LEARNING OUTCOME

2

After completing this lecture…

The students should be able to:

• Understand the behavior of open channel flow under various conditions

• Learn the basic theories that govern the design of open channels and hydraulic structures

• Apply the basic theories to derive various formula used in the design calculation of hydraulic structures such as weir/notches

References:

Fluid Mechanics With Engineering Applications, 10TH ED, By E. Finnemore and Joseph Franzini, Mcgraw Hills

OPEN CHANNEL HYDRAULICS

An open channel is the one in which stream is not complete enclosed by solid boundaries and therefore has a free surface subjected only to atmosphere pressure.

The flow in such channels is not caused by some external head, but rather only by gravitational component along the slope of channel. Thus open channel flow is also referred to as free surface flow or gravity flow.

Examples of open channel are

• Rivers, canals, streams, & sewerage system etc

3


Flow conditions

Uniform flow:

Non-uniform flow

� For uniform flow through open channel, dy/dl is equal to zero. Howeverfor non-uniform flow the gravity force and frictional resistance are not inbalance. Thus dy/dl is not equal to zero which results in non-uniformflow.

� There are two types of non-uniform flows.

� In one the changing condition extends over a long distance and this iscalled gradually varied flow.

� In the other the change may occur over very abruptly and the transitionis thus confined to a short distance. This may be designated as a localnon-uniform flow phenomenon or rapidly varied flow. 4


Characteristics of Uniform flow

1Z

g

V

2

2

1

Datum

So

1y

2Z

g

V

2

2

2

2y

HGL

EL

Water

Level

Sw

S

∆L

∆x

For Uniform Flow : y1=y2 and V12/2g=V2

2/2g

Hence the line indicating the bed of the channel, water surface profile and energy line are parallel to each other.For θ being very small (say less than 5 degree) i.e ∆x=∆L

So=Sw=S5

So= Slope of Channel Bed

(Z1-Z2)/(∆x)= -∆Z/∆x

Sw= Slope of Water Surface

[(Z1+y1)-(Z2+y2)]/∆x

S= Slope of Energy Line

[(Z1+y1+V12/2g)-(Z2+y2+V2

2/2g)]/∆L= hl/∆L


Energy Equation:

[(Z1+y1+V12/2g)=(Z2+y2+V2

2/2g)+HL

Let’s assume two section close to each other (neglecting head loss) and take bed of channel as datum, above equation can be rewritten as

y1+V12/2g=y2+V2

2/2g

E1=E2

Where E1 and E2 are called specific energy at 1 and 2.

( )Vy

B

VBy

B

AV

B

Qqwhere

yE

yE

yg

q

gV

====

+=

+=

2

2

2

2

2

Specific Energy at a section in an open channel is the energy with reference to the bed of the channel.

• Mathematically;

Specific Energy = E = y+V2/2g

For a rectangular Channel

BQqwhereyEyg

q/2

2

2=+= As it is clear from E~y diagram drawn for constant

discharge for any given value of E, there would betwo possible depths, say y1 and y2. These two depthsare called Alternate depths.

However for point C corresponding to minimumspecific energy Emin, there would be only onepossible depth yc. The depth yc is know as criticaldepth.

The critical depth may be defined as depthcorresponding to minimum specific energy dischargeremaining Constant. 6

OPEN CHANNEL HYDRAULICSTYPES OF FLOW IN OPEN CHANNELS

Subcritical, Critical and Supercritical Flow. These are classified with Froude number.

Froude No. (Fr). It is ratio of inertial force to gravitational force of flowing fluid. Mathematically, Froude no. is

If ; Fr. < 1, Flow is subcritical flowFr. = 1, Flow is critical flowFr. > 1, Flow is supercritical flow

gh

VFr =

Where, V is average velocity of flow, h is depth of flow and g is gravitational acceleration

Alternatively:If y>yc , V<Vc Deep Channel

Sub-Critical Flow, Tranquil Flow, Slow Flow.and y<yc , V>Vc Shallow Channel

Super-Critical Flow, Shooting Flow, Rapid Flow, Fast Flow.

7


Critical depth for rectangular channels: Critical depth for non rectangular channels:

• T is the top width of channel

ycyg

Q

T

A

=

=

23( ) 3/12

g

q

cy =

c

y

cc yyEE c

23

2min =+==

8

OPEN CHANNEL HYDRAULICSCHEZY’S AND MANNING’S EQUATIONS

Chezy’s Equation � Manning’s Equation

2/13/21oSR

nV =

( ) 2/13/23 1/ oSAR

nsmQ =

( ) 2/13/2486.1oSAR

ncfsQ =

SI

BG

oRSCV =

oRSCAQ =

Value of C is determine from respective BG or SI Kutter’sformula.

C= Chezy’s Constant

A= Cross-sectional area of flow A= Cross-sectional area of flow

By applying force balance along the direction of flow in an open channel

having uniform flow, the following equations can be derived. Both of the

equations are widely used for design of open channels.

9

OPEN CHANNEL HYDRAULICSENERGY EQUATION FOR GRADUALLY VARIED

FLOW.

( )

( ) ( )

profilesurfacewateroflengthLWhere

SS

EEL

LSLSEE

Now

forL

ZZ

X

ZZS

L

hS

hZZg

Vy

g

Vy

o

o

o

oL

L

=∆

−

−=∆

∆+∆−=

<∆

−≈

∆

−=

∆=

+−−+=+

)1(

6,

22

21

21

2121

21

2

22

2

11

θ

An approximate analysis of gradually varied, non uniform flow can be achieved by considering a length of stream consisting of a number of successive reaches, in each of which uniform occurs. Greater accuracy results from smaller depth variation in each reach.

The Manning's formula (or Chezy’sformula) is applied to average conditions in each reach to provide an estimate of the value of S for that reach as follows;

3/4

22

2/13/21

m

m

mm

R

nVS

SRn

V

=

=

2

2

21

21

RRR

VVV

m

m

+=

+=

In practical, depth range of the interest is divided into small increments, usually equal, which define the reaches whose lengths can be found by equation (1)

10

OPEN CHANNEL FLOWWATER SURFACE PROFILES IN GRADUALLY

VARIED FLOW.

3/10

3/10

22

3/10

22

2/13/5

2/13/2

1

1

=∴

=

=

=

=

≈

y

y

S

S

y

qnS

channel

gulartanrecinflowuniformFor

y

qnS

orSyn

q

orSyn

V

yR

channelgulartanrecwideaFor

o

o

o

o

21 F

SS

dx

dy o

−

−=

Consequently, for constant q and n,when y>yo, S<So, and the numeratoris +ve.

Conversely, when y<yo, S>So, andthe numerator is –ve.

To investigate the denominator weobserve that,

if F=1, dy/dx=infinity;

if F>1, the denominator is -ve; and

if F<1, the denominator is +ve.11

++=

++=++=

2

2

2

22

1

2

22

gy

q

dx

d

dx

dy

dx

dZ

dx

dH

gy

qyZ

g

vyZH

Rectangular

channel !!

OPEN CHANNEL FLOW

WATER SURFACE PROFILES IN GRADUALLY

VARIED FLOW

12

FLOW OVER HUMP

For frictionless two-dimensional flow, sections 1 and 2 in Fig are related by continuity and energy:

Eliminating V2 between these two gives a cubic polynomial equation for the water depth y2 over the hump.

1 1 2 2

2 2

1 21 2

2 2

v y v y

v vy y Z

g g

=

+ = + +

2 23 2 1 12 2 2

2

12 1

02

2

v yy E y

g

vwhere E y Z

g

− + =

= + −

y2y1

y3

Z

V1V2

1 2 3

This equation has one negativeand two positive solutions if Z isnot too large.It’s behavior is illustrated byE~y Diagram and dependsupon whether condition 1 isSubcritical (on the upper) orSupercritical (lower leg) of theenergy curve.

B1=B2

Hump is a streamline construction provided at the bed of the

channel. It is locally raised bed.

13

FLOW OVER HUMP

Damming Action

y1=yo, y2>yc, y3=yoy1=yo, y2>yc, y3=yo

y1=yo, y2=yc, y3=yo y1>yo, y2=yc, y3<yo

ycy1

y3

Z

Z=Zc

y1

Z<<Zc

y2 y3

Z

Z<Zc

y2

y1

y3

Z

Z>Zc

Afflux=y1-yo

y3

yc

yoZ

y1

14

FLOW THROUGH CONTRACTION

When the width of the channel is reduced while the bed remains flat, thedischarge per unit width increases. If losses are negligible, the specificenergy remains constant and so for subcritical flow depth will decreasewhile for supercritical flow depth will increase in as the channel narrows.

B1 B2

y2yc

y1( )

1 1 1 2 2 2

2 2

1 21 2

1 2

2 2 2 2 2 2

2 2

1 1

'

2 2

Using both equations, we get

2Q=B y v =B y

1

Continuity Equation

B y v B y v

Bernoulli s Equation

v vy y

g g

g y y

B y

B y

=

+ = +

−

−

15

FLOW THROUGH CONTRACTION

If the degree of contraction and the flow conditions are such that upstream flow is subcritical and free surface passes through the critical depth yc in the throat.

ycyc

y1

( )

3/ 2

2

2sin

3

2 12

3 3

1.705

c c c c c c

c

c

Q B y v B y g E y

ce y E

Therefore Q B E g E

Q BE in SI Units

= = −

=

=

=

B1 Bc

y2yc

y1

16

HYDRAULICS JUMP OR STANDING WAVE

Hydraulics jump is local non-uniform flow phenomenon resultingfrom the change in flow from super critical to sub critical. In suchas case, the water level passes through the critical depth andaccording to the theory dy/dx=infinity or water surface profileshould be vertical. This off course physically cannot happen andthe result is discontinuity in the surface characterized by a steepupward slope of the profile accompanied by lot of turbulence andeddies. The eddies cause energy loss and depth after the jump isslightly less than the corresponding alternate depth. The depthbefore and after the hydraulic jump are known as conjugatedepths or sequent depths.

y

y1

y2

y1

y2

y1 & y2 are called

conjugate depths

17

CLASSIFICATION OF HYDRAULIC JUMP

Classification of hydraulic jumps: (a) Fr =1.0 to 1.7: undular jumps; (b) Fr =1.7 to 2.5: weak jump; (c) Fr =2.5 to 4.5: oscillating jump; (d) Fr =4.5 to 9.0: steady jump; (e) Fr =9.0: strong jump.

18

CLASSIFICATION OF HYDRAULIC JUMP

Fr1 <1.0: Jump impossible, violates second law ofthermodynamics.

Fr1=1.0 to 1.7: Standing-wave, or undular, jump about 4y2 long; lowdissipation, less than 5 percent.

Fr1=1.7 to 2.5: Smooth surface rise with small rollers, known as aweak jump; dissipation 5 to 15 percent.

Fr1=2.5 to 4.5: Unstable, oscillating jump; each irregular pulsationcreates a large wave which can travel downstream for miles,damaging earth banks and other structures. Not recommended fordesign conditions. Dissipation 15 to 45 percent.

Fr1=4.5 to 9.0: Stable, well-balanced, steady jump; bestperformance and action, insensitive to downstream conditions.Best design range. Dissipation 45 to 70 percent.

Fr1>9.0: Rough, somewhat intermittent strong jump, but goodperformance. Dissipation 70 to 85 percent.

19

USES OF HYDRAULIC JUMP

Hydraulic jump is used to dissipate or destroy the energy of waterwhere it is not needed otherwise it may cause damage tohydraulic structures.

It may also be used as a discharge measuring device.

It may be used for mixing of certain chemicals like in case of watertreatment plants.

20

EQUATION FOR CONJUGATE DEPTHS

1

2

F1 F2

y2y1

So~0

1 2 2 1

1

2

( )

Resistance

g f

f

Momentum Equation

F F F F Q V V

Where

F Force helping flow

F Force resisting flow

F Frictional

Fg Gravitational component of flow

ρ− + − = −

=

=

=

=

Assumptions:

1. If length is very small frictional resistance may be neglected. i.e (Ff=0)

2. Assume So=0; Fg=0

Note: Momentum equation may be stated as sum of all external forces is equal to rate of change of momentum.

L

21


Let the height of jump = y2-y1

Length of hydraulic jump = Lj

2 1

1 1 2 2 2 1

1 1 1 2 2 2

1 2 ( )

( )

Depth to centriod as measured

from upper WS

.1

Eq. 1 stated that the momentum flow rate

plus hydrostatic force is the same at both

c c

c

c c

F F Q V Vg

h A h A Q V Vg

h

QV h A QV h A eqg g

γ

γγ γ

γ γγ γ

− = −

− = −

=

+ = + ⇒

2 2

1 1 2 2

1 2

sections 1 and 2.

Dividing Equation 1 by and

changing V to Q/A

.2c c m

Q QA h A h F eq

A g A g

γ

+ = + = ⇒

2

;

Specific Force=

: Specific force remains same at section

at start of hydraulic jump and at end of hydraulic

jump which means at two conjugate depths the

specific force is constant.

m

Where

QF Ahc

Ag

Note

= +

( )

( )( )

2 2 2 2

1 21 2

1 2

2 22 2

1 2

1 2

22 2

2 1

1 2

2

2 12 1 2 1

1 2

Now lets consider a rectangular channel

2 2

.32 2

1 1 1

2

1

2

y yq B q BBy By

By g By g

y yq qeq

y g y g

qy y

g y y

or

y yqy y y y

g y y

∴ + = +

+ = + ⇒

− = −

−= − +

22


2

2 11 2

1 1 2 2

2 2

1 1 2 11 2

3

1

22

1 2 2

1 1 1

2

22 21

1 1

2

1

.42

Eq. 4 shows that hydraulic jumps can

be used as discharge measuring device.

Since

2

2

0 2 N

y yqy y eq

g

q V y V y

V y y yy y

g

by y

V y y

gy y y

y yF

y y

y

y

+ = ⇒

= =

+ ∴ =

÷

= +

= + −

( )

2

1

212 1

1 1 4(1)(2)

2(1)

1 1 82

N

N

F

yy F

− ± +=

= − ± +

( )

( )

212 1

221 2

Practically -Ve depth is not possible

1 1 8 .52

1 1 8 .52

N

N

yy F eq

Similarly

yy F eq a

∴ = − + + ⇒

= − + + ⇒

23

LOCATION OF HYDRAULIC JUMPS

Change of Slope from Steep to Mild

Hydraulic Jump may take place

1. D/S of the Break point in slope y1>yo1

2. The Break in point y1=yo1

3. The U/S of the break in slope y1<yo1

So1>Sc

So2<Sc

yo1

y2yc

Hydraulic Jump

M3

y1

24

LOCATION OF HYDRAULIC JUMPS

Flow Under a Sluice Gate

So<Sc

yo yc ys

y1 y2=yo

L Lj

Location of hydraulic jump where it starts isL=(Es-E1)/(S-So)

Condition for Hydraulic Jump to occurys<y1<yc<y2

Flow becomes uniform at a distance L+Lj from sluice gate whereLength of Hydraulic jump = Lj = 5y2 or 7(y2-y1)

25

NOTCHES AND WEIRS

26

NOTCHES AND WEIRS

27

NOTCHES AND WEIRS

Notch. A notch may be defined as an opening in the side of a tank or vessel such that the liquid surface in the tank is below the top edge of the opening.

A notch may be regarded as an orifice with the water surface below its upper edge. It is generally made of metallic plate. It is used for measuring the rate of flow of a liquid through a small channel of tank.

Weir: It may be defined as any regular obstruction in an open stream over which the flow takes place. It is made of masonry or concrete. The condition of flow, in the case of a weir are practically same as those of a rectangular notch.

Nappe: The sheet of water flowing through a notch or over a weir

Sill or crest. The top of the weir over which the water flows is known as sill or crest.

Note: The main difference between notch and weir is that the notch is smaller in size compared to weir.

28

CLASSIFICATION OF NOTCHES/WEIRS

Classification of Notches

1. Rectangular notch

2. Triangular notch

3.Trapezoidal Notch

4. Stepped notch

� Classification of Weirs

� According to shape

� 1. Rectangular weir

� 2. Cippoletti weir

� According to nature of discharge

� 1. Ordinary weir

� 2. Submerged weir

� According to width of weir

� 1. Narrow crested weir

� 2. Broad crested weir

� According to nature of crest

� 1. Sharp crested weir

� 2. Ogee weir

29

DISCHARGE OVER RECTANGULAR

NOTCH/WEIR

Consider a rectangular notch or weir provided in channel carrying water as shown in figure.

Figure: flow over rectangular notch/weir

H=height of water above crest of

notch/weir

P =height of notch/weir

L =length of notch/weir

dh=height of strip

h= height of liquid above strip

L(dh)=area of strip

Vo = Approach velocity

Theoretical velocity of strip

neglecting approach velocity =

Thus,

discharge passing through strips

=

gh2

velocityArea×

30

DISCHARGE OVER RECTANGULAR NOTCH/WEIR

Where, Cd = Coefficient of discharge

LdhA

ghv

strip

strip

=

= 2( )ghLdhdQ 2=

Therefore, discharge of strip

In order to obtain discharge over whole area we must integrate above eq. from h=0 to h=H, therefore;

2/3

0

23

2

2

LHgQ

dhhLgQ

H

=

= ∫

2/323

2LHgCQ dact =

Note: The expression of discharge (Q) for rectangular notch and sharp

crested weirs are same.

thactd QQC /=Q

31


Dimensional analysis of weir lead to the following conclusion

Comparing this with previously derived expression, we conclude that Cd depend on weber number, W, Reynold’s number, R, and H/P.

It has been found that H/P is most important of these.

2/3,, LHgP

HRWQ

= φ

T. Rehbook of the karlsruhe hydraulics laboratory in Germany provided following expressions for Cd

p

H

HC

p

H

HC

d

d

08.01000

1605.0

08.0305

1605.0

++=

++=In BG units: H & P in ft

In SI units: H & P in m

32


For convenience the formula of Q is expressed as

Where, Cw the coefficient of weir, replaces

Using a value of 0.62 for Cd, above equation can be written as

These equations give good results for H/P>0.4 which is well within operating range.

2/323

2LHgCQ dact =

2/3LHCQ wact =

gCd 23

2

2/3

2/3

83.1

32.3

LHQ

LHQ

act

act

=

=In BG units:

In SI units:

33

RECTANGULAR WEIR WITH END CONTRACTIONS

When the length L of the crest of a rectangular weir less than the width the channel, the nappe will have end contractions so that its width is less than L.

Hence for such a situations, the flow rate may be computed by employing corrected length of crest, Lc, in the discharge formula

Lc=(L-0.1nH)

Where, n is number of end contractions.

Francis formula

34

NUMERICAL PROBLEMS

A rectangular notch 2m wide has a constant head of 500mm. Find the discharge over the notch if coefficient of discharge for the notch is 0.62.

35

NUMERICAL PROBLEMS

A rectangular notch has a discharge of 0.24m3/s, when head of water is 800mm. Find the length of notch. Assume Cd=0.6

36

DISCHARGE OVER TRIANGULAR NOTCH (V-

NOTCH)In order to obtain discharge over whole area we must integrate above equation from h=0 to h=H, therefore;

( ) ( )( )( )

( ) ( ) dhhhHgQ

ghhHdhQ

H

H

∫

∫

−=

−=

0

0

2/tan22

22/tan2

θ

θ

( ) ( )

( )

=

−= ∫

2/5

0

2/32/1

15

42/tan22

2/tan22

HgQ

dhhHhgQ

H

θ

θ

( )[ ]2/52/tan215

8HgQ θ=

( )[ ]2/52/tan215

8HgCQ dact θ=

37

NUMERICAL PROBLEMS

Find the discharge over a triangular notch of angle 60o, when head over triangular notch is 0.2m. Assume Cd=0.6

38

NUMERICAL PROBLEMS

During an experiment in a laboratory, 0.05m3 of water flowing over a right angled notch was collected in one minute. If the head over sill is 50mm calculate the coefficient of discharge of notch.

Solution:

Discharge=0.05m3/min=0.000833m3/s

Angle of notch, θ=90o

Head of water=H=50mm=0.05m

Cd=?

39

NUMERICAL PROBLEMS

A rectangular channel 1.5m wide has a discharge of 0.2m3/s, which is measured in right-angled V notch, Find position of the apex of the notch from the bed of the channel. Maximum depth of water is not to exceed 1m. Assume Cd=0.62

Width of rectangular channel, L=1.5m

Discharge=Q=0.2m3/s

Depth of water in channel=1m

Coefficient of discharge=0.62

Angle of notch= 90o

Height of apex of notch from bed=Depth of water in channel-

height of

water over V-notch

=1-0.45= 0.55m 40

BROAD CRESTED WEIR

A weir, of which the ordinary dam isan example, is a channel obstructionover which the flow must deflect.

For simple geometries the channeldischarge Q correlates with gravityand with the blockage height H towhich the upstream flow is backedup above the weir elevation.

Thus a weir is a simple but effectiveopen-channel flow-meter.

Figure shows two common weirs,sharp-crested and broad-crested,assumed. In both cases the flowupstream is subcritical, acceleratesto critical near the top of the weir,and spills over into a supercriticalnappe. For both weirs the dischargeq per unit width is proportional tog1/2H3/2 but with somewhat differentcoefficients Cd.

B

41

BROAD CRESTED WEIR

Z>Zc

Vcy1

B

BGing

VHLCQ

SIing

VHLCQ

dact

dact

2/32

2/32

dact

1

209.3

27.1

QCQ Since

channel ofwidth L

crestover HeadH

Q/LyapproachofveloctyV

+=

+=

=

=

=

==

32

3c

2c

cc

2

222

2

22

L

23

2

V

g

VLVLyQ :Since

23

2

22

2

2

22 :flow criticalFor

22

h ignoringby equation energy Applying

+=

===

+=

+=+∴

=

++=++

g

VHg

g

LQ

g

LVc

g

VHgV

g

V

g

V

g

VH

y

g

V

g

VyZ

g

VZH

c

cc

cc

cc

42

BROAD CRESTED WEIRCOEFFICIENT OF DISCHARGE, CD ALSO CALLED WEIR DISCHARGE

COEFFICIENT, CW

Cw depends upon Weber number W, Reynolds number R and weirgeometry (Z/H, B, surface roughness, sharpness of edges etc).

It has been found that Z/H is the most important.

The Weber number W, which accounts for surface tension, is importantonly at low heads.

In the flow of water over weirs the Reynolds number, R is generallyhigh, so viscous effects are generally insignificant. For Broad crestedweirs Cw depends on length, B. Further, it is considerably sensitive tosurface roughness of the crest.

Z>Zc

Vc

B

2/32

2

+=

g

VHLCQ wact

2/32

3

23

2

+

=

g

VHg

g

LCQ dact

43

BROAD CRESTED WEIRCOEFFICIENT OF DISCHARGE, CD ALSO CALLED WEIR DISCHARGE

COEFFICIENT, CW

44

PROBLEM 1

45

PROBLEM 2

46

THANK YOU

47

chapter 2 open channel hydraulics

Engineering