Chapter 13: Open Channel Flow

1
Chapter 13 Open Channel Flow

• Eric G. Paterson
• Department of Mechanical and Nuclear Engineering
• The Pennsylvania State University
• Spring 2005

2
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
3
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

4
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.

5
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.

6
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

7
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.

8
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!)

9
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.

10
Froude Number and Wave Speed

• OC flow is also classified by the Froude number
• Resembles classification of compressible flow
  with respect to Mach number

11
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

12
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

13
Froude Number and Wave Speed

• Continuity equation (b width)
• Momentum equation

14
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

15
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

16
Specific Energy

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

17
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.

18
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

19
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

20
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

21
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

22
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

23
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
24
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

25
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

26
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

27
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

28
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

29
Gradually Varied Flow

• 12 distinct configurations for surface profiles
  in GVF.

30
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

31
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

32
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

33
Rapidly Varied Flow and Hydraulic Jump

• Continuity equation
• X momentum equation
• Substituting and simplifying

Quadratic equation for y2/y1
34
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

35
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

36
Rapidly Varied Flow and Hydraulic Jump

• Experimental studies indicate that hydraulic
  jumps can be classified into 5 categories,
  depending upon the upstream Fr

37
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.

38
Flow Control and MeasurementUnderflow 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
39
Flow Control and MeasurementUnderflow 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)

40
Flow Control and MeasurementOverflow 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.

41
Flow Control and MeasurementBroad-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

42
Flow Control and MeasurementSharp-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

43
Flow Control and MeasurementSharp-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