chapter 13: open channel flow

Chapter 13: Open Channel Flow

Eric G. PatersonDepartment of Mechanical and Nuclear Engineering

The Pennsylvania State University

Spring 2005

Chapter 13: Open Channel FlowME33 : Fluid Flow 2

Note to InstructorsThese 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 PatersonPenn State, University ParkAugust 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 pipesLearn the different flow regimes in open channels and their characteristicsPredict if hydraulic jumps are to occur during flow, and calculate the fraction of energy dissipated during hydraulic jumpsLearn 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.



In an open channel, Velocity is zero on bottom and sides of channel due to no-slip conditionVelocity is maximum at the midplane of the free surfaceIn most cases, velocity also varies in the streamwise directionTherefore, the flow is 3DNevertheless, 1D approximation is made with good success for many practical problems.



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



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.



Like pipe flow, OC flow can be laminar, transitional, or turbulent depending upon the value of the Reynolds number

Where = density, = dynamic viscosity, = kinematic viscosityV = average velocity

Rh = Hydraulic Radius = Ac/pAc = cross-section areaP = wetted perimeterNote that Hydraulic Diameter was defined in pipe flows as Dh = 4Ac/p = 4Rh (Dh is not 2Rh, BE Careful!)



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



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



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



Continuity equation (b = width)

Momentum equation



Combining the momentum and continuity relations and rearranging gives

For shallow water, where y << 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 z=0 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 usefulEasy 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.


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


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



Example: Rectangular ChannelCross section area, Ac = ybPerimeter, 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



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



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



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



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 asSteep : yn < yc

Critical : yn = yc

Mild : yn > yc

Horizontal : S0 = 0

Adverse : S0 < 0

Initial depth is given a number1 : y > yn

2 : yn < y < yc

3 : y < yc



12 distinct configurations for surface profiles in GVF.



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 gatesWeirsWaterfallsAbrupt changes in cross section

Often characterized by significant 3D and transient effects

BackflowsSeparations



Consider the CV surrounding the hydraulic jumpAssumptions

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



Continuity equation

X momentum equation

Substituting and simplifying

Quadratic equation for y2/y1



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



Often, hydraulic jumps are avoided because they dissipate valuable energyHowever, in some cases, the energy must be dissipated so that it doesn’t cause damageA measure of performance of a hydraulic jump is its fraction of energy dissipation, or energy dissipation ratio



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 valvesIn OC flows, flow rate is controlled by partially blocking the channel.

Weir : liquid flows over deviceUnderflow gate : liquid flows under device

These devices can be used to control the flow rate, and to measure it.


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


Flow Control and MeasurementUnderflow Gate

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)

Schematic of flow depth-specific

energy diagram for flow through

underflow gates


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.


Flow Control and MeasurementBroad-Crested Weir

Flow over a sufficiently high obstruction in an open channel is always criticalWhen placed intentionally in an open channel to measure the flow rate, they are called weirs


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


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

chapter 13: open channel flow

Documents

fluid flow

open channel flowme33

classification of open

open channel floweric

flow depth

open channels

flow rates

channel flowsin