1. Basic Hydraulics: Open Channel Flow – I

2. 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).

3. Open Channel Nomenclature
Elevation (profile) of some open channel

4. 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)

5. Open Channel Nomenclature
Cross Sections
Natural Cross Section
Engineered Cross Section

6. Open Channel Nomenclature
Cross Section Geometry and Measures
Flow area (all the “blue”)
Wetted perimeter
Topwidth
Flow depth
Thalweg (path along bottom of channel)

7. 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

8. 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

9. 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.

10. Cross Section Geometry
Rectangular Channel
Depth-Area
Depth-Topwidth
Depth-Perimeter

11. Cross Section Geometry
Trapezoidal Channel
Depth-Area
Depth-Topwidth
Depth-Perimeter

12. 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

13. Cross Section Geometry
Circular Channel (Conduit with Free-Surface)
Contact Angle:
Depth-Area:
Depth-Topwidth:
Depth-Perimeter:

14. Cross Section Geometry
Irregular Cross Section
Use tabulations for the hydraulic calculations

15. Cross Section Geometry
Irregular Cross Section – Depth-Area
Depth A4 A3 A2 A1
Area A1
A1+A2
A1+A2+A3

16. Cross Section Geometry
Irregular Cross Section – Depth-Area
Depth T4 T3 T2 T1 T1
Topwidth
T1+T2
T1+T2+T3

17. Cross Section Geometry
Irregular Cross Section – Depth-Perimeter
Depth P4 P3 P2 P1
Perimeter
P1+P2 P1
P1+P2+P3

18. Flow Direction/ Cross Section Geometry
Convention is to express station along a section with respect to “looking downstream”
Left bank is left side of stream looking downstream (into the diagram)
Right bank is right side of stream looking downstream (into the diagram)
Left Bank
Right Bank
Flow Direction

19. Energy Equation in Open Channel Flow
α = velocity head correction factor

20. Energy Equation in Open Channel Flow
When velocity is nearly uniform across the channel the correction factor is usually treated as unity (α = 1)
Hence, the energy equation is typically written as

21. 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

22. 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
The sum of these two “energy” components is the total dynamic head (usually just “total head”)

23. Hydraulic Grade Line
The hydraulic grade line is coincident with the water surface.
It represents the static head at any point along the channel.

24. Specific Energy
Total energy is the sum of potential and kinetic components:
Energy relative to the bottom of the channel is called the specific energy (at a section)

25. Specific Energy
Relationship of Total and Specific Energy (at two different sections) y2 y1

26. Specific Energy Calculations
For example
Rectangular channel,
Q=100cfs; B=10 ft So
Table shows values. Plot on next page = specific energy diagram

27. Specific Energy Diagram

28. 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)?

29. Flow Classification by Critical Depth
Subcritical flow –Water depth is above critical depth (velocity is less than the velocity at critical depth)
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) Q3 Q2 Q1

30. 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.

31. Flow profiles in Long Channels
Flow is
subcritical.
Flow is at
critical depth.
supercritical.

32. Conveyance
The cross sectional properties can be grouped into a single term called conveyance
Manning’s equation becomes
Units of conveyance are CFS

33. 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

34. 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 root-finding tool (i.e. Excel Goal Seek/Solver) than to work the algebra because of the exponentation of the geometric variables.

35. 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)
where Q = design flow (cfs); n = Manning’s roughness coefficient ; Sx = pavement cross slope; S = friction slope; d = normal depth (ft).