1 Hidayat Jamal & Zulhilmi ismail Hydraulics & Hydrology Department, FKA, UTM Chapter 2
1
Hidayat Jamal & Zulhilmi
ismail
Hydraulics & Hydrology
Department, FKA, UTM
Chapter 2
2
Uniform flow
■ Equations are developed for steady-state conditions
– Depth, discharge, area, velocity all constant along
channel length
■ Rarely occurs in natural channels (even for constant
geometry) since it implies a perfect balance of all
forces
■ Two general equations in use: Chezy and Manning
formulas
3
Or in general, uniform flow
- Energy slope = Bed slope or dH/dx = dz/dx
- Water surface slope = Bed slope = dy/dz = dz/dx
- Velocity and depth remain constant with x
4
Q = C
V = C
y = C
S0 = C
A = C
B = C
n = C
UNIFORM FLOW
Normal depth implies that flow rate, velocity,
depth, bottom slope, area, top width, and roughness
remain constant within a prismatic channel as
shown below
5
6
General Flow Equation
Q = Av
Flow rate (m3/s)
Avg. velocity of
flow at a cross-
section (m/s)Area of the
cross-section
(m2)
7
Chezy’s Formula
Antoine Chezy, in 1769
A French civil engineer developed the
Chézy equation, which relates the uniform
flow velocity to channel roughness,
hydraulic radius, and bed slope.
R = hydraulic radius
So = slope of the channel
C = coefficient depending upon the
various characteristics of the
channel and their comparison with
those of another similar channel
oRSACQ
8
Other Flow Formulas Two other flow formula defined more accurately the value
for the coefficient C in the Chezy formula.
(a) Bazin formula:
R
mC
1
87
Values of m to be used in the Bazin formula for determining C in the chezy formula.
9
Values of n to be used in the Kutter’s nformula for determining C in the chezy formula.
))00155.0
23(1
00155.0123
R
nx
S
SnC
o
o
(b) Kutter formula:
10
Manning Formula
Robert Manning, 1885;
Developed Manning formula used for
open channel flow conditions.
R = hydraulic radius
So = slope of the energy gradient
n = a roughness coefficient
2 13 2
oAR SQ
n
(Refer to the Table)
A and R = function of y
11
Table of Manning’s n Roughness
CoefficientType of Channel and Description Minimum Normal Maximum
Streams
Streams on plain
Clean, straight, full stage, no rifts or deep pools 0.025 0.03 0.033
Clean, winding, some pools, shoals, weeds & stones 0.033 0.045 0.05
Same as above, lower stages and more stones 0.045 0.05 0.06
Sluggish reaches, weedy, deep pools 0.05 0.07 0.07
Very weedy reaches, deep pools, or floodways 0.075 0.1 0.15
with heavy stand of timber and underbrush
Mountain streams, no vegetation in channel, banks
steep, trees & brush along banks submerged at
high stages
Bottom: gravels, cobbles, and few boulders 0.03 0.04 0.05
Bottom: cobbles with large boulders 0.04 0.05 0.07
12
Channel Conditions Values
Material Involved Earth n0 0.025
Rock Cut 0.025
Fine Gravel 0.024
Coarse Gravel 0.027
Degree of irregularity Smooth n1 0.000
Minor 0.005
Moderate 0.010
Severe 0.020
Variations of Channel Cross
Section Gradual n2 0.000
Alternating Occasionally 0.005
Alternating Frequently 0.010-0.015
Relative Effect of Obstructions Negligible n3 0.000
Minor 0.010-0.015
Appreciable 0.020-0.030
Severe 0.040-0.060
Vegetation Low n4 0.005-0.010
Medium 0.010-0.025
High 0.025-0.050
Very High 0.050-0.100
Degree of Meandering Minor m5 1.000
Appreciable 1.150
Severe 1.300
n = (n0 + n1 + n2 +
n3 + n4 ) m5
Values for the computation of the
roughness coefficient (Chow,
1959)
13
Values of Manning’s n according
to MASMA
Open Channel (Surface Cover of Finishing) Manning’s n (Minimum)
Manning’s n (Maximum)
Grass Swales
Short grass cover 0.030 0.035
Tall grass cover 0.035 0.053
Lined Drains
Concrete
Troweled finished 0.011 0.015
Off form finished 0.013 0.018
Stone Pitching
Dressed stones in Mortar 0.015 0.017
Random stones or rubble masonry 0.020 0.035
Rock riprap 0.025 0.030
Brickwork 0.012 0.018
Precast masonry blockwork 0.012 0.015
14
Manning’s Roughness (n)
Roughness coefficient (n) is a function of:
– Channel material
– Surface irregularities
– Variation in shape
– Vegetation
– Flow conditions
– Channel obstructions
– Degree of meandering
15
Conveyance Factor, K
K, the capacity of the channel to carry flow
Simply calculated from Chezy or Manning formulae
oRSACQ Chezy Formula:
where
Manning formulae:
where
RACK
oSARn
Q 321
n
ARK
32
16
Channel Section Factor, Z
Or,
32
ARZ
oS
nQnKAR 3
2
Z, the characteristics of channel geometry
for Manning
12Z AR for Chezy
for Manning
12
o
K QAR
C C S for Chezy
17
Flow Rate Per Unit Width, q
For rectangular channel only
B
B
y
unit as or
m
sm /3
msm ./3
Basic, Q = Av
q = vy
18
Very Wide Channel
Shallow flow depth compared to the channel width
Very wide channel;
So, remember
yB
yR
19
Calculation of Normal
Depth, yo
Can be calculated using either:
(i) Trial and error method
(ii) Graphical method
(iii) Charts
B
yo
B
yo
Z
1
20
Trial and error method
Given the flow condition (channel dimension,roughness, flow rate and bed slope)
Normal depth is calculated using the Manning orChezy flow formula by trial and error method
Graphical method
A plot of y vs Z (where ) is made
Normal depth is when
32
ARZ
oS
nQZ
21
Chart: Curve for Determining
Normal Depth (Chow, 1959)
10
Rectangularz = 0.5
z = 1.0
z = 1.5z = 2.0z = 2.5z = 3.0z = 3.5
0.0001 0.001 0.01 0.1 1 100.01
0.1
1
0.01
0.1
1
0.0001 0.001 0.01 0.1 10
Circular
y
1
zB
y
NOTE :
1
0
32
S
nQAR
38
32
38
32
Φ
ARor
B
AR
Φ
y
or
B
y
o
o
22
Best Hydraulic Section(BHS)
For a given Q, there are many channel shapes. Thereis the need to find the best proportions of B and y whichwill make the discharge is maximum.
Using Chezy's formula:
Flow rate:
For a rectangular Channel: P = B +2y
A = By and therefore: B = A/yi.e. P = A/y + 2y
oRSCV
)1( oo SP
AACRSACQ
B
y
23
For a given Area (A), Q will be maximum when P isminimum (from equation 1)
Differentiate P with respect to y
For minimum P i.e. Pmin ,
A = 2y2 ,
Since A = By ie. By = 2y2 ie. B = 2y
i.e. for maximum discharge, R = y/2
22
y
A
dy
dP
022
y
A
dy
dP
24
Area of cross section, A = By + zy2
Width, B = A/y - zy ----------------------(1)
Wetted Perimeter, P = B + 2y ( 1 + z2 )1/2
P = A/y - zy + 2y(1 + z2 ) 1/2---------------------(2)
Differentiate P with respect to y
B
y
Z
1
For a Trapezoidal Section
L
25
1) ; you got or
- Top water width (T) is twice the side length (L)
- Hydraulic Radius;
2) for ; you got
- The side slope is 60⁰ or B = L
0dy
dP 212 zyT LT 2
0dz
dP
2
yR
3
1z
26
Best Hydraulic Section
27
Why BHS???
Called “Most efficient cross-section”
Efficient cross-section lead in- Economical designed in reduce material to
construct the channel with minimum wettedparameter
- Convey maximum designed discharge
So, what channel shape is most efficient??
28
1) A smooth concrete-lined trapezoidal channel (n =
0.011) was constructed earlier in a development project.
The channel bottom width is 10 meter, total channel
depth (including freeboard of 300 mm) is 2.3 meter, side
slopes z = 2, and longitudinal slope is 0.0001. Determine
the discharge, Q.
2) If the channel has to be re-designed as the Best
Hydraulic Section (z = 0.58) to convey a similar
discharge, find the new size of the trapezoidal channel.
Maintain the type of lining material and longitudinal
slope for the channel. Sketch your result.
Example
29
Solution
10m
PAR /
2.3
m
n = 0.011
y = 2.3 - 0.3
= 2.0m
z = 2
So = 0.0001
A = (B +zy)y
2)2(210
228m
212 zyBP 2)2(1)2(210
m94.18
94.18/28
478.1
30
n
SARQ
o3
2
011.0
0001.0)478.1(28 32
Q
sm /03.33 3
Re-designed as the Best Hydraulic
Section as
z = 0.58
n = 0.011
So = 0.0001
Q = 33.03 m3/s
2732.1 y
22 58.0152.1 yy
From T = 2L
Then A = By +zy2
2122 zyzyB
yyB )58.0(2)58.0(12 2
yB 152.1
Cont…
31
212 zyBP 21.152 2 1 (0.58)y y
y464.3
PAR /
y
y
464.3
732.1 2
y5.0
n
SARQ
o3
2
0001.0
)011.0(03.333
2
AR
my 72.3
33.3632
AR
33.36)5.0(732.1 32
2 yy
299.3338
y
)72.3(152.1B
mB 29.4
Cont…
32
Final Design Diagram
y = 3.72mz = 0.58 Total depth
= 4.02m
B = 4.29m
T = 8.61m
T = B + 2zy = 4.29 + 2 x 0.58 x 3.72 = 8.61 m
Cont…
33
The Design of Circular Culverts-
Optimum Water Depth
y optimum for max discharge
34
y optimum for max velocity
35
Design of Erodible Channels
(Earthen Channels)