1. Uniform Open Channel Flow – Ch 7
Uniform flow - Manning’s Eqn in a 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

2. 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
Q = C
V = C
y = C
S0 = C
A = C
b = C
n = C
Wetted perimeter P remains constant A P

3. Uniform Flow Force Balance

4. Uniform Open Channel Flow – Chezy and Manning’s Eqn.
Hydrostatic, Friction, and Weight all act together.
Derivation of these eqns requires a force balance (x)
Actual forces (F = hydrostatic) are summed across C.V.

5. Chezy and Manning’s Eqn.
Since hydrostatic forces are equal, and
Slopes are very mild
2. Define R = A/P, the hydraulic radius

6. Chezy and Manning’s Eqn.
Finally, we can equate the two eqns for shear stress
C = Chezy Coefficient (1768) in Paris
Manning was an Irish Eng and 1889 developed his EQN.

7. Manning’s OCF Equation
Manning’s Eqn for velocity or peak flow rate
where
n = Manning’s roughness coefficient
R = hydraulic radius = A/P
S = channel slope

8. Uniform Open Channel Flow – Brays B.
Brays Bayou
Concrete Channel

9. 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

10. Optimal Channels - Max R and Min P

11. Water surface slope = Bed slope = dy/dz = dz/dx
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 H

12. Bernoulli Flow Eqn H

13. Critical Depth and Flow
My son Eric

15. 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

16. 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

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

18. 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