1. Open channel hydraulics
CE 3372 – Lecture 13

2. Outline Flow Terminology Energy Equation Critical Depth/Flow
Flow Profiles for GVF
Manning’s Equation
Understand relationships

3. Open Channel Design Concepts
Interest to engineers:
Water surface elevation (WSE) (minimize impact/reduce floods)
Discharge –Depth relationships
Channel design

4. Conduits whose upper boundary of flow is the liquid surface.
Open Channels
Conduits whose upper boundary of flow is the liquid surface.
Canals, streams, bayous, rivers are common examples of open channels.
Storm sewers and sanitary sewers are special cases of open channels; in some parts of a sewer system these channels may be operated as pressurized pipes, either intentionally or accidentally.
Similar to closed, there’s multiple dimensions, but 1D is simplest to analyse.

5. TYPES OF FLOW
Steady Flow – flow, depth and velocity may differ from point to point but remain constant over time
Unsteady Flow – flow, depth, and velocity is a function of time
Uniform Flow – occurs in prismatic channels when flow depths are equal no change in velocity within the channel: Q, y, A, S are all constant
Non-uniform Flow – velocity is not the same at every point
Temporal
the friction slope Sf (slope of the EGL) would be the same as the bottom slope S0.
Draw on board the diagram
Steady+unsteady are temporal with time!!
And uniform is spatial!!
Spatial

7. Open Channel Nomenclature
Flow depth is the depth of flow at a cross-section measured from the channel bottom. y

8. Open Channel Nomenclature
Elevation of the channel bottom is the elevation at a cross- section measured from a reference datum (typically MSL). y z
Datum

9. Open Channel Nomenclature
Slope of the channel bottom, So, is called the topographic slope or channel slope. y z So 1
Datum

10. Open Channel Nomenclature
Slope of the water surface is the slope of the HGL, or slope of WSE (water surface elevation).
HGL
Swse y 1 z So 1
Datum

11. Open Channel Nomenclature
Slope of the energy grade line (EGL) is called the energy or friction slope.
EGL
HGL Sf
V2/2g 1
Q=VA
Swse y 1 z So 1
Datum

12. Steady NON-uniform Flow
Based on cross-sections:
Section 1 is upstream
Section 2 is downstream
Sketch of steady flow in a channel

13. Steady Flow The energy grade line (EGL) is: z_head + P_head + v_head
The hydraulic grade line (HGL) is at the water surface
Energy grade line (EGL)
Hydraulic grade line (HGL)
velocity
Profile grade line is the channel bottom
Sketch of steady flow in a channel
pressure (depth)
The head loss is depicted as the difference
between a horizontal zero-loss energy grade line
and the energy grade line
elevation
PRESSURE IS CALLED DEPTH!!!!!!!

14. Like closed conduits, the various terms are part of mass, momentum, and energy balances.
Unlike closed conduits, geometry is flow dependent, and the pressure term is replaced with flow depth.

15. Energy relationships Energy equation for closed conduits
Energy equation for open conduits
There is usually a kinetic energy correction factor (alpha) before the velocity terms but we assume that = 1 for turbulent flow

16. Specific energy
The sum of the depth of flow + velocity head (Head relative to the channel bottom)
For a given discharge, the SE can be calculated for various flow depths including critical depth
SE is important to find flow depths

17. Occurs when dE/dy = 0 and Fr = 1
Critical depth
Depth of flow for a given discharge, where the specific energy is at a minimum
Occurs when dE/dy = 0 and Fr = 1
It is important to calculate yc in order to determine if the flow in the channel will be subcritical or supercritical
Can be found through Specific Energy Diagram
What?
Why is it important? How to find it?de/dy = 0 = KE = PE

18. Plug & chug. Solve for y 3 roots –1 negative = 2 depths
There are 3 flows here. Increase to the right
top part of the curve approaches the E = y line and the bottom part of the curve tends toward the x-axis.
critical energy or minimum energy, Ec and the corresponding critical depth value, yc
The critical depth has a Froude number equal to one and corresponds to the minimum energy a flow can possess for a given discharge.
IMP NOTE: Alternate depths for a single specific energy
critical depth value mentioned in the E–y diagram section above is mathematically represented by the ratio of the fluid velocity to the velocity of a small amplitude gravity wave.
Alternate Depths:
Plug & chug. Solve for y 3 roots –1 negative = 2 depths
A = By y
Q=qy where q is the discharge/unit width of channel B

19. Open channel flow is also classified by the Froude number
Open channel flows
Open channel flow is also classified by the Froude number
Critical depth, yc occurs at Fr = 1
critical depth value mentioned in the E–y diagram section above is mathematically represented by the ratio of the fluid velocity to the velocity of a small amplitude gravity wave.

20. Open channel flows
Subcritical flow
Low velocities, Fr < 1
Disturbance travels upstream
y > yc
Supercritical flow
High velocities, Fr > 1
Disturbances travel downstream
y < yc
Arbitrary cross sections are handled by numerical integration, regular geometries are considerably simpler.

21. Arbitrary cross-section
Critical Flow T dy
Has a minimum at yc y A dA P
dE/dy for a minimum, gradient must vanish.
Yellow = variation of energy with respect to depth in discharge form
More general definition of Fr

22. Critical Flow – Rectangular channelyc Ac
Only for rectangular channels!
Given the depth we can find the flow!

23. Critical Flow: Rectangular Channels
velocity head =
0.5 (depth)

24. Open channel flows
Similar to pipe flow, open channel flow can be classified into
which is dependent on Reynolds number
Arbitrary cross sections are handled by numerical integration, regular geometries are considerably simpler.
Imagine being a single drop of water

25. Open channel flows
Where V = average velocity
Rh = hydraulic radius
v = kinematic viscosity
Arbitrary cross sections are handled by numerical integration, regular geometries are considerably simpler.

26. Area represents cross sectional area of the fluid
Wetted perimeter does not include the free surface
These equations are found err where. Just know how to get them.

27. Rectangular Conduit

28. Trapezoidal Channel common geometry
Engineered (improved) natural channels are reasonably well approximated by trapezoidal equations
the geometry is important in drainage engineering

29. Circular Conduit
Sweep angle definition matters, book uses 2a.

30. Varied Flow
Gradually varied flow – change in flow depth moving upstream or downstream is gradual
Rapidly varied flow – change in flow depth occurs over a very short distance
Ex: waterfall, hydraulic jumps, etc.
RVF is outside the scope of this course.
Assumptions of velocity on the sides?

31. Gradually Varied Flow
Equation relating slope of water surface, channel slope, and energy slope:
Discharge and
Section Geometry
Gives a relationship for rate oof chNge of depth
So and Sf are positive when sloping down in direction of flow
y is measured from channel bottom
dy/dx =0 means water depth is _______
Variation of
Water Surface Elevation
Discharge and
Section Geometry

32. Gradually Varied Flow
Procedure to find water surface profile is to integrate the depth taper with distance:

33. yc < y < yn OR yn < y < yn
Flow profiles
SLOPE
DEPTH RELATIONSHIP
Steep
yn < yc
Critical
yn = yc
Mild
yn > yc
Horizontal
S0 = 0
Adverse
S0 < 0
PROFILE TYPE
DEPTH RELATIONSHIP
Type-1
y > yc AND y > yn
Type -2
yc < y < yn OR yn < y < yn
Type -3
y < yc AND y < yn

34. Purpose?

35. MANning equation (1891)
Depth-Discharge Calculator for any open channel implements Manning's equation
The equation is the U.S. customary version
A drainage engineer in the US should memorize this equation!

36. Values of Manning n
n = f(surface roughness, channel irregularity, stage...)
d in ft
d = median size of bed material

37. Summary of open channels
All the complications of pipe flow plus ______________________
Importance of Froude Number
Fr>1 decrease in E gives increase in y
Fr<1 decrease in E gives decrease in y
Fr=1 standing waves (also min E given Q)
free surface location