uniform open channel flow manning’s eqn for velocity or flow where n = manning’s roughness...
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Uniform Open Channel Flow
Manning’s Eqn for velocity or flow
€
v =
1
n
R
2 / 3
S S.I. units
v =
1 . 49
n
R
2 / 3
S English units
where n = Manning’s roughness
coefficient R = hydraulic radius = A/PS = channel slope
Q = flow rate (cfs) = v A
Brays Bayou
Concrete Channel
Uniform Open Channel Flow – Brays B.
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
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 flowFr < 1 for subcritical flowFr > 1 for supercritical flow
Critical Flow in Open Channels
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 + α v
2
/ 2 g
( )
dH
dx
=
dz
dx
+
dy
dx
+
α
2 g
dv
2
dx
⎛
⎝
⎜
⎞
⎠
⎟
Where H = total energy headz = elevation head,
αv2/2g = velocity head
Thus,
Replace terms for various values of S and So. Let v = q/y = flow/unit width - solve for dy/dx
– S = − S
o
+
dy
dx
1 −
q
2
gy
3
⎡
⎣
⎢
⎤
⎦
⎥
since v = q / y
1
2 g
d
dx
v
2
[ ]=
1
2 g
d
dx
q
2
y
2
⎡
⎣
⎢
⎤
⎦
⎥
= −
q
2
g
1
y
3
⎡
⎣
⎢
⎤
⎦
⎥
dy
dx
Given the Fr number, we can solve for the slope of the water surface - dy/dx
Fr
2
= v
2
/ gy
( )
dy
dx
=
S
o
− S
1 − v
2
/ gy
=
S
o
− S
1 − Fr
2
where S = total energy slopeSo = bed slope, dy/dx = water surface slope
Note that the eqn blows up when Fr = 1 or So = S
Now apply Energy Eqn. for a reach of length L
y
1
+
v
1
2
2 g
⎡
⎣
⎢
⎤
⎦
⎥
= y
2
+
v
2
2
2 g
⎡
⎣
⎢
⎤
⎦
⎥
+ S − S
o
( )L
L =
y
1
+
v
1
2
2 g
⎡
⎣
⎢
⎤
⎦
⎥
− y
2
+
v
2
2
2 g
⎡
⎣
⎢
⎤
⎦
⎥
S − S
0
This Eqn is the basis for the Standard Step Method to compute water surface profiles in open channels
Backwater Profiles - Compute Numerically
Routine Backwater Calculations1. 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
4. Calculate Manning’s
5. Calculate L = ∆X from first equation
6. X = ∑∆Xi for each stream reach (SEE SPREADSHEET)
€
S =nVm
1.49Rm
23
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
2
€
L =
y1 + v12
2g
⎛
⎝ ⎜
⎞
⎠ ⎟−
y2 + v22
2g
⎛
⎝ ⎜
⎞
⎠ ⎟
S − S0
Watershed Hydraulics
Main Stream
Tributary
Cross Sections
Cross Sections
A
B
C
D
QA
QD
QC
QB
Bridge Section
Bridge
Floodplain
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 settings288 Crossing
The Floodplain
Top Width
Floodplain Determination
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.
The Woodlands
HEC RAS Cross Section
3-D Floodplain