Title: Open Channel Flow
1Open Channel Flow
- Uniform flow - Mannings 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
2Uniform Open Channel Flow
Mannings Eqn for velocity or flow
where n Mannings roughness coefficient R
hydraulic radius A/P S channel slope
Q flow rate (cfs) v A
3Uniform Open Channel Flow Brays B.
Brays Bayou
Concrete Channel
4Normal 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
5Normal 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
Q C V C y C S0 C A C B C n
C
61
a
z
Common Geometric Properties
Cot a z/1
7Optimal Channels - Max R and Min P
8H 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
9My son Eric
Critical Depth and Flow
10Critical 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 0. q const E y q2/2gy2
y
Min E Condition, q C
E
11-
- 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
12Y vs E
E y q2/2gy2 q const
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14 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 lt 1 for subcritical flow Fr gt 1 for
supercritical flow
Critical Flow in Open Channels
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16Non-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.
17Non-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
18Non-Uniform Open Channel Flow
Lets evaluate H, total energy, as a function of
x.
Take derivative,
Where H total energy head z elevation
head, ?v2/2g velocity head
19Replace 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
20Given 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
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22Yn gt Yc
Uniform Depth
Mild Slopes where - Yn gt Yc
23Now 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
24Backwater Profiles - Mild Slope Cases
?x
25Backwater Profiles - Compute Numerically
Compute y3 y2 y1
26Routine 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
27Backwater Calculations (contd)
- Prepare a table of values
- Calculate Vm (V1 V2) / 2
- Calculate Rm (R1 R2) / 2
- Calculate Mannings
- Calculate L ?X from first equation
- X ??Xi for each stream reach (SEE
SPREADSHEETS)
Energy Slope Approx.
28100 Year Floodplain
Bridge
D
QD
Tributary
Floodplain
C
QC
Main Stream
Bridge Section
B
QB
A
QA
Cross Sections
Cross Sections
29The Floodplain
Top Width
30Floodplain Determination
31The 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.
32HEC 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.
33HEC - 2
Orientation - looking downstream
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38River
Multiple Cross Sections
39HEC RAS (River Analysis System, 1995)
40HEC RAS Bridge CS
41HEC RAS Input Window
42HEC RAS Profile Plots
433-D Floodplain
44HEC RAS Cross Section Output Table
45Brays 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