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

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

Uniform Open Channel Flow Brays B.

Brays Bayou

Concrete Channel

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

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

1

a

z

Common Geometric Properties

Cot a z/1

Optimal Channels - Max R and Min P

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

My son Eric

Critical Depth and Flow

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

- 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

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.

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

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

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

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

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

Backwater Profiles - Mild Slope Cases

?x

Backwater Profiles - Compute Numerically

Compute y3 y2 y1

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

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.

100 Year Floodplain

Bridge

D

QD

Tributary

Floodplain

C

QC

Main Stream

Bridge Section

B

QB

A

QA

Cross Sections

Cross Sections

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 (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.

HEC - 2

Orientation - looking downstream

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River

Multiple Cross Sections

HEC RAS (River Analysis System, 1995)

HEC RAS Bridge CS

HEC RAS Input Window

HEC RAS Profile Plots

3-D Floodplain

HEC RAS Cross Section Output Table

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