Open Channel Hydraulics
Open Channel Flows •
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Refers to flows whose top surface is exposed to atmospheric pressure Examples –
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Rivers and streams, irrigation canals, sewer lines that flow partially full, storm drains, street gutters, etc.
Applications –
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Design and analysis of channels, sewer lines, storm drains, gutters Predicting water surface profile for streams and rivers
Open Channel Flows Figure 5.2 (p. 87) Haden-Rhodes Aqueduct, Central Arizona Project. (Courtesy of the U.S. Bureau of Reclamation. 1985, photograph by Joe Madrigal Jr.)
Definitions Types of flow •
Based on variation of flow depth with space –
Uniform flow -> Flow depth does not change with space
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Nonuniform flow -> Flow depth changes with space •
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Gradually varied flow Rapidly varied flow
Based on variation of flow depth with time –
Steady flow -> Flow depth doesn’t change with time
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Unsteady flow -> Flow depth changes with time
Combinations –
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Steady uniform flow -> will be covered Steady nonuniform flow -> will be partially covered
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Unsteady uniform flow -> unrealistic
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Unsteady nonuniform flow -> advanced topic
Rapidly Varied Flow and Gradually Varied Flow
Definitions Channel Types •
Natural Channel -> developed by natural processes and has not been significantly improved by humans –
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E.g., river channels
Artificial Channel -> refers to all channels which have been developed by human –
E.g., storm drains, gutters, navigation channels, power and irrigation channels, aqueducts, etc
Definitions Artificial Channels •
Prismatic Channel -> has constant cross-sectional shape and bottom slope.
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Noprismatic Channel -> cross sectional shape and/or slope
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changes Canal -> rather long channel of mild slope
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• •
Flume -> a channel build above the ground surface to convey flow across a depression (e.g., river crossing). Channels in lab are also referred to as flume Chute -> channel with a steep slope Drop -> has also steep slope but is much shorter than chute Culvert -> used to convey water under roads
Definitions •
Depth of Flow -> vertical distance from the bottom of channel to the water surface
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Stage -> elevation of water surface relative to a datum
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Top Width -> width of channel section at the water
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surface Flow Area -> cross-sectional area of the flow normal to the flow direction
Wetted Perimeter -> Total length of the interface between the water and the channel boundary Hydraulic Radius -> ratio of flow area to wetted perimeter Hydraulic Depth -> ratio of flow area to top width
θ = 2 cos −1 1 −
2y
D0
Steady Uniform Flow •
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•
•
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Occurs when depth of flow does not vary along the length of the channel
Normal depth -> depth of flow in uniform open channel flow Flow velocity is the same along the length of the channel Channel bed slope (S0) is the same as water surface slope (Sf) Steady uniform flow assumption is commonly used to design channels
Uniform Flow
The objective in open channel flow hydraulics is to relate flow rate or velocity to depth of flow, slope of the channel, channel shape, and channel roughness
Chezy Equation •
Developed based on momentum equation for fully turbulent flows (common in open channel)
V = C RS o where C = Chezy coefficient R = Hydraulic radius (m, ft) S0 = Slope of the channel (bed slope) (ft/ft, m/m) V = Mean flow velocity (m/s, ft/s)
Manning Equation •
Manning developed an empirical relation for Chezy Coefficient (C) C=
K n
=> V =
R
1/ 6
K n
R
2/3
S
1/ 2 0
=> Q =
K n
AR 2 / 3 S 01/ 2
where K = Unit constant (1 for SI units, and 1.49 for US units) n = Manning coefficient ( See Table 5.1.1) R = Hydraulic radius (m, ft) 2
2
A = Cross-sectional area (m , ft ) S0 = Slope of the channel (bed slope) ( ft/ft, m/m) V = mean flow velocity (m/s, ft/s) Q = flow rate (m3/s, ft3/s)
Manning Equation •
Manning equation is valid for fully turbulent flow
•
Criteria for fully turbulent flow
n 6 RS f ≥ 1.9 × 10 −13 (R in feet) n 6 RS f ≥ 1.1× 10 −13
(R in meter)
Manning Equation for Circular Channels
Example
Example
Composite Channels
Example
Best Hydraulic Sections •
•
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A maximum conveyance channel (i.e., a channel section with the smallest wetted perimeter for a given channel section area, resistance and slope) The sections may not be practical due to construction difficulties The concept is valid only for nonerodible channels
Best Hydraulic Sections
Example
Design of Open Channels •
Artificial channels may be lined or unlined (not common)
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Basic design principles for open channels –
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If possible, use best hydraulic section (efficient section)
Velocity should not be too low to avoid sedimentation (2.5 ft/s or higher is usually acceptable) Shear stress should not be too high to avoid erosion (scouring) for unlined and flexible-lined (e.g., vegetation) channels
Design of Open Channels Basic design principles •
Side slope should not be too steep to ensure stability –
Max. of 3H:1V for roadside unlined channels(FHWA)
Max of 1.5H:1V for concrete lined channels (USBR) Longitudinal slope should minimize excavation cost while meeting minimum velocity requirement –
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Freeboard -> vertical distance between the water surface and the top of the channel as a safety factor to account for design uncertainties
Design of Lined Channels •
Channel linings are used to control erosion
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Linings are classified as rigid and flexible
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Rigid linings crack when deflected and include: concrete, stone masonry, grouted riprap and asphalt
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Flexible linings are able to conform to changes in channel shape and include: riprap, vegetation, and gravel
Rigid-Lined Channels •
Advantages Transport water at high velocity to reduce construction and excavation costs –
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Decrease seepage loss Decrease operation and maintenance cost
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Ensure stability of channel slopes
All high flow velocity channels must be rigid-lined (ASCE 1992)
Design of Rigid-Lined Channels Design Steps •
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For the specified lining, select n (see Table 15.3.1), side slope (z), and bottom slope (S0) Determine the section factor (i.e., AR2/3) Determine dimensions (e.g., y,and b, orpracticability D) based on the section channel factor, hydraulic efficiency Check the minimum velocity to avoid sedimentation Add freeboard to the depth of channel calculated Minimum freeboard = 0.3 m (1 ft) for small channels (USBR) FB = Cy1/2 for larger channels. C varies from 0.7 (1.2) for small channels of 0.6 m3/s (20 cfs) to 0.9 (1.6) for larger channels of 85 m3/s (3000 cfs) or greater (USBR) –
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Example
Design of Flexible-Lined Channels •
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Flexible-lined channels can conform to change in channel slope (e.g., vegetative lining, rock riprap) Advantages for stormwater conveyance –
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Permit infiltration and exfiltration Filter out contaminants Provide greater energy dissipation Allow flow condition that is ecologically more desirable Less expensive
Design of Flexible-Lined Channels •
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Choose a flexible lining from Table 15.3.4 and note its permissible shear stress p Select channel shape, slope and design discharge Use Table 15.3.1 for nonvegetative linings to find Manning n based on assumed range of depth. For vegetative linings use Table 15.3.5 to determine the retardant class, and determine n as a function of R. comp) from Manning Calculate flow depth ( y equation
Design of Flexible-Lined Channels Manning n for vegetative lining is given as (US units) n=
k1
(ac + k 2 )
where k1 = R1/6, R is hydraulic radius in ft k2 = 19.97log(R1.4S00.4), S0 is channel bed slope ac = 15.8, 23.0, 30.2, 34.6, and 37.7 for retardance classes A, B, C, D, and E respectively
Design of Flexible-Lined Channels Manning n for vegetative lining is given as (SI units) n=
k1
(ac + k 2 )
where k1 = 1.22R1/6, R is hydraulic radius in meter k2 = 19.97log(R1.4S00.4), S0 is channel bed slope ac = 30.2, 37.4, 44.6, 49, and 52.1 for retardance classes A, B, C, D, and E respectively
Design of Flexible-Lined Channels •
Compute shear stress (τdes) and compare it with τp. If τdes < τp , the lining is acceptable. Otherwise, choose another lining. τ des = γRS 0
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Determine freeboard (FB) (ASCE, 1992) FB = 0.152 +
V2 2g
FB in m, V in m/s, and g in m/s2
Example
Example
Specific Energy •
Total head at a location in an open channel flow is P γ
•
+
V2
+z
2g
Total head above the channel bottom (i.e.,z = 0), and assuming pressure is hydrostatic, V2
Q2
E = y + 2 g = y + 2 gA2 •
where y is flow depth
E is known as specific energy
Specific Energy y1 +
V1
2
2g
= y2 +
V22 2g
=
Specific Energy Diagram
E
+ y
Q22 2 gA 2
Specific Energy
Specific Energy •
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The lower portion of the curve asymptotically approaches the abscissa, where y = 0 The upper portion of the curve asymptotically 0
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approaches a 45 line (i.e., E = y) For any level of specific energy other than the minimum, two depths (alternate depths) exist The curve passes through a minimum energy state where dE/dy is zero and where the depth assumes a critical value, yc
Specific Energy
Example
Critical Flow •
Occurs when specific energy (E) is minimum for a given discharge (dE/dy = 0) Q2
E = y+
2
2 gA
dE dy
=> •
= 1− Q 2T gA3
Q 2 dA gA3 dy
= 1,
dA
= 0, =>
dy
V2 g
=
=T
A T
These equations can be velocity used to determine criticaltwo depth and/or critical
Critical Flow V2 g (A / T ) =>
V gD
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•
= 1,
D = A/T
=1
Froude number (Fr) =
V gD
Fr is a dimensionless ratio of inertial forces to gravity forces, and is used to determine whether flow is critical, subcritical or supercritical.
Critical Flow •
If Fr < 1, => Flow is subcritical • •
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If Fr > 1, => Flow is supercritical • • •
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Deep (y > yc) and tranquil flow Flow is controlled from downstream and upstream Depth is less than yc Flow is shallow and rapid Flow is upstream controlled
If Fr = 1, => Flow is critical • •
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Flow is unstable and it occurs at transitions Channels should be designed for flow depth that is well above or well below yc It is recommended that Fr > 1.4 or Fr< 0.6 for designs
Critical Flow
Critical Flow •
For rectangular channels 2
QT gA3
2
=1
=>
Q b 3
g (byc )
= 1,
q=
Q b
1/ 3
q2 => yc = g Ec = •
3 2
yc
For circular channels 1.01 Q 0.26 D g 2
0.25
y = c
if 0.2D ≤ yc ≤ 0.85D
where D (diameter), Q and g are in ft, cfs and ft/s2, respectively
Example
Example
Hydraulic Jump •
Occurs where supercritical flow changes to subcritical flow –
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E.g., downstream of sluice gates, at the foot of spillways, when a steep channel suddenly becomes flat
Hydraulic jump causes significant energy loss Momentum equation is used to analyze hydraulic jumps
Typical Applications of Hydraulic Jump •
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Energy dissipation in flows over dams, weirs, etc Reduction of uplift pressure under structures by raising water depth on apron of structures To maintain high water levels in channels for diversions
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Mixing of chemicals
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Aeration of flows, etc.
Hydraulic Jump Momentum equation for turbulent flows
∑F
x
= ρQ(V2, x − V1, x )
∑Fx = resultant of forces in the x direction
(include hydrostatic force, gravity force in the flow direction, and shear force) ρ = fluid density V2 = velocities at the downstream section V1 = velocities at the upstream section
Hydraulic Jump For short, horizontal reach, ignoring shear force and gravity force in the x-direction,
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Q2 gA1 •
+Ay = 1
1
Specific Force (F)
Q2
+A y
gA2 F=
2
Q2 gA
2
+ Ay
where y is distance from water surface to the centroid
Hydraulic Jump
y1 and y2 are known as sequent depths
Hydraulic Jump For rectangular channels –
Relationship between sequent depths y2
=
1
=
1
y1 y1 y2 –
2 −1+ 1+ 8 r1 F 2
2 − 1 + 1 + 8 Fr 2 2
Energy loss due to hydraulic jump −
3
2 1 ∆E = E1 − E 2 = ( y4 y yy ) 1 2
Hydraulic Jump Approximate lengths of jump for rectangular channels Fr1
Lj/y2
2 3
4.4 5.25
4
5.8
5
6.0
6
6.1
7
6.15
8
6.15
Adapted from USBR (1955)
Example
Example
