Transcript ME33: Fluid Flow Lecture 1: Information and Introduction

Chapter 13: Open Channel Flow
Eric G. Paterson
Department of Mechanical and Nuclear Engineering
The Pennsylvania State University
Spring 2005
Note to Instructors
These slides were developed1, during the spring semester 2005, as a teaching aid
for the undergraduate Fluid Mechanics course (ME33: Fluid Flow) in the Department of
Mechanical and Nuclear Engineering at Penn State University. This course had two
sections, one taught by myself and one taught by Prof. John Cimbala. While we gave
common homework and exams, we independently developed lecture notes. This was
also the first semester that Fluid Mechanics: Fundamentals and Applications was
used at PSU. My section had 93 students and was held in a classroom with a computer,
projector, and blackboard. While slides have been developed for each chapter of Fluid
Mechanics: Fundamentals and Applications, I used a combination of blackboard and
electronic presentation. In the student evaluations of my course, there were both positive
and negative comments on the use of electronic presentation. Therefore, these slides
should only be integrated into your lectures with careful consideration of your teaching
style and course objectives.
Eric Paterson
Penn State, University Park
August 2005
1 This
Chapter was not covered in our class. These slides have been developed at the request of McGraw-Hill
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Chapter 13: Open Channel Flow
Objectives
Understand how flow in open channels
differs from flow in pipes
Learn the different flow regimes in open
channels and their characteristics
Predict if hydraulic jumps are to occur
during flow, and calculate the fraction of
energy dissipated during hydraulic jumps
Learn how flow rates in open channels
are measured using sluice gates and
weirs
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Chapter 13: Open Channel Flow
Classification of Open-Channel Flows
Open-channel flows are
characterized by the
presence of a liquid-gas
interface called the free
surface.
Natural flows: rivers,
creeks, floods, etc.
Human-made systems:
fresh-water aqueducts,
irrigation, sewers,
drainage ditches, etc.
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Chapter 13: Open Channel Flow
Classification of Open-Channel Flows
In an open channel,
Velocity is zero on bottom and sides of
channel due to no-slip condition
Velocity is maximum at the midplane of the
free surface
In most cases, velocity also varies in the
streamwise direction
Therefore, the flow is 3D
Nevertheless, 1D approximation is made with
good success for many practical problems.
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Chapter 13: Open Channel Flow
Classification of Open-Channel Flows
Flow in open channels is
also classified as being
uniform or nonuniform,
depending upon the
depth y.
Uniform flow (UF)
encountered in long
straight sections where
head loss due to friction
is balanced by elevation
drop.
Depth in UF is called
normal depth yn
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Chapter 13: Open Channel Flow
Classification of Open-Channel Flows
Obstructions cause the flow depth to vary.
Rapidly varied flow (RVF) occurs over a short distance
near the obstacle.
Gradually varied flow (GVF) occurs over larger distances
and usually connects UF and RVF.
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Chapter 13: Open Channel Flow
Classification of Open-Channel Flows
Like pipe flow, OC flow can be laminar,
transitional, or turbulent depending upon the
value of the Reynolds number
Where
 = density,  = dynamic viscosity,  = kinematic viscosity
V = average velocity
Rh = Hydraulic Radius = Ac/p
Ac = cross-section area
P = wetted perimeter
Note that Hydraulic Diameter was defined in pipe flows as
Dh = 4Ac/p = 4Rh (Dh is not 2Rh, BE Careful!)
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Chapter 13: Open Channel Flow
Classification of Open-Channel Flows
The wetted perimeter
does not include the
free surface.
Examples of Rh for
common geometries
shown in Figure at the
left.
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Chapter 13: Open Channel Flow
Froude Number and Wave Speed
OC flow is also
classified by the
Froude number
Resembles
classification of
compressible flow
with respect to Mach
number
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Chapter 13: Open Channel Flow
Froude Number and Wave Speed
Critical depth yc occurs at Fr = 1
At low flow velocities (Fr < 1)
Disturbance travels upstream
y > yc
At high flow velocities (Fr > 1)
Disturbance travels downstream
y < yc
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Chapter 13: Open Channel Flow
Froude Number and Wave Speed
Important parameter in study
of OC flow is the wave speed
c0, which is the speed at
which a surface disturbance
travels through the liquid.
Derivation of c0 for shallowwater
Generate wave with plunger
Consider control volume (CV)
which moves with wave at c0
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Chapter 13: Open Channel Flow
Froude Number and Wave Speed
Continuity equation (b = width)
Momentum equation
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Chapter 13: Open Channel Flow
Froude Number and Wave Speed
Combining the momentum and continuity relations and
rearranging gives
For shallow water, where y << y,
Wave speed c0 is only a function of depth
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Chapter 13: Open Channel Flow
Specific Energy
Total mechanical energy of the
liquid in a channel in terms of
heads
z is the elevation head
y is the gage pressure head
V2/2g is the dynamic head
Taking the datum z=0 as the
bottom of the channel, the
specific energy Es is
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Chapter 13: Open Channel Flow
Specific Energy
For a channel with constant
width b,
Plot of Es vs. y for constant V
and b
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Chapter 13: Open Channel Flow
Specific Energy
This plot is very useful
Easy to see breakdown of Es into pressure (y)
and dynamic (V2/2g) head
Es   as y  0
Es  y for large y
Es reaches a minimum called the critical point.
There is a minimum Es required to support the
given flow rate.
Noting that Vc = sqrt(gyc)
For a given Es > Es,min, there are two different
depths, or alternating depths, which can occur
for a fixed value of Es
A small change in Es near the critical point
causes a large difference between alternate
depths and may cause violent fluctuations in flow
level. Operation near this point should be
avoided.
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Chapter 13: Open Channel Flow
Continuity and Energy Equations
1D steady continuity equation can
be expressed as
1D steady energy equation
between two stations
Head loss hL is expressed as in
pipe flow, using the friction factor,
and either the hydraulic diameter
or radius
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Chapter 13: Open Channel Flow
Continuity and Energy Equations
The change in elevation head can be written in terms
of the bed slope 
Introducing the friction slope Sf
The energy equation can be written as
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Chapter 13: Open Channel Flow
Uniform Flow in Channels
Uniform depth occurs
when the flow depth (and
thus the average flow
velocity) remains
constant
Common in long straight
runs
Flow depth is called
normal depth yn
Average flow velocity is
called uniform-flow
velocity V0
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Chapter 13: Open Channel Flow
Uniform Flow in Channels
Uniform depth is maintained as long as the slope,
cross-section, and surface roughness of the channel
remain unchanged.
During uniform flow, the terminal velocity reached, and
the head loss equals the elevation drop
We can the solve for velocity (or flow rate)
Where C is the Chezy coefficient. f is the friction
factor determined from the Moody chart or the
Colebrook equation
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Chapter 13: Open Channel Flow
Best Hydraulic Cross Sections
Best hydraulic cross
section for an open
channel is the one
with the minimum
wetted perimeter for a
specified cross
section (or maximum
hydraulic radius Rh)
Also reflects economy
of building structure
with smallest
perimeter
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Chapter 13: Open Channel Flow
Best Hydraulic Cross Sections
Example: Rectangular Channel
Cross section area, Ac = yb
Perimeter, p = b + 2y
Solve Ac for b and substitute
Taking derivative with respect to
To find minimum, set derivative to zero
Best rectangular channel has
a depth 1/2 of the width
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Chapter 13: Open Channel Flow
Best Hydraulic Cross Sections
Same analysis can be
performed for a trapezoidal
channel
Similarly, taking the derivative
of p with respect to q, shows
that the optimum angle is
For this angle, the best flow
depth is
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Chapter 13: Open Channel Flow
Gradually Varied Flow
In GVF, y and V vary slowly,
and the free surface is stable
In contrast to uniform flow, Sf 
S0. Now, flow depth reflects
the dynamic balance between
gravity, shear force, and
inertial effects
To derive how how the depth
varies with x, consider the total
head
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Chapter 13: Open Channel Flow
Gradually Varied Flow
Take the derivative of H
Slope dH/dx of the energy line is equal to negative of the
friction slope
Bed slope has been defined
Inserting both S0 and Sf gives
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Chapter 13: Open Channel Flow
Gradually Varied Flow
Introducing continuity equation, which can be written as
Differentiating with respect to x gives
Substitute dV/dx back into equation from previous slide,
and using definition of the Froude number gives a
relationship for the rate of change of depth
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Chapter 13: Open Channel Flow
Gradually Varied Flow
This result is important. It
permits classification of liquid
surface profiles as a function of
Fr, S0, Sf, and initial conditions.
Bed slope S0 is classified as
Steep : yn < yc
Critical : yn = yc
Mild : yn > yc
Horizontal : S0 = 0
Adverse : S0 < 0
Initial depth is given a number
1 : y > yn
2 : yn < y < yc
3 : y < yc
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Chapter 13: Open Channel Flow
Gradually Varied Flow
12 distinct
configurations for
surface profiles in
GVF.
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Chapter 13: Open Channel Flow
Gradually Varied Flow
Typical OC system
involves several
sections of different
slopes, with
transitions
Overall surface profile
is made up of
individual profiles
described on previous
slides
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Chapter 13: Open Channel Flow
Rapidly Varied Flow and Hydraulic
Jump
Flow is called rapidly
varied flow (RVF) if the
flow depth has a large
change over a short
distance
Sluice gates
Weirs
Waterfalls
Abrupt changes in cross
section
Often characterized by
significant 3D and
transient effects
Backflows
Separations
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Chapter 13: Open Channel Flow
Rapidly Varied Flow and Hydraulic
Jump
Consider the CV
surrounding the
hydraulic jump
Assumptions
1. V is constant at sections
(1) and (2), and 1 and 2
1
2. P = gy
3. w is negligible relative to
the losses that occur
during the hydraulic jump
4. Channel is wide and
horizontal
5. No external body forces
other than gravity
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Chapter 13: Open Channel Flow
Rapidly Varied Flow and Hydraulic
Jump
Continuity equation
X momentum equation
Substituting and simplifying
Quadratic equation for y2/y1
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Chapter 13: Open Channel Flow
Rapidly Varied Flow and Hydraulic
Jump
Solving the quadratic equation and keeping only the
positive root leads to the depth ratio
Energy equation for this section can be written as
Head loss associated with hydraulic jump
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Chapter 13: Open Channel Flow
Rapidly Varied Flow and Hydraulic
Jump
Often, hydraulic jumps
are avoided because they
dissipate valuable energy
However, in some cases,
the energy must be
dissipated so that it
doesn’t cause damage
A measure of
performance of a
hydraulic jump is its
fraction of energy
dissipation, or energy
dissipation ratio
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Chapter 13: Open Channel Flow
Rapidly Varied Flow and Hydraulic
Jump
Experimental
studies
indicate that
hydraulic
jumps can be
classified into
5 categories,
depending
upon the
upstream Fr
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Chapter 13: Open Channel Flow
Flow Control and Measurement
Flow rate in pipes and ducts is
controlled by various kinds of
valves
In OC flows, flow rate is controlled
by partially blocking the channel.
Weir : liquid flows over device
Underflow gate : liquid flows under
device
These devices can be used to
control the flow rate, and to
measure it.
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Chapter 13: Open Channel Flow
Flow Control and Measurement
Underflow Gate
Underflow gates are located at
the bottom of a wall, dam, or
open channel
Outflow can be either free or
drowned
In free outflow, downstream
flow is supercritical
In the drowned outflow, the
liquid jet undergoes a hydraulic
jump. Downstream flow is
subcritical.
Free outflow
Drowned outflow
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Chapter 13: Open Channel Flow
Flow Control and Measurement
Underflow Gate
Schematic of flow depth-specific
energy diagram for flow through
underflow gates
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Es remains constant for
idealized gates with
negligible frictional effects
Es decreases for real
gates
Downstream is
supercritical for free
outflow (2b)
Downstream is subcritical
for drowned outflow (2c)
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Chapter 13: Open Channel Flow
Flow Control and Measurement
Overflow Gate
Specific energy over a bump at station 2 Es,2 can be
manipulated to give
This equation has 2 positive solutions, which depend
upon upstream flow.
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Chapter 13: Open Channel Flow
Flow Control and Measurement
Broad-Crested Weir
Flow over a
sufficiently high
obstruction in an open
channel is always
critical
When placed
intentionally in an
open channel to
measure the flow
rate, they are called
weirs
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Chapter 13: Open Channel Flow
Flow Control and Measurement
Sharp-Crested V-notch Weirs
Vertical plate placed in a
channel that forces the
liquid to flow through an
opening to measure the
flow rate
Upstream flow is
subcritical and becomes
critical as it approaches
the weir
Liquid discharges as a
supercritical flow stream
that resembles a free jet
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Chapter 13: Open Channel Flow
Flow Control and Measurement
Sharp-Crested V-notch Weirs
Flow rate equations can be derived using energy
equation and definition of flow rate, and experimental
for determining discharge coefficients
Sharp-crested weir
V-notch weir
where Cwd typically ranges between 0.58 and 0.62
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Chapter 13: Open Channel Flow