open channel flow...fiu, department of civil and environmental engineering cwr 3201 fluid mechanics,...
TRANSCRIPT
![Page 1: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/1.jpg)
FIU, Department of Civil and Environmental Engineering
CWR 3201 Fluid Mechanics, Fall 2018
Arturo S. Leon, Ph.D., P.E., D.WRE
Open-Channel Flows
![Page 2: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/2.jpg)
Learning Objectives
1. Describe various types of open-channel
flows
2. Use energy and momentum principles
for rapidly varied flow configurations
3. Sketch water surface profiles
![Page 3: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/3.jpg)
Animations of Unsteady Open Channel Flows
� Emergency water releases at 25 dams https://www.youtube.com/watch?v=o3E4s59OSLQ
� Road Collapse- Maine 2008https://www.youtube.com/watch?v=NTbhyHNA1Vc
![Page 4: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/4.jpg)
What is Open-channel Flow?
![Page 5: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/5.jpg)
Types of Open-channel
Canal: A canal is usually a long and mild-sloped channel built in the ground
![Page 6: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/6.jpg)
Chute: A chute is a channel with a steep slope
Types of Open-channel (Cont.)
![Page 7: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/7.jpg)
Drop: A drop is a channel with a sudden change in elevation
Types of Open-channel (Cont.)
![Page 8: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/8.jpg)
Culvert: A culvert is a covered channel flowing usually
partly full.
Types of Open-channel (Cont.)
![Page 9: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/9.jpg)
Natural channel: A natural channel has irregular geometry. Examples include, rivers and creeks.
Types of Open-channel (Cont.)
![Page 10: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/10.jpg)
FLOW IN OPEN CHANNEL
STEADY FLOW UNSTEADY FLOW
UNIFORM FLOW NON-UNIFORM FLOW
RAPIDLY VARIED FLOW
GRADUALLY VARIED FLOW
TEMPORAL (Time)
SPATIAL (Space)
Classification of open-channel flows
![Page 11: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/11.jpg)
Classification of open-channel flows
![Page 12: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/12.jpg)
Wave speed in open channel flows
![Page 13: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/13.jpg)
Propagation of a disturbance in subcritical, critical and supercritical flows
![Page 14: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/14.jpg)
Froude Number:
When Fr > 1, the flow possesses a relatively high velocity and
shallow depth; on the other hand, when Fr < 1, the velocity is
relatively low and the depth is relatively deep.
![Page 15: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/15.jpg)
Uniform Flow
Arturo S. Leon, Ph.D., P.E., D.WRE
![Page 16: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/16.jpg)
Cross-section Representation
A composite section
![Page 17: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/17.jpg)
Regular cross sections
![Page 18: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/18.jpg)
The Chezy-ManningEquation
The depth associated with uniform flow is designated y0; it is
called either uniform depth or normal depth.
Equation for Uniform Flow
Uniform flow occurs in a channel when the depth and velocity do
not vary along its length
Where:
c1 = 1 for SI units and c1 = 1.49 for English units.
n = Manning roughness coefficient
A = Hydraulic area
R = Hydraulic radius
S0 = slope of the channel bottom
![Page 19: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/19.jpg)
Average values of the Manning Coefficient, n
![Page 20: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/20.jpg)
The Most Efficient Section (or best hydraulic cross section)
The Most Efficient cross-section is defined as the section of maximum
flow rate (Q) for a constant hydraulic area (A), slope (So), and roughness
coefficient (n). Alternatively, the Most Efficient cross-section can be
defined as the section of minimum hydraulic area (A) for a constant flow
rate (Q).
![Page 21: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/21.jpg)
The best hydraulic cross-section for
various shapes
![Page 22: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/22.jpg)
Example:The following data are obtained for a particular reach of the Provo River in Utah: A = 183 ft2, free-surface width = 55 ft, average depth = 3.3 ft, Rh = 3.32 ft, V = 6.56 ft/s, length of reach = 116 ft, and elevation drop of reach = 1.04 ft. Determine (a) the Manning coefficient, n, and (b) the Froude number of the flow.
![Page 23: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/23.jpg)
![Page 24: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/24.jpg)
Example:At a given location, under normal conditions a river flows with a Manning coefficient of 0.030, and a cross section as indicated in part (a) of the figure below. During flood conditions at this location, the river has a Manning coefficient of 0.040 (because of tress and brush in the floodplain) and a cross section as shown in part (b) of the figure below. Determine the ratio of the flowrate during flood conditions to that during normal conditions.
![Page 25: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/25.jpg)
![Page 26: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/26.jpg)
Energy concepts
Arturo S. Leon, Ph.D., P.E., D.WRE
![Page 27: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/27.jpg)
Total energy: The sum of the vertical distance to the channel
bottom measured from a horizontal datum, the depth of flow, and
the kinetic energy head.
Energy is actually an
energy head.
hL is the head loss.
10.4 Energy Concepts
![Page 28: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/28.jpg)
Specific energy: Measurement
of energy relative to the bottom of the channel.
Specific discharge: The total
discharge divided by the channel
width (valid only for a rectangular channel).
10.4 Energy Concepts
Specific energy diagram
![Page 29: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/29.jpg)
Critical depth
For a rectangular channel (q = Q/b)
For any cross-section:
![Page 30: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/30.jpg)
Energy Equation in Transitions
The condition of choked flow or a choking condition implies that
minimum specific energy exists within the transition.
![Page 31: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/31.jpg)
Flow Choking1E
For a rectangular channel:
cE
For a non-rectangular channel:
hc
![Page 32: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/32.jpg)
Example:Consider a channel where the upstream velocity is 5.0 m/s
and the upstream flow depth is 0.6 m. The flow then passes
over a bump 15 cm in height.
(a) Compute the flow depth and velocity on the crest of the
bump.
(b) Compute the maximum allowable bump height that keeps
water from backing up upstream.
![Page 33: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/33.jpg)
![Page 34: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/34.jpg)
![Page 35: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/35.jpg)
Example:Compute the critical depth in a trapezoidal channel for a flow of 30
m3/s. The channel bottom width is 10 m, side slopes are 2H:1V.
![Page 36: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/36.jpg)
![Page 37: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/37.jpg)
Momentum Concepts
Arturo S. Leon, Ph.D., P.E., D.WRE
![Page 38: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/38.jpg)
Hydraulic Jump
Hydraulic jump: https://www.youtube.com/watch?v=cRnIsqSTX7Q
![Page 39: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/39.jpg)
Low head dams:
https://www.youtube.com/watch?v=XsYgODmmiAM
![Page 40: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/40.jpg)
10.5 Momentum Concepts
Fig. 10.13 Channel flow over an obstacle: (a) idealized flow; (b) control volume
Linear momentum equation is:
For a rectangular section:
Let’s define M (momentum
function) as:
![Page 41: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/41.jpg)
Momentum function M for various channels
![Page 42: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/42.jpg)
10.5 Momentum Concepts
Also called
sequent depths
![Page 43: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/43.jpg)
Hydraulic Jump in a rectangular channel (Cont.)
![Page 44: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/44.jpg)
Classificationof Hydraulic
Jumps
![Page 45: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/45.jpg)
Example:Under appropriate conditions, water flowing from a faucet, onto a flat plate, and over the edge of the plate can produce a circular hydraulic jump as shown in the figure below. Consider a situation where a jump forms 3.0 in from the center of the plate with depths upstream and downstream of the jump of 0.05 in and 0.20 in, respectively. Determine the flow rate from the faucet.
![Page 46: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/46.jpg)
![Page 47: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/47.jpg)
![Page 48: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/48.jpg)
Gradually varied flow
Arturo S. Leon, Ph.D., P.E., D.WRE
![Page 49: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/49.jpg)
Gradually varied Flows
Differential Equation for Gradually Varied Flow
Where:
S = total energy slope
So = bed slope
Fr = Froude Number
![Page 50: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/50.jpg)
Does water depth increase or decrease in x direction?
Is positive or negative? dy
dx
Assuming a wide rectangular channel:
![Page 51: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/51.jpg)
Classification of Surface Profiles
![Page 52: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/52.jpg)
Examples of Gradually Varied Flows
Typical surface configurations for nonuniform depth flow with a mild
slope
![Page 53: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/53.jpg)
Example:Sketch the water surface profile for the two-reach open-channel
system below. A gate is located between the two reaches and the
second reach ends with a sudden fall.
![Page 54: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/54.jpg)
Sketch the water surface profile for the open-channel system
below.
Example:
![Page 55: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/55.jpg)
Sketch the water surface profile for the open-channel system below.
Example:
![Page 56: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/56.jpg)
Sketch the water surface profile for the open-channel system below.
Example:
![Page 57: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/57.jpg)
Sketch the water surface profile for the open-channel system below.
Example:
![Page 58: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/58.jpg)
Numerical Analysis of Water Surface ProfilesRegardless of the type of method follows these steps:
1. The channel geometry, channel slope S0, roughness coefficient
n, and discharge Q are given or assumed.
2. Determine normal depth y0 and critical depth yc.
3. Establish the controls (i.e., the depth of flow) at the upstream
and downstream ends of the channel reach.The computation procedure is to determine the depth at a section a distance DL away from a section with a known depth.To find y0
To find yc
Energy slope s
![Page 59: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/59.jpg)
Q
“Standard Step” Method This method solves sequentially for y1, y2, y3, … starting at the
control section (upstream or downstream end) with known water depth. The computation procedure is to determine the depth at a section a distance ∆x away from a section with a known depth.
Step size (∆x) must be small enough so that changes in water depth aren’t very large. Otherwise estimates of the friction slope and the velocity head are inaccurate
![Page 60: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/60.jpg)
“Standard Step” Method (cont.)
![Page 61: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/61.jpg)
“Standard Step” Method (cont.)
Solve sequentially for unknown water depth (y) starting at the
control section. The computation procedure is to determine the
depth at a section a distance ∆x away from a section with a known depth.
In general:For Subcritical flow:
For Supercritical flow:
![Page 62: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/62.jpg)
Example:A rectangular concrete-lined channel (n = 0.015) has a constant
bed slope of 0.0001 and a bottom width of 40 m. A control gate at
the dam increased the depth at the dam to 12 m when the
discharge is 300 m3/s. Compute the water surface profile from the
dam up to 200 km upstream of the dam. (See Excel spreadsheet
for rectangular channels).
![Page 63: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/63.jpg)
Solution
The first step is to calculate the critical and normal depths.
yo is computed using the Chezy-Manning formula
yo = 4.65 m
yc is computed using the critical flow condition:
yc = 1.79 m
Because y > yo > yc, the profile is M1
![Page 64: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/64.jpg)
Show exercises using Excel for rectangular channels
![Page 65: Open Channel Flow...FIU, Department of Civil and Environmental Engineering CWR 3201 Fluid Mechanics, Fall 2018 Arturo S. Leon, Ph.D., P.E., D.WRE Open-Channel Flows Animations of Unsteady](https://reader034.vdocuments.site/reader034/viewer/2022042020/5e77d5f0faf32d4c131d496a/html5/thumbnails/65.jpg)
Show examples using Annel2