open channel hydralics

1. CHAPTER 2: REVIEW OF OPEN CHANNEL
HYDRAULICS AND THEORY OF
DISCHARGE MEASURING STRUCTURES
DR. MOHSIN SIDDIQUE
ASSISTANT PROFESSOR
1
0401544-HYDRAULIC STRUCTURES
University of Sharjah
Dept. of Civil and Env. Engg.

2. LEARNING OUTCOME
2
After completing this lecture…
The students should be able to:
• Understand the behavior of open channel flow under
various conditions
• Learn the basic theories that govern the design of open
channels and hydraulic structures
• Apply the basic theories to derive various formula used in
the design calculation of hydraulic structures such as
weir/notches
References:
Fluid Mechanics With Engineering Applications, 10TH ED, By E.
Finnemore and Joseph Franzini, Mcgraw Hills

3. OPEN CHANNEL HYDRAULICS
An open channel is the one in which stream is not complete
enclosed by solid boundaries and therefore has a free surface
subjected only to atmosphere pressure.
The flow in such channels is not caused by some external head,
but rather only by gravitational component along the slope of
channel. Thus open channel flow is also referred to as free
surface flow or gravity flow.
Examples of open channel are
• Rivers, canals, streams, & sewerage system etc
3

4. OPEN CHANNEL HYDRAULICS
Flow conditions
Uniform flow:
Non-uniform flow
For uniform flow through open channel, dy/dl is equal to zero. However
for non-uniform flow the gravity force and frictional resistance are not in
balance. Thus dy/dl is not equal to zero which results in non-uniform
flow.
There are two types of non-uniform flows.
In one the changing condition extends over a long distance and this is
called gradually varied flow.
In the other the change may occur over very abruptly and the transition
is thus confined to a short distance. This may be designated as a local
non-uniform flow phenomenon or rapidly varied flow.
4

5. OPEN CHANNEL HYDRAULICS
Characteristics of Uniform flow
1Z
g
V
2
2
1
Datum
So
1y
2Z
g
V
2
2
2
2y
HGL
EL
Water
Level
Sw
S
∆L
∆x
For Uniform Flow : y1=y2 and V1
2/2g=V2
2/2g
Hence the line indicating the bed of the channel, water surface profile and
energy line are parallel to each other.
For θ being very small (say less than 5 degree) i.e ∆x=∆L
So=Sw=S
5
So= Slope of Channel Bed
(Z1-Z2)/(∆x)= -∆Z/∆x
Sw= Slope of Water Surface
[(Z1+y1)-(Z2+y2)]/∆x
S= Slope of Energy Line
[(Z1+y1+V1
2/2g)-(Z2+y2+V2
2/2g)]/∆L
= hl/∆L

6. OPEN CHANNEL HYDRAULICS
Energy Equation:
[(Z1+y1+V1
2/2g)=(Z2+y2+V2
2/2g)+HL
Let’s assume two section close to each
other (neglecting head loss) and take
bed of channel as datum, above equation
can be rewritten as
y1+V1
2/2g=y2+V2
2/2g
E1=E2
Where E1 and E2 are called specific
energy at 1 and 2.
( ) Vy
B
VBy
B
AV
B
Q
qwhere
yE
yE
yg
q
g
V
====
+=
+=
2
2
2
2
2
Specific Energy at a section in an open
channel is the energy with reference to
the bed of the channel.
• Mathematically;
Specific Energy = E = y+V2/2g
For a rectangular Channel
BQqwhereyE yg
q
/2
2
2
=+=
As it is clear from E~y diagram drawn for constant
discharge for any given value of E, there would be
two possible depths, say y1 and y2. These two depths
are called Alternate depths.
However for point C corresponding to minimum
specific energy Emin, there would be only one
possible depth yc. The depth yc is know as critical
depth.
The critical depth may be defined as depth
corresponding to minimum specific energy discharge
remaining Constant. 6

7. OPEN CHANNEL HYDRAULICS
TYPES OF FLOW IN OPEN CHANNELS
Subcritical, Critical and Supercritical Flow. These are classified
with Froude number.
Froude No. (Fr). It is ratio of inertial force to gravitational force of
flowing fluid. Mathematically, Froude no. is
If ; Fr. < 1, Flow is subcritical flow
Fr. = 1, Flow is critical flow
Fr. > 1, Flow is supercritical flow
gh
V
Fr =
Where, V is average velocity of flow, h is depth of flow and g is
gravitational acceleration
Alternatively:
If y>yc , V<Vc Deep Channel
Sub-Critical Flow, Tranquil Flow, Slow Flow.
and y<yc , V>Vc Shallow Channel
Super-Critical Flow, Shooting Flow, Rapid Flow, Fast Flow.
7

8. OPEN CHANNEL HYDRAULICS
Critical depth for rectangular channels: Critical depth for non rectangular channels:
• T is the top width of channel
ycy
g
Q
T
A
=






=
23
( ) 3/12
g
q
cy =
c
y
cc yyEE c
2
3
2min =+==
8

9. OPEN CHANNEL HYDRAULICS
CHEZY’S AND MANNING’S EQUATIONS
Chezy’s Equation Manning’s Equation
2/13/21
oSR
n
V =
( ) 2/13/23 1
/ oSAR
n
smQ =
( ) 2/13/2486.1
oSAR
n
cfsQ =
SI
BG
oRSCV =
oRSCAQ =
Value of C is determine from
respective BG or SI Kutter’s
formula.
C= Chezy’s Constant
A= Cross-sectional area of flow A= Cross-sectional area of flow
By applying force balance along the direction of flow in an open channel
having uniform flow, the following equations can be derived. Both of the
equations are widely used for design of open channels.
9

10. OPEN CHANNEL HYDRAULICS
ENERGY EQUATION FOR GRADUALLY VARIED
FLOW.
( )
( ) ( )
profilesurfacewateroflengthLWhere
SS
EE
L
LSLSEE
Now
for
L
ZZ
X
ZZ
S
L
h
S
hZZ
g
V
y
g
V
y
o
o
o
o
L
L
=∆
−
−
=∆
∆+∆−=
<
∆
−
≈
∆
−
=
∆
=
+−−+=+
)1(
6,
22
21
21
2121
21
2
2
2
2
1
1
θ
An approximate analysis of gradually
varied, non uniform flow can be
achieved by considering a length of
stream consisting of a number of
successive reaches, in each of which
uniform occurs. Greater accuracy
results from smaller depth variation in
each reach.
The Manning's formula (or Chezy’s
formula) is applied to average
conditions in each reach to provide
an estimate of the value of S for that
reach as follows;
3/4
22
2/13/21
m
m
mm
R
nV
S
SR
n
V
=
=
2
2
21
21
RR
R
VV
V
m
m
+
=
+
=
In practical, depth range of the
interest is divided into small
increments, usually equal, which
define the reaches whose
lengths can be found by equation
(1)
10

11. OPEN CHANNEL FLOW
WATER SURFACE PROFILES IN GRADUALLY
VARIED FLOW.
3/10
3/10
22
3/10
22
2/13/5
2/13/2
1
1






=∴
=
=
=
=
≈
y
y
S
S
y
qn
S
channel
gulartanrecinflowuniformFor
y
qn
S
orSy
n
q
orSy
n
V
yR
channelgulartanrecwideaFor
o
o
o
o
2
1 F
SS
dx
dy o
−
−
=
Consequently, for constant q and n,
when y>yo, S<So, and the numerator
is +ve.
Conversely, when y<yo, S>So, and
the numerator is –ve.
To investigate the denominator we
observe that,
if F=1, dy/dx=infinity;
if F>1, the denominator is -ve; and
if F<1, the denominator is +ve.
11






++=
++=++=
2
2
2
22
1
2
22
gy
q
dx
d
dx
dy
dx
dZ
dx
dH
gy
q
yZ
g
v
yZH
Rectangular
channel !!

12. OPEN CHANNEL FLOW
WATER SURFACE PROFILES IN GRADUALLY
VARIED FLOW
12

13. FLOW OVER HUMP
For frictionless two-dimensional
flow, sections 1 and 2 in Fig are
related by continuity and energy:
Eliminating V2 between these two
gives a cubic polynomial equation
for the water depth y2 over the
hump.
1 1 2 2
2 2
1 2
1 2
2 2
v y v y
v v
y y Z
g g
=
+ = + +
2 2
3 2 1 1
2 2 2
2
1
2 1
0
2
2
v y
y E y
g
v
where E y Z
g
− + =
= + −
y2
y1
y3
Z
V1
V2
1 2 3
This equation has one negative
and two positive solutions if Z is
not too large.
It’s behavior is illustrated by
E~y Diagram and depends
upon whether condition 1 is
Subcritical (on the upper) or
Supercritical (lower leg) of the
energy curve.
B1=B
2
Hump is a streamline construction provided at the bed of the
channel. It is locally raised bed.
13

14. FLOW OVER HUMP
Damming
Action
y1=yo, y2>yc, y3=yo
y1=yo, y2>yc, y3=yo
y1=yo, y2=yc, y3=yo y1>yo, y2=yc, y3<yo
yc
y1
y3
Z
Z=Zc
y1
Z<<Zc
y2 y3
Z
Z<Zc
y2
y1
y3
Z
Z>Zc
Afflux=y1-yo
y3
yc
yo
Z
y1
14

15. FLOW THROUGH CONTRACTION
When the width of the channel is reduced while the bed remains flat, the
discharge per unit width increases. If losses are negligible, the specific
energy remains constant and so for subcritical flow depth will decrease
while for supercritical flow depth will increase in as the channel narrows.
B1 B2
y2
yc
y1
( )
1 1 1 2 2 2
2 2
1 2
1 2
1 2
2 2 2 2 2 2
2 2
1 1
'
2 2
Using both equations, we get
2
Q=B y v =B y
1
Continuity Equation
B y v B y v
Bernoulli s Equation
v v
y y
g g
g y y
B y
B y
=
+ = +
 
 
− 
 
  −    
15

16. FLOW THROUGH CONTRACTION
If the degree of contraction and the flow conditions are such that
upstream flow is subcritical and free surface passes through the
critical depth yc in the throat.
ycyc
y1
( )
3/2
2
2
sin
3
2 1
2
3 3
1.705
c c c c c c
c
c
Q B y v B y g E y
ce y E
Therefore Q B E g E
Q BE in SI Units
= = −
=
 
=  
 
=
B1 Bc
y2
yc
y1
16

17. HYDRAULICS JUMP OR STANDING WAVE
Hydraulics jump is local non-uniform flow phenomenon resulting
from the change in flow from super critical to sub critical. In such
as case, the water level passes through the critical depth and
according to the theory dy/dx=infinity or water surface profile
should be vertical. This off course physically cannot happen and
the result is discontinuity in the surface characterized by a steep
upward slope of the profile accompanied by lot of turbulence and
eddies. The eddies cause energy loss and depth after the jump is
slightly less than the corresponding alternate depth. The depth
before and after the hydraulic jump are known as conjugate
depths or sequent depths.
y
y1
y2
y1
y2
y1 & y2 are called
conjugate depths
17

18. CLASSIFICATION OF HYDRAULIC JUMP
Classification of hydraulic jumps:
(a) Fr =1.0 to 1.7: undular jumps;
(b) Fr =1.7 to 2.5: weak jump;
(c) Fr =2.5 to 4.5: oscillating jump;
(d) Fr =4.5 to 9.0: steady jump;
(e) Fr =9.0: strong jump.
18

19. CLASSIFICATION OF HYDRAULIC JUMP
Fr1 <1.0: Jump impossible, violates second law of
thermodynamics.
Fr1=1.0 to 1.7: Standing-wave, or undular, jump about 4y2 long; low
dissipation, less than 5 percent.
Fr1=1.7 to 2.5: Smooth surface rise with small rollers, known as a
weak jump; dissipation 5 to 15 percent.
Fr1=2.5 to 4.5: Unstable, oscillating jump; each irregular pulsation
creates a large wave which can travel downstream for miles,
damaging earth banks and other structures. Not recommended for
design conditions. Dissipation 15 to 45 percent.
Fr1=4.5 to 9.0: Stable, well-balanced, steady jump; best
performance and action, insensitive to downstream conditions.
Best design range. Dissipation 45 to 70 percent.
Fr1>9.0: Rough, somewhat intermittent strong jump, but good
performance. Dissipation 70 to 85 percent.
19

20. USES OF HYDRAULIC JUMP
Hydraulic jump is used to dissipate or destroy the energy of water
where it is not needed otherwise it may cause damage to
hydraulic structures.
It may also be used as a discharge measuring device.
It may be used for mixing of certain chemicals like in case of water
treatment plants.
20

21. EQUATION FOR CONJUGATE DEPTHS
1
2
F1 F2
y2y1
So~0
1 2 2 1
1
2
( )
Resistance
g f
f
Momentum Equation
F F F F Q V V
Where
F Force helping flow
F Force resisting flow
F Frictional
Fg Gravitational component of flow
ρ− + − = −
=
=
=
=
Assumptions:
1. If length is very small frictional resistance may be neglected. i.e (Ff=0)
2. Assume So=0; Fg=0
Note: Momentum equation may be stated as sum of all external forces is
equal to rate of change of momentum.
L
21

22. EQUATION FOR CONJUGATE DEPTHS
Let the height of jump = y2-y1
Length of hydraulic jump = Lj
2 1
1 1 2 2 2 1
1 1 1 2 2 2
1 2 ( )
( )
Depth to centriod as measured
from upper WS
.1
Eq. 1 stated that the momentum flow rate
plus hydrostatic force is the same at both
c c
c
c c
F F Q V V
g
h A h A Q V V
g
h
QV h A QV h A eq
g g
γ
γ
γ γ
γ γ
γ γ
− = −
− = −
=
+ = + ⇒
2 2
1 1 2 2
1 2
sections 1 and 2.
Dividing Equation 1 by and
changing V to Q/A
.2c c m
Q Q
A h A h F eq
A g A g
γ
+ = + = ⇒
2
;
Specific Force=
: Specific force remains same at section
at start of hydraulic jump and at end of hydraulic
jump which means at two conjugate depths the
specific force is constant.
m
Where
Q
F Ahc
Ag
Note
= +
( )
( )( )
2 2 2 2
1 2
1 2
1 2
2 22 2
1 2
1 2
2
2 2
2 1
1 2
2
2 1
2 1 2 1
1 2
Now lets consider a rectangular channel
2 2
.3
2 2
1 1 1
2
1
2
y yq B q B
By By
By g By g
y yq q
eq
y g y g
q
y y
g y y
or
y yq
y y y y
g y y
∴ + = +
+ = + ⇒
 
− = − 
 
 −
= − + 
 
22

23. EQUATION FOR CONJUGATE DEPTHS
2
2 1
1 2
1 1 2 2
2 2
1 1 2 1
1 2
3
1
2
2
1 2 2
1 1 1
2
22 2
1
1 1
2
1
.4
2
Eq. 4 shows that hydraulic jumps can
be used as discharge measuring device.
Since
2
2
0 2 N
y yq
y y eq
g
q V y V y
V y y y
y y
g
by y
V y y
gy y y
y y
F
y y
y
y
+ 
= ⇒ 
 
= =
+ 
∴ =  
 
÷
 
= +  
 
 
= + − 
 
( )
2
1
21
2 1
1 1 4(1)(2)
2(1)
1 1 8
2
N
N
F
y
y F
− ± +
=
= − ± +
( )
( )
21
2 1
22
1 2
Practically -Ve depth is not possible
1 1 8 .5
2
1 1 8 .5
2
N
N
y
y F eq
Similarly
y
y F eq a
∴ = − + + ⇒
= − + + ⇒
23

24. LOCATION OF HYDRAULIC JUMPS
Change of Slope from Steep to Mild
Hydraulic Jump may take place
1. D/S of the Break point in slope y1>yo1
2. The Break in point y1=yo1
3. The U/S of the break in slope y1<yo1
So1>Sc
So2<Sc
yo1
y2
yc
Hydraulic Jump
M3
y1
24

25. LOCATION OF HYDRAULIC JUMPS
Flow Under a Sluice Gate
So<Sc
yo
yc
ys
y1 y2=yo
L Lj
Location of hydraulic jump where it starts is
L=(Es-E1)/(S-So)
Condition for Hydraulic Jump to occur
ys<y1<yc<y2
Flow becomes uniform at a distance L+Lj from sluice gate
where
Length of Hydraulic jump = Lj = 5y2 or 7(y2-y1)
25

26. NOTCHES AND WEIRS
26

27. NOTCHES AND WEIRS
27

28. NOTCHES AND WEIRS
Notch. A notch may be defined as an opening in the side of a tank or vessel
such that the liquid surface in the tank is below the top edge of the opening.
A notch may be regarded as an orifice with the water surface below its
upper edge. It is generally made of metallic plate. It is used for measuring
the rate of flow of a liquid through a small channel of tank.
Weir: It may be defined as any regular obstruction in an open stream over
which the flow takes place. It is made of masonry or concrete. The condition
of flow, in the case of a weir are practically same as those of a rectangular
notch.
Nappe: The sheet of water flowing through a notch or over a weir
Sill or crest. The top of the weir over which the water flows is known as sill
or crest.
Note: The main difference between notch and weir is that the notch is
smaller in size compared to weir.
28

29. CLASSIFICATION OF NOTCHES/WEIRS
Classification of Notches
1. Rectangular notch
2. Triangular notch
3.Trapezoidal Notch
4. Stepped notch
Classification of Weirs
According to shape
1. Rectangular weir
2. Cippoletti weir
According to nature of
discharge
1. Ordinary weir
2. Submerged weir
According to width of weir
1. Narrow crested weir
2. Broad crested weir
According to nature of crest
1. Sharp crested weir
2. Ogee weir
29

30. DISCHARGE OVER RECTANGULAR
NOTCH/WEIR
Consider a rectangular notch or weir provided in channel carrying
water as shown in figure.
Figure: flow over rectangular notch/weir
H=height of water above crest of
notch/weir
P =height of notch/weir
L =length of notch/weir
dh=height of strip
h= height of liquid above strip
L(dh)=area of strip
Vo = Approach velocity
Theoretical velocity of strip
neglecting approach velocity =
Thus,
discharge passing through strips
=
gh2
velocityArea×
30

31. DISCHARGE OVER RECTANGULAR NOTCH/WEIR
Where, Cd = Coefficient of discharge
LdhA
ghv
strip
strip
=
= 2
( )ghLdhdQ 2=
Therefore, discharge of strip
In order to obtain discharge over
whole area we must integrate
above eq. from h=0 to h=H,
therefore;
2/3
0
2
3
2
2
LHgQ
dhhLgQ
H
=
= ∫
2/3
2
3
2
LHgCQ dact =
Note: The expression of discharge (Q) for rectangular notch and sharp
crested weirs are same.
thactd QQC /=Q
31

32. DISCHARGE OVER RECTANGULAR NOTCH/WEIR
Dimensional analysis of weir lead to the following conclusion
Comparing this with previously derived expression, we
conclude that Cd depend on weber number, W, Reynold’s
number, R, and H/P.
It has been found that H/P is most important of these.
2/3
,, LHg
P
H
RWQ 





= φ
T. Rehbook of the karlsruhe hydraulics laboratory in Germany
provided following expressions for Cd
p
H
H
C
p
H
H
C
d
d
08.0
1000
1
605.0
08.0
305
1
605.0
++=
++=In BG units: H & P in ft
In SI units: H & P in m
32

33. DISCHARGE OVER RECTANGULAR NOTCH/WEIR
For convenience the formula of Q is expressed as
Where, Cw the coefficient of weir, replaces
Using a value of 0.62 for Cd, above equation can be written as
These equations give good results for H/P>0.4 which is well within
operating range.
2/3
2
3
2
LHgCQ dact =
2/3
LHCQ wact =
gCd 2
3
2
2/3
2/3
83.1
32.3
LHQ
LHQ
act
act
=
=In BG units:
In SI units:
33

34. RECTANGULAR WEIR WITH END CONTRACTIONS
When the length L of the crest of a rectangular weir less than the
width the channel, the nappe will have end contractions so that its
width is less than L.
Hence for such a situations, the flow rate may be computed by
employing corrected length of crest, Lc, in the discharge formula
Lc=(L-0.1nH)
Where, n is number of end contractions.
Francis formula
34

35. NUMERICAL PROBLEMS
A rectangular notch 2m wide has a constant head of 500mm.
Find the discharge over the notch if coefficient of discharge
for the notch is 0.62.
35

36. NUMERICAL PROBLEMS
A rectangular notch has a discharge of 0.24m3/s, when
head of water is 800mm. Find the length of notch.
Assume Cd=0.6
36

37. DISCHARGE OVER TRIANGULAR NOTCH (V-
NOTCH)
In order to obtain discharge over
whole area we must integrate
above equation from h=0 to h=H,
therefore;
( ) ( )( )( )
( ) ( ) dhhhHgQ
ghhHdhQ
H
H
∫
∫
−=
−=
0
0
2/tan22
22/tan2
θ
θ
( ) ( )
( ) 





=
−= ∫
2/5
0
2/32/1
15
4
2/tan22
2/tan22
HgQ
dhhHhgQ
H
θ
θ
( )[ ]2/5
2/tan2
15
8
HgQ θ=
( )[ ]2/5
2/tan2
15
8
HgCQ dact θ=
37

38. NUMERICAL PROBLEMS
Find the discharge over a triangular notch of angle 60o, when
head over triangular notch is 0.2m. Assume Cd=0.6
38

39. NUMERICAL PROBLEMS
During an experiment in a laboratory, 0.05m3 of water flowing over
a right angled notch was collected in one minute. If the head over
sill is 50mm calculate the coefficient of discharge of notch.
Solution:
Discharge=0.05m3/min=0.000833m3/s
Angle of notch, θ=90o
Head of water=H=50mm=0.05m
Cd=?
39

40. NUMERICAL PROBLEMS
A rectangular channel 1.5m wide has a discharge of 0.2m3/s,
which is measured in right-angled V notch, Find position of the
apex of the notch from the bed of the channel. Maximum depth of
water is not to exceed 1m. Assume Cd=0.62
Width of rectangular channel, L=1.5m
Discharge=Q=0.2m3/s
Depth of water in channel=1m
Coefficient of discharge=0.62
Angle of notch= 90o
Height of apex of notch from bed=Depth of water in channel-
height of
water over V-notch
=1-0.45= 0.55m 40

41. BROAD CRESTED WEIR
A weir, of which the ordinary dam is
an example, is a channel obstruction
over which the flow must deflect.
For simple geometries the channel
discharge Q correlates with gravity
and with the blockage height H to
which the upstream flow is backed
up above the weir elevation.
Thus a weir is a simple but effective
open-channel flow-meter.
Figure shows two common weirs,
sharp-crested and broad-crested,
assumed. In both cases the flow
upstream is subcritical, accelerates
to critical near the top of the weir,
and spills over into a supercritical
nappe. For both weirs the discharge
q per unit width is proportional to
g1/2H3/2 but with somewhat different
coefficients Cd.
B
41

42. BROAD CRESTED WEIR
Z>Zc
Vc
y1
B
BGin
g
V
HLCQ
SIin
g
V
HLCQ
dact
dact
2/3
2
2/3
2
dact
1
2
09.3
2
7.1
QCQSince
channelofwidthL
crestoverHeadH
Q/LyapproachofveloctyV








+=








+=
=
=
=
==
3
2
3
c
2
c
cc
2
222
2
22
L
23
2
V
g
V
LVLyQ:Since
23
2
22
2
2
22
:flowcriticalFor
22
hignoringbyequationenergyApplying


















+=
===








+=
+=+∴
=
++=++
g
V
Hg
g
L
Q
g
L
Vc
g
V
HgV
g
V
g
V
g
V
H
y
g
V
g
V
yZ
g
V
ZH
c
cc
cc
c
c
42

43. BROAD CRESTED WEIR
COEFFICIENT OF DISCHARGE, CD ALSO CALLED WEIR DISCHARGE
COEFFICIENT, CW
Cw depends upon Weber number W, Reynolds number R and weir
geometry (Z/H, B, surface roughness, sharpness of edges etc).
It has been found that Z/H is the most important.
The Weber number W, which accounts for surface tension, is important
only at low heads.
In the flow of water over weirs the Reynolds number, R is generally
high, so viscous effects are generally insignificant. For Broad crested
weirs Cw depends on length, B. Further, it is considerably sensitive to
surface roughness of the crest.
Z>Zc
Vc
B
2/3
2
2 







+=
g
V
HLCQ wact
2/3
2
3
23
2








+








=
g
V
Hg
g
L
CQ dact
43

44. BROAD CRESTED WEIR
COEFFICIENT OF DISCHARGE, CD ALSO CALLED WEIR DISCHARGE
COEFFICIENT, CW
44

45. PROBLEM 1
45

46. PROBLEM 2
46

47. THANK YOU
47