Hydraulic Jump

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.

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

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,

Where hc is the depth to the centroid of the end area.

1221 21VVQ

gAhAhF ccx

Rearranging the previous equation

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

2211 21AhQV

gAhQV

g cc

This can be applied to any shape of cross section.

consttAhAg

QF cm

2

For a rectangular cross section,

and the discharge Q in terms of discharge per unit width can be given by Q=bq. So

byAyhc ,2/

constty

gy

qy

gy

qfm

22

22

2

221

1

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.

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

This is then transmitted to the specific energy diagram to determine the value ‘cd’ of V2

2/2g. The value of V1

2/2g 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

The above expression is same as the relation for critical depth

2

22 y

gy

qfm

3/12

2

02

2

g

qy

y

gy

q

dy

dfm

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

so

))(( 121221

22 yyyyyy

221

21

2 yyyy

g

q

32

22

1

31

21

2

811

2

811

2

gy

qyy

gy

qyy

Flow measuring structures

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.

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.

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.

Sharp crested weir

A weir having a horizontal or nearly horizontal crest sufficiently long in the direction of flow

These support the falling jet over their crest in the longitudinal direction.

Discharge over a rectangular weir can be calculated by using the formula

Cd is the coefficient of discharge and L is the length of crest.

2/323

2LHgCQ d

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)

hg

VH

2000

2

)(2

2

2

hHgV

hHg

V

The discharge over weir at B is given by

Q=Cd*Area of Flow*Velocity

Substituting the value of V

)(2 hHghLCQ d

(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

322 hHhgLCQ d

Hh

hH

hH

hHh

hHhdh

d

dh

dQ

3

2

32

032

032

0)(

2

32

Substituting the values in (1)

2/3max 2385.0 HgLCQ d

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

Triangular or V-notch weir

The fundamental equation for discharge over a triangular v-notch weir is given by

2/5

2tan2

15

8HgCQ d

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.

Velocity of approach

In deriving formula for discharge over broad crested weir, we have ignored velocity of approach but in actual practice velocity of approach of water is sure to affect the discharge over the weir.

Let a = x- sectional area of channel on u/s side of weir

Additional height / head due to velocity of approach

A

QVa

g

VaHa

2

2

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

)(2385.0 2

3

2

3

max HaHtgLCQ d

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.

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:-

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.

Venturi meter

Venturi Flume

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 1-1 thus depth at section 2-2 decreases

energy eq. between sec 1-1 &2-2 (as z1=z2)

g

vh

g

vh

22

22

2

21

1

Since discharge at section 1-1&2-2is continuous therefore

A1 V1 = A2 V2 (Eq.of cont.)

)1(22

21

22

21 g

v

g

vhh

21

22

222

1

1

221

a

vav

a

vav

Substituting this value in eq.

21

22

21

22

21

22

22

22

21

22

22

22

21

2

1222

1

2

a

aa

g

v

a

a

g

v

g

v

ga

va

g

vhh

Let h = difference of head at 1-1&2-2

21

22

21

22

2 a

aa

g

vh

ghaa

av 2)( 2

221

212

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

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.

Modular Venturi flume