sab 2513 hydraulic chapter 3
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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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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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Critical Flow Relationships:Rectangular Channels
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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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Critical Flow Relationships:Rectangular Channels
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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Analysis of Flow across aWeir in a Rectangular Channel
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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Analysis of Flow across aWeir in a Rectangular Channel
Effect of a weir on the water level as explained using the
specific energy diagram
A l i f Fl
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Analysis of Flow across aWeir in a Rectangular Channel
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
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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
CHANNEL CONSTRICTION
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CHANNEL CONSTRICTION CASE 2
CHANNEL CONSTRICTION
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CHANNEL CONSTRICTION CASE 3
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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