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Chapter 13: Open Channel Flow Bryan Pearce Department of Civil and Environmental Engineering University of Maine Fall 2009

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Page 1: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

Chapter 13: Open Channel Flow

Bryan PearceDepartment of Civil and Environmental Engineering

University of MaineFall 2009

Page 2: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

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

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 2

Learn how flow rates in open channels are measured using sluice gates and weirs

Page 3: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

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.

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 3

drainage ditches, etc.

Page 4: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

Classification of Open-Channel Flows

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 3D

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 4

Therefore, the flow is 3DNevertheless, 1D approximation is made with good success for many practical problems.

Page 5: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

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 y n

( “Normal refers to uniform”)

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 5

Page 6: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

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.

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 6

Page 7: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

Classification of Open-Channel Flows

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

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 7

Where ρ = density, µ = dynamic viscosity, ν = kinematic viscosityV = average velocityRh = Hydraulic Radius = A c/p

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

Page 8: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

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.

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 8

Page 9: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

Froude Number and Wave Speed

OC flow is also classified by the Froude number

Resembles classification of compressible flow with respect to

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 9

compressible flow with respect to Mach number

Page 10: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

Froude Number and Wave Speed

Critical depth yc occurs at Fr = 1

At low flow velocities (Fr < 1)

2

22 1

cc A

Q

gg

Vy ==

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 10

At low flow velocities (Fr < 1)Disturbance travels upstreamy > yc

At high flow velocities (Fr > 1)Disturbance travels downstreamy < yc

Page 11: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

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 c for shallow-water

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 11

Derivation of c0 for shallow-water Generate wave with plungerConsider control volume (CV) which moves with wave at c0

Page 12: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

Froude Number and Wave Speed

Continuity equation (b = width)

Momentum equation

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 12

Momentum equation

b

Page 13: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

Froude Number and Wave Speed

Combining the momentum and continuity relations and rearranging gives

For shallow water, where δy << y, (Wave speed c0 is only a function of depth, c0is both phase and group velocity.)

Shallow water is typically defined as the wavelengt h, λ λ λ λ > 20y

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 13

Shallow water is typically defined as the wavelengt h, λ λ λ λ > 20y

For “deep water” λ λ λ λ < 2y. C= λλλλ /T=gT/2π π π π (Phase velocity)

In between is “transitional. C= λλλλ /T=gT/2π[π[π[π[tanh(2 ππππy/λλλλ)] (Phase velocity)

Page 14: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

Froude Number and Wave Speed

How long will it take a tsunami generated in the Aleutian Islands to reach Hawaii?

Assume: The

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 14

Andr1.movDart_04.swf

Assume: The average depth of Pacific is about 13,000 feet & it is about 4000 km.

Page 15: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

Froude Number and Wave Speed

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 15

Indo_gl2.mov

Page 16: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

Froude Number and Wave Speed

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 16

Discuss Refraction and Shoaling

Page 17: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

Froude Number and Wave Speed

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 17

Page 18: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

Specific Energy

Total mechanical energy of the liquid in a channel in terms of heads

z is the elevation heady is the gage pressure headV2/2g is the dynamic head

Taking the datum z=0 as the bottom of the channel, the specific energy E is

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 18

the channel, the specific energy Es is

Page 19: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

Specific Energy

For a channel with constant width b,

Plot of Es vs. y for constant V and b

ybVVAQ c ===V&

22

2

2 ygb

QyEs +=

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 19

222 ygbyEs +=

2

2

2 :get weand

Q/bq case,lower use common to isit channelsr rectangulafor that Note

gy

qyEs +=

=

Page 20: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

Specific Energy

This plot is very usefulEasy to see breakdown of Es into pressure (y) and dynamic (V2/2g) headEs → ∞ as y → 0Es → y for large yEs reaches a minimum called the critical point.

There is a minimum Es required to support the given flow rate.Noting that Vc = sqrt(gyc)

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 20

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.

Page 21: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

1D steady continuity equation can be expressed as

1D steady energy equation between two stations

Continuity and Energy Equations

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 21

Head loss hL is expressed as in pipe flow, using the friction factor, and either the hydraulic diameter or radius

Page 22: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

Continuity and Energy Equations

The change in elevation head can be written in terms of the bed slope α

Introducing the friction slope Sf

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 22

The energy equation can be written as

Page 23: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

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

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 23

Flow depth is called normal depth yn (as in regular not perp.)

Average flow velocity is called uniform-flow velocity V0

Page 24: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

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 is reached, and the head loss equals the elevation drop

We can the solve for velocity (or flow rate)

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 24

Where C is the Chezy coefficient. f is the friction factor determined from the Moody chart or the Colebrook equation

Page 25: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

Uniform Flow in Channels

Antoine Chezy came up with this relationship in about 1769 when given the task of getting water to a palace near Paris.

Values of the “Chezy” Coefficient are tabulated.Many people, especially in Europe, use the Chezy formula.About one hundred years later an Irishman saw fit to mess with success (who wants a French formula?) and came up with “Manning’s” equation, by finding a formula for C in terms of a different fudge factor, n.

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 25

finding a formula for C in terms of a different fudge factor, n.

“American’s”, generally but not all, use Manning’s “n”.Values of Manning’s “n” as well as “C” are tabulated.

sftssmaWhere

SRn

aAorSR

n

aR

n

aC hhh

/49.1/)2808.3(/1

QV

3/13/13/1

2/1

0

3/22/1

0

3/2

0

6/1

===

===

Page 26: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

Uniform Flow in Channels

sftorsmaRRn

aAorRR

n

ahh /49.1/1QV 3/13/12/1

03/22/1

03/2

0 ===

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 26

Water is flowing in a weedy excavated earth channel of trapezoidal cross section with a bottom width of 0.8 m, trapezoid angle of 60°, and a bed slope angle of 0.3 °, as shown. If the flow depth is measured to be 0.52 m, determine the flow rate of water through the channel. What would your answer be if the bed slope angle were 1°?

Note: from chart , n=0.03

Q=0.6 m3/s for 0.3°, & 1.1 m 3/s for 1.0°.Review Solution

Page 27: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-6 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

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 27

Page 28: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-6 Best Hydraulic Cross Sections

Example: Rectangular ChannelCross section area, Ac = ybPerimeter, p = b + 2ySolve Ac for b and substitute

Taking derivative with respect to y

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 28

To find minimum, set derivative to zero

Best rectangular channel has a depth 1/2 of the width

Page 29: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-6Best Hydraulic Cross Sections

Similar analysis can be performed for a trapezoidal channel yielding

For θ=90o this results in y=b/2 as before.

Plugging into the Rh formulas yields Rh=y/2 (for any θ, Rh=y/2 yields the best cross section).

Similarly, taking the derivative of p with respect to q, shows that the optimum angle is (a hexagon).

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 29

q, shows that the optimum angle is (a hexagon).

For this angle (θ=60o ), the best flow depth is (half a hexagon):

Page 30: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-6 Best Hydraulic Cross Sections

Water is to be transported at a rate of 2 m3/s in uniform flow in an asphalt lined open channel. The bottom slope is 0.001. Determine the dimensions of the best cross section if the channel is (a) rectangular and (b) trapezoidal.Note: n=0.016 from table of experimental values.

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 30

Page 31: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-6 Best Hydraulic Cross Sections

Water is to be transported at a rate of 2 m3/s in uniform flow in an asphalt lined open channel. The bottom slope is 0.001. Determine the dimensions of the best cross section if the channel is (a) rectangular and (b) trapezoidal.Note: n=0.016

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 31

Page 32: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-7 Gradually Varied Flow

In GVF, y and V vary slowly, and the free surface is stableIn contrast to uniform flow, Sf ≠ S0. Now, flow depth reflects the dynamic balance between gravity, shear force, and inertial effectsTo derive how how the depth varies with x, consider the total head

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 32

Page 33: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-7 Gradually Varied Flow

Take the derivative of H

Slope dH/dx of the energy line is equal to negative of the friction slope

dx

dV

g

V

dx

dy

dx

dz

dx

dH b ++=

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 33

Bed slope has been defined

Inserting both S0 and Sf gives

Page 34: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-7 Gradually Varied Flow

Introducing continuity equation, which can be written as

Differentiating with respect to x gives

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 34

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

Page 35: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-7 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 asSteep : yn < yc

Critical : yn = yc

Mild : yn > yc

Horizontal : S0 = 0Adverse : S < 0

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 35

Adverse : S0 < 0

Initial depth is given a number1 : y > yn

2 : yn < y < yc

3 : y < yc

Page 36: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-7 Gradually Varied Flow

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 36

Page 37: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-7 Gradually Varied Flow

Typical OC system involves several sections of different slopes, with transitionsOverall surface profile is made up of individual profiles described on previous slides

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 37

Page 38: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-8 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

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 38

Page 39: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-8 Rapidly Varied Flow and Hydraulic Jump

Consider the CV surrounding the hydraulic jumpAssumptions

1. V is constant at sections (1) and (2), and β1 and β2 ≈ 1

2. P = ρgy3. τw is negligible relative to the losses that

occur during the hydraulic jump4. Channel is wide and horizontal5. No external body forces other than gravity

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 39

Page 40: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-8 Rapidly Varied Flow and Hydraulic Jump

Continuity equation

X momentum equation

Substituting and simplifying

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 40

Quadratic equation for y2 / y1

Page 41: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-8 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

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 41

Head loss associated with hydraulic jump

Page 42: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-8 Rapidly Varied Flow and 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 damage

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 42

damageA measure of performance of a hydraulic jump is its fraction of energy dissipation, or energy dissipation ratio

Page 43: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-8 Rapidly Varied Flow and Hydraulic Jump

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 43

Page 44: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-9 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.

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 44

Page 45: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-9 Flow Control and Measurement Underflow Gate

Underflow gates are located at the bottom of a wall, dam, or open channelOutflow can be either free or drownedIn free outflow, downstream flow is supercriticalIn the drowned outflow, the liquid jet undergoes a hydraulic jump. Downstream flow is subcritical.

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 45

Free outflowDrowned outflow

Page 46: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-9 Flow Control and Measurement Underflow Gate

Ct coefficien discharge Add

22

and VV :assume andfriction Ignore

22

2. and 1 pointsbetween BernoulliApply

121

22

1221

22

22

21

11

gyVyg

Vyy

g

Vy

P

g

Vy

P

=→=

<<<<

++=++γγ

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 46

gate. theof bottom theofheight thea and

width,channel theis b where2CQ flow,for or

2C

Ct coefficien discharge Add

1d

1d2

d

)(

gyab

gyV

contractavenaandfriction

==

Note that Cd for free outflows is between 0.5 and 0.6.

Cd for various outflows can be picked off chart of experimental values.

Page 47: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-9 Flow Control and Measurement Underflow Gate

Es remains constant for idealized gates with negligible frictional effectsEs decreases for real gatesDownstream is supercritical for free outflow (No Friction) (2a)Downstream is supercritical for free outflow (With Friction – Real) (2b)Downstream is subcritical for drowned outflow (2c)

Schematic of flow depth-specificenergy diagram for flow throughunderflow gates

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 47

Page 48: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-9 Flow Control and Measurement Free Outflow

Water discharged through a sluice gate undergoes a hydraulic jump. Find y2 and V2. Assume conservation of energy between 1 and 2, or E1=E2.

y1 = 8 ft

Sluicegate

y2

y3a =1 ft

Hydraulicjump

For free outflow, we only need the depth ratio y1/a to determine the discharge coefficient. The corresponding discharge coefficient is found from Fig. 13-38 to be Cd= 0.58. Then the discharge rate becomes:

8ft 1

ft 81 ==a

y

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 48

= 0.58. Then the discharge rate becomes:

/sft 13.16 3=== ft) 8)(ft/s (32.22ft) ft)(1 1(58.02 21gybaCdQ

ft 042.8ft)] ft)(8 (1)[ft/s 2(32.2

)/sft (13.16ft 8

)(22 22

23

21

2

1

21

11 =+=+=+=byg

yg

VyEs

Q

122

2

2

22

22 )(22 ss Ebyg

yg

VyE =+=+= Q

ft 042.8)]ft)( (1)[ft/s 2(32.2

)/sft (13.162

22

23

2 =+y

y

Page 49: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-9 Flow Control and Measurement Free Outflow

y1 = 8 ft

Sluicegate

y2

y3a =1 ft

Hydraulicjump

8ft 1

ft 81 ==a

y

ft 042.8)]ft)( (1)[ft/s 2(32.2

)/sft (13.162

22

23

2 =+y

y

Solve: y2=0.601 feet

ft/s 21.9====ft) ft)(0.601 (1

/sft 16.13 3

22 byA

Vc

QQ 97.4ft) )(0.601ft/s (32.2

ft/s 9.21Fr

22

22 ===

gy

V

Examine flow after the jump.

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 49

ft 3.94=

×++−=

++−= 22

223 97.4811ft) 601.0(5.0Fr8115.0 yy

ft/s 3.34=== )ft/s 9.21(ft 3.94

ft 601.02

3

23 V

y

yV

Examine flow after the jump.

ft 93.3)ft/s 2(32.2

ft/s) (3.34-ft/s) (21.9ft) (3.94-ft) 601.0(

2 2

2223

22

32 =+=−

+−=g

VVyyhL

Find the difference in energy before and after the jump to calculate headloss.

Page 50: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-9 Flow Control and Measurement Example 13-8 Drowned Outflow

Water is released from a 3-m-deep reservoir into a 6-m-wide open channel through a sluice gate with a 0.25-m-high opening at the channel bottom. The flow depth after all turbulence subsides is measured to be 1.5 m. Determine the rate of discharge,Q.

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 50

Page 51: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-9 Flow Control and Measurement Overflow Gates – Bumps and Weirs

:is change theand

2

ERemember

22

.elevations bottom downstream and upstream theare

& note andfriction Ignore

2

s

22

22

21

11

21

g

Vy

g

Vyz

g

Vyz

zz

bb

bb

+=

++=++

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 51

figure. in theshown curve on theliemust Qconstant afor depths flow All

2E

b, idth,constant wFor EEE

:is change theand

22

2

s

2211

12s2s1s

ygb

Qy

VbyVbyQ

zzz bbb

+=

==

∆=−=−=∆

Page 52: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-9 Flow Control and Measurement Bump

height". bump"or given afor y find want to weand

Vknown ,known b,constant with channelr rectangula a have We

22

11

bzy

∆cal.supercriti one and lsubcritica one

solutions real twohas yfor equation cubic The 2

When the bump gets high enough we reach critical flow. We cannot increase the flow without increasing the energy. If we continue to raise the bump the flow will “Choke”.

(The water level upstream must rise to increase the energy.)V

yzV

y

y

VyVVyVy

zg

Vyz

g

Vy bb E

22E

22

21

2

1122211

2s2

22

22

21

1s1

++∆=+

=→=

∆+=++∆=+=

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 52

02

)E(

2E

2E

2

21

212

22s132

22

21

21

22s1

22

21

21

22s1

21

1

=+∆−−

+=∆−

++∆==+

g

Vyyzy

gy

Vyyz

gy

Vyyz

g

Vy

b

b

b

to increase the energy.)

Draw bump and EGL

g

Vyz

g

Vy b 22

222

11 ++∆=+

Page 53: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-9 Flow Control and Measurement Broad-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

22

221 ++=++

g

VPy

g

VPH c

wcw

g

VHbg

g

VHbg

)2

(3

2CQ

(Ideal) )2

(3

2Q

2/32

1

2/3

2/1broadwd,Weir-BC

2/32

1

2/3

2/1Weir-BC

+

=

+

=

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 53

widthchannel AQ

)2

(3

22

3

222

weir over the critical is flow The22

2/32/1

cWeir

2

1

21

21

2

====

+=

=+→+=+

=

b

ybggybyV

g

VHy

y

g

VH

g

gyy

g

VH

gyVgg

cccc

c

ccc

cc

wPH

gHbg

/1

65.0C

)2

(3

CQ

broadwd,

broadwd,Weir-BC

+=

+

=

Can sometimes neglect V1.

For Lw<2H the flow may not reach Vc.

For Lw>12H the flow may be < Vc.

The best weir has 2H<Lw>12H .

wd=weir discharge coefficient - empirical

Can you examine these limits in the lab?

Page 54: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-9 Flow Control and Measurement Sharp-Crested

Vertical plate placed in a channel that forces the liquid to flow through an opening or over the weir to measure the flow rateUpstream flow is subcritical and becomes critical as it approaches the weirLiquid discharges as a supercritical flow stream that resembles a free jet

22 uV 2

thenH2

and P If2

1w <<>>

g

VH

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 54

])2

()2

[()2(3

22Q

limits.) assigningin ion approximat (NoteQ. find toHh to0h from u Integrate

222

22

2/32

12/32

12/121

Hh

0h

Ideal

2

212

22

21

22

21

g

V

g

VHgbdhVghb

Vghug

uh

g

Vg

uhPH

g

VPH ww

−+=+=

==

+=→+−=

+−+=++

∫=

=

2 for 0897.0598.0

)()2(3

2Q

)()2(3

2Q

,

2/32/1,Real

2/32/1Ideal

≤+=

=

=

wwrecwd

recwd

P

H

P

HC

HgbC

Hgb

Page 55: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-9 Flow Control and Measurement V-Notch

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 55

58.0)20145 & 0.2m(HFor

62.058.0

22

tan15

8Q

)V neglectedagain have we(Hered)complicate more is (Integral

Notch Weir-V afor similar is Process

,oo

,

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1

≈⇒<<><<

=

triwd

triwd

//triwd

C

(H)g))(θ

(C

Page 56: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-9 Flow Control and Measurement Bump – Example 13-9

Water flowing in a wide horizontal open channel encounters a 15-cm-high bump at the bottom of the channel. If the flow depth is 0.80 m and the velocity is 1.2 m/s before the bump, determine if the water surface isdepressed over the bump

channel.r rectangula afor :NOTE

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 56

3/12

2

3222/322

)(

)(

channel.r rectangula afor :NOTE

gb

Qy

gybygbQ

gybyVAQ

c

cc

ccccc

=

==

==

Page 57: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-9 Flow Control and Measurement Bump – Example 13-9

2

1122211

22

212

2

221

11

VVVV and

2723.015.0

873.0/81.9*2

)/2.1(80.0

2

y

yyy

g

VymmEE

msm

smm

g

VyE

ss

s

=→=+==−=

=+=+=

V 22 y

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 57

We get three solutions:0.59m, 0.36m, and -0.22m. How do we know that 0.59m is the solution?

0047.0723.0

047.064.0*073.0

*2

V723.0

23

222

2

22

221

2

22

222

1

=+−

+=+=+=

yy

yy

y

mmy

yg

yym

Page 58: Pearce (2009-Lecture)-Open Channel Flow-University of Maine-USA

13-9 Flow Control and Measurement Sharp-Crested Weir Example 13-10

The flow rate of water in a 5-m-wide horizontal open channel is being measured with a 0.60-m-high sharp-crested rectangular weir of equal width. If the water depth upstream is 1.5 m, determine the flow rate of water

Evaluate assumption of V12/2g<<H

Chapter 13: Open Channel FlowCIE350 : Hydraulics (13g) 58

0.077m is 8.6 percent of the weir head, which is significant. When the upstream velocity head is considered, the flow rate becomes 10.2 m3 /s, which is about 10 percent higher than the value determined. Therefore, it is good practice to consider the upstream velocity head unless the weir height Pw is very large relative to the weir head H.