1

M6a: O

pen C

hannel Flo

w

(Mannin

g’s

Equation, P

art

ially

Flo

win

g P

ipes, and S

pecific

Energ

y)

Robert

Pitt

Univ

ers

ity o

f A

labam

a

and

Shirle

y C

lark

Penn S

tate

-H

arr

isburg

Chin

2000

; F

igure

4.1

Ste

ady N

on-U

niform

Flo

w in a

n O

pen C

hannel

Continuity E

quation: V

1A

1= V

2A

2

Chin

2000

; F

igure

4.2

Ste

ady, N

on-U

niform

Flo

w in a

n O

pen C

hannel

The m

om

entu

m e

quation c

an b

e u

sed to d

eri

ve the e

xpre

ssio

n for

shear str

ess:

fo

RS

γτ

=γ

= s

pecific

weig

ht of

wate

r (6

2.4

lbs/f

t3)

R =

hydra

ulic

radiu

s (

ft)

Sf

= h

ydra

ulic

slo

pe

(ft

/ft)

(slo

pe o

f th

e e

nerg

y g

rad

e lin

e, or

the

fri

ction s

lope)

Mannin

g C

oeff

icie

nts

for

Open C

hannels

(Table

4.1

, C

hin

2000)

SI units (m

/s; m

)

U.S

. C

usto

mary

units (ft/s

ec; ft)

Only

valid for hydra

ulically r

ough flo

w, w

hen:

21

32

1fS

Rn

V=

21

32

486

.1

fS

Rn

V=

13

610

9.1

−≥

xRS

nf

(SI units)

Where

Sfis

the s

lope o

f th

e

energ

y g

rade lin

e (fr

iction

slo

pe). If th

e c

hannel slo

pe is

used, th

en im

plies u

niform

flow

, w

hic

h is rare

.

2

Mannin

g’s

Equation

Exam

ple

:

•D

ete

rmin

e the flo

w r

ate

in a

recta

ngula

r concre

te c

hannel

with a

wid

th o

f 3 m

and a

HG

L s

lope o

f 0.0

01 m

/mw

hen

the d

epth

of flow

is 1

.5 m

. A

ssum

e n

= 0

.014.

Giv

en:

n =

0.0

14 (

concre

te c

ha

nne

l)

L =

3 m

(w

idth

of

channel)

w =

1.5

m (d

epth

of

flow

)

sf=

0.0

01 m

/m

Mannin

g’s

Equation

•U

se t

he M

annin

g's

equation:

Need A

(cro

ss-s

ectional are

a o

f flow

):

A =

Lw

Substitu

ting:

A =

(3 m

)(1.5

m)

A =

4.5

m2

21

32

fS

RnA

Q=

Mannin

g’s

Equation

•N

eed R

(hydra

ulic

radiu

s):

R =

A/P

•N

eed P

, th

e w

etted p

erim

ete

r (n

ote

d o

n d

raw

ing b

y

thic

ker

lines). P =

L +

2w

Substitu

ting:

P =

(3 m

) +

2(1

.5 m

)

P =

6 m

Substitu

ting into

equation f

or

hydra

ulic

radiu

s:

R =

A/P

R =

(4.5

m2)/

(6 m

)

R =

0.7

5 m

Mannin

g’s

Equation

•S

ubstitu

ting into

Mannin

g's

equation:

sec

/39

.8

014

.0

)/

001

.0(

)75

.0

)(5.

4(

3

2/

13/

22

mQ

mm

mm

Q

==

3

Mannin

g’s

Equation

Exam

ple

:

•G

iven a

V-s

haped c

hannel w

ith a

HG

L s

lope o

f 0.0

01, a

top w

idth

of

10 feet, a

nd a

depth

of

5 feet, d

ete

rmin

e the

velo

city o

f flow

usin

g the M

annin

g’s

equation. F

ind the

dis

charg

e in b

oth

ft3

/sec (

cfs

) and m

3/s

ec (

cm

s).

Mannin

g’s

Equation

Substitu

ting:

R =

(25 ft2

)/14.1

4 ft

R =

1.7

7 ft

Assum

e t

hat

the c

hannel is

concre

te-lin

ed w

ith a

M

annin

g’s

n o

f 0.0

15.

Q =

114.9

cfs

= 1

15 ft3

/sec(0

.3048 m

/ft)

3=

32.5

m

3/s

ec

2/

13/

22

2/

13/

2

)001

.0(

)77

.1

)(25

(015

.0

49

.1

49

.1

ftft

Q

SAR

nQ

=

=

Mannin

g’s

Equation

Exam

ple

:

•F

ind the d

imensio

ns o

f a r

ecta

ngula

r concre

te c

hannel to

carr

y a

flo

w o

f 150 m

3/s

ec, w

ith a

HG

L s

lope o

f 0.0

15 a

nd

a m

ean v

elo

city o

f 10.2

m/s

ec.

Giv

en:

Q =

150 m

3/s

ec

V =

10.2

m/s

ec

Sf=

0.0

15

Assum

e: M

annin

g’s

n =

0.0

13 (

co

ncre

te c

hanne

l)

Mannin

g’s

Equation

•H

ave e

very

thin

g n

eeded t

o s

olv

e M

annin

g’s

equation for

the h

ydra

ulic

radiu

s, R

:

mR

mR

SVn

R

SVn

R

SRn

V

13

.1

)015

.0(

013

.0

sec)(

/2.

10

(

1

2/

3

2/

1

2/

3

2/

1

2/

1

3/

2

2/

13/

2

=

=

=

=

=

4

Mannin

g’s

Equation

By d

efinitio

n, R

= c

ross-s

ectional are

a o

f flow

/wetted p

eri

mete

r

By the C

ontin

uity E

qu

atio

n:

[]

1.13m

Depth

2

Base

th)

(Base)(D

epR

:ng

Substituti

flow)

of

2(D

epth

channel)

of

(B

ase

P

flow)

of

epth

channel)(D

of

(B

ase

A

=+

=

+==

))(

(7.

14

sec

/2.

10

sec

/150

))(

(

23

Depth

Base

mA

mmA

Depth

Base

VQA

===

==

Mannin

g’s

Equation

•H

ave tw

o e

quations a

nd t

wo u

nkno

wns:

Tw

o p

ossib

le s

olu

tions to q

uadra

tic (

both

are

corr

ect)

:

Base =

2.9

mD

epth

of

Flo

w =

5.0

4 m

Base =

10.1

mD

epth

of

Flo

w =

1.4

6 m

07.

14

13

2

13

27.

14

0.13

27.

14

0.13

2

27.14

13

.1

7.14

))(

(7.

14

2

2

2

2

=+

−

=+

=+

=+

+==

=

Depth

Depth

Depth

DepthDepth

Depth

mDepth

Base

Depth

Base

mm

Depth

Base

Depth

Base

m

Fig

ure

5-8

(C

ho

w 1

959)

can b

e u

sed to s

ignific

antly s

hort

en th

e

calc

ula

tion e

ffort

for

the d

esig

n o

f channels

. T

his

fig

ure

is u

sed to

calc

ula

te th

e n

orm

al d

epth

(y)

of a c

hanne

l based o

n th

e c

hann

el

sid

e s

lopes a

nd k

no

wn f

low

an

d c

hanne

l chara

cte

ristics, usin

g

the M

an

nin

g’s

eq

uation in the f

ollo

win

g f

orm

:

5.0

32

49

.1

S

nQ

AR

=

Initia

l chan

ne

l chara

cte

ristics that m

ust be k

now

inclu

de: z (

the s

ide

slo

pe),

an

d b

(th

e c

ha

nne

l botto

m w

idth

, assum

ing a

tra

pezoid

). It is

easy to e

xam

ine s

evera

l diffe

rent channel options (

z a

nd b

) b

y

calc

ula

ting the n

orm

al d

epth

(y)

for

a g

iven p

eak d

ischarg

e r

ate

,

channe

l slo

pe,

and r

ough

ness. T

he m

ost pra

ctical chan

ne

l can th

en b

e

sele

cte

d f

rom

the a

ltern

atives.

5

Com

posite M

annin

g’s

n E

stim

ate

, Exam

ple

4.3

(C

hin

2000)

A flo

odpla

in (next slide) can b

e d

ivid

ed into

seven s

ections a

s s

how

n

belo

w. U

se the v

arious form

ula

in the table

to e

stim

ate

the c

om

posite

roughness v

alu

e for th

is c

hannel.

Flo

odpla

in s

how

ing s

even s

epara

te s

ections c

orr

espondin

g to

diffe

rent valu

es o

f n.

Fig

ure

4.4

, C

hin

200

0

Each s

ection h

as the follow

ing g

eom

etr

ic c

hara

cte

ristics:

These v

alu

es a

re u

sed w

ith the p

rior equations to result in the

follow

ing e

stim

ate

s for M

annin

g’s

n:

The e

stim

ate

s o

f th

e c

om

posite n

valu

es c

an v

ary

consid

era

bly

,

resultin

g in s

imilar diffe

rences is p

redic

ted d

ischarg

es.

6

As s

how

n e

arlie

r, the M

annin

g’s

equation c

an

als

o b

e a

lso u

sed to p

redic

t flow

s in p

ipes.

Dra

inage s

yste

ms a

re typic

ally d

esig

ned a

s

open c

hannel flow

s in c

ircula

r pip

es, although

oth

er

cro

ss-s

ectional shapes a

re u

sed.

Chart

s o

r ta

ble

s c

an b

e u

sed to h

elp

pre

dic

t th

e

flow

conditio

ns in these s

yste

ms w

hen the

pip

es a

re n

ot flow

ing full.

Sew

ers

Flo

win

g P

art

ly F

ull

Fro

m: M

etc

alf

and

Ed

dy, In

c.

and

Ge

org

e

Tchob

an

oglo

us.

Wastewater

Engineering:

Collection and

Pumping of

Wastewater.

McG

raw

-Hill

,

Inc.

198

1.

Sew

ers

Flo

win

g P

art

ly F

ull

Fro

m: M

etc

alf a

nd

Ed

dy,

Inc.

an

d G

eorg

e

Tchob

an

oglo

us. Wastewater Engineering:

Collection and Pumping of Wastewater.

McG

raw

-Hill

, In

c.

198

1.

Sew

ers

Flo

win

g P

art

ly F

ull

Exam

ple

:

•D

ete

rmin

e the d

epth

of flow

and v

elo

city in a

sew

er

with a

dia

mete

r of

300 m

m h

avin

g a

HG

L s

lope o

f

0.0

05 m

/mw

ith a

n n

valu

e o

f 0.0

15 w

hen

dis

charg

ing 0

.01 m

3/s

ec.

Giv

en:

D =

300 m

m

Sf=

0.0

05 m

/m

n =

0.0

15

Q =

0.0

1 m

3/s

ec

7

Sew

ers

Flo

win

g P

art

ly F

ull

•U

se t

he m

odifie

d M

annin

g’s

equation f

or

part

ly f

ull

se

wers

: 0526

.0

'

)/

005

.0(

)3.

0(

sec)

/01

.0

)(015

.0(

'

:

'

:Re

'

2/

13/

8

3

2/

13/

8

2/

13/

8

===

=

K

mm

m

mK

ng

Substituti

SD

nQ

K

arranging

SD

nKQ

Se

wers

Flo

win

g P

art

ly F

ull

•U

sin

g T

able

2-5

(equation in t

erm

s o

f

dia

mete

r of

pip

e):

Clo

se t

o K′=

0.0

534

There

fore

, d/D

= 0

.28

•S

ubstitu

ting:

d/(

0.3

m)

= 0

.28

Depth

of

flo

w,

d =

0.0

84 m

Se

wers

Flo

win

g P

art

ly F

ull

•T

o c

alc

ula

te v

elo

city a

t depth

of

wate

r of

84 m

m, need t

o u

se c

ontinuity e

quatio

n:

Q =

VA

•U

sin

g M

ann

ing

’s p

art

ial flo

w d

iagra

m

(assum

ing a

consta

nt

n):

At

d/D

= 0

.28

Sew

ers

Flo

win

g P

art

ly F

ull

Fro

m: M

etc

alf a

nd

Ed

dy,

Inc.

an

d G

eorg

e T

chob

an

oglo

us. Wastewater Engineering:

Collection and Pumping of Wastewater.

McG

raw

-Hill

, In

c.

1981

.

d/D

= 0

.28

A/A

full= 0

.22

8

Sew

ers

Flo

win

g P

art

ly F

ull

•U

sin

g M

annin

g’s

part

ial flow

dia

gra

m

(assum

ing a

consta

nt

n):

At d/D

= 0

.28, A

/Afu

ll=

0.2

2

•C

alc

ula

te A

full.

()

2

22

0707

.0

3.0

44

mA

mD

A

full

full

=

=

=

ππ

Sew

ers

Flo

win

g P

art

ly F

ull

•S

ubstitu

ting:

•S

ubstitu

ting into

the c

ontinuity e

quation:

2

2

0156

.0

0707

.0

22

.0

mA

m

A

A

A full =

==

sec

/641

.0

)0156

.0(

sec

/01

.0

23

mV

mV

mVA

Q

=

=

=

In-C

lass P

roble

m (Part

ially

Flo

win

g S

ew

er)

•D

ete

rmin

e the d

epth

of flow

and v

elo

city in a

sew

er

with a

dia

mete

r of

600 m

m h

avin

g a

HG

L

slo

pe o

f 0.0

05 m

/mw

ith a

n n

valu

e o

f 0.0

13

when d

ischarg

ing 0

.055 m

3/s

ec.

Pra

suh

n 1

98

7

Rem

em

ber th

e p

roble

m h

avin

g tw

o “

corr

ect”

answ

ers

:

The s

pecific

energ

y d

iagra

m is u

sed to d

ete

rmin

e the m

ost likely

wate

r depth

.

9

Typic

al Specific

Energ

y D

iagra

m

(Fig

ure

4.5

, C

hin

2000)

g

Vy

E2

2

α+

=

If w

ate

r de

pth

is d

eep

er

than

th

e

cri

tical depth

(y c

), th

en t

he

flo

w is

subcri

tical.

If the w

ate

r d

epth

is s

hallo

wer

tha

n

the c

ritical de

pth

, th

en t

he f

low

is

superc

ritical.

When the w

ate

r de

pth

is c

lose t

o

cri

tical, s

mall

chan

ges in s

pecific

energ

y r

esults in larg

e d

epth

chan

ges,

resultin

g in

unsta

ble

and

excessiv

e w

ave a

ction.

Fr

should

be <

0.8

6 o

r >

1.1

3 t

o

pre

ven

t th

is.

Pra

suh

n 1

98

7

In t

he s

ubcri

tical zo

ne

, th

e

wate

r d

epth

com

pon

en

t is

much larg

er

than t

he

velo

city

hea

d

In t

he s

up

erc

ritical

zo

ne

, th

e v

elo

city h

ea

d

com

pon

en

t is

much

larg

er

than t

he w

ate

r

dep

th.

Pra

suh

n 1

98

7

In this

exam

ple

, th

e u

pstr

eam

flo

w is s

ubcritical (d

eeper th

an the c

ritical

depth

). W

hen the w

ate

r appro

aches a

n u

pw

ard

ste

p, th

e s

pecific

energ

y

calc

ula

tion r

esults in tw

o p

ossib

le w

ate

r depth

solu

tions. The c

orr

ect

dow

nstr

eam

wate

r depth

solu

tion is o

n the s

am

e lim

b o

f th

e s

pecific

energ

y

dia

gra

m a

s the u

pstr

eam

wate

r depth

. In

this

case, th

e flo

w is s

till s

ubcritical,

although the w

ate

r depth

actu

ally d

ecre

ases (but it c

annot pass thro

ugh the

critical depth

valu

e).

10

Pra

suh

n 1

98

7

In this

exam

ple

, th

e u

pstr

eam

flo

w is s

uperc

ritical (s

hallow

er th

an the c

ritical

depth

). W

hen the w

ate

r appro

aches a

n u

pw

ard

ste

p, th

e s

pecific

energ

y

calc

ula

tion r

esults in tw

o p

ossib

le w

ate

r depth

solu

tions. The c

orr

ect

dow

nstr

eam

wate

r depth

solu

tion is o

n the s

am

e lim

b o

f th

e s

pecific

energ

y

dia

gra

m a

s the u

pstr

eam

wate

r depth

. In

this

case, th

e flo

w is s

till

superc

ritical: the w

ate

r depth

incre

ases (but it c

annot pass thro

ugh the

critical depth

valu

e).

In a

recta

ngula

r channel, the c

ritical depth

can b

e e

asily c

alc

ula

ted

usin

g a

unit w

idth

flo

w rate

:

bQq=

31

2

=gq

yc

cc

yE

23=

Where

b is the w

idth

of th

e recta

ngula

r channel

The c

ritical flow

depth

can then b

e c

alc

ula

ted a

s:

and the m

inim

um

specific

energ

y in a

recta

ngula

r channel is

there

fore

:

Exam

ple

Pro

ble

m: D

ete

rmin

e the D

ow

nstr

eam

Wate

r

Depth

when A

ffecte

d b

y C

hannel B

ottom

Ris

e (ex. 7-3

,

Pra

suhn 1

987)

Dete

rmin

e the d

ow

nstr

eam

depth

in a

horizonta

l re

cta

ng

ula

r cha

nn

el

in w

hic

h the b

ott

om

ris

es 0

.5 f

t, if th

e s

tead

y f

low

dis

charg

e is 3

00 c

fs,

the c

hann

el w

idth

is 1

2 f

t, a

nd the u

pstr

eam

depth

is 4

ft.

ftcfs

ft

ft

bQq

/25

12

sec

/300

3

==

=

The d

ischarg

e p

er

unit w

idth

is:

()

ftft

ftcfs

gqyc

69

.2

sec

/2.

32

/25

31

2

2

31

2

=

=

=

The c

ritical depth

is there

fore

:

()

()(

)ft

ftft

ftcfs

ftgy

qy

H61

.4

24

sec

/2.

32

2

/25

42

2

2

2 1

2

101

=+

=+

=

2 2

2

201

02

2gy

qy

zH

H+

=∆

−=

The u

pstr

eam

depth

is there

fore

subcritical.

11

()

()(

)ft

ftft

ftcfs

ftgy

qy

H61

.4

24

sec

/2.

32

2

/25

42

2

2

2 1

2

101

=+

=+

=

2 2

2

201

02

2gy

qy

zH

H+

=∆

−=

()

()2 2

22

202

sec

/2.

32

2

/25

11

.4

5.0

61

.4

yft

ftcfs

yft

ftft

H+

==

−=

The u

pstr

eam

specific

energ

y is c

alc

ula

ted t

o b

e:

and the c

orr

espo

ndin

g d

ow

nstr

ea

m s

pecific

energ

y is:

Fro

m the s

ubcritical positio

n o

n the s

pecific

energ

y

dia

gra

m, th

e d

epth

and the w

ate

r surf

ace e

levation w

ill

both

decre

ase d

ow

nstr

eam

over

the “

bum

p”

in the

channel bottom

. T

here

fore

, y

2m

ust be g

reate

r th

an y

c

and less than y

1-∆

z:

2.6

9 ft <

y2

< 3

.5 f

t

Solv

ing the e

quation b

y ite

ration w

ithin

this

range r

esults

in the s

olu

tion o

f y

2=

3.0

9 f

t. T

he trial solu

tions c

an b

e

used to d

raw

in the s

pecific

energ

y d

iagra

m.

In-C

lass P

roble

m

Dete

rmin

e t

he d

ow

nstr

eam

depth

in a

horizonta

l re

cta

ngula

r channel in

wh

ich t

he

bott

om

ris

es 0

.75 f

t, if

the s

tead

y f

low

dis

charg

e is 5

50 c

fs,

the c

hannel w

idth

is

5 f

t, a

nd t

he u

pstr

eam

depth

is 6

ft. A

lso

dra

w t

he s

pecific

en

erg

y d

iagra

m f

or

this

pro

ble

m.

“C

hoke”

What happens w

hen ∆

z is incre

ased to a

gre

ate

r and

gre

ate

r valu

e

under

subcritical conditio

ns? A

s ∆

zin

cre

ases, H

o2

must als

o c

ontinue to

decre

ase. T

here

fore

, y

2decre

ases a

s w

ell.

The lim

it is r

each

ed

for

subcritical flow

when y

2equals

the c

ritical depth

at

whic

h p

oin

t th

e

transitio

n b

eco

mes a

“choke.”

A f

urt

her

incre

ase in ∆

z r

esults in

the

impossib

le s

ituation

where

Ho

2is

less than

Ho

min

(the

re w

ould

be n

o

positiv

e s

olu

tion to H

o2):

the

upstr

ea

m f

low

has insuff

icie

nt energ

y t

o

pass thro

ugh t

he tra

nsitio

n a

t th

e s

pecifie

d d

ischa

rge.

The f

low

will

no

t cease

, but

will

adju

st

itself to e

ither

a lo

wer

dis

charg

e o

r

an incre

ase in s

pecific

energ

y.

The f

low

will

lik

ely

not ch

ange d

ue to

upstr

ea

m f

low

sourc

es. T

he u

pstr

ea

m f

low

will

incre

ase

both

its

upstr

ea

m d

ep

th a

nd s

pe

cific

energ

y b

y m

ean

s o

f a g

entle s

well

or

a

series o

f sm

all

waves that

travel upstr

ea

m.

The n

ew

upstr

ea

m d

epth

will

be s

uch tha

t th

e f

low

can just p

ass the t

ransitio

n a

nd

y2

will

eq

ual y

c, and

Ho

2w

ill e

qual H

om

in. H

o1

will

exceed H

om

inb

y t

he v

alu

e o

f ∆

z . T

his

transitio

n is c

alle

d a

choke

sin

ce the c

ritical dep

th p

revails

rega

rdle

ss o

f

the incre

ase in

upstr

ea

m e

nerg

y.

12

During s

uperc

ritical flow

conditio

ns, th

e f

low

behavio

r is

diffe

rent as ∆

z incre

ases. A

choke o

ccurs

when the

min

imum

specific

energ

y is r

eached. H

ow

ever,

when

additio

na

l ∆

z o

ccurs

, a s

urg

e (

wall

of

wate

r) m

oves

upstr

eam

. W

hen e

quili

briu

m is r

eached, th

e s

uperc

ritical

flow

will

ha

ve b

een r

epla

ced b

y the id

entical subcri

tical flow

dis

cussed a

bove, and the tra

nsitio

n w

ill c

ontinue t

o a

ct as a

choke.

(sum

mari

zed f

rom

Pra

suhn

1987)