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