11 Open Channel

8/18/2019 11 Open Channel

1/62

Monroe L. Weber-Shirk School of Civil and

Environmental Engineering

Open Channel Flow

http://ceeserver.cee.cornell.edu/mw24/Default.htmhttp://www.cee.cornell.edu/index.cfmhttp://www.cee.cornell.edu/index.cfmhttp://www.cee.cornell.edu/index.cfmhttp://www.cee.cornell.edu/index.cfmhttp://www.cee.cornell.edu/index.cfmhttp://www.cee.cornell.edu/faculty/info.cfm?abbrev=faculty&shorttitle=bio&netid=mw24http://www.cornell.edu/http://www.cee.cornell.edu/index.cfmhttp://www.cee.cornell.edu/index.cfmhttp://www.cee.cornell.edu/index.cfmhttp://www.cee.cornell.edu/index.cfmhttp://www.cee.cornell.edu/index.cfmhttp://ceeserver.cee.cornell.edu/mw24/Default.htm


2/62

Open Channel Flow

Liqid !water" flow with a #### ########

!interface between water and air"

relevant for

natral channel$% river$& $tream$

engineered channel$% canal$& $ewer

line$ or clvert$ !partiall' fll"& $torm drain$

of intere$t to h'dralic engineer$ location of free $rface

velocit' di$tribtion

di$charge - $tage !######" relation$hip$

optimal channel de$ign

free surface

depth

http://e/CEE%20331/Lectures/Viscous%20Flow%20open%20channel.ppt


3/62

(opic$ in Open Channel Flow

)niform Flow *i$charge-*epth relation$hip$

Channel tran$ition$ Control $trctre$ !$lice gate$& weir$+" ,apid change$ in bottom elevation or cro$$ $ection

Critical& Sbcritical and Spercritical Flow

'dralic mp /radall' 0aried Flow

Cla$$ification of flow$

Srface profile$

normal depth


4/62

Cla$$ification of Flow$

Stead' and )n$tead'

Stead'% velocit' at a given point doe$ not change with

time

)niform& /radall' 0aried& and 1onniform )niform% velocit' at a given time doe$ not change

within a given length of a channel

/radall' varied% gradal change$ in velocit' with

di$tance

Laminar and (rblent

Laminar% flow appear$ to be a$ a movement of thin

la'er$ on top of each other

(rblent% packet$ of liqid move in irreglar path$

( Temporal)

(Spatial)


5/62

Momentm and Energ'

Eqation$

Con$ervation of Energ'

2lo$$e$3 de to conver$ion of trblence to heat

$efl when energ' lo$$e$ are known or $mall ############

M$t accont for lo$$e$ if applied over long di$tance$ ###############################################

Con$ervation of Momentm 2lo$$e$3 de to $hear at the bondarie$

$efl when energ' lo$$e$ are nknown ############

Contractions

Expansion

We need an equation for losses


6/62

/iven a long channel ofcon$tant $lope and cro$$$ection find the relation$hip

between di$charge and depth 4$$me

Stead' )niform Flow - ### #############

pri$matic channel !no change in ######### with di$tance"

)$e Energ' and Momentm& Empirical or *imen$ional4nal'$i$5

What control$ depth given a di$charge5

Wh' doe$n6t the flow accelerate5

Open Channel Flow%

*i$charge7*epth ,elation$hip

8

no accelerationgeometr'

Force balance

4


7/62

Stead'-)niform Flow% Force

9alance

θ

W

θ

W sin θ

∆x

a

b

c

d

Shear force

Energy grade line

Hydraulic grade lin

Shear force :########

;$in =∆−∆ x P x A oτ θ γ ;$in =∆−∆ x P x A oτ θ γ

θ γ τ $in

P

Ao = θ γ τ $in

P

Ao =

h, :

P

Ah

, :

P

Aθ

θ

θ $in

co$

$in≅=S θ

θ

θ $in

co$

$in≅=S

W cos θ

g

V

<

<

g

V

<

<

Wetted perimeter : ##

/ravitational force : ########

Hydraulic radius

Relationship beteen shear and !elocity" ##############

τo8 ∆ =

8

γ 4 ∆= $inθ

Turbulence


8/62

$eometric parameters ###################

################### ###################

Write the functional relationship

%oes &r a'ect shear" #########

P

A Rh =

P

A Rh ='dralic radi$ ! Rh"

Channel length !l "

,oghne$$ !ε"

Open Condit$%

*imen$ional 4nal'$i$

f & &,e& ph h

l C r

R R

e * +- .

F ,M, Wf & &,e& ph h

l C r

R R

e * +- .

F ,M, W

V Fr

yg

* 1o>


9/62

8re$$re Coefficient for Open

Channel Flow5

<

<C

V

p p

ρ

∆−=

<

<C

V

p p

ρ

∆−=

<

<C

V

ghl hl =

<

<C

V

ghl hl =

<

<C

f

f

S

gS l

V *

<

<C

f

f

S

gS l

V *

l f h S l *l f h S l *

l h p γ =∆− l h p γ =∆−8re$$re Coefficient

ead lo$$ coefficient

Friction $lope coefficient

!Energ' Lo$$ Coefficient"

Friction $lope

Slope of E/L


10/62


f & & ,e f S

h h

l C

R R

e * +- .

f & &,e f S

h h

l C

R R

e * +- .

f

h

S

RC

l l*

f

hS

RC

l l*

<

< f h gS l R

V l

l*<< f h gS l R

V l

l*< f h gS R

V l

*< f h gS R

V l

*<

f h

g V S R

l*

< f h

g V S R

l*

f &,e f S

h h

l C R R

e * +- .f & ,e

f S

h h

l C R R

e * +- .Head loss ∝ length of channel

f & ,e f

h

S h

RC

l R

el

* *

+ ,- .f &,e

f

h

S h

RC

l R

el

* *

+ ,- .

<

<

f

f

S

gS l C

V *

<

<

f

f

S

gS l C

V *

(li/e f in %arcy0Weisbach)


11/62

Che?' eqation !@AB"

Dntrodced b' the French engineer 4ntoine

Che?' in @AB while de$igning a canal for

the water-$ppl' $'$tem of 8ari$

h f V C R S * h f V C R S *

@;FCFB; s

m

s

m @;FCFB;

s

m

s

mhere C * Che1y coe2cient

here 34 is for rough and 564 is for smooth

also a function of R (li/e f in %arcy0Weisbach)

< f h

g V S R

l*

< f h

g V S R

l*compare


12/62

*arc'-Wei$bach eqation !@G;"

here d78 * roc/ si1e larger than

789 of the roc/s in a random

&or roc/0bedded streams

f * %arcy0Weisbach friction factor

<

G

@

@.< + ,- .? @

<

G

@

@.< + ,- .? @

G

G

8

<

d

d

d

A Rh =

==π

π

G

G

8

<

d

d

d

A Rh =

==π

π

<

<l

l V h f

d g

*<

<l

l V h f

d g

*<

G <l

h

l V h f

R g

*<

G <

l

h

l V h f

R g

*

<

G < f

h

l V S l f

R g *

<

G < f

h

l V S l f

R g *

<

f h

V S R f

g *

<

f h

V S R f

g *

f h

g V S R

f *

f h

g V S R

f *


13/62

Manning Eqation !@I@"

Aost popular in BS for open channels

(English system)

@7<

o


14/62

0ale$ of Manning n

Lined Canals n

Cement plaster 0.011Untreated gunite 0.016Wood, planed 0.012Wood, unplaned 0.013Concrete, trowled 0.012

Concrete, wood forms, unfinished 0.015Rubble in cement 0.020Asphalt, smooth 0.013Asphalt, rough 0.016

Natural Channels

Gravel beds, straight 0.025Gravel beds plus large boulders 0.040

Earth, straight, with some grass 0.026Earth, winding, no vegetation 0.030Earth , winding with vegetation 0.050

Lined Canals n

Cement plaster 0.011Untreated gunite 0.016Wood, planed 0.012Wood, unplaned 0.013Concrete, trowled 0.012

Concrete, wood forms, unfinished 0.015Rubble in cement 0.020Asphalt, smooth 0.013Asphalt, rough 0.016

Natural Channels

Gravel beds, straight 0.025Gravel beds plus large boulders 0.040

Earth, straight, with some grass 0.026Earth, winding, no vegetation 0.030Earth , winding with vegetation 0.050

d * median si1e of bed

material

n *f(surface

roughnessLchannelirregularityLstage)

B7@

;H.; d n = B7@

;H.; d n =

B7@;H@.; d n = B7@;H@.; d n = d in

ftd in m


15/62

(rape?oidal Channel

*erive 8 : f!'" and 4 : f!'" for a

trape?oidal channel

ow wold 'o obtain ' : f!J"5

15

b

y z y yb A


16/62

Flow in ,ond Condit$

−=r

yr arcco$θ

−=r

yr arcco$θ

( )θ θ θ co$$in< −= r A ( )θ θ θ co$$in< −= r A

θ $in


17/62

Open Channel Flow% Energ'

,elation$


18/62

Energ' relation$hip$

< <

@ @ < <@ @ < <

< < L

p V p V z z h

g g a a

g g< < * < < <

< <

@ <@ <

< <o f

V V y S x y S x

g g

< % < * < < % Turbulent Oo (α ≅ 5)

1 0 measuredfrom hori1ontal

datum

y 0 depth of Oo

Pipe Oo

Energy Equation for Gpen Channel &lo

< <

@ <@ <

< <

o f

V V y S x y S x

g g

< < % * < < %

&rom diagram on pre!iousslide


19/62

Specific Energ'

(he $m of the depth of flow and the

velocit' head i$ the $pecific energ'%

g

V y E

<

<

+=

Qf channel bottom is hori1ontal and nohead loss


20/62

Specific Energ'

Qn a channel ith constant dischargeL


21/62

;

@

<

H

G

B

A

C

I

@;

; @ < H G B A C I @;

E

'

Specific Energ'% Slice /ate

<

<


22/62

;

@

<

H

G

; @ < H G

E

'

Specific Energ'% ,ai$e the Slice

/ate

<

<


23/62

;

@

<

H

G

; @ < H G

E

'

Specific Energ'% Step )p

ShortL smooth step ith rise ∆y in channel

∆y

@


24/62

8

4

Critical Flow

T

dy

y

T*surfaceidth

&ind critical depthL yc

<

<


25/62

Critical Flow%

,ectanglar channel

yc

T

Mc

H

<

@

c

c

gA

T Q=

qT Q = T y A cc =

H

<

HH

H<

@

cc gy

q

T gy

T q==

H7@

<

= g

q yc

H

c gyq =

Gnly for rectangular channels

cT T =

$i!en the depth e can nd the Oo


26/62

Critical Flow ,elation$hip$%

,ectanglar Channel$

H7@<

=

g

q yc cc

yV q =

= g

yV y

cc

c


27/62

Critical Flow

Characteri$tic$

)n$table $rface

Serie$ of $tanding wave$ Occrrence

9road cre$ted weir !and other weir$"

Channel Control$ !rapid change$ in cro$$-$ection"

Over fall$

Change$ in channel $lope from mild to $teep

)$ed for flow mea$rement$

########################################### Bnique relationship beteen depth and discharge

%i2cult to measure depth

;

@

<

H

; @ < H G

E

'


28/62

9road-cre$ted Weir

H

P

yc

E

Hc gyq = HcQ b gy*

E ycH

<=

H7 <H7


29/62

9road-cre$ted Weir% E=ample

Calclate the flow and the depth p$tream.

(he channel i$ H m wide. D$ appro=imatel'

eqal to E5

46

yc

E

Kroad0crestedeir

yc*4J m

Solution

ow do 'o find flow5####################

ow do 'o find 5######################

Critical flow relation

Energ' eqation

http://www.eng.vt.edu/fluids/msc/gallery/waves/sinkb.htm


30/62

'dralic mp

)$ed for energ' di$$ipation

Occr$ when flow tran$ition$ from

$percritical to $bcritical ba$e of $pillwa'

We wold like to know depth of water

down$tream from Kmp a$ well a$ thelocation of the Kmp

Which eqation& Energ' or Momentm5

http://www.eng.vt.edu/fluids/msc/gallery/waves/sinkb.htm


31/62

'dralic mp>


32/62

'dralic mp

y5

y

I

E$IhI

Conser!ation ofAomentum


33/62

'dralic mp%

ConKgate *epth$

Auchalgebra

&or a rectangular channel ma/e the folloingsubstitutions

%y A = @@V %yQ =

@@

@

V Fr

gy* &roude number

( )


34/62

'dralic mp%

Energ' Lo$$ and Length

Fo general theoretical solution

Experiments sho


35/62

/radall' 0aried Flow

< <

@ <@ <

< <o f

V V y S x y S x

g g < < % * < < % Energy equation for

non0uniformL steadyOo

@< y ydy −=

<

< f o

V dy d S dx S dx

g

( )< < *+ ,- .

8

4

T

dy

y

dy

dxS

dy

dxS

g

V

dy

d

dy

dyo f =+

+<

<

( )

< <

< @< @< <

o f

V V S dx y y S dx g g

( )* 0 < 0


36/62


<

H

<

H

<

<


37/62



38/62

Srface 8rofile$

Mild $lope !'n'c"

in a long channel $bcritical flow will occr

Steep $lope !'n'c"

in a long channel $percritical flow will occr

Critical $lope !'n:'c"

in a long channel n$table flow will occr

Hori?ontal $lope !So:;" 'n ndefined

Adver$e $lope !So;"

'n ndefined

Note: These slopes are f(Q)


39/62

Formal depth

Steep slope (S)

Hydraulic Zump

Sluice gate

Steep slope

Gbstruction

Srface 8rofile$


40/62

More Srface 8rofile$

S4 0 Sf 5 0 &r dydx

5 < < <

< 0 0

J 0 0 < ;

@

<

H

G

; @ < H G

E

'

yn

yc


41/62

*irect Step Method

xS g

V y xS

g

V y f o ∆++=∆++


42/62

*irect Step Method

Friction Slope

< <

G7H f

h

n V S

R

*

< <

G7H


43/62

*irect Step

Limitation% channel m$t be ######### !$o thatvelocit' i$ a fnction of depth onl' and not a fnctionof ="

Method identif' t'pe of profile !determine$ whether ∆' i$ or -"

choo$e ∆' and th$ 'n@ calclate h'dralic radi$ and velocit' at 'n and 'n@ calclate friction $lope 'n and 'n@

calclate average friction $lope

calclate ∆=

prismatic


44/62

*irect Step Method

:!/@B-/@"7!!F@F@B"7


45/62

Standard Step

/iven a depth at one location& determine the depth at a$econd location

Step $i?e !∆=" m$t be $mall enogh $o that change$ in

water depth aren6t ver' large. Otherwi$e e$timate$ of thefriction $lope and the velocit' head are inaccrate

Can $olve in p$tream or down$tream direction p$tream for $bcritical

down$tream for $percritical

Find a depth that $ati$fie$ the energ' eqation

xS g

V y xS

g

V y

f o ∆++=∆++


46/62

What crve$ are available5

S

S*

+s there a cure 'etween yc an- yn that

-ecreases in -epth in the upstrea -irection.

;.;

;.<

;.G

;.B

;.

@.;

@.<

@.G

;@;@


47/62

Wave Celerit'

<

@

<

@ gy F ρ = ( )


48/62

Wave Celerit'%

Momentm Con$ervation

( ) ( ) ( )[ ]&&&r V V V V V V V y F −−−+−= δ ρ

( ) V V V y F &r δ ρ −=

( )[ ] ( ) V V V y y y y g & δ ρ δ ρ −=+−


49/62

Wave Celerit'

( ) ( ) ( )&& V V V y yV V y −++=− δ δ

&&& yV yV V yV y yV yV yV yV δ δ δ δ δ −−+++=−

( ) y

yV V V &δ δ −−=

( ) V V V y g & δ δ −=−

( ) y

yV V y g &

δ δ


50/62

Wave 8ropagation

Spercritical flow

c0

wave$ onl' propagate down$tream water doe$n6t 2know3 what i$ happening down$tream

######### control

Critical flow

c:0

Sbcritical flow

c0

wave$ propagate both p$tream and down$tream

upstream

Mo$t Efficient 'dralic


51/62

Mo$t Efficient 'dralic

Section$

4 $ection that give$ ma=imm di$charge for a$pecified flow area Minimm perimeter per area

1o frictional lo$$e$ on the free $rface

4nalog' to pipe flow

9e$t $hape$

be$t be$t with < $ide$

be$t with H $ide$

Wh' i$n6t the mo$t efficient


52/62

Wh' i$n t the mo$t efficient

h'dralic $ection the be$t de$ign5

Ainimum area * least exca!ation only if topof channel is at grade

Cost ofliner

Complexity of form or/

Erosion constraint 0 stability of sidealls

Open Channel Flow *i$charge


53/62

Open Channel Flow *i$charge

Mea$rement$

*i$charge

Weir broad cre$ted

$harp cre$ted

trianglar

0entri Flme

Spillwa'$

Slice gate$

0elocit'-4rea-Dntegration


54/62


55/62

Smmar'

4ll the complication$ of pipe flow pl$

additional parameter... #################

0ario$ de$cription$ of head lo$$ termChe?'& Manning& *arc'-Wei$bach

Dmportance of Frode 1mber

Fr@ decrea$e in E give$ increa$e in '

Fr@ decrea$e in E give$ decrea$e in '

Fr:@ $tanding wave$ !al$o min E given J"

Method$ of calclating location of free $rface

free surface location

;

@

<

H

G

; @ < H G

E

'


56/62

9road-cre$ted Weir% Soltion

46

yc

E

Kroad0crested

eir

yc*4J m

H

c gyq =

( )H< H.;"7.I! m smq =

smq 7@GG.;


57/62

Slice /ate

m

54 cm

S * ;.;;E

Sluice gatereser!oir


58/62

Smmar'7Overview

Energ' lo$$e$


Empirical

f h

g V S R

f *

f h

g V S R

f *

@7<

o


59/62

Energ' Eqation

Specific Energ'

(wo depth$ with $ame energ'>

ow do we know which depth

i$ the right one5

D$ the path to the new depth

po$$ible5

< <

@ <@ <

< <o f

V V y S x y S x

g g < < % * < < %

<


60/62

;

@

<

H

G

; @ < H G

E

'

Specific Energ'% Step )p

ShortL smooth step ith rise ∆y in channel

∆y

@


61/62

Critical *epth

Minimm energ' for qWhen

When kinetic : potential>Fr:@

Fr@ : Spercritical

Fr@ : Sbcritical

;=

dy

dE

;

@

<

H

G

; @ < H G

E

'

g V y cc


62/62

What ne=t5

Water $rface profile$

,apidl' varied flow

4 wa' to move from $percritical to $bcritical flow!'dralic mp"

/radall' varied flow eqation$Srface profile$

*irect $tep

Standard $tep

Date post:	07-Jul-2018
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