Monroe L. Weber-Shirk School of Civil and

Environmental Engineering

Open Channel FlowOpen Channel Flow

Open Channel FlowOpen Channel Flow Liquid (water) flow with a ____ ________

(interface between water and air) relevant for

natural channels: rivers, streams engineered channels: canals, sewer

lines or culverts (partially full), storm drains

of interest to hydraulic engineers location of free surface velocity distribution discharge - stage (______) relationships optimal channel design

Liquid (water) flow with a ____ ________ (interface between water and air)

relevant for natural channels: rivers, streams engineered channels: canals, sewer

lines or culverts (partially full), storm drains

of interest to hydraulic engineers location of free surface velocity distribution discharge - stage (______) relationships optimal channel design

free surfacefree surface

depthdepth

Topics in Open Channel FlowTopics in Open Channel Flow

Uniform Flow Discharge-Depth relationships

Channel transitions Control structures (sluice gates, weirs…) Rapid changes in bottom elevation or cross section

Critical, Subcritical and Supercritical Flow Hydraulic Jump Gradually Varied Flow

Classification of flows Surface profiles

Uniform Flow Discharge-Depth relationships

Channel transitions Control structures (sluice gates, weirs…) Rapid changes in bottom elevation or cross section

Critical, Subcritical and Supercritical Flow Hydraulic Jump Gradually Varied Flow

Classification of flows Surface profiles

normal depthnormal depth

Classification of FlowsClassification of Flows Steady and Unsteady

Steady: velocity at a given point does not change with time

Uniform, Gradually Varied, and Nonuniform Uniform: velocity at a given time does not change

within a given length of a channel Gradually varied: gradual changes in velocity with

distance

Laminar and Turbulent Laminar: flow appears to be as a movement of thin

layers on top of each other Turbulent: packets of liquid move in irregular paths

Steady and Unsteady Steady: velocity at a given point does not change with

time

Uniform, Gradually Varied, and Nonuniform Uniform: velocity at a given time does not change

within a given length of a channel Gradually varied: gradual changes in velocity with

distance

Laminar and Turbulent Laminar: flow appears to be as a movement of thin

layers on top of each other Turbulent: packets of liquid move in irregular paths

(Temporal)(Temporal)

(Spatial)(Spatial)

Momentum and Energy Equations

Momentum and Energy Equations

Conservation of Energy “losses” due to conversion of turbulence to heat useful when energy losses are known or small

____________

Must account for losses if applied over long distances _______________________________________________

Conservation of Momentum “losses” due to shear at the boundaries useful when energy losses are unknown

____________

Conservation of Energy “losses” due to conversion of turbulence to heat useful when energy losses are known or small

____________

Must account for losses if applied over long distances _______________________________________________

Conservation of Momentum “losses” due to shear at the boundaries useful when energy losses are unknown

____________

ContractionsContractions

ExpansionExpansion

We need an equation for lossesWe need an equation for losses

Given a long channel of constant slope and cross section find the relationship between discharge and depth

Assume Steady Uniform Flow - ___ _____________ prismatic channel (no change in _________ with distance)

Use Energy and Momentum, Empirical or Dimensional Analysis?

What controls depth given a discharge? Why doesn’t the flow accelerate?

Given a long channel of constant slope and cross section find the relationship between discharge and depth

Assume Steady Uniform Flow - ___ _____________ prismatic channel (no change in _________ with distance)

Use Energy and Momentum, Empirical or Dimensional Analysis?

What controls depth given a discharge? Why doesn’t the flow accelerate?

Open Channel Flow: Discharge/Depth Relationship

Open Channel Flow: Discharge/Depth Relationship

PP

no accelerationno accelerationgeometrygeometry

Force balanceForce balance

AA

Steady-Uniform Flow: Force Balance

Steady-Uniform Flow: Force Balance

W

W sin

x

a

b

c

d

Shear force

Energy grade line

Hydraulic grade line

Shear force =________

0sin xPxA o 0sin xPxA o

sin P

Ao sin

P

Ao

hR = P

AhR =

P

A

sin

cos

sin S

sin

cos

sin S

W cos

g

V

2

2

g

V

2

2

Wetted perimeter = __

Gravitational force = ________

Hydraulic radiusHydraulic radius

Relationship between shear and velocity? ______________

oP x

P

A x sin

TurbulenceTurbulence

Geometric parameters ___________________ ___________________ ___________________

Write the functional relationship

Does Fr affect shear? _________

Geometric parameters ___________________ ___________________ ___________________

Write the functional relationship

Does Fr affect shear? _________

P

ARh

P

ARh Hydraulic radius (Rh)Hydraulic radius (Rh)

Channel length (l)Channel length (l)

Roughness ()Roughness ()

Open Conduits:Dimensional Analysis

Open Conduits:Dimensional Analysis

f , ,Re,ph h

lC r

R Reæ ö

= ç ÷è øF ,M, Wf , ,Re,p

h h

lC r

R Reæ ö

= ç ÷è øF ,M, W

VFr

yg=No!No!

Pressure Coefficient for Open Channel Flow?

Pressure Coefficient for Open Channel Flow?

2

2C

V

pp

2

2C

V

pp

2

2C

V

ghlhl

2

2C

V

ghlhl

2

2C

f

fS

gS l

V= 2

2C

f

fS

gS l

V=

l fh S l=l fh S l=

lhp lhp Pressure CoefficientPressure Coefficient

Head loss coefficientHead loss coefficient

Friction slope coefficientFriction slope coefficient

(Energy Loss Coefficient)(Energy Loss Coefficient)

Friction slopeFriction slope

Slope of EGLSlope of EGL

Dimensional AnalysisDimensional Analysis

f , ,RefS

h h

lC

R Reæ ö

= ç ÷è øf , ,Re

fSh h

lC

R Reæ ö

= ç ÷è ø

f

hS

RC

ll=

f

hS

RC

ll=

2

2 f hgS l RV l

l=2

2 f hgS l RV l

l=2 f hgS R

Vl

=2 f hgS R

Vl

=2

f h

gV S R

l=

2f h

gV S R

l=

f ,RefS

h h

lC

R Reæ ö

= ç ÷è øf ,Re

fSh h

lC

R Reæ ö

= ç ÷è øHead loss length of channelHead loss length of channel

f ,Ref

hS

h

RC

l Re

læ ö

= =ç ÷è øf ,Re

f

hS

h

RC

l Re

læ ö

= =ç ÷è ø

2

2f

fS

gS lC

V= 2

2f

fS

gS lC

V=

(like f in Darcy-Weisbach)

Chezy equation (1768)Chezy equation (1768)

Introduced by the French engineer Antoine Chezy in 1768 while designing a canal for the water-supply system of Paris

Introduced by the French engineer Antoine Chezy in 1768 while designing a canal for the water-supply system of Paris

h fV C R S= h fV C R S=

150 < C < 60s

m

s

m 150 < C < 60

s

m

s

mwhere C = Chezy coefficientwhere C = Chezy coefficient

where 60 is for rough and 150 is for smoothalso a function of R (like f in Darcy-Weisbach)

where 60 is for rough and 150 is for smoothalso a function of R (like f in Darcy-Weisbach)

2f h

gV S R

l=

2f h

gV S R

l=compare

Darcy-Weisbach equation (1840)Darcy-Weisbach equation (1840)

where d84 = rock size larger than 84% of the rocks in a random sample

where d84 = rock size larger than 84% of the rocks in a random sample

For rock-bedded streamsFor rock-bedded streams

f = Darcy-Weisbach friction factorf = Darcy-Weisbach friction factor

2

84

1

1.2 2.03log h

fRd

=æ öé ù

+ ê úç ÷è øë û

2

84

1

1.2 2.03log h

fRd

=æ öé ù

+ ê úç ÷è øë û

4

4

P

2

d

d

d

ARh

4

4

P

2

d

d

d

ARh

2

2l

l Vh f

d g=

2

2l

l Vh f

d g=

2

4 2lh

l Vh f

R g=

2

4 2lh

l Vh f

R g=

2

4 2fh

l VS l f

R g=

2

4 2fh

l VS l f

R g=

2

8f h

VS R f

g=

2

8f h

VS R f

g=

8f h

gV S R

f=

8f h

gV S R

f=

Manning Equation (1891)Manning Equation (1891)

Most popular in U.S. for open channels Most popular in U.S. for open channels

(English system)(English system)

1/2o

2/3h SR

1

nV 1/2

o2/3h SR

1

nV

1/2o

2/3h SR

49.1

nV 1/2

o2/3h SR

49.1

nV

VAQ VAQ

2/13/21oh SAR

nQ 2/13/21

oh SARn

Q very sensitive to nvery sensitive to n

Dimensions of Dimensions of nn??Dimensions of Dimensions of nn??

Is Is nn only a function of roughness? only a function of roughness?Is Is nn only a function of roughness? only a function of roughness?

(MKS units!)(MKS units!)

NO!NO!

T /L1/3T /L1/3

Bottom slopeBottom slope

Values of Manning nValues of Manning nLined Canals n

Cement plaster 0.011 Untreated gunite 0.016 Wood, planed 0.012 Wood, unplaned 0.013 Concrete, trowled 0.012 Concrete, wood forms, unfinished 0.015 Rubble in cement 0.020 Asphalt, smooth 0.013 Asphalt, rough 0.016

Natural Channels Gravel beds, straight 0.025 Gravel beds plus large boulders 0.040 Earth, straight, with some grass 0.026 Earth, winding, no vegetation 0.030 Earth , winding with vegetation 0.050

Lined Canals n Cement plaster 0.011 Untreated gunite 0.016 Wood, planed 0.012 Wood, unplaned 0.013 Concrete, trowled 0.012 Concrete, wood forms, unfinished 0.015 Rubble in cement 0.020 Asphalt, smooth 0.013 Asphalt, rough 0.016

Natural Channels Gravel beds, straight 0.025 Gravel beds plus large boulders 0.040 Earth, straight, with some grass 0.026 Earth, winding, no vegetation 0.030 Earth , winding with vegetation 0.050

d = median size of bed materiald = median size of bed material

n = f(surface roughness, channel irregularity, stage...)

n = f(surface roughness, channel irregularity, stage...)

6/1038.0 dn 6/1038.0 dn

6/1031.0 dn 6/1031.0 dn d in ftd in ftd in md in m

Trapezoidal ChannelTrapezoidal Channel

Derive P = f(y) and A = f(y) for a trapezoidal channel

How would you obtain y = f(Q)?

Derive P = f(y) and A = f(y) for a trapezoidal channel

How would you obtain y = f(Q)?

z1

b

yzyybA 2 zyybA 2

byzyP 2/1222 byzyP 2/1222

bzyP 2/1212 bzyP 2/1212

2/13/21oh SAR

nQ 2/13/21

oh SARn

Q

Use Solver!Use Solver!

Flow in Round ConduitsFlow in Round Conduits

r

yrarccos

r

yrarccos

cossin2 rA cossin2 rA

sin2rT sin2rT

y

T

A

r

rP 2 rP 2

radiansradians

Maximum discharge when y = ______0.938d0.938d

( ) ( )sin cosr rq q=( ) ( )sin cosr rq q=

Open Channel Flow: Energy Relations

Open Channel Flow: Energy Relations

2g

V21

1

2g

V22

2

xSo

2y1y

x

L fh S x= Denergy grade linehydraulic grade line

velocity head

Bottom slope (So) not necessarily equal to surface slope (Sf)

water surfacewater surface

Energy relationshipsEnergy relationships

2 21 1 2 2

1 1 2 22 2 L

p V p Vz z h

g ga a

g g+ + = + + +

2 21 2

1 22 2o f

V Vy S x y S x

g g+ D + = + + D

Turbulent flow ( 1)

z - measured from horizontal datum

y - depth of flow

Pipe flow

Energy Equation for Open Channel Flow

2 21 2

1 22 2o f

V Vy S x y S x

g g+ + D = + + D

From diagram on previous slide...

Specific EnergySpecific Energy

The sum of the depth of flow and the velocity head is the specific energy:

The sum of the depth of flow and the velocity head is the specific energy:

g

VyE

2

2

If channel bottom is horizontal and no head loss21 EE

y - potential energy

g

V

2

2

- kinetic energy

For a change in bottom elevation1 2E y E- D =

xSExSE o f21

y

Specific EnergySpecific Energy

In a channel with constant discharge, Q

2211 VAVAQ

2

2

2gA

QyE

g

VyE

2

2

where A=f(y)

Consider rectangular channel (A=By) and Q=qB

2

2

2gy

qyE

A

B

y

3 roots (one is negative)

q is the discharge per unit width of channel

0123456789

10

0 1 2 3 4 5 6 7 8 9 10

E

y

Specific Energy: Sluice GateSpecific Energy: Sluice Gate

2

2

2gy

qyE

2

2

2gy

qyE

1

2

21 EE

sluice gatey1

y2

EGL

y1 and y2 are ___________ depths (same specific energy)

Why not use momentum conservation to find y1?

q = 5.5 m2/sy2 = 0.45 mV2 = 12.2 m/s

E2 = 8 m

alternatealternateGiven downstream depth and discharge, find upstream depth.

0

1

2

3

4

0 1 2 3 4

E

ySpecific Energy: Raise the Sluice

GateSpecific Energy: Raise the Sluice

Gate

2

2

2gy

qyE

1 2

E1 E2E1 E2

sluice gate

y1

y2

EGL

as sluice gate is raised y1 approaches y2 and E is minimized: Maximum discharge for given energy.

0

1

2

3

4

0 1 2 3 4

E

y

Specific Energy: Step UpSpecific Energy: Step UpShort, smooth step with rise y in channel

y

1 2E E y= +D

Given upstream depth and discharge find y2

0

1

2

3

4

0 1 2 3 4

E

y

Increase step height?

PPAA

Critical FlowCritical Flow

T

dy

y

T=surface width

Find critical depth, yc

2

2

2gA

QyE

0dy

dE

TdydA 0dEdy

= =

3

2

1c

c

gA

TQ

Arbitrary cross-section

A=f(y)

2

3

2

FrgA

TQ 22

FrgA

TV

dA

AD

T= Hydraulic Depth

2

31Q dAgA dy

-

0

1

2

3

4

0 1 2 3 4

E

y

yc

Critical Flow:Rectangular channel

Critical Flow:Rectangular channel

yc

T

Ac

3

2

1c

c

gA

TQ

qTQ TyA cc

3

2

33

32

1cc gy

q

Tgy

Tq

3/12

g

qyc

3cgyq

Only for rectangular channels!

cTT

Given the depth we can find the flow!

Critical Flow Relationships:Rectangular Channels

Critical Flow Relationships:Rectangular Channels

3/12

g

qyc cc yVq

g

yVy ccc

223

g

Vy cc

2

1gy

V

c

cFroude numberFroude number

velocity head =

because

g

Vy cc

22

2

2

cc

yyE Eyc

3

2

forcegravity force inertial

0.5 (depth)0.5 (depth)

g

VyE

2

2

Kinetic energyPotential energy

Critical FlowCritical Flow

Characteristics Unstable surface Series of standing waves

Occurrence Broad crested weir (and other weirs) Channel Controls (rapid changes in cross-section) Over falls Changes in channel slope from mild to steep

Used for flow measurements ___________________________________________

Characteristics Unstable surface Series of standing waves

Occurrence Broad crested weir (and other weirs) Channel Controls (rapid changes in cross-section) Over falls Changes in channel slope from mild to steep

Used for flow measurements ___________________________________________Unique relationship between depth and discharge

Difficult to measure depth

0

1

2

3

4

0 1 2 3 4

E

y

Broad-crested WeirBroad-crested Weir

H

Pyc

E

3cgyq 3

cQ b gy=

Eyc

3

2

3/ 23/ 22

3Q b g Eæö=

è ø

Cd corrects for using H rather than E.

3/12

g

qyc

Broad-crestedweir

E measured from top of weirE measured from top of weir

3/ 223dQ C b g Hæ ö=

è ø

Hard to measure yc

Broad-crested Weir: ExampleBroad-crested Weir: Example

Calculate the flow and the depth upstream. The channel is 3 m wide. Is H approximately equal to E?

Calculate the flow and the depth upstream. The channel is 3 m wide. Is H approximately equal to E?

0.5yc

E

Broad-crestedweir

yc=0.3 m

Solution

How do you find flow?____________________

How do you find H?______________________

Critical flow relation

Energy equation

Hydraulic JumpHydraulic Jump

Used for energy dissipation Occurs when flow transitions from

supercritical to subcritical base of spillway

We would like to know depth of water downstream from jump as well as the location of the jump

Which equation, Energy or Momentum?

Used for energy dissipation Occurs when flow transitions from

supercritical to subcritical base of spillway

We would like to know depth of water downstream from jump as well as the location of the jump

Which equation, Energy or Momentum?

Hydraulic Jump!Hydraulic Jump!

Hydraulic JumpHydraulic Jump

y1

y2

L

EGLhL

Conservation of Momentum

221121 ApApQVQV

2

gyp

A

QV

222211

2

2

1

2 AgyAgy

A

Q

A

Q

xx ppxx FFMM2121 xx ppxx FFMM

2121

12

11 AVM x 12

11 AVM x

22

22 AVM x 22

22 AVM x

sspp FFFWMM 2121 sspp FFFWMM 2121

Hydraulic Jump:Conjugate DepthsHydraulic Jump:

Conjugate Depths

Much algebra

For a rectangular channel make the following substitutionsByA 11VByQ

11

1

VFr

gy= Froude number

21

12 811

2Fr

yy

valid for slopes < 0.02

Hydraulic Jump:Energy Loss and Length

Hydraulic Jump:Energy Loss and Length

No general theoretical solution

Experiments show

26yL 204 1 Fr

•Length of jump

•Energy Loss

2

2

2gy

qyE

LhEE 21

significant energy loss (to turbulence) in jump

21

312

4 yy

yyhL

algebra

for

Gradually Varied FlowGradually Varied Flow2 2

1 21 22 2o f

V Vy S x y S x

g g+ + D = + + D Energy equation for

non-uniform, steady flow

12 yydy

2

2 f o

Vdy d S dx S dx

g

æ ö+ + =ç ÷è ø

PPAA

T

dy

y

dy

dxS

dy

dxS

g

V

dy

d

dy

dyof

2

2

( )2 2

2 12 1 2 2o f

V VS dx y y S dx

g g

æ ö= - + - +ç ÷è ø

Gradually Varied FlowGradually Varied Flow

2

3

2

3

2

2

22

2

2

22Fr

gA

TQ

dy

dA

gA

Q

gA

Q

dy

d

g

V

dy

d

fo SSFrdx

dy 21

dy

dxS

dy

dxS

g

V

dy

d

dy

dyof

2

2

dy

dxS

dy

dxSFr of 21

Change in KEChange in PEChange in KEChange in PE

We are holding Q constant!

21 Fr

SS

dx

dy fo

Gradually Varied FlowGradually Varied Flow

21 Fr

SS

dx

dy fo

Governing equation

for gradually varied flow

Gives change of water depth with distance along channel Note

So and Sf are positive when sloping down in direction of flow

y is measured from channel bottom dy/dx =0 means water depth is constant

Gives change of water depth with distance along channel Note

So and Sf are positive when sloping down in direction of flow

y is measured from channel bottom dy/dx =0 means water depth is constant

yn is when o fS S=

Surface ProfilesSurface Profiles Mild slope (yn>yc)

in a long channel subcritical flow will occur

Steep slope (yn<yc) in a long channel supercritical flow will occur

Critical slope (yn=yc) in a long channel unstable flow will occur

Horizontal slope (So=0) yn undefined

Adverse slope (So<0) yn undefined

Mild slope (yn>yc) in a long channel subcritical flow will occur

Steep slope (yn<yc) in a long channel supercritical flow will occur

Critical slope (yn=yc) in a long channel unstable flow will occur

Horizontal slope (So=0) yn undefined

Adverse slope (So<0) yn undefined Note: These slopes are f(Q)!

Normal depth

Steep slope (S2)

Hydraulic Jump

Sluice gateSteep slope

Obstruction

Surface ProfilesSurface Profiles

21 Fr

SS

dx

dy fo

S0 - Sf 1 - Fr2 dy/dx

+ + +

- + -

- - + 0

1

2

3

4

0 1 2 3 4

E

y

yn

yc

More Surface ProfilesMore Surface Profiles

S0 - Sf 1 - Fr2 dy/dx

1 + + +

2 + - -

3 - - + 0

1

2

3

4

0 1 2 3 4

E

y

yn

yc

21 Fr

SS

dx

dy fo

Direct Step MethodDirect Step Method

xSg

VyxS

g

Vy fo

22

22

2

21

1

of SS

g

V

g

Vyy

x

22

22

21

21

energy equation

solve for x

1

1

y

qV

2

2

y

qV

2

2

A

QV

1

1

A

QV

rectangular channel

prismatic channel

Direct Step MethodFriction Slope

Direct Step MethodFriction Slope

2 2

4/3fh

n VS

R=

2 2

4/32.22fh

n VS

R=

2

8fh

fVS

gR=

Manning Darcy-Weisbach

SI units

English units

Direct StepDirect Step

Limitation: channel must be _________ (so that velocity is a function of depth only and not a function of x)

Method identify type of profile (determines whether y is + or -) choose y and thus yn+1

calculate hydraulic radius and velocity at yn and yn+1

calculate friction slope yn and yn+1 calculate average friction slope calculate x

Limitation: channel must be _________ (so that velocity is a function of depth only and not a function of x)

Method identify type of profile (determines whether y is + or -) choose y and thus yn+1

calculate hydraulic radius and velocity at yn and yn+1

calculate friction slope yn and yn+1 calculate average friction slope calculate x

prismaticprismatic

Direct Step MethodDirect Step Method

=(G16-G15)/((F15+F16)/2-So)

A B C D E F G H I J K L My A P Rh V Sf E Dx x T Fr bottom surface0.900 1.799 4.223 0.426 0.139 0.00004 0.901 0 3.799 0.065 0.000 0.9000.870 1.687 4.089 0.412 0.148 0.00005 0.871 0.498 0.5 3.679 0.070 0.030 0.900

=y*b+y^2*z

=2*y*(1+z^2)^0.5 +b

=A/P

=Q/A

=(n*V)^2/Rh^(4/3)

=y+(V^2)/(2*g)

of SS

g

V

g

Vyy

x

22

22

21

21

Standard StepStandard Step

Given a depth at one location, determine the depth at a second location

Step size (x) must be small enough so that changes in water depth aren’t very large. Otherwise estimates of the friction slope and the velocity head are inaccurate

Can solve in upstream or downstream direction upstream for subcritical downstream for supercritical

Find a depth that satisfies the energy equation

Given a depth at one location, determine the depth at a second location

Step size (x) must be small enough so that changes in water depth aren’t very large. Otherwise estimates of the friction slope and the velocity head are inaccurate

Can solve in upstream or downstream direction upstream for subcritical downstream for supercritical

Find a depth that satisfies the energy equation

xSg

VyxS

g

Vy fo

22

22

2

21

1

What curves are available?What curves are available?

S1

S3

Is there a curve between yc and yn that decreases in depth in the upstream direction?

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

05101520

distance upstream (m)

elev

atio

n (m

)

bottom

surface

yc

yn

Wave CelerityWave Celerity

21

2

1gyF 2

2

2

1yygF

2221

2

1yyygFFFr

F1

y+yV+VV

Vw

unsteady flow

y y y+yV+V-VwV-Vw

steady flow

V+V-VwV-Vw

F2

Wave Celerity:Momentum Conservation

Wave Celerity:Momentum Conservation

wwwr VVVVVVVyF

VVVyF wr

VVVyyyyg w 22

2

1

VVVyyyg w 22

1

VVVyg w

y y+yV+V-VwV-Vw

steady flow

12 MMFr yVVM w2

1 Per unit widthPer unit width

Wave CelerityWave Celerity

ww VVVyyVVy

www yVyVVyVyyVyVyVyV

y

yVVV w

VVVyg w

y

yVVyg w

2

2wVVgy wVVc gyc

Mass conservation

y y+yV+V-VwV-Vw

steady flow

Momentum

c

VFr

yg

V

Wave PropagationWave Propagation

Supercritical flow c<V waves only propagate downstream water doesn’t “know” what is happening downstream _________ control

Critical flow c=V

Subcritical flow c>V waves propagate both upstream and downstream

Supercritical flow c<V waves only propagate downstream water doesn’t “know” what is happening downstream _________ control

Critical flow c=V

Subcritical flow c>V waves propagate both upstream and downstream

upstreamupstream

Most Efficient Hydraulic Sections

Most Efficient Hydraulic Sections

A section that gives maximum discharge for a specified flow area Minimum perimeter per area

No frictional losses on the free surface Analogy to pipe flow Best shapes

best best with 2 sides best with 3 sides

A section that gives maximum discharge for a specified flow area Minimum perimeter per area

No frictional losses on the free surface Analogy to pipe flow Best shapes

best best with 2 sides best with 3 sides

Why isn’t the most efficient hydraulic section the best design?

Why isn’t the most efficient hydraulic section the best design?

Minimum area = least excavation only if top of channel is at gradeMinimum area = least excavation only if top of channel is at grade

Cost of linerCost of liner

Complexity of form workComplexity of form work

Erosion constraint - stability of side wallsErosion constraint - stability of side walls

Open Channel Flow Discharge Measurements

Open Channel Flow Discharge Measurements

Discharge Weir

broad crested sharp crested triangular

Venturi Flume Spillways Sluice gates

Velocity-Area-Integration

Discharge Weir

broad crested sharp crested triangular

Venturi Flume Spillways Sluice gates

Velocity-Area-Integration

Discharge MeasurementsDischarge Measurements

Sharp-Crested Weir

Triangular Weir

Broad-Crested Weir

Sluice Gate

Sharp-Crested Weir

Triangular Weir

Broad-Crested Weir

Sluice Gate

5/ 282 tan

15 2dQ C g Hqæ ö=

è ø

3/ 223dQ C b g Hæ ö=

è ø

3/ 222

3 dQ C b gH=

12d gQ C by gy=

Explain the exponents of H!

SummarySummary

All the complications of pipe flow plus additional parameter... _________________

Various descriptions of head loss term Chezy, Manning, Darcy-Weisbach

Importance of Froude Number Fr>1 decrease in E gives increase in y Fr<1 decrease in E gives decrease in y Fr=1 standing waves (also min E given Q)

Methods of calculating location of free surface

All the complications of pipe flow plus additional parameter... _________________

Various descriptions of head loss term Chezy, Manning, Darcy-Weisbach

Importance of Froude Number Fr>1 decrease in E gives increase in y Fr<1 decrease in E gives decrease in y Fr=1 standing waves (also min E given Q)

Methods of calculating location of free surface

free surface locationfree surface location

0

1

2

3

4

0 1 2 3 4

E

y

Broad-crested Weir: SolutionBroad-crested Weir: Solution

0.5yc

E

Broad-crestedweir

yc=0.3 m

3cgyq

32 3.0)/8.9( msmq

smq /5144.0 2

smqLQ /54.1 3Eyc

3

2myE c 45.0

2

32

1 2 0.95E E y m= +D =

21

2

112gy

qyE

435.05.011 myH935.01 y

2

1 1212

qE y

gE- @

Sluice GateSluice Gate

2 m

10 cm

S = 0.005

Sluice gatereservoir

Summary/OverviewSummary/Overview

Energy lossesDimensional AnalysisEmpirical

Energy lossesDimensional AnalysisEmpirical

8f h

gV S R

f=

8f h

gV S R

f=

1/2o

2/3h SR

1

nV 1/2

o2/3h SR

1

nV

Energy EquationEnergy Equation

Specific Energy Two depths with same energy!

How do we know which depth is the right one?

Is the path to the new depth possible?

Specific Energy Two depths with same energy!

How do we know which depth is the right one?

Is the path to the new depth possible?

2 21 2

1 22 2o f

V Vy S x y S x

g g+ + D = + + D

2

22q

ygy

= +g

VyE

2

2

2

22Q

ygA

= +

0

1

2

3

4

0 1 2 3 4

E

y

0

1

2

3

4

0 1 2 3 4

E

y

Specific Energy: Step UpSpecific Energy: Step UpShort, smooth step with rise y in channel

y

1 2E E y= +D

Given upstream depth and discharge find y2

0

1

2

3

4

0 1 2 3 4

E

y

Increase step height?

Critical DepthCritical Depth

Minimum energy for qWhenWhen kinetic = potential!Fr=1

Fr>1 = SupercriticalFr<1 = Subcritical

Minimum energy for qWhenWhen kinetic = potential!Fr=1

Fr>1 = SupercriticalFr<1 = Subcritical

0dy

dE

0

1

2

3

4

0 1 2 3 4

E

y

g

Vy cc

22

2

3

TQ

gA=

3c

q

gy=c

c

VFr

y g=

What next?What next?

Water surface profiles Rapidly varied flow

A way to move from supercritical to subcritical flow (Hydraulic Jump)

Gradually varied flow equations Surface profiles Direct step Standard step

Water surface profiles Rapidly varied flow

A way to move from supercritical to subcritical flow (Hydraulic Jump)

Gradually varied flow equations Surface profiles Direct step Standard step