d Plume '4 F i g u r e F o r m u l a s o f K a l i n s k e , . E i n s t e i n and M e y e r - P e t e r and M u l l e r f o r u n i f o r m m a t e r i a l w i t h o u t bed r i p p l e s 1 C h a r t f o r r i p p l e c o e f f i c i e n t |.' 2 R e l a t i o n s h i p between t h e r i p p l e c o e f f i c i e n t and t h e p a r a m e t e r Y = A d/RI 3 Proposed s i m p l i f i c a t i o n o f t h e t r a n s p o r t f o r m u l a . R e s u l t s of measurements 4 Diagram o f measurements on t h e *Vaal ( R i v e r ) 5

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

T h e r e a r e many r e a s o n s why a g r e a t d e a l of e f f o r t shoxild he expended toward e s t a b l i s h i n g a good f o r m u l a f o r c a l c u l a t i n g t r a n s p o r t e d bed m a t e r i a l .

I n p r a c t i c e , c o m p r e h e n s i v e knowledge of the l a w s of b e d l o a d t r a n s p o r t i s n e c e s s a r y , among o t h e r t h i n g s , to p e r m i t the s t u d y of the c o n s e q u e n c e s of man's d i s t u r b i n g n a t u r a l w a t e r c o u r s e s .

F o r l a b o r a t o r y s t u d i e s a good b e d l o a d f o r m u l a i s i n d i s p e n s a b l e f o r c a l c u l a t i o n of s c a l e r a t i o s f o r movable bed models.

The f o r m u l a s employed f o r t r a n s p o r t of bed m a t e r i a l may be d i v i d e d i n t o :

( 1 ) Formulas p r i n c i p a l l y d e r i v e d by t h e o r y and

( 2 ) F o r m u l a s p r i m a r i l y based on the r e s u l t s of measurement of t r a n s p o r t e d m a t e r i a l s i n w h i c h the r e l a t i o n s h i p s between the c o e f f i c i e n t s g o v e r n i n g t r a n s p o r t a r e , f o r the most p a r t , e s t a b l i s h e d e m p i r i c a l l y .

The r e c e n t f o r m u l a s of KALINSI^E ( l 9 4 7 ) and EIMSTEIN ( l 9 5 0 ) may be c o n s i d e r e d as the most r e p r e s e n t a t i v e of the f i r s t group,

the f o r m u l a of MEYER-PETER and I.IULLER e m i n a t i n g from the L a b o r a t o r y of E.T.H. a t ZURICH has been c h o s e n a s an example from the group of e m p i r i c a l f o r m u l a s .

I n the f o l l o w i n g d i s c u s s i o n the t h r e e a r e compared and examined on the b a s i s of t h e i r a p p l i c a b i l i t y . A d e t a i l e d c r i t i c i s m i s not atternptedbecause the f o r m u l a s of KALIKSKE and of IffiYER-PETER and MULLER have a l r e a d y been c o v e r e d i n a p r e v i o u s p u b l i c a t i o n ( R e f e r ence 4 ) *

-As a r e s u l t of t h a t s t u d y a f o r m u l a i s g i v e n f o r t h e p r o p e r a p p r o x i m a t i o n , w i t h i n p r a c t i c a l l i m i t s , w h i c h i s based on the f o r m u l a s d i s c u s s e d and a l s o on a c e r t a i n number of measurements made i n t h e

2»

"branches o f t h e IffllN]!) i n H o l l a n c l and i n t h e L a b o r a t o r y a t D£LFT. 2 • j^oino:..rison__o_f j u i e F o r m u l a s o f K a l i n s k e . . E i n s t e i n and M e y e r - I ' e t e r ancl M u l l e r a. K a l i n s k e ' s f o r m u l a (PLeference l ) T l i i s f o r m u l a i s p r i n c i p a l l y based on Mie o b s e r v a t i o r : 3 o f V / h i t e ( R e f e r e n c e 5 ) ^ c o i i c e r n i n g t h e c r i t i c a l s h e a r a l o n g t h e bed f o r w h i c h a g r a i n s t a r t s t o move, and on t h e k n o w l e d g e o f t h e t u r b u l e n t v e l o c i t y d i s t r i b u t i o r ; made n e a r t h e bed-I n i t s d i i n e n s i o i J . e s s f o r m , t h e f o r m u l a i s ; a y W • ^ • 5 « ( - ^ ) ( 1 ) where T i s t h e b e d l o a d e x p r e s s e d i n u n i t s o f v o l u m e p e r u n i t o f t i m e and p e r u n i t o f w i d t h d = t h e d i a m e t e r o f i.he g r a i n s E = t h e h y d r a u l i c r a d i u s I = the s l o i i e o f t l i e e n e r g y g r a d i e n t T c = t h e c r i t i c a l b o t t o m s l i e a r o = t h e r' .an v a l u e o f s l i e a r on t h e b o t t o m t h a t i s p r e s e n t T 0 i s a c o m n l i c a t e c ! f u i - c t i o i i o f t h a t K a l i n s k e h a s T-O d e v e l o p e d uy c o n s i d e r i n g t h e l a w o f d i f f e r e n c e a c c o r d i n g t o Gauss f o r t h e f l u c t u a t i o n of i l m v e l o c i t y a l o n g t h e hod. A c r a p h o f t h e f u n c t i o n i s g i v e n i n t"ne p u b l i c a t i o n b;/ K a l i n c k e , A c c o r d i n g t o t h e r e s u l t s o f W h i t e , t h e q u o t i e n t — m a y be v / r i t t e n ::KS f o l l o v / s i - f t h e pLeynoldB number f o r a ;';rain i s g r e a t e r t h a n 3-5^

4 ^ - = 0 . 1 . - f ( 2 )

s The t o t a l v o l u m e o f idio g r a i n i s luiown» a.ccount has n o t t h e n heen t a l : e n o f t h e v o l u m e o f t h e p o r e s w h i c h must be added t o T f o r f i n d i n g t h e v o l u m e o f a d e p o s i t e d mass, f o r e x a m r l e .

where A = — ' "^^e r e l a t i v e o e n s i t y o f t h e s o l i d g r a i n s u n d e r w a t e r . By t h e a i d o f t h i s f o r m u l a , v/e may w r i t e ; _ ^ T ( A d ) 0 (^) ( l a ) o r X = F l ( Y ) ( I h ) A ] , a r e c o n s t a n t s ,

X

=

T/ö.~'^'^g^^''l!i^^'^'

a d i m e n s i o n l e s s e x p r e s s i o n o f t r a j n s p o r t , Y = Ad/Ri, a p a r a m e t e r w h i c h denends on t h e i n t e n s i t y o f t r a n s p o r t , h. E i n s t e i n ' s f o r m u l a ( R e f e r e n c e 2) I n t h e most g e n e r a l f o r m , t h i s f o r m u l a i s w r i t t e n :

f\\

-^ 2 - = -^ _ _ 1 _

L^'

d t '^2^

^ J -

Y - 1/po ( 3 ) i n w h i c h po, A2, B2 = c o n s t a n t s , t = t h e v a r i a h l e o f i n t e g r a t i o n , i B = t h e f r a c t i o n o f t h e hot]load, h a v i n g t h e s i z e o f g r a i n s c o n s i d e r e d , i . ^ = t h e f r a c t i o n o f m a t e r i a l o f w h i c h t h e h e d i s c o n s t i t u t e d , h a v i n g a s i z e o f g r a i n s c o n s i d e r e d , Y = Y'f(ö, d, K ) o r f ( 6 , d, K ) and i s a f u n c t i o n m w h i c h t h e s i z e a n a l y s i s o f t h e m i x t u r e , and o f t h e c u r r e n t n e a r t h e hed i s e x p r e s s e d . I f t h e s i z e o f t h e g r a i r j s i s u n i f o r m and i f t h e i s h y d r a u -l i c a -l -l y r o u g h , Y i s e q u a -l t o Y'. DC R' i s t h e p a r t o f t h e h y d r a u l i c r f i d i i i s , R, v/hic)i i s due t o t h e r o u g h n e s s o f t h e g r a i n s . E i n s t e i n e x p r e s s e s a l s o t h e i n -f l u e n c e o -f t h e i r r e g u l a r i t i e s o -f t h e b e d . I n p a r a g r a p h 4, t h e manner i n w h i c h R' i s d e t e r a i n e d w i l l he shown.

I n t h e ctise v/here t h e hed i s smooth, w i t h o u t r i p p l e s o r b a r s and. w i t h o u t l a t e r a l r e s i s t a n c e s , R' must he e q u a l t o R. The form.ula o f E i n s t e i n i s t h e n ;

4 .

X = F2 ( Y ) ( 3 a ) P o r a m i x t u r e o f g r a i n s o f d i f f e r e n t d i a m e t e r s v/hen t h e

hed i s i n waves and t h e t h i c k n e s s of the l a m i n a r s u b l a y e r i s not n e g l i g i b l e , i t i s a l s o n e c e s s a r y to a p p l y a c e r t a i n number of

c o r r e c t i o n s which may he found from the f o u r graphs. The e q u n t i o n , ( 3 a ) , i s a l s o g i v e n i n g r a p h i e form. E i n s t e i n , i n t h e d e r i v a t i o n of t h i s f o r m u l a , has c o n s i d e r e d t u r b u l e n c e a s a s t a t i s t i c a l problem. c. Formula of Meyer-Peter_ajjd_MujJ,e^ T h i s f o r m u l a i s w r i t t e n a s f o l l o w s ; I ' r e p r e s e n t s t h e p a r t of t h e s l o p e , I , u s e d f o r o v e r

-coming the roughness of t h e bed, w h i l e t h e r e m a i n d e r — t h a t i s t o say, I _ I » — w i l l be u s e d i n overcoming the roughness of t h e w a l l s t h a t c a u s e d t h e r i p p l e s , e t c .

U s i n g the f o r m u l a of Chezy, one i s a b l e t o w r i t e c o n s e c u -t i v e l y :

V = CVE I and ( 5 )

V = C' KÉf" ( 5 a ) Then, I ' = (^,) I ( 6 )

C, the a c t u a l c o e f f i c i e n t of Chezy, might be c a l c u l a t e d w i t h t h e E q u a t i o n ( 5 ) from d a t a o b t a i n e d from t h e w a t e r c o u r s e , w h i l e , w i t h the a i d of f o r m u l a s f o r roughness, C' i s d e t e r m i n e d from v a l u e s of E, k ( r o u g h n e s s of the bed) and e v e n t u a l l y of 6, the t h i c k n e s s of t h e l a m i n a r s u b l a y e r . M e y e r - P e t e r and M u l l e r u s e the f o r m u l a of S t r i c k l e r ;

C' = 26 ( f ) • ( 7 ) 1/6

I n the h y d r a u l i c a l l y rough rangs, t h i s f o r m u l a d i f f e r s l i t t l e from the l o g a r i t h m i c f o r m u l a . ( R e f e r e n c e 6)

g

I n our o p i n i o n , t h i s f o r m u l a s h o u l d he p r e f e r r e d b e c a u s e i t i s v a l i d f o r a g r e a t e r range of c o n d i t i o n s .

I t appears, however, t h a t the exponent 2 i n E q u a t i o n ( 6 ) i s not i n a c c o r d a n c e m t h expèriments made a t Z u r i c h . A c c o r d i n g t o M e y e r - P e t e r and M u l l e r , an exponent 3 / 2 — t h a t i s to say, 75 p e r c e n t

of the t h e o r e t i c a l v a l u e g i v e s b e t t e r r e s u l t s . E q u a t i o n ( 6 ) , then, may be w r i t t e n s

I ' = ( § . ) ^ / ' l ( 6 a ) Here a l s o I ' must be e q u a l to I when the bed i s smooth,

w i t h o u t r i p p l e s o r dunes, or, i n g e n e r a l , when the t r a n s p o r t i s s m a l l .

T h e r e f o r e , X = - 0 . 0 4 7 ) ^ ^ ^ (^a)

o r X = P3 ( Y ) ( 4 b ) d. Comparison o f t h e t h r e e f o r m u l a s

I n comparing the t h r e e f o r m u l a s ( l b ) , ( 3 a ) and ( 4 b ) , i t may be seen t h a t f o r c e r t a i n c i r c u m s t a n c e s , e a c h may he w r i t t e n a s a r a t i o between the same d i m e n s i o n l e s s p a r a m e t e r s . These f u n c t i o n s a r e shorn g r a p h i c a l l y i n F i g u r e 1.

I t a p p e a r s t h a t the f o r m u l a s of E I N S T E I N and MEYER-PETER and MULLER a g r e e q u i t e w e l l f o r v a l u e s of "X" g r e a t e r than O.OO3, w h i c h i s when the b e d l o a d i s not v e r y s m a l l . F o r s m a l l e r v a l u e s of "X" the c u r v e from ZURICH a p p r o a c h e s a s y m p t o t i c a l l y the h o r i z o n t a l l i n e , Y = 21.3. T h i s i s b e c a u s e of the c o n c e p t i o n of a c r i t i c a l v a l u e of the mean boundary s h e a r . T h i s vd.ll be r e f e r r e d to i n p a r a g r a p h 5. Pigu.re 1 shows t h a t the KALINSl-CE formula, when i t i s a p p l i e d f o r u n i f o r m m a t e r i a l s i n w h i c h r i p p l e s a r e not formed, g i v e s v a l u e s

f o r "X" o r f o r bedload t r a n s p o r t of about l / 2 t h e v a l u e s o f t h e o t h e r two f o r m u l a s . T h i s comes from t h e f a c t t h a t KALIITSKK does n o t a p p l y i n h i s f o r m u l a a s p e c i a l r i p p l e c o e f f i c i e n t f o r t h e r e d u c t i o n of t h e f o r c e due t o f r i c t i o n c a u s e d by bed r i p p l e s . I n p a r a g r a p h 4, t h i s w i l l be examined a g a i n more c l o s e l y . 3• C h a r a c t e r i s t i c D i a m e t e r s o f G r a i n s The q u e s t i o n a r i s e s — w h a t i s t h e v a l u e of t h e d i a m e t e r o f t h e g r a i n s t o he u s e d i n t h e p r e v i o u s l y mentioned f o r m u l a s when m i x t u r e s of m a t e r i a l s of d i f f e r e n t g r a i n d i a m e t e r s a r e i n v o l v e d i n s t e a d of u n i f o r m m a t e r i a l s ? One must d i s t i n g u i s h betv/een t h e c o m p o s i t i o n of t h e bed m a t e r i a l s and the c o m p o s i t i o n of t h e m a t e r i a l s i n movement w h i c h a s E i n s t e i n has p r o p e r l y s a i d a r e not a l w a y s t h e same.

Por t h e v a l u e of roughness of t h e g r a i n s on t h e bed ( E q u a t i o n s 7 and 7 a ) MEYER-PETER and MULLER t a k e d 90, w h i c h i s t h e d i a m e t e r

of w h i c h 90 p e r c e n t of t h e m i x t u r e i s e q u a l o r s m a l l e r i n g r a i n s i z e . EINSTEIN u s e s a v a l u e d65 w h i c h i s t h e d i a m e t e r of w h i c h 65 p e r c e n t of t h e m i x t u r e i s e q u a l o r s m a l l e r i n g r a i n s i z e . I n t h f

J^ether-l a h d a t h e b e s t r e s u J^ether-l t s have been a t t a i n e d w i t h d 90, a v a J^ether-l u e t h a t was a l s o t h e r e s u l t o f a s p e c i a l s t u d y .

EINSTEIN, i n h i s l a t e s t p u b l i c a t i o n , no l o n g e r a p p l i e d t h e bedload t r a n s p o r t f o r m u l a on t h e m i x t u r e i n i t s e n t i r e l y , but on e a c h f r a c t i o n a l p a r t , where c o n s i d e r a t i o n s h o u l d be g i v e n t o r e c i p r o c a l i n f l u e n c e . However i n our o p i n i o n , t h i s r e f i n e m e n t i s not i n p r o p o r -t i o n w i -t h -t h e p r e c i s i o n -t h a -t a c a l c u l a -t i o n of -t r a n s p o r -t i s a b l e -t o g i v e now. However, i f i t i s a p p l i e d t o t h e e n t i r e m i x t u r e , i t i s p r e f e r a b l e t o e x p r e s s more o r l e s s t h e form of t h e c u r v e o f s i ^ v e s i z e i n t h e v a l u e of t h e d i a m e t e r of t h e g r a i n s . We b e l i e v e t h a t t h e b e s t means of e x p r e s s i n g t h i s i s t o i n t r o d u c e t h e v a l u e o f KREY ( R e f e r e n c e 7 ) :

d

i n w h i c h "d" i s t h e mean dia.meter i n a p o r t i o n "p".

Por m i x t u r e s of sand i n the Plolland r i v e r s t h a t v a l u e c o r r e s p o n d s t o dg^ t o dg^.

MEYEE-PETER and MULLER have a l s o u s e d t h e e x p r e s s i o n

"d " i n d e v e l o p i n g t h e r e s u l t s of t h e i r o h s e r v a t i o n s . KALINSTCE recom¬ mends t h e d i a m e t e r d^^, w h i l e E i n s t e i n , i n a p r e c e d i n g p u b l i c a t i o n ( R e f e r e n c e 8 ) , employs d^^ t o d^^ as a c h a r a c t e r i s t i c d i a m e t e r .

4« R i p p l e C o e f f i c i e n t s .

a. T r a n s p o r t of hed m a t e r i a l and l o s s of energy

I t i s w e l l known t h a t t h e bedload t r a n s p o r t i s o b s t r u c t e d by t h e r i p p l e s on t h e bed o r by t h e dunes. A p a r t of t h e energy i s expended by overcoming t h e r e s i s t a n c e of t h e form of the r i p p l e s , s i n c e , b e h i n d t h e r i p p l e s , t h e c u r r e n t i s d e t a c h e d from t h e bed, and h e r v o r t i c e s a r e formed, w i t h t h e consequent t r a n s f o r m a t i o n of energy i n t o t u r b u l e n c e ajid, f i n a l l y , i n t o h e a t . T h i s t r a n s f o r m a -t i o n of energy -t a k e s p l a c e a -t s u c h a d i s -t a n c e from -t h e bed i n w h i c h the t r a n s p o r t c o n s i d e r e d i s produced. T h e r e f o r e , i t i s c o n s i d e r e d t o c o n t r i b u t e l i t t l e o r n o t h i n g t o t h a t t r a n s p o r t .

T h i s i s q u i t e d i f f e r e n t from t h e l o s s of energy w h i c h i s c a u s e d by t h e pure roughness of t h e g r a i n s and w h i c h i s d i r e c t l y r e l a t e d t o t r a n s p o r t .

I t i s för t h i s r e a s o n t h a t i t a p p e a r s p r o p e r t o d i v i d e the mean shear,"^o ,, e q u a l t o pgRI» i n t o two p a r t s .

F o r example , s i m i l a r t o t l i e r c s i s t c u i c e i n a p i p e h e i n r d i v i d e d i n t o f - ' • i c t i o n r e s i s t a . n c e and i n t o f o r m r e s i s t a n c e caused by t'-o o n t r n n o e l o s suddon e n l r r g o i n e n t j and b y t h e l o s s due t o c u r v e n \-atbf^r t h a n a '-eneral e x p r e s s i o n o f l o s s o f 'i.ead due t o t u r b u J e n t a g i t a t i o n .

The coel'f i c i p - " t t } i a t i r ' ' : t n c vr-.iich p a r t o f T'Q s h o u l d

i j . 'JoGTfioient;-.' -ovi r i m j l e r , a o c o r d i n g t o S i m - . t o i n r-nö. acoorO.iii• t o feoyer-Feter niA M u l l e r

. ' i i i n s t e i n i n t r o d u c e d a ri7?j>le c o e r f i c i e n t Tjy r e . l u c t i o n o f t h e h y d r a u l i c r a d i u s , 'R, v / h i l e h e y e r - P e t e r i\m- K u i l e r i n t r o d u c e t h e r i p p l e c o e f f i c i e n t h y r e d u o t i o n o f t h e s l o p e I . S i n c e i t S/J a l w a y s t h a v a l u e pgRI -.yhn c h c o n t r o l s OI.R oedload t r a n s p o r t , i t makes . L i t t l e dille.i.'-snce VL^OH «leciont i s r e d u c e d . J t i s o n l y n e c e s s a r y t o o o i n t

o u t t h a t a r e d u c - i i o n o-'- t h e h y d r a u l i c r a d i u s i s more d i f f i c u l t t o i m a g i n e th.r.n a r e d u c t i o n o f Vae slo;">e.

The rr:inner i n w h i c h txie c o e f f i c i e n t o f r e d u c t i o n i s d o t s x -mined i s n e a r l y t h e sfei..e i n b o t h o a s e s . One c a l c u l a t e s m a i n l y t h e h y d r a u l i c r a d i u s , H' ( E i n s t e i n ) o r t h u 3 l o p e , I ' ( Z u r i o h ) , w h i c h r e s u l t s f r o m f o r m u l a s o f r e s i s t a n c e , f o r t h e v e l o c i t y o b s e r v e d and a t t h e i n t r o d u c t i o n o f k a r a irie;asure o f r o u g h n e s s . T h i s v a l u e o f E' c r , i l l t h e S w i s s f o r u i u l a , ' , w h i c l i i s s m a l l e r t h a n t h e o r i f ? i n a l v a l u G , i a u s e d F o r m u l a s ( 3 ) and ( 4 ) f o r c a l c u l a t i o n o f oedload t r t u r n . ' O r t , Tn o r d e r t o 1 e:riidt r c o m p a r i s o n o f t h e d i f i e r o n t n r o o e s a e s , t h e r e i s r e p r s s e n t o d b e l o w t h e f o r r . s o f tyy f o r m u l a s as v / ^ l l as t h e a p p l i c a t i o n o f t h e r e s i s t a n c e f o r n i u l a o f S t r i c k l e r w l i i c h h a s t h e a p p l i c a t i o n 01 t i i e l o g a r i t h m i c r e s i s t a n c e f o r m u l a . An han boon na.id a l r e a d y , t h e two f o r i r u l a s rvive v e r y n e a r l y t h e samo r e s u l t s f o r t h e h y d r a u 1 i c a l l y r o u g h c o n d i t i o n s . E i n s t e i n M e y e r - P e t e r and fuller

26k-}'6.«2/3jl/2 (.y 26 ^/^B^/^f^ ^ 2 ^^^^

v = - 1 / 0 ^ 2 / 3 , 1 / 2 ^^^y R'= ( | ) '/^R

(10)

T. =

( | ) V 3

I I n t h e s e e q u a t i o n s , ir. r e i a - e s e n t s t h e r o u g h n e s s o f t h e g r a i n s (dJ'O) and Ï., t h e e q u i v a l e n t roughness o f t h e w a l l s , and i n c l u d e d t h e r i p i D l e s , d u n o s , e t c .

Ci;J<^: y

y

^ i/ \

^ y'--/y^ dK ^ ^ /V ^ T' ^ ' '

As h a s heeYi s a i d i n P t i r a g r a p h 2, t h e e x p e r i m e n t s made a t Z u r i o h iiave d e m o n c t r a t e d t h a t t h e t l i e o r e t i c a l e x p o n e n t 1/3 i n E q u a t i o n ( l 2 ) i s t o o l a r g e : a v a l u e o f 0,75 l / 3 = l / 4 j g i v i n g b e t t e r r e s u l t s . k V 4 I ' = ^ ^ ^''^^ t t h i s g i v e s t h e same r e d u c t i o n o f t h e s h e a r g i l l as t h a t v/hich E i m s t e i n f o u n d i n u s i n g t h e h y d r a u l i c r a d i u s ( E q u a t i o n l O ) , +

\

I n a p p l y i n g t h e l o g a r i t h m i c f o r m u l a , t h e s i m i l a r i t y o f t h e bedJ.otid foruiula.s i s n o t n u i t e so a p p n x e n t . 12R' V = 1 0 ïï^r l o g ( - - t - ) ( 1 4 ) 21 V = 18 ym^' l o g ( " ^ - - ) ( 1 5 a ) V = G/RÏ ( I 5 b ) i ' ^ ( — - i r - i ^ T i p - ) ) ' • ^ ( 1 6 ) k A g a i n , a c c o r d i i i ^ t o t h e e x p e r i m e n t s made a t Z u r i c h , t h e exj'Onent 2 i n E q u a t i o n (ló) d e d u c e d J'roiii th.e above c o n s i d e r a t i o n ,

\ \^ s h o u l d be m o d i f i e d by 0,75 x 2, sh.ichi i s etpuo,! t o 3 / ? . + I f one i n t r o d u c e s t h i s r e d u c t i o n b y means o f E q u a t i o n ( l O ) o r E q u a t i o n ( l 3 ) and taJvoa t h i s v a l u e o f K o f t h e E q u a t i o n (9t>) or ( l i b ) , t h e n one o b t a i n s t h e f o l l o w i n g r e s u l t s : c g R ' l = pgRI' . pg(v/2ó)V2j,V4il/4 A p p a r e n t l y t h e h y d r a u l i c r a d i u s , H, i s e l i m i n a t e d f r o m t h e b e d l o a d f o r i n u l a i f t h i s r e d u c t i o n i s a p p l i c a b l e , V-Vt^k ^/V" ' y

I.-'y

or

I n E q u a t i o n ( I 4 ) o f E i n n t e i n , R' a p p e a r s i m p l i c i t l y as t h e o n l y unknown. The v a l u e R' must he f o u n d by t r i a l . E i n s t e i n and B q r h a r o s s a ( R e f e r e n c e 9) have d e t e r m i n e d f o r t h a t f o r m u l a t h e v a l u e o f R' f o r a c e r t a i n number o f fijnerican r i v e r s , and t h e y have g i v e n t h e r e s u l t s i n g r a i l i i o f o r m . These c u r v e s show t h a t R' a p p r o a c h e s R when t h e d i s c h a r g e s a r e l a r g e . T l i i s , t h e a u t h o r s e x p l a i n e d , i n p a s s i n g , t h a t t h e r i v e r s meender a g r e a t d e a l a t s m a l l d i s c h a r g e s . 'A'hen t h e d i s c h a r g e s a r e l a r g e r , a n d t h e b e d l o a d movement i s i n c r e a s e d , t h e r i v e r w i l l t a k o n a s t r a i g h t e r c o u r s e and e n c o u n t e r l e s s s u p p l e m e n t a r y r e s i s t a n c e . n T h i s i s a case s i m i l i a r - i n p r i n c i p l e t o t h a t i n w h i c h bed r i p p l e s i n c r e a s e w i t h d i s c h a r g e m a k i n g t h e s u p p l e m e n t a r y r e s i s t a n c e l a r g e r and l a r g e r . M e a s u r e m e n t s on H o l l a n d r i v e r s a,nd i n t h e l a b o r a t o r y d e m o n s t r a t e t h i s f a c t . . £ 2 : l i l i 4 £ £ ^ ^ : ^ l i ^ _ i ^ c o e f f i c i e n t . W i t h r e s p e c t t o t h e e q u a l i t y o f h y d r a u l i c r o u g h n e s s o f t h e T o ^ a - i t h m i c f o r m u l a f o r r e s i s t a n c e and t h e a p p r o x i m a t i o n a c c o r d i n g t o S t r i c k l e r , t h e c o n c o r d i a i c e i n d i c a t e d above f o r E q u a t i o n s ( l O ) and ( 1 3 ) must a l s o e x i s t f o r E q u a t i o n s ( I 4 ) and ( 1 7 ) . T h i s a y p e a r s t o be p e r f e c t l y t r u e . The v a l u e o f R'/H, a c c o r d i n g co E q u a t i o n ( I 4 ) i.s t h e same as t h a t

o f i ' / l , a c c o r d i n g t o E q u a t i o n ( l ? ) . However, E q u a t i o n ( I 4 ) has t h e p r a c t i c a l i n c o n v e n i e n c e t h a t R' doe.s n o t a p p e a r e x p l i c i t l y . E q u a t i o n ( l ? ) l e a d s t o s i m p l e c a l c u l a t i o n s cuid i s t h e r e f o r e p r e f e r a b l e . One ca.n w r i t e R'/R I ' / l = ^ , w h e r e i s t h e r i p p l e c o e f f i c i e n t w i t h w h i c h t h e mean s h e a r , x^^, e q u a l t o p g R I must be m u l t i p l i e d i n o r d e r t o e l i m i n a + ?^,in t h e t r a n s p o r t f o r m u l a , t h e i n f l u e n c e o f t h e r o u g h n e s s o f t h e w a l l s and o f t h e b e d .

Then, a c c o r d i n g t o E q u a t i o n ( l ? ) C ( 1 8 )

1Ö l o g ( 22K7k|

^ v i t h u n i t e i n m e t e r s , k i l o g r a m s , s e c o n d s , and C = v / f t i , t h e a p p a r e n t c o e f f i c i e n t o f Chezy. R = t h e a p p a r e n t h y d r a u l i c r a d i u s , k = t h e r o u g h n e s s o f t h e g r a i n s Y/hich, a c c o r d i n g t o P a r a g r a p h 3, i s e q u a l t o dQO

Prom E q u a t i o n ( l 8 ) h a s heen made a g r a p h ( F i g u r e 2) w h i c h

gXYQs

t h e v a l u e o f p as a f u n c t i o n o f C amd o f R/k. The i n c o n v e n i e n c e o f E q u a t i o n ( l 8 ) i s t h a t m.ust he c a l c u l a t e d f r o m t h e a p p a r e n t v a l u s o f C as i t p r e s e n t s i t s e l f i n n a t u r e , w h i c h must t h e n he c a l c u l a t e d f r o m t h e measurements o f v e l o c i t y , s l o p e and c r o s s s e c t i o n . A f o r e c a s t o f t r a n s p o r t o f s a n d i n new s i t u a t i o n o f a w a t e r c o u r s e w i t h o u t i n t r o d u c t i o n o f o t h e r f a c t o r s i s t h e n i m p o s s i b l e . However, as E i n s t e i n and B a r b a r o s s a ( R e f e r e n c e 9) h a v e s t a t e d , f o r banks o f s a n d , t h e r i p p l e h e i g h t may, a c c o r d i n g l y , be r e l a t e d t o t r a n s p o r t w h i c h , i n t u r n , i s a f u n c t i o n o f t h e q u a n t i t y = A d / i - i R l . y^ C o n s e q u e n t l y , ^ = f , (A d / u R l ) d g ) / Xy 2

and f o r t h e same r e a s o n ]x = f,^ (A d/^iRI) ^ , (20) I t a p p e a r s t h a t t h e r e i s , i n e f f e c t , a n e t r e l a t i o n s l i i p a c c o r d i n g t o E q u a t i o n (20), f o r a s t r a i g h t - ^ w a t e r c o u r s e w h e r e t h e i n f l u e n c e o f r o u g h n e s s o f t h e l a t e r a l w a l l s on t h e s l o p e may he e l i m i n a t e d . I n Fig-ure 3-a, t h a t r e l a t i o n s h i p , as i n d i c a t e d a c c o r d i n g t o t h e s t u d i e s made a t D e l f t , ( s e e a p p e n d i x ) , h a s been i n d i c a t e d by a s t r a i g h t l i n e . When t h e v a l u e s o f A d / P I a r e l a r g e — t h a t i s t o s a y , when t h e t r a n s p o r t i s s m a l l and t h e b e d smooth — p, must a p p r o a c h u n i t y because t h e r o u g h n e s s o f t h e b e d , K, i s n e a r l y e q u a l t o t h e r o u g h n e s s o f t h e g r a i n s , k.

; i 2 ;

F o r n a t u r a , ! w a t e r c o u r s e s w i t h r e v e t m e n t s , d i l c e s , g r o i n s , and o t h e r a r t i f i c i a l o b s t r u c t i o n s , t h e v a l u e o f u, i f Zi d/RI i s l a r g e , must alwaj^s be a l i t t l e l e s s t h a n 1 , because t h e n t h e e q u i v a l e n t

r o u g h n e s s , K, alv/ays r e m a i n s l a r g ; e r t h a n t h e r o u g h n e s s o f t h e g r a i n s , k. O n l y i n t h e case where one succeeds i n e l i m i n a t i n g t h e i n f l u e n c e o f bhe g r o i n s , e t c . , on the s l o p e , c o u l d one a g a i n f i n d t h e r e l a t i o n - r s h i p o f 11 and A d / R l f r o m l a b o r a t o r y s t u d i e s .

F o r t h e H o l l a n d r i v e r s , such a r e l a t i o n s h i p a p p e a r s t o

a c t u a l l y e x i s t . I n F i g u r e 3-b, t h e v a l u e s o f have been drawn such t h a t t h e y f o l l o w the measurenients o f t r a n s p o r t made i n t h e Waal a.nd i n t h e Lower R h i n e R i v e r s ( see a p p e n d i x ) .

vVith a s i m i l a r r e l a t i o n s h i p betv/een u and A d / R l , i t i s e q u a l l y p o s s i b l e o t p r e d i c t f o r a c e r t a i n d i s c h a r g e t h e t o t a l r o u g b n e s s , and c o n s e q u e n t l y t h e v a l u e o f C f r p m Chezy.

u. I n f l u e n c e o f r i p p l e s i n K a l i n s k e ' s f o r m u l a

K a l i n s k e d i d n o t a p p l y c o r r e c t i o n s f o r t a k i n g i n t o a c c o u n t t h e i n f l u e n c e o f r i p p l e s , e t c . , on b e d l o a d t r a n s p o r t . S i n c e , as t h e a u t h o r has d e m o n s t r a t e d , h j s f o r m u l a i s i n c o n c o r d n.ricu w i t h a oci..a.u) number o f o b s e r v a t i o n oi d i f f e r e n t o r i g i n , t h e a a p p o s i x i o n luost ue a c c e p t e i ] t i i a t the i n f l u e n c e on r i p p l e s i s i m p l i c i t l y c o n t a i a o d i n t i i e f o r m u l a . T h i s i s a l s o p o s s i b l e f o r t h e two f o r r i r u l a s d i s c u s s e d u n l e s s one a c c e p t s a c o n s t a n t r e l a t i o n s h i p between t h e r i p p l e c o e f f i c i e n t , ( i , and t h e d i m e n s i o n l e s s p a r a m e t e r , Y - .,d/RI: f o r e x a m p l e , t h e s t r a i g h t l i n e i n F i g u r e 3-a, D o i n g t h i s , one t a k e s i n t o a c c o u n t o n l y t h e n a t u r a l f o r m a t i o n o f t h e r i i p u . e s ai,d does n o t liave l o c o r i s i d e r a l l t h e o t h e r sup])le!'ientai-y r e - . i i s t a n c e s s u c h ua sand cind g r a . v e l s , u a r s , g r o i n s , e t c .

On t h e c o n t r a r y , i n a d m i t t i n g , f o r e x a m p l e , t h a t t h e f o r m u l a o f E i n s t e i n i..ast be e x a c t w i t h o u t t h e i n f l u e n c e o f t l i c r i n p l e s , one i s a b l e tö d e t e r i i i i n e f r o m F i g u r e 1 , t h e i r b o v e - m e n t i o n e d r e l a t i o n s h i p betv/een u and Y, whic!! i s i i i . p l l c i t l y c o n t a i n e d , i n t l i e f o r m u l a o f

' - 13 K a l i n s k e . Then i t i s a m a t t e r o n l y o f d i v i d i n g t h e o r d i n a t e n o f t h e c u r v e s o f K a l i n s k e hy t h o s e o f E i n s t e i n t o d e t e r m i n e v a l u e s o f Y. FiÉ^re 3 c g i v e s t h e r e s u l t s . T h e r e i s an agreement w i t h t h e s t r a i g h t l i n e w h i c h f o l l o w s t h e t e s t made a t D e l f t and a t Z u r i c h ( R e f e r e n c e 3, F i g u r e 1 2 ) . g ^ l ^ l ^ a l V e l o c i t y o f _ C i j ^ t i c a l _ S ^ ^ t h e B e g i n n i n g - o f T r a n s p o r t , I n most o f t h e f o r m u l a d e r i v e d i n t h e p a s t , one i s a h l e t o f i n d a c r i t i c a l v a l u e o f v e l o c i t y o r t h e mean s h e a r , To , w h i c h w o u l d he c h a r a c t e r i s t i c f o r t h e b e g i n n i n g o f t h e t r a n s p o r t . I t i s o n l y i n t h e f o r m u l a b a s e d on t h e modern t h e o r i e s o f t u r b u l e n c e , such as t h e f o r m u l a s d i s c u s s e d o f K a l i n s k e and o f E i n s t e i n , t h a t one does n o t e n c o u n t e r such a v a l u e . T h i s v a l u e i s r e p l a c e d by t h e p r o b e b i l i t y o f movement w h i c h a l w a y s d i m i n i s h e s i n p r o p o r t i o n as becomes s m a l l e r .

T h i s i s a c c o r d w i t h what one i s a b l e t o see i n l a b o r a t o r y s t u d i e s c o n c e r n i n g t h e commencement o f t r a n s p o r t . I n e f f e c t , t h e r e a r e a l w a y s some g r a i n s t h a t a r e i n movement because o f a s m a l l amount o f t u r b u l e n c e , and t h e r e f o r e an a b s o l u t e r e p o s e o f bed m a t e r i a l s i s n o t q u i t e a t t a i n a b l e . A l s o , t h e l i m i t e i n t h e a b o v e - m e n t i o n e d f o r m u l a s a r e o f t e n a r b i t r a r y , and b e s i d e s a r e d i f f i c u l t t o d e t e r m i n e .

M e y e r - P e t e r and M u l l e r i n t r o d u c e t h e same t w o l i m i t e , one f o r t h e a b s o l u t e r e p o s e and t h e o t h e r f o r commencement o f t r a n s p o r t . A l s o , i t i s n e c e s s a r y i n t h e b e d l o a d t r a n s p o r t f o r m u l a s t o g i v e p r e f e r e n c e t o t h e p r o b i l i t y o f movement t h a t i s i n a c c o r d a n c e w i t h n a t u r e . I t i s c h i e f l y f o r t h e v e r y s m a l l b e d l o a d s t h a t one o b t a i n s t h e most e x a c t r e s u l t s by t h e use o f a c r i t i c a l v a l u e . However, i n c e r t a i n c a s e s i t may be u s e f u l — f o r e x a m p l e , when s t a b l e c a n a l s a r e p r o p o s e d — t o be a h l e t o i n s e r t a v a l u e o f v e l o c i t y o r s h e a r f o r w h i c h t h e r e w i l l n e v e r be any a p p r e c i a b l e b e d l o a d t r a n s p o r t .

14

/9 I t i s r e i n a i k a b l e t h a t mcst o f t h e e x p e r i m e n t s e x p r e s s e d t h a t l i m i t by t h e parame'ter, Y = a d / R I , w h i c h p l a y s such an i m p o r t a n t r o l e i ^ t h e t r a n s p o r t f o r m u l a s discussed.,

Ti.son ( R e f e r e n c e lü) has a s s e u b l e d a c e r t a i n number o f d a t a f r o m d i f f e r e n t s o u r c e s , and he has e s t a b l i s h e d t h e i n f l u e n c e o f R,,, t h e R e y n o l d s number f o r a g r a i n :

R = d K 5 ^

. J L - - - ( 2 1 )

iïhen R has a v a l u e b e t w e e n 10 and 100, t i i e v a l u e o f Y f o r t h e p r a c t i c a l b e g i n n i n g o f t r a n s p o r t a p p e a r s t o l i e betv/een 50 and 25: i t has a v a l i i e b e t w e e n 3'.' and 20.

I>ieyer-Reter and M u l l e r ( R e f e r e n c o 3 ) g i v e t h e v a l u e o f "i = 33 f o r

t h e a b s o l u t e b e g i n n i n g o f movement. ^ 6. E m p i r i c a l S i r r p l i f i c a t i o n P r o p o s e d f o r %he F o r m u l a s .Disciijig.edj^

The t h r e e f o r m u l a s d i s c u s s e d above a r e a l l somewhat i n c o n v e n i e n t f o r p r a c t i c a l a _ r p l i c a t i o n . F i g u r e 1 shows, h o w e v e r , t h a t t h e c u r v e s o f E i n s t e i n and o f Iv e y e r - P e t e r and M u l l e r rray be a p p r o x i m a t e d w i t h s u f J ' i c i e n t a c c u r a c y b y a s i m p l e f o r m u l a . ^. ^ ,^yl/2 e-0.27Y ( 2 2 ) When t h e v a l u e o f Y a r e l a r g e , t h i s cui-ve d i f f e r s f r o m t h e one d e v e l o p e d a t Z u r i c h because i n t h i s r a n g e t h e s m a l l e r c r i t i c a l v a l u e does n o t e n t e r i n . A t thie sairie tin;e one may f i n d w U h t h a t c u r v e g r e a t e r v a l u e s o f t r a n s p o r t t h a n by t h e uso o f E i n s t e i t i ' s c u r v e , w h i c h a g a i n c o r r e s p o n d s t o t h e r e s u l t s o f o h a e r v a t i o n ( s e e R e i e r e n c e 4 ) • The m i d d l e r^art wf t h e c u r v e , f o r t h n v a l u e o f Y b e t w e e n 3 and I 6 , i s p r a c t i c a l l y i d e n t i c a l t e b o t h t h e E i n s t e i n and Z u r i c h c u r v e s .

15 I f t h e v a l u e s o f Y a r e s m a l l e r t h a n 3, t h e a p p r o x i m a t e o u r v e t e n d s t o g i v e t o o s m a l l v a l u e s o f t r a n s p o r t . I n t h i s r e g i o n where b e d l o a d t r a n s p o r t i s v e r y i n t e n s i v e , i t o a n n o t be s a i d , v / i t h few o b s e r v a t i o n s a v a i l a b l e , t h e s e f o r m u l a s g i v e t h e e x a o t b e d l o a d t r a n s p o r t b e c a u s e some s u s p e n d e d l o a d may be i n c l u d e d . ' As has been e x p l a i n e d i n P a r a g r a p h 4 a , i t i s p r e f e r a b l e a t t h e DELFT l a b o r a t o r y t o u t i l i z e t h e r i p p l e f a c t o r o f J^EYEH-PETER and DULLER when a p p l y i n g t h e f o r m u l a o f l o g a r i t h m i c r e s i s t a n c e : 3/2 ( I 8 a ) 18 l o g 12R ^-0 + 3 ^/ygy H a v i n g t a k e n i n t o a c c o u n t .ae i n f l u e n c e o f t h e r i p p l e s , i t t h e n f o l l o w s : . , x - l / 2 - 0.27 ' ( 2 3 ) X = S (Y/^i) ' e T h i s e q u a t i o n may a l s o be w r i t t e n y = 5e - ^ - ^ ^ ^ ^ p R i ( ^ 3 . ; d ^ V T T R i T h i s f o r m u l a g i v e s a s i m p l e s t r a i g h t l i n e when t h e d i m e n s i o n l e s s t e r n on t h e l e f t i s p l o t t e d as an a h c i s s a on a l o g a r i t h m i c s c a l e and t h e l a r g e v a l u e Y/'p =Ad/RIp as t h e o r d i n a t e on a l i n e a r s c a l e ( F i g u r e 4 ) . 7 , C o n c l u s i o n s a. I t a p p e a r s t h a t each o f t h e t h r e e f o r m u l a s can be r e d u c e d t o a r a t i o b e t w e e n d i m e u s i o n l e s s p a r a m e t e r s w h i c h a r e t h e same f o r each f o r t i . u l a . The f o r m u l a s a p p e a r t o be n e a r l y t h e same. h. K a l i n s k e ' s f o r m u l a can o n l y be a p p l i e d by means o f a d i s g r a m b e c a u s e i t s c o n s t r u c t i o n i s c o m p l i c a t e d . c. F o r t h e same r e a s o n E i n s t e i n ' s f o r m u l a can o n l y be u s e d v / i t h a d i a g r a m . E i n s t e i n a p p l i e s a r e d u c t i o n o f t h e b o u n d a r y s h e a r 'to i n o r d e r t o o b t a i n t h a t p a r t o f w h i c h i s c h a r a c t e r i s t i c f o r xhe t r a n s p o r t m a t e r i a l . The v a l u e o f t h a t c o e f f i c i e n t i s e q u a l t o t h a t o f

LEYER-PETER and FULLER, w h e r e a s t h e f o r m o f t h e l a t t e r l e a d s t o s i m p l a r and more r a p i d l y made c a l c u l a t i o n s .

d. E i n s t e i n does n o t recommend t h e a p p l i c a t i o n ' d . ' t h e t r a n s p o r t f o r m u l a on a m i x t u r e o f g r a i n s i n i t s e n t i r e t y , because he f o u n d t h a t i t i s n o t e x a c t enough. A c c o r d i n g t o h i i i ' t h e c a l c u l a t i o n o f t r a n s p o r t o f each s e p e r a t e f r a c t i o n o f l u a t e r i a l , b y a p p l y i n g t h e c o r r e c t i v e c o e f f i c i e n t f o r t h e i r m u t u a l i n f l u e n c e , l e a d s t o n u c h g r e a t e r a c c u r a c y . I t i s q u e s t i o n a b l e i f p r e s e n t k n o w l e d g e o f t h e phenomenon t h a t i s a t o u r coiiimand j u s t i f i e s t h e s e c o m p l i c a t e d c a l c u l a t i o n s . F o r i n s t a n c e , t h e u n c e r t a i n t y i n t h e d e t e r i i . i n a t i o n o f t h e d i a m e t e r o f t h e g r a i n s l e a d s one t o b e l i e v e t h a t t h e a t t a i n e d p r e c i s i o n i s o n l y a p p a r e n t .

e. IvJEYER - PETER and MULLER u s e i n t h e i r p r i m a r i l y e m p i r i c a l f o r m u l a a l i n i i t o f t h e mean t r a c t i v e f o r c e t h a t w i l l be c h a r a c t e r i s t i c f o r t h e b e g i n n i n g o f t r a n s p o r t . Because o f t h e e f f e c t o f t u r b u l e n c e t h a t i s a l w a y s f o u n d - f n w a t e r c o u r s e s t r a n S i . ' O r t i n g m a t e r i a l s , t h a t l i m i t i s t o t a l l y f i c t i t i o u s . The f o r m i u l a l e a d s t o f a l s e v a l u e s f o r v e r y s m a l l bed loa,d t r a n s p o r t . A s i d e f r o m t h i s , t h e f o r m u l a g i v e s good r e s u l t s , and i t s c o n s t r u c t i o n i s s i m p l e . The r e d u c t i o n o f t h e mean t r a c t i v e f o r c e i s made i n an a c c e p t a b l e ma.nnêr. f . F o r s t r a i g h t c h a n n e l s v / i t h o n l y t h e r e s i s t a n c e o f t h e r i p p l e s and t h e r o u g h n e s s o f t h e g r a i n s , t h e r i p p l e c o e f f i c i e n t a p p e a r s t o depend u n i q u e l y on t h e p a r a m e t e r A d . T h a t f u n c t i o n t a k e s o t a e r v a l u e s when t h e R I

w a t e r c o u r s e i o c u r v e d and when t h e r e a r e b a r s o r v/hen t h e r e a r e o t h e r o b s t a c l e s . The f u n c t i o n i s f u n d a m e n t a l l y a n o t h e r t h a n t h a t v;bich E i n s t e i n has f o u n d f o r t h e A m e r i c a n r i v e r s , i n w h i c h t h e ood w i t l i s m a l l d i s c h a r g e s meanders b e t w e e n t h e b a n k s , s t r a i g h t e n s a t t h e h i g h f l o w s and becomoc l e s s r o u g h . g. The t h r e e f o r m u l a s d i s c u s s e d a g r e e very w i l . B e s i d e s t h e y iha.y be a p p r o x i m a t e d w i t h s u f f i c i e n t p r e c i s i o n by t h e s i m u l e f o r m u l a d v T T ^ T ^ ^ n, ' ^

17

w h i c h c o v e r s f o r t h e most p a r t t h e e x p e r i m e n t a l c u r v e o f //'EYER-PETER and iv.ULLER d e r i v e d f r o m a l a r g e number - f l a b o r a t o r y o b s e r v a t i o n s and w i c h i s i n s a t i s f a c t o r y a c c o r d a n c e w i t h a c e r t a i n number o f o b s e r v a t i o n s i n t j i e f i e l d .

APPENDIK A.

D e s c r i p t i o n o f o b s e r v a t i o n s i n t h e r i v e r s o f H o l l a n d .

The measurements w e r e made w i t h t l i e a i i p a r a t u s f o r m e a s u r i n g t r a n s p o r t c a l l e d ARA'HEI^ ( B T L I A ) w h i c l i has a l r e a d y been d e s c r i b e d

e l s e w h e r e ( R e f e r e n c e 1 1 and 1 2 ) . I t i s p o s s i b l e t o measure w i t h t h i s

e q u i p m e n t t h e m a t t e r c a r r i e d i n t h e f i r s t 0 . 1 m e t e r ( 3 . 9 3 i n c h e s )

above t h e b e d .

Ten o b s e r v a t i o n s v/ere made a t t h e same p o i n t . Then t h e

m e a s u r e m e n t s were r e p e a t e d a t a p o i n t 1 0 m e t e r s ( 3 3 f e e t ) u p s t r e a m a n d

1 0 m e t e r s ( 33 f e e t ) d o w n s t r e a m f o r t h e i n i t i a l p o i n t , so t h a t t h e

ari;ount o f t r a n s p o r t a t a s i n g l e p o i n t i n t h e t r a n s v e r s e p r o f i l e was

d e t e r m i n o d b y 30 o o s e r v a t i o n s , I<"igure 5 . The t r a n s v e r s e p r o f i l e

was m e a s u r e d c o m p l e t e l y w i t h a d i s t a n c e o f 4^. m e t e r s b e t w e e n t w o

p o i n t s a l o n g t h e a x i s o f idie r i v e r . Ow.--: ' o m p l e t e o b s e r v a t i o n t o o k 2 d a y s

on t h e Lower iïhine and 3 n^^ys on "the Waal.,

The v a r i a t i o n i n t h e 1 0 mpp-.u^^omcnts a t a s i n g l e p o i n t a r e

somewhat l a r g e . V a l u e s f r o m 0 t o 5OO p e r c e n t o f t h e mean v a l u e a r e

f o u n d . The v a r i a t i o n i n t h e mean v a l u e s f o r t h e t h r e e p o i n t s l o c a t e d

one b e h i n d t h e o t h e r a r e no more t h a n 5 0 p e r c e n t i n t h e n o r m a l c a s e s .

The mean t r a n s p o r t o v e r t h e e n t i r e r i v e r s was c a l c u l a t e d w i t h

t h e a i d o f a d i a g r a m , Eig-ure 5 . The v a l u e s A,B,C, e t c . a r e t h e mean

v a l u e s c a l c u l a t e d f r o m t h e 30 o b s e r v a t i o n s a t h a n d . Tho a r e a o f t h e

d i a g r a m d i v i d e d oy che e f f e c t i v e w i d t h b , g i v e s t h e v a l u e T, t h e meam bed l o a d t r a n s p o r t p e r u n i t o f w i d t h f o r t h e c r o s s - s e c t i o n .

T h i s e f f e c t i v e w i d t h i s u s e d because i t i s c l e a r t h a t t h e u s e o f t h e t o t a l w i d t h B g i v e s r e s u l t s t h a t a r e n o t e x a c t . I t i s known

t h a t i n t h a t v a l u e o f b , d e t e r m i n e d f r o m i a d i a g r a m s u c h as F i g u r e 5» some u n c e r t a i n t y must be a c k n o w l e d g e d . I t was n o t p o s s i b l e t o f i n d a b e t t e r m e t h o d . T a b l e s 1 and 2 g i v e p o s t w a r d a t a f o r t h e Lower R h i n e and t h e V/aal. I n a d d i t i o n t o t h e t r a n s p o r t T, t h e f o l l o w i n g v a l u e s were g i v e n ; The mean d e p t h h , t h e s l o p e , t h e d i s c h a r g e q p e r u n i t o f v / i d t h , t h e v a r i a t i o n h i n t h e l e v e l o f w a t e r d u r i n g t h e o b s e r v a t i o n , t h e e f f e c t i v e v / i d t h b, t h e c a l c u l a t e d v a l u e s o f

. . . ,

hys

- ^

T/d^/ g p R I and Y /p = A d ^ / [ i . R I , and t h e r i p p l e c o e f f i c i e n t . • D u r i n g each measurements o f t r a n s p o r t , t h e c r o s s - s e c t i o n was m.easured w i t h an y l t r a - s o n i c s o u n d e r . The d e p t h - h - i s t h e mean

d e p t l i a l o n g t h e a f f e c t i v e w i d t h b, F i g u r e 5«

The v a l u e g i v e n f o r t h e d i a m e t e r s o f t h e g r a i n s a r e t h e mean o f a v e r y l a r g e number o f m e a s u r e m e n t s . D u r i n g t h e o b s e r v a t i o n s o f b e d l o a d t r a n s p o r t , some s a m p l e s f r o m t h e bed were t a k e n a t each p o i n t o f m e a s u r e m e n t s : t h a v a l u e g i v e n o f d ^ ^ o f a l l t h e s a m p l e s .

, A l l t h e s a m p l e s t a k e n w i t h t h e BTMA ( S a m p l e r ) have a l s o been s i f t e d : t h e v a l u e g i v e n f o r d i s l i k e w i s e t h e mean o f d o f a l l t h e s a m p l e s .

APPEMDIX B.

D e s c r i p t i o n o f measurements made a t t h e H y d r a u l i c L a b o r a t o r y a t DELFT.

A c e r t a i n number o f measurements o f t r a n s p o r t were made i n a g l a s s s i d e d f l u m e 15 m e t e r s ( 5 0 f e e t ) l o n g , 0 . 5 m e t e r

( 1 . 6 5 f e e t ) w i d e , and 0 . 6 m e t e r ( 2 f e e t ) deep, w i t h l i g h t m a t e r i a l s ( g r o u n d pumice s t o n e and g r o u n d b a k e l i t e ) . The m a i n o b j e c t o f t h e s e s t u d i e s was t o e x a m i n e t h e s e m a t e r i a l s t o d e t e r m i n e t h e i r a d a p t a b i l i t y t o use i n m o d e l s "with m o v a b l e b e d s .

19

The t r a n s p o r t e d m a t e r i a l s were r e c o v e r e d a t t h e d o w n s t r e a m and o f t h e f l u m e s i n a d t e p p i t . By w e i g h i n g t h e r e c o v e r e d m a t e r i a l s d u r i n g each s t u d y , t h e amount o f t r a u i s p o r t was d e t e r m i n e d . Ta.hles 3 and 4 g i v e t h e r e s u l t s o f t h e s e o b s e r v a t i o n s .

L I S T OF REFERENCES *

1 . K a l i n s k e , A. A., Movement of sediment a s bed l o a d i n r i v e r s , T r a n s . Am. Geonhys. Union, V o l . 2 8 , P t . 4 , pp 615-620, I 9 4 7

2 . E i n s t e i n , H. A., The bed l o a d f u n c t i o n f o r sediment t r a n s p o r -t a -t i o n i n open c h a n n e l f l o w s , S o i l Cons- S e r v i c e , U. S. Dep-t. of A g r i c u l t u r e , Techn. B u l l . No. 1 2 6 , Washington, I 9 5 O

3 . Meyer-Peter, E . e t M u l l e r , R. Formulas f o r B e d - l o a d T r a n s p o r t , Deuxieme r e u n i o n de 1 'A s s o c i a t i o n I n t e r n a t i o n a l e de R e c h e r c h e s H y d r a u l i q u e s , Stockholm, I 9 4 8 4 - Elzerman, J . J . e t P r i j l i n k , H. C , P r e s e n t s t a t e of t h e i n v e s t i -g a t i o n s on b e d - l o a d movement i n H o l l a n d , I X me Assemblee de I'U.G.G.I., B r u x e l l e s , I 9 5 I

5 . White, C. M., E q u i l i b r i u m of g r a i n s on the bed of a stream, P r o c . R. S o c , London, V 174A, pp 3 2 2 - 3 3 8 , I94O

6. T h i j s s e , J . Th., F o r m u l a s f o r t h e f r i c t i o n head l o s s a l o n g con-d u i t w a l l s u n con-d e r t u b u l e n t f l o w . T r o i s i e m e r e u n i o n con-de 1' A s s o c i a -t i o n I n -t e r n a -t i o n a l e de R e c h e r c h e s H y d r a u l i q u e s , Grenoble, 1949 7 . Krey, H. D., M o d e l l v e r s u c h e f u r e i n e n F l u s z mit s t a r k e r

Geschiebe-bewegung ohne e r k e n n b a r e Bankwanderung, B e r l i n , I 9 3 5

8 . E i n s t e i n , H. A., F o r m u l a s f o r t h e t r a n p o r t a t i o n of bed l o a d , T r a n s , A.S.C.E., V o l . 1 0 7 , pp 5 6 1 , 1947

9 . E i n s t e i n , H, A. e t B a r b a r o s s a , N, L,, R i v e r C h a n n e l Roughness, P r o c . A,S,C.E,, V o l . 7 7 - sep, No. 7 8 , I 9 5 1

10. T i s o n , L. J . , R e c h e r c h e s s u r l a t e n s i o n l i m i t e d ' e n t r a i n e m e n t des m a t e r i a u x c o n s t u t i f s du l i t , Ann. Soc. S c i e n t i f i q u e , B r u x e l l e s pp 1 6 3 - 1 8 1 , 1 9 4 7

1 1 . E i n s t e i n , H. A., D e t e r m i n a t i o n of r a t e s of b e d - l o a d movement, P r o c . Fed. I n t . Agency S e d i m e n t a t i o n Conf., Washington I 9 4 8 1 2 . SchaanJc, E. M. H., M i t t e i l u n g e n u b e r den G e s c h i e b e f anger

"Arnhem" P r e m i e r e r e u n i o n de 1' A s s o c i a t i o n I n t . de Rech. Hydr., P 1 1 3 , B e r l i n , 1937

L I S T OF SYMBOLS U S E D — C o n t i n u e d Symljol Dimension D e s c r i p t i o n R R e y n o l d s number f o r t h e g r a i n s 2 -1 T I t T r a n s p o r t e d m a t e r i a l i n u n i t s o f volume p e r u n i t o f w i d t h and p e r u n i t o f time t V a r i a b l e o f i n t e g r a t i o n X D i m e n s i o n l e s s parameter T / d V 2 , 1 / 2 Y D i m e n s i o n l e s s parameter ii d / R I . A R e l a t i v e d e n s i t y P i - P P T)o C o n s t a n t 0 K a l i n s k e ' s f u n c t i o n \x R i p p l e c o e f f i c i e n t I ' R' 2 - 1 V I t K i n e m a t i c v i s c o s i t y p ml D e n s i t y o f t h e t r a n s p o r t i n g l i q u i d pi ml D e n s i t y o f t h e t r a n s p o r t e d g r a i n s —1 —2

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