University of Dundee

OPEN CHENNEL FLOWS

Fluid Mechanics – CE31003

Prof: P A Davies

Prabu Rengarajan

Student id : 120021583

14.11.2012

OPEN CHENNEL FLOWS

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Content:

1. Introduction

2. Test Results

3. Discussion

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1. Introduction

Open channel flow is the flow of a liquid in conduit with a free surface. There are

examples such, both artificial (flumes, spillways, canals, weirs, drainage) and natural (streams, rivers,

flood plains).

The purpose of this experiment is to determine the flow characteristics and estimate

discharge coefficient by method of sharp crested weir and broad crested weir.

Sharp crested weir and Broad crested weir are an obstruction in an open channel flow that

water must flow over the weir. They are used to find flow rate and head on weir.

Sharp crested weir is used to measure the discharge in small open channels which is accuracy

is needed. Broad crested weirs are strong solid structures. It is usually made of reinforced concrete. It

is used to measure to discharge of rivers. It has advantage that it operates more effectively with

higher downstream water levels than sharp crested weir.

In an open-channel flow, a supercritical flow can change into a sub-critical flow by passing

through a hydraulic jump. The upstream flow has high velocity and is low height. The downstream

flow has low velocity and is deep.

2. Test Results

(I) Sharp crested weir:

Depth (Y.crest) at weir crest is obtained from experiments with increasing of discharge of

flow and Upstream depths (H). The values are tabulated in below.

Table1 : Sharp crested weir

Flow rate(Q) Upstream Depth H

Q ± 0.1(l/s) Q ± 0.1 (m^3/s) log (Q) H ± 0.1(mm) H(m) log (H)

2.5 0.0025 -2.6020600 67.50 0.0675 -1.1706962

2.0 0.0020 -2.6989700 58.30 0.0583 -1.2343314

1.5 0.0015 -2.8239087 47.70 0.0477 -1.3214816

1.0 0.0010 -3.0000000 36.60 0.0366 -1.4365189

0.5 0.0005 -3.3010300 22.80 0.0228 -1.6420652

Y.crest, Depth at weir crest Errors of Flow rate (Q)

± 0.1mm m Q + 0.1 (m^3/s) log (Q+0.1) Q - 0.1 (m^3/s) log (Q-0.1)

44.50 0.04450 0.0026 -2.5850267 0.0024 -2.6197888

38.40 0.03840 0.0021 -2.6777807 0.0019 -2.7212464

32.00 0.03200 0.0016 -2.7958800 0.0014 -2.8538720

24.70 0.02470 0.0011 -2.9586073 0.0009 -3.0457575

14.90 0.01490 0.0006 -3.2218487 0.0004 -3.3979400

H+0.1(m) log (H+0.1) H-0.1(m) log (H-0.1)

0.0676 -1.1700533 0.0674 -1.1713401

0.0584 -1.2335872 0.0582 -1.2350770

0.0478 -1.3205721 0.0476 -1.3223930

0.0367 -1.4353339 0.0365 -1.4377071

0.0229 -1.6401645 0.0227 -1.6439741

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Figure1. Upstream depth Log (H) Versus Flow rate Log (Q)

According to the table1, the values are plotted graphically as Log (H) in x- axis and Log (Q) in y-axis as

shown in Figure1. In the same way, the errors of flow discharge (Q ± 0.1) and depth (H ± 0.1) are

plotted in order to determine precision value of n and k here. From this graph, we obtained three

linear equation as in form of y = nx + logK, where “n” is gradient of line and “logK” is intercept of line.

Three straight lines such as best fit line, Upper extreme line, lower extreme line have been obtained

from this graph. The value of K and n have been determined from these three equations.

The value of n is 1.4831 ± 0.21 and the value of K is 0.135925 ± 0.0966 from experiments.

The co-efficient of discharge (Cw) can be calculated from below equations (1) and (2) using of n and K

values.

Q = Cw (2/3) (2g)1/2

B H 3/2

…………(1)

Q = K Hn

……….. (2)

The sample calculation of finding co-efficient of discharge (Cw) can be done in below for sharp

crested weir. The average value of Cw for sharp crested is 0.607 from experiments. The sample

calculation for finding Cw has been done in below by choosing best line equation.

y = 1.4831x - 0.8667

y = 1.72x - 0.5704

y = 1.2807x - 1.1213

-3.50

-3.00

-2.50

-2.00

-1.80 -1.55 -1.30 -1.05

Log(H) vs Log(Q)

Log(H+0.1) vs

Log(Q+.01)

Log(H-0.1) vs Log(Q-

0.1)

Linear (Log(H) vs

Log(Q))

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The values Y.crest obtained from experiments is plotted with theoretical value (2H/3) in

below.

Figure2. Y.crest versus 2H/3

(II) Broad crested weir:

The depth above weir crest(Ym) are obtained from experiments in order to

increasing of discharge of flow and Upstream depths (H). The values are tabulated in below.

Table 3: Broad crested weir

Flow rate(Q) Upstream Depth H

Q ± 0.1(l/s) Q ± 0.1 (m^3/s) log (Q) H ± 0.1(mm) H(m) log (H)

2.5 0.0025 -2.602060 72.0 0.0720 -1.142668

2.0 0.0020 -2.698970 61.4 0.0614 -1.211832

1.5 0.0015 -2.823909 51.9 0.0519 -1.284833

1.0 0.0010 -3.000000 39.1 0.0391 -1.407823

0.5 0.0005 -3.301030 24.0 0.0240 -1.619789

Ym, Depth above weir Error of Flow rate (Q)

± 0.1mm m Q + 0.1 (m^3/s) log (Q+0.1) Q - 0.1 (m^3/s) log (Q-0.1)

47.3 0.0473 0.0026 -2.5850 0.0024 -2.6198

40.5 0.0405 0.0021 -2.6778 0.0019 -2.7212

33.9 0.0339 0.0016 -2.7959 0.0014 -2.8539

25.8 0.0258 0.0011 -2.9586 0.0009 -3.0458

15.8 0.0158 0.0006 -3.2218 0.0004 -3.3979

H+0.1(m) log (H+0.1) H-0.1(m) log (H-0.1)

0.0721 -1.1421 0.0719 -1.1433

0.0615 -1.2111 0.0613 -1.2125

0.0520 -1.2840 0.0518 -1.2857

0.0392 -1.4067 0.0390 -1.4089

0.0241 -1.6180 0.0239 -1.6216

Table2:

Crest(m) 2H/3 (m)

0.0445 0.0450

0.0384 0.0389

0.0320 0.0318

0.0247 0.0244

0.0149 0.0152 0.000

0.020

0.040

0.060

0.00 0.02 0.04 0.06

Y.crest Vs 2H/3

Y.crest Vs

2H/3

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Figure3. Upstream depth Log (H) Versus Flow rate Log (Q)

According to the table3, the values are plotted graphically as Log (H) in x- axis and Log (Q) in y-axis as

shown in Figure1. In the same way, the errors of flow discharge (Q ± 0.1) and depth (H ± 0.1) are

plotted in order to determine precision value of n and k here. From this graph, we obtained three

linear equation as in form of y = nx + logK, where “n” is gradient of line and “logK” is intercept of line.

Three straight lines such as best fit line, Upper extreme line, lower extreme line have been obtained

from this graph. The value of “K” and “n” has been determined from these three equations.

The value of n is 1.4691 ± 0.21 and the value of K is 0.1185 ± 0.08 from experiments.

The co-efficient of discharge (Cw) can be calculated from below equations (1) and (2) using of n and K

values.

Q = Cw (2/3) (2g)1/2

B H 3/2

…………(1)

Q = K Hn

……….. (2)

The sample calculation of finding co-efficient of discharge (Cw) can be done in below for broad

crested weir. The average value of Cw for sharp crested is 0.55 from experiments. The sample

calculation for finding Cw has been done in below by choosing best line equation.

y = 1.4691x - 0.9263

y = 1.265x - 1.175

y = 1.6995x - 0.6421

-4.00

-3.50

-3.00

-2.50

-2.00

-1.80 -1.55 -1.30 -1.05

Log(H) vs Log(Q)

Log(H+0.1) vs

Log(Q+.01)

Log(H-0.1) vs

Log(Q-0.1)

Linear (Log(H) vs

Log(Q))

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The values depth above weir (Ym) obtained from experiments are plotted with theoretical value

Critical depth (Yc) in below.

Figure4. Crest versus 2H/3

(III) Hydraulic jump:

The upstream depth and downstream depth are obtained from experiments by

increasing of discharge of flow. The values are tabulated in below.

Table5 Hydraulic jump experiment

Flow rate(Q) Upstream Depth y1 Downstream Depth y2

Q ± 0.1 (l/s) Q ± 0.1 (m^3/s) H ± 0.1(mm) H(m) ± 0.1mm m

2.5 0.0025 15.3 0.0153 107.4 0.1074

2.0 0.0020 15.3 0.0153 84.3 0.0843

1.5 0.0015 15.3 0.0153 61.1 0.0611

1.0 0.0010 15.3 0.0153 39.6 0.0396

Error of Flow rate (Q) Error of Upstream Depth (y1) Error of Downstream Depth y2

Q + 0.1 (m^3/s) Q - 0.1 (m^3/s) y1+0.1(m) y1-0.1(m) y2+0.1(m) y2-0.1(m)

0.0026 0.0024 0.0154 0.0152 0.1075 0.1073

0.0021 0.0019 0.0154 0.0152 0.0844 0.0842

0.0016 0.0014 0.0154 0.0152 0.0612 0.0610

0.0011 0.0009 0.0154 0.0152 0.0397 0.0395

V1=Q/By1 Fr1=V1/(gy1)^0.5 (1+8Fr1^2)^0.5 2y2/y1

Best fit line

2.055329 5.30519 15.0386 14.0392

1.644264 4.24415 12.0459 11.0196

1.233198 3.18312 9.0586 7.9869

0.822132 2.12208 6.0849 5.1765

V1=Q/By1 Fr1=V1/(gy1)^0.5 (1+8Fr1^2)^0.5 2y2/y1

Upper

extreme line

2.123663 5.463748 15.4861 13.9610

1.715266 4.413027 12.5219 10.9610

1.306869 3.362306 9.5625 7.9481

0.898473 2.311586 6.6142 5.1558

V1=Q/By1 Fr1=V1/(gy1)^0.5 (1+8Fr1^2)^0.5 2y2/y1

Lower

extreme line

1.986097 5.143328 14.5819 14.1184

1.572327 4.071801 11.5601 11.0789

1.158557 3.000275 8.5448 8.0263

0.744786 1.928748 5.5462 5.1974

Table4:

Depth above

weir Ym(m)

Critical depth

Yc(m)

0.0473 0.0477

0.0405 0.0401

0.0339 0.0331

0.0258 0.0252

0.0158 0.0159

0.0000

0.0200

0.0400

0.0600

0 0.02 0.04 0.06

Ym Vs Yc

Ym Vs Yc

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Figure5. 2y2/y1 versus (1+8Fr^2) ^0.5

According to the table5, the values are plotted graphically as 2y2/y1 in x- axis and (1+8Fr^2) ^0.5 in y-

axis as shown in Figure5. From this graph, we obtained three linear equation as in form of y= nx + C,

where “n” is slope of line and “C” is intercept of line. The value of C and n have been determined

from this graph. The theoretical value of n is 1 and intercept is -1.The average value of n is 1.0074 ±

0.11 and Intercept C is 0.9308 ± 0.85 from experiments.

3. Discussion

The average theoretical value of n is 1.5 for sharp crested weir and broad crested weir. The value of n

has been found out significant difference between experimental result and theoretical value of 1.5.

The value of n with limits of error satisfies the theoretical value. The errors can be obtained by

measuring of discharge Q on flow meter scale or measuring of flow depth H on point gauge scale.

The reason could be also that the gap between channel and crest plate is not properly sealed.

The value for coefficient of discharge (Cw) for sharp crested weir can be calculated as 0.607 for best

line. The coefficient of discharge (Cw) for broad crested weir is 0.55. Typical values of Cw are 0.60-

0.81 for sharp crested weir and 0.45-0.58. The mean values of coefficient of discharge (Cw) with

limits of errors of both broad crested and sharp crested agree with the theoretical values of Cw.

In sharp crested weir, the value of Y.crest from experimental has been compared with calculated

(2H/3) theoretical value. There is a very tiny difference between values of Y-crest values from

experimental to theoretical value. Error may be occurred due to parallax in reading the gauge scale.

In broad crested weir, the value of Depth above weir (ym) from experimental has been compared

with calculated Critical depth (Yc) theoretical value. There is a very small difference between values

of Depth above weir (Ym) values from experimental to theoretical value.

y = 1.0074x + 0.9308

y = 0.8931x + 1.9722

y = 1.1289x - 0.2741

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

18.00

0.00 5.00 10.00 15.00

2y2/y1 Vs (1+8Fr^2)^0.5

Upper extreme line

Lower extreme line

Linear (2y2/y1 Vs

(1+8Fr^2)^0.5)

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The flow may not have been stabilized when the readings were taken and reaction time error when

using the stop watch.

Within limits of error, discharge coefficient (Cw) is directly influenced by the flow rate. When

increasing flow rate; discharge of coefficient is also increased.

According to values of depth ratio and Froude number, the flow is super critical. The value of

gradient n and intercept with limits of error satisfies the theoretical value of 1 and -1 respectively.

There may be human error and instrument error also. The human error may be, errors occurred by

taking erroneous reading of depths and in operation of slice gates. Error may be leakage from flume

and frictional forces also had some effect on the experiment.