dynamics of fluid flow

NPTEL Course Developer for Fluid Mechanics Dr. Niranjan SahooModule 04; Lecture 31 IIT-Guwahati

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DYMAMICS OF FLUID FLOW

FLOW OVER NOTCHES AND WEIRS

Continued……

Velocity of Approach

It is defined as the velocity with which the flow approaches/reaches the notch/weir before

it flows past it. The velocity of approach for any horizontal element across the notch

depends only on its depth below the free surface. In most of the cases such as flow over a

notch/weir in the side of the reservoir, the velocity of approach may be neglected. But,

for the notch/weir placed at the end of the narrow channel, the velocity of approach to the

weir will be substantial and the head producing the flow will be increased by the kinetic

energy of the approaching liquid. Thus, if aV is the velocity of approach, then the

additional head aH due to velocity of approach, acts on the water flowing over the notch

or weir. So, the initial and final height of water over the notch/weir will be aH H and

aH respectively. It may be determined by finding the discharge over the notch/weir

neglecting the velocity of approach i.e.

aQVA (1)

where Q is the discharge over the notch/weir and A is the cross-sectional area of

channel on the upstream side of the weir/notch. Additional head corresponding to the

velocity of approach will be,2.

2a

aVHg

(2)

and being the kinetic energy correction factor to allow for the non-uniformity of

velocity in the cross-section of the channel.

For example, the discharge over a rectangular notch/weir of width B

32

3322

2 . . 2 . ....without velocity of approach32 . . 2 . ....with velocity of approach3

d

d a a

Q C B g H

C B g H H H

(3)


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Empirical formula for discharge over rectangular weirs

A rectangular weir is frequently used for measuring the rate of flow of water in channels.

However, many researchers have conducted number of experimental investigations and

proposed some empirical relations commonly used for rectangular weirs. Some of them

are described below.

(a) Francis’ formula: It is one of the most commonly used formula for computing the

discharge over a sharp or narrow crested weirs with and without end contractions. Based

on this formula, the discharge is expressed by,

33221.84 0.1 a a aQ B n H H H H H

(4)

where n is the number of end contractions.

(b) Bazin’s formula: Based on this formula, the discharge over a rectangular weir is

given by,

32. 2 . . aQ m g B H H (5)

where 0.0030.405a

mH H

is the Bazin coefficient

(c) Rehbock’s formula: Based on the experiments conducted by Rehbock, the following

empirical formula is proposed;

3 22 0.0010.605 0.08 . 2 . .3

HQ g B HZ H

(6)

where z is the crest height in meters.

Sharp-Crested Weirs

A sharp-crested weir is essentially a vertical sharp-edged flat plate placed across the

channel in a way such that the fluid must flow across the sharp edge and drop into the

pool downstream of the weir plate as shown in Fig. 1. The specific shape of the flow area

in the plane of the weir plate may be of rectangular/triangular/trapezoidal type.

The main forces governing flow over a weir are gravity and inertia. The gravity

accelerates the fluid from its free surface elevation upstream of the weir to a larger

velocity as it flows down the hill formed by the nappe. Although viscous and surface


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tension effects are usually of secondary importance, such effects cannot be entirely

neglected. Generally, appropriate experimentally determined coefficients such as

Francis, Bazin’s and Rehbock’s formulae are used to account for these effects.

H

z

QNappe

Weir plate

Fig. 1: Sharp-crested weir geometry.

Broad-Crested Weirs

Broad-crested weirs differ from thin-plate and narrow-crested weirs by the fact that

different flow pattern is developed. Experimental investigations have shown that if the

length of the crest of the weir 0.625 i.e. 1.6w wL H H L , the jet of water touches

only the upstream edge and flows clear of the downstream. Weirs falling under these

classes are called “thin-plate weirs”. On the other hand, if 0.5 1.6wH L , the jet of

water remains in contact with the entire crest and these weirs are called “narrow-crested

weirs”. In both the cases, the flow pattern is similar corresponding to that of a rectangular

notch/weir.

H

z

V1

Water level

Lw

B

h

(1)(2)

v

Fig. 2: Broad-crested weir geometry.

A “broad-crested weir” is a structure in an open channel that has a crest above which

the fluid pressure may be considered hydrostatic. The typical configuration is shown in


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Fig. 2. Broad-crested weirs are operated in the range, 0.08 0.5wH L so that nearly

uniform critical flow is achieved in the short reach above the weir block. For long weir

blocks 0.08wH L , head losses across the weir can not be neglected. On the other

hand, for short weir blocks 0.5wH L , the streamlines of the flow over the weir

block are not horizontal. Although, broad-crested weirs can be used in channels of any

cross-sectional shape, but our attention will be limited to rectangular channels.

Consider a broad-crested weir with length, width and height of crest as wL , B and

z . Referring to the Fig. 2, Bernoulli’s equation can be applied between sections ‘a-a’

upstream of the weir and section ‘b-b’ over the weir i.e.2 2

2 2aV vH z h zg g

(7)

or, if the upstream velocity head is negligible, then

2

or 22vH h v g H hg

(8)

The discharge over the broad-crested weir is given by,

. 2d wQ C L h g H h (9)

In order to measure the discharge over the broad-crested weir, two heads

i.e. andH h need to be measured. However, experiments have shown that the flow

adjusts itself to have maximum discharge for the available head H . The downstream

head over the weir can be computed mathematically by differentiating Eq. (9) with

respect to h and equating it to zero i.e.

. . 2 02

2or,3

d wdQ hC L g H hdh H h

h H

(10)

This value of h is known as critical depth. In other words, the discharge over the broad-

crested weir is maximum when the critical depth of flow occurs over the surface of the

weir crest. The maximum discharge over the weir corresponding to critical depth will be,


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1322

max2 1. . 2 . 1.7 .3 3d w d wQ C L H g H C L H

(11)

If the “velocity of approach” aV is considered, then the above equation can be modified

as,3

2 2

max 1.7 .2

ad w

VQ C L Hg

(12)

Submerged Weirs

When the water level on the downstream of the weir is above the crest of the weir, then

the weir is said to be submerged weir as shown in the Fig. 3. These weirs, constructed

across the rivers have larger discharging capacity compared with freely discharging weirs

and hence become more useful in discharging water during floods.

H1

Va

Water level

(1)(2)

H2

Lw

Fig. 3: Submerged weir geometry.

As shown in Fig. 3, the discharge over the submerged weir may be obtained by

dividing it into two parts;

the portion between the upstream and downstream water surfaces is treated as a

free weir 1Q

the portion between the downstream water surface and crest of the weir is treated

as drowned orifice 2Q

If aV is the velocity of approach, 1 2andH H are the heads on the upstream and

downstream of the weir and wL is the length of the weir, then


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3 3

2 22 2

1 1 1 22 2 .3 2 2

a ad w

V VQ C g L H Hg g

(13)

22 2 2 1 2. 2d w aQ C L H g H H V

(14)

where 1 2andd dC C are the discharge coefficients for the free and drowned portion

respectively.

Submergence ratio (SR) and Modular limit

Submergence ratio may be defined as the ratio of heads available on the downstream side

of the weir to the head available in the upstream side. Mathematically,

22

1

SR

2a

HVH

g

(15)

Both sharp and broad crested weirs are susceptible to submergence depending on the

values of “SR”. The sharp crested weirs behave, as a free weir only up to “SR” value of

0.66 and the corresponding values for broad crested weirs are 0.85. This is because of the

fact that the flow conditions are such that the downstream water level is held away from

the crest and hence it does not affect the upstream flow conditions. The limiting value of

“SR” up to which any submerged weir may behave, as free weir is known as “Modular

Limit”.

Ogee spillway, Siphon spillway and Proportional/Sutro weirs

Ogee spillway

A spillway is a portion of a dam over which the excess water, which cannot be stored in

the reservoir formed on the upstream of the dam, flows to the downstream side. The

profile of an ogee spillway conforms to the shape of the nappe of the sharp-crested weir

of the same height as spillway and under the same head as shown in Fig. 4. The main

advantage of providing such a shape for the spillway is that the flowing sheet of water

remains in contact with the surface of the spillway and thereby preventing negative

pressure being developed on the downstream side. This condition will be fulfilled as long


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as the head over the crest of the spillway is equal to or less than the designed head. In the

Fig. 4, if H W is less than 0.75, the discharge can be taken as that of a rectangular

weir.

Siphon spillway

It is essentially an Ogee weir provided with an airtight cover and large rectangular section

pipe connecting the upstream and downstream water surfaces. It allows the discharge of

water at a controlled rate. It has the following advantages over Ogee spillway;

Operating head and hence discharge is comparatively more.

Since the crest of a siphon spillway can be raised, so it allows a greater amount of

water to be stored in the reservoir.

H

Fig. 4: Ogee spillway and / or Siphon spillway.

Proportional/Sutro weir

The discharge over a weir is mainly proportional to the pressure head above the crest. For

most of the weirs it is expressed as, nQ H , where 3 2n for rectangular weir and

5 2n for a triangular weir. In Proportional/Sutro weir as shown in Fig. 5, the discharge

varies linearly with the H .

H

a

L

Water level

Fig. 5: Proportional weir.


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The discharge through this weir is given by,

12. . 2 .

3daQ C L g a H

(16)

where dC is coefficient of discharge (0.6 to 0.65), andL a are the width and height of

the rectangular shaped aperture that forms the base of the weir.

Example 1

It is proposed to use a notch for measuring the water flow from a reservoir. It is estimated

that the error in measuring the head above the bottom of the notch could be 1.5mm. For a

discharge of 0.3m3/s, determine the percentage error, which may occur, using a right-

angled triangular notch with coefficient of discharge of 0.6.

Solution:

For a V-notch,528 . 2 tan

15 2dQ C H g

Taking 00.6 and 90dC ,5 52 28 900.6 2 9.81 tan 1.417

15 2Q H H

When 30.3m s 0.5374mQ H

Now325 2.51.417

2Q QHH H

Or, 2.5 2.5 0.0015 100 0.7%0.5374

Q HQ H

Example 2

The stream of water from a waterfall of height 40m approaches a weir where the

measured head is recorded as 0.3m. The length of the weir is 3m and the velocity of

approach is 1.2m/s. Determine, the power available at the waterfall. Use Bazin’s formula

with 1.5 for the flow over the weir.

Solution:

According to Bazin’s formula, 32. 2 . . aQ m g B H H


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where 0.0030.405a

mH H

Now, 22 1.5 1.2. 0.11m2 2 9.81

aa

VHg

and 0.3mH , 3mB

So, 0.0030.405 0.4960.3 0.11

m

3

320.496 2 9.81 3 0.3 0.11 1.7 m sQ

Power available at the fall = . . . 1000 9.81 1.7 40kW 667kW1000 1000g Q H

Example 3

A rectangular channel 6m wide carries 168 lits/min at a depth of 0.9m. What height of a

rectangular weir must be installed to double the depth? Discharge coefficient of weir may

be taken as 0.85.

Solution:

The discharge for a broad crested weir is given by,3

2 21.7 .

2a

d wVQ C L H

g

Here, 3 3168m min 2.8m sQ ; 6mwL ; 0.85dC

Then,2 2

2 3 32.8 0.47m2 1.7 1.7 0.85 6

a

d w

V QHg C L

The depth of the flow required = 2 0.9 = 1.8m

The velocity of approach is given by,

2

2.8 0.26m s6 1.8 6 1.8

0.0034m20.47 0.0034 0.4666m

a

aa

QV

Vhg

H

Height of the broad crested weir = 1.8 – 0.4666 = 1.3334m.


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EXERCISES

1. A triangular notch is used to measure flow in a channel under a head of 0.2m. If the

discharge is to be measured within 3% accuracy, what is the maximum velocity of

approach that can be neglected?

2. Water flows through a rectangular channel 1m wide and 0.5m deep and then over a

sharp Cipolletti weir of crest length of 0.6m. If the water level in the channel is 0.225m

above the weir crest, calculate the discharge over the weir. Take 0.6dC and make

correction for velocity of approach.

3. A rectangular notch of crest width 0.4m is used to measure the flow of water in a

rectangular channel of 0.6m wide and 0.45m deep. If the water level in the channel is

0.25m above the weir crest, find the discharge in the channel. For the notch, assume

0.6dC and take velocity of approach into account.

4. For the stepped notch shown below, find the discharge if 0.6dC for all the sections.

0.4m0.8m

1.2m

0.15m0.3m

0.5m

5. A discharge of 3.6m3/min was measured over a right-angled notch. An error of 0.15cm

was mead while measuring the head over the notch. Determine the percentage of error in

discharge if the coefficient of discharge for the notch is 0.6.

6. The head of water over a triangular notch of angle 600 is 50cm and the coefficient of

discharge is 0.62. The flow measured by it is to be within the accuracy of 1.2%. Find

the limiting values of the head.

7. Determine the discharge over 1.5m high sharp-crested weir fixed across 2m wide

rectangular channel when the head over the weir is 0.05m.

8. In a 5m wide rectangular channel with 1.2m depth of flow, a sharp-crested weir of

2.5m length and 0.6m height is fixed symmetrically across the channel width. If it flows

free, determine the discharge.


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9. A sharp crested weir of 1m height is fixed across 5m wide channel as shown in the

following figure. The depth of flow on the upstream and downstream sides of the weir is

1.5m and 1.2m respectively. Find the discharge over the weir and compare the discharge

with that of a submerged weir.

H1

Water level

y1 = 1.5m

H2

W = 1 my2 = 1.2m

10. A river 30m wide and 3m deep has a mean velocity of 1.2m/s. Find the height of a

weir to raise the water level by 1m.

11. A spillway 40m long having discharge coefficient 1.8 permits a maximum discharge

90m3/s from a storage reservoir. It is proposed to replace the spillway by a siphon

spillway of section 0.75m1.5m with operating head 8m and discharge coefficient 0.64.

Find the number of siphons required and the amount of extra water stored, if the siphons

have a priming depth of 0.15m. Take the average surface area of the reservoir as

5105m2.

12. Determine the discharge over an ogee spillway of 150m length under a head of 1.5m.

What will be the depth of the flow at the toe of the dam if the height of the dam is 50m.

13. What will be the head required to carry a discharge of 2.75m3/s through a 2m wide

gate at 0.3m opening under free flow conditions?

dynamics of fluid flow

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