Open Channel FlowOpen Channel Flow

April 18, 2023April 18, 2023

Hydraulic radiusHydraulic radius

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 = ________

oP x

P

x sin

Dimensional analysisDimensional analysisRelationship between shear and velocity? ___________________

Geometric parameters ___________________ ___________________ ___________________

Write the functional relationship

Geometric parameters ___________________ ___________________ ___________________

Write the functional relationship

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

Re, , ,ph h

lC f

R Reæ ö

= ç ÷è øF,M,WRe, , ,p

h h

lC f

R Reæ ö

= ç ÷è øF,M,W

glVFglVF

ReVlrm

=

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

f2C

f

V

lgSS

2

f2C

f

V

lgSS

lShl f lShl f

lhp lhp Pressure CoefficientPressure Coefficient

Head loss coefficientHead loss coefficient

Friction slope coefficientFriction slope coefficient

(Energy Loss Coefficient)(Energy Loss Coefficient)

Friction slopeFriction slope

The friction slope is the slope of the EGL. The friction slope is The friction slope is the slope of the EGL. The friction slope is the same as the bottom slope (the same as the bottom slope (SSoo) for steady, uniform flow.) for steady, uniform flow.

Dimensional AnalysisDimensional Analysis

f, ,ReS

h h

lC f

R Reæ ö

= ç ÷è øf, ,ReS

h h

lC f

R Reæ ö

= ç ÷è ø

l

RC hSf

l

RC hSf

l

R

V

lgS h

2

f2 l

R

V

lgS h

2

f2

hRgS

V f2

hRgSV f2 hRS

gV f

2

hRS

gV f

2

f,ReS

h h

lC f

R Reæ ö

= ç ÷è øf,ReS

h h

lC f

R Reæ ö

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

f,Reh

Sh

RC f

l Re

læ ö

= =ç ÷è øf,Reh

Sh

RC f

l Re

læ ö

= =ç ÷è ø

2

f2f

V

lgSCS

2

f2f

V

lgSCS

(like f in Darcy-Weisbach)

2

f 2lh

l Vh S l

R gl= =

2

f 2lh

l Vh S l

R gl= =

Chezy formula

Open Channel Flow FormulasOpen Channel Flow Formulas

VAQ VAQ

2/13/21oh SAR

nQ 2/13/21

oh SARn

Q

1/2o

2/3h SR

1n

V 1/2o

2/3h SR

1n

V

hSRg

V

2 hSRg

V

2

Dimensions of n?

Is n only a function of roughness?

hSRCV hSRCV

Manning formula (MKS units!)

NO!

T /L1/3

Manning FormulaManning Formula

The Manning n is a function of the boundary roughness as well as other geometric parameters in some unknown way... ____________________ _______________________________

Hydraulic radius for wide channelsHydraulic radius for wide channels

The Manning n is a function of the boundary roughness as well as other geometric parameters in some unknown way... ____________________ _______________________________

Hydraulic radius for wide channelsHydraulic radius for wide channels

1/2o

2/3h SR

1n

V 1/2o

2/3h SR

1n

V

RAPh A bh

P b h 2

Rbhb hh 2

Channel curvature (bends)Cross section geometry

1 2

P1 < P2

Rh1 > Rh2

Why Use the Manning FormulaWhy Use the Manning Formula

Tradition Natural channels are geometrically complex and

the errors associated with using an equation that isn’t dimensionally correct are small compared with our inability to characterize stream geometry

Measurement errors for Q and h are large We only ever deal with water in channels, so we

don’t need to know how other fluids would respond

Tradition Natural channels are geometrically complex and

the errors associated with using an equation that isn’t dimensionally correct are small compared with our inability to characterize stream geometry

Measurement errors for Q and h are large We only ever deal with water in channels, so we

don’t need to know how other fluids would respond

Values of Manning nValues of Manning nLined Canals n

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

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

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

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

The worst channel has…

Roughness at many scales!

Example: Manning FormulaExample: Manning Formula

What is the flow capacity of a finished concrete channel that drops 1.2 m in 3 km?

What is the flow capacity of a finished concrete channel that drops 1.2 m in 3 km?

11

22

3 m3 m

1.5 m1.5 m

solution

Depth as f(Q)Depth as f(Q)

Find the depth in the channel when the flow is 5 m3/s

Hydraulic radius is function of depth Area is a function of depth Can’t solve explicitly Use trial and error or solver

Find the depth in the channel when the flow is 5 m3/s

Hydraulic radius is function of depth Area is a function of depth Can’t solve explicitly Use trial and error or solver

QnAR Sh o

1 2 3 1 2/ /

SummarySummary

Open channel flow equations can be obtained in a similar fashion to the Darcy-Weisbach equation (based on dimensional analysis)

The dimensionally incorrect Manning equation is the standard in English speaking countries

Open channel flow equations can be obtained in a similar fashion to the Darcy-Weisbach equation (based on dimensional analysis)

The dimensionally incorrect Manning equation is the standard in English speaking countries

Turbulent Flow Losses in Open Conduits

Turbulent Flow Losses in Open Conduits

Maximum shear stress

No shear stress

Hydraulic JumpHydraulic Jump

in in out outV y V y=in in out outV y V y=

yy11yy11

2 2

2 2in out

in out L

V Vy y h

g g+ = + +

2 2

2 2in out

in out L

V Vy y h

g g+ = + +

yy22

2 2

2 2in out

L

in out

V Vp py y h

g gg gæ ö æ ö

+ + = + + +ç ÷ ç ÷è ø è øEnergyEnergy

MassMass Per unit widthPer unit width

Unknown lossesUnknown losses

cs1cs1

cs2cs2

Hydraulic JumpHydraulic Jump

221122

212

1 ypypyVyV 221122

212

1 ypypyVyV

yy11yy11

yy22

22

22

21

22

212

1

yyyVyV

22

22

21

22

212

1

yyyVyV

gyVyy

y 12

12

112

222

gyVyy

y 12

12

112

222

MomentumMomentum

Much algebra...Much algebra...

ssxppxx FFFMMxx

2121 ssxppxx FFFMMxx

2121

21

1

yp

21

1

yp

ExampleExample

QnAR Sh o

1 2 3 1 2/ /

Q m m1

0 0129 0 927 0 00042 2 3 1 2

.. .

/ /c ha f a f

RAP

mh 0 927.A m m m m m 3 15 3 15 9 2afa f a fa f. .

P m m m m 3 2 3 15 9 712 2 2af a f a f. .

n 0 012.

Q m s 14 3 3. /

Smm0

123000

0 0004 .

.

Grand Coulee DamGrand Coulee Dam

http://users.owt.com/chubbard/gcdam/html/gallery.html

Columbia Basin ProjectColumbia Basin Project

The Columbia Basin Project is a major water resource development in central Washington State with Grand Coulee Dam as the project's primary feature. Water stored behind Grand Coulee Dam is lifted by giant pumps into the Banks Lake Feeder Canal and then into Banks Lake. The water stored in Banks Lake is used to irrigate 0.5 million acres of land stretching 125 miles from Grand Coulee Dam.

The Columbia Basin Project is a major water resource development in central Washington State with Grand Coulee Dam as the project's primary feature. Water stored behind Grand Coulee Dam is lifted by giant pumps into the Banks Lake Feeder Canal and then into Banks Lake. The water stored in Banks Lake is used to irrigate 0.5 million acres of land stretching 125 miles from Grand Coulee Dam.

PumpsPumps

At the time of original construction the pumping plant contained six 65,000 horsepower pumps. In 1973 work began on extending the plant. The pump bay was doubled in length to the south and six 67,500 horsepower pump/generators were added (the last in 1983) providing 12 pumps in all.

Each pump lifts water from Lake Roosevelt up through a 12 foot diameter discharge pipe to the feeder canal above. For most of their length the discharge pipes are buried in the rocky cliff to the west but at the top of the hill they emerge and can be seen as 12 silver pipes leading to the headworks of the feeder canal. The original pumps can supply water to the feeder canal at a rate of 1,600 cubic feet of water a second while the newer units can supply 2,000 cubic feet of water a second. They also have the advantage of being reversible. During times of peak power need the new pumps can be reversed thus turning them into generators. Water flows back down through the outlet pipes, through the generators and into Lake Roosevelt. When operating in this mode each pump can produce 50 megawatts of electrical power.

At the time of original construction the pumping plant contained six 65,000 horsepower pumps. In 1973 work began on extending the plant. The pump bay was doubled in length to the south and six 67,500 horsepower pump/generators were added (the last in 1983) providing 12 pumps in all.

Each pump lifts water from Lake Roosevelt up through a 12 foot diameter discharge pipe to the feeder canal above. For most of their length the discharge pipes are buried in the rocky cliff to the west but at the top of the hill they emerge and can be seen as 12 silver pipes leading to the headworks of the feeder canal. The original pumps can supply water to the feeder canal at a rate of 1,600 cubic feet of water a second while the newer units can supply 2,000 cubic feet of water a second. They also have the advantage of being reversible. During times of peak power need the new pumps can be reversed thus turning them into generators. Water flows back down through the outlet pipes, through the generators and into Lake Roosevelt. When operating in this mode each pump can produce 50 megawatts of electrical power.

Grand Coulee Feeder CanalGrand Coulee Feeder Canal

The Grand Coulee Feeder Canal is a concrete lined canal which runs from the outlet of the pumping plant discharge tubes to the north end of Banks Lake. The original canal was completed in 1951 but has since been widened to accommodate the extra water available from the six new pump/generators added to the pumping plant. The canal is 1.8 miles in length, 25 feet deep and 80 feet wide at the base. It has the capacity to carry 16,000 cubic feet of water per second.

The Grand Coulee Feeder Canal is a concrete lined canal which runs from the outlet of the pumping plant discharge tubes to the north end of Banks Lake. The original canal was completed in 1951 but has since been widened to accommodate the extra water available from the six new pump/generators added to the pumping plant. The canal is 1.8 miles in length, 25 feet deep and 80 feet wide at the base. It has the capacity to carry 16,000 cubic feet of water per second.

Columbia Basin Irrigation Project Columbia Basin Irrigation Project

Unsteady Hydraulics!Unsteady Hydraulics!

The base width of the feeder canal was increased from 50 to 80 feet; however, the operating capacity remained at 16,000 cubic feet per second. Water depth was reduced from 25 to about 20 feet to safely accommodate wave action when the water flow is reversed as the pump-generators are changed from pumping to generating and vice-versa.

The base width of the feeder canal was increased from 50 to 80 feet; however, the operating capacity remained at 16,000 cubic feet per second. Water depth was reduced from 25 to about 20 feet to safely accommodate wave action when the water flow is reversed as the pump-generators are changed from pumping to generating and vice-versa.

GatesGates

GatesGates

Banks LakeBanks Lake