FUNDAMENTALS OF FLUID MECHANICS Chapter 10 Flow in Open ...

1

FUNDAMENTALS OFFUNDAMENTALS OFFLUID MECHANICSFLUID MECHANICS

Chapter 10Chapter 10Flow in Open ChannelsFlow in Open Channels

JyhJyh--CherngCherng ShiehShiehDepartment of BioDepartment of Bio--Industrial Industrial MechatronicsMechatronics Engineering Engineering

National Taiwan UniversityNational Taiwan University

2

MAIN TOPICSMAIN TOPICS

General Characteristics of OpenGeneral Characteristics of Open--Channel FlowChannel FlowSurface WavesSurface WavesEnergy ConsiderationsEnergy ConsiderationsUniform Depth Channel FlowUniform Depth Channel FlowGradually Varies FlowGradually Varies FlowRapidly Varies FlowRapidly Varies Flow

3

IntroductionIntroduction

Open channel flow involves the flows of a liquid in a channel orOpen channel flow involves the flows of a liquid in a channel orconduit that is not completely filled.conduit that is not completely filled.There exists a free surface between the flowing fluid (usually wThere exists a free surface between the flowing fluid (usually water) ater) and fluid above it (usually the atmosphere).and fluid above it (usually the atmosphere).The main deriving force is the fluid weightThe main deriving force is the fluid weight--gravity forces the fluid gravity forces the fluid to flow downhill.to flow downhill.Under steady, fully developed flow conditions, the component if Under steady, fully developed flow conditions, the component if the the weight force in the direction of flow is balanced by the equal aweight force in the direction of flow is balanced by the equal and nd opposite shear force between the fluid and the channel surface.opposite shear force between the fluid and the channel surface.

4

Open Channel Flow vs. Pipe FlowOpen Channel Flow vs. Pipe Flow

There can be no pressure force driving the fluid through the chaThere can be no pressure force driving the fluid through the channel nnel or conduit.or conduit.For steady, fully developed channel flow, the pressure distributFor steady, fully developed channel flow, the pressure distribution ion within the fluid is merely hydrostatic.within the fluid is merely hydrostatic.

5

Examples of Open Channel FlowExamples of Open Channel Flow

The natural drainage of water through the numerous creek and rivThe natural drainage of water through the numerous creek and river er systems.systems.The flow of rainwater in the gutters of our houses.The flow of rainwater in the gutters of our houses.The flow in canals, drainage ditches, sewers, and gutters along The flow in canals, drainage ditches, sewers, and gutters along roads.roads.The flow of small rivulets, and sheets of water across fields orThe flow of small rivulets, and sheets of water across fields orparking lots.parking lots.The flow in the chutes of water rides.The flow in the chutes of water rides.

6

Variables in OpenVariables in Open--Channel FlowChannel Flow

CrossCross--sectional shape.sectional shape.Bends.Bends.Bottom slope variation.Bottom slope variation.Character of its bounding surface.Character of its bounding surface.Most openMost open--channel flow results are based on correlation channel flow results are based on correlation obtained from model and fullobtained from model and full--scale experiments.scale experiments.Additional information can be gained from various analytical Additional information can be gained from various analytical and numerical efforts.and numerical efforts.

7

General Characteristics of General Characteristics of OpenOpen--Channel FlowChannel Flow

8

Classification of OpenClassification of Open--Channel FlowChannel Flow

For openFor open--channel flow, the existence of a free surface allows channel flow, the existence of a free surface allows additional types of flow.additional types of flow.The extra freedom that allows the fluid to select its freeThe extra freedom that allows the fluid to select its free--surface surface location and configuration allows important phenomena in openlocation and configuration allows important phenomena in open--channel flow that cannot occur in pipe flow.channel flow that cannot occur in pipe flow.The fluid depth, y, varies with time, t, and distance along the The fluid depth, y, varies with time, t, and distance along the channel, x, are used to classify openchannel, x, are used to classify open--channel flow:channel flow:

9

Classification Classification -- Type IType I

Uniform flow (UF): The depth of flow does not vary along the Uniform flow (UF): The depth of flow does not vary along the channel (channel (dy/dxdy/dx=0).=0).NonuniformNonuniform flows:flows:

►►Rapidly varying flows (RVF): Rapidly varying flows (RVF): The flow depth changes The flow depth changes considerably over a relatively considerably over a relatively short distance dy/dx~1.short distance dy/dx~1.

►►Gradually varying flows (GVF): Gradually varying flows (GVF): The flow depth changes slowly The flow depth changes slowly with distance with distance dy/dxdy/dx <<1.<<1.

10

Classification Classification -- Type IIType II

Laminar flow: Re < 500.Laminar flow: Re < 500.Transitional flow: Transitional flow: Turbulent flow: Re > 12,500.Turbulent flow: Re > 12,500.

μρ= /RVR heV is the average velocity of the fluid.V is the average velocity of the fluid.RRhh is the hydraulic radius of the channel.is the hydraulic radius of the channel.

Most openMost open--channel flows involve water (which has a fairly small channel flows involve water (which has a fairly small viscosity) and have relatively large characteristic lengths, it viscosity) and have relatively large characteristic lengths, it is is uncommon to have laminar openuncommon to have laminar open--channel flows.channel flows.

11

Classification Classification -- Type IIIType III

Critical Flow: Froude number Fr =1.Critical Flow: Froude number Fr =1.SubcriticalSubcritical Flow: Froude number Fr <1.Flow: Froude number Fr <1.Supercritical Flow: Froude number Fr >1.Supercritical Flow: Froude number Fr >1.

lg/VrF =

12

Surface WaveSurface Wave

13

Surface WaveSurface Wave

The distinguishing feature of flows involve a free surface (as iThe distinguishing feature of flows involve a free surface (as in n openopen--channel flows) is the opportunity for the free surface to distorchannel flows) is the opportunity for the free surface to distort t into various shapes.into various shapes.The surface of a lake or the ocean is usually distorted into eveThe surface of a lake or the ocean is usually distorted into everr--changing patterns associated with surface waves.changing patterns associated with surface waves.

14

Kinds of Surface WaveKinds of Surface Wave

Some of the surface waves are very high, some barely ripple the Some of the surface waves are very high, some barely ripple the surface; some waves are very long, some are short; some are surface; some waves are very long, some are short; some are breaking wave that form white caps, others are quite smooth.breaking wave that form white caps, others are quite smooth.

small amplitude small amplitude FiniteFinite--sized solitarysized solitaryContinuous sinusoidal shape Continuous sinusoidal shape

15

small amplitudesmall amplitude Wave Speed Wave Speed 1/51/5

Consider a single elementary wave of small height, by Consider a single elementary wave of small height, by δδyy, is , is produced on the surface of a channel by suddenly moving the produced on the surface of a channel by suddenly moving the initially stationary end wall with speed initially stationary end wall with speed δδVV..

16

small amplitude small amplitude WaveWave Speed Speed 2/52/5

The water in the channel was stationary at the initial time, t=0The water in the channel was stationary at the initial time, t=0..A stationary observer will observe a single wave move down the A stationary observer will observe a single wave move down the channel with a wave speed c, with no fluid motion ahead of the channel with a wave speed c, with no fluid motion ahead of the wave and a fluid velocity of wave and a fluid velocity of δδVV behind the wave.behind the wave.The motion is unsteady.The motion is unsteady.For a observer moving along the channel with speed c, the flow wFor a observer moving along the channel with speed c, the flow will ill appear steady.appear steady.To this observer, the fluid velocity will be on theTo this observer, the fluid velocity will be on the observerobserver’’s s right and to the left of the observer.right and to the left of the observer.

i-cVvv

=iV)(-cVvv

δ+=

Momentum Equation + Continuity Equation Momentum Equation + Continuity Equation

17


With the assumption of uniform oneWith the assumption of uniform one--dimensional flow, the dimensional flow, the continuity equation becomescontinuity equation becomes

Similarly, the momentum equationSimilarly, the momentum equation

b)yy)(Vc(cyb δ+δ+−=−

yVy

yV)yy(c

δδ

=δ

δδ+=⇒

yy <<δ

[ ]c)Vc(bcyb)yy(21by

21FF

b)yy(21AyFby

21AyF

2212

211c1

222c2

−δ−ρ=δ+γ−γ=−

δ+γ=γ=γ=γ=

(1)(1)

18


cg

yV=

δδ

⇒

yy)y( 2 δ<<δ

(2)(2)

(1)+(2)(1)+(2) gyc =

Energy Equation + Continuity Equation Energy Equation + Continuity Equation

(3)(3)

19


The single wave on the surface is seen by an observer moving witThe single wave on the surface is seen by an observer moving with h the wave speed, c.the wave speed, c.Since the pressure is constant at any point on the free surface,Since the pressure is constant at any point on the free surface, the the Bernoulli equation for this frictionless flow isBernoulli equation for this frictionless flow is

The continuity equationThe continuity equation

Combining these two equations and using the fact V=cCombining these two equations and using the fact V=c

0yg

VVttanconsyg2

V2

=δ+δ

→=+

0yVVyttanconsVy =δ+δ→=

gyc =

20

finitefinite--sized solitarysized solitary Wave SpeedWave Speed

More advanced analysis and experiments show that the wave speed More advanced analysis and experiments show that the wave speed for finitefor finite--sized solitary wavesized solitary wave

2/1

yy1gyc ⎟⎟⎠

⎞⎜⎜⎝

⎛ δ+= gy

yy1gyc

2/1

>⎟⎟⎠

⎞⎜⎜⎝

⎛ δ+=

The larger the amplitude, the faster the wave travel.The larger the amplitude, the faster the wave travel.

21

ContinuousContinuous sinusoidal shapesinusoidal shape Wave Speed Wave Speed 1/21/2

A more general description of wave motion can be obtained by A more general description of wave motion can be obtained by considering continuous (not solitary) wave of sinusoidal shape.considering continuous (not solitary) wave of sinusoidal shape.

By combining waves of various wavelengths, By combining waves of various wavelengths, λλ, and amplitudes, , and amplitudes, δδy.y.The wave speed varies with both the wavelength and fluid depth The wave speed varies with both the wavelength and fluid depth asas

2/1y2tanh

2gc ⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛λπ

πλ

=& (4)(4)

22

ContinuousContinuous sinusoidal shapesinusoidal shape Wave Speed Wave Speed 2/22/2

2/1y2tanh

2gc ⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛λπ

πλ

=&

Wave speed as a function of wavelength.Wave speed as a function of wavelength.

πλ

=⇒

=⎟⎠⎞

⎜⎝⎛

λπ

⇒∞→λ

2gc

1y2tanhy

gyc

y2y2tanh0y

=⇒

λπ

=⎟⎠⎞

⎜⎝⎛

λπ

⇒→λ

Deep layerDeep layer

Shallow layerShallow layer

23

Froude Number Effects Froude Number Effects 1/31/3

Consider an elementary wave Consider an elementary wave travellingtravelling on the surface of a fluid.on the surface of a fluid.If the fluid layer is stationary, the wave moves to the right wiIf the fluid layer is stationary, the wave moves to the right with th speed c relative to the fluid and stationary observer.speed c relative to the fluid and stationary observer.When the fluid is flowing to the left with velocity V.When the fluid is flowing to the left with velocity V.

If V<c, the wave will travel to the right with a speed of cIf V<c, the wave will travel to the right with a speed of c--V.V.If V=c, the wave will remain stationary.If V=c, the wave will remain stationary.If V>c, the wave will be washed to the left with a speed of VIf V>c, the wave will be washed to the left with a speed of V--c.c.

c/Vg/VrF == l

FroudeFroude NumberNumber

Is the ratio of the fluid Is the ratio of the fluid velocity to the wave speed.velocity to the wave speed.

24


When a wave is produced on the surface of a moving stream, as When a wave is produced on the surface of a moving stream, as happens when a rock is thrown into a river.happens when a rock is thrown into a river.

If V=0, the wave speeds equally in all directions.If V=0, the wave speeds equally in all directions.If V<c, the wave can move upstream. Upstream locations are If V<c, the wave can move upstream. Upstream locations are said to be in hydraulic communication with the downstream said to be in hydraulic communication with the downstream locations. Such flow conditions, V<c, or Fr<1, are termed locations. Such flow conditions, V<c, or Fr<1, are termed subcriticalsubcritical..If V>c, no upstream communication with downstream locations. If V>c, no upstream communication with downstream locations. Any disturbance on the surface downstream from the observer Any disturbance on the surface downstream from the observer will be washed farther downstream. Such conditions, V>c, or will be washed farther downstream. Such conditions, V>c, or Fr>1, are termed supercritical.Fr>1, are termed supercritical.

25


If V=c or Fr=1, the upstream propagating wave remains If V=c or Fr=1, the upstream propagating wave remains stationary and the flow is termed critical.stationary and the flow is termed critical.

26

Energy ConsiderationsEnergy Considerations

27

Energy Considerations Energy Considerations 1/31/3

x and y are taken as the x and y are taken as the distance along the distance along the channel bottom and the channel bottom and the depth normal to the depth normal to the bottom.bottom.

l21

0zzS −

= The slope of the channel bottom or bottom The slope of the channel bottom or bottom slope is constant over the segmentslope is constant over the segment

Very small for most openVery small for most open--channel flows.channel flows.

28


With the assumption of a uniform velocity profile across any secWith the assumption of a uniform velocity profile across any section tion of the channel, the oneof the channel, the one--dimensional energy equation becomedimensional energy equation become

(5)(5)L2

222

1

211 hz

g2Vpz

g2Vp

+++γ

=++γ

hL is the head loss due to viscous effects between sections (1) and (2).

(5)(5)

2211

o21y/py/p

Szz=γ=γ

=− l

L

22

2o2

11 h

g2V

ySg2

Vy ++=++ l (6)(6)

29


(6)(6)

l/hS Lf =

l)of

21

22

21 SS(g2VVyy −+

−=− (7)(7)

(7)(7)

For a horizontal channel bottom (S0=0) and negligible head loss (Sf=0)

g2VVyy

21

22

21−

=−

30

Specific Energy Specific Energy 1/41/4

Define specific energy, EDefine specific energy, E

g2VyE

2+=

l)SS(EE of21 −+=

(8)(8)

(9)(9)

2211 zEzE +=+(9)(9) The sum of the specific energy and The sum of the specific energy and the elevation of the channel bottom the elevation of the channel bottom remains constant.remains constant.This a statement of the Bernoulli This a statement of the Bernoulli equation.equation.

Head losses are negligible, Head losses are negligible, SSff=0=0

12o zzS −=− l

31


If the crossIf the cross--sectional shape is a rectangular of width bsectional shape is a rectangular of width b

For a given channelFor a given channelb= constantb= constantq = constantq = constantE = E (y) E = E (y) Specific energy diagramSpecific energy diagram

2

2

gy2qyE += (10)(10)

Where q is the Where q is the flowrateflowrate per unit width, q=Q/b=per unit width, q=Q/b=Vyb/bVyb/b==VyVy

32


2

2

gy2qyE +=

For a given q and E, equation (10) is a For a given q and E, equation (10) is a cubic equation with three solutions , cubic equation with three solutions , yysupsup, , yysubsub, and , and yynegneg..If E >If E >EEminmin, two solutions are positive , two solutions are positive and and yynegneg is negative (has no physical is negative (has no physical meaning and can be ignored).meaning and can be ignored).These two depths are term alternative These two depths are term alternative depths.depths.

(10)(10)

33


Approach y=EApproach y=EVery deep and very slowlyVery deep and very slowly

E > E > EEminminTwo possible depths of flow, Two possible depths of flow, one one subcriticalsubcritical and the other and the other supercriticalsupercritical

subsup

subsup

VV

yy

>

<

Approach y=0Approach y=0Very shallow and Very shallow and very high speedvery high speed

34

Determine Determine EEminmin

To determine the value of To determine the value of EEminmin3/12

c3

2

gqy0

gyq1

dydE0

dydE

⎟⎟⎠

⎞⎜⎜⎝

⎛=⇒=−=⇒=

1FrgyyqV

2y3E cc

cc

cmin =⇒==⇒=

(11)(11)

Sub. (11) into (10)Sub. (11) into (10)

1.1. The critical conditions (Fr=1) occur at the location of The critical conditions (Fr=1) occur at the location of EEminmin..2.2. Flows for the upper part of the specific energy diagram Flows for the upper part of the specific energy diagram

are are subcriticalsubcritical (Fr<1)(Fr<1)3.3. Flows for the lower part of the specific energy diagram are Flows for the lower part of the specific energy diagram are

supercritical (Fr>1) supercritical (Fr>1)

35

Example 10.1 Specific Energy diagram Example 10.1 Specific Energy diagram --QualitativeQualitative

Water flows under the sluice gate in a constant width rectangulaWater flows under the sluice gate in a constant width rectangular r channel as shown in Fig. E10.1a. Describe this flow in terms of channel as shown in Fig. E10.1a. Describe this flow in terms of the the specific energy diagram. Assume specific energy diagram. Assume inviscidinviscid flow.flow.

36

Example 10.1 Example 10.1 SolutionSolution1/21/2

InviscidInviscid flow flow SSff=0=0Channel bottom is horizontal zChannel bottom is horizontal z11=z=z22 (or S(or Soo=0)=0)EE11=E=E22 qq11=q=q22

The specific energy diagram for this flow is as shown in Fig. E1The specific energy diagram for this flow is as shown in Fig. E10.1b.0.1b.

The The flowrateflowrate can remain the same for this channel even if the can remain the same for this channel even if the upstream depth is increased. This is indicated by depths yupstream depth is increased. This is indicated by depths y11’’ and yand y22’’in Fig E10.1c.in Fig E10.1c.To remain same To remain same flowrateflowrate, the distance between the bottom of the , the distance between the bottom of the gate and the channel bottom must be decreased to give a smaller gate and the channel bottom must be decreased to give a smaller flow area (yflow area (y22’’ < y< y22 ), and the upstream depth must be increased to ), and the upstream depth must be increased to give a bigger head (y give a bigger head (y 11’’ > y> y11 ).).

37


On the other hand, if the gate remains fixed so that the downstrOn the other hand, if the gate remains fixed so that the downstream eam depth remain fixed (ydepth remain fixed (y22’’’’ = y= y22 ), the ), the flowrateflowrate will increase as the will increase as the upstream depth increases to y upstream depth increases to y 11’’’’ > y> y11..

q”>q0

38

Example 10.2 Specific Energy diagram Example 10.2 Specific Energy diagram ––QuantitativeQuantitative

Water flows up a 0.5Water flows up a 0.5--ftft--tall ramp in a constant width rectangular tall ramp in a constant width rectangular channel at a rate of q = 5.75 ftchannel at a rate of q = 5.75 ft22/s as shown in Fig. E10.2a. (For now /s as shown in Fig. E10.2a. (For now disregard the disregard the ““bumpbump””) If the upstream depth is 2.3 ft, determine the ) If the upstream depth is 2.3 ft, determine the elevation of the water surface downstream of the ramp, yelevation of the water surface downstream of the ramp, y22 + z+ z22. . Neglect viscous effects.Neglect viscous effects.

39


With SWith S0 0 ll =z=z11--zz22 and and hhLL=0, conservation of energy requires that=0, conservation of energy requires that

4.64Vy90.1

zg2

Vpzg2

Vp

22

2

2

222

1

211

+=⇒

++γ

=++γ

s/ft75.5Vy

VyVy2

22

1122

=⇒

=

0513.0y90.1y 22

32 =+−

(10.2(10.2--1)1)

The continuity equationThe continuity equation

(10.2(10.2--2)2) ft466.0yft638.0ft72.1y

2

2−=

=

40


The corresponding elevations of the free surface are eitherThe corresponding elevations of the free surface are either

ft14.1ft50.0ft638.0zyft22.2ft50.0ft72.1zy

22

22=+=+

=+=+

Which of these flows is to be expected?Which of these flows is to be expected?This can be answered by use of the specific energy diagram This can be answered by use of the specific energy diagram obtained from Eq.(10), which for this problem isobtained from Eq.(10), which for this problem is

2y513.0yE +=

The diagram is shown in Fig.E10.2(b).The diagram is shown in Fig.E10.2(b).

41


42


The upstream condition corresponds to The upstream condition corresponds to subcriticalsubcritical flow; the flow; the downstream condition is either downstream condition is either subcriticalsubcritical or supercritical, or supercritical, corresponding to points 2 or 2corresponding to points 2 or 2’’..

Note that since ENote that since E11=E=E22+(z+(z22--zz11)=E)=E22+0.5 ft, it follows that the +0.5 ft, it follows that the downstream conditions are located to 0.5 ft to the left of the downstream conditions are located to 0.5 ft to the left of the upstream conditions on the diagram.upstream conditions on the diagram.

…….. The surface elevation is.. The surface elevation is

ft22.2ft50.0ft72.1zy 22 =+=+

43

Channel Depth Variations Channel Depth Variations 1/31/3

Consider gradually varying flows.Consider gradually varying flows.For such flows, For such flows, dy/dxdy/dx<<1, and it is reasonable to impose the one<<1, and it is reasonable to impose the one--dimensional velocity assumption.dimensional velocity assumption.At an section the total headAt an section the total head

and the energy equationand the energy equation

zyg2

VH2

++=

L21 hHH +=

dxdz

dxdy

dxdV

gVzy

g2V

dxd

dxdH 2

++=⎟⎟⎠

⎞⎜⎜⎝

⎛++=

The slop of the energy lineThe slop of the energy line

fL S

dxdh

dxdH

== oSdxdz

=

44


dxdy

yV

dxdy

yq

dxdV

yqV

2−=−=⇒=

dxdyF

dxdy

gyV

dxdV

gV 2

r

2−=−=

(12)(12)of

0L

SSdxdy

dxdV

gV

Sdxdy

dxdV

gV

dxdh

−=+

++=

For a given For a given flowrateflowrate per unit width, q, in a rectangular channel of per unit width, q, in a rectangular channel of constant width b, we haveconstant width b, we have

(12)(12) (13)(13)

45


)F1(SS

dxdy

2r

of

−

−=Sub. (13) into (12)Sub. (13) into (12) (14)(14)

Depends on the local slope of the channel bottom, the Depends on the local slope of the channel bottom, the slope of the energy line, and the slope of the energy line, and the FroudeFroude number.number.

46

Uniform Depth Channel FlowUniform Depth Channel Flow

47

Uniform Depth Channel Flow Uniform Depth Channel Flow 1/31/3

Many channels are designed to carry fluid at a uniform depth allMany channels are designed to carry fluid at a uniform depth allalong their length.along their length.

Irrigation canals.Irrigation canals.Nature channels such as rivers and creeks.Nature channels such as rivers and creeks.

Uniform depth flow (Uniform depth flow (dy/dxdy/dx=0) can be accomplished by adjusting the =0) can be accomplished by adjusting the bottom slope, Sbottom slope, S00, so that it precisely equal the slope of the energy , so that it precisely equal the slope of the energy line, line, SSff..A balance between the potential energy lost by the fluid as it cA balance between the potential energy lost by the fluid as it coasts oasts downhill and the energy that is dissipated by viscous effects (hdownhill and the energy that is dissipated by viscous effects (head ead loss) associated with shear stress throughout the fluid.loss) associated with shear stress throughout the fluid.

48


Uniform flow in an open channel.Uniform flow in an open channel.

49


Typical velocity and shear stress distributions in an open channTypical velocity and shear stress distributions in an open channel: el: ((aa) velocity distribution throughout the cross section. () velocity distribution throughout the cross section. (bb) shear ) shear stress distribution on the wetted perimeter.stress distribution on the wetted perimeter.

50

The The ChezyChezy & Manning Equation & Manning Equation 1/61/6

Control volume for uniform flow in an open channel.Control volume for uniform flow in an open channel.

51


0)VV(QF 12x =−ρ=∑0sinWPFF w21 =θ+τ−− l (15)(15)

Under the assumption of steady uniform flow, the x component of Under the assumption of steady uniform flow, the x component of the momentum equationthe momentum equation

where Fwhere F11 and Fand F22 are the hydrostatic pressure forces across either end are the hydrostatic pressure forces across either end of the control volume.of the control volume.P is wetted perimeter.P is wetted perimeter.

52


yy11=y=y22 FF11=F=F22

(15)(15)

PARAW

PsinW

0sinWP

hw

w

=γ=θ

=τ

=θ+τ−

ll

l

oho

w SrRP

SA=

γ=τ

l

l(16)(16)

Wall shear stress is proportional to the dynamic pressureWall shear stress is proportional to the dynamic pressure2

V2

w ρ∝τ

2VK

2

w ρ=τ

K is a constant dependent upon the roughness of the pipeK is a constant dependent upon the roughness of the pipe

(Chapter 8)(Chapter 8)

53


(16)(16) (17)(17)

C is termed the C is termed the ChezyChezy coefficientcoefficient

Was developed in 1768 by A. Was developed in 1768 by A. ChezyChezy (1718(1718--1798), a French 1798), a French engineer who designed a canal for the Paris water supply.engineer who designed a canal for the Paris water supply.

oh

2SrR

2VK =ρ ohSRCV =

ChezyChezy equationequation

(17)(17) 2/1oSV ∝

hRV ∝

ReasonableReasonable

3/2hRV ∝

Manning EquationManning Equation

54


In 1889, R. manning (1816In 1889, R. manning (1816--1897), an Irish engineer, developed the 1897), an Irish engineer, developed the following somewhat modified equation for openfollowing somewhat modified equation for open--channel flow to channel flow to more accurately describe the more accurately describe the RRhh dependence:dependence:

Manning equationManning equation(18)(18)

n is the Manning resistance coefficient.n is the Manning resistance coefficient.Its value is dependent on the surface material Its value is dependent on the surface material of the channelof the channel’’s wetted perimeter and is s wetted perimeter and is obtained from experiments.obtained from experiments.It has the units of s/mIt has the units of s/m1/31/3 or s./ftor s./ft1/31/3

nSRV

2/1o

3/2h=

55


(19)(19) Where Where κκ=1 if SI units are =1 if SI units are used, used, κκ=1.49 if BG units =1.49 if BG units are used.are used.

2/1o

3/2h SR

nV κ=(18)(18)

(20)(20)2/1o

3/2h SAR

nQ κ=

5/25/3

2/1o

3/2

2/1o

3/52/1

o

3/2P

kSnQA

PSA

nS

PAA

nQ

⎥⎥⎦

⎤

⎢⎢⎣

⎡=⇒

κ=⎟

⎠⎞

⎜⎝⎛κ

=

The best hydraulic cross section is defined as the section of miThe best hydraulic cross section is defined as the section of minimum nimum area for a given area for a given flowrateflowrate Q, slope, SQ, slope, Soo, and the roughness coefficient, n., and the roughness coefficient, n.

PAR h = constant

A channel with minimum A is one with a minimum P.A channel with minimum A is one with a minimum P.

56

Value of the Manning Coefficient, nValue of the Manning Coefficient, n

57

Uniform Depth ExamplesUniform Depth Examples

58

Example 10.3 Uniform Flow, Determine Example 10.3 Uniform Flow, Determine Flow RateFlow Rate

Water flows in the canal of trapezoidal cross section shown in FWater flows in the canal of trapezoidal cross section shown in Fig. ig. E10.3a. The bottom drops 1.4 ft per 1000 ft of length. DetermineE10.3a. The bottom drops 1.4 ft per 1000 ft of length. Determine the the flowrateflowrate if the canal is lined with new smooth concrete. Determine if the canal is lined with new smooth concrete. Determine the Froude number for this flow. the Froude number for this flow.

59


(20)(20) κκ=1.49 if BG units are used.=1.49 if BG units are used.

ft25.3P/ARft6.27)ft40sin/5(2ft12P

ft8.89ft40tan5ft5)ft5(ft12A

h

2

===°+=

=⎟⎠⎞

⎜⎝⎛

°+=

2/1o

3/2h SAR

nQ κ=

cfs915n98.10)0014.0()ft25.3)(ft8.89(

n49.1Q 2/13/22 ===

From Table 10.1, n=0.012From Table 10.1, n=0.012

804.0...gyVFrs/ft2.10A/QV =====

60


61

Example 10.4 Uniform Flow, Determine Example 10.4 Uniform Flow, Determine Flow DepthFlow Depth

Water flows in the channel shown in Fig. E10.3 at a rate o Q = 1Water flows in the channel shown in Fig. E10.3 at a rate o Q = 10.0 0.0 mm33/s. If the canal lining is weedy, determine the depth of the flo/s. If the canal lining is weedy, determine the depth of the flow.w.

62

Example 10.4 Example 10.4 SolutionSolution

y=1.50 my=1.50 m

66.3y11.3y66.3y19.1

PAR

66.3y11.340siny266.3P

y66.3y19.1A

2

h

2

++

==

+=⎟⎠⎞

⎜⎝⎛

°+=

+=

...03.00.1SAR

n10Q 2/1

o3/2

h =κ

==

From Table 10.1, n=0.030From Table 10.1, n=0.030

0)66.3y11.3(515)y66.3y19.1( 252 =+−+

63

Example 10.5 Uniform Flow, Maximum Example 10.5 Uniform Flow, Maximum Flow RateFlow Rate

Water flows in a round pipe of diameter D at a depth of 0 Water flows in a round pipe of diameter D at a depth of 0 ≤≤ y y ≤≤ D, D, as shown in Fig. E10.5a. The pipe is laid on a constant slope ofas shown in Fig. E10.5a. The pipe is laid on a constant slope of SS00, , and the Manning coefficient is n. At what depth does the maximumand the Manning coefficient is n. At what depth does the maximumflowrateflowrate occur? Show that for certain occur? Show that for certain flowrateflowrate there are two depths there are two depths possible with the same possible with the same flowrateflowrate. Explain this behavior.. Explain this behavior.

64



θθ−θ

==θ

=

θ−θ=

4)sin(D

PAR

2DP

)sin(8

DA

h

2

2/1o

3/2h SAR

nQ κ=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

θ

θ−θκ= 3/2

3/5

3/2

3/82/1

o)sin(

)4(8DS

nQ

This can be written in terms of the flow depth by usingThis can be written in terms of the flow depth by using

)]2/cos(1[2Dy θ−=

65


D938.0ywhenQQ max ==

A graph of A graph of flowrateflowrate versus flow depth, Q = versus flow depth, Q = Q(yQ(y), has the ), has the characteristic indicated in Fig. E10.5(b).characteristic indicated in Fig. E10.5(b).

The maximum The maximum flowrateflowrate occurs when y=0.938D, or occurs when y=0.938D, or θθ=303=303ºº

66

Example 10.6 Uniform Flow, Effect of Example 10.6 Uniform Flow, Effect of Bottom SlopeBottom Slope

Water flows in a rectangular channel of width b = 10 m that has Water flows in a rectangular channel of width b = 10 m that has a a Manning coefficient of n = 0.025. Plot a graph of Manning coefficient of n = 0.025. Plot a graph of flowrateflowrate, Q, as a , Q, as a function of slope Sfunction of slope S00, indicating lines of constant depth and lines of , indicating lines of constant depth and lines of constant Froude number.constant Froude number.

67


(19)(19)

)y2b(by

PARy10byA h +

====

2/1o

3/22/1

o3/2

h Sy210

y10025.00.1SR

nV ⎟⎟

⎠

⎞⎜⎜⎝

⎛+

=κ

=

3/42

o

2/1o

3/22/1

y5y5yFr00613.0S

Sy210

y10025.00.1Fr)gy(

⎟⎟⎠

⎞⎜⎜⎝

⎛ +=⇒

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

(10.6(10.6--1)1)

(10.6(10.6--2)2)

68


For given value of Fr, we pick For given value of Fr, we pick various value of y, determine the various value of y, determine the corresponding value of Scorresponding value of Soo from from EqEq(10.6(10.6--2)2), and then calculate Q=VA, , and then calculate Q=VA, with V from either with V from either EqEq (10.6(10.6--1)1) or or V=(gy)V=(gy)1/21/2FrFr. .

The results are indicated in The results are indicated in Fig. E10.6Fig. E10.6

69

Example 10.7 Uniform Flow, Variable Example 10.7 Uniform Flow, Variable RoughnessRoughness

Water flows along the drainage canal having the properties shownWater flows along the drainage canal having the properties shown in in Fig. E10.7a. If the bottom slope is SFig. E10.7a. If the bottom slope is S0 0 = 1 ft/500 ft=0.002, estimate = 1 ft/500 ft=0.002, estimate the the flowrateflowrate when the depth is y = 0.8 ft + 0.6 ft = 1.4 ft.when the depth is y = 0.8 ft + 0.6 ft = 1.4 ft.

70

Example 10.7 Example 10.7 SolutionSolution

2/1o

3/2hi

ii SRA

n49.1Q

i=

321 QQQQ ++=

s/ft8.16...Q 3==

71

Example 10.8 Uniform Flow, Best Example 10.8 Uniform Flow, Best Hydraulic Cross SectionHydraulic Cross Section

Water flows uniformly in a Water flows uniformly in a rectangular channel of width rectangular channel of width b and depth y. Determine the b and depth y. Determine the aspect ratio, aspect ratio, b/yb/y, for the best , for the best hydraulic cross section.hydraulic cross section.

72



)Ay2(Ay

)y2b(A

)y2b(by

PAR

y2bPbyA

2h+

=+

=+

==

+==

2/1o

3/2h SAR

nQ κ=

2/1o

3/2

2 S)Ay2(

AyAn

Q ⎟⎟⎠

⎞⎜⎜⎝

⎛

+

κ=

2/3

2/1o

22/5

SnQK)Ay2(KyA ⎟

⎟⎠

⎞⎜⎜⎝

⎛

κ=+=

constantconstant

73


The best hydraulic section is the one that gives the minimum A The best hydraulic section is the one that gives the minimum A for all y. That is, for all y. That is, dA/dydA/dy = 0.= 0.

ky4A

dydAy4KAy

dydAA

250

dydA

2/5

2/52/3

=⇒

⎟⎟⎠

⎞⎜⎜⎝

⎛+=+⇒=

2y/b =The rectangular with the best hydraulic cross section twice The rectangular with the best hydraulic cross section twice as wide as it is deep, oras wide as it is deep, or

byy2 2 =

74


The best hydraulic cross section for other shapesThe best hydraulic cross section for other shapes

75

Gradually Varied FlowGradually Varied Flow

1dxdy

<<

76

Gradually Varied Flow Gradually Varied Flow 1/21/2

Open channel flows are classified as uniform depth, gradually Open channel flows are classified as uniform depth, gradually varying or rapidly varying.varying or rapidly varying.If the channel bottom slope is equal to the slope of the energy If the channel bottom slope is equal to the slope of the energy line, line, SSoo==SSff, the flow depth is constant, , the flow depth is constant, dy/dxdy/dx=0.=0.

The loss in potential energy of the fluid as it flows downhill iThe loss in potential energy of the fluid as it flows downhill is s exactly balanced by the dissipation of energy through viscous exactly balanced by the dissipation of energy through viscous effects.effects.

If the bottom slope and the energy line slope are not equal, theIf the bottom slope and the energy line slope are not equal, the flow flow depth will vary along the channel.depth will vary along the channel.

77

Gradually Varied Flow Gradually Varied Flow 2/22/2

)F1(SS

dxdy

2r

of

−

−= (14)(14)

The sign of The sign of dy/dxdy/dx, that is, whether the flow depth , that is, whether the flow depth increase or decrease with distance along the channel increase or decrease with distance along the channel depend on depend on SSff --SSoo ad 1ad 1--FrFr22

78

Classification of Surface Shapes Classification of Surface Shapes 1/31/3

The character of a gradually varying flow is often classified inThe character of a gradually varying flow is often classified in terms terms of the actual channel slope, Sof the actual channel slope, Soo, compared with the slope required to , compared with the slope required to produce uniform critical flow, Sproduce uniform critical flow, Sococ..The character of a gradually varying flow depends on whether theThe character of a gradually varying flow depends on whether thefluid depth is less than or greater than the uniform normal deptfluid depth is less than or greater than the uniform normal depth, h, yynn..

12 possible surface configurations12 possible surface configurations

79


Fr<1 : y>Fr<1 : y>yyccFr>1 : y<Fr>1 : y<yycc

80

Examples of Gradually Varies Flows Examples of Gradually Varies Flows 1/51/5

Typical surface configurations for nonuniform depth flow with a mild slope. S0 < S0c.

DropDrop--down profiledown profile

Backwater curveBackwater curve

81


Typical surface configurations for Typical surface configurations for nonuniformnonuniform depth flow with a depth flow with a critical slope. Scritical slope. S00 = = SS0c0c..

82


Typical surface Typical surface configurations for configurations for nonuniformnonuniform depth depth flow with a steep flow with a steep slope. Sslope. S00 > > SS0c0c..

83


Typical surface configurations for Typical surface configurations for nonuniformnonuniform depth flow with a depth flow with a horizontal slope. Shorizontal slope. S00 =0.=0.

84


Typical surface configurations for Typical surface configurations for nonuniformnonuniform depth flow with a depth flow with a adverse slope. Sadverse slope. S00 <0.<0.

85


The free surface is relatively free to conform to the shape thatThe free surface is relatively free to conform to the shape thatsatisfies the governing mass, momentum, and energy equations.satisfies the governing mass, momentum, and energy equations.The actual shape of the surface is often very important in the dThe actual shape of the surface is often very important in the design esign of openof open--channel devices or in the prediction of flood levels in channel devices or in the prediction of flood levels in natural channels.natural channels.

☯☯The surface shape, y=The surface shape, y=y(xy(x), can be calculated by solving ), can be calculated by solving the governing differential equation obtained from a the governing differential equation obtained from a combination of the Manning equation (20) and the combination of the Manning equation (20) and the energy equation (14).energy equation (14).

Numerical techniques have been developed and Numerical techniques have been developed and used to predict openused to predict open--channel surface shapes.channel surface shapes.

86

Rapidly Varied FlowRapidly Varied Flow

1~dxdy

87

Rapidly Varied FlowRapidly Varied Flow

Rapidly varied flow: flow depth changes occur over a relatively Rapidly varied flow: flow depth changes occur over a relatively short distance.short distance.

Quite complex and difficult to analyze in a precise fashion.Quite complex and difficult to analyze in a precise fashion.Many approximate results can be obtained by using a simple Many approximate results can be obtained by using a simple oneone--dimensional model along with appropriate experimentally dimensional model along with appropriate experimentally determined coefficients when necessary.determined coefficients when necessary.

88

Occurrence of Rapidly Varied Flow Occurrence of Rapidly Varied Flow 1/21/2

Flow depth changes significantly un a short distance: The flow Flow depth changes significantly un a short distance: The flow changes from a relatively shallow, high speed condition into a changes from a relatively shallow, high speed condition into a relatively deep, low speed condition within a horizontal distancrelatively deep, low speed condition within a horizontal distance of e of just a few channel depths.just a few channel depths.

Hydraulic Jump

89

Occurrence of Rapidly Varied Flow Occurrence of Rapidly Varied Flow 2/22/2

Sudden change in the channel geometry such as the flow in an Sudden change in the channel geometry such as the flow in an expansion or contraction section of a channel.expansion or contraction section of a channel.

Rapidly varied flow may occur in a Rapidly varied flow may occur in a channel transition section.channel transition section.

90

Example of Rapidly Varied Flow Example of Rapidly Varied Flow 1/21/2

The scouring of a river bottom in the neighborhood of a bridge pThe scouring of a river bottom in the neighborhood of a bridge pier.ier.

The complex threeThe complex three--dimensional flow structure around a bridge pier.dimensional flow structure around a bridge pier.

Responsible for Responsible for the erosion near the erosion near the foot of the the foot of the bridge pier.bridge pier.

91

Example of Rapidly Varied Flow Example of Rapidly Varied Flow 2/22/2

Many openMany open--channel channel flowflow--measuring devicesmeasuring devices are based on are based on principles associated with rapidly varied flows.principles associated with rapidly varied flows.

BroadBroad--crested weirs.crested weirs.SharpSharp--crested weirs.crested weirs.Critical flow flumes.Critical flow flumes.Sluice gates.Sluice gates.

92

Hydraulic JumpHydraulic Jump

93

The Hydraulic Jump The Hydraulic Jump 1/61/6

Under certain conditions it is possible that the fluid depth wilUnder certain conditions it is possible that the fluid depth will l change very rapidly over a short length of the channel without achange very rapidly over a short length of the channel without any ny change in the channel configuration.change in the channel configuration.

Such changes in depth can be approximated as a discontinuity in Such changes in depth can be approximated as a discontinuity in the free surface elevation (the free surface elevation (dy/dxdy/dx==∞∞).).

This near discontinuity is called a This near discontinuity is called a hydraulic jumphydraulic jump....

94


A simplest type of hydraulic jump in a horizontal, rectangular cA simplest type of hydraulic jump in a horizontal, rectangular channel.hannel.

Assume that the flow at sections (1) and (2) is nearly uniform, Assume that the flow at sections (1) and (2) is nearly uniform, steady, steady, and oneand one--dimensional.dimensional.

95


The x component of the momentum equationThe x component of the momentum equation)VV(byV)VV(QFF 1211221 −ρ=−ρ=−

2/byApF 2111c1 γ==

2/byApF 2222c2 γ==

(21)(21))VV(gyV

2y

2y

211

22

21 −=−

The conservation of mass equationThe conservation of mass equation (22)(22)QbVybVy 2211 ==

The energy equationThe energy equation L

22

2

21

1 hg2

Vyg2

Vy ++=+ (23)(23)

The head loss is due to the violent turbulent mixing and dissipaThe head loss is due to the violent turbulent mixing and dissipation.tion.

96


(21)+(22)+(23)(21)+(22)+(23) Nonlinear equationsNonlinear equations

One solution is yOne solution is y11=y=y22, V, V11=V=V22, , hhLL=0=0Other solutions?Other solutions?

(21)+(22)(21)+(22)

0F2yy

yy 2

1r1

22

1

2 =−⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎟⎠⎞

⎜⎝⎛ +±−= 2

1r1

2 F81121

yy

1

11

gyVFr =

⎟⎠⎞

⎜⎝⎛ ++−= 2

1r1

2 F81121

yy

)yy(gy

yVVy

yVgyV

2y

2y

212

121

12

111122

21 −=⎟⎟

⎠

⎞⎜⎜⎝

⎛−=−

SolutionsSolutions (24)(24)

97


(23)(23) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−+−=

2

2

121r

1

2

1

Lyy1

2F

yy1

yh

(25)(25)

(24)+(25)(24)+(25) Depth ratio and dimensionless head loss across a Depth ratio and dimensionless head loss across a hydraulic jump as a function of upstream hydraulic jump as a function of upstream FroudeFroudenumber.number.

98


The head loss is negative if FrThe head loss is negative if Fr11<1.<1.Violate the second law of Violate the second law of thermodynamicsthermodynamics

Not possible to produce a Not possible to produce a hydraulic jump with Frhydraulic jump with Fr11<1.<1.

99

Classification of Hydraulic Jump Classification of Hydraulic Jump 1/21/2

The actual structure of a hydraulic jump is a complex function oThe actual structure of a hydraulic jump is a complex function of Frf Fr11, , even though the depth ratio and head loss are given quite accuraeven though the depth ratio and head loss are given quite accurately tely by a simple oneby a simple one--dimensional flow analysis.dimensional flow analysis.A detailed investigation of the flow indicates that there are A detailed investigation of the flow indicates that there are essentially five type of surface and jump conditions.essentially five type of surface and jump conditions.

100

Classification of Hydraulic Jump Classification of Hydraulic Jump 2/22/2

101

Hydraulic Jump Variations Hydraulic Jump Variations 1/21/2

Hydraulic jumps can occur in a variety o channel flow Hydraulic jumps can occur in a variety o channel flow configurations, not just in horizontal, rectangular channels as configurations, not just in horizontal, rectangular channels as discussed above.discussed above.Other common types of hydraulic jumps include those that occur iOther common types of hydraulic jumps include those that occur in n sloping channels and the submerged hydraulic jumps that can occusloping channels and the submerged hydraulic jumps that can occur r just downstream of a sluice gate.just downstream of a sluice gate.

102

Hydraulic Jump Variations Hydraulic Jump Variations 2/22/2

Hydraulic jump variations:Hydraulic jump variations:((aa) jump caused by a change ) jump caused by a change in channel slope, in channel slope, ((bb) submerged jump) submerged jump

103

Example 10.9 Hydraulic JumpExample 10.9 Hydraulic Jump

Water on the horizontal apron of the 100Water on the horizontal apron of the 100--ftft--wide spillway shown in wide spillway shown in Fig. E10.9a has a depth o 0.60 ft and a velocity of 18 ft/s. DetFig. E10.9a has a depth o 0.60 ft and a velocity of 18 ft/s. Determine ermine the depth, ythe depth, y22, after the jump, the Froude numbers before and after , after the jump, the Froude numbers before and after the jump, Frthe jump, Fr11 and Frand Fr22, and the power dissipated, P, and the power dissipated, Pdd, with the jump., with the jump.

104


10.4)ft60.0)(s/ft2.32(

s/ft18gyVFr

21

11 ===

32.5...F81121

yy 2

1r1

2 ==⎟⎠⎞

⎜⎝⎛ ++−=

Conditions across the jump are determined by the upstream Conditions across the jump are determined by the upstream FroudeFroude numbernumber

(24)(24)

ft19.3)ft60.0(32.5y2 ==

Since QSince Q11=Q=Q22, or V, or V22=(y=(y11VV11)/y)/y22=3.39ft/s=3.39ft/s

105


334.0)ft19.3)(s/ft2.32(

s/ft39.3gyVFr

21

22 ===

L11Ld hVbyQhP γ=γ=

The poser dissipated, Pd, by viscous effects within the jump canThe poser dissipated, Pd, by viscous effects within the jump canbe determined from the head lossbe determined from the head loss

ft26.2...g2

Vyg2

Vyh22

2

21

1L ==⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟

⎟⎠

⎞⎜⎜⎝

⎛+=(23)(23)

hp277s/lbft1052.1...hVbyQhP 5L11Ld =⋅×=γ=γ=

106


22

2

21121

y81.1y

gy2qyE

s/ft8.10VybQqqq

+=+=

=====

Various upstream depthVarious upstream depth

107

Weirs and GateWeirs and Gate

108

WeirWeir

A weir is an obstruction on a channel bottom over which the fluiA weir is an obstruction on a channel bottom over which the fluid d must flow.must flow.Weir provides a convenient method of determining the Weir provides a convenient method of determining the flowrateflowrate in in an open channel in terms of a single depth measurement.an open channel in terms of a single depth measurement.

109

SharpSharp--Crested Weir Crested Weir 1/41/4

A sharpA sharp--Crested weir is essentially a verticalCrested weir is essentially a vertical--edged flat plate placed edged flat plate placed across the channel.across the channel.The fluid must flow across the sharp edge and drop into the poolThe fluid must flow across the sharp edge and drop into the pooldownstream of the weir plate.downstream of the weir plate.

110

SharpSharp--Crested Weir Crested Weir -- Geometry Geometry 2/42/4

SharpSharp--crested weir plate geometry: (a) rectangular, (b) triangular, (ccrested weir plate geometry: (a) rectangular, (b) triangular, (c) ) trapezoidal. trapezoidal.

111

SharpSharp--Crested Weir Crested Weir –– FlowrateFlowrate 3/43/4

Assume that the velocity profile upstream of the weir plate Assume that the velocity profile upstream of the weir plate is uniform and that the pressure within the is uniform and that the pressure within the nappenappe is is atmosphere.atmosphere.

Assume that the Assume that the fluid flows fluid flows horizontally over the horizontally over the weir plate with a weir plate with a nonuniformnonuniform velocity velocity profile.profile.

112

SharpSharp--Crested Weir Crested Weir –– FlowrateFlowrate 4/44/4

With PWith PBB=0, the Bernoulli equation for flow along the arbitrary =0, the Bernoulli equation for flow along the arbitrary streamline Astreamline A--B indicated can be written asB indicated can be written as

g2VpH

g2Vpz

21

w

21A

A ++=+γ

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

g2Vhg2u

21

2

g2u)hPH(z

g2Vp 2

2wA

21A +−+=++

γ(26)(26)

Since the total head for any particle along the vertical sectionSince the total head for any particle along the vertical section (1) is (1) is the samethe same

(26)(26) ∫∫=

===

Hh

0h22 dhudAuQ l (27)(27)

113

Rectangular Weir Rectangular Weir –– FlowrateFlowrate 1/21/2

For a rectangular weir, For a rectangular weir, ll=b=b

(28)(28)

(28)(28) (29)(29)

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛+=⎟

⎟⎠

⎞⎜⎜⎝

⎛+= ∫

2/321

2/321

H

0

2/121

g2V

g2VHbg2

32dh

g2Vhbg2Q

Hg2

V21 <<

2/332 bHg2Q =

2/332

wr bHg2CQ =

Because of the numerous approximations made to obtain Because of the numerous approximations made to obtain EqEq. (29). (29)

(30)(30)

114

Rectangular Weir Rectangular Weir –– FlowrateFlowrate 2/22/2

CCwrwr is the rectangular weir coefficient. is the rectangular weir coefficient. CCwrwr is function of Reynolds number (viscous effects), Weber is function of Reynolds number (viscous effects), Weber number (surface tension effects), H/number (surface tension effects), H/PPww (geometry effects).(geometry effects).

In most practical situations, the Reynolds and Weber number In most practical situations, the Reynolds and Weber number effects are negligible, and the following correction can be usedeffects are negligible, and the following correction can be used..

(31)(31)⎟⎟⎠

⎞⎜⎜⎝

⎛+=

wwr P

H075.0611.0C

115

Triangular Weir Triangular Weir –– FlowrateFlowrate 1/21/2

An experimentally determined An experimentally determined triangular weir coefficient, Ctriangular weir coefficient, Cwtwt, is , is used to account for the real world effects neglected in the analused to account for the real world effects neglected in the analysis ysis so thatso that

(32)(32)

For a triangular weirFor a triangular weir

2/5Hg22

tan158Q θ

=

2tan)hH(2 θ−=l

Hg2

V21 <<

2/5wt Hg2

2tan

158CQ θ

=

116

Triangular Weir Triangular Weir –– FlowrateFlowrate 2/22/2

Weir coefficient for Weir coefficient for triangular sharptriangular sharp--crested crested weirsweirs

117

About About NappeNappe

Flow conditions over a weir without a free Flow conditions over a weir without a free nappenappe: (: (aa) plunging ) plunging nappenappe, , ((bb) submerged ) submerged nappenappe..

FlowrateFlowrate over a weir depends on whether the over a weir depends on whether the napplenapple is free or is free or submerged.submerged.

FlowrateFlowrate will be different for these situations will be different for these situations than that give by than that give by EqEq. (30) and (32).. (30) and (32).

118

BroadBroad--Crested Weir Crested Weir 1/31/3

A broadA broad--crested weir is a structure in an open channel that has a crested weir is a structure in an open channel that has a horizontal crest above which the fluid pressure may be considerehorizontal crest above which the fluid pressure may be considered d hydrostatic.hydrostatic.

119


Generally, these weirs are restricted to the range 0.08 < H/Generally, these weirs are restricted to the range 0.08 < H/LLww < 0.50.< 0.50.For long weir block (H/For long weir block (H/LLww < 0.08), head losses across the weir < 0.08), head losses across the weir cannot be neglected.cannot be neglected.For short weir block (H/For short weir block (H/LLww > 0.50), the streamlines of the flow over > 0.50), the streamlines of the flow over the weir are not horizontal.the weir are not horizontal.Apply the Bernoulli equationApply the Bernoulli equation

g2VPy

g2VPH

2c

wc

21

w ++=++

g2V

g2VVyH

2c

21

2c

c =−

=−

If the upstream velocity head is negligibleIf the upstream velocity head is negligible

120


H32y

2yyH c

cc ==−

cc2 gyVV ==SinceSince

2/32/3

2/3ccc22 H

32gbygbVbyVbyQ ⎟⎠⎞

⎜⎝⎛====The The flowrateflowrate isis

Again an empirical Again an empirical broadbroad--crested weir coefficient, crested weir coefficient, CCwbwb, is used to , is used to account for the real world effects neglected in the analysis so account for the real world effects neglected in the analysis so thatthat

2/32/3

wb H32gbCQ ⎟⎠⎞

⎜⎝⎛= 2/1

w

wb

PH1

65.0C

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=(33)(33) (34)(34)

121

Example 10.10 SharpExample 10.10 Sharp--Crested and broadCrested and broad--Crested WeirsCrested Weirs

Water flows in a rectangular channel of width b = 2 m with Water flows in a rectangular channel of width b = 2 m with flowrateflowratebetween between QQminmin = 0.02 m= 0.02 m33/s and /s and QQmaxmax = 0.60 m= 0.60 m33/s. This /s. This flowrateflowrate is to is to be measured by using (a) a rectangular sharpbe measured by using (a) a rectangular sharp--crested weir, (b) a crested weir, (b) a triangular sharptriangular sharp--crested weir with crested weir with θθ=90=90ºº, or (c) a broad, or (c) a broad--crested crested weir. In all cases the bottom of the flow area over the weir is weir. In all cases the bottom of the flow area over the weir is a a distance Pdistance Pww = 1 m above the channel bottom. Plot a graph of Q= = 1 m above the channel bottom. Plot a graph of Q= Q(H) for each weir and comment on which weir would be best for Q(H) for each weir and comment on which weir would be best for this application.this application.

122


For the rectangular weir with For the rectangular weir with PPww=1.=1.

2/3

2/332

w

2/332

wr

H)H075.0611.0(91.5Q

bHg2PH075.0611.0bHg2CQ

+=

⎟⎟⎠

⎞⎜⎜⎝

⎛+==(30)+(31)(30)+(31)

For the triangular weirFor the triangular weir

(32)(32) 2/5wt

2/5wt HC36.2Hg2

2tan

158CQ =

θ=

123


For the broadFor the broad--crested weircrested weir

(33)+(34)(33)+(34) 2/32/3

2/1

w

2/32/3

wb H32gb

PH1

65.0H32gbCQ ⎟

⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=⎟⎠⎞

⎜⎝⎛=

2/32/1 H

)H1(22.2Q

+=

124


125

Underflow Gates Underflow Gates 1/41/4

A variety of underflow gate structure is available for A variety of underflow gate structure is available for flowrateflowratecontrol at the crest of an overflow spillway, or at the entrancecontrol at the crest of an overflow spillway, or at the entrance of an of an irrigation canal or river from a lake.irrigation canal or river from a lake.

Three variations of underflow gates: (Three variations of underflow gates: (aa) vertical gate, () vertical gate, (bb) radial gate, () radial gate, (cc) drum gate.) drum gate.

126


The flow under the gate is said to be free outflow when the fluiThe flow under the gate is said to be free outflow when the fluid d issues as a jet of supercritical flow with a free surface open tissues as a jet of supercritical flow with a free surface open to the o the atmosphere.atmosphere.In such cases it is customary to write this In such cases it is customary to write this flowrateflowrate asas

1d gy2aCq = (35)(35)

Where q is the Where q is the flowrateflowrate per unit width.per unit width.

The discharge coefficient, The discharge coefficient, CCdd, is a function of contraction coefficient, , is a function of contraction coefficient, CCcc = y= y22/a, and the depth ration y/a, and the depth ration y11/a./a.

127


Typical discharge coefficients for underflow gatesTypical discharge coefficients for underflow gates

128


Drowned outflow from a sluice gate.Drowned outflow from a sluice gate.

The depth downstream of the gate is controlled by some downstreaThe depth downstream of the gate is controlled by some downstream m obstacle and the jet of water issuing from under the gate is oveobstacle and the jet of water issuing from under the gate is overlaid rlaid by a mass of water that is quite turbulent.by a mass of water that is quite turbulent.

129

Example 10.11 Sluice gateExample 10.11 Sluice gate

Water flows under the sluice gate shown in Fig. E10.11. The Water flows under the sluice gate shown in Fig. E10.11. The channel width is b = 20 ft, the upstream depth is ychannel width is b = 20 ft, the upstream depth is y11= 6 ft, and the = 6 ft, and the gate is a = 1.0 ft off the channel bottom. Plot a graph of gate is a = 1.0 ft off the channel bottom. Plot a graph of flowrateflowrate, Q, , Q, as a function of yas a function of y33..

130


(35)(35) cfsC393gy2baCbqq d1d ===

(Figure 10.29)(Figure 10.29)

cfs220cfs)56.0(393Q ==

Along the vertical line yAlong the vertical line y11/a=6./a=6.For yFor y33=6 ft, =6 ft, CCdd=0=0

The value of The value of CCdd increases as yincreases as y33/a decreases, reading a maximum of/a decreases, reading a maximum ofCCdd=0.56 when y=0.56 when y33/a=3.2. Thus with y/a=3.2. Thus with y33=3.2a=3.2ft=3.2a=3.2ft

For yFor y33 < 3.2 ft the < 3.2 ft the flowrateflowrate is independent of yis independent of y33, and the outflow , and the outflow is a free outflow.is a free outflow.

131


The The flowrateflowrate forfor3.2ft 3.2ft ≤≤ yy3 3 ≤≤ 6ft6ft

FUNDAMENTALS OF FLUID MECHANICS Chapter 10 Flow in Open ...

Documents

Transcript of FUNDAMENTALS OF FLUID MECHANICS Chapter 10 Flow in Open ...