open chnallle notes for b.tech students..

depth
Open Channel Flow
 Liquid (water) flow with a ____ ________
(interface between water and air)
 relevant for
natural channels: rivers, streams
engineered channels: canals, sewer
lines or culverts (partially full), storm drains
 of interest to hydraulic engineers
location of free surface
velocity distribution
discharge - stage (______) relationships
optimal channel design
free surface

Topics in Open Channel Flow
 Uniform Flow
Discharge-Depth relationships
 Channel transitions
Control structures (sluice gates, weirs…)
Rapid changes in bottom elevation or cross section
 Critical, Subcritical and Supercritical Flow
 Hydraulic Jump
 Gradually Varied Flow
Classification of flows
Surface profiles
normal depth

Classification of Flows
 Steady and Unsteady
Steady: velocity at a given point does not change with
time
 Uniform, Gradually Varied, and Rapidly Varied
Uniform: velocity at a given time does not change
within a given length of a channel
Gradually varied: gradual changes in velocity with
distance
 Laminar and Turbulent
Laminar: flow appears to be as a movement of thin
layers on top of each other
Turbulent: packets of liquid move in irregular paths
(Temporal)
(Spatial)

Momentum and Energy
Equations
 Conservation of Energy
“losses” due to conversion of turbulence to heat
useful when energy losses are known or small
____________
Must account for losses if applied over long distances
_______________________________________________
 Conservation of Momentum
“losses” due to shear at the boundaries
useful when energy losses are unknown
____________
Contractions
Expansion
We need an equation for losses

 Given a long channel of
constant slope and cross
section find the relationship
between discharge and depth
 Assume
Steady Uniform Flow - ___ _____________
prismatic channel (no change in _________ with distance)
 Use Energy, Momentum, Empirical or Dimensional
Analysis?
 What controls depth given a discharge?
 Why doesn’t the flow accelerate?
Open Channel Flow:
Discharge/Depth Relationship
P
no acceleration
geometry
Force balance
A
l
d
hl
4
0

 


Steady-Uniform Flow: Force
Balance

W

W sin 
Dx
a
b
c
d
Shear force
Energy grade line
Hydraulic grade line
Shear force =________
0
sin 
D

D x
P
x
A o





 sin
P
A
o 
h
R
=
P
A



sin
cos
sin


S
W cos 
g
V
2
2
Wetted perimeter = __
Gravitational force = ________
Hydraulic radius
Relationship between shear and velocity? ___________
oP D x
P
A Dx sin
Turbulence
o h
R S
t g
=

Pressure Coefficient for Open
Channel Flow?
2
2
C
V
p
p

D


2
2
C
V
ghl
hl

2
2
C f
f
S
gS l
V
=
l f
h S l
=
l
h
p 

D

Pressure Coefficient
Head loss coefficient
Friction slope coefficient
(Energy Loss Coefficient)
Friction slope
Slope of EGL

Chezy Equation (1768)
Introduced by the French engineer Antoine
Chezy in 1768 while designing a canal for
the water-supply system of Paris
h f
V C R S
=
150
<
C
<
60
s
m
s
m
where C = Chezy coefficient
where 60 is for rough and 150 is for smooth
also a function of R (like f in Darcy-Weisbach)
2
f h
g
V S R
l
=
compare
0.0054 > > 0.00087
l
4 h
d R

For a pipe
0.022 > f > 0.0035

Manning Equation (1891)
 Most popular in U.S. for open channels
(English system)
1/2
o
2/3
h S
R
1
n
V 
1/2
o
2/3
h S
R
49
.
1
n
V 
VA
Q 
2
/
1
3
/
2
1
o
h S
AR
n
Q  very sensitive to n
Dimensions of
Dimensions of n
n?
?
Is
Is n
n only a function of roughness?
only a function of roughness?
(MKS units!)
NO!
T /L1/3
Bottom slope

Values of Manning n
Lined Canals n
Cement plaster 0.011
Untreated gunite 0.016
Wood, planed 0.012
Wood, unplaned 0.013
Concrete, trowled 0.012
Concrete, wood forms, unfinished 0.015
Rubble in cement 0.020
Asphalt, smooth 0.013
Asphalt, rough 0.016
Natural Channels
Gravel beds, straight 0.025
Gravel beds plus large boulders 0.040
Earth, straight, with some grass 0.026
Earth, winding, no vegetation 0.030
Earth , winding with vegetation 0.050
d = median size of bed material
n = f(surface
roughness,
channel
irregularity,
stage...)
6
/
1
038
.
0 d
n 
6
/
1
031
.
0 d
n  d in ft
d in m

Trapezoidal Channel
Derive P = f(y) and A = f(y) for a
trapezoidal channel
How would you obtain y = f(Q)?
z
1
b
y
z
y
yb
A 2


2
/
1
3
/
2
1
o
h S
AR
n
Q 

Flow in Round Conduits







 

r
y
r
arccos

 


 cos
sin
2

 r
A

sin
2r
T 

y
T
A
r

r
P 2

radians
Maximum discharge
when y = ______
0.938d
( )( )
sin cos
r r
q q
=

-3
2
0 1 2 3 4 5
depth
[m
]
v(y)[m/s]
Velocity Distribution
  0
1
1 ln
y
v y V gdS
d

 
  
 
 
1 ln
y
d
- =
At what elevation does the
velocity equal the average
velocity?
For channels wider than 10d
0.4
k » Von Kármán constant
V = average velocity
d = channel depth
1
y d
e
= 0.368d
0.4d
0.8d
0.2d
V

Open Channel Flow: Energy
Relations
2g
V2
1
1

2g
V2
2
2

x
SoD
2
y
1
y
x
D
L f
h S x
= D
______
grade line
_______
grade line
velocity head
Bottom slope (So) not necessarily equal to EGL slope (Sf)
hydraulic
energy

Energy Relationships
2 2
1 1 2 2
1 1 2 2
2 2 L
p V p V
z z h
g g
a a
g g
+ + = + + +
2 2
1 2
1 2
2 2
o f
V V
y S x y S x
g g
+ D + = + + D
Turbulent flow (  1)
z - measured from
horizontal datum
y - depth of flow
Pipe flow
Energy Equation for Open Channel Flow
2 2
1 2
1 2
2 2
o f
V V
y S x y S x
g g
+ + D = + + D
From diagram on previous slide...

Specific Energy
The sum of the depth of flow and the
velocity head is the specific energy:
g
V
y
E
2
2


If channel bottom is horizontal and no head loss
2
1 E
E 
y - _______ energy
g
V
2
2
- _______ energy
For a change in bottom elevation
1 2
E y E
- D =
x
S
E
x
S
E o D


D
 f
2
1
y
potential
kinetic

Specific Energy
In a channel with constant discharge, Q
2
2
1
1 V
A
V
A
Q 

2
2
2gA
Q
y
E 

g
V
y
E
2
2

 where A=f(y)
Consider rectangular channel (A = By) and Q = qB
2
2
2gy
q
y
E 

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

0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
E
y
Specific Energy: Sluice Gate
2
2
2gy
q
y
E 

1
2
2
1 E
E 
sluice gate
y1
y2
EGL
y1 and y2 are ___________ depths (same specific energy)
Why not use momentum conservation to find y1?
q = 5.5 m2/s
y2 = 0.45 m
V2 = 12.2 m/s
E2 = 8 m
alternate
Given downstream depth and discharge, find upstream depth.
vena contracta

0
1
2
3
4
0 1 2 3 4
E
y Specific Energy: Raise the Sluice
Gate
2
2
2gy
q
y
E 

1 2
E1  E2
sluice gate
y1
y2
EGL
as sluice gate is raised y1 approaches y2 and E is minimized:
Maximum discharge for given energy.

NO! Calculate depth along step.
0
1
2
3
4
0 1 2 3 4
E
y
Step Up with Subcritical Flow
Short, smooth step with rise Dy in channel
Dy
1 2
E E y
= + D
Given upstream depth and discharge find y2
Is alternate depth possible? __________________________
0
1
2
3
4
0 1 2 3 4
E
y
Energy conserved

0
1
2
3
4
0 1 2 3 4
E
y
Max Step Up
Short, smooth step with maximum rise Dy in channel
Dy
1 2
E E y
= + D
What happens if the step is
increased further?___________
0
1
2
3
4
0 1 2 3 4
E
y
y1 increases

P
A
Critical Flow
T
dy
y
T=surface width
Find critical depth, yc
2
2
2gA
Q
y
E 

0

dy
dE
dA =
0
dE
dy
= =
3
2
1
c
c
gA
T
Q

Arbitrary cross-section
A=f(y)
2
3
2
Fr
gA
T
Q

2
2
Fr
gA
T
V

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

Critical Flow:
Rectangular channel
yc
T
Ac
3
2
1
c
c
gA
T
Q

qT
Q  T
y
A c
c 
3
2
3
3
3
2
1
c
c gy
q
T
gy
T
q


3
/
1
2









g
q
yc
3
c
gy
q 
Only for rectangular channels!
c
T
T 
Given the depth we can find the flow!

Critical Flow Relationships:
Rectangular Channels
3
/
1
2









g
q
yc c
c y
V
q 









g
y
V
y
c
c
c
2
2
3
g
V
y
c
c
2

1

g
y
V
c
c
Froude number
velocity head =
because
g
V
y c
c
2
2
2

2
c
c
y
y
E 
 E
yc
3
2

force
gravity
force
inertial
0.5 (depth)
g
V
y
E
2
2


Kinetic energy
Potential energy

Critical Depth
Minimum energy for a given q
Occurs when =___
When kinetic = potential! ________
Fr=1
Fr>1 = ______critical
Fr<1 = ______critical
dE
dy
0
1
2
3
4
0 1 2 3 4
E
y
2
2 2
c c
V y
g
=
3
T
Q
gA
=
3
c
q
gy
=
c
c
V
Fr
y g
=
0
Super
Sub

Critical Flow
 Characteristics
Unstable surface
Series of standing waves
 Occurrence
Broad crested weir (and other weirs)
Channel Controls (rapid changes in cross-section)
Over falls
Changes in channel slope from mild to steep
 Used for flow measurements
___________________________________________
Unique relationship between depth and discharge
Difficult to measure depth
0
1
2
3
4
0 1 2 3 4
E
y
0

dy
dE

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