sediment transport in rivers flow of water in a curved

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Therefore the edge was made oblique, with the highest point near the inner side ... helical velocity components and the depth-averaged flow field resulting from ... in which Q is the total discharge, B is the channel width, I is the slope of .... s i g n i f i c a n t d i f f e r e n c e s seem to occur (see Figures 4 and 5 ) : i n the lower ...
sediment transport in rivers

flow of water in a curved open channel with a fixed uneven bed

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

report on experimental and theoretical investigations

R 657-VI

nil.

• El

i-h-

.-a-

M 1415 part II November 1978

toegepast onderzoek waterstaat

I

CONTENTS

L I S T OF SYMBOLS

SUMMARY page 1

Introduction.

1

2

Experimental

2

2.1

Channel

2.2

Flow c o n d i t i o n s

2

2.3

Measured d a t a

3

3

Results

4

3.1

E l a b o r a t i o n of t h e measured d a t a

4

3.2

V e r t i c a l d i s t r i b u t i o n of t h e m a i n v e l o c i t y

3.3

V e r t i c a l d i s t r i b u t i o n of t h e h e l i c a l v e l o c i t y

3.4

Depth-averaged v e l o c i t y

3.5

I n t e n s i t y of the secondary

3.6

Water s u r f a c e c o n f i g u r a t i o n and energy

4

Conclusions

set-up

geometry

2

field

component component

6 7 8

flow

9 head

10

12

REFERENCES

TABLES

FIGURES

APPENDIX A

Instrumentation

APPENDIX B

Measuring

APPENDIX C

E l a b o r a t i o n of measured

procedure data

L I S T OF TABLES

I

Summary of

measurements

II

V e r t i c a l s grouped a c c o r d i n g t o d e p t h of f l o w and Chêzy's

III

Depth-averaged

IV

Water s u r f a c e

V

E n e r g y head

quantities elevation

factor

L I S T OF

FIGURES

1

C h a n n e l geometry

2

Bed

3

Depth-averaged i n f l o w v e l o c i t y

4

S i m i l a r i t y of the v e r t i c a l d i s t r i b u t i o n of t h e main v e l o c i t y

5

A p p l i c a b i l i t y of the l o g a r i t h m i c m a i n v e l o c i t y

profile

6

M a i n v e l o c i t y p r o f i l e s compared w i t h p l a n e bed

data

7

S i m i l a r i t y of the v e r t i c a l d i s t r i b u t i o n of t h e h e l i c a l v e l o c i

8

C o m p a r i s o n between the d e p t h - a v e r a g e d v e l o c i t i e s

9

C o m p a r i s o n between i n t e n s i t i e s of the s e c o n d a r y

10

Secondary flow

11

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

12

E n e r g y head

13

Transverse

c o n f i g u r a t i o n of the w a t e r

14

Transverse

c o n f i g u r a t i o n of t h e e n e r g y head

elevation distribution

i n s e c t i o n D^ ( p l a n e

bed)

surface

flow

Constants

i n t h e e l a b o r a t i o n of t h e measured d a t a ,

e q u a l to h/R and t a n a Channel

theoretically

respectively

width

S l o p e of t h e c a l i b r a t i o n c u r v e of t h e v e l o c i t y m e t e r Value

of t h e v e l o c i t y c o r r e s p o n d i n g

Value

of c ^ a t c a l i b r a t i o n

to zero p u l s e

frequency

temperature

C o e f f i c i e n t of p r o p o r t i o n a l i t y i n t h e d i r e c t i o n measurement C o e f f i c i e n t of p r o p o r t i o n a l i t y i n the temperature L o c a l v a l u e of C h e z y ' s r o u g h n e s s Value

factor

of C a t t h e r e f e r e n c e d e p t h o f f l o w h ^

Depth of f l o w s c a l e Energy

c o r r e c t i o n of c ^

i n the numerical

s i m u l a t i o n of the flow

head

Mean v a l u e of e 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 V e r t i c a l d i s t r i b u t i o n f u n c t i o n of t h e main v e l o c i t y

component

V e r t i c a l d i s t r i b u t i o n f u n c t i o n of t h e h e l i c a l v e l o c i t y component V e r t i c a l d i s t r i b u t i o n f u n c t i o n of the flow d e v i a t i o n angle F r o u d e number b a s e d

((f)-a)

on t h e o v e r a l l mean v e l o c i t y and d^

A c c e l e r a t i o n due t o g r a v i t y L o c a l depth of flow Reference depth of f l o w f o r Chezy's Number o f t h e g r i d p o i n t

( i = 1 near

L o n g i t u d i n a l s l o p e of t h e energy Nikuradse

factor t h e bed)

line

r o u g h n e s s l e n g t h ( e q u i v a l e n t sand

Number o f p u l s e s c o u n t e d

roughness)

by t h e v e l o c i t y m e t e r d u r i n g an o b s e r v a t i o n

Number of g r i d p o i n t s i n a v e r t i c a l Discharge Local radial Radius

coordinate

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

field

Time-mean r e a d i n g o f t h e p o t e n t i o m e t e r

d u r i n g f l o w d i r e c t i o n measurements

R e f e r e n c e r e a d i n g of t h e p o t e n t i o m e t e r

d u r i n g f l o w d i r e c t i o n measurements

D u r a t i o n of 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 E s t i m a t e o f t h e mean v e l o c i t y when d e t e r m i n i n g velocities

i n t h e lower p a r t s o f a v e r t i c a l

L o c a l time-mean h e l i c a l v e l o c i t y component

C from t h e measured

L I S T OF SYMBOLS

(continued)

Vj^^^

Secondary flow

v'^^j^

Normalized h e l i c a l v e l o c i t y

V . _main v . main Vm a m .

L o c a l time-mean main v e l o c i t y component ^ ^ D e p t h - a v e r a g e d v a l u e of v ^ ° mam E s t i m a t e o f t h e d e pf t h - a v e r a g& e d v e l o c i t yj i n t h e e l a b o r a t i o n o f t h e measured

v'

. mam

intensity component

data

N o r m a l i z e d main v e l o c i t y

v^

L o c a l time-mean r a d i a l v e l o c i t y

v^

D e p t h - a v e r a g e d v a l u e of v ^

Vj.

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

v^

D e p t h - a v e r a g e d v a l u e of v ^

v^ ^ _tot v^ ^ tot Vmam. .

L o c a l time-mean t o t a l

velocity ^ Depth-averaged v a l u e of v ^ * tot Main v e l o c i t y^ i n gB r i d• pf o i n t is, c a l c u l a t e d

y

Transverse coordinate

z

V e r t i c a l d i s t a n c e t o t h e bed

z

s

Ze 's z* o

component

component

from measured d a t a

( y = 0 a t t h e l e f t bank)

Water s u r f a c e e l e v a t i o n mean v a i u e or z i n cne r e i e v a n c c r o s s - s e c c i o n s L e v e l a t w h i c h t h e l o g a r i t h m i c v e l o c i t y d i s t r i b u t i o n becomes to

zero

a

D i r e c t i o n of t h e d e p t h - a v e r a g e d v e l o c i t y

X]^

Dynamic v i s c o s i t y

of w a t e r

Dynamic v i s c o s i t y o f w a t e r

vector

a t the measuring a t the c a l i b r a t i o n

temperature temperature

K

Von Karman's c o n s t a n t

(j)

L o c a l time-mean f l o w a n g l e w i t h r e s p e c t t o t h e c h a n n e l

(()'

L o c a l time-mean f l o w angle, r e a d

(j)'

R e f e r e n c e f l o w a n g l e , r e a d from t h e f l o w d i r e c t i o n m e t e r

axis

from t h e f l o w d i r e c t i o n m e t e r

/

equal

SUMMARY

T h i s r e p o r t d e s c r i b e s p h y s i c a l and n u m e r i c a l e x p e r i m e n t s water

i n a curved flume,

c o n s i s t i n g of a 38 m l o n g s t r a i g h t

a 90° bend w i t h a r a d i u s of c u r v a t u r e of 50 m was

formed by

s c a l e bed

two v e r t i c a l

s i d e w a l l s and

6 m and

the depth

mean v a l u e of about 0.2

m.

s e c t i o n f o l l o w e d by

( s e e F i g u r e 1 ) . The

cross-section

a c o n c r e t e bed w i t h a s c h e m a t i z e d

c o n f i g u r a t i o n as c a n be e x p e c t e d

t h e f l u m e was

on t h e s t e a d y f l o w of

large-

i n a n a t u r a l r i v e r bend. The w i d t h

of f l o w v a r i e d between 0.05

m and 0.4

m,

of

with a

Measurements were t a k e n a t two d i s c h a r g e s : 0.463 m^/s

and 0.232 m^/s ( y i e l d i n g mean v e l o c i t i e s

of about 0.4

m/s

and 0.2

m/s

respec-

tively) .

During

the experiments

t h e f o l l o w i n g phenomena were

investigated:

a. t h e v e r t i c a l d i s t r i b u t i o n of t h e h o r i z o n t a l v e l o c i t y components v m a i n f l o w d i r e c t i o n ) and v,

. ( i n the mam direction);

^ ( p e r p e n d i c u l a r t o t h e main f l o w

hei

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 of t h e d e p t h - a v e r a g e d v e l o c i t y and of t h e s e c o n d a r y c. the water The

flow;

s u r f a c e c o n f i g u r a t i o n and

t h e d i s t r i b u t i o n of t h e energy

model of s t e a d y 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 F l u i d M e c h a n i c s of the D e l f t U n i v e r s i t y of T e c h n o l o g y

The v e r t i c a l d i s t r i b u t i o n s of t h e main v e l o c i t y

by

t h e l o g a r i t h m i c p r o f i l e used

head.

mathematical

i n t h e L a b o r a t o r y of

(de V r i e n d ,

1976

and

1977).

t u r n e d out to be h i g h l y s i m i l a r

t h e f l o w f i e l d , b u t t h e y a l l were somewhat f l a t t e r

The v e r t i c a l

intensity

and

e x p e r i m e n t a l r e s u l t s have b e e n compared w i t h the r e s u l t s of a

throughout

the

i n the mathematical

d i s t r i b u t i o n s of t h e h e l i c a l v e l o c i t y

than p r e d i c t e d

model. (i.e.

a g r e e d w e l l w i t h the t h e o r e t i c a l c u r v e i n t h e d e e p e r

t h e shape of the c u r v e )

p a r t s of t h e bend, but i n

t h e s h a l l o w e r p a r t s t h e measured d a t a were too i n a c c u r a t e to draw c o n c l u s i o n s . The

observed depth-averaged

t h e p r e d i c t e d one:

velocity field

the observed

showed an i m p o r t a n t d e v i a t i o n from

s h i f t i n g of t h e v e l o c i t y maximum towards

the

o u t e r w a l l i n the bend was

c o n s i d e r a b l y s t r o n g e r . T h i s phenomena, w h i c h was

encountered

experiments

during e a r l i e r

must be a t t r i b u t e d

observed

The

which

flow, which i s

model.

i n t e n s i t y of t h e s e c o n d a r y

p r e d i c t e d one,

(de V r i e n d and Koch, 1977) ,

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

not i n c o r p o r a t e d i n t h e m a t h e m a t i c a l The

w i t h a p l a n e bed

also

f l o w was

c o n s i d e r a b l y s t r o n g e r than

i s i n a c c o r d a n c e w i t h t h e r e s u l t s of the p l a n e bed

computed c o n f i g u r a t i o n s of t h e w a t e r

s u r f a c e and

t h e energy

the

experiments.

surface deviated

from t h e measured d a t a i n t h a t r e s p e c t l a r g e r , and a d i f f e r e n t t r a n s v e r s e found. An i m p o r t a n t p a r t o f t h e s e e n c e s b e t w e e n t h e computed

t h a t t h e l o n g i t u d i n a l head l o s s e s were

slope

of t h e w a t e r s u r f a c e

d i f f e r e n c e s c a n be e x p l a i n e d

i n t h e bend from t h e

and t h e measured d e p t h - a v e r a g e d v e l o c i t y

was

differ-

fields.

FLOW OF WATER I N A CURVED OPEN CHANNEL WITH A F I X E D UNEVEN BED

1

Introduction

T h i s r e p o r t i s a c o n t i n u a t i o n o f t h e r e p o r t R657-V/M 1415, in

P a r t I , "Flow o f w a t e r

a c u r v e d c h a n n e l w i t h a f i x e d p l a n e b e d " (de V r i e n d and Koch, 1 9 7 7 ) . The i n -

v e s t i g a t i o n s d e s c r i b e d i n b o t h r e p o r t s were e x e c u t e d w i t h i n the a p p l i e d r e s e a r c h group f o r sediment

transport i n rivers,

i n which R i j k s w a t e r s t a a t D i r e c t o r a t e f o r

Water Management and R e s e a r c h , 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 D e l f t U n i v e r s i t y of T e c h n o l o g y r e s e a r c h program TOW

collaborate. This project

i s incorporated i n the b a s i c

( T o e g e p a s t Onderzoek W a t e r s t a a t : A p p l i e d R e s e a r c h Water-

s t a a t ) . F u r t h e r i n f o r m a t i o n on t h e background o f t h e e x p e r i m e n t s

i s given i n

Part I .

P a r t I d e s c r i b e s experiments bed

plane

and a r e c t a n g u l a r c r o s s - s e c t i o n . The p r e s e n t r e p o r t g i v e s t h e r e s u l t s of a

series a

i n a wide s h a l l o w c u r v e d c h a n n e l w i t h a f i x e d

of experiments

executed

i n t h e same flume w i t h a f i x e d uneven bed h a v i n g

l a r g e - s c a l e c o n f i g u r a t i o n a s i n a n a t u r a l r i v e r bend. F o r t h i s t y p e of e x p e r i -

ment l i t t l e

d a t a have been r e p o r t e d i n l i t e r a t u r e

and

1970).

The

r e s u l t s of the experiments

( d e V r i e n d , 1976; Y e n , 1967

have b e e n compared w i t h t h e r e s u l t s of a mathemat-

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 open c h a n n e l s , d e v e l o p e d of

F l u i d M e c h a n i c s o f t h e D e l f t U n i v e r s i t y of T e c h n o l o g y

a comparison

i s thought

a t the Laboratory

(de V r i e n d , 1976).

t o be even more i m p o r t a n t f o r t e s t i n g

Such

the mathematical

model t h a n a c o m p a r i s o n w i t h t h e r e s u l t s o f t h e e a r l i e r f l a t bed e x p e r i m e n t s , a s t h e c h a n n e l c o n f i g u r a t i o n was c l o s e t o t h e n a t u r a l one.

The of

experiments Technology

have b e e n e x e c u t e d by Mr. H.J. de V r i e n d o f t h e D e l f t

and by Mr. F.G. Koch of 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 .

University

-2-

2

Experimental

2.1

set-up

C h a n n e l geometry

During

the experiments,

t i o n ) and

d e s c r i b e d i n t h i s r e p o r t , the flow

t h e e l e v a t i o n of t h e f r e e w a t e r

w i t h a f i x e d uneven bed

having

( v e l o c i t y and

direc-

s u r f a c e were measured i n a c u r v e d

flume

a l a r g e - s c a l e c o n f i g u r a t i o n as i n a n a t u r a l r i v e r

bend.

The

geometry of t h e c h a n n e l

of a 38 m l o n g s t r a i g h t c u r v a t u r e of 50 m

i s shown i n F i g u r e s 1 and

c h a n n e l w i d t h was

v a r i e d as shown i n F i g u r e 2a. I n the s t r a i g h t

longitudinal

plan-view c o n s i s t e d

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 of

( s e e F i g u r e 1 ) . The

B ^ ) t h e c h a n n e l was

2. The

6 m and

the bed e l e v a t i o n

s e c t i o n ( c r o s s - s e c t i o n s A^, A j , A^,

p r i s m a t i c , w i t h a p a r a b o l i c shape of t h e bed

and

s l o p e ( s e e F i g u r e 2 b ) . Between c r o s s - s e c t i o n s BQ and

a zero

C^ the

bed

e l e v a t i o n g r a d u a l l y changed from a p a r a b o l i c c r o s s - s e c t i o n to a c r o s s - s e c t i o n w i t h a p o i n t bar near

t h e i n n e r w a l l and

a deeper channel near

the outer w a l l

(see

c r o s s - s e c t i o n C^ i n F i g u r e 2 a ) . A l l s u b s e q u e n t c r o s s - s e c t i o n s ( C j , D^, Dj and had

t h e same c o n f i g u r a t i o n as CQ, b u t were s l o p i n g downward w i t h a s l o p e of

3 * 10""* a l o n g

2.2

E^)

Flow

the channel a x i s

(see F i g u r e 2b).

conditions

Two

s e r i e s of measurements were c a r r i e d out, one w i t h a d i s c h a r g e of 0.463 m'/s,

and

the other with

a d i s c h a r g e of 0.232 m^/s. D u r i n g

both s e r i e s

s u r f a c e e l e v a t i o n a t the u p s t r e a m boundary of t h e c h a n n e l was so t h a t the depth of f l o w i n t h e c h a n n e l a x i s was average and

v e l o c i t i e s of about 0.4

m/s

and

0.2 m/s

the

kept

about 0.26 m,

water

constant,

(yielding

r e s p e c t i v e l y ) . The

discharge

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 r e g u l a t e d w i t h a movable R o m i j n - w e i r

and

a t a i l - g a t e as d e s c r i b e d i n P a r t I .

The w a t e r

i n f l o w a t t h e u p s t r e a m end

of t h e c h a n n e l was

a d j u s t e d i n such a way

t h a t t h e d i s t r i b u t i o n of t h e d e p t h - a v e r a g e d v e l o c i t y i n s e c t i o n Aj agreed a t h e o r e t i c a l d i s t r i b u t i o n . The all

l a t t e r was

d e r i v e d by a p p l y i n g C h e z y ' s law t o

s t r e a m l i n e s , s u p p o s i n g t h e c h a n n e l to be

l o n g i t u d i n a l s l o p e of t h e e n e r g y l i n e

s t r a i g h t and

that i s constant

prismatic, with

distribution.

h above t h e bed,

was

a

i n a cross-section. Figure

3 shows t h a t f o r Q = 0.463 m^/s the mean v e l o c i t y , e s t i m a t e d by measured a t 0.4

with

i n good agreement w i t h t h i s

the

velocity

theoretical

-3-

At

t h e downstream end of t h e f l u m e t h e w a t e r

When t h e edge of t h i s t h e i n n e r w a l l due

tail

g a t e was

l e v e l was

kept h o r i z o n t a l ,

to t h e uneven drawdown c a u s e d by

r e g u l a t e d by a t a i l

t h e f l o w was t h e bed

so t h a t t h e f l o w p a t t e r n i n s e c t i o n E ^ , and p a r t i c u l a r l y

a n g l e a , was

2.3

about t h e same a s i n s e c t i o n

Measured

The

drawn towards

configuration.

T h e r e f o r e t h e edge was made o b l i q u e , w i t h the h i g h e s t p o i n t n e a r wall,

gate.

the i n n e r s i d e the mean f l o w

Dj.

data

experimental

t e s t i n g of 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

the

f o l l o w i n g phenomena: -

The v e r t i c a l d i s t r i b u t i o n s of t h e h o r i z o n t a l v e l o c i t y and h e l i c a l

8, f o r 'a summary of t h e t e s t s

see Table 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 s of 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

secondary

flow)

(T7 and

components (main

flow i n t e n s i t y

flow

the

( T 8 ) ; and

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 of the w a t e r

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

the

energy

head ( T 9 ) .

In

order

averaged

to 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 about t h e s e phenomena, t h e

time-

magnitude and d i r e c t i o n of t h e v e l o c i t y v e c t o r were measured 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 , the water

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

using s t a t i c

tubes

The m e a s u r i n g

The la),

measured i n s e l e c t e d v e r t i c a l s

procedure

used

13 e q u i d i s t a n t v e r t i c a l s l b ) , and

i s d e s c r i b e d and m o t i v a t e d

i n each c r o s s - s e c t i o n

(numbered 1 to 13,

a number of g r i d p o i n t s p e r v e r t i c a l v a r y i n g from 2 to

g r i d p o i n t s i n a v e r t i c a l was

C r o s s - s e c t i o n AQ, s i t u a t e d

0.01

see 10

The minimum d i s t a n c e

m.

c l o s e to t h e u p s t r e a m boundary, t u r n e d out to 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 the g r i d there.

i n Appendix B.

(numbered Aj to E ^ ; see F i g u r e

a c c o r d i n g to the d e p t h o f f l o w i n t h e r e l e v a n t v e r t i c a l . between two

grid,

( s e e Appendix A ) .

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

Figure

of t h i s

and

as the f l o w was

not y e t f u l l y

developed

-4-

3

Results

3.1

E l a b o r a t i o n of t h e measured

The

e l a b o r a t i o n procedure

measured v e l o c i t i e s dicular The

data

d e s c r i b e d i n Appendix C was

i n t o a main and

to s p l i t up

a h e l i c a l component, p a r a l l e l

to t h e s t r e a m l i n e s of t h e d e p t h - a v e r a g e d f l o w f i e l d

m a i n o u t l i n e s of t h i s p r o c e d u r e

the r e c t a n g u l a r channel d a t a to p r e v e n t rule

used

one

1 9 7 8 ) , by

r e p l a c i n g the a v e r a g i n g

i n t h e v e r t i c a l by a normalized

this

later in this

elaboration are dealt with

the e l a b o r a t i o n procedure,

made

to depend on

must be g i v e n . The

however, c a n o n l y be used

t h e l o c a l d e p t h of f l o w h,

r o u g h n e s s of t h e c h a n n e l

e q u i v a l e n t r o u g h n e s s l e n g t h of t h e bed i n the c h a n n e l . I n a d d i t i o n ,

be

r o u g h n e s s l e n g t h k ~ 1 * 10"^ m;

and

resulting

from

i f Chezy's c o n s t a n t C

shows no

expected

B e c a u s e i t must

a r e l a t i o n between C and large variations,

c a n be assumed to be

t h e bed may

main

fitting

Chapter.

i s g i v e n , some a t t e n t i o n must be p a i d to t h i s c o n s t a n t f i r s t .

5 * 10""* m),

for

(de V r i e n d and Koch, 1 9 7 7 ) , but an a t t e m p t was

the d e p t h - a v e r a g e d f l o w f i e l d

(Nikuradse

used

of t h e m a i n s o u r c e s of e r r o r s , v i z . , t h e t r a p e z o i d a l i n t e g r a t i o n

(de V r i e n d ,

expected

perpen-

respectively.

h e l i c a l v e l o c i t y components and

be

and

a g r e e w i t h t h o s e of t h e p r o c e d u r e

of t h e measured d a t a to g i v e n t h e o r e t i c a l c u r v e s . The

As

the

constant

so

h

the

everywhere

t o be h y d r a u l i c a l l y

rough

viscous sublayer thickness

so t h a t t h e f o l l o w i n g l o g a r i t h m i c r e l a t i o n between C and

the

l o c a l d e p t h of f l o w h c a n be d e r i v e d from t h e W h i t e - C o l e b r o o k f o r m u l a f o r

shallow

channels:

C = C^ + 18 ^°log(h/h^) ,

(1)

i n which C

d e n o t e s t h e v a l u e of C a t t h e r e f e r e n c e d e p t h of f l o w h . T h e r e o ^ o v a r i o u s p o s s i b i l i t i e s to e s t i m a t e C^, such a s : -

The

White-Colebrook formula

= 18 ^ " l o g i f ^

f o r a h y d r a u l i c a l l y rough

bed:

,

(2)

where k d e n o t e s t h e e q u i v a l e n t sand r o u g h n e s s a c c o r d i n g k = -

1 * 10"^ m and h^ = 0.25

m,

t h i s y i e l d s C^ = 62

C h e z y ' s law a p p l i e d to t h e s t r a i g h t (1) t h i s

yields:

are

to N i k u r a d s e .

For

mVs.

s e c t i o n upstream. I n combination

wi t h

-5-

/ • ( f ) * -logCf) d(|) c

Q

_ _ _ J

'^

1p0

/ (f) 0

o

O

/ (f) d(f)

d(|)

O

0

i n which Q i s the t o t a l d i s c h a r g e ,

o

B i s the c h a n n e l w i d t h , I

i s the s l o p e

of

s t h e energy l i n e and axis

y i s a h o r i z o n t a l coordinate

(y = 0 a t t h e l e f t bank, y = B a t t h e r i g h t b a n k ) . F o r

e x p e r i m e n t s the v a l u e s and

of C

the

channel

present

found i n t h i s way were 58 m^/s f o r Q = 0.463 m^/s

the measured 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 i n s e m i -

logarithmic plots, estimating secondary flow

e l a b o r a t i o n b a s e d on

d (v . ) mam ,.10-,

d('"log where v

I—.

_

• g

z^

KG

-)

. mam

the

t o t a l v e l o c i t y ( i f the

the main v e l o c i t y obtained

yields a straight

l i n e under a s l o p e

estimated

and

T h i s method was

Koch,

the l o g a r i t h m i c

+ ^ KG

So C (and

from the s l o p e of a s t r a i g h t l i n e

to y i e l d C = 70 m V s a t the h i g h e r

. = V (1 + ^ main ^ KG

of

mam 0.435 '

i s the mean v e l o c i t y i n the r e l e v a n t v e r t i c a l . •'

c h a r g e (de V r i e n d

from

V

t h r o u g h the p l o t t e d d a t a .

Fitting

the

trapezoidal integration. A logarithmic v e l o c i t y

t h i s way

t h r o u g h ( 1 ) , C^) c a n be

data

t h e m a i n v e l o c i t y by

i s r e l a t i v e l y weak) o r by

distribu tion plotted

V

to the

61 m V s f o r Q = 0.232 m V s .

G r a p h i c a l r e p r e s e n t a t i o n of

an

perpendicular

applied

and

to the r e c t a n g u l a r

C = 50 m V s a t the

lower

hence, ' fitted channel dis-

1977).

curve

In -) h^

(5)

t h r o u g h the main v e l o c i t i e s measured i n the l o w e r p o i n t s

of a v e r t i c a l ,

s t a r t i n g from the a s s u m p t i o n t h a t the v e l o c i t y d i s t r i b u t i o n i s l o g a r i t h m i c %)

n e a r the bed. V and

A d o p t i n g a l e a s t - s q u a r e s method to d e t e r m i n e the

i n each v e r t i c a l ,

G

can be d e r i v e d

present

experiments t h i s y i e l d e d v a l u e s i of about 70 m^/s.

is

V

. . mam

of G

from the f o r e g o i n g one

assumed c l o s e to the bed

t h e unknown c o n s t a n t

the

O

K.U

T h i s method d i f f e r s

constants

form G t h r o u g h ( 1 ) . I n

i n the s t r a i g h t c h a n n e l s e c t i o n o

i n t h a t the l o g a r i t h m i c d i s t r i b u t i o n

r a t h e r t h a n i n the whole v e r t i c a l .

v figures

i n ( 5 ) i n s t e a d of

Gonsequently,

the v e r t i c a l mean v e l o c i t y

-6-

W i t h o u t d i s c u s s i n g t h e a c c u r a c y o f t h e v a r i o u s methods, C h a s been c h o s e n i ° 60 m / s f o r b o t h d i s c h a r g e s and h ^ = 0.25 m.

3.2

V e r t i c a l d i s t r i b u t i o n of t h e main v e l o c i t y component

I n order

t o make t h e v e r t i c a l d i s t r i b u t i o n s of t h e m a i n v e l o c i t y c o m p a r a b l e from

vertical

to v e r t i c a l , v

V

. mam

=

V' . mam

=

. h a s been n o r m a l i z e d mam ^ ( s e e Appendix C ) : tot

by i t s v e r t i c a l mean v a l u e ^

.

(6)

v^^^

I n accordance with

the r e c t a n g u l a r channel

case, t h i s normalized

main v e l o c i t y

d i s t r i b u t i o n h a s b e e n more c l o s e l y examined w i t h r e l a t i o n t o two q u e s t i o n s

that

a r e i m p o r t a n t i n t h e development of a m a t h e m a t i c a l model of c u r v e d

flow

(de V r i e n d ,

1976 and

channel

1977):

t o what e x t e n t -

i s v ' . s i m i l a r from v e r t i c a l t o v e r t i c a l ? and mam c a n t h e d i s t r i b u t i o n be 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 ? So

-;ain

=

' ^ - ^ ' - é

0



Compared w i t h t h e r e c t a n g u l a r c h a n n e l

c a s e , however, c o m p l i c a t i o n s

arise

here

from t h e v a r i a t i o n of t h e d e p t h of f l o w , l e a d i n g to v a r i a t i o n s i n C ( f o r t h e p r e s e n t v a r i a t i o n of h, C v a r i e s between 40 and 65 m^/s, i f C^ = 60 m^/s and h ^ = 0.25 m). As v ^ ^ ^ ^ i s most l i k e l y

t o depend on C, t h i s

i m p l i e s t h a t , as l o n g

as t h e r e l a t i o n s h i p between v ' . and C has n o t been e s t a b l i s h e d ( c f . ( 7 ) ) , ^ main v //» o n l y v e r t i c a l s w i t h a p p r o x i m a t e l y t h e same v a l u e s of C, i . e . , t h e same depth of f l o w , c a n be c o n s i d e r e d

when i n v e s t i g a t i n g

have b e e n grouped a c c o r d i n g of v e r t i c a l s

s i m i l a r i t y . Therefore the v e r t i c a l s

t o t h e i r d e p t h of f l o w i n T a b l e I I ,

and a s e l e c t i o n

from e a c h group was u s e d t o i n v e s t i g a t e s i m i l a r i t y .

I n F i g u r e 4, r e p r e s e n t i n g

t h e main v e l o c i t y d i s t r i b u t i o n s i n t h r e e c l a s s e s of C

(and h ) f o r t h e two d i s c h a r g e s , no s i g n i f i c a n t d i f f e r e n c e s between t h e v a r i o u s 1

verticals

i n a c l a s s c a n be o b s e r v e d , e x c e p t

the main v e l o c i t y near straight

may be f o r C = 60 - 65 m^/s, where

t h e bed tends t o be h i g h e r

s e c t i o n ( F i g u r e s 4c and

i n t h e bend t h a n

i n the

4f). KC

F i g u r e 5 shows t h e d i s t r i b u t i o n s of t h e q u a n t i t y -j=^ ^'^main ~ ' ^ verticals

various

of e a c h of t h e c r o s s - s e c t i o n s A j , C^ and D^, A c c o r d i n g to t h i s

t h e r e i s no o r h a r d l y any s i g n i f i c a n t d i f f e r e n c e between t h e v e r t i c a l s cross-section.

figure,

of one

-7-

Hirtien c o m p a r i n g t h e m a i n v e l o c i t y p r o f i l e s w i t h

the logarithmic d i s t r i b u t i o n ( 7 ) ,

s i g n i f i c a n t d i f f e r e n c e s seem t o o c c u r

( s e e F i g u r e s 4 and 5 ) : i n t h e l o w e r p a r t

of a v e r t i c a l

a r e higher

t h e measured v e l o c i t i e s

t h a n p r e d i c t e d by ( 7 ) , and i n

t h e upper p a r t , e s p e c i a l l y c l o s e t o t h e w a t e r s u r f a c e , they a r e l o w e r . these d i f f e r e n c e s occur to be t h e l a r g e s t

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

i n t h e bend and a t t h e h i g h e r

discharge

Although

they

tend

(see Figure 5 ) .

A c c o r d i n g to F i g u r e 4, t h e dependency o f t h e above-mentioned d i f f e r e n c e s on t h e p l a c e i n t h e f l u m e (bend o r s t r a i g h t it

i s concluded

section) hardly violates s i m i l a r i t y .

Hence

t h a t a s i m i l a r i t y a p p r o x i m a t i o n c a n be made f o r t h e main v e l o c i t y .

Regarding the a p p l i c a b i l i t y of the l o g a r i t h m i c curve, the measured d a t a and ( 7 ) g i v e r i s e

t h e d i f f e r e n c e s between

to r e s e r v a t i o n s not only f o r the present

c a s e o f an uneven bed, b u t a l s o f o r a p l a n e bed, where t h e s e d i f f e r e n c e s a r e q u a l i t a t i v e l y and q u a n t i t a t i v e l y Koch,

3.3

The

t h e same ( F i g u r e 6; s e e a l s o de V r i e n d and

1977).

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

helical velocities

i n the v a r i o u s v e r t i c a l s o f a c l a s s o f C and h a r e made

c o m p a r a b l e by n o r m a l i z i n g

v^^-j^ by

"hel i n w h i c h v ^ ^ ^ i s d e t e r m i n e d f o r e a c h v e r t i c a l by f i t t i n g , \ e l

z/h I n ( f )

^.J

K?

Z*/h o

z

^

, "^^h^ 1

/ „

z/h I n ^ z

KC

Z^/h

h

o

(J)

, 1

the t h e o r e t i c a l

curve

^

'^W

h

KG

O

w i t h — = exp(™ 1 0

) and v ' . a s g i v e n i n ( 7 ) , t o t h e measured h e l i c a l /. main ^ g v e l o c i t i e s ( s e e Appendix C ) . F i g u r e 7 shows t h e h e l i c a l v e l o c i t i e s n o r m a l i z e d i n t h i s way f o r t h r e e c l a s s e s i of C and f o r t h e two d i s c h a r g e s . E x c e p t measured d a t a a r e w i d e l y in a v e r t i c a l

f o r C = 50 - 55 m V s , where t h e

s c a t t e r e d due t o t h e s m a l l number o f m e a s u r i n g

points

and t h e s m a l l v a l u e s o f t h e h e l i c a l v e l o c i t y component, t h e

measured d a t a l i e w i t h i n a n a r r o w band and no s i g n i f i c a n t 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 class.

occur

-8-

I n a d d i t i o n , the d a t a except

l i e c l o s e to the t h e o r e t i c a l

f o r C = 6 0 - 6 5 m^/s, where

s m a l l e r near

the normalized

F i g u r e 5 ) , the " s o u r c e "

of

main v e l o c i t y

-

of

( R o z o v s k i i , 1 9 6 1 ; de V r i e n d ,

conclusions The

l a r g e r near

the bed

t h e s t r e a m l i n e s of 1 9 7 3 and

drawn from t h e s e r e s u l t s

helical velocity

and with

from t h e l o g a r i t h m i c c u r v e

t h e s e c o n d a r y c i r c u l a t i o n b e i n g - J - C^^^) oZ Kg

d e n o t e s t h e r a d i u s of c u r v a t u r e

The

| v ^ ^ j ^ | t e n d s t o be

representing ( 9 ) ,

t h e s u r f a c e . T h e s e d e v i a t i o n s a r e i n q u a l i t a t i v e agreement

t h e d e v i a t i o n s of

field

curves

(see

i f R

2

3

distance f r o m outer wall ( m )

A) CROSS - SECTIONS 20 1,4 5 nn fron 1 inner \/vall

o 15 )*: E c

o

10

•5 -o

5

/

0

-5

^

I- 5 5 n-1 frc>m 0 uter wal

-10

Ao

•15

0

Al 10

- > (:ros£>-sec tions Co ^1 Bl A? Bo, 20

>

30

40

50

60

70

Do 80

90

Dl

Eo

100 110

longitudinal distance along channel axis

(m)

(B) LONGITUDINAL SECTIONS

BED ELEVATION

DELFT

HYDRAULICS

LABORATORY/DELFT

UNIVERSITY O F

TECHNOLOGY

^

R657/M1415 FIG,

2

AcooioNio^.i. 10 Aiis^j:nAiNn ijnaa/A^io.iVcMonvi somv^iciAii i-n.-ici NOII.flillMJ.SKI A J I 3 0 1 3 A M O I JNI

a39Vd3AV-Hld3a

1-9 1

O

o

o

o o o k) u) :^

-vl

bl

p

p

p

p 00

O (D

I I I

to

i s

CTl

i

cn

2

0 > + A X V

vertical

(V'main-1)

^ 0 71 uneven bed ( Q = 0.463 m^/s) Do 4 J ^ 0 ^ 1 plane bed Do 4J

(Q = 0.61 m3/s)

theory

MAIN VELOCITY PROFILES WITH PLANE BED DATA DELFT

HYDRAULICS LABORATORY/DELFT

COMPARED

UNIVERSITY O F

TECHNOLOGY

T 2 -1 ,T y - i

jF?65y/M1415

6

1.0-

1.0

1.0—r

Njx:

N|^

I

I • I

•1

T 8-2 Q =0.232 m 3 / s cn

2

-73

-5.0 -2.5 0 2.5 5.0 7.5 normalized helical velocity (v'hei = ^ ^ e l ) > Vhel

SIMILARITY DISTRIBUTION

OF THE

O F THE

normalized helical velocity

VERTICAL

HELICAL

-7.5

-7.5