The document discusses principles of energy and momentum in open channel flow. It defines specific energy as the total energy of water at a cross-section, and critical depth as the depth corresponding to minimum specific energy for a given discharge. Critical flow occurs when the Froude number equals 1. For a rectangular channel, the critical depth can be calculated as a function of discharge. Flow can be subcritical or supercritical depending on whether the depth is more or less than critical depth. The concepts are applied to analyze flow over humps, through contractions, and over weirs.

1. Chapter 6
Energy and Momentum Principles
in open channel flow
1
Hydraulics
Er. Binu Karki
Lecturer

2. Specific Energy and Critical Depth Basic
Definitions
2
c Head
c Energy per unit weight
c Energy Line
c Line joining the total head at different positions
c Hydraulics Grade Line
c Line joining the pressure head at different
positions

3. Specific Energy and Critical Depth
Basic Definitions
c Open Channel Flow
Z1
V2
1
2g
Datu
m
S
o
y1
Z2
V2
2
2g
y2
HG
L
E
L
Water
Level
V
3
V
 hl
2
2
2 g
 Z  y 2 2
2
1
2 g
Z 1  y 1 

4. Specific Energy
In a channel with constant discharge, Q
2211 VAVAQ 
2
2
2gA
Q
yE 
g
V
yE
2
2
 where
A=f(y)
Consider rectangular channel (A = By) and Q = qB
2
2
2gy
q
yE 
A
B
y
3 roots (one is negative)
q is the discharge per unit width of
channel
How many possible depths given a specific energy? _____
2

5. P
A
Critical Flow
T
dy
y
T=surface width
Find critical depth, yc
2
2
2gA
Q
yE 
0
dy
dE
dA =0
dE
dy
= =
3
2
1
c
c
gA
TQ

Arbitrary cross-section
A=f(y)
2
3
2
Fr
gA
TQ

2
2
Fr
gA
TV

dA
A
D
T
= Hydraulic Depth
2
3
1
Q dA
gA dy
-
0
1
2
3
4
0 1 2 3 4
E
y
yc
Tdy
More general definition of Fr

6. Critical Flow:
Rectangular channel
yc
T
Ac
3
2
1
c
c
gA
TQ

qTQ  TyA cc 
3
2
33
32
1
cc gy
q
Tgy
Tq

3/1
2









g
q
yc
3
cgyq 
Only for rectangular channels!
cTT 
Given the depth we can find
the flow!

7. Critical Flow Relationships:
Rectangular Channels
3/1
2









g
q
yc cc yVq 









g
yV
y
cc
c
22
3
g
V
y
c
c
2

1
gy
V
c
c
Froude number
velocity head =
because
g
Vy cc
22
2

2
c
c
y
yE  Eyc
3
2

forcegravity
forceinertial
0.5 (depth)
g
V
yE
2
2

Kinetic energy
Potential energy

8. c Slopes in Open Channel
Flow
c So= Slope of Channel Bed = (Z1-Z2)/(Δx)= -ΔZ/Δx
c Sw= Slope of Water Surface= [(Z1+y1)-(Z2+y2)]/Δx
c S= Slope of Energy Line= [(Z1+y1+V1
2/2g)-
(Z2+y2+V2
2/2g)]/Δx
Specific Energy and Critical Depth Basic
Definitions
8
= hl/ΔL

9. Specific Energy and Critical Depth Basic
Definitions
c Slopes in Open Channel
Flow
Z1
V2
1
2g
Datu
m
S
o
y1
Z2
V2
2
2g
y2
HG
L
E
L
Wate
r
Level
S
w
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
6

10. Specific Energy and Critical Depth
(Rectangular Channels)
10
c Specific Energy
c Specific Energy at a section in an open channel is the energy
with reference to the bed of the channel.
c Mathematically;
Specific Energy = E = y+V2/2g
For a rectangular Channel
q = Discharge per unit width m3/s
per m
B
Q AV ByV
B B B
  Vywhere q 
E  y 
E  y  2
V 2
2 g y
2g
q2
yDatu
m

11. Specific Energy and Critical Depth
11
c 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.c However
fo
r
point C
corresponding to minimum
specific energy
Emin,
there
woul
d
be only one possible depth yc. The
depth yc is know as critical depth.
c
The critical depth may be defined as
depth corresponding to minimum
specific energy discharge remaining
Constant.
c E~y Diagram or E-Diagram
Static Head
Line
2q
2g y2
where q  Q / BE  y 

12. Specific Energy and Critical Depth
12
c
Fo
r
y>yc ,V<Vc Deep Channel
c Sub-Critical Flow,Tranquil Flow, Slow
Flow.
y<yc ,V>Vc
Shallow
Channelc For
c Super-Critical Flow, Shooting Flow, Rapid Flow, Fast
Flow.

13. Specific Energy and Critical Depth
Relationship Between Critical Depth and Specific Energy
for rectangular channels
2
yc
2g 2
Vc

From eq.(3)

V y
g
y
g
q
c g
g
dE
dy
dE
dy
cc
c
Vc y c
2g
3
1/3
1/3
q23
gy3
2gy3
2g
22
2 2
2 2
2
2gy2
y3
c
y c
2
(1)
 
 Vc
yc   
y   
yc  ( )
 1 q 2
 0
1 2q 2
 y  q 2
E  y  V 2
Qq Vc yc
Q y yc
(2)
(5)2
3
min
c c
yc
2
E  y
E  Ec  yc 
(3)
(4)
13
2
1
c
gy
yc
22 g
Vc
Vc

Froude
no
=1 !!
From eq.(1)

14. Specific Energy and Critical Depth
14
Relationship Between Critical Depth and Specific Energy
c Froude’s number may be
numerically calculate as
gA3
Q2
T
DF2
F
T
A2
g A
r
r
F

A
T

Q

V
gD
2
Therefore
for Critical flow
dE
dy gA
g 
yyc
T

 A3
Q2

Q2
dE
 0
dy
1 3 T
dy
Since
gA3
dy
dA Tdy
2
Q2
2gA2
dE
 1
Q dA
Eq.(1) E  y 
r

15. Critical depth computation

18. Depth Discharge diagram

19. Momentum equation and specific
force
1
2

22. Flow Over Hump
c Hump:
is a streamline construction provided at the bed of the
channel.
It is locally raised bed.
Let’s examine the case of hump in a rectangular
channel. We will neglect the head loss.

23. Flow Over Hump
c For frictionless two-dimensional
flow, sections 1 and 2 in Fig are
related by continuity and energy:
1 2
v2
v2
2g 2g
v1y1  v2 y2
1
 y  2
 y  Z
B1=
B2
E1 E2+Z
• A hump of any height “Z” would cause the lowering of the water surface
over the hump in case of subcritical flow in channel.
• It is also clear that a gradual increase in the height of hump “Z” would
cause a gradual reduction in y2 value.
• That height of hump which is just causing the flow depth over hump
equal to yc is know as critical height of hump Zc.

24. Flow Over Hump
• Further increase in Z (>Zc) would cause the flow depth y2
remaining equal yc thus causing the water surface over the
hump to rise.
• This would further cause an increase in the depth of water
upstream of the hump which mean that water surface upstream
of the hump would rise beyond the previous value i.e y1>yo. This
phenomenon of rise in water surface upstream with Z>Zc is
called damming action and the resulting increase in depth
upstream of the hump i.e y1-yo is known as Afflux.

25. Flow Through Contraction
B
1
B
2
y
2y
c
1 2
1 2
2
v v
2g 2g
c 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.
Continuity Equation
B1y1v1  B2 y2v2
y   y 
E1=E2
y1

26. Flow Through Contraction
c 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.
y
c
y
c
y
1
3
3
c
c
3
since
Therefore
in SI Units
Q  Bc ycvc  Bc yc 2g E  yc 
y 
2
E
2g
1
EQ  B
2
E  
 
Q 1.705BE3/2
B1
B
c
y1 y2
yc

27. Broad Crested Weir
• Weir: It is a channel
obstruction over which the
flow must deflect. eg:
ordinary dam.
• 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

28. Broad Crested Weir

29. Venturi Flume
Ordinary Flume
y
2y
c
y
1
H = y2-
y1