2. 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
Open Channel Flow
Geometric Parameters :
1) Hydraulic Radius (Rh)
2) Channel Length (l)
3) Roughness (e)
P
A
Rh
3. 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
Gradually varied: gradual changes in velocity with
distance
Laminar and Turbulent
Laminar: No mixing of layers.
Turbulent: Mixing of layers
TYPES OF FLOWS
4. Chezy Equation (1768)
Introduced by the French engineer Antoine
Chezy in 1768 while designing a canal for
the water-supply system of Paris
h fV C R S
where C = Chezy coefficient
4 hd R
For a pipe
C=(8g/f)^(1/2)
5. Robert –Manning improved Work
Upon Chezy’s Equation and gave:
V(m/s)=1/n[Rh]^(2/3) S^(1/2)
V(ft/s)=1.486/n [Rh]^(2/3) S^(1/2)
These are emprical formulas
Where “n” is manning co-efficent
6. 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
n = f(surface roughness,
channel irregularity,
stage...)
6/1
031.0 dn
6/1
038.0 dn
d in ft
d in m
d = median size of bed material
7. Trapezoidal Channel
Derive P = f(y) and A = f(y) for a
trapezoidal channel
z
1
b
y
2/13/2
1
oh SAR
n
Q
!
zyybA 2
1/ 222
2P y yz b
1/ 22
2 1P y z b
8. m
.
Flow in Round Conduits
r
yr
arccos
cossin2
rA
sin2rT
y
T
A
r
rP 2
radians
Maximum discharge
when y = ______0.938d
sin cosr r
10. 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
11. 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 cV y
g
3
T
Q
gA3
c
q
gy
c
c
V
Fr
y g
0
Super
Sub
12. 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
13. Water Surface Profiles:
Putting It All Together
2 m
10 cm
Sluice gatereservoir
1 km downstream from gate there is a broad crested
weir with P = 1 m. Draw the water surface profile.
14. Wave Celerity
1
21
2pF gy 2
21
2pF g y y
1 2
221
2p pF F g y y y
Fp1
y+yV+VV
Vw
unsteady flow
y y y+yV+V-VwV-Vw
steady flow
V+V-VwV-Vw
Fp2
1 21 2 p p ssM M W F F F
Per unit width
15. Wave Celerity:
Momentum Conservation
1 2 w w wy V V V V V V VM M
1 2 wy V V VM M
VVVyg w y y+yV+V-VwV-Vw
steady flow
yVVM w
2
1 Per unit width2 w wM V V V V V y
Now equate pressure and momentum
1 2
221
2p pF F g y y y
2 2 21
2
2 wg y y y y y y V V V
16. Wave Celerity
ww VVVyyVVy
www yVyVVyVyyVyVyVyV
y
y
VVV w
VVVyg w
y
y
VVyg w
2
2
wVVgy wVVc gyc
Mass conservation
y y+yV+V-VwV-Vw
steady flow
Momentum
c
V
Fr
yg
V
17. Wave Propagation
Supercritical flow
c<V
waves only propagate downstream
water doesn’t “know” what is happening downstream
_________ control
Critical flow
c=V
Subcritical flow
c>V
waves propagate both upstream and downstream
upstream
18. Discharge Measurements
Sharp-Crested Weir
V-Notch Weir
Broad-Crested Weir
Sluice Gate
5/ 28
2 tan
15 2
dQ C g H
3/ 2
2
3
dQ C b g H
3/ 22
2
3
dQ C b gH
12d gQ C by gy
Explain the exponents of H! 2V gH
19. Summary (1)
All the complications of pipe flow plus
additional parameter... _________________
Various descriptions of energy loss
Chezy, Manning, Darcy-Weisbach
Importance of Froude Number
Fr>1 decrease in E gives increase in y
Fr<1 decrease in E gives decrease in y
Fr=1 standing waves (also min E given Q)
free surface location
0
1
2
3
4
0 1 2 3 4
E
y
20. Summary (2)
Methods of calculating location of free
surface (Gradually varying)
Direct step (prismatic channel)
Standard step (iterative)
Differential equation
Rapidly varying
Hydraulic jump
2
1 Fr
SS
dx
dy fo
23. Energy Equation
Specific Energy
Two depths with same energy!
How do we know which depth
is the right one?
Is the path to the new depth
possible?
2 2
1 2
1 2
2 2o f
V V
y S x y S x
g g
2
2
2
q
y
gyg
V
yE
2
2
2
2
2
Q
y
gA
0
1
2
3
4
0 1 2 3 4
E
y
24. What next?
Water surface profiles
Rapidly varied flow
A way to move from supercritical to subcritical flow
(Hydraulic Jump)
Gradually varied flow equations
Surface profiles
Direct step
Standard step
25. Mild Slope
If the slope is mild, the depth is less than the
critical depth, and a hydraulic jump occurs,
what happens next?
Rapidly varied flow!
When dy/dx is large
then V isn’t normal to cs
Hydraulic jump! Check conjugate depths
0
0
0
0
0
0
0
0510152025303540
distance upstream (m)
bottom
surface
yc
yn
27. Open Channel Reflections
Why isn’t Froude number important for describing
the relationship between channel slope, discharge,
and depth for uniform flow?
Under what conditions are the energy and
hydraulic grade lines parallel in open channel
flow?
Give two examples of how the specific energy
could increase in the direction of flow.