Download DES601 - Hour 13

Basic Hydraulics:
Open Channel Flow – I
Open Channel Definitions
• Open channels are conduits whose upper boundary of
flow is the liquid surface
• Canals, rivers, streams, bayous, drainage ditches are
common examples of open channels.
• Storm and sanitary sewers are are also open channels
unless they become surcharged (and thus behave like
pressurized systems).
Open Channel Nomenclature
• Elevation (profile) of some open channel
Open Channel Nomenclature
• Flow profile related variables
•
•
•
•
•
•
Flow depth (pressure head)
Velocity head
Elevation head
Channel slope
Water slope (Hydraulic grade line)
Energy (Friction) slope (Energy grade line)
Open Channel Nomenclature
• Cross Sections
Natural Cross Section
Engineered Cross Section
Open Channel Nomenclature
• Cross Section Geometry and Measures
•
•
•
•
Flow area (all the “blue”)
Wetted perimeter
Topwidth
Flow depth
• Thalweg (path along bottom of channel)
Open Channel Steady Flow
• For any discharge (Q) the flow at any section is
described by:
• Flow depth
• Mean section velocity
• Flow area (from the cross section geometry)
• Depth-area, depth-topwidth, depth-perimeter are
used to estimate changes in depth (or flow) as one
moves from section to section
Open Channel Steady Flow Types
• The flow-depth, depth-area, etc. relationships are
non-unique, flow “type” is relevant
•
•
•
•
Uniform (normal)
Sub-critical
Critical
Super-critical
Cross Section Geometry
• Normal, Critical, Sub-, Super-Critical flow all depend
on channel geometry.
• Engineered cross sections almost exclusively use just
a handful of convenient geometry (rectangular,
trapezoidal, triangular, and circular).
• Natural cross sections are handled in similar fashion
as engineered, except numerical integration is used
for the depth-area, topwidth-area, and perimeter-area
computations.
Cross Section Geometry
• Rectangular Channel
• Depth-Area
A(y)  By
• Depth-Topwidth

T(y)  B
• Depth-Perimeter

Pw (y)  B  2y


Cross Section Geometry
• Trapezoidal Channel
• Depth-Area
A(y)  y(B  my)
• Depth-Topwidth
T(y)  B  2my
• Depth-Perimeter
Pw (y)  B  2y 1 m 2
Cross Section Geometry
• Triangular Channels
• Special cases of
trapezoidal channel
• V-shape; set B=0
• J-shape; set B=0, use ½
area, topwidth, and
perimeter
Cross Section Geometry
• Circular Channel (Conduit with Free-Surface)
• Contact Angle:  (y)  2cos1(1
2y
)
D
D2 

 
A(y)


sin(
)cos(
)

• Depth-Area:
4 2
2
2 


• Depth-Topwidth: T(y)  Dsin( )
2

• Depth-Perimeter: Pw (y)  D
2


Cross Section Geometry
• Irregular Cross Section
• Use tabulations for the
hydraulic calculations
Cross Section Geometry
• Irregular Cross Section – Depth-Area
Depth
A4
A3
A2
A1
A1+A2
A1+A2+A3
Area
A1
Cross Section Geometry
• Irregular Cross Section – Depth-Area
Depth
T4
T3
T2
T1
T1+T2
T1+T2+T3
Topwidth
T1
Cross Section Geometry
• Irregular Cross Section – Depth-Perimeter
Depth
P4
P3
P2
P1
P1+P2
P1+P2+P3
Perimeter
P1
Flow Direction/ Cross Section
Geometry
• Convention is to express
station along a section with
respect to “looking
downstream”
Left Bank
• Left bank is left side of
stream looking downstream
(into the diagram)
• Right bank is right side of
stream looking downstream
(into the diagram)
Right Bank
Flow Direction
Energy Equation in Open Channel Flow
• Energy equation:
y1  z1 
v
2
1 1
2g
 y 2  z2 
 = velocity head correction factor
 2v 2
2g
2
 hL

Energy Equation in Open Channel Flow
• When velocity is nearly uniform across the
channel the correction factor is usually treated as
unity ( = 1)
• Hence, the energy equation is typically written as
2
1
2
v
v2
y1  z1 
 y 2  z2 
 hL
2g
2g
Potential Energy
• In pressurized systems the potential energy is the
sum of the pressure and elevation head.
• In open channels, elevation is taken at the bottom
of the channel, the analog to pressure is the flow
depth.
• Thus the potential energy is the sum of elevation
and flow depth
Static Head (Potential) = y  z

Kinetic Energy
• Kinetic energy is the energy of motion; in
pressurized as well as open channel systems, this
energy is represented by the velocity head
v2
Kinetic (Velocity) Head =
2g
• The sum of these two “energy” components is the
total dynamic head (usually just “total head”)

v2
Total Head = y  z 
2g
Hydraulic Grade Line
• The hydraulic grade line is coincident with the water
surface.
• It represents the static head at any point along the
channel.
Specific Energy
• Total energy is the sum of potential and kinetic
components:
2
v
E total  z  y 
2g
• Energy relative to the bottom of the channel is called
the specific energy (at a section)

v2
E specific  y 
2g

Specific Energy
• Relationship of Total and Specific Energy (at two
different sections)
v2
E total  z  y 
2g
y2
E specific  y 
y1

v2
2g
Specific Energy Calculations
• For example
• Rectangular channel,
• Q=100cfs; B=10 ft
Q
100 cfs
10
V 

A (10 ft )(y ft ) y
• So


v2
(10 / y) 2
100
E specific  y 
y
y
2g
2g
2gy 2
• Table shows values. Plot on next
page = specific energy diagram
Specific Energy Diagram
Critical Depth
• Specific energy relationship has a minimum point
• Flow at specified discharge cannot exist below
minimum specific energy value
• Depth associated with minimum energy is called
“critical depth”
• Critical depth (if it occurs) is a “control section” in a
channel
• What is the value of critical depth for the case shown
in the previous diagram (and table)?
Flow Classification by Critical Depth
• Subcritical flow –Water
depth is above critical depth
(velocity is less than the
velocity at critical depth)
Q3
• Supercritical flow – Water
depth is below critical depth
(velocity is greater than the
velocity at critical depth)
• Critical flow – Water depth
is equal to critical depth. 1
to 1 depth-discharge at
critical (dashed line)
Q2
Q1
Flow Classification by Critical Depth
• Classification important in water surface profile (HGL)
estimation and discharge measurement.
• Water can exist at two depths except at critical depth
• Critical depth important in measuring discharge
• Sub- and Super-Critical classification determine if the
controlling section is upstream or downstream.
• Sub- and Super-Critical classification determines if computed
HGL will be a front-water or back-water curve.
Conveyance
• The cross sectional properties can be grouped into a
single term called conveyance
1.49 2 / 3
K
AR
n
• Manning’s equation becomes

Q  KS1/0 2
• Units of conveyance are CFS

Normal depth
• Normal depth is another flow condition where the
slope of the energy grade line, channel bottom, and
the slope of the hydraulic grade line are all the same
• Manning’s equation assuming normal depth is
1.49 2 / 3 1/ 2
Q
AR S0
n

A
Pw
Normal depth
• To use need depth-area, depth-perimeter information
from channel geometry.
• Then can rearrange (if desired) to express normal
depth in terms of discharge, and geometry.
• Computationally more convenient to use a rootfinding tool (i.e. Excel Goal Seek/Solver) than to
work the algebra because of the exponentation of
the geometric variables.
Normal depth
• For example, TxDOT HDM Eq 10-1 is one such
Manning’s equation, rearranged to return normal
depth in a triangular section (J-shape)
 QnS x 
d  1.24 1/ 2 
 S

3/8
Sx
d

where Q = design flow (cfs);
n = Manning’s roughness

coefficient ; Sx = pavement cross slope; S = friction
slope; d = normal depth (ft).