1. CH 7 - Open Channel Flow Brays Bayou Concrete Channel Uniform & Steady
Non-uniform and Steady
Non-Uniform and Unsteady
Concrete Channel

2. Open Channel Flow
Uniform & Steady - Manning’s Eqn in prismatic channel - Q, v, y, A, P, B, S and roughness are all constant
Critical flow - Specific Energy Eqn (Froude No.)
Non-uniform flow - gradually varied flow (steady flow) - determination of floodplains
Unsteady and Non-uniform flow - flood waves

3. Uniform Open Channel Flow
Manning’s Eqn for velocity or flow
where
n = Manning’s roughness coefficient
R = hydraulic radius = A/P
S = channel slope
Q = flow rate (cfs) = v A

4. Normal depth is function of flow rate, and
geometry and slope. One usually solves for normal
depth or width given flow rate and slope information B b

5. Normal depth implies that flow rate, velocity, depth,
bottom slope, area, top width, and roughness remain
constant within a prismatic channel as shown below
UNIFORM FLOW B y
Q = Const
V = C
y = C
S0 = C
A = C
B = C
n = C V A So

6. Common Geometric Properties
1 + Z2 1 a z
Common Geometric Properties
Cot a = z/1

7. Optimal Channels - Max R and Min P

8. Water surface slope = Bed slope = dy/dz = dz/dx
H = z + y + v2/2g = Total Energy
E = y + v2/2g = Specific Energy
 often near 1.0 for most channels
Energy Coeff.
a = S vi2 Qi
V2 QT H
Uniform Flow
Energy slope = Bed slope or dH/dx = dz/dx
Water surface slope = Bed slope = dy/dz = dz/dx
Velocity and depth remain constant with x

9. Critical Depth and Flow
My son Eric
Critical Depth and Flow

10. Critical depth is used to characterize channel flows -- based on addressing specific energy E = y + v2/2g :
E = y + Q2/2gA2 where Q/A = q/y and q = Q/b
Take dE/dy = (1 – q2/gy3) and set = q = const
E = y + q2/2gy2 y
Min E Condition, q = C E

11. For a rectangular channel bottom width b,
Solving dE/dy = (1 – q2/gy3) and set = 0.
For a rectangular channel bottom width b,
1. Emin = 3/2Yc for critical depth y = yc
yc/2 = Vc2/2g
yc = (Q2/gb2)1/3
Froude No. = v/(gy)1/2
We use the Froude No. to characterize critical flows

12. Y vs E
E = y + q2/2gy2
q = const

14. Critical Flow in Open Channels
In general for any channel shape, B = top width
(Q2/g) = (A3/B) at y = yc
Finally Fr = v/(gy)1/2 = Froude No.
Fr = 1 for critical flow
Fr < 1 for subcritical flow
Fr > 1 for supercritical flow
Critical Flow in Open Channels

16. Non-Uniform Open Channel Flow
With natural or man-made channels, the shape, size, and slope may vary along the stream length, x. In addition, velocity and flow rate may also vary with x. Non-uniform flow can be best approximated using a numerical method called the Standard Step Method.

17. Non-Uniform Computations
Typically start at downstream end with known water level - yo.
Proceed upstream with calculations using new water levels as they
are computed.
The limits of calculation range between normal and critical depths.
In the case of mild slopes, calculations start downstream.
In the case of steep slopes, calculations start upstream.
Calc. Q
Mild Slope

18. Non-Uniform Open Channel Flow
Let’s evaluate H, total energy, as a function of x.
Take derivative,
Where H = total energy head
z = elevation head,
v2/2g = velocity head

19. Replace terms for various values of S and So
Replace terms for various values of S and So. Let v = q/y = flow/unit width - solve for dy/dx, the slope of the water surface

20. Given the Froude number, we can simplify and solve for dy/dx as a fcn of measurable parameters
*Note that the eqn blows up when Fr = 1 and goes to
   zero if So = S, the case of uniform OCF.
where S = total energy slope
So = bed slope,
dy/dx = water surface slope

22. Mild Slopes where - Yn > Yc
Uniform Depth
Mild Slopes where - Yn > Yc

23. Now apply Energy Eqn. for a reach of length L
This Eqn is the basis for the Standard Step Method Solve for L = Dx to compute water surface profiles as function of y1 and y2, v1 and v2, and S and S0

24. Backwater Profiles - Mild Slope Casesx

25. Backwater Profiles - Compute Numerically
y y2 y1

26. Routine Backwater Calculations
Select Y1 (starting depth)
Calculate A1 (cross sectional area)
Calculate P1 (wetted perimeter)
Calculate R1 = A1/P1
Calculate V1 = Q1/A1
Select Y2 (ending depth)
Calculate A2
Calculate P2
Calculate R2 = A2/P2
Calculate V2 = Q2/A2

27. Backwater Calculations (cont’d)
Prepare a table of values
Calculate Vm = (V1 + V2) / 2
Calculate Rm = (R1 + R2) / 2
Calculate Manning’s
Calculate L = ∆X from first equation
X = ∑∆Xi for each stream reach (SEE SPREADSHEETS)
Energy Slope Approx.

28. 100 Year Floodplain Bridge D QD Floodplain C QC Bridge Section B QB A
Tributary
Floodplain C QC
Main Stream
Bridge Section B QB A QA
Cross Sections
Cross Sections

29. The Floodplain
Top Width

30. Floodplain Determination

31. The Woodlands
The Woodlands planners wanted to design the community to withstand a 100-year storm.
In doing this, they would attempt to minimize any changes to the existing, undeveloped floodplain as development proceeded through time.

32. HEC RAS (River Analysis System, 1995)
HEC RAS or (HEC-2)is a computer model designed for
natural cross sections in natural rivers. It solves the
governing equations for the standard step method,
generally in a downstream to upstream direction. It can
Also handle the presence of bridges, culverts, and
variable roughness, flow rate, depth, and velocity.

33. HEC - 2
Orientation - looking downstream

38. River
Multiple Cross Sections

39. HEC RAS (River Analysis System, 1995)

40. HEC RAS Bridge CS

41. HEC RAS Input Window

42. HEC RAS Profile Plots

43. 3-D Floodplain

44. HEC RAS Cross Section Output Table

45. Brays Bayou-Typical Urban System
Bridges cause unique problems in hydraulics
Piers, low chords, and top of road is considered
Expansion/contraction can cause hydraulic losses
Several cross sections are needed for a bridge
288 Bridge causes a 2 ft
Backup at TMC and is being replaced by TXDOT
288 Crossing