1. Basic Hydraulics: Channels Analysis and design – I

2. Terminology

3. Manning’s equation
Manning’s equation originally developed for open channel flow (by an accountant, no less!)
Usually written as
v = (1.486/n) R2/3Sf1/2
where v = velocity (ft/sec); n = Manning’s coefficient (also called Manning’s n); R = hydraulic radius (A/P in ft); P = wetted perimeter (ft); Sf = slope of energy gradient line (ft/ft) = hL / L
Tables of n values available for various surfaces.
Rearrange Manning's equation to solve for Sf = hL / L (head loss per unit length):
(hL / L) = [Vn / (1.486 R2/3)]2

4. Hydraulic radius
The hydraulic radius (Rh) is the cross sectional area of the flow divided by the wetted perimeter. For a circular pipe flowing full, the hydraulic radius is one-fourth of the diameter. For a wide rectangular channel, the hydraulic radius is approximately equal to the depth.
A = cross–sectional area of the flowing fluid; P = wetted perimeter.

5. Slope-area method
Provides a simple relationship for relating water surface elevation to discharge at a particular channel section.
Often used for calculating tailwater at culvert outlets and storm drain outlets.
TxDOT Hydraulic Design Manual suggests using this procedure for small stream crossings or situation for which no unusual flow characteristics are anticipated.
If crossing is an important one, the Hydraulic Design Manual recommends using a “backwater method.” (More on that later.)

6. Slope-area data needs
Channel cross section: Choose a typical cross section downstream from crossing
Channel roughness
Channel slope
Use average bed slope near site
Find from surveys or topographic maps
Use Manning’s equation to calculate water surface elevation as a function of discharge

7. Hydraulic depth
The Froude Number (Fr) represents the ratio of inertial force to gravitational force and is calculated by:
where dm is the hydraulic mean depth and is defined by dm = A/T where A is the cross-sectional area of the flow and T is the channel top width at the water surface.

8. Backwater (Frontwater) Methods
Computing water surface profiles in cases more complex than slope-area situations uses the energy equation to estimate the water surface elevation at different sections from a known location.
The plot of the elevations is usually called a back-water curve for M1,M2, and S1 curves and a front-water curve for M3, S2, and S3 curves.
The variable step method is illustrated by an example to familiarize the participant with the mechanics of the method, however for practical applications use of specialized software is recommended (HEC-RAS, WSPRO, SWMM, etc.)

9. Backwater (Frontwater) Methods
Backwater methods start with the specific energy at two cross sections

10. Backwater (Frontwater) Methods
Then the bottom elevations are included as is the head loss
Therefore the total head at both sections are equal

11. Backwater (Frontwater) Methods
Next incorporate the channel bottom slopes and the energy grade line slope to replace the elevations in terms of these slopes

12. Backwater (Frontwater) Methods
Now use some calculus and section geometry to convert into discharge, area, and depth

13. Backwater (Frontwater) Methods
Now use some calculus and section geometry to convert into discharge, area, and depth

14. Backwater (Frontwater) Methods
Finally insert the substitutions and the result is an equation that relates depth taper to channel geometry and specific energy

15. Backwater (Frontwater) Methods
“Integrating” the GVF equation from a known section forward in space or backward in space produces the front- or back-water curve (water surface profile)
Gravity
Depth
Friction
Geometry
Position

16. Backwater (Frontwater) Methods
Recall the original expression of the Froude Number
Notice the area and topwidth are incorporated, some algebra and an alternate expression is

17. Backwater (Frontwater) Methods
Now we have the relationships for computing a water surface profile from some known condition
Such computation involves:
Select a location where depth is known (or assumed)
Determine the slope designation, and profile type (M1, S2, etc.)
Use the designation to decide if integration is downstream or upstream.

18. Backwater (Frontwater) Methods
Once these steps are completed, the simplest method is the constant depth change, variable distance method.
The energy equation is rearranged to solve for the spatial step as

19. Backwater (Frontwater) Methods
This form of the equation suggests the following algorithm
Start at the known section, Q must be specified.
Calculate specific energy for the starting section (section 1)
Calculate friction slope at the section (Manning’s equation solved for slope is typically used)

20. Backwater (Frontwater) Methods
This form of the equation suggests the following algorithm
Change the depth slightly, use that value as the depth at section 2
Calculate specific energy at section 2

21. Backwater (Frontwater) Methods
This form of the equation suggests the following algorithm
Calculate friction slope at section 2
Compute average friction slope for the reach

22. Backwater (Frontwater) Methods
This form of the equation suggests the following algorithm
Solve for the distance to section 2
Move to next section and repeat.

23. Backwater Example
The figure below is a backwater curve in a rectangular channel with discharge over a dam (somewhere to the right of the figure)

24. Backwater Example
The channel is 5 meters wide, bottom slope is 0.001, Manning’s n is 0.02 and channel discharge is 55.4 cubic meters per second
Our goal is to compute the water surface profile and locate the distance upstream where the water flow depth is nearly at normal depth for the channel

25. Backwater Example
The channel is 5 meters wide, bottom slope is 0.001, Manning’s n is 0.02 and channel discharge is 55.4 cubic meters per second
Select a location where depth is known (or assumed)
Use the pool on the right y= 8 meters
We will call this location x=0

26. Backwater Example
The channel is 5 meters wide, bottom slope is 0.001, Manning’s n is 0.02 and channel discharge is 55.4 cubic meters per second
Determine the slope designation, and profile type (M1, S2, etc.)
Compute normal depth in the channel from Manning’s equation, yn = 5 meters
Compute critical depth in the channel by setting Froude number to unity and solving for depth, yc = 2.3 meters

27. Backwater Example
The channel is 5 meters wide, bottom slope is 0.001, Manning’s n is 0.02 and channel discharge is 55.4 cubic meters per second
Determine the slope designation, and profile type
yn = 5 meters, yc = 2.3 meters, y0=8.
Using the slope designation the channel is Mild slope.
Using the type designation the curve will be a Type 1.
Thus this is an M1 curve.

28. Backwater Example
The channel is 5 meters wide, bottom slope is 0.001, Manning’s n is 0.02 and channel discharge is 55.4 cubic meters per second
Use the designation to decide if integration is downstream or upstream.
Curve is M1
Downstream control
Integrate upstream (-x direction)

29. Backwater Example
Use Excel to build a spreadsheet to facilitate the computations. A portion of such a sheet is shown

30. Backwater Methods
The variable step method was illustrated by an example to familiarize the participant with the mechanics of the method.
The example geometry is intentionally simple, and other simplifications are imbedded into the example.

31. Backwater Methods
For practical applications use of specialized software is recommended (HEC-RAS, WSPRO, SWMM, etc.)
These program are computationally similar to the method presented herein
These programs have user interfaces to facilitate the data entry
These programs allow spatial locations to be fixed and depths to be computed, which is far more practical for engineering application.