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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

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

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.

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.

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

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.

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!)

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.

Froude Number and Wave Speed

- OC flow is also classified by the Froude number
- Resembles classification of compressible flow

with respect to Mach number

Froude Number and Wave Speed

- Critical depth yc occurs at Fr 1
- At low flow velocities (Fr lt 1)
- Disturbance travels upstream
- y gt yc
- At high flow velocities (Fr gt 1)
- Disturbance travels downstream
- y lt yc

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 shallow-water
- Generate wave with plunger
- Consider control volume (CV) which moves with

wave at c0

Froude Number and Wave Speed

- Continuity equation (b width)
- Momentum equation

Froude Number and Wave Speed

- Combining the momentum and continuity relations

and rearranging gives - For shallow water, where ?y ltlt y,
- Wave speed c0 is only a function of depth

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 z0 as the bottom of the

channel, the specific energy Es is

Specific Energy

- For a channel with constant width b,
- Plot of Es vs. y for constant V and b

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 gt 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.

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

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

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

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

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

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

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

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

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

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

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 lt yc
- Critical yn yc
- Mild yn gt yc
- Horizontal S0 0
- Adverse S0 lt 0
- Initial depth is given a number
- 1 y gt yn
- 2 yn lt y lt yc
- 3 y lt yc

Gradually Varied Flow

- 12 distinct configurations for surface profiles

in GVF.

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

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

Rapidly Varied Flow and Hydraulic Jump

- Consider the CV surrounding the hydraulic jump
- Assumptions
- V is constant at sections (1) and (2), and ?1 and

?2 ? 1 - P ?gy
- ?w is negligible relative to the losses that

occur during the hydraulic jump - Channel is wide and horizontal
- No external body forces other than gravity

Rapidly Varied Flow and Hydraulic Jump

- Continuity equation
- X momentum equation
- Substituting and simplifying

Quadratic equation for y2/y1

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

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 doesnt cause damage - A measure of performance of a hydraulic jump is

its fraction of energy dissipation, or energy

dissipation ratio

Rapidly Varied Flow and Hydraulic Jump

- Experimental studies indicate that hydraulic

jumps can be classified into 5 categories,

depending upon the upstream Fr

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.

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

Flow Control and Measurement Underflow Gate

Schematic of flow depth-specific energy diagram

for flow through underflow gates

- 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)

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.

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

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

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