Transcript Document

Uniform Open Channel Flow
Manning’s Eqn for velocity or flow
1 2 /3
v R
S S.I. units
n
1.49 2 /3
v
R
S English units
n
where
n = Manning’s roughness coefficient
R = hydraulic radius = A/P
S = channel slope
Q = flow rate (cfs) = v A
Uniform Open Channel Flow – Brays B.
Brays Bayou
Concrete Channel
Normal depth is function of flow rate, and geometry
and slope. Can solve for flow rate if depth and
geometry are known.
Critical depth is used to characterize channel flows -based on addressing specific energy:
E = y + Q2/2gA2
where Q/A = q/y
Take dE/dy = (1 – q2/gy3) = 0.
For a rectangular channel bottom width b,
1. Emin = 3/2Yc for critical depth y = yc
2. yc/2 = Vc2/2g
3. yc = (Q2/gb2)1/3
Critical Flow in Open Channels
In general for any channel, 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
Optimal Channels
Non-uniform Flow
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.
H  z  y   v / 2g 
2
dH dz dy  dv 
    
dx dx dx 2 g  dx 
2
Thus,
Where
H = total energy head
z = elevation head,
v2/2g = velocity head
Replace terms for various values of S and So.
Let v = q/y = flow/unit width - solve for dy/dx
dy  q 
–S   So  1 3  since v = q / y
dx  gy 
2
1 d 2
1 d q  q
v 

 2   
2g dx
2g dx y 
g
2
2
1 dy
 3 
y dx
Given the Fr number, we can solve for
the slope of the water surface - dy/dx
Fr   v / gy
2
2
dy
So  S
So  S


2
2
dx 1  v / gy 1  Fr
Note that the eqn blows up when Fr = 1 or So = S
where
S = total energy slope
So = bed slope,
dy/dx = water surface slope
Now apply Energy Eqn. for a reach of length L
v1 
v 2 
y1    = y2    S  S o L
2 g 
2 g 
2
2
 v1  
v2 
y 1  2 g  y 2  2g 

 

L
S  S0
2
2
This Eqn is the basis for the Standard Step Method
to compute water surface profiles in open channels
Backwater Profiles - Compute Numerically
Routine Backwater Calculations
1. Select Y1 (starting depth)
2. Calculate A1
(cross sectional area)
3. Calculate P1
(wetted perimeter)
4. Calculate R1 = A1/P1
5. Calculate V1 = Q1/A1
6. Select Y2 (ending depth)
7. Calculate A2
8. Calculate P2
9. Calculate R2 = A2/P2
10. Calculate V2 = Q2/A2
Backwater Calculations (cont’d)
1. Prepare a table of values
2. Calculate Vm = (V1 + V2) / 2
3. Calculate Rm = (R1 + R2) / 2


nV
m
 Manning’s
4. Calculate S  
2

3 
1.49Rm 
2
y1  v12  y 2  v 22 

 

 2g   2g 
5. Calculate L = ∆X from first equation L 
S  S0

6. X = ∑∆Xi for each stream reach (SEE SPREADSHEET)

Watershed Hydraulics
Bridge
D
Floodplain
Tributary
C
QD
QC
Main Stream
Bridge Section
B
QB
A
Cross Sections
QA
Cross Sections
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
Critical in urban settings
288 Crossing
The Floodplain
Top Width
Floodplain Determination
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.
HEC RAS Cross Section
3-D Floodplain