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Most important phenomena in Hydraulics
Occurs when supercritical flow has its velocity
reduced to subcritical
The result is marked by a discontinuity of
surface characterised by a steep upward
slope of the water surface broken throughout
by violent turbulences known as Hydraulic
Jump.
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This hydraulic jump is said to be the
dissipater of the surplus energy of water.
After the hydraulic jump the water flows with
a greater depth and less velocity.
The use of this phenomenon is sometimes
made in hydraulic structures such as spillway
of a dam to minimize the erosive power of
supercritical flow
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Relationships for the calculation of depth of flow
before and after the jump are needed.
For channels of gradual slope (less than 0.05) the
gravity component of the weight is very small
and can be neglected without introducing
significant error.
Frictional forces can also be neglected because of
shorter length of the channel and therefore the
only significant forces are the hydrostatic forces.

Applying momentum equation between
section 1 and 2 of the figure and using the
relation for end forces,
F
x

 hc1 A1  hc2 A2 

g
QV2  V1 
Where hc is the depth to the centroid of the
end area.

Rearranging the previous equation

g

QV1  hc1 A1 

g
QV2  hc2 A2
This states that the momentum plus the
pressure force on the cross sectional area is
constant, or dividing by γ and observing that
V=Q/A
Fm
Q2

 Ahc  constt
Ag
This can be applied to any shape of cross
section.

For a rectangular cross section,
hc  y / 2, A  by
and the discharge Q in terms of discharge per
unit width can be given by Q=bq. So
2
fm
2
1
2
2
2
q
y
q
y




 constt
gy1
2
gy2
2



A curve of different values of fm for different
values of y is plotted.
The loss of energy due to the jump will not
affect the force so fm before and after the
jump is the same
So the vertical line on the fm curve shows two
conjugate depths y1 and y2.
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These
depths
represent
possible
combinations of depth that could occur
before and after the jump.
So in the figure, the line for initial water level
y1 intersects the fm curve at ‘a’ giving a value
of fm which must be the same after the jump.
The vertical line ‘ab’ then fixes the value of y2
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This is then transmitted to the specific energy
diagram to determine the value ‘cd’ of
V22/2g. The value of V12/2g is the vertical
distance ef.
For a rectangular channel, it can be shown
that the minimum value of fm occurs at the
same depth as the minimum value of E.
Differentiating fm with respect to ‘y’ and
equating to zero
2
fm
y2
q


gy
2
dfm
q2
2y


0
dy
gy
2
1/ 3
q 
y 
 g 



2

The above expression is same as the relation
for critical depth
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So for a given ‘q’ the minimum value of fm
occurs at the same depth as does the
minimum value of E.
When the rate of flow and depth before or
after the jump are given, the equation for fm
becomes a cubic equaiton when solving for
other depth
This can be reduced to a quadratic equation
by observing that
y22  y12  ( y2  y1 )( y2  y1 )

so
q2
y1  y2
 y1 y2
g
2

 1 


y2 
 1 
y1 
2 

y1
y2 
2




8q 2 

1
gy23 

8q 2
1
gy13
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Weirs are overflow structures build across
channels to measure the volumetric rate of
flow of water.
It is an obstruction that causes the liquid to
rise behind the weir and flow over it.
The crest of the measuring weir is normally
perpendicular to the direction of flow.
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Constructed from metal, masonry or concrete
etc.
Weirs are identified by the shape of their
opening or notch
The edge of the opening can be sharp or it
can be broad, consequently, the weirs will
weather be sharp crested or broad crested.
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A weir with a sharp upstream corner is known
as sharp crested weir
The weir plate thickness at the crest edges
should be from 0.03 to 0.08 inches.
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A weir having
horizontal crest
direction of flow
a horizontal or nearly
sufficiently long in the
These support the falling jet over their crest
in the longitudinal direction.
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Discharge over a rectangular weir can be
calculated by using the formula
2
3/ 2
Q  Cd
2 g LH
3

Cd is the coefficient of discharge and L is the
length of crest.
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Consider a broad-crusted wire as shown
above.
Let A and B are u/s and d/s ends of the weir.
Let L is length and Cd is coefficient of
discharge.
H is the head at u/s and h is at d/s of weir.
Let V is velocity of water at B now applying
energy eq. at A&B (considering the approach
velocity to be zero)
2
V
00 H  0
h
2g
2
V
 H h
2g
 V  2 g ( H  h)

The discharge over weir at B is given by
Q=Cd*Area of Flow*Velocity

Substituting the value of V
Q  Cd hL 2g (H  h)
Q  C d L 2 g Hh  h
2


3
(1)
If Cd, L and g are constant, then discharge
will be maximum when (Hh2 – h3) is
maximum.
Differentiating with respect to h and equating
to zero
dQ d
2
3

( Hh  h )  0
dh dh
2
2 Hh  3h  0
2 H  3h  0
2 H  3h
2
h H
3

Substituting the values in (1)
Qmax  0.385Cd L 2g H
3/ 2

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For relatively small flows, the rectangular weir
could be quite narrow.
In this case the value of H over the weir will
be so small that the nappe will not stay clear
rather it will cling to the plate
Thus the use of triangular weir comes into
practice as it function for relatively low flows

The fundamental equation for discharge over
a triangular v-notch weir is given by
8
 5/ 2
Q  Cd
2 g tan H
15
2



Sometimes a weir is provided in a stream or a
river (in storage reservoir vel. of app is taken
as zero) to measure the flow of water.
In such a case, the water approaching the
u/s of weir has got some velocity of approach
and is assumed to be uniform over the whole
weir.
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In deriving formula for discharge
crested weir, we have ignored
approach but in actual practice
approach of water is sure to
discharge over the weir.
over broad
velocity of
velocity of
affect the
Let a = x- sectional area of channel on u/s
side of weir
Q
Va 
A
Additional height / head due to velocity of
approach
2
Va
Ha 
2g

Thus if velocity of approach is also
considered then total height above the weir is
Ht = H + Ha H = Ht – Ha
Discharge over the weir will then be given by
3
2
3
2
Qmax  0.385Cd L 2 g ( Ht  Ha )
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In every hydraulic structure (i. e. Bridge, regulator etc)
which is constructed across an open channel, a few
opening are left to allow the water to pass.
If the total width of all these openings is practically
the same as that of the channel, such a structure is
called full width or unflumed structure.
But generally the total width of such a structure is
kept much less than the width of the channel to keep
economy in its construction and to increase its utility.
Such a structure whose width is less than the width of
channel is called flumed structure.
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A flumed structure used for the measurement
of the quantity of water is called
venturiflume.
The difference between venturimeter and
venturiflume is that in venturimeter the flow
is under pressure whereas in a venturiflume
the flow is under gravity and pressure is
atmospheric.
Following two types of venturiflume are
important from subject point of view:-
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Non-modular venturiflume
Modular venturiflume
A non-modular venturiflume consists of
Convergent cone
It is short part of the channel which
converges from b1 to b2. It is also called
inlet of the venturiflume.
Throat
It is the small portion of the channel in
which width b2 is kept constant
Divergent cone
It is the small portion of the channel which
diverges from width b2 to again width b1.
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The liquid while flowing through venturiflume
is accelerated between section 1-1 and 2-2
so the velocity at 2-2 is more than that at 11 thus depth at section 2-2 decreases
energy eq. between sec 1-1 &2-2 (as z1=z2)
2
1
2
v
v2
h1 
 h2 
2g
2g
2
2
v2
v1
h1  h2 

 (1)
2g
2g


Since discharge at
continuous therefore
section
A1 V1 = A2 V2 (Eq.of cont.)
a2 v2
v1 
a1
2
v1 
2
a2 v2
a1
2
2
1-1&2-2is

Substituting this value in eq.
2
2
2
v2
a v
h1  h2 

2g
a12
v22

2g
 a12  a22

2

a
1





2
2
2

1
v
v2
a2 



1 2 

2g
2g
2g 
a1 

2
2
2
Let h = difference of head at 1-1&2-2
v2 a  a
h
2
2 g a1
2
2
1
2
2
2
1
a
v2  ( 2
)2 gh
2
a1  a2
2
Discharge through venturiflume is
Q = coeff of venturiflume xa2 v2
Q = Ca2v2
Value of C depends on smoothness of surface
of bed/sides and round ness of cornors of
V.F= It ranges 0.95 to 1.0
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It is the type of V. F. in which the width of
throat is decreased to such an extent that the
depth of water in throat is equal to critical
depth.
The velocity of flow through the throat
corresponding to critical depth is also critical.