SAB 2513 Hydraulic Chapter 3

7/31/2019 SAB 2513 Hydraulic Chapter 3

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2

Non-Uniform flow

Non-uniform flow, So = Sw = Si Uniform flow, So = Sw = Si

y1 = y2 Water depth must be specified at selected section

2g

V211

2g

V222

oS

2y

1ywS

iS

1z

1H1

E

Section 1 Section 2


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Energy in Open Channel

g

vyz 2

2

Total Energy, H (m)

g

vyzH

2

2

1111

z = potential energy or potential head

y = hydrostatic energy or hydrostatic head

= kinetic energy or kinetic head

= Coriolis coefficient (value between 1.0 to 1.36)

Normally use = 1.0

g

v

2

2

Energy at section 1 is thus


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Specific Energy

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

As know, v = Q/A

g

v

yE 2

2

y - hydrostatic energy

gv2

2

- kinetic energy

2

2

2gA

QyE


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Curve for

different, higher

Q.


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(A = By) and Q = qB

2

2

2gy

qyE

A

B

y

q is the discharge per unit width of channel

SPECIAL CASE: Rectangular channel,

22

22

2 ygBqByE


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In understanding Non-uniform flow phenomena(1) A plot of flow depth (y) vs. specific Energy (E)

- Constant discharge (Q or q)

- Call Specific Energy Diagram

(2) A plot of flow depth (y) vs. discharge (Q or q)

- Constant specific energy

- Call Discharge Diagram

Why Specific Energy

Equation is important???


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ysub and ysuperare alternate depths (same specific energy)

Relationship y-E(constant Q or q)

yc = critical depth

Subcritical flow, ysub

Supercritical flow, ysuper

cyy

cyy

Specific Energy Diagram


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Relationship y-q(constant E )

For rectangular channel only

yc = critical depth

ysub & ysuper = alternate depth

Discharge Diagram


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10

State of Flow Characteristics

Critical Flow, yc Fr = 1 or y = yc

Subcritical (y1 or ysub) Fr < 1 or y1 > yc

Supercritical (y2 or ysuper) Fr > 1 or y2 < yc

So Remember!!


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Critical Depth


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Critical Flow

Characteristics

Unstable surface

Series of standing waves

Occurrence

Broad crested weir (and other weirs) Channel Constriction (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


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Critical Flow

Find critical depth, yc

; 2

2

2gA

QyE 0

dy

dE

Froude number, Fr = 1 Specific energy is minimum for a given discharge

0

1

2

3

4

0 1 2 3 4

E

y

dy

dA

gA

Q

dy

dE3

2

1

dy

dA

gA

v2

1


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P

A

Critical Flow

T

dy

y

T = surface width

dyTdA .Arbitrary cross-section

dA

The differential water area near the surface

(see Figure)

T

AD

gA

Tv

dy

dE2

1

and

gD

v

dy

dE 21


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Critical Flow

0

dy

dE

At critical state of flow,

12

gD

v

22

2D

g

v

Well known as , means at critical flow Fr = 11gD

v


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Critical Flow

A

Qv By substituting and

TA

AQ

g 21

21

2

2

T

AD

Therefore, general equation for critical flow:

13

2

c

cgA

TQ(any cross-section channel)


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Critical Flow:Rectangular channel

yc

Tc

Ac

3

2

1

c

c

gA

TQ

cc ByA

13

2

cgBy

BQ

3/12

g

qyc

for rectangular

channel

cTT

From general equation,

;

Then,

;

So, or

B

B

Qq

3

2

g

qyc


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Critical Flow Relationships:Rectangular Channels

32

cgyq

When E = Emin, critical depth, y = yc

differentiating

When E = Emin,

or

2

2

2gy

qyE

3

2

1gy

q

dy

dE

cyydy

dE ,0

3

2

g

qyc Specific Energy Diagram


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32

cgyq

Sub. Into the energy eqn. at the point of critical flow:

; ;

2

2

min

2 cc

gy

qyE

cyE 5.1min

cyy minEE

2

3

min2 c

cc

gy

gyyE

cc yyE 5.0min cyE

2

3min

or


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Discharge diagram y vs. q for constant E

For constant E, q maximum at critical flowi.e

at q = qmax

2

2

2gy

qyE

)(2 22 yEgyq

Discharge Diagram

)(22 yEgyq

0dy

dq

cyE 5.13

max cgyq and


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Summary of Critical Flow in Open Channels

(1) General equation during critical condition

- ALL channel cross-section shapes;

- For rectangular channel;

13

2

c

c

gATQ

;32

g

qyc ;

3

2min

Eyc

3

2

max

g

qyc


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Summarycont

(2) Specific Energy during critical condition(constant Q or q)

- ALL channel cross-section shapes;

- For RECTANGULAR channel;

3

2

g

qyccyEE 5.1min

2

22

min22 c

cc

cgAQy

gvyE

where


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Summarycont

(3) Flow rate per unit width, q (constant E) ismaximum during critical flow condition

- For RECTANGULAR channel only;

Eyc3

2where

and3

max cgyq


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Summarycont

(4) Froude Number is 1 during critical flow

22

2D

g

v

gD

vFr


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Analysis of Flow across aWeir in a Rectangular Channel

What is a Weir? Structure placed across the channel to obstruct the

uniform flow and still allows water to flow over it

Propose mainly to control flow in the open channel

By ensuring a control section is formed over the weir for

all ranges of discharges in the channel.

Effectiveness of weir depends on the channel discharge

(Q or q) range and it height.


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This section will look into the analysis of weir inRECTANGULAR channel

The weir will raise the bed level by its height (Z)

Specific energy defined as the energy measured from the

channel bed

Over the weir structure, the specific energy (E) is

reduced by the amount Z without any change to the flowrate (q constant)


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Effect of a weir on the water level as explained using the

specific energy diagram

A l i f Fl


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For analysis purpose, consider:

- Rectangular channel of constant width (B m)- Carrying a constant discharge (Q m3/s) giving q = Q/B- Flowing at a normal depth (yo m)- Weir height (Z m) is placed across the channel

- Four representative channel cross-sections are definedas marked as:

0 --- very far upstream of the weir

1 --- just behind (upstream) of the weir

2 --- above the weir3 --- just after (downstream) of the weir


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WITHOUT WEIR

29

Uniform flow condition- WITHOUT WEIR and channel isprismatic


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WEIR CASE 1


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WEIR CASE 2


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WEIR CASE 3


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Summary of Energy and Flow Depth Weir Case

Approaching flow is subcritical and uniform

Given Q, B and normal flow depth, yo

(1) Energy of approaching flow

or (rectangular channel ONLY)

g

vy

gA

QyE

ooo 22

2

2

2

2

2

2gy

qyE oo


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Summarycont

(2) Critical weir height, Zc

- (a) Critical flow depth: yc using Chart or graphical

or for rectangular channel

- (b) Minimum specific energy: any cross-section

or for rectangular channel

- (c) Critical weir height:

g

vyE cc

2

2

min

13

2

c

c

gA

TQ

;3

2

g

q

yc

minEEZ oc

cyE 5.1min


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Summarycont

(3) Compared actual weir height, Z to critical weir height, Zc

- (a) CASE 1: (Weir is drowned)

[calculate y2 from E2=Eo- Z (y2 is still subcritical)]

- (b) CASE 2: (Weir is controlling)[calculate y3 from E3=Eo (y3 is still supercritical & alternate

depth of y1)]

- (c) CASE 3: (Weir is controllingbut backwater effect is formed)

c

o

o

o

yy

EZEE

EEE

yy

2

min2

31

1

ZEE

EEE

yyy

o

o

o

2

31

31

cyy

EE

ZEEE

2

min2

min31

[calculate y1 & y3 from E1=E3=Emin+ Z (y1 & y3 is alternate depth]

cZZ

cZZ

cZZ

Analysis of Flow across a


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Analysis of Flow across aChannel Constriction in aRectangular Channel

What is a Channel Constriction? Structure reduced width placed across the channel to

control the flow and still allows water to flow over it but at

an increased velocity and q.

Effectiveness of CC depends on the channel discharge

(Q or q) range and the width of the channel constriction

(Bf).

Normally does not raise the bed level.

The discharge diagram (y vs. q with E constant) is relevant


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Summary of Energy and Flow Depth Channel Constriction

Approaching flow is subcritical and uniform

Given Q, B and normal flow depth, yo

Width at channel constriction = Bfwhere Bf< B

Critical depth at channel = yc

Critical depth at channel constriction = ycf


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Analysis of Flow across a ChannelConstriction in Rectangular Channel

For analysis purpose, consider:

- Rectangular channel of constant width (B m)- Carrying a constant discharge (Q m3/s) giving q = Q/Bm3/s.m

- Flowing at a normal depth (yo m) & subcritical

- Channel constriction width (Bfm) is placed- Bf< B and therefore qf> q- Four representative channel cross-sections is defined as

marked as:

0 --- very far upstream of the channel constriction

1 --- just behind (upstream) of the channel constriction2 --- above the channel constriction

3 --- just after (downstream) of the channel constriction

Analysis of Flow across a


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yChannel Constriction in aRectangular Channel

Effect of a channel constriction on the water level as

explained using the discharge diagram

WITHOUT CHANNEL


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WITHOUT CHANNELCONSTRICTION

Uniform flow condition- WITHOUT CHANNELCONSTRICTION and channel is prismatic

CHANNEL CONSTRICTION


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CHANNEL CONSTRICTION CASE 1



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(1) Energy of approaching flow

or (rectangular channel ONLY)

g

vy

gA

QyE ooo

22

2

2

2

2

2

2gy

qyE oo

Summarycont


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Summarycont

(2) Critical channel width, Bc

- (a) Minimum specific energy = Emin = Eo

hence

- (b) Maximum flow rate at this energy,

- (c) Critical channel width,

cfyE 5.1min

maxq

QBc

min3

2Ey

cf

3

max cfgyq


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Summarycont

(3) Compared Bf to critical channel width, Bc- (a) CASE 1: (CC is not controlling)

[calculate yf from Ef=Eo and q=qf(yf is still subcritical)]

- (b) CASE 2: (CC is controlling)[calculate y3 from E3=Eo and discharge=q (y3 is still

supercritical & alternate depth of y1)]

- (c) CASE 3: (CC is controlling but

backwater effect is formed) calculate E

cf

f

of

o

yy

EE

EEEE

yy

2

min

31

1

of

o

EEEE

yyy

31

31

minEEf

[calculate y1 & y3 from E1=E3=E and q (y1 & y3 is alternate depth]

cf BB

cf BB

cf BB

min

min 5.1

EE

yE cf

Home ork


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A rectangular channel of width 3.5m wide and conveys water withdischarge of 17.5m3/s at a depth of 2.0m. A hydraulic structure is

constructed at the downstream of the channel and the channel width is

reduced to 2.5m. Assume the constriction to be horizontal and the flow to

be frictionless. Determine;

(i)state of flow,

(ii) water depths just before, just after and at the constriction,

(iii) sketch the flow profile along the channel. Show the important valuesin your sketch.

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Homework

Critical Section in Open


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Critical Section in OpenChannels

Critical section channel cross-section havecritical condition

If this condition exists throughout the channel

flow in channel is called critical flow.

If channel flow is uniformAND critical, y = yo = yc

A channel critical flow has a bed slope (So)called critical bed slope(Sc)i.e So = Sc


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Cont

If Soyc, vSc subcritical flow, yvc


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Flow Control

Defined as a channel cross-section where theflow depth can be determined conclusively

At control section, the stage-discharge

relationship is established and easily determined

At critical section for example, by using critical

flow relationships, q can be calculated easily from

the depth

Examples of Control


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Examples of ControlSections

SAB 2513 Hydraulic Chapter 3

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