Open Channel Flow

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Open 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

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Uniform 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
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Uniform Open Channel Flow Brays B.
Brays Bayou
Concrete Channel
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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
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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
Q C V C y C S0 C A C B C n
C
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1
a
z
Common Geometric Properties
Cot a z/1
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Optimal Channels - Max R and Min P
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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
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My son Eric
Critical Depth and Flow
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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 0. q const E y q2/2gy2
y
Min E Condition, q C
E
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• 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

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Y vs E
E y q2/2gy2 q const
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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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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.
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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
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Non-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
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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
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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
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Yn gt Yc
Uniform Depth
Mild Slopes where - Yn gt Yc
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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
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Backwater Profiles - Mild Slope Cases
?x
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Backwater Profiles - Compute Numerically
Compute y3 y2 y1
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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

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Backwater 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.
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100 Year Floodplain
Bridge
D
QD
Tributary
Floodplain
C
QC
Main Stream
Bridge Section
B
QB
A
QA
Cross Sections
Cross Sections
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The Floodplain
Top Width
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Floodplain Determination
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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.

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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.
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HEC - 2
Orientation - looking downstream
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River
Multiple Cross Sections
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HEC RAS (River Analysis System, 1995)
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HEC RAS Bridge CS
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HEC RAS Input Window
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HEC RAS Profile Plots
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3-D Floodplain
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HEC RAS Cross Section Output Table
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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